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Abstract
A nondivergent barotropic mode1 and a shallow-water model are presented that exploit a high-order shape-preserving scheme for the advection of vorticity or potential vorticity as well as tracers. The dissipation associated with the advection scheme is found to be due to the spreading of features as they are advected across a finite-resolution grid. The strength and scale selectivity of this dissipation are quantified using some simple tests. The resolved tracer variance and (potential) enstrophy are not conserved by the advection scheme; this can be interpreted as a cascade to unresolved scales. In simulations of turbulent mixing, satisfactory cascade of tracer variance, energy, and (potential) enstrophy are obtained without the need for any special parameterization of the cascade to unresolved scales.
Abstract
A nondivergent barotropic mode1 and a shallow-water model are presented that exploit a high-order shape-preserving scheme for the advection of vorticity or potential vorticity as well as tracers. The dissipation associated with the advection scheme is found to be due to the spreading of features as they are advected across a finite-resolution grid. The strength and scale selectivity of this dissipation are quantified using some simple tests. The resolved tracer variance and (potential) enstrophy are not conserved by the advection scheme; this can be interpreted as a cascade to unresolved scales. In simulations of turbulent mixing, satisfactory cascade of tracer variance, energy, and (potential) enstrophy are obtained without the need for any special parameterization of the cascade to unresolved scales.
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. One is the dissipation inherent in the advection scheme, which is estimated to be significantly stronger than the dissipation usually imposed in climate models of comparable resolution. The other is the grid structure, which leads to conspicuous symmetry errors in test cases where the true solution is symmetrical. The symmetry errors appear to arise because the hexagonal grid boxes are not perfectly regular but are somewhat distorted, particularly in certain regions of the grid, leading to larger truncation errors in the advection scheme in those regions.
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. One is the dissipation inherent in the advection scheme, which is estimated to be significantly stronger than the dissipation usually imposed in climate models of comparable resolution. The other is the grid structure, which leads to conspicuous symmetry errors in test cases where the true solution is symmetrical. The symmetry errors appear to arise because the hexagonal grid boxes are not perfectly regular but are somewhat distorted, particularly in certain regions of the grid, leading to larger truncation errors in the advection scheme in those regions.
Abstract
A shallow water model is used to simulate a case of planetary wave breaking in the lower winter stratosphere. The simulation is diagnosed in terms of zonal mean mass and zonal momentum budgets, and also in terms of potential vorticity (PV) contour mass and circulation budgets. The time evolution of the PV contour diagnostics depends only on nonconservative processes such as diabatic heating, friction, and irreversible small-scale mixing;transient but essentially reversible events such as a temporary displacement of the vortex from the pole are effectively filtered out. The PV contour diagnostics show unambiguously and quantitatively aspects of the evolution such as shrinking of the vortex and sharpening of the vortex edge. The cross-contour mass flux gives a radically different view of meridional transport from that given by the mass-weighted Eulerian mean poleward velocity, both in terms of its qualitative behavior and in terms of the physical mechanisms that cause it. The PV contour diagnostics can be used to define a balanced, zonally symmetric state, whose evolution can be compared directly with that of the Eulerian zonal mean state. A new expression is presented for finite-amplitude wave activity in terms of the PV contour diagnostics. Wave activity diagnostics for the wave-breaking simulation are shown. There are large differences between the zonal mean wave activity flux and its small-amplitude approximation, the Eliassen–Palm (EP) flux; some of the implications for interpreting EP flux diagnostics in the stratosphere are discussed.
Abstract
A shallow water model is used to simulate a case of planetary wave breaking in the lower winter stratosphere. The simulation is diagnosed in terms of zonal mean mass and zonal momentum budgets, and also in terms of potential vorticity (PV) contour mass and circulation budgets. The time evolution of the PV contour diagnostics depends only on nonconservative processes such as diabatic heating, friction, and irreversible small-scale mixing;transient but essentially reversible events such as a temporary displacement of the vortex from the pole are effectively filtered out. The PV contour diagnostics show unambiguously and quantitatively aspects of the evolution such as shrinking of the vortex and sharpening of the vortex edge. The cross-contour mass flux gives a radically different view of meridional transport from that given by the mass-weighted Eulerian mean poleward velocity, both in terms of its qualitative behavior and in terms of the physical mechanisms that cause it. The PV contour diagnostics can be used to define a balanced, zonally symmetric state, whose evolution can be compared directly with that of the Eulerian zonal mean state. A new expression is presented for finite-amplitude wave activity in terms of the PV contour diagnostics. Wave activity diagnostics for the wave-breaking simulation are shown. There are large differences between the zonal mean wave activity flux and its small-amplitude approximation, the Eliassen–Palm (EP) flux; some of the implications for interpreting EP flux diagnostics in the stratosphere are discussed.
