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

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 piecewise‐constant 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. Copyright © 2005 Royal Meteorological Society