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Abstract
Numerical simulations are conducted to examine the role played by different amplification mechanisms in the development of large-amplitude mountain waves. It is shown that when the static stability has a two-layer structure, the nonlinear response can differ significantly from the solution to the equivalent linear problem when the parameter Nh/U is as small as 0.3. In the cases where the nonlinear waves are much larger than their linear counterparts, the highest stability is found in the lower layer and the flow resembles a hydraulic jump. Simulations of the 11 January 1972 Boulder windstorm are presented which suggest that the transition to supercritical flow, forced by the presence of a low-level inversion, plays an essential role in triggering the windstorm. The similarities between breaking waves and nonbreaking waves which undergo a transition to supercritical flow are discussed.
Abstract
Numerical simulations are conducted to examine the role played by different amplification mechanisms in the development of large-amplitude mountain waves. It is shown that when the static stability has a two-layer structure, the nonlinear response can differ significantly from the solution to the equivalent linear problem when the parameter Nh/U is as small as 0.3. In the cases where the nonlinear waves are much larger than their linear counterparts, the highest stability is found in the lower layer and the flow resembles a hydraulic jump. Simulations of the 11 January 1972 Boulder windstorm are presented which suggest that the transition to supercritical flow, forced by the presence of a low-level inversion, plays an essential role in triggering the windstorm. The similarities between breaking waves and nonbreaking waves which undergo a transition to supercritical flow are discussed.
Abstract
A new diagnostic equation is presented which exhibits many advantages over the conventional forms of the anelastic continuity equation. Scale analysis suggests that use of this “pseudo-incompressible equation” is justified if the Lagrangian time scale of the disturbance is large compared with the time scale for sound wave propagation and the perturbation pressure is small compared to the vertically varying mean-state pressure. No assumption about the magnitude of the perturbation potential temperature or the strength of the mean-state stratification is required.
In the various anelastic approximations, the influence of the perturbation density field on the mass balance is entirely neglected. In contrast, the mass-balance in the “pseudo-incompressible approximation” accounts for those density perturbations associated (through the equation of state) with perturbations in the temperature field. Density fluctuations associated with perturbations in the pressure field are neglected.
The pseudo-incompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. As the stability increases, the pseudo-incompressible approximation gives a more accurate result. The pseudo-incompressible equation, together with the unapproximated momentum and thermodynamic equations, forms a closed system of governing equations that filters sound waves. The pseudo-incompressible system conserves an energy form that is directly analogous to the total energy conserved by the complete compressible system.
The pseudo-incompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. The pseudo-incompressible equation also permits the diagnostic calculation of the vertical velocity in adiabatic flow. The pseudo-incompressible equation might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the three-dimensional velocity field and very coarse resolution (single sounding) thermodynamic data.
Abstract
A new diagnostic equation is presented which exhibits many advantages over the conventional forms of the anelastic continuity equation. Scale analysis suggests that use of this “pseudo-incompressible equation” is justified if the Lagrangian time scale of the disturbance is large compared with the time scale for sound wave propagation and the perturbation pressure is small compared to the vertically varying mean-state pressure. No assumption about the magnitude of the perturbation potential temperature or the strength of the mean-state stratification is required.
In the various anelastic approximations, the influence of the perturbation density field on the mass balance is entirely neglected. In contrast, the mass-balance in the “pseudo-incompressible approximation” accounts for those density perturbations associated (through the equation of state) with perturbations in the temperature field. Density fluctuations associated with perturbations in the pressure field are neglected.
The pseudo-incompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. As the stability increases, the pseudo-incompressible approximation gives a more accurate result. The pseudo-incompressible equation, together with the unapproximated momentum and thermodynamic equations, forms a closed system of governing equations that filters sound waves. The pseudo-incompressible system conserves an energy form that is directly analogous to the total energy conserved by the complete compressible system.
The pseudo-incompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. The pseudo-incompressible equation also permits the diagnostic calculation of the vertical velocity in adiabatic flow. The pseudo-incompressible equation might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the three-dimensional velocity field and very coarse resolution (single sounding) thermodynamic data.
