A Derivation of Radiative Equilibrium Model Equations

Transcription

1 MER/OCN 465: Computer Programming with Applications in Meteorology and Oceanography Dr. Dave Dempsey Dept. of Geosciences, FU A Derivation of Radiative Equilibrium Model Equations Underlying Assumptions Our radiative equilibrium model is based on the principle of conservation of energy, which can be written as: Rate at which an object s = (um of the object s (um of its sinks heat content changes w/r/t time sources of heat) of heat) his equation represents a budget of heat for any particular object. In this relation, any energy source is expressed as the rate at which the object gains heat by some mechanism, such as conduction; absorption of radiative energy; or transformation of latent heat into sensible heat when water freezes, condenses, or deposits inside the object. An energy sink is expressed as the rate at which the object loses heat by some mechanism, such as conduction; emission of radiative energy; or transformation of sensible heat into latent heat when water melts, evaporates, or sublimates. If the sum of heat sources equals the sum of the sinks, then the heat budget is balanced, and the object s heat content is at equilibrium that is, not changing with time. In our case, we apply the conservation of energy to each individual part of an earth system consisting of layers of air in the atmosphere that encircle the globe (forming stacked, spherical shells) and to the earth s surface (the surface of a sphere). o simplify our model while retaining enough significant physics for it still to offer us physical insight, we make two serious assumptions: 1. he heat budget of each part of the system is at equilibrium. 2. Absorption and emission of radiative energy are the only mechanisms by which the atmospheric layers and earth s surface gain and lose heat. hese assumptions allow us to write the conservation of energy for each atmospheric layer and for the surface as: (1) 0 = (um of the rates at which a (Rate at which the layer layer absorbs radiative energy emits radiative energy) emitted by various objects)

2 he Radiative Energy ink By assumption, the only sink of heat for each layer is the emission of radiative energy. If we represent the temperature of any particular layer using a single (globally averaged) value, then we can relate the layer s global- average radiative emission flux (E) to its absolute temperature () by applying the tefan- Boltzmann law: (2) E = εσ 4 where σ is the tefan- Boltzmann constant and ε is the layer s emissivity (integrated over the range of wavelengths that the object emits in our case, mostly longwave infrared (LWIR), also called terrestrial, radiation). he emissivity is a fraction between 0 and 1. he rate at which an atmospheric layer emits radiative energy is simply its emission flux multiplied by its total surface area (Atot). ince an atmospheric layer emits both upward from its top side and downward from its bottom side, the total surface area must account for both. ince we are treating each layer as a spherical shell encircling the earth, and each shell is quite thin (because the atmosphere itself is quite thin compared to the radius of the earth), the top and bottom of each have about the same surface area. Hence, the total surface area is approximately Atot = 2 A = 2 4πRe 2, where A is the surface area of the top or bottom of a layer and Re is the radius of the earth. he radiative emission rate of any atmospheric layer is therefore: (3.1) Atot E = (2 A) Ε = (2 A) εσ 4 = (2 4πRe 2 ) εσ 4 In contrast, the earth s surface has only one side, which has a total area of A=4πR e2, so the total rate at which it loses heat by radiative emission is given by: (3.2) A Es = A εsσs 4 = (4πRe 2 ) εsσs 4 where Es is the global- average radiative emission flux from the surface, εs is the emissivity of the surface (averaged over longwave infrared wavelengths), and s is the (global average) surface temperature.

