Primitive Equation Models

The filtered and the simplified models, discussed in the foregoing sections, looked promising for a while but obviously had their limitations so far as weather forecasts were concerned, simply because the real atmosphere and its behaviour are more complicated. So, the demand for better forecast naturally called for use of more complete and original hydrodynamical equations which at one time were dubbed as primitive equations. The primitive equation model of the atmosphere consisted of the three prognostic equations (two consisting of the x and y components of the momentum equation and the third the thermodynamic energy equation) and the three diagnostic equations, viz., the hydrostatic equation, the continuity equation and the equation of state. The six equations constitute a closed set of equations in the six dependent variables u, v, m, O, a and 0 and, therefore, were amenable to solution. However, the equations are nonlinear, so could not be solved by any known analytical methods. Attempts were, therefore, made to solve them by numerical methods with suitable boundary conditions. As already stated in the introduction, the first attempt to do so, though with discouraging result, was by Lewis Fry Richardson in England in 1922.

17.6.1 PE Model in Sigma Co-ordinates

Since atmospheric observations are recorded at pressure surfaces, it is advantageous to work with pressure co-ordinates. Other advantages of using a pressure co-ordinate system are that the density does not appear explicitly in the pressure gradient term and that the continuity equation has a simple form. Also, the sound waves are completely filtered.

However, there is a problem in specifying the lower boundary condition at the ground with the pressure co-ordinate. The assumption usually made is that the vertical velocity a0 is zero at the lower boundary z0 where the pressure p0 is treated as constant at 1000 mb. It is well-known that this assumption is not valid because of the uneven topography of the earth's surface. Also, pressure at the lower boundary does not remain constant but varies with time.

To overcome this difficulty, a modified version of the isobaric co-ordinate system, in which the vertical co-ordinate is pressure, p, normalized with surface pressure (ps), is used and is known as the a-sigma system. Thus, a = p/ps

In the sigma system, the lower boundary is defined by a = 1, and the upper boundary by a = 0, so the vertical a-velocity at the lower as well as the upper boundary is zero at all times, even over a high mountain. That is, da/dt = 0, at both a = 1 and a = 0.

We now transform the primitive equations from the p to the a co-ordinate system as follows. The equations in the p co-ordinate system, neglecting friction, are:

where d/dt = d/dt + V • V + ad/dp, and the horizontal Del operator refers to an isobaric surface.

Continuity equation

Thermodynamic energy equation cpd(ln 9)/dt = dS/dt (17.6.5)

We now apply the transformation formula (11.6.8) to the momentum equation (17.6.1) and obtain dV/dt + f kxV = -VO + (o/ps)VpsdO/do (17.6.6)

where V is now applied to the o constant surface and the total differential is given by d/dt = d/dt + V - V + (do/dt)d/do

Similarly, by applying (11.6.8) to the equation of continuity (17.6.2), we first transform the divergence term to the o-constant surface

To transform the term dm/dp to the o co-ordinate, we note that ps does not depend on o, so we may write the continuity equation in the form ps(V-V)p + dm/do = 0 (17.6.8)

Now, m is related to the sigma vertical velocity do/dt by the relation m = dp/dt = d(ops)/dt = ps do/dt + o dps/dt = psdo/dt + o(dps/dt + V - Vps)

Differentiating the above with respect to o, we obtain dm/do = psd (do/dt)/do + (dps/dt + V-Vps)o + odV/do-Vps (17.6.9)

The Eq. (17.6.9) combined with (17.6.7) and (17.6.8), after re-arrangement, yields the transformed continuity equation

With the aid of the equation of state (17.6.4) and the Poisson's relation, T = 9(p/p0)K, the hydrostatic equation(17.6.3) in sigma co-ordinate system is dO/do = -RT/o = -(R9/o)(p/p0)K (17.6.11)

where p0 = 1000 hPa.

We now expand the total derivative on the left-side of the thermodynamic energy equation (17.6.5) to write it in the form d9/dt + V-V9 + (do/dt)d9/do = (9/cp)dS/dt (17.6.12)

If we now multiply (17.6.10) by 9, multiply (17.6.12) by ps, and add the results, we get the thermodynamic energy equation in the sigma co-ordinates in flux-form d(ps9)/dt + V-(ps9V) + d{ps9(do/dt)}/do = (ps9/cp )dS/dt (17.6.13)

A similar transformation of the x and y components of the momentum equation (17.6.6) gives them in flux forms d (psu)/dt + V-(psuV) + d{psu(do/dt)}/dt - f psv = -psdO/dx - R9(p/p0)Kdps/dx (17.6.14)

d (psv)/dt + V-(psvV) + d{psv(do/dt)}/dt + f psu = -psdO/dy - R9(p/p0)Kdps/dy (17.6.15)

The five equations (17.6.10), (17.6.11), (17.6.13), (17.6.14) and (17.6.15) contain the six dependent scalar variables, u, v, do/dt, O, 9, ps. This means that one more equation is required for a closed system of equations. The additional equation is the pressure tendency equation which is obtained by vertical integration of the continuity equation (17.6.10) with the boundary condition, do/dt = 0, at o = 1, 0. Thus, the sixth equation is,

TheEq. (17.6.16) states that the rate of change of surface pressure at a given point on the sigma surface is simply the total mass convergence in the overlying column of air of unit cross-section of the surface. With the inclusion of (17.6.16), we have now a complete set of prediction equations which can be put into a suitable finite-difference form and numerically integrated in time for different forecast periods.

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