Transcript Visser 3 Turbulence - Universitetet i Bergen

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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
Cascade of inertia (mechanical energy)
Mechanical
Energy In
Re 
UL

small scale
Re  1
Re  1
large scale
Turbulence
Heat
Mechanical
Energy Out
Kolmogorov spectra theory
Energy cascade
Conserve angular
momentum (w) and kinetic
energy (1/2 u2)
0 =L
Andrei
Kolmogorov
1=L/2
.......
T 
TL  L / uL

u
3
 u  

1/ 3
2
/
3 1/ 4




   
1/ 3
  
 
2
9
u  C 
5
E(k)
energy density spectrum, E(k)
(L3/T2)
Kolmogorov spectra theory

1
3
 2/3
2/3
k5/3
viscous
sub-range
2p/L
2p/
wave number, k (2p/ℓ)
inertial
sub-range
k
Governed by 2 parameters
viscosity 
dissipation rate 
Kolmogorov spectra measured in nature
k
m2/s3 = W/kg
Turbulent
dissipation
rate is
becoming a
routine
physical
measurement
Microstructure
Shear
Probe
Sinks freely through the water column
Measuring turbulence in nature
Measuring turbulence in nature
Dissiption rate
varies vertically
Yamazaki et al 2002 The Sea v 12
Dissiption rate
varies in time
105
10-3
104
10-4
103
10-5
102
10-6
101
10-7
100
10-8
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18
20
22
dissipation rate (m 2/s3)
(wind speed) 3 (m3/s3)
Measuring turbulence in nature
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October 1998
Visser et al, Mar Biol 2001
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
10-6
to
10-4
m2/s3
surface
>10-3 m2/s3 tidal currents
damping
in
thermocline
internal waves
units: m3/s3 = W/kg = 104 cm2/s3
bottom
friction
Modelling turbulence in nature
turbulence closure schemes
Tidal currents
Oliver Ross, Thesis, SOC 2002
Turbulent dispersion
How 2 particles move relative to each other
x
could be molecules
could be organisms
scale dependent
what are the statistics of the variance of the
interparticle separation xx
For a diffusive process
x2 = 2 D t
Turbulent dispersion
log10 Scale (m)
-4
-2
0
2
Molecular
diffusion
Turbulent
straining
Richardson’s
law
ξ2  t
ξ 2  et
ξ2  t3
Batchelor scale
Kolmogorov scale
phytoplankton
hetertrophic protists
adult copeods
larval fish
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6
horizontal
-6
vertical
-8
Turbulent
eddy
diffusion
ξ2  t
Integral length scale
Turbulent dispersion: Richardsons law (inertial subrange)
x1
ℓ
xn
x0
xN
ℓ4/3
D (cm2/s)
x2
Scale dependent
Diffusivity =
the time rate of change of x2
10m
1km
100km
Relative motion and turbulence
Turbulence increases the relative motion of particles
Richardson's law for scales within the inertial subrange
w(x) = a ( x)1/3
also for scales within the viscous subrange
w(x) = g x = ( / )1/2 x
Relative motion and turbulence
The stucture function
Velocity difference (arbitrary scale)
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Viscosity
dominates
3  0.25

   
 
velocity difference ~ x
1
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)
Encounter rate and turbulence (1) The Up Side
perception distance
Z=Cb=pC
Rothschild & Osborn, J Plankton Res 1988
R2
(u2
+
v2
+
2w2)1/2
Evans, J Plankton Res 1989
prey
predator
u
R
b
is the encounter kernel
≈ maximum clearance rate
w
v
turbulent velocity scale
w = a ( R)1/3
Visser & MacKenzie, J Plankton Res 1998
Encounter rate and turbulence (1) The Up Side
Encounter rate
increase due to
turbulence
component due
solely to behaviour
turbulent dissipation rate
Encounter rate and turbulence (2) Ingestion rate
Encounter rate is not the same as ingestion rate
Ingestion rate
Functional response
Z
bC
I

1  Z 1  bC
-1
concentration
increases
turbulent dissipation rate

is handling time
Encounter rate and turbulence from the lab
Acartia tonsa feeding on ciliates
Clearance rate, cm3 / day
1000
800
Observed
Predicted
what happens here ?
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400
200
0
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
Dissipation rate, cm2s-1
101
102
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
Encounter rate and turbulence from the lab
Saiz, Calbet & Broglio Limnol Oceanogr 2003
Filteri
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Encounter rate and turbulence from the field
0
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17
Some species appear to
be impeded by turbulence
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Residual filtering index
Calanus
finmarchicus
October
1998
f = gut content/ambient chl
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Filtering index, f
18
15
10
5
0
-5
-7
10
-6
-5
-4
-3
log10()
5
0
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October 1998
Visser, Saito, Saiz & Kiørboe, Mar Biol (2001)
depth of
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Encounter rate and turbulence from the field
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20
18
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October 1998
Oithona similis
Some species appear to
migrate vertically to
mitigate the effects of
strong turbulence
depth of center of mass (m)
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-7
-6
-5
-4
-3
dissipation rate
log10() (m2/s3)
Visser, Saito, Saiz & Kiørboe, Mar Biol (2001)
Encounter rate and turbulence: Factors effecting detection
a
u
w
Reaction (detection) distance is a function of:
• Predator size b and sensitivity s
v
• Prey size a, velocity u and mode of motion
• Turbulence  vrs signal strength
Encounter rate and turbulence: Signal to noise
radius
a
velocity
U
Self-propelled body at
low Reynolds number
u(r) = U(a/r)2
Reaction (detection)
distance in still water
R0  a(U/s)1/2
r
2b
Visser, Mar Ecol Prog Ser 2001
Signal to noise ratio
Reaction (detection) distance
in turbulent waters
u (r )  s