3 ources of Radiative Energy Any particular atmospheric layer (or the surface) call it the nth layer absorbs radiative energy that is emitted from either (a) the sun, or (b) other parts of our earth system. he rate at which the nth layer absorbs energy from one of these sources depends on several factors. If the source is the sun, the absorption rate depends on the following factors: (.1) he rate at which solar energy strikes the earth at the top of the atmosphere. (.2) he fraction of the energy that transmits through intervening layers 1 through n- 1. (.3) he fraction of the remaining, transmitted energy that the nth layer absorbs. If the source is terrestrial radiation emitted by another layer or the surface (call it the mth layer), the absorption rate depends on these factors: (.1) he rate at which the mth layer emits terrestrial radiation toward the nth layer. (.2) he fraction of the emitted energy that is transmitted through intervening layers. (.3) he fraction of the remaining, transmitted energy that the nth layer absorbs. First consider the factor (.1). At the top of the atmosphere directly facing the sun, the flux of solar radiative energy () is the rate at which solar radiative energy passes through a unit surface area directly facing the sun. Because the earth s orbit is slightly elliptical, the distance between the earth and the sun varies by ±1.5% or so and varies by a modest amount accordingly. he average value of over the course of a year, the solar constant (0), is about 1370 W/m 2. For simplicity, we ll assume that 0. Because the earth is a sphere, at any particular time there is only one place where a horizontal surface at the top of the atmosphere directly faces the sun, so we can t calculate the total rate at which solar energy strikes the earth by multiplying the solar constant by the earth s surface area, or even by half the earth s surface area (taking into account the fact that the sun illuminates only half of the earth at any time). Rather, we need to recognize that the earth intercepts solar radiation at exactly the same rate as would a projection of the earth onto a plane directly facing the sun. he area of this projection is just the area of a circle with radius equal to the earth s radius.

4 his area is just the cross- sectional area of the earth, Axs = πre 2. Hence, the rate at which solar radiation strikes the earth is: (.4.1) Axs = (πre 2 ) olar energy arriving at the top of the atmosphere passes through a succession of atmospheric layers, each of which absorbs and reflects some of it and transmits the rest. he rate at which solar energy passes through (that is, is transmitted by) any particular layer is simply the product of the transmissivity of that layer and the rate at which energy strikes the layer. he rate at which solar energy is transmitted by a series of n- 1 layers is the product of the transmissivities of those layers, τi, and the rate at which energy strikes the first (topmost) one as given by (.4.1): (.4.2) $ & % n"1! i # i=1 ' ) [Axs ] = &#! i ) [(πre 2 ) ] ( i=1 (where the symbol Π is a shorthand representing a product, just as Σ is a shorthand representing a sum). Finally, the rate at which the nth layer absorbs solar energy is the product of the solar absorptivity of the nth layer, an, and the rate at which solar energy strikes the nth layer as given by (.4.2): (.4.3) an &#! i ) [Axs ] = an &#! i ) [(πre 2 ) ] i=1 (Note that when n = 1, the topmost layer, there are no intervening layers, and the index, i, on the product term starts at 1 and ends at n- 1 = 0. In such cases the product is defined to be 1, and there are no transmissivities in the expression.) (o simplify things in this model, we ll assume that all solar radiation reflected by a layer passes through the layers above and out to space unaffected. ) Now consider the rate at which the nth layer absorbs terrestrial radiation emitted by the mth layer. For simplicity, assume that the mth layer lies above the nth layer (so that m < n). We start with the factor (.1), which is just the rate at which the mth layer emits radiative energy from the side facing the nth layer (in this case, downward): i=1 (.4.1) A Em = (4πRe 2 ) Em = (4πRe 2 ) εmσm 4

5 errestrial radiation emitted from the mth layer toward the nth layer passes through a succession of atmospheric layers, each of which might absorb some of it and transmit the rest (because none is reflected). he rate at which terrestrial radiation passes through (that is, is transmitted by) any particular layer is simply the product of the transmissivity of the layer and the rate at which terrestrial radiation strikes the layer. he rate at which terrestrial radiation is transmitted by a series of n- m- 1 layers (the number of layers between the nth and mth layers, from layer m+1 through layer n- 1) is the product of the transmissivities of those layers, τi, and the rate at which the mth layer emits terrestrial radiation downward as given by (.4.1): (.4.2) $ & % n"1! i # i= m +1 ' ) [A Em] = &#! i ) [A εmσm 4 ] ( % i= m +1 ( = &#! i ) [(4πRe 2 ) εmσm 4 ] i= m +1 Finally, the rate at which the nth layer absorbs the terrestrial radiation striking it from the mth layer is the product of the terrestrial absorptivity of the nth layer, an, and the rate at which terrestrial radiation from the mth layer strikes the nth layer as given by (.4.2): (.4.3.1) an &#! i ) [A Em] = an &#! i ) [A εmσm 4 ] % i= m +1 ( % i= m +1 ( = an &#! i ) [(4πRe 2 ) εmσm 4 ] (Note that when n = m+1 there are no intervening layers, and the index (i) on the product term starts at m+1 and ends at n- 1 = m+1-1 = m, which is less than where it started. In such cases the product is defined to be 1, and there are no transmissivities in the expression.) If the mth layer lies below the nth layer (so that m > n), then (.4.3.1) is: (.4.3.2) an m!1 # & %"! i ( [A Em] = $ m"1 ' $ i=n!1 ' an &#! i ) [A εmσm 4 ] % i= n"1 ( $ m"1 ' = an &#! i ) [(4πRe 2 ) εmσm 4 ] i= m +1 i= n"1

