Oceanography And Marine Biology

OCEANOGRAPHY AND MARINE BIOLOGY

AN ANNUAL REVIEW Volume 25

OCEANOGRAPHY AND MARINE BIOLOGY AN ANNUAL REVIEW Volume 25

HAROLD BARNES, Founder Editor MARGARET BARNES, Editor The Dunstaffnage Marine Research Laboratory Oban, Argyll, Scotland

ABERDEEN UNIVERSITY PRESS

FIRST PUBLISHED IN 1987 This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” This book is copyright under the Berne Convention. All rights reserved. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright Act, 1956, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, electrical, chemical, mechanical, optical, photocopying, recording or otherwise, without the prior permission of the copyright owner. Enquiries should be addressed to the Publishers. © Aberdeen University Press 1987 British Library Cataloguing in Publication Data Oceanography and marine biology: an annual review.—Vol. 25 1. Oceanography—Periodicals 2. Marine biology—Periodicals 551.46′005 GC1 ISBN 0-203-40068-2 Master e-book ISBN

ISBN 0-203-70892-X (Adobe eReader Format) ISBN 0-08-035066-8 (Print Edition) ISSN 0075-3218

PREFACE

1987 sees the publication of the 25th volume of Oceanography and Marine Biology: An Annual Review. The series was started in 1963 by Dr Harold Barnes because he thought the time was appropriate to draw together the work being done in all the marine sciences. The aim was stated in the preface to volume 1: “… to consider annually basic aspects of the field, returning to each at appropriate intervals, to deal with subjects of especial and topical importance, and to add new ones as they arise.” As far as possible this aim has been fulfilled. In the 25 years 254 articles have been published involving over 300 authors, some of whom have contributed more than once. At no time has there been any controversy between the editor and authors; the willingness with which they have acceded to editorial requests has always been appreciated and has made the editor’s task a pleasure rather than a tedious trial. The Annual Review has also been well served by its publishers who have attended to its production with meticulous care and so far have always managed to maintain the annual schedule. Manuscripts continue to be submitted to this series; many experts are still willing and even anxious to accept invitations to contribute to it. The desire to publish in it must reflect its importance and value to marine scientists in general. During the years since the death of Harold Barnes in 1978 I have been fortunate in having the advice of many friends and colleagues including, in particular, Drs A.D.Ansell, R.N.Gibson, and T.H.Pearson. Their help has been, and still is, greatly appreciated. It is hoped that this series of Annual Reviews will continue for many more years to fulfil the aims of its founder editor.

CONTENTS

PREFACE

iv

Phytoplankton Dynamics in Marginal Ice Zones WALKER O.SMITH JR

1

Sampling and the Description of Spatial Pattern in Marine Ecology N.L.ANDREWAND B.D.MAPSTONE

26

Flumes: Theoretical and Experimental Considerations for Simulation of Benthic Environments ARTHUR R.M.NOWELL AND PETER A.JUMARS

70

Larval Settlement of Soft-sediment Invertebrates: the Spatial Scales of Pattern explained by Active Habitat Selection and the emerging Rôle of Hydrodynamical Processes CHERYL ANN BUTMAN

89

Aplysia: its Biology and Ecology THOMAS H.CAREFOOT

139

A Review of the Comparative Anatomy of the Males in Cirripedes WALTRAUD KLEPAL

250

The Benguela Ecosystem. Part IV. The Major Fish and Invertebrate Resources R.J.M.CRAWFORD , L.V.SHANNON AND D.E.POLLOCK

305

The Association between Gobiid Fishes and Burrowing Alpheid Shrimps ILAN KARPLUS

458

The Ecological Impact of Salmonid Farming in Coastal Waters: A Review R.J.GOWEN AND N.B.BRADBURY

508

AUTHOR INDEX

520

SYSTEMATIC INDEX

547

SUBJECT INDEX

559

PHYTOPLANKTON DYNAMICS IN MARGINAL ICE ZONES WALKER O.SMITH, JR Botany Department and Graduate Program in Ecology, University of Tennessee, Knoxville, Tennessee 37996, U.S.A.

INTRODUCTION The marginal ice zone is an oceanographic front in which a transition from dense (those waters completely covered with ice) pack ice to one completely free of ice occurs (Fig. 1). The marginal ice zone is dynamic, responding rapidly to physical forcing; hence, the transition from 10/10 ice cover to open water can be abrupt or occur over hundreds of kilometres. The position of the ice edge can vary widely, with mesoscale variations occurring over the time scale of days and large-scale changes occurring seasonally (see Fig. 2). Furthermore, significant interannual variations occur (Niebauer, 1980; Zwally et al., 1983) which are related to global variations in air-sea interactions. An understanding of the processes which create changes in the ice-edge position will allow successful modelling of the marginal ice zone (both its physical and biological processes) as well as prediction of spatial and temporal variations in its position and gradients of ice concentration within it. This in turn will allow human activities in the region (offshore petroleum exploration, transportation, fisheries development) to increase. Polar regions are characterized by environmental extremes. An obvious major seasonal change occurs in the quantity of incident light, with long periods of darkness followed by a rapid change to continuous irradiance. As a result of the low amounts of incident radiation received annually, the local heat budget of polar regions is negative (i.e. a net flux of heat occurs to the atmosphere), which results in very low seawater temperatures, often near the freezing point. Both light, by virtue of its direct and indirect effects on phytoplankton growth, and temperature, via its overall control of microbial growth (Eppley, 1972), strongly influence autotrophic processes in polar regions. Despite the constraints that these factors (and potentially others) place on phytoplankton growth, they in and of themselves do not produce an environment conceptually different from temperate regions. Recent investigations into the physical and biological characteristics of marginal ice zones have, however, clearly shown that the marginal ice zone possesses attributes which are unique and which generate features which distinguish the ice-edge region from either polar or temperate areas. It has been noted for many years that the ice edge is a locus for activity of all trophic levels. For example, in the Antarctic, krill (Euphausia superba) seem to be associated with the marginal ice zone (Marr, 1962), as are many species of whales which feed upon krill (Mackintosh, 1970). Many species of birds are highly concentrated in the marginal ice zone, such as penguins (Spheniscidae) and snow petrels (Pagodroma nivea) (Ainley & Jacobs, 1981). In the Arctic polar bears frequent the ice edge, feeding on seals which surface in the leads during summer. The causes of the elevated higher trophic level biomass and activity in the marginal ice zone can be related to food abundance, using ice as a refugia from predation, and using ice as a nesting and breeding site (Ainley & Jacobs, 1981). Because the food web at the ice edge is ultimately

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WALKER O.SMITH, JR

Fig. 1.—An aerial image of the marginal ice zone in the Fram Strait: the image was constructed by photography from repeated airplane overpasses; each ‘strip’ is approximately 10 km wide; note the irregular features along the ice edge, including a prominent eddy, ice bands, and meanders; the centre of the eddy is located at approximately 78°48′ N: 2°37′ W, and the mosaic is orientated in a north-south direction; composite provided by Dr R.Shuchman, Environmental Research Institute of Michigan, Ann Arbor, Michigan.

PHYTOPLANKTON IN MARGINAL ICE ZONES

3

dependent on the primary producers, such elevated standing stocks and activities of higher trophic levels must be supported by enhanced primary production within the marginal ice zone. Carbon fixation is conducted by algae attached to the ice and by phytoplankton. Ice algae contribute biogenic material early in the growing season and can reach large concentrations in a spatially confined zone, but the annual primary production associated with the ice when compared with that of the water column plankton is low (Whitaker, 1977; Garrison, Sullivan & Ackley, 1986). Ice-edge phytoplankton blooms (defined as standing stocks greater than those present in the absence of ice) have clearly been shown to be consistent features of marginal ice zones (Hart, 1934; Marshall, 1957; El-Sayed, 1971; McRoy & Goering, 1974; Alexander & Niebauer, 1981; El-Sayed & Taguchi, 1981; Schandelmeier & Alexander, 1981; Smith & Nelson, 1985a, b; Smith, Smith, Codispoti & Wilson, 1985). It is the purpose of this paper to discuss the features of marginal ice zones which make them unique among polar systems and how the oceanographic processes that occur within them influence phytoplankton growth and accumulation. PHYSICAL-BIOLOGICAL INTERACTIONS WITHIN THE MARGINAL ICE ZONE The major feature of the marginal ice zone is the physical presence of ice. The effects of ice are numerous. For example, during its seasonal advance and retreat, the freeze-thaw cycle changes the surface layer’s salinity and density, thereby changing the stability of the upper water column. Melting ice has been shown to create large density gradients in the Arctic (e.g. Marshall, 1957; McRoy & Goering, 1974; Alexander & Niebauer, 1981; Smith et al., 1985) and the Antarctic (e.g. Jacobs, Gordon & Ardai, 1979; Jacobs, Huppert, Holdsworth & Drewry, 1981; Smith & Nelson, 1985a); in contrast, freezing ice and subsequent brine rejection leads to instability and deep vertical mixing (Foster, 1968; Matthews, 1981). Changes in the stability of the upper layers of the ocean can have a rapid impact on plankton growth. The effect of stability on phytoplankton growth was first described by Riley (1942) and Sverdrup (1953), who mathematically derived the relationship between critical depth and the depth of vertical mixing to predict the onset of a temperate spring phytoplankton bloom. At a receding ice edge, low salinity (hence low density) melt-water is released, and the vertical stratification is greatly increased (Marshall, 1957; Alexander & Niebauer, 1981; Smith & Nelson, 1985a). This maintains the phytoplankton in the upper portion of the water column and provides sufficient light for growth to occur. In contrast, increased vertical mixing caused by brine rejection should decrease phytoplankton growth; such an effect has never been directly observed. The effect of melting ice on vertical stability appears to be the major factor in the initiation of a phytoplankton bloom within the marginal ice zone. Ice also modifies the quantity of surface irradiance which reaches the water column. Ice within the marginal ice zone can be either annual (produced within the past winter season) or multi-year ice (present for more than one winter). The optical characteristics of the two types are substantially different, but both greatly attenuate light. In general, pack ice (including any snow cover layer) will absorb from 80–99–95% of surface irradiance (Maykut & Grenfell, 1975; Sullivan, Palmisano & SooHoo, 1984). Any microbial community on or within the ice will further contribute to light attenuation. Despite the fact that the compensation light intensity of phytoplankton photosynthesis in polar waters is low (less than 1 µE·m−2·s−1; El-Sayed, Biggs & Holm-Hansen, 1983; Palmisano et al., 1985), phytoplankton photosynthesis in areas of heavy ice cover is severely restricted by the attenuation of light by sea ice per se. Ice can also influence the properties of the water column by modifying the air-sea interaction. For example, ice shields the water column from the wind’s energy, thereby reducing vertical mixing in areas with extensive ice cover. In areas with less ice cover, the ice moves more rapidly in response to wind than does the ocean’s upper layer, since the frictional drag on the rougher surface is greater than that on the

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ocean. Another effect of ice is the generation of ice-edge upwelling. Upwelling has been observed within the marginal ice zone in the Bering Sea (Alexander & Niebauer, 1981) and in the East Greenland Sea (Buckley et al., 1979; Johannessen et al., 1983), although never in the Southern Ocean. The proximate cause of iceedge upwelling appears to be wind-driven off-ice Ekman transport (Johannessen et al., 1983; Niebauer & Alexander, 1985), and analytical (Gammelsrød, Mork & Røed, 1975) and numerical (Niebauer, 1982) models confirm this. Røed & O’Brien (1983), however, developed a model with a movable ice sheet which predicted surface convergence (rather than divergence) under the same conditions as the models of Gammelsrød et al. and Niebauer. This discrepancy may be the result of the sensitivity of all the models to various physical coefficients describing air-sea-ice interactions (Røed, 1983). Regardless of the conditions which give rise to upwelling, it is clear that such an event would have a significant impact on phytoplankton growth in the marginal ice zone of the Arctic by injecting nutrients into the euphotic zone, particularly during the summer months after nutrients had been depleted. Any occurrence of ice-edge upwelling in the marginal ice zone of the Southern Ocean, however, might be counter-productive, in that nutrients are rarely limiting, even in the densest blooms (Nelson & Smith, 1986). Upwelling in the Antarctic would reduce vertical stability caused by ice-melt, increase vertical mixing, and thereby result in a reduced light environment for phytoplankton. Therefore, if ice-edge upwelling were to occur in the Antarctic, a decrease in phytoplankton productivity and biomass might result. The ice edge also plays a rôle in the formation of eddies. They have been frequently observed in the marginal ice zone of the Fram Strait region (Johannessen et al., 1987; Shuchman et al., 1987; Manley et al., 1987), and there have been a number of mechanisms proposed to explain their genesis. For example, Hakkinen (1986) modelled the generation of eddies via a variable ice cover along an ice edge, where eddies are generated by a differential wind-induced circulation of water and ice. Topographical anomalies also have been shown to induce eddy formation (Smith, Morison, Johannessen & Untersteiner, 1984), and because the bottom topography in the vicinity of the ice edge of the Fram Strait is complex, eddies are frequently shed. Eddies are important biologically to the marginal ice zone of the Fram Strait for a number of reasons. First, they transfer heat into and out of the ice, increasing the rate of ice ablation and altering the stability of the water column. Secondly, they can move parcels of water under the ice (Manley et al., 1987), thereby reducing the amount of light available in the water column and light-dependent phytoplankton productivity. Finally, eddies induce vertical motion within their boundaries. The location within the eddy of the vertical motion is dependent on the direction of horizontal flow (cyclonic vs. anticyclonic) and whether the eddy is accelerating or decelerating, but in a manner similar to ice-edge upwelling, eddies will inject nutrients into the euphotic zone, and result in a significant stimulation of phytoplankton growth and accumulation during certain periods of the year. FACTORS INFLUENCING PHYTOPLANKTON GROWTH IN THE MARGINAL ICE ZONE Because marginal ice zones occur in different oceanic basins, each with unique physical, chemical, and biological features, it is difficult to generalize among the various ice-edge systems. Therefore, a comparative analysis of different marginal ice zones may be instructive. Four regions with well studied ice edges are included in this analysis: the Bering and Chuckchi Seas, the Fram Strait and the Barents Sea, the Weddell Sea, and the Ross Sea.

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5

MARGINAL ICE ZONES OF THE ARCTIC Bering and Chuckchi Seas The Bering Sea is a shallow continental shelf which is normally completely covered during the winter, but completely ice free by early June (Fig. 2). In some years the ice cover retreats over 1000 km (Niebauer, 1982), with the mean distance of melt-back along 170° W being 920 km (Konishi & Saito, 1974). The retreat occurs over a period of three months, so that the mean rate of ice retreat is 10 km per day. There are also variations in rate of retreat during the three months; mean distances of retreat are 135 km in March, 460 in April, and 325 in May (Konishi & Saito, 1974). It has been shown that a large phytoplankton bloom occurs within the Bering Sea marginal ice zone (Alexander & Niebauer, 1981; Schandelmeier & Alexander, 1981), and that this bloom is related to the creation of a vertically stable region in the vicinity of the receding ice. Extremely high concentrations of phytoplankton biomass occur in the bloom (chlorophyll a concentrations of more than 25 µg·l−1) and completely deplete the nitrate within the euphotic zone (Alexander & Niebauer, 1981). The bloom appears to be restricted to the zone of increased stability, but the distance of this stratified region is variable, ranging from 25 km (Alexander & Niebauer, 1981) to 100 km from the ice edge (Niebauer, Alexander & Cooney, 1981). The difference in horizontal extent is a function of the rate of meltwater input, the wind direction and speed, and the local current patterns. Given that there is nutrient depletion within the bloom, further new production (sensu Dugdale & Goering, 1967; Eppley & Peterson, 1979) requires introduction of nitrate into the surface waters. This apparently is accomplished by two mechanisms in the Bering Sea. The first is ice-edge upwelling (Alexander & Niebauer, 1981; Niebauer, 1983; Niebauer & Alexander, 1985), and the second is deep vertical mixing, often enhanced at the 50 m isobath by tidal forces (Niebauer & Alexander, 1985). Without the introduction of nitrate into the euphotic zone after the ice-edge bloom, significant primary production would not be expected to be sustained. That is, the ice-edge bloom would change the strength and timing of primary productivity, but would not increase the total annual carbon fixation. It is clear that mixing and upwelling do occur; their frequency is, however, unknown, and hence an estimate of their impact on phytoplankton blooms in the marginal ice zone is not yet possible. McRoy & Goering (1976) estimated the importance of primary production within the ice-edge system, and concluded that it accounts for approximately 40% of the annual production on the continental shelf. More recently, Alexander (unpubl.) has measured primary production within the marginal ice zone (Table I); her data show the intensity of the bloom during ice retreat and also the expected seasonal cycle in the open waters. When the ice-edge and open-water productivity is integrated through the entire year, approximately 50% of the total production is contributed by the ice-edge system (Table I). Recent estimates of primary productivity in the western portion of the Bering Sea by Sambrotto, Goering & McRoy (1984) indicated that production in that portion of the continental shelf had been significantly under-estimated because the stimulation of growth by nutrient input via cross-shelf flow had not been taken into account. Regardless of the absolute level of open-water production, it is clear that ice-edge phytoplankton blooms in the Bering Sea provide a highly significant contribution to the annual rates of carbon fixation in the region. It is also possible to estimate the impact of marginal ice zone-related blooms at one location by combining data on the extent of the bloom, the duration, the area covered, and the primary productivity within the bloom (Smith & Nelson, 1986). Because the magnitude of the production is variable through time (Table I), the localized impact is also variable. At a location on the outer shelf, primary production is limited to only the early melt-back period (March), and hence production is small (about 0·32 g C·m−2·day −1).

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Fig. 2.—The average maximum and minimum extent of pack ice in the Northern and Southern Hemispheres: data for Southern Hemisphere ice limits from Zwally et al. (1983); data for Northern Hemisphere after Dietrich, Kalle, Krauss & Siedler (1980).

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TABLE I Primary productivity of open water locations and within the ice-edge phytoplankton bloom: values in brackets are linear extrapolations; data are means of 14C-productivity measurements taken on the continental shelf south of 60° N; data by courtesy of V.Alexander, University of Alaska Primary productivity Ice edge Daily C·m−2·day−1)

Month

(g

January February March April May June July August September October November December Total (g C·m−2·yr−1)

– – 0·32 3·16 6·60 – – – – – – –

Open ocean Monthly (g

C·m−2)

– – 4·03 94·80 204·60 – – – – – – – 309·32 (50·1%)

Daily (g

C·m−2·day−1)

[0·10] [0·11] 0·13 2·05 4·00 1·50 [1·10] 0·69 [0·48] [0·27] 0·06 [0·07]

Monthly (g C·m−2) [2·85] [3·15] 4·03 61·50 124·00 61·50 [33·95] 21·39 [14·40] [8·37] 1·80 [2·40] 308·44 (49·9%)

Because ice retreat occurs at a rate of 10 km·day−1 and the maximum extent of the bloom is 100 km (Niebauer et al., 1981), a bloom would be maintained at one location at most for ten days, and primary productivity contributed by the bloom would be 7·35 g C·m−2. This calculation assumes that the major factor in initiating and sustaining ice-edge blooms is the introduction of melt-water from the retreating ice edge (Alexander & Niebauer, 1981; Smith & Nelson, 1986). Similarly, for a mid-shelf location influenced by the ice-edge bloom in April, the primary productivity at one point would be 20·6 g C·m−2, and for a shallow inner-shelf area which supports a bloom in May, the primary productivity would equal 63·0 g C·m −2. Given that total production on the continental shelf is estimated to be at least 300 g C·m−2·yr−1 (Sambrotto et al., 1984), ice-related production is a modest source of organic matter on the outer shelf. Clearly, as the bloom intensifies it becomes more significant to the yearly production for a single location, and the major impact occurs in areas uncovered during late spring. Significant interannual variations in the extent and concentration of ice cover in the Bering Sea have been reported by Niebauer (1980). These are apparently related to basin-wide air-sea interaction anomalies, in that during years when significant ENSO (El Niño-Southern Oscillation) events occur (Barber & Smith, 1981), ice cover is much less extensive than in ‘normal’ years. During ‘warm’ years the Aleutian low pressure system shifts southward, bringing warmer air temperatures over the Bering Sea which to a large degree restrict the development of ice. A one- or two-year time lag occurs between an ENSO event and the Bering Sea temperature anomaly, indicating that such large scale interactions are of substantial magnitude (Niebauer, 1984). Because the ice-edge blooms in the Bering Sea are related to vertical stability induced by melt-water, it is clear that during warm years the effect of these blooms on the annual productivity cycle is lessened; the magnitude of the interannual productivity fluctuations is, however, unknown.

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The ice-edge system of the Chuckchi Sea has been investigated to a lesser extent than that of the Bering Sea. Significant ice-melt does not begin until June, and large portions of the region remain ice-covered the entire year. Those areas that are uncovered during ice-melt exhibit strong stratification (Fleming & Heggarty, 1966; Hameedi, 1978), and nutrient levels are extremely low during the growing season. For example, nitrate was virtually undetectable within the euphotic zone during the summer at stations near the ice edge (Hameedi, 1978), and surface chlorophyll a levels were moderate (ranging from 0·3–0·8 µg·l−1). An extreme chlorophyll maximum was observed at nearly all stations in the marginal ice zone, being 1–2 orders of magnitude greater than surface levels. This maximum was associated with the base of the pycnocline; in view of the surface layer’s depletion of inorganic nitrogen, the maximum probably formed from surface biomass production and subsequent passive sinking of intact, metabolically active cells. The mean integrated euphotic zone chlorophyll a value was 147·1 mg·m−2, which represents nearly the entire water column in this shallow continental shelf region. Primary productivity ranged from 150 to 6857 mg C·m−2·day−1, with a mean of 1914 mg C·m−2·day−1 (all values reported in Hameedi, 1978, were doubled, since the original data were given on a half-day basis). These values are among the largest ever reported for a marginal ice zone and may be partially reflective of the elevated temperatures at which incubations were conducted. For example, one station had water column temperatures ranging from 0·9 to –1·6°C, but productivity incubations were conducted at 10·2°C. In view of the susceptibility of polar phytoplankton to significant influences of temperature (Neori & Holm-Hansen, 1982), the productivity data of Hameedi (1978) should be viewed with caution until substantiated. From Strait and Barents Sea Although geographically close, the marginal ice zones of the Fram Strait and the Barents Sea are in many ways dissimilar. For example, the Barents Sea is quite shallow (less than 100 m), and its southern portion is permanently ice-free due to the advection into the area of warm (greater than 3°C) North Atlantic water (Sverdrup, Johnson & Fleming, 1943). It also is influenced by freshwater input from rivers. The Fram Strait region, however, is isolated from continental influences and is at least 2000 m deep in most places. The bottom topography is complex and apparently generates a quasi-permanent eddy centred around the Malloy Deep (Wadhams & Squire, 1983; Smith et al., 1984). The area is influenced not only by the northerly flowing North Atlantic water but also by the southerly outflow from the Arctic Basin, the East Greenland Current (Paquette, Bourke, Newton & Perdue, 1985). The boundary between these waters, the Polar Front, is a region of rapidly changing temperature and also separates distinct biological communities (Smith et al., 1985). The marginal ice zone of each region is also different. The ice edge in the Barents Sea begins breakup in May and retreats approximately 475 kilometres by early August (Rey & Loeng, 1985); ice retreat is controlled by localized solar heating. The Fram Strait marginal ice zone is relatively invariant in space, being controlled by the position of the Polar Front (Paquette et al., 1985). When ice is blown or advected over the warmer North Atlantic water, melting occurs independent of the time of year; similarly, if the ice retreats so that it covers only waters originating from the East Greenland Current, little ablation will occur. The distance between the mean minimum and maximum extent of ice in the Fram Strait at 80° N is 120 km (Vinje, 1977). During September-October primary production within the marginal ice zone in the Barents Sea averaged 23·9 mg C·m−2·h−1 (equivalent to 287 mg C·m−2·day−1 if a 12-h day-length is assumed; Heimdal, 1983). During this study the surface nitrate values were generally low, ranging from 0·1–1·6 µM; phosphate and silicate were also low. All stations exhibited marked vertical stability, with the pycnocline generally located near 20 m, although there was some variation in the depth of the mixed layer (from about 12–25 m).

