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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.
Abstract
This paper presents results of experiments designed to test the importance of arithmetic precision in short-term forecasts and long-term climate simulations with the NCAR global circulation model. It is expected that the next-generation computers will have a sizable speed gain when using lower-precision arithmetic. To determine how precision affects the model results, we compare several short- and long-term experiments using 48-bit mantissa arithmetic (normal for the CDC 6600 and 7600 computers) with corresponding experiments using 24- and 21-bit mantissa arithmetic. The errors due to the lower precision are much smaller than typical observational errors. In addition, it appears that in the short-term experiments the rapid error growth of the model dominates the round-off error accumulation resulting from the lower-precision arithmetic. Therefore, the lower precision used by the next-generation computers should not have a detrimental effect on short-term forecast accuracy. The long-term climate simulation experiments indicated a very similar conclusion. Even though there were some differences in the results of the experiments, climate indicators such as zonal wind, zonal temperature or eddy transport are quite similar.
Abstract
This paper presents results of experiments designed to test the importance of arithmetic precision in short-term forecasts and long-term climate simulations with the NCAR global circulation model. It is expected that the next-generation computers will have a sizable speed gain when using lower-precision arithmetic. To determine how precision affects the model results, we compare several short- and long-term experiments using 48-bit mantissa arithmetic (normal for the CDC 6600 and 7600 computers) with corresponding experiments using 24- and 21-bit mantissa arithmetic. The errors due to the lower precision are much smaller than typical observational errors. In addition, it appears that in the short-term experiments the rapid error growth of the model dominates the round-off error accumulation resulting from the lower-precision arithmetic. Therefore, the lower precision used by the next-generation computers should not have a detrimental effect on short-term forecast accuracy. The long-term climate simulation experiments indicated a very similar conclusion. Even though there were some differences in the results of the experiments, climate indicators such as zonal wind, zonal temperature or eddy transport are quite similar.
Abstract
The authors compare short forecast errors and the balance of terms in the moisture and temperature prediction equations that lead to those errors for the Community Atmosphere Model versions 2 and 3 (CAM2 and CAM3, respectively) at T42 truncation. The comparisons are made for an individual model column from global model forecasts at the Atmospheric Radiation Measurement Program (ARM) Southern Great Plains site for the April 1997 and June–July 1997 intensive observing periods. The goal is to provide insight into parameterization errors in the CAM, which ultimately should lead to improvements in the way processes are modeled. The atmospheric initial conditions are obtained from the 40-yr ECMWF Re-Analysis (ERA-40). The land initial conditions are spun up to be consistent with those analyses. The differences between the model formulations that are responsible for the major differences in the forecast errors and/or parameterization behaviors are identified. A sequence of experiments is performed, accumulating the changes from CAM3 back toward CAM2 to demonstrate the effect of the differences in formulations.
In June–July 1997 the CAM3 temperature and moisture forecast errors were larger than those of CAM2. The terms identified as being responsible for the differences are 1) the convective time scale assumed for the Zhang–McFarlane deep convection, 2) the energy associated with the conversion between water and ice of the rain associated with the Zhang–McFarlane convection parameterization, and 3) the dependence of the rainfall evaporation on cloud fraction. In April 1997 the CAM2 and CAM3 temperature and moisture forecast errors are very similar, but different tendencies arising from modifications to one parameterization component are compensated by responding changes in another component to yield the same total moisture tendency. The addition of detrainment of water in CAM3 by the Hack shallow convection to the prognostic cloud water scheme is balanced by a responding difference in the advective tendency. A halving of the time scale assumed for the Hack shallow convection was compensated by a responding change in the prognostic cloud water. Changes to the cloud fraction parameterization affect the radiative heating, which in turn modifies the stability of the atmospheric column and affects the convection. The resulting changes in convection tendency are balanced by responding changes in the prognostic cloud water parameterization tendency.
Abstract
The authors compare short forecast errors and the balance of terms in the moisture and temperature prediction equations that lead to those errors for the Community Atmosphere Model versions 2 and 3 (CAM2 and CAM3, respectively) at T42 truncation. The comparisons are made for an individual model column from global model forecasts at the Atmospheric Radiation Measurement Program (ARM) Southern Great Plains site for the April 1997 and June–July 1997 intensive observing periods. The goal is to provide insight into parameterization errors in the CAM, which ultimately should lead to improvements in the way processes are modeled. The atmospheric initial conditions are obtained from the 40-yr ECMWF Re-Analysis (ERA-40). The land initial conditions are spun up to be consistent with those analyses. The differences between the model formulations that are responsible for the major differences in the forecast errors and/or parameterization behaviors are identified. A sequence of experiments is performed, accumulating the changes from CAM3 back toward CAM2 to demonstrate the effect of the differences in formulations.
