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Abstract
Due to its prohibitive computational cost, variational nonlinear normal mode initialization has received little interest during the last 10 years. Recently, soon after the introduction of the framework now called implicit nonlinear normal mode initialization, an efficient reformulation of variational nonlinear normal mode initialization using the implicit technique was demonstrated in the context of a barotropic finite-element regional model. This scheme allowed full variation of the variational weights at a low computational cost. To complement this previous work, the same variational approach for a global spectral shallow-water model is presented here. Similar results regarding the controlling and balancing aspects of the scheme are illustrated. The special form taken by the variational scheme in the context of height-constrained initialization is reconsidered after establishing the relationship between the implicit schemes and quasi-geostrophic theory on an f plane. Possible extensions of the method are mentioned at the end of the paper.
Abstract
Due to its prohibitive computational cost, variational nonlinear normal mode initialization has received little interest during the last 10 years. Recently, soon after the introduction of the framework now called implicit nonlinear normal mode initialization, an efficient reformulation of variational nonlinear normal mode initialization using the implicit technique was demonstrated in the context of a barotropic finite-element regional model. This scheme allowed full variation of the variational weights at a low computational cost. To complement this previous work, the same variational approach for a global spectral shallow-water model is presented here. Similar results regarding the controlling and balancing aspects of the scheme are illustrated. The special form taken by the variational scheme in the context of height-constrained initialization is reconsidered after establishing the relationship between the implicit schemes and quasi-geostrophic theory on an f plane. Possible extensions of the method are mentioned at the end of the paper.
Abstract
The degree of imbalance forced by deep convection in a three-dimensional variational analysis scheme (3DVAR) is examined. Simulated surface precipitation rates are used with various degree of errors together with different background atmospheric fields. Dynamical imbalances are defined only for the fastest timescales associated with gravity waves and diagnosed according to the implicit normal mode framework. Local measures of ageostrophic perturbations are also considered over rainy areas. Slow timescale perturbations on internal gravitational modes introduced by convection during 3DVAR are also monitored using temporal evidence of their presence. The diagnostic quantities used here are more appropriate for deep vertical scales where Machenhauer's balance scheme can be justified but must be used with care when discussing possible imbalance for shallow vertical gravity modes (especially for mesoscale data assimilation). These diagnostic measures are not actually used as an explicit constraint in the 3DVAR analysis but only serve as diagnostic tools. These measures are not meant as optimal penalty terms to be used in variational analysis schemes however since this aspect is not considered in this study.
It is found that gravity wave imbalance is introduced early in the minimization process when no balance constraint is imposed (other than the simple geostrophic constraint used in the background error statistics). Precipitation observations localized over a restricted horizontal domain are sufficient to trigger non-negligible imbalances. A challenging issue is the introduction by 3DVAR of slow timescale internal modes that significantly differ from those already present in the background trajectory. Whether these oscillations need to be controlled in some ways in order to ensure that the variational adjustment of convective forcing leads to slow timescales within some neighborhood of those of the background trajectory remains an open question. Traditional normal mode tools as those used in implicit normal mode initialization (especially for the first two internal vertical modes) can be used for such constraining problems in principle. For operational applications, the now widely used digital time-filtering approach presumably would need some extension in order to achieve the same controlling effect on slow timescales.
Abstract
The degree of imbalance forced by deep convection in a three-dimensional variational analysis scheme (3DVAR) is examined. Simulated surface precipitation rates are used with various degree of errors together with different background atmospheric fields. Dynamical imbalances are defined only for the fastest timescales associated with gravity waves and diagnosed according to the implicit normal mode framework. Local measures of ageostrophic perturbations are also considered over rainy areas. Slow timescale perturbations on internal gravitational modes introduced by convection during 3DVAR are also monitored using temporal evidence of their presence. The diagnostic quantities used here are more appropriate for deep vertical scales where Machenhauer's balance scheme can be justified but must be used with care when discussing possible imbalance for shallow vertical gravity modes (especially for mesoscale data assimilation). These diagnostic measures are not actually used as an explicit constraint in the 3DVAR analysis but only serve as diagnostic tools. These measures are not meant as optimal penalty terms to be used in variational analysis schemes however since this aspect is not considered in this study.
