Update: I changed the title, to be a bit less of a redneck fisherman playing theoretical physicist.
Update 2: For some reason I have had a few comments mentioning that my blog is about Hydrogen. I do despise paying $5 a gallon at the dock, Hydrogen is not political, it is efficient, though a bear to transport, so you need a big honking boat pulling truck to haul the gas. If it matters, I don't think there is a big hydrogen to be in the pocket of :)
The flow of heat is a flux. As heat flows through any media, energy is lost to that media unless the heat coefficient of that media is perfect, or the coefficient of heat flux is equal to one. Heat has three basic methods of flux, conductive, convective (which we will latent) and radiative. Each of these three means of heat flux have varying coefficients related to their media and the temperature difference between the point of origin of the flux and the end point the heat sink. A change in flux relative to temperature takes the form dF/dT, where F is in units of W/m-2 and T is in unit Kelvin. For radiative heat flow, Stefan-Boltzmann has proven the F=a*b*(T)^4, Where a, is assumed to be a constant of emissivity and b =5.67*e-08.
Solving that formula for a one degree change from a temperature of 250K yields, a*567e-8*(251)^4/250 -a*5.67e-8*(T)^4 = a*0.2573 Since the value a, varies, we can include in its value the remainder .0073 to simplify the solution to the change in flux from 250K to 251K is equal to alpha*0.25 or alpha/4, so if flux increases by 4 temperature increase by 1, dF/dT=alpha*4*F/T, alpha is the coefficient of heat flow through a medium, similar to impedance in an electrical circuit.
For purely radiative flow, alpha is approximately equal to 0.926, with the inclusion of the remainder used to simplify, Alpha = 0.9333 or 0.93, a unit-less value, resulting in units for dF/dT of W/Km-2. alpha is considered the emissivity of the medium of heat flux source. In a medium that interacts with the flux, the term alpha times the emissivity of the media is equal to e, the effective emissivity. For this example, assume this is true, the actual value can be easily estimate using the NASA Energy Budget, below which I have modified to include an estimate greenhouse flux.
Update: This is the revised modification of the NASA Energy Budget Drawing, showing a better visualization of the atmospheric effect. The former did have an error.
For a combined flux flow, The sum of the flows is equal to 4(aFc+bFl+eFr)/T, for conductive, latent(convective) and radiative flux. where the source of the heat fluxes is a common temperature T.
For the coefficient a=alpha*heat flow coefficient of conduction in the media, b=alpha*heat flow coefficient of convection and e=alpha*media emissivity.
So this simple equation describes the combined flow of fluxes in a media with the different effects of the media on the individual flux.
Both aFc and bFl are limited by their media. aFc, decreases with distance as a function of the change in its media known commonly as resistance. aFl decreases with distance as a function of temperature. eFr decreases with distances as a function of the media's emissivity.
At the surface, three equal fluxes will result in warming of the media proportional to flux and heat flow coefficient. If 100Wm-2 of conductive flux leaves the surface it will decrease to zero at an altitude relative to the decrease in pressure with height. The conductive flux from a source with an initial T will decrease to zero at some temperature (T-4aFc/T), i.e., the conductive flux will transition to radiative flux, and be lost to space. As this flux transistions, energy is added to the system, which becomes either latent or radiant heat which in turn transition back to conductive, for conversation of energy, Latent heat absorbed at the surface rises with reduced local pressure, gaining and losing sensible heat and radiant heat until condensation, where that energy, known as the heat of evaporation is lost to the atmosphere as conductive and radiant heat. The latent heat can be regained with vaporization or lose more heat with the freezing known as the heat of fusion. Radiative flux resisted by the emissivity of the increases as the emissivity reduces and the radiant energy release of the conductive and convected increases. The combined fluxes at the surface equals the radiative flux at the top of the atmosphere.where the emissivity of space approaches unity. All of these interactions between the energy fluxes warm the atmosphere near the surface and cool it at the top.
Conductive flux warms the surface because of its resistance to flow, convective flux cools the surface by transporting absorbed energy from the surface to an altitude, Radiant flux warms the surface due to the initial high resistance and cools the atmosphere as emissivity decreases with respect to out and increases with respect to in or down. Work is being preformed on the system at varying locations at varying rates. The heat as a result of this warm, generates the atmospheric effect which is a heat source at a distance with a different coefficient of heat flux.
For conductive flux, the resistance to flow increases with altitude until all of the flux has transitioned into radiative flux, this point, which I will call the effective temperature and pressure of conduction, is located fairly low in the atmosphere. Latent heat is absorbed at the surface and transported via convection until the reduction in temperature precipitates all of the moisture. This point I will call, the effective temperature and pressure of convective heat. For radiant heat, the flux uniformly increases to a value equal to the total of all surface fluxes less the energy retained in the atmosphere as latent and sensible heats, potential energies. Once radiant heat reaches its maximum value, i.e. both conductive and convective have transitioned in to radiant flux, i will call this the effective temperature and pressure of radiant flux, approximately located the tropopause. So we have an atmosphere with three effective temperatures, conductive, convective and radiant at three different temperatures and pressures. The pressures correspond to altitudes.
