Tracking a Chemical Spill Down a River

A pulse of chemical dumped into a river drifts downstream with the current and spreads out as it goes. Its concentration is a bell curve whose center moves at the river's speed and whose width grows like the square root of time.

How it works

Advection–diffusion: transport plus the heat equation

In one sentence

The advection–diffusion equation $C_t + vC_x = DC_{xx}$ on the whole line with an instantaneous release of mass $M$ at $x = 0$ has the solution $C = \frac{M}{\sqrt{4\pi Dt}}e^{-(x-vt)^2/(4Dt)}$ : the heat kernel, the fundamental solution of the heat equation, carried along the characteristics $x - vt = \text{const}$ .

General formula

{\htmlData{sym=0}{C}}_{\htmlData{sym=4}{t}} + {\htmlData{sym=1}{v}}\,{\htmlData{sym=0}{C}}_{\htmlData{sym=5}{x}} = {\htmlData{sym=2}{D}}\,{\htmlData{sym=0}{C}}_{{\htmlData{sym=5}{x}}{\htmlData{sym=5}{x}}}, \qquad {\htmlData{sym=0}{C}}({\htmlData{sym=5}{x}}, {\htmlData{sym=4}{t}}) = \frac{\htmlData{sym=3}{{M}}}{\sqrt{4\pi {\htmlData{sym=2}{D}}{\htmlData{sym=4}{t}}}}\,e^{-\frac{({\htmlData{sym=5}{x}} - {\htmlData{sym=1}{v}}{\htmlData{sym=4}{t}})^2}{4{\htmlData{sym=2}{D}}{\htmlData{sym=4}{t}}}}

Two things happen at once: the river carries the cloud along, and mixing smears it out. Ride along with the current and you see just the heat equation.

What the symbols mean

$C$
Concentration
Milligrams of chemical per liter of river water.
$v$
Current speed
How fast the river carries the chemical downstream.
$D$
Dispersion coefficient
How fast turbulent mixing spreads the chemical out.
$M$
Amount spilled
The total amount released, which the plume carries along.
$t$
Time
Hours since the spill.
$x$
Position
Kilometers downstream of the spill.

The current carries the chemical downstream at speed $v$ : that's the transport term $v\,C_x$ , which on its own would slide the pulse along unchanged.
Turbulent mixing spreads it like heat in a rod: the diffusion term $D\,C_{xx}$ , where $D$ is the dispersion coefficient.
For a sudden release of $M$ , the solution is the heat equation's bell-shaped kernel carried along the characteristics $x - vt = \text{const}$ .
The width grows like $\sqrt{2Dt}$ and the peak falls like $1/\sqrt{t}$ , while the total amount (the area under the curve) stays $M$ .

Learn the math

First-Order PDEs and CharacteristicsPDE · Lesson 2, Foundations The Heat EquationPDE · Lesson 4, Foundations The Fourier Transform and the Infinite RodPDE · Lesson 17, Beyond the Basics