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- Author or Editor: Dale R. Durran x
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Abstract
A simple class of barotropic Rossby waves are shown to propagate westward as the graphically balanced meridional windfield periodically reverses in response to a small meridional pressure gradient arising from the latitudinal variation of the Coriolis parameter.
Abstract
A simple class of barotropic Rossby waves are shown to propagate westward as the graphically balanced meridional windfield periodically reverses in response to a small meridional pressure gradient arising from the latitudinal variation of the Coriolis parameter.
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.
