The importance of friction in mountain wave drag amplification by Scorer parameter resonance

A mechanism for amplification of mountain waves, and their associated drag, by parametric resonance is investigated using linear theory and numerical simulations. This mechanism, which is active when the Scorer parameter oscillates with height, was recently classified by previous authors as intrinsically nonlinear. Here it is shown that, if friction is included in the simplest possible form as a Rayleigh damping, and the solution to the Taylor-Goldstein equation is expanded in a power series of the amplitude of the Scorer parameter oscillation, linear theory can replicate the resonant amplification produced by numerical simulations with some accuracy. The drag is significantly altered by resonance in the vicinity of n/l_0 = 2, where l_0 is the unperturbed value of the Scorer parameter and n is the wave number of its oscillation. Depending on the phase of this oscillation, the drag may be substantially amplified or attenuated relative to its non-resonant value, displaying either single maxima or minima, or double extrema near n/l_0 = 2. Both non-hydrostatic effects and friction tend to reduce the magnitude of the drag extrema. However, in exactly inviscid conditions, the single drag maximum and minimum are suppressed. As in the atmosphere friction is often small but non-zero outside the boundary layer, modelling of the drag amplification mechanism addressed here should be quite sensitive to the type of turbulence closure employed in numerical models, or to computational dissipation in nominally inviscid simulations.

Mountain waves in two-layer sheared flows: Critical-level effects, wave reflection, and drag enhancement

Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

Asymptotic gravity wave drag expressions for non-hydrostatic rotating flow over a ridge

Asymptotic expressions are derived for the mountain wave drag in flow with constant wind and static stability over a ridge when both rotation and non-hydrostatic effects are important. These expressions, which are much more manageable than the corresponding exact drag expressions (when these do exist) are found to provide accurate approximations to the drag, even when non-hydrostatic and rotation effects are strong, despite having been developed for cases where these effects are weak. The derived expressions are compared with approximations to the drag found previously, and their asymptotic behaviour in various limits is studied.

Internal wave drag in stratified flow over mountains on a beta plane

The impact of the variation of the Coriolis parameter f on the drag exerted by internal Rossby-gravity waves on elliptical mountains is evaluated using linear theory, assuming constant wind and static stability and a beta-plane approximation. Previous calculations of inertia-gravity wave drag are thus extended in an attempt to establish a connection with existing studies on planetary wave drag, developed primarily for fluids topped by a rigid lid. It is found that the internal wave drag for zonal westerly flow strongly increases relative to that given by the calculation where f is assumed to be a constant, particularly at high latitudes and for mountains aligned meridionally. Drag increases with mountain width for sufficiently wide mountains, reaching values much larger than those valid in the non-rotating limit. This occurs because the drag receives contributions from a low wavenumber range, controlled by the beta effect, which accounts for the drag amplification found here. This drag amplification is shown to be considerable for idealized analogues of real mountain ranges, such as the Himalayas and the Rocky mountains, and comparable to the barotropic Rossby wave drag addressed in previous studies.

A linear model of gravity wave drag for hydrostatic sheared flow over elliptical mountains

An analytical model of orographic gravity wave drag due to sheared flow past elliptical mountains is developed. The model extends the domain of applicability of the well-known Phillips model to wind profiles that vary relatively slowly in the vertical, so that they may be treated using a WKB approximation. The model illustrates how linear processes associated with wind profile shear and curvature affect the drag force exerted by the airflow on mountains, and how it is crucial to extend the WKB approximation to second order in the small perturbation parameter for these effects to be taken into account. For the simplest wind profiles, the normalized drag depends only on the Richardson number, Ri, of the flow at the surface and on the aspect ratio, γ, of the mountain. For a linear wind profile, the drag decreases as Ri decreases, and this variation is faster when the wind is across the mountain than when it is along the mountain. For a wind that rotates with height maintaining its magnitude, the drag generally increases as Ri decreases, by an amount depending on γ and on the incidence angle. The results from WKB theory are compared with exact linear results and also with results from a non-hydrostatic nonlinear numerical model, showing in general encouraging agreement, down to values of Ri of order one.

