Energy balance

ENERGY BALANCE (HEAT BALANCE)

When solar radiation enters Earth’s environment, it provides energy for maintenance and dynamic functions of different components of global environment. The continuous maintenance of particular physical and chemical states of matter in atmosphere, hydrosphere, lithosphere and biosphere requires energy provided by solar radiation. Further, various dynamic changes in these states such as air and water movements, changes in the state of water from vapor to liquid to solid and vice-versa and the activities of living organisms are found to occur. These changes are possible only through the expenditure of energy provided by solar radiation. The energy of Earth’s surface radiation balance is expended on heating of atmosphere through turbulent heat conductivity, on evaporation of water, on heat exchange with deeper layers of hydrosphere and lithosphere etc. and photosynthesis in biosphere. In general, the quantitative characteristics of all forms of transformations of solar energy on the Earth’s surface are represented in the equation of global energy (heat) balance. This equation includes the algebraic sum of flows of energy reaching and leaving the Earth’s surface. This sum is always zero according to the law of conservation of energy. The energy balance and radiation-balance at Earth’s surface are linked together.

The equations representing energy balance may be compiled for various volumes and surfaces of atmosphere, hydrosphere and lithosphere. However, in the studies of global environment, equations are often employed for an imaginary column whose upper end is at the upper boundary of atmosphere and which passes through atmosphere deep below Earth’s surface. Three equations of energy (heat) balance describing global energy balance are:

(a) Energy balance equation of Earth’s surface

(b) Energy balance equation of Earth-atmosphere system

(c) Energy balance equation of atmosphere.

(a) Energy balance equation of Earth’ surface

Major elements of this equation are (Fig. 1):

(i) Radiation balance (R) i.e. radiation flux, which is considered positive in value when it describes inflow of energy (heat) from above to underlying Earth’s surface.

(ii) Turbulent energy (heat) flow (P) from underlying Earth’s surface to atmosphere.

(iii) Underground energy (heat) flow (A) from Earth’s surface to deeper layers of hydrosphere or lithosphere.

(iv) Energy (heat) expenditure on evaporation (or release of heat in condensation) (LE) where L is latent heat of vaporization and E is rate of evaporation.

With the above elements, energy balance equation of Earth’s surface is given as:

R = LE + P + A

The elements of energy balance not included in the above equation are:

(i) Energy expenditure on melting of ice or snow on surface (or inflow of heat from freezing of water)

(ii) Energy expenditure associated with friction of air currents, ocean waves produced by winds and ocean tides

Figure-1. Components of the energy balance of Earth-Atmosphere-Hydro-Lithosphere system.

(iii) Energy (heat) flows transferred by precipitation whenever their temperature is not equal to that of underlying surface

(iv) Energy expenditure on photosynthesis

(v) Energy (heat) inflow from oxidation of biomass.

With the addition of these elements also, comprehensive energy balance equation of Earth’s surface may be obtained.

The magnitude of underground energy (heat) flow (A) may be obtained from the energy balance equation of a vertical column whose upper base is at Earth’s surface and lower base at the depth below ground surface where heat flow is negligible (Fig. 2). Since heat flow from depths of Earth’s crust is negligible, vertical flow of heat at the lower base of column may be assumed to be zero. The equation for A is given as:

A = Fo + B

where, B represents the changes in heat content inside the column over a given period of time and Fo is the inflow of heat produced by horizontal heat exchange between the column being considered and the surrounding space of hydrosphere or lithosphere. Fo is equal to the difference between amounts of heat entering and leaving through vertical walls of column.

In lithosphere, Fo usually becomes negligible due to low heat conductivity of soil. Thus for land A = B and since over a period of whole year, upper layers of soil are neither heated nor cooled, A = B = 0.

Fo becomes large in case of water bodies having currents with a large horizontal heat conductivity determined by macroturbulence. In case of closed water bodies taken as a whole whose depth and area are large, values of A and B are close. It is because heat exchange between such bodies of water and the ground are usually negligible. However, in specific sectors of oceans, seas and lakes, magnitudes of A and B may be substantially different. The average yearly value of heat exchange of an active surface with lower is not zero but is equal to the quantity of heat received or lost due to currents and macroturbulence i.e. A = Fo.

