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Abstract
A modification is introduced in a semi-implicit version of a grid point model of the shallow water equations. The new model is simpler, runs one-third, and after 5 days of integration, the forecasts differ by less than 1 m.
Abstract
A modification is introduced in a semi-implicit version of a grid point model of the shallow water equations. The new model is simpler, runs one-third, and after 5 days of integration, the forecasts differ by less than 1 m.
Abstract
It has been demonstrated previously by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-implicit semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. Rivest et al. recommended using a second-order instead of a first-order semi-implicit off-centering to eliminate the spurious resonances, the former being more accurate. The present study shows by a linear one-dimensional analysis that a first-order semi-implicit off-centering can be used more effectively to eliminate the spurious resonances when combined with a spatially averaged Eulerian instead of a semi-Lagrangian treatment of mountains. The analysis reveals that the resonance is much less severe with the spatially averaged Eulerian treatment of mountains and, hence, can be suppressed with a weaker first-order off-centering. This combination could represent a valid alternative to second-order off-centering that needs extra time levels. The study also reveals that a serious truncation error is present in the neighborhood of the twin resonances when a semi-Lagrangian treatment of mountains is used. With the spatially averaged Eulerian treatment of the mountains the numerical solution filters the corresponding waves. These various points are illustrated with both barotropic and baroclinic semi-implicit semi-Lagrangian spectral models. An important feature of the baroclinic model formulation is the inclusion of topography in the basic-state solution that is used for the semi-implicit treatment of the gravity-wave-producing terms. In tests run from real data it appears that, in current three-time-level models, simply changing from the semi-Lagrangian to the spatially averaged Eulerian treatment of mountains is sufficient to significantly reduce the topographic resonance problem, permitting the use of larger time steps that produce acceptable time truncation errors without provoking the fictitious numerical amplification of short-scale waves.
Abstract
It has been demonstrated previously by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-implicit semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. Rivest et al. recommended using a second-order instead of a first-order semi-implicit off-centering to eliminate the spurious resonances, the former being more accurate. The present study shows by a linear one-dimensional analysis that a first-order semi-implicit off-centering can be used more effectively to eliminate the spurious resonances when combined with a spatially averaged Eulerian instead of a semi-Lagrangian treatment of mountains. The analysis reveals that the resonance is much less severe with the spatially averaged Eulerian treatment of mountains and, hence, can be suppressed with a weaker first-order off-centering. This combination could represent a valid alternative to second-order off-centering that needs extra time levels. The study also reveals that a serious truncation error is present in the neighborhood of the twin resonances when a semi-Lagrangian treatment of mountains is used. With the spatially averaged Eulerian treatment of the mountains the numerical solution filters the corresponding waves. These various points are illustrated with both barotropic and baroclinic semi-implicit semi-Lagrangian spectral models. An important feature of the baroclinic model formulation is the inclusion of topography in the basic-state solution that is used for the semi-implicit treatment of the gravity-wave-producing terms. In tests run from real data it appears that, in current three-time-level models, simply changing from the semi-Lagrangian to the spatially averaged Eulerian treatment of mountains is sufficient to significantly reduce the topographic resonance problem, permitting the use of larger time steps that produce acceptable time truncation errors without provoking the fictitious numerical amplification of short-scale waves.
Abstract
This study examines the linearization properties of a simplified planetary boundary layer parameterization based on the vertical diffusion equations, in which the exchange coefficients are a function of the local Richardson number and wind shear. Spurious noise, associated with this parameterization, develops near the surface in the tangent linear integrations. The origin of this problem is investigated by examining the accuracy of the linearization and the numerical stability of the scheme used to discretize the vertical diffusion equations. The noise is primarily due to the linearization of the exchange coefficients when the atmospheric state is near neutral static stability and when a long time step is employed. A regularization procedure based on the linearization error and a criterion for the numerical stability is proposed and tested. This regularization is compared with those recently adopted by Mahfouf, who neglects the perturbations of the exchange coefficients, and by Janisková et al., who reduce the amplitude of those perturbations when the Richardson number is in the vicinity of zero.
