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- Author or Editor: Dale R. Durran x
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It is demonstrated that the inertial oscillation is not produced exclusively by “inertial forces,” and that the inertial oscillation appears as oscillatory motion even when viewed from a nonrotating frame of reference. The component of true gravity parallel to the geopotential surfaces plays a central role in forcing the inertial oscillation, and in particular it is the only force driving the oscillation in the nonrotating reference frame.
It is demonstrated that the inertial oscillation is not produced exclusively by “inertial forces,” and that the inertial oscillation appears as oscillatory motion even when viewed from a nonrotating frame of reference. The component of true gravity parallel to the geopotential surfaces plays a central role in forcing the inertial oscillation, and in particular it is the only force driving the oscillation in the nonrotating reference frame.
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.
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.
Abstract
The dynamics of a prototypical atmospheric bore are investigated through a series of two-dimensional numerical simulations and linear theory. These simulations demonstrate that the bore dynamics are inherently finite amplitude. Although the environment supports linear trapped waves, the supported waves propagate in roughly the opposite direction to that of the bore. Qualitative analysis of the Scorer parameter can therefore give misleading indications of the potential for wave trapping, and linear internal gravity wave dynamics do not govern the behavior of the bore. The presence of a layer of enhanced static stability below a deep layer of lower stability, as would be created by a nocturnal inversion, was not necessary for the development of a bore. The key environmental factor allowing bore propagation was the presence of a low-level jet directed opposite to the movement of the bore. Significant turbulence developed in the layer between the jet maximum and the surface, which reduced the low-level static stability behind the bore. Given the essential role of jets and thereby strong environmental wind shear, and given that idealized bores may persist in environments in which the static stability is constant with height, shallow-water dynamics do not appear to be quantitatively applicable to atmospheric bores propagating against low-level jets, although there are qualitative analogies.
Abstract
The dynamics of a prototypical atmospheric bore are investigated through a series of two-dimensional numerical simulations and linear theory. These simulations demonstrate that the bore dynamics are inherently finite amplitude. Although the environment supports linear trapped waves, the supported waves propagate in roughly the opposite direction to that of the bore. Qualitative analysis of the Scorer parameter can therefore give misleading indications of the potential for wave trapping, and linear internal gravity wave dynamics do not govern the behavior of the bore. The presence of a layer of enhanced static stability below a deep layer of lower stability, as would be created by a nocturnal inversion, was not necessary for the development of a bore. The key environmental factor allowing bore propagation was the presence of a low-level jet directed opposite to the movement of the bore. Significant turbulence developed in the layer between the jet maximum and the surface, which reduced the low-level static stability behind the bore. Given the essential role of jets and thereby strong environmental wind shear, and given that idealized bores may persist in environments in which the static stability is constant with height, shallow-water dynamics do not appear to be quantitatively applicable to atmospheric bores propagating against low-level jets, although there are qualitative analogies.
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.
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.
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.
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.
Abstract
The physical reason for quasi-geostrophic vertical motion is reviewed. Various techniques for estimating synoptic-scale vertical motion are examined, and their utility (or lack thereof) is illustrated by a case study. The Q-vector approach appears to provide the best means of calculating vertical motions numerically. The vertical motion can be estimated by eye with reasonable accuracy by examining the advection of vorticity by the thermal wind or by examining the relative wind and the isobar field on an isentropic chart. The traditional form of the omega equation is not well suited for practical calculation.
Abstract
The physical reason for quasi-geostrophic vertical motion is reviewed. Various techniques for estimating synoptic-scale vertical motion are examined, and their utility (or lack thereof) is illustrated by a case study. The Q-vector approach appears to provide the best means of calculating vertical motions numerically. The vertical motion can be estimated by eye with reasonable accuracy by examining the advection of vorticity by the thermal wind or by examining the relative wind and the isobar field on an isentropic chart. The traditional form of the omega equation is not well suited for practical calculation.
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Abstract
The development of rotor flow associated with mountain lee waves is investigated through a series of high-resolution simulations with the nonhydrostatic Coupled Ocean–Atmospheric Mesoscale Prediction System (COAMPS) model using free-slip and no-slip lower boundary conditions. Kinematic considerations suggest that boundary layer separation is a prerequisite for rotor formation. The numerical simulations demonstrate that boundary layer separation is greatly facilitated by the adverse pressure gradients associated with trapped mountain lee waves and that boundary layer processes and lee-wave-induced perturbations interact synergistically to produce low-level rotors. Pairs of otherwise identical free-slip and no-slip simulations show a strong correlation between the strength of the lee-wave-induced pressure gradients in the free-slip simulation and the strength of the reversed flow in the corresponding no-slip simulation.
Mechanical shear in the planetary boundary layer is the primary source of a sheet of horizontal vorticity that is lifted vertically into the lee wave at the separation point and carried, at least in part, into the rotor itself. Numerical experiments show that high shear in the boundary layer can be sustained without rotor development when the atmospheric structure is unfavorable for the formation of trapped lee waves. Although transient rotors can be generated with a free-slip lower boundary, realistic rotors appear to develop only in the presence of surface friction.
In a series of simulations based on observational data, increasing the surface roughness length beyond values typical for a smooth surface (z 0 = 0.01 cm) decreases the rotor strength, although no rotors form when free-slip conditions are imposed at the lower boundary. A second series of simulations based on the same observational data demonstrate that increasing the surface heat flux above the lee slope increases the vertical extent of the rotor circulation and the strength of the turbulence but decreases the magnitude of the reversed rotor flow.
