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- Author or Editor: Robert Sadourny x
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Abstract
A class of conservative finite-difference approximations of the primitive equations is given for quasi-uniform spherical grids derived from regular polyhedrons. The earth is split into several contiguous regions. Within each region, a coordinate system derived from central projections is used, instead of the spherical coordinate system, to avoid the use of inconsistent boundary conditions at the poles. The presence of artificial internal boundaries has no effect on the conservation properties of the approximations. Examples of conservative schemes, up to the second order in the case of a cube, are given. A selective damping operator is needed to remove the two-grid interval waves generated by the existence of internal boundaries.
Abstract
A class of conservative finite-difference approximations of the primitive equations is given for quasi-uniform spherical grids derived from regular polyhedrons. The earth is split into several contiguous regions. Within each region, a coordinate system derived from central projections is used, instead of the spherical coordinate system, to avoid the use of inconsistent boundary conditions at the poles. The presence of artificial internal boundaries has no effect on the conservation properties of the approximations. Examples of conservative schemes, up to the second order in the case of a cube, are given. A selective damping operator is needed to remove the two-grid interval waves generated by the existence of internal boundaries.
Abstract
Two simple numerical models of the shallow-water equations identical in all respects but for their con-servation properties have been tested regarding their internal mixing processes. The experiments show that violation of enstrophy conservation results in a spurious accumulation of rotational energy in the smaller scales, reflected by an unrealistic increase of enstrophy, which ultimately produces a finite rate of energy dissipation in the zero viscosity limit, thus violating the well-known dynamics of two-dimensional flow. Further, the experiments show a tendency to equipartition of the kinetic energy of the divergent part of the flow in the inviscid limit, suggesting the possibility of a divergent energy cascade in the physical system, as well as a possible influence of the energy mixing on the process of adjustment toward balanced flow.
Abstract
Two simple numerical models of the shallow-water equations identical in all respects but for their con-servation properties have been tested regarding their internal mixing processes. The experiments show that violation of enstrophy conservation results in a spurious accumulation of rotational energy in the smaller scales, reflected by an unrealistic increase of enstrophy, which ultimately produces a finite rate of energy dissipation in the zero viscosity limit, thus violating the well-known dynamics of two-dimensional flow. Further, the experiments show a tendency to equipartition of the kinetic energy of the divergent part of the flow in the inviscid limit, suggesting the possibility of a divergent energy cascade in the physical system, as well as a possible influence of the energy mixing on the process of adjustment toward balanced flow.
Abstract
The hexagonal grid based on a partition of the icosahedron has distinct geometrical qualities for the mapping of a sphere and also presents some indexing difficulties. The applicability of this grid to the primitive equations of fluid dynamics is demonstrated, and a finite-difference approximation of these equations is proposed. The basic variables are the mass fluxes from one hexagonal cell to the next through their common boundary. This scheme conserves the total mass, the total momentum, and the total kinetic energy of the fluid as well as the total squared vorticity of a nondivergent flow. A computational test was performed using a hexagonal grid to describe space periodic waves on a nonrotating plane. The systematic variation of total kinetic and potential energy is less than 10−5 after 1,000 time steps.
Abstract
The hexagonal grid based on a partition of the icosahedron has distinct geometrical qualities for the mapping of a sphere and also presents some indexing difficulties. The applicability of this grid to the primitive equations of fluid dynamics is demonstrated, and a finite-difference approximation of these equations is proposed. The basic variables are the mass fluxes from one hexagonal cell to the next through their common boundary. This scheme conserves the total mass, the total momentum, and the total kinetic energy of the fluid as well as the total squared vorticity of a nondivergent flow. A computational test was performed using a hexagonal grid to describe space periodic waves on a nonrotating plane. The systematic variation of total kinetic and potential energy is less than 10−5 after 1,000 time steps.
