MARINE ECOLOGY PROGRESS SERIESMar Ecol Prog Ser

Vol. 294: 295–309, 2005 Published June 9

INTRODUCTION

Ever since Ernst Haeckel put to rest Viktor Hen-sen’s notion that plankton are uniformly distributed inthe ocean, we have been searching for ways to mea-sure, characterize, and understand the dynamicsunderlying plankton patchiness. Plankton patchinesshas been shown to be a ubiquitous and importantfeature of oceanic ecosystems. Many organisms havebeen shown to exploit patches of food, and patch for-mation may be necessary for the sexual encountersamong individuals of relatively rare species. If we canunderstand the physical and biological dynamics con-trolling plankton patchiness, we may gain some pre-dictive ability concerning the scales, intensities, dura-

tions, and ecological importance of this planktonicvariability.

One powerful tool for quantifying the spatial scalesand intensities of planktonic variability is Fourier orspectral analysis. In spectral analysis, a data series isdecomposed into a series of sines and cosines withincreasingly small wavelengths (high wave numbers)or short periods (high frequencies). The amplitude ofthe sine or cosine gives the intensity of variability atthat scale. Commonly, the log of the spectral density,the variance or ‘energy’ given by the amplitudesquared, of the property is plotted versus the log of thewavenumber or frequency. This is the spectrum.

In physical systems, the slope of the velocity spec-trum is generally characteristic of the underlying

© Inter-Research 2005 · www.int-res.com*Email: pfranks@ucsd.edu

REVIEW

Plankton patchiness, turbulent transport and spatial spectra

Peter J. S. Franks*

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 92093-0218, USA

ABSTRACT: Spatial variability of plankton is a well-known, though poorly characterized and under-stood, phenomenon. Several studies suggest that patches of plankton are essential to the growth andsurvival of some planktonic species; thus, any ability to predict the spatial patterns of plankton wouldsignificantly enhance our understanding of ocean ecosystems. Furthermore, models that accurately de-scribe planktonic patchiness could permit a mechanistic understanding of the dynamics structuring theplankton. One such model is based on plankton being mixed in 3D isotropic turbulence, giving a k –5/3

slope to the spatial spectrum of plankton variability. Many papers have been published that (oftenfavorably) compare their data to this k –5/3 model. Using a selection of papers published since 1993,I show, however, that the data in many of these studies have not been gathered on scales that allow com-parison to the k –5/3 model. I then discuss whether planktonic data can even be used to test such a model:the model is continuum-based, while plankton are quanta at scales relevant to the turbulent scales. Still,many data sets resolving planktonic patchiness show a k –5/3 slope of the spectrum, or at least show abiological spectrum with a different slope than the temperature, salinity, or velocity spectra. I discuss sev-eral aspects of biological data that might consistently bias the spectral slopes of planktonic spectra, andconclude that the spectral slope does not contain enough information to distinguish among alternatemodels of plankton patchiness. While this paper is not a critique of the k –5/3 model or of the spectralmethod, it is a critique of the use of spectra of biological properties to test the k –5/3 model. I hope that bypointing out some of the pitfalls of spectral analysis of biological data we might gain better insights intothe physical-biological interactions creating microscale patchiness in the ocean.

KEY WORDS: Turbulence · Patchiness · Spectrum · Inertial subrange · Plankton · Variability

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Mar Ecol Prog Ser 294: 295–309, 2005

dynamics. One particularly well-known example is thewavenumber spectrum of 3D turbulence. For a wave-number k (the wavenumber is 2π/wavelength) thevelocity spectrum has a shape given by k –5/3, as dis-cussed in more detail below. Models of plankton inphysical flows have been developed that predictplanktonic variability to have a spectrum with a slopeof k –β, where β depends on the types of physicaldynamics, and the relative strengths of physical andbiological forces. In particular, one of the most well-known models predicts that, in a turbulent flow, plank-ton should show a spectrum that has some portion thatfalls as k –5/3. If we could gather appropriate data to testthese models, we might gain an important element ofunderstanding and predictive ability of planktonicpatchiness in the ocean. The dynamics that determinethe model spectral slope might reasonably be inferredto be controlling the spectral slope of the field data. Ifthis is so, we now have a mechanistic understanding ofthe dynamics controlling planktonic variability.

A great deal of work has been undertaken in the last3 decades formulating spectral models of planktonpatchiness, and in particular gathering field data totest these models. As new instruments have beendeveloped, new and better-resolved data sets havebeen generated. Comparison of these data to the spec-tral models has yielded interesting insights into thedynamics structuring the plankton. However, in thedecades since the models were developed, the com-parison of models and data has often been performeduncritically. In other words, do the data really repre-sent a test of the model, and are the data appropriate tocompare to the model? Below, I demonstrate that thedata are often inappropriate for comparison to certainmodels, as they have been gathered on the wrong spa-tial scales. Furthermore, a comparison of spectralslopes of models and data does not represent a strongtest of the model. There are a number of factors thatcan control the spectral slope of the data that cannot beaccounted for in the model, and multiple combinationsof dynamics can generate the same spectral slope.Here, I discuss several potential properties of data setsthat are generally ignored in analyses of biologicaldata, but which may significantly alter the spectralslope, giving spurious correlations with models.

