Coupling plankton population models to hydrodynamical studies

Paul R. Holland AMlMA Centre for Polar Observation and Modelling, Department of Space and Climate Physics,

University College London, Gower Street, London, WClE 6BT, UK. Abstract dimensions allows us to take tlicsc well-uridcrstood

Tlrc. />oprlltrtiotr c i ~ ~ t r t r t ~ r i t ~ . ~ (?/ /)l(ttrlitotr trrtp .strotr,yl~~ (~(/i.t.t(,tl l?. tlrc parameterisations of biological processes and extend them to tlie

I I ~ ~ l r o p l ~ ~ . v i c . t r l c~orrt1itiotr.v itr i1~lrit.11 tlrc'l. rc~.vit/~~. Itr t1ri.s trrtic.lc~ I study of ncw phenomena. sucli as horizontal currents. eddying, I i s v i t 1 1 ~ I I 1 1 t ~ t i t t i ~ t / i t / I (1 t ~ t t ~ t r and turbulence. tlrc~thotl , / i ) r c~or~plirrg />ltrrrktotr po/>rrltrtiotr tiroclc~ls to tirrt lrcric~r~l .srr~r/it*.s ( I / ' t11c ,/7o 11. rrrr(1 trrrtr.s/?ort itr ~ ~ t r i 9 i r o t r t ~ r ( ~ ~ i t ( t l ./71ri(/ l)o(/it,.s. 1 thcvr gi1.t. r r r r t,.vtrtlr/>/c* (!/'.vrrt*lr trtr c~t.o/o.qit-rrI SIII(/I ' it1 Ltrkc Brrik-trl. Sihc~rirr.

Introduction

T he steady increase in the power of computers has m, 'I d e a rnngc of. Iluid problems amenable to the techniques of Co~iiputotion;~l Fluid Dynamics (CFD) . A p:irticular

advance is the ability to niodel environmental flows (such as those in rivers. lakes ; ~ n d oceans) with ;I grid resolution whicli pcrmits the study of relatively small-scale phcnomcna. This enables the modcller to examine tlie importance of flows to local- ised populntions of plankton. microscopic organisms which exist virtually cvcrywlicrc in tl;~tural fluid bodics.

Phytoplankton are tlie tiny plants which grow in the surface layers of ;I water body, and arc visible as a green sludge in over- nutrilicd gnrdcn ponds. They rcquirc sunlight and nutrients to grow. and are tlic primary producer of energy in an aquatic eco- system. Zooplankton arc microscopic animals which feed on tlie pliytoplonkton and ci~cli other. Both classifications arc extrcmcly

N-P-Z modelling A class of multidimensional modcls i n common us;tgc arc the so- called N-P-Z models. which have n t hree-component ccosystcni fknturing state vnri:lblcs for a univers;~l limiting nutrient (N). pliytopl:~nkton (P) , who iltilise this nutrient. and ;I herbivorous zooplankton ( Z ) who feed on the phytoplankton. Although defining single compartments for phytoplankton and zooplank- ton. with accompanying avcragcd valucs for growth. niortality. etc.. is an extremely rough approximation, sucli models can still bc of great use in answering well-delined ecological questions 161. All tlircc components are expressed in units 01' clicniicnl conccn- tration (e.g. mg-at Nitrogen I- ' . where I g-at = I ;~ to~i i ic mass unit = mass of 1 Hydrogen atom) and are evaluated ;1t each cell of the computational grid used to discrctise the governing rlilkrcnti;~l equations. In the very simplest N-P-Z modcls the nutrient is sup- plied by a mortality rate of both pl;~nkton species and this is sulf?cicnt to close thc systcni of cqu;~tions.

The threc components are each governed by a scalar transport equ:~tion of the model in qucstion. such as the two-dimensional advcction-difl'itsion equation for an incompressible Iluid:

general in that tlicy represent org;inisms of many different aC' dC aC ( ) ( , a(. ) spccics and classes. ranging over scales from unicellular to tlic - +rl- +l \ ' - = - h' - + - h' - + S,

at a.v d: d.v d.v d: 2: ( I

barely-visible. The modern practice of coupling plankton population models

to C F D in two or three dimensions h:is precipitated a significant advance in our undcrst;~nding ol' tlic intcrnction bctween tlie physics and ccology of natural water bodies and is extremely uscfi~l in addressing a range of practical problems. Determining ninximum rcle;~sc concentrations of nutrient-rich cflluent (sucli as sewage sludge or fertiliser runofl' from fields) can help to avoid provoking massive plilnkton blooms. which have been rcsponsi- blc for upsetting tlic balance of aquatic ecosystems by releasing

