A variational approach to numerical weather analysis and prediction

Abstract

New spatial variational principles and corresponding Weber's transformations for the adiabatic motion of a perfect fluid conditions of hydrostatic balance and geostrophic equilibrium are presented. A variational principle for barotropic flow is shown. The div-invariance of the Lagrangian density is used to determine the canonical form of the energy-momentum tensor for the Herivel-Lin principle stated in rotating coordinates. An alternative formulation of the Herivel-Lin principle is presented. Weber's transformation for this and principle due to Reid are derived. A first time integral of the vorticity equation is examined. The formulation of finite-difference analogues of the Euler-Lagrange equations is systematized and new conservation law analogues are derived. The Sasaki diagnostic technique is expanded to yield a generalized "balance" equation. The basic constraints are hydrostatic equilibrium and the conservation of horizontal velocity divergence. A surface friction term is included. The used of variational differences permits the use of techniques designed for elliptic systems. Computational results indicate that the constraint leads to a suitable state of balance away from the surface boundary layer. A seven-level model is used to make a one-half hour forecast. The forecast is used to find the Lagrangian multiplier fields. Although masked by an unstable difference analogue, it is found that the velocity field may be recovered effectively from these potential fields.