Abstract
A lateral diffusion scheme designed to efficiently parameterize the subgrid scale lures associated with barotropic and baroclinic transients is presented and tested on a quasi-geostrophic, two layer model, with law-scale thermal forcing. The scheme is based on formal energy conservation and potential enstrophy dissipation. At very coarse resolutions, where the cutoff scale is of the order of the internal radius of deformation, the diffusion scheme is shown to produce a realistic amount of potential-to-kinetic energy conversions, and realistic amplitudes of the large-scale barotropic modes.
Abstract
A lateral diffusion scheme designed to efficiently parameterize the subgrid scale lures associated with barotropic and baroclinic transients is presented and tested on a quasi-geostrophic, two layer model, with law-scale thermal forcing. The scheme is based on formal energy conservation and potential enstrophy dissipation. At very coarse resolutions, where the cutoff scale is of the order of the internal radius of deformation, the diffusion scheme is shown to produce a realistic amount of potential-to-kinetic energy conversions, and realistic amplitudes of the large-scale barotropic modes.
Abstract
The variability of atmospheric flow is analyzed by separating it into an internal part due to atmospheric dynamics only and an external (or forced) part due to the variability of sea surface temperature forcing. The two modes of variability are identified by performing an ensemble of seven independent long-term simulations of the atmospheric response to observed SST (1970–1988) with the LMD atmospheric general circulation model. The forced variability is defined from the analysis of the ensemble mean and the internal variability from the analysis of deviations from the ensemble mean. Emphasis is put on interannual variability of sea level pressure and 5OO-hPa geopotential height for the Northern Hemisphere winter. In view of the large systematic errors related to the relatively small number of realizations, unbiased variance estimators have been developed. Although statistical significance is not reached in some extratropical regions, large significant extratropical responses are found at the North Pacific-Alaska sector for SLP and over western Canada and the Aleutians for 5OO-hPa geopotential height. The influence of SST variations on internal variability is also examined by using a 7-year simulation using the climatological SST seasonal cycle. It is found that interannual SST changes strongly influence the geographical distribution of internal variability; in particular, it tends to increase it over oceans. Patterns of internal and external variability of the 5OO-hPa geopotential height are further examined by using EOF decompositions showing that the model realistically simulates the leading observed variability modes. The geographical structure of internal variability patterns is found to be similar to that of total variability, although similar modes tend to evolve rather differently in time. The zonally symmetric seesaw dominates the internal variability for both observed and climatologically prescribed SST. The Pacific-North American (PNA) and Western Pacific (WP) patterns, on the other hand, are the dominant modes associated with patterns of SST variability: the latter is related to Atlantic anomalies, while the former responds to both El Niño events and extratropical forcing.
Abstract
The variability of atmospheric flow is analyzed by separating it into an internal part due to atmospheric dynamics only and an external (or forced) part due to the variability of sea surface temperature forcing. The two modes of variability are identified by performing an ensemble of seven independent long-term simulations of the atmospheric response to observed SST (1970–1988) with the LMD atmospheric general circulation model. The forced variability is defined from the analysis of the ensemble mean and the internal variability from the analysis of deviations from the ensemble mean. Emphasis is put on interannual variability of sea level pressure and 5OO-hPa geopotential height for the Northern Hemisphere winter. In view of the large systematic errors related to the relatively small number of realizations, unbiased variance estimators have been developed. Although statistical significance is not reached in some extratropical regions, large significant extratropical responses are found at the North Pacific-Alaska sector for SLP and over western Canada and the Aleutians for 5OO-hPa geopotential height. The influence of SST variations on internal variability is also examined by using a 7-year simulation using the climatological SST seasonal cycle. It is found that interannual SST changes strongly influence the geographical distribution of internal variability; in particular, it tends to increase it over oceans. Patterns of internal and external variability of the 5OO-hPa geopotential height are further examined by using EOF decompositions showing that the model realistically simulates the leading observed variability modes. The geographical structure of internal variability patterns is found to be similar to that of total variability, although similar modes tend to evolve rather differently in time. The zonally symmetric seesaw dominates the internal variability for both observed and climatologically prescribed SST. The Pacific-North American (PNA) and Western Pacific (WP) patterns, on the other hand, are the dominant modes associated with patterns of SST variability: the latter is related to Atlantic anomalies, while the former responds to both El Niño events and extratropical forcing.