It is found that gravity wave imbalance is introduced early in the minimization process when no balance constraint is imposed (other than the simple geostrophic constraint used in the background error statistics). Precipitation observations localized over a restricted horizontal domain are sufficient to trigger non-negligible imbalances. A challenging issue is the introduction by 3DVAR of slow timescale internal modes that significantly differ from those already present in the background trajectory. Whether these oscillations need to be controlled in some ways in order to ensure that the variational adjustment of convective forcing leads to slow timescales within some neighborhood of those of the background trajectory remains an open question. Traditional normal mode tools as those used in implicit normal mode initialization (especially for the first two internal vertical modes) can be used for such constraining problems in principle. For operational applications, the now widely used digital time-filtering approach presumably would need some extension in order to achieve the same controlling effect on slow timescales.
Abstract
It is shown that implicit normal mode initialization can be combined with a variational technique, in order to control the relative magnitudes of the changes to the analyzed mass and wind fields. Since the initialization procedure is expressed entirely in physical space, the use of locally varying weights in the variational integral becomes more straightforward than in previous efforts to combine variational methods with normal mode initialization.
We present details of the application to a finite-element model of the shallow water equations on a stereographic projection. It is demonstrated that the use of variational initialization can change the slowly evolving component of the subsequent forecast, as well as eliminate the unrealistic fast component.
Abstract
It is shown that implicit normal mode initialization can be combined with a variational technique, in order to control the relative magnitudes of the changes to the analyzed mass and wind fields. Since the initialization procedure is expressed entirely in physical space, the use of locally varying weights in the variational integral becomes more straightforward than in previous efforts to combine variational methods with normal mode initialization.
We present details of the application to a finite-element model of the shallow water equations on a stereographic projection. It is demonstrated that the use of variational initialization can change the slowly evolving component of the subsequent forecast, as well as eliminate the unrealistic fast component.
Abstract
Advanced operational four-dimensional variational data assimilation (4DVAR) schemes include a linearized version of moist convective parameterization and its adjoint. At the Meteorological Service of Canada, work is underway to implement 4DVAR for both global and regional operational data assimilation. Moreover, the Kain– Fritsch moist convective parameterization scheme is currently under operational testing for global and regional weather forecasting. Consequently, tangent linear and adjoint versions of this convective scheme have been developed. Sources of nonlinearities and accuracy of the tangent linear approximation of the convective scheme itself were examined. The procedure to test this latter aspect uses Monte Carlo simulations based on background error covariances from the operational three-dimensional variational data assimilation (3DVAR) system at the Canadian Meteorological Centre. It is shown that for a critical level of amplitudes of vertical perturbations of temperature or moisture greater than typically 0.1 K or 0.1 g kg−1, the tangent linear approximation becomes inaccurate (e.g., typical perturbation response having the wrong sign and amplitude errors larger than 100%). For such perturbation amplitudes, there is a rapid increase of convective points where the tangent linear convective approximation is very strongly in error. Deactivation of the Kain–Fritsch scheme becomes frequent and a significant source of invalid tangent linear approximation for input perturbations exceeding typically 0.3 K or 0.3 g kg−1. Potential implications of this study for linearized moist convection in the context of 4DVAR and moist singular vector computation are discussed.
Abstract
Advanced operational four-dimensional variational data assimilation (4DVAR) schemes include a linearized version of moist convective parameterization and its adjoint. At the Meteorological Service of Canada, work is underway to implement 4DVAR for both global and regional operational data assimilation. Moreover, the Kain– Fritsch moist convective parameterization scheme is currently under operational testing for global and regional weather forecasting. Consequently, tangent linear and adjoint versions of this convective scheme have been developed. Sources of nonlinearities and accuracy of the tangent linear approximation of the convective scheme itself were examined. The procedure to test this latter aspect uses Monte Carlo simulations based on background error covariances from the operational three-dimensional variational data assimilation (3DVAR) system at the Canadian Meteorological Centre. It is shown that for a critical level of amplitudes of vertical perturbations of temperature or moisture greater than typically 0.1 K or 0.1 g kg−1, the tangent linear approximation becomes inaccurate (e.g., typical perturbation response having the wrong sign and amplitude errors larger than 100%). For such perturbation amplitudes, there is a rapid increase of convective points where the tangent linear convective approximation is very strongly in error. Deactivation of the Kain–Fritsch scheme becomes frequent and a significant source of invalid tangent linear approximation for input perturbations exceeding typically 0.3 K or 0.3 g kg−1. Potential implications of this study for linearized moist convection in the context of 4DVAR and moist singular vector computation are discussed.
