## Two Level Primitive Equation Model

In the quasi-geostrophic model that we discussed in Sect. 17.3, the static stability o was assumed to be constant. But it is well-known that it varies in space and time depending upon the dynamical and thermodynamical processes in the atmosphere. We, therefore, need to take its variability into consideration in the primitive equation prediction models by computing d0/do at each time step. This requires temperature to be predicted in at least two levels, instead of one as it was in the case of the two-level quasi-geostrophic model. As in the two-parameter baroclinic model (Fig. 17.1), the atmosphere is divided into two layers by boundary surfaces labelled 0, 2, and 4, corresponding to sigma surfaces 0, 1/2 and 1 respectively The vertical differencing scheme in the case of the two-level primitive equation model, usually adopted, is illustrated in Fig. 17.2.

The momentum equations (17.6.14 and 17.4.6) and the thermodynamic energy equation (17.6.13) are applied at the levels 1 and 3, corresponding to sigma surfaces V2 and 3/4 which lie at the centers of the two layers. The vertical differencing for the momentum equations involves u2, v2, and do2 /dt, while that for the thermodynamic energy equation involves 02 and do2/dt. The variables u2, v2 and 02 are obtained by linear interpolation. For example, 02 = (0i + 03)/2. The field of do2/dt may be obtained diagnostically by using the continuity equation (17.6.10) for levels 1 and 3 with the vertical derivatives computed by centered differencing. This means that for level 1, we take the difference between levels 0 and 2, and for level 3 we take the difference between levels 2 and 4.

The continuity equations at the two levels are then dps/dt + V-(psV1)+2ps d02/dt = 0 (17.6.17)

Levels &o=0

Levels &o=0

LOWER BOUNDARY

Fig. 17.2 Vertical differencing scheme in the two-level primitive equation model

LOWER BOUNDARY

Fig. 17.2 Vertical differencing scheme in the two-level primitive equation model

The addition of (17.6.17) and (17.6.18) gives us the finite-difference form of the surface pressure tendency equation dps/dt = —V-{ps(V1 + V3)/2} (17.6.19)

while the difference between the equations yields the diagnostic equation for do2/dt which is d02/dt = —V-{ps(V1 — V3)}/4ps (17.6.20)

We now need to express O1 and O3 in terms of 01 and 03 using the hydrostatic relationship (17.6.11) in order to complete the specification of all the dependent variables. However, according to Arakawa (1972), the finite difference equations will have the same energy conservation properties as the original differential equations only if the variables O1 and O3 in (17.6.11) are expressed in a special form dO/dpK = —cp9(p0)—K (17.6.21)

Applying (17.6.21) at level 2, we get

O1 — O3 = — cp92{(p! /p0)K — (p3/p0 )K} (17.6.22)

Next we rewrite (17.6.11) by using the ideal gas law d(oO)/do = O — poa (17.6.23)

By integrating (17.6.23) with respect to o from o = 1 to o = 0, we obtain

O4 = {(O3 — ps o3a3) + (O1 — pso1a0}/2 (17.6.24)

where we have used the values (oO) = O4 at o = 1, and (oO) = 0 at o = 0. We now solve (17.6.22) and (17.6.24) for O1 and O3 to obtain

O1 = O4 + ps(o3a3 + o1a1)/2 + cp92{(p3/p0)K — (p1/p0)K}/2 (17.6.25) O3 = O4 + ps(o3a3 + o1a1)/2 — cp92{(p3/p0)K — (p1/p0)K}/2 (17.6.26)

This completes the formulation of the two-level primitive equation model.

17.6.3 Computational Procedure

The computational procedure in the case of the two-level primitive equation model consists of the following main steps:

1. Write suitable finite difference analogs of the momentum and thermodynamic energy equations at levels 1 and 3 and the surface pressure tendency equation at level 4.

2. Use the prediction equations to obtain the tendencies of V1; V3, 01; 03, and ps fields.

3. Extrapolate the tendencies ahead using a suitable time differencing scheme. Usually, the first time step is forward and the subsequent ones centered.

