2D Inversion
We begin with the “primitive equations” which make only
the hydrostatic, beta-plane assumptions. We write them in a
coordinate system (x, y, ξ) where ξ is a function of pressure.
From 12.802 “coordinates” notes, we can write the mo­
mentum, mass, and thermodynamic equations as
D
u + f k̂ × u = −∇ϕ
Dt
(p.1)
∂
ρc
ϕ=g
ρ
∂ξ
(p.2)
∇·u+
1 ∂
(ρc ω) = 0
ρc ∂ξ
∂
∂
η + u · ∇η + ω η = 0 ∂t
∂ξ
(p.3)
(p.4)
where ϕ is the geopotential height of a surface of constant ξ
(coincident with a surface of constant pressure). The entropy is
η, and ρc is the density associated with the coordinate change
∂p
= −gρc
∂ξ
The PV is
q =
1
(∇3 × u + f k̂) · ∇3 η
ρc
1
(pv)
Ocean form:
In the Boussinesq approximation, ρ = ρ0 (1 + σ), ρc = ρ0
and we simply use η ∝ −σ. Then
∂
φ = −gσ
∂ξ
(oc − hyd)
and
q = −(∇3 × u + f k̂) · ∇3 σ
(oc − pv)
Atmospheric form:
Here we choose ξ so that ρc /ρ = θ/θ0 . Using the relation­
ship between θ, ρ, p
θ
=
θ0
�
ρ
ρ0
�−1 �
�
we have
ξ = Hs
Then
γ−1
1−
γ
�
p
p0
p
p0
�1/γ
q
�(γ−1)/γ �
∂
θ
φ=g
∂ξ
θ0
q=
1
(∇3 × u + f k̂) · ∇3 θ
θ0
2
(atm − hyd)
(atm − pv)
2D forms
Now we assume the fields are independent of x. Then
q = (f −
∂u ∂b
∂u ∂b
)
+
∂y ∂z
∂z ∂y
(where we’ve used the buoyancy b for either −σ or θ/θ0 and z
�
for ξ). Using the geostrophic equation f u = − ∂φ
∂y and hydro­
�z 2
�
static equation b = ∂φ
,
where
b
=
N + b , gives
∂z
∂ 2 φ
N 2 ∂ 2 φ
1 ∂ 2 φ ∂ 2 φ
1
2
q = fN + f
+
+
−
∂z 2
f ∂y 2
f ∂y 2 ∂z 2
f
�
∂ 2 φ
∂y∂z
�2
This is the form we wish to invert for φ and therefore u and
b .
As a basically elliptic equation, we need to specify condi­
tions on the four boundaries of the domain. On the top and
�
bottom, we specify b = ∂φ
∂z ; on the left and right, we have a
number of choices – and they do matter.
3
MIT OpenCourseWare
http://ocw.mit.edu
12.804 Large-scale Flow Dynamics Lab
Fall 2009
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