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Chlorophyll a concentrations were always less than 2·0 µg·l−1. Because pre-bloom concentrations of nitrate in the region are approximately 10 µM (Anderson & Dyrssen, 1981), it is obvious that substantial production and export had occurred prior to the study. No discernible increases in nutrient concentrations (via ice-edge upwelling), phytoplankton biomass or primary production were noted within the marginal ice zone, although the upward movement of isopycnals was observed in at least one section from which biological data were taken (Johannessen et al., 1983). Surface productivity and chlorophyll values were noted to range from 0·8–57·8 mg C·m−3·day−1 and 0.2–1.6 µg·l−1, respectively, in the western Barents Sea (Vedernikov & Solov’yeva, 1972); however, the stations occupied were very close to the coast and may have been influenced by continental processes. Rey & Loeng (1985) also studied the production within the marginal ice zone of the Barents Sea. They found that primary productivity was greatest during the spring (1465 mg C·m−2·day−1) but decreased to much lower rates in summer (239–495 mg C·m−2·day−1) as a result of the restriction of nutrient input due to strong vertical stability. No biological manifestations of upwelling were noted, probably because the sampling period encompassed only eight days in the period May through October. A phytoplankton bloom associated with the ice edge was observed and found to follow the retreat of the ice through the entire summer; the quantity and vertical distribution of phytoplankton biomass was, however, much different in the summer than in the spring, in that the summer distribution was characterized by marked subsurface maxima. This study was one of the few that was able to follow the temporal progression of a bloom associated with the marginal ice zone, and confirmed for this region the overriding importance of vertical stability not only to the timing of bloom initiation but on the limitation of primary production by restriction of nutrient influx. The Fram Strait marginal ice zone is, as stated, different in many respects from that of the Barents Sea. It is, however, similar in that initial nutrient concentrations prior to a spring bloom are alike (Anderson & Dyrssen, 1981), chlorophyll a concentrations within the bloom are somewhat similar (Smith et al., 1985; Smith, Baumann, Wilson & Aletsee, 1987), and the degree of vertical stability is similar in both. Smith et al. (1985) studied the distribution of nutrients and phytoplankton in relation to physical processes and found that nitrate concentrations were reduced in the euphotic zone to below 0·3 µM when adequate light was available for phytoplankton growth. Chlorophyll a concentrations reached a maximum of 11 µg·l−1. Based on the observed chlorophyll-nitrate relationship, a number of stations appeared anomalous (i.e. ‘excess’ chlorophyll a was present relative to the nitrate levels found). This suggested the introduction of nitrate into surface waters by some mechanism which subsequently stimulated phytoplankton growth and/or accumulation. In a study at the same location approximately one year later, Smith et al. (unpubl.) found slightly elevated levels of chlorophyll a in the same area, and that these were associated with lowered concentrations of nitrate (Fig. 3). That area is the site of a quasi-permanent eddy described by Wadhams & Squire (1983) centred in the vicinity of the Malloy Deep (a topographic depression). Because eddies can impart significant vertical motion within their structure, it was concluded that the increased chlorophyll was a result of growth enhanced by eddy-induced vertical flux of nutrients. Furthermore, eddies are a common feature of the marginal ice zone in the Fram Strait (Shuchman et al., 1987), apparently being generated not only via conservation of vorticity (i.e. induced by topographic anomalies; Smith et al., 1984) but by winds (Hakkinen, 1986), and variations in ice concentration (Hakkinen, 1986). The duration of the transient eddies in the vicinity of the ice edge appears to be at least 20 days (Shuchman et al., 1987), which is definitely long enough for an increase in phytoplankton growth and biomass to occur. The spatial extent of the eddies is variable. The Malloy Deep eddy appears to be approximately 100 km in diameter at the surface (although its shape is by no means symmetrical; Fig. 3), and its horizontal motion extends to the bottom (Johannessen et al., 1987). The transient eddies, those associated with the ice edge or formed by instabilities generated by

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the Malloy Deep flow, are smaller (about 30 km in diameter) and may have their vertical and horizontal motions confined to the upper 200 m (Manley et al., 1987). Determinations of primary production in the Fram Strait region are scarce and the seasonal cycle of production is poorly known. Smith et al. (1987) found large variations in productivity across the ice edge, with the largest values (1718 mg C·m−2·day−1) occurring within 10 km of the ice edge. The mean productivity was 426 mg C·m−2·day−1, with minimum values of 29 mg C·m−2·day−1 at a station 60 km from the ice edge in waters totally covered with ice and 178 mg C·m−2·day−1 at a station 50 km from the ice edge in open water. Mesoscale variations in phytoplankton biomass were also noted (Fig. 4) and appeared to be in part related to the movement of the ice edge through time. The data do, however, clearly show that the ice edge consistently is a locus of phytoplankton biomass in the region. Interannual variations in phytoplankton productivity in the Fram Strait marginal ice zone (as well as all ice-edge systems) are basically unknown, despite the well-known yearly variations in ice extent and concentration (e.g. Zwally et al., 1983). Two studies completed in 1983 and 1984 (Smith et al., 1985; Smith et al., 1987), however, suggest the potential for significant differences between years. The 1983 study took place in July-August, whereas the 1984 study was conducted in June-July, so that some seasonal bias in the comparison of data is unavoidable. Chlorophyll a concentrations within the Malloy Deep eddy in 1983 were greater than 300 mg·m−2; in 1984 integrated chlorophyll a values never exceeded 100 mg·m−2 in the same region (both data sets integrated through 100 m). Nutrient values were slightly higher in 1984, but still were less than 1 µM in surface waters during both years. The major difference noted was a difference in temperature; temperatures in 1983 within the eddy were greater than 4°C, whereas in 1984 temperatures were less than 2°C. Vertical density gradients were large in both years. It is unclear whether this difference was related to differences between the two years or whether it was a manifestation of a seasonal effect. Fig. 3 cont. The Fram Strait region, like the Bering Sea, is interesting in that ice-melt results in active phytoplankton growth which ultimately becomes nutrient-limited, and that further new production is dependent on introduction of nitrate into the euphotic zone. Therefore, new productivity is dependent on physical processes such as upwelling or eddy formation, and the annual primary productivity budget, particularly during the summer (and presumably nutrient-limited) months, is dependent on the frequency of these physical events. Because the ice completely disappears by June in the Bering Sea, other processes not related to the presence of ice such as tidal mixing and storm events must replenish nutrients. In the Fram Strait ice-related processes, however, occur throughout the entire growing season. A complete understanding of the phytoplankton dynamics of the Fram Strait will require knowledge of the duration, extent, and magnitude of vertical fluxes generated by the important physical processes (eddies, ice-edge upwelling) found in the marginal ice zone of this region. MARGINAL ICE ZONES OF THE SOUTHERN OCEAN Weddell Sea The Weddell Sea is completely ice-covered for much of the year, with the pack ice extending to near the Antarctic Convergence. The circulation within the Weddell Sea is essentially a large cyclonic gyre, and significant mixing with other water masses (Drake Passage water, Scotia Sea water) occurs at the gyre’s northern extension. Ice retreat begins in October, with ice-melt and compaction occurring at the northern edge and in some years in the centre of the gyre (Zwally et al., 1983). At the minimum extent of the ice

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Fig. 3.—The distribution of temperature (a), σt (b), nitrate (c), and chlorophyll a (d) in a section taken across the Fram Strait: the influence of the eddy centred over the Malloy Deep can be seen in the density, nutrient and chlorophyll distributions (from Smith et al., unpubl.).

during February, ice remains along the coast of the Antarctic Peninsula (see Fig. 2), but the rest of the region is generally ice-free. The biology of the Weddell Sea marginal ice zone has been investigated since the early DISCOVERY expeditions (Hart, 1934, 1942). Hart (1942) separated various phytoplankton taxa into different groups, one of which included ice-edge forms. He also noted that this group included species whose relative contribution to the seasonal phytoplankton maximum was greatest. Thus, even early descriptive works indicated the biological importance of marginal ice zones. In 1968 an extensive bloom within the pack ice was observed by El-Sayed (1971). Chlorophyll a concentrations of 190 µg·l−1 were measured at the surface, and the bloom was overwhelmingly comprised of the diatom Thallasiosira tumida. The area covered by the

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bloom was estimated to be 15500 km2. Unfortunately, only two stations were occupied which had chlorophyll a values at the surface greater than 2·0 µg·l−1, and water column characteristics were reported for only one of those. It is clear that the accumulation was not due to in situ growth, since nitrate concentrations even if quantitatively converted into phytoplankton biomass could not result in such large chlorophyll a levels. There must have been some concentration of cells from the ice and/or water column during the study. It is most probable that the massive accumulations of diatoms were the result of freezing ice which concentrated the algae from within the top 10 m of the water column by the mechanism proposed by Garrison, Ackley & Buck (1983). Even if ice crystals concentrated all the cells from the upper 10 m to the top few cm of the water column, the surface layer still must have had a chlorophyll a concentration of approximately 15–20 µg·l−1. It should also be noted that El-Sayed (1971) reported large concentrations of pancake ice in the area studied, indicating that water was actively freezing. Therefore the bloom was indeed unusual in its mode of formation, but its importance to seasonal production was probably small.

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Fig. 4.—The distribution of integrated (from 0–100 m) chlorophyll a in the marginal ice zone of the Fram Strait, June, 1984: the ice-edge position indicated by a dashed line; the marginal ice zone is the major site of growth and accumulation of phytoplankton biomass in this region.

The first detailed study of the marginal ice zone in the Southern Ocean was conducted in 1977 by ElSayed & Taguchi (1981). The cruise investigated the variations in phytoplankton biomass and productivity along the ice edge in relation to local physical and chemical factors. Large differences in autotrophic biomass and productivity were noted between the northern (north of 72° S) and southern sectors; specifically, mean integrated chlorophyll a concentrations were seven times greater in the southern sector (31·6 compared with 4·4 mg·m−2), and primary productivities were four times as great (0·41 compared with 0·10 mg C·m−2·day−1). No difference between the degree of ice cover between the two sectors was observed, and the vertical stability profiles showed no obvious systematic difference. The large-scale biological differences were attributed to water column stability, grazing, and proximity to land masses. During the same study, an analysis of the quantity, distribution, and taxonomic composition of ice-algae was also completed (Ackley, Buck & Taguchi, 1979). In general, low amounts of algal biomass within ice were found; maximum chlorophyll a concentrations were 4·3 µg·l−1 (that value represents the integrated pigment concentration for an entire ice core in which subsamples had been melted and quantified). The potential contribution of ice-algae to overall autotrophic biomass appeared to be larger in the northern portion of the Weddell Sea, not as a result of increased growth and production of ice-algae, but rather as a result of lower levels of phytoplankton biomass observed within the water column. El-Sayed & Taguchi (1981) did, however, describe the potential for biological interactions between the communities in the ice and those in the water column. The phytoplankton dynamics of the Weddell Sea marginal ice zone has recently been studied as part of AMBRIEZ (Antarctic Marine Ecosystem Research at the Ice-Edge Zone). The purpose of this project is to describe the seasonal patterns at various trophic levels within one marginal ice zone. In November– December, 1983, the biomass and productivity of phytoplankton was determined in transects normal to the ice edge to determine the level of activity at the onset of ice retreat. In March, 1986, similar transects were occupied to compare the standing stocks and growth of an ice edge which is advancing with one which had been retreating (Fig. 5). The first cruise started approximately six weeks prior to the maximum solar angle (most direct radiation), and the second approximately nine weeks after the equinox. Vertical density differences observed during the 1983 cruise were not as great as those in 1986 (Nelson et al., 1987; Muench

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& Husby, unpubl.). The mean depth of the mixed layer in spring was 55 m; in autumn it averaged 35 m. In fact, the vertical stratification observed in the autumn was nearly constant and independent of ice cover (Fig. 5), whereas the mixed layer depths in spring were highly variable, ranging from 15 to 130 m, and spatially correlated with the retreating ice edge. These conditions persisted despite (or as a result of) the surface freezing and melting that occurred during the autumn and spring, respectively. Chlorophyll a concentrations were much greater in spring than in autumn, with mean euphotic zone values in the spring of 3·1 µg·l−1 (maximum value 12·9 µg·l−1), and in the autumn the euphotic zone average being 0·18 µg·l−1 (maximum observed value 1·80 µg·l−1). Maximum standing stocks in the spring occurred in the vicinity of the ice edge, but no clear spatial correlation between the vertical stability and chlorophyll distributions in autumn was found. Primary productivity values paralleled the biomass distribution (Fig. 5), with integrated (through the depth to which 0·1% light penetrated) means for spring and autumn being 571 and 200 mg C·m −2·day−1, respectively. Thus, the conceptual model that an ice-edge phytoplankton bloom is initiated by melt-water induced stratification and is dissipated by the breakdown of this stability appears only partially correct for the Weddell Sea ice edge, in that strong stability was still observed in autumn but elevated phytoplankton standing stocks were absent. The water column characteristics were qualitatively similar to those found in the Scotia Sea (Rönner, Sörensson & Holm-Hansen, 1983; Glibert, Biggs & McCarthy, 1982), where mixed layer depths of approximately 50 m occurred in the absence of ice. The low biomass and primary productivity encountered in autumn, 1986 may have resulted from low ambient light conditions, substantial grazing pressure or the dependence on ammonium as a nitrogen source for growth; regardless of the cause(s), it is clear that an ice-edge bloom in this region did not persist into the austral autumn. Ross Sea The Ross Sea is hydrographically less complex than the Weddell Sea in that it is somewhat more isolated from the large-scale circulation patterns of the Fig. 5 cont. Fig. 5 cont. entire Southern Ocean. Water movement within the Ross Sea can be described as a cyclonic gyre (Ainley & Jacobs, 1981), with water flowing along the coast of Victoria Land towards the equator and along the Ross Ice Shelf towards Victoria Land. The entire sea is ice-covered until late October, when open-water areas begin to appear along the edge of the Ross Ice Shelf at about 175° E (Zwally et al., 1983). The open areas continue to expand equatorward and towards Victoria Land until the entire region is ice-free east of 160° E. Ainley & Jacobs (1981) reported that in December areas with little or no ice exhibited the colouration and light attenuation coefficients characteristic of water with large accumulations of phytoplankton (no chlorophyll a measurements were made), and found that these bloom areas were confined to shelf waters which had been recently uncovered via ice-melt. Holm-Hansen, El-Sayed, Franceschini & Cuhel (1977) occupied a number of stations in the Ross Sea, but none of them can be considered to be within the marginal ice zone. Their data do provide, however, a good estimate of the background (i.e. areas not recently influenced by ice) biomass and productivity in the Antarctic region. Mean integrated (through 200 m) chlorophyll a and carbon assimilation values were found to be 10·4 mg·m−2 (range 0·73–30·8 mg·m−2; n=16) and 140 mg C·m−2·day−1 (range 40–294 mg C·m−2·day −1; n=23), respectively. El-Sayed, Biggs & Holm-Hansen (1983) investigated the nutrient and phytoplankton distributions in the vicinity of the Ross Ice Shelf during January but, again, these locations cannot be considered to be part of the marginal ice zone at the time of the study because any effects of the pack ice had been dissipated. The mean integrated (through the euphotic zone) chlorophyll a value was 19·1 mg·m−2,

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Fig. 5.—The distribution in 1983 (a, c, e) and 1986 (b, d, f) of density (as sigma-t), chlorophyll a, and primary productivity from sections normal to the ice edge in the Weddell Sea: the 1983 section was conducted in the austral spring (November) and was orientated along a north-south axis at about 61° S; the 1986 section was completed in autumn (March) and was orientated in an east-west direction at 64° S: data for a and c from Nelson et al. (1987); data in b courtesy of Drs R.Muench and D. Husby; data in d, e and f from W.Smith (unpubl.).

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slightly higher than that reported by Holm-Hansen et al. (1977). El-Sayed & Turner (1977) reported a mean of 20·2 mg·m−2 for chlorophyll a concentrations within the euphotic zone in the Ross Sea.

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The distribution of phytoplankton biomass, productivity, and nutrient uptake at the ice edge off Victoria Land in January–February, 1983 was investigated by Smith & Nelson (1985a, b). Transects occupied

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normal to the ice edge found that a large freshwater lens was present at the surface which created a large density gradient within the water column, reducing the mixed layer depth to less than 20 m. The spatial distribution of phytoplankton biomass (as evidenced by chlorophyll a, particulate carbon, particulate nitrogen, biogenic silica, and cell numbers) was highly correlated with the spatial extent of the zone of reduced salinity (and density). At locations removed from the ice-melt influence, vertical stability and phytoplankton biomass decreased markedly. The bloom was confined to an area about 250 km from the ice edge. Some variations in biomass along the ice edge were noted (Smith & Nelson, 1985b) and were attributed to fast-ice effects on the source water. The stratification induced by ice-melt was observed in all transects occupied. Because a number of estimates of biomass were measured concurrently in this study, it provided insights into what may be unusual adaptations of some species of Antarctic phytoplankton. Chlorophyll concentrations in the mixed layer of the bloom averaged 3·7 µg·l−1 (Smith & Nelson, 1986). Particulate carbon levels for the same samples averaged 39·5 µmol·l−1; therefore, the carbon/chlorophyll (w/w) ratio was 118. Such a value is extremely high when compared with temperate and tropical systems, and may reflect the low light conditions within this bloom (due to self-shading). Other studies, however, have also observed elevated C/Chl. ratios (Li, 1980; Sakshaug & Holm-Hansen, 1984), so it is possible that this is an adaptation to the low temperatures encountered in polar waters. The biogenic silica concentrations (a measure of the diatom, silicoflagellate, and radiolarian opaline material) were the highest ever measured in any ocean, averaging 24·4 µmol·l−1 within the bloom’s mixed layer (Smith & Nelson, 1986); in fact, the concentrations observed were higher by a factor of two than the previous maxima (Smith & Nelson, 1985a). The biogenic material was composed almost entirely of diatoms (Wilson, Smith & Nelson, 1986). The silica/carbon molar ratio was 0·60; normal oceanic diatoms have ratios of approximately 0·12 (Brzezinski, 1985). Therefore, the bloom contained only moderate levels of chlorophyll, very high levels of particulate carbon, and massive amounts of biogenic silica. If observations had consisted of only chlorophyll, an erroneous impression of the magnitude of the bloom would have been obtained. Microscopic examination of whole water samples indicated that extremely low levels of recognizable detrital material were present; nearly all of the observable particles were intact diatoms. Primary productivity within the bloom averaged 960 mg C·m−2·day−1 (Wilson et al., 1986), and outside of the high stability region it averaged 617 mg C·m−2·day−1. Integrated (through the euphotic zone) chlorophyll a concentrations for the bloom stations averaged 63·7 mg·m−2 (Smith & Nelson, 1985b). Species composition within and outside the bloom was similar (Smith & Nelson, 1985a), and growth rates within and outside the bloom were also similar. This was interpreted as evidence that at the edge of the bloom, the depth of vertical mixing increased, distributing the biomass through the entire water column, and hence over short time periods no difference in species composition would be noted. Furthermore, growth rates were similar because both areas were low-light environments; within the bloom the high standing stocks of phytoplankton caused self-shading, and outside the bloom deeper vertical mixing (the average mixed layer depth outside the bloom was 40 m) reduced the average light environment encountered by the phytoplankton. Few studies have investigated the form of inorganic nitrogen removed by phytoplankton within marginal ice zones of the Southern Ocean. Outside the marginal ice zone it has been found that nitrate provides anywhere from 5 to 50% of the total nitrogen required for growth (Rönner et al., 1983). It is this percentage of primary production that can be converted to higher trophic level biomass and is available for export (Dugdale & Goering, 1967; Eppley & Peterson, 1979). In view of the massive amounts of nitrate that are almost always present in surface waters in the Antarctic, such extensive use of recycled nitrogen (ammonium) is somewhat surprising, although it has been suggested on the basis of nutrient utilization

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patterns (Kamykowski & Zentara, 1985). Nelson & Smith (1986) measured ammonium and nitrate uptake within the marginal ice zone of the Ross Sea and found that on average 65% (range 32–95%) of the primary production was supported by nitrate. Such extensive amounts of new production (i.e. exportable to higher trophic levels or as particulate flux out of the euphotic zone) are among the highest measured values to date in the ocean, and may help explain why marginal ice zones are often the site of large accumulations of herbivores (krill, copepods, baleen whales, salps), birds (penguins, fulmars, petrels), fishes (myctophids), and marine mammals (seals, whales). IMPACT OF ICE-ALGAL COMMUNITY ON ICE-EDGE PHYTOPLANKTON BLOOMS Numerous species of algae grow on and within the pack ice of both polar regions (Horner, 1976; Horner & Schrader, 1982; Palmisano & Sullivan, 1983; Garrison, Sullivan & Ackley, 1986). It is now clearly established that the ice-algae begin active growth prior to any phytoplankton bloom by virtue of their relative position in the light field (Matheke & Horner, 1974), and upon the disintegration of the ice, the algae are released into the water column. It has been suggested that the ice-algae, by virtue of their extreme biomass (chlorophyll a concentrations within a small section of ice often are greater than 100 µg·1−1; Bunt & Lee, 1970; Whitaker, 1977; Palmisano & Sullivan, 1983), can give rise to bloom conditions within the marginal ice zone (Meguro, Ito & Fukushima, 1967). Recent observations have indicated that ice-algae sink very rapidly (Horner & Schrader, 1982) and that the residence time for most species within the water column is short. The fact that most ice-algae do not remain suspended for long time periods does not, however, preclude selected species from actively growing in the highly stratified upper layers of marginal ice zones. Wilson et al. (1986) found that Nitzschia curta, a pennate diatom usually found as a member of the ice-algal community, dominated the bloom in the western Ross Sea. N. curta also occurred in the ice, but it was not as common in the ice as it was in the water. Furthermore, micro-autoradiographic studies confirmed that it was actively photosynthesizing in the water column. Wilson et al. (1986) suggested that N. curta was released during ice-melt, selectively seeded the stratified surface waters, and accumulated to produce the large biomass observed in the study. Garrison & Buck (1985) used a statistical approach to determine the similarity in taxonomic composition of ice-algal communities and water column phytoplankton. They found that there was a significant overlap in taxa between ice samples and plankton taken from water directly below the ice, and suggested that the under-ice pelagic populations are derived from those in the ice. In a similar fashion, phytoplankton are concentrated from the water column during the formation of frazil ice (Garrison et al., 1983). Ice in the Weddell Sea consists of 50–70% frazil ice (Clarke & Ackley, 1984) which forms as the surface loses heat to the atmosphere i.e. where polynias and leads remain open due to wind action. Therefore, substantial amounts of frazil ice can form in energetic environments. As ice crystals form, they rise to the surface (their formation is limited to approximately the upper 10 m), scavenging and concentrating algae as they rise. In such conditions a strong species similarity would be expected. During conditions of ice retreat, a bloom forms however, in the wake of the ice; that is, there should be a spatial separation of the ice and the maximum concentration of bloom biomass. The magnitude of the spatial separation should depend on the rate of ice retreat and water-column seeding. In the Bering Sea, where rates of retreat are rapid, ice-algal communities and bloom communities are strikingly dissimilar (Schandelmeier & Alexander, 1981), whereas in the Ross Sea there seemed to be more overlap (Wilson et al., 1986). The degree to which icealgae serve as innocula for the surface waters undoubtedly depends not only on the input rate of algal cells, but on the. physical characteristics of the ice and the surrounding oceanographic environment.

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PARTITIONING OF PRIMARY PRODUCTIVITY WITHIN MARGINAL ICE ZONES Ice-edge phytoplankton blooms are conspicuous mesoscale features of marginal ice zones, but an assessment of their importance to food-web dynamics and biogeochemical cycles has just begun. Just as numerous physical, chemical, and biological differences exist among the marginal ice zones studied to date, there is no reason to expect that the fate of ice-edge primary productivity will be the same among these various systems. The Bering Sea ice-edge bloom appears to contribute to a demersal food web. Little phytoplankton biomass appears to be utilized by herbivores within the water column during the life of the bloom (Alexander, 1980; Alexander & Niebauer, 1981), but rather sinks to the benthos where it is ingested, remineralized and/or resuspended during storm events. Walsh et al. (1985) suggest that a significant portion of primary production of the Bering Sea is advected and deposited in the sediments of the Chuckchi Sea, but the proportion of the ice-edge production being deposited is unknown. In the Fram Strait region, total annual marginal ice zone primary production appears to be less than in other systems, and zooplankton (large calanoids such as Calanus hyperboreus and C. finmarchicus) appear to remove 80% of the daily primary productivity despite their relatively low biomass (S.Smith, pers. comm.). The region also supports an active pelagic microbial food web (nanoplankton, bacterioplankton, and heterotrophic flagellates). Some biogenic material is lost from the euphotic zone via sinking of particles. Wefer & Honjo (1985) found that at 2100 m the maximum vertical flux occurred in September to October and was not temporally correlated with the maximum period of surface productivity. They also found that the flux of organic matter in autumn reached 7 mg·m−2·day−1, which is approximately 2·4% of the surface production. During the spring and summer seasons, a much smaller percentage of organic matter reaches the deep sea, indicating that surface production is being efficiently cycled at most times of the year. The Ross Sea marginal ice zone may be quite unusual among ice-edge systems in its manner of partitioning of biogenic material. There are large deposits of biogenic particulates in the region, and it has been assumed that these deposits reflect a large surface productivity (Noriki, Harada & Tsunogai, 1985). These deposits of siliceous material are large enough to make the area one of the largest locations for silica removal in the global silica budget (DeMaster, 1981). Carbon productivity for much of the growing season is low, with the exception of the ice-edge bloom which presumably occurs when- and wherever vertical stability results from melting ice. When the annual bloom-related carbon production is calculated from the data of Wilson, Smith & Nelson (1986) and compared with the total yearly production calculated by HolmHansen et al. (1977), the bloom supplies about 67% of the annual (bloom+non-bloom) productivity. When compared with carbon contents and deposition rates of the sediments below (Dunbar, Anderson & Domack, 1985; Ledford-Hoffman, DeMaster & Nittrouer, 1986), it is clear that most (about 90%) of the surface organic production is remineralized within the water column or at the surface of the sediments. When a similar calculation is made for biogenic silica, nearly all of the silica produced at the surface is being deposited in the benthos. This implies that opaline sediment accumulation in the Ross Sea occurs not because of extraordinarily large surface production but by a decoupling of organic and siliceous cycles either within the water column during particle flux or at the sediment surface, and that the marginal ice zone has an extremely important rôle in the global biogeochemical cycle of silica. Micro-paleontological data also indicate the importance of the ice edge to the benthos in the Ross Sea. Nitzschia curta, the overwhelmingly dominant species in the ice-edge bloom (Wilson et al., 1986), is also the dominant form found in the sedimentary record (Truesdale & Kellogg, 1979). Furthermore, it has been deposited consistently throughout the past 18000 years, indicating that the surface bloom is indeed a predictable feature. The contribution of N. curta increases in the biogenic sediments closer to the coast of Victoria Land (Truesdale & Kellogg, 1979), which is consistent with the idea that an ice-edge bloom is

