Copepod aggregations:influences of physics and collective behavior
Glenn R. Flierl and Nicholas W. Woods
J. Stat. Phys., 158, 665-698
1000

1050-5-10-10-50510

Motivation: copepods and whales
Right whales congregate in Cape Cod Bay during the spring and feed on the zooplankton (primarily Calanus copepods) blooming at that time. Most years, the copepods form
very intense patches along tidal fronts off Provincetown or in the Great South Channel,
and the whales cruise along these fronts.
Graphics: map Cape Cod Bay Gulf of Maine [Manning, et al, 2009]1

Copepods: Calanus Finmarchicus• Life History: (Tarrant et al., 2008; Durbin et al., 1997)- Pre-adults overwinter in “diapause”- Molt into adults and mate, hatch and molt through 13 stages- One or two generations per year
- Eggs and nauplii stages are an important source of food for cod and haddock(Kane, 1984)- Later copepodite stages are the primary prey of right whales (Wishner et al.,1995) Graphics: Prey species calanus [CCS] calanus [JHU, Lab Exp
Fluid Dyn]
• Swimming ability: (Buchanan et al., 1982; Epstein and Beardsley, 2001; Genin etal., 2005; Lenz et al., 2004)- Typical: 0.1-10cm/s- Escape: Up to 80cm/s• Patchiness: (Wishner et al., 1995; Beardsley et al., 1996; Baumgartner and Mate,
2003, Epstein, 1995)- SCOPEX study in the Great South Channel suggested densest patches coincidewith salinity front.
- Densest patches order 100-1000m across, 1-10m vertical, hours to days(?)- Peak densities 103 to 104 copepods/m3 . 10-1000 times background
Whales:
Copepod aggregations are a food source for many species, including the endangeredright whales (Eubalena glacialis), which return to the southwestern GOM every spring toforage.

Graphics:

Predator species

right whale2

underwaterscale

Outline:• Vertical swimming and physics• Horizontal convergence• Social behavior• Turbulent stirring• Combined effects

Vertical swimming and physics• Eulerian: b(x, z, t)∂b + ∇i (Ui b − Kij ∇j b) = 0∂tU and Kij include both fluid motions, directed swimming, and drifts associated withvariations in random motion.

• Concentration can only increase if biological velocities in U are convergent.
• Dominant directed motion is vertical swimming ws (z, t)∂∂∂b + ∇ · (ub − K∇b) +([w + ws ]b − Kv b) = 0∂t∂z∂z(with u and ∇ being horizontal).
• strong swimming and randomness with preferred depth∂∂(ws b − Kv b) ≃ 0∂z∂z∂
ln F = ws /Kv∂z⇒ b = b′ (x, t)F (z) withR• Depth integrated equation ( dz F ≡ 1)
∂ ′b + ∇ · (ũb′ − K̃∇b′ ) = 0∂t,ũ =Zdz F u ,K̃ =Velocities ũ are divergent and have an associated stretching ss = −∇ · ũ =3ZF∂w∂zZdz F K

If F is very sharply peaked, the velocities and stretching are just the horizontal flows
at the preferred depth; in any case, we’ll drop the tilde and prime and useand ∂w∂z∂
b + ∇ · (ub − K∇b) = 0∂tor∂b + u · ∇b − ∇ · K∇b = sb∂t

Later the social movement will be included in u.
ln b = s along-track integrals

Graphics:

LagrangianD/Dt

Steady state patches
F = ub − K∇b = ẑ × ∇χGain insight from K = const. and u = −∇φ∇ ln b = −1
∇φKb = b0 exp(−φ/K)

Linear geometry1) Pure strain u = −U (x/L), φ = 12 U L(x/L)22) Localized front u = U0 − U tanh(x/L), φ = −U0 x + U L ln[cosh(x/L)]
2a)U0 > U then ub − K∇b = (U0 − U )b−∞ (χ 6= 0)
3) Localized velocities u = −U (x/L) exp(−x2 /2L2 ), φ = −U L exp(−x2 /2L2 ) Graphics: one d purestrain localfront u0=0.2 u0=0.5 u0=1.2 localvel