Thus for average yearly period, energy balance equation of Earth’s surface will be:

(i) For land: R = LE + P

(ii) For ocean: R = LE + P + Fo

(iii) For deserts (where evaporation is almost zero): R = P

(iv) For global oceans as a whole (where redistribution of heat by currents is compensated and is zero): R = LE + P

(b) Energy balance equation of Earth-atmosphere system

This equation can be derived by considering the inflow and expenditure of energy in a vertical column passing downwards from the top of atmosphere to that level in hydrosphere or lithosphere at which noticeable daily or seasonal fluctuations of temperature stop (Figure-1). Energy (heat) flow through the lower base of this column is practically zero.

Energy balance equation of Earth-atmosphere system may given as:

Rs = Fs + L(E - r) + Bs

All the terms on the right-hand side of equation are assumed positive in value when they describe expenditure of energy (heat). The elements of the equations are as discussed below:

(i) Radiation balance of Earth-atmosphere system (Rs): It describes the energy (heat) exchange between the vertical column under consideration and the outer space and is equal to the difference between the amounts of total solar radiation absorbed by the entire column and the total long-wave radiation from column to outer space. It is considered positive when it describes inflow of energy (heat) into the Earth-atmosphere system.

(ii) Total horizontal heat transfer (Fs): It occurs through the sides of the column under consideration and is given as:

Fs = Fo + Fa

where, Fo = horizontal heat transfer through sides of the column in the atmosphere and Fa = horizontal heat transfer through the sides of column in the hydrosphere or lithosphere. Value of Fa is similar to that of Fo and describes the difference of inflow and expenditure of heat in the column of air resulting from atmospheric advection and macroturbulence.

(iii) Heat transfers in change of the state of water: Heat balance of column is also influenced by sources of heat (both positive and negative) that are located within the column itself. These include the inflow and expenditure of heat due to changes in state of water, especially by evaporation and condensation.

Over sufficiently homogeneous surfaces during long periods, the average difference in the magnitudes of condensation and evaporation of water drops in atmosphere is equal to the sum of precipitation (r) and the inflow of heat is equal to Lr. Corresponding component in the energy balance represents the difference between heat inflow from condensation and its expenditure in the evaporation of drops. It may differ from Lr in conditions of rugged surfaces and also in individual short periods of time.

The difference between heat expenditure on evaporation the surface of water bodies, soils and vegetation and heat inflows from condensation on these surfaces are equal to LE.

The overall influence of condensation and evaporation on the column’s energy balance may be approximated in terms L(r -E).

(iv) Changes in the heat content within the column: This change over the period being referred is represented by component Bs in the energy balance equation.

Remaining components of the balance such as heat inflow from dissipation of mechanical energy, difference between heat expenditure and inflow on melting and formation of ice, difference between heat expenditure on photosynthesis inflow from oxidation of biomass etc. are very small and may be neglected.

Consideration of different components of energy balance equation under different conditions shows that:

(i) For an average yearly period, magnitude of Bs is apparently close to zero and the equation simplifies to:

Rs = Fs + L(E - r)

(ii) For the land conditions, the equation becomes:

Rs = Fa + L(E - r)

(iii) For the entire globe, E = r over a period of one year and horizontal inflow of heat into the atmosphere and hydrosphere is apparently zero. Thus the energy balance equation of Earth-atmosphere system for the Earth as a whole simplifies to:

Rs = 0

(c) Energy balance equation of atmosphere

This equation may be obtained by either

(i) Summing up the corresponding flows of heat or

(ii) As difference between members in the heat balance equation for the Earth-atmosphere system and in that for Earth’s surface.

Assuming that atmospheric radiation balance is given by:

Ra = Rs - R

and changes in the heat content of atmosphere (Ba) are given by:

Ba = Bs - B

it can be seen that:

Ra = Fa - Lr - P + Ba

and for an average yearly period, equation is:

Ra = Fa - Lr - P

DISTRIBUTION OF ENERGY BALANCE COMPONENTS

Distribution of energy balance components of Earth’s surface

Important components of energy balance of Earth’s surface which show geographical differences in their values are heat expenditure on evaporation, turbulent heat exchange and redistribution of heat through atmospheric and oceanic currents.