6 he Radiative Equilibrium Equations ubstituting Eq. (3.1) for each atmospheric layer, or Eq.(3.2) for the surface, together with Eqs.(.4.3), (.4.3.1), and (.4.3.2), into the right- hand side of Eq.(1), gives us Eqs.(4), a set of radiative equilibrium equations, one for each of N atmospheric layers plus the surface (where subscript s for surface has been replaced by subscript N+1). he layers are numbered, starting at the topmost layer, from 1 to N+1, and all quantities are subscripted accordingly. Rewriting the Equations for olution by Gaussian Elimination If we interpret i 4 as the unknowns in Eqs.(4), then Eqs.(4) constitute a set of simultaneous, linear equations with constant coefficients. he constant coefficients that multiply the unknowns ( i 4 ) are determined by the tefan- Boltzmann constant (! ), the surface area of the earth (A), and radiative parameters for LWIR radiation of the various layers in particular, the absorptivities ( a i ), transmissivities (! i ), and emissivities (! i ) for LWIR radiation. he nonhomogeneous (solar absorption) terms depend on the flux of solar radiative energy at the top of the atmosphere (), the cross- sectional area of the earth (Axs), and radiative parameters for solar radiation of the various layers in particular, the absorptivities ( a i ) and transmissivities (! i ) for solar radiation. However, Eqs.(4) are not written in a way that is easiest to interpret as simultaneous, linear equations amenable to solution by Gaussian elimination. o do that, we want to take the following steps: (1) Leave the nonhomogeneous (solar absorption) terms on the right- hand side of Eqs.(4). (2) Move all terms that contain i4 on the right- hand side of Eqs.(4) (which describe emission and absorption fluxes of LWIR radiation) to the left- hand side. (3) Note that the temperature of each layer (N+1 total) is represented among the N+1 terms on the left- hand side of each of the equations in Eqs.(4). Reorganize the terms so that the first column contains all of the terms with the temperature of the first layer (1) in them, the second column contains all of the terms with 2 in them and in general the nth column contains all of the terms with n in them. (4) Furthermore, we note that the surface area of the earth (A) and the tefan- Boltzmann constant (! ) are common factors in every term on the left- hand side, and the cross- sectional area of the earth, Axs, a common factor in the nonhomegeous terms on the right- hand side, is equal to A/4. Hence, we can simplify the equations somewhat by dividing all of the equations by A!.

7 aking steps (1) through (4) above results in Eqs.(5). We can prepare Eqs.(5) equations further for solution by Gaussian elimination as follows: (1) Kirchoff s Law states that for any particular wavelength, a good absorber is also a good emitter. More specifically, an object s absorptivity and emissivity at any particular wavelength are equal. For simplicity, in our model we treat LWIR wavelengths as one group of wavelengths (without making distinctions within the group) and solar radiation as a second group (again making no distinctions within the group), and the layers emit essentially nothing but LWIR radiation, so it follows that! n = a n for each layer, n. We will make this substitution in the equations. (2) From the Principle of Conservation of Energy, we can say that for any layer n, a n +! n +! n = 1 (where! is the albedo of the layer for any particular wavelength), and this applies to each wavelength separately. From this relationship we can deduce the following: a. ince nothing on earth reflects LWIR radiation, if follows that! n = 0, and so we can write! n = 1! a n. hat is, for LWIR radiation, the only radiative parameter that we have to specify for any layer is the absorptivity. (From that we can calculate the LWIR transmissivity of the layer.) b. For the earth s surface in particular (layer N+1), we can go even further. ince the surface is not transparent to LWIR radiation at all (! N+1 = 0 ), we can simply set a N+1 = 1. c. he earth s surface is not transparent to solar radiation, either (! N+1 = 0 ), so we can write a N+1 = 1!! N+1. hat is, the only radiative parameter for solar radiation that we have to specify at the earth s surface is the albedo. (From that we can calculate the solar absorptivity of the surface.) d. Finally, for atmospheric layers, we can write! n = 1! a n!" n. hat is, for the atmospheric layers, the only radiative parameters for solar radiation that we have to specify are the solar absorptivity and the albedo. (From those we can calculate the solar transmissivity of each layer.)

