Quasi geostrophic dynamics near the equator
V. Resmi
Outline
Effect of Earth’s Rotation
Shallow Water Equations
Geostrophic balance
Quasi-geostrophic dynamics
Coriolis force
~f = −2Ω
~ × ~u
Resolve to components at latitude φ:
(0, f cos(φ), f sin(φ))
φ - latitude.
(1)
For small excursions around φ0
Taylor expand f around φ0
For example, the vertical component,
fz = f sin(φ0
For small deviations,
fz
= 2Ω sin (φ0 ) + 2Ω cos(φ − φ0 )(φ − φ0 ) + O (φ − φ0 )2
y
= 2Ω sin (φ0 ) + 2Ω cos(φ − φ0 ) + O (φ − φ0 )2
R
Approximations to f
f-plane
f = 2Ω sin(φ0 ) = f0
I
tangent plane at φ0 .
I
not valid for large length scales in meridianal plane.
I
not valid near the equator.
β-plane
f
= 2Ω sin(φ0 ) + 2Ω cos(φ0 )
= f0 + βy .
y
R
Shallow Water Equation with rotation
linearlized around (0, 0, H0 )
∂u
∂η
− fv = −g
∂t
∂x
∂η
∂v
+ fu = −g
∂t
∂y
∂η
∂u ∂v
+ H0
+
= 0.
∂t
∂x
∂y
Geostrophic balance
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No acceleration.
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Pressure gradient = Coriolis force.
g ∂η
f ∂y
g ∂η
f ∂x
u = −
v =
∂u ∂v
+
= 0.
∂x
∂y
Plane wave solutions in f-plane, f = f0
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geostrophic mode,
σ=0
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external inertial-gravity waves,
q
σ = ± f02 + kx2 + ky2 gH0
Quasi-geostrophic dynamics - evolve Geostrophic mode
Assumptions
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Rotation is significant
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Motion is slow compared to the wavespeed
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height perturbations are small compared to the mean height
Steps:Non-dimensionalization.
∂u
∂η
− f0 v = g .
∂t
∂x
u → U ũ
η → N η̃
x
→ Lx̃
Steps: Identify small, non-dimensional parameters
∂u
1
−Γ ∂η
−
v= 2
,
∂t
Ro
Fr ∂x
where
Ro =
Fr
=
Γ =
u
f0 L
u
c
N
,
H0
Steps:What are the strengths of the parameters?
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Rotation is significant, Ro < 1.
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Motion is slow compared to the wavespeed, Fr < 1
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height perturbations are small compared to the mean height,
Γ<1
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the easiest choice, Ro = Fr = Γ = < 1.
Steps:Rewrite in terms of small parameter, ∂u v
−1 ∂η
−
=
∂t
∂x
∂v
u
−1 ∂η
+
=
∂t
∂y
1
∂u ∂v
∂η
+ 1+
+
= 0.
∂t
∂x
∂y
Steps:Expand in terms of u = u (0) + u (1) + 2 u (2) + . . .
v = v (0) + v (1) + 2 v (2) + . . .
η = η (0) + η (1) + 2 η (2) + . . .
Balancing terms at O
1
∂eta(0)
∂x
∂eta(0)
= −
∂y
v (0) =
u (0)
∂u (0) ∂v (0)
+
∂x
∂y
geostrophic balance
= 0.
Balancing terms at O (1)
D ∆η (0) − η (0)
= 0.
Dt
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evolution of the geostrophic mode, σ = 0
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inertial-gravity waves are filtered out.
In β-plane where |f0 | > β
Dg ∆ψ −
ψ
L2D
Dt
a material conservation law.
+ βψ
= 0.
Linearizing with respect to the base state of rest,
∂ ∆ψ − Lψ2
∂ψ
D
+β
= 0.
∂t
∂x
Plane wave solution is given by
ψ ∼ ψˆk e i(kx−σt) ,
σ=
Rossby wave
−βkx0
.
K 2 + 1/L2D
Quasi geostrophic theory near the equator?
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f-plane approximation is not valid
f0 = 2Ω sin(φ0 ) = 0.
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β-plane approximation is possible
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Problem: |βy | > |f0 |.
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Nature of solution is different from mid-latitude β-plane where
|βy | < |f0 |.
It is possible to write a relation
σ=0
q
σ = ± H0 g (kx2 + ky2 ) + β 2 y 2
What are the valid assumptions for a QG theory near the equator?