Gerry Browning writes:
The Correct System of Equations for Climate and Weather Models
The system of equations numerically approximated by both weather and climate models is called the hydrostatic system. Using a scale analysis for mid-latitude large scale motions in the atmosphere (motions with a horizontal length scale of 1000 km and time scale of a day), Charney (1948) showed that hydrostatic balance, i.e., balance between the vertical pressure gradient and gravitational force, is satisfied to a high degree of accuracy by these motions. As the fine balance between these terms was difficult to calculate numerically and to remove fast vertically propagating sound waves to allow for numerical integration using a larger time step, he introduced the hydrostatic system that assumes exact balance between the vertical pressure gradient and the gravitational force. This system leads to a columnar (function of altitude) equation for the vertical velocity called Richardson’s equation.
A scale analysis of the equations of atmospheric motion assumes that the motion will retain those characteristics for the period of time indicated by the choice of the time scale (Browning and Kreiss, 1986). This means that the initial data must be smooth (have spatial derivatives on the order of 1000 km) that lead to time derivatives on the order of a day. To satisfy the latter constraint, the initial data must satisfy the elliptic constraints determined by ensuring a number of time derivatives are of the order of a day. If all of these conditions are satisfied, then the solution can be ensured to evolve smoothly, i.e., on the spatial and time scales used in the scale analysis. This latter mathematical theory for hyperbolic systems is called “The Bounded Derivative Theory” (BDT) and was introduced by Professor Kreiss (Kreiss, 1979, 1980).
Instead of assuming exact hydrostatic balance (leads to a number of mathematical problems discussed below), Browning and Kreiss (1986) introduced the idea of slowing down the vertically propagating waves instead of removing them completely, thus retaining the desirable mathematical property of hyperbolicity of the unmodified system. This modification was proved mathematically to accurately describe the large scale motions of interest and, subsequently, also to describe smaller scales of motion in the mid-latitudes (Browning and Kreiss, 2002). In this manuscript, the correct elliptic constraints to ensure smoothly evolving solutions are derived. In particular the elliptic equation for the vertical velocity is three dimensional, i.e., not columnar, and the horizontal divergence must be derived from the vertical velocity in order to ensure a smoothly evolving solution.
It is now possible to see why the hydrostatic system is not the correct reduced system (the system that correctly describes the smoothly evolving solution to a first degree of approximation). The columnar vertical velocity equation (Richardson’s equation) leads to columnar heating that is not spatially smooth. This is called rough forcing and leads to the physically unrealistic generation of large amounts of energy in the highest wave numbers of a model (Browning and Kreiss, 1994; Page, Fillion, and Zwack, 2007). This energy requires large amounts of nonphysical numerical dissipation in order to keep the model from becoming unstable, i.e., blowing up. We also mention that the boundary layer
interacts very differently with a three dimensional elliptic equation for the vertical velocity than with a columnar equation (Gravel, Browning, and Kreiss).
Browning, G. L., and H.-O. Kreiss 1986: Scaling and computation of smooth atmospheric motions. Tellus, 38A, 295–313.
——, and ——, 1994: The impact of rough forcing on systems with multiple time scales. J. Atmos. Sci., 51, 369-383
——, and ——, 2002: Multiscale bounded derivative initialization for an arbitrary domain. J. Atmos. Sci., 59, 1680-1696.
Charney, J. G., 1948: On the scale of atmospheric motions. Geofys.Publ., 17, 1–17.
Kreiss, H.-O., 1979: Problems with different time scales for ordinary differential equations. SIAM J. Num. Anal., 16, 980–998.
——, 1980: Problems with different time scales for partial differential equations. Commun. Pure Appl. Math, 33, 399–440.
Gravel, Sylvie et al.: The relative contributions of data sources and forcing components to the large-scale forecast accuracy of an operational model. This web site
Page, Christian, Luc Fillion, and Peter Zwack, 2007: Diagnosing summertime mesoscale vertical motion: implications for atmospheric data assimilation. Monthly Weather Review, 135, 2076-2094.