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P = epsilon * sigma * T^4

epsilon = emissivity
sigma = stefan's constant
T = absolute temperature
P = power radiated / unit area (W/m^2)

do simple example - double P
and double double epsilon
what value do you have to change T by for the result to balance?
HINT = T doesn't have to change at all in the example.

What's being absorbed additionally by the ocean? That additional 3.7W/m^2 for a doubling means only that the surface emissions to space are down by 3.7W/m^2. For there to be extra energy in the earth/atmospheric system for the ocean to absorb, there must be a temperature increase somewhere. The 3.7W/m^2 additional absorption means that epsilon increased due to reciprocity. When you apply stefan's law with the new larger epsilon value, P increases meaning that more energy is being radiated outward by the atmospheric section that absorbed the 3.7W/m^2 in the first place. The question becomes whether T (of the atmosphere) needs to increase or decrease to fine tune the new balance.

If the 3.7W/m^2 were an increase due to insolation where emissivity didn't change, then stefan's law dictates that T must increase so that P outputs an additional 3.7W/m^2 to regain balance. It's a rather small change though since it's to the 4th power. But this is not relevent to ghgs as ghgs change emissivity.

it's not about bouncing around. Emission/absorption occur simultaneously. Absorption doesn't mean blocking totally. And this is still with radiative only - ignoring convective and conductive acitivities.

Gases tend to emit in lines. However, at the fluid densities of the troposphere, the factors of molecules bouncing around enter in. I think you might be basically right on what you're trying to say in the last paragraph. In dense atmosphere, some of the absorbed energy is reradiated as is and some is internalized in molecular vibrations and some is kinetic, bouncing the molecule around into others - creating heat. Any wavelength can have that effect. Your microwave heats your tea water by exciting some rotational mode of liquid h2o. If you put that microwave in a walkin freezer and tried to melt an icecube - nothing would happen to the icecube's temperature.

Planck's law is a more detailed accounting of what's going on than stefan's law. It breaks it up by wavelength. It's also the beginning of the transition of classical physics to quantum physics and has the introduction of the photon. It forms a curve that falls off at short wavelengths quickly and has a long tail out towards the longer wavelengths. It's basic shape is the same no matter what the temperature - anything above absolute 0 radiates. The peak location and height (and total energy) do depend upon the temperature. Wien's displacement shows the peak wavelength depends linearly upon the temperature while stefan's law shows the total radiated energy depends upon T^4.

Except in the upper atmosphere which is full of ions and is essentially a vacuum where molecules don't collide frequently things become line absorptions and emissions and laser type phenomenon can occur (stimulated emission - 1 photon triggers a second as it passes by), one sees much more of the classical physics, planck's law, stefan's law etc. Hence, absorbed lines are spread into bands and emissions depend upon temperature in the classical stefan, planck realm. Just because an object has to be 1000C to glow visibly doesn't mean it's not putting out near infrared at lower temperatures. This is classical - not quantum stuff - think averages of quadtrillion billions of molecules giving a classical value not individual molecules doing some thing or another weird quantum factors.

I'm not sure what your question on emissivity is or which statement you mean. Emissivity is not just a single number as it varies with wavelength. It does have an average value that most who deal with it use as an 'engineering' assumption. However, it is a function of wavelength. Again, at lower atmospheric levels, you have bands not lines involved due to the smearing out of the quantum effects into the classical realm. That is the lines have line width and groups of many lines form the bands. Perhaps you are asking if these are what increases in emission rates if they increase in absorption rates?