Steady solutions exist with b → 0 for cases 1, 2 but are proportional to theupstream value for 2a, 3. U0 = U does not have steady solutions (diffusivelygrowing region). The amplitude depends on the Pèclet number of the convergent
flow φ/K.
Graphics:

t 1/2

developmentmax

4

2D steady
Decompose the velocitiesu = uψ − ∇φThere is a solution b = b(φ)∇ ln b = −1
∇φK(φ)

if K, φ, and ψ contours are parallel.
Ekman transport from bottom friction or eddy-wind effect (stress related to W − u)indeed has the divergent flow perpendicular to the rotational flow. For a bottom boundary
layer, linear Ekman theory gives φ ≃ 21 ψ.
Time dependence and spatial inhomogeneity
Time scales:• Homogenization along ψ lines• Initial growth (exponential)• linear growth as b fluxes in• equilibration when outward diffusive flux compensates Graphics:
flow maximum

example swirling

Buoyant plume
Buoyant fresh plumes from snowmelt run down the coast, enhancing the normal
coastal jet. Can they “snowplow” up enough copepods?

Interior plume-relative circulation may be either sign.

562d "plow"layer
constant depth

5max concentration

432

10020

40

60

80time (hours)

100120140

Physics would need to act on long times, gather copepods from large areas,and have small turbulent losses

6Social behavior
Desirable features of the social grouping model we shall consider include:1) Spontaneous break-out of patchiness. Not surprisingly, this tends to occur at thesensing scale.
2) Shear resistant. Patches should be maintained under reasonable shear or strain in thefluid flow.
3) Sticky. If two patches are brought near enough by the eddies or other motions, theyshould coalesce to form a larger patch which can hold together.
4) Potential for large-scale patches. If patches begin to coagulate, but then break apartagain, dense aggregations on a scale much bigger than the sensing scale are unlikely.

For the nth organism,
dX(n) = V(n) dt
dV(n) = −r(V(n) − u − ub )dt + dW
ub [b]: preferred swimming velocity
dW: a random increment hdWi = 0, hdWi dWj i = 2Kb r 2 dt δij
Kb [b]: biological diffusivity associated with random swimming
Translates to1∂b + ∇ · ub + ub b − ∇(rKb )b − (κ + Kb )∇b = 0∂tr
• Kb = const.
• ub = −∇φb
RR
- Local density: β(x) = dx′ w(x − x′ )b(x′ )- Weighting function w(r) = πa3 2 (1 − r 2 /a2 )2 (r ≤ a)
- Potential: φb = −U aβ/(1 + β)7Stability of uniform statesFor b = b + A(t) exp(ık · x), we have∂
A = |k|2 (bΦ(|k|) − Kb )A∂twithUa
Φ(|k|) =(1 + b)2ZZdx′ w(|x′ |) exp(ık · x)

implying instability forPeb>1(1 + b)2with P e = U a/Kb the Pèclet number for the organism’s taxis velocity.
Graphics:
critical curves linear log example
Note: both low and high densities are stable; thus a large, dense patch will
not break up by the behavioral processes.
Patch sizesβb = b0 exp(−φb /Kb ) = b0 exp P e1+β

Iteration: b → β → exp(P e β/[1 + β]) → b0 [normalization] → b Graphics:
max b area
Shear:
u = Us cos(k0 y) Graphics:

shear merger

8examplepatch size

More complex flows
Unlike the purely physical problems where φ can become large and positive, andexp(−φ/K) can decay towards zero, the social model has φb always negative so that insteady state b ≥ b0 > 0. b0 is related to the far-field value:b∞b∞ = b0 exp P e
1 + b∞b0 ≃ b∞ /(1 + P e b∞ )b ∼ b∞ + Ce−kx• Each patch has an exponentially small field which attracts other patches.
• But it will also tunnel out along axes of divergencephi, iter = 650
010-0.2

5phi

-0.4

0-0.6-5-0.8-10-10-50

5-1-1510-10-50510

15xFig. 20: The potential φphys + φb ; the highest values are at the north and south,
and it gets more negative to the east and west as shown on the east-west crosssection.
Stirring
θx1
Us hθy1ψ = √ cos(kx +) cos(ky +)33k 7+ cos(2kx + θx2 ) cos(3ky + θy2 ) + cos(3kx + θx3 ) cos(2ky + θy3 )ik = 2π/W , θxj and θyj : independent random walks with δθ = 0.02 for a time step of 2−9 .