1. Heat expenditure on evaporation: The magnitudes of evaporation from land surface and the oceans in the vicinity of coastlines, differ significantly. This may apparently be explained

(i) differences in the value of possible evaporation on land and on ocean and

(ii) the influence of insufficient moisture in many land areas which limits the intensity evaporation processes and of heat expenditure on evaporation.

At extratropical latitudes, absolute value of heat expenditure on evaporation generally decreases with increasing latitudes. However, major non-zonal changes on land and ocean alter this pattern. In tropical latitudes, distribution of heat expenditure on evaporation is quite complex. Compared to high-pressure regions, its value declines somewhat in the ocean regions adjoining the Equator.

In the oceans, maximum mean latitudinal heat expenditure on evaporation occurs within high-pressure belts. At 50-70 degrees where radiation balances of land and oceans are approximately same, the heat expenditure on evaporation is substantially larger for oceans. This is evidently due to large expenditure of heat brought by ocean currents. In oceans, distribution of warm and cold currents is principal cause of the non-zonal changes in heat expenditure on evaporation. All the major warm currents increase heat expenditure substantially while cold currents reduce it. This may be clearly seen in regions influenced by warm currents like Gulf stream and Kuroshio by old currents like those of Canary Islands, Bengal, California, Peru and Labrador. The yearly evaporation from ocean surface at a particular latitude may change by several time depending on the increase or decrease in water temperature brought about by the currents. In addition, non-zonal in the values of heat expenditure on evaporation and so of evaporation from oceans are also influenced by conditions of atmospheric circulation determining wind velocity and the annual humidity deficit over the oceans. The ocean surfaces have somewhat higher radiation balance than land surfaces and evaporating surfaces may additionally receive a large quantity of heat energy through redistribution of heat by ocean currents. Therefore, evaporation from ocean surface in tropical areas corresponds to a layer of water more than two meters thick.

Table-1. Average values of Earth’s surface energy balance components at various latitudes (kcal/sq. cm/year)

Latit-sude (in degr-ees)	R	LE	P	A	R	LE	P	R	LE	P	A
	Ocean	Ocean	Ocean	Ocean	Land	Land	Land	Earth	Earth	Earth	Earth
70-60 N	23	33	16	-26	20	14	6	21	20	9	-8
60-50 N	29	39	16	-26	30	19	11	30	28	13	-11
S	28	31	8	0	49	25	24	72	60	12	0
50-40 N	51	53	14	-16	45	24	21	48	38	17	-7
S	57	55	9	-7	41	21	20	56	53	9	-6
40-30 N	83	86	13	-16	60	23	37	73	59	23	-9
S	82	80	9	-7	62	28	34	80	74	12	-6
30-20 N	113	105	9	-1	69	20	49	96	73	24	-1
S	101	100	7	-6	70	28	42	94	83	15	-4
20-10 N	119	99	6	14	71	29	42	106	81	15	10
S	113	104	5	4	73	41	32	104	90	11	3
10-0 N	115	80	4	31	72	48	24	105	72	9	24
S	115	84	4	27	72	50	22	105	76	8	21

On the land, mean latitudinal value of heat expenditure on evaporation is maximum at equator. These values change within the subtropical high-pressure belts. In both hemispheres, a certain increase in evaporation occurs with increase in latitudes though the increase is more pronounced in Northern Hemisphere. This increase is due to increased precipitation as compared with arid zones at lower latitudes. The distribution of heat expenditure on evaporation from land surface deviates from zonal pattern even more than from oceans. This is due to very great influence of climatic moisture conditions on evaporation. In regions of sufficient soil moisture found at high latitudes and in humid regions at middle and tropical altitudes, heat expenditure on evaporation and the evaporation are governed largely by balance. In regions of insufficient moisture, evaporation is reduced due to insufficient soil moisture while in desert and semi-desert areas, evaporation is almost equal to low yearly total precipitation. Highest heat expenditure on evaporation occurs in certain equatorial regions where in case of abundant moisture and large inflows of heat, it exceeds 60 kcal/sq. cm/year. This corresponds to yearly evaporation of layer of water more than one meter thick.

