Gerry Browning of CIRA has contributed a post today discussing climate models. If you go to Google Scholar and search “Browning Kreiss”, you will get a list of formidable papers on numerical questions. Gerry has tried to distill the issues for a wider audience here.
Recent awards include the NOAA Environmental Research Laboratories’ Outstanding Scientific Paper Award for: Browning, G.L. and H.O. Kreiss, “The Role of Gravity Waves in Slowly Varying in Time Mesoscale Motions,” Journal of the Atmospheric Sciences , 54, 9 1166 – 1184 (1997).
Here is Gerry’s citation as winner of a 2002 Research Initiative Award from the Cooperative Institute for Research in the Atmosphere:
In a series of papers, Browning and Professor Heinz Kreiss, a colleague and mentor, have extended Kreiss’ Bounded Derivate Theory (BDT) to multiscale flows in the atmosphere and oceans. There are many ramifications of this new theory. The first is that the well posed system introduced by Browning & Kreiss as a replacement for the ill-posed primitive equations (used in all models for large-scale atmospheric flows) also accurately describes both the dominant and gravity wave portions of all remaining atmospheric flows, i.e., the new system is the only well-posed multiscale system that accurately describes all atmospheric motions. The second is that the reduced system clearly indicates what balances are appropriate for all diabatic cases, i.e., it is the only method that has provided a hot start initialization for cases where the heating is the controlling influence on the solution. The impact of this theory is being felt in many areas, and Browning’s cuttingedge research and the potential for breakthrough applications in numerical modeling
An Introduction to Climate and Weather Models
Although there remains residual debate about the validity of various time dependent systems used to describe fluid motions, a number of these systems are in general use in both the engineering and scientific communities, e.g. the viscous, compressible Navier-Stokes equations, the Euler equations of gas dynamics (essentially the inviscid, compressible Navier-Stokes equations), and the magneto-hydrodynamic (plasma) equations. The continuum behavior of specific solutions of these systems can sometimes be understood by considering special cases (such as the propagation of sound) that lead to simpler systems that are more amenable to classical analysis. Sometimes simplifications of these systems are also made to make the numerical approximations of specific solutions of the continuum system on a computer more tractable. There can be two problems associated with either type of simplification.
Although the original systems usually have known mathematical properties, e.g. the Navier-Stokes equations are a quasilinear differential system, the simplifications sometimes can lead to systems with unknown or even bad mathematical properties. A well known example of an equation with bad mathematical properties is the heat equation run backwards in time. A small perturbation of the initial conditions for this equation can lead to instantaneous, unbounded growth and time dependent systems that exibit this type of behavior are called ill-posed systems. It is quite surprising how often simplifications that have been made in practice have led to this type of problem. Therefore, any simplification of the original continuum equations should be checked to ensure that the simplified system accurately approximates the continuum solution of interest and is properly posed
Difference Methods for Initial-Value Problems: Richtmyer and Morton
Numerical Modeling Considerations
Once the continuum system to be approximated has been determined to be properly posed, it can be approximated by a number of numerical methods, but all must be both accurate (consistent) and stable for the method to converge to the continuum solution of the initial-value problem (see Lax Equivalence Theorem in above reference). The accuracy of the numerical method determines how fast the numerical method will converge to the continuum solution, e.g. a fourth order method will take fewer mesh points than a second order method (assuming both are stable). However, the numerical accuracy can be reduced by a number of factors, e.g. errors in the approximations of the continuum equations or errors in the model.
There can be two other significant problems with numerical models. If there are any boundaries present, those boundaries must be dealt with very carefully both in the continuum system and in the numerical model. This is an extremely delicate process and if handled improperly can reduce the accuracy of a numerical method and even lead to an incorrect solution. The other major problem is that the solution of the continuum system may contain a complete spectrum of waves, but a numerical model can only compute a finite part of that spectrum. This is a typical and very serious problem. Henshaw, Kreiss, and Reyna have determined the minimal scale that will be produced by the nonlinear, incompressible, Navier-Stokes equations for a given viscosity coefficient (molasses has a large viscosity and air a very small one).
Convergent numerical solutions have shown that the estimates of this scale are extremely accurate. If the numerical model does not resolve the correct number of waves indicated by the estimate, the model blows up. If the model resolves the number of waves indicated by the estimate, the numerical method will converge to the continuum solution for long periods of time. Thus, if a numerical model is unable to resolve the spectrum of the continuum solution, the model is forced to artificially increase the viscosity coefficient or use a numerical method that has nonphysical viscosity built into the method.
Analysis of Numerical Methods: Isaacson and Keller
Time Dependent Problems and Difference Methods: Gustafsson, Kreiss, and Oliger
Initial-Boundary Value Problems and the Navier Stokes Equations: Kreiss and Lorenz
Large-Scale Weather Prediction Models
Given the above brief introduction to time dependent partial differential equations and numerical methods for those systems, we can now discuss large-scale weather prediction and climate models.
Clearly, the atmosphere contains motions of many spatial and time scales and no numerical model can hope to resolve all of those motions. For large-scale motions in the midlatitudes above the turbulent lower boundary layer, the inviscid, unforced Euler equations on a rotating sphere can be scaled under certain mathematical assumptions. For these motions, the vertical acceleration term is approximately 6 orders of magnitude smaller than the remaining terms in the time dependent equation for the vertical velocity and thus are typically neglected in large-scale weather prediction models leaving only the hydrostatic balance terms. The resulting system is sometimes referred to as the primitive or hydrostatic equations. The neglect of the vertical acceleration term made the equations tractable for computing, i.e. the inclusion of the vertical acceleration term would have required too small a time step to satisfy the stability criterion mentioned above, but altered the mathematical properties of the original system.
After the derivation of the hydrostatic equations, approximations of the turbulent boundary layer, eddy viscosity (much larger than the true atmospheric viscosity and sometimes even of a different type, e.g. hyperviscosity), and all kinds of approximations to various
atmospheric phenomena (parameterizations) are added onto the hydrostatic equations.
In the fall of 2001, Sylvie Gravel (RPN) ran a series of tests on the Canadian large-scale operational large-scale numerical weather prediction model. The parameterizations could all be turned off and the turbulent boundary layer approximation greatly simplified without significant difference in the 36 hour model forecast. However, the large-scale weather prediction model quickly started to deviate from the observations at a later time and only by updating the winds in the jet stream every 12 hours did the model stay on track. (The satellite data did not help the model forecast unless there was also radiosonde data available at the same site.) A simple change in the data assimilation program based on the Bounded Derivative Theory had a substantial impact on the forecast and the Canadian global weather prediction model continues to perform better than the NOAA global weather prediction model even though the latter model employs a more accurate numerical method.
Browning and Kreiss, 1986: Scaling and Computation of Smooth Atmospheric Motions
Tellus, 38A, 295-313 (and Charney reference therein)
Browning and Kreiss, 2002: Multiscale Bounded Derivative Initialization for an Arbitrary Domain, JAS, 59, 1680-1696
The updating discussed above is not possible in a climate model and because climate models use even a coarser mesh than a large-scale weather prediction model, they must use an effectively larger viscosity than a global weather prediction model. Recently (BAMS, 2004), it has been shown that a climate model also deviates from reality in a matter of hours because of the errors in the parameterizations (not unexpected based on result above) and over longer periods of time the effectively larger viscosity causes the numerical solution to produce a spectrum quite different than the real atmosphere unless forced in a nonphysical manner.