Wind flows over heated terrain with a central plateau, producing thermal boundary layers, convective plumes, and flow separation. The simulation uses the Lattice Boltzmann Method (D2Q9) with thermal coupling on the GPU via WebGPU compute shaders. The central variable is potential temperature θ, a direct measure of specific entropy: s = cp ln(θ) + const. Five visualization modes (velocity, vorticity, temperature, θ, entropy production σ) reveal different aspects of the flow. Live readouts track Re, Pr, Ri, Nu, and Be, while the Pr-entropy chart plots viscous vs. thermal entropy production against the Prandtl number.
The Richardson number Ri = g · β · ΔT · L / U² compares buoyancy forces (heated land driving convection) to inertial forces (wind shear). When Ri > 0.25, stratification suppresses vertical mixing and the flow remains laminar. When Ri < 0.25, shear overcomes buoyancy and turbulent mixing develops. This is the fundamental stability criterion for atmospheric boundary layers over heated surfaces.
Try it: raise Land Temp and reduce Wind speed to increase Ri. Watch convective plumes appear in Vorticity view.Potential temperature θ = T · (ρ₀/ρ)(γ-1)/γ adjusts measured temperature for adiabatic compression or expansion. In the atmosphere, θ increases with height in stable conditions and is constant (well-mixed) in convective conditions. Since entropy s = cp ln(θ), surfaces of constant θ are isentropes. The θ view reveals the entropy structure of the flow directly.
Try it: switch to θ view. Uniform color means well-mixed (high entropy production). Sharp gradients mean stratified (low mixing).Entropy is produced by two mechanisms: viscous dissipation σv = μΦ/T (kinetic energy degraded to heat) and heat transfer across temperature gradients σq = k|∇T|²/T². The Bejan number Be = σq/(σq + σv) quantifies which mechanism dominates. Be near 1 means thermal gradients drive irreversibility; Be near 0 means viscous friction dominates.
Try it: switch to σ view. Bright spots near the surface show thermal entropy production. Bright spots behind hills show viscous entropy from flow separation.The Nusselt number Nu = hL/k measures convective heat transfer relative to pure conduction. Nu = 1 means conduction only; larger Nu means convection enhances surface heat exchange. For flow over a flat plate, Nu ~ Re1/2 Pr1/3 (Blasius solution). Terrain features disrupt the boundary layer, locally increasing Nu at hilltops and in separation zones. Higher Nu means more entropy production at the surface.
Try it: increase Wind speed to raise Re and watch Nu grow. The thermal boundary layer thins, steepening temperature gradients at the surface.The Reynolds number Re = U · L / ν compares inertial forces to viscous forces. Low Re (laminar): viscosity smooths out disturbances. High Re (turbulent): inertia amplifies them. In this simulation, L is the domain width and U is the inlet velocity. As Re increases, wake structures behind terrain features become increasingly chaotic and the boundary layer transitions from smooth to turbulent. Re also sets the scale for Nu and Ri.
Try it: raise Wind speed or lower Viscosity to increase Re. Watch the wake behind the plateau transition from steady recirculation to unsteady vortex shedding.The Prandtl number Pr = ν / α is the ratio of momentum diffusivity (viscosity) to thermal diffusivity. For air, Pr ≈ 0.71, meaning heat diffuses slightly faster than momentum. When Pr < 1, the thermal boundary layer is thicker than the velocity boundary layer. When Pr > 1 (oils, viscous fluids), the thermal layer is thinner. Pr governs the shape of the Pr-entropy chart: the crossover point where viscous and thermal entropy production balance shifts with Pr.
Try it: adjust Viscosity and Diffusivity independently. Watch the Pr value in the readouts and the Pr-entropy chart update. The Pr = 1 reference line marks equal diffusion rates.The Eckert number Ec = U² / (cp · ΔT) compares kinetic energy of the flow to its thermal energy. Small Ec means temperature differences dominate the energy budget (typical for atmospheric flows). Large Ec means flow kinetic energy is significant relative to thermal energy, so viscous dissipation can appreciably heat the fluid. Ec appears in the energy equation as the coupling coefficient between mechanical and thermal dissipation, scaling how much viscous friction contributes to entropy production.
Try it: increase Wind speed while keeping Land Temp low. Ec grows, and the viscous component of entropy production (seen in the σ view and Pr-entropy chart) becomes more prominent.The draggable sun controls the angle of incoming solar radiation. Rays arrive as parallel beams (the sun is far away), and the terrain surface receives heat proportional to the dot product of the surface normal with the sun direction: Qsolar ∝ max(n̂ · ŝ, 0.15). South-facing slopes (facing the sun) receive more radiation than north-facing slopes in shadow. This aspect-dependent heating creates differential thermal forcing, driving asymmetric convection cells and slope winds. The 0.15 floor represents diffuse sky radiation that reaches shaded surfaces.
Try it: drag the sun to one side. Watch convective plumes strengthen on the sunlit face of the plateau while the shadowed side stays cooler and more stable.Nick Thompson, weather research · Stephen Guerin, simulation and visualization · built with love and Claude Code