Air flowing over a ridge is pushed into standing waves downwind. Change the wind, the stability and the ridge, and see where the waves climb, where they are trapped near the ground, where lenticular clouds form, and where a sailplane would find lift.
Waves of horizontal wavenumber k can travel upward only where l is larger than k. Where l drops below the ridge's 1/a (dashed), they are reflected: that is trapping.
With the current ridge and stability, the wind at the ground is varied from 5 to 30 m/s: stronger wind lengthens the waves and weakens the lift for a given ridge until the ridge is no longer a steep obstacle (N·h/U small).
The science
Linear mountain waves. For steady, small-amplitude, two-dimensional flow with wind U(z), buoyancy frequency N(z) and no rotation, the vertical velocity of a Fourier mode eikx obeys the Taylor-Goldstein equation w″ + (l² − k²) w = 0 with Scorer parameter l² = N²/U² − U″/U. Here U is linear in height (U″ = 0), so l = N/U. The streamline displacement is δ̂ = ŵ/(ikU). The ground condition is δ̂(0) = ĥ(k) with ĥ = π h a e−ka for the Witch-of-Agnesi ridge h(x) = h a²/(x² + a²). Each mode is integrated from a radiation condition at 14 km down to the ground (RK4); modes are summed over k, δ = (1/π) Re ∫ ĥ T eikx dk. The vertical wind is w = U ∂δ/∂x.
Regimes. Where k < l, the wave propagates upward with vertical wavelength 2π/√(l² − k²), which for wide ridges is 2πU/N (hydrostatic limit). Where k > l the wave decays with height as e−κz, κ = √(k² − l²). If l is larger in a lower layer than aloft, waves with l₂ < k < l₁ are trapped; for two layers (N₁ below height zi, N₂ above, uniform wind) the trapped wavenumbers satisfy m cos(m zi) + κ sin(m zi) = 0, m = √(l₁² − k²), κ = √(k² − l₂²). These are the long, regular lee-wave trains that glider pilots use.
Momentum flux and drag. With u′ = −∂(Uδ)/∂z the wave momentum flux ρ∫u′w dx is constant with height for non-dissipating waves (Eliassen-Palm) and equals the drag of the ridge on the flow. It is shown here in N per metre of ridge with ρ = 1.2 kg/m³.
Clouds. Cloud is drawn where the vertical displacement δ in the chosen 1 km thick moist layer exceeds the lift required for saturation (a single number you set); real clouds depend on humidity profiles and microphysics. Lenticular clouds are the cloud caps on wave crests.
Honest simplifications. Linear theory: no breaking waves, rotors or turbulence, valid for N·h/U well below 1 (a warning is shown otherwise). Boussinesq: air density is taken as constant, whereas in the real atmosphere wave amplitude grows with height as density falls. No Coriolis force, no moisture in the dynamics, and a weak Rayleigh friction (decay length set by the slider) gives the trapped waves a finite length and keeps the integral finite. Wind shear is linear, N has two levels. The ridge is infinitely long. Trees, glider and clouds are not to scale; the vertical scale is exaggerated 4 times. A glider sink rate of 0.7 m/s is a typical value for a high-performance sailplane, not a specific aircraft.
Educational model, not a flight-planning tool.