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The Numerical Solution of the One-Dimensional Advection–Diffusion Equation in Layered Coordinates

William K. DewarDepartment of Oceanography and Supercomputer Computations Research Institute, The Florida State University, Tallahassee, Florida

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Trevor J. McDougallCSIRO Marine Research, Hobart, Tasmania, Australia

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Abstract

The numerical solution of the vertical advection–diffusion equation in layered coordinates is revisited. The objectives of this work are to propose a generalization of the discontinuous layered representation of the ocean tracer field to higher-order, smoother representations (while retaining the quasi-Lagrangian character of the coordinate) and compare the solutions generated by several approaches in order to illustrate their respective advantages and disadvantages. The one-dimensional advection–diffusion equation is chosen as a test bed for layered coordinates because ocean simulation for climatic purposes requires the inclusion of dianeutral diffusive processes.

The layered approach is generalized by replacing the traditional stack of well-mixed layers by stacks of piecewise smooth profiles. All the well-known properties of quasi-Lagrangian coordinates are retained. Comparisons of the quasi-Lagrangian solutions with coarse- and fine-resolution fixed grid solutions illustrates the efficiency of the adaptive, quasi-Lagrangian coordinate.

Corresponding author address: Dr. William K. Dewar, Department of Oceanography (4320), The Florida State University, Tallahassee, FL 32306-4320.

Email: dewar@ocean.fsu.edu

Abstract

The numerical solution of the vertical advection–diffusion equation in layered coordinates is revisited. The objectives of this work are to propose a generalization of the discontinuous layered representation of the ocean tracer field to higher-order, smoother representations (while retaining the quasi-Lagrangian character of the coordinate) and compare the solutions generated by several approaches in order to illustrate their respective advantages and disadvantages. The one-dimensional advection–diffusion equation is chosen as a test bed for layered coordinates because ocean simulation for climatic purposes requires the inclusion of dianeutral diffusive processes.

The layered approach is generalized by replacing the traditional stack of well-mixed layers by stacks of piecewise smooth profiles. All the well-known properties of quasi-Lagrangian coordinates are retained. Comparisons of the quasi-Lagrangian solutions with coarse- and fine-resolution fixed grid solutions illustrates the efficiency of the adaptive, quasi-Lagrangian coordinate.

Corresponding author address: Dr. William K. Dewar, Department of Oceanography (4320), The Florida State University, Tallahassee, FL 32306-4320.

Email: dewar@ocean.fsu.edu

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