*
w (b,  )
1/ 2


U

R ( )  a *
 w (b,  )  s 
  1/ 6
Laboratory study of Acartia tonsa feeding on ciliate
Strombidium sulcatum under turbulent conditions
-0.9
-1.3
Coefficients:
intercept = 1.343
slope = 0.167
r² = 0.965
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-1.4
6
-1.0
Log10(R)
-1.1
-1.2

-1.5
agitation rate
-1.6
-1.7
-3
-2
-1
0
Log10()
observed clearence rate bo
and solving
bo = p R2 (v2 + 2 a2 ( R)2/3)1/2
1
2
Detection distance
dependence on
turbulent dissipation
rate
R   1/6
Saiz E, Kiørboe T, 1995. Mar Ecol Prog Ser
Encounter rate and turbulence: Dome - shape
Ingestion rate
Increased
ingestion rate
due to more
encounters
Decreased
ingestion rate
due to impaired
detection –
caputre efficiency
turbulence
Behavioural shifts
Dome – shaped
response
Active avoidance of high turbulence zones
Change of feeding mode with turbulence
Interaction specific
Modelling turbulent diffusion: random walk
zn+1 = r (2 d D)1/2
how much light a
phytoplankton cell receives
depth
zn
r is a random number such that
mean(r) = 0
variance(r) = 1
d is the time step between evaluations
D is the diffusivity
Modelling turbulent diffusion: random walk
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Depth(m)
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Time(hours)
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1000
Modelling turbulent diffusion: random walk
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-5
Depth(m)
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100
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Time(hours)
700
800
900
1000
Modelling turbulent diffusion: what can go wrong
diffusivity
distribution
vertical random walk
depth
z (t  d)  z (t )


 r 2dD  z  t  
1/ 2
distribution predicted by
C
  2  DC 
t
Unmixes an initially uniform distribution
Visser 1997
Modelling turbulent diffusion: corrected for accumulation
diffusivity
distribution
vertical random walk
z (t  d)  z (t )


depth
 r 2dD  z  t  
1/ 2
D
d
z
vertically uniform
distribution as predicted
by diffusion equn.
C
    DC 
t
Visser 1997
Turbulence and distribution patterns
A blob of ink in a stirred fluid
time
Length of filament ~ exp(g t)
Variance2 ~ t to t3
Turbulence and distribution patterns
Distribution of solutes
Plankton distribution
100’s km
Diffusion is useful in describing the probability of a distribution
BUT
Any given distribution does not look diffusive
Photo: Alice Alldredge
100’s µm
Cascade and dissipation of variance
For a passive tracer
Cascade of
variance
Folding and
stretching
Diffusion:
dissipation of
variance
Passive tracer: molecular diffusion
Biologically active tracer: mortality & motility
Diffusion vrs stirring
Patchiness and growth
Advection-diffusion-reaction
C
    DC    b    C
t
reproduction
b=
mortality
C
    DC    b    C
t
 C(x,y,t)  uniform
Pair correlation by birth and death
Young et al 2001
Patchiness and growth
0.10
final: rmean = 0.0058
frequency
0.08
0.06
initial: rmean = 0.0112
0.04
poisson: rmean = (4 C)-1/2 = 0.0112
0.02
0.00
0.00
0.01
0.02
0.03
0.04
nearest neighbour separation
0.05
Patchiness and functional group
Large scale
gradients
“dissipation”
variance
k-5/3
Passive tracer
Phytoplankton
Zooplankton
length scale
Motility: swimming vrs turbulence
Memory: growth rate vrs turbulence
Increasing
small scale
variance
(patchiness)
Turbulence and swimming
Strong swimmers can remain in patches
in the face of increasing turbulence.
Maar et al 2003, L & O
Swimming ability
Weak swimmers become more
dispersed as turbulence increases
Turbulence, population dynamics + patchiness
P
Chaotically stirred ocean
Z
Simple Nutrient Phytoplankton Zooplankton model
 Complex spatial patterns
Nutrients
Phytoplankton
N(background)
N
Zooplankton
Abraham, Nature 1998
Turbulence, population dynamics + patchiness
variance
Large scale
gradients
close together "now"
k-5/3
length scale
large separation
Slow process → high variance
Memory
”inertia”
Fast process → low variance
Abraham, Nature 1998
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.