When the sizes of the atmospheric state perturbations are 1 m s−1 for the winds and 1 K for the temperature, which is the typical size of analysis increments, regularizations proposed here and by Janisková et al. perform similarly and are slightly better than neglecting the perturbations of the exchange coefficients. On the other hand, when the state perturbations are much smaller (e.g., 3 orders of magnitude smaller), the linearization becomes accurate and a regularization is no longer necessary, as long as the time step is short enough to avoid numerical instability. In this case, the regularization proposed here becomes inactive while the others introduce unnecessary errors.
Abstract
This study examines the linearization properties of a simplified planetary boundary layer parameterization based on the vertical diffusion equations, in which the exchange coefficients are a function of the local Richardson number and wind shear. Spurious noise, associated with this parameterization, develops near the surface in the tangent linear integrations. The origin of this problem is investigated by examining the accuracy of the linearization and the numerical stability of the scheme used to discretize the vertical diffusion equations. The noise is primarily due to the linearization of the exchange coefficients when the atmospheric state is near neutral static stability and when a long time step is employed. A regularization procedure based on the linearization error and a criterion for the numerical stability is proposed and tested. This regularization is compared with those recently adopted by Mahfouf, who neglects the perturbations of the exchange coefficients, and by Janisková et al., who reduce the amplitude of those perturbations when the Richardson number is in the vicinity of zero.
When the sizes of the atmospheric state perturbations are 1 m s−1 for the winds and 1 K for the temperature, which is the typical size of analysis increments, regularizations proposed here and by Janisková et al. perform similarly and are slightly better than neglecting the perturbations of the exchange coefficients. On the other hand, when the state perturbations are much smaller (e.g., 3 orders of magnitude smaller), the linearization becomes accurate and a regularization is no longer necessary, as long as the time step is short enough to avoid numerical instability. In this case, the regularization proposed here becomes inactive while the others introduce unnecessary errors.
Abstract
The semi-implicit algorithm, originally developed by Robert for an economical integration of the primitive equations in large-scale models of the atmospheric, is here generalized in order to integrate the fully compressible, nonhydrostatic equations. We show that there is little computational overhead associated with the integration of the full, and hence presumably more correct, set of equations that do not invoke the hydrostatic assumption to exclude the high frequency, vertically propagating acoustic modes.
Abstract
The semi-implicit algorithm, originally developed by Robert for an economical integration of the primitive equations in large-scale models of the atmospheric, is here generalized in order to integrate the fully compressible, nonhydrostatic equations. We show that there is little computational overhead associated with the integration of the full, and hence presumably more correct, set of equations that do not invoke the hydrostatic assumption to exclude the high frequency, vertically propagating acoustic modes.
Abstract
A four-dimensional variational (4DVAR) data assimilation problem may be constrained so that the solution closely fits the observations but is balanced. In this way, the processes of data analysis and initialization are combined. The method of initialization considered here, digital filtering, is widely used in weather forecasting centers. The digital filter was found to control high-frequency noise when implemented as a strong or as a weak constraint in the context of a global shallow water model. Implementation of a strong constraint did not result in a recovery of small scales although some recovery of intermediate scales did occur. Implementation of a weak constraint as a penalty method with a single fixed value of the penalty parameter resulted in analyses that were smooth, but depended upon the choice of the parameter. With a parameter value that was too large, the divergent kinetic energy spectrum of the analysis was excessively damped in the large scales. The rotational kinetic energy spectrum was also affected by the choice of penalty parameter. Both types of constraint were found to adequately control gravity wave noise although caution is advised in choosing the penalty parameter for the simple penalty term method.
Abstract
A four-dimensional variational (4DVAR) data assimilation problem may be constrained so that the solution closely fits the observations but is balanced. In this way, the processes of data analysis and initialization are combined. The method of initialization considered here, digital filtering, is widely used in weather forecasting centers. The digital filter was found to control high-frequency noise when implemented as a strong or as a weak constraint in the context of a global shallow water model. Implementation of a strong constraint did not result in a recovery of small scales although some recovery of intermediate scales did occur. Implementation of a weak constraint as a penalty method with a single fixed value of the penalty parameter resulted in analyses that were smooth, but depended upon the choice of the parameter. With a parameter value that was too large, the divergent kinetic energy spectrum of the analysis was excessively damped in the large scales. The rotational kinetic energy spectrum was also affected by the choice of penalty parameter. Both types of constraint were found to adequately control gravity wave noise although caution is advised in choosing the penalty parameter for the simple penalty term method.
