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R. J. Purser

Abstract

This paper is concerned with the fundamental role played by contact transformations and their corresponding generating functions in determining the structure and dynamical properties of a very general class of semigeostrophic theories possessing a Hamiltonian of the kind discovered by Salmon. It is shown that each member of this class of theories is associated with a self-adjoint tendency equation.

To illustrate the utility of the contact transformation concept, a new vortex theory is constructed by imposing upon the generating function a radial scaling symmetry, equivalent dynamically to the very reasonable constraint that small balanced perturbations about any state of solid-body rotation are simulated accurately. The new theory is manifestly a consistent generalization of the existing axisymmetric-vortex form of semigeostrophic dynamics and preserves the important conservation laws of mass, energy, potential vorticity, and angular momentum. In a sensitive idealized test, the new theory is shown to give reasonably accurate simulations of barotropic instability.

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R. J. Purser

Abstract

For semigeostrophic (SG) theories derived from the Hamiltonian principles suggested by Salmon it is known that a duality exists between the physical coordinates and geopotential, on the one hand, and isentropic geostrophic momentum coordinates and geostrophic Bernoulli function, on the other hand. The duality is characterized geometrically by a contact structure. This enables the idealized balanced dynamics to be represented by horizontal geostrophic motion in the dual coordinates while the mapping back to physical space is determined uniquely by requiring each instantaneous state to be the one of minimum energy with respect to volume-conserving rearrangements within the physical domain.

It is found that the generic contact structure permits the emergence of topological anomalies during the evolution of discontinuous flows. For both theoretical and computational reasons it is desirable to seek special forms of SG dynamics in which the structure of the contact geometry prohibits such anomalies. It is proven in this paper that this desideratum is equivalent to the existence of a mapping of geographical position to a Euclidean domain, combined with some position-dependent additive modification of the geopotential, which results in the SG theory being manifestly Legendre transformable from this alternative representation to its associated dual variables.

Legendre-transformable representations for standard Boussinesq f-plane SG theory and for the axisymmetric gradient-balance version used to study the Eliassen vortex are already known and exploited in finite element algorithms. Here, two other potentially useful classes of SG theory discussed in a recent paper by the author are reexamined: (i) the nonaxisymmetric f-plane vortex and (ii) hemispheric (variable f) SG dynamics. The authors find that the imposition of the natural dynamical and geometrical symmetry requirements together with the requirement of Legendre transformability makes the choice of the f-plane vortex theory unique. Moreover, with modifications to accommodate sphericity, this special vortex theory supplies what appears to be the most symmetrical and consistent formulation of variable-f SG theory on the hemisphere. The Legendre-transformable representations of these theories appear superficially to violate the original symmetry of rotation about the vortex axis. But, remarkably, this symmetry is preserved provided one interprets the metric of the new representation to be a pseudo-Euclidean Minkowski metric. Rotation invariance of the dynamical formulation in physical space is then perceived as a formal Lorentz invariance in its Legendre-transformable representation.

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M. J. P. Cullen and R. J. Purser

Abstract

The Lagrangian form of the semigeostrophic equations has been shown to possess discontinuous solutions that have been exploited as a simple model of fronts and other mesoscale flows. In this paper, it is shown that these equations can be integrated forward in time for arbitrarily long periods without breaking down, to give a “slow manifold” of solutions. In the absence of moisture, orography and surface friction, these solutions conserve energy, despite the appearance of discontinuities.

In previous work these solutions have been derived by making finite parcel approximations to the data. This paper shows that there is a unique solution to the equations with general piecewise smooth data, to which the finite parcel approximation converges. It is also shown that the time integration procedure is well defined, and that the solutions remain bounded for all finite times.

