Urban Heat Island Edge Effect: 2D Diffusion Model

Diurnal Cycle

Time of day 12:00

Auto-advance day (~40s/cycle)

City Design

Tree canopy / park cover 28%

Cool-roof & pavement adoption 0%

Horizontal heat mixing D 0.40 cell²/hr

Species

Thermal tolerance limit 29°C

Readouts

Mean city temp

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Population alive

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In refugia now

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Elapsed time

0.0 d

Park edge-warming Δ (diffused − isolated)

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Hot-tile edge-cooling Δ (isolated − diffused)

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Cool tile (near rural baseline) Moderate heat Severe heat-island tile Agent in cool refugia Agent under heat stress

How this 2D model differs from the 3D one

The 3D sim relaxes every tile's temperature toward its own solar target independently — no tile ever "feels" its neighbors. This 2D companion adds the physics that model leaves out: real heat conducts and advects horizontally between adjacent surfaces, which is exactly why parks measurably cool the streets just beyond their own boundary (the documented "park cool island" edge effect, typically reported to extend on the order of a park-width beyond the tree line). Each tile now solves a coupled reaction-diffusion equation instead of a pointwise one:

solar(t) = ΔT_max(tile) · max(0, sin(π·(t−6)/13))     for 6h ≤ t ≤ 19h
∂H/∂t    = (solar(t) − H)/τ(tile)  +  D·∇²H            (Fickian horizontal mixing)
∇²H[i,j] ≈ H[i−1,j]+H[i+1,j]+H[i,j−1]+H[i,j+1] − 4H[i,j]   (5-point stencil, zero-flux edges)
T(tile)  = T_rural + H

A second, invisible field runs alongside the visible one with D = 0 — the same per-tile relaxation the 3D sim uses, with no coupling to neighbors. Comparing the two live gives an honest measurement of the edge effect: park tiles bordering hot pavement come out warmer in the diffused field than in the isolated one (heat bleeding in), while pavement tiles bordering a park come out cooler than isolated (heat bleeding out). Those two deltas are the readouts below — verified numerically offline, they stay positive and shrink toward zero as the mixing slider D is turned down, exactly as a diffusion term should behave.

Pollinator agents sample the diffused field with bilinear interpolation (not nearest-tile lookup) and step down the locally interpolated gradient — negative thermotaxis — biased by random wander, and accumulate heat stress above the tolerance limit exactly as in the 3D sim:

stress(t) = stress(t−dt) + max(0, T − T_tolerance)·dt
recovery  = stress − 0.5·dt      if T ≤ T_tolerance
death     when stress exceeds a fixed lethal dose (6 degree-hours)

The bottom panel is a live cross-section: it samples the diffused temperature (solid) and the isolated no-diffusion baseline (dashed) along the grid's middle row, so you can watch the diffused curve visibly smear across park/pavement boundaries that the dashed curve treats as sharp steps — that smearing is the edge effect, drawn directly from the numbers.

12:00 · Day 0

Mid-row transect — solid = diffused, dashed = isolated (no mixing)

Urban Heat Island Edge Effect: Diffusion-Coupled Microclimate (2D)