I begin by describing the 3D model of the inertialsubrange of turbulence, followed by a discussion ofmodels of passive and reactive tracers in this flow.These models are the most commonly used for compar-ison to biological data, and it is worth understandingtheir genesis and limitations. I then analyze the datafrom 12 recently published papers to explore whetherthe data should be compared to 3D turbulence models,and whether the data test the models. Furthermore, itis useful to ask whether biological data can be used to

test continuum models of planktonic distributions inisotropic turbulence. I then explore 2 alternate spectralmodels of plankton patchiness: (1) internal waves and(2) geostrophic turbulence. This is followed by ananalysis of particular features of data sets that mightgive unrepresentative spectral slopes. It is my hopethat by becoming more aware of the pitfalls of spectralanalysis we might use it more critically and in conjunc-tion with other analysis techniques to explore andunderstand plankton patchiness in the oceans.

THE k –5/3 INERTIAL SUBRANGE MODEL

In 1941, Kolmogorov put forward the hypothesis of aninertial subrange for 3D isotropic turbulence at highReynolds numbers. He hypothesized that between thelarge scales at which turbulence is generated, and thesmallest scales at which turbulence is dissipated, thereshould exist a range over which turbulent eddies losetheir energy to successively smaller eddies, with littleloss of energy to heat, until the smallest eddies are dis-sipated by viscosity. The only factors governing the ve-locities of the water motions at a given scale in this iner-tial subrange are the rate at which turbulent kineticenergy is put into the system (parameterized as the rateof turbulent kinetic energy being dissipated, ε, [m2 s–3]),and the length scale (given as the wavenumber, k,[m–1]). Based on this assumption, dimensional analysisleads to the energy spectrum given by:

E(k) = αε2/3k –5/3 (1)

where α is a scaling parameter. Thus, within the iner-tial subrange, the spectrum should fall off with increas-ing wavenumber with a slope of k –5/3. A great deal ofdata from the atmosphere, ocean, and laboratory hasconfirmed Kolmogorov’s hypothesis, and it is nowconsidered a theory rather than a model.

The inertial subrange is constrained at the largestand smallest spatial scales. At the large scales, the tur-bulent eddies must work against stratification in caus-ing mixing. Stratification is given by the buoyancyfrequency, N (s–1):

(2)

where ρ is the density, g the acceleration due to gravity,and z the vertical coordinate. If the stratification is toostrong (large N), the mixing cannot overcome it, andthe turbulence will not be isotropic (e.g. Gargett et al.1984). The largest turbulent eddies become flattenedand anisotropic. The largest turbulent scales are usu-ally parameterized by the Ozmidov scale, Lo, or thebuoyancy wavenumber, kb:

(3)L k No b= =− −1 3ε

Ng

z2 = −

ρ∂ρ∂

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In unusual situations such as deep convective mixingor exceptionally strong tidal mixing, Lo can be as largeas 100 m (ε ~ 10–5 m2 s–3, N ~ 10–3 s–1). In most of theocean, most of the time, however, the largest turbulentscales are between about 10 and 100 cm (ε ~ 10–9 to10–8 m2 s–3, N ~ 10–4 to 10–3 s–1). In wind-mixed layers,ε tends to be between 10–8 and 10–6 m2 s–3 for winds of

Mar Ecol Prog Ser 294: 295–309, 2005

Eis(k) = B χε–1/3k –5/3 (7)

Here, like the passive tracer discussed by Batchelor(1959), the plankton variability should show a k –5/3

dependence on wavenumber.The third regime was found at the smallest scales

(high wavenumbers), where a viscous-convective sub-range might exist:

Ehw(k) = Cχ(v/ε)1/2k –1 (8)

where C is a scaling parameter. The spectral slopes ofthe 3 predicted regimes were thus k–1 for the low-wavenumber, biologically dominated regime, k –5/3 forthe inertial subrange, and k –1 in the viscous-convec-tive regime at high wavenumbers.

Powell & Okubo (1994) built on the earlier work ofDenman by including plankton–plankton interactions(e.g. grazing) in their spectral model of plankton patch-iness. They showed that in 3D turbulence, their modelreproduced those of Denman if the plankton had nointeractions with each other. In particular, the spec-trum had a region of k –5/3 slope in the inertial subrangeof turbulence. However, if planktonic interactions

were included, the spectral slope could be almost any-thing, including having discontinuities in the turbulentinertial subrange. This may have been the first sugges-tion that the spectral slope of plankton distributions didnot have enough information to accurately diagnosethe underlying dynamics.

TESTS OF THE k –5/3 INERTIAL SUBRANGE PLANKTON MODEL

Since Denman & Platt’s (1976) prediction of a k –5/3

inertial subrange in the spectrum of plankton embed-ded in a 3D turbulent field, many studies have beenpublished (and continue to be published) comparingtheir data to the k –5/3 spectral model. It is important toask, however, whether these data constitute a strongtest of the k –5/3 model, or, indeed, whether the datashould be compared to the k –5/3 model at all.

