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- Author or Editor: DAVID WILLIAMSON x

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## Abstract

Discrete approximations to hyperbolic partial differential equations governing frictionless two-dimensional fluid flow are developed in Cartesian geometry for use over arbitrary triangular grids. A class of schemes is developed that conserves mass, momentum, and total energy. The terms of the governing equations are also approximated individually and their truncation error is examined. For test integrations, the schemes are applied to an equilateral triangular (homogeneous) grid on a beta plane. In one case, the same scheme is integrated over a square grid for comparison between four- and six-point differences. Both second- and fourth-order schemes are integrated and compared with a fine resolution solution.

## Abstract

Discrete approximations to hyperbolic partial differential equations governing frictionless two-dimensional fluid flow are developed in Cartesian geometry for use over arbitrary triangular grids. A class of schemes is developed that conserves mass, momentum, and total energy. The terms of the governing equations are also approximated individually and their truncation error is examined. For test integrations, the schemes are applied to an equilateral triangular (homogeneous) grid on a beta plane. In one case, the same scheme is integrated over a square grid for comparison between four- and six-point differences. Both second- and fourth-order schemes are integrated and compared with a fine resolution solution.

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## Abstract

Simulations are compared to determine the effect of the details of the coupling of the parameterization suite with the dynamical core on the simulated climate. Simulations based on time-split and process-split couplings are compared to a simulation with the original version of the NCAR Community Climate Model–3 (CCM3), which is a mixture of the two approaches. In the *process-split* coupling, the two components are based on the same state and their tendencies are added to produce the updated state. In the *time-split* coupling, the two components are calculated sequentially, each based on the state produced by the other. Overall the differences between simulations produced with the various coupling strategies are relatively small. Thus, with the time step used in the CCM3, the different time truncation errors introduced by the different coupling strategies have less effect on simulations than other arbitrary aspects of the model design. This does not imply that the time truncation errors are insignificant, just that they are similar in the cases examined here. There are, however, regions where the differences are statistically significant. The differences in the thermal balance are analyzed in these regions. The most notable differences occur between the time-split case and CCM3 over regions of Antarctica. In summer, although the temperature difference near the surface is modest, the balance of terms in the two cases is very different, with a difference in sign in the sensible heat flux between the two cases. In winter, the parameterization terms have a very strong grid-scale structure associated with parameterized clouds forming predominantly at a single grid level. The dynamics is unable to respond with a grid-scale structure. This draws into question whether the vertical resolution is adequate to properly model the physical processes.

## Abstract

Simulations are compared to determine the effect of the details of the coupling of the parameterization suite with the dynamical core on the simulated climate. Simulations based on time-split and process-split couplings are compared to a simulation with the original version of the NCAR Community Climate Model–3 (CCM3), which is a mixture of the two approaches. In the *process-split* coupling, the two components are based on the same state and their tendencies are added to produce the updated state. In the *time-split* coupling, the two components are calculated sequentially, each based on the state produced by the other. Overall the differences between simulations produced with the various coupling strategies are relatively small. Thus, with the time step used in the CCM3, the different time truncation errors introduced by the different coupling strategies have less effect on simulations than other arbitrary aspects of the model design. This does not imply that the time truncation errors are insignificant, just that they are similar in the cases examined here. There are, however, regions where the differences are statistically significant. The differences in the thermal balance are analyzed in these regions. The most notable differences occur between the time-split case and CCM3 over regions of Antarctica. In summer, although the temperature difference near the surface is modest, the balance of terms in the two cases is very different, with a difference in sign in the sensible heat flux between the two cases. In winter, the parameterization terms have a very strong grid-scale structure associated with parameterized clouds forming predominantly at a single grid level. The dynamics is unable to respond with a grid-scale structure. This draws into question whether the vertical resolution is adequate to properly model the physical processes.

