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Toward More Accurate Wave-Permeable Boundary Conditions

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  • 1 Department of Atmospheric Sciences, University of Washington, Seattle, Washington
  • | 2 Department of Mechanical Engineering, University of Washington, Seattle, Washington
  • | 3 Department of Atmospheric Sciences, University of Washington, Seattle, Washington
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Abstract

This paper investigates several fundamental aspects of wave-permeable, or “radiation,” lateral boundary conditions. Orlanski (1976) proposed that approximate wave-permeable boundary conditions could be constructed by advecting disturbances out of the domain at a phase speed c*, which was to be calculated from the values of the prognostic variable near the boundary. Rigorous justification for this approach is possible for one-dimensional shallow-water flow. It is shown, however, that the floating c* approach gives poor results in the one-dimensional shallow-water problem because all accuracy in the c* calculations is eventually destroyed by the positive feedback between errors in c* and (initially small) errors in the prognostic fields at the boundary. Better results were achieved by using fixed values of c*. In our test cases, an externally specified c* could deviate from the true phase speed U + c by 40%–60% and still yield better results than schemes in which c* was calculated at the boundary.

In order to examine the effects of wave dispersion on the question of whether c* should be fixed or calculated, tests were conducted with a two-level shallow-water model. Once again, the simulations with fixed c* were distinctly superior to those in which c* was calculated at the boundary. A reasonable, though nonoptimal, value for the fixed c* was the phase speed of the fastest wave.

Wave dispersion is, however, not the only factor that makes it difficult to specify wave-permeable boundary conditions. Two-dimensional shallow-water waves are nondispersive, but their trace velocities along the x and y axes are functions of wavenumber. As a consequence, the simple radiation boundary condition appropriate for one-dimensional shallow-water flow is just an approximation for two-dimensional flow. Engquist and Majda ( 1977) developed improved boundary conditions for the two-dimensional problem by constructing approximate “one-way equations.” In this paper, the approach of Engquist and Majda is used to construct second-order one-way wave equations for situations with nonzero mean flow. The new boundary condition is tested against several alternative schemes and found to give the best results. The new boundary condition is particularly recommended for situations where waves strike the boundary at nonnormal angles of incidence.

Abstract

This paper investigates several fundamental aspects of wave-permeable, or “radiation,” lateral boundary conditions. Orlanski (1976) proposed that approximate wave-permeable boundary conditions could be constructed by advecting disturbances out of the domain at a phase speed c*, which was to be calculated from the values of the prognostic variable near the boundary. Rigorous justification for this approach is possible for one-dimensional shallow-water flow. It is shown, however, that the floating c* approach gives poor results in the one-dimensional shallow-water problem because all accuracy in the c* calculations is eventually destroyed by the positive feedback between errors in c* and (initially small) errors in the prognostic fields at the boundary. Better results were achieved by using fixed values of c*. In our test cases, an externally specified c* could deviate from the true phase speed U + c by 40%–60% and still yield better results than schemes in which c* was calculated at the boundary.

In order to examine the effects of wave dispersion on the question of whether c* should be fixed or calculated, tests were conducted with a two-level shallow-water model. Once again, the simulations with fixed c* were distinctly superior to those in which c* was calculated at the boundary. A reasonable, though nonoptimal, value for the fixed c* was the phase speed of the fastest wave.

Wave dispersion is, however, not the only factor that makes it difficult to specify wave-permeable boundary conditions. Two-dimensional shallow-water waves are nondispersive, but their trace velocities along the x and y axes are functions of wavenumber. As a consequence, the simple radiation boundary condition appropriate for one-dimensional shallow-water flow is just an approximation for two-dimensional flow. Engquist and Majda ( 1977) developed improved boundary conditions for the two-dimensional problem by constructing approximate “one-way equations.” In this paper, the approach of Engquist and Majda is used to construct second-order one-way wave equations for situations with nonzero mean flow. The new boundary condition is tested against several alternative schemes and found to give the best results. The new boundary condition is particularly recommended for situations where waves strike the boundary at nonnormal angles of incidence.

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