. toxins or consuming large portions of the available oxygen as

( thcydeconipose [I]. Alter~i;~tively predicting blooms in the ocean may one day enable lislicrmcn and ecologists to sitrmisc tlie likely

: location of fish and other organisms higher up in the food chain [2]. In other situations tlie modelled plankton distributions can be used to infer details of Ilow regimes. as in tlic case of assimila- tion of remotely-scnsed ocean colour data into physical ocean models [3]. 0cc;lnic plankton play a vital role in the cycle of various important elements. sucli as Carbon. and should thcrc- fore bc resolved in long-term coupled models of the Earth's climate [4].

Plankton models in onc (vertical) dimension liavc been in development for decades and have addressed a wide range of sci- entific questions [S]. Tlie inclusion of one or two horizontal

where C'= K! PI Z i s the component of interest. h',, and h', are hor- izontal and vcrt ia~l eddy dill'ilsivitics (rcprcscnting turbulent mixing on a scale finer than that of tlie grid). t is time and 11. 11-arc velocities in tlie .v. : plane. S, represents n series of source tcrms which account for the interactions between plankton components.

Tlie model used :IS an example here [7] consists of tlic following source tcrms:

where (l represents depth and dcscriptions of the v;~rious niodcl parameters and thcir units are givcn in table I. A sclicni;~tic diagram of the model described by (2) - (5) is shown in ligurc 1. The zooplankton grazing ratc I is givcn by

Mathematics TODAY DFCEMRER 2003 185 I

Example Application - the thennal bar in Lake Baikal

TIic mc~dcl described above h;~a I>cc~i si~cccssfi~lly i~scd to invcsti- gate the localised plankton bloom \vliich accompi~nics the Spring tlicrnial bar in Lake R;~ikal. Sibcri;~ [7]. The thermal bar is n do\vn\vclling plume 01' dense water \vliich 1i)rnls ivlicn the shal- lo\vs of :I lake u,arm or cool through 4°C. the tcmpcniturc of ninsimuni density for fresh w;\tcr. Water o n each sidc of the plume is either warmer or cooler than 4°C ;tnti is tlicrcfi)rc lighter and does not sink. resulting in n localised pl i~mc \vhich the lior- izontal surf;rcc flows converp. The tlicrmal b ; ~ r moves to\v;~rd ~ l i c open I;tkc ;IS the surl';~cc Ilc;tt Iluu continues. ; ~ n d ligurc ! slio\\,s ;I sn;ipsliot 01' ;I simul;~tctl tr;l\,clling tlicrm;~l h;tr ;tl'~cr 20 cl;~ys 01' \\,;~rniing [ 7 ) . .l'lic physics ; ~ n d ecology 01' tlii.; plicnomc-

I I I - 1 1 1 1 c l I I I ~ I I I . I I l I l 11011 tile Sllb.ieet reeellt revie\v tile ; l l l t l l O r [S]. n o asymptotic \~;iluc li)r incrci~sing p1iytoplanktc)n conccntril- ~h~ spring t ~ c r I l l ~ , l bar of ~~k~ 1 3 ~ i k ~ l i s kllo\4,n t o h;l\rc tions hcc;~usc the zooplankton adapt their feeding rate to tlie i I l lpor tan t interncl inn \4,itll i t s pl;lnkton poplll;ltions 191. \\.ith li)otl ;~bund;~ncc. \~;irious species reacting dillkrcntly to the Ilo\v p;~ttcrns gcncr-

;itciI. The grcr\r 111 o f ;\I1 tli;ttariis. to v;~ryit i~tlc~rc.cs. is Ih t~~ id to he promoted inshore 01' ;11ic1 ;I[ the position 01' (hc. thcrni;~l h r .

' ',,g M;~\inltrrn pliytopl;~nkton 2 d a y , sro\t lh ralc

'l I . i ~ l i t extinction coellicicnt 0.1 111 l

R,,, M;~xim~~m insestio~l r.:lte of 0.5 d a y 1 /oopl;ln kton

A IVIC\ . cons1;ltit 0.5 nig-at N I - '

111, %oopl;~nkton rnort;~lity r.;ltc 02 d a y '

I t can he seen t1i;lt phytopl;~nkton photosynthesis (the first term on the right-li;~nd sidc ot'ccli~atic)n (2)) responds to t\ilo lim- iting I.:~ctors in this Ibrmul;~tion: ni~tricnt sliort;~ge and lack of light. Growth follo\\,s an exponential decrease with depth due to tlic ;~ttcnu;~tion of ;rv;~il;~blc light \vhilc tlic Micliaclis-Mcntcn r;~tio (lirst usctl in the tlescription 01' enzyme c;~t;~lysis) describes the clkct of nutrient supply and limitation. The exact response of this li)rmul;~tion to \.;~ri;~tion in .V ; ~ n d c l is slio\vn in ligi~rc 2. li>r the lixcil v;~luc I' = 1.

