Mode decomposition as a methodology for developing convective‐scale representations in global models

SUMMARY Mode decomposition is proposed as a methodology for developing subgrid-scale physical representations in global models by a systematic reduction of an originally full system such as a cloud-resolving model (CRM). A general formulation is presented, and also discussed are mathematical requirements that make this procedure possible. Features of this general methodology are further elucidated by the two specific examples: mass fluxes and wavelets. The traditional mass-flux formulation for convective parametrizations is derived as a special case from this general formulation. It is based on the decomposition of a horizontal domain into an approximate sum of piecewise-constant segments. Thus, a decomposition of CRM outputs on this basis is crucial for their direct verification. However, this decomposition is mathematically not well-posed nor unique due to the lack of admissibility. A classification into cloud types, primarily based on precipitation characteristics of the atmospheric columns, that has been used as its substitute, does not necessarily provide a good approximation for a piecewiseconstant segment decomposition. This difficulty with mass-flux decomposition makes a verification of the formulational details of parametrizations based on mass fluxes by a CRM inherently difficult. The wavelet decomposition is an alternative possibility that can more systematically decompose the convective system. Its completeness and orthogonality also allow a prognostic description of a CRM system in wavelet space in the same manner as is done in Fourier space. The wavelets can, furthermore, efficiently represent the various convective coherencies by a limited number of modes due to their spatial localizations. Thus, the degree of complexity of the wavelet-based prognostic representation of a CRM can be extensively reduced. Such an extensive reduction may allow its use in place of current cumulus parametrizations. This wavelet-based scheme can easily be verified from the full original system due to its direct reduction from the latter. It also fully takes into account the multi-scale nonlinear interactions, unlike the traditional mass-flux-based schemes.

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