Heat Balance of the Earths Surface Upward and Downward Transfer of Heat

9.1 Introduction: General Considerations

It is estimated that of the total solar energy that is intercepted by the earth at the top of the atmosphere, only about half reaches the earth's surface and is absorbed by it as heat energy. When heated, the surface emits its own radiation. We showed in Fig. 6.2 that radiation from bodies at the temperatures of the earth's surface and atmosphere lies in the longwave part of the spectrum with maximum emissions at wavelengths between about 10|J. and 15^. It so happens that a fraction of this longwave radiation moving upward is absorbed by some gases present in the atmosphere, such as water vapour and carbon dioxide, which have strong absorption bands in the infrared part of the spectrum. The absorption leads to warming of the gases which then emit their characteristic longwave radiation, a part of which is sent back to the earth. Thus, the net radiative heating of the earth's surface depends on three factors:

(a) Heat gained from the incoming solar radiation, I,

(b) Heat lost or emitted by longwave radiation, E, and

(c) Heat gained from atmospheric gases as downward longwave radiation, G.

Concurrently, the earth's surface loses sensible heat (Hs) through vertical exchanges with the atmosphere above and the ground or water below. Over a water surface, considerable heat (He) is lost by evaporation.

So, the temperature of the earth's surface is determined by the following balance relation:

where

I is the net solar radiation absorbed by the earth's surface, E the longwave radiation emitted by the surface,

G the longwave radiation returned to the earth's surface by the atmospheric gases (mainly water vapour and carbon dioxide),

Hs the sensible heat lost to the atmosphere above and soil or water below, and He the heat lost by evaporation of water from the surface.

An accurate determination of the heat balance of the earth's surface using (9.1.1) is well-nigh impossible, because of the great inhomogeneity of the earth's surface and our incomplete knowledge and understanding of the various heat transfer processes.

About two-thirds of the earth's surface consists of oceans and one-third land. Though the major parts of the oceans appear as water surfaces, the polar regions are largely frozen and appear as solid ice surfaces. Then, there are the warm and cold ocean currents which cover wide areas and introduce large inhomogeneity in ocean surface temperatures. The land surface also is highly uneven, in fact more so than the ocean surface, with high snow-clad mountains over several areas flanked by warm valleys, vast deserts and lands co-existing with equatorial rainforests and rivers and in-land lakes, and so on. To date, inspite of numerous field experiments and laboratory and theoretical studies, the values of the co-efficients of heat exchanges with the environment under different stability conditions are only approximately known. So, when we talk of the heat balance of the earth's surface, the existing inhomogeneity of the surface and the above-mentioned uncertainties in the values of the exchange co-efficients must be borne in mind. However, notwithstanding these limitations, we may, perhaps, consider some broad aspects of the heat transfer processes outlined in the relation (9.1.1).We start off with the radiative processes.

But, before we consider the actual atmosphere, let us ask: What would have been the heat balance at the surface of our planet if it did not have an atmosphere? Although some may regard the question as hypothetical, it helps one to understand what role the existing atmosphere plays in the heat balance and what it means to life on earth.

9.2 Heat Balance on a Planet Without an Atmosphere

The hypothetical scenario is depicted in Fig. 9.1.

On an earth without an atmosphere, the solar radiation would impinge directly at the earth's surface and be absorbed by it to raise its temperature. Let S be a measure of the energy received per unit area of a surface held perpendicular to the incoming radiation per unit time, which, in this case, will be equal to the solar constant, S0.

S cxS

Fig. 9.1 Radiative heating and cooling of a planet without an atmosphere

S cxS

Radiation intercepted by the earth then amounts to nr2 S, where r is the mean radius of the earth. A fraction of this incident radiation, a, will be reflected from the surface and returned to space. The balance, S (1 - a), will be absorbed by the surface of the earth. As a result of this absorption, let the surface temperature be raised to T. The surface will then give out longwave radiation which, according to Stefan-Boltzmann law, will amount to o T4 per unit area of the earth's surface.

The balance between the incoming and the outgoing radiation may then be given by the relation

If it is assumed that the albedo of the earth, a, in the hypothetical case was no different from the present value of about 30%, then substituting the known values of S and o in (9.2.2), we get a value of about 255 K for T, which is about 33 °C colder than the average temperature on the present earth. So, the question is: what made up this deficiency in the earth's surface temperature? The answer clearly is the present-day atmosphere with one of its remarkable property, popularly known as the greenhouse effect.In the following section, we look into some aspects of this property of our atmosphere.

