Abstract
A black planet irradiated by a sun serves as the archetype for a simple radiating two-layer system admitting of a continuum of steady states under steadfast insolation. Steady entropy production rates may be calculated for different opacities of one of the layers, explicitly so for the radiative interactions, and indirectly for all the material irreversibilities involved in maintaining thermal uniformity in each layer. The second law of thermodynamics is laid down in two versions, one of which is the well-known Clausius-Duhem inequality, the other being a modern version known as the entropy inequality. By maximizing the material entropy production rate, a state may be selected that always fulfills the Clausius-Duhem inequality. Some formally possible steady states, while violating the latter, still obey the entropy inequality. In terms of Earth’s climate, global entropy production rates exhibit extrema for any “greenhouse effect”. However, and only insofar as the model be accepted as representative of Earth’s climate, the extrema will not be found to agree with observed (effective) temperatures assignable to both the atmosphere and surface. This notwithstanding, the overall entropy production for the present greenhouse effect on Earth is very close to the maximum entropy production rate of a uniformly warm steady state at the planet’s effective temperature. For an Earth with a weak(er) greenhouse effect the statement is no longer true.
Highlights
In theories of thermomechanical behavior of a material continuum, conservation laws are on the whole not sufficient to constrain the constitutive equations that relate dependent fields
It is known that in thermodynamic theories of linear constitutive equations, irreversible processes can be written as bilinear expressions in “fluxes” and “thermodynamic forces”, the former being considered as the response to the latter
By confining myself to the model’s abstract thermodynamics I wish to draw explicitly but three partly alternate conclusions: (1) The model as defined is inadmissible and even impossible to realize, because for some eligible surface temperatures Ts it would have to be attended by negative material entropy production rates
Summary
In theories of thermomechanical behavior of a material continuum, conservation laws are on the whole not sufficient to constrain the constitutive equations that relate dependent fields. Further principles are Entropy 2014, 16 called for in restricting possible relationships Among the former, the second law of thermodynamics, in the guise of the Clausius-Duhem inequality, plays a prominent role in thermomechanical theories [1,2]. The Clausius-Duhem inequality is an entropy imbalance that defines an entropy production rate, a genuine creation of entropy, so it cannot be negative whenever irreversible processes take place within the system to which it is applied. It is this rate that is laid down in any investigation purporting to evince its maximum in some natural processes. We calculate the radiation sources that follow from the temperature field and, by so doing, will be in a position to determine, if only indirectly, the rate of entropy production of the model
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