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Abstract
An analysis of variational data assimilation schemes for linear dynamical forecast models shows that the penalty functional must include an explicit contribution from the initial conditions in order to ensure a unique, low-noise forecast. The noise level is related to the effective number of data being assimilated.
Abstract
An analysis of variational data assimilation schemes for linear dynamical forecast models shows that the penalty functional must include an explicit contribution from the initial conditions in order to ensure a unique, low-noise forecast. The noise level is related to the effective number of data being assimilated.
Abstract
In this work the performance of ensembles generated by commonly used methods in a nonlinear system with multiple attractors is examined. The model used here is a spectral truncation of a barotropic quasigeostrophic channel model. The system studied here has 44 state variables, great enough to exhibit the problems associated with high state dimension, but small enough so that experiments with very large ensembles are practical, and relevant probability density functions (PDFs) can be evaluated explicitly. The attracting sets include two stable limit cycles.
To begin, the basins of attraction of two known stable limit cycles are characterized. Large ensembles are then used to calculate the evolution of initially Gaussian PDFs with a range of initial covariances. If the initial covariances are small, the PDF remains essentially unimodal, and the probability that a point drawn from the initial PDF lies in a different basin of attraction from the mean of that PDF is small. If the initial covariances are so large that there is significant probability that a given point in the initial ensemble does not lie in the same basin of attraction as the mean, the initial Gaussian PDF will evolve into a bimodal PDF. In this case, graphical representation of the PDF appears to split into two distinct regions of relatively high probability.
The ability of smaller ensembles drawn from spaces spanned by singular vectors and by bred vectors to capture this splitting behavior is then investigated, with the objective here being to see how well they capture multimodality in a highly nonlinear system. The performance of similarly small random ensembles drawn without dynamical constraints is also evaluated.
In this application, small ensembles chosen from subspaces of singular vectors performed well, their weakest performance being for an ensemble with relatively large initial variance for which the Gaussian character of the initial PDF remained intact. This was the best case for the bred vectors because of their tendency to align tangent to the attractor, but the bred vectors were at a disadvantage in detection of the tendency of an initially Gaussian PDF to evolve into a bimodal one, as were the unconstrained ensembles.
Abstract
In this work the performance of ensembles generated by commonly used methods in a nonlinear system with multiple attractors is examined. The model used here is a spectral truncation of a barotropic quasigeostrophic channel model. The system studied here has 44 state variables, great enough to exhibit the problems associated with high state dimension, but small enough so that experiments with very large ensembles are practical, and relevant probability density functions (PDFs) can be evaluated explicitly. The attracting sets include two stable limit cycles.
To begin, the basins of attraction of two known stable limit cycles are characterized. Large ensembles are then used to calculate the evolution of initially Gaussian PDFs with a range of initial covariances. If the initial covariances are small, the PDF remains essentially unimodal, and the probability that a point drawn from the initial PDF lies in a different basin of attraction from the mean of that PDF is small. If the initial covariances are so large that there is significant probability that a given point in the initial ensemble does not lie in the same basin of attraction as the mean, the initial Gaussian PDF will evolve into a bimodal PDF. In this case, graphical representation of the PDF appears to split into two distinct regions of relatively high probability.
The ability of smaller ensembles drawn from spaces spanned by singular vectors and by bred vectors to capture this splitting behavior is then investigated, with the objective here being to see how well they capture multimodality in a highly nonlinear system. The performance of similarly small random ensembles drawn without dynamical constraints is also evaluated.
In this application, small ensembles chosen from subspaces of singular vectors performed well, their weakest performance being for an ensemble with relatively large initial variance for which the Gaussian character of the initial PDF remained intact. This was the best case for the bred vectors because of their tendency to align tangent to the attractor, but the bred vectors were at a disadvantage in detection of the tendency of an initially Gaussian PDF to evolve into a bimodal one, as were the unconstrained ensembles.
Abstract
Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified.
In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution.
Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented.
Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.
Abstract
Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified.
In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution.
Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented.
Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.
Abstract
Cumulus formation and convection initiation are examined near a cold front–dryline “triple point” intersection on 24 May 2002 during the International H2O Project (IHOP). A new Lagrangian objective analysis technique assimilates in situ measurements using time-dependent Doppler-derived 3D wind fields, providing output 3D fields of water vapor mixing ratio, virtual potential temperature, and lifted condensation level (LCL) and water-saturated (i.e., cloud) volumes on a subdomain of the radar analysis grid. The radar and Lagrangian analyses reveal the presence of along-wind (i.e., longitudinal) and cross-wind (i.e., transverse) roll circulations in the boundary layer (BL). A remarkable finding of the evolving radar analyses is the apparent persistence of both transverse rolls and individual updraft, vertical vorticity, and reflectivity cores for periods of up to 30 min or more while moving approximately with the local BL wind. Satellite cloud images and single-camera ground photogrammetry imply that clouds tend to develop either over or on the downwind edge of BL updrafts, with a tendency for clouds to elongate and dissipate in the downwind direction relative to cloud layer winds due to weakening updrafts and mixing with drier overlying air. The Lagrangian and radar wind analyses support a parcel continuity principle for cumulus formation, which requires that rising moist air parcels achieve their LCL before moving laterally out of the updraft. Cumuli form within penetrative updrafts in the elevated residual layer (ERL) overlying the moist BL east of the triple point, but remain capped by a convection inhibition (CIN)-bearing layer above the ERL. Dropsonde data suggest the existence of a convergence line about 80 km east of the triple point where deep lifting of BL moisture and locally reduced CIN together support convection initiation.
Abstract
Cumulus formation and convection initiation are examined near a cold front–dryline “triple point” intersection on 24 May 2002 during the International H2O Project (IHOP). A new Lagrangian objective analysis technique assimilates in situ measurements using time-dependent Doppler-derived 3D wind fields, providing output 3D fields of water vapor mixing ratio, virtual potential temperature, and lifted condensation level (LCL) and water-saturated (i.e., cloud) volumes on a subdomain of the radar analysis grid. The radar and Lagrangian analyses reveal the presence of along-wind (i.e., longitudinal) and cross-wind (i.e., transverse) roll circulations in the boundary layer (BL). A remarkable finding of the evolving radar analyses is the apparent persistence of both transverse rolls and individual updraft, vertical vorticity, and reflectivity cores for periods of up to 30 min or more while moving approximately with the local BL wind. Satellite cloud images and single-camera ground photogrammetry imply that clouds tend to develop either over or on the downwind edge of BL updrafts, with a tendency for clouds to elongate and dissipate in the downwind direction relative to cloud layer winds due to weakening updrafts and mixing with drier overlying air. The Lagrangian and radar wind analyses support a parcel continuity principle for cumulus formation, which requires that rising moist air parcels achieve their LCL before moving laterally out of the updraft. Cumuli form within penetrative updrafts in the elevated residual layer (ERL) overlying the moist BL east of the triple point, but remain capped by a convection inhibition (CIN)-bearing layer above the ERL. Dropsonde data suggest the existence of a convergence line about 80 km east of the triple point where deep lifting of BL moisture and locally reduced CIN together support convection initiation.