9

10-1

stirring
diffusion

10-2

variance

10-3

10-4

10-5

10-6

10-7
0

Graphics:

stirring

50

100t

150

200

stirring plus social

All together now...
Central point1) Weak convergence and swirl
φ = −0.1ψ

,11
ψ = Uv Lv exp(− [x/Lv ]2 − [y/Lv ]2 )222) Social groupingφ ← φ + φb,

βφb = −U a1+β,β=ZZdx′ w(x − x′ )b(x′ )3) Stirringψ ← −Uf y + ψstirGraphics:

centerphysical,

socialφ ← φ − 0.1ψstirmax b

Front
1) Frontal flow and convergencev = V0 sech2 (x/Lv ) ,3) Stirring only in ψ, not φ.
Graphics: front social

φ = −0.1V0 Lv sech2 (x/Lv )fraction in front

10Summary• Physical convergence is probably not strong enough, extensive enough, steady enoughto enhance the concentration that dramatically• Social behavior can produce large densities but on small scales; however some formsof behavior allow patches to merge into larger, stable aggregations.
• Stirring feeds small patches into the region of convergence where they can merge withthe larger aggregations and, to some degree, resist detrainment• The resulting dense aggregations are large compared with the se4nsing scale but small
compared to the convergent regions (part of making b much larger than physics alone).
• In the ocean, this is probably involves a whole chain of merging processes, from individuals to social patches, then groups combining to larger groups, ... while theconvergence acts overall.

11References[1] Manning, J.P., McGillicuddy, D.J., Pettigrew, N.R., Churchill, J.H. and Incze, L.S., 2009: Drifterobservations of the gulf of maine coastal current, Cont. Shelf Res., 29, 835-845. (p1, Manning, et al,2009)[2] Tarrant, A.M., Baumgartner, M.F., Verslycke, T., Johnson, C.L., 2008: Diﬀerential gene expression
in diapausing and active Calanus finmarchicus (Copepoda), Mar. Ecol. Prog. Ser., 355, 193-207.
(p2, Tarrant et al., 2008)[3] Durbin, E.G., Runge, J.A., Campbell, R.G., Garrahan, P.R., Casas, M.C., Plourde, S., 1997: Latefall-early winter recruitment of Calanus finmarchicus on Georges Bank, Mar. Ecol. Prog. Ser., 151,103-114. (p2, Durbin et al., 1997)[4] Kane, J., 1984: The feeding habits of co-occuring cod and haddock larvae from Georges Bank, Mar.
Ecol. Prog. Ser., 16, 9-20. (p2, Kane, 1984)[5] Wishner, K.F., Schoenherr, J.R., Beardsley, R. and Chen, C., 1995: Abundance, distribution andpopulation structure of the copepod Calanus finmarchicus in a springtime right whale feeding areain the southwestern Gulf of Maine, Cont. Shelf Res., 15, 475-507. (p2, Wishner et al., 1995)[6] Buchanan, C., Goldberg, B. and McCartney, R., 1982: A laboratory method for studying zooplankton swimming behaviors, Hydrobiologia, 94, 77-89. (p2, Buchanan et al., 1982)[7] Epstein, A.W. and Beardsley, R.C., 2001: Flow-induced aggregation of plankton at a front: a 2-deulerian model study, Deep Sea Res. II, 48, 395-418. (p2, Epstein and Beardsley, 2001)[8] Genin, A., Jake, J.S., Reef, R., Ricther, C. and Franks, P.J.S., 2005: Swimming against the flow: A
mechanism of zooplankton aggregation, Science, 308, 860-862. (p2, Genin et al., 2005)[9] Lenz, P., Hower, A. and Hartline, D., 2004: Force production during pereiopod power strokes incalanus finmarchicus, J. Mar. Syst., 49, 133-144. (p2, Lenz et al., 2004)[10] Beardsley, R.C., Epstein, A.W., Chen, C., Wishner, K.F., Macaulay, M.C., Kenney, R.D., 1996:
Spatial variability in zooplankton abundance near feeding right whales in the Great South Channel,Deep-Sea Res. II, 43, 1601-1625. (p2, Beardsley et al., 1996)[11] Baumgartner, M.F., and Mate, B.R., 2003: Summertime foraging ecology of North Atlantic rightwhales, Mar. Ecol. Prog. Ser., 264, 123-135. (p2, Baumgartner and Mate, 2003)[12] Epstein, A.W., 1995: Physical processes and zooplankton distribution in the great south channel: Observational and numerical studies, Ph.D. Thesis, MIT/WHOI Joint Program in Physical
Oceanography. (p2, Epstein, 1995)