Resonant gravity-wave drag enhancement in linear stratified flow over mountains

High-drag states produced in stratified flow over a 2D ridge and an axisymmetric mountain are investigated using a linear, hydrostatic, analytical model. A wind profile is assumed where the background velocity is constant up to a height z1 and then decreases linearly, and the internal gravity-wave solutions are calculated exactly. In flow over a 2D ridge, the normalized surface drag is given by a closed-form analytical expression, while in flow over an axisymmetric mountain it is given by an expression involving a simple 1D integral. The drag is found to depend on two dimensionless parameters: a dimensionless height formed with z_1, and the Richardson number, Ri, in the shear layer. The drag oscillates as z_1 increases, with a period of half the hydrostatic vertical wavelength of the gravity waves. The amplitude of this modulation increases as Ri decreases. This behaviour is due to wave reflection at z_1. Drag maxima correspond to constructive interference of the upward- and downward-propagating waves in the region z < z_1, while drag minima correspond to destructive interference. The reflection coefficient at the interface z = z_1 increases as Ri decreases. The critical level, z_c, plays no role in the drag amplification. A preliminary numerical treatment of nonlinear effects is presented, where z_c appears to become more relevant, and flow over a 2D ridge qualitatively changes its character. But these effects, and their connection with linear theory, still need to be better understood.

The effect of wind shear and curvature on the gravity wave drag produced by a ridge

The analytical model proposed by Teixeira, Miranda, and Valente is modified to calculate the gravity wave drag exerted by a stratified flow over a 2D mountain ridge. The drag is found to be more strongly affected by the vertical variation of the background velocity than for an axisymmetric mountain. In the hydrostatic approximation, the corrections to the drag due to this effect do not depend on the detailed shape of the ridge as long as this is exactly 2D. Besides the drag, all the perturbed quantities of the flow at the surface, including the pressure, may be calculated analytically.

An analytical model of mountain wave drag for wind profiles with shear and curvature

An analytical model is developed to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain over which there is a stratified flow with a velocity profile that varies relatively slowly with height. The model is linear with respect to the perturbations induced by the mountain, and solves the Taylor–Goldstein equation with variable coefficients using a Wentzel–Kramers–Brillouin (WKB) approximation, formally valid for high Richardson numbers, Ri. The WKB solution is extended to a higher order than in previous studies, enabling a rigorous treatment of the effects of shear and curvature of the wind profile on the surface drag. In the hydrostatic approximation, closed formulas for the drag are derived for generic wind profiles, where the relative magnitude of the corrections to the leading-order drag (valid for a constant wind profile) does not depend on the detailed shape of the orography. The drag is found to vary proportionally to Ri21, decreasing as Ri decreases for a wind that varies linearly with height, and increasing as Ri decreases for a wind that rotates with height maintaining its magnitude. In these two cases the surface drag is predicted to be aligned with the surface wind. When one of the wind components varies linearly with height and the other is constant, the surface drag is misaligned with the surface wind, especially for relatively small Ri. All these results are shown to be in fairly good agreement with numerical simulations of mesoscale nonhydrostatic models, for high and even moderate values of Ri.

Drag produced by trapped lee waves and propagating mountain waves in a two-layer atmosphere

The surface drag force produced by trapped lee waves and upward propagating waves in non-hydrostatic stratified flow over a mountain ridge is explicitly calculated using linear theory for a two-layer atmosphere with piecewise-constant static stability and wind speed profiles. The behaviour of the drag normalized by its hydrostatic single-layer reference value is investigated as a function of the ratio of the Scorer parameters in the two layers l_2/l_1 and of the corresponding dimensionless interface height l_1 H, for selected values of the dimensionless ridge width l_1 a and ratio of wind speeds in the two layers. When l_2/l_1 → 1, the propagating wave drag approaches 1 in approximately hydrostatic conditions, and the trapped lee wave drag vanishes. As l_2/l_1 decreases, the propagating wave drag progressively displays an oscillatory behaviour with l_1 H, with maxima of increasing magnitude due to constructive interference of reflected waves in the lower layer. The trapped lee wave drag shows localized maxima associated with each resonant trapped lee wave mode, occurring for small l_2/l_1 and slightly higher values of l_1 H than the propagating wave drag maxima. As l1a decreases, i.e. the flow becomes more non-hydrostatic, the propagating wave drag decreases and the regions of non-zero trapped lee wave drag extend to higher l_2/l_1. These results are confirmed by numerical simulations for l_2/l_1 = 0.2. In parameter ranges of meteorological relevance, the trapped lee wave drag may have a magnitude comparable to that of propagating wave drag, and be larger than the reference single-layer drag. This may have implications for drag parametrization in global climate and weather-prediction models.

