Environmental Lapse Rate

A lapse rate is one subtraction and one division, and the only trick in it is the sign convention. The rate is written as the temperature BELOW minus the temperature ABOVE, so that ordinary air, which cools as you climb, comes out positive. Two thermometers reading 20 °C at the surface and 13.5 °C on a mast 1000 m higher give $\Gamma = (20 - 13.5)/1000 = 0.0065$ °C per metre, which the whole meteorological literature writes as 6.5 °C/km. That number is not an accident of the example. ICAO fixed 6.5 °C/km as the troposphere of the standard atmosphere in 1952, and every pressure altimeter in the world is calibrated against it. Note that only the DIFFERENCE of the two temperatures enters, so a Celsius interval and a kelvin interval are the same thing and the conversion cancels itself.

The rate becomes useful only when it is compared against the rate a parcel of air would cool at if it were lifted with no heat exchange at all. That one is not measured, it is derived. A rising parcel expands and does work against the surrounding pressure, and the energy comes out of its own heat content, so $\Gamma_d = g/c_p = 9.80665/1005 = 0.00976$ K/m, or 9.8 °C/km. Below that value the atmosphere is stable: a lifted parcel cools faster than its surroundings, becomes denser, and sinks back where it came from. Above it the parcel stays warmer than the air around it and keeps climbing, which is instability. Saturated air releases latent heat as it rises and so cools more slowly, roughly 5.4 °C/km in warm air and nearer 8 in cold air, which is why the band between 5.4 and 9.8 is called conditionally unstable.

Everything a plume does follows from that comparison. In unstable air the plume loops, dragged up and down by convective eddies, and a loop that touches down produces a brief ground-level concentration far above anything an hourly average would suggest. In neutral air it cones, spreading symmetrically, which is the case the Gaussian model describes best because it is the case the Gaussian model was fitted to. Under an inversion it fans, flattening into a thin ribbon that can travel many kilometres almost undiluted while the ground underneath sees essentially nothing. The dangerous part is what happens next. When morning sun heats the surface and erodes the inversion from below, the entire night's ribbon is mixed down at once. That is fumigation, and it produces the highest short-term ground-level concentrations most sources ever cause.

Two mistakes are worth naming. The first is measuring the gradient across too thin a layer near the ground, where on a sunny afternoon the bottom few metres can be superadiabatic by tens of degrees per kilometre while the air a hundred metres up is perfectly neutral. A lapse rate is only meaningful over the layer the plume actually occupies. The second is treating the lapse rate as the whole of stability. The Pasquill class that supplies the dispersion coefficients is set by wind speed and solar radiation as well, and a strong wind mixes mechanically no matter what the temperature profile says. Use the lapse rate to know which regime you are in, then take the class from the standard insolation and wind table rather than from the gradient alone.