To make the point that the issues that I discuss beloware current, I chose to analyze 12 papers publishedsince 1993 in which a time-series or a spatial series ofplankton distributions were compared to the k –5/3

model (Tsuda et al. 1993, Mountain & Taylor 1996,Seuront et al. 1996a,b, 1999, 2002, Wiebe et al. 1996,Seuront & Lagadeuc 1997, 2001, Lovejoy et al. 2001a,b,Pershing et al. 2001). In some cases, the comparison ofthe data to the k –5/3 model is quite ancillary to the mainpoint of the paper, and any criticisms implied by thefollowing analyses do not detract from the centralpoints of their work. The data to be explored havebeen gathered in the open ocean and in coastalregions, with most of the data from coastal regions.Several of the papers discuss data gathered onGeorges Bank, an area that is particularly turbulentdue to tidal mixing (e.g. Yoshida & Oakey 1996).

The papers to be analyzed include data collected byremote sensing, vertical profiling, horizontal transects,anchor stations, and sampling while drifting. The datainclude samples of dissolved nutrients, chlorophyll afluorescence, and zooplankton. Zooplankton datawere obtained using an optical particle counter oracoustic backscatter. For data collected as a timeseries, I have used the authors’ own conversions tospace (frequency to wavenumber).

To compare the planktonic data to the k –5/3 model,the data must resolve the appropriate spatial scales, theinertial subrange. This inertial subrange lies betweenthe Ozmidov scale, Lo, at the large end, and theKolmogorov scale, η, at the small end. Except in partic-ularly intense turbulence, the inertial subrange usuallylies between scales of about 1 and 100 cm. In Fig. 1, Ihave plotted the range of wavenumbers/spatial scalesresolved by the various sampling programs, as well asan estimated range for the inertial subrange. To accu-

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Fig. 1. Summary of data from 12 manuscripts published since1993, indicating the range of spatial scales sampled (thickgray lines), the likely spatial range of the inertial subrange of3D isotropic turbulence (thick black lines), and the waterdepth (arrows). In most cases there is little overlap betweenthe scales sampled and the scales of the inertial subrange. Fordata sampled in time, I have used the authors’ conversions

to space

Franks: Turbulent transport and plankton patchiness

rately estimate the inertial subrange would requireknowledge of ε, N and ν, the dissipation rate of turbu-lent kinetic energy, the buoyancy frequency (the verti-cal density gradient), and the viscosity, respectively.Unfortunately, these values are not included in most ofthe papers. Given this lack of information, I estimatedthe inertial subrange to be 1 to 100 cm in most cases,and 0.3 to 5000 cm in the well-mixed waters overGeorges Bank. In these tidally mixed waters, the turbu-lence mixes the water column from top to bottom, giv-ing a maximal turbulent eddy size the same size as thewater depth (about 50 m) (but see Pershing et al. 2001for an alternate explanation of the vertical circulation).This estimation of the location of the inertial subrangemay be generous, based on the estimates of Gargett etal. (1984) who suggested that the inertial subrangewould be found between about 10kb ≤ k ≤ 0.1ks (i.e. amuch smaller spatial range than kb to ks), though thisestimate is for stratification-limited turbulence and maynot be valid for turbulence influenced by the surface orbottom of the ocean.

It is quite apparent from Fig. 1 that, in most cases, theinertial subrange lies well outside the range of sampledspatial scales, or only overlaps slightly with the data. Itis only in cases of extreme vertical mixing (e.g. GeorgesBank) that the sampled scales overlap with the spatialscales of the turbulent inertial subrange by more than adecade. Interestingly, in these cases, the spectral slopeof the fluorescence was statistically indistinguishablefrom k –5/3 (e.g. Mountain & Taylor 1996).

It is possible that my estimates of the range of the in-ertial subrange were too conservative. Perhaps the tur-bulent eddies were much larger than I allowed. Themost unconservative estimate of the size of the largesteddies that could still be 3-dimensionally isotropic (sta-tistically identical in every direction) is the depth of thewater column; no isotropic eddy can be larger than thewater column depth. Plotted in Fig. 1 is the water col-umn depth given in each of the published studies. Inseveral cases, the sampling is too coarse to resolve eveneddies that would mix the water from top to bottom; inmost cases, the sampling would resolve only about 1decade of the largest possible inertial subrange. Withthese sampling constraints, there is little possibility ofresolving the inertial subrange of turbulence. If theinertial subrange is not resolved by the sampling, it isinappropriate to compare the data to the k –5/3 model.

Dimensionality

The Kolmogorov model of the inertial subrange isinherently 3D; isotropic turbulence implies that thestructures (velocity, temperature, passive tracers) arestatistically identical in every direction. However, we

typically sample only along 1 dimension, or at the most2 dimensions (as in remote sensing). Is it reasonable tocompare data gathered in 1 dimension to a 3D model?

The problem with sampling a 3D field with a 1Dsampling device is that the energy (variance, spectraldensity) of high-wavenumber features can be aliasedinto lower wavenumbers. A 2D analog of this problemcan be illustrated with 1D sampling of a wave field(Fig. 2). Samples taken along a transect perpendicularto the crests of the waves will record an accurate wave-length and amplitude (energy, variance, or spectraldensity) for the waves. A transect taken at an angle tothe crests will always record a longer wavelength(lower wavenumber), but the same amplitude. In theresulting spectrum, this energy will appear at a lowerwavenumber than it should.

