The omega equation and potential vorticity

Two fundamental perspectives on the dynamics of midlatitude weather systems are provided by potential vorticity (PV) and the omega equation. The aim of this paper is to investigate the link between the two perspectives, which has so far received very little attention in the meteorological literature. It also aims to give a quantitative basis for discussion of quasi-geostrophic vertical motion in terms of components associated with system movement, maintaining a constant thermal structure, and with the development of that structure. The former links with the isentropic relative-flow analysis technique. Viewed in a moving frame of reference, the measured development of a system depends on the velocity of that frame of reference. The requirement that the development should be a minimum provides a quantitative method for determining the optimum system velocity. The component of vertical velocity associated with development is shown to satisfy an omega equation with forcing determined from the relative advection of interior PV and boundary temperature. The analysis carries through in the presence of diabatic heating provided the omega equation forcing is based on the interior PV and boundary thermal tendencies, including the heating effect. The analysis is shown to be possible also at the level of the semi-geostrophic approximation. The analysis technique is applied to a number of idealized problems that can be considered to be building blocks for midlatitude synoptic-scale dynamics. They focus on the influences of interior PV, boundary temperature, an interior boundary, baroclinic instability associated with two boundaries, and also diabatic heating. In each case, insights yielded by the new perspective are sought into the dynamical behaviour, especially that related to vertical motion. Copyright © 2003 Royal Meteorological Society

Potential energy for slantwise parcel motion

Two formulations for the potential energy for slantwise motion are compared: one which applies strictly only to two-dimensional flows (SCAPE) and a three-dimensional formulation based on a Bernoulli equation. The two formulations share an identical contribution from the vertically integrated buoyancy anomaly and a contribution from different Coriolis terms. The latter arise from the neglect of (different) components of the total change in kinetic energy along a trajectory in the two formulations. This neglect is necessary in order to quantify the potential energy available for slantwise motion relative to a defined steady environment. Copyright © 2000 Royal Meteorological Society.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.