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Control Of Gravitational Oscillations in Variational Data Assimilation

X. ZouSupercomputer Computations Research Institute, Florida State University, Tallahassee, Florida

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I. M. NavonDepartment of Mathematics and Supercomputer Computations, Research Institute, Florida State, University, Tallahassee, Florida

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J. SelaNOAA, NMC, World Weather Building, Washington, D.C.

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Abstract

Variational four-dimensional data assimilation, combined with a penalty method constraining time derivatives of the surface pressure, the divergence, and the gravity-wave components is implemented on an adiabatic version of the National Meteorological Center's 18-level primitive equation spectral model with surface drag and horizontal diffusion. Experiments combining the Machenhauer nonlinear normal-mode initialization procedure and its adjoint with the variational data assimilation are also presented. The modified variational data-assimilation schemes are tested to assess how well they control gravity-wave oscillations.

The gradient of a penalized cost function can be obtained by a single integration of the adjoint model. A detailed derivation of the gradient calculation of different penalized cost functions is presented, which is not restricted to a specific model.

Numerical results indicate that the inclusion of penalty terms into the cost function will change the model solution as desired. The advantages of the use of simple penalty terms over penalty terms including the model normal modes results in a simplification of the procedure, allowing a more direct control over the model variables and the possibility of using weak constraints to eliminate the high-frequency gravity-wave oscillations. This approach does not require direct information about the model normal modes. One of the encouraging results obtained is that the introduction of the penalty terms does not slow the convergence rate of the minimization process.

Abstract

Variational four-dimensional data assimilation, combined with a penalty method constraining time derivatives of the surface pressure, the divergence, and the gravity-wave components is implemented on an adiabatic version of the National Meteorological Center's 18-level primitive equation spectral model with surface drag and horizontal diffusion. Experiments combining the Machenhauer nonlinear normal-mode initialization procedure and its adjoint with the variational data assimilation are also presented. The modified variational data-assimilation schemes are tested to assess how well they control gravity-wave oscillations.

The gradient of a penalized cost function can be obtained by a single integration of the adjoint model. A detailed derivation of the gradient calculation of different penalized cost functions is presented, which is not restricted to a specific model.

Numerical results indicate that the inclusion of penalty terms into the cost function will change the model solution as desired. The advantages of the use of simple penalty terms over penalty terms including the model normal modes results in a simplification of the procedure, allowing a more direct control over the model variables and the possibility of using weak constraints to eliminate the high-frequency gravity-wave oscillations. This approach does not require direct information about the model normal modes. One of the encouraging results obtained is that the introduction of the penalty terms does not slow the convergence rate of the minimization process.

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