4. Use the new values of the dependent variables to diagnostically determine do2/dt, ®i and 03.

5. Repeat steps 2 and 4 until forecasts are obtained for the desired period.

In applying the above computational scheme, it must be borne in mind that the equations in the sigma co-ordinate system contain the mechanism to generate the fast-moving sound and gravity waves that interfere with the slow-moving meteorological waves and produce computational instability. To prevent the generation of the unwanted waves, it is necessary to keep the time increment for extrapolation small enough to satisfy the condition c(At)/d = 1/V2 (17.6.27)

where c is the speed of the fastest moving sound waves 330ms-1), d is the horizontal grid distance and At is the time increment. This condition is known in literature as the Courants-Friederichs-Lewy (CFL) condition. For this reason, in order to avoid computational instability, the time increment in primitive equation prediction model must be considerably less than that in quasi-geostrophic models.

### 17.7 Present Status of NWP

So far, we have traced only the early developments of the subject of NWP before, say, 1970. Since then, the field has expanded so much that it is practically impossible to cover all aspects of the later developments in this brief survey. Computing machines are now millions of times faster; a development which has allowed meteorologists to take up models with much higher spatial resolution over a global domain both in the horizontal and the vertical than before. Data coverage has improved a great deal. Apart from conventional observational network, geostationary and polar -orbiting satellites now regularly report data from different parts of the global atmosphere. Four-dimensional data assimilation schemes have been devised which prepare initial data directly for integration through a 6-hr data assimilation-prediction scheme. With improved computational schemes in place (at many centers, finite-difference models have been replaced by spectral models), the global forecast centers are now able to issue extended-range forecasts of mass and flow fields over the globe for periods ranging from a day to a week or more.

Verification statistics of present-day numerical forecasts appear to suggest that by and large these improved numerical models perform better than those based on persistence or other statistical or subjective methods, especially at extended ranges. For example, according to a study by Leith (1978), quoted by Holton (1979), the root-mean-square error in a 24-hr forecast of 500 mb height was 40 m by the primitive equation model as against 65 m by persistence, the initial height error being 20 m (now less than 10 m). With extension of the forecast period to 48 h, the errors in the two methods grew to about 60 m and 95 m respectively. Experiments on how the growth of errors limits the inherent predictability of the atmosphere have been carried out by a number of workers using primitive equation forecast models. Results of these experiments appear to indicate that the theoretical limit for any useful forecasts of synoptic-scale motion is probably one to two weeks.

Actual predictive skill of present-day models falls far short of this theoretical limit due to various sources of error, some of which are computational and some physical. Among the main sources of error are the replacement of derivatives of the dependent variables by finite-differences, inadequate representation of mountain effects, boundary layer processes, frictional dissipation, cloud and precipitation, and horizontal and vertical resolution of different scales of motion. Further, it is to be understood that the above error statistics apply only to forecasts of variables of the atmosphere which are continuous functions of space and time, such as pressure, temperature, geopotential height, wind components, etc. They do not apply to forecasts of actual weather phenomena experienced in daily life, such as fog, cloud, rain, thunderstorms, tornadoes, etc, which may or may not occur during the forecast period. However, practicing forecasters give due weightage to NWP model output statistics and interpret them suitably along with other guiding materials available to them in assessing the probability of occurrence or non-occurrence of any of these weather elements.

That is how the art or science of weather forecasting stands to-day. However, there is no doubt that attempts to develop numerical models with better coverage of initial data over land and ocean, improved numerical techniques, and higher resolution in space and time have led to better understanding of the dynamics of the atmosphere during the last several decades and the trend will undoubtedly continue.

So far as long-range forecasting is concerned, there has been increasing realization in recent years of the role played by oceans in atmospheric circulation and the atmosphere in ocean circulation. This has led to development of interacting coupled ocean-atmosphere models at many of the world meteorological centers (e.g. Saha et al., 2006) and the trend holds out a great promise for future climate forecasts.