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initially seeded by ice-algae but the taxonomic composition of the bloom will change in space and time as a result of selective growth of released species. Concentrations of herbivores, particularly euphausids, and other pelagic higher trophic levels may be somewhat less in the Ross Sea than in the Weddell Sea (Marr, 1962; Everson, 1977). It is possible that a large proportion of the biogenic material is utilized in the benthos rather than the water column. Sedimenttrap collections indicate that much of the material involved in vertical flux is whole phytoplankton cells rather than faecal material (Dunbar et al., 1985); therefore, the Ross Sea may function like the Bering Sea and support an active demersal food web. Little quantitative data exist (other than the geochemical evidence of siliceous deposition) to substantiate this hypothesis. The dependence of the Weddell Sea food web on the marginal ice zone production also has not been quantified. Much greater standing stocks of krill (Euphausia superba) have, however, been observed in the South Atlantic sector of the Southern Ocean than any other (Marr, 1962), including massive swarms in the vicinity of Elephant Island (Shulenberger, Wormuth & Loeb, 1984). The numbers of pelagic birds, penguins, and marine mammals also seems to be greater in the Weddell Sea than in other regions (Everson, 1977; Ainley, O’Connor & Boekelheide, 1984; Fraser & Ainley, 1986), but the degree of dependence on production associated with the marginal ice zone is unknown. Extremely high rates of biogenic silica accumulation have been measured in the South Atlantic sector near the northern limit of the Antarctic Convergence (DeMaster, 1981); the rate of organic matter deposition is unknown. Studies of the food web of the Weddell Sea ice-edge system are at present in progress, so that the uncertainties involved in the trophic relationships of this area should become resolved. CONCLUSIONS Marginal ice zones are unusual regions within polar seas; they not only have a distinctive physical setting, being covered by a moving ice pack during portions of the year, but have a characteristic biological system associated with them. The ice edge is a physically dynamic region, with eddies, fronts, jets, and other mesoscale features often present. They are also the sites of certain oceanographic processes such as ice melting or freezing, and upwelling, all of which have immediate impacts on biological communities. Communities in physically dynamic habitats are often coupled to the energy provided by the physical processes in the environment (Margalef, 1978); those in marginal ice zones, and phytoplankton assemblages in particular, are no exception. Recently developed techniques for the study of phytoplankton distributions need to be applied to marginal ice zones. For example, the Coastal Zone Color Scanner (CZCS) has been used in temperate and tropical regions to document the meso- and large-scale distributions of chlorophyll in the ocean. Although problems remain in its application within regions with a high solar angle, preliminary studies of ice-edge systems have been made (Maynard, 1986). Further use of airborne and satellite-based imaging techniques will greatly stimulate traditional biological oceanographic studies, as they have in other regions. Continuous monitoring of biological properties (e.g. optical transmission, fluorescence, bioluminescence) should also be utilized and will enable researchers to have a greater appreciation of the spatial and temporal variations of these properties in marginal ice zones, as well as the interactions between physical and biological processes in these systems. The phytoplankton assemblages of ice-edge regions are often much more productive than those removed from the ice edge or those within the heavy ice pack, and hence a potential exists for the marginal ice zone to be a locus of activity for the entire food web and for the biogeochemical cycles of the region. In many ice-edge systems this has been shown to be true, but the physical and biological processes differ among ice-

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edge systems. Much remains to be learnt about the trophic structure within these ecologically important regions. For example, the annual progression of primary production in a pelagic environment has not been well documented, so that the quantitative impact of marginal ice zones on carbon cycles cannot be accurately estimated. The relationship of the water column plankton to the ice-algal community of the pack ice remains elusive, as does the quantitative significance of nanoplankton and microbial grazing. Finally, the relationship of ice cover (and the biology of the marginal ice zones) to large-scale variations in global circulation is unknown, despite evidence that such teleconnections exist. Given that so much about marginal ice zones remains to be learnt, future research efforts directed at understanding these regions will undoubtedly contribute much to our knowledge of the structure and function of polar ocean systems. ACKNOWLEDGEMENTS Financial support for research in marginal ice zones has been provided by the Office of Naval Research and the Division of Polar Programs, National Science Foundation. Many colleagues and students have collaborated on various aspects of this research and have contributed generously to the synthesis of concepts presented in this manuscript. I would especially like to acknowledge Drs D.Nelson, H.J.Niebauer, S.Smith, and L.Codispoti. REFERENCES Ackley, S.F., Buck, K.R. & Taguchi, S., 1979. Deep-Sea Res., 26, 269–282. Ainley, D.G. & Jacobs, S.S., 1981. Deep-Sea Res., 28, 1173–1185. Ainley, D.G., O’Connor, E.F. & Boekelheide, E.F., 1984. Ornithol. Monogr. 32, Amer. Ornithol. Union, Washington, D.C., 97 pp. Alexander, V., 1980. Cold Reg. Science Tech., 2, 157–178. Alexander, V. & Niebauer, H.J., 1981. Limnol. Oceanogr., 26, 1111–1125. Anderson, L. & Dyrssen, D., 1981. Oceanol. Acta, 4, 305–311. Barber, R.T. & Smith, R.L., 1981. In, Analysis of Marine Ecosystems, edited by A.R.Longhurst, Academic Press, New York, pp. 31–68. Brzezinski, M.A., 1985. J. Phycol., 21, 347–357. Buckley, J.R., Gammelsrød, R., Johannessen, J.A., Johannessen, O.M. & Røed, L.P., 1979. Science, 203, 165–167. Bunt, J.S. & Lee, C.C., 1970. J. mar. Res., 28, 304–320. Clarke, D.B. & Ackley, S.F., 1984. J. geophys. Res., 89, 2087–2095. DeMaster, D.J., 1981. Geochim. Cosmochim. Acta, 45, 1715–1732. Dietrich, G., Kalle, K., Krauss, W. & Siedler, G., 1980. General Oceanography. J. Wiley & Sons, New York, 626 pp. Dugdale, R.C. & Goering, J.J., 1967. Limnol. Oceanogr., 12, 196–206. Dunbar, R.B., Anderson, J.B. & Domack, E.W., 1985. In, Oceanology of the Antarctic Continental Shelf, edited by S.S.Jacobs, American Geophysical Union, Washington, D.C., pp. 291–312. El-Sayed, S.Z., 1971. In, Biology of the Antarctic Seas, edited by G.Llano & I. Wallen, American Geophysical Union, Washington, D.C., pp. 301–312. El-Sayed, S.Z., Biggs, D.C. & Holm-Hansen, O., 1983. Deep-Sea Res., 30, 871–886. El-Sayed, S.Z. & Taguchi, S., 1981. Deep-Sea Res., 28, 1017–1032. El-Sayed, S.Z. & Turner, J.T., 1977. In, Polar Oceans, edited by M.Dunbar, Arctic Institute of North America, Montreal, pp. 463–503. Eppley, R.W. 1972. Fish. Bull. NOAA, 70, 1063–1085. Eppley, R.W. & Peterson, B.J., 1979. Nature, Land., 282, 677–680. Everson, I., 1977. The Living Resources of the Southern Ocean. FAO, Rome, 156 pp.

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Oceanogr. Mar. Biol. Ann. Rev., 1987, 25, 39–90 Margaret Barnes, Ed. Aberdeen University Press

SAMPLING AND THE DESCRIPTION OF SPATIAL PATTERN IN MARINE ECOLOGY N.L.ANDREW and B.D.MAPSTONE* Institute of Marine Ecology, Zoology Building, School of Biological Sciences, University of Sydney, Sydney, N.S.W. 2006, Australia

INTRODUCTION The description of pattern is of fundamental importance in ecology. Irrespective of the field of study, all marine ecologists are faced with the problem of establishing and quantifying patterns in nature. Observed patterns are the building blocks of the models from which we generate hypotheses, both about the patterns themselves and about processes that may govern them. Predictive hypotheses about processes suggested to explain observed patterns are tested by experiments and thus experiments are dependent on the patterns we perceive. More subtly, our perceptions of patterns often colour the sorts of questions we ask, thereby canalizing the design of experiments and the sorts of answers we get from them. Furthermore, observed patterns frequently provide the context within which the results of experiments are interpreted. Information on the distribution and abundance of organisms is often the sole basis for ecological and management decisions. The accurate and precise description of pattern is, therefore, essential to most aspects of ecology. Our reading of the literature suggests that the adequacy of sampling methods and designs has not often been demonstrated. Most studies apparently proceed more by custom and tradition than by careful consideration of potential biases and problems inherent in sampling different organisms. Certain procedures or methods are so popular in some fields that they have become “standards”; a good example is the use of the 0·25 m2 quadrat in intertidal studies. In other fields enormously divergent procedures have been used, apparently without justification. For example, in studies of tropical fish, a great variety of methods, ranging m transects, have been used to quantify from counts per unit time of observation to counts per abundances of fish. Many authors and/or editors of journals may consider demonstrations of the adequacy of sampling methods to be too preliminary or mundane to be incorporated into papers. Hence, this information may lie unpublished in filing cabinets or in inaccessible theses or reports. Alternatively, the appropriateness of the design may be determined completely informally, with intuition and experience guiding the design in an unstructured manner. Uncritical acceptance of standard techniques is, however, apparently widespread. Popular methods were often used despite overwhelming evidence of great mprecision and consequent low reliability and repeatability (see Downing, 1979; Resh, 1979). There are obviously many different methods of gathering data on the abundance of organisms. These can vary from the use of microscopy to count bacteria, to the use of cores to collect meiofauna, to aerial surveys of cetaceans. Each method has a special set of conditions and problems. Some methods are definitely more

*Order of authorship determined alphabetically.

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limiting than others: it is much easier to estimate the abundance of algae on a shore than it is to count deepsea fish. Despite the many and varied methods used, the data so collected have two features in common: (1) all are subject to problems of inaccuracy and imprecision arising from the application of the sampling methods; and (2) all ecologists are constrained by time and funding, which restrict the placement and number of samples taken. Although many problems of sampling are specific to the particular methods used, there are considerations in design that have broad generality. Our aim in this review is to assess how marine ecologists have addressed the description of patterns in the arrangement and abundance of organisms and to discuss principles that have general applicability across all disciplines. We shall discuss two aspects of the description of pattern. The term “spatial pattern” will be used to refer to the arrangement of individuals within an area and “patterns in abundance” will refer to the abundance of organisms within and among areas. We have not used the more common term dispersion in our discussion of the arrangement of organisms in space to avoid confusion with the mathematical usage of the term. The term dispersion is reserved to describe the spread, or variability, of estimates about their mean (Sokal & Rohlf, 1981). We shall discuss the design of sampling programmes involving univariate data gathered to answer specific questions about spatial pattern or patterns in abundance of organisms. For example, how many bivalves are there in an estuary and are they non-randomly arranged? Are there more in one estuary than in others along the coast? We offer nothing more novel than the synthesis and interpretation of a large and scattered literature. We hope that this synthesis will promote the careful design of sampling programmes that are tailored to the demands of particular studies. While the review focuses on how marine ecologists have approached the description of pattern, relevant literature about terrestrial or freshwater organisms will be discussed where necessary; the principles are, after all, the same. For convenience, we shall concentrate most of our discussion on the estimation of abundance, but the discussion may be applied to most other variables, e.g. estimates of biomass or behavioural data. Two different aims may be identified when discussing the determination of abundance. One is to estimate the number of organisms in one area, and the second is to examine differences among areas. The design of sampling programmes to fulfil these different aims follows different guidelines. The procedures that best estimate the total number of organisms in an area may require uneven replication within sub-areas, adjusted to the exact requirements of the area (stratified sampling). On the other hand, when comparing estimates among areas, the requirements of the analysis are often best met by having equal numbers of replicates in each area (balanced multi-stage sampling). We discuss the best ways to analyse data collected for these different purposes in the relevant sections. Because we are primarily interested in univariate data sets gathered to answer the sorts of questions posed above, we have not considered multivariate techniques such as classification and ordination. Such techniques are more exploratory or hypothesis-generating in nature and are concerned with answering fundamentally different questions from those that will be discussed here (Green, 1979). Studies concerned with the description of whole communities or species diversity, and measures of association between species are also outside the scope of this review. We introduce sampling programmes designed to assess the abundance of organisms via a discussion of some general considerations such as the formulation of objectives and the interaction between statistical adequacy and biological realism. A discussion of the concepts of accuracy and precision leads on to more specific problems such as the determination of appropriate sizes of sampling units, replication, optimization procedures and the design of single-stage programmes to estimate the number of organisms in one area. This is then extended to more complex designs such as stratified random sampling and multi-stage

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sampling. Power analysis is reviewed as a tool for designing sampling programmes and for analysing the sensitivity of tests. In most marine studies, the description of spatial pattern entails the demonstration of non-random pattern. The various indices used to detect non-random pattern are reviewed along with their application to marine studies. The techniques used to describe non-random pattern in more complex situations, and in more detail, are then briefly outlined. In reviewing the literature we were primarily interested in those papers that concentrated on the description of spatial pattern or patterns of abundance, and especially those that were concerned with procedural aspects, i.e., studies that introduced new tests or justified the use of methods. Journals were searched from 1965 (or their inception) to June 1986; a list of the journals searched is provided in Table I. We have made selective forays into other journals and the earlier literature where necessary, usually as a result of citations from papers found in the above search. We have deliberately avoided citing unpublished papers and those in the “grey” literature, such as local or internal reports, because these are generally not available to most readers. We apologize to anyone who feels a paper of his/hers has been unjustly omitted. TERMS AND CONCEPTS In general we shall keep to a minimum the use of mathematics in this review. There are, however, a number of mathematical terms used that should be explicitly defined to avoid confusion. As far as possible we shall maintain consistency with terminology already used in the ecological literature. The term “population” will generally be used in a biological sense to refer to the local assemblage of organisms being studied (Caughley, 1977; Elliott, 1977). Where “population” is used in a statistical sense, clear qualifiers will be used to avoid ambiguity (e.g. population of values from which sample data are drawn). TABLE I List of journals searched American Naturalist Annual Review of Ecology and Systematics Canadian Journal of Fisheries and Aquatic Sciences (formerly Journal of the Fisheries Research Board of Canada) Coral Reefs Ecology Ecological Monographs Estuarine and Coastal Shelf Science (formerly Estuarine and Coastal Marine Science) Journal of Animal Ecology Journal of Applied Ecology Journal of Ecology Journal of Experimental Marine Biology and Ecology Journal of the Marine Biological Association of United Kingdom Journal of Wildlife Management Limnology and Oceanography Marine Biology

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Marine Ecology Progress Series Oceanography and Marine Biology: an Annual Review Oecologia (Berlin) Oikos

It is unusual in biological, especially ecological, research to be able to count or measure the entire population being studied (Cochran & Cox, 1957; Cochran, 1963; Southwood, 1966; Elliott, 1977; Seber, 1982). Consequently, data are collected from a subset, or “sample”, of the population and inferences about the population are made based on these sample data. The size of the sample is usually very small relative to the size of the population (Caughley, 1977; Elliott, 1977; Green, 1979). In those cases where a relatively large proportion (>5–10%: Cochran, 1963; Elliott, 1977; Green, 1979; Seber, 1982) of the population is being measured, corrections for finite sampling may be necessary, details of which can be found in one of the many texts that treat sampling design (Cochran, 1963; Scheaffer, Mendenhall & Ott, 1979; Snedecor & Cochran, 1980). We use the term “sampling method” to describe the procedures used to obtain samples (e.g. coring, visual census, vertically hauled plankton nets). The actual device, implement or operation used to obtain mm core, m transect, 250each reading in a sample will be called the “sampling unit” (e.g. mm diameter plankton net). The way in which sampling units are allocated in space and/or time will determine the sampling design. To illustrate: a population of meiofauna might be sampled by taking cores (the sampling method) using a simple random sampling design, with ten replicate cores of 20 mm diameter mm depth (the sample unit) being collected. In this example, each core might be subsampled to economize counting time, in which case a similar statement could be made about the method, design, unit size, and replication of the sub-sampling procedure. “Variables” are the characteristics of the organisms or population being measured. Examples would include standard length, test diameter, density or TABLE II A list of symbols and their meaning as used throughout this review Symbol

Meaning

Description

n x µ

Sample size Variate Population mean Sample mean Population variance Sample variance Population standard deviation Sample standard deviation Standard error

Number of replicate units taken Single measure of variable, e.g. number of individuals in quadrat Mean of the population Mean of sample: estimator of µ Variance of the population Variance of sample: estimator of σ2 Standard deviation of population Standard deviation of sample: estimator of σ Standard deviation of

σ2 s2 σ s SE

distance from nearest neighbour. “Parameters” are mathematical or numerical values used to describe some characteristics of the entire population of the variable being measured (Winer, 1971; Scheaffer et al., 1979; Snedecor & Cochran, 1980). In most situations, the values of parameters are never known since they are the exact or “true” measures of an entire population of values. Normally we only have a sample of those values.

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For example, the true mean and variance of the weights of a population of snails are parameters of the population. Estimates of parameters are represented by the corresponding values obtained from a sample of the population; e.g. the arithmetic values of the mean and variance of the weights of a sample of snails taken from the above population. Winer (1971) and Elliott (1977) refer to estimates obtained from samples as “statistics”. We shall use “statistics” in two specific senses: sample or descriptive statistics being synonymous with the sample estimates, as in the above usage; and “test statistic” meaning the derived numeric value (e.g. F-value, t-value) used in a statistical test of a hypothesis about the data. Clearly, when an entire population is measured, the sample will be the population, and the parameters and their estimates will be identical. As noted above, this is unlikely in ecological contexts. The symbols and their meanings used throughout this review are given in Table II. Note that we have listed only the arithmetic mean as a measure of location and standard deviation, variance and standard error as the associated measures of dispersion. These are, by far, the most often used parameters or estimators in the ecological literature. ESTIMATING ABUNDANCE Virtually all ecologists are faced at some stage with the task of estimating abundances of organisms in the field. In recent years there have appeared a few publications specifically concerned with the design of sampling programmes and/or experimental studies (Holme & McIntyre, 1971; Caughley, 1977; Green, 1979; Underwood, 1981; Hurlbert, 1984; Underwood & Denley, 1984; Millard, Yearsley & Lettenmaier, 1985; Millard & Lettenmaier, 1986). These recent publications differ from excellent works such as Cochran & Cox (1957), Cochran (1963), Winer (1971), Scheaffer et al. (1979), Snedecor & Cochran (1980) and Seber (1982) in that the former were more concerned with ecological problems than the latter texts, which tended to concentrate on statistical procedures, agricultural experiments or sample survey design for human populations. Our purpose in this section is to outline the procedures that have been, or might be, followed by researchers intending to sample marine organisms. This will mostly involve reiterating the points made in the above publications. Green (1979), in particular, has thoroughly outlined the proper development of sampling programmes for environmental impact studies. Our discussion will largely parallel his “ten principles”. QUESTIONS, SCALE AND GENERAL CONSIDERATIONS As Green (1979) has pointed out, the interpretation of the results of a sampling exercise will be only as clear as the statement of the objectives of the study. It cannot be emphasized too strongly that before a sampling programme can be designed efficiently, all the questions to be addressed must be stated clearly and explicitly. Ambiguity in the intentions can only result in corresponding weaknesses in the design of the programme and the results that come from it. For example, if the objective of a study is to estimate the total number of holothurians on a particular coral reef, then a stratified sampling design should be used. If, however, the objective is to compare abundances among reefs, then a multi-stage sampling design should be used. It is most important to identify the sources of variation that might influence the data and, hence, the perception of pattern. Sources of variation in sampling programmes are often associated with phenomena operating at various scales, or can be estimated at several scales. For example, Caffey (1985) studied the recruitment of barnacles at several sites on each of several rocky shores at several locations along the New South Wales coast and thus accounted for variation at four spatial scales. The spatial scale at which a study

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is done can have major consequences for both the results obtained and their subsequent interpretation (recent references include Harris, 1980; Denman & Gargett, 1983; Dayton & Tegner, 1984; Irish & Clarke, 1984; Millard et al., 1985; Wiens, 1986; Wiens, Addicott, Case & Diamond, 1986). Perceptions of “scale” are entrenched in our perception of problems, and subsequent generation of hypotheses. As Resh (1979) and Harris (1980) pointed out, however, we must deal not only with the convenient scales of our own perceptions and methods, but also with the functional scales at which the organisms are likely to respond to their environment. Clearly, it is logistically impossible to deal with all scales and/or sources of variation and careful choices have to be made. Results cannot be interpreted at scales, or for sources of variation, that the sampling programme did not address. Careful consideration has to be given also to the nature of the factors being investigated: are the questions being posited for only specific situations (e.g. on a particular coral reef) or is more generality required (the whole Great Barrier Reef)? Resolution of this question influences not only the formulation of testable hypotheses and the interpretation of results, but also the selection of study conditions (such as locations): studies of a single coral reef in one region cannot be used to infer patterns for the whole Great Barrier Reef. Resolution will come partially from a greater attention to the framing of questions and partly from a recognition that many ecological questions need answering at more than one spatial scale. An explicit statement of the questions to be addressed by a sampling programme will not always be straightforward. The questions must be expressed sufficiently clearly that specific null hypotheses can be stated. The hypotheses to be tested will incorporate the original objectives of the study and decisions about the scales of interest. They will strictly determine the subsequent design of the sampling programme. Once a hypothesis is stated, the researcher has implicitly defined the ‘right’ and ‘wrong’ scales at which sampling should be done, so statement must be clear and explicit. Decisions about the appropriate spatial scales and sources of variation have to be made from such intangibles as intuition and experience, in addition to existing natural history and distributional data. These decisions will be easier if the objectives are clearly stated. Incorrect decisions are likely to result in a design that is inappropriate to the objectives of the study and inaccurate perceptions of real patterns. At this stage of the design process, the researcher should have a clear concept of the general structure of the sampling programme. It is imperative to anticipate the analysis of the data and to check that the proposed programme conforms to the assumptions of those analyses. For example, if a programme is designed with the expectation that a three-way Analysis of Variance will be used to analyse the results, the form of the Analysis of Variance—the F ratios and their degrees of freedom—should be examined before the data are collected. If, for example, a fully factorial analysis is anticipated in which two of the factors are considered random factors, then there will be no valid F test for the main effect of the fixed factor, unless at least one of the first order interactions containing that factor is clearly non-significant (P>0·25, by convention, Winer, 1971) so that a pooled denominator mean square can be calculated (see Winer, 1971; Snedecor & Cochran, 1980; Underwood, 1981). Changing a priori the designation of one of the random factors to ‘fixed’ may correct the problem of analysis but will also fundamentally alter the original hypotheses, the way in which sampling is done and the interpretation of the results (Underwood, 1981). Problems of this kind must be sorted out a priori rather than a posteriori. As Green (1979) has commented, the time for statistical advice is when the programme is being designed, not after the data are in hand and one is wondering what can be done with them. The investigator must next give careful consideration to the way in which samples will be collected. In many situations, the identity and spacing of the basic unit of replication is relatively clear e.g. discrete units such as patch reefs, tide-pools or islands. This is less clear, however, in situations where the sample unit is arbitrary (e.g. a quadrat) and the area being studied cannot be divided, a priori, into natural units. For

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example, if a large and relatively uniform area has to be arbitrarily divided, then how far apart should the basic units be: 1, 10, 100 or 1000 metres? The determination of the spatial scales at which a study will be done involves an explicit judgement about the relative importance of variation at different scales. Variability among these scales, and its importance, will vary with the species and physical environment studied but can be estimated before the main study by doing multi-stage pilot studies. Replication at all levels to be considered is essential (Green, 1979; Hurlbert, 1984). Furthermore, Green (1979), Bernstein & Zalinski (1983), Hurlbert (1984), and Underwood & Denley (1984), among others, have all discussed several aspects of confounding in experimental or sampling designs. All have emphasized, that data must measure variability within as well as among the levels of a sampling programme and that the sources of variation about which hypotheses are posed must be unambiguously identifiable in the data. Hurlbert (1984) presented an extensive discussion of the ways in which designs have often been confounded. In general, random collection of sample units within the lowest level (smallest source of variation considered) of a design is recommended to ensure independence of the data and to avoid systematic error. True randomization is often difficult to implement but great care should be exercised in the substitution of ‘haphazard’ sampling for random sampling: ‘haphazard’ sampling may introduce subtle, unanticipated biases. Millard et al. (1985) have clearly demonstrated the effects of non-independence of sample data. It is also important to consider the proportion of an area or population that will be sampled by some chosen number of replicates, particularly where random samples are to be taken from an area or set of areas on a number of occasions. If the proportion of the local population or area sampled each time (the sampling fraction) is greater than 5–10%, then the estimate of abundance may need to be adjusted for finite sampling (Cochran, 1963; Elliott, 1977; Seber, 1982). Where the sampling fraction is larger, there may be a high probability of samples substantially overlapping and introducing unplanned confounding in the data. Replicates will not be independent and variability will be under-estimated. When comparisons of abundances are of interest, it is also desirable that the sampling programme remains balanced—that is, that replication at each level of the programme is consistent across all treatments. Comparative analyses of unbalanced data are possible for many situations, but in general the results of unbalanced analyses are less reliable than those of a corresponding balanced analysis (Bradley, 1968). When an initially balanced sampling programme becomes unbalanced it may be preferable to rebalance the data set by omitting some values prior to analysis. When the objective of a sampling programme is to estimate the total number of organisms in an area (e.g. the population of cockles in a bay), however, the best design may be an unbalanced stratified sampling design. Where sampling is designed to measure an effect of some perturbation or treatment, it is essential that control situations are also sampled so that the effects of treatments can be distinguished from stochastic processes or other, unspecified events that affect only some sampled areas (Green, 1979; Millard & Lettenmaier, 1986). Bernstein & Zalinski (1983) have elaborated on Green’s statements and have emphasized the need to consider various scales of change in both control and treatment conditions. Finally, after the sampling programme is completed, the data should be examined to ensure that they conform to the assumptions of the proposed statistical analyses and they should be transformed if necessary. This subject has been covered repeatedly, both in the statistical literature (Winer, 1971; Snedecor & Cochran, 1980; Sokal & Rohlf, 1981) and in the ecological literature (Downing, 1979, 1980, 1981; Green, 1979; L.R.Taylor, 1980; W. D.Taylor, 1980; Underwood, 1981; Morin, 1985) and will not be laboured here.

