# Environment of Earth

## March 11, 2008

### WATER BALANCE OF EARTH

Filed under: Environment — gargpk @ 1:02 pm
Tags: ,

The Earth’s hydrological cycle includes exchange of water in its various forms, between the hydrosphere, atmosphere, upper layers of lithosphere and the biosphere (living organisms). The corresponding process is described in terms of the equations of a water balance defined for each of the individual component of environment.

# Water balance of land surface

Water balance equation for land surface may be given as:

r = E + fw + G

where, r = precipitation; E = evaporation at Earth’s surface which is equal to the difference between evaporation and condensation at Earth’s surface; fw = surface runoff; G = flow of moisture from Earth’s surface to deeper layers.

## The above equation of water balance of land surface equates to zero the algebraic sum of all types of water inflows and expenditures, in solid, liquid and gaseous states entering a horizontal sector of surface from the surrounding space a specified interval of time. This equation is generally used in a slightly modified form. If water balance for a vertical column passing through the upper layers of lithosphere and reaching the depth at which moisture exchange with deeper layer stops is taken into consideration the vertical flow of water in lithosphere (G) is given as:

###### G = fp + b

where, fp = run-off within the soil; b = change in water content in upper layers of lithosphere.

Further, the full run-off i.e. run-off normal (f) is given as:

f = fp + fw

where, fp = run-off within the soil; fw = surface run-off.

Taking the equations for G and f into consideration, the water balance equation for land surface becomes:

r = E + f + b

Water balance of water bodies

The above water balance equation can also be used to calcu­late water balances of water bodies and also of individual sectors within such bodies. In such cases, f will describe the total redistribution of water along the horizontal plane during the time interval under consideration both within the water body itself and in layers of underlying soil (in cases where significant redistribution of moisture at such levels does occur). Similarly, b for a closed water body will also be equal to overall change in the water content both within the body of water itself and in underlying layers in those cases in which there are perceptible changes in moisture content. In many cases, b usually describes changes in levels. For an average period of one year, b is often quite negligible and the water balance equation becomes:

r = E + f

For land surfaces having no run-off, e.g. deserts, f = 0 and the water balance equation takes the form:

r = E

Water balance of atmosphere

At a particular place in the atmosphere, the water is brought by evaporation from the Earth’s surface and is re­moved by precipitation or air currents and horizontal turbulent exchange.

The water balance equation for atmosphere is, therefore, obtained by summing all the categories of inflow and expenditure of moisture within a vertical column passing through the atmosphere and is given as:

###### E = r + Ca + ba

where, Ca = quantity of moisture received or lost by vertical column due to air currents and horizontal turbulent exchange; ba = change in quantity of water in the column.

Since the atmosphere can retain only very small quantities of water in all its various states, value of ba is usually much less than other quantities. Its average value over a period of one year is always close to zero and the water balance equation becomes:

###### E = r + Ca

The atmosphere receives a considerable amount of moisture through transpiration of plants which usually represents several tens percent of total evaporation. Therefore, transpiration may exert a considerable influence on the atmos­pheric water exchange and hence, on the volume of precipitation. In studying this influence principles governing atmos­pheric water exchange are important. These principles have been formulated in quantitative theory of water exchange discussed below.

1. Flow of water vapor brought by air currents to a territory: For a territory whose average linear scale is L, this flow may be as­sumed to be wu where w = atmospheric moisture content in the windward side of area and u = average velocity of air flows carrying water vapor over the territory. There will be change in w along air current’s path according to the difference between precipitation (r) and evaporation (E).

2. Flow of water vapor carried away by air currents from the territory: This flow from the territory will be:

wu – (r -E)L.