Hinze (1959) presented a transformation for un-aliasing the energy from a 1D spectrum to allow com-parison to a 3D spectral model. For a given 1D wave-number spectrum E1D(k), the 3D spectrum E3D(k) canbe obtained by:

(9)

Thus, multiplying the slope of the 1D spectrum by thewavenumber gives the 3D spectrum. For a 1D spec-trum that is proportional to k –n, the 3D spectrum is:

(10)

giving a 3D spectral slope that is the same as the 1Dspectrum. Thus, a spectrum that slopes as k –5/3 in 1Dshould have the same slope in 3D. However, this onlyworks if the 1D spectrum has no curvature in log-logspace; that is, there is only 1 n (n is any number) and itis constant for the entire range of wavenumbers. As anexample of what can happen when the 1D spectrumhas curvature, I performed this transformation on the

E k kk

k nkn n3D( ) = − =− −∂

∂

E k kk

E k3 1D D( ) = − ( )∂

∂

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Fig. 2. 1D sampling through a 2D wave field; sampling alongPath 1 (perpendicular to the crests) gives the correct wave-number (the wavelength is indicated by the distance betweenthe dots on the path). Sampling along Path 2 (at an angle tothe crests) gives a lower wavenumber (dots are further apart),but the same amplitude as Path 1. In the spectrum, the waveenergy for Path 2 will have been aliased into a lowerwavenumber (longer wavelength) than Path 1, which has the

correct wavenumber

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spatial spectrum of fluorescence given in Franks &Jaffe (2001, their Fig. 11) (Fig. 3). Even though the 1Dspectrum monotonically decreases with increasingwavenumber (though with no suggestion of a k –5/3

sloping region), the transformation to 3 dimensionsgives a spectrum that increases over much of thewavenumber range. The transformation has removedspectral energy from the low wavenumbers and trans-ferred it to higher wavenumbers, completely changinghow we might interpret the spectral slopes. Even inisotropic turbulence, this transformation will tend toreduce the range of wavenumbers over which we findan inertial subrange by removing spectral energy fromthe low-wavenumber end of the spectrum (e.g. Gar-gett et al. 1984).

Can the inertial subrange model be tested withbiological data?

The analyses presented above suggest that fewdata sets exist that can actually test the k –5/3 model ofplankton patchiness in 3D isotropic turbulence, sincemost data sets do not resolve the appropriate (small)spatial scales. However, an interesting problem ariseswhen the small scales are resolved by appropriateinstruments; plankton occur as quanta. Plankton arenot a continuum as the k –5/3 model requires. Plank-ters are individuals, and when they are viewed onscales around the high-wavenumber end of the

inertial subrange, they appear as quanta rather thanas a continuum.

To illustrate this problem, I use an image of phyto-plankton fluorescence gathered with an imaging fluo-rometer, similar to the OSST (Franks & Jaffe 2001). Thefluorescence of phytoplankton was stimulated by athin (6.5 mm) sheet of green laser light, and the fluo-rescence recorded by a sensitive CCD camera with abandpass filter centered at 685 nm. The camera andlaser were mounted on a free-falling vehicle, andsequences of images acquired 80 cm below the vehicleas it sank at about 10 cm s–1, 10 km off the coast of San

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Fig. 3. An example of a conversion from a 1D spectrum(E1D(k), thick line) to a 3D spectrum (E3D(k), thin line) usingEq. (9). This conversion removes the low-wavenumber energyaliased by 1D sampling of high-wavenumber variability in anisotropic 3D field. In this case, the 1D spectrum monotonicallydecreased with wavenumber, while the 3D spectrumincreased over most of its wavenumber range. Data from

Franks & Jaffe (2001)

Fig. 4. A 24 × 13 cm section from a 32 × 32 cm image of phyto-plankton fluorescence taken with an imaging fluorometer atabout 35 m depth, 10 km off the coast of San Diego, CA, USA.The local chlorophyll a concentration was about 0.6 µg l–1.The bar at the right indicates the relative fluorescence of theobjects. The image has been corrected for spreading of the in-cident laser beam. The resolution of the image is 300 ×300 µm; the large fluorescent objects are large diatom chains,

other phytoplankton, and aggregates

Franks: Turbulent transport and plankton patchiness

Diego, CA, USA. The image shown in Fig. 4 was cap-tured at the chlorophyll maximum layer (~0.6 µg l–1 chla) at a depth of about 35 m. The image has been cor-rected for the spreading of the incident laser beam,and a 24 × 13 cm portion of the total image wasextracted for analysis. The resolution of the image is300 × 300 µm, and individual phytoplankton cells canbe clearly seen in the image (see also Zawada 2002).

The image of Fig. 4 is large enough (24 × 13 cm) toresolve the high-wavenumber end of the inertial sub-range, should it exist, suggesting that we might expectto see a k –5/3 spectral slope of the phytoplankton fluo-rescence. To test this hypothesis, I calculated spectrafrom 400 rows and columns of the image, each row orcolumn being 256 pixels (7.7 cm) long. I then averagedall the rows and all the columns (Fig. 5) to reduce theconfidence limits of the spectrum. It is clear from thehorizontal and vertical average spectra of Fig. 5 thatthere is no k –5/3 region of the spectrum; in fact, thespectra have no slope at all. The spectrum is ‘white’.Why is this?

