Lake-Effect Snow Lab: Cold Air over a Warm Lake, Fetch, Heat Flux and Upslope Snow Belt

Presets

Live readouts

Graphs

Boundary-layer top and cloud base along the fetch over the lake

Surface heat fluxes along the fetch

Snowfall rate along the wind through the click point

Storm total along the wind (white) and maximum total over time (violet)

Numerical checks

running...

How to use

Cold, dry air blows across a lake that is still unfrozen and therefore warm. The lake feeds the air with heat and water vapour, the air layer deepens, cloud forms, and snow falls on the downwind shore, strongest where hills lift the cloud. The map shows the lake (dark blue), land, cloud, radar-style snowfall rate and the snow that has already piled up. Click the map to move the orange transect line: the graphs follow it. Try: (1) the preset Frozen lake shuts the machine off; (2) turn the wind round with Wind blows from and watch the band move and weaken when the fetch over the lake gets short; (3) raise the air temperature and see the cloud thin out and the snow turn to rain near 0 C; (4) switch off the hills and compare the snow belt. Space pauses, R resets. The storm clock runs 2 to 240 minutes per second.

The science

Why does a lake make snow? When air much colder than the water crosses a lake, the surface heats it from below (sensible heat) and evaporates water into it (latent heat). The heated layer becomes unstable, mixes up to a growing depth, and its moisture condenses at the top as cloud. The cloud sheds snow, mostly after the air has reached the downwind shore and been lifted by hills. SI units, T in kelvin inside the formulas.

1. Surface fluxes (bulk formulas). With wind speed U, air density ρ = 1.2 kg/m³, exchange coefficient C (about 1.2 to 1.6 × 10^-3) and ice-free fraction (1 − ice):

H = ρ c_p C U (T_s − θ_m), E = ρ L_v C U (q_s(T_s) − q_m)

where q_s comes from the Magnus form of the Clausius-Clapeyron law, e_s = 6.112 exp(17.62 t / (243.12 + t)) hPa, so a 10 K warmer lake can hold roughly twice as much vapour at saturation.

2. Slab mixed layer along the fetch. Follow a column of air moving at speed U (time t = x / U). Its depth h, potential temperature θ_m and humidity q_m obey (Lilly 1968; Tennekes 1973):

dh/dt = w_e = A F_θ / Δθ, dθ_m/dt = (F_θ + w_eΔθ) / h, dq_m/dt = (F_q + w_eΔq − P/ρ) / h

F_θ = H / (ρc_p) and F_q = E / (ρL_v) are the kinematic fluxes, Δθ is the temperature jump at the top of the layer (which grows with the stable air above, γ = dθ/dz), Δq the moisture jump (dry air above), A about 0.2 the entrainment ratio and P the snow fall-out. The model is integrated with fourth-order Runge-Kutta in 1-km steps up to 320 km.

3. Cloud. Cloud forms where the layer is deeper than the lifting condensation level z_L (found by bisection of q_s(θ_m − Γ_d z, p(z)) = q_m); Espy's rule z_L = 125 m/K (T − T_d) is the cross-check. Above z_L the liquid water grows adiabatically along a moist adiabat:

LWP = ∫_{z_L}^h ρ [ q_m − q_s(T(z), p(z)) ] dz, P = LWP / τ, τ = cloud-to-snow time

Snow then falls out exponentially: LWP(d) = LWP₀ exp(−d / (U τ)) at distance d inland, which sets the width of the snow belt (U τ is 20 to 70 km).

4. Hills. Where the ground rises along the wind, the layer is lifted at w = U · (dz/ds), cools, and extra water condenses:

P_oro = ε_o ρ w h (−dq_s/dz), −dq_s/dz = q_s (L_vΓ_m / (R_vT²) − g / (R_dT))

Snow depth rate is the water-equivalent rate times the snow-to-liquid ratio (about 10 to 20 for cold, dry snow). The map reads the fetch for every pixel by tracing the wind upstream over the lake, so a lake elongated along the wind makes the strongest band.

Rule of thumb in the readout. Forecasters ask for a lake minus 850 hPa temperature difference of at least 13 K (Holroyd 1971) and a fetch of more than about 80 km; both appear in the live readout.

Checks (run in the page). (1) The integral of the surface flux plus entrainment heating equals the change of hθ_m (energy budget of the slab, independent trapezoid sum against the Runge-Kutta result); (2) the same for moisture including snow fall-out; (3) to (5) Magnus saturation pressure against reference table values at 0, 20 and −10 C; (6) lifting condensation level by bisection against Espy's rule.

Simplifications. One slab per wind line, wind steady and uniform, no convective rolls or mesoscale vortices, no Coriolis force, terrain is an invented Great-Lakes-like map (not a real lake). Adiabatic cloud water, one conversion time for all precipitation, and a constant exchange coefficient (real values rise with instability). The absolute snowfall depends on those closures: read the numbers as typical orders of magnitude, not a forecast. Real lake-effect events reach several cm/h in narrow bands, and often more than a metre in a day when a band stalls over one place.