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- Author or Editor: Laura L. Ehret x
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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
The optimization method proposed in this paper is for determining open boundary conditions from interior observations. Unknown open boundary conditions are represented by an open boundary parameter vector (B), while known interior observational values are used to form an observation vector (O). For a hypothetical B* (generally taken as the zero vector for the first time step and as the optimally determined B at the previous time step afterward), the numerical ocean model is integrated to obtain solutions (S*) at interior observation points. The root-mean-square difference between S* and O might not be minimal. The authors change B* with different increments δ B. Optimization is used to get the best B by minimizing the error between O and S.
The proposed optimization method can be easily incorporated into any ocean models, whether linear or nonlinear, reversible or irreversible, etc. Applying this method to a primitive equation model with turbulent mixing processes such as the Princeton Ocean Model (POM), an important procedure is to smooth the open boundary parameter vector. If smoothing is not used, POM can only be integrated within a finite period (45 days in this case). If smoothing is used, the model is computationally stable. Furthermore, this optimization method performed well when random noise was added to the “observational” points. This indicates that real-time data can be used to inverse the unknown open boundary values.
Abstract
The optimization method proposed in this paper is for determining open boundary conditions from interior observations. Unknown open boundary conditions are represented by an open boundary parameter vector (B), while known interior observational values are used to form an observation vector (O). For a hypothetical B* (generally taken as the zero vector for the first time step and as the optimally determined B at the previous time step afterward), the numerical ocean model is integrated to obtain solutions (S*) at interior observation points. The root-mean-square difference between S* and O might not be minimal. The authors change B* with different increments δ B. Optimization is used to get the best B by minimizing the error between O and S.
The proposed optimization method can be easily incorporated into any ocean models, whether linear or nonlinear, reversible or irreversible, etc. Applying this method to a primitive equation model with turbulent mixing processes such as the Princeton Ocean Model (POM), an important procedure is to smooth the open boundary parameter vector. If smoothing is not used, POM can only be integrated within a finite period (45 days in this case). If smoothing is used, the model is computationally stable. Furthermore, this optimization method performed well when random noise was added to the “observational” points. This indicates that real-time data can be used to inverse the unknown open boundary values.
