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Dale R. Durran

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.

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Dale R. Durran

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¯uw′〉, 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¯uw′〉 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¯uw′〉.

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.

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Dale R. Durran

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.

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Dale R. Durran

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.

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Kevin R. Haghi
and
Dale R. Durran

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.

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Dale R. Durran
and
Joseph B. Klemp

Abstract

Expressions are derived for the Brunt- Väisälä frequency Nm , in a saturated atmosphere, which are analogous to commonly-used formulas for the dry Brunt- Väisälä frequency. These formulas are compared with others which have appeared in the literature, and the derivation by Lalas and Einaudi (1974) is found to be correct. The simplifying assumptions, implicit in derivations by Dudis (1972) and Fraser et al. (1973) are clarified. Numerical examples are presented which suggest that these incomplete formulations for Nm are reasonably accurate approximations, except when the saturated static stability is small. A new formula expressing Nm in terms of moist conservative variables is presented, and an accurate approximation is also given which may be useful when evaluating Nm .

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Dale R. Durran
and
Joseph B. Klemp

Abstract

The effects of latent heat release on the dynamics of mountain lee waves are examined with the aid of two-dimensional numerical simulations, for several situations in which the Scorer parameter has a nearly two-layer vertical structure. Changes in the moisture in the lowest layer are found to produce three fundamentally different behaviors: 1) resonant waves in an absolutely stable environment are distorted and untrapped by an increase in moisture; 2) resonant waves in a conditionally unstable layer are destroyed by an increase in moisture; and 3) resonant waves in a moist environment are detuned by a decrease in moisture. Changes in the humidity in the upper layer are found to amplify or damp the wave response, depending on the depth of the lower layer. In most situations, the wave response is significantly more complicated than that predicted by simply replacing the dry stability with an equivalent moist stability in the saturated layer.

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Louisa B. Nance
and
Dale R. Durran

Abstract

The accuracy of three anelastic systems (Ogura and Phillips; Wilhelmson and Ogura; Lipps and Hemler) and the pseudo-incompressible system is investigated for small-amplitude and finite-amplitude disturbances. Based on analytic solutions to the linearized, hydrostatic mountain wave problem, the accuracy of the Lipps and Hemler and pseudo-incompressible systems is distinctly superior to that of the other two systems. The linear dispersion relations indicate the accuracy of the pseudo-incompressible system should improve and the accuracy of the Lipps and Hemler system should decrease as the waves become more nonhydrostatic.

Since analytic solutions are not available for finite-amplitude disturbances, five nonlinear, nonhydrostatic numerical models based on these four systems and the complete compressible equations are constructed to determine the ability of each “sound proof” system to describe finite-amplitude disturbances. A comparison between the analytic solutions and numerical simulations of the linear mountain wave problem indicate the overall quality of the simulations is good, but the numerical errors are significantly larger than those associated with the pseudo-incompressible and Lipps and Hemler approximations. Numerical simulations of flow past a steady finite-amplitude heat source for an isothermal atmosphere and an atmosphere with an elevated inversion indicate the Lipps and Hemler and pseudo-incompressible systems also produce the most accurate approximations to the compressible solutions for finite-amplitude disturbances.

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Rajul E. Pandya
and
Dale R. Durran

Abstract

The dynamical processes that determine the kinematic and thermodynamic structure of the mesoscale region around 2D squall lines are examined using a series of numerical simulations. The features that develop in a realistic reference simulation of a squall line with trailing stratiform precipitation are compared to the features generated by a steady thermal forcing in a “dry” simulation with no microphysical parameterization. The thermal forcing in the dry simulation is a scaled and smoothed time average of the latent heat released and absorbed in and near the leading convective line in the reference simulation. The mesoscale circulation in the dry simulation resembles the mesoscale circulation in the reference simulation and around real squall lines; it includes an ascending front-to-rear flow, a midlevel rear inflow, a mesoscale up- and downdraft, an upper-level rear-to-front flow ahead of the thermal forcing, and an upper-level cold anomaly to the rear of the thermal forcing. It is also shown that a steady thermal forcing with a magnitude characteristic of real squall lines can produce a cellular vertical velocity field as the result of the nonlinear governing dynamics. An additional dry simulation using a more horizontally compact thermal forcing demonstrates that the time-mean thermal forcing from the convective leading line alone can generate a mesoscale circulation that resembles the circulation in the reference simulation and around real squall lines.

The ability of this steady thermal forcing to generate the mesoscale circulation accompanying squall lines suggests that this circulation is the result of gravity waves forced primarily by the low-frequency components of the latent heating and cooling in the leading line. The gravity waves in the dry and reference simulation produce a perturbed flow that advects diabatically lifted air from the leading line outward. In the reference simulation, this leads to the development of leading and trailing anvils, while in the dry simulation this produces a pattern of vertically displaced air that is similar to the distribution of cloud in the reference simulation. Additional numerical simulations, in which either the thermal forcing or large-scale environmental conditions were varied, reveal that the circulation generated by the thermal forcing shows a greater sensitivity to variations in the thermal forcing than to variations in the large-scale environment. Finally, it is demonstrated that the depth of the thermal forcing in the leading convective line, not the height of the tropopause, is the primary factor determining the height of the trailing anvil cloud.

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Peter P. Miller
and
Dale R. Durran

Abstract

The influence of terrain asymmetry on the development and strength of downslope windstorms was examined through the numerical simulation of three basic atmospheric configurations: 1) flow beneath a mean-state critical layer, 2) flow in the presence of breaking waves and 3) flow in a two-layer atmosphere without wave breaking or a mean-state critical layer. When a mean-state critical layer was present in the flow and the wind speed and stability beneath that critical layer were essentially constant, the maximum downslope wind speed was nearly independent of mountain asymmetry. Such insensitivity to mountain shape is consistent with hydraulic theory and supports the idea that there is a close mathematical analog between stratified flow beneath a mean-state critical layer and conventional shallow-water hydraulic theory. When downslope winds were generated by breaking waves, and the upstream stability and wind speed were constant with height, the dependence of lee-slope velocities on terrain asymmetry remained weak. When downslope winds were produced in a two-layer atmosphere without wave breaking or a mean-state critical layer, the flow exhibited a noticeable, but not dominating, sensitivity to mountain asymmetry.

The preceding results were obtained from simulations without surface friction. When a surface friction parameterization was included in the numerical model, the sensitivity of the downslope wind speed to mountain asymmetry was significantly enhanced. It appears that in the surface friction simulations, the most significant shape parameter is not mountain asymmetry per se, but simply the steepness of the lee slope, with steep lee slopes being most favorable for strong winds.

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