In June–July 1997 the CAM3 temperature and moisture forecast errors were larger than those of CAM2. The terms identified as being responsible for the differences are 1) the convective time scale assumed for the Zhang–McFarlane deep convection, 2) the energy associated with the conversion between water and ice of the rain associated with the Zhang–McFarlane convection parameterization, and 3) the dependence of the rainfall evaporation on cloud fraction. In April 1997 the CAM2 and CAM3 temperature and moisture forecast errors are very similar, but different tendencies arising from modifications to one parameterization component are compensated by responding changes in another component to yield the same total moisture tendency. The addition of detrainment of water in CAM3 by the Hack shallow convection to the prognostic cloud water scheme is balanced by a responding difference in the advective tendency. A halving of the time scale assumed for the Hack shallow convection was compensated by a responding change in the prognostic cloud water. Changes to the cloud fraction parameterization affect the radiative heating, which in turn modifies the stability of the atmospheric column and affects the convection. The resulting changes in convection tendency are balanced by responding changes in the prognostic cloud water parameterization tendency.
Abstract
Detailed comparisons are made between the climate simulated by a seasonal version of the NCAR Community Climate Mode) (CCM1) at 12 level, R15 spectral resolution, and that revealed by ECMWF operational analyses over 1980–86 truncated to a similar resolution. A variety of circulation statistics are presented to reveal the spatial character and seasonality of CCM1 biases in temperatures, winds, and wave flux quantities. CCM1 biases are typical of current climate models run at similar resolution. Interrelationships between the above biases are a focus of this study, in particular using wave-mean flow interaction diagnostics.
CCM1 exhibits a westerly zonal wind bias in the tropics and a lack of westerlies in the high latitude Southern Hemisphere (SH). The tropical zonal mean meridional circulation (Hadley cell) in the model is approximately a factor of two too weak. The poleward eddy heat flux is accurately simulated, but the poleward eddy momentum flux is severely underestimated, particularly in the SH. There is a resulting excessive large-scale wave drag in the model extratropical upper troposphere, in qualitative agreement with the weak model high latitude westerlies (and temperature bias patterns). Conversely, the model tropical zonal wind bias does not appear to be related to influences by large-scale waves. Wave flux biases are compared for stationary and transient statistics; model stationary waves are in good agreement with observations, while the largest relative momentum flux error is found for higher frequency transient waves.
Abstract
Detailed comparisons are made between the climate simulated by a seasonal version of the NCAR Community Climate Mode) (CCM1) at 12 level, R15 spectral resolution, and that revealed by ECMWF operational analyses over 1980–86 truncated to a similar resolution. A variety of circulation statistics are presented to reveal the spatial character and seasonality of CCM1 biases in temperatures, winds, and wave flux quantities. CCM1 biases are typical of current climate models run at similar resolution. Interrelationships between the above biases are a focus of this study, in particular using wave-mean flow interaction diagnostics.
CCM1 exhibits a westerly zonal wind bias in the tropics and a lack of westerlies in the high latitude Southern Hemisphere (SH). The tropical zonal mean meridional circulation (Hadley cell) in the model is approximately a factor of two too weak. The poleward eddy heat flux is accurately simulated, but the poleward eddy momentum flux is severely underestimated, particularly in the SH. There is a resulting excessive large-scale wave drag in the model extratropical upper troposphere, in qualitative agreement with the weak model high latitude westerlies (and temperature bias patterns). Conversely, the model tropical zonal wind bias does not appear to be related to influences by large-scale waves. Wave flux biases are compared for stationary and transient statistics; model stationary waves are in good agreement with observations, while the largest relative momentum flux error is found for higher frequency transient waves.
Abstract
The large-scale transient components of atmospheric flow have been studied for many years. Observational studies indicate that large amplitude regularly westward propagating waves appear episodically in the atmosphere. These waves have spatial structures and frequencies enticingly similar to those of external Rossby modes obtained theoretically from the linearized baroclinic equations.
In the present study, the episode of 10–28 January 1979 is studied by expanding global objective analyses into the normal modes of a global baroclinic quasi-geostrophic model linearized about the observed nonseparable zonal mean wind field. Several coherent regularly propagating waves are found. One of the most significant of these (the R3 1 or 16 day mode) is examined in detail. Although the observed structure is similar in some ways to the theoretically derived structure, there are important discrepancies. These discrepancies are examined more closely using upper air soundings from Antarctica.
The experimental results suggest that for the episode of 10–28 January 1979 the identification of the observed regularly propagating 16 day wave with the theoretically derived R3 1 mode is doubtful and more sophisticated explanations may be required.
