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Derivation of the Quasigeostrophic Model

A geostrophic meteorological model is one in which the horizontal wind vector in the equations defining the model is replaced everywhere by its geostrophic approximation. In a quasigeostrophic model the horizontal wind vector in some instances is replaced by the geostrophic approximation but not in others. In the standard quasigeostrophic model it is also assumed that the Coriolis parameter f is constant except where its derivative ∂f/∂y=β is taken. (Justification 1) This is known as the beta-plane approximation.

In the following ∇ represents the horizontal gradient operator. Symbols in red represent vectors and symbols i,j and k represent unit vectors.

The assumption of constant Coriolis parameter f means that the geostrophic wind approximation used in the model is

Vg = (1/f0)k×∇φ

where f0 is the value of the Coriolis parameter at midlatitude of the study area. Thus the geostrophic wind used in the quasigeostrophic model is an approximation to the exact geostrophic wind. (Since the geostrophic wind is used as an approximation of the true wind, the quasigeostrophic wind may or may not be a worse approximation of the true wind than is the exact geostrophic wind.)

The Eulerian acceleration operator is

(D/Dt) = (∂/∂t) + V·(∇ )

but in the quasigeostrophic model this is replaced by

Dg/Dt = ∂/∂t + Vg·∇
In particular
DgVg/Dt = ∂Vg/∂t + Vg·∇Vg

The Momentum Equations

The horizontal momentum vector equation is:

DV/Dt = fk×V + ∇φ

The RHS of the above equation is

fk×V + ∇φ = (f0+βy)k×(Vg+Va)+∇φ

But ∇φ= -f0k×Vg so the above reduces upon expansion to

DV/Dt = f0k·Va + βy(k×Vg) + βy(k×Va).

The last term in the above, βy(k×Va), is of a low order of magnitude than the other terms and may be neglected. Justification 2

When the Eulerian acceleration DV/Dt is replaced by the geostrophic acceleration DgVg/dt the momentum equation is then

DgVg/dt = f0k·Va + βy(k×Vg)

The Continuity Equation

In pressure coordinates the continuity equation is

∇·V + ∂ω/∂p = 0,
or, in terms of the components of V
∇·Vg + ∇·Va + ∂ω/∂p = 0

But ∇·Vg = 0. ( Justification 3) This means that the continuity equation for the quasigeostrophic model reduces to

∇·Va + ∂ω/∂p = 0.


The Thermodynamic Equation

The general thermodynamic equation is

∂T/∂t +V·∇T - Spω = J/cp

where Sp is the stability parameter and is equal to -T(∂ln(θ)/∂p).

The temperature field T is decomposed into two components: the barotropic field T0(p) which is a function of pressure alone and the deviational field Td(x,y,p,t). The magnitude of the gradient of Td with respect to pressure is much smaller than the magnitude of the gradient of T0 with respect to pressure. (Justification 3)

The stability parameter Sp may be expressed as

Sp = -T(∂ln(θ)/∂p) = (-R(T)/p)(p/R)(∂ln(θ)/∂p)
or
= [(-RT/p)(∂ln(θ)/∂p)](p/R) = σp/R

where σ = -(RT/p)(∂ln(θ)/∂p).

The potential temperature θ is also decomposed into its barotropic and deviational components and, because of the assumption concerning the gradients,

σ = -(RT0/p)(∂ln(θ0)/∂p)

Since ∇T0=0, the thermodynamic equation for the quasigeostrophic model reduces to

DgTd/Dt - (σp/R)ω = J/cp



Justification 1: The justification for treating the Coriolis parameter f as a constant

On a sphere f=2Ωsin(φ) so

β=∂f/∂y = (∂f/∂φ)/(∂y/∂φ)
= 2Ωcos(φ)/a

where a is the radius of the Earth. Thus

βΔy/f = (2Ωcos(φ)(L/a)(2Ωsin(φ))
= cot(φ)(L/a)

where L is the scale of a meteorological disturbance.

Since cot(φ) is on the order of unity, for the midlatitudes this means that the deviation of f from a constant is on the order of (L/a), which is a lower order of magnitude than unity. Hence the Coriolis parameter can be reasonably approximated at the midlatitudes as a constant.



Justification 2

The ratio of the third term to the first term is equal to (βy/f0) which from the previous justification is a lower order of magnitude than unity.

The ratio of the third term to the second term is equal to |Va|/|Vg|, which is a lower order of magnitude than unity.


Justification 3

The quasigeostrophic wind vector Vg is (1/f0)k×∇φ. A straight forward computation gives

Vg = (1/f0)[-(∂φ/∂/y)i + (∂φ/∂/x)j]

The divergence of any vector V=ui+vj is (∂u/∂x)+(∂v/∂y), therefore

∇·Vg = (1/f0)[-(∂2φ/∂x∂/y)+(∂2φ/∂y∂/x)] = 0

because of the equality of the cross derivatives.

An alternate version of the above justification starts with the generic formula for the divergence of a vector cross product:

∇·(A×B) = B·(∇×A) - A·(∇×B)

For

Vg = (1/f0)k×∇φ
this means
∇·Vg = (1/f0)[∇Φ·(∇×k) - ∇Φ·(∇×∇Φ)]

The term in the bracket, ∇×∇Φ, is alway the zero vector because the curl of a gradient always vanishes. The first term in the bracket is also the zero vector if the local vertical unit vector k is treated as a constant vector. In spherical coordinates this vector is not constant in direction although in rectangular coordinates it is.

Thus

∇·Vg = 0.



Justification 4

The assumption that |dT0/dp| >> |dTd/dp| is based upon the notion that T0 includes all of the sytematic dependence of temperature upon pressure and thus Td cannot contain any systematic pressure dependence.


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