The ccologic;~l niodcl is couplctl to ;I liydrodyn;~mic;~l 1brrnlll;t- lion \vliicli has si~ccessl*i~lly predicted tlie tr;~nsicnt thermal struc- ture and position of. the Spring thermal bar [7]. This ;illo\vs the plankton bloonis ;it the position of the tlicrm;~I bar to be qualita- tively predicted in the presence of accuriitc Ilo\\, ; ~ n d density con- ditions. It can be seen that p o p ~ ~ l ; ~ t i s n gro\vth is strongly promotcd j ~ ~ s t inslinrc of the moving thermal b ;~ r (ligurc 3). \\.hieh agrees \vit l i obscrvntions t;~kcn in L;~kc I3;tik;1l (01.

Tliis result is of great v;rluc bcc;~use ;I study of the various ternis in the ccosystcni modcl can subsccli~c~itly rcvc;~l ;I lot of L I S ~ I ' I I I inft>rm;~tion about plankton bcli;n,iour \\~itliin this bloom. For example tlie nite of pliotosyntlicsis in the hIoi>ni is fi>und to be fiiirly constant through0111 thcsimulation. rc\,c;~ling tIi;it tliclinii- tation of gro\vtli due to depletion of nutrients is b;~lanccd by o continual influx of photosynthesising pliytoplnnkton carried by the converging si~rlitcc Ilo\vs [7]. Examining the vari;~hlcs cli~anti- Vying photosyntlicsis in cclil;~tio~i (1). I! ,V;~ntl ( l , i t isclc;~r th;~t the blooni occurs at the position where the si~rI~;~cc 11o\vs con\.erge bcc;~usc the clkcts of nutrient li~iiit;~tion thcrc (lot\ '2') arc coun- tered by a n inllux of pliytopl:~nkton (high 1') and ;~bund;~nt light (low (l). In tlie ori_cini~l paper [7] this ;irgument is supported by a

. ,...., h , , . 8 , I . , . L ,,, X I. I., ' ' ..,

- , q - 1 , . , . -"?$ ,

Malhrm.ltirs TODAY DECFMRER 200:i 186

detailed analysis of' c:~cli biological interaction tcrm. which is omitted here in the interest of brevity.

The study also fi11ds that the plankton model is I'i~ndnnicntally dependant upon the vertical dilhsion tcrm in the plankton trans- port equation ( I ) . and in particular upon the value of K, which quantifies the amount of* vertical turbulent mixing. Wlicn applied to 1;irgc hotlies of water. h', is I I S I I ; I I I ~ modelled by :I

simple f.i~nction of the vertical density structure of the fluid. as buoy;lncy efkcts niay help to generate or suppress turbulence. The study shows that plankton growth near the surfi~ce is strongly responsive to the value of K,. as blooms can be trapped near tlie surl*:lcc Ibr sm:~ll K, or diflhscd away from the growth region rapidly when K, is large. In this way tlic plankton responds to the lake's density structure. which is predicted by the two- dimensional flows of' the model.

Discussion In this ;\rticIe I II;IVC attcmptcd to gi\.e a flavour of the coupled pliysic;~l-ecological modelling which is now possible using com- putational techniques. ant1 I have given a brief' example of such an application which has been pcrli~rnied on :in ordinary desktop coniputcr.

I t should be noted that the formulation 1 have used as an example here is at tlic simpler end of the spectrum of plankton models. I t is possible to adopt modcls with many more state vari- ables. representing dillkrent species of plankton and nutrient types and ;~ddition;llly mt)dclling such factors as Oxygen concen- trations. the bcntliic (bottom-dn.elling) communities. and even the elrects of higher organisms such as fish. However. extending models in this \\ay c:r~i be probleniatic due to uncertainty in the mathcm;itic;~l rcprcscntations of ecological processes and tlie increasing number of poorly-known parameters which arise fro111 such increased cc>mplexity. The model used :IS an example here is mcrcly the lirst step in a series of attaiipts to capture matl~ematically tlic interaction between the physics and ecology of tlie thermal bar circulation [101.