Further, the patterns of seasonal heat expenditure on evaporation in extratropical latitudes are different on land and oceans. On the land, this expenditure and evaporation decreases substantially during cold season and depending on moisture conditions, attains a maximum at the beginning or in middle of warm season. In contrast, evaporation from oceans usually increases in cold season due to greater difference in temperature of water and air at that time which increases difference in concentration of water vapor on the surface of water and in air. In addition, in many oceanic regions average wind velocities are greater in cold seasons and this also increases evaporation.

2. Turbulent heat exchange: The value of turbulent heat exchange is positive heat is released by Earth’s surface into air and is negative when heat is received by Earth’s surface from atmosphere during the year. Over a year, all the land surfaces except Antarctica and larger part of ocean surfaces release heat into the atmosphere.

In oceans, turbulent heat exchange gradually increases towards higher latitudes. Its magnitude is not large for greater part of ocean surfaces and usually does not exceed 10-20% of the magnitudes of principal components of energy balance equation. Large absolute values of turbulent heat flow, exceeding 30-40 kcal/cm²/year, occur in regions of powerful warm currents e.g. Gulf Stream. Here water is on average warmer than air and at higher latitudes where sea is still free from ice. Cold oceanic currents reduce temperature of water, reduce turbulent heat flow from ocean surface to the atmosphere and increase it in reverse direction.

On land, turbulent heat flow decreases towards higher latitudes. Its maximum value occurs within high-pressure belts which declines somewhat near Equator and sharply decreases at high latitudes. Magnitude of turbulent heat -exchange on continents is greatly influenced by climatic moisture conditions. In arid regions, turbulent heat flow from land surface into the atmosphere is much higher than in humid regions. Highest expenditure of turbulent heat flows on land is found in tropical deserts where it may exceed 60 kcal/sq. cm/year. In humid regions, especially in regions at middle latitudes, heat expenditure through turbulent flows is usually much lower.

The very different patterns of change in turbulent heat exchange on land and in oceans reflect differences in the mechanisms of air mass transformation on the surfaces of continents and oceans.

3. Heat redistribution through water currents: In the heat balance of oceans, inflow or expenditure of energy owing to horizontal exchanges primarily through oceanic currents is very important. A large quantity of heat is redistributed in oceans between tropical and extratropical latitudes. Both warm and cold currents play important role in redistribution of heat in oceans. Regions of increased positive values of that particular component of heat balance (reflecting outflow of heat from ocean surface to lower layers) correspond with regions of cold currents and the regions of reduced negative values correspond with warm currents. Such correspondence is observed for major warm currents e.g. Gulf Stream, Kuroshio and Southwest Pacific Stream as well as for cold currents e.g. Canary Islands, Bengal, California and Peru. Ocean currents carry away heat mainly from a zone ranging from 20 degrees N latitude to 20 degrees S latitude. Maximum of heat absorbed is slightly shifted to the north of Equator. Further, the heat is carried to higher latitudes and expended in the region of 50 degrees to 70 degrees N latitude where warm currents are especially strong.

Studies of Strokina (1963,1969) concerning changes in heat content of ocean’s upper layers over a year have shown that these changes may attain significant magnitudes which are quite comparable with changes in magnitudes of the main components of heat balance. Greatest yearly changes in heat content of ocean’s upper layers (over 25 kcal/sq. cm/year) are observed in Northwestern regions of Pacific ocean and adjoining areas.

Distribution of energy balance components of Earth-atmosphere system

Data for average yearly conditions show that relative proportions of various components of energy balance of Earth-atmosphere change perceptibly at various latitudes.

In equatorial zone, the large inflow of radiation energy is further increased by addition of a substantial inflow of heat produced by changes in state of water through condensation and evaporation. These sources of heat produce large expenditure of heat on atmospheric and oceanic advection. A relatively narrow zone adjoining Equator is and extremely important source of energy for these advection conditions.