3. The overall flow of water vapor carried over the territory: This is made of two subflows:

(i) Flow of external (advective) water vapor: It is the moisture originating outside the territory by evaporative processes and brought to the territory. On the windward side of area, it is equal to wu while on leeward side (as flow leaves the area’s boundaries), it will be wu – raL where, ra precipitation produced by this advective water vapor in area. On average such flow over a territory is given by:

wu – 0.5 raL

(ii) Flow of local water vapor: It is the moisture originating within the area by local evaporation. In areas having plant cover, it consists mainly of transpiration. This local flow is equal to zero on windward side and (E – rl)L on leeward side where rl = total precipitation produced from this local water vapor in the area. On average such flow over a territory is given by:

0.5(E – rl)L

Thus, the overall flow of water vapor over the territory produced by these two subflows together is given by:

wu – 0.5(r -E)L

and,

r = ra + rl

The molecules of water vapor of external origin and local origin are mixed in the atmosphere during turbulent exchange. Therefore, the ratio of total precipitation derived from external water vapor and that derived from local water vapor is equal to the ratio of the quantity of corresponding molecules of water vapor in the atmosphere. In short, it may be assumed that:

ra/rl = [wu – 0.5 raL] / [0.5(E – rl)L]

This gives two equations:

(a) ra = r / [1 + EL/2wu];

(b) rl = r / [1 + 2wu/EL]

Coefficient of water exchange (K): This is equal to the ratio of total precipitation ® to precipitation from advec­tive moisture (ra). It is determined from the equation for ra and given as:

K = r/ra = 1 + [EL/2wu]

The above equation shows that:

(i) The K depends on the factors determining water (vapor) balance in the atmosphere.

(ii) Relations in formulae for ra, rl and K depend on the size of territory under consideration. With increase in scale L, rl and K increase but ra decreases. For sufficiently territories, actual dependence of K on their size is not linear. For large territories, average rate of transfer of water vapor (u) somewhat declines due to curvature of air particle trajectories.

Contribution of local evaporation to total precipita­tion

1. Direct contribution: Data of water exchange over all the continents of Earth given by O. A. Drozdov and associates is given in the following table. The data shows that the princi­pal source of water vapor for precipitation over continents is advective water vapor, which originates as oceanic evapora­tion. Owing to large-scale transfer of oceanic water vapor in the Earth’s atmosphere, direct contribution of local evapora­tion processes from the surface of continents to the total volume of precipitation is relatively modest, especially for the territories of dimensions less than 1 million square kilometers. However, influence of local evaporation on total precipitation is not limited to changes in the components of atmospheric water exchange only but includes indirect influence also.

2. Indirect influence: The indirect influence of local evaporation on totalprecipitation derives from the linkage between the volume of precipitation and the relative humidity atmosphere.

Table-4: Atmospheric water exchange over continents.

 Continent r (sq. km /yr) ra (sq. km /yr) rl (sq. km /yr) K Europe 7540 5310 2230 1.42 Asia 33240 18360 14880 1.81 Africa 21400 15080 6330 1.42 N. America 16200 9790 6360 1.65 S. America 28400 16900 11500 1.68 Australia 3470 3040 430 1.14

Studies of Drozdov (1963) have shown that volume of total precipitation may be established by following semi-empirical formula:

r = aw f(h)

where, r = total precipitation; h = average relative humidity within atmospheric layer upto 7 km altitude; w = atmospheric moisture content; a = proportionality coefficient which is equal to 1 when total precipitation of 0.1 w per day falls for 100% h.

Empirical function f(h) is generally similar for many re­gions of middle latitudes. The formula may be employed to establish the effect of local evaporation processes on total precipitation. The values of w and h are then determined according to the influence of local evaporation on the over­all quantity of water vapor being carried over the given territory. In this connection, it becomes clear that the influence of local evaporation on total precipitation within specific limits of air humidity may be several times larger than the contribution of local evaporation to overall volume of atmospheric water vapor.

Studies of Drozdov have shown that without the large role that is often played by the indirect influence of local evaporation on precipitation, the continental regions at distance from ocean would have been transformed into deserts.