Abstract
Some basic aspects related to the problem of incorporating moist convective processes in a variational data assimilation framework are considered. The methodology is based on inverse problem theory and is formulated in its simplest context where the adjustment of temperature and humidity fields take place only in the vertical. In contrast to previous studies on the subject, the impact of error statistics from prior information and data sources of information is clarified. The accuracy of linearization of convection operators and the resulting impact in a minimization procedure are examined. The former was investigated using Monte Carlo methods. Versions of two schemes are examined: the Kuo–Anthes scheme and the relaxed Arakawa–Schubert scheme (RAS).
It is found, in general, that for nonpathological convective points (i.e., points where convection always remains active during the minimization process), a significant adjustment of convection (and precipitation rate) is realizable within the range of realistic background temperature and specific humidity errors and precipitation rate observation errors. Typically, three to five iterations are sufficient for convergence of the variational algorithm for both convective schemes. The degree of nonlinearity of both schemes appears comparable. The vertical correlation length for the background error temperature field is shown to produce a strong interaction with the RAS scheme in the minimization process where significantly different vertical structures of analysis increments for the temperature field are generated in the vicinity of a critical value of the correlation length.
Abstract
Some basic aspects related to the problem of incorporating moist convective processes in a variational data assimilation framework are considered. The methodology is based on inverse problem theory and is formulated in its simplest context where the adjustment of temperature and humidity fields take place only in the vertical. In contrast to previous studies on the subject, the impact of error statistics from prior information and data sources of information is clarified. The accuracy of linearization of convection operators and the resulting impact in a minimization procedure are examined. The former was investigated using Monte Carlo methods. Versions of two schemes are examined: the Kuo–Anthes scheme and the relaxed Arakawa–Schubert scheme (RAS).
It is found, in general, that for nonpathological convective points (i.e., points where convection always remains active during the minimization process), a significant adjustment of convection (and precipitation rate) is realizable within the range of realistic background temperature and specific humidity errors and precipitation rate observation errors. Typically, three to five iterations are sufficient for convergence of the variational algorithm for both convective schemes. The degree of nonlinearity of both schemes appears comparable. The vertical correlation length for the background error temperature field is shown to produce a strong interaction with the RAS scheme in the minimization process where significantly different vertical structures of analysis increments for the temperature field are generated in the vicinity of a critical value of the correlation length.
Abstract
Recent studies have demonstrated that variational nonlinear normal-mode initialization can be efficiently implemented in the context of shallow-water models, provided one uses a physical space formulation. The implicit nonlinear normal-mode initialization (INMI) technique provides essentially the same balancing benefit as standard ”explicit“ nonlinear NMI but does not require the explicit computation of the linear free modes of the model. This allows variational initialization with arbitrary horizontal variation of the weights that specify the changes to the analyzed fields during initialization. As a consequence, the variational extension of INMI allows more flexibility to control the relative adjustment of mass and wind fields over data-rich and data-poor regions.
The purpose of this paper is to demonstrate the feasibility of variational implicit normal-mode initialization (VINMI) for multilevel models. This new scheme is illustrated on the presently operational Canadian baroclinic regional finite-element (RFE) model. It is shown that the VINMI scheme efficiently controls the relative magnitude of the changes to the analyzed mass and wind fields during the balancing (initialization) processes. A comparison is also made of the impact of the VINMI scheme versus that of the presently operational unconstrained version of the initialization scheme (INMI). Future development and applications of the method are discussed at the end of the paper.
