Mid-ocean Geostrophic Turbulence. prototype: two-dimensional turbulence. Continuously stratified case. Energy transfer is to low. WKB form. Energy transfer is to low. Energy transfer is toward the equator and into high vertical mode. - PowerPoint PPT Presentation
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Mid-ocean Geostrophic Turbulence
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∂∂t∇ 2ψ +
∂ ψ ,∇ 2ψ( )
∂ x, y( )= 0
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⇒dE
dt=
dZ
dt= 0
€
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
€
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
€
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.