## Quantifying the limits of convective parameterizations : a statistical characterization of simulated cumulus convection

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Jones, Todd R.

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This study reviews and characterizes such departures from convective quasiequilibrium, that is, fluctuations about the equilibrium state that are found to be present in convective simulations of a cloud-resolving model (CRM) that lead to effects of nonequilibrium and non-deterministic behavior. Information about such behavior is hypothesized to be important to the development of stochastic convective parameterizations that are employed to introduce temporal variability into general circulation models (GCMs) in an informed manner to improve the statistics of various fields. Following a process similar to the methods used by Xu et al. (1992) the response of the statistical characteristics of a variety of convection-related parameters to an imposed periodic largescale forcing is determined in terms of a variety of convective variables. More specifically, a number of CRM simulations are employed to address various issues, among which are how the response varies with changes in the length of the forcing period, how the response varies with computational domains of varying sizes, and how different points in the response to a cyclical forcing compare to the response to a comparable timeaveraged constant forcing. Additionally, this thesis includes the results of the CRM’s participation in the TWP-ICE intercomparison study that was used to validate the model. As a control, the model is subjected to a series of constant forcing simulations for a variety of large-scale forcing magnitudes, which provides a cloud-resolving model representation of statistical equilibrium. It is shown that the coefficient of variation is not independent of forcing magnitude as may be expected, particularly for forcing magnitudes that are very small or very large. Seemingly minor variations are expected to bias aspects of GCM simulations in ways that can alter the representation of statistical (periodic) features. With the application of a periodic forcing at varying period lengths and consideration of the simulation results on a series of subdomain sizes, it is shown that there is a considerable range of responses to a given large-scale forcing that are dependent upon the rate of change in the forcing and in the size of the averaging domain. Specifically, the analyses show that the more slowly a forcing varies, the more it is well approximated by an equilibrium assumption. The point at which the transition between being an acceptable approximation of equilibrium or not occurs is approximately located where the timescale of the variation of the large-scale forcing is greater than 30 hours. Convective activity is also found to be dependent upon whether the large-scale forcing was increasing or decreasing and also the rate at which it was doing so. Further, nondeterministic variability for a given situation is much greater at smaller domain sizes; this is the problem of insufficient sample size, which is one that grows in importance as grid spacing in GCMs approaches the lower tens of kilometers. This relationship is best described by a logarithmic function of the domain area. Based on the statistics for the weather regime presented in this thesis, the best recommendation is that the line between determinism and non-determinism should be drawn such that the considered area (grid box area) is no smaller than about half the domain size used in this thesis. This is equivalent to about 33,000 km2 or grid spacing slightly larger than 180 km. Cummulatively, the results of the performed experiment highlight both the complexity underlying the development of, and the need for, stochastic convective parameterization. Classical assumptions about quasi-equilibrium are never exact and break down altogether when the time scale for changes in the resolved-scale weather is near or less than the convective adjustment time. This is a problem that is made more severe in newer high-resolution models (e.g. Arribas 2004; Bechtold et al. 2008) just because shorter time scales are inherent in convective systems with smaller spatial scales. The basics of a statistical approach for the development of a stochastic parameterization are outlined.