Abstract
Expressions are derived for the local pseudomomentum density in two-dimensional compressible stratified flow and are compared with the expressions for pseudomomentum in two-dimensional Boussinesq and anelastic flow derived by Shepherd and by Scinocca and Shepherd. To facilitate this comparison, algebraically simpler expressions for the anelastic and Boussinesq pseudomomentum are also obtained. When the vertical wind shear in the reference-state flow is constant with height, the Boussinesq pseudomomentum is shown to reduce to a particularly simple form in which the pseudomomentum is proportional to the perturbation vorticity times the fluid-parcel displacement. The extension of these compressible pseudomomentum diagnostics to viscous flow and to three-dimensional flows with zero potential vorticity is also discussed.
An expression is derived for the pseudomomentum flux in stratified compressible flow. This flux is shown to simultaneously satisfy the group-velocity condition for both sound waves and gravity waves in an isothermal atmosphere with a constant basic-state wind speed. Consistent with the earlier results of Andrews and McIntyre, it is shown that for inviscid flow over a topographic barrier, the pseudomomentum flux through the lower boundary is identical to the cross-mountain pressure drag—provided that the flow is steady and that the elevation of the topography returns to its upstream value on the downstream side of the mountain.
Abstract
Expressions are derived for the local pseudomomentum density in two-dimensional compressible stratified flow and are compared with the expressions for pseudomomentum in two-dimensional Boussinesq and anelastic flow derived by Shepherd and by Scinocca and Shepherd. To facilitate this comparison, algebraically simpler expressions for the anelastic and Boussinesq pseudomomentum are also obtained. When the vertical wind shear in the reference-state flow is constant with height, the Boussinesq pseudomomentum is shown to reduce to a particularly simple form in which the pseudomomentum is proportional to the perturbation vorticity times the fluid-parcel displacement. The extension of these compressible pseudomomentum diagnostics to viscous flow and to three-dimensional flows with zero potential vorticity is also discussed.
An expression is derived for the pseudomomentum flux in stratified compressible flow. This flux is shown to simultaneously satisfy the group-velocity condition for both sound waves and gravity waves in an isothermal atmosphere with a constant basic-state wind speed. Consistent with the earlier results of Andrews and McIntyre, it is shown that for inviscid flow over a topographic barrier, the pseudomomentum flux through the lower boundary is identical to the cross-mountain pressure drag—provided that the flow is steady and that the elevation of the topography returns to its upstream value on the downstream side of the mountain.
Abstract
Numerical simulations are examined in order to determine the local mean flow response to the generation, propagation, and breakdown of two-dimensional mountain waves. Realistic and idealized cases are considered, and in all instances the pressure drag exerted by flow across an O(40 km) wide mountain fails to produced a significant net mean flow deceleration in the O(400 km) region surrounding the mountain. The loss of momentum in the local patches of decelerated flow that appear in regions of wave overturning directly above the mountain is approximately compensated by momentum gained in other nearby patches of accelerated flow. The domain-average mean flow deceleration in the O(400 km) domain is not determined solely by the divergence of the horizontally averaged momentum flux, 〈ρ¯u′w′〉, because differences in the upstream and downstream values of ρu2 +p provide nontrivial contributions to the total domain-averaged momentum budget. As confirmed by additional simulations in an O(1000 km) wide periodic domain, terrain-induced perturbations in the pressure and horizontal velocity fields are rapidly transmitted hundreds of kilometers away from the mountain and distributed as a small-amplitude signal over a very broad area far away from the mountain. These results suggest that both 〈ρ¯u′w′〉 and information about the wave-induced horizontal momentum fluxes need to be parameterized in order to completely define the local subgrid-scale forcing associated with mountain wave propagation and breakdown. The forcing for the globally averaged mean flow deceleration can, nevertheless, be determined solely from the vertical divergence of 〈ρ¯u′w′〉.
