Diurnal temperature variation in the boundary layer

of the atmosphere within which diurnal variation of various atmospheric parameters (temperature, humidity, wind speed, etc) is well defined is called boundary layer of the atmosphere

The height of the boundary layer depends upon static stability of the atmosphere.

Very stable atmosphere

Very unstable atmosphere

In tropics, where the atmosphere can be…

extremely unstable, the top of the boundary layer can reach 3000 m.

observed near the sunrise or a bit later.

Top of the boundary layer

As soon as the solar radiation comes to the surface, the latter starts warming.

Due to molecular diffusion, the heat is transferred to the thin layer of air.

Later the heat is transferred upward due to eddy mixing from one layer to that above of it and so on.

Since for the heat transfer up some period of time is needed, there is delay with temperature grow in every following layer

because each layer takes some fraction of heat from the ascending heat flux. That makes temperature variation at each following layer smaller and smaller too.

Suppose, there are n layers, the lowest of them (adjacent to the surface) is the well known “surface layer”, where the heat flux is quasi- constant, or well call it zero layer (Q0).

The following layers are 1, 2, … n. In every of these layers the heat influx is smaller than in the lower one.
It means that ∆T values will decrease with height.

amplitude will become smaller, and, at the boundary layer top, it will be close to zero.

with height. At the altitude of about 1.5 km it is 6 – 7 times less than near the surface.
Near the top of the boundary layer the amplitude can be very complicated with 2 or 3 maxima.

Reasons

Observations are not accurate enough.
Fluctuation of eddy intensity.

Near the surface, variation is rather significant. Therefore, comparably small fluctuation of eddy intensity do not seriously distort the diurnal temperature variation rate. At the top of the boundary layer, situation is different. Here, the amplitude is much smaller than at the surface, and even not significant eddy activity fluctuation may result in distortion of the diurnal variation rate.

the cloud amount always results in decrease of the diurnal temperature variation amplitude

Wind

Wind makes the eddy mixing more intensive, and by this way it diminishes the variation amplitude. Along with this, it is associated with temperature advection that also distorts the rate of the variation.

Warm advection

No advection

Cold advection

Starting moment of advection

the second order

Boundary conditions

At the given boundary conditions, the solution of the equation will be:

where the amplitudes are of the same value.

Adopting as a boundary layer altitude z*, where the amplitude 100 times less than that at the surface, we obtain:

If A0=10°C, A(z*)=0.1°C

Thus, the key role of the amplitude variation with height belongs to K value. On the base of that value, one can calculate the boundary layer altitude

It shows clearly the role of the K value in determination of the z* value.
The normal K value is known to be from 1 to 5 m²/s. However, some cases were reported with K>100 m²/s. This kind of a situation is associated with extremely well developed convection, for instance, in tropics. At these cases the diurnal temperature variation can be observed through the whole troposphere.

in July, and minimal temperature in January or February.
The annual temperature variation amplitude decreases with height in the same manner as the diurnal one does.
Π1=24 hours is the period of one Earth’s spin.
Π2=24·365.25 hours is the period of annual Earth’s rotation around the Sun.

Assuming normal condition (K=5 m²/s), the annual temperature variation spreads over a layer of about 32 km (whole troposphere and significant part of the stratosphere). We know a little about K value variation with height except the fact that it increases within the surface layer. There are some evidences that it remains almost constant in the boundary layer. As to the free atmosphere, we can judge about it from indirect evidences such as comparison observed and calculated parameters.

variation, above else, depends on Earth's surface temperature variation.
The rate of the heat propagation is a finite value.
The extreme temperatures are to occur the later, the higher altitude is.
Suppose t1 is the time the earth’s surface temperature reaches its maximum; t2 is the respective time for the altitude z.

Lag time

Phase velocity:

In case K=5 m²/s

radiation. The intensity of the effective radiation, in turn, depends on the properties of the soil and state of the sky. Significant fall of the temperature occurs at cloudless sky condition.

Effective radiation

Cooling of the surface and adjacent air

Inversion layer formation

Weakening the eddy exchange intensity within the inversion layer

Heat flux decreasing by 3 – 5 times.

Further cooling of the surface and adjacent air

Possibility for a frost formation

Brent’s formula