Momentum Fluxes of Gravity Waves Generated by Variable Froude Number Flow over Three-Dimensional Obstacles

Abstract

Fully nonlinear mesoscale model simulations are used to investigate the momentum fluxes of gravity waves that emerge at a “far-field” height of 6 km from steady unsheared flow over both an axisymmetric and elliptical obstacle for nondimensional mountain heights ĥm = Fr−1 in the range 0.1–5, where Fr is the surface Froude number. Fourier- and Hilbert-transform diagnostics of model output yield local estimates of phase-averaged momentum flux, while area integrals of momentum flux quantify the amount of surface pressure drag that translates into far-field gravity waves, referred to here as the “wave drag” component. Estimates of surface and wave drag are compared to parameterization predictions and theory. Surface dynamics transition from linear to high-drag (wave breaking) states at critical inverse Froude numbers Frc−1 predicted to within 10% by transform relations. Wave drag peaks at Frc−1 < ĥm ≲ 2, where for the elliptical obstacle both surface and wave drag vacillate owing to cyclical buildup and breakdown of waves. For the axisymmetric obstacle, this occurs only at ĥm = 1.2. At ĥm ≳ 2–3 vacillation abates and normalized pressure drag assumes a common normalized form for both obstacles that varies approximately as ĥm−1.3. Wave drag in this range asymptotes to a constant absolute value that, despite its theoretical shortcomings, is predicted to within 10%–40% by an analytical relation based on linear clipped-obstacle drag for a Sheppard-based prediction of dividing streamline height. Constant wave drag at ĥm ∼ 2–5 arises despite large variations with ĥm in the three-dimensional morphology of the local wave momentum fluxes. Specific implications of these results for the parameterization of subgrid-scale orographic drag in global climate and weather models are discussed.

The physics of orographic gravity wave drag

“…Eckermann et al [152] evaluated the dependence of the momentum fluxes onh for flow over 3D obstacles, essentially confirming the findings of Ólafsson and Bougeault [150] and Bauer et al [56] for the surface drag. They additionally showed that the drag vacillates for flow across an elongated obstacle in the 0.7 ≤h < 3 range, due to cyclical buildup and breakdown of wave activity, with the amplitude of these vacillations decreasing ash increases.…”

“…They additionally showed that the drag vacillates for flow across an elongated obstacle in the 0.7 ≤h < 3 range, due to cyclical buildup and breakdown of wave activity, with the amplitude of these vacillations decreasing ash increases. Ath > 2 − 3, Eckermann et al [152] noted that the drag decreased smoothly proportionally toh −1/3 to values lower than 1, and the dimensional drag approached a constant value at highh, a constraint they suggested implementing in drag parametrizations.…”

The Impact of Small-Scale Topography on the Dynamical Balance of the Ocean

“…The clearest illustration of this point is provided by the poorly known extent to which (8) and other equivalent corrections adequately capture the form drag associated with highly nonlinear high-drag states, occurring for Fr 21 values of O(1) (e.g., Eckermann et al 2010). Such high-drag states are characterized by hydraulic jumps, intense internal wave breaking, and vortex shedding at and around topographic obstacles, and may be largely described in terms of internal wave dynamics (Peltier and Clark 1979;Bacmeister and Pierrehumbert 1988;Welch et al 2001).…”

“…Such high-drag states are characterized by hydraulic jumps, intense internal wave breaking, and vortex shedding at and around topographic obstacles, and may be largely described in terms of internal wave dynamics (Peltier and Clark 1979;Bacmeister and Pierrehumbert 1988;Welch et al 2001). Their effective drag has been found to exceed linear theory predictions by O(10%-100%), depending on various aspects of topographic configuration (Welch et al 2001;Wells et al 2008;Eckermann et al 2010). Expression (8) may thus provide a lowerbound estimate of jt iw j for Fr 21 ; 1.…”

The Impact of Finite-Amplitude Bottom Topography on Internal Wave Generation in the Southern Ocean

“…There is a rich atmospheric literature suggesting that wave radiation by finite-amplitude 3D topography is strongly suppressed compared to the 2D limit (e.g., Miranda and James 1992;Baines and Smith 1993;Welch et al 2001;Epifanio and Durran 2001;Eckermann et al 2010). While the flow can become blocked by topography both in 2D and 3D, in 3D the flow can also split and go around topographic obstacles rather than over them (Fig.…”