Stochastic parametrisations are used in weather and climate models to improve the representation of unpredictable unresolved processes. When compared to a deterministic model, a stochastic model represents `model uncertainty, i.e., sources of error in the forecast due to the limitations of the forecast model. We present a technique for systematically deriving new stochastic parametrisations or for constraining existing stochastic approaches. A high-resolution model simulation is coarse-grained to the desired forecast model resolution. This provides the initial conditions and forcing data needed to drive a Single Column Model (SCM). By comparing the SCM parametrised tendencies with the evolution of the high resolution model, we can estimate the error in the SCM tendencies that a stochastic parametrisation seeks to represent. We use this approach to assess the physical basis of the widely used Stochastically Perturbed Parametrisation Tendencies (SPPT) scheme. We find justification for the multiplicative nature of SPPT, and for the use of spatio-temporally correlated stochastic perturbations. We find evidence that the stochastic perturbation should be positively skewed, indicating that occasional large-magnitude positive perturbations are physically realistic. However other key assumptions of SPPT are less well justified, including coherency of the stochastic perturbations with height, coherency of the perturbations for different physical parametrisation schemes, and coherency for different prognostic variables. Relaxing these SPPT assumptions allows for an error model that explains a larger fractional variance than traditional SPPT. In particular, we suggest that independently perturbing the tendencies associated with different parametrisation schemes is justifiable, and would improve the realism of the SPPT approach.
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