John Thuburn

A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids

A numerical scheme applicable to arbitrarily-structured C-grids is presented for the nonlinear shallow-water equations. By discretizing the vector-invariant form of the momentum equation, the relationship between the nonlinear Coriolis force and the potential vorticity flux can be used to guarantee that mass, velocity and potential vorticity evolve in a consistent and compatible manner. Underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field. The numerical scheme also exhibits conservation of total energy to within time-truncation error. Simulations of the standard shallow-water test cases confirm the analysis and show convergence rates between 1st- and 2nd-order accuracy when discretizing the system with quasi-uniform spherical Voronoi diagrams. The numerical method is applicable to a wide class of meshes, including latitude-longitude grids, Voronoi diagrams, Delaunay triangulations and conformally-mapped cubed-sphere meshes.

Stratospheric Influence on Tropopause Height: The Radiative Constraint

Abstract Earlier theoretical and modeling work introduced the concept of a radiative constraint relating tropopause height to tropospheric lapse rate and other factors such as surface temperature. Here a minimal quantitative model for the radiative constraint is presented and used to illustrate the essential physics underlying the radiative constraint, which involves the approximate balance between absorption and emission of thermal infrared (IR) radiation determining tropopause temperature. The results of the minimal model are then extended in two ways. First, the effects of including a more realistic treatment of IR radiation are quantified. Second, the radiative constraint model is extended to take into account non-IR warming processes such as solar heating and dynamical warming near the tropopause. The sensitivity of tropopause height to non-IR warming is estimated to be a few kilometers per K day−1, with positive warming leading to a lower tropopause. Sensitivities comparable to this are found in GCM...

Numerical representation of geostrophic modes on arbitrarily structured C-grids

A C-grid staggering, in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces (or edges in two dimensions) is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes. However, the discretization of the Coriolis terms is non-trivial. For constant Coriolis parameter, the linearized shallow water equations support geostrophic modes: stationary solutions in geostrophic balance. A naive discretization of the Coriolis terms can cause geostrophic modes to become non-stationary, causing unphysical behaviour of numerical solutions. Recent work has shown how to discretize the Coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the Coriolis terms remain energy conserving. In this paper this result is extended to arbitrarily structured C-grids. An explicit formula is given for constructing an appropriate discretization of the Coriolis terms. The general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal C-grid and the spherical longitude-latitude C-grid. Numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal-pentagonal and triangular geodesic C-grids on the sphere.

GCM Tests of Theories for the Height of the Tropopause

The sensitivity of the tropopause height to various external parameters has been investigated using a global circulation model (GCM). The tropopause height was found to be strongly sensitive to the temperature at the earth’s surface, less sensitive to the ozone distribution, and hardly sensitive at all to moderate changes in the earth’s rotation rate. The strong sensitivity to surface temperature occurs through changes in the atmospheric moisture distribution and its resulting radiative effects. The radiative and dynamical mechanisms thought to maintain the tropopause height have been investigated in some detail. The assumption that the lower stratosphere is close to radiative equilibrium leads to an easily computed relationship between tropospheric lapse rate and tropopause height. This relationship was found to hold well in the GCM in the extratropics away from the winter pole. Possible reasons for the breakdown of the relationship in the Tropics and near the winter pole are discussed. Simple relationships predicted by two different baroclinic adjustment theories, between parameters such as potential temperature gradients, the Coriolis parameter, and tropopause height, were examined. When some of these parameters were changed explicitly in GCM experiments, the remaining parameters, determined internally by the GCM, did not respond in the predicted way. These results cast doubt on the relevance of baroclinic adjustment to the height of the tropopause.

A PV-Based Shallow-Water Model on a Hexagonal-Icosahedral Grid

Abstract A new global shallow-water model has been developed. It uses a hexagonal–icosahedral grid, potential vorticity as a prognostic variable, and a conservative, shape-preserving scheme for advection of mass, potential vorticity, and tracers. A semi-implicit time scheme is used so that the maximum time step for stable integrations is limited by the advection speed rather than the gravity wave phase speed. This combination of numerical methods avoids some of the major problems of more traditional numerical methods, such as pole problems, and spurious oscillations and negatives in advected quantities. Sample results from a standard set of test cases are presented to illustrate the model’s performance. In a pure advection test case the model’s advection scheme shows good isotropy and phase-speed properties, but it is a little diffusive. In the remaining test cases the model’s overall accuracy is comparable to that of other gridpoint models for which results are available. Two sources of error are noted. ...

