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The Step-Mountain Coordinate: Model Description and Performance for Cases of Alpine Lee Cyclogenesis and for a Case of an Appalachian Redevelopment

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  • 1 Geophysical Fluid Dynamics Program, Princeton University, Princeton, New Jersey
  • | 2 Department of Physics and Meteorology, University of Belgrade, Yugoslavia
  • | 3 National Meteorological Center, NWS, NOAA, Washington, DC
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Abstract

The problem of the pressure gradient force error in the case of the terrain-following (sigma) coordinate does not appear to have a solution. The problem is not one of truncation error in the calculation of space derivatives involved. Thus, with temperature profiles resulting in large errors, an increase in vertical resolution may not reduce and is even likely to increase the error. Therefore, an approach abandoning the sigma system has been proposed. It involves the use of “step” mountains with coordinate surfaces prescribed to remain at fixed elevations at places where they touch (and define) or intersect the ground surface. Thus, the coordinate surfaces are quasi-horizontal, and the sigma system problem is not present. At the same time, the simplicity of the sigma system is maintained.

In this paper, design of the model (“silhouette” averaged) mountains, properties of the wall boundary condition, and the scheme for calculation of the potential to kinetic energy conversion are presented. For an advection scheme achieving a strict control of the nonlinear energy cascade on the semistaggered grid, it is demonstrated that a straightforward no-slip wall boundary condition maintains conservation properties of the scheme with no vertical walls, which are important from the point of view of the control of this energy cascade from large to small scales. However, with that simple boundary condition considered, momentum is not conserved. The scheme conserving energy in conversion between the potential and kinetic energy, given earlier for the one-dimensional case, is extended to two dimensions.

Results of real data experiments are described, testing the performance of the resulting “Step-mountain” model. An attractive feature of a step-mountain (“eta”) model is that it can easily be run as a sigma system model, the only difference being the definition of ground surface grid point values of the vertical coordinate. This permits a comparison of the sigma and the eta formulations. Two experiments of this kind have been made, with a model version including realistic steep mountains (steps at 290, 1112 and 2433 m). They have both revealed a substantial amount of noise resulting from the sigma, as compared to the eta, formulation. One of these experiments, especially with the step mountains, gave a rather successful simulation of the perhaps difficult “historic” Buzzi–Tibaldi case of Genoa lee cyclogenesis. A parallel experiment showed that, starting with the same initial data, one obtains no cyclogenesis without mountains. Still, the mountains experiment did simulate the accompanying midtropospheric cutoff, a phenomenon that apparently has not been reproduced in previous simulations of mountain-induced Genoa lee cyclogeneses.

For a North American limited area region, experimental step-mountain simulations were performed for a case of March 1984, involving development of a secondary storm southeast of the Appalachians. Neither the then operational U.S. National Meteorological Center's Limited Area Forecast Model (LFM) nor the recently introduced Nested Grid Model (NGM) were successful in simulating the redevelopment. On the other hand, the step-mountain model, with a space resolution set up to mimic that of NGM, successfully simulated the ridging that indicates the redevelopment.

Abstract

The problem of the pressure gradient force error in the case of the terrain-following (sigma) coordinate does not appear to have a solution. The problem is not one of truncation error in the calculation of space derivatives involved. Thus, with temperature profiles resulting in large errors, an increase in vertical resolution may not reduce and is even likely to increase the error. Therefore, an approach abandoning the sigma system has been proposed. It involves the use of “step” mountains with coordinate surfaces prescribed to remain at fixed elevations at places where they touch (and define) or intersect the ground surface. Thus, the coordinate surfaces are quasi-horizontal, and the sigma system problem is not present. At the same time, the simplicity of the sigma system is maintained.

In this paper, design of the model (“silhouette” averaged) mountains, properties of the wall boundary condition, and the scheme for calculation of the potential to kinetic energy conversion are presented. For an advection scheme achieving a strict control of the nonlinear energy cascade on the semistaggered grid, it is demonstrated that a straightforward no-slip wall boundary condition maintains conservation properties of the scheme with no vertical walls, which are important from the point of view of the control of this energy cascade from large to small scales. However, with that simple boundary condition considered, momentum is not conserved. The scheme conserving energy in conversion between the potential and kinetic energy, given earlier for the one-dimensional case, is extended to two dimensions.

Results of real data experiments are described, testing the performance of the resulting “Step-mountain” model. An attractive feature of a step-mountain (“eta”) model is that it can easily be run as a sigma system model, the only difference being the definition of ground surface grid point values of the vertical coordinate. This permits a comparison of the sigma and the eta formulations. Two experiments of this kind have been made, with a model version including realistic steep mountains (steps at 290, 1112 and 2433 m). They have both revealed a substantial amount of noise resulting from the sigma, as compared to the eta, formulation. One of these experiments, especially with the step mountains, gave a rather successful simulation of the perhaps difficult “historic” Buzzi–Tibaldi case of Genoa lee cyclogenesis. A parallel experiment showed that, starting with the same initial data, one obtains no cyclogenesis without mountains. Still, the mountains experiment did simulate the accompanying midtropospheric cutoff, a phenomenon that apparently has not been reproduced in previous simulations of mountain-induced Genoa lee cyclogeneses.

For a North American limited area region, experimental step-mountain simulations were performed for a case of March 1984, involving development of a secondary storm southeast of the Appalachians. Neither the then operational U.S. National Meteorological Center's Limited Area Forecast Model (LFM) nor the recently introduced Nested Grid Model (NGM) were successful in simulating the redevelopment. On the other hand, the step-mountain model, with a space resolution set up to mimic that of NGM, successfully simulated the ridging that indicates the redevelopment.

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