Towards Climate-Dependent Sub-Grid-Scale Parameterizations in Atmospheric Models

Climate system models use a multitude of parameterization schemes for small-scale processes. These should respond to externally forced climate variability in an appropriate manner so as to reflect the response of the parameterized process to a changing climate. Indications are that they might not satisfy this condition to full satisfaction. The most attractive route to master the challenge of achieving such a behavior would be provided by theoretical understanding sufficiently deep to enable the à-priori design of climate-sensitive parameterization schemes.  An alternative path might be helpful when the parameter tuning involved in the development of a scheme is objective enough so that these parameters can be described as functions of the statistics of the climate system. Provided that the dynamics of the process yields a sufficiently smooth probability distribution, and that the external forcing is not too strong, the quasi-Gaussian fluctuation-dissipation theorem (qg-FDT, Risken 1984) might be a tool to predict from the statistics of a system (e.g. the atmosphere) how an objectively tuned parameterization should respond to external forcing (e.g. by anomalous sea-surface temperatures). First promising steps in following these strategies will be described in two parts of the talk. (1) The qg-FDT approach has been examined within the framework of a toy atmosphere, to be simulated by a low-order model (i.e. a toy atmosphere model) with realistic internal variability. At sufficiently weak (but yet realistic) forcing strength use of the qg-FDT is found to systematically improve the agreement between the responses of model and atmosphere, respectively (Achatz et al 2013). Encouraging results on the application of this technique to an increasingly complex atmospheric setting will be reported as well. (2) The application of stochastic-mode reduction (Majda et al 2003) to obtain local stochastic SGS schemes could be another interesting route. Starting from a high-resolution finite-difference discretization of the equations of a dynamical system, this approach is based on splitting the model variables into fast, small-scale and slow, large-scale modes by averaging over neighboring grid cells. After that, the fast modes are eliminated by applying a stochastic mode reduction procedure. The new parameterization has so far been applied to a one-dimensional turbulent-flow system (Burgers equation). It is shown to compare favorably to traditional approaches on SGS turbulence parameterizations (Dolaptchiev et al 2013a,b).

Achatz, U., U. Löbl, S. Dolaptchiev and A. Gritsun, 2013: Fluctuation-Dissipation Supplemented by Nonlinearity: A Climate-Dependent Sub-Grid-Scale Parameterization in Low-Order Climate Models. J. Atmos Sci., 70, 1833-1846

Dolaptchiev, S.I., Achatz U. and I. Timofeyev , 2013a: Stochastic closure for local averages in the finite-difference discretization of the forced Burgers equation. Theor. Comput. Fluid Dyn. 27, 297-317

Dolaptchiev, S.I., Timofeyev, I. and Achatz, U., 2013b: Subgrid-scale closure for the inviscid Burgers-Hopf equation, Commun. Math. Sci., 11, 757–777

Majda, A., Timofeyev, I., and E. Vanden-Eijnden, 2003: Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci., 60, 1705–1722

Risken, H., 1984: The Fokker-Plank Equation: Methods of Solution and Applications. Springer-Verlag, 474 pp