Mid-ocean Geostrophic Turbulence

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∂∂t∇ 2ψ +

∂ ψ ,∇ 2ψ( )

∂ x, y( )= 0

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⇒dE

dt=

dZ

dt= 0

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E =1

2dx dy∫∫ v ⋅v =

1

2dx dy∫∫ ∇ψ ⋅∇ψ = dk

0

∞

∫ E k( )

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Z =1

2dx dy∫∫ ζ 2 =

1

2dx dy∫∫ ∇ 2ψ( )

2= dk

0

∞

∫ k 2E k( )

prototype: two-dimensional turbulence

Continuously stratified case

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∂q

∂t+

∂ ψ ,q( )∂ x,y( )

= 0, q =∂ 2ψ

∂x 2+

∂ 2ψ

∂y 2+

f 2

N 2

∂ 2ψ

∂z2+ f

Energy transfer is to low

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ktotal ≡ kx2 + ky

2 + kn2

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kn =f

N

nπ

H=

1

n - th deformation radius

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WKB form

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∂q

∂t+

∂ ψ ,q( )∂ x,y( )

= 0, q =∂ 2ψ

∂x 2 +∂ 2ψ

∂y 2 +βy( )

2

N 2

∂ 2ψ

∂z2 + f

Energy transfer is to low

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ktotal ≡ kx2 + ky

2 + kn y( )2

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kn y( ) =βy

N

nπ

H=

1

n - th deformation radius

Energy transfer is toward the equator and into high vertical mode.

The energy in mode n shows an equatorial peak of width W,determined by

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ky = kn

€

1

Wn

=βWn

N

nπ

H⇒ Wn =

NH

β nπ≡ equatorial deformation radius

i.e.

“Global characteristics of ocean variability estimatedfrom regional TOPEX/POSEIDON altimeter measurements”

D. Stammer, J. Phys. Oc. 1997

Eddy kinetic energy at the sea surface (cm/sec)**2

U. Send, C. Eden & F. Schott“Atlantic equatorial deep jets…” J. Phys. Oc. 2002

Zonal flow (cm/sec) along the equator from six cruises.

Six-layer QG model

1982

kinetic energy in modes 0 to 5 as a function of y

Six-layer QG model

mode: 0 3 5

Two-layer QG flow

3 quadratic invariants: energy & potential enstrophy of each layer

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E, Z1 (top), Z2 (bottom)

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Typically only E and Z1 + Z2 are used.

This is not wrong, but it is incomplete. Not all the information is being used.

To see what phenomena are being missed, consider caseswith a very high degree of asymmetry between the layers.

Two-layer flow with the lower layer at rest

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∂∂t

q +∂ ψ ,q( )∂ x, y( )

+ β∂ψ

∂x= 0 , q =∇ 2ψ − kR

2ψ

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E = dk∫ E k( ), Z = dk∫ k 2 + kR2

( )E k( )

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Energy moves to lowest k 2 + kR2 ≡ keff

2

Work of Jurgen Theiss (“Rhines latitude”)

Problem with this model: the lower layer does not remain at rest.

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∂q1

∂t+ J ψ1,q1( ) + β

∂ψ1

∂x= S1,

Two-layer QG flow

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∂q2

∂t+ J ψ 2,q2( ) + β

∂ψ 2

∂x= S2,€

q1 =∇ 2ψ1 + F1 ψ 2 −ψ1( ),

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q2 =∇ 2ψ 2 + F2 ψ1 −ψ 2( ),€

F1 =f0

2

′ g H1

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F2 =f0

2

′ g H 2

top

bottom

2 sources of asymmetry:

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δ ≡H1

H1 + H2

<<1,

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S2 << S1

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Suppose β = S2 = 0. Then q2 ≡ 0 consistently.

Consistent equivalent barotropic dynamics

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∂q1

∂t+ J ψ1,q1( ) = S1

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∇2ψ 2 + F2 ψ1 −ψ 2( ) = 0 ⇒ k 2 + F2( )ψ 2 = F2ψ1

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q1 =∇ 2ψ1 + F1 ψ 2 −ψ1( ) ⇒ q1 = − k 2 + F1( )ψ1 + F1ψ 2

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⎫ ⎬ ⎭q1 = −

k 2 k 2 + F1 + F2( )

k 2 + F2( )ψ1 ≡ −keff

2ψ1

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E = dk∫ E k( )

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Z = dk∫ keff2E k( ) = dk∫

k 2 k 2 + kR2

( )

k 2 + δkR2

( )E k( )

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KE1 k( ) =k 2 + F2( )

k 2 + F1 + F2( )E k( ), KE2 k( ) =

F1F2

k 2 + F2( ) k 2 + F1 + F2( )E k( ),

APE k( ) =F1k

2

k 2 + F2( ) k 2 + F1 + F2( )E k( )

1976

Guidance from equilibrium stat-mech (equipartition theory)

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E k( ) =k

C1 + C2keff2

But beta doesn’t actually vanish…… or does it?

2-layer QG with a zonal mean flow

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∂q1′

∂t+ U1

∂q1′

∂x+ β1

∂ψ1′

∂x+ J ψ1

′,q1′ ⎛

⎝ ⎜ ⎞

⎠ ⎟= 0

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∂q2′

∂t+ U2

∂q2′

∂x+ β 2

∂ψ 2′

∂x+ J ψ 2

′,q2′ ⎛

⎝ ⎜ ⎞

⎠ ⎟= 0

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β1 = β + F1 U

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β2 = β − F2 U

Baroclinic instability requires opposite signs.

Therefore the threshold for instability is:

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β2 = 0 ⇒ U = β /F2 (atmospheric case) allows q2 ≡ 0

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β1 = 0 ⇒ U = −β /F1 (tropical ocean) allows q1 ≡ 0

A view of the ocean

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β2 = 0 (the atmospheric case) has been studied a lot

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β1 = 0 is not simply the flip of this, because H1 << H2

initial stream function

1000 sq km = 512 x 512, rdef = 40 km, depth ratio=10

tropical extra-tropical

upper layer

lower layer

urms = 10 km/day

The two cases of neutral stability

Stream function at 100 days

upper

lower

tropical extra-tropical

4.02 k/d

8.47 k/d

Rhines scale = 88 km 27 km

7.20 k/d

0.51 k/d

Potential vorticity at 100 days

upper

lower

tropical extratropical

Stream function at 150 days

upper

lower

tropical extra-tropical

6.42 k/d

1.18 k/d

Rhines scale = 77 km 25 km

6.52 k/d

0.49 k/d

Second experiment: lower layer initially at rest

Potential vorticity at 150 days

upper

lower

tropical extratropical