Abstract
The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by , who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, this study is extended by considering the impact of temporally accumulating first-order linearization errors on the limit of validity of the tangent linear approximation due to the use of small but finite perturbations. The results of this paper are based on the examination of the passive advection problem. In particular, the impact of using incorrect interpolation schemes is studied as a function of scale and Courant number.
For a constant zonal wind leading to an integral value of the Courant number, the first-order linearization errors are seen to amplify linearly in time and to resemble the second-order derivative of the advected field for linear interpolation and the fourth-order derivative for cubic Lagrange interpolation. Solid-body rotation experiments on the sphere show that in situations where linear interpolation results in accurate integrations, the limit of validity of the TLM is nevertheless reduced. First-order cubic Lagrange linearization errors are smaller and affect small scales. For this to happen requires a wind configuration leading to a persistent integral value of the Courant number. Regions where sharp gradients of the advected tracer field are present are the most sensitive to this error, which is nevertheless observed to be small. Finally, passive tracers experiments driven by winds obtained from a shallow-water model integration confirm that higher-order interpolation schemes (whether correct or not) give similar negligible linearization errors since the probability of having the upstream point being located exactly on a grid point is vanishingly small.
Abstract
The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by , who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, this study is extended by considering the impact of temporally accumulating first-order linearization errors on the limit of validity of the tangent linear approximation due to the use of small but finite perturbations. The results of this paper are based on the examination of the passive advection problem. In particular, the impact of using incorrect interpolation schemes is studied as a function of scale and Courant number.
For a constant zonal wind leading to an integral value of the Courant number, the first-order linearization errors are seen to amplify linearly in time and to resemble the second-order derivative of the advected field for linear interpolation and the fourth-order derivative for cubic Lagrange interpolation. Solid-body rotation experiments on the sphere show that in situations where linear interpolation results in accurate integrations, the limit of validity of the TLM is nevertheless reduced. First-order cubic Lagrange linearization errors are smaller and affect small scales. For this to happen requires a wind configuration leading to a persistent integral value of the Courant number. Regions where sharp gradients of the advected tracer field are present are the most sensitive to this error, which is nevertheless observed to be small. Finally, passive tracers experiments driven by winds obtained from a shallow-water model integration confirm that higher-order interpolation schemes (whether correct or not) give similar negligible linearization errors since the probability of having the upstream point being located exactly on a grid point is vanishingly small.
Abstract
A four-dimensional variational data assimilation (4DVAR) scheme has recently been implemented in the medium-range weather forecast system of the Meteorological Service of Canada (MSC). The new scheme is now composed of several additional and improved features as compared with the three-dimensional variational data assimilation (3DVAR): the first guess at the appropriate time from the full-resolution model trajectory is used to calculate the misfit to the observations; the tangent linear of the forecast model and its adjoint are employed to propagate the analysis increment and the gradient of the cost function over the 6-h assimilation window; a comprehensive set of simplified physical parameterizations is used during the final minimization process; and the number of frequently reported data, in particular satellite data, has substantially increased. The impact of these 4DVAR components on the forecast skill is reported in this article. This is achieved by comparing data assimilation configurations that range in complexity from the former 3DVAR with the implemented 4DVAR over a 1-month period. It is shown that the implementation of the tangent-linear model and its adjoint as well as the increased number of observations are the two features of the new 4DVAR that contribute the most to the forecast improvement. All the other components provide marginal though positive impact. 4DVAR does not improve the medium-range forecast of tropical storms in general and tends to amplify the existing, too early extratropical transition often observed in the MSC global forecast system with 3DVAR. It is shown that this recurrent problem is, however, more sensitive to the forecast model than the data assimilation scheme employed in this system. Finally, the impact of using a shorter cutoff time for the reception of observations, as the one used in the operational context for the 0000 and 1200 UTC forecasts, is more detrimental with 4DVAR. This result indicates that 4DVAR is more sensitive to observations at the end of the assimilation window than 3DVAR.
