Turbulence and encounter

1. Physics of small organisms in fluids Turbulence and encounter

2. Turbulence: an intuitive understanding

3. Found in all fluids at a variety of scales Magnetically stirred fluid in the lab Soap bubble Coccolithophore bloom in the North Sea Circulation in the atmosphere of Jupiter

4. Cascade of inertia (mechanical energy) Mechanical Energy In small scale Turbulence large scale Heat Mechanical Energy Out

5. Andrei Kolmogorov Kolmogorov spectra theory Energy cascade Conserve angular momentum (w) and kinetic energy (1/2 u2) =L =L/2 .......

6. Kolmogorov spectra theory energy density spectrum, E(k) (L3/T2) Governed by 2 parameters viscosity n dissipation rate e wave number, k (2p/ℓ)

7. Kolmogorov spectra measured in nature kh

8. Measuring turbulence in nature e m2/s3 = W/kg Turbulent dissipation rate is becoming a routine physical measurement Sinks freely through the water column Microstructure Shear Probe

9. Measuring turbulence in nature Dissiption rate varies vertically Yamazaki et al 2002 The Sea v 12

10. Measuring turbulence in nature Dissiption rate varies in time Visser et al, Mar Biol 2001

11. Measuring turbulence in nature Vertical structure Typical values of dissipation rate wind surface waves <10-10 m2/s3 deep ocean 10-8 m2/s3 themocline damping in thermocline 10-6 to 10-4 m2/s3 surface >10-3 m2/s3 tidal currents internal waves units: m3/s3 = W/kg = 104 cm2/s3 bottom friction

12. Modelling turbulence in nature turbulence closure schemes e D Tidal currents Oliver Ross, Thesis, SOC 2002

13. Turbulent dispersion How 2 particles move relative to each other could be molecules could be organisms scale dependent x what are the statistics of the variance of the interparticle separation  For a diffusive process 2 = 2 D t

14. Turbulent dispersion log10 Scale (m) -8 -6 -4 -2 0 2 4 6 Turbulent eddy diffusion horizontal Molecular diffusion Turbulent straining Richardson’s law vertical Batchelor scale Kolmogorov scale Integral length scale phytoplankton hetertrophic protists adult copeods larval fish

15. x1 ℓ x2 ℓ4/3 xN x0 xn D (cm2/s) Scale dependent 10m 1km 100km Turbulent dispersion: Richardsons law (inertial subrange) Diffusivity = the time rate of change of x2

16. Relative motion and turbulence Turbulence increases the relative motion of particles Richardson's law for scales within the inertial subrange w(x) = a (ex)1/3 also for scales within the viscous subrange w(x) = gx = (e / n)1/2x

17. Relative motion and turbulence The stucture function 10 Viscosity dominates velocity difference ~ x 1 Velocity difference (arbitrary scale) Inertia dominates velocity difference ~ x1/3 In nature 1 to 0.1 cm Kolmogorov scale 0.1 0.1 1 10 100 Separation distance (units of Kolmogorov scale)

18. Encounter rate and turbulence (1) The Up Side perception distance Rothschild & Osborn, J Plankton Res 1988 Z = Cb = pCR2 (u2 + v2 + 2w2)1/2 Evans, J Plankton Res 1989 prey predator b is the encounter kernel ≈ maximum clearance rate u w R turbulent velocity scale v w = a (eR)1/3 Visser & MacKenzie, J Plankton Res 1998

19. Encounter rate and turbulence (1) The Up Side increase due to turbulence Encounter rate component due solely to behaviour turbulent dissipation rate

20. Encounter rate and turbulence (2) Ingestion rate Encounter rate is not the same as ingestion rate t is handling time Functional response t -1 Ingestion rate concentration increases turbulent dissipation rate

21. Acartia tonsa feeding on ciliates 1000 Observed 800 Predicted what happens here ? 600 Clearance rate, cm3 / day 400 200 0 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 2 -1 Dissipation rate, cm s Encounter rate and turbulence from the lab

22. Encounter rate and turbulence (3) The Down Side Turbulence interferes with the remote detection ability of organisms hydromechanical chemical Turbulence sweeps prey out of the detection zone before organísms can capture them Turbulence interferes with the structure and efficiency of feeding currents

23. Encounter rate and turbulence from the lab Saiz, Calbet & Broglio Limnol Oceanogr 2003

24. 20 15 Filtering index, f 10 5 0 16 17 18 19 20 21 22 23 24 25 October 1998 Encounter rate and turbulence from the field Some species appear to be impeded by turbulence Calanus finmarchicus f = gut content/ambient chl Visser, Saito, Saiz & Kiørboe, Mar Biol (2001)