Abstract
The large-scale transient components of atmospheric flow have been studied for many years. Observational studies indicate that large amplitude regularly westward propagating waves appear episodically in the atmosphere. These waves have spatial structures and frequencies enticingly similar to those of external Rossby modes obtained theoretically from the linearized baroclinic equations.
In the present study, the episode of 10–28 January 1979 is studied by expanding global objective analyses into the normal modes of a global baroclinic quasi-geostrophic model linearized about the observed nonseparable zonal mean wind field. Several coherent regularly propagating waves are found. One of the most significant of these (the R3 1 or 16 day mode) is examined in detail. Although the observed structure is similar in some ways to the theoretically derived structure, there are important discrepancies. These discrepancies are examined more closely using upper air soundings from Antarctica.
The experimental results suggest that for the episode of 10–28 January 1979 the identification of the observed regularly propagating 16 day wave with the theoretically derived R3 1 mode is doubtful and more sophisticated explanations may be required.
Abstract
Numerical models for the prediction of atmospheric motions are described by a finite number of coupled ordinary differential equations. We formally solve the initial-value problem for small-amplitude perturbations on some basic state as described by the prediction system. The solution and hence initial conditions are expressed as a sum over the normal modes of oscillation of the perturbation equations. The question as to which modes describe the evolution of meteorologically significant information may be answered for models which are used not only for prediction but also for climate simulation. Those modes which have a much larger amplitude in noisy real data than in climate simulation studies can be filtered from the initial data. The expansion of grid-point data into the normal modes of a model thus allows filtering in a more selective and rational fashion than has been possible using classical initialization procedures. Such an expansion also allows comparison of numerical simulation studies with spectral studies of actual free modes in the atmosphere. As an example of the model expansion procedure, we describe the application of a finite-difference approximation to the dynamics of a two-layer ocean model on a rotating sphere. In the limit of infinitesimal grid interval, the expansion of initial data is given by the Hough functions of tidal theory. For a finite grid interval, it is necessary to consider not only modes related to the Hough modes but also computational modes specific to the finite-difference equations employed. Examples of the eigenfrequencies and eigenfunctions for a basic state at rest are compared with those obtained assuming a basic state with a latitudinally varying zonal wind.
Abstract
Numerical models for the prediction of atmospheric motions are described by a finite number of coupled ordinary differential equations. We formally solve the initial-value problem for small-amplitude perturbations on some basic state as described by the prediction system. The solution and hence initial conditions are expressed as a sum over the normal modes of oscillation of the perturbation equations. The question as to which modes describe the evolution of meteorologically significant information may be answered for models which are used not only for prediction but also for climate simulation. Those modes which have a much larger amplitude in noisy real data than in climate simulation studies can be filtered from the initial data. The expansion of grid-point data into the normal modes of a model thus allows filtering in a more selective and rational fashion than has been possible using classical initialization procedures. Such an expansion also allows comparison of numerical simulation studies with spectral studies of actual free modes in the atmosphere. As an example of the model expansion procedure, we describe the application of a finite-difference approximation to the dynamics of a two-layer ocean model on a rotating sphere. In the limit of infinitesimal grid interval, the expansion of initial data is given by the Hough functions of tidal theory. For a finite grid interval, it is necessary to consider not only modes related to the Hough modes but also computational modes specific to the finite-difference equations employed. Examples of the eigenfrequencies and eigenfunctions for a basic state at rest are compared with those obtained assuming a basic state with a latitudinally varying zonal wind.
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 0monotonicity condition produces the best solution to our test cases. The C 1monotonic 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 0forms. 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 0monotonicity condition produces the best solution to our test cases. The C 1monotonic 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 0forms. 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
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.
Abstract
In Part I of this paper we review initialization methods for numerical weather prediction models, leading up to the development of schemes based on the normal modes of the forecast model. We present the derivation of the normal modes of ECMWF's multilevel global grid-point model, and compare the horizontal normal modes with those obtained using alternative finite-difference schemes. The impact of stability-enhancing Fourier filtering procedures on the normal modes is also discussed. Finally in Part I we apply linear normal mode initialization to a nine-level version of the model with 3.75° horizontal resolution. The application of nonlinear normal mode initialization to this model is presented in Part II.
Abstract
In Part I of this paper we review initialization methods for numerical weather prediction models, leading up to the development of schemes based on the normal modes of the forecast model. We present the derivation of the normal modes of ECMWF's multilevel global grid-point model, and compare the horizontal normal modes with those obtained using alternative finite-difference schemes. The impact of stability-enhancing Fourier filtering procedures on the normal modes is also discussed. Finally in Part I we apply linear normal mode initialization to a nine-level version of the model with 3.75° horizontal resolution. The application of nonlinear normal mode initialization to this model is presented in Part II.