At higher latitudes upto 30-40 degrees, a positive radiation balance that decreases with increasing latitude is accompanied by substantial expenditures of energy on water exchange. In most parts of that zone, energy of radiation balance is almost equal to heat expended on water exchange and very little heat is redistributed through air and water currents.

At latitudes above 40 degrees, a zone of negative radiation balance is found. Its absolute value increasing at higher latitudes. The negative radiation balance of that zone is compensated by inflow of heat brought by air and water currents. Proportions of those components within that balance which compensate for the deficiency of radiation energy vary at different latitudes. For the belt between 40--60 degrees, excess energy released in condensation of water is major source of heat while inflow of heat redistributed by ocean currents is also important. At higher latitudes, especially in polar regions, heat inflow from condensation is very small and influence of ocean currents is either absent (in South polar zone) or is weak due to permanent ice cover (in North polar zone). At these latitudes, redistribution of heat through atmospheric circulation is major source of heat.

The average values of various components of energy balance of Earth-atmosphere system over six-month periods at various latitudes have been studied (Table-2). These show that magnitude of radiation absorbed by Earth-atmosphere system (Qa) is not the only factor determining the magnitude of outgoing long-wave radiation at the top of atmosphere (Is). For middle and high latitudes during October to March in Northern Hemisphere and for high latitudes in Southern Hemisphere throughout the entire year, the main source of heat is heat-transfer from lower latitudes through atmospheric circulation.

Distribution of energy balance components of atmosphere

The average radiation balance of atmosphere at various latitudes changes less than other components of heat balance. The large absolute negative values for the atmospheric radiation-balance observed at all latitudes are compensated largely by inflows of heat from condensation. The role of heat from Earth’s surface through turbulent heat exchange is less important though the influence is quite perceptible.

Distribution of components of energy balance of whole Earth

Depending on the relative proportions of land and ocean areas in particular zones, mean latitudinal distribution of the components of energy balance of Earth as a whole is characterized by patterns typical of continents or by the patterns typical of oceans. Average values of energy balance components for individual continents and oceans (Table-3) show that in three continents (Europe, North America and South America) greater share of energy radiation balance is expended on evaporation. In the remaining three continents (Asia, Africa and Australia) where dry climates prevail, opposite is true.

Energy balance components of three oceans show little difference from each other. For each ocean the sum of heat expenditure on evaporation and turbulent heat exchange is close to the magnitude of radiation balance. This means that the heat exchange among different oceans resulting from currents does not exert any substantial influence on the heat balances of individual oceans.

The values of the components of energy balance for Earth a whole show that in oceans approximately 90% of the energy of radiation balance is expended on evaporation and only 10% on direct turbulent heating of atmosphere. These magnitudes are nearly same on land. For Earth as a whole, 83% of the energy of radiation balance is expended on evaporation 17% on turbulent heat exchange.

The values of the components of energy balance for the Earth as a whole are shown in Figure-2. Overall yearly flux of solar radiation entering outer boundary of troposphere is approximately 1000 kcal/sq. cm. Due to the spherical shape of Earth, about 25% of this yearly flux (i.e. 250 kcal/sq.cm), passes through a unit surface of the upper boundary of troposphere. Assuming that Earth’s albedo (As) is 0.33, short-wave radiation absorbed by Earth represented by Qs(1-As) is approximately 167 kcal/sq. cm/year. Out of this, short-wave radiation reaching Earth’s surface is 126 kcal/sq. cm/year. Average value of albedo at Earth’s surface (A) is 0.14. This takes into account the differences in value of incoming solar radiation in various regions. Thus the amount of short-wave radiation absorbed at Earth’s surface, represented by Q(1-A), is 108 kcal/sq. cm/year and 18 kcal/sq. cm/year is reflected back from the surface. The atmosphere absorbs about 59 kcal/sq. cm/year which is substantially less than that absorbed at Earth’s surface. Since radiation balance of Earth’s surface (R) is 72 kcal/sq. cm/year, average effective radiation of Earth’s surface (I) comes to be 360 kcal/sq. cm/year. Overall value of Earth’s long-wave radiation (Is) is quite close to 167 kcal/sq. cm/year. The ratio I/Is much less than