9.3 Heat Balance on a Planet with an Atmosphere: The Greenhouse Effect

The presence of an atmosphere changes the scenario depicted in the foregoing section. In the atmosphere, there are some gases, such as water vapour, carbon dioxide and ozone, which, though they exist in very small and variable proportions, play very important roles in the heat budget of the earth's surface, because they are almost transparent to the incoming shortwave radiation but absorb a large fraction of the outgoing longwave radiation emitted by the earth's surface after the latter is heated by the shortwave radiation of the sun. The absorbed energy raises the temperature of the gases which then give out their own characteristic longwave radiation a part of which, moving downward, is returned to the earth. Thus, the presence of the gases prevents the earth's surface from being continually cooled by the outgoing longwave radiation. The process is somewhat similar to that which occurs in a greenhouse which allows the shortwave solar radiation to come in during daytime and warm up the enclosure, but does not allow the longwave radiation from inside to leave the enclosure with the result that the inside remains warm enough for the in-house plants to survive especially in cold climates where the outside may be extremely cold. For this reason, the atmospheric gases which absorb the outgoing longwave radiation from the earth's surface and re-radiate a part of it back to the earth's surface are called greenhouse gases and the process the greenhouse effect. The process is explained more fully in the next sub-section.

9.3.1 The Greenhouse Effect

The working of a layer of greenhouse gas in the atmosphere may be compared to that of a slab of glass placed horizontally a little above the earth's surface and exposed to the incoming solar radiation. It is well-known that glass is transparent to radiation of wavelengths shorter than about 4^ but partially opaque to radiation of longer wavelengths. Likewise, the gas layer is transparent to shortwave solar radiation moving downward but partially opaque to longwave radiation moving upward from the earth's surface (Fig. 9.2).

In Fig. 9.2, let I denote the net downward-moving solar radiation which after passing through the gas layer warms up the earth's surface to a temperature T. The earth's surface then emits longwave radiation at this temperature which is given by the Stefan-Boltzmann's law,

Where U denotes the upward flux of the longwave radiation

Now, if ax is the absorption coefficient of the gas for wavelength X, a fraction axU of the upward-moving flux is absorbed by the layer, while the remainder, U (1 — ax), leaves it. The absorbed energy warms up the gas which then emits its own radiation, approximately half of which moves upward and half downward. Let the part moving upward or downward be denoted by G. Then, for equilibrium, the upward- and downward-moving radiations must balance. Thus,

From (9.3.2), we get G = axU/2, and U = I/(1 — ax/2). Substituting the value of U from (9.3.1) in (9.3.2), we get

If we assume that the layer absorbs all the longwave radiation that enters it from below, i.e., if the absorption co-efficient ax is unity, it is easy to see from (9.3.3) that

Fig. 9.2 Illustrating the greenhouse effect

the presence of the greenhouse gas enhances the ground temperature by about 19%. Laboratory experiments show that water vapour has a very strong absorption band in the longwave part of the spectrum. So if this gas is present in sufficient concentration in the atmosphere as with a cloudy sky, it would absorb practically all the outgoing longwave radiation and re-radiate it back to the earth's surface to warm it up or reduce its cooling. This may explain why on a cloudy day the warm and sultry weather that one experiences persists throughout the night when radiational cooling of the surface is prevented by downward longwave radiation from the clouds.

The greenhouse effect occurs in the atmosphere over both land and ocean and is not unique to earth alone. It occurs on other planets as well which have atmospheres with gases which can absorb longwave radiation emitted by the planet's surface. For example, on Mars which has a very tenuous atmosphere of carbon dioxide and dust particles, the greenhouse warming is about 5 K. On the planet Venus which has a much denser atmosphere of carbon dioxide, the greenhouse warming amounts to about 500 K, a temperature too high to support life of the kind we know on our planet. It is believed that when the planets formed, our earth had nearly as much CO2 in its atmosphere as the planet Venus, but most of this CO2 was absorbed by the earth's oceans and used up in the formation of rocks in the earth's crust, thus leaving its concentration to the present level.

In the present terrestrial atmosphere, greenhouse warming is caused naturally to the extent of about 90% by water vapour (H2 O) which exists in its three phases and the rest by carbon dioxide (CO2) and ozone (O3). According to Hettner's experiments, as quoted by Simpson (1928), water vapour absorbs longwave radiation strongly over a very wide range of wavelengths: for example, in bands centered at 1.37|| 1.84| and 2.66|; a very intense band centered at 6.26|; and a very wide band extending from about 14| upwards to almost 80|. Carbon dioxide has a narrow intense absorption band centered at 14.7| and extending from about 12| to 16.3|. Ozone has a strong absorption band at 9-10|. There is a growing concern among humankind, supported by several scientific studies, that the emission of increasing amount of CO2 into the terrestrial atmosphere by industrial activity and burning of fossil fuels, etc., on earth may slowly enhance global warming to such an extent that it will one day produce catastrophic effects on this planet, such as melting of the arctic sea ice, flooding of all low-lying areas by raising the mean sea level, changing global climate, etc. Given this bleak scenario, it stands to reason that strict measures should be taken by all concerned to control the release of greenhouse gases especially CO2 into the atmosphere to avert the catastrophe.