Abstract
The development of rotor flow associated with mountain lee waves is investigated through a series of high-resolution simulations with the nonhydrostatic Coupled Ocean–Atmospheric Mesoscale Prediction System (COAMPS) model using free-slip and no-slip lower boundary conditions. Kinematic considerations suggest that boundary layer separation is a prerequisite for rotor formation. The numerical simulations demonstrate that boundary layer separation is greatly facilitated by the adverse pressure gradients associated with trapped mountain lee waves and that boundary layer processes and lee-wave-induced perturbations interact synergistically to produce low-level rotors. Pairs of otherwise identical free-slip and no-slip simulations show a strong correlation between the strength of the lee-wave-induced pressure gradients in the free-slip simulation and the strength of the reversed flow in the corresponding no-slip simulation.
Mechanical shear in the planetary boundary layer is the primary source of a sheet of horizontal vorticity that is lifted vertically into the lee wave at the separation point and carried, at least in part, into the rotor itself. Numerical experiments show that high shear in the boundary layer can be sustained without rotor development when the atmospheric structure is unfavorable for the formation of trapped lee waves. Although transient rotors can be generated with a free-slip lower boundary, realistic rotors appear to develop only in the presence of surface friction.
In a series of simulations based on observational data, increasing the surface roughness length beyond values typical for a smooth surface (z 0 = 0.01 cm) decreases the rotor strength, although no rotors form when free-slip conditions are imposed at the lower boundary. A second series of simulations based on the same observational data demonstrate that increasing the surface heat flux above the lee slope increases the vertical extent of the rotor circulation and the strength of the turbulence but decreases the magnitude of the reversed rotor flow.
Abstract
The generation of nonstationary trapped mountain lee waves through nonlinear wave dynamics without any concomitant change in the background flow is investigated by conducting two-dimensional mountain wave simulations. These simulations demonstrate that finite-amplitude lee-wave patterns can exhibit temporal variations in local wavelength and amplitude, even when the background flow is perfectly steady. For moderate amplitudes, a nonlinear wave interaction involving the stationary trapped wave and a pair of nonstationary waves appears to be responsible for the development of nonstationary perturbations on the stationary trapped wave. This pair of nonstationary waves consists of a trapped wave and a vertically propagating wave, both having horizontal wavelengths approximately twice that of the stationary trapped wave. As the flow becomes more nonlinear, the nonstationary perturbations involve a wider spectrum of horizontal wavelengths and may dominate the overall wave pattern at wave amplitudes significantly below the threshold required to produce wave breaking. Sensitivity tests in which the wave propagation characteristics of the basic state are modified without changing the horizontal wavelength of the stationary trapped wave indicate these nonstationary perturbations are absent when the background flow does not support nonstationary trapped waves with horizontal wavelengths approximately twice that of the stationary trapped mode. These sensitivity tests also show that a second nonstationary trapped wave can assume the role of the nonstationary vertically propagating wave when the Scorer parameter in the upper layer is reduced below the threshold that will support the vertically propagating wave. In this case, a resonant triad composed of three trapped waves appears to be responsible for the development of nonstationary perturbations.
The simulations suggest that strongly nonlinear wave dynamics can generate a wider range of nonstationary trapped modes than that produced by temporal variations in the background flow. It is suggested that the irregular variations in lee-wave wavelength and amplitude observed in real atmospheric flows and the complex fluctuations above a fixed point that are occasionally found in wind profiler observations of trapped lee waves are more likely to be generated by nonlinear wave dynamics than changes in the background flow.
Abstract
The generation of nonstationary trapped mountain lee waves through nonlinear wave dynamics without any concomitant change in the background flow is investigated by conducting two-dimensional mountain wave simulations. These simulations demonstrate that finite-amplitude lee-wave patterns can exhibit temporal variations in local wavelength and amplitude, even when the background flow is perfectly steady. For moderate amplitudes, a nonlinear wave interaction involving the stationary trapped wave and a pair of nonstationary waves appears to be responsible for the development of nonstationary perturbations on the stationary trapped wave. This pair of nonstationary waves consists of a trapped wave and a vertically propagating wave, both having horizontal wavelengths approximately twice that of the stationary trapped wave. As the flow becomes more nonlinear, the nonstationary perturbations involve a wider spectrum of horizontal wavelengths and may dominate the overall wave pattern at wave amplitudes significantly below the threshold required to produce wave breaking. Sensitivity tests in which the wave propagation characteristics of the basic state are modified without changing the horizontal wavelength of the stationary trapped wave indicate these nonstationary perturbations are absent when the background flow does not support nonstationary trapped waves with horizontal wavelengths approximately twice that of the stationary trapped mode. These sensitivity tests also show that a second nonstationary trapped wave can assume the role of the nonstationary vertically propagating wave when the Scorer parameter in the upper layer is reduced below the threshold that will support the vertically propagating wave. In this case, a resonant triad composed of three trapped waves appears to be responsible for the development of nonstationary perturbations.
The simulations suggest that strongly nonlinear wave dynamics can generate a wider range of nonstationary trapped modes than that produced by temporal variations in the background flow. It is suggested that the irregular variations in lee-wave wavelength and amplitude observed in real atmospheric flows and the complex fluctuations above a fixed point that are occasionally found in wind profiler observations of trapped lee waves are more likely to be generated by nonlinear wave dynamics than changes in the background flow.