Abstract
Recent studies have demonstrated that variational nonlinear normal-mode initialization can be efficiently implemented in the context of shallow-water models, provided one uses a physical space formulation. The implicit nonlinear normal-mode initialization (INMI) technique provides essentially the same balancing benefit as standard ”explicit“ nonlinear NMI but does not require the explicit computation of the linear free modes of the model. This allows variational initialization with arbitrary horizontal variation of the weights that specify the changes to the analyzed fields during initialization. As a consequence, the variational extension of INMI allows more flexibility to control the relative adjustment of mass and wind fields over data-rich and data-poor regions.
The purpose of this paper is to demonstrate the feasibility of variational implicit normal-mode initialization (VINMI) for multilevel models. This new scheme is illustrated on the presently operational Canadian baroclinic regional finite-element (RFE) model. It is shown that the VINMI scheme efficiently controls the relative magnitude of the changes to the analyzed mass and wind fields during the balancing (initialization) processes. A comparison is also made of the impact of the VINMI scheme versus that of the presently operational unconstrained version of the initialization scheme (INMI). Future development and applications of the method are discussed at the end of the paper.
Abstract
Some problems posed by the coupling of moist-convective and stratiform precipitation processes for variational assimilation of precipitation-rate data are examined in a 1D-Var framework. Background-error statistics and vertical resolution are chosen to be representative of current operational practice. Three advanced parameterization schemes for moist-convection are studied: the relaxed Arakawa–Schubert (RAS) scheme, Tiedtke’s mass-flux scheme (operational at the European Centre for Medium-Range Weather Forecasts), and the Betts–Miller scheme. Both fractional-stepping and process-splitting approaches for combining physical processes are examined. The behavior of the variational adjustment for background profiles of temperature and specific humidity in the neighborhood of saturation is of particular interest.
In the 1D-Var context examined here, it is demonstrated that the introduction of the stratiform precipitation process can have a negative impact on the minimization in the sense that, even when only slight supersaturation occurs, the minimization is controlled by the stratiform precipitation process at the expense of convective precipitation. This is generally true in process-splitting mode and conditionally true in fractional-stepping mode. The net result in such cases is an adjustment to the wrong type of precipitation over convective regions. In some of the cases examined (1D-Var with the RAS scheme, for instance), it is preferable to deactivate the stratiform precipitation process and to explicitly control the degree of supersaturation during the adjustment of convection. Evaporation of precipitation in subsaturated layers also appears as an important factor influencing the partition of precipitation. The method of fractional stepping appears less problematical compared to the process-splitting approach.
These results also indicate the need for a detailed examination of the partition of precipitation between convective and stratiform type in more sophisticated 3D/4D-Var data assimilation systems, and for a better combined parameterization of the two physical processes.
Abstract
Some problems posed by the coupling of moist-convective and stratiform precipitation processes for variational assimilation of precipitation-rate data are examined in a 1D-Var framework. Background-error statistics and vertical resolution are chosen to be representative of current operational practice. Three advanced parameterization schemes for moist-convection are studied: the relaxed Arakawa–Schubert (RAS) scheme, Tiedtke’s mass-flux scheme (operational at the European Centre for Medium-Range Weather Forecasts), and the Betts–Miller scheme. Both fractional-stepping and process-splitting approaches for combining physical processes are examined. The behavior of the variational adjustment for background profiles of temperature and specific humidity in the neighborhood of saturation is of particular interest.
In the 1D-Var context examined here, it is demonstrated that the introduction of the stratiform precipitation process can have a negative impact on the minimization in the sense that, even when only slight supersaturation occurs, the minimization is controlled by the stratiform precipitation process at the expense of convective precipitation. This is generally true in process-splitting mode and conditionally true in fractional-stepping mode. The net result in such cases is an adjustment to the wrong type of precipitation over convective regions. In some of the cases examined (1D-Var with the RAS scheme, for instance), it is preferable to deactivate the stratiform precipitation process and to explicitly control the degree of supersaturation during the adjustment of convection. Evaporation of precipitation in subsaturated layers also appears as an important factor influencing the partition of precipitation. The method of fractional stepping appears less problematical compared to the process-splitting approach.
