Physics of Climatology

A viable simplified climate model using a three-layer representation of the earth is described. The distinct layers correspond to the earth’s surface, lower atmosphere, and upper atmosphere. By considering the fact that any inequality of inward and outward energy fluxes would adjust to equilibrium according to blackbody laws, it is possible to generate layer-centric equations of climate variables. Accounting for incoming radiative fluxes in the UV spectrum and emitted radiative fluxes in the IR spectrum, most of the thermodynamics can be accounted for. With the inclusion of a parameter for nonradiative flux, a sufficient model is formed. By mapping these variables to an 864-cell grid, the computational power of Matlab can be quickly used to produce significant results. This model has the merits of being mathematically undemanding and potentially very accurate, thus justifying its appeal. Introduction During the course of the semester, I worked toward the development of an effective climate model. Using accepted and observed quantities, in conjunction with equilibrium and thermodynamics equations, the interrelationships of climactic variables were analyzed for consistency with the model. The procedure I took started with the establishment of a variable set, which was then followed by an investigation of the preestablished two-layer model, and then a progression into a three-layer model with twopart averaging for trends. Two-Layer Model The two-layer model strives to represent the thermodynamics of earth using distinct layers representing earth’s surface and its atmosphere. The fluxes concerned are shown in the diagram below. Incoming radiation from the sun in the UV spectrum (So) is distributed to the atmosphere and the surface, with some being reflected back into outer space. The IR blackbody radiation of the atmosphere is emitted equally into outer space and onto the surface according to the product of its ideal flux (Sa) and the emmissivity (ε – denoted e in diagram). Similarly, the IR blackbody radiation of the surface is transmitted to the atmosphere according to the product of its ideal flux (Se) and the emmissivity. The non-radiative flux (Snr) caused by conduction and convection exchanges energy between the surface and the atmosphere and is taken to be upward since most regions have this characteristic. By examining the equilibrium condition with the fluxes outlined on the above diagram, Schimdt-Helfer-Knox arrived at the following equations in “Development of an elementary climate model. II. Two-Layer cellular case”. (1a) 2εSa – εSe = ASo + Snr (1b) –εSa + Se = BSo Snr After acquiring the observed variables of surface temperature, surface reflectivity, and cloud cover, with the addition of global constants, the value of the remaining variables can be determined by the flux equilibrium equations. The temperature of the atmosphere is now calculated for each cell. Equations 1a and 1b are added to cancel the less-absolute non-radiative flux parameter: ASo+BSo=(1-ε)Se+εSa, and solving for Sa yields: Sa = [ASo + BSo – Se + εSe] / ε Since all the parameters on the right hand side are already determined, the global average, weighted for cell areas, turns out to be 251.887 W/m. In the process of computing the two-layer parameters, a significant discrepancy was found between the standard model number of paper 1 and the calculated value for the fraction of UV solar insolation absorbed by the atmosphere (denoted A). The value calculated from observed variables and flux equilibrium equations was about half of what the standard model predicted (0.0337 vs. 0.0664). This difference could potentially propagate through other calculations in an effectual manner when this parameter is not being added to a larger value. Three-Layer Model The three-layer model extends the two-layer model by distinguishing between the lower and upper atmospheres. These two parts are appropriately separated because the difference in temperature caused by the proximity to the surface or outer space has a large impact on their blackbody characteristics. The basic variables from the two-layer model carry over, but the new lower atmosphere flux (Sb) is unknown and the non-radiative flux parameter (Snr) is challenged anew. By examining the equilibrium condition with the fluxes outlined on the above diagram, Knox and Helfer arrived at the following equations in “Development of an elementary climate model. III. Three-Layer case”. (1) εSa + εSb – εSe = ASo + Snr (2) –εSb + Se = BSo – Snr Finding a relationship between Sa and Sb The first objective was to estimate the flux of the lower atmosphere assuming a linear relationship to the flux of the upper atmosphere and a value of 102 W/m for Snr. To accomplish this, equation (2) was solved for Snr, and Sb was replaced by the assumed linear function, with c as the constant of proportionality (Sb=c*Sa): Snr = BSo + εcSa – Se Then a binary search was executed on c until Snr was 102. The c value needed to obtain this result was 7/6. Therefore, it appears that making the assumption that the flux out of the lower atmosphere is 7/6 of the flux out of the upper atmosphere is consistent with the three-layer model. However, this is merely an estimation as there is nothing in physics to allow us to conclude such a strong statement as a linear relationship. Calculating c (Sb/Sa) This section provides methods for calculating the ratio of flux out of the bottom of the atmosphere (Sb) to flux out of the top of the atmosphere (Sa), which is labeled c. The relevant equations are listed below. (1) Snr = εcSa + BSo – Se (2) Sa = [(A+B)So – (1–ε)Se]/ ε (3) Snr = c[(A+B)So – (1–ε)Se] + BSo – Se (4) c = [Snr – BSo + Se]/[(A+B)So – (1–ε)Se] The first two equations come from version (b) of the three-layer model. The third results from substituting (2) into (1). The fourth comes by solving (3) for the variable c. The general strategy is to assume a value of Snr and solve equation (4). All the other variables on the right hand side of this formula have been calculated in keeping with the standard model. After the value for c is found, the implied value of Sb is calculated (see Appendix 3) and compared with the expected value of 360 W/m. This value comes from the fact that the observed flux of 324 W/m will be the ideal radiative flux times the emmisivity ε=0.9, so the value sought for the ideal flux Sb will be 324/0.9=360. To proceed, the value of Snr is taken to be 102. Method 1: Using Global Averages. This method computes a globally constant c with only values that are globally averaged i.e. there are no cell dependent variables thus the calculation can be done on any calculator with merely the inputs given in Appendix 1. A straightforward evaluation of (4) with averaging of the form: c=[Snr – + ] / [( + ) – (1–ε)] yields a value for c of 1.3519. Method 2a: Using cell data – c is a global constant. This method utilizes the cell dependent variables to calculate c, then averages the cell dependent c values to find a globally constant c. Using equation (4), a matrix of c values was formed by entering this equation into Matlab. Then a weighted global average of these values was taken. (See Appendix 2). The result was = 1.2867. Method 2b: Using cell data – c is a function of cell. This method utilizes the cell dependent variables to calculate c, and then keeps a matrix for c as a cell dependent variable for further computations. This method is the same as method 2a, except it omits the averaging at the end, preventing c from taking on a single value, instead it’s a cell dependent matrix so Sb is determined by Sb= i.e. the 2 line in Appendix 2 is replaced by c=c_matrix. To test the correctness of the value for c, the implied value for Sb was compared to the formerly predicted value of 360 Watts/m. By Calculation in Appendix 3: Method 1: Sb = 319.144 11.3% difference Method 2a: Sb = 303.762 15.6% difference Method 2b: Sb = 275.147 23.6% difference Therefore, Method 1 appears to be the most promising. This is surprising because the methods that retain the most data, and hence would be expected to be the most accurate, turn out to be further off.