Abstract
Using a variational technique, middle atmosphere gravity wave drag (GWD) is estimated from Met Office middle atmosphere analyses for the year 2002. The technique employs an adjoint model of a middle atmosphere dynamical model to minimize a cost function that measures the differences between the model state and observations. The control variables are solely the horizontal components of GWD; therefore, the minimization determines the optimal estimate of the drag. For each month, Met Office analyses are taken as the initial condition for the first day of the month, and also as observations for each successive day. In this way a three-dimensional GWD field is obtained for the entire year with a temporal resolution of 1 day. GWD shows a pronounced seasonal cycle. During solstices, there are deceleration regions of the polar jet centered at about 63° latitude in the winter hemisphere, with a peak of 49 m s−1 day−1 at 0.24 hPa in the Southern Hemisphere; the summer hemisphere also shows a deceleration region but much weaker, with a peak of 24 m s−1 day−1 centered at 45° latitude and 0.6 hPa. During equinoxes GWD is weak and exhibits a smooth transition between the winter and summer situation. The height and latitude of the deceleration center in both winter and summer hemispheres appear to be constant. Important longitudinal dependencies in GWD are found that are related to planetary wave activity; GWD intensifies in the exit region of jet streaks. In the lower tropical stratosphere, the estimated GWD shows a westward GWD descending together with the westward phase of the quasi-biennial oscillation. Above, GWD exhibits a semiannual pattern that is approximately out of phase with the semiannual oscillation in the zonal wind. Furthermore, a descending GWD pattern is found at those heights, similar in magnitude and sign to that in the lower stratosphere.
Abstract
Using a variational technique, middle atmosphere gravity wave drag (GWD) is estimated from Met Office middle atmosphere analyses for the year 2002. The technique employs an adjoint model of a middle atmosphere dynamical model to minimize a cost function that measures the differences between the model state and observations. The control variables are solely the horizontal components of GWD; therefore, the minimization determines the optimal estimate of the drag. For each month, Met Office analyses are taken as the initial condition for the first day of the month, and also as observations for each successive day. In this way a three-dimensional GWD field is obtained for the entire year with a temporal resolution of 1 day. GWD shows a pronounced seasonal cycle. During solstices, there are deceleration regions of the polar jet centered at about 63° latitude in the winter hemisphere, with a peak of 49 m s−1 day−1 at 0.24 hPa in the Southern Hemisphere; the summer hemisphere also shows a deceleration region but much weaker, with a peak of 24 m s−1 day−1 centered at 45° latitude and 0.6 hPa. During equinoxes GWD is weak and exhibits a smooth transition between the winter and summer situation. The height and latitude of the deceleration center in both winter and summer hemispheres appear to be constant. Important longitudinal dependencies in GWD are found that are related to planetary wave activity; GWD intensifies in the exit region of jet streaks. In the lower tropical stratosphere, the estimated GWD shows a westward GWD descending together with the westward phase of the quasi-biennial oscillation. Above, GWD exhibits a semiannual pattern that is approximately out of phase with the semiannual oscillation in the zonal wind. Furthermore, a descending GWD pattern is found at those heights, similar in magnitude and sign to that in the lower stratosphere.
Abstract
Discretizations of the linearized shallow-water equations on a spherical C grid are considered. Constraints on the schemes' coefficients that ensure conservation of mass, angular momentum, and energy are derived. These results generalize previously published results to the case of nonuniform and rotated grids (but are restricted to the linearized equations). Grids with υ stored at the poles and grids with u and h stored at the poles are both considered. Energy conservation is shown to be problematic for grids with u and h at the poles.
It is also shown that an inappropriate averaging of the Coriolis terms leads to a misrepresentation of the Rossby modes with shortest meridional scale. The appropriate averaging is shown to be compatible with the constraints required for conservation, and, indeed, the energy-conserving averaging of the Coriolis terms improves the dispersion properties of Rossby modes.
Abstract
Discretizations of the linearized shallow-water equations on a spherical C grid are considered. Constraints on the schemes' coefficients that ensure conservation of mass, angular momentum, and energy are derived. These results generalize previously published results to the case of nonuniform and rotated grids (but are restricted to the linearized equations). Grids with υ stored at the poles and grids with u and h stored at the poles are both considered. Energy conservation is shown to be problematic for grids with u and h at the poles.
It is also shown that an inappropriate averaging of the Coriolis terms leads to a misrepresentation of the Rossby modes with shortest meridional scale. The appropriate averaging is shown to be compatible with the constraints required for conservation, and, indeed, the energy-conserving averaging of the Coriolis terms improves the dispersion properties of Rossby modes.
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 experiments in which imposed changes in the ozone distribution or in the driving of the stratospheric residual mean meridional circulation lead to changes in tropopause height. In the Tropics the influence of the stratospheric circulation is found to extend down at least as far as the main convective outflow level, some 5 km below the temperature minimum.
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 experiments in which imposed changes in the ozone distribution or in the driving of the stratospheric residual mean meridional circulation lead to changes in tropopause height. In the Tropics the influence of the stratospheric circulation is found to extend down at least as far as the main convective outflow level, some 5 km below the temperature minimum.
Abstract
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.
Abstract
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.