Effect of plant cover on precipitation

The precipitation in an area also depends on the plant cover which influences the quantity of local evaporation. It has been observed that when mossy swamps are drained and a forest cover develops in such area, total evaporation appears to increase. The reverse effect is observed when lowland grassy swamps are drained. It may be assumed that the replacement of various types of plant cover changes total evaporation by about 10%.

With the use of formula r = aw f(h) it is indicated that influence of evaporation on precipitation depends on the size of area in which evaporation changes. Within area of linear scale of the order of 1,000 km, the influence of fluctuations in evaporation on precipitation is negligible. However, this influence may be substantial in case of changes in evapora­tion over entire continents or large parts of continents.

It is possible that during Lower Palaeozoic era when plant cover existed on continents, the ability of upper layers of lithosphere to retain water was quite limited. Under such conditions, evaporation from land would be low and run-off high. Since this condition of water balance components existed over large areas of Earth, it exerted substan­tial influence on precipitation on continents. With appear­ance of plant cover on land, soil formation increased which caused increase in evaporation and decrease in run-off. Thus with increasing land area coming under plant cover, volume of atmospheric precipitation also increased. This in turn con­tributed to further increase in spread of plant cover within continents. Thus a positive feedback relation exists between living organisms and their environment which has contributed to spread of plant cover over much of land surface. However, this feedback relation could alter precipitation only in areas having favourable conditions of atmospheric circula­tion.

Water balance of whole Earth

If water balance of Earth as a whole is considered, the horizontal redistribution of water is of no importance and the water balance equation simplifies to:

r = E

The average yearly values of precipitation, evaporation and run-off for different continents and oceans is given in following Table-5. Features of water balance of Earth can be discussed on the basis of these values.

(1) For each continent, evaporation is equal to the differ­ence between precipitation and run-off.

(2) The ratio of evaporation to run-off differs greatly on various continents. In Australia evaporation is close to precipitation. In all other continents except Africa, evaporation is less than about 66% of the sum of precipitation.

(3) The difference between evaporation from the surface of World Ocean and precipitation is equal to the river run-off from continents into the oceans.

(4) For the individual ocean, evaporation is equal to the sum of the river run-off into it and horizontal transfer of water from other oceans through global oceanic circulation process­es. It is difficult to determine the magnitude of horizontal transfer through direct methods because it represents difference between two small values of inflow and outflow of water. Determination of both values is subject to significant er­rors. It may be simpler to estimate the exchange of water between oceans as a residual element in the water balance. The accuracy of estimates in this case is also limited.

(i) For Atlantic Ocean sum of precipitation and run-off is less than the magnitude of evaporation. It is clear that this ocean receives water from other oceans including Arctic Ocean where evaporation is substantially smaller than the sum of precipitation and river run-off.

(ii) In Indian Ocean, sum of precipitation and run-off is somewhat smaller than the magnitude of evaporation while in the Pacific Ocean, the sum of precipitation and run-off is larger than evaporation, which reflects a transfer of water into other oceans.

(5) Values of precipitation and evaporation for land and oceans indicate that for the Earth as a whole, the magnitude of precipitation over a year is 113 cm, which is equal to that of evaporation.

Table-5: Precipitation, evaporation and run-off on continents & oceans.

 Continents/Ocean Precipitation (cm/year) Evaporation (cm/year) Run-off (cm/year) Europe 77 49 28 Asia 63 37 26 Africa 72 58 14 N. America 80 47 33 S. America 160 94 66 Australia 45 41 4 All land 80 48.5 31.5 Atlantic 101 136 23 Pacific 146 151 8 Indian 132 142 8 World ocean 127 140 13

(6) Calculations of the values of precipitation and evapora­tion for various latitudinal zones show that the inflow of water vapor into the atmosphere from evaporation may be both larger and smaller in different latitudinal zones than its expenditure on precipitation. The source of water vapor in atmosphere is provided by high-pressure zones where evapora­tion is much more than precipitation. That surplus water is expended in zones adjacent to the Equator and, also at middle and high latitudes where precipitation is much more than evaporation.