Tlic most important consider;~tion is to lirid accurate results \vliicli are o f relevance to tlie real world. regardless of tlic com- plexity of' the model used to obtain them. With :I host of parame- ters to be set in each ecosystem model. increasing rapidly with every new sopliistication. i t is the lack of' in-situ hydrobiological

da ta required to v;~lidatc these parameters. rather than computing power. which now limits tlie numerical morlclling 01' pl:u~kton cco1ogy.U

Acknowledgements The author would likc to csprcss liis r l ~ a ~ i k s to Vinccn~o I3ottc and Anthony Kay Ihr permitting the use of tlicir ligurcs and liw many i~sefi~l comments on this article.

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X Iloll:~nd. I'.K.. ; lnd Kit?. A.. 200.7. :\ rc\ic\\ 01' the pli!\t~.\ itttd ccoloptc;tl iniplic;~tio~i\ <>I' tlie tlicrni:~l h;tr circ~tI;ttt~~ii. I . I I I I I IO/~I ,~~~ (1. 3 7 . l 5 l - l f 2

0 Lihlit~\li\\.;~y. Y.V.. Ku/niiti;~. A.Y.. I'ot>e~iihtti;~. I.(;.. I'o~\~tiiht~i. \:l ... ;11ic1

SIittii;tr:~e\. M.Y. . 1'1'10. 'l'lic d t \ t r ih~t r to~i 01' 1It;t1<11ii\ 11c;tr ;I tIicrti~;tl h:tr 111 I . ; t hc I3itik;1l. . / ~ ~ ~ r r r ~ ~ r / ~ ~ / (;rc,crr 1 , ~ t l . ~ ~ ~ R~,t,~rr<./i. 22:5-14.

10 Illdl:lnd. I'R.. K;!!. A . and Hortc. V.. 2rlo3. Nunicrtc:tl ~iio(lclli~ig 111' tlic ~ I ~ ~ r t i i ; t l

h ; ~ r :tnd it.; ccolo~ic;~l c~>n\cq~tc~icc.; In ;I r i \cr-~loi~~t~i; t tccI I:thc. . / t , ~ r r t ~ ~ r l r,/ \I,~rrirt. S~.\rctir\. 43:hl-S I .

Paul Holland is a post-doctoral Research Fellow at the Centre for Polar Observation and Modelling. University College London. His research interests lie in numerical modelling of fluid dynamics and plankton ecology. and he is currently involved in the modelling of frazil ice formation in gravity currents under ice shelves as part of the Autosub Under-Ice NERC thematic programme.

Pensions By Dickens T J Fairclough CMath MlMA

Wickfield & Heep punctual especially in such inclement \ \~c:~t~icr if I m;ly t;lkc the

T heclock wasstriking tlir appointed hour on a drizzly Deccm- liberty of saying. her morning as 1)rnlid C'c>ppcrlicld tapped liglitly on the 'Good Morning Heep. Mr. Micawbcr'. sitid Coppcrficltl. counting-house \vindow. Above his head and below the nodding to each as he entered the chamber. removing liis stovc-

three brass balls a painted wooden sign proclaimed " Wick field & pi pc hat to shake the raindrops from i t and sitting down as he Hecp - Independent Fitinncinl Advisers". Inside Uriall Heep spoke. 'l have an appointment with Mr. Wicklicld. plc;~sc be so jumped down from liis stool and thrust aside thegreat fat book in good as to tell him that I've arrived.' '13lcss you sir' rcplictl I-iccp which liis lank fi>relinpr Iind been tracing each line as he reitd. 'Rut unli>rtunatcly Mr. Wicklicld is inclisposed totl:~y ;tntl h;ls Hurrying over to the door he unctuously admitted his visitor asked me to stand in for him unless of' course you have any objcc- with a grinding of palms and liis customary two bows and tions Master Copperfield'? - I should say Mister. hut tlic other scrape. comes so nnturnl.'said Uriah

'Good morning Master Coppcrlield niy dear Sir and most '1 suppose not' replied David i~nentliusi;tstic;~lly. 'Very wcll welcome I'm sure to my cr.,. I shoi~ld say of course o,,r ilmble then let's get down to business straight :iw;~y sli;~ll we? I've establislinic~it. I t is indeed the mark 01' a gentleman to be so rcclucsted this meeting because I want some lin:~ncial :tdvicc.

Malhcmaticc TODAY DFCEMRER 200:1 187