## Quasi Geostrophic Models

A model suited for prediction of synoptic-scale motions in middle and high latitudes is the quasi-geostrophic model which utilizes the isobaric vorticity equation (17.2.4) and the thermodynamic energy equation (12.2.8). In this model, the following simplifications and approximations are made in the vorticity equation (17.2.4):

(a) The vertical advection and the twisting terms (the 2nd and the 4th terms on the right-hand side of the equation) are neglected;

(b) The relative vorticity is neglected compared to the Coriolis parameter in the divergence term;

(c) The horizontal velocity is replaced by the geostrophic wind velocity in the horizontal advection term;

(d) The relative vorticity is replaced by the geostrophic vorticity; and

(e) The beta-plane approximation is applied to the Coriolis parameter.

The Coriolis parameter f about a given latitude is approximated by the P-plane approximation:

f = f0 + Py, where f0 is the Coriolis parameter at the given latitude, P = (df/dy) at is regarded as constant, and y is the distance from the given latitude

For synoptic-scale midlatitude disturbances, Py < f0 Hence, in the geostrophic model, f is replaced by f0. Applying all the above-mentioned approximations, the quasi-geostrophic vorticity equation reduces to dZg/dt = -Vg-V(Zg + f) - foV-V (17.3.1)

where Zg = V20/f0 and Vg = k x VO/f0, have both constant f0.

It is important to note that in the divergence term of (17.3.1), we have not replaced the wind by its geostrophic value. This step is necessary to retain vertical velocity in the model, since divergence of the geostrophic wind with a constant f0 is zero. It is the departure of the wind from its geostrophic value that produces divergence for vertical motion. For, if, in isobaric co-ordinates, we take V' as the ageostrophic wind,

With this substitution for divergence, (17.3.1) reduces to dZg/dt = -Vg • V(Zg + f) + f0 da/dp (17.3.3)

Since Zg and Vg are both functions of O, (17.3.3) offers a better method of computing the m-field from observations of O than the continuity equation (12.8.2), because in midlatitudes both O and dO/dt can be determined with somewhat greater accuracy than the wind at isobaric surfaces.

In a similar manner, using the quasi-geostrophic wind in the horizontal advec-tion term and neglecting the role of diabatic heating in midlatitude synoptic-scale motion systems, we can approximate the thermodynamic energy equation (12.2.8) to the form d(-dO/dp)/dt = -Vg • V(—dO/dp)+om (17.3.4)

where the static stability o is assumed constant and, on the basis of the hydrostatic approximation, a = —dO/dp, and the equation of state, pa = RT, we have replaced Tbyp (—dO/dp)/R.

Thus, the quasi-geostrophic vorticity equation (17.3.3) and the hydrostatic ther-modynamic energy equation (17.3.4) each contains only two dependent variables O and m,and, therefore, together constitute a closed set of prediction equations in O and m.

From these two equations, we can eliminate m and obtain an expression for the geopotential tendency, dO/dt. We can also manipulate the two equations suitably to compute the vertical velocity, m, yielding the so-called omega equation.

The expressions for the geopotential tendency and the vertical motion, derived in this manner, are as follows:

The geopotential tendency equation,

{V2 + (f0/o)d2/dp2}dO/dt = — f0Vg-V{(1/fc)V2O + f}

The omega equation,

{V2 + (f2/o)d2/dp2}m =(f0/o)d[Vg-V{(1/f0)V2O + f}]/dp

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