A column of 800 pixels (fluorescence intensity versusdistance) arbitrarily chosen from Fig. 4 is plotted inFig. 6. It can be seen that the fluorescence is domi-nated by a few very intense spikes at the locations oflarge cells. These spikes behave as delta functions inthe Fourier transform that generates the spectrum. TheFourier transform of a delta function is a constant inwavenumber space; that is, equal energy (variance,spectral density) at all wavenumbers. Thus, the few,

large, intensely fluorescent cells cause the spectrum tobe flat, or white. As an example of this, I created 256point data series with 20 randomly placed delta func-tions. I then calculated the spectrum of the series, andaveraged the spectra of 400 such series (Fig. 7). Thespectrum, as that of Fig. 5, is white.

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Fig. 5. Horizontal and vertical spatial spectra calculated fromthe image of Fig. 4. 256 pixel spectra were calculated for 400rows or columns of the image, and then averaged to obtain 1vertical and 1 horizontal spectrum. Even though the high-wavenumber end of the inertial subrange would probablyhave been resolved in such an image, the spectra are white(flat) because of the spikes of fluorescence caused by large

phytoplankton cells and aggregates (see Fig. 6)

Fig. 6. An example of the fluorescence versus distance alonga column of pixels taken from the image in Fig. 4. Note thelarge isolated spikes of fluorescence caused by individual cells

Fig. 7. (a) A 256 point data series composed of 20 delta func-tions. (b) The average of 400 spectra made up of data seriessuch as that shown in (a). Compare the shape of this spectrum

to those of Fig. 5. Units for all axes are arbitrary

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As plankton are quanta, the only way to obtain anunderlying k –5/3 spectrum from a data series of plank-ton is to average over such a large volume that individ-ual plankters no longer contribute to the variance. Theplankton can then be viewed as a continuum. Howlarge a volume must we average over? As given inSiegel (1998), we will have about a 1% counting errorwhen 400 cells are counted. Thus, to minimize the con-tribution of individuals, we must average over a vol-ume of water that contains about 400 of the types oforganisms whose individual fluorescence is intenseenough to cause large quantum changes in the localfluorescence. Large, intensely fluorescent phytoplank-ton cells might have a concentration of 1 to 10 cells ml–1,giving an averaging volume of 40 to 400 ml. The lengthscale that is associated with this volume depends onthe instrument, but would usually be >10 cm, andprobably >50 cm in most cases. The length scales aremuch longer when counting zooplankton, as they aremuch rarer. These length scales suggest that we can-not resolve the high-wavenumber end of the inertialsubrange using plankton, and that we can only resolvethe low-wavenumber end when it occurs at relativelylarge scales (>1 m). This implies that we may only beable to find a turbulence-induced k –5/3 spectral slopeof plankton in extremely turbulent environments; forexample, in the tidally mixed waters over GeorgesBank. In more quiescent environments, we may beable to resolve a k –5/3 spectrum of plankton if theplankton are dominated by abundant small cells; i.e.,no spikes of fluorescence due to large individuals. Oneconclusion is that you need to know what you are sam-pling. Even a few spikes will cause a spectrum tobecome whiter (as discussed in more detail below).

ALTERNATE MODELS OF PLANKTON PATCHINESS

As shown above, most studies of plankton patchinessresolve scales that are larger than the inertial subrangeof turbulence. It is, therefore, useful to explore thephysical dynamics that might dominate at these scales,and to investigate the expected spectral slopes of atracer embedded in these flows. The 2 types of physi-cal dynamics I will consider are internal waves and 2Dgeostrophic turbulence.

Internal waves

Internal waves dominate the motions of stratifiedwaters at frequencies lower than the buoyancy fre-quency N, and higher than the Coriolis frequency f.Motions at higher frequencies than N are turbulent,

and motions at lower frequencies than f tend to be in ageostrophic balance. The spatial scales of internalwaves range from ~1 m to ~5 km. Internal waves canpropagate in any direction, with lower-frequency wavespropagating vertically, and high-frequency wavespropagating horizontally (as a surface gravity wave).

The spectral slope of temperature variance in a well-developed internal wave field depends on the way inwhich it is measured (Phillips 1980). In horizontal tran-sects, the wavenumber spectrum of temperature vari-ance should fall as k –5/2. In vertical profiling, the fre-quency spectrum should fall as ω–3, where ω is thefrequency. Hence, the sampling scheme is an impor-tant determinant of the spectral slope, and transforma-tions from frequency to wavenumber (time to space)based on, for example, ambient horizontal currentsmay give incorrect results.

For horizontal transects, a k –5/2 slope is significantlysteeper than a k –5/3 slope, and should be distinguish-able with sufficiently well-resolved data series. Thesteeper slope implies that there is relatively less vari-ability at small scales in an internal wave field com-pared to 3D turbulence.

Piontkovski et al. (1997) calculated spectral slopesfor temperature, phytoplankton (fluorescence), andzooplankton (pump samples), sampled with 5 km reso-lution along transects in the Indian and AtlanticOceans, and the Mediterranean Sea. From the spectralslopes of k –2 to k –3, they concluded that the variabilitywas driven by the low-frequency internal wave field,though patchiness created by geostrophic turbulencewould also be consistent with these results (see below).