Abstract
A four-dimensional variational data assimilation (4DVAR) scheme has recently been implemented in the medium-range weather forecast system of the Meteorological Service of Canada (MSC). The new scheme is now composed of several additional and improved features as compared with the three-dimensional variational data assimilation (3DVAR): the first guess at the appropriate time from the full-resolution model trajectory is used to calculate the misfit to the observations; the tangent linear of the forecast model and its adjoint are employed to propagate the analysis increment and the gradient of the cost function over the 6-h assimilation window; a comprehensive set of simplified physical parameterizations is used during the final minimization process; and the number of frequently reported data, in particular satellite data, has substantially increased. The impact of these 4DVAR components on the forecast skill is reported in this article. This is achieved by comparing data assimilation configurations that range in complexity from the former 3DVAR with the implemented 4DVAR over a 1-month period. It is shown that the implementation of the tangent-linear model and its adjoint as well as the increased number of observations are the two features of the new 4DVAR that contribute the most to the forecast improvement. All the other components provide marginal though positive impact. 4DVAR does not improve the medium-range forecast of tropical storms in general and tends to amplify the existing, too early extratropical transition often observed in the MSC global forecast system with 3DVAR. It is shown that this recurrent problem is, however, more sensitive to the forecast model than the data assimilation scheme employed in this system. Finally, the impact of using a shorter cutoff time for the reception of observations, as the one used in the operational context for the 0000 and 1200 UTC forecasts, is more detrimental with 4DVAR. This result indicates that 4DVAR is more sensitive to observations at the end of the assimilation window than 3DVAR.
Abstract
As a second step in the development of the Canadian Regional Data Assimilation System following Fillion et al., this study extends the approach to the four-dimensional variational data assimilation (4D-Var) context. Emphasis is first put on illustrating the importance of controlling lateral boundary conditions (LBCs). The use in the minimization of a horizontal grid over a domain exceeding the horizontal grid of the high-resolution nonlinear model is then proposed. The authors examine the performance of this 4D-Var formulation as an upcoming upgrade to the currently operational regional three-dimensional variational data assimilation (3D-Var) system. Forecast verifications against radiosonde data for 118 winter cases and 118 summer cases were performed. Results indicate a slight positive impact up to 48 h against North American radiosondes, but with a significant positive impact (especially for winds) at mid- and high latitudes during the summer. Accumulated precipitation scores over 24 h, whether during the first or second day of the forecasts, are slightly improved. The regional 4D-Var analysis system described in this study can run within current real-time “regional run” allocation for operations at the Canadian Meteorological Center (CMC). Future improvements of this system are briefly mentioned especially regarding the upcoming computer upgrade at CMC.
Abstract
As a second step in the development of the Canadian Regional Data Assimilation System following Fillion et al., this study extends the approach to the four-dimensional variational data assimilation (4D-Var) context. Emphasis is first put on illustrating the importance of controlling lateral boundary conditions (LBCs). The use in the minimization of a horizontal grid over a domain exceeding the horizontal grid of the high-resolution nonlinear model is then proposed. The authors examine the performance of this 4D-Var formulation as an upcoming upgrade to the currently operational regional three-dimensional variational data assimilation (3D-Var) system. Forecast verifications against radiosonde data for 118 winter cases and 118 summer cases were performed. Results indicate a slight positive impact up to 48 h against North American radiosondes, but with a significant positive impact (especially for winds) at mid- and high latitudes during the summer. Accumulated precipitation scores over 24 h, whether during the first or second day of the forecasts, are slightly improved. The regional 4D-Var analysis system described in this study can run within current real-time “regional run” allocation for operations at the Canadian Meteorological Center (CMC). Future improvements of this system are briefly mentioned especially regarding the upcoming computer upgrade at CMC.
Abstract
A modified semi-Lagrangian scheme is proposed in the context of semi-implicit forecast models to reduce the important distortion of topographically forced waves that is produced when the Courant–Friedrichs–Lewy (CFL) number is greater than 1. The improved semi-Lagrangian scheme combines the original semi-implicit formulation and the spatial averaging of all nonlinear terms. The impact of the spatial averaging is assessed in two baroclinic forecast models: a global spectral model and a regional gridpoint model. The modified semi-implicit semi-Lagrangian scheme is shown to improve short- and medium-range forecasts, and to increase the efficiency of the models by reducing the number of interpolations by 20%–40%.
