Abstract
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 temperature θ_{e} and proxy equivalent potential temperature tθ_{e} vanish 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 the presence of the random component.
The focus of the application in this study assesses the capabilities of several models to simulate the conservation of moist entropy and reversibility of moist-adiabatic processes over a period of 10 days. The assessment includes four different versions of the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM) and the University of Wisconsin (UW) hybrid isentropic-sigma (θ–σ) model. The assessment from the 10-day simulations focuses on the temporal evolution of the global sum of squares of the differences of equivalent potential temperature and its trace and the three components. In the case of all models expressed in sigma coordinates, the global sum of squares as simulated exceeds the global sum of squares from the UW θ–σ model. The partitioning into three components of variance revealed different structures of average differences resulting from errors in vertical exchange, and also different magnitudes of the random component among the CCM models. In contrast, the component sum of squares in the UW θ–σ model simulation was minimal, except for small global and area average differences stemming from transport across the interface between the isentropic and sigma domains of the model in the low troposphere. The empirical relative frequency distribution for the pure error differences in the UW θ–σ model tends to equilibrate and be triangular in form as would be expected from statistical theory in which the random variate is given by the difference of two variates, each of which is drawn from a uniform distribution of random errors.
In conclusion, the combination of the methods developed in the earlier study and this paper provides a robust strategy for the global assessments of numerical accuracies in simulating reversibility within weather and climate predictions throughout the model domain globally and also regionally.
Corresponding author address: Dr. Donald R. Johnson, Space Science and Engineering Center, University of Wisconsin—Madison, 1225 W. Dayton St., Madison, WI 53706. Email: donj@ssec.wisc.edu