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ACCURACY AND PRECISION In most standard English dictionaries, accuracy and precision are defined in terms of each other and are often noted as synonyms (Gove, 1969; Sykes, 1982). In scientific and engineering fields, however, the terms have come to represent quite different concepts (Cochran & Cox, 1957; Cochran, 1963; Lafedes, 1978; Sokal & Rohlf, 1981; Lincoln, Boxhall & Clark, 1982). The difference between common and technical usage has apparently led to inconsistent use of the two terms in much of the ecological literature. Clear distinction between the terms seems, however, to be both useful and desirable. Accuracy is defined as the closeness of a measurement or estimate to the true value of the variable being measured or parameter being estimated (Cochran & Cox, 1957: Cochran, 1963; Sokal & Rohlf, 1981; Lincoln et al., 1982). Thus, accuracy refers to the location of an estimate relative to the location of the true value. For example, if the true mean standard length (SL) of a population of fish is 43·7 mm, then a sample mean of 43·6 mm SL is a more accurate estimate of the population mean than a sample mean of 40·5 mm SL. A method that gives estimates that are repeatedly and predictably inaccurate is said to be biased. Bias, then, is the systematic deviation of an estimate from the true value and is caused by artefacts of the method used to obtain the estimate. Precision refers to the degree of concordance among a number of measurements or estimates for the same population (Cochran & Cox, 1957; Cochran, 1963; Sokal & Rohlf, 1981; Lincoln et al., 1982). Precision is reflected by the variability of an estimate. In the above example, if a number of samples of the population of fish returned a set of estimates of mean SL that ranged from 40·3 mm to 40·7 mm, then those estimates would have greater precision than a set with range 46·5 mm SL-52·0 mm SL. Note that accuracy cannot generally be inferred from precision. The preceding two sets of samples were both inaccurate (because the true mean was 43·7 mm) but the first was far more precise than the second. Clearly, then, estimates can be inaccurate but precise, both accurate and precise and so on. Precision and accuracy are truly synonymous only when no methodological biases exist (Cochran & Cox, 1957). The terms have been used in this sense (e.g. Winer, 1971; Underwood, 1981; Seber, 1982) but it is unlikely that bias will be absent in biological research and we will assume that some bias is always, at least potentially, present. We thus use the terms strictly and separately. Two other terms that have often been used with accuracy and precision are reliability and repeatability. These terms are apparently less clearly defined than accuracy and precision but have sometimes been used as synonyms for accuracy or precision, respectively (Cochran & Cox, 1957; Cochran, 1963). We shall use the terms reliability and repeatability to describe something of both accuracy and precision. Repeatability will be used to refer to the degree to which a sampling design or method can be expected to give consistently estimates with the same accuracy and precision. Reliability refers to the confidence that we can have in a given method or procedure: a reliable estimate would be both accurate and precise and an unreliable estimate would be neither. Accuracy and methods Inaccuracy in estimates can be attributed to two main sources: (1) inappropriate design of the sampling programme; and (2) biases inherent in the sampling methods. The first of these sources arises because the design of a sampling programme is inappropriate to the question(s) being investigated (see pp. 44–47). The second is a source of error that is systematically implicated in all sampling programmes. All estimators are dependent on the methods used to obtain the sample data and will thus incorporate the biases of those methods. Bias may arise from many sources, including errors made by observers, loss of organisms from sample units, over- or under-estimation of the areas sampled, disturbance caused by sampling, avoidance of

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the sampling device by organisms, and others. Our main interest in this section is not the cause of methodological biases, but the ways in which researchers have investigated the accuracy of their methods. Although many authors have apparently ignored questions of accuracy, several have emphasized the importance of methodological bias in determining the results of sampling programmes (Wiebe & Holland, 1968; McIntyre, 1971; Wiebe, 1971; Russell, Talbot, Anderson & Goldman, 1978; Resh, 1979; Southward & Barry, 1980; Sale, 1980; Sale & Douglas, 1981; Weinberg, 1981; Omori & Hamner, 1982; Fairweather & Underwood, 1983; Sale & Sharp, 1983; Kennelly & Underwood, 1984, 1985; Pot, Noakes, Ferguson & Coker, 1984). Practical tests of accuracy will be specific to each methodology, and will usually rely on attempts to assess absolute abundance by an independent method. This independent assessment must be accurate, otherwise one is only comparing two sampling methods, both with unknown biases (Kinzie & Snider, 1978). There have been a number of reports where accuracy has been thoroughly tested, usually by knowing a priori the absolute abundance of organisms in the sampled population or enumerating completely a population after sampling (Haury, 1973; Weinberg, 1981; Pihl & Rosenberg, 1982; Bell, Craik, Pollard & Russell, 1985; Andrew & Stocker, 1986). In many systems, such as plankton or bacterial communities, the accuracy of sampling and/or processing methods is extremely difficult, if not impossible, to assess because direct, independent enumeration of populations is impossible. In these, and many less problematic instances, the relative accuracy of different methods has been assessed by sampling the same populations by different methods and then comparing the values of the estimates obtained. Where estimates differ one method can only be said to be more accurate than another on the basis of intuition, auxiliary knowledge, and argument rather than on unequivocal evidence that it gave a truer account of the population. When methodologies have been compared, it has often been argued that the greatest estimate of abundance is the most accurate (Leatherwood, Gilbert & Chapman, 1978; Sale & Douglas, 1981; Sale & Sharp, 1983; Stretch, 1985; Gray & Bell, 1986). This argument is based on the assumption that you do not count what is not present, so over-estimation is extremely unlikely (Caughley, 1977). Exclusion or loss of organisms from samples is far more likely than inclusion. Some authors, however, have pointed out that boundary effects, where the inclusion of individuals in a sampling unit such as a quadrat is somewhat subjective, often lead to over-estimation (references in Downing & Anderson, 1985; Downing & Cyr, 1985). For sessile or slow-moving organisms, such boundary effects can be avoided by procedures such as including all ‘borderline’ individuals on two sides of a quadrat and excluding them along the other two sides. The problem is more difficult to circumvent when the organisms being counted move rapidly across sample unit boundaries and the observer has to make immediate decisions about whether an organism was inside or outside the unit at the instant when it was first sighted. Ideally, inclusions and exclusions of such marginal individuals should be equi-probable but inclusion seems more likely than exclusion (references in Downing & Anderson, 1985). For example, when visually censusing fish along belt transects, inclusion of individuals that are seen to swim into the transect is more likely than exclusion of fish that are observed leaving the transect. A number of recent studies have attempted either to quantify absolute accuracy or quantitatively to compare the accuracy of alternative methods. Jones (1974) showed that direct counts of bacteria using epifluorescence techniques varied dramatically when the bacteria were treated with different stains. Youngbluth (1982) came to a similar conclusion about the effect of the design of emergence traps on estimates of abundance of demersal zooplankton. Mackay, Cooling & Berrie (1984) found considerable disparities among five methods of estimating primary production in estuarine angiosperms. Gray & Bell (1986) reported that catches of vagile macrofauna from sea-grass meadows varied greatly with the method used (poison compared with beam trawl). There have been several comparisons of methods for assessing the

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abundance of fish, either destructively or non-destructively, and all report major differences in the estimates returned by different methods (Russell et al., 1978; Brock, 1982; DeMartini & Roberts, 1982; Pot et al., 1984; Kimmell, 1985). By contrast, Bouchon (1981) found that quadrat and line transect methods gave similar estimates of coral cover and diversity. Nie & Vijverberg (1985) found no differences between catches of plankton from a Friedinger sampler and those from a Schindler sampler. Leatherwood et al. (1978) obtained similar, although very imprecise, estimates of numbers of dolphins using different methods of aerial survey. They noted that estimates varied with observers for all methods. Kennelly & Underwood (1984) found no difference between estimates of abundance of epilithic micro-organisms obtained in situ using an underwater microscope and from chips of substratum examined with superior microscopes in the laboratory. The underwater microscope, however, detected only about 25% of the micro-organisms that could be collected with a small suction sampler (Kennelly & Underwood, 1985). In a few instances, computer simulations have been used to model the sampling characteristics of different methods and consequently determine their biases and accuracy. Kinzie & Snider (1978) found that four visual survey techniques commonly used in field studies of corals, consistently gave inaccurate results when simulated with coral populations with varying characteristics. Wiebe & Holland (1968) and Wiebe (1971) found in simulations that estimates of the abundance of plankton were affected significantly by both the diameter of the plankton nets used and the distance over which the nets were towed. Furthermore, several of the more inaccurate estimates were more precise than most accurate estimates, illustrating that precision cannot be used to infer accuracy (e.g. Keast & Harker, 1977). Unless such simulations are constructed on a sound knowledge about the distribution and behaviour of the organisms in natural conditions and of the properties of the sampling device, extrapolations from computer to nature should be treated cautiously. Wiebe & Holland (1968) found good agreement between the precision of their simulated samples and that of several field studies and Wiebe (1972) tested the predictions of his earlier simulation study in the field. Despite the problems involved a number of researchers have attempted to assess absolute accuracy. Van Vleet & Williams (1980) tested the accuracy of a number of methods of sampling organic films on the sea surface by sampling, in the laboratory, surfaces that had been contaminated only with measured quantities of known compounds. Haury (1973) experimentally seeded a Longhurst-Hardy plankton recorder during operation with known quantities of real and model organisms in order to quantify biases arising from the variable residence time of plankton in the catch net leading the recorder. Weinberg (1981) tested the accuracy of seven methods of surveying coral communities by estimating abundances of corals by each method in an area that had previously been mapped in detail. The methods differed greatly in accuracy. An example of careful assessment of bias and accuracy of a sampling method has been presented by Pihl & Rosenberg (1982). By sampling known populations of vagile decapods and flatfish within fenced areas of a shallow sediment flat, Pihl & Rosenberg were able to compare the estimates of abundance obtained from the use of their drop trap with the known true values. They assumed that sampling fenced organisms was equivalent to sampling unfenced organisms. They attempted to address this problem by observing the responses of many individual organisms to the movements of the trap carriers and the operation of the trap. In most cases, the true assessment of accuracy will be very problematic and almost always extremely expensive in time and effort. In many studies, it may be unwarranted because knowledge of absolute abundance is not of primary interest (Caughley, 1977). Where comparisons are the aim of a study, estimates of relative abundance are often sufficient, although such comparisons assume equal sampling bias for all sites or conditions being compared. Unknown variations in the bias of a method with its use in different circumstances (e.g. habitats) is particularly problematic and difficult to assess. For example, artefacts of

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visual censuses of fish may change with changes in the complexity of the habitat surveyed. Similarly, Kennelly & Underwood (1985) found their suction sampler to be reliable when sampling clean, hard substrata of low complexity but to be of little use on more complex substrata such as silted surfaces or articulated coralline algae. Although estimates that are known to be inaccurate may be useful when accurate ones are not obtainable, some knowledge of the accuracy of the sampling methods used is always desirable. PRECISION The usefulness of an estimate is dependent on its precision as well as its accuracy (Cochran, 1963; Caughley, 1977; Green, 1979). Unlike accuracy, precision can be assessed relatively easily from characteristics of the sample data. Consequently, the well-established statistical tools for assessing precision are not specific to methodologies and can be considered as general principles. Precision is a function of the variance of the sample estimate: precision increases as the variance of the estimate decreases (Cochran & Cox, 1957; Cochran, 1963; Elliott, 1977; Eberhardt, 1978a, b). Precision is related to the confidence we have in the estimate, such as mean abundance. An important distinction must be drawn here between variation among sample data and the variation of the sample estimate, by which precision is measured. Variation in the sample data is an estimate of the variation in the arrangement of organisms in the real world. The spatial arrangement of individuals is inherently variable in space and time. In most populations, organisms are not regularly or randomly distributed but aggregated (Southwood, 1966; Pielou, 1969; Elliott, 1977; Downing, 1979; Resh, 1979). As we will discuss later (see p. 70), the variation among sample data depends on the size of the sampling unit relative to the scale of aggregation. Variations in the numbers of organisms in sampling units represent real phenomena and, if the methods used are reasonably reliable, sample variance should be relatively invariate with changes in sample size (=number of replicates). Variation in the sample estimate is, by contrast, implicitly dependent on the size of the sample. Consider the earlier example of estimating the mean size of fish. If a large number of fish (say 200) were measured for each repeated sample, then the sample means are more likely to be similar than if each of the samples included only 20 fish. Precision is, therefore, a characteristic of the sampling procedure rather than a reflection of some characteristic of the population being sampled. The Standard Error (SE) of a sample mean is the estimated standard deviation of a population of sample means from samples of that size. It is estimated from the standard deviation (s) of the sample data by:

where n is the sample size. Thus, for any s, which will be determined by the spatial arrangement of organisms and the size of units used to sample them, SE will decrease as replication increases. It follows that precision is a function of the SE of a sample. Precision has, however, been measured in terms of several sample statistics. The variance/mean ratio has sometimes been used in this context in addition to its use as a measure of the spatial arrangement of organisms (see p. 71). The ‘coefficient of of the data is also an expression often used in discussions of precision (Wiebe & variation’ Holland, 1968; Wiebe, 1971; Elliott, 1977; Resh, 1979; Pringle, 1984). These measures are not indicators of precision but of the relative variability of the sample data, standardized for the magnitude of the sample mean (Snedecor & Cochran, 1980). In other studies, precision is appropriately measured by comparing the 95% confidence intervals of the sample mean to that mean (Resh, 1979) or, more commonly, by the ratio (Southwood, 1966; Downing, 1979; Pihl & Rosenberg, 1982; Pringle, 1984; Downing & Anderson,

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1985; Downing & Cyr, 1985; Morin, 1985). Unfortunately, the ratio has also been referred to as the ‘coefficient of variation’ (Irish & Clarke, 1983; Pringle, 1984) and in several papers it was not clear or was used to calculate precision. whether Precision is inversely related to the values of the ratios used to measure it. Precision is great when the SE is small relative to the mean and precision decreases as the ratio increases. The precision required for an estimate should be set a priori by the researcher and will be largely determined by the questions being to be 0·5 whilst another may demand greater precision, for asked. One researcher may require The important point to note here, is that by setting desired precision, the researcher example can then design a sampling programme to achieve that precision as economically as possible. OPTIMIZATION AND PILOT STUDIES It is usually desirable to describe patterns with the maximal precision and resolution possible with the available resources. Almost all studies are constrained by logistic and economic considerations (Cochran, 1963; Holme & McIntyre, 1971; Lewis, 1976; Saila, Pikanowski & Vaughan, 1976; Green, 1979; Resh, 1979; Morin, 1985; Millard & Lettenmaier, 1986). Optimization of the design of sampling programmes is achieved by determining the most efficient allocation of resources—i.e., minimizing decreases in precision and/or resolution imposed by cost or by logistical constraints (Cochran, 1963; Saila et al., 1976; Scheaffer, Mendenhall & Ott, 1979; Irish & Clarke, 1984; Downing & Anderson, 1985; Downing & Cyr, 1985; Morin, 1985). The notion of optimizing the design of sampling methods is certainly not new (see references in Cochran, 1963; Cochran & Cox, 1957; Snedecor & Cochran, 1980). Statistical methods for choosing appropriate replication and cost-efficient allocation of resources have been available for several decades. In this section we shall review the use of optimization procedures for designing sampling programmes in marine studies. Apart from considerations, such as ensuring that sampling programmes are appropriate to the question(s) being asked, there are two main procedural questions that need to be addressed empirically if a sampling programme is to be efficiently designed: (1) how big should each sample unit be, and (2) how many replicates are needed? For all sampling designs, optimization procedures require estimates of variances and/ or means before the main programme has commenced. These estimates can be obtained from three sources: (1) pilot studies; (2) previous studies; and (3) published data. If pilot studies are not possible, or if the same organisms have been studied in the same system then one of the other two methods may be used to obtain estimates of mean abundance, spatial pattern and variance. Clearly, recent prior studies done in the same system are far more likely to give useful estimates than studies done in a different system. Numerous authors have emphasized that pilot studies are preferable to the other two sources of estimates (Cochran, 1963; Gray, 1971; Holme & McIntyre, 1971, Hulings & Gray, 1971; Downing, 1979; Green, 1979; Resh, 1979; Underwood, 1981; Downing & Anderson, 1985; Downing & Cyr, 1985). Green (1979, p. 31) made the point emphatically: “Those who skip this step because they do not have enough time, usually end up losing time.” We shall discuss the implementation of pilot studies and draw attention to some of the problems likely to be encountered. In the first instance (see below) we shall assume that sample units are placed randomly over an area to obtain a single estimate for a population—Simple Random Sampling. We shall then discuss the more complex Stratified and Multi-stage designs.

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The size and number of sampling units The two aspects of sampling that most affect the precision of sample estimates are the size of the sampling units and the number of replicates collected (Cochran, 1963; Greig-Smith, 1964; Southwood, 1966; Snedecor & Cochran, 1980; Elliott, 1977; Green, 1979). The size and shape of the sampling units used have been repeatedly shown, both theoretically and empirically, to have a great impact on the precision of an estimate (Wiebe & Holland, 1968; Stoddart, 1969; Gray, 1971; McIntyre, 1971; Wiebe, 1971; Elliott, 1977; Resh, 1979; Sale & Douglas, 1981; Sale & Sharp, 1983; Pringle, 1984; Downing & Anderson, 1985 and references therein; Downing & Cyr, 1985; Morin, 1985). The choice of the size of the sample unit is fundamentally related to characteristics of the population being sampled such as spatial arrangement. This is particularly true when organisms are aggregated. Units that are smaller than or equal to the scale of aggregation will often give more variable estimates of density than those that are large with respect to the scale of aggregation of the organisms (Smith, 1938; Pechanec & Stewart, 1940; Wiebe & Holland, 1968; Wiebe, 1971; Elliott, 1977; Watling, Kinner & Maurer, 1978; Green, 1979; Helshe & Ritchey, 1984). Estimates of abundance will be most variable when the size of the sample unit is approximately equal to the average distance between aggregations (Southwood, 1966; Elliott, 1977) (see p. 71). This occurs because these and smaller units are likely either to miss aggregations and so contain few or no organisms or to include an aggregation, and contain many organisms. Larger units will be likely to include part of at least one aggregation and so very small or zero counts are unlikely to occur. Thus, estimates of average abundance obtained from large sampling units will be less affected by the patterns in the spatial arrangement of the organisms. Consequently, for a given sample size, the precision of a sample estimate is likely to increase with increasing size of sampling units (Elliott, 1977; Downing, 1979; Resh, 1979). The rate of increase in precision with increasing size of sample unit will initially be great but will rapidly decline as the size of the unit exceeds the average distance between aggregations in the population. Resh (1979), however, cautioned that this trend is dependent on sample units being located wholly within habitats or other physical clines. If sampling units correspond to the scale of such factors as natural micro-habitat units, great variability may result from factors not simply related to the spatial pattern of the organisms. The shapes of sampling units may also affect the precision of estimates of abundance. Where boundary effects are important, the amount of boundary relative to the area or volume of the sample unit should be minimized (references in Downing & Anderson, 1985). Boundary effects are a function of the shape as well as the size of the sampling unit and may be particularly important where the shape and size of sampling units correspond to those of aggregations of organisms or topographic features to which organisms respond (Resh, 1979). Furthermore, the minimum linear dimension of a unit may be more critical than the other linear dimensions, area or volume. For example, Caughley (1977) and Sale & Sharp (1983) have demonstrated the importance of the widths of transects when using transect survey techniques for visual censuses. Hargrave & Burns (1979 and references therein) found that both diameter and aspect (diameter, depth of sampler) of sediment traps significantly affected their performance and that the effects varied with the turbulence of the water in which they were used. Four characteristics of the size and shape of sample units can be considered: length, area, volume, and time. The relative importance of each of these measurements to the precision and accuracy of the units will depend on the particular application. In general it would seem expedient to compare sampling units of a number of sizes at the beginning of a study (Southwood, 1966). Where there is some prior knowledge of the spatial arrangement of individuals in the population (see p. 70), the smallest unit considered should, if feasible, be larger than the average spacing among aggregations. Where the spatial arrangement of the organisms is unknown or unimportant, then the smallest unit should be at least one order of magnitude larger than the size of the largest organism being

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counted; Green (1979) suggested at least 20 times larger. The largest unit tested should be as large as logistic and cost constraints will allow, with the provisos that it is not so large that it approximates or exceeds the physico-environmental scales of interest or causes overlap among replicates when sampling a limited area. It is difficult to suggest how many sizes of sampling unit should be compared, but three should be the minimum and more than six or seven is likely to be too costly for a pilot study. In the simplest case (simple random sampling), an equal number of (at least three) replicate sampling units of each size should be collected randomly within the area to be studied. The randomization of all units within one area is intended to minimize the chance that systematic biases caused by variations within the area will affect only the replicates of a unit of particular size. For example, if all units of one size are placed in one area and all units of another size are placed in another area, then the effects of unit size will be confounded with pre-existing differences between the areas (Hurlbert, 1984). The analysis of pilot data should consider two aspects of the sample units tested—their relative accuracy and their relative precision. Relative accuracy is assessed by comparing the estimates of mean density (number of organisms/size of unit) obtained from the units of various sizes. Significant differences among the standardized means indicate differences in the relative accuracies of at least some of the unit sizes tested, although it will not necessarily be clear which unit size is the most accurate unless the true density of the population is known. Differences among means do not necessarily indicate differences in absolute accuracy; it is possible that two significantly different means are equally inaccurate, one being larger than the true population mean and the other being smaller by about the same amount. Decisions about which size of sample unit is the most accurate can only be made in the same way as decisions about the accuracy of different sampling methods (see pp. 48–50). Sale & Sharp (1983) studied the effects of changing the width of transects used in visual censuses of coral-reef fish. They found significant linear relationships between width of transect and estimated abundances of a number of species and advocated the use of such relationships to predict the expected true density of fish. Using a separate method (which they argued was more accurate than censusing along transects) they independently estimated the abundance of one of the species and compared this estimate with the ‘true’ value predicted from their regression of abundance on width of transect. There was no significant difference between the two values, but nor was there a difference between the independent estimate and the estimate obtained from the narrowest transect used for that species. can be calculated A number of avenues can be followed to compare precision. In the simplest case, for units of each size and the values compared. The unit with the smallest value will give the most precise estimates. When desired precision (p) is set,

By re-arrangement of terms, necessary replication (n) can be calculated from:

In their detailed computer simulations, Wiebe & Holland (1968) and Wiebe (1971) considered the implications of sampling patchily distributed plankton populations with nets of various sizes towed over a number of distances. They concluded that, in general, nets of larger diameter gave estimates of abundance that were both more accurate and more precise than did smaller nets. They also found that increasing the length of the tow significantly increased both accuracy and precision for all net sizes. In both papers,

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however, it was emphasized that both precision and accuracy increased with the size of net and length of tow at a decreasing rate. Thus, the most efficient sample units are not always the largest. For example, precision increased by about 26% with an increase in net diameter from 25 to 100 cm, but a further fourfold increase in the diameter of nets caused only an additional 8·2% improvement in precision. Pihl & Rosenberg (1982) took increasing After arbitrarily setting their desired precision at numbers of samples with a sampling unit of set size until they attained the required precision for estimates of abundance for the four species they studied (two fish and two crustaceans). Although they found that only 10, 15, 20, and 25 samples would be required for each of the four species, respectively, they conservatively chose to use 30 replicates in their main study. This study represents an example of the empirical rather than mathematical determination of appropriate sample size. With more complex but more potentially informative designs, a number of estimates of precision can be obtained for units of all sizes and the mean precisions compared using analysis of variance with the replicate measures of precision as data. This is possible where the pilot study is repeated at a number of similar sites or over a short time or when sufficient replicates of each sized unit are collected to allow subdivision of the data into ‘b’ subsets of n/b replicates for each size of unit (Scheffe, 1959). Measures of precision are calculated for each data set for each size of unit and the values analysed. Significant differences among the means of estimates of precision indicate better average precision for one or more unit sizes than for others. If the costs of using different sized units are similar (e.g. it might be only marginally more time consuming to count urchins in a 1 m2 quadrat than in a 0·25 m2 quadrat), then the analysis could (greatest precision) is the best choice if costs are stop here. The size of unit with the smallest value for not considered because fewer units of that size than of the other sizes will be required for any chosen level of precision. When differences in the costs among units are negligible, the cost of sampling is proportional only to sample size. In such a case, the choice of the size of the unit to be used is greatly simplified and the unit with the smallest CV should be used since that unit will yield greatest precision for any sample size. Green (1979) suggests that when precision is the same for units of all sizes then larger rather than smaller units should be favoured. It is unlikely, however, that all units will cost the same to use and hence efficiency (cost of attaining required precision) will be likely to favour the choice of one unit over others. For any size of sampling unit, precision will increase with sample size because the standard error and confidence intervals decrease with increasing replication. Increased precision is offset by the increased cost (time, effort, money) of obtaining and processing large samples. As with increases in the size of sampling units, the rate of increase in precision with increasing sample size is initially great but declines as sample size becomes large (Cochran & Cox, 1957; Cochran, 1963; Green, 1979; Scheaffer, Mendenhall & Ott, 1979). The declining rate of increase in precision suggests a situation of diminishing return for effort. It is usually economically desirable to use the smallest possible size of unit, but the cost of obtaining sufficient of them to obtain a required precision must also be considered. The total cost of sampling will be the cost per unit multiplied by the number of units required, plus any overhead costs, such as getting to the study site. To , fewer larger units will usually be required than smaller obtain a given precision, expressed as, say, units. In studies from several fields, it has been found that it is generally more economical to use many smaller sample units than a few larger ones (Gerard & Berthet, 1971; Elliott, 1977; Downing, 1979; Pringle, 1984; Downing & Anderson, 1985; Downing & Cyr, 1985; Morin, 1985). This result is not, however, universal and several authors have recommended the use of larger rather than smaller sampling units (Dennison & Hay, 1967; Wiebe & Holland, 1968; Gray, 1971; Wiebe, 1971; Kenchington, 1978). The relative economy of using sample units of various sizes should not be the sole arbiter of optimality. Particular biological, behavioural, physical or other factors will also be important in choosing units of appropriate, optimum size. For example, observer fatigue may increase bias and decrease precision when

