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Global thermodynamics, the Clausius inequality, and entropy radiation
Abstract
Global steady state conditions on entropy radiation are compared to a global inequality on energy radiation analogous to the Clausius inequality of classical thermodynamics. It is demonstrated that these conditions are simultaneously valid and have similar origins. They are discussed in terms of their connection to the second law of thermodynamics, and the distinction between the role of energy radiation and entropy radiation.
... Although each perspective provides a consistent description of the climate system, the magnitude of the entropy importṠ e and the implied irreversible entropy productionṠ i differs greatly between the planetary, material, and transfer definitions, and previous authors have disagreed on which perspective is most relevant for studies of the climate (e.g., Essex, 1984Essex, , 1987Goody, 2000). Following Goody (2000) and a number of other authors (e.g. ...
... This definition is appealing, because it gives the radiant entropy flux as being equal to the loss of entropy by the emitting object, but it fails to account for the irreversible nature of spontaneous emission and absorption represented by the 4/3 factor in (8) (Feistel, 2011). Studies such as Peixoto et al. (1991) therefore do not fully account for the irreversible entropy production associated with radiative processes and they underestimate the planetary entropy production rate (Essex, 1984(Essex, , 1987Stephens and O'Brien, 1993). Such studies nevertheless remain relevant, because, as will be shown below, entropy production associated with radiative transfer does not affect the material entropy budget, and conclusions regarding material entropy production are unaffected whether one uses (11) or (8) to estimate radiant entropy fluxes. ...
... For the parameters chosen, the one-layer model gives a material entropy production rate ofṠ mat i = 39 mW m −2 K −1 . Given the assumptions of the model, this is a rather crude estimate of the material entropy production in the climate system, but it highlights the small magnitude ofṠ mat i compared to the total irreversible production rateṠ i , implying that the bulk of the irreversible entropy production in the climate system occurs due to radiative processes (Essex, 1984(Essex, , 1987Goody, 2000;Li et al., 1994;Stephens and O'Brien, 1993). ...
The second law of thermodynamics implies a relationship between the net entropy export by the Earth and its internal irreversible entropy production. The application of this constraint for the purpose of understanding Earth's climate is reviewed. Both radiative processes and material processes are responsible for irreversible entropy production in the climate system. Focusing on material processes, an entropy budget for the climate system is derived which accounts for the multi-phase nature of the hydrological cycle. The entropy budget facilitates a heat-engine perspective of atmospheric circulations that has been used to propose theories for convective updraft velocities, tropical cyclone intensity, and the atmospheric meridional heat transport. Such theories can only be successful, however, if they properly account for the irreversible entropy production associated with water in all its phases in the atmosphere. Irreversibility associated with such moist processes is particularly important in the context of global climate change, for which the concentration of water vapor in the atmosphere is expected to increase, and recent developments toward understanding the response of the atmospheric heat engine to climate change are discussed. Finally, the application of variational approaches to the climate and geophysical flows is briefly reviewed, including the use of equilibrium statistical mechanics to predict behavior of long-lived coherent structures, and the controversial maximum entropy production principle.
... Its major principles were however enunciated by Planck [60,71] over a century ago, and further developed over the past century (e.g. [78][79][80][81][82][83][84][85][86]). However, there still remains widespread confusion in its calculation, over choices of symbols and preferred parameters, and even in the most appropriate theoretical approach 7 . ...
... Similarly, we can consider the specific entropy intensity or entropy radiance L ν (W K −1 m −2 s sr −1 ) of radiation. This is given by [71,78,79,[81][82][83][84][85]: ...
... whereupon (43) reduces to |j S,ν | = 4 3 k SB T 3 , where k SB is the Stefan-Boltzmann constant [81,82]. We can now construct the local entropy production as the sum of non-radiative (material) and radiative components [81][82][83][84][85][86]:σ ...
This chapter concerns "control volume analysis", the standard engineering
tool for the analysis of flow systems, and its application to entropy balance
calculations. Firstly, the principles of control volume analysis are enunciated
and applied to flows of conserved quantities (e.g. mass, momentum, energy)
through a control volume, giving integral (Reynolds transport theorem) and
differential forms of the conservation equations. Several definitions of steady
state are discussed. The concept of "entropy" is then established using Jaynes'
maximum entropy method, both in general and in equilibrium thermodynamics. The
thermodynamic entropy then gives the "entropy production" concept. Equations
for the entropy production are then derived for simple, integral and
infinitesimal flow systems. Some technical aspects are examined, including
discrete and continuum representations of volume elements, the effect of
radiation, and the analysis of systems subdivided into compartments. A Reynolds
decomposition of the entropy production equation then reveals an "entropy
production closure problem" in fluctuating dissipative systems: even at steady
state, the entropy production based on mean flow rates and gradients is not
necessarily in balance with the outward entropy fluxes based on mean
quantities. Finally, a direct analysis of an infinitesimal element by Jaynes'
maximum entropy method yields a theoretical framework with which to predict the
steady state of a flow system. This is cast in terms of a "minimum flux
potential" principle, which reduces, in different circumstances, to maximum or
minimum entropy production (MaxEP or MinEP) principles. It is hoped that this
chapter inspires others to attain a deeper understanding and higher technical
rigour in the calculation and extremisation of the entropy production in flow
systems of all types.
