Numerical Uncertainties in the Simulation of Reversible Isentropic Processes and Entropy Conservation

Abstract

A challenge common to weather, climate, and seasonal numerical prediction is the need to simulate accurately reversible isentropic processes in combination with appropriate determination of sources/sinks of energy and entropy. Ultimately, this task includes the distribution and transport of internal, gravitational, and kinetic energies, the energies of water substances in all forms, and the related thermodynamic processes of phase changes involved with clouds, including condensation, evaporation, and precipitation processes. All of the processes noted above involve the entropies of matter, radiation, and chemical substances, conservation during transport, and/or changes in entropies by physical processes internal to the atmosphere. With respect to the entropy of matter, a means to study a model?s accuracy in simulating internal hydrologic processes is to determine its capability to simulate the appropriate conservation of potential and equivalent potential temperature as surrogates of dry and moist entropy under reversible adiabatic processes in which clouds form, evaporate, and precipitate. In this study, a statistical strategy utilizing the concept of ?pure error? is set forth to assess the numerical accuracies of models to simulate reversible processes during 10-day integrations of the global circulation corresponding to the global residence time of water vapor. During the integrations, the sums of squared differences between equivalent potential temperature ?e numerically simulated by the governing equations of mass, energy, water vapor, and cloud water and a proxy equivalent potential temperature t?e numerically simulated as a conservative property are monitored. Inspection of the differences of ?e and t?e in time and space and the relative frequency distribution of the differences details bias and random errors that develop from nonlinear numerical inaccuracies in the advection and transport of potential temperature and water substances within the global atmosphere. A series of nine global simulations employing various versions of Community Climate Models CCM2 and CCM3?all Eulerian spectral numerics, all semi-Lagrangian numerics, mixed Eulerian spectral, and semi-Lagrangian numerics?and the University of Wisconsin?Madison (UW) isentropic-sigma gridpoint model provides an interesting comparison of numerical accuracies in the simulation of reversibility. By day 10, large bias and random differences were identified in the simulation of reversible processes in all of the models except for the UW isentropic-sigma model. The CCM2 and CCM3 simulations yielded systematic differences that varied zonally, vertically, and temporally. Within the comparison, the UW isentropic-sigma model was superior in transporting water vapor and cloud water/ice and in simulating reversibility involving the conservation of dry and moist entropy. The only relative frequency distribution of differences that appeared optimal, in that the distribution remained unbiased and equilibrated with minimal variance as it remained statistically stationary, was the distribution from the UW isentropic-sigma model. All other distributions revealed nonstationary characteristics with spreading and/or shifting of the maxima as the biases and variances of the numerical differences of ?e and t?e amplified.