Geostrophic turbulence

At scales larger than the baroclinic Rossby radius ofdeformation (~10 km in most of the ocean), and timescales longer than f –1, motions of the water are largelyhorizontal and approximately 2D. The ambient frontsand eddies create a large-scale turbulence known as2D ‘geostrophic’ turbulence (e.g. Kraichnan 1967,Charney 1971). This turbulence is fundamentally dif-ferent than 3D turbulence, since tilting and stretchingof vertical vortices are not possible in 2 dimensions.

In geostrophic turbulence, the velocity spectrum ispredicted to fall as k –3, while a passive tracer is pre-dicted to have a k –1 spectral slope (Kraichnan 1967,Charney 1971, Bennett & Denman 1985). Over thelong-time scales (>f –1) of geostrophic turbulence,plankton can no longer be considered passive tracers,as their growth and mortality create structures in theenvironment. As shown by Abraham (1998), thegrowth rates of the plankton are a strong determinantof the spectral slope; fast-growing organisms (e.g.

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phytoplankton) tend to have spectral slopes near k –3,while slower-growing organisms (e.g. crustacean zoo-plankton) have spectral slopes near k –1. The ratio ofthe biological timescale to the timescale of the flowdetermines the spectral slope (e.g. Abraham & Bowen2002), with slower-growing organisms behaving morelike a passive tracer; higher variance at highwavenumbers (more structure at small scales).

Powell & Okubo (1994) modeled both passive andreactive constituents of 2D and 3D turbulence. Theyfound that non-reactive tracers behaved as predicted(k –1 spectral slope in 2D geostrophic turbulence).However, when the organisms were allowed to inter-act, the spectrum tended to flatten, becoming whiter,with relatively more variance at small scales. Wash-burn et al. (1998) calculated spectral slopes for temper-ature, salinity, chlorophyll fluorescence, and beamattenuation coefficient from data collected along tran-sects in the subarctic North Atlantic. They found fluo-rescence and salinity to be coherent at scales >7 km,with a spectral slope of k –2. At higher wavenumbers,the fluorescence spectrum fell off as k –3, while thesalinity spectrum continued as k –2. They suggested

that at low wavenumbers the variability was a result ofhorizontal stirring (geostrophic turbulence), while athigher wavenumbers, non-conservative biological pro-cesses dominated. Smith et al. (1988) analyzed satelliteimages of phytoplankton pigment and found spectralslopes of about k –3 offshore, k –2.2 inshore, and k –1

slopes at length scales

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internal wave field, we might find the fluorescencespectrum to be flatter, perhaps closer to k –5/3 thank –5/2.

A further problem with using fluorescence as a proxyfor patchiness of phytoplankton biomass is that fluores-cence is a complicated, non-linear and time-dependentfunction of the cell’s light history. At time scales shorterthan about 15 min, the fluorescence yield (fluores-cence as a function of irradiance) decreases withincreasing irradiance above about 100 µEin m–2 s–1 dueto non-photochemical quenching (e.g. Morrison 2003and references therein). Cells that are moved upward,due to mixing or high-frequency internal waves, andthen are moved down again will have a lower intensityof fluorescence than cells that were stationary in thelight field. Thus, high-frequency vertical motions of

the water can create horizontal variations in fluores-cence that are uncoupled from variations in biomass,or even the present location of the isopycnals (e.g.Lennert-Cody & Franks 2002). Over longer time scales,the fluorescence yield changes due to photoadapta-tion, a process that may also introduce spatial varia-bility of fluorescence to an otherwise homogeneousbiomass distribution.

Spikes

As discussed above, plankton is made up of individu-als, some of which can be quite large or intensely fluo-rescent. A few large individuals in a time series of fluo-rescence, for example, will create a time series

dominated by isolated spikes.These isolated spikes behave asdelta functions, and the Fouriertransform of a delta function is aconstant; that is, a white spectrum.Thus, the inclusion of a few spikesin the data will tend to shift any un-derlying spectral distribution to-wards a flat (white) spectrum.

To illustrate this effect, I createda synthetic data set from 30 sinewaves whose amplitude wasdetermined by a k –5/2 spectralslope, characteristic of internalwaves. The sine waves wereassigned a randomly chosen phase(the relative positions of the crestsof the waves), and a data set 256points long was created. To thisdata set, I added 40 randomly-placed spikes that had an ampli-tude 0, 1, 2, or 5× the amplitude ofthe largest (lowest wavenumber)

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Fig. 9. (a) Synthetic data sets made of30 sine waves with random phase andan underlying k –5/2 spectrum. Addedto these data are delta functions ofsize (top-to-bottom) 0, 1, 2 and 5× theamplitude of the lowest wavenumbersine wave. (b) k –5/2 spectral slope (grayline) and spectrum calculated fromdata series in (a) (black line). The spec-tral slope of the data is calculated bylinear regression of the log of the spec-trum and the log of the wavenumber.Note that the high-wavenumber com-ponent of the spectrum is most stronglyaffected by the inclusion of spikes inthe data. Units for all axes are arbitrary

Franks: Turbulent transport and plankton patchiness

sine wave (Fig. 9). I then calculated the spectral slopeof the data sets after removing any linear trend. Thereis a clear tendency for the spikes to lower the slope ofthe spectrum, from k –5/2 with no spikes, to ~k –2/3 whenthe spikes were 5× the amplitude of the largest sinewave.