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large numbers of organisms are being counted (Caughley, 1977; Green, 1979; Kirchman, Sigda, Kapuscinski & Mitchell, 1982). It is also likely that small quadrats will be searched more thoroughly than large quadrats. Wiebe (1971) discusses, but does not analyse, the possible effects of costs and logistics, such as ship time and sorting time, on the choice of optimum size sampling units for studies of plankton. He also re-emphasizes suggestions by other authors that, in such studies, the choice of larger rather than smaller nets is likely to be an important means of decreasing avoidance of nets by real zooplankton in the field, an important source of inaccuracy in estimates of abundance of plankton. Such considerations will be specific to different fields. Where both cost and precision vary with unit size it has been repeatedly shown that efficiency is not always simply a function of the size of sampling units (Downing, 1979; Pringle, 1984; Downing & Anderson, 1985; Downing & Cyr, 1985; Morin, 1985). The estimate with the smallest variation will often be from the largest and probably the most costly unit. The efficiency (cost of obtaining a desired precision) of units of various sizes can be compared by cost-benefit analyses (Cochran, 1963; Saila et al., 1976; Snedecor & Cochran, 1980; Underwood, 1981; Irish & Clarke, 1984). In this case the cost-benefit analysis is the determination of the cost of using sufficient units of each size to estimate abundance with the desired precision. From the equation on page 55, number, n, of units of size u necessary for desired precision, p, is:

The toptal costs, Ct, of sampling with units of size u will be the sum of the overhead costs, co, and nu times the cost per unit, cu:

Evaluation of this expression for units of each size allows a direct comparison of their efficiency and, hence, allows the choice of optimum size of unit for the population being sampled. If overhead costs are constant and independent of size of unit, then only the last term in the above equation need be calculated. Note also that if a general cost function is desired, cost per unit can be expressed as a function of one or or size of unit (Downing & Anderson, 1985; Downing & Cyr, 1985; Morin, more variables, such as 1985). In most fields of ecology the relationships among the size of sampling units, replication, and efficiency have been examined only infrequently. Pringle (1984) compared the sampling efficiency (=cost per unit precision) of six quadrat sizes (0·25 m2−4·0 m2) used for estimating the biomass of a seaweed and found that although replication required for given precision varied linearly with size of sample units (quadrats), the relationship between total sampling time and size of quadrat was curvilinear. The relationship between cost and benefit (in terms of precision) was not a simple linear function of the size and number of quadrats used. Intermediate sized quadrats (1·0 m2 and 1.56 m2) were least efficient whilst the smallest quadrat tested was the most efficient. This means that it is more economical to take a larger number of the smallest units than a smaller number of larger units to estimate biomass with any given level of precision. Several Canadian scientists have recently reported similar investigations of optimal numbers and sizes of sampling units for the study of benthic organisms in freshwater lakes and streams. Downing (1979) and Morin (1985) analysed published data for a number of benthic animals from lakes and large rivers, and from streams, respectively. Downing & Anderson (1985) analysed original and published data for estimated

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biomass of macrophytes; Downing & Cyr (1985) did likewise for epiphytic invertebrates. These authors derived empirical relationships between variance, mean abundance and the size of sample units based on the principles of Taylor’s power law (see p. 75) and the expected change in variance with changes in unit size. The relationships were developed by multiple regression of the logarithm of variance on the logarithms of mean density and sample unit size. In all cases the relationships were highly significant and accounted for between 88 and 95% of the variation in the large sets of data fitted. These empirically derived relationships and the terms re-arranged to give an were then substituted into the formula for precision, measured as expression for n (sample size) in terms of mean abundance, size of sample unit and desired precision. The conclusions from these syntheses of large bodies of published data were relatively consistent across all of the studies. In all cases, the sample size required to achieve a given level of precision was a tight function of population density and the size of the sample units. The sample size needed to achieve the desired precision declined as the density of the population being sampled increased and as larger sample units were used. In both cases, the rate of change in replication necessary for desired precision also declined. W.D. Taylor (1980) has criticized the approach of Downing (1979) arguing that conclusions taken from the synthesis of such a variety of species, systems, and methods are of doubtful use for studies of single species. For further comment see also Downing (1980, 1981) and L.R.Taylor (1980). When functions describing the costs of obtaining and processing samples were applied to the precision formulae in the above four studies the results diverged. Downing (1979) and Downing & Anderson (1985) concluded that it was always more efficient to use many small units than fewer large ones. In the first study, the cost was estimated by the total area of sediment that would have to be collected and sorted for the required replication of units of a given size. In the second study, costs per sample were measured when data were being collected for comparisons of performance of different sized sample units (quadrats). Downing & Cyr (1985) measured the costs of collecting and processing samples using quadrats of five different sizes and concluded that a quadrat of intermediate size (500 cm2) was the most efficient. Morin (1985) concluded that when population density was great, small sample units were most efficient but that larger units were more cost effective for sampling populations of low density. These studies illustrate the importance of replication and the size of sample units to sampling efficiency and precision. The relationships identified by Downing (1979), Downing & Anderson (1985), Downing & Cyr (1985), and Morin (1985) are useful indicators for the calculation of optimum sample size but encompass data from a wide range of taxa, methods, and locations. As these authors stress, pilot studies are the best means of fine tuning methods, size of sample units, and sample size for a particular study. Stratified sampling In the preceding section, we discussed optimization procedures on the assumption that one sample was to be collected to give a single estimate for a population. Estimates from simple random sampling are most likely to be reliable when the characteristics (e.g. aggregation, density) of the population are relatively consistent throughout the area sampled. Often the population or area being sampled is obviously not homogeneous and such characteristics as density and patterns of aggregation vary from place to place. In such cases, overall abundance or average population density can be estimated more precisely if a number of subsections of the population are sampled separately. In other words, various strata of the environment or population are identified and the organisms sampled in a way that is appropriate for each stratum. This is referred to as stratified sampling (Cochran, 1963; Scheaffer et al., 1979; Snedecor & Cochran, 1980). As with singlestage sampling, the objective is the estimation of the total number of organisms in the whole area (e.g. the

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number of scallops in a bay). Precise estimates of parameters are obtained from the weighted combination of the estimates from all strata. There are several ways to design a stratified sampling programme. In the simplest case (Stratified Simple Random Sampling) little is known about the population and strata are identified on the basis of some auxiliary variable(s) (e.g. depth, type of sediment or habitat). Equal numbers of sample units are then collected randomly from each stratum. Alternatively, the total number of replicates feasible for the programme is allocated to the strata in proportion to their areas (Proportional Stratified Sampling). In both procedures, the researcher need know nothing about the population prior to the main programme, but requires a map of the area to be sampled. In practice, optimal sampling procedures may differ among strata. The most precise estimates of the total population will be obtained when sampling is optimal in all strata. To obtain approximately similar precision for all strata, sampling effort is weighted among strata in proportion to the variance of the data obtained from each stratum. Strata with very variable population densities are more intensively sampled than those with relatively homogeneous data. Here, the allocation of samples is proportional to characteristics of the subpopulations, rather than the areas of the strata. Sample allocation can be optimized according to three sets of criteria: (1) minimize the variance of the estimate for fixed total cost; (2) minimize the total cost of obtaining an estimate with a given variance, where costs per unit are equal for all strata; and (3) minimize the total cost of obtaining an estimate with a given variance, where costs per unit vary among strata. Formulae and full discussions of these may be found in Cochran (1963), Scheaffer et al. (1979), and Snedecor & Cochran (1980). To obtain estimates of identical precision from each stratum, each stratum should be treated as a single sample and the replication needed to give desired precision calculated as discussed on pages 52–58. In this case (Stratified Sampling with Optimal Allocation), the total number of replicates is the sum of the numbers of replicates from each stratum. If this total effort is beyond the limits of the programme, then one of the proportional allocation procedures must be used or the level of precision relaxed. It may also be desirable to optimize sampling methods and/or the size of sampling units for each stratum separately, provided that this does not introduce biases that differ among strata. For optimal allocation or proportional allocation based on variance, estimates of variation are required prior to the main study. These are best obtained from pilot studies in which a small sample is recorded from each stratum. Cochran (1963) has pointed out that stratification with sample allocation based only on auxiliary variables often fails to give estimates that are more precise than those from a Simple Random Sample design. Stratified Simple Random Sampling is the stratified design most commonly used in marine ecological studies. It is common, for example, to stratify sampling by habitat. In most cases, however, this design is used to obtain data for comparison of abundances in different habitats rather than to estimate the total number of organisms in all habitats sampled. Other stratified sampling designs, better suited to estimating total numbers, have apparently not been common in marine studies. Cuff & Coleman (1979) used a Proportional Stratified Sampling design to estimate the abundances of molluscs, polychaetes and crustaceans in Western Port Bay, Australia. They allocated samples according to the areas of the strata and used the data to calculate the expected variance of the estimated populations of the three taxa for three

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designs of stratified sampling and for Simple Random Sampling. They found that the precision of the estimates was not improved by stratifying the sampling and concluded that stratification was generally no better than Simple Random Sampling. Green (1980) critically assessed Cuff & Coleman’s study and argued that they were not justified in extending their results, from only one study in one bay, to general recommendations for the design of future sampling programmes elsewhere. The failure of stratification to increase precision may well have arisen from the way in which Cuff & Coleman divided the bay into strata. More informed allocation may well have produced a more precise estimate than Simple Random Sampling (Cochran, 1963). Furthermore, Cuff & Coleman pooled counts of species and analysed the data by phyla. Green (1980) noted that this level of taxonomic resolution may be an inappropriate basis for stratification because it is unlikely that all species in a phylum will respond to environmental variables in the same way and have similar patterns of abundance. Heisig & Hoenig (1986) described a method of optimizing sampling design for estimating secondary productivity from size-frequency data. The allocation of effort among different times of the year (=strata) was optimized according to the expected abundance of organisms at each time. Multi-stage sampling Many studies are more concerned with examining patterns in abundance than with estimating the total number of organisms in an area. It may be important to assess whether a population can be considered homogeneous at various spatial scales, or whether differences in population parameters occur among, for example, different habitats. These situations have been referred to as mensurative experiments (Hurlbert, 1984). The simplest design for such comparisons is a stratified Simple Random Sampling design. In most instances, stratified sampling with proportional or optimal allocation of replicates would be more efficient for obtaining a single estimate for a population. Because of the unequal sample sizes, however, such designs are not as well suited to statistical comparisons among strata as Stratified Simple Random Sampling. Stratified designs can involve two or more hierarchically arranged levels of replication. Cuff & Coleman (1979), for example, collected replicate grab samples at each of several stations within each stratum of a bay. When replication is balanced among strata and at each level within strata, these hierarchical, multi-stage designs are perhaps the most powerful designs for comparing estimates. In general this class of designs is concerned with the estimation of differences among means for a single variable. The analysis of this type of data is often best accomplished through techniques of analysis of variance. No other analyses can formally test for both hierarchical and interactive effects within the same design. Formal introductions to these techniques of analysis may be gained from standard texts such as Winer (1971), Snedecor & Cochran (1980), and Sokal & Rohlf (1981). Underwood (1981) gave an introduction to techniques of analysis of variance and reviewed their use in marine ecology. Multi-stage designs allow the researcher to estimate variability at different scales within strata (such as habitats) and provide a stronger basis for generalization. For example, in the investigation of patchiness in the abundance of plankton, replicate samples should be taken in close proximity at sites separated by several kilometres, with several sites being sampled at stations separated by hundreds of kilometres. Another example might be the comparison of numbers of fish in kelp forests with the numbers in sea-grass beds. In order to draw general conclusions it would be necessary to sample each habitat at more than one location (separated by perhaps several kilometres) and to sample more than one site within each habitat at each location. The problem with complex designs such as these, is how to allocate effort efficiently among the various levels of the programme—how many locations, sites, and replicates give optimum precision to the estimates of abundances of fish in the two habitats.

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Methods are readily available for such multi-stage cost-benefit analyses (see Cochran, 1963; Snedecor & Cochran, 1980; Sokal & Rohlf, 1981). Saila et al. (1976) and Underwood (1981) discussed the procedures and advantages of multi-stage cost-benefit analysis. Based on the hierarchical (or nested) analysis of variance, this form of cost-benefit analysis uses estimates of the proportion of variation explained by each level of sampling (such as locations, sites, and replicates), and the expected costs of replication at each level, to allocate sampling effort among levels so that overall variation of estimates of means is minimized. Bernstein & Zalinski (1983) have recommended the use of these procedures for planning environmental monitoring studies. Perhaps the most frequent application of multi-stage sampling is in studies of micro-organisms where hierarchical subsampling is employed to decrease the effort and expense of counting the large numbers of organisms collected in each field sample. Subsampling is routinely used in studies of plankton, benthic infauna, and bacteria. Well-developed and tested methods of subsampling and accurately splitting the samples into subunits exist, but there has been little attention to optimization. Subsampling has been done mostly by traditional rather than tested practices (Alden, Dahiya & Young, 1982; Kirchman et al., 1982; Montagna, 1982; Irish & Clarke, 1984). Kirchman et al. (1982) used cost-benefit analyses to determine the optimum number of subsamples, aliquots (=filters) and fields of view to be taken when using hierarchical Subsampling to estimate bacterial abundance in bodies of water (note that they refer to the field samples as subsamples of the water body). Irish & Clarke (1984) used cost-benefit procedures to determine optimal designs for sampling, and Subsampling, a tarn to estimate chlorophyll a concentrations and abundance of phytoplankton. Kirchman et al. (1982) found that by optimizing the design of the Subsampling procedures, “accuracy” (actually precision) of the estimates of abundance for the water bodies sampled increased by 20–50% compared with the traditionally used designs with only marginal increases in costs. They emphasize that most customary sampling designs fail to estimate variation at all levels (e.g. among fields or among filters) of the Subsampling procedure and, in the worst cases, cannot provide any statement of precision for the estimate of the number of bacteria in the sample. Montagna (1982) also found that the optimum design for Subsampling sediment samples to count bacteria involved replication of subsamples, filters, and fields of view. He also pointed out that the customary practice of counting 15–20 fields of view on only one filter per sample was inadequate. Venrick (1971) has also stressed that it is important to estimate the amount of variation at each level of Subsampling procedures. She gave a full explanation of the components of variation at each level and used a computer simulation to examine the relative effects of different subsample fractions and replication at each level on the estimation of the variation in the estimate of abundance at the highest level. She did not, however, deal with cost-benefit analyses. In other fields, spatial and/or temporal variation have only infrequently been investigated usefully by explicitly incorporating spatial scale into the design of sampling programmes. Replicates within levels (representing different scales) of hierarchical sampling programmes have often been combined with the result that variation at those scales could not be estimated (but see Lee & McAlice, 1979; Schwiegert & Sibert, 1983; Hankin, 1984; Kennelly & Underwood, 1984, 1985; Schwiegert, Haegele & Stocker, 1985; Andrew, 1986). Schwiegert & Sibert (1983) examined optimum designs for sampling Pacific herring from commercial catches to estimate the age structure and abundance of the fished population. The optimal designs varied among age classes of fish and among the different fishing ports sampled. In some instances, the analyses indicated that up to 100 fish should have been taken from several subsamples of each load that entered a port. In other cases, replication of subsamples was unnecessary and the optimal design was a two-stage

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design (a number of fish taken from several replicate loads). For further discussion, see Smith (1984) and Schwiegert & Sibert (1984). Schwiegert et al. (1985) compared the precision of estimates of the abundance of herring spawn derived from several two-stage designs and a simple random sampling design. They found that carefully designed two-stage sampling gave more precise estimates than Simple Random Sampling but that other two-stage designs were no better than Simple Random Sampling. Hankin (1984) also compared a number of methods of allocating samples among levels of a two-stage sampling design implemented to estimate the abundance of fish in freshwater streams. He found that the most efficient design was one in which primary units (sections of stream) were variable in size and corresponded to discrete habitat units, such as riffles or pools. Random secondary units were then taken within each of these primary units. Hankin’s study essentially represented an empirical demonstration of the advantages of carefully designed stratified sampling programmes for estimating the total abundance of organisms in a heterogeneous habitat. Saila, Pikanowski & Vaughan (1976) used data from a prior survey in single and multi-stage cost-benefit analyses to estimate optimal designs for sampling several species of benthic invertebrates in the New York Bight. Kennelly & Underwood (1984, 1985) used pilot studies and cost-benefit analyses to derive the optimal replication at three spatial scales for sampling micro-organisms in a temperate sub-littoral kelp forest. All of the above studies have used the minimization of sample variance, and maximization of precision of estimated means, as the criterion of ‘benefit’ in cost-benefit analyses. Millard & Lettenmaier (1986) have discussed the use of multi-stage cost-benefit analyses based on balancing statistical power against cost rather than simply considering cost-precision relationships. They argue that in many situations optimal design based on the precision of means is inadequate since no account is taken of the likelihood that a hypothesized effect will be detected. Several authors have emphasized that patterns and processes can be better understood if estimates of variability at various scales are explicitly included in the design of sampling programmes. Caffey (1985) is a good example of such a study. The shortage of studies that have done this perhaps reflects attitudes of research workers to variability and its manifestation at various scales of investigation. As with more simple designs, pilot studies are required for optimization procedures for stratified and multi-stage sampling. Ideally, different sized units and methods should be examined in a number of the habitats, sites, etc., involved in these designs, although it may not always be possible to do this at all levels of the intended programme. Tests done under the extremes of density, habitat complexity or other independent variables may suffice. In pilot studies for multi-stage sampling it is important that each level of replication in the proposed programme be addressed by replication at corresponding levels in the pilot study. Note that cost-benefit analyses in this context can only be applied to hierarchically arranged factors. Replication at higher level orthogonal factors, when not fixed implicitly by the objectives of the study, can only be designated after estimates of replication at all lower levels have been calculated (Underwood, 1981). For example, in the earlier example of estimating the abundance of fish in kelp forests and in sea-grass beds, cost-benefit analyses would allow the determination of optimal numbers of replicate transects and sites in each habitat at each location. The number of locations to be sampled can then be calculated by dividing the total resources (time, money, etc.) available by the amount of resources required to sample the optimum number of sites and transects in each habitat. When the number of treatments in orthogonal or higher level factors is predetermined by the objectives of the programme, the total affordable cost is divided equally among the orthogonal treatments. The replication of nested factors within those treatments is then determined by cost-benefit analysis based on the sub-divided cost limits (Schwiegert et al., 1985).

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POWER ANALYSIS Most sampling programmes are suboptimal because of logistic and/or cost constraints. Procedures such as cost-benefit analysis will ensure that the best use is made of limited resources, but the question remains whether the design is good enough to do the job asked of it. Recognition of this problem makes it desirable to know, a priori, how large a difference among means could be detected or, a posteriori, how sensitive a test was. Power analysis makes such statements possible. Statistical power is defined as the probability that a test will lead to the correct rejection of the null hypothesis; that is, that the null will be rejected when the alternative is true (Sokal & Rohlf, 1981). Power is, the probability of falsely retaining conceptually and probabilistically, the complement of Type II error (Yamane, 1967; Winer, 1971; Cohen, 1977; Sokal & a null hypothesis, and accordingly is expressed as Rohlf, 1981). A clear and extensive treatment of power analysis is given by Cohen (1977) who has provided tests, F tests, analysis of tables for the determination of power for correlation coefficients, t tests, variance for fixed effects and their interactions (for a limited number of designs) and analysis of covariance. A more statistically demanding treatment may be found in Winer (1971), who also considered the power of tests involving random factors in analysis of variance. Recent discussions in the ecological literature include Underwood (1981), Bernstein & Zalinski (1983), Toft & Shea (1983), and Rotenberry & Wiens (1985). Cohen (1977) has identified four parameters of statistical inference useful in power analysis : (1) power, (3) sample size (n), and (4) effect size. Knowledge of any three will allow the (2) significance level determination of the fourth (Cohen, 1977). The relationship between these parameters provides the research worker with the opportunity to exercise greater control over the design and analysis of sampling programmes and allows better and more careful interpretation of results (Bernstein & Zalinski, 1983; Toft & Shea, 1983). Power analysis is the analysis of the sensitivity of tests of null hypotheses against specified, quantified alternatives. It may be used either in the design phase of a programme or as a post hoc test of the adequacy of a design (Winer, 1971; Cohen, 1977; Sokal & Rohlf, 1981; Underwood, 1981; Bernstein & Zalinski, 1983; Toft & Shea, 1983; Rotenberry & Wiens, 1985; Millard & Lettenmaier, 1986). For example, the researcher may wish to know what sample size is required to detect an effect of given size subject to chosen probabilities of both Type I and Type II error. Alternatively, where a test failed to reject the null hypothesis, the sensitivity of the test can be examined: was the design too weak to detect anything but very large differences (Toft & Shea, 1983)? Despite this great potential, power analysis has remained a much under-utilized technique in ecological research (Toft & Shea, 1983). Several reasons for this lack of use have been suggested. Toft & Shea (1983) have argued that it is a reflection of our pre-occupation with Type I error and also a lack of awareness of Type II error and its importance. The reluctance (or inability) of ecologists to specify exact alternative hypotheses has been cited by Underwood (1981) and Rotenberry & Wiens (1985) as another possible reason for the lack of use of power analysis. By convention, we accept a 5% chance of committing Type I error without any real idea of the probability of Type II error. It is usually argued that this is acceptable because it is better to know the probability that an effect will be incorrectly claimed than it is to know the probability of not demonstrating a real effect. This is not necessarily so, particularly when strong assertions are made from the lack of a statistically significant result—when a null hypothesis is retained (Toft & Shea, 1983). These authors have argued that where a positive conclusion is drawn from a ‘negative’ (non-significant) statistical result, it should be exposed to the same restrictive standards as a positive conclusion drawn from a ‘positive’ (significant) statistical result (see also Rotenberry & Wiens, 1985).

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In many situations, Type II error may have ramifications at least as important as Type I error, yet Type II error is rarely considered. As an ecological example, consider the situation in which research workers wish to assess the effects of commercial fishing on the numbers of adult fish. Areas that are regularly fished are sampled and mean abundance of the fished species in these areas is compared with that in areas that have been protected from fishing. A Type I error (asserting that fishing has depleted stocks when, in fact, it has not) is to be guarded against because such a decision would subject the fishermen to the imposition of strict quotas and the under-utilization of the resource. Further consideration, however, reveals that a Type II error is potentially at least as dangerous. The incorrect assertion that the stock is being fished at or below its sustainable yield (that is, existing fishing has no significant biological effect on stock size) could result in the total collapse of the fishery with consequences more severe than the imposition of quotas. Knowledge of the power of tests is particularly important in the description of pattern. Experiments test hypotheses concerning processes that might explain patterns. The pattern, or observation is taken for granted. In contrast, when hypotheses about patterns are tested, the observation itself is being questioned. The test is not concerned with why there are more gastropods here than over there, but rather whether it is true that there are more here than there. Incorrectly claiming no difference in abundance between the two areas will mean that the ecological processes influencing the density of gastropods will be assumed to be the same in both places. This could seriously delay understanding the processes that influence the number of gastropods in the two areas. of falsely In such cases, knowledge of is perhaps more important than knowing the probability rejecting the null hypothesis. Certainly, serious consideration should be given to the relative weighting of Type I and Type II errors. Setting power at a given level makes an explicit statement about the relative costs of Type I and Type II error (Winer, 1971; Cohen, 1977; Underwood, 1981; Toft & Shea, 1983). Winer (1971) has noted that the conventions of using 0·05 and 0·01 as acceptable probabilities of Type I error have little scientific or logical basis. Greater consideration of the relative costs of Type I and II error may lead to the de-sanctification of 0·05 and 0·01 as inviolate standards of Type I error. is not a The relationship between power and the more familiar probability of making a Type I error power will increase with simple one. If the size of an effect is held constant then, for a fixed value of increasing sample size. Conversely, if power and effect are fixed, then α decreases as sample size increases. That is, Type I error is less likely with larger samples than with smaller samples when power is set to a desired level. If sample size is fixed, as is often the case because of cost constraints, α and power will be directly related. More stringent requirements for power will be accompanied by decreased probability of power and effect size Type II error but increased probability of Type I error. The relations among for a two sample case are shown in Figure 1. Power analysis for planning If power analysis is to be used in the design stage of a sampling programme, then several alternative and n. The approaches may be taken. One is to determine the power of a test given effect size, are familiar and directly determination of appropriate sample size and significance level for Type I error under the research worker’s control. The determination of an appropriate effect size is much less clear, and great caution should be exercised when deciding the size of effect that is considered important (Cohen, 1977; Underwood, 1981; Rotenberry & Wiens, 1985). Effect size is defined as the standard deviation of means divided by a common standard deviation (Cohen, 1977). Where only two means are compared, effect size is the difference between the two means, expressed as a proportion of one of them. Where more than two means are being compared, effect size has no immediately intuitive meaning because a significant

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result can be the product of differences among one or several of the means. Furthermore, the way in which similar means may be grouped and the number of groups may affect the interpretation of effect size and the power of the test. Winer (1971) and Cohen (1977) have provided various procedures by which an appropriate effect size can be calculated with reference to existing information. An appropriate effect size may be determined from previous studies in the system. Pilot studies should provide an estimate of error variance which can be used to calculate the effect size that a future sampling programme would be expected to detect. In such a case, after setting power, the only unknown becomes the numerator in the effect size ratio. Sweatman (1985) used the magnitude of differences among means that were significantly different in a similar study as the effect size criterion for calculating the power of non-significant results. In addition, Cohen has provided operational definitions of small, medium, and large effect sizes. He stressed, however, that these are relative measures, the specific value of which will be unique to the investigation. What constitutes a small effect in one system might be of overwhelming biological importance in another (Cohen, 1977; Underwood, 1981). It should be remembered that power analysis is a statistical procedure. Even if a sampling programme has great power, say 0·95 for all tests, it will be of little use if it is set within an unrealistic biological context. Millard & Lettenmaier (1986) have discussed the use of power analysis in designing environmental monitoring programmes. They advocate the a priori statement of an effect size considered to be important, possibly by legislation, and the use of cost-benefit analyses to design programmes that optimize the power of detecting an effect of that size. Unfortunately, they only consider the power of detecting main effects in complex designs. It is often the case that the interaction of these main effects (e.g. Event status×Time) are of greater interest in detecting environmental change than the main effects alone (Green, 1979; Bernstein & Zalinski, 1983, but see Hurlbert, 1984). Rotenberry & Wiens (1985) have argued that the suitability of a design be determined, not from the calculation of power for a given effect size, and n, but from the calculation of detectable effect size from and n. Decisions about the adequacy of the design would then be made after consideration of known a priori, the suitability of that effect size to the biological hypothesis being tested. By assigning subjectivity is shifted from the critical, a priori estimation of an unknown effect size, to the evaluation of suitability of a known one. That is, rather than stating an explicit, numeric alternative hypothesis, the relative importance of Type I and Type II error are stated and the worker decides upon the suitability of a calculated effect size that is expected to be detectable. Post hoc power analysis The second major use of power analysis is as a post hoc procedure to determine the sensitivity of an already completed test. The analysis here is essentially dependent on decisions already made by the research worker when the sampling programme was designed. Consequently, post hoc power analysis can only indicate the probability of erroneous retention of the null hypothesis; i.e. it can only be meaningfully applied to nonsignificant results. The only three published instances of post-hoc power analysis that we encountered (Sweatman, 1985; Andrew, 1986; Doherty & Sale, 1986) all sought to explain non-significant results. In both Andrew (1986) and Doherty & Sale (1986) the analyses revealed important deficiencies in design. A non-significant F ratio had 70–77% chance of being in error. Power analysis should not be used to justify the post hoc elevation of non-significant trends in the data to ‘pseudo-significance’, rather, it should point towards better design for future studies. It seems, therefore, that while power analysis opens the door to greater control over sampling design, the research worker should tread carefully. The greater control brings with it a greater responsibility in terms of

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Fig. 1.—The relations among Power Effect Size (E), Sample Size (n), Type I and Type II error for a hypothetical two-sample situation. In each diagram, the two curves represent the distribution of the means of samples of size n taken from two populations. and are the grand means of the two sets of estimates and the best estimates of the population means. If the null-hypothesis is true, sample means greater than will result in the incorrect rejection of a true null-hypothesis (Type I error). Thus as a percentage of all values under curve i, is represented by the area under curve i to the right of in each diagram. Conversely, if the null-hypothesis is false, values of less than will result in the incorrect retention of the null-hypothesis (Type II error). In this case, the area under curve ii to the left of in each diagram will represent (as percentage of all results described by curve ii). The area under the remainder of curve ii (to the right of ) represents the correct rejection of the (false) null-hypothesis, i.e. Power It can be seen from diagram B, that increasing a with unchanged n and effect size, results in decrease in and increased power. Diagrams C-E show the effects of increasing sample size (n) and so increasing the precision of the estimates In C, is arbitrarily fixed at the same level as in A, with the result that decreases and power increases, and power are both fixed in D as in A and, clearly, declines. Finally, in diagram E, and power are set at the same level as in A and the detectable effect size is seen to diminish considerably.