These results also indicate the need for a detailed examination of the partition of precipitation between convective and stratiform type in more sophisticated 3D/4D-Var data assimilation systems, and for a better combined parameterization of the two physical processes.
Abstract
This study examines the modification to the balance properties of the analysis increments in a global three-dimensional variational data assimilation scheme when using flow-dependent background-error covariances derived from an operational ensemble Kalman filter instead of static homogenous and isotropic background-error covariances based on lagged forecast differences. It is shown that the degree of balance in the analysis increments is degraded when the former method is used. This change can be attributed in part to the reduced degree of rotational balance found in short-term ensemble Kalman filter perturbations as compared to lagged forecast differences based on longer-range forecasts. However, the use of a horizontal and vertical localization technique to increase the rank of the ensemble-based covariances are found to have a significant deleterious effect on the rotational balance with the largest detrimental impact coming from the vertical localization and affecting particularly the upper levels. The examination of the vertical motion part of the analysis increments revealed that the spatial covariance localization technique also produces unrealistic vertical structure of vertical motion increments with abnormally large increments near the surface. A comparison between the analysis increments from the ensemble Kalman filter and from the ensemble-based three-dimensional variational data assimilation (3D-Var) scheme showed that the balance characteristics of the analysis increments resulting from the two systems are very similar.
Abstract
This study examines the modification to the balance properties of the analysis increments in a global three-dimensional variational data assimilation scheme when using flow-dependent background-error covariances derived from an operational ensemble Kalman filter instead of static homogenous and isotropic background-error covariances based on lagged forecast differences. It is shown that the degree of balance in the analysis increments is degraded when the former method is used. This change can be attributed in part to the reduced degree of rotational balance found in short-term ensemble Kalman filter perturbations as compared to lagged forecast differences based on longer-range forecasts. However, the use of a horizontal and vertical localization technique to increase the rank of the ensemble-based covariances are found to have a significant deleterious effect on the rotational balance with the largest detrimental impact coming from the vertical localization and affecting particularly the upper levels. The examination of the vertical motion part of the analysis increments revealed that the spatial covariance localization technique also produces unrealistic vertical structure of vertical motion increments with abnormally large increments near the surface. A comparison between the analysis increments from the ensemble Kalman filter and from the ensemble-based three-dimensional variational data assimilation (3D-Var) scheme showed that the balance characteristics of the analysis increments resulting from the two systems are very similar.
Abstract
A detailed examination of the Jacobian matrix of sensitivities of the prognostic cloud scheme operational at the European Centre for Medium-Range Weather Forecasts (ECMWF) is presented. These Jacobians exhibit sensitivities of output variables (e.g., cloud condensate) to small input perturbations on temperature and moisture. The coupling of the cloud scheme with the ECMWF convective mass-flux scheme is considered. The sensitivity of the cloud scheme is split into all its contributing parts in order to extract the dominant terms. A selection of contrasted convective cases normally present in a regular operational forecast is considered. Some comparisons are made with a simpler diagnostic cloud scheme.
It is shown that the main contributing terms to the sensitivity in cloud condensate l are the following for cases of deep convection: detrainment from moist convection, evaporation processes, and conversion of cloud water into rain. The structure of the Jacobians in terms of temperature and moisture perturbations in such cases is strongly dominated by the structure of the Jacobians of the convective mass flux at cloud base. Because of this dominance, the Jacobians from a simpler diagnostic cloud scheme have similarities in shape with those produced by the prognostic scheme. For shallow convection cases, the same terms as for deep convective cases are important (convective effects still dominate), but now erosion of clouds becomes significant. Jacobians of integrated l produced by the diagnostic and prognostic cloud schemes here are also similar, still because of the dominance of convective effects. When moist convection plays a negligible role, the dominant terms are the condensation/evaporation effects and the conversion of cloud water into rain. Larger differences are found then between the Jacobians of the prognostic and diagnostic cloud schemes. In such situations, the sensitivity of l with respect to vertical motion plays an equally important role compared to temperature and humidity.
