Radiative Equilibrium Models. Solar radiation reflected by the earth back to space. Solar radiation absorbed by the earth

Transcription

1 I. The arth as a Whole (Atmosphere and Surface Treated as One Layer) Longwave infrared (LWIR) radiation earth to space by the earth back to space Incoming solar radiation Top of the Solar radiation absorbed by the earth Atmosphere and the earth s surface treated as one layer The equilibrium (balanced) heat budget equation for the earth as a whole (from the Law of Conservation of nergy) can be written (where T! global, long-term average absolute temperature of the earth): DT Dt! "T "t! 0 # (rate of solar absorption by the earth) (rate of LWIR radiative emission by the earth to space) ( )! 4! R 2! or 0! a S!! R 2! S 0 where: S 0! average intensity (flux) of solar radiation arriving at the top of the earth s on a surface directly facing the sun (the solar constant) =1368 W/m 2 R! the radius of the earth;! R 2 is the effective area that the earth presents to the direct rays of the sun; and 4! R 2 is the total surface area of the earth a S! solar absorptivity of the earth (the fraction of incoming solar radiation that the earth absorbs) 1

2 We can modify the equilibrium heat budget equation for the earth as a whole in two ways. First, we recognize that the earth is not at all transparent, so that all solar radiation striking the earth is either reflected back to space or absorbed. If the albedo of the planet is!, then it follows that a S = 1!!. Second, we can relate the radiative emission flux of the earth to its absolute temperature, T, using the Stefan-Boltzmann Law: =! "T 4 (where!! emissivity of the earth, which is the fraction of perfect blackbody emission that the earth actually emits; and!! Stefan-Boltzmann constant, a physical constant with a value of (W/m 2 )/K 4 ). Substituting these two new relations into the long-term, global average (approximately balanced) heat budget equation gives: 0! ( 1"! ) # (! R 2 # S 0 ) " 4! R 2 4 # (!!T ) We can divide the entire equation by the surface area of the earth, 4! R 2, to eliminate the size of the earth from the relation entirely: 4 0! ( 1"! ) # ( S 0 4) " " #T Since we know that S 0 =1368 W/m 2, and a S = 1!!! = 0.7, and!! 1 for the earth as a whole, and! = (W/m 2 )/K 4, then we can solve this equation for what the long-term, global average absolute temperature of the planet as a whole (T ) would have to be so that the planet s heat budget is balanced. It does not depend on the size of the earth only on the solar constant (which depends mostly on the distance between the planet and the sun and a little on solar output, which varies by very small amounts), and on the absorptivity of the planet (which depends on the albedo, or reflectivity, of the planet, which varies mostly with the extent of cloudiness in the ). 2

3 II. The Atmosphere and the arth s Surface Treated as Separate Layers surface directly to space to space by the to space by the surface to space Incoming solar radiation Top of the The ( Layer #1) LWIR from the surface, absorbed by the Solar radiation absorbed by the earth s surface emitted downward by the, absorbed by the surface Solar radiation absorbed by the surface The earth s surface ( Layer #2) 3

4 The equilibrium (balanced) heat budget equations for the and for the earth s surface (ignoring non-radiative exchanges of heat between the surface and the, namely (a) conduction and (b) evaporation from the surface followed by condensation to form clouds in the ): Atmosphere ( Layer 1 ): 0! (rate of solar absorption) + (rate of absorption of LWIR (rate of LWIR emission surface ) by the ) or 0! a 1 S!! R 2! S 0 ( ) + a T 1! 4! R 2 4 (! " 2 #T 2 ) 2! 4! R 2 4 (! " 1 #T 1 ) where a S! solar absorptivity, T! absolute temperature,!! emissivity, a T! terrestrial (LWIR) absorptivity, and the subscripts 1 and 2 refer to Layer #1 (the ) and Layer #2 (the earths surface), respectively. The factor of 2 in the last term accounts for the emission of by the both upward and downward. The Surface ( Layer #2 ): 0! (rate of solar + (rate of absorption of LWIR (rate of LWIR emission absorption) emitted downward by the ) by the surface) or 0! a 2 S!! 1 S! " R 2! S 0 ( ) + a T 2! 4! R 2 4 (! " 1 #T 1 ) 4! R 2 4! " 2 #T 2 where! S! solar transmissivity (the fraction of radiation striking a layer that the layer transmits through, unaffected), and subscripts 1 and 2 refer to Layer #1 (the ) and Layer #2 (the earths surface), respectively. 4

5 As with the simpler, one-layer model for the earth as a whole, we can divide each of the two equations above by the surface area of the earth, 4! R 2, to eliminate the size of the earth from the relations entirely: and 0! a S 1 " ( S 0 4) + a T 1 "! 2 "T # 2 "! 1 "T 1 0! a S 2 "! S 1 " ( S 0 4) + a T 2 "! 1 "T #! 2 "T 2 If we know the physical parameters of the two layers (the solar and terrestrial absorptivities, the emissivities, and the solar transmissivity) and the physical constants (Stefan-Boltzmann constant and the solar constant), then these equations represent a pair of simultaneous, linear equations with constant coefficients in the variables T 1 4 and T 2 4. We can solve for these, take the fourth root of each, and get the absolute temperatures that each layer must have if their heat budgets are to be radiatively balanced. 5