In this paper we consider the time evolution of vortices simulated by the method of contour dynamics. Special attention is being paid to the Hamiltonian character of the governing equations and in particular to the conservational properties of numerical time integration for them. We assess symplectic and non-symplectic schemes. For the former methods, we give an implementation which is both efficient and yet effectively explicit. A number of numerical examples sustain the analysis and demonstrate the usefulness of the approach.

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SUMMARYA method is presented to manually adjust the analysis of a numerical weather-forecasting model and study the impact of the adjustment on the ensuing forecast. The adjustment is made in terms of potential vorticity. The prime motivation for using potential vorticity is that within the constraints of hydrostatic and geostrophic balance the pressure, wind and temperature fields are adjusted whenever the potential vorticity field is adjusted. This is a consequence of the invertibility principle of potential vorticity. Another motivation is that potential vorticity usually correlates well with water vapour satellite images at midlatitudes. Obvious mismatches between the two fields are therefore indicative of errors and may, in actual practice, be a reason to adjust the analysis. The method is based on three-dimensional variational data-assimilation and utilizes the special properties of potential vorticity.The potential vorticity of the analysis is manually adjusted by means of a graphical interface. The resulting modified potential vorticity field is treated as information that is to be incorporated into a new modified analysis. For this purpose we use three-dimensional variational data-assimilation, in which the observation term in the cost function is replaced by a potential vorticity term. In analogy with the observation term the potential vorticity term measures the difference between the modified potential vorticity and the potential vorticity of the model state. The extent to which the modified analysis reproduces the modified potential vorticity depends on the relative weights of the potential vorticity term and the background term in the cost function. An example is given of a modified analysis and its effect on a 24-hour forecast, using a limited-area weather-forecasting model.

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