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SUMMARYWe study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical geopotential and shallow atmosphere approximations in addition to the hydrostatic approximation. As is well known, a consistent application of the shallow atmosphere approximation requires omission of those Coriolis terms that vary as the cosine of latitude and of certain other terms in the components of the momentum equation. An approximate model is here regarded as consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity, and (following R. Müller) if its momentum component equations have Lagrange's form. Within these criteria, four consistent approximate global models, including the HPEs themselves, are identified in a heightcoordinate framework. The four models, each of which includes the spherical geopotential approximation, correspond to whether the shallow atmosphere and hydrostatic (or quasi-hydrostatic) approximations are individually made or not made. Restrictions on representing the spatial variation of apparent gravity occur. Solution methods and the situation in a pressure-coordinate framework are discussed.

Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.

† The contributions of these authors were written in the course of their employment at the Met Office, UK and are published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.Three data assimilation methods are compared for their ability to produce the best analysis: (i) 4DVar, four-dimensional variational data assimilation using linear and adjoint models with either a (perfect) 3D climatological background-error covariance or a 3D ensemble background-error covariance; (ii) EDA, an ensemble of 4DEnVars, which is a variational method using a 4D ensemble covariance; and (iii) the deterministic ensemble Kalman filter (DEnKF, also using a 4D ensemble covariance).The accuracy of the deterministic analysis from each method was measured for both perfect and imperfect toy model experiments. With a perfect model, 4DVar with the climatological covariance is easily beaten by the ensemble methods, due to the importance of flow-dependent background-error covariances. When model error is present, 4DVar is more competitive and its relative performance is improved by increasing the observation density. This is related to the model error representation in the background-error covariance.The dynamical time-consistency of the 4D ensemble background-error covariance is degraded by the localization, since the localization function and the nonlinear model do not commute. As a result, 4DVar with the ensemble covariance performs significantly better than the other ensemble methods when severe localization is required, i.e. for a small ensemble.