Abstract
A detailed examination of the Jacobian matrix of sensitivities of the prognostic cloud scheme operational at the European Centre for Medium-Range Weather Forecasts (ECMWF) is presented. These Jacobians exhibit sensitivities of output variables (e.g., cloud condensate) to small input perturbations on temperature and moisture. The coupling of the cloud scheme with the ECMWF convective mass-flux scheme is considered. The sensitivity of the cloud scheme is split into all its contributing parts in order to extract the dominant terms. A selection of contrasted convective cases normally present in a regular operational forecast is considered. Some comparisons are made with a simpler diagnostic cloud scheme.
It is shown that the main contributing terms to the sensitivity in cloud condensate l are the following for cases of deep convection: detrainment from moist convection, evaporation processes, and conversion of cloud water into rain. The structure of the Jacobians in terms of temperature and moisture perturbations in such cases is strongly dominated by the structure of the Jacobians of the convective mass flux at cloud base. Because of this dominance, the Jacobians from a simpler diagnostic cloud scheme have similarities in shape with those produced by the prognostic scheme. For shallow convection cases, the same terms as for deep convective cases are important (convective effects still dominate), but now erosion of clouds becomes significant. Jacobians of integrated l produced by the diagnostic and prognostic cloud schemes here are also similar, still because of the dominance of convective effects. When moist convection plays a negligible role, the dominant terms are the condensation/evaporation effects and the conversion of cloud water into rain. Larger differences are found then between the Jacobians of the prognostic and diagnostic cloud schemes. In such situations, the sensitivity of l with respect to vertical motion plays an equally important role compared to temperature and humidity.
Abstract
The differences in the balance characteristics between dry and precipitation areas in estimated short-term forecast error fields are investigated. The motivation is to see if dry and precipitation areas need to be treated differently in atmospheric data assimilation systems. Using an ensemble of lagged forecast differences, it is shown that perturbations are, on average, farther away from geostrophic balance over precipitation areas than over dry areas and that the deviation from geostrophic balance is proportional to the intensity of precipitation. Following these results, the authors investigate whether some improvements in the coupling between mass and rotational wind increments over precipitation areas can be achieved by using only the precipitation points within an ensemble of estimated forecast errors to construct a so-called diabatic balance operator by linear regression. Comparisons with a traditional approach to construct balance operators by linear regression show that the new approach leads to a gradually significant improvement (related to the intensity of the diabatic processes) of the accuracy of the coupling over precipitation areas as judged from an ensemble of lagged forecast differences. Results from a series of simplified data assimilation experiments show that the new balance operators can produce analysis increments that are substantially different from those associated with the traditional balance operator, particularly for observations located in the lower atmosphere. Issues concerning the implementation of this new approach in a full-fledged analysis system are briefly discussed but their investigations are left for a following study.
Abstract
The differences in the balance characteristics between dry and precipitation areas in estimated short-term forecast error fields are investigated. The motivation is to see if dry and precipitation areas need to be treated differently in atmospheric data assimilation systems. Using an ensemble of lagged forecast differences, it is shown that perturbations are, on average, farther away from geostrophic balance over precipitation areas than over dry areas and that the deviation from geostrophic balance is proportional to the intensity of precipitation. Following these results, the authors investigate whether some improvements in the coupling between mass and rotational wind increments over precipitation areas can be achieved by using only the precipitation points within an ensemble of estimated forecast errors to construct a so-called diabatic balance operator by linear regression. Comparisons with a traditional approach to construct balance operators by linear regression show that the new approach leads to a gradually significant improvement (related to the intensity of the diabatic processes) of the accuracy of the coupling over precipitation areas as judged from an ensemble of lagged forecast differences. Results from a series of simplified data assimilation experiments show that the new balance operators can produce analysis increments that are substantially different from those associated with the traditional balance operator, particularly for observations located in the lower atmosphere. Issues concerning the implementation of this new approach in a full-fledged analysis system are briefly discussed but their investigations are left for a following study.
