Euler-Lagrange equations for the spectral element shallow water system

Julia C. Levin, Dale B. Haidvogel, B. Chua, A. F. Bennett, M. Iskandarani

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)348-377
Number of pages30
JournalOcean Modelling
Issue number3-4
StatePublished - 2006


  • 4DVAR variational data assimilation
  • Euler-Lagrange equations
  • Inverse ocean model
  • Spectral element method
  • Twin experiment

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Oceanography
  • Geotechnical Engineering and Engineering Geology
  • Atmospheric Science


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