(7) The difference between precipitation and evaporation is also equal to the difference between the inflow of water vapor into the atmosphere and the outflow resulting from the horizontal air movement. The large difference in these two in many regions indicates the importance of the transfer of atmos­pheric water vapor in the formation of precipitation. The influence of transfer of water vapor on the volume of precipitation has been examined in the studies of hydrological cycle in the atmosphere.

(8) The components of continental and oceanic water balance are not constant and change as a result of climatic fluctua­tions and other factors. Since changes in the components of water balance over a year are small in comparison with their absolute values, they may be neglected in determining the corresponding magnitudes. On the other hand, data on changes in water balance components may be very important in the studies of the evolution of hydrosphere.

(9) During glaciation periods, large masses of water used up in formation of continental ice sheets and volume of water in the World Ocean had changed. At the termination of last glaciation, approximately 20 thousand years ago, the level of World Ocean was lower by about 100 meters than it is today. Subsequently the level gradually increased and reached the present level about 5,000 years ago and has not significantly changed since then.

(10) Observations have shown that in the 20th century, level of World Ocean has increased by about 15 cm while reserves of subsoil water on land and volume of water in many lakes have declined. Though sources of water contributing to rise in water level in World Ocean are not absolutely clear, melting of ice covers, loss of groundwater and decrease of water in lakes on land are responsible for the increase in volume of oceanic water.

# (a) Run-off normal (f): It is the volume of water flowing off on the average during a year from a unit of land surface in form of various horizontal flows;

(b) Coefficient of run-off (f/r): It is the ratio of run- normal to total yearly precipitation.

(c) Radiation index of dryness (R/Lr): It is the ratio of the total radiation energy received to total energy expended on precipitation.

Evaporation is one the major processes of transformation of solar energy at Earth’s surface and it very much affects the yearly run-off. Thus, run-off normal and coefficient of run-off are linked to principal components of energy balance. Following general considerations apply to relation between elements of energy and water balance on land:

(i) Average total evaporation from land surface (E) depends on the quantity of precipitation r and inflow of solar energy. Evaporation increases with increase in and radiation balance r. If soil is dry, total precipitation is caught by molecular forces on soil particles and eventually expended on evaporation. In such conditions (e.g. in deserts), run-off coefficient (f/r) approaches zero.

Since the average dryness of soil increases when inflow of radiation energy increases and amount of precipitation decreases. Thus:

f/r —> 0 or E/r —-> 1 when R/Lr —->

(ii) With decrease in value of R/Lr, value of E/ralso declines and certain run-off appears. For sufficiently high value of total precipitation and sufficiently small value of total inflow of radiation, a state of full moistening of the upper layers of soil will be achieved. In such cases, the maximum portion of heat energy available from all sources will be expended on evaporation. The value of this expenditure may be calculated by considering the valve nature of the latent heat exchange between underlying surface and the atmosphere.

# LE  R when R/Lr  0

Linkage equation for water and energy (heat) balance: The relation between water balance and energy (heat) balance of the land may be described by the equation:

E/r = (R/Lr)

The form of the function for R/Lr 0 and for R/r is determined by respective conditions considered above in (i) and (ii).

The most important consideration related to this equation is that the magnitude of possible evaporation at a locality is determined by the radiation balance corresponding to the conditions of sufficient moisture at which vegetation can exist. The values of radiation balance at a locality differ in conditions of sufficient and insufficient moisture. The reasons for this variation may be summarized as below:

(a) In moist regions of humid climate, albedo of Earth’s surface shows little change with change in humidity. The average difference between temperatures of Earth’s surface and air is relatively small in the presence of sufficient moisture. Therefore, magnitude of evaporation may be approximately inferred from the radiation balance corresponding to the actual state of Earth’s surface.