The effective conductive temperature is equal to the dry adiabatic lapse rate and its effective temperature is the potential temperature of air at that altitude.
The effective temperature of latent is equal to the moist adiabatic lapse rate and located above the effective conductive temperature and pressure.
The effective temperature of radiant flux is at an altitude above the latent effective temperature and pressure. All of the three effective temperatures interact as each has a temperature and pressure different than the others. The net impact of the interaction of fluxes at equilibrium is that the total heat zthe atmosphere gained is equal to the potential temperature of conductive effective temperature and pressure, as can be determined by Poisson's equation.
Stage 2 Estimating the greenhouse down welling Flux
The simplest estimate is to determine the atmospheric impact on the individual flux values, by dividing the values by their respective coefficients, Fc=24/0.33=72.7, Fl=79/1.09=72 and the Fr component absorbed by the atmosphere 72-21=51/.825=61.8, the sum, 72.7+72+61.8=206.5. This estimate does not include the radiation absorbed in the atmosphere by the radiant energy through the atmospheric window.
The 21Wm-2 through the atmospheric window, while it does not transfer significant energy to the atmosphere, still experiences scattering and some attenuation due to the atmosphere, The estimated TOA emissivity of the atmosphere is ~0.61, so the surface equivalent source for the 21Wm-2 would be 21/0.61=34.4Wm-2.
When combined, 206.6+34.4=240Wm-2, the value of the radiation at the TOA. The radiative flux through the atmospheric window does not get a completely free pass. Water and ice in the atmosphere interact with portions of this radiation, so the true value of the down welling radiation would be between the 206 and 240 values. The radiation in the atmospheric window does interact with the atmosphere. Since the radiation window down is more opaque than up, only 39% is felt at the surface, 21*.39=8.8, so 206.5+8.8=215 Wm-2 is a more accurate estimate of the atmosphere effect in W/m-2.
If you consider the value provide by K&T, 321Wm-2, if it were applied at the top of the atmosphere, 321*.61 would equal 192.6, had K&T not miscalculated the amount of radiation absorb by the atmosphere, which is 51 per NASA and ~26per K&T, the value would be 192.6+25=217.6Wm-2.
If the logic is followable, one can see that the formula accurately predicts energy interactions in the atmosphere, even with approximated values for a, b and e, based on the NASA flux values. But is there any reason that the 216 range for DWLR should be used instead of 321 per K&T or the 155 common used in literature? Yes, because the atmospheric effect is based on a physical property, the adiabatic lapse rate. The work done by the competing upwelling and down welling fluxes it to maintain the potential energy of the atmosphere, it lapse rate. The conductive heat flux from the surface energizes the molecules in the atmosphere causing then to rise from the surface to a point where the force of gravity stops their accent. If the atmosphere collapsed the heat generate would equal the atmospheric effective temperature.
Ohms Law and Atmospheric Thermal Balance.
As I mentioned in the beginning, the conductive flux resistance is equal to the potential energy divided by the flux, similar to Ohms law, were V=IR. R,the conductive resistance of the atmosphere is equal to the potential 288-216 divided by thermal current or flux 24, 288-216/24=3. the coefficient for a in aFc =1/R=0.33 and I have not named that value. For bFl, Rl=(288-216)/79=0.91 b=1/Rl=1.097. For the radiative values, Rr=(288-216)/87=1.21 e=1/1.21=0.825. Why did I chose 87 when the NASA drawing shows 79? Because it appears to have a small error. 79 plus the portion of the radiation through the atmospheric window is the correct value, 87-79=8. Now of course this is all still numerology, or is it?
Greenhouse Down Welling Radiation and its Relationship to Potential Temperature.
The value of the DWLR is 216, which I can calculate an effective temperature for using the S-B equation of 248 degrees not using the 0.926 indicated in the S-B equation, that is include in the coefficients. If I calculate the potential temperature of this value using the dry adiabatic lapse rate to determine the approximate pressure, 273-248= degrees or -25 degrees at~ 600 mb the answer is 287.2, very close to the surface temperature of 288K Still Numerology?
The values for the coefficients a, b and e are estimated for the NASA drawing using the modified Flux field equation. They are close, but not exact and they are also variable as each is dependent on the others. Hopefully this little exercise shows that there is some value in the equation and that 216 is a very reasonable estimate for DWLR with physical meaning pertinent to understanding out atmosphere.
Update: http://en.wikipedia.org/wiki/Relativistic_heat_conduction
So why is a simple equation that works not used? I'll never know. Perhaps it is just better science to over complicate things. That's why there are so many engineers that are skeptical.
Please Note that this is consuming too much of my time, donation would be greatly appreciate in the tip jar. An a proof reader to clean up my poor attempt desperately needed.
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