Abstract

Numerical mountain wave simulations have documented that intense lee-slope winds frequently arise when wave-overturning occurs above the mountain. Explanations for this amplification process have been proposed by Clark and Peltier in terms of a resonance produced by linear-wave reflections from a self-induced critical layer, and by Smith in terms of solutions to Long's equation for flow beneath a stagnant well-mixed layer. In this paper, we evaluate the predictions of these theories through numerical mountain-wave simulations in which the level of wave-overturning is fixed by a critical layer in the mean flow. The response of the simulated flow to changes in the critical-layer height and the mountain height is in good agreement with Smith's theory. A comparison of Smith's solution with shallow-water theory suggests that the strong lee-slope winds associated with wave-overturning are caused by a continuously stratified analog to the transition from subcritical to supercritical flow in conventional hydraulic theory.

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.

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.

Abstract

Expressions are derived for the Brunt- Väisälä frequency N_m , 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 N_m are reasonably accurate approximations, except when the saturated static stability is small. A new formula expressing N_m in terms of moist conservative variables is presented, and an accurate approximation is also given which may be useful when evaluating N_m .

Abstract

The flow of a nonrotating atmosphere with uniform stratification and wind speed past an isolated three-dimensional topographic obstacle is investigated with a nonhydrostatic numerical model having a free-slip lower boundary. When the mountain is sufficiently high, the transient development of a quasi-steady flow occurs in two phases. During the first phase, which occurs over a dimensionless time of O(1), the flow is essentially inviscid and adiabatic, and potential vorticity (PV) is conserved. The transient evolution of the flow during the second phase, which occurs over a dimensionless time of O(10) to O(100), is controlled by dissipation and is accompanied by the generation of PV anomalies.

Two cases are examined in which the flow is forced to remain left–right symmetric with respect to the axis of the incident flow. In the first, the dimensionless mountain height NH/U is 1.5, and gravity waves break over the mountain. In the second, NH/U = 3, and a quasi-steady recirculating wake containing a doublet of positive and negative vortices develops in the lee. In both cases potential vorticity anomalies are generated by dissipation, although the sources of dissipation are different in each case. The net effect of the dissipation on the PV budget is, nevertheless, similar as may be understood from the generalized Bernoulli theorem that equates the generation of potential vorticity fluxes to the development of a Bernoulli function gradient on quasi-steady isentropic surfaces. In these experiments a Bernoulli deficit develops either from strong localized dissipation in the wave-breaking region (dominant for NH/U = 1.5), or as the result of weak dissipation throughout the elongated wake (dominant for NH/U = 3).

Oscillating von Kármán vortex streets appear if the flows are allowed to develop asymmetries with respect to the axis of the incident flow. It is shown that the transition into the vortex shedding regime is associated with an absolute instability of the symmetrical wake, which feeds upon the shear present at the edges of the wake. The most unstable global normal mode is diagnosed numerically and shows strong similarities with the corresponding mode in shallow-water theory. The doubling time of the instability is a few hours, which is consistent with the rapid formation of observed atmospheric vortex streets. The individual vortices in the fully developed vortex street are quasi-balanced warm-core vortices that are associated with both PV and surface potential temperature anomalies.

Abstract

The impact of mean-flow variability on finite-amplitude trapped mountain lee waves is investigated by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows. The lee-wave patterns generated by these time-dependent flows depend on two factors: 1) the degree to which the transition in the background flow changes the amplitude of the stationary trapped lee wave and 2) the difference between the group velocities of the trapped waves generated before and after the transition. When the transition in the background flow significantly reduces the amplitude of the stationary lee wave, the lee-wave pattern generated prior to the transition gradually drifts downstream away from the mountain or back over the mountain, depending on the sign of this wave packet’s group velocity after the transition. When the transition in the background flow changes the resonant wavelength while leaving the lee-wave amplitude relatively unchanged, the lee-wave train develops either 1) a smooth transition in horizontal wavelength or 2) a region of irregular variations in wavelength and amplitude, depending on the difference between the group velocities of the waves generated before and after the transition. Although linear theory is able to predict how changes in the background flow will affect the group velocities of the trapped waves, it is not able to predict whether the temporal variations in the large-scale environmental flow will amplify or dampen the resonant waves when the waves are no longer linear.

Regions of irregular variations in wavelength and amplitude may develop when stationary trapped waves generated after a transition in the background flow overtake the trapped waves generated before the transition. The fluctuations in the vertical velocities associated with such numerically simulated lee waves are compared with wind profiler observations. Estimates of the time required for the trapped waves generated after the transition to overtake those generated before the transition suggest that the temporal changes in the background flow required to qualitatively reproduce the observed vertical velocity variations are not likely to occur on a realistic timescale. In addition, the observed temporal variations in lee-wave vertical velocities appear to be the superposition of at least two distinct frequencies, whereas the temporal variations in the simulated waves are dominated by one distinct frequency.

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.

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.

Abstract

The influence of gravity waves generated by surface stress and by topography on the atmospheric kinetic energy (KE) spectrum is examined using idealized simulations of a cyclone growing in baroclinically unstable shear flow. Even in the absence of topography, surface stress greatly enhances the generation of gravity waves in the vicinity of the cold front, and vertical energy fluxes associated with these waves produce a pronounced shallowing of the KE spectrum at mesoscale wavelengths relative to the corresponding free-slip case. The impact of a single isolated ridge is, however, much more pronounced than that of surface stress. When the mountain waves are well developed, they produce a wavenumber to the −5/3 spectrum in the lower stratosphere over a broad range of mesoscale wavelengths. In the midtroposphere, a smaller range of wavelengths also exhibits a −5/3 spectrum. When the mountain is 500 m high, the waves do not break, and their KE is entirely associated with the divergent component of the velocity field, which is almost constant with height. When the mountain is 2 km high, wave breaking creates potential vorticity, and the rotational component of the KE spectrum is also strongly energized by the waves. Analysis of the spectral KE budgets shows that the actual spectrum is the result of continually shifting balances of direct forcing from vertical energy flux divergence, conservative advective transport, and buoyancy flux. Nevertheless, there is one interesting example where the −5/3-sloped lower-stratospheric energy spectrum appears to be associated with a gravity-wave-induced upscale inertial cascade.

Abstract

The interaction of a midlatitude cyclone with an isolated north–south mountain barrier is examined using numerical simulation. A prototypical cyclone develops from an isolated disturbance in a baroclinically unstable shear flow upstream of the ridge, producing a cold front that interacts strongly with the topography. The structure and evolution of the lee waves launched by the topography are analyzed, including their temporal and their north–south variation along the ridge. Typical mountain wave patterns are generated by a 500-m-high mountain, but these waves often exhibit significant differences from the waves produced in 2D or 3D simulations with steady large-scale-flow structures corresponding to the instantaneous conditions over the mountain in the evolving flow. When the mountain height is 2 km, substantial wave breaking occurs, both at low levels in the lee and in the lower stratosphere. Despite the north–south uniformity of the terrain profile, large north–south variations are apparent in wave structure and downslope winds. In particular, for a 24-h period beginning after the cold front passes the upstream side of the ridge toward the south, strong downslope winds occur only in the northern half of the lee of the ridge. Just prior to this period, the movement of the cold front across the northern lee slopes is complex and accompanied by a burst of strong downslope winds and intense vertical velocities.