8 (3) From the point of view of the Gaussian elimination algorithm, the constant coefficients determine the solution, while the unknowns i 4 ( ) act merely as placeholders. imilarly, the + and = signs play no active role in the solution, once the coefficients are organized properly. Hence, we will drop these components of the equations, which simplifies them visually. Applying (1) (3) above to Eqs.(5) produces Eqs.(6). (he non- homogeneous coefficients in column N+1 are segregated from the other coefficients for visual convenience.) Once the constant coefficients in Eqs.(6) are calculated, the equations are now in a form that we can solve efficiently using Gaussian elimination. ome observations about the coefficients in Eqs.(6): (1) he main diagonal terms are all of the form 2a k (k=1,n), while the last one is simply a N+1. hese terms originated as radiative emission fluxes, and since each atmospheric layer emits from both top and bottom while the surface only emits from one side, the atmospheric layers had the factor of 2 multiplying them while the surface emission flux term didn t. (2) he terms off of the main diagonal originated as LWIR radiative absorption fluxes. Each depended in part on (a) the absorptivity of the absorbing layer; and (b) the emissivity of the emitting layer, which is equal to the absorptivity of the emitting layer (according to Kirchoff s Law). ince each row of the coefficient matrix represents an absorbing layer and each column represents an emitting layer, each coefficient off of the main diagonal contains the absorptivity for the row in which it lies (representing the layer doing the absorbing) and the absorptivity of the column in which it lies (representing the layer doing the emitting). hese two absorptivity factors act like bookends around the product of the transmissivities of the layers lying between the two layers (if any). he transmissivity of each intervening layer, i, has been replaced by 1! a i. he farther any given coefficient lies from the main diagonal (either vertically or horizontally the distance is the same), the more layers lie between the emitting layer (the column) and the absorbing layer (the row), and hence the more transmissivities appear in the product. he terms immediately next to the main diagonal represent absorption by one layer of LWIR radiation emitted by the layer next to it, so there are no intervening layers and hence no transmissivities in the corresponding coefficient.

9 Now, since the each absorbing layer is also an emitting layer, and the same set of layers lie between the each pair of absorbing and emitting layers regardless of which one we consider to be the absorbing and which the emitting layer, we should not be surprised to discover that the coefficient j!1 Ak,j =!a k " 1! a i=k+1( i ) a j (coefficient for absorption by the kth layer of LWIR radiation emitted by the jth layer) is the same as the coefficient k!1 Aj,k =!a j " 1! a i i= j+1( ) a k (coefficient for absorption by the jth layer of LWIR radiation emitted by the kth layer). (Note that the coefficients themselves are not the radiative absorption and emission fluxes; those also involve the absolute 4 i, as well as the tefan- Boltzmann constant.) (3) ince Aj,k and Ak,j are located symmetrically across the main diagonal from each other (in a direction perpendicular to the main diagonal), it follows that this matrix is symmetric (that is, Ak,j = Aj,k for k, j = 1, N). hat is, the coefficients above the main diagonal are mirror images across the main diagonal of the coefficients below the main diagonal. We take advantage of this when computing the coefficients: we compute the coefficients on one side of the main diagonal and simply set the coefficients on the other side equal to them. (4) Each solar absorption coefficient (in column N+1) is closely related to the one above it- - the only differences are (a) the solar absorptivity (the leading factor) and (b) an additional solar transmissivity factor (because solar radiation must pass through one more layer than the layer above, namely the layer above). ince computing any particular coefficient goes a long way toward computing the next one, we can take advantage of this to make our coefficient- calculating code more efficient.