River Heat Budget: Riparian Shade, Cold Tributary & Trout Stress Reach Lab

Follow a 10 km river reach through a summer day. Sun, sky, air and wind add or remove heat; the current carries it downstream; trees shade the water and a cold tributary dilutes it. Where does the water pass the limit that stresses trout?

Scenario

Clear-cut section (1.5 to 4.5 km)

Weather and place

The river

Probe along the river

Where the heat goes

Temperature along the river now, °C

Outlet temperature over the day, °C

Heat fluxes at mid-reach, W/m²

Experiment: how much shade is enough?

Press the button: the same day is simulated 11 times with canopy cover 0, 10 ... 100 %.

Missions

Independent numerical checks

Also run outside the browser: node src/check.js.

Educational model with typical, illustrative coefficients [assumption]: 1-D reach, uniform width/depth from simple hydraulic geometry, no bed conduction or groundwater seepage except the one tributary, clear-sky solar attenuation law, Brunt sky emissivity, linear wind function. The river is drawn about 4 times wider than its modelled width so it stays visible. Not a regulatory temperature model.

Controls

Drag / one finger: orbit; wheel / pinch: zoom. Aerial, Reach, Tributary and Map set the view.
The time slider scrubs the day (the simulation is precomputed for the current settings); Play day runs it. Any slider recomputes three simulated days and shows the last, periodic one.
The probe slider reads the water temperature at any point along the reach.

The science

Heat budget. The water surface gains H = H_sw + H_lw + H_e + H_c (W/m²). Short-wave: H_sw = (1 − 0.06) S · (1 − 0.75 C^3.4) (1 − s) with the clear-sky value S = 1361 sinα · 0.74^1/sinα, solar elevation α from latitude, day of year and hour angle, cloud fraction C and shade fraction s. Long-wave: sky and canopy emit toward the water, H_lw = 0.97(ε_s(1 − s) + 0.97 s) σ T_a⁴ − 0.97 σ T_w⁴ with the Brunt emissivity ε = 0.51 + 0.066 √e_a (e_a in hPa). Evaporation: H_e = −(3.0 + 1.5 W)(e_s(T_w) − e_a), with condensation heating the water when the air is moist and warm. Conduction follows from the Bowen ratio R_b = 0.61 (P/1013) (T_w − T_a)/(e_s − e_a), H_c = R_b H_e.

Transport. Each 100 m cell obeys ∂T/∂t + u ∂T/∂x = H / (ρ c_w h), solved with a first-order upwind scheme (stable for uΔt/Δx < 1). Width, depth and speed come from a simple hydraulic geometry in the discharge Q (w = 10 (Q/5)^0.5 m, h = 0.5 (Q/5)^0.4 m). At the tributary the flows and heats mix: T = (Q T + Q_t T_t)/(Q + Q_t). Low flow means a thin water column that heats fast: the same sun warms a creek more than a big river.

Shade. Canopy cover removes direct sunlight and part of the sky long-wave, and shelters the surface from wind. A clear-cut section removes most of it for 3 km. The sweep finds the cover at which the daily maximum stays under the trout limit (a typical ~20 C stress threshold, [assumption], varies by species and life stage).

Checks. Solar geometry (solstice noon elevation, equinox day length), the Stefan-Boltzmann handbook value, Magnus vapour pressure, the scheme against the analytic exponential relaxation, tributary mixing, and an energy ledger of the whole reach over the final day.

Not modelled: bed conduction, hyporheic and groundwater exchange along the reach, channel meanders changing exposure, topographic shading, changing flow through the day, sediment, ice, and the biology beyond a simple threshold.