Spikes in plankton could arise from any number ofcauses that have nothing to do with mixing of a dis-solved tracer in 3D isotropic turbulence. Fluorescencespikes could be created by large cells, aggregates,pieces of seaweed, zooplankton guts, or even some-thing getting temporarily jammed in the instrument.Careful examination of the raw data, and in particularvisual examination of water samples, is essential forassessing whether the spikes should be interpreted or

not. The occurrence of spikes and the consequent dif-ferences between the temperature and planktonicspectra may have little to do with the relative strengthof the underlying physical and biological dynamics,the usual explanation for such differences.

Steps

A step or front is a sudden change in the local aver-age value of a property. Steps and fronts are commonfeatures of the ocean, and occur in both physical andbiological variables, though not always coincidently.The spectrum of a step gives a k –2 slope. Any data setcontaining steps (sharp fronts), regardless of its under-

lying physically driven spectralslope, will tend toward a spectralslope of k –2.

An example of the influence ofsteps on the spectral slope can beeasily formulated. I took 30 sinewaves with random phase, all withequal amplitudes (an underlyingflat, white spectrum). I then added 4randomly placed steps of randomsign (positive or negative steps),with amplitudes of 0, 1, 10 or 20× theamplitude of the sine waves(Fig. 10). The spectra calculatedfrom these data series clearly showthe tendency of the steps to force thespectral slope toward k–2. With steps10× the amplitude of the underlyingsine waves, the spectral slope is al-ready k –1, though the steps are noteasy to pick out in the data.

Steps or fronts in the data couldthus make an underlying whitespectrum tend toward k –5/3, or aspectrum of a tracer in an internalwave field (k –5/2) tend toward k –2

or k –5/3. Washburn et al. (1998) rec-

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Fig. 10. (a) Synthetic data sets made of30 sine waves with random phase andan underlying k0 (white) spectrum.Added to these data are steps of size(top-to-bottom) 0, 1, 10 and 20× the am-plitude of the sine waves. (b) k0 spectralslope (gray line) and spectrum calcu-lated from data series in (a) (black line).Spectral slope of data is calculated bylinear regression of the log of the spec-trum and the log of the wavenumber.

Units for all axes are arbitrary

Mar Ecol Prog Ser 294: 295–309, 2005

ognized this effect of steps, and removed a large stepfrom their data prior to calculating the spectral slopes(which ended up being about k –2 in any case).

Trends

In a fine paper that presaged this one by almost 2decades, Armi & Flament (1985) discuss the effect oflinear trends in a data series. They showed that a peakmade up of 2 linear ramps (increasing and decreasing)has a Fourier transform that gives a spectrum that fallsoff as k –4. A data series with such a linear trend wouldhave a slope tending towards k –4.

Following the approach of the pre-vious sections, I made a syntheticdata series of 30 sine waves withrandom phase, all with the sameamplitude (an underlying whitespectrum). I then added a linearramp (Fig. 11) to the data series, thepeak of the ramp being 0, 2, 5, or 10×the amplitude of the sine waves. Theresulting spectra clearly show theeffects of the ramp (Fig. 12), eventhough the ramp is difficult to see inthe data. As the ramp gets taller, thespectrum falls off more steeply,reaching k –2 (for the low-wave-number part of the spectrum) whenthe ramp is 10× the sine wave ampli-tude. There is also a marked changein the slope of each spectrum fromnegative slopes at low wavenumbersto zero slope at higher wave num-bers. The break occurs at the samescale as the ramp width (120 points).

The inclusion of a ramp-like trendin the data could easily drive an

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Fig. 12. (a) Synthetic data sets made of 30sine waves with random phase and anunderlying k0 (white) spectrum. Added tothese data are ramps of amplitude (top-to-bottom) 0, 2, 5 and 10× the amplitudeof the sine waves (see Fig. 11). (b) k0

spectral slope (gray line) and spectrumcalculated from data series in (a) (blackline). The spectral slope of the data is cal-culated by linear regression of the log ofthe spectrum and the log of the wave-number for only the lowest 5 wave-numbers. Note that it is the lowestwavenumbers that are most stronglyaffected by the presence of a ramp.

Units for all axes are arbitrary

Fig. 11. Linear ramps used to explore the effects of this fea-ture on the spectral slope of the data. Four ramps were used(bottom-to-top): 0, 2, 5, and 10× the amplitude of the sine

waves (see Fig. 12). Units for both axes are arbitrary

Franks: Turbulent transport and plankton patchiness

underlying white spectrum to have a k –5/3 slope, par-ticularly at the lower wavenumbers. The importantquestion to ask is how the ramp in the data got there. Isit a result of the instrument, the sampling, or theunderlying physical and biological dynamics?

What does k –5/3 look like?

I have given 4 distinct ways of generating a k –5/3

spectrum: (1) a combination of sine waves of the appro-priate amplitudes, (2) a combination of sine waves andspikes in the data, (3) a combination of sines and steps,and (4) a combination of sines and linear ramps. Toshow how different these data series look, I created

synthetic data sets, each of which has a k –5/3 spectralslope (Fig. 13). These data sets were made up of(1) pure sine waves with amplitudes giving a k –5/3

slope, (2) a set of sine waves with a white spectrumplus 7 steps of random sign and magnitude, (3) a set ofsine waves with a k –5/2 spectrum (characteristic ofinternal waves) with randomly placed spikes of ran-dom magnitude, and (4) a random combination of steps(k –2) and spikes (k0). All these data sets have a k –5/3

spectrum, yet they look very different when plotted asthe raw data series.