Most previous results on the finite parcel solutions are restricted to the case of a Boussinesq atmosphere on an f-plane with rigid-wall boundary conditions. In this paper the results are extended to non-Boussinesq fluids, free-surface and periodic boundary conditions, and variable Coriolis parameter. Previous work on a version of the theory for axisymmetric flows is extended to approximately axisymmetric flows. The behavior of the equations on a sphere and the effects of external forcing are discussed.

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M. J. P. Cullen and R. J. Purser

Abstract

The Lagrangian conservation law form of the semi-geostrophic equations used by Hoskins and others is studied further in two and three dimensions. A solution of the inviscid equations containing discontinuities corresponding to atmospheric fronts is shown to exist for all time under fairly general conditions, and to be unique if the potential vorticity is required to be nonnegative. Computational results show that this solution agrees with high resolution solutions of the viscous semi-geostrophic equations. The solution, however, disagrees with that obtained from the two-dimensional viscous primitive equations. An important aspect of the difference is that the semi-geostrophic solutions allow the front to propagate into the interior of the fluid while the primitive equation solutions do not. This is discussed. If correct, it may indicate a tendency for a separation effect in the atmosphere where frictional effects are present.

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R. J. Purser and L. M. Leslie

Abstract

It is shown both in theory and in practice that the accuracy of time-split finite-difference methods may be increased by the very simple device of incrementally adding a portion of the advection term in the adjustment step of the method.

In the first part of this study, the theory of the approach is described, and large gains in accuracy are demonstrated by comparison with an exact solution to a forced oscillatory system.

The technique is then applied to a considerable number (100) of 48-h forecasts with a barotropic numerical weather prediction (NWP) model using 500-hPa data on the Australian region NWP domain. It is found that the incrementally split scheme is more accurate than the conventional split scheme, which generates more noise and has unrealistically larger values of the root-mean-square ageostrophic wind. A further advantage of the new scheme is that it has superior initialization properties.

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R. James Purser and M. J. P. Cullen

Abstract

The semigeostrophic equations are a filtered approximation to the primitive equations that have been applied with considerable success to the study of atmospheric frontogenesis and, increasingly, to the study of the influence of orography on the balanced component of flow. In the simplest Boussinesq and f-plane form of these equations the requirement that the instantaneous solutions should not be symmetrically unstable can be expressed geometrically as the need for a certain potential related to the geopotential field to be a convex function. Conversely, any convex function can be associated directly with a stable solution of the semigeostrophic equations. This property of convexity is sufficient to allow a natural mapping from any instantaneous solution to a conjugate solution in which the roles of cyclonic and anticyclonic regions are reversed. In terms of this surprising duality,a solution possessing a mature surface front is associated with a conjugate solution whose flow is blocked or separated by orography, allowing some general conclusions concerning the appearance of surface fronts to be translated directly to corresponding statements concerning orographic blocking, and vice versa.

It is noted that instantaneous semigeostrophic solutions can be defined by a minimum energy principle, which also provides a physicgl interpretation of the convex potentials we introduce. Due to this variational principle it is possible to show that the typical onset of frontogenesis or of orographic separation from initially smooth data can be viewed in terms of the geometrical singularities of elementary catastrophe theory.

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R. J. Purser and L. M. Leslie

Abstract

Having recently demonstrated that significant enhancement of forecast accuracy in a semi-Lagrangian model results from the application of high-order time integration methods to the second-derivative form of the equations governing the trajectories, the authors here extend the range of available methods by introducing a class of what they call “generalized Lorenz” (GL) schemes. These explicit GL schemes, like Lorenz’s “N-cycle” methods, which inspired them, achieve a high formal accuracy in time for linear systems at an economy of storage that is the theoretical optimum. They are shown to possess robustly stable and consistent semi-implicit modifications that allow the deepest (fastest) gravity waves to be treated implicitly, so that integrations can proceed efficiently with time steps considerably longer than would be possible in an Eulerian framework.

