Diffusion through a stable layer

E. Linacre


The discussion in Section 7.6 of static stability in the atmosphere shows how vertical motion of air is suppressed within an inversion. So the vertical exchange of momentum, heat, moisture and pollutants by eddy diffusion is suppressed, and therefore large vertical gradients of these quantities result. Eddy diffusion involves transport between interlocking loops of movement of various sizes, in all three dimensions, i.e. turbulence. Nevertheless, the transfer of these quantities by molecular diffusion still occurs. Molecular diffusion (or conduction) is much less rapid than eddy diffusion, but it is proportional to the gradient of the quantity (moisture, momentum ), so it can become important in a well-defined inversion.

Some insight into atmospheric conduction through an inversion is obtained by considering a nocturnal radiation inversion at the ground. The ground cools by the net loss of longwave radiation energy to space, so that the surface temperature becomes lower than the screen temperature. Turbulent mixing wanes and the air becomes calm and stratified, as can be seen when a thin layer of fog forms over a meadow. Consequently sensible heat is partly conducted through the air down to the ground. There is also a flow of latent heat when the surface cools below the dewpoint temperature of the air, for that creates a gradient of water-vapour concentration; the result is dew on the ground.

We start by examining the terms of the energy balance -

input - loss of storage = output

In this case, the input to the surface consists of three downwards factors, longwave sky radiation Rld, sensible heat diffusing down from the air H, and latent heat from dew formation L.E. Heat conducted through the ground G is the change of storage; G<0 in this case since the flow is upwards. The only output from the surface is the longwave radiation upwards Rlu. So we have the following energy balance (all fluxes in W/m2)-

Rld + H + L.E - G = Rlu

Hence, (Rlu - Rld) + G = H + L.E

The terms on the left of this equation can be estimated, giving an indication of the magnitude of H, the flux of sensible heat down through the radiation inversion.


The value of Rlu is given by the Stefan-Boldtzmann equation in terms of the surface temperature, here assumed close to the screen temperature T. An empirical expression for Rld given by Linacre (1) is as follows, for a cloudless sky -

Rld = 208 + 6.T W/m2

Then evaluation of (Rlu - Rld) for various temperatures T shows that the net longwave radiation flux (under a clear sky) varies as in Table 1.

Table 1

T: C
























(Rlu - Rld)









Table 1 shows that the net longwave flux is remarkably constant when the sky is clear and the relative humidity above the radiation inversion constant (around 50%). It is about 96 W/m2, say.

Some indication of the magnitude of the term G in the energy-balance equation can be derived from Fig. 3.16 in the textbook. There is an average fall of temperature through the top 45 cm of the ground by 3.2 K between 1 pm and 5 am. (The temperature variation in deeper layers is negligible.) That implies an average conduction of heat to the surface G of about 57 W/m2, assuming a soil density of 2.7 kg/m3 and specific heat of 0.84 kJ/(kg.K) (2).This is comparable with the range of values quoted in the literature for bare ground (1).

So the heat flux from the air to the surface, within the nocturnal ground inversion, is (96 - 57), i.e. about 39 W/m2. This is made up of parallel fluxes of sensible heat H and latent heat L.E. The proportions of each depend on the water-vapour pressure of the air, which partly governs dewfall, i.e. L.E (1). A typical figure for this flux is below 17 W/m2 (Section 4.7), i.e. less than half of the approximate overall flux of heat.

A more accurate figure for H could be obtained by measuring the net longwave radiation flux by means of a net radiometer, and using a heat-flux plate to measure G near the surface. Dewfall could be measured by the increase of weight of ground at the surface.



  1. Linacre, E.T. 1992. Climate Data and Resources (Routledge) 366pp.
  2. List, R.J. 1949. Smithsonian Meteorological Tables (Smithsonian Institution, Washington) 527pp.