Abstract
We hypothesize that the convective atmospheric boundary layer is marginally stable when the damping effects of turbulence are taken into account. If the effects of turbulence are modeled as an eddy viscosity and diffusivity, then an idealized analysis based on the hypothesis predicts a well-known scaling for the magnitude of the eddy viscosity and diffusivity. It also predicts that the marginally stable modes should have vertical and horizontal scales comparable to the boundary layer depth. A more quantitative numerical linear stability analysis is presented for a realistic convective boundary layer potential temperature profile and is found to support the hypothesis.
Abstract
We hypothesize that the convective atmospheric boundary layer is marginally stable when the damping effects of turbulence are taken into account. If the effects of turbulence are modeled as an eddy viscosity and diffusivity, then an idealized analysis based on the hypothesis predicts a well-known scaling for the magnitude of the eddy viscosity and diffusivity. It also predicts that the marginally stable modes should have vertical and horizontal scales comparable to the boundary layer depth. A more quantitative numerical linear stability analysis is presented for a realistic convective boundary layer potential temperature profile and is found to support the hypothesis.
Abstract
Currently, most operational forecasting models use latitude–longitude grids, whose convergence of meridians toward the poles limits parallel scaling. Quasi-uniform grids might avoid this limitation. Thuburn et al. and Ringler et al. have developed a method for arbitrarily structured, orthogonal C grids called TRiSK, which has many of the desirable properties of the C grid on latitude–longitude grids but which works on a variety of quasi-uniform grids. Here, five quasi-uniform, orthogonal grids of the sphere are investigated using TRiSK to solve the shallow-water equations.
Some of the advantages and disadvantages of the hexagonal and triangular icosahedra, a “Voronoi-ized” cubed sphere, a Voronoi-ized skipped latitude–longitude grid, and a grid of kites in comparison to a full latitude–longitude grid are demonstrated. It is shown that the hexagonal icosahedron gives the most accurate results (for least computational cost). All of the grids suffer from spurious computational modes; this is especially true of the kite grid, despite it having exactly twice as many velocity degrees of freedom as height degrees of freedom. However, the computational modes are easiest to control on the hexagonal icosahedron since they consist of vorticity oscillations on the dual grid that can be controlled using a diffusive advection scheme for potential vorticity.
Abstract
Currently, most operational forecasting models use latitude–longitude grids, whose convergence of meridians toward the poles limits parallel scaling. Quasi-uniform grids might avoid this limitation. Thuburn et al. and Ringler et al. have developed a method for arbitrarily structured, orthogonal C grids called TRiSK, which has many of the desirable properties of the C grid on latitude–longitude grids but which works on a variety of quasi-uniform grids. Here, five quasi-uniform, orthogonal grids of the sphere are investigated using TRiSK to solve the shallow-water equations.
Some of the advantages and disadvantages of the hexagonal and triangular icosahedra, a “Voronoi-ized” cubed sphere, a Voronoi-ized skipped latitude–longitude grid, and a grid of kites in comparison to a full latitude–longitude grid are demonstrated. It is shown that the hexagonal icosahedron gives the most accurate results (for least computational cost). All of the grids suffer from spurious computational modes; this is especially true of the kite grid, despite it having exactly twice as many velocity degrees of freedom as height degrees of freedom. However, the computational modes are easiest to control on the hexagonal icosahedron since they consist of vorticity oscillations on the dual grid that can be controlled using a diffusive advection scheme for potential vorticity.
Abstract
A vertical discretization of the primitive equations in a general vertical coordinate is described that enables a primitive equation model to use terrain-following sigma levels near the ground and isentropic levels higher up, with a smooth transition region in between. Therefore, it combines many of the advantages of the computational efficiency of σ coordinates and the predictive and diagnostic potential of θ coordinates, and should be particularly useful for general circulation models to be used for studies of stratosphere-troposphere exchange and middle-atmosphere transport of trace gases. It is shown that the semi-implicit time scheme can be used in a straightforward manner with this discretization. A discussion is given of how to optimize the transition from sigma levels to isentropic levels so as to avoid model levels crossing each other. A numerical problem caused when very shallow, very strong inversions occur in the temperature field is countered by a form of vertical-scale selective dissipation. Baroclinic wave life cycles and full general circulation simulations have been successfully performed with a modified version of the European Centre for Medium-Range Weather Forecasts model.
Abstract
A vertical discretization of the primitive equations in a general vertical coordinate is described that enables a primitive equation model to use terrain-following sigma levels near the ground and isentropic levels higher up, with a smooth transition region in between. Therefore, it combines many of the advantages of the computational efficiency of σ coordinates and the predictive and diagnostic potential of θ coordinates, and should be particularly useful for general circulation models to be used for studies of stratosphere-troposphere exchange and middle-atmosphere transport of trace gases. It is shown that the semi-implicit time scheme can be used in a straightforward manner with this discretization. A discussion is given of how to optimize the transition from sigma levels to isentropic levels so as to avoid model levels crossing each other. A numerical problem caused when very shallow, very strong inversions occur in the temperature field is countered by a form of vertical-scale selective dissipation. Baroclinic wave life cycles and full general circulation simulations have been successfully performed with a modified version of the European Centre for Medium-Range Weather Forecasts model.
