Mixing and Turbulent Diffusion in the Turbulent Diffusion in the Ocean. f Mixing in Water Mass...
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Mixing and Turbulent Diffusion in the Ocean
ρ Heat Salt Nutrients Oxygen Nitrogen Plankton Trace Metals Many others
Mixed Layer
Pycnocline
Bottom Mixed Layer
What gets mixed?
How and why do they get mixed? Why is it important?
Upper Ocean Turbulent Mixing
T-REMUS observations during the LOCO 2006 experiment of Beta 700/Chlorophyll_a, a surrogate for bottom particulates. Figure shows bottom particles being swept up into the bottom mixed layer
2 D Ocean Turbulence
Sea surface chlorophyll distribution derived from sea surface color in the western Sargasso Sea on May 27, 2007
0
2
0 0 0 0 "
2 2 2
0 0 0 0 0
1 D Problem Let X =0 & 0
' '( , ') '
( ') " ' '( , ") '( , ') " ' ( " ')
( ') { " ' ( " ')} ( ') {2 " ( )} 2 ( ') (1 ) ( )
Ca
t
t t t t
t t t t t
u
x u X t dt
x dt dt u X t u X t dt dt R t t
u dt dt t t u dt d t u d t τρ τρ τ τ ρ τ
=
= ⇒
< >= < >= −
=< > − =< > = < > −
∫
∫ ∫ ∫ ∫
∫ ∫ ∫ ∫ ∫ 2 2 2
2 2
se I: ( ') ( ')
Case II: ( ') 2 ( ') 2 u
I I
x u t x u t t
τ τ
τ τ τ κ
=< >
>> < >= < > =
Lagrangian Approach to Turbulent Diffusion
Example: Ocean Subsurface floats, Small particles in lab
X
2kt
t=0 u
0
( , ') ' tdxu x X u X t dt
dt = ⇒ = + ∫
κ Diffusivity
Start t=0
End t
Turbulent Diffusion as a Random Walk Process
One dimensional Case
1 2 3 i 2
2 2 1 2 3
2 2 2 2 1 2 3
2 2 2 2
2
... where
0 & { < 0 for and < }
< ( ... )
( ) ( ) ( ) ... ( )
Thus < { } { }
Let { }=2 Si
n
i i j i i
n
n
x x x x x x l x x x i j x x l
x x x x x x x x x
l lx nl n t t t t
l t
κ
= + + + = ±
= > = ≠ >=
⇒ > = < + + + >
= < > + < > + < > + < >
>= = ∆ = ∆ ∆
∆ 2 2nce < =2 ( ') Ix u tτ> < > ⇒
l
2 2< 2 where ( ') :
I
I
x t u Note t
κ κ τ τ
>= =< > = ∆
“x” has Gaussian Statistics
Ψ
2
22
2 2 2
1 ( )Probability Density Function exp( ) 22
( ) ( )
Note 1
x x
x x dx x
x x dx x x
dx
ψ ψ σπσ
ψ
σ ψ
ψ
∞
∞
∞
∞
∞
∞
− = = = −
=< >=
=< − >= −
= =
∫
∫
∫
2 2 tσ κ=
x σ+ 2x σ+x
1 σ 68%
2 σ 95%
2
2 0t x ρ ρκ∂ ∂− =
∂ ∂
1D Diffusion Equation 2
Solution ( )exp( ) 44
( )
M x ktkt
x dx M
ρ π
ρ ∞
−∞
= −
=∫
2σ
2 2
2 2
2 2
( ) 0
exp( ) 4 4
t x z
M x y kt kt
ρ ρ ρκ
ρ π
∂ ∂ ∂ − + =
∂ ∂ ∂
+ = −
2D Isotropic Diffusion Equation
Solution
2 2 2
2 2 2
2 2 2
3 2
( ) 0
exp( ) 4[4 ]
t x y z
M x y z ktkt
ρ ρ ρ ρκ
ρ π
∂ ∂ ∂ ∂ − + + =
∂ ∂ ∂ ∂
+ + = −
3D Isotropic Diffusion Equation
Solution
Green chlorophyll; Blue turbulent dissipation rate
3 D Ocean Turbulence
z2 2 2z tσ κ=
h2 2 2h tσ κ=
2 2 2
2 2 2
2 2 2
3 4 2
3 Anisotrpic Diffusion Equation
( ) 0
vertical diffusivity horizontal(x,y) diffusivity
1 exp { } 4 4(4 )
z h
z
h
h zh z
D
t z x z
x y z k t k t
ψ ψ ψ ψκ κ
κ κ
ψ π σ σ
∂ ∂ ∂ ∂ − − + =
∂ ∂ ∂ ∂ = =
+ = − +
Turbulent Diffusion in the Ocean
f
Mixing in Water Mass Formation and Transport
Water is cooled by air sea exchange
Dense water sinks and flows toward open ocean
Actual path near Antarctica
Evolution of a patch of particles (plankton, detritus,others) In an evolving turbulent filed
Diffusion Stretching Rotating Compression
Density surface
Homework Question: Suppose particles of various sizes on the sea floor are swept up by the bottom flow into the turbulent bottom boundary layer. What size particles do you expect to remain the longest in the turbulent field and undergo diffusion? Explain.
Role of Shear in Stretching and Dispersion Current Shear
U U-αz
t =time L tzα=
zk 2
2
1[1 ( ) ] 12
1[1 ( ) ] 12
h z
h z
t
k t k
σ α σ
α
2 2= +
⇒ = +
tzα
L stretching distance
Shear Dispersion
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