Table 2. Mean latitudinal values of energy balance components of Earth-atmosphere system for six-month periods (kcal/sq. cm/year) April-September

Latitude (in degrees)	Qa	Fa	Fo	Bs	Is
80-90 N	7.8	-4.5	0.0	0.8	11.5
S	0.0	-8.8	0.0	0.0	8.8
70-80 N	8.2	-4.4	0.0	0.8	11.8
S	0.2	-9.8	0.0	0.0	10.0
60-70 N	11.5	-1.8	-0.4	1.2	12.5
S	1.0	-9.5	0.0	0.8	11.3
50-60 N	14.6	0.0	-1.3	2.8	13.1
S	3.3	-3.9	-2.6	-2.5	12.3
40-50 N	16.9	1.0	-2.0	3.9	14.0
S	5.9	-1.9	-1.6	-3.5	12.9
30-40 N	19.2	1.4	-1.7	4.3	15.2
S	8.9	-0.4	-0.4	-4.2	13.9
20-30 N	20.0	2.0	-0.4	2.9	15.5
S	13.0	1.4	0.3	-3.4	14.7
10-20 N	19.7	2.4	0.9	1.4	15.0
S	16.2	2.6	0.9	-2.5	15.2
0-10 N	18.4	1.4	2.2	-0.1	14.9
S	18.0	2.3	2.2	-1.5	15.0

October- March

80-90 N	0.1	-9.1	0.0	-0.8	10.0
S	3.4	-6.9	0.0	0.0	10.3
70-80 N	0.5	-9.3	0.0	-0.8	10.6
S	4.4	-6.5	0.0	0.0	10.9
60-70 N	1.8	-6.7	-1.5	-1.2	11.2
S	8.1	-4.3	0.0	0.8	11.6
50-60 N	4.0	-4.7	-0.5	-2.8	12.0
S	12.8	-2.6	0.6	2.5	12.3
40-50 N	6.5	-2.9	0.6	-3.9	12.7
S	15.9	0.0	-0.6	3.5	13.0
30-40 N	9.5	-0.9	0.7	-4.3	14.0
S	18.9	2.0	-1.4	4.2	14.1
20-30 N	13.7	0.9	0.6	-2.9	15.1
S	20.6	3.5	-0.3	2.5	15.0
10-20 N	17.0	1.8	1.0	-1.4	15.6
S	20.7	3.5	-0.3	2.5	15.0
0-10 N	18.7	2.1	1.4	0.1	15.1
S	19.7	2.4	0.8	1.5	15.0

Table-3. Energy balance of continents and oceans (kcal/sq. cm/year)

Continent	R	LE	P	Ocean	R	LE	P
Europe	39	24	15	Atlantic	82	72	8
Asia	47	22	25	Pacific	86	78	8
Africa	68	26	42	Indian	85	77	7
N.America	40	23	17	S.America	70	45	25
Australia	70	22	48

the ratio Q(1-A)/Qs(1-As). This difference shows that greenhouse effect has very large influence on the thermal processes of Earth. Due to this effect, Earth receives about 72 kcal/sq.cm/year of radiation energy. This energy is partly expended on evaporation of water (LE = 60 kcal/sq. cm/year) and partly returned to the atmosphere by turbulent heat losses (P = 12 kcal/sq. cm/year). Thus, the energy balance of atmosphere has following components:

(i) Heat inflow from absorbed short-wave radiation = 59 kcal/sq. cm/year

(ii) Heat inflow from condensation of water vapor (Lr) = 60 kcal/sq. cm/year

(iii) Heat inflow from turbulent heat losses at the Earth’s surface = 12 kcal/sq. cm/year

(iv) Heat expenditure on effective radiation into outer space (Is - I) = 131 kcal/sq. cm/year.

The last figure corresponds to the sum of first three components of energy balance.

SOLAR RADIATION AND PLANETARY TEMPERATURE

The temperature of a planet irradiated by solar radiation can be estimated by balancing the amount of radiation absorbed (Ra) against the amount of outgoing radiation (Ro). The Ra will be the product of:

(i) Solar irradiance (I)

(ii) Area of planet. Area of planet relevant for such calculations is the area of the planet as seen by incoming radiation which is given by r² where r = radius of the planet.

