Mixing and Turbulent Diffusion in the Ocean

ρHeat SaltNutrientsOxygenNitrogenPlanktonTrace MetalsMany 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 upinto 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 dtτρ τρ τ τ ρ τ

=

= ⇒

< >= < >= −

=< > − =< > = < > −

∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫2 2 2

2 2

se I: ( ') ( ')

Case II: ( ') 2 ( ') 2u

I I

x u tx 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

Startt=0

End t

Turbulent Diffusion as a Random Walk Process

One dimensional Case

1 2 3 i2

2 21 2 3

2 2 2 21 2 3

2 22 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 lx x x i j x x l

x x x x xx x x x

l lx nl n t tt t

lt

κ

= + + + = ±

= > = ≠ >=

⇒ > = < + + + >

= < > + < > + < > + < >

>= = ∆ =∆ ∆

∆2 2nce < =2 ( ') Ix u tτ> < > ⇒

l

2 2< 2 where ( '):

I

I

x t uNote t

κ κ ττ

>= =< >= ∆

“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 ψσπσ

ψ

σ ψ

ψ

∞

∞

∞

∞

∞

∞

−= = = −

=< >=

=< − >= −

= =

∫

∫

∫

2 2 tσ κ=

x σ+ 2x σ+x

1 σ 68%

2 σ 95%

2

2 0t xρ ρκ∂ ∂

− =∂ ∂

1D Diffusion Equation2

Solution( )exp( )44

( )

M xktkt

x dx M

ρπ

ρ∞

−∞

= −

=∫

2σ

2 2

2 2

2 2

( ) 0

exp( )4 4

t x z

M x ykt kt

ρ ρ ρκ

ρπ

∂ ∂ ∂− + =

∂ ∂ ∂

+= −

2D Isotropic Diffusion Equation

Solution

2 2 2

2 2 2

2 2 2

32

( ) 0

exp( )4[4 ]

t x y z

M x y zktkt

ρ ρ ρ ρκ

ρπ

∂ ∂ ∂ ∂− + + =

∂ ∂ ∂ ∂

+ += −

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 diffusivityhorizontal(x,y) diffusivity

1 exp { }4 4(4 )

z h

z

h

h zh z

D

t z x z

x y zk t k t

ψ ψ ψ ψκ κ

κκ

ψπ 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

DiffusionStretching RotatingCompression

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

UU-αz

t =timeL tzα=

zk2

2

1[1 ( ) ]12

1[1 ( ) ] 12

h z

h z

t

k t k

σ α σ

α

2 2= +

⇒ = +

tzα

L stretchingdistance

Shear Dispersion