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## Abstract

Conservative finite-difference approximations are developed for the primitive barotropic model over a spherical geodesic grid. Truncation error considerations show that the grid resolution must be at least as fine as 2½° in order for the error not to dominate the mass flux calculations. When the fine resolution is used, the approximations are seen to be quite good. Comparisons are made with schemes in use today with approximately the same resolution applied to the same initial condition.

## Abstract

Conservative finite-difference approximations are developed for the primitive barotropic model over a spherical geodesic grid. Truncation error considerations show that the grid resolution must be at least as fine as 2½° in order for the error not to dominate the mass flux calculations. When the fine resolution is used, the approximations are seen to be quite good. Comparisons are made with schemes in use today with approximately the same resolution applied to the same initial condition.

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## Abstract

A series of 5-day forecasts is made with the National Center for Atmospheric Research (NCAR) Global Circulation Model (GCM) starting from the National Meteorological Center (NMC) analysis of 0000 GMT 11 January 1973. The six-layer model, formulated with height as a vertical coordinate, is integrated without mountains. Hemispheric forecasts are made with 5°, 2½° and 1¼° horizontal resolution and second- and fourth-order horizontal, centered finite-difference approximations. Integrations are also carried out with two types of horizontal diffusion. The first, similar to that usually used in the NCAR GCM, has the form ∇ṁ*K*∇, where *K* is a nonlinear coefficient dependent on the deformation. The second, more scale selective, is of the form ∇^{2}
*K*
^{2}, where the nonlinear coefficient *K* is the same as in the first type.

The eddy kinetic energy and 6 km pressure patterns of the forecasts are examined in detail. The fourth-order approximations result in the same general kinetic energy characteristics as the second-order, with slightly more aliasing. The ∇ṁ*K*∇ diffusion produces excessive damping of all scales especially with the 5° grid, whereas the more scale-selective ∇^{2}
*K*∇^{2} diffusion controls the small scales with less damping of the baroclinic scales. The 6 km pressure patterns are examined in terms of root-mean-square errors in spherical harmonic spectral bands with the contributions from both phase and amplitude errors separated. Fourth-order accuracy results in improvements in the phases of the shorter waves (wavenumber 10–18), but not in the largest waves where the second-order approximations have sufficient accuracy. The improvement in the amplitude of the largest scales (wavenumbers 1–3) with the finer resolution can be attributed to the accompanying decreased diffusion rather than more accurate approximations. This amplitude improvement in the largest scales is also seen in 5° forecasts which use a smaller diffusion coefficient, or more scale-selective diffusion, and is accompanied by improvement in the phase of the small scales through more accurate advection.

## Abstract

A series of 5-day forecasts is made with the National Center for Atmospheric Research (NCAR) Global Circulation Model (GCM) starting from the National Meteorological Center (NMC) analysis of 0000 GMT 11 January 1973. The six-layer model, formulated with height as a vertical coordinate, is integrated without mountains. Hemispheric forecasts are made with 5°, 2½° and 1¼° horizontal resolution and second- and fourth-order horizontal, centered finite-difference approximations. Integrations are also carried out with two types of horizontal diffusion. The first, similar to that usually used in the NCAR GCM, has the form ∇ṁ*K*∇, where *K* is a nonlinear coefficient dependent on the deformation. The second, more scale selective, is of the form ∇^{2}
*K*
^{2}, where the nonlinear coefficient *K* is the same as in the first type.

The eddy kinetic energy and 6 km pressure patterns of the forecasts are examined in detail. The fourth-order approximations result in the same general kinetic energy characteristics as the second-order, with slightly more aliasing. The ∇ṁ*K*∇ diffusion produces excessive damping of all scales especially with the 5° grid, whereas the more scale-selective ∇^{2}
*K*∇^{2} diffusion controls the small scales with less damping of the baroclinic scales. The 6 km pressure patterns are examined in terms of root-mean-square errors in spherical harmonic spectral bands with the contributions from both phase and amplitude errors separated. Fourth-order accuracy results in improvements in the phases of the shorter waves (wavenumber 10–18), but not in the largest waves where the second-order approximations have sufficient accuracy. The improvement in the amplitude of the largest scales (wavenumbers 1–3) with the finer resolution can be attributed to the accompanying decreased diffusion rather than more accurate approximations. This amplitude improvement in the largest scales is also seen in 5° forecasts which use a smaller diffusion coefficient, or more scale-selective diffusion, and is accompanied by improvement in the phase of the small scales through more accurate advection.