Tests of the GL methods are conducted using an ensemble of 360 forecast cases over the Australian region at high spatial resolution, verifying at 48 h against a control forecast employing time steps sufficiently short to render time truncation errors negligible. Compared with the performance of the best alternative semi-Lagrangian treatment of equivalent storage economy (a quasi-second-order generalized Adams–Bashforth method), our new GL methods produce significant improvements both in formal accuracy and in actual forecast skill.

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L. M. Leslie and R. J. Purser

Abstract

Traditional finite-difference numerical forecast models usually employ relatively low-order approximations on grids staggered in both the horizontal and the vertical. In a previous study, Purser and Leslie (1988) demonstrated that high-order differencing on an unstaggered horizontal grid led to improved forecast accuracy. The present investigation has two aims. The first aim simply is to extend the earlier work to a three-dimensional formulation, by using high-order horizontal numerics in a time-split, three-dimensional, semi-Lagrangian model on a grid that is unstaggered in both the horizontal and vertical. The choice of an unstaggered grid is very effective in a semi-Lagrangian model as it ensures that a single set of interpolations suffices for all variables at each advection step. The second aim specifically is to increase the accuracy of the vertical discretization of key quantities such as the vertically integrated divergence, and the computation of the geopotential from the hydrostatic equation. Errors introduced in these terms potentially can have a large impact on the forecast accuracy. The increased accuracy also serves to mitigate any possible deterioration that might result from the adoption of a vertically unstaggered grid.

It is shown over a four-month period of daily 24-h forecasts that the use of vertical quadrature techniques, on the aforementioned terms, based on layer integrals of high-order interpolating Lagrange polynomials, leads to a significant reduction of about 5% in the root-mean-square errors of the geopotential and wind fields. A much greater improvement in model performance is found in the forecasts of vertical velocity and precipitation fields, as they are more sensitive to the new vertical discretization. Moreover, these gains are obtained at minimal computational cost both in time and storage.

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R. J. Purser and H-L. Huang

Abstract

An attempt is made to formulate consistent objective definitions of the concept of “effective data density” applicable both in the context of satellite soundings more generally in objective data analysis. The definitions based upon various forms of Backus-Gilbert “spread” functions are found to be seriously misleading in satellite soundings where the model resolution function (expressing the sensitivity of retrieval or analysis to changes in the background error) features sidelobes. Instead, estimates derived by smoothing the trace components of the model resolution function are proposed. The new estimates are found to be more reliable and informative in simulated satellite retrieval problems and, for the special case of uniformly spaced perfect observations, agree exactly with their actual density. The new estimates integrate to the “degrees of freedom for signal,” a diagnostic that is invariant to changes of units or coordinates used.

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R. J. Purser and L. M. Leslie

Abstract

An efficient method is proposed for performing the grid interpolations required at each advective time step of a multilevel, limited-area semi-Lagrangian model. The distinctive feature of the method is that it is composed of a cascade of one-dimensional interpolations of entire fields of data through a sequence of intermediate grids. These intermediate grids are formed as hybrid combinations of the standard model grid coordinates, which, together with the Lagrangian coordinates, are delineated by the origins of the trajectories that characterize the semi-Lagrangian method.

When applied at Nth-order accuracy, the cascade method requires only O(N) operations compared with O(N 3) for the conventional three-dimensional interpolation methods, making the adoption of high-order schemes attractive.

The technique was tested on a large number (100) of 48-h forecasts and was found to be as accurate as the conventional interpolation procedures based on point-by-point Cartesian products of one-dimensional inter-polators. However, the cascade interpolation technique was 2.9, 6.1, and 10.2 times as fast as the conventional interpolation scheme for fourth-, sixth-, and eighth-order, respectively.

We observe that the cascade method is equally applicable to the problem of interpolation from the grid of termini of forward trajectories to the standard model grid, for which there is no obvious counterpart in the Cartesian product method. Our technique therefore opens the way to a whole new class of high-order accurate semi-Lagrangian methods that incorporate the use of forward trajectories as part of the time-stepping process.

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