9.4 Vertical Transfer of Radiative Heating - Diurnal Temperature Wave

An interesting case of radiative transfer of heat in the atmosphere is the vertical movement of the diurnal cycle of solar heating from the earth's surface upward. The appropriate transfer or diffusion equation which may be found in any standard text-book on mathematical physics is dT/dt = KRd2T/dz2 (9.4.1)

where T is temperature at height z and time t, and KR is co-efficient of radiative diffusivity.

The boundary condition to solve (9.4.1) is given by

where T0 is the mean temperature at the earth's surface, A the amplitude and p the frequency of the temperature wave.

The solution of (9.4.1) consistent with the boundary equation (9.4.2) is

where we have assumed a vertical lapse rate of temperature, P, and b is a constant which is given by b2 = p/2 Kr, where p = 2n/(24 x 60 x 60)

The diurnal variation of temperature at any height z is then given by (9.4.3) in which the amplitude A falls off exponentially with height and the time of the maximum temperature at height z occurs bz/p seconds later than at the surface. An estimated value of Kr in the atmosphere, as given by Brunt (1944), is 1.3 x 103.

9.5 Sensible Heat Flux

The earth's surface whether land or water always tends to remain in approximate thermal equilibrium with its environment, immediately above and below the surface. So, whenever there is a gradient of temperature between the surface and the environment above or below, there is a flux of heat flowing down the temperature gradient. In the case of land, heat may flow both upward into the atmosphere and downward into the soil or earth's crust. Over the ocean, heat may flow in the same way upward into the atmosphere and downward into the water. We first consider vertical transfer of sensible heat into the atmosphere, from both land and sea.

9.5.1 Vertical Transfer of Sensible Heat into the Atmosphere

The sensible heat flux, Hs, which gives the amount of heat flowing upward across unit area of a surface per unit time at a height z near surface, is usually governed by the flux-gradient relationship which may be stated as:

where p is air density, cp is specific heat of air at constant pressure, KH is co-efficient of diffusivity of heat, T is temperature, —dT/dz is vertical gradient of temperature with height z, and r is dry adiabatic lapse rate of temperature.

The upward flux at height z + 5z is then given by

where Hs denotes the sensible heat flux at height z.

The difference in flux between the two levels, - (dHs/dz) 5z, is then the heat which is retained in the layer 5z and which warms it up at the rate p cp (dT/dt) 5z. Using the value of Hs from (9.5.1) and assuming that p and KH do not vary appreciably with height, we arrive at the vertical heat transfer Eq. (9.5.3)

which is known as the Fourier equation for heat transfer.

Equation (9.5.3), though derived under assumptions which may not strictly hold in the real atmosphere, is quite general and may be used to solve different types of heat transfer problems with suitable boundary conditions, provided we choose proper values for the co-efficient of thermal diffusivity, KH.

In the atmosphere, heat transfer by molecular conduction or diffusion may be ruled out, since it is an extremely slow and inefficient process. So, heat is transferred mostly by radiation, turbulent motion and convection. The co-efficient of heat transfer by these latter processes is several orders of magnitude greater than that in molecular conduction. Brunt (1944) gives relative estimates for approximate values of the co-efficient of heat diffusivity in the different cases: 0.16 for molecular conduction, 1.3 x 103 for radiative diffusivity and 1 x 105 for convective or turbulent transfer. These are values derived under idealized conditions and may often vary in the real atmosphere. However, they give us an idea of the relative effectiveness of the different transfer processes. They appear to confirm that one of the most effective mechanism of heat transfer in the atmosphere is by turbulent motion or eddies, provided we use appropriate value for the co-efficient KH.

In turbulence, KH is written for l'w', where l' is a linear dimension of the eddy, w' the vertical component of the eddy velocity and the overbar denotes the mean value over a period of time. However, an important point in vertical transfer of heat by eddies needs to be emphasized. It may be seen from (9.5.1) that the upward heat flux across unit area of a horizontal surface at height z per unit time is

where r is the dry adiabatic lapse rate and - dT/dz the prevailing lapse rate of temperature. The above expression is positive or negative, according as the actual lapse rate is greater or less than the dry adiabatic lapse rate. In other words, the heat flux is upward only when the atmosphere is vertically unstable. When the atmosphere is stable and the heat flux is downward, the eddy motion will warm up the lower layer and cool the upper layer till the lapse rate increases to the dry adiabatic lapse rate.