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the realistic setting of ß and effect size. The more specifically the hypotheses are stated, the greater the usefulness of the technique. It should be noted that tests of power are inherently one-tailed; tests have no power if any effect occurred that was opposite to that proposed (Cohen, 1977). Clearly in two-tailed tests of significance, Type II error is possible at either tail of the null distribution, but cannot occur at both. Consequently, the power of a test is meaningful only when it is discussed with respect to a single alternative hypothesis. This point has been treated more fully by Sokal & Rohlf (1981). CONCLUDING REMARKS It is important to know the limitations of any sampling programme. The concept of a pilot study is one of doing a smaller, cheaper version of the anticipated main study so that the main study can be optimized. It would, therefore, be pointless expending a large portion of the total budget, time, and resources on a pilot study and subsequently severely constraining the main programme. The amount of time and resources spent on the pilot study will be specific to each project and generalizations cannot be made. The cost of a pilot study, however, will generally be outweighed in the long run by the improved efficiency, accuracy, and precision in the main sampling programme (Green, 1979). Pilot studies along the lines of those suggested on pp. 44–52 could also be designed to compare methods of sampling, with different methods being substituted for the different sized units. More elaborately, various dimensions of units (e.g. core diameter and depth, transect length and width) can be varied orthogonally in pilot studies to establish not only the optimum size (volume or area) but also the optimum dimensions of sampling units. Different methods can also be compared across a range of unit sizes by testing unit size and method orthogonally in the one pilot study. Although orthogonally testing a suite of variables is more powerful and informative, to do so may prove prohibitively expensive. Many possibilities exist, but cost will limit the extent of a pilot study. It will generally be cheaper and often sufficient to evaluate methods, unit size and replication at various levels in a set of sequential, separate pilot studies than in a single big one. Some arguments against pilot studies have been made. Smith (1984) argued that estimates of variance obtained in pilot studies are frequently biased by small sample sizes (Cochran, 1963) and that the apparent benefits of pilot studies may be misleading when a population is sampled at a number of times: a design that is optimal at one time may be very poor at another time because of changes in the characteristics of the population being sampled. Schwiegert & Sibert (1984) replied that some indication of possible improvements in design is better than consistently sampling in ignorance. If temporal variation in population characteristics is likely to cause changes in the optimum design of a programme, then small pilot studies may be required at the beginning of each sampling occasion (Kennelly & Underwood, 1985). With respect to the problem of biased estimates caused by small sample sizes, Schwiegert & Sibert (1984) pointed out that Cochran (1963) detailed methods of estimating confidence limits for sample sizes chosen on the basis of pilot studies. In general, precision remains relatively stable for sample sizes within these confidence limits. In the absence of careful design procedures, there is the risk that methods and designs for sampling will be chosen simply by reference to previous studies. In some cases, such as successive studies of a system, this may be justified. In others, however, it leads down the potentially blind alley of standardization by convention. If a large number of independent studies come to similar conclusions and widespread standardization of sampling procedures in some systems is a result, then the use of standard procedures is to be encouraged. Standardization for its own sake, against evidence of variation in the usefulness of methods and designs when used under differing conditions, is, however, to be discouraged (Gray, 1971; Loya, 1978; Omori & Hamner, 1982).

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From our discussion of power analysis, it should be clear that much is to be gained from the use of power analysis to examine the likely usefulness of the planned sampling programme and analyses. With data from the pilot studies, the proposed analyses can be thoroughly tested for their sensitivity with the expected nature of the data. It is at this stage that careful decisions have to be made about the relative importance of Type I and Type II errors. It may be that the desired levels of these errors are unattainable simultaneously with the design and replication proposed for the main sampling programme. In such a case, critical reexamination of the sampling design, of the desired levels for Type I and Type II errors, of costs of increasing replication (to increase power), and even of the basic questions being asked will be necessary if the programme is going to fulfil its intended purpose. The adjustments that have to be made will be entirely specific to each study. SPATIAL PATTERN The arrangement of organisms in space may range from aggregated (contagious, over-dispersed, clumped) through a random pattern to being, more rarely, regular (under-dispersed, uniform). In this context it is worth repeating Cassie’s (1963) explanation of the terms over- and under-dispersion. The terms refer to the distribution of density estimates about the mean rather than the spatial arrangement of individuals. In an aggregated population there will be a greater occurrence of small and large densities, hence the frequency distribution would be over-dispersed. In contrast, if the biological population were uniformly arranged in space, then density estimates would be less variable and their frequency distribution under-dispersed. Knowledge of the spatial patterns of organisms is a necessary prerequisite in establishing what processes might underlie the observed arrangement of organisms. If, for example, individuals were regularly spaced throughout an area then competition might be invoked and tested for experimentally. If they were aggregated, then a range of processes leading to that arrangement might be tested. The extent to which the arrangement of organisms deviates from a random pattern is also implicitly considered in the design of sampling programmes used to estimate their abundance. Clark & Evans (1954, p. 446) have provided a definition of spatial randomness: “In a random distribution of a set of points on a given area, it is assumed that any point has had the same chance of occurring on any sub-area as any other point, that any sub-area has had the same chance of receiving a point as any other subarea of that size, and that the placement of each point has not been influenced by that of any other point.” In this review, we are concerned only with the description of pattern and do not wish to infer processes to account for any identified patterns. The first step in the description of the spatial pattern of a population is to test the null hypothesis of spatial randomness. A null hypothesis of spatial randomness is advantageous because tests of this hypothesis are two-tailed: rejection unambiguously suggests one of the two possible alternatives, either clumped or regular. If the test fails to reject the null, then little more needs to be said about the arrangement of those organisms: random is random. The need for more detailed description, or explanation, only arises if there is evidence of non-random pattern. Furthermore, although the great majority of populations in nature are clumped, random and regular populations have both been described, (e.g. Holme, 1950; Barnes & Marshall, 1951; Stimson, 1974; Jumars, 1975a; Seapy & Kitting, 1978; Ebert & McMaster, 1981). The null hypothesis that the population under study may be random is realistic. DETECTING NON-RANDOM PATTERN Indices of spatial pattern fall into two general categories: those derived from estimates gained from some chosen sample unit, such as quadrats or cores; and those based on measurements of distances between

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organisms. We shall consider several of the most commonly used indices from each category and discuss are based on estimates the limitations of their use. The coefficient of dispersion and Morisita’s (1959) derived from unit sampling and Clark & Evans’ (1954, 1979) R, Pielou’s (1959) and Mountford’s (1961) and Johnson & Zimmer’s (1985) I are based on estimates of distances between organisms. Measures from sample units Variance/mean ratio (Coefficient of dispersion). The variance/mean ratio is the simplest index of spatial pattern and perhaps the one most commonly used in the marine ecological literature. It is based on the characteristic that, for a population of individuals that are randomly arranged in space, the variance of a sample will equal the mean density. Because equality of the variance and mean is a defining characteristic of the Poisson distribution, deviation from this expectation provides the basis for a test of spatial randomness (Pielou, 1969). Under the null hypothesis of random spatial pattern therefore, the estimated variance (s2) and obtained from a sample of randomly placed units should be equal. mean The variance/mean ratio may be tested for significant departure from the expectation (under a Poisson process) of unity by calculation of its confidence intervals (Greig-Smith, 1964). Examples of this approach include Kosler’s (1968) investigation of the patterns of spatial pattern of meiofauna in the Baltic Sea and Dayton’s (1973) analysis of the rocky intertidal alga, Postelsia palmaeformis. More usually, departures from unity are tested for statistical significance after conversion to the statistic:

which is compared with a distribution with degrees of freedom (Southwood, 1966; Pielou, 1969; Fisher, 1970). Tests of significant departure of the variance/mean ratio from unity have been used in many studies of species ranging from polychaetes and molluscs (Rosenberg, 1974) to asteroids (Scheibling, 1980). In several papers reviewed it was unclear whether the variance/mean ratio was incorrectly compared distribution or was first converted to its test statistic I. Partial explanation for this may lie directly with a in semantic confusion, as the terms “variance/mean ratio”, “coefficient of dispersion” and “index of dispersion” appear to have slipped into synonymy (e.g. Southwood, 1966; Elliott, 1977; Diggle, 1983). The test statistic as used above (Fisher, term “index of dispersion (I)” has traditionally meant the 1970). The “coefficient of dispersion”, as originally proposed by Blackman (1942, p. 352), referred to the variance/mean ratio: “This estimate of dispersion, which might be termed a ‘reduced index of dispersion’, since it is the index of dispersion divided by the degrees of freedom, has been called by Clapham (1936) ‘relative variance’.” Blackman, therefore, clearly distinguished the “coefficient of dispersion” from the “index of dispersion”, as have many authors since then (e.g. Cassie, 1963; Greig-Smith, 1964; Pielou, 1969; Jumars, 1975a). The distinction between these terms should be maintained. Several authors have investigated the power of the “coefficient of dispersion” to detect non-random pattern (Bateman, 1950; Kathirgamatamby, 1953; Diggle, 1979; Perry & Mead, 1979; Helshe & Ritchey, 1984). In general, this statistic was a powerful test for non-randomness, even when the organisms were only mildly aggregated. The “coefficient of dispersion” was not, however, as powerful in detecting regular patterns. Bateman (1950) has shown that the “coefficient of dispersion” provided a powerful test of randomness only when sample size was greater than five. Bateman (1950) considered that if the mean density of organisms per sample unit was less than one, then the results of the test were invalid. Means of

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between one and five organisms per unit produced conservative results, there being only a small probability of rejecting the null hypothesis of random spatial pattern (see also Cassie, 1963). This caution seems to be supported by results of studies in which “coefficients of dispersion” based on mean densities of less than five were tested for significant departure from unity (e.g. Holme, 1950; Barnes & Marshall, 1951; Kosler, 1968; Gage & Geekie, 1973a). In these studies there was a strong trend towards detection of significant departure from randomness with increasing density, and in many samples with a mean density of less than five there was no significant evidence for non-random pattern. The variance/mean ratio is a useful descriptive statistic. It is perhaps best regarded as an exploratory descriptive device, capable of detecting non-random pattern, but subject to a number of limitations. Its mean, and n make it of little use as a comparative test statistic when strong dependence on unit size, these variables differ among samples (Elliott, 1977). Goodness-of-fit tests. A test for non-random pattern may also be made by directly comparing the observed frequency distribution of estimates of density to the Poisson distribution with the same mean. The test, the more recently recommended comparison is made by a goodness-of-fit test such as the traditional G test, or their nonparametric equivalent, the Kolmogorov-Smirnov test (Sokal & Rohlf, 1981). Gage & Geekie (1973a) analysed a number of data sets from an investigation of benthic fauna with both the test and tests rejected the null hypothesis in a the Kolmogorov-Smirnov test. They found that, although the two similar percentage of tests the two tests produced conflicting results for some data sets. Fifteen of 43 tests test were not significant by the Kolmogorov-Smirnov test. Gage & judged to be significant by the Geekie argued that it is generally assumed that discrepancies arise in such cases from the different power of test, size classes with low expected frequency have to be combined, thus decreasing the two tests. In the the degrees of freedom of the test. Because such pooling is not required for the Kolmogorov-Smirnov test, it would not be similarly affected at the tails of the distribution. This index is based on an analysis of the proportion of the total number of organisms Morisita’s index found in each replicate sample unit. The index is calculated from:

where Σx is the sum of individuals found in all replicates (Morisita, 1959, 1962, 1971; Southwood, 1966; for maximum regularity and n when all Elliott, 1977). Its value can range between individuals are in the same replicate (maximum aggregation). It has a value of one for a random pattern, is, therefore, strongly dependent on sample size (Green, 1966; Elliott, 1977). The significance of may be tested against the F dis-tribution by calculating

(Southwood, 1966). Alternatively, the test statistic

is distributed with degrees of freedom (Elliott, 1977). Morisita (1959), Hairston, Hill & Ritte (1971), and Elliott (1977) have investigated the behaviour of under differing spatial patterns and found it to be strongly influenced by the size of sampling unit. For an aggregated pattern with individuals randomly gives a large and constant value with increasing unit size until the size of the arranged within clumps,

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sample unit approximates the size of the clumps. Beyond this it declines towards one. If individuals are will increase with increasing unit size before uniformly arranged within clumps then the value of declining as above. The index gave unpredictable results when applied to a pattern that was known to be has been shown to be relatively invariant under conditions of uniform (see Fig. 18 in Elliott, 1977). different abundances (Morisita, 1959; Gage & Geekie, 1973b; Elliott, 1977). is relatively invariant with Gage & Geekie (1973b) and Colby & Fonesca (1984) used the fact that differing densities in their comparisons of spatial pattern between samples of infaunal molluscs and sand crabs, respectively. Interestingly, Gage & Geekie (1973b) found considerable concordance between the “index of dispersion” (I) and in the detection of significant departures from randomness in a number of was found to be slightly more conservative than I, but both tests gave more significant their samples, goodness-of-fit test or the Kolmogorov-Smirnov test. results than tests of a Poisson model using either a Other indices based on sample units. Other indices derived from measures of variance of density estimated with sample units have been discussed in several texts (Greig-Smith, 1964; Southwood, 1966; Pielou, 1969; Elliott, 1977). These indices have received little use in the marine ecological literature and will not be discussed here. Most are variants of the variance/mean ratio and are subject to the same limitations in interpretation (Elliott, 1977). One variant that deserves some mention is the index proposed by Green (1966). The index is given by:

and is valued at for maximum regularity, 0 for random pattern, and 1 for maximum and and may, therefore, be used as a aggregation (Green, 1966). Cx is independent of variations in n, comparative index even when these values vary between samples (Green, 1966). The usefulness of Cx as a comparative statistic is limited because there is at present no test of significance for departures from randomness. Hogue (1978) and Findlay (1981) have used Cx as a descriptive statistic in studies of pattern in benthic meiofauna. Jumars (1975a) has extended the use of the “index of dispersion” to consider the spatial pattern of more than one species. By comparing the summed “indices of dispersion” with the “index of dispersion” calculated from the summed data, differences in the spatial pattern of the species could be detected. If all species are similarly arranged then the two will be similar. If species tend to be segregated in sample units then they will differ. Two studies of deep-sea meiofauna have found evidence for patchiness in the arrangement of species using this test (Jumars, 1975b; Bernstein & Meador, 1979). Cautions for the use of indices based on sample units. The widespread use of these indices probably stems from the fact that data obtained to estimate abundance can be used to test for non-random pattern. Although this simplicity may be attractive, several points should be kept in mind when interpreting results. The first and most important point is that, for aggregated populations, the results gained may be an unpredictable consequence of the size and shape of the sampling units used (Skellam, 1952; Southwood, 1966; Payandeh, 1970; Perry & Mead, 1979; Diggle, 1983; Helshe & Ritchey, 1984). The observed pattern may vary from random, to aggregated and finally regular depending on the scale at which the population was sampled relative to the scale at which it was aggregated. This sampling artefact may arise from at least two sources. The relationship between variance and mean may change due to variations in abundance alone or through variations in the variance without attendant shifts in abundance, or both. Perry & Mead (1979) have shown that the capacity of significance tests to detect significant departures from random pattern for the variance/mean ratio increases with increasing density, and suggested that this is more the result of an increase in variance with increasing density than it is due to increases in density per se.

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For aggregated populations of many different types of organisms, there is strong empirical evidence that the variance of a sample increases disproportionately with increases in its mean (Clapham, 1936; Clark & Milne, 1955; Taylor, 1961, 1971; Kosler, 1968; Gage & Geekie, 1973a; Gagnon & Lacroix, 1982). This relationship is known as Taylor’s power law, and is essentially a statement that the spatial behaviour of organisms is density-dependent (Taylor, Woiwod & Perry, 1978). Under these conditions, as the size of the sampling unit changes, aggregation, as estimated by a unit-based index may change as a function of mean density, even if the variance remains constant. Alternatively, apparent constancy of the variance/mean ratio may arise from changes in both elements of the ratio acting to cancel out each other (Downing, 1979). Indices based on distances between organisms From the above discussion, it will be clear that in the description of the arrangement of organisms in space, there are constraints imposed by the use of artificially defined sampling units. The use of quadrats or other sampling units imposes an order on the community being sampled that may mask true spatial relations among organisms. Consideration of these problems led to the development, by plant ecologists in the 1950s, of indices based on distances between organisms rather than on estimates of abundance gained from unit-based sampling. The data collected are, therefore, free from artefacts introduced by the sampling units. The general technique has been termed plotless sampling, or nearest-neighbour (N-N) analysis. Clark and Evans’ R. The earliest index based on distances between organisms to gain common usage was that proposed by Clark & Evans (1954). The measure is based on the fact that under the null hypothesis of random spatial pattern the expected mean distance between nearest neighbours (rE) can be calculated from knowledge of the density of organisms. where is an independent estimate of the density of organisms. The index R is the ratio of the observed mean distance (r0) and the expected number. R is tested for significance by the standard normal variate Z:

where

N being the total number of individuals in area A. R has been used in several studies of spatial pattern in marine organisms. An early example is Connell’s (1963) laboratory study of the spatial arrangement of colonizing Erichthonius braziliensis, a tube-dwelling amphipod. Immediately after settlement no non-random pattern was discernible, but then a regular pattern became established. Other examples include Stimson’s (1974) analysis of the spatial arrangement of corals and Wilson’s (1976) study of the bivalve Tellina tennis. Clark & Evans (1979) have extended their original index to include considerations of the spatial pattern of organisms in k dimensions. Practically, this allows the analysis of patterns along a line (one dimension) and in three dimensions and removes the restrictive assumption that organisms be positioned in two dimensions only. The extension of an analysis of spatial pattern to three dimensions might be useful in the study of infauna, or organisms inhabiting branching corals. Rohlf & Archie (1978) briefly discussed practical means of plotting the positions of organisms in three dimensions. The three dimensional form of R has apparently not yet been used in marine ecology.

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Pielou’s and Mountford’s α Pielou (1959) has developed a different measure derived from the distribution of distances between randomly chosen points and the nearest individual rather than N-N distances. If individuals in a population are randomly arranged, then the distance between random points and nearest neighbours will be the same as N-N distances. The calculation of Pielou’s index requires a set of such distances and the density of the population. The index is given by:

where d is the density of organisms and is the mean, squared distance between random points and values greater than one indicating nearest-neighbours. Random pattern will be indicated by clumpedness, and values less than one indicating regularity. may be tested for significant departure from 1 by the test statistic

where n is the number of distances measured. A calculated value greater than the tabulated at a significance level of (usually) 0·05 indicates significant aggregation, while a value less than the (usually) 0·95 tabulated value indicates a regular pattern. Mountford (1961) has demonstrated that when density is estimated rather than known absolutely (by enumeration of the entire population) then may result in excessive numbers of erroneous departures from randomness, compared with that expected for a randomly arranged population. He provided a corrected version of the test statistic for that takes into account the estimation (rather than absolute knowledge) of density. has been used to detect non-random spatial pattern in studies of intertidal gastropods and subtidal sea urchins and gastropods (Underwood, 1976; Andrew & Stocker, 1986; Choat & Andrew, 1986). Mountford’s corrected test statistic was not used in these studies because the whole population used in the calculation of was counted. Density was therefore known, not estimated. Johnson and Zimmer’s I. Johnson & Zimmer (1985) have recently proposed a new index of spatial pattern (I). This index seems to offer several advantages over previous indices because all that is needed for its calculation is a set of distances between random points and the nearest individual. The calculation of I does not require an independent estimate of density. It can be used to consider the arrangement of organisms in three dimensions, as can R. When applied to sets of real data Johnson & Zimmer’s index compared favourably with results given by Fisher’s “index of dispersion” and Pielou’s and Mountford’s (Johnson & Zimmer, 1985). When tested with simulated data, it proved to be more powerful than in detecting regular patterns and an aggregated pattern that followed the Negative Binomial distribution. Although the behaviour of I under different spatial patterns and sampling properties has yet to be fully explored, it would seem to have great potential. No accounts of its use have so far appeared in the marine ecological literature. Cautions for the use of indices based on distances. The three indices considered are based on the assumption that an infinite population is being sampled, i.e. there are no boundaries to the area. Edge or boundary effects will arise if the individual (or random point) from which distances are measured is close to the edge of the area considered because there is a higher probability that its nearest neighbour will be outside the area of study. This is less likely to be true if the base individual were in the centre of the area. Selection of the closest individual within the area may therefore introduce bias by over-estimating the distance to nearest-neighbour.