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## Abstract

The linear stability condition for explicit, second-order, centered difference approximations to the shallow-water equations on a uniform latitude-longitude spherical grid is determined. The grid has points at the poles and on the equator. Two modifications involving Fourier filtering in the longitudinal direction are considered to permit a longer stable time step. In the first, the prognostic variables are filtered, while in the second the zonal pressure gradient in the zonal momentum equation and the zonal divergence in the continuity equation are filtered. Both modifications permit approximately the same time step when the wavenumber cutoff is the same. Although filtering can he performed at all latitudes, the most efficient procedure involves filtering only near the poles. To allow a reasonable time step, all but approximately the six longest waves must be removed at the first row of points next to the pole. This filtering next to the pole determines a time step such that additional filtering is necessary at only a few latitudes, usually poleward of 80°.

The stability condition is also found for second- and fourth-order approximations on a shifted mesh in which the first grid points are a half-grid interval away from the pole. Only filtering of the prognostic variables is considered. As in the previous case, to allow a reasonable time step, only the longest waves can be. retained next to the pole. This filtering determines a time step such that filtering is not required at all latitudes. When such filtering is performed, the second- and fourth-order schemes have approximately the same stable time step. The modifications to the free oscillations of the model caused by the filtering are also discussed.

## Abstract

The linear stability condition for explicit, second-order, centered difference approximations to the shallow-water equations on a uniform latitude-longitude spherical grid is determined. The grid has points at the poles and on the equator. Two modifications involving Fourier filtering in the longitudinal direction are considered to permit a longer stable time step. In the first, the prognostic variables are filtered, while in the second the zonal pressure gradient in the zonal momentum equation and the zonal divergence in the continuity equation are filtered. Both modifications permit approximately the same time step when the wavenumber cutoff is the same. Although filtering can he performed at all latitudes, the most efficient procedure involves filtering only near the poles. To allow a reasonable time step, all but approximately the six longest waves must be removed at the first row of points next to the pole. This filtering next to the pole determines a time step such that additional filtering is necessary at only a few latitudes, usually poleward of 80°.

The stability condition is also found for second- and fourth-order approximations on a shifted mesh in which the first grid points are a half-grid interval away from the pole. Only filtering of the prognostic variables is considered. As in the previous case, to allow a reasonable time step, only the longest waves can be. retained next to the pole. This filtering determines a time step such that filtering is not required at all latitudes. When such filtering is performed, the second- and fourth-order schemes have approximately the same stable time step. The modifications to the free oscillations of the model caused by the filtering are also discussed.

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## Abstract

Conventional procedures designed to balance global initial data for primitive equation forecast models often result in unrealistic large-amplitude, high-frequency oscillations during the initial stages of the forecasts. In an attempt to reduce these oscillations, Dickinson and Williamson (1972) proposed a method to initialize data by expanding the data into the normal modes or free oscillations of the linearized version of the forecast model. Once the data are expanded into the normal modes, the modal amplitudes thought to be erroneously large can be reduced or set to zero. This procedure is tested here with the shallow water equations. In the first set of one-day forecasts performed, the method eliminates the large-amplitude, high-frequency waves which occur when using analyzed heights and winds for initial data by removing the gravity waves and computational Rossby waves from the initial data. The standard deviation of the error and the *S*
_{1} skill score show substantial improvement in the filtered case. This improvement is a result of the smoothing due to the initial filtering rather than an improvement in the forecast of the waves retained. When included, the gravity waves do not interact significantly with the Rossby waves during the one-day forecast.