1. In regions of dry climate, albedo and temperature surface changes with increasing humidity. The Earth’s surface temperature approaches the air temperature in condition of sufficient moisture. Evidently, the magnitude of evaporation has to be determined from the value of radiation balance corresponding to the albedo of a moist surface and to to a temperature at Earth’s surface that is equal to air temperature.

Comparison of R/Lr and E/r values for observed data of river basins on different continents was made was made (Fig. 18). This has allowed verification of the important considerations concerning relations between E/r and R/Lr for both small and large values of R/Lr, The ratio E/r for low values of R/r is represented as straight line OA according to condition given in (ii). Similarly, the ratio E/r for large values of R/Lr is represented as straight line AB according to condition given in (i). The experimental points representing values of E/r are calculated from water balance data by averaging E/r for specific intervals of R/Lr. The experimental points clearly show a smooth transition from line OA to line OB, which as expected, are limiting values for the relation between E/r and R/Lr.

## Important features of linkage equation

(a) The figure shows that the relationship on which linking of energy and water balances is based is largely determined by two limiting conditions. One condition corresponds to the valve mechanism of turbulent heat exchange in the layer adjoining Earth while second condition corresponds obviously the small value of coefficient of run-off (f/r) in dry climates. The corresponding relation for most of the interval of changes in the parameters of the linkage equation remains close to a boundary condition. Therefore, selection of one or the other interpolation function for passing from first to second condition is not particularly significant.

Thus despite semi-empirical nature of the linkage equation it can be justified mainly with the general considerations. Further, this equation represents a supplementary relationship, which is independent of the energy balance and water balance equations.

(b) The above linkage relation may be analytically represent­ed by employing the following equation which represents the relation between average yearly evaporation and the radiation balance:

E = /Rr/L th Lr/R (1 – ch R/Lr + sh R/Lr)

where, th is a function for hyperbolic tangent, ch and sh are hyperbolic cosine and sine respectively.

Since run-off normal (f) is equal to difference between precipitation and evaporation (f = r – E), equations representing f and f/r in terms of above equation will be:

f = r – /Rr/L th Lr/R (1 – ch R/Lr + sh R/Lr)

f/r = 1 – /R/L th Lr/R (1 – ch R/Lr + sh R/Lr)

These relations between components of energy and water balance have been verified in a number of studies.

(c) The linkage equation makes it possible to represent the relation of run-off and evaporation to total yearly precipitation and the radiation balance in a general form. The regularity explains a number of empirical relations between run-off and precipitation found in various studies.

(d) The linkage equation also makes possible to express dependence of run-off on precipitation for values of radia­tion balance corresponding to various localities. Curve A in figure represents the relation of run-off normal and yearly precipitation calculated for average conditions in European lowlands using the equation. Curve B represents the empirical relation found by H. Keller (1906) on the basis of observation at West European rivers and curve C represents the empirical relation found by D. L. Sokolovsky (1936) on the basis of observation on East European rivers. Correspondence of empirical curves B and C with calculated curve A verifies the universal nature of the linkage equation.

(e) In calculation of average empirical dependence of run-off on precipitation in data from various regions, considerable dispersion in the dependence has been observed. The linkage equation makes it possible to explain this dispersion. There is substantial variability of radiation balance in middle latitudes. This variability causes the run-off of basins with large radiation balance (in more southerly regions) to be much smaller than that of basins with low radiation balance (in more northerly regions) for equal total precipitation values. This also influences the rate of change in run-off corresponding to increases in precipitation (df/dr). The linkage equation indicates that this rate should be larger in northern basins than in southern basins. Empirical data amply confirms this prediction of linkage equation.

(f) The calculations of run-off based on linkage equation correspond with the observed data. This confirms the decisive role of climatic factors, particularly of energy factors, in total yearly run-off in large river basins with areas comparable with geographical zones. In case of small areas river run-off may change substantially as a result of local conditions of non-climatic nature.