25. Encounter rate and turbulence from the field Oithona similis Some species appear to migrate vertically to mitigate the effects of strong turbulence Visser, Saito, Saiz & Kiørboe, Mar Biol (2001)

26. Encounter rate and turbulence: Factors effecting detection a u • Reaction (detection) distance is a function of: • Predator size b and sensitivity s • Prey size a, velocity u and mode of motion • Turbulence e vrs signal strength w v

27. velocity U Self-propelled body at low Reynolds number u(r) = U(a/r)2 radius a r 2b Reaction (detection) distance in still water R0a(U/s)1/2 Visser, Mar Ecol Prog Ser 2001 Encounter rate and turbulence: Signal to noise Signal to noise ratio Reaction (detection) distance in turbulent waters

28. Coefficients: intercept = -1.343 slope = -0.167 r² = 0.965 -0.9 -1.0 -1.1 -1.2 Log10(R) -1.3 -1.4 agitation rate -1.5 Detection distance dependence on turbulent dissipation rate -1.6 -1.7 -3 -2 -1 0 1 2 Log10(e) observed clearence rate bo and solving bo = p R2 (v2 + 2 a2 (eR)2/3)1/2 R e-1/6 Laboratory study of Acartia tonsa feeding on ciliate Strombidium sulcatum under turbulent conditions Saiz E, Kiørboe T, 1995. Mar Ecol Prog Ser

29. Ingestion rate Interaction specific Behavioural shifts turbulence Active avoidance of high turbulence zones Change of feeding mode with turbulence Encounter rate and turbulence: Dome - shape Increased ingestion rate due to more encounters + Decreased ingestion rate due to impaired detection – caputre efficiency - Dome – shaped response =

30. Modelling turbulent diffusion: random walk zn+1 = r (2 dD)1/2 how much light a phytoplankton cell receives zn r is a random number such that mean(r) = 0 variance(r) = 1 depth d is the time step between evaluations D is the diffusivity

31. Modelling turbulent diffusion: random walk Depth(m) Time(hours)

32. Modelling turbulent diffusion: random walk Depth(m) Time(hours)

33. Modelling turbulent diffusion: what can go wrong diffusivity distribution vertical random walk depth distribution predicted by Unmixes an initially uniform distribution Visser 1997

34. Modelling turbulent diffusion: corrected for accumulation diffusivity distribution vertical random walk depth vertically uniform distribution as predicted by diffusion equn. Visser 1997

35. Turbulence and distribution patterns A blob of ink in a stirred fluid time Length of filament ~ exp(g t) Variance2 ~ t to t3

36. Turbulence and distribution patterns Distribution of solutes Plankton distribution 100’s km Photo: Alice Alldredge 100’s µm Diffusion is useful in describing the probability of a distribution BUT Any given distribution does not look diffusive

37. Cascade of variance Folding and stretching Diffusion: dissipation of variance Cascade and dissipation of variance For a passive tracer Passive tracer: molecular diffusion Biologically active tracer: mortality & motility

38. Diffusion vrs stirring

39. Patchiness and growth Advection-diffusion-reaction reproduction mortality b=m  C(x,y,t)  uniform Pair correlation by birth and death Young et al 2001

40. Patchiness and growth final: rmean = 0.0058 initial: rmean = 0.0112 poisson: rmean = (4 C)-1/2 = 0.0112

41. Large scale gradients “dissipation” k-5/3 Passive tracer variance Increasing small scale variance (patchiness) Phytoplankton Zooplankton length scale Motility: swimming vrs turbulence Memory: growth rate vrs turbulence Patchiness and functional group

42. Turbulence and swimming Weak swimmers become more dispersed as turbulence increases Swimming ability Strong swimmers can remain in patches in the face of increasing turbulence. Maar et al 2003, L & O

43. Turbulence, population dynamics + patchiness P Z Chaotically stirred ocean Simple Nutrient Phytoplankton Zooplankton model N N(background)  Complex spatial patterns Nutrients Phytoplankton Zooplankton Abraham, Nature 1998

44. Large scale gradients k-5/3 variance length scale Turbulence, population dynamics + patchiness close together "now" backwards in time large separation Memory ”inertia” Slow process → high variance Fast process → low variance Abraham, Nature 1998

45. Summary statements Turbulence is an important environmental variable effecting the interaction of plankton. There are both positive effects (encounter rate) and negative effects (sensory impairment) leading to a general dome-shaped response curve. Because turbulence varies greatly in the vertical direction, some plankton can mitigate the negative effects of turbulence by migrating downwards. Chaotic stirring together with population dynamics generate complex spatial structures.