Abstract
A modified semi-Lagrangian scheme is proposed in the context of semi-implicit forecast models to reduce the important distortion of topographically forced waves that is produced when the Courant–Friedrichs–Lewy (CFL) number is greater than 1. The improved semi-Lagrangian scheme combines the original semi-implicit formulation and the spatial averaging of all nonlinear terms. The impact of the spatial averaging is assessed in two baroclinic forecast models: a global spectral model and a regional gridpoint model. The modified semi-implicit semi-Lagrangian scheme is shown to improve short- and medium-range forecasts, and to increase the efficiency of the models by reducing the number of interpolations by 20%–40%.
Abstract
On 15 March 2005, the Meteorological Service of Canada (MSC) proceeded to the implementation of a four-dimensional variational data assimilation (4DVAR) system, which led to significant improvements in the quality of global forecasts. This paper describes the different elements of MSC’s 4DVAR assimilation system, discusses some issues encountered during the development, and reports on the overall results from the 4DVAR implementation tests. The 4DVAR system adopted an incremental approach with two outer iterations. The simplified model used in the minimization has a horizontal resolution of 170 km and its simplified physics includes vertical diffusion, surface drag, orographic blocking, stratiform condensation, and convection. One important element of the design is its modularity, which has permitted continued progress on the three-dimensional variational data assimilation (3DVAR) component (e.g., addition of new observation types) and the model (e.g., computational and numerical changes). This paper discusses some numerical problems that occur in the vicinity of the Poles where the semi-Lagrangian scheme becomes unstable when there is a simultaneous occurrence of converging meridians and strong wind gradients. These could be removed by filtering the winds in the zonal direction before they are used to estimate the upstream position in the semi-Lagrangian scheme. The results show improvements in all aspects of the forecasts over all regions. The impact is particularly significant in the Southern Hemisphere where 4DVAR is able to extract more information from satellite data. In the Northern Hemisphere, 4DVAR accepts more asynoptic data, in particular coming from profilers and aircrafts. The impact noted is also positive and the short-term forecasts are particularly improved over the west coast of North America. Finally, the dynamical consistency of the 4DVAR global analyses leads to a significant impact on regional forecasts. Experimentation has shown that regional forecasts initiated directly from a 4DVAR global analysis are improved with respect to the regional forecasts resulting from the regional 3DVAR analysis.
Abstract
On 15 March 2005, the Meteorological Service of Canada (MSC) proceeded to the implementation of a four-dimensional variational data assimilation (4DVAR) system, which led to significant improvements in the quality of global forecasts. This paper describes the different elements of MSC’s 4DVAR assimilation system, discusses some issues encountered during the development, and reports on the overall results from the 4DVAR implementation tests. The 4DVAR system adopted an incremental approach with two outer iterations. The simplified model used in the minimization has a horizontal resolution of 170 km and its simplified physics includes vertical diffusion, surface drag, orographic blocking, stratiform condensation, and convection. One important element of the design is its modularity, which has permitted continued progress on the three-dimensional variational data assimilation (3DVAR) component (e.g., addition of new observation types) and the model (e.g., computational and numerical changes). This paper discusses some numerical problems that occur in the vicinity of the Poles where the semi-Lagrangian scheme becomes unstable when there is a simultaneous occurrence of converging meridians and strong wind gradients. These could be removed by filtering the winds in the zonal direction before they are used to estimate the upstream position in the semi-Lagrangian scheme. The results show improvements in all aspects of the forecasts over all regions. The impact is particularly significant in the Southern Hemisphere where 4DVAR is able to extract more information from satellite data. In the Northern Hemisphere, 4DVAR accepts more asynoptic data, in particular coming from profilers and aircrafts. The impact noted is also positive and the short-term forecasts are particularly improved over the west coast of North America. Finally, the dynamical consistency of the 4DVAR global analyses leads to a significant impact on regional forecasts. Experimentation has shown that regional forecasts initiated directly from a 4DVAR global analysis are improved with respect to the regional forecasts resulting from the regional 3DVAR analysis.