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Sinclair (1985) has shown that boundary effects may have a large impact on R in the above form, biasing results in favour of regularity by over-estimating rE. He went on to discuss a correction factor proposed by Donnelly (1978, in Sinclair, 1985) that counters this bias. Sinclair demonstrated that when the corrected test indicated a significantly regular pattern, with a probability of Type I error of 0·05, the probability of Type I error in the uncorrected test was actually 0·40. This enormous discrepancy arose from a population bounded within a square, which is the most common shape of quadrat. The magnitude of this bias will depend on the shape of the area considered with longer boundaries relative to the area enclosed meaning a greater bias. An alternative method of avoiding biases introduced by edge effects is to create a buffer, or border, around the edges of the area. Individuals in this area may be included as neighbours but may not be used as base individuals. Although this tactic was suggested by Clark & Evans (1954) in their original paper it has rarely been used (Sinclair, 1985). An appropriate size for such a buffer may be calculated as that distance within which a large proportion of N-N distances fall. Anderson & Kendziorek (1982) set that proportion to be 0·9 and found the required distance (d) by solving the equation

where is the estimated density within the area. The formula may be used for any proportion by appropriate substitution. Similarly, Harvey, Ryland & Hayward (1976, their p. 101) set the size of the buffer around a circular area “…such that individuals lying at the perimeter in a randomly distributed population would have a neighbour nearer than the edge of the disk on 95% of occasions…”. They arrived at that distance by solving the formula

for the square of the required width of the buffer, is the density. They noted that for aggregated populations, this distance would be an over-estimate. Several studies, e.g. Underwood (1976) and Andrew & Stocker (1985) have used buffers to avoid biases introduced by edge effects. Kinzie & Snider (1978) and Simberloff (1979) have considered the effect that the assumption of point processes has on the results gained from analyses of spatial pattern based on N-N distances. Both papers have demonstrated that the assumption that individuals were points in space, i.e. they had no area (or volume), can cause misleading results. The magnitude of the errors introduced will increase with the size of the individuals relative to the distances between them. Simberloff (1979) provided solutions to this problem for Clark & Evans’ R. He provided an approximate algorithm to correct this bias for situations in which the average diameter of organisms (assuming they are circular) is less than half of the expected mean N-N distance. Simberloff suggested the use of simulation procedures to estimate the magnitude of the bias for organisms with diameters greater than half the expected mean N-N distance. Simberloff’s consideration of these biases was extended to the three dimensional form of R. Simberloff re-analysed Connell’s (1963) data, taking into account the size of the amphipods, and although Connell’s interpretations were confirmed, the probability that the significant regularity was an error was increased. A preliminary re-analysis of Stimson’s (1974) and Wilson’s (1976) data suggested that results from both studies may have been similarly affected by the assumption that the individuals under study had no area. The expected mean N-N distance in the two studies was dependent on the density of the organisms, and ranged between 2·9 and 6·7 cm and between 16·7 and 28·8 cm, respectively. The magnitude of bias introduced would depend on the size of the organisms; the larger the organisms relative to these N-N

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distances the greater the bias. Although the sizes of the corals and bivalves were not given, it would appear likely that, at least at the higher densities considered, the expected mean N-N distances would have been under-estimated, therefore favouring regularity. Stimson found that tests of R consistently indicated significant regularity. Interestingly this result was in contrast to a random pattern as indicated by the “index of dispersion” (but see p. 73 for limitations of that index). Wilson did not reject the null hypothesis of random pattern for all densities measured in the field. In the laboratory there was some evidence for aggregation but only in the lower two-thirds of the density range considered. In the absence of size estimates, the extent to which the statistical significance of these results, and interpretation, are in error, is a matter of conjecture. Anderson & Kendziorek (1982) have made use of the modified form of R in their detailed study of the spacing patterns of tube-dwelling polychaetes. They found significant evidence for regularity in the spacing of individuals. Pielou (1959) has demonstrated the importance of randomly selecting the base individual when using Clark & Evans’ R. If individuals are chosen by some short-cut method, such as choice of random areas followed by random selection of individuals within those areas, then the results of the test will be biased in favour of regularity. Assuming an aggregated population, randomly choosing small areas, and then individuals within them, would decrease the probability of selecting individuals within aggregations. True randomization would require the identification of all the organisms in the population and the selection of a sample using random numbers—a task that would be enormous if not impossible in many instances. Clark & Evans (1979) noted that this was one of the major practical limitations of their technique. Pielou (1969) has also pointed out an important distinction between those indices based on N-N distances, such as R, and those derived from estimation of distances between random points and nearest-neighbours. Pielou has identified two aspects of spatial pattern: intensity and grain. The intensity of a pattern may be thought of as the variability in density from place to place (Pielou, 1969). The grain of spatial pattern may be thought of as the way that variability in density is arranged in space. For example, a coarse-grained pattern may take the form of aggregations that are widely spaced with large areas of low density between aggregations. Alternatively, if there are large fluctuations in density over short distances, then the pattern may be considered to be fine-grained. The grain of a pattern may be independent of its intensity (Pielou, 1969). Indices of spatial pattern based on N-N distances measure only the intensity of pattern (Pielou, 1969). This is because they are derived from distances between organisms and in an aggregated population the majority of distances measured will be within clumps. The contribution from individuals outside clumps will be less than that from isolated individuals if random point to nearest-neighbour distances were measured. The latter indices are therefore influenced by the grain of the pattern to a greater degree than the N-N indices (Pielou, 1969). Advantages of the various indices It is not easy to compare the relative merits of tests for non-random spatial pattern based on numbers of individuals in sampling units with those based on measurements of distances. Both have advantages, whose relative worth will depend on the situation considered. Where distances between individuals can be measured, the distance techniques would seem to be preferable because they are less subject to artefacts introduced by the sampling units. For many research workers, however, the ability to measure distances among individuals may be an unobtainable luxury, e.g. those working on plankton ecology or on deep-sea fishes. Measures based on numbers of individuals per sample unit are the only option open under such conditions. Interpretations should always be constrained by the limits of the technique of sampling and

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analysis. If indices based on data from sampling units are used, then explicit statements should be made about the scale at which any deviation from random pattern was observed. Detecting spatial pattern in heterogeneous areas The discussion so far has assumed that samples were taken from homogeneous areas, i.e. the probability of an individual occurring at any point was constant throughout the area. An ecological problem that has generated some recent work concerns the determination of boundaries of occurrence of species along gradients. The assumption of homogeneity clearly does not apply, so different probability rules are necessary. Pielou & Routledge (1976) applied Pielou’s (1975, as quoted in Pielou & Routledge, 1976) test for the independent assortment of boundaries to the occurrence of plant species along a gradient in a salt marsh community. The test is based on the determination of the probability of a boundary (i.e. presence and absence of a species in adjacent quadrats) occurring along a continuous transect of quadrats running along the gradient. Gardiner & Haedrich (1978) and Underwood (1978) have independently shown that the probability theory used by Pielou to establish the expected occurrence of boundaries within quadrats biased the test in favour of regularity (but see Pielou, 1979). A corrected version of the test based on probability rules that distinguish among species was provided in both papers. Gardiner & Haedrich applied this modified test to an analysis of boundaries of the occurrence of deep-sea fauna along a single transect. Underwood (1978) considered the problem of species’ boundaries in replicated transects on a rocky intertidal shore. Chaloupka & Hall (1984) also examined the occurrence of species’ boundaries with increasing height up intertidal shores. These authors have extended the Gardiner & Haedrich and Underwood test to consider situations in which the number of species boundaries per quadrat is restricted. Abel, Williams, Sammarco & Bunt (1983) have applied the same probability rules as Pielou & Routledge for the determination of the probability of a species boundary occurring in any given quadrat, but have constrained the number of boundaries that can occur within a single quadrat. They applied the test to a study of the distribution of corals in the Caribbean Sea. DESCRIBING NON-RANDOM PATTERN The indices so far described are sufficient only to detect deviations from spatial randomness, and provide no information on the details of those non-random patterns. For example, they cannot detect any underlying periodicity or constancy to the pattern or whether it takes the form of a mosaic of patches of high and low density; Jumars, Thistle & Jones (1977) provide an illustration of this point. Furthermore, the indices give no indication of scales at which pattern might be found other than the scale at which the data used to calculate the index are gathered (Hill, 1973). In Pielou’s (1969) terminology, the indices tell us only about the intensity (variability in density from place to place) of pattern and, although some are influenced by grain (Pielou, 1969), they say nothing explicit about the grain of a non-random pattern. Additional levels of structure may overlay pattern at the lowest level, i.e. aggregations of aggregations or large scale uniformity in the data. Pattern may be most evident at the finest scale (among-individual), and the knowledge that non-random pattern exists at that scale may often be all that is required (see earlier references). In some instances, however, this is not sufficient, and the investigator may wish to describe pattern more comprehensively. The description of pattern and the detection of higher orders of pattern are more difficult than demonstrations of non-randomness. The data required are more exhaustive and the analyses used more

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complex (and consequently more open to mis-interpretation). Two techniques that have been widely used in the ecological literature are Spectral Analysis and Pattern Analysis. Both require intensive, systematic sampling, but only techniques based on this sort of data will provide extensive information on the nature of pattern in a community. The point estimates introduced by Jumars et al. (1977) provide some information based on a limited amount of systematic sampling, but if detailed descriptions of non-random pattern are required then extensive systematic sampling is required. Of course the optimal, but rarely possible, solution is to map the positions of all organisms in the entire area. Although detailed discussions of Spectral Analysis and Pattern Analysis will not be attempted here, a very brief introduction to the literature is appropriate. Spectral Analysis describes observed patterns by fitting a linear combination of wave forms to variability in data collected as a series of counts along a transect (Ripley, 1978). The use of Spectral Analysis in ecology has been well reviewed by Platt & Denham (1975), Fasham (1978), and Ripley (1978). Despite the enthusiasm Platt & Denham (1975) had for the future of the technique as a source of a new theoretical framework for ecological research, the only area in which it has been used extensively in the marine ecological literature has been in the study of patchiness of plankton. Fasham (1978) has reviewed this literature and so we shall not discuss its use in this field here. Ripley (1978) cautioned that some experience is required if Spectral Analysis is to be interpreted correctly. Pattern Analysis, or “contiguous quadrat analysis”, has been used extensively by ecologists studying terrestrial plants (Greig-Smith, 1952; Kershaw, 1957; see Greig-Smith, 1979, for review), yet has been virtually ignored by marine ecologists. The following brief discussion is intended more as an introduction to the literature than a review of usage. As first outlined by Greig-Smith (1952), Pattern Analysis involved the comparison of counts from contiguous quadrats of increasing size. The area under study was divided into a grid of contiguous quadrats and density estimated within each. Numbers from adjacent quadrats were then summed, producing estimates from quadrats double the original size, which were then doubled, and so on. Thus the size of quadrats considered increased as powers of two. Variances of the mean counts for each size of quadrat were then plotted against quadrat size. Peaks in the plots occur at scales at which there is great variability in estimates of density. These peaks are interpreted as being the scales at which non-random pattern occurred. Extensive descriptions of the technique may be found in Greig-Smith (1964, 1979). This original formulation has been shown to have several limitations (Thompson, 1958; Pielou, 1969; Kershaw, 1970; Errington, 1973; Mead, 1974; Usher, 1975; Ludwig & Goodall, 1978; Upton, 1984). The most serious among these arose from the non-independence of estimates of variance among sizes of quadrat. This reduced the usefulness of the technique because the F ratios used to test for the significance of peaks were based on non-independent mean square estimates, and therefore were invalid. Other problems lay in the critical dependence of the starting point of the transect on the estimation of the scale at which the pattern (s) occurred. This was especially a problem when there was a periodic pattern of variability in density along a transect. If the transect started in an area of low density, then the estimated distance between peaks in variance would differ from that given if the start coincided with a peak. Another problem was that the length of the transect increased by powers of two and, therefore, there were often relatively few data points to analyse (Pielou, 1969; Goodall, 1974). There have been a number of derivations of the original technique of Pattern Analysis proposed to circumvent the problems posed above, but none has been entirely successful. Mead (1974) modified the test test for anomalous groupings (i.e. groupings of high to compare quadruplet groupings of quadrats by a or low density). Upton (1984) has subsequently demonstrated that Mead’s test is subject to the same problem of critical dependence on starting point as the original formulation.

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The only study we came across that had used classical Pattern Analysis in a marine ecological study was Grassle’s (1973) study of pattern in a coral community. Our failure to find more studies using this technique seems likely to be more a reflection of our limited search than a true picture of the popularity of the technique. It seems improbable that Grassle’s paper is the only instance where some form of Pattern Analysis has been applied to a marine problem, particularly in the phycological literature. Fitting contagious distributions Where significant departures from random spatial pattern have been demonstrated, many different statistical distributions have been fitted to summarize data. In particular, the negative binomial distribution, has proved popular as a model for contagious populations, e.g. Gärdefors & Orrhage (1968), Oviatt & Nixon (1973), Fasham, Angel & Roe (1974), and Todd (1978). See Cassie (1963) for a review. The negative binomial distribution is described by two parameters, mean density and the exponent k (Elliott, 1977; Taylor, Woiwod & Perry, 1979, and references therein). The reciprocal of k has been used as a measure of clumping of individuals: as the population tends to a random (Poisson) distribution 1/k tends to zero. Taylor et al. (1978, 1979) have investigated the utility of the negative binomial distribution as a model for the spatial pattern of organisms. In order for k to be a useful descriptor of spatial arrangement, it must behave in a predictable way when density and the degree of aggregation changes. It fails to do this. Taylor et al.’s (1979) analysis suggests that k has a rather dim future as a measure of dispersion. They found that even assuming that the negative binomial distribution adequately fitted the data, it had a number of severe limitations in its application to ecological problems. Numerous authors have cautioned against drawing ecological conclusions from the apparent fit of statistical models such as the negative binomial (Pielou, 1969; Bliss, 1971; Sokal, p. 374 in Taylor, 1971; Taylor et al., 1978, 1979; Todd, 1978). Although the data may fit a given distribution, it cannot then be assumed that the mathematical processes that determined that particular function are adequate descriptors of processes that determined the spatial arrangement of the organisms. Goodall (1974) suggested that the fitting of such contagious models has not been helpful in generating interesting and testable hypotheses about the arrangement of organisms in space. Taylor et al. (1979, their p. 301) concluded that the study of aggregation might best proceed if “based solidly on real data unprejudiced by preconceptions from models having little correspondence with reality”. CONCLUDING REMARKS If statements about spatial patterns are based on counts from a sample of randomly placed units, then only limited statements can be made about the intensity of the non-random pattern (Pielou, 1969). Non-random pattern can be claimed with more confidence from sets of distances between organisms, but unless the data contain detailed information about the spatial relations among individuals, i.e. the positions of organisms are mapped, then little more than a demonstration of non-random pattern is possible. Rapid methods for plotting the positions of organisms in the field may be found in Underwood (1977), Rohlf & Archie (1978), and Weinberg (1981). Ripley (1978) and Diggle (1983) have provided detailed discussions of the types of analyses that may be applied to mapped patterns. A review of the literature suggests that data of this completeness have only rarely been collected in marine systems. The overwhelming majority of studies concerning the spatial patterns of organisms seem content with testing for non-random pattern. As models attempting to explain the organization of communities become more complex, information more detailed

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than demonstrations of non-random pattern will be required to test hypotheses related to the spatial relations of organisms. GENERAL CONCLUSIONS It will be evident that there are no easy prescriptive solutions to the problems of designing a sampling programme. To paraphrase Williams (1978), slightly out of context, there are only vague answers to such precise questions as: where?, how big? and how many? The best way to sample the abundance of an organism has to be determined for each individual study. We have reviewed the application of sampling procedures in the marine ecological literature and found considerable evidence of standardization of methodologies and of adherence to customary ways of doing things. Following a discussion of well-established criteria used to design better, more efficient sampling programmes, we have joined others (e.g. Underwood, 1981; Hurlbert, 1984) in seeking to attract attention to the advantages to be gained from good design. The reappraisal of methodologies continues to be a topic of much research and discussion in other fields. For example, in the literature on wildlife management there are many papers dealing with the types of issues we have discussed here. The great logistic difficulties facing many studies in marine ecology should prompt similar discussion in this field (see also Omori & Hamner, 1982). Our recommendations should not be misconstrued as a plea for the complete breakdown of established techniques. Hopefully, popular techniques will prove to have a sound basis justifying their common usage. The important point is that their use should depend upon their demonstrated suitability to each study, rather than simply on convention. It may be argued that the use of different sampling units will make comparative work difficult because the estimates will be expressed in different units. The ability to standardize estimates of abundance among samples from units of different sizes, circumvents this objection. The standardization is, however, subject to several assumptions and limitations. Assuming that different estimates of mean abundance are equally accurate and, therefore, that sampling units of different size will yield proportional results, data may be scaled to a different size of unit. Variance estimates can be scaled as the square of the change in unit size, i.e. if the unit has to be doubled to the standard unit then the variance of the mean should be increased fourfold. The description of spatial pattern from estimates based on units cannot be standardized so simply. As discussed previously, if the population is aggregated, the relationship between the mean and variance of a sample will vary unpredictably with the size of the sample unit. Unless the relationship between the variance and mean is known for the density range considered, the degree to which variance will change with a given change in mean cannot be calculated. Measures of spatial pattern derived from adjusted estimates of abundance will, therefore, generally be invalid. We suggest that the advantages to be gained from designing programmes so as to gain accurate and precise estimates, with minimal cost, far outweigh any disadvantages. It should be borne in mind that, given the evidence we have presented, the advantages of using a standardized method for comparative work may be illusory because estimates of abundance may be poor and, hence, of little comparative value in any event. If no statements about the accuracy and precision of estimates are made in studies that are to be compared then it can only be assumed that the estimates are equally reliable and therefore truly comparable. We do not suggest that extensive discussions of sampling methodologies be published in every paper. All that is required is a brief justification of why the organisms were sampled the way they were. Such brief statements will also be an aid to those who are designing similar programmes by providing a range of

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possible methods to be considered. Toft & Shea (1983) and Underwood (1981) have made similar pleas for the publication of power of tests, or at least the provision of sufficient details for others to calculate power. Details of replication and sources of variation are now generally required in describing experimental studies. There is no lesser need in the description of sampling programmes. Ecologists have long recognized the importance of justifying their sampling methods. The German naturalist Victor Hensen, one of the first ecologists to estimate the abundance of plankton, was concerned with establishing the validity and reproducibility of his results (Hensen, 1911, 1912, as quoted in Lussenhop, 1974). The development of more sophisticated methodologies for the collection of data and the increasingly large array of statistical procedures for their analysis make the need to evaluate the accuracy and precision of the sampling even greater today. Fortunately, the tools to do that evaluation have also been developed considerably since Hensen’s day, so it is also easier. Optimization techniques, such as cost-benefit analysis and power analysis, put the modern marine ecologist in a good position to design sampling programmes that provide accurate and precise descriptions of natural systems. Our impression is that the description of pattern has become the poor cousin of experimental work in ecological studies. We do not take issue with the need for manipulative experiments to investigate processes in ecology. Experiments, however, will be difficult to design and interpret without extensive and reliable natural history and distributional information. We hope our discussion will prompt a greater attention to how marine ecologists describe spatial pattern. ACKNOWLEDGEMENTS We thank A.J.Underwood for his advice and extensive criticisms of the manuscript. We are grateful to J.H.Choat, P.G.Fairweather, K.McGuiness, P.F.Sale and H.P.A.Sweatman for discussion and constructive comments on the manuscript. We hope they will accept the path we chose through their sometimes conflicting criticisms. We are especially indebted to L.J.Stocker for her many contributions, in formulating ideas, giving us advice and encouragement, and in the preparation of the manuscript. Funding for this project was provided by a Commonwealth Scholarship and Fellowship Plan Award (to N.L.A.), a Commonwealth Postgraduate Research Award, and grants from the Great Barrier Reef Marine Park Authority and from the Australian Coral Reef Society (to B.D.M.). We are grateful for the support provided by the School of Biological Sciences, University of Sydney. REFERENCES Abel, D.J., Williams, W.T., Sammarco, P.W. & Bunt, J.S., 1983. Mar. Ecol. Prog. Ser., 12, 257–265. Alden, R.W., Dahiya, R.C. & Young, R.J., 1982. J. exp. mar. Biol. Ecol., 59, 185– 206. Anderson, D.J. & Kendziorek, M., 1982. J. exp. mar. Biol. Ecol., 58, 193–205. Andrew, N.L., 1986. J. exp. mar. Biol. Ecol., 97, 63–79. Andrew, N.L. & Stocker, L.J., 1986. J. exp. mar. Biol. Ecol., 100, 11–23. Barnes, H. & Marshall, S.M., 1951. J. mar. biol. Ass. U.K., 30, 233–263. Bateman, G.I., 1950. Biometrika, 37, 59–63. Bell, J.D., Craik, G.J.S., Pollard, D.A. & Russell, B.C., 1985. Coral Reefs, 4, 41– 44. Bernstein, B.B. & Meador, J.P., 1979. Mar. Biol., 51, 179–183. Bernstein, B.B. & Zalinski, J., 1983. J. Environ. Mgmt, 16, 35–43. Blackman, G.E., 1942. Ann. Bot. N.S., 6, 351–370. Bliss, C.I., 1971. In, Statistical Ecology, Vol 1, Spatial Patterns and Statistical Distributions, edited by G.P.Patil et al., Pennsylvania State University Press, Pennsylvania, pp. 311–335.

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Oceanogr. Mar. Biol. Ann. Rev., 1987, 25, 91–112 Margaret Barnes, Ed. Aberdeen University Press

FLUMES: THEORETICAL AND EXPERIMENTAL CONSIDERATIONS FOR SIMULATION OF BENTHIC ENVIRONMENTS* ARTHUR R.M.NOWELL and PETER A.JUMARS School of Oceanography, University of Washington, Seattle, WA 98195, U.S.A.

INTRODUCTION Flumes have been used to transport water since Roman times. The word flume is derived from Latin “fluere… to flow” and today means “an inclined channel for conveying water from a distance”. Flumes have been extensively used in sedimentary geology and civil engineering for approximately a century to examine the modes and rates of sediment transport. Their use in benthic biology resulted from the recognition that a variety of processes of biological interest were strongly influenced by water motions at the boundary with solid surfaces and that these processes in the bottom boundary layer can be modelled in flumes. The benthic boundary layer in the transport of scalar quantities such as nutrients plays an important rôle not only to benthic biologists and geochemists, but also to sedimentary geologists and boundary-layer fluid mechanicians. Our primary objective will be to review the characteristics of such flows and what constraints must then be placed on the design and operation of laboratory models of benthic boundary layers irrespective of the discipline to which the flume experiments are applied. As the application of flume techniques to biological problems in particular is not straightforward, our goal in this review is to present the fluid dynamic issues and compromises that govern the applicability and operation of laboratory flumes. Previous articles on flumes have focused on engineering considerations and construction techniques (Williams, 1971), on the differing types of flumes that are routinely used in sedimentary geology (Middleton & Southard, 1984), and on descriptions of specific flumes (for example, Vogel & LaBarbara, 1978; Nowell, Jumars & Eckman, 1981). Use of open-channel flumes in sediment transport studies ranges from early work by Osborne Reynolds in the 1880s, to the classical work by Gilbert (1914) and Shields (1935) to more recent studies by Guy et al. (1966), Grass (1971), and Sumer & Deigaard (1981). All these studies were concerned with the complex case of boundary-layer flow over a deformable (soft-sediment) bed, and the response of the bed to fluid forcing. All used fresh water and paid no attention to biota. Apart from the effort involved in filling the flumes, there were few limits on the sizes of the devices. Thus flumes used by these authors ranged up to 80 m in length. Once one incorporates a concern for the biota, especially for either controlling microbial abundance or for implanting a community of macrofauna and meiofauna, a severe pragmatic limitation is imposed on the size of the device. We take it as given that the purpose of a flume is not only to simulate realistic field conditions near the sea bottom, but also to simplify them so that the flow characteristics can be summarized in a small number

*University of Washington Contribution No. 1649.

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of parameters (i.e. those presented by Nowell & Jumars, 1984). Only then can the results in terms of effects of changes in those parameters be both replicated and generalized. There is no benthic flow environment simple enough to be characterized by a single variable, such as flow velocity (volumetric discharge divided by cross-sectional area or streamwise velocity at a given depth) along a flume or in nature, so there is no point in reviewing biological observations made where this was the only variable measured or used in matching field and laboratory flow regimes. The epitome of irreproducibility in this context is the taking of small-scale physical (e.g. Statzner & Holm, 1982) or chemical (e.g. Jørgensen & Revsbech, 1985) measurements about organisms without measuring or controlling the larger-scale fluid motions that determine these small-scale patterns. Nowell & Jumars (1984) review the important parameters, but do not show how to control them in a laboratory setting. Rules of thumb in flume design have been given previously in the marine ecological literature (Jumars & Nowell, 1984; Muschenheim, Grant & Mills, 1986) but their derivations have not. Our primary aim in this review is to expose the principles underlying the rules of thumb utilized in flume design, and to do so highly selectively. The problem of flume design can be broken into four parts, i.e. producing specific entrance conditions, providing certain exit conditions, driving the flow (including, in some cases, return of fluid with or without its particle load to the flume entrance), and lastly but most important tailoring the flow in the test section. Flows in the entrance, exit and drive sections are complex, defying simple and general parameterization, but conditions there are for the most part irrelevant to biological measurements of interest. Furthermore, the miniaturization often required of flumes for biological purposes usually makes no special demands here; the rules of thumb are reasonably independent of physical scale. For these reasons, we give brief treatment of these design considerations which are often satisfied with simple, empirically derived guidelines. Control of fluid dynamics in the test section is another matter. Here is where the serious and inevitable compromises of flume scale must be made. It is impossible to make them intelligently without understanding the underlying principles. We review those principles (of conservation of mass and momentum) by recourse to the basic Navier-Stokes equations, to which most oceanographers have at one or several times been exposed. Very readable introductions to such material, which focus on phenomenological understanding rather than mathematics, are presented by Bradshaw (1971), Francis (1975), van Dyke (1982), and Allen (1985). Finally, we exercise those principles by applying them to specific biological problems, including ones where it becomes obvious that laboratory flume work is impossible or impractical. We limit our scope explicitly to flumes for simulation of bottom boundary layers. Specifically we are excluding from consideration flow devices where the flow goes into solid-body rotation, such models generally being used to simulate geostrophic flows. We also exclude enclosed water tunnels for, although such devices are useful for measuring drag on bodies (e.g. fish fins or model submarine hulls) suspended far from a boundary, they are not useful without extensive corrections in simulating bottom boundary layers or in making drag measurements on objects attached to the bed or sidewalls. Thus, our major emphasis will be on flumes which permit us to study small-scale, viscous, and turbulent flows, most often flows with a single dominant velocity gradient. We consider in this review flows with a free surface, and in which stratification and rotation play negligible rôles. While the oceans are clearly stratified, rotating shear flows, the flow close to a body, and flows on the scales of importance to a single organism, or a local segment of a community, may be viewed as dominated by the relative balance of inertial, viscous, and body forces alone. As Shakespeare points out about life, and as more emphatically noted in “Trobriand Cricket”, by our entrances and exits shall we be remembered. The entrance conditions as well as the exit conditions can affect the nature of flow in the test section of a flume. What specific characteristics of fluid motion we need are specified by the question that we are trying to answer. How well we can answer the question will be

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determined by how well we can tailor the boundary-layer flow in our flume to match the field situation. Our ability to mimic a field situation in the laboratory depends on selection of an appropriate flume design. ENTRANCE CONDITIONS The entrance to the flume is crucial for the development of simple flows. The problem can be viewed as getting the flow to ‘forget’ its recent history. Because the water usually is delivered by a pipeline which most likely has bends in it, the flow entering the flume often will have strong circulation from going round a curve. In addition the flow may have to diverge as it enters the channel. As an example of these problems consider the flow entering a flume from a 20 cm (diam.) pipe having just turned through 90° and diverging rapidly into a channel that is 30 cm wide. Clearly the flow from the pipe will form a centre jet which will take many pipe diameters to decay (based on many observations downstream of obstacles the flow will ‘remember’ for about 20 times the critical length scale which in this case would be pipe diameter). We could, therefore, not expect to develop a simple boundary layer for over 4 m. A smooth, gradual divergence at the entrance is required to avoid separation of the flow. Experience from the design of ship hulls shows that expansion angles exceeding 7° often show flow separation (Chang, 1970), so this angle should not be exceeded without detailed examination of the consequences. Rapid expansions (e.g. O’Brien, Tay & Zwart, 1986) lead to free jets and complex flows that are virtually impossible to describe in detail, again diminishing the value of bringing the problem into the laboratory. One common solution to the problem of removing the effects of the pipe flow and upstream delivery of the fluid is to force it through a diffuser aimed at the far upstream wall of the flume. The motion is broken into small scales and the flow distributed across the full width of the flume. Honeycomb grids downstream of the diffuser will break up remaining scales of motion that are slightly bigger than the grid scale, but such grids cannot markedly affect flow non-uniformity on the scale of the whole channel diameter. The use of grids for generating small-scale turbulence has a long and well-recorded history (Laws & Livesy, 1978) and it has become almost a shibboleth of flume building to include a screen at the entrance. Laws & Livesy (1978) detail the calculations required to estimate the effects of screens on the flow, but we note that in essence screens only help to tailor an already well-cut flow; they cannot generate smooth boundary layer velocity profiles out of non-uniform jets. Moreover in free-surface flows such as flumes, the more dense the screen, the bigger the pressure drop across it; consequently, a very dense or a very long section of mesh will result in a downstream hydraulic jump that propagates surface waves downstream, causing non-uniformity in the boundary layer. As a general rule the tubes comprising the mesh should be about 20 times longer than the mesh diameter in order to apply a sufficiently strong strain to the fluid to alter the incoming turbulence. Elegant mesh designs have been developed for wind tunnels in order to pre-shape the flow into a boundarylayer profile, but such designs require considerable care and many hours of computing. Furthermore, they are well suited to but a single flow rate. If a fully rough-turbulent boundary layer is desired, its development can be accelerated or ‘tripped’ by placing small-scale roughness elements across the flume bed on the downstream side of the honeycomb grid. EXIT CONDITIONS The exit problem can be viewed as not letting the flow know what is coming, i.e. of making a smooth exit without breaking cadence. At the exit of the test section the flow can either fall freely, be retained by a weir, or be directed through tail gates. In the case of a weir, especially if the flow is supercritical (with respect to Froude number defined below), upstream effects are clearly visible. The flow is suddenly deepened having