A simpler description of the local mean flow response to gravity wave propagation and breakdown may be obtained using pseudomomentum diagnostics. When the velocities in the unperturbed cross-mountain flow are positive, the vertical pseudomomentum flux is negative and may be regarded as an upward flux of negative pseudomomentum whose source is the cross-mountain pressure drag. In regions where the waves are steady and not undergoing dissipation the horizontal average of the vertical pseudomomentum flux is constant with height. The sinks for this flux are located in the regions of wave dissipation. Unlike the conventional perturbation momentum, the pseudomomentum perturbations generated by breaking mountain waves are all negative. According to the pseudomomentum viewpoint, the signature of gravity wave drag is a secular increase in the strength of the negative pseudomomentum anomalies generated by wave dissipation. In contrast to the behavior of the perturbation momentum, the average rate of pseudomomentum loss in an O(400 km) domain surrounding the mountain is a significant fraction of the total decelerative forcing provided by the cross-mountain pressure drag. Since pseudomomentum is a second-order quantity that decays rapidly upstream and downstream of the mountain, the horizontally averaged pseudomomentum budget can be closed in open domains of reasonable finite size without explicitly accounting for the pseudomomentum fluxes through the lateral boundaries, and thus, the temporal changes in the horizontally averaged pseudomomentum can be determined solely from the divergence of the vertical pseudomomentum flux.
Momentum and pseudomomentum perturbations in trapped mountain lee waves are also investigated. These waves generate nontrivial domain-averaged pseudomomentum perturbations in the low-level flow and should be considered an important potential source of low-level gravity wave drag. These waves are, however, inviscid, and the pseudomomentum perturbations do not grow as a result of dissipation but rather as a result of wave transience through the continued downstream expansion of the wave train.
Abstract
Numerical simulations are examined in order to determine the local mean flow response to the generation, propagation, and breakdown of two-dimensional mountain waves. Realistic and idealized cases are considered, and in all instances the pressure drag exerted by flow across an O(40 km) wide mountain fails to produced a significant net mean flow deceleration in the O(400 km) region surrounding the mountain. The loss of momentum in the local patches of decelerated flow that appear in regions of wave overturning directly above the mountain is approximately compensated by momentum gained in other nearby patches of accelerated flow. The domain-average mean flow deceleration in the O(400 km) domain is not determined solely by the divergence of the horizontally averaged momentum flux, 〈ρ¯u′w′〉, because differences in the upstream and downstream values of ρu2 +p provide nontrivial contributions to the total domain-averaged momentum budget. As confirmed by additional simulations in an O(1000 km) wide periodic domain, terrain-induced perturbations in the pressure and horizontal velocity fields are rapidly transmitted hundreds of kilometers away from the mountain and distributed as a small-amplitude signal over a very broad area far away from the mountain. These results suggest that both 〈ρ¯u′w′〉 and information about the wave-induced horizontal momentum fluxes need to be parameterized in order to completely define the local subgrid-scale forcing associated with mountain wave propagation and breakdown. The forcing for the globally averaged mean flow deceleration can, nevertheless, be determined solely from the vertical divergence of 〈ρ¯u′w′〉.
A simpler description of the local mean flow response to gravity wave propagation and breakdown may be obtained using pseudomomentum diagnostics. When the velocities in the unperturbed cross-mountain flow are positive, the vertical pseudomomentum flux is negative and may be regarded as an upward flux of negative pseudomomentum whose source is the cross-mountain pressure drag. In regions where the waves are steady and not undergoing dissipation the horizontal average of the vertical pseudomomentum flux is constant with height. The sinks for this flux are located in the regions of wave dissipation. Unlike the conventional perturbation momentum, the pseudomomentum perturbations generated by breaking mountain waves are all negative. According to the pseudomomentum viewpoint, the signature of gravity wave drag is a secular increase in the strength of the negative pseudomomentum anomalies generated by wave dissipation. In contrast to the behavior of the perturbation momentum, the average rate of pseudomomentum loss in an O(400 km) domain surrounding the mountain is a significant fraction of the total decelerative forcing provided by the cross-mountain pressure drag. Since pseudomomentum is a second-order quantity that decays rapidly upstream and downstream of the mountain, the horizontally averaged pseudomomentum budget can be closed in open domains of reasonable finite size without explicitly accounting for the pseudomomentum fluxes through the lateral boundaries, and thus, the temporal changes in the horizontally averaged pseudomomentum can be determined solely from the divergence of the vertical pseudomomentum flux.
Momentum and pseudomomentum perturbations in trapped mountain lee waves are also investigated. These waves generate nontrivial domain-averaged pseudomomentum perturbations in the low-level flow and should be considered an important potential source of low-level gravity wave drag. These waves are, however, inviscid, and the pseudomomentum perturbations do not grow as a result of dissipation but rather as a result of wave transience through the continued downstream expansion of the wave train.
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.