Additional experiments are performed to examine the effect on the one-day forecast of removing the small-scale Rossby waves from the initial data. In general, except for the smallest longitudinal-scale Rossby waves, removal of these modes degrades the forecasts. A third set of forecasts examines the effect of the large-scale gravity waves on the forecast. The largest latitudinal-scale gravity waves have little effect on the forecast skill scores; they neither improve nor degrade the forecast with the shallow water equations. Inclusion of the medium-and smaller-scale gravity waves in the initial data degrades the forecasts. Several forecasts are repeated with the mean depth decreased. The conclusions with respect to the modal filtering are unchanged although the impact of the filtering is less dramatic in these cases. The results are also insensitive to the particular longitudinal filtering used near the poles to allow longer time steps.

## Abstract

Conventional procedures designed to balance global initial data for primitive equation forecast models often result in unrealistic large-amplitude, high-frequency oscillations during the initial stages of the forecasts. In an attempt to reduce these oscillations, Dickinson and Williamson (1972) proposed a method to initialize data by expanding the data into the normal modes or free oscillations of the linearized version of the forecast model. Once the data are expanded into the normal modes, the modal amplitudes thought to be erroneously large can be reduced or set to zero. This procedure is tested here with the shallow water equations. In the first set of one-day forecasts performed, the method eliminates the large-amplitude, high-frequency waves which occur when using analyzed heights and winds for initial data by removing the gravity waves and computational Rossby waves from the initial data. The standard deviation of the error and the *S*
_{1} skill score show substantial improvement in the filtered case. This improvement is a result of the smoothing due to the initial filtering rather than an improvement in the forecast of the waves retained. When included, the gravity waves do not interact significantly with the Rossby waves during the one-day forecast.

Additional experiments are performed to examine the effect on the one-day forecast of removing the small-scale Rossby waves from the initial data. In general, except for the smallest longitudinal-scale Rossby waves, removal of these modes degrades the forecasts. A third set of forecasts examines the effect of the large-scale gravity waves on the forecast. The largest latitudinal-scale gravity waves have little effect on the forecast skill scores; they neither improve nor degrade the forecast with the shallow water equations. Inclusion of the medium-and smaller-scale gravity waves in the initial data degrades the forecasts. Several forecasts are repeated with the mean depth decreased. The conclusions with respect to the modal filtering are unchanged although the impact of the filtering is less dramatic in these cases. The results are also insensitive to the particular longitudinal filtering used near the poles to allow longer time steps.

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## Abstract

The differences in the polar lower-troposphere temperature simulated by semi-Lagrangian and Eulerian approximations are examined and their cause is identified. With grids having 8–10 layers below 500 mb, semi-Lagrangian simulations are colder than Eulerian by 2–4 K in the region poleward of 60°N and below 400 mb in winter. Diagnostic calculations with the NCAR CCM3 show that the semi-Lagrangian dynamical approximations tend to produce a cooling relative to the Eulerian at the 860-mb grid level. The difference occurs over land and sea ice where an inversion forms in the atmosphere with its top at the 860-mb grid level. The source of the difference is shown to be the different way the vertical advection approximations treat vertical structures found at the tops of marginally resolved inversions when the vertical velocity is reasonably vertically uniform surrounding the top of the inversion. The Eulerian approximations underestimate the cooling that should occur at the top of the inversion. This is also verified with diagnostic calculations on a grid with substantially increased resolution below 800 mb. On this grid, the adiabatic tendency differences between semi-Lagrangian and Eulerian approximations are small and the two approximations produce the same simulated lower-tropospheric temperature, which is also the same as that produced by the semi-Lagrangian approximations on the coarse grid. Compared to the NCEP reanalysis, the low vertical resolution Eulerian simulated temperature looks better than the semi-Lagrangian, but those approximations produce that “better” simulated temperature by an incorrect mechanism. For practical applications, the Eulerian approximations require higher vertical resolution below 800 mb than usually used today in climate models, but the semi-Lagrangian approximations are adequate on these coarser grids.

