Turbulence: an intuitive understanding

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

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

Andrei Kolmogorov Energy cascade Conserve angular momentum (  ) and kinetic energy (1/2 u 2 ) =L =L/ Kolmogorov spectra theory

wave number, k (2  /ℓ) energy density spectrum, E(k) (L 3 /T 2 ) Governed by 2 parameters viscosity dissipation rate  Kolmogorov spectra theory

kk Kolmogorov spectra measured in nature

Turbulent dissipation rate is becoming a routine physical measurement m 2 /s 3 = W/kg Microstructure Shear Probe Sinks freely through the water column Measuring turbulence in nature

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

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

>10 -3 m 2 /s 3 tidal currents < m 2 /s 3 deep ocean m 2 /s 3 themocline to m 2 /s 3 surface Typical values of dissipation rate units: m 3 /s 3 = W/kg = 10 4 cm 2 /s 3 wind surface waves bottom friction internal waves damping in thermocline Vertical structure Measuring turbulence in nature

Oliver Ross, Thesis, SOC 2002 turbulence closure schemes Tidal currents Modelling turbulence in nature

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

log 10 Scale (m) Molecular diffusion vertical horizontal Turbulent eddy diffusion Richardson’s law Kolmogorov scaleIntegral length scaleBatchelor scale Turbulent straining phytoplankton hetertrophic protists adult copeods larval fish Turbulent dispersion

D (cm 2 /s) 10m1km100km ℓ 4/3 ℓ 11 22 nn NN x0x0 Diffusivity = the time rate of change of     Scale dependent Turbulent dispersion: Richardsons law (inertial subrange)

Relative motion and turbulence Turbulence increases the relative motion of particles Richardson's law for scales within the inertial subrange also for scales within the viscous subrange w(  ) =  (   ) 1/3 w(  ) =   = (  / ) 1/2 

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

Encounter rate and turbulence (1) The Up Side u v R w Rothschild & Osborn, J Plankton Res 1988 Evans, J Plankton Res 1989 Visser & MacKenzie, J Plankton Res 1998 w =  (  R) 1/3 Z = C  =  C R 2 (u 2 + v 2 + 2w 2 ) 1/2  is the encounter kernel ≈ maximum clearance rate turbulent velocity scale prey predator perception distance

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

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

Acartia tonsa feeding on ciliates Dissipation rate, cm 2 s Clearance rate, cm3 / day Observed Predicted what happens here ? Encounter rate and turbulence from the lab

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

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

Visser, Saito, Saiz & Kiørboe, Mar Biol (2001) Calanus finmarchicus Some species appear to be impeded by turbulence Encounter rate and turbulence from the field October Filtering index, f f = gut content/ambient chl

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

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

Signal to noise ratio Reaction (detection) distance in turbulent waters r velocity U radius a 2b2b Self-propelled body at low Reynolds number u(r) = U(a/r) 2 Reaction (detection) distance in still water Visser, Mar Ecol Prog Ser 2001 R 0  a(U/s) 1/2 Encounter rate and turbulence: Signal to noise

Log 10 (  ) Log 10 (R) Laboratory study of Acartia tonsa feeding on ciliate Strombidium sulcatum under turbulent conditions Saiz E, Kiørboe T, Mar Ecol Prog Ser observed clearence rate  o and solving  o =  R 2 (v  2 (  R) 2/3 ) 1/2 agitation rateCoefficients: intercept =  slope =  r² = R    1/6 Detection distance dependence on turbulent dissipation rate

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

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

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

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

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

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

time Length of filament ~ exp(  t) Variance 2 ~ t to t 3 Turbulence and distribution patterns A blob of ink in a stirred fluid

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

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

Diffusion vrs stirring

Patchiness and growth Pair correlation by birth and death Advection-diffusion-reaction Young et al 2001 reproductionmortality  =   =    C(x,y,t)  uniform

final: r mean = initial: r mean = poisson: r mean = (4 C) -1/2 = Patchiness and growth

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

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

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

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

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.