Published In

Monthly Weather Review

Document Type

Article

Publication Date

11-1-1994

Subjects

Hydrology -- Data processing, Hydrologic models -- Evaluation, Inverse problems (Differential equations)

Abstract

Practical hydrostatic ocean models are often restricted to statically stable configurations by the use of a convective adjustment. A common way to do this is to assign an infinite boat conductivity to the water at a given level if the water column should become statically unstable. This is implemented in the form of a switch. When a statically unstable configuration is detected, it is immediately replaced with a statically stable one in which heat is conserved. In this approach, the model is no longer governed by a smooth set of equations, and usual techniques of variational data assimilation must be modified. In this note, a simple one-dimensional diffusive model is presented. Despite its simplicity, this model captures the essential behavior of the convective adjustment scheme in a widely used ocean general circulation model. Since this simple model can be derived from the more complex general circulation model, it then follows that many of the properties of the constrained system can be observed in this very simple scalar ordinary differential equation with a constraint on the solution. Techniques from the theory of optimal control are used to find solutions of a simple formulation of the variational data assimilation problem in this simple case. The optimal solution involves the solution of a nonlinear problem, even when the unconstrained dynamics are linear. In cases with discontinuous dynamics, one cannot define the adjoint of the linearized system in a straightforward manner. The very simplest variational formulation is shown to have nonunique stationary points and undesirable physical consequences. Modifications that lead to better behaved calculations and more meaningful solutions are presented. Whereas it is likely that the underlying principles from control theory are applicable to practical ocean models, the technique used to solve the simple problem may be applicable only to steady problems. Derivation of suitable techniques for initial value problems will involve a major research effort.

Description

This is the publisher's final PDF. © Copyright 1994 American Meteorological Society (AMS). Permission to use figures, tables, and brief excerpts from this work in scientific and educational works is hereby granted provided that the source is acknowledged. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act September 2010 Page 2 or that satisfies the conditions specified in Section 108 of the U.S. Copyright Act (17 USC §108, as revised by P.L. 94-553) does not require the AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a web site or in a searchable database, or other uses of this material, except as exempted by the above statement, requires written permission or a license from the AMS. Additional details are provided in the AMS Copyright Policy, available on the AMS Web site located at (http://www.ametsoc.org/) or from the AMS at 617-227-2425 or copyrights@ametsoc.org.

Persistent Identifier

http://archives.pdx.edu/ds/psu/8378

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