Morning Glory Roll Cloud (2D)

What is a Morning Glory? A long roll cloud, sometimes 1000 km long, that moves at about 10-15 m/s over the Gulf of Carpentaria (northern Australia) in the early morning. Sea-breeze fronts collide over Cape York and push a pile of cool, moist air ahead of them. That air slides along the nocturnal inversion as a train of solitary waves; the leading crest lifts the moist air until it saturates and rolls into a cloud tube.

Two-layer model. A cool inversion layer of depth h₁ lies under a deeper neutral layer h₂. Their potential temperatures differ by Δθ, so gravity acting across the interface is reduced:

g' = g Δθ / θ₀,   c₀ = √( g' h₁ h₂ / (h₁ + h₂) )

Korteweg-de Vries equation for the interface displacement η(x,t): η_t + (c₀ + α η) η_x + β η_xxx = 0 with α = 3 c₀ (h₂ - h₁) / (2 h₁ h₂) and β = c₀ h₁ h₂ / 6. The nonlinear term steepens the front, the dispersive term spreads short waves; their balance is the solitary wave

η = a sech²( (x - V t) / W ),   V = c₀ + α a / 3,   W = √( 12 β / (α a) )

Taller waves are narrower and faster. A smooth pile of cold air splits into a train of such waves (an undular bore), the tallest in front, exactly as observed. The page solves the equation with a pseudo-spectral method (FFT, integrating-factor Runge-Kutta 4, 2/3 de-aliasing) on a periodic 80 km track.

Roll cloud. In the lower layer the vertical displacement at height z is ζ = η z / h₁. An unsaturated parcel lifted by ζ cools by about (Γ_d - Γ_dew) ζ ≈ 8 K/km relative to its dew point, so it condenses once ζ exceeds z_sat = ΔT_d / 8 K km^-1 (125 m per kelvin of dew-point depression). The cloud base is z = z_sat h₁ / η and its top is the interface.

Ground signature. Hydrostatically Δp = ρ g' η (about 1 hPa for 300 m) and mass conservation in the cold layer gives a surface wind jump u = c η / (h₁ + η), a few m/s in the propagation direction.

Honest simplifications. Two homogeneous layers with a sharp interface and a rigid, flat top; weakly nonlinear and weakly dispersive (KdV is best for k h₂ < 1 and a < h₁); one-dimensional propagation (the lateral curve of the crest is drawn, not simulated); no friction, rotation, moisture feedback or latent heating; the cloud is a diagnostic of lifting, not a cloud-physics model. The 3D view stretches the vertical axis (slider) and enlarges the sailplane and the station so they can be seen.

Checks in this page. (1) a numerical solitary wave travels at the analytic speed with unchanged shape; (2) the invariants ∫η dx and ∫η² dx stay constant in the live run; (3) the KdV dispersion agrees with the exact two-layer dispersion relation c² = g' / (k (coth kh₁ + coth kh₂)).