Generalized large-scale semigeostrophic approximations for the f-plane primitive equations

Sergiy Vasylkevych, Marcel Oliver

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)
240 Downloads (Pure)


We derive a family of balance models for rotating stratified flow in the primitive equation (PE) setting. By construction, the models possess conservation laws for energy and potential vorticity and are formally of the same order of accuracy as Hoskins' semigeostrophic equations. Our construction is based on choosing a new coordinate frame for the PE variational principle in such a way that the consistently truncated Lagrangian degenerates. We show that the balance relations so obtained are elliptic when the fluid is stably stratified and certain smallness assumptions are satisfied. Moreover, the potential temperature can be recovered from the potential vorticity via inversion of a non-standard Monge–Ampère problem which is subject to the same ellipticity condition. While the present work is entirely formal, we conjecture, based on a careful rewriting of the equations of motion and a straightforward derivative count, that the Cauchy problem for the balance models is well posed subject to conditions on the initial data. Our family of models includes, in particular, the stratified analog of the L 1 balance model of Salmon.
Original languageEnglish
Number of pages18
JournalJournal of Physics A: Mathematical and Theoretical
Issue number18
Publication statusPublished - 24 Mar 2016


  • geostrophic balance, primitive equations, strati fi ed fl ow, semi- geostrophic equations, Euler – Poincare equations, variational asymptotics


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