An Idealised Test Case For Assessing The Linearization of Tracer Transport Schemes in NWP Models

The linearized version of a Numerical Weather Prediction (NWP) model, which consists of its tangent linear model (TLM) and adjoint, has a number of important applications in atmospheric modelling. As such it is important that the linearized version of the NWP model can provide an accurate representation of the perturbation growth that occurs in the nonlinear model and does not introduce spurious instability. A suite of test cases, built upon existing frameworks, are developed to assess the accuracy of the linearization of the tracer transport component of the NWP model. Deformation velocities are prescribed that return the tracer back to the initial conditions, thus providing an analytical solution. A selection of smooth and discontinuous tracers and tracer perturbations are used. Example results are shown using second-order and third-order tracer transport schemes, both with and without nonlinear flux limiters. Metrics are offered for assessing the skill of the linearization and predicting when problems will occur. For the example schemes used the results show that linearizations of the nonlinear flux-limited transport schemes behave poorly due to the presence of unstable modes. Some linearized model implementation strategies are offered for situations where the nonlinear scheme should not be linearized.