## Abstract

The differences in the polar lower-troposphere temperature simulated by semi-Lagrangian and Eulerian approximations are examined and their cause is identified. With grids having 8–10 layers below 500 mb, semi-Lagrangian simulations are colder than Eulerian by 2–4 K in the region poleward of 60°N and below 400 mb in winter. Diagnostic calculations with the NCAR CCM3 show that the semi-Lagrangian dynamical approximations tend to produce a cooling relative to the Eulerian at the 860-mb grid level. The difference occurs over land and sea ice where an inversion forms in the atmosphere with its top at the 860-mb grid level. The source of the difference is shown to be the different way the vertical advection approximations treat vertical structures found at the tops of marginally resolved inversions when the vertical velocity is reasonably vertically uniform surrounding the top of the inversion. The Eulerian approximations underestimate the cooling that should occur at the top of the inversion. This is also verified with diagnostic calculations on a grid with substantially increased resolution below 800 mb. On this grid, the adiabatic tendency differences between semi-Lagrangian and Eulerian approximations are small and the two approximations produce the same simulated lower-tropospheric temperature, which is also the same as that produced by the semi-Lagrangian approximations on the coarse grid. Compared to the NCEP reanalysis, the low vertical resolution Eulerian simulated temperature looks better than the semi-Lagrangian, but those approximations produce that “better” simulated temperature by an incorrect mechanism. For practical applications, the Eulerian approximations require higher vertical resolution below 800 mb than usually used today in climate models, but the semi-Lagrangian approximations are adequate on these coarser grids.

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## Abstract

A semi-Lagrangian version of the National Center for Atmospheric Research Community Climate Model is developed. Special consideration is given to energy consistency aspects. In particular, approximations are developed in which the pressure gradient in the momentum equations is consistent with the energy conversion term in the thermodynamic equation. In addition, consistency between the discrete continuity equation and the vertical velocity ω in the energy conversion term of the thermodynamic equation is obtained. Simulated states from multiple-year simulations from the semi-Lagrangian and Eulerian versions are compared. The principal difference in the simulated climate appears in the zonal average temperature. The semi-Lagrangian simulation is colder than the Eulerian at and above the tropical tropopause. The terms producing the thermodynamic balance are examined. It is argued that the semi-Lagrangian scheme produces less computational smoothing of the temperature at the tropopause than the first-order finite-difference vertical advection approximations in the Eulerian version. Thus, by decreasing this particular computational error, the semi-Lagrangian produces less computational warming at the tropical tropopause. The net result is a colder tropical tropopause.

## Abstract

A semi-Lagrangian version of the National Center for Atmospheric Research Community Climate Model is developed. Special consideration is given to energy consistency aspects. In particular, approximations are developed in which the pressure gradient in the momentum equations is consistent with the energy conversion term in the thermodynamic equation. In addition, consistency between the discrete continuity equation and the vertical velocity ω in the energy conversion term of the thermodynamic equation is obtained. Simulated states from multiple-year simulations from the semi-Lagrangian and Eulerian versions are compared. The principal difference in the simulated climate appears in the zonal average temperature. The semi-Lagrangian simulation is colder than the Eulerian at and above the tropical tropopause. The terms producing the thermodynamic balance are examined. It is argued that the semi-Lagrangian scheme produces less computational smoothing of the temperature at the tropopause than the first-order finite-difference vertical advection approximations in the Eulerian version. Thus, by decreasing this particular computational error, the semi-Lagrangian produces less computational warming at the tropical tropopause. The net result is a colder tropical tropopause.

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## Abstract

The more attractive one dimensional, shape-preserving interpolation schemes as determined from a companion study are applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry. Hermite cubic and a rational cubic are considered for the interpolation form. Both require estimates of derivatives at data points. A cubic derivative form and the derivative estimates of Hyman and Akima are considered. The derivative estimates are also modified to ensure that the interpolant is monotonic. The modification depends on the interpolation form.