The two‐day wave in a middle atmosphere GCM

A strong signal of the two-day wave is diagnosed in a middle atmosphere GCM, with characteristics very similar to those of the observed two-day wave. It results from an instability, but its global structure shows similarities to a Rossby-gravity planetary normal mode. It has a remarkable potential vorticity structure in wave 3 and sometimes wave 4 and higher wavenumbers. It is shown that gravity wave drag is important in maintaining the unstable zonal mean state.

Sensitivity of mesospheric mean flow, planetary waves, and tides to strength of gravity wave drag

A global circulation model which extends from the surface to 125 km is used to study how the strength of gravity wave drag affects the dynamics of the mesosphere. The strength of gravity wave drag has a strong influence on the zonal mean state of the mesosphere, in particular the magnitude and variability of the summer mesopause temperature and the shear on the top of the mesospheric jets. This change in the zonal mean state strongly affects the susceptibility of the mesosphere to baroclinic and/or barotropic instability and hence the formation of the 2-day wave. The 2-day wave, in turn, interacts nonlinearly with the diurnal tide, producing secondary waves and a reduction in amplitude of the diurnal tide. Previous studies with quasi-linear mechanistic tidal models have captured some semiannual variation in tidal amplitude through direct interactions between tides and gravity waves. Our fully nonlinear global circulation model results support an alternative explanation in terms of interactions between planetary waves and tides.

Some conservation issues for the dynamical cores of NWP and climate models

The rationale for designing atmospheric numerical model dynamical cores with certain conservation properties is reviewed. The conceptual difficulties associated with the multiscale nature of realistic atmospheric flow, and its lack of time-reversibility, are highlighted. A distinction is made between robust invariants, which are conserved or nearly conserved in the adiabatic and frictionless limit, and non-robust invariants, which are not conserved in the limit even though they are conserved by exactly adiabatic frictionless flow. For non-robust invariants, a further distinction is made between processes that directly transfer some quantity from large to small scales, and processes involving a cascade through a continuous range of scales; such cascades may either be explicitly parameterized, or handled implicitly by the dynamical core numerics, accepting the implied non-conservation. An attempt is made to estimate the relative importance of different conservation laws. It is argued that satisfactory model performance requires spurious sources of a conservable quantity to be much smaller than any true physical sources; for several conservable quantities the magnitudes of the physical sources are estimated in order to provide benchmarks against which any spurious sources may be measured.

Numerical wave propagation on the hexagonal C-grid

Inertio-gravity mode and Rossby mode dispersion properties are examined for discretizations of the linearized rotating shallow water equations on a regular hexagonal C-grid in planar geometry. It is shown that spurious non-zero Rossby mode frequencies found by previous authors in the f-plane case can be avoided by an appropriate discretization of the Coriolis terms. Three generalizations of this discretization that conserve energy even for non-constant Coriolis parameter are presented. A quasigeostrophic @b-plane analysis is carried out to investigate the Rossby mode dispersion properties of these three schemes. The Rossby mode dispersion relation is found to have two branches. The primary branch modes are good approximations, in terms of both structure and frequency, to corresponding modes of the continuous governing equations, and offer some improvements over a quadrilateral C-grid scheme. The secondary branch modes have vorticity structures approximating those of small-scale modes of the continuous governing equations, suggesting that the hexagonal C-grid might have an advantage in terms of resolving extra Rossby modes; however, the frequencies of the secondary branch Rossby modes are much smaller than those of the corresponding continuous modes, so this potential advantage is not fully realized.

Poleward heat transport by the atmospheric heat engine

The atmospheric heat transport on Earth from the Equator to the poles is largely carried out by the mid-latitude storms. However, there is no satisfactory theory to describe this fundamental feature of the Earths climate. Previous studies have characterized the poleward heat transport as a diffusion by eddies of specified horizontal length and velocity scales, but there is little agreement as to what those scales should be. Here we propose instead to regard the baroclinic zone—the zone of strong temperature gradients and active eddies—as a heat engine which generates eddy kinetic energy by transporting heat from a warmer to a colder region. This view leads to a new velocity scale, which we have tested along with previously proposed length and velocity scales, using numerical climate simulations in which the eddy properties have been varied by changing forcing and boundary conditions. The experiments show that the eddy velocity varies in accordance with the new scale, while the size of the eddies varies with the well-known Rhines β-scale. Our results not only give new insight into atmospheric eddy heat transport, but also allow simple estimates of the intensities of mid-latitude storms, which have hitherto only been possible with expensive general circulation models.