## Mass budget

Let us start by deriving expressions describing the mass budget in separate subsystems of the climatic system. Define the mass mA of the atmospheric column with unit cross-section as m4

where pA is the air density, ps is the surface atmospheric pressure, £ is the height of the underlying surface elevations, g is gravity and z is the vertical coordinate oriented upward.

Let us integrate the continuity equation over the whole thickness of the atmosphere, and take advantage of the condition by which the vertical mass flux approaches zero at a sufficiently large height in the atmosphere, and also of the kinematic condition: w = dc,/dt + (vV)C — (P — E)/p/li at the underlying surface (here w and v are the vertical component and horizontal vector of wind velocity, P is the precipitation rate, £ is the evaporation rate, V is the gradient operator on the surface of a sphere with radius a and t is time).

As a result we have

(Iffl

ot where vA is the vertically averaged horizontal vector of wind velocity.

Similarly, the conservation equations for water mass in the ocean, as well as those for sea ice and land substance, become

(JTH

ot uTTl

ot where the first terms on the left-hand side represent the rate of change of mass m referring to a unit of surface area; the second terms represent the horizontal divergence of the integral mass transport; vQ, v, and vL represent the horizontal vectors of current velocity in the ocean, and of sea ice drift, and of river and ground water movement; £, is the rate of sea ice formation (or melting); the symbol A signifies mass averaging; and subscripts O, I and L refer to the ocean, sea ice and land.

We note that the integral mass transport mLvL within the limits of the land is realized solely by the river and ground waters (river and underground run-off); the remaining part of the land, according to the condition vL = 0, does not contribute to mLvL.

Let us integrate Equation (2.2.1) over the surface of the entire Earth, Equation (2.2.3) over the surface of the sea ice, and Equation (2.2.4) over the surface of the land. After that we designate the total (river + underground) run-off into the ocean, relating to unit surface area, as {£R}. Then, taking into account the continuity of the integral mass transport at the land-ocean boundary, we obtain

| K}/l - {P - E}fh + {£r}/l = 0, ot where /G = sQ/s, /, = s,/s, /L = sL/s are the ratios of the ocean area (sG), sea ice area (s,) and land area (sL) to the area s of the Earth's surface (or, otherwise, the ocean, sea ice and land fractions) interrelated as fQ + ft + fL = 1; braces signify area averaging over the entire Earth's area.

Adding these expressions we obtain f- (M + {mo}fo + W/i + K}/l) = o. (2.2.5)

Equation (2.2.5) describes the obvious fact that for time scales which are much smaller than the lifetime of the continental ice sheets (for such time scales the continental ice sheets may be considered as part of the land) the mass of the atmosphere-ocean-sea-ice-land system remains constant in time.

Let us turn back to Equation (2.2.1) and integrate it over longitude. As a result we obtain

2na cos <p — [mA] H---MmTA = — 2na cos <p([P] — [£])> (2.2.6)

dt a d(p where

mAvAa cos cp dX

o is the meridional mass transport in the atmosphere, vA is the vertically averaged meridional component of wind velocity, <p is the latitude, X is the longitude, square brackets signify zonal averaging.

Let us remember here that atmospheric mass is determined unambiguously by the value of surface atmospheric pressure and that, according to Oort (1983), the annual mean surface atmospheric pressures in the Northern and Southern Hemispheres are 983.6 and 988 hPa (or 2.569 x 1018 and 2.581 x 1018 kg). This suggests that an annual mean mass transport from the Southern to the Northern Hemisphere must exist, and this transport must be controlled by the different intensity of hydrologic cycles in both hemispheres (see Equation (2.2.6)). But the existence of atmospheric mass transport from the Southern Hemisphere to the Northern Hemisphere presupposes a reverse compensational mass transport in other subsystems of the climatic system. Figure 2.1 Seasonal variations of the difference between actual (not referred to the sea level) surface atmospheric pressure and its annual mean value, according to Oort (1983): (1) the Northern Hemisphere; (2) the Southern Hemisphere; (3) the Earth as a whole.

Figure 2.1 Seasonal variations of the difference between actual (not referred to the sea level) surface atmospheric pressure and its annual mean value, according to Oort (1983): (1) the Northern Hemisphere; (2) the Southern Hemisphere; (3) the Earth as a whole.

We postpone further discussion of this subject until Section 2.4 and direct our attention here to Figure 2.1, which shows variations in the difference between actual (not reduced to sea level) surface atmospheric pressure and its annual mean value. One can see that atmospheric mass undergoes distinct seasonal oscillations and these oscillations are out of phase for the Southern and Northern Hemispheres. This last fact is usually associated with mass transport from the summer to the winter hemisphere (see Oort, 1983). Thus, it is implicitly anticipated that sources and sinks of mass on the right-hand side of Equation (2.2.6) counterbalance each other separately in each hemisphere, or because they are too small they can be ignored (excluded from the kinematic condition at the underlying surface, see above). As applied to the annual mean conditions, this assumption is equivalent to an absence of mass transport across the equator, and, hence, to the equality of surface atmospheric pressures in both hemispheres. The controversy surrounding this point is obvious.

The primary source of energy for the climatic system is an insolation determined by the so-called solar constant (flux of solar radiation at an average distance of the Earth from the Sun; the most probable value of this flux is in the range from 1368 to 1377 W/m2), as well as by the Earth's axis tilt, by the orbit eccentricity and by the longitude of perigee. Variations in the last three astronomical parameters have time scales of the order of tens of thousands of years (see above); that is why they have no real effect on changes in the climatic system on the time scales of decades that we are interested in.

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