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

Abstract

SUMMARYHigh-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 z 1 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 physics of orographic gravity wave drag

“…While in flow over an axisymmetric mountain, the locations of drag maxima were found to be accurately predicted by linear theory, in 2D flow intermediate maxima tended to disappear, confirming the results of Miranda and Valente [97]. For the same 2D flow as considered by Teixeira et al [96], Teixeira and Miranda [98] showed that wave breaking for Re c < 9/4 behaves differently from wave breaking in an atmosphere with constant parameters, occurring not directly over the mountain, but in regions displaced both vertically and horizontally upstream and downstream. Teixeira et al [100] showed that differences in the drag behavior for wind profiles where the wind varies linearly near the surface and is constant aloft depend (in hydrostatic conditions) on whether critical levels exist, and are located above or below the shear discontinuity.…”

“…As in Leutbecher [94], these effects are attributable to directional wave dispersion. Using linear theory and for the same wind profile over both 2D and axisymmetric mountains, Teixeira et al [96] derived analytical expressions for the drag, the former of which can be written as:…”

Estimation of the Friction Velocity in Stably Stratified Boundary-Layer Flows Over Hills

“…This new function ensures that u (x, z) → 0 as z → 0 and that, as in the HLR solution, u (x, z) matches with the middle layer solution as z increases. Weng (1989) validated this modified solution against observations from the Askervein Hill Project reported by Taylor and Teunissen (1985) (which has been used as a test case for numerous authors-see, for example, Lopes et al 2007) and obtained significant improvement by comparison with the original HLR model.…”

“…This model has been used previously by Teixeira et al (2005Teixeira et al ( , 2008 to assess the behaviour of analytical mountain wave drag predictions by comparison with numerical results. The FLEX model includes a set of physical and numerical features that make it able to simulate mesoscale flows over arbitrary orography for any value of the static stability.…”

Aviation Turbulence

“…This may lead to a region of wave breaking and turbulent mixing, which is characterized by Ri < 1/4 Teixeira et al 2005;. In the vicinity of a critical level, nonlinearity needs to be considered since the flow is highly nonlinear there.…”