Sea Breeze & Land Breeze: 2D Vorticity Circulation Solver

Time & Conditions

Time of day 12:00

Land–sea thermal contrast 1.0×

Prevailing wind (sea→land +) 0.0 m/s

Live Readouts (measured from the solved field)

Transition

Land temp

15.0°C

Sea temp

15.0°C

ΔT (land − sea)

0.0°C

Surface wind u (solved)

0.0 m/s

Circulation |ψ|max

0 m²/s

Peak vorticity |ζ|max

0×10⁻³ 1/s

How this 2D model differs from the 3D version

The 3D simulator plugs land/sea temperature straight into a prescribed analytic streamfunction ψ(x,y)=A·sin(kₓx)·sin(k_yy) and recomputes it from scratch every frame — the flow shape never has memory. This 2D version instead numerically integrates the real governing equations of a Boussinesq thermal-circulation cell on a finite-difference grid, so the circulation has to build up (and decay) with real physical inertia:

∂ζ/∂t + u∂ζ/∂x + w∂ζ/∂y = ∂b/∂x + ν∇²ζ      (baroclinic vorticity generation)
∇²ψ = −ζ                                        (Poisson solve every substep, ψ=0 walls)
u = ∂ψ/∂y,  w = −∂ψ/∂x                          (velocity is exactly divergence-free)
∂b/∂t + u∂b/∂x + w∂b/∂y = κ∇²b + relax(b_surface)

Buoyancy b = g·(T−T₀)/T₀ is forced only in a thin near-surface layer, relaxing toward the same diurnal land/sea temperature curves as the 3D model (land heats/cools faster — lower heat capacity). A horizontal buoyancy gradient generates vorticity baroclinically — rising over the warmer surface, sinking over the cooler one — and the streamfunction is solved by Jacobi relaxation from that vorticity every physical timestep, guaranteeing the resulting wind field is mass-conserving by construction, not by design of the shape.

Because the flow has inertia, dragging the time slider produces a visible lag before the wind field catches up to the new forcing — a genuine dynamical effect the 3D model's instantaneous algebra cannot show. The surface wind and circulation strength readouts are measured directly from the solved ψ/ζ/u fields, not derived from a separate closed-form formula.

The prevailing wind slider still adds a uniform background advection on top (same simplification as the 3D model) — real sea breezes only survive when this synoptic wind is weak.

12:00

Warm rising air (w>0) Cool sinking air (w<0)