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Dale R. Durran

Abstract

The third-order Adams–Bashforth method is compared with the leapfrog scheme. Like the leapfrog scheme, the third-order Adams–Bashforth method is an explicit technique that requires just one function evaluation per time step. Yet the third-order Adams–Bashforth method is not subject to time splitting instability and it is more accurate than the leapfrog scheme. In particular, the O[(Δt)4] amplitude error of the third-order Adams–Bashforth method can be a marked improvement over the O[(Δt)2] amplitude error generated by the Asselin-filtered leapfrog scheme—even when the filter factor is very small. The O[(Δt)4] phase-speed errors associated with third-order Adams–Bashforth time differencing can also be significantly less than the O[(Δt)2] errors produced by the leapfrog method. The third-order Adams–Bashforth method does use more storage than the leapfrog method, but its storage requirements are not particularly burdensome. Several numerical examples are provided illustrating the superiority of third-order Adams–Bashforth time differencing. Other higher-order alternatives to the Adams–Bashforth method are also surveyed. A discussion is presented describing the general relationship between the truncation error of an ordinary differential solver and the amplitude and phase-speed errors that develop when the scheme is used to integrate oscillatory systems.

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J. Brent Bower
and
Dale R. Durran

Abstract

Wind profiler data from Lay Creek, Colorado, along with stability data from the Lander and Grand Junction rawinsonde observations, were examined in an attempt to link various parameters in the upstream flow to the onset of strong downslope winds in Boulder. Some correlation was found between the occurrence of high surface winds at Boulder and the upstream wind direction, upper tropospheric wind shear and the vertical phase shift across the troposphere. However, these parameters alone were not able to distinguish between windstorm and nonwindstorm events. It is likely that the remaining ambiguity could be eliminated with information on the location and strength of inversions in the upstream flow.

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Dale R. Durran
and
Daniel B. Weber

Abstract

The factor that contribute to the development of the sharp edge along the poleward boundary of the jet-stream cirrus are examined in three cast studies, using wind field and thermodynamic information from the FGGE dataset as input to a numerical model. The model generated a cloud field that satisfactorily reproduced the cirrus cloud distributions shown on satellite photos. Trajectory calculations, together with an examination of the vertical velocity field, suggest that the cloud boundary is not directly produced by differential vertical motions (with sinking on the clear side of the cloud edge and rising motion on the cloudy side). In each case, significant ascent was found in the clear air on the poleward side of the cloud boundary. The cloud. boundary appears to develop when a preexisting moisture gradient experiences relatively uniform lifting. Differential vertical motions (together with horizontal confluence) were found to play a significant role in generating the initial moisture gradient.

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Robert L. Grossman
and
Dale R. Durran

Abstract

Seven-year averaged values of percent frequency of occurrence of highly reflective cloud for the months June, July, and August indicate that offshore convection is a major component of the cloudiness of the southwest monsoon. Principal areas of convection occur off of the western coats of India, Burma, Thailand, and the Philippines. This study concentrates on the area upstream of the Western Ghats Mountains of India. Analysis of a special boundary layer mission flown during the WMO/ICSU Summer Monsoon Experiment leads us to believe that partial deceleration of the monsoon flow by upstream blocking effects of the mountains initiates and maintains a vertical and horizontal motion field that could support the observed convection. Data obtained on this mission allow a large-scale momentum budget computation for the subcloud layer, which shows pressure deceleration to be significant. The budget, dominated by advection, predicts an increase of average wind speed which is observed. The pressure deceleration result is further explored by applying an idealized monsoon flow to an analytical, nonliner, two-dimensional mountain-flow interaction model using a smoothed profile of the Western Ghats Mountains. The model qualitatively agrees with aircraft observations taken in the subcloud layer, and predicts large vertical wind shears over the coastal area and mountain crest which would inhibit deep convection. These shears are confirmed by earlier observations.

