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Finding the limits of convective parameterizations

Todd R. Jones and David A. Randall

1. Introduction
Convective parameterizations that have proven useful in climate and numerical weather prediction models of the past are becoming more and more incompatible with the advanced capabilities of today's models. As computing power has continued to increase, there has been a significant shift in the structure of these models toward finer grid resolution and better representation of smaller-scale physical atmospheric processes. The older parameterizations that provide some statistical representation of convection without explicitly resolving the convective motions make use of a number of assumptions that do not necessarily apply to all models.

For instance, quasi-equilibrium (QE) closure-based parameterizations are expected to apply to a large ensemble of clouds under slowly changing weather conditions. There is an assumed separation of scales: there must be a large number of clouds that are individually much smaller than the area under consideration, and the large-scale processes that force the creation of convective clouds must vary so slowly that the clouds appear to react nearly instantaneously to the forcing. The QE assumption therefore tends to break down under rapidly changing conditions, such as those that vary on the order of a cloud lifetime, or when the area considered, such as within one model grid cell, is too small to provide an adequate sample of the cloud field. Both of these conditions exist in higher-resolution models where grid cells are small and convective systems with smaller spatial scales tend to have shorter time scales. This is all in addition to the inherently unpredictable nature of small-scale motions, the uncertainty of which grows to affect larger scales of motion. Because of this sensitive dependence on initial conditions, the grid cell-averaged precipitation rate and other statistics are intrinsically uncertain (e.g., Hohenegger and Schar 2007). This is a fundamental issue affecting all convective parameterizations.

2. Test Design
Many have recognized the potential for problems with using these assumptions in newer models (e.g., Xu et al. 1992). In order to improve understanding of non-equilibrium and non-deterministic cumulus convection, we have quantitatively explored the capabilities and limitations of cumulus parameterization as applied to models with grid spacing on the order of 20- 50 km, or even finer. We first define QE by examining fluctuations about an equilibrium state in constantly forced simulations of a three-dimensional cloud-resolving model (the VVM, Jung and Arakawa 2008). Then those random fluctuations are compared to the random fluctuations about an ensemble mean of simulations that were created to determine how the response to prescribed periodic large-scale forcing changes with the period of the forcing and the size of the averaging domain. Figure 1 shows example output from such a simulation with a 24 hour forcing period.

Figure 1. 24 hour forcing period simulation.

3. Results
Variability around the mean response from a constant forcing was found to be on the order of 10% of the mean, defining what one should expect from true equilibrium. This coefficient of variation (COV, noise-to-signal ratio) is nearly independent of the magnitude of the forcing.

The more slowly the forcing varies, the better the response is approximated by the equilibrium solution (variation on the order of 10% of the mean and close to being in phase with the forcing). For deep convection, QE is viable when the time scale for the variation of the large-scale forcing is greater than about 30 h (Figure 2). Lags between the forcing and the response arise from a number of factors, including mesoscale organization that allows the convection to become self-sustaining.

For a given weather regime, nondeterministic variability is much stronger with smaller domain sizes; this is the problem of insufficient sample size, which grows in importance as grid spacing decreases. The COV was found to follow a logarithmic function of the domain area.

4. Conclusions
Our results suggest that QE applies for domain sizes of 256 km or greater, and for forcing periods of 30 h or greater
(Figure 2). Even then, a deterministic parameterization will have "expected errors" on the order of 10%. Use of QE convective parameterizations outside of these forcing period
and domain limits will be detrimental to model simulations.


References

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