Is missing orographic gravity wave drag near 60°S the cause of the stratospheric zonal wind biases in chemistry–climate models?

Nearly all chemistry–climate models (CCMs) have a systematic bias of a delayed springtime breakdown of the Southern Hemisphere (SH) stratospheric polar vortex, implying insufficient stratospheric wave drag. In this study the Canadian Middle Atmosphere Model (CMAM) and the CMAM Data Assimilation System (CMAM-DAS) are used to investigate the cause of this bias. Zonal wind analysis increments from CMAMDAS reveal systematic negative values in the stratosphere near 608S in winter and early spring. These are interpreted as indicating a bias in the model physics, namely, missing gravity wave drag (GWD). The negative analysis increments remain at a nearly constant height during winter and descend as the vortex weakens, much like orographic GWD. This region is also where current orographic GWD parameterizations have a gap in wave drag, which is suggested to be unrealistic because of missing effects in those parameterizations. These findings motivate a pair of free-runningCMAMsimulations to assess the impact of extra orographicGWDat 608S. The control simulation exhibits the cold-pole bias and delayed vortex breakdown seen in the CCMs. In the simulation with extra GWD, the cold-pole bias is significantly reduced and the vortex breaks down earlier. Changes in resolved wave drag in the stratosphere also occur in response to the extra GWD, which reduce stratospheric SH polar-cap temperature biases in late spring and early summer. Reducing the dynamical biases, however, results in degraded Antarctic column ozone. This suggests that CCMs that obtain realistic column ozone in the presence of an overly strong and persistent vortex may be doing so through compensating errors.

Sensitivity of simulated climate to conservation of momentum in gravity wave drag parameterization

The Canadian Middle Atmosphere Model is used to examine the sensitivity of simulated climate to conservation of momentum in gravity wave drag parameterization. Momentum conservation requires that the parameterized gravity wave momentum flux at the top of the model be zero and corresponds to the physical boundary condition of no momentum flux at the top of the atmosphere. Allowing momentum flux to escape the model domain violates momentum conservation. Here the impact of momentum conservation in two sets of model simulations is investigated. In the first set, the simulation of present-day climate for two model-lid height configurations, 0.001 and 10 hPa, which are identical below 10 hPa, is considered. The impact of momentum conservation on the climate with the model lid at 0.001 hPa is minimal, which is expected because of the small amount of gravity wave momentum flux reaching 0.001 hPa. When the lid is lowered to 10 hPa and momentum is conserved, there is only a modest impact on the climate in the Northern Hemisphere; however, the Southern Hemisphere climate is more adversely affected by the deflection of resolved waves near the model lid. When momentum is not conserved in the 10-hPa model the climate is further degraded in both hemispheres, particularly in winter at high latitudes, and the impact of momentum conservation extends all the way to the surface. In the second set of simulations, the impact of momentum conservation and model-lid height on the modeled response to ozone depletion in the Southern Hemisphere is considered, and it is found that the response can display significant sensitivity to both factors. In particular, both the lower-stratospheric polar temperature and surface responses are significantly altered when the lid is lowered, with the effect being most severe when momentum is not conserved. The implications with regard to the current round of Intergovernmental Panel on Climate Change model projections are discussed.