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to go over the gate, and as a consequence a hydraulic jump may occur just upstream of the gate, or a backwater slope is developed (Henderson, 1966). But upstream effects also exist in subcritical situations and the weir causes the flow to diverge from the bed upstream of the gate. This effect clearly imposes extra length on our flume as it reduces the length of the useful test section. Similarly, a free overfall has an upstream effect at subcritical Froude numbers as the flow close to the bed is accelerated on approaching the overfall. As a compromise to the two extremes of a weir and a free-overfall, louvred gates, much like venetian blinds (set either vertically or horizontally) provide a minimum of upstream interference. A good picture of such gates is provided by Yalin (1977). DRIVING THE FLOW Flumes are used because we wish to control the mass or momentum flux or the boundary stress. Such control is achieved by manipulating discharge (volume per cross-sectional area per time) through the flume. Discharge is regulated either by using a pump and returning the fluid to the entrance, or by using a constanthead tank. For the types of velocities that are most commonly needed in oceanographic studies, a constanthead tank provides an inexpensive and effective method of providing a regulated, constant discharge. Pumps are often necessary for high discharges but require considerable attention to avoid problems of unsteadiness at low velocities (for example picking up the blade frequency from the impellor) and varying discharge at high rates when small voltage changes may cause large changes in effective discharge rate. Most flows of concern have boundary layer Reynolds numbers (free stream velocity times boundary layer thickness divided by kinematic viscosity) greater than 105, and because the fluid is well mixed through the return pipes and pumps we rarely have to worry about fluid stratification. At low-flow Reynolds numbers, however, fluid convection can cause numerous problems in the laboratory. Low Reynolds numbers of settling particles, such as larvae, provide a challenging design problem. Obtaining a totally still body of water is more complex than usually imagined and many settling columns are plagued with problems of convection. Although the velocities of convection are low, so are the velocities of larval settling. Temperature changes can also occur in flumes due to heating of the water as it passes through the pump; because of the high thermal capacity of water this only becomes a problem in very small flumes when short recirculation times may occur. Temperature variations should always be monitored for small changes in temperature can often markedly affect velocity sensors using heat transfer rate to infer velocity. TEST SECTIONS At last the stage is set. Well characterized, reproducible flow in the test section is the goal for the design of any flume. The test section should not be considered as the distance left over after we have moved far enough away from the entrance to avoid non-uniformity or advective effects and far enough upstream to miss the results of the flow leaving the flume. Rather the test section should be long enough to allow simulation of the flow field of interest. Unfortunately, unless one has carried out the calculations as to one’s needs before one either builds the flume or tries to use an already existing device, one may end up with no useful test section at all. MASS AND MOMENTUM BALANCE To obtain a valid simulation of any flow there is a hierarchy of similarity that needs to be considered. At the simplest level we wish to maintain geometric similarity; that is, the shape of the boundary (the bed in most

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cases) should be the same from field to laboratory. Because the perimeter of a body in a flow where the noslip condition (zero velocity immediately at the body’s surface) applies is also a streamline, we will maintain kinematic similarity if we keep the velocity the same between laboratory and field. We usually wish, however, not only to maintain geometric and kinematic similarity but also to model the fluxes of material and momentum, in other words we wish to maintain similarity of forces, or dynamic similarity. To do so reliably we shall treat the equations that express the balance of forces within the flow; we shall use Newton’s second law. The governing equations are the continuity equation for an incompressible fluid (1) and the conservation of momentum equation (2) which may be written out in its scalar form as

The velocity is represented by ui, which has three components u, v and w, t is time, p pressure, g gravity, is density and and v are the dynamic and kinematic viscosity (and ) The terms on the left of the Navier-Stokes Equation 2 represent, respectively, the local and advective rate of change of momentum. They are balanced on the right hand side of the equation, respectively, by the pressure gradient, the viscous forces and the body force (which we take to be gravity). Equation 2 is just the familiar force balance F=ma written for a fluid parcel. Another way to conceptualize each term is as a force (MLT−2) per unit volume (L −3) of the water parcel, again yielding the proper units for each term (ML−2T−2). General solutions of these equations are not available, and simplification of them is achieved only if we can apply certain restrictions. To understand the physics represented by these equations we want to transform each of the terms into a nondimensional form. Let us define

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where L is an independent length scale, V is an independent velocity scale and is an independent time for dimensional uniformity. We substitute these terms into the scale and we divide the pressure by Equation 2:

and multiplying through by L/V2

Each of the terms is now dimensionless, whereas in Equation 2 it had the dimension of mass/unit volume through multiplying the various times acceleration. There are four dimensionless groups, labelled terms. Term one, a frequency parameter, is called the Strouhal number, and is used as a measure of the unsteadiness of the flow. Most commonly it is used to relate periodic vortex shedding from cylinders to the velocity incident upon and diameter of the cylinder (Tritton, 1977), but its general interpretation is as a measure of unsteadiness. When the Strouhal number is zero the flow is steady and values rarely exceed unity for even the most unsteady cases. For example, in a tidal flow in an estuary where the tidal frequency seconds, the maximum velocity of order 50 cm·s-1 and the flow depth say 10 m, the Strouhal is thus modelling such a boundary-layer flow in the laboratory could be done using a number is quasi-steady flow especially if we were interested in the response of the bed, which is nearly instantaneous to changes in stress. As another example, consider shallow-water wind waves in the intertidal at a depth of 1 m where the mean flow velocity may be of order 20 cm·s−1, and wind waves may have a period of 5 s. Such parameters will result in a Strouhal number of 1·0. Clearly it would be unwise to attempt to model in the laboratory diffusional processes in such regions without including the unsteady term. By keeping the Strouhal number approximately the same between laboratory and field we ensure that we are correctly reproducing the kinematics of interest from the field. Term two is the ratio of gravitational to inertial forces, and is the inverse of the Froude number. It is important whenever we must consider free-surface effects, i.e. when the boundary layer extends through the complete flow depth, or when surface waves enter the problem. The Froude number, which as Lu (1977) notes is named after William Froude who ‘retired’ at age 36 to do research on ship rolling and resistance, also represents the ratio of the mean flow velocity to the phase speed of a shallow-water wave. As such it distinguishes between two regimes. Subcritical or tranquil flow has a Froude number less than unity, and the flow is deep and slow; information and boundary layer effects from downstream may be transmitted upstream by surface waves (viz. Lu, 1977). When the Froude number is greater than one, supercritical or shooting flow exists and information is transmitted only downstream. The Froude number is thus crucial when we attempt to model flows in estuarine and intertidal areas. Flows over most tidal flats are subcritical most of the time, although occasionally in run-off channels the flow is critical or supercritical, and standing waves may be observed at the transition.

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The third term is the familiar Reynolds number (actually its inverse) which is the ratio of inertial to viscous forces. The ratio of terms two and four in Equation 2 namely the inertial to viscous terms may be written as

which dimensionally is

We see that changing the Reynolds number then represents changing the balance of force from two terms in the Navier Stokes equation. In order to model flows dynamically we must maintain similarity of forces; hence all the dimensionless groups must be kept numerically similar from field to laboratory. The fourth dimensionless group, called the Euler number, is a measure of the pressure gradient, and is of consequence mainly in flows where gravity plays only a small rôle and the flow is being driven by a pressure drop, such as in flow through a horizontal pipe. We thus have to consider three dimensionless groups when we attempt to model flow in the laboratory. If we consider only steady flow (no changes with mean flow parameters with time, however), then we can dispense with consideration of the Strouhal number. Flumes are traditionally very long relative to their widths, and customarily are operated with flow depths which are shallow relative to their widths. The reasons for this design can be seen if we take the Navier Stokes equations and simplify them. Let us consider a steady flow with no gradients in the cross-stream direction, much like the flow in a wide, straight river or the flow over a wide sand flat. With no gradients in the y-direction the steady-flow equations reduce to conservation equations for mass (3) and, dividing through by and ignoring gravity, for momentum (4) (3)

(4)

If we consider a flow over a flat bed and apply the no-slip condition at the wall (at z=0, U=0) then the only geometric length scale is the distance x from the leading edge. Observation of such flows shows that the is very small, so that If we let the x component of velocity be thickness of the boundary layer of order u, and d/dx be of order 1/x then du/dx is of order u/x and from the continuity equation dw/dz must be of the same order. Now as w is smaller than u and since d/dz is larger than d/dx then the order of dw/dz may

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be met by considering w to be of order order

and d/dz of order

77

Then using the symbol 0( ) to mean of

we consider the order of the terms in the steady two-dimensional equations as (5)

(6) Equations 5 and 6 are termed the thin shear layer equations (Cebeci & Bradshaw, 1973) and can be solved to give the growth rates of boundary layers. They can be integrated over the boundary layer thickness to give the flux of momentum to the bed, that is the momentum integral yields the flux of momentum to the bed, or the stress acting on the bed. Now the two acceleration terms are of the same order in each equation, but the first viscous term in Equation 5 is very much smaller than the second viscous term so can be neglected. Since fluid parcels may be accelerated in boundary layers, and since strong viscous effects exist (having imposed the no-slip condition at the wall) the dominant viscous term is assumed to be of the same order of magnitude as the inertial terms, i.e.

This relation can be rearranged so that we see

Thus from a purely order-of-magnitude analysis we deduce that a viscous boundary layer thickness increases All the terms in Equation 6 are smaller than in Equation 5 and so to order may be ignored. as The shear-layer equations then become (7)

(8) The exact growth rate of a boundary layer can be predicted by using Equations 7 and 8 and solving for any imposed velocity profile, as done in many textbooks (cf. Shames, 1967) exactly for laminar flow and in an

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approximate manner for turbulent flows. The resulting growth laws show that boundary layers are very thin relative to their lengths of development, and thus we need flumes that are long in order to develop boundary layers of reasonable thickness. Table I gives some typical thickness for viscous and turbulent boundary layers for commonly used velocities. Because our flume must have sides, the numerical artifice of saying that we have purely two-dimensional flow is not empirically obtainable. Boundary layers grow just as quickly on the sides of the flume as on the bed, as long as the roughness is the same. Because in the field the assumption of two-dimensional flow may often be fulfilled, we must attempt to make our flumes as wide as feasible. Increasing the width of the flume puts immediate demands on the pumping capacity, because the boundary layer must be fairly thick in TABLE I Upper part gives boundary layer thickness (in cm) for various lengths of channel, lower part gives corresponding boundary layer Reynolds numbers : the viscous cases, in which Re <3,000, was computed using the exact Blasius solution to Equations (7) and (8), namely for the turbulent cases, the approximation was used; note that in some cases the boundary layer goes turbulent at a considerable distance down channel; also note the caveat on using Re=3,000, mentioned in the text; *calculated using turbulent flow formula Free-stream velocity cm s−1

Downstream distances (x)

50

100

200

500

1000

1 5 10 20 50 1 5 10 20 50

3·3 1·5 1·1 0·73 1·6* 328 734 1038 1467 7826

4·6 2·1 1·5 3·3* 2·7* 464 1037 1467 6546 13625

6·6 2·9 2·1 5·7* 4·7* 656 1450 2075 11398 23724

10·4 4·6 3·3 11·9* 9·9* 1037 2320 3281 23723 49379

14·7 6·6 23·7* 20·6* 17·2* 1467 3281 23724 41306 85973

order to accommodate, for example, macrofauna. How narrow can the flume be? Clearly, if the flume were twice as wide as deep, we would have flow at the midpoint of the flume width that was equidistant from each solid surface, which certainly is not a two-dimensional approximation. None the less, many if not most flow devices are less than twice as wide as deep (e.g. McIntire, Garrison, Phinney & Warren, 1964) and the measurements and corrections required to back-calculate bottom effects without sidewalls are considerable (e.g. Grant, Boyer & Sanford, 1982). In addition, because we only have three solid surfaces and one liquid-gas interface we shall develop asymmetries in the resistance to flow, resulting in secondary (non-downstream) circulation across the channel. To simplify the situation as much as possible we want the flume wide so that we are very much closer to the bed than to the sides in the measurement region of interest. A minimum value for the width-to-depth ratio then is 5, to reduce in the channel centre the effects of secondary circulation and the boundary layers from the sidewalls to less than 5% of the mean velocity. It is wise to try to get a ratio of 10:1, but designing and operating flumes is full of compromises between conflicting dimensionless groups and pragmatic limitations such as pump capacity. As an example of the need to compromise we usually want the Reynolds number to be high, and because we have only a finite depth of water in our flume we usually increase the velocity. But as we increase the

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Reynolds number we are also increasing the Froude number, and we usually have to compromise to a lower Reynolds number in order to keep the Froude number in the correct range (i.e. in the region of subcritical flow). So far in dealing with the equations we have said (except for the Strouhal number) nothing about boundary conditions or whether the flow is viscous or turbulent. While there is no single Reynolds number which delineates the two flow regimes, it is easy to observe when the flow is turbulent. Boundary layers over natural sediment surfaces are virtually always turbulent: as a rough guideline, good to a factor of 5, turbulence occurs when the mean flow Reynolds number exceeds 3000. This number can only be a guideline because it is chiefly the boundary condition, namely the roughness which determines the onset of turbulence for any imposed mean velocity. To see where turbulence is generated we take the Navier-Stokes equations, perform Reynolds averaging on them and then identify the extra term in the equation. Formally, we say that the instantaneous velocity is made up of a mean and a fluctuating component where the mean of the fluctuations is zero,

The overbar indicates mean quantities and the prime indicates fluctuating quantities. When we introduce these terms into the Navier-Stokes equations and average (cf. Hinze, 1975 or Tennekes & Lumley, 1972) we obtain

Comparing this equation with Equation 2 there is one extra term, which may be identified as a turbulent advection term, that is, it is identical to the mean advection term next to it in the equation but applies only to the fluctuating velocities. By using Equation 1 and the rule of differentiation of a product we may, however, rewrite the equation as

The new term, now on the right hand side, is a stress, and is called the Reynolds stress. It is diffusing momentum within the flow and is apparent to us by rapid mixing and nearly random motion. By analogy to the viscous stress, we write the term in the form of a viscosity multiplied by a mean velocity gradient. The concept of the eddy viscosity coefficient (which has the same dimensions as the coefficient for viscous momentum diffusion, i.e. L2T−1) has much appeal for boundary layers because it is simple to compute, intuitively easy to understand and above all works remarkably well. Thus, turbulent flows have more stresses, resulting in greater resistance to flow, and to higher mixing. This higher mixing means that we mix higher-momentum fluid closer to the bed, so that boundary layers in turbulent flows exhibit stronger velocity gradients than those in laminar flows. In order to use the eddy viscosity concept we need to specify a length (L) times a velocity scale (LT−1). The length scale for the size of the eddies responsible for mixing is regarded as the same size as distance to the bed (z), that is the largest vertical scale of eddy that could be accommodated, so

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where is von Karman’s constant (0·41) and the velocity scale is taken as the square root of the boundary stress, divided by density, that is

so our eddy viscosity in the lower 30% of the boundary layer is given as

A more general formulation valid throughout the boundary layer is given by Businger & Arya (1974) and Long (1981). To examine the flow close to the bed we can use the parameters that we have already defined to characterize the velocity. The velocity gradient close to the bed must depend on the turbulent velocity scale, Thus, we write the gradient as represented dimensionally by u*, and the mixing length

and if we integrate this equation we obtain

This relationship is called the law of the wall and describes the velocity profile in the region close to the wall where the momentum is transferred by the turbulent stresses. Because the left-hand side is dimensionless, the right-hand side must also be dimensionless, and thus z0 is a length scale, termed the roughness length. The law of the wall clearly applies only when we are far enough away from the bed that viscosity is not the main mechanism responsible for transferring momentum. The scales of the viscous sublayer, when the viscous stresses dominate and where the velocity profile has the form

have been given in Jumars & Nowell (1984). A full description of the velocity profiles is given in Dyer (1986). In order of distance away from the wall there is an inner region, usually millimetres thick called the viscous sublayer, a region above this called the logarithic layer which extends to approximately 30% of the boundary layer thickness, and an outer flow region termed the velocity defect region (Coles, 1956). Within the viscous sublayer, there is an even thinner region called the diffusive sublayer. While the viscous sublayer is the region where the flux of momentum is determined by the velocity gradient times the viscosity, the diffusive sublayer is the region where a scalar quantity such as oxygen is diffused by molecular diffusion. As the molecular diffusion coefficient for any chemical species is smaller than the molecular diffusion coefficient for momentum (which is the dynamic viscosity), the diffusive sublayer will be thinner than the viscous sublayer. While momentum can be transferred from molecule to molecule,

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chemical properties are transferred only via the migration of chemical species, making chemical transfer inherently less efficient than momentum transfer. OTHER NECESSARY SCALINGS There are three other dimensionless groups that must be considered if we are to model flow over a deformable bed, i.e. where the sediment on the bed can be moved either as bedload or as suspended load. Clearly the roughness of the bed will affect the local transfer of momentum to the bed, and the scale of the particles, i.e. the sediment grain size (ks), can be considered one such appropriate scale. The shear velocity, which is the square root of the boundary shear stress (the force acting on the bed), gives us the appropriate velocity scale, and we can create a roughness Reynolds number,

which is a measure of the bed and its interaction with the flow. When the roughness Reynolds number is less than about 3 the particles on the bed are subsumed within a viscous sublayer, and the flow is termed hydraulically smooth or smooth-turbulent. When the roughness Reynolds number is greater than about 70 the flow is called hydraulically rough or rough-turbulent and there is no continuous viscous sublayer. Under such circumstances the diffusion of nutrients or chemical species will be determined entirely by the turbulent eddy viscosity and will occur many times faster than when there is a continuous viscous sublayer. As with the Froude number there are classes of flow; at a minimum it is necessary to ensure our laboratory roughness Reynolds number falls in the same class as that from the field. Once we are concerned with moving material we need to consider the dimensionless force required to move the particles, namely

and the ratio of the settling velocity (ws) of the particle to the boundary shear velocity (u*). This ratio is termed the Rouse number; small values (1/20) indicate rapid diffusion of the particles throughout the boundary layer, whereas large values (around unity) indicate strong concentration gradients of larger material close to the bed. The significance of these terms has been outlined by Jumars & Nowell (1984) and is discussed in detail by Southard, Boguchwal & Romia (1980). There are two further complications that must be considered in modelling the natural marine environment, namely non-uniform roughness and wave effects. Rarely in the field will the bed be composed of a semi-infinite flat bed of uniform roughness; patchiness on many scales is more usual. When one wants to study, for example, flow around seagrasses one is faced with asking what characteristics one wishes to simulate. It is unlikely that there will be a flume wide enough to accommodate a natural patch, and if we make the patch the same width as the flume we have forced the flow to go over and through the seagrass, whereas in the field it is likely that horizontal cross-stream deflection of momentum is very important. Modelling flows over non-uniform roughness requires special care. Work by Antonia & Luxton (1971) and references covered in Eckman & Nowell (1984) and Paola, Gust & Southard (1986) indicate the types of problem that must be addressed. An abrupt change in bottom roughness results in an internal boundary layer growing from the leading edge of the roughness. This horizontal non-uniformity in the vertical direction means that we must exercise great care to ensure first that our measurements in the vertical refer to local

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effects and not to effects advected from upstream (Townsend, 1969; Peterson, 1971). Secondly, we must now maintain similarity of two sets of dimensionless groups, those referring to the upstream conditions and those referring to the local bed. This double requirement is especially important with reference to the Froude number. Rarely is the marine environment devoid of waves, yet for many problems we can deal with steady flow as a first approximation. Especially in the intertidal and on the continental shelf, however, waves are a zero order effect and we must determine the appropriate dimensionless groups. In a series of papers Grant & Madsen (1979, 1984, 1986) have very elegantly demonstrated the mechanics of the wave boundary layer, and shown that one can model the combined effects of waves and currents on sediment movement as a large steady stress. Although this work includes neither strong local acceleration (which may be important on sloping beaches) nor breaking waves, the work is of great general relevance especially to soft-bottom environments. Grant & Madsen demonstrate how the waves and currents interact and show how to use simple dimensionless scalings based on the maximum combined wave and current stress to parameterize the effects of waves. Strict analog modelling of such flows in the laboratory requires a very different type of flume from that used to model steady, unidirectional flow (Jonsson, 1966; Long, 1981). In summary, theoretical considerations lead us to suggest that for boundary layer flows over a modifiable where is the boundary layer bed we will require that the boundary layer Reynolds number is the free stream velocity, the Froude number the Strouhal number thickness and where n is the frequency, the roughness Reynolds number the Shields parameter and the all be comparable between the laboratory and field. In addition, we shall Rouse parameter need more dimensionless groups if the bed is non-uniform, or if we wish to consider the combined effects of a steady current with waves superimposed. The thin shear layer equations give us guidelines on the scale of the flume we shall require. If we are concerned only with small obstacles on the bed, we have no further concerns other than how to measure the flow, but often the body we want to place in the flow is of a predetermined size. Target organisms rarely come in a complete size range and thus if we wish to examine flow around, for example, a bryozoan there will be a minimum size for the organism. Such a size will not only determine how thick the boundary layer flow over it must be in order to maintain appropriate dynamic similarity, but in the flume we also have to be concerned about flow blockage. If the obstacle occupies more than about 35% of the flow depth (derived from potential flow theory) in the laboratory the flow pattern may be very different from that in the field. If the flow pattern about a seagrass bed is to be studied we have to be careful not to constrain the flow by forcing it to go through the seagrass bed in the flume whereas in the field the flow might diverge around the patch. EXPERIMENTAL CONSIDERATIONS This extensive list of factors that must be considered leads to two obvious but often ignored conclusions: (1) there is no such item as a universally useful flume; and (2) we must define the problem that we wish to study in the laboratory quite specifically before we can identify the crucial elements in the flume that will permit our laboratory measurements to simulate accurately the field environment. While many flumes exist at present it is unwise a priori to expect them to be suitable for specific questions. For example, the flume described by Vogel & LaBarbara (1978) is totally unsuited for boundary layer work because of its size, its three dimensional flow, and its entrance conditions, but is suitable for making drag measurements on bodies far from a solid surface. As a second example, the flume described by Nowell, Jumars & Eckman (1981), and the similar flume described by Muschenheim, Grant & Mills (1986)

FLUMES: SIMULATION OF BENTHIC ENVIRONMENTS

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are completely inadequate to undertake continuous sediment-transport measurements but are appropriate for making initial-motion measurements and studying organism responses to boundary-layer fluid stresses. As two further examples, the annular flume described by Taghon, Nowell & Jumars (1984) is suitable for transport distance questions, but it is unsuited for precise sediment transport rate problems due to the inherent secondary circulation in an annulus; the flume described by Nowell, Jumars, Self & Southard (in press) is suitable for sediment transport rate measurements under steady and tidal flows, but cannot readily simulate wave effects. Most water tables and fluid containers are complex geometric shapes and so we can expect that any flow within them will have complex velocity fields. Hubbe (1981) provides a useful summary of flow in commonly used laboratory containers and gives approximations for back-calculating approximate values of boundary shear stress. Such methods are clearly approximations but prove very useful when trying to put previous work into its appropriate fluid mechanical context. A series of questions may help to distinguish the properties of the flume that is needed. (1) Is the problem one where unsteadiness dominates or can a quasi-steady approximation be used (Strouhal number)? (2) What are the field values of the Reynolds number and Froude number and can they be appropriately matched in the flume? Do we need other scaling groups such as the roughness Reynolds number, Shields parameter or Rouse number? (3) Are we dealing with an initial motion problem or do we have to maintain constancy of transport not only of the fluid but also the sediment or some other diffused quantity such as larvae (determining the drive or recirculation mechanism)? (4) Is the region of concern approximately uniform or is patchiness important (one set of scalings or many)? (5) What measurement accuracy do we need from the laboratory to answer the question we are posing? FOUR CLASSES OF EXAMPLES There are three major categories of flow channel, namely a straight-through tilting channel where the flow and the sediment may either flow through just once or be recirculated via a pump (Middleton & Southard, 1984), a closed horizontal system such as the race-track flume described by Nowell et al. (in press), and a Ushaped channel for studying oscillatory boundary layers (Jonsson, 1966). Within each category there are wide ranges of flume size and configuration, but the parameters listed will permit us to select not only the type of flume that is appropriate for the problem, but also to decide whether the flumes available fulfil the required scaling lengths. In this section we shall use four examples to highlight some of the questions and challenges involved in designing a laboratory flume study. The problems cover a wide range and end with an example in which it is easier to remain in the field than to try to simulate field conditions in the laboratory. We take first a simple problem to illustrate the use of the equations outlined above. Recent studies (Miller & Jumars, 1986; Hannan, pers. comm.) indicate that feeding rate is dramatically altered for some deposit-feeding polychaetes when their faecal pellets are removed from their feeding ambits. At what shear velocities (u*) are the pellets eroded from the bed? We assume a bed without ripples. As a caveat we emphasize that preliminary observations of the organism under current regimes in the flume will be required not only to determine a realistic bed roughness, but also to permit one to specify when or if the pellets are deposited. For example, Pseudopolydora kempi japonica releases the pellets