Taylor (1915) applied the heat transfer Eq. (9.4.3) to a case in which a mass of air flowing over a warm land at surface temperature To moved over a cold sea with surface temperature Ts. Suppose the warm air had a vertical lapse rate P at the time it entered the sea and the drop in temperature (To - Ts) was all of a sudden at time t = 0. The problem was to solve the heat transfer equation with the specified boundary conditions in order to compute the height to which the effect of the surface cooling will extend after time t. The solution, which may be found in any standard text-book (e.g., Heat conduction by Ingersoll et al., 1948) or any other book on heat transfer, is given by z/V4Kh t

T(z, t)=To — pz + (To — Ts){1 — (2/n) f exp(— |2)d|} (9.5.4)

In (9.5.4), the term within the second bracket is unity at z = 0, and falls to a value 0.1 at a height given by z/^ (4Kh t) = 1.2. Taylor assumed that the surface cooling had no appreciable effect beyond a height given by z = 2/Kh t (9.5.5)

Taylor tested his theory of the vertical diffusion of heat by turbulence by measuring temperatures at different heights and at different distances from the coastline over the Great Banks of Newfoundland where a layer of marked temperature inversion existed over the cold sea with a temperature distribution approaching the dry adiabatic lapse rate at higher levels. Knowing the height of the inversion layer at a given distance and the time the air has flown over the sea to reach there, he computed the value of KH to be of the order of 103cm2 s \ He, however, noted that the turbulent transfer in the inversion layer which was vertically stable was highly subdued and felt that with a larger lapse rate of temperature and stronger turbulence, the value of KH should increase.

Taylor derived (9.5.5) on the supposition that the surface temperature dropped suddenly to Ts at the coastline. Later, he changed this boundary condition and assumed that the surface temperature drops gradually over the sea at the uniform rate of n° degree per unit time, so that the surface temperature is given by T0-nt. Introducing the revised boundary condition, he found that the temperature at height z at time t is given by (9.5.6):

T(z, t)=T0 — pz — nt[(1 + z2/2 Kh t){1 — (2/^%) J exp (—|2)d|}

The term which multiplies nt in (9.5.6) is unity at the surface and falls to a value of 0.1 at z/(a/4Kh t) = 0.8. The height reached by the temperature drop in this case is, therefore, less than in the case of the sudden drop. However, an approximation can still be made and we may write z2 = 4 Kh t

This case also gives a value of KH of the order of 103 cm2 s 1. Thus, both (9.5.5) and (9.5.6) emphasize the fact that even in the stable atmosphere over the Great Banks, heat transfer by turbulent motion is much more effective than molecular motion. In more normal situation, or when a cold continental airmass enters a warm sea, such as the sea of Japan during the northern winter, where there is considerable turbulence present, the co-efficient of heat diffusivity may be of the order of

105 cm2 s or even greater.

Another rather complex case of airmass modification through vertical heat transfer is observed over the Arabian sea during the northern summer, in which a hot continental airmass from east Africa with a surface temperature of over 40 °C first flows over a cold sea swept by the Somali current where the temperature often drops to about 20 °C or below and then enters a warm region of the eastern Arabian sea with temperatures around 30 °C. A strong low-level temperature inversion develops over the western part of the sea and makes the atmosphere vertically very stable with little or no clouds in the sky. But as the airmass flows over the warmer part of the sea, considerable turbulence and convection currents develop which weaken the inversion and transfer heat and moisture rapidly from the sea surface to a height of 2-3 km. With inversion gone, towering clouds develop and a lot of precipitation occurs over the eastern part of the sea.

9.6 Evaporation and Evaporative Heat Flux from a Surface

The process of loss of water in the form of vapour from a wetted land surface, or from a water surface, such as that of ponds, rivers, lakes or open oceans, whenever the vapour pressure of the air above the surface falls short of its saturation value at the temperature of the underlying surface is called evaporation. The evaporative flux, E, is the amount of water lost per unit area of the surface per unit time and may be easily derived using the equation of state for water vapour (4.3.1) and expressed by the relation

where KD is the co-efficient of eddy diffusivity for water vapour, p is the density of air, e is the ratio of the molecular weights of water vapour to dry air (e= 0.625), p is the atmospheric pressure, e is the vapour pressure of water, and z is a small height above the surface.

The evaporative heat flux, He, is then given by

where L is the latent heat of vaporization of water at the temperature of the underlying surface.

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