Angular momentum conservation and gravity wave drag parameterization: implications for climate models

The robustness of the parameterized gravity wave response to an imposed radiative perturbation in the middle atmosphere is examined. When momentum is conserved and for reasonable gravity wave drag parameters, the response to a polar cooling induces polar downwelling above the region of the imposed cooling, with consequent adiabatic warming. This response is robust to changes in the gravity wave source spectrum, background flow, gravity wave breaking criterion, and model lid height. When momentum is not conserved, either in the formulation or in the implementation of the gravity wave drag parameterization, the response becomes sensitive to the above-mentioned factors—in particular to the model lid height. The spurious response resulting from nonconservation is found to be nonnegligible in terms of the total gravity wave drag–induced downwelling.

On the tropospheric response to anomalous stratospheric wave drag and radiative heating

Observational and numerical evidence suggest that variability in the extratropical stratospheric circulation has a demonstrable impact on tropospheric variability on intraseasonal time scales. In this study, it is demonstrated that the amplitude of the observed tropospheric response to vacillations in the stratospheric flow is quantitatively similar to the zonal-mean balanced response to the anomalous wave forcing at stratospheric levels. It is further demonstrated that the persistence of the tropospheric response is consistent with the impact of anomalous diabatic heating in the polar stratosphere as stratospheric temperatures relax to climatology. The results contradict previous studies that suggest that variations in stratospheric wave drag are too weak to account for the attendant changes in the tropospheric flow. However, the results also reveal that stratospheric processes alone cannot account for the observed meridional redistribution of momentum within the troposphere.

Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part I: gravity wave forcing

Parameterization schemes for the drag due to atmospheric gravity waves are discussed and compared in the context of a simple one-dimensional model of the quasi-biennial oscillation (QBO). A number of fundamental issues are examined in detail, with the goal of providing a better understanding of the mechanism by which gravity wave drag can produce an equatorial zonal wind oscillation. The gravity wave–driven QBOs are compared with those obtained from a parameterization of equatorial planetary waves. In all gravity wave cases, it is seen that the inclusion of vertical diffusion is crucial for the descent of the shear zones and the development of the QBO. An important difference between the schemes for the two types of waves is that in the case of equatorial planetary waves, vertical diffusion is needed only at the lowest levels, while for the gravity wave drag schemes it must be included at all levels. The question of whether there is downward propagation of influence in the simulated QBOs is addressed. In the gravity wave drag schemes, the evolution of the wind at a given level depends on the wind above, as well as on the wind below. This is in contrast to the parameterization for the equatorial planetary waves in which there is downward propagation of phase only. The stability of a zero-wind initial state is examined, and it is determined that a small perturbation to such a state will amplify with time to the extent that a zonal wind oscillation is permitted.

Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part II: combined effects of gravity waves and equatorial planetary waves

This study examines the effect of combining equatorial planetary wave drag and gravity wave drag in a one-dimensional zonal mean model of the quasi-biennial oscillation (QBO). Several different combinations of planetary wave and gravity wave drag schemes are considered in the investigations, with the aim being to assess which aspects of the different schemes affect the nature of the modeled QBO. Results show that it is possible to generate a realistic-looking QBO with various combinations of drag from the two types of waves, but there are some constraints on the wave input spectra and amplitudes. For example, if the phase speeds of the gravity waves in the input spectrum are large relative to those of the equatorial planetary waves, critical level absorption of the equatorial planetary waves may occur. The resulting mean-wind oscillation, in that case, is driven almost exclusively by the gravity wave drag, with only a small contribution from the planetary waves at low levels. With an appropriate choice of wave input parameters, it is possible to obtain a QBO with a realistic period and to which both types of waves contribute. This is the regime in which the terrestrial QBO appears to reside. There may also be constraints on the initial strength of the wind shear, and these are similar to the constraints that apply when gravity wave drag is used without any planetary wave drag. In recent years, it has been observed that, in order to simulate the QBO accurately, general circulation models require parameterized gravity wave drag, in addition to the drag from resolved planetary-scale waves, and that even if the planetary wave amplitudes are incorrect, the gravity wave drag can be adjusted to compensate. This study provides a basis for knowing that such a compensation is possible.