Abstract
In order to represent in a most adequate way the various feedback mechanisms that govern the atmosphere–ocean or atmosphere–surface couplings, a “delocalized physics” method is introduced, in which the subgrid-scale physical parameterizations of the atmospheric model are computed on a grid defined independently of the grid where adiabatic dynamics are computed. This “physical grid,” which can be irregular and time dependent, is defined by juxtaposing the surface grids constructed independently for the ocean, land, and sea-ice models. The delocalization method allows taking into account the nonlinearities of vertical fluxes due to heterogeneities in the finescale surface properties that are not resolved by the adiabatic atmospheric dynamics calculations. The impact of delocalizing the physics to fit a given higher resolution surface grid is tested first on experiments where the global atmosphere is forced by observed ocean temperatures. The comparison demonstrates a significant improvement of the predominant variability mode of the outgoing longwave radiation field, corresponding mainly to the seasonal cycle. The delocalized physics method is then tested on an experiment where the General circulation model of the Laboratoire de Météorologie Dynamique is coupled to the Laboratoire d’Océanographie Dynamique et de Climatologie tropical Pacific ocean model. Delocalized physics allow surface fluxes to respond better to fine sea surface temperatures structures like the Legekis waves produced by the ocean model.
Abstract
In order to represent in a most adequate way the various feedback mechanisms that govern the atmosphere–ocean or atmosphere–surface couplings, a “delocalized physics” method is introduced, in which the subgrid-scale physical parameterizations of the atmospheric model are computed on a grid defined independently of the grid where adiabatic dynamics are computed. This “physical grid,” which can be irregular and time dependent, is defined by juxtaposing the surface grids constructed independently for the ocean, land, and sea-ice models. The delocalization method allows taking into account the nonlinearities of vertical fluxes due to heterogeneities in the finescale surface properties that are not resolved by the adiabatic atmospheric dynamics calculations. The impact of delocalizing the physics to fit a given higher resolution surface grid is tested first on experiments where the global atmosphere is forced by observed ocean temperatures. The comparison demonstrates a significant improvement of the predominant variability mode of the outgoing longwave radiation field, corresponding mainly to the seasonal cycle. The delocalized physics method is then tested on an experiment where the General circulation model of the Laboratoire de Météorologie Dynamique is coupled to the Laboratoire d’Océanographie Dynamique et de Climatologie tropical Pacific ocean model. Delocalized physics allow surface fluxes to respond better to fine sea surface temperatures structures like the Legekis waves produced by the ocean model.
Abstract
A finite difference scheme is developed for numerical integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid covering the sphere. The grid is made by dividing the 20 triangular faces of an icosahedron into smaller triangles, the vertices of which are the grid points. Each grid point is surrounded by six neighboring points, except the 12 vertices of the icosahedron which are surrounded by five points. The difference scheme for the advection of vorticity exactly conserves total vorticity, total square vorticity, and total kinetic energy. A numerical test is made, with a stationary Neamtan wave as the initial condition, by integrating over 8 days with 1-hr. time steps and a grid of 1002 points for the sphere. There is practically no distortion of the waves over the 8 days, but there is a phase displacement error of about 1° of long. per day toward the west.
Abstract
A finite difference scheme is developed for numerical integration of the nondivergent barotropic vorticity equation with an icosahedral-hexagonal grid covering the sphere. The grid is made by dividing the 20 triangular faces of an icosahedron into smaller triangles, the vertices of which are the grid points. Each grid point is surrounded by six neighboring points, except the 12 vertices of the icosahedron which are surrounded by five points. The difference scheme for the advection of vorticity exactly conserves total vorticity, total square vorticity, and total kinetic energy. A numerical test is made, with a stationary Neamtan wave as the initial condition, by integrating over 8 days with 1-hr. time steps and a grid of 1002 points for the sphere. There is practically no distortion of the waves over the 8 days, but there is a phase displacement error of about 1° of long. per day toward the west.