When the lifting predicted by the model is applied to mean dropwindsonde soundings, well upstream of the coast, for days with and without offshore convection, deep convection is predicted for the mean sounding associated with offshore convection. The mean sounding for days without deep convection shows more offshore lifting is needed to produce convection; even if the lifting were applied, the convection would not be very deep due to a cooler surface layer and a dry layer above the boundary layer which may have originated from the desert areas to the west and/or upper tropospheric downward motion. We conclude that the mountains, though not very high, play an important role in overall monsoon convection for India. It is suggested that, given the climatic character of offshore monsoon convection, interaction of the low-level flow with the western coastal mountains of India, Burma, Thailand, and the Philippines should be considered a factor in monsoon climatology.

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Dale R. Durran
and
Joseph B. Klemp

Abstract

A two-dimensional, nonlinear, nonhydrostatic model is described which allows the calculation of moist airflow in mountainous terrain. The model is compressible, uses a terrain-following coordinate system, and employs lateral and upper boundary conditions which minimize wave reflections.

The model's accuracy and sensitivity are examined. These tests suggest that in numerical simulations of vertically propagating, highly nonlinear mountain waves, a wave absorbing layer does not accurately mimic the effects of wave breakdown and dissipation at high levels in the atmosphere. In order to obtain a correct simulation, the region in which the waves are physically absorbed must generally be included in the computational domain (a nonreflective upper boundary condition should be used as well).

The utility of the model is demonstrated in two examples (linear waves in a uniform atmosphere and the 11 January 1972 Boulder windstorm) which illustrate how the presence of moisture can influence propagating waves. In both cases, the addition of moisture to the upstream flow greatly reduces the wave response.

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Joseph B. Klemp
and
Dale R. Durran

Abstract

A radiative upper boundary condition is proposed for numerical mesoscale models which allows vertically propagating internal gravity waves to pass out of the computational domain with minimal reflection. In this formulation, the pressure along the upper boundary is determined from the Fourier transform of the vertical velocity at that boundary. This boundary condition can easily be incorporated in a wide variety of models and requires little additional computation. The radiation boundary condition is derived from the linear, hydrostatic, Boussinesq equations of motion, neglecting Coriolis effects. However, tests of this radiation boundary condition in the presence of nonhydrostatic, Coriolis, nonlinear and non-Boussinesq effects suggest that it would be effective in many mesoscale modeling applications.

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Lucas M. Harris
and
Dale R. Durran

Abstract

Most mesoscale models can be run with either one-way (parasitic) or two-way (interactive) grid nesting. This paper presents results from a linear 1D shallow-water model to determine whether the choice of nesting method can have a significant impact on the solution. Two-way nesting was found to be generally superior to one-way nesting. The only situation in which one-way nesting performs better than two-way is when very poorly resolved waves strike the nest boundary. A simple filter is proposed for use exclusively on the coarse-grid values within the sponge zone of an otherwise conventional sponge boundary condition (BC). The two-way filtered sponge BC gives better results than any of the other methods considered in these tests. Results for all wavelengths were found to be robust to other changes in the formulation of the sponge boundary, particularly with the width of the sponge layer. The increased reflection for longer-wavelength disturbances in the one-way case is due to a phase difference between the coarse- and nested-grid solutions at the nested-grid boundary that accumulates because of the difference in numerical phase speeds between the grids. Reflections for two-way nesting may be estimated from the difference in numerical group velocities between the coarse and nested grids, which only becomes large for waves that are poorly resolved on the coarse grid.

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Dale R. Durran
and
Peter N. Blossey

Abstract

Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fast-wave–slow-wave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on Adams methods or on backward differencing. Two new families of methods are proposed that have good stability properties in fast-wave–slow-wave problems: one family is based on Adams methods and the other on backward schemes. Here the focus is primarily on four specific schemes drawn from these two families: a pair of Adams methods and a pair of backward methods that are either (i) optimized for third-order accuracy in the explicit component of the full IMEX scheme, or (ii) employ particularly good schemes for the implicit component. These new schemes are superior, in many respects, to the linear multistep IMEX schemes currently in use.

The behavior of these schemes is compared theoretically in the context of the simple oscillation equation and also for the linearized equations governing stratified compressible flow. Several schemes are also tested in fully nonlinear simulations of gravity waves generated by a localized source in a shear flow.

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Dale R. Durran
,
Ming-Jen Yang
,
Donald N. Slinn
, and
Randy G. Brown

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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