Modeling Vertical Variation in Air Temperature

2.3 Modeling Vertical Variation in Air Temperature

The theory of turbulent transport, which we study in Ch. 7, specifies the shape of the temperature profile over a uniform surface with steady-state

conditions. The temperature profile equation is:

where is the mean air temperature at height is the apparent aerodynamic surface temperature,

is a roughness parameter for heat transfer, H is the sensible heat flux from the surface to the air,

is the volumetric specific heat of air (1200 J

at

C and sea level),

0.4 is von Karman's constant, and u* is the friction velocity (related to the friction or drag of the stationary surface on the moving air). The reference level from which is measured is always somewhat arbitrary, and the correction factor d , called the zero-plane displacement, is used to adjust for this. For a flat, smooth surface, d

For a uniformly vegetated surface,

where h is canopy height. We derive Eq. (2.1) in Ch. 7, but use it here to interpret the shape of the temperature profile and extrapolate temperatures measured at one height to other heights. The following points can be made.

and d

vs. is a straight line).

1. The temperature profile is logarithmic (a plot of

Modeling Vertical Variation in Air Temperature

H is negative (heat flux to- ward the surface) and decreases with height when is positive. During the day, sensible heat flux is generally away

2. Temperature increases with height when

the surface so T decreases with height.

3. The temperature gradient at a particular height increases in magnitude as the magnitude of

H increases, and decreases as wind or turbulence increases.

Example 2.1. The following temperatures were measured above a 10 cm high alfalfa crop on a clear day. Find the aerodynamic surface temperature,

Height (m) 0.2 0.4 0.8 1.6 Temperature (C)

Solution. It can be seen from Eq. (2.1) that T = To when the term

is zero, which happens when z = d + since

= = 0. If -

is plotted versus T (normally the independent variable is plotted on the abscissa or horizontal axis, but when the independent variable is height, it is plotted on the ordinate or vertical axis) and ex-

trapolate to zero, the intercept will be For a 10 cm (0.1 m) high canopy,

0.002 m, and d 0.06 m. The following can therefore be computed:

Height (m) 0.2 0.4 0.8 1.6 Temp. (C)

The Figure for Example 2.1 on the following page is plotted using this data, and also shows a straight line fitted by linear least squares through the data points that is extrapolated to zero on the log-height scale. The intercept is at 34.6" C, which is the aerodynamic surface temperature.

Example 2.2. The mean temperature at

2 m above the soil surface is 3" C. At a height of 1 m, the temperature is C. If the crop below the point where these temperatures are measured is 50 cm tall, will the crop experience a temperature below freezing?

22 Temperature

20 25 30 35 Temperature (C)

Solution. This problem could be solved by plotting, as we did

1, or it could be done algebraically. Here we use algebra. The constants, in Eq. (2.1) are the same for all heights. For convenience, we represent them by the symbol A. Equation (2.1) can then be written for each height as

Subtracting the second equation from the and solving for A gives

A -2.25" C. Substituting this back into either equation gives

-8.6" C. Knowing these, now solve for where h =

0.5 m:

So the top of the canopy is below the freezing temperature. These two examples illustrate how temperatures can be interpolated

or extrapolated. In each case, two temperatures are required, in addition to information about the height of roughness elements at the surface. Typically temperature is measured at a single height. From Eq.

it is clear that additional information about the sensible heat flux density

H and the wind would be needed to extrapolate a single temperature measurement. This is taken up later when we have the additional tools needed to model heat flux.

Soil Temperature Changes with Depth and Time