Abstract
The inhibition of baroclinic instability in low-resolution quasi-geostrophic models is studied using a simplified second-order closure approximation. It is shown that this inhibition, which results in systematic overestimation of baroclinic energy and systematic underestimation of barotropic energy, is the result of spurious energy dissipation due to the use of inadequate lateral diffusion. Consequently, a possible methodology for the parameterization of baroclinic instability in low-resolution models is proposed, based on the construction of diffusion operators able to dissipate potential enstrophy while conserving energy exactly.
Abstract
The inhibition of baroclinic instability in low-resolution quasi-geostrophic models is studied using a simplified second-order closure approximation. It is shown that this inhibition, which results in systematic overestimation of baroclinic energy and systematic underestimation of barotropic energy, is the result of spurious energy dissipation due to the use of inadequate lateral diffusion. Consequently, a possible methodology for the parameterization of baroclinic instability in low-resolution models is proposed, based on the construction of diffusion operators able to dissipate potential enstrophy while conserving energy exactly.
Abstract
Simple second-order closure models of quasi-geostrophic turbulence are derived, applying either to two-layer flows within isentropic boundaries, or to Eady-type frontogenesis with vanishing potential vorticity; homogeneity and horizontal isotropy are used as simplifying assumptions. Long-term numerical integrations of the two models are performed to obtain the structure of regime flows under stationary large-scale baroclinic forcing. The various cascade processes and the corresponding inertial ranges are discussed and visualized, showing characteristic differences between fully developed baroclinic instability and the linear theory. Further applications of such models may include studies of truncation effects on the efficiency of baroclinic instability.
Abstract
Simple second-order closure models of quasi-geostrophic turbulence are derived, applying either to two-layer flows within isentropic boundaries, or to Eady-type frontogenesis with vanishing potential vorticity; homogeneity and horizontal isotropy are used as simplifying assumptions. Long-term numerical integrations of the two models are performed to obtain the structure of regime flows under stationary large-scale baroclinic forcing. The various cascade processes and the corresponding inertial ranges are discussed and visualized, showing characteristic differences between fully developed baroclinic instability and the linear theory. Further applications of such models may include studies of truncation effects on the efficiency of baroclinic instability.
Abstract
The classical phenomenological relations between dispersion laws, second-order structure functions and energy spectra are reexamined from a more quantitative standpoint. It is shown that when a nonlocal energy spectrum (steeper than k −3) is substituted into the relation giving structure functions or dispersion laws, an infrared divergence occurs so that the structure functions or dispersion laws at inertial-range separation are not dominated by contributions from the inertial range spectrum; they saturate and become independent of spectral steepness. It follows that the spectral steepness of real flows in the enstrophy inertial range must be extremely difficult to estimate from correlation or dispersion measurements alone. This might explain why the existence of steep spectra, speculated on the basis of numerical modeling, has not been confirmed by real flow measurements.
Abstract
The classical phenomenological relations between dispersion laws, second-order structure functions and energy spectra are reexamined from a more quantitative standpoint. It is shown that when a nonlocal energy spectrum (steeper than k −3) is substituted into the relation giving structure functions or dispersion laws, an infrared divergence occurs so that the structure functions or dispersion laws at inertial-range separation are not dominated by contributions from the inertial range spectrum; they saturate and become independent of spectral steepness. It follows that the spectral steepness of real flows in the enstrophy inertial range must be extremely difficult to estimate from correlation or dispersion measurements alone. This might explain why the existence of steep spectra, speculated on the basis of numerical modeling, has not been confirmed by real flow measurements.
