Euler–Lagrange equations for the spectral element shallow water system |
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| Authors: | JC Levin DB Haidvogel B Chua AF Bennett M Iskandarani |
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| Institution: | aInstitute of Marine and Coastal Sciences, Rutgers University, 71 Dudley Rd., New Brunswick, NJ 08901, USA;bCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, USA;cRosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149, USA |
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| Abstract: | We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with which elemental assembly and mass averaging are applied in the forward and in the adjoint systems. Our derivation can be extended to obtain an adjoint for any Galerkin finite element and spectral element system.We begin the derivations using a linear wave equation in one dimension. We then apply our technique to a two-dimensional shallow water ocean model and test it on a classic double-gyre problem. The spectral element forward and adjoint ocean models can be used in a variety of inverse applications, ranging from traditional data assimilation and parameter estimation, to the less traditional model sensitivity and stability analyses, and ensemble prediction. Here the Euler–Lagrange equations are solved by an indirect representer algorithm. |
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| Keywords: | Spectral element method Euler– Lagrange equations Inverse ocean model 4DVAR variational data assimilation Twin experiment |
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