Water temperature

In plain words

Water holds a lot of heat, so a lake changes temperature slowly. Day by day it moves toward the air temperature: when the air is colder than the water the lake cools, and the larger the difference, the faster it cools. That is Newton's law of cooling. Wind speeds it up: it mixes the water and carries heat away from the surface faster, so the app lets the rate grow with wind speed.

The model is fitted to Lake St. Clair itself: Selfridge ANGB (KMTC) daily mean air temperature and wind vs NOAA GLSEA daily lake-average temperature, October 2014-2025, n=299 days. Before a plan is built, it carries the latest measured lake temperature forward day by day to the plan's date.

The heat-balance model

$T_n$: lake-average water temperature on morning $n$, F. $T_{\text{air}}$: the day's mean air temperature, (high + low) / 2, at Selfridge, F. $U$: the day's mean wind at the shore station, mph. Fitted for Lake St. Clair (lake record):

With a wind of 8 mph, $k = 0.0135 + 0.0058 \times 8 = 0.0599$ per day: the lake closes 6.0% of the gap between water and air each day. In steady weather half the gap is gone after $\ln 0.5 / \ln(1 - k) = 11.2$ days. In continuous form this is $dT/dt = k\,(T_{\text{air}} - T)$, with the solution $T(t) = T_{\text{air}} + (T_0 - T_{\text{air}})\, e^{-kt}$ and a time constant $1/k = 16.7$ days.

The same form, relaxation toward an equilibrium temperature with an exchange coefficient that grows with wind speed, is the equilibrium-temperature method of Edinger, Duttweiler and Geyer (1968). The app uses the daily mean air temperature in place of the equilibrium temperature Working rule: a simplification; the fit absorbs the difference on October days.

Weather for each day

• Days already past: observed highs, lows and wind at Selfridge.
• The next 7 days: the National Weather Service forecast for the shore point.
• After day 7: NOAA 1991-2020 daily normal high and low for Selfridge (station USW00014804) and a wind of 8 mph Starting value: near the October mean at Selfridge (Wind over water). The app labels these days as climatology-driven.

Sensitivity: an error of 1.5 F in the air temperature moves the projection by $k \times 1.5 = 0.09$ F on the first day and $1.5\,(1 - (1 - k)^7) = 0.53$ F after 7 days: the error grows more slowly than 7 times the first day's because the model keeps closing the gap.

Source: SPEC 5.2; reference/python/projection.py and wx_analysis.py. Engine: projectWaterTemp in packages/engine/src/waterTemp.ts.

Golden run

The trip analysis ran the model from 64.346 F on 2026-09-29 with the NWS forecast and then Selfridge normals. The engine reproduces it; largest difference 0.0005 F against a tolerance of 0.01 F (computed now):

The 80% band

The heat balance gives the central value; history gives the spread. Across 31 years (GLSEA 1995-2025) the Oct 7-11 lake temperature averaged 60.9 F with a standard deviation of 3.9 F. Knowing the temperature on Sept 29 narrows that, by a straight-line fit (least squares) over the same years:

The 80% band is the central value $\pm 1.28\, s$; $1.28$ is the 90th percentile of the standard normal distribution, so 10% of outcomes fall below the band and 10% above it. Here $1.28 \times 2.79 = 3.6$ F. For 2026, with 64.35 F on Sept 29: $60.9 \pm 3.6$ F (reference run: 57.3-64.5 F).

The app does the same for every day: the 1995-2025 climatology (mean, 10th and 90th percentile for each day of the year) and the regression slope and residual SD for each month and each lag of 1-30 days, computed from GLSEA when the data are loaded. The Water conditions screen shows the projection inside this band.

Source: SPEC 5.2; data/golden/engine_golden.json (regression); reference/python/temps.py. Engine: linearFit, band80, Z80 in waterTemp.ts.

Known bias

The model runs warm in strong wind. A hindcast of Sept 18-29, 2026 was up to 2.6 F warm during a three-day northeast blow and 0.6 F warm at the end (SPEC 5.2). The coefficients were fitted on October days only. Task 1.8 refits $k_0$ and $k_w$ by month on full-year data.

Calculator

Constants

Sources

• NOAA GLERL, Great Lakes Surface Environmental Analysis (GLSEA), daily lake-average temperature, 1995-2026.
• Iowa Environmental Mesonet, Selfridge ANGB (KMTC) daily summaries; National Weather Service forecasts; NOAA NCEI U.S. Climate Normals 1991-2020.
• Edinger, J.E., Duttweiler, D.W. and Geyer, J.C. (1968). The response of water temperatures to meteorological conditions. Water Resources Research 4: 1137-1143.
• Links: References.