(iii) Absorbed fraction of radiation. The fraction of radiation that is absorbed is given by (1 - A where A = albedo of planet. This albedo represents the fraction of radiation that is reflected back from the planet.

Figure. 2. Energy balance of the Earth. (Components values in kcal/cm²/year).

Thus the energy absorbed by the planet will be:

Ra = I ^ (1 - A)

Intensity of outgoing radiation of a body is given by Stefan-Boltzmann Law i.e.

Io =  T⁴

where = Stefan-Boltzmann Constant = 5.6 x 10^-8 Wm^-2 K^-4. The total energy radiated by the planet will be the product of the intensity of outgoing radiation (Io) and the area of the whole planetary surface giving out radiation (4  r ²). Thus, the outgoing radiation (Ro) from the planet is given by:

Ro = 4r² T⁴

Effective planetary temperature

Since Ra = Ro i.e. system is assumed to be in steady-state where radiation absorbed and outgoing radiation are equal, an expression for the effective planetary temperature (Te) can be obtained from the above equation and it may be given by:

Te = [I - (1 - A)/4 ]^0.25

In this expression of effective planetary temperature, effect of atmosphere has not been taken into account. For Earth, solar irradiance (I) at the top of atmosphere is about 1.4 x 10³/m²/s and albedo of Earth as a whole is about 0.33. From these values, the calculated equilibrium temperature of Earth comes to be 254 K. However, the actual observed average ground level temperature of Earth is about 288 K. This higher effective temperature of Earth from the calculated value is due to the greenhouse-effect of atmosphere.

The black-body spectrum of Earth at 288 K shows that radiation from Earth is of much longer wavelength and is at much lower intensity than radiation from Sun. The absorption spectrum of Earth’s atmosphere overlaps fairly well with the solar emission spectrum. Except for a very narrow window in the absorption bands, much of the long-wave radiations from Earth correspond with the region of absorption in the atmosphere. This means that much of the incoming radiation reaches the Earth’s surface while the outgoing thermal radiation is largely absorbed by the atmosphere rather than being lost to space. Thus, the effect of atmosphere is to trap the outgoing thermal radiation. This effect is termed green-house effect.. The thermal radiation i.e. the heat trapped by the atmosphere due to green-house effect is responsible for the effective temperature of Earth being higher than the temperature calculated without taking into account the effect of atmosphere. In general, absorption of re-emitted long-wave radiation and vertical mixing processes determine the temperature profile of the lower part of atmosphere (troposphere) which in turn determine the Earth’s temperature.

Optical depth of atmosphere and Earth’s surface temperature

The atmosphere is not transparent to the outgoing long-wave radiation and much of this radiation is absorbed in the lower part of the atmosphere, which is warmer than the upper parts. Simple radiative equilibrium models have been developed for Earth and to account for this effect, these models divide the atmosphere into layers that are just thick enough to absorb the outgoing radiation. These atmospheric layers are said to be optically thick and the atmosphere is discussed in terms of its optical depth based on the number of these atmospheric layers of different optical thickness. Earth’s atmosphere is sometimes said to have two layers while that of planet Venus has almost 70 layers which are largely due to enormous amount of CO₂ in the atmosphere of Venus. The radiation equilibrium model indicates that the effective planetary temperature (Te) is thus related to ground-level planetary temperature (Tg) by the equation:

Tg⁴ = (1 - )Te⁴ (where  = optical depth of atmosphere)

The optical depth of atmosphere increases with increase in atmospheric concentrations of carbon dioxide and water vapor because both these are principal atmospheric absorbers of outgoing long-wave radiation. With increasing concentrations of CO₂ in lower layers of atmosphere, other such gases that are responsible for radiating heat to outer space are pushed to slightly higher and colder levels of atmosphere. The radiating gases will radiate heat less efficiently because they are colder at higher altitudes. Thus, the atmosphere becomes less efficient radiator of heat and this results in rise of atmospheric temperature. This rise in atmospheric temperature, in turn, leads to more evaporation and increase in atmospheric water vapor, which is a greenhouse gas and further increases the absorption of outgoing long-wave thermal radiation. This positive feedback results in further increase in atmospheric temperature. The model also suggests that increase in atmospheric CO₂ is associated with decrease in temperatures of upper (stratospheric).