Three methods are used to apply the interpolators to two-dimensional semi-Lagrangian advection. The first consists of fractional time steps or time splitting. The method has noticeable displacement errors and larger diffusion than the other methods. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. This approach is examined for the Hermite cubic interpolant with cubic derivative estimates and produces very good results. The additional complications expected in extending to it three dimensions and the lack of corresponding two-dimensional forms for the rational cubic led to the consideration of the third approach—a tensor product form of monotonic one-dimensional interpolants. Although a description of the properties of the implied interpolating surface is difficult to obtain, the results show this to be a viable approach. Of the schemes considered, the Hermic cubic coupled with the Akima derivative estimate modified to satisfy a *C*
^{0}monotonicity condition produces the best solution to our test cases. The *C*
_{1}monotonic forms of the Hermite cubic have serious differential phase errors that distort the test patterns. The *C*
_{1} forms of the rational cubic do not show this distortion and produce virtually the same solutions as the corresponding *C*
_{0}forms. The second best scheme (or best *C*
_{1} continuity is desired) is the rational cubic with Hyman derivative approximations modified to satisfy *C*
_{1} monotonicity condition.

The two-dimensional interpolants are easily applied to spherical geometry using the natural polar boundary conditions. No problems are evident in advecting test shapes over the poles. A procedure is also introduced to calculate the departure point in spherical geometry. The scheme uses local geodesic coordinate systems based on each arrival point. It is shown to be comparable in accuracy to the one proposed Ritchie, which uses a Cartesian system in place of the local geodesic system.

## Abstract

The more attractive one dimensional, shape-preserving interpolation schemes as determined from a companion study are applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry. Hermite cubic and a rational cubic are considered for the interpolation form. Both require estimates of derivatives at data points. A cubic derivative form and the derivative estimates of Hyman and Akima are considered. The derivative estimates are also modified to ensure that the interpolant is monotonic. The modification depends on the interpolation form.

Three methods are used to apply the interpolators to two-dimensional semi-Lagrangian advection. The first consists of fractional time steps or time splitting. The method has noticeable displacement errors and larger diffusion than the other methods. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. This approach is examined for the Hermite cubic interpolant with cubic derivative estimates and produces very good results. The additional complications expected in extending to it three dimensions and the lack of corresponding two-dimensional forms for the rational cubic led to the consideration of the third approach—a tensor product form of monotonic one-dimensional interpolants. Although a description of the properties of the implied interpolating surface is difficult to obtain, the results show this to be a viable approach. Of the schemes considered, the Hermic cubic coupled with the Akima derivative estimate modified to satisfy a *C*
^{0}monotonicity condition produces the best solution to our test cases. The *C*
_{1}monotonic forms of the Hermite cubic have serious differential phase errors that distort the test patterns. The *C*
_{1} forms of the rational cubic do not show this distortion and produce virtually the same solutions as the corresponding *C*
_{0}forms. The second best scheme (or best *C*
_{1} continuity is desired) is the rational cubic with Hyman derivative approximations modified to satisfy *C*
_{1} monotonicity condition.

The two-dimensional interpolants are easily applied to spherical geometry using the natural polar boundary conditions. No problems are evident in advecting test shapes over the poles. A procedure is also introduced to calculate the departure point in spherical geometry. The scheme uses local geodesic coordinate systems based on each arrival point. It is shown to be comparable in accuracy to the one proposed Ritchie, which uses a Cartesian system in place of the local geodesic system.

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## Abstract

A unified analysis-initialization technique is introduced and tested in the framework of the shallow water equations. It consists of iterating multivariate optimal interpolation and nonlinear normal mode initialization. For extratropical regions, it is shown that such a technique produces an analysis consistent with observational errors and in nonlinear balance. The linear errors of multivariate optimal interpolation associated with geostrophically related covariances are eliminated.

## Abstract

A unified analysis-initialization technique is introduced and tested in the framework of the shallow water equations. It consists of iterating multivariate optimal interpolation and nonlinear normal mode initialization. For extratropical regions, it is shown that such a technique produces an analysis consistent with observational errors and in nonlinear balance. The linear errors of multivariate optimal interpolation associated with geostrophically related covariances are eliminated.