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Thomas E. Rosmond

Abstract

A linear stability model for mesoscale cellular convection in the atmosphere is developed. The model includes a forcing term which is a parameterization of the net heating due to small-scale cumulus convection. A sub-cloud and cloud layer are defined, with the forcing term having non-zero values only in the cloud layer. Positive static stability is assumed in both layers so that the only source of buoyant energy is the forcing term.

The parameterization of latent heating due to cumulus convection is accomplished by assuming that the heating is proportional to the cloud-environment temperature difference and the vertical flux of moisture by the perturbation vertical velocity.

Normal mode horizontal dependence and exponential time dependence is assumed for the vertical velocity, and the forcing term is defined as proportional to the vertical velocity at the interface between the two layers. The solutions in the two layers are matched across the interface and the particular solution associated with the forcing term is expressed in terms of the arbitrary constants contained in the homogeneous solutions. This yields a homogeneous solution matrix which is solved.

Solutions are found for a wide range of values of atmospheric static stability, system depth, mean temperature and relative humidity, as well as varying degrees of anisotropy of the eddy mixing coefficients. The observed flattening of atmospheric cells, with diameter-to-depth ratios an order of magnitude greater than predicted by the stability analysis of classical Rayleigh convection, is duplicated by the model. Anisotropy of the eddy mixing coefficients is not a requirement for flattened cells in the model. The choice of boundary conditions is also of minor importance in producing cell flattening. Growth rates and preferred cell diameters are most sensitive to the relative humidity and static stability of the atmosphere. These two parameters represent, respectively, the source and sink of buoyant energy in the model. Positive static stability is responsible for cell flattening because it suppresses very strongly the relatively large vertical velocities associated with smaller cells.

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Kevin Judd
,
Carolyn A. Reynolds
,
Thomas E. Rosmond
, and
Leonard A. Smith

Abstract

This paper investigates the nature of model error in complex deterministic nonlinear systems such as weather forecasting models. Forecasting systems incorporate two components, a forecast model and a data assimilation method. The latter projects a collection of observations of reality into a model state. Key features of model error can be understood in terms of geometric properties of the data projection and a model attracting manifold. Model error can be resolved into two components: a projection error, which can be understood as the model’s attractor being in the wrong location given the data projection, and direction error, which can be understood as the trajectories of the model moving in the wrong direction compared to the projection of reality into model space. This investigation introduces some new tools and concepts, including the shadowing filter, causal and noncausal shadow analyses, and various geometric diagnostics. Various properties of forecast errors and model errors are described with reference to low-dimensional systems, like Lorenz’s equations; then, an operational weather forecasting system is shown to have the same predicted behavior. The concepts and tools introduced show promise for the diagnosis of model error and the improvement of ensemble forecasting systems.

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