When we calculate a spectrum of a data series, wethrow away some essential information; the phaserelationships of the various Fourier components mak-ing up the spectrum (see also Armi & Flament 1985, for

further analysis of this issue). Thedata sets of Fig. 13 all have the samespectrum, but very different phaserelationships of the various sinesand cosines making up the Fourierdecomposition of the data. It is thisrelative phase that gives the dataset its visual character. It is vitallyimportant, then, to spend some timelooking at the raw data to try toidentify the features of the data thatmight be driving the spectral slope,and ask whether these features areincluded in the dynamics of themodel that the data are being com-pared to. Large spikes in the fluo-rescence data may indicate rare andrandom pieces of detritus that can-not be accommodated in a contin-uum model of a reactive tracer in aturbulent flow. Large steps in thefluorescence may indicate changesin species composition with conse-quent changes in the fluorescenceyield, but may have little to do withthe underlying physical flows. Ingeneral, there is not enough infor-mation in the spectral slope tounambiguously identify the dynam-

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Fig. 13. Synthetic data series, all with ak–5/3 spectral slope. (a) 30 sine waves ofrandom phase, (b) 30 sine waves with ak0 (white) spectrum with 7 randomlyplaced steps of random sign and magni-tude, (c) 30 sine waves with a k–5/2

(internal wave) spectrum with 40 ran-domly placed spikes, and (d) a combina-tion of steps (k–2) and spikes (k0). Units

for both axes are arbitrary

Mar Ecol Prog Ser 294: 295–309, 2005

ics that created the patterns. The spectral slope is not astrong test of any model of physical-biological inter-actions, and should only be used as a supplement toother careful analyses.

CONCLUSIONS

Theoretical models predicting the spectrum ofplankton patchiness offer the potential of a mechanis-tic understanding of the physical and biologicaldynamics structuring the distributions. The k –5/3

model of plankton advected in a 3D isotropic turbu-lent flow is often the basis for comparison to fielddata. However, an analysis of recent publicationssuggests that most field data do not resolve theappropriate spatial scales to allow comparison of thefield data to the k –5/3 model. Furthermore, it may beimpossible in most cases to gather planktonic data fortesting the spectral slope in the inertial subrange ofturbulence because of the quantum character of theplankton. Plankton are not a continuum as modeled,but are individuals. To remove the effect of individu-als from the data (since individuals would tend tomake the spectrum flatter and more white), someaveraging must be performed. The scales for averag-ing will depend on the types of organisms present, sosome knowledge of the composition of the plankton isnecessary. In many cases, the averaging scales willbe so large that the inertial subrange cannot beresolved with planktonic data. It is only when theeffects of individual plankton are undetectable, andthe inertial subrange is sufficiently large (e.g. intenseturbulence and low stratification) that planktonic datacan be robustly compared to the k –5/3 spectral model.

Most planktonic data are gathered at spatial scalesthat are more appropriate for comparison to theinternal wave (k –5/2 for transects) or geostrophic tur-bulence (k –3 to k –1 for reactive tracers) spectral mod-els. Still, the biological spectra often differ from thesemodels. These differences could arise through thebiological dynamics that are usually invoked. How-ever, other factors such as spikes in the data, sharpsteps in the data, layers of plankton, or linear rampsin the data can confound the interpretation of theunderlying spectrum. Calculation of the spectrumnecessitates throwing away important informationconcerning the phase relationships of the Fouriercomponents of the data. It is the phase, not theamplitude, which determines much of the visualcharacter of the data. The data should be carefullyexamined for such features, and decisions madeabout whether their effects on the spectral slope areconsistent with the model that the data are beingcompared to. The spectral slope, by itself, does not

contain enough information to unambiguously iden-tify the dynamics underlying the observations, oreven to distinguish among different spectral models.It is important to be critical when comparing modelsand data: (1) Does the model contain the appropriatedynamics? (2) Do the data test the model? (3) Are thedata of the correct dimensionality for comparison tothe model? Careful consideration of these factorsmay help us develop new and powerful insights intothe dynamics coupling physics and biology in theocean.

Acknowledgements. This paper has been a long time brew-ing, and has benefited from discussions with many people.While I acknowledge their considerable help in synthesizingthese ideas, I must emphasize that I am solely culpable for anymistakes, distortions, misstatements etc., that may appear inthis manuscript. In particular, I wish to thank K. Fisher, J.Jaffe, D. Rudnick, J. Pringle, W. Munk, C. Garrett, A. Gargett,E. Boss, and H. Yamazaki for fruitful discussions and occa-sional flashes of insight. This work was supported by NSFgrants DBI-9871359 and OCE02-20379, and ONR grantN00014-03-1-0391.

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Editorial responsibility: Otto Kinne (Editor-in-Chief), Oldendorf/Luhe, Germany

Submitted: July 15, 2004; Accepted: January 3, 2005Proofs received from author(s): May 20, 2005