It seems that climate modelers are dealing with a quandary: How can we improve on the unsatisfactory results from climate modeling?
A.Continue tweaking models using classical maths though they depend on climate being in quasi-equilibrium; or,
B.Start over from scratch applying non-equilibrium maths to the turbulent climate, though this branch of math is immature with limited expertise.
In other words, we are confident in classical maths, but does climate have features that disqualify it from their application? We are confident that non-equilibrium maths were developed for systems such as the climate, but are these maths robust enough to deal with such a complex reality?
It appears that some modelers are coming to grips with the turbulent quality of climate due to convection dominating heat transfer in the lower troposphere. Heretofore, models put in a parameter for energy loss through convection, and proceeded to model the system as a purely radiative dissipative system. Recently, it seems that some modelers are striking out in a new, possibly more fruitful direction. Herbert et al 2013 is one example exploring the paradigm of non-equilibrium steady states (NESS). Such attempts are open to criticism from a classical position, but may lead to a breakthrough for climate modeling.
That is my layman’s POV. Here is the issue stated by practitioners, more elegantly with bigger words:
“In particular, it is not obvious, as of today, whether it is more efficient to approach the problem of constructing a theory of climate dynamics starting from the framework of hamiltonian mechanics and quasi-equilibrium statistical mechanics or taking the point of view of dissipative chaotic dynamical systems, and of non-equilibrium statistical mechanics, and even the authors of this review disagree. The former approach can rely on much more powerful mathematical tools, while the latter is more realistic and epistemologically more correct, because, obviously, the climate is, indeed, a non-equilibrium system.”
Lucarini et al 2014
Here’s how Herbert et al address the issue of a turbulent, non-equilibrium atmosphere. Their results show that convection rules in the lower troposphere and direct warming from CO2 is quite modest, much less than current models project.
“Like any fluid heated from below, the atmosphere is subject to vertical instability which triggers convection. Convection occurs on small time and space scales, which makes it a challenging feature to include in climate models. Usually sub-grid parameterizations are required. Here, we develop an alternative view based on a global thermodynamic variational principle. We compute convective flux profiles and temperature profiles at steady-state in an implicit way, by maximizing the associated entropy production rate. Two settings are examined, corresponding respectively to the idealized case of a gray atmosphere, and a realistic case based on a Net Exchange Formulation radiative scheme. In the second case, we are also able to discuss the effect of variations of the atmospheric composition, like a doubling of the carbon dioxide concentration.
The response of the surface temperature to the variation of the carbon dioxide concentration — usually called climate sensitivity — ranges from 0.24 K (for the sub-arctic winter profile) to 0.66 K (for the tropical profile), as shown in table 3. To compare these values with the literature, we need to be careful about the feedbacks included in the model we wish to compare to. Indeed, if the overall climate sensitivity is still a subject of debate, this is mainly due to poorly understood feedbacks, like the cloud feedback (Stephens 2005), which are not accounted for in the present study.”
Vertical Temperature Profiles at Maximum Entropy Production with a Net Exchange Radiative Formulation
Herbert et al 2013
In this modeling paradigm, we have to move from a linear radiative Energy Budget to a dynamic steady state Entropy Budget. As Ozawa et al explains, this is a shift from current modeling practices, but is based on concepts going back to Carnot.
“Entropy of a system is defined as a summation of “heat supplied” divided by its “temperature” [Clausius, 1865].. Heat can be supplied by conduction, by convection, or by radiation. The entropy of the system will increase by equation (1) no matter which way we may choose. When we extract the heat from the system, the entropy of the system will decrease by the same amount. Thus the entropy of a diabatic system, which exchanges heat with its surrounding system, can either increase or decrease, depending on the direction of the heat exchange. This is not a violation of the second law of thermodynamics since the entropy increase in the surrounding system is larger.
Carnot regarded the Earth as a sort of heat engine, in which a fluid like the atmosphere acts as working substance transporting heat from hot to cold places, thereby producing the kinetic energy of the fluid itself. His general conclusion about heat engines is that there is a certain limit for the conversion rate of the heat energy into the kinetic energy and that this limit is inevitable for any natural systems including, among others, the Earth’s atmosphere.
Thus there is a flow of energy from the hot Sun to cold space through the Earth. In the Earth’s system the energy is transported from the warm equatorial region to the cool polar regions by the atmosphere and oceans. Then, according to Carnot, a part of the heat energy is converted into the potential energy which is the source of the kinetic energy of the atmosphere and oceans.
Thus it is likely that the global climate system is regulated at a state with a maximum rate of entropy production by the turbulent heat transport, regardless of the entropy production by the absorption of solar radiation This result is also consistent with a conjecture that entropy of a whole system connected through a nonlinear system will increase along a path of evolution, with a maximum rate of entropy production among a manifold of possible paths [Sawada, 1981]. We shall resolve this radiation problem in this paper by providing a complete view of dissipation processes in the climate system in the framework of an entropy budget for the globe.
The hypothesis of the maximum entropy production (MEP) thus far seems to have been dismissed by some as coincidence. The fact that the Earths climate system transports heat to the same extent as a system in a MEP state does not prove that the Earths climate system is necessarily seeking such a state. However, the coincidence argument has become harder to sustain now that Lorenz et al.  have shown that the same condition can reproduce the observed distributions of temperatures and meridional heat fluxes in the atmospheres of Mars and Titan, two celestial bodies with atmospheric conditions and radiative settings very different from those of the Earth.”
THE SECOND LAW OF THERMODYNAMICS AND THE GLOBAL CLIMATE SYSTEM: A REVIEW OF THE MAXIMUM ENTROPY PRODUCTION PRINCIPLE
Hisashi Ozawa et al 2003