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The ability of soils to store heat absorbed from solar radiation depends on the heat capacity of the soil, which may be written

in terms of the heat capacity of dry soil (Csdry typically in the range 2.0-2.5 x 106 J m-3 K-1) and of water [Cw = 4.2 x 106 J m-3 K-1 or roughly half of this value for ice (frozen soil)]. The mixing ratios (by volume) ms and mw are much more variable. For soils with different air content, ms may be 0.2-0.6 (low for peat, medium for clay and high for sand). The moisture content mw spans the range from zero to the volume fraction otherwise occupied by air, ma (equal to about 0.4 for clay and sand, but 0.9 for peat) (Sellers, 1965). In temperate climates an average value of mw for the upper soil layer is around 0.16 (Geiger, 1961).

Since the absorption of radiation takes place at the soil surface, the storage of heat depends on the competition between downward transport and long-wavelength re-radiation plus evapotranspiration and convective transfer. The thermal conductivity of the soil, A(z), is defined as the ratio between the rate of heat flow and the temperature gradient, at a given depth z,

(Note that by the sign convention previously adopted, z is negative below the soil surface.) Figure 2.103 gives a rough idea of the dependence of A on the moisture content for different soil types.

If the downward heat transport can be treated as a diffusion process, equation (2.27) is valid. Taking the partial derivative of (2.37) with respect to depth (-z), and assuming that k and A are independent of z (homogeneous soil), d Ez = A dT d z k dt

Together with the heat transport equation d Ez = C dT d z s dt

(cf. section 2.C) in the absence of heat sources, heat sinks and a fluid velocity in the soil), this implies the relation k = A/Cs. (2.38)

Figure 2.104 shows an example of the variation in the monthly average temperature as a function of depth in the soil. The time variation diminishes with increasing depth, and the occurrence of the maximum and the mini mum is increasingly delayed. These features are direct consequences of the diffusion approach, as one can easily see, e.g. by assuming a periodic temperature variation at the surface (Carslaw and Jaeger, 1959),

Figure 2.103. Heat conductivity, X, for various soil types, as a function of water content by volume, mw (based on Sellers, 1965).
Figure 2.104. Seasonal variations of soil temperature at various depths measured at St Paul, Minnesota. In winter, the average air temperature falls appreciably below the top soil temperature (z = - 0.01 m), due to the soil being insulated by snow cover (based on Bligh, 1976).

With this boundary condition, (2.27) can be satisfied by

and the corresponding heat flux is from (2.37)

rn C

The amplitude of the varying part of the temperature is seen from (2.40) to drop exponentially with depth (—z), as the heat flux (2.41) drops to zero. For a fixed depth, z, the maximum and minimum occur at a time which is delayed proportional to \z\, with the proportionality factor (&> Cs / 2A) 1/2. The maximum heat flux is further delayed by one-eighth of the cycle period (also at the surface).

The approximation (2.39) can be used to describe the daily as well as the seasonal variations in T and Ez. For the daily cycle, the surface temperature is maximum some time between 1200 and 1400 h. This temperature peak starts to move downward, but the highest flux away from the surface is not reached until 3 h later, according to (2.41). While the temperature peak is still moving down, the heat flux at the surface changes sign some 9 h after the peak, i.e. in the late evening. Next morning, 3 h before the peak temperature, the heat flux again changes sign and becomes a downward flux. Of course, the strict sine variation may only be expected at equinoxes.

Geothermal heat fluxes

In the above discussion, the heat flux of non-solar origin, coming from the interior of the Earth, has been neglected. This is generally permissible, since the average heat flux from the interior is only of the order of 3 x 1012 W, or about 8 x 10—2 W m—2. Locally, in regions of volcanoes, hot springs, etc., the geothermal heat fluxes may be much larger. However, it has been estimated that the heat transfer through the Earth's surface by volcanic activity only contributes under 1% of the average flux (Gutenberg, 1959). Equally small or smaller is the amount of heat transmitted by seismic waves, and most of this energy does not contribute any flux through the surface.

Although the distribution of heat generation within the solid Earth is not directly measurable, it may be estimated from assumptions on the composition of various layers of the Earth's interior (cf. Lewis, 1974 and Fig. 2.3). Thus, one finds that most of the heat generation takes place within the rocks present in the Earth's crust. The source of heat is the decay of radioactive elements, chiefly potassium, uranium and thorium. The estimated rate of generation (see e.g. Gutenberg, 1959) is roughly of the same order of mag z z nitude as the outward heat flux through the surface, although there is considerable uncertainty about this.

It is believed that the temperature gradient is positive inwards (but of varying magnitude) all the way to the centre, so that the direction of heat transport is predominantly outwards, with possible exceptions in regions of convection or other mass movement.

A more detailed discussion of the nature of the geothermal energy flux is deferred to section 3.5.2.

Temperature gradients also exist in the oceans, but it will become clear that they are primarily maintained not by the heat flux from the Earth's interior, but by extraterrestrial solar radiation coupled with conditions in the atmosphere and in the surface layers of the continents.

Momentum exchange processes between atmosphere and oceans

The mixing and sinking processes sketched above constitute one of the two sources of oceanic circulation. The other one is the momentum exchange between the atmospheric circulation (winds) and the oceans. The interaction between wind and water is complex, because a large fraction of the kinetic energy is transformed into wave motion rather than directly into currents. Although the roughness length over water [z0 of (2.32), (2.33)] is very small, a strong wind is capable of raising waves to a height of about 10 m.

The wind stress acting on the ocean surface may be taken from (2.33),

where pa is the density of air, K is von Karman's constant and z1 is a reference height. When the height variation of the horizontal wind velocity V is logarithmic as in (2.33), t is independent of height. This is true for the lower part of the turbulent layer (the Prandtl layer), within which z1 must be taken, and the value of t found in this way will remain constant through the laminar boundary layer, i.e. the force exerted on the ocean surface can be determined from measurements in a height z1 of, say, 10 m.

However, the approach may be complicated by the motion of the water surface. If waves are propagating along the water surface in the direction of the wind, one may argue that the stress should be proportional to the square of the relative velocity (V - Uw)2, where Uw is the wave velocity, rather than to the square of V alone. Further, the roughness parameter z0 may no longer be a constant, but it may depend on the wave velocity as well as on the wave amplitude. Given that the waves are formed as a result of the wind stress t, this implies a coupled phenomenon, in which t initiates wave motion, which again modifies t as a result of a changed surface friction, etc. There is experimental evidence to suggest that z0 is nearly constant except for the low est wind velocities (and equal to about 6 x 10-3 m; Sverdrup, 1957).

The mechanism by which the surface stress creates waves is not known in detail, nor is the distribution on waves and currents of the energy received by the ocean from the winds, or any subsequent transfer (Pond, 1971). Waves may be defined as motion in which the average position of matter (the water "particles") is unchanged, whereas currents do transport matter. This may also be viewed as a difference in scale. Waves play a minor role in the atmosphere, but are important in the hydrosphere, particularly in the upper water layers, but also as internal waves. The reason is, of course, the higher density of water, which allows substantial potential energy changes to be caused by modest lifting heights.

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