Vertical heat transport and Earth’s surface temperature

Simple models of radiation balance of atmosphere do not take into account various other processes that transport heat vertically in the atmosphere and, therefore, overestimate the surface temperature of Earth. Convection is major process of vertical heat transport and is very important in lowering the surface temperature. Convection occurs because warm air is lighter than cool air and so rises upwards carrying heat from Earth’s surface to the upper atmosphere. As warm air rises up, it expands due to fall in pressure and work done in expansion causes it to cool adiabatically. Thus,

C_v T = - P V (where Cv = molar heat capacity at constant volume)

Ideal gas equation PV = RT takes the differential form P dV + V dP = R dt which may be rearranged in incremental form as:

- P V + R  = V P

This equation may be combined with equation C_v T = - P V using the fact that C_p - C_v = R, where C_p = molar heat capacity at constant pressure = 29.05 J/mol/K. This results in following equation:

C_p T = (C_v + R) T = - P V + R T = V P = (RT/P) P .............(a)

It can be shown that P/P = - M_mg z /RT where M_m = mean molecular weight of air = 0.028966 kg/mol; g = acceleration due to gravity = 9.8065 m/s/s; z = altitude. This gives:

RT/P = - Mmg z/ P.......................................................(b)

Substitution of the above equation (b) in equation (a) gives:

C_p T = - M_mgz

or, T/ z = - (M_mg/Cp)

For Earth’s atmosphere, the lapse rate ( T/ z) works out to be -9.8 k/km for dry air. However, the air is usually wet and as it rises up, it releases latent heat so the measured lapse rate is -6.5 K/km.

If atmospheric temperature falls much less slowly with height than the lapse rate (or even rises with height) then inversion conditions exist and air is very stable with respect to vertical convective mixing. Conversely, if temperature falls very rapidly with height, at a rate greater than lapse rate, then the atmosphere is unstable and convective mixing will be active.

Short-wave radiation and temperature

The discussion till now has assumed total transparency of atmosphere to incoming solar radiation. Though it is true for visible range of radiation, it is not true for ultra-violet region of the solar spectrum. Though the amount of such short-wave radiation is very small, it has important consequences for the temperature of Earth-atmosphere system.

Various ultra-violet wavelengths are absorbed in the atmosphere at different heights. At just over 40 km, absorption of ultra-violet radiation by ozone results in considerable warming of stratosphere and in this zone, temperature rises with altitude. Average temperature of stratosphere is 250 K. Considering it to be a black-body radiator, maximum power radiation would be expected at 11.5 m. This value is very close to absorption band of carbon dioxide which means that this gas also plays important role in stratospheric temperature. Increase in concentration of carbon dioxide in stratosphere might allow more effective radiation from stratosphere and, therefore, its cooling. This effect is quite opposite to that noted for troposphere.

Further, at the altitude of thermosphere, atmosphere is very thin. In this zone, molecules are exposed to unattenuated solar radiation of extremely short wavelength i.e. of high energy. This radiation arises from the outer region of Sun. At wavelengths below 50 nm, effective emission temperature exceeds 10,000 K. High-energy solar protons of such wavelengths are absorbed by gas molecules giving them high transitional energies i.e. high temperatures. The energies may be large enough to dissociate oxygen and nitrogen. Temperatures in thermosphere undergo wide variations depending upon the state of Sun. During solar disturbances, output of high-energy protons is very much enhanced that results in very high atmospheric temperatures. Temperature in this zone may further be increased by another mechanism. The temperature is normally defined in terms of transitional energy but absorption and emission of radiation occur through vibrational and rotational changes. In upper atmosphere, the frequency of molecular collisions is relatively low and so exchange of translational, vibrational and rotational energies is infrequent. Hence the cooling of thermosphere by re-radiation is very inefficient. The temperature of thermosphere increases with height so it is also stable against convection. Heat can be lost only by very inefficient diffusion processes and as a result, thermospheric temperatures are extremely high.