Mixing and Turbulent Diffusion in the Turbulent Diffusion in the Ocean. f Mixing in Water Mass...

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Transcript of Mixing and Turbulent Diffusion in the Turbulent Diffusion in the Ocean. f Mixing in Water Mass...

  • 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 ∞

    −∞

    = −

    =∫

  • 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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