Reservoir Stratification - 2D Depth-Time Temperature Map, Thermocline and Turnover

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Live readouts

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Numerical check

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Performance graph

How to use

A one-dimensional reservoir is heated by sunlight and exchanges heat with the air through its surface. The big colour panel shows water temperature against depth for the last two years; the white tick marks the thermocline (steepest density gradient). The right panel shows the temperature profile (orange) and the buoyancy frequency N² (blue); the strip below shows surface, bottom and air temperature.

Watch the cycle: spring warming builds a warm surface layer, summer wind and light set how deep it gets, autumn cooling deepens the mixed layer until turnover, and in cold winters the surface can drop below 4 C (lighter, inverse stratification).

Try: raise the wind and see the thermocline sink; make the water murky (Kd 2) so the heat stays near the surface; shallow 10 m reservoirs mix fully; deepen to 60 m and the bottom stays near 4 C.

The science

Vertical heat balance (z downward, constant area, no inflows):

ρ c_p ∂T/∂t = ∂/∂z( ρ c_p K(z,t) ∂T/∂z ) + ∂I/∂z (absorbed light)

Surface boundary: Q_s = 0.4 Q_sw(t) − h (T_s − T_air(t)), the remaining 60% of shortwave penetrates as I(z) = 0.6 Q_sw e^{−K_d z}. Seasonal forcing: T_air = T_m + T_a sin(2π (d − 110)/365), Q_sw = Q₀(1 + 0.8 sin(2π (d − 80)/365)).

Water density uses the Martin-McCutcheon fit, ρ(T) = 1000 [1 − (T + 288.9414)(T − 3.9863)² / (508929.2 (T + 68.12963))], with its maximum at 3.98 C. Stratification strength is the buoyancy frequency N² = (g/ρ₀) ∂ρ/∂z (z downward) and the Schmidt stability S = g ∫ (z − z_v)(ρ − ρ̄) dz (J/m²).

Turbulent diffusivity: K = K_{molecular+background} + 2·10⁻⁵ U² e^−z/z_w / (1 + 3000 N²) with z_w = 1.5 + 0.4 U metres: wind mixing decays with depth and is suppressed by stratification. The equation is solved with implicit (backward Euler) diffusion, then a convective adjustment mixes any statically unstable layers, as in simple 1-D lake models.

Honest simplifications: constant surface area (no hypsography), no inflows/outflows or river plumes, no ice cover, no evaporation or longwave detail (folded into h), K is a simple empirical closure (not a k-epsilon model), 0.5 m grid with a 30 min step, fresh water only. It illustrates the physics of stratification and turnover, not a calibrated model of a specific reservoir.

Keyboard, touch and accessibility

Space pauses or resumes the simulation.
R resets it to its initial state.
C re-runs the numerical check.
All sliders are native range inputs and work with arrow keys, Home and End; on touch devices drag on the canvas where the sim supports pointer input.
Rendering is capped at a device pixel ratio of 2 so phones stay smooth; the frame-time graph above shows the cost of the current settings (green line is 16.7 ms, i.e. 60 frames per second).

What is checked and why you can trust it

Every time the page loads, and whenever you press Re-run check, the simulation evaluates its own mathematics against an independent reference, such as a closed-form formula, a finite-difference derivative or an exhaustive count. Each line above reports PASS or FAIL together with the measured numbers, so you can see by how much it agrees. Results are also published on window.__msChecks for automated testing. The check never uses the same code path as the animation it validates, otherwise it would only confirm itself.

How to read a failure: a FAIL line means the measured discrepancy exceeded the stated tolerance for the current slider values. Try Reset, then move the sliders back toward their defaults; if a check still fails at the defaults it is a bug worth reporting. Tolerances are chosen from the numerical method itself, never tuned to make a test pass: round-off level (1e-9) for exact algebra, about 1e-3 for finite differences and Monte Carlo estimates with a few hundred thousand samples, and a documented margin for stochastic learning runs.

Reproducibility: the particle and pen simulations use the browser random generator for initial positions, so exact pixels differ between visits, but the checks depend only on mathematics and not on those positions. Fixed wave tables (curl noise) and fixed grammars (L-system) are deterministic.

This is an educational model. It shows the mechanism clearly, but it is deliberately simplified; the science panel lists what was left out.