Numerical Uncertainties in Simulation of Reversible Isentropic Processes and Entropy Conservation: Part II

• Published in: Journal of climate 2002-07, Vol.15 (14), p.1777-1804
• Main Authors: Johnson, Donald R., Lenzen, Allen J., Zapotocny, Tom H., Schaack, Todd K.
• Format: Article
• Language: English
• Subjects: Advection; Atmospheric models; Climate models; Clouds; Earth, ocean, space; Entropy; Exact sciences and technology; External geophysics; Frequency distribution; Geophysics. Techniques, methods, instrumentation and models; Modeling; Simulations; Statistical analysis; Statistics; Temperature; Temperature distribution; Water vapor
• ISSN: 0894-8755; 1520-0442
• DOI: 10.1175/1520-0442(2002)015<1777:NUISOR>2.0.CO;2

The objectives of this study are 1) to provide the framework for an in-depth statistical analysis of the numerical uncertainties in the simulation of conservation of entropy, potential vorticity, and like properties under appropriate modeling constraints, and 2) to illustrate the discriminating nature of the analysis in an application that isolates internal numerical inaccuracies in the simulation of reversible atmospheric processes. In an earlier study the authors studied the pure error sum of squares function as a quadratic measure of uncertainties by summing the squared differences between equivalent potential temperature as simulated by the nonlinear governing equations for mass, energy, water vapor, and cloud water and its counterpart simulated as a trace constituent. Within the experimental design to examine a model’s capabilities to conserve the moist entropy, the continuum equations demand that the differences between equivalent potential temperatureθₑand proxy equivalent potential temperaturetθₑvanish at all discrete model information points throughout the 10-day simulation. The differences that develop provide a measure of numerical inaccuracies in the simulation of reversibility. In this extension of the earlier study, the first consideration is to examine zonal–vertical cross sections of the differences, relative frequency distributions of the differences, and the vertical structure of systematic differences. Subsequently, through an analysis of variance, the sum of squares is partitioned into three components: the squared deviations of differences from an area mean difference, the square of the deviation of the mean difference from the global mean difference, and the square of the global mean difference. In the situation where biases vanish in all three components, a theoretical development based on the uniqueness of a distribution with its moment-generating function suggests that the nearer the empirical relative frequency distribution of pure error differences is to the classical triangular distribution of the differences of two random variates, the closer the model’s simulation is to the optimum accuracy feasible in ensuring reversibility and appropriate conservation of moist entropy. A final consideration is to place the random and systematic components of differences within a probability perspective in which the normal distribution is utilized to assess whether the magnitude of the average difference exceeds that expected to develop from