... This insight into the role of radiation in producing entropy instigated the division of the climate into nonradiative (material) and radiative sub-systems and the separate tabulations of the entropy produced in each (Essex 1987;Goody and Abdou 1996;Goody 2000). Paltridge's meridional view was mapped in higher-dimensional models to the material entropy production rate, which includes contributions from horizontal and vertical sensible and latent heating (as established by Pujol and Llebot (1999)) as well as all other non-radiative processes. ...
... The photon gas permeating the atmosphere is considered part of the surroundings and radiative heating and cooling supply the cross-boundary fluxes of energy (Bannon 2015). The local temperature distribution of the matter is unchanging at steady state and so the net heating of each parcel by material processes must be balanced by radiative cooling to space or within the climate system, and vice versa (Essex 1987;Goody 2000). ...
There is ongoing interest in the global entropy production rate as a climate diagnostic and predictor, but progress has been limited by ambiguities in its definition; different conceptual boundaries of the climate system give rise to different internal production rates. Three viable options are described, estimated and investigated here, two of which -- the material and the total radiative (here 'planetary') entropy production rates -- are well-established and a third which has only recently been considered but appears very promising. This new option is labelled the 'transfer' entropy production rate and includes all irreversible processes that transfer heat within the climate, radiative and material, but not those involved in the exchange of radiation with space. Estimates in three model climates put the material rate in the range 27-48 mW/mK, the transfer rate 67-76 mW/mK, and the planetary rate 1279-1312 mW/mK. The climate-relevance of each rate is probed by calculating their responses to climate changes in a simple radiative-convective model. An increased greenhouse effect causes a significant increase in the material and transfer entropy production rates but has no direct impact on the planetary rate. When the same surface temperature increase is forced by changing the albedo instead, the material and transfer entropy production rates increase less dramatically and the planetary rate also registers an increase. This is pertinent to solar radiation management as it demonstrates the difficulty of reversing greenhouse gas-mediated climate changes by albedo alterations. It is argued that the transfer perspective has particular significance in the climate system and warrants increased prominence.
... As evident from Equation (68), the entropy irradiance is a property of the radiation itself, distinct from the entropy produced by its conversion to heat. Thus in the presence of electromagnetic radiation, in addition to the non-radiative (material) component in Equation (4), there is a separate radiative component to the local entropy production [142][143][144][145]147]:σ ...
The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form Πinfo=U/c, where U is a characteristic flow velocity and c is a signal velocity or wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless groups in science and engineering are defined by geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. In this Part II, the information-theoretic definition is applied to a number of flow systems with wave phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves, capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate Mach, Euler, Froude, Rossby or other dimensionless number(s)—including new groups for internal gravity, inertial and electromagnetic waves—to classify their flow regimes. For flows with wave dispersion, the coexistence of different celerities for individual waves and wave groups—each with a distinct information-theoretic group—is shown to imply the existence of more than two information-theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses are shown to provide a more complete understanding of frictional behavior and sharp transitions in compressible and open channel flows, as well as the transport of entropy by electromagnetic radiation. The analyses significantly extend the applications of entropic similarity for the analysis of flow systems with wave propagation.
... However, we would argue that the definition as given here is more fundamental. The material entropy production rate avoids dependence on spectral properties of radiation and so the net heating of each parcel by material processes must be balanced by radiative cooling to 245 space or within the climate system, and vice versa (Essex 1987;Goody 2000). 246 The view that motivates this approach is that these material (or 'molecular') processes -such as In our EBM (Figure 1), the material entropy production rate can be directly specified as: ...
There is ongoing interest in the global entropy production rate as a climate diagnostic and predictor, but progress has been limited by ambiguities in its definition; different conceptual boundaries of the climate system give rise to different internal production rates. Three viable options are described, estimated and investigated here, two of which – the material and the total radiative (here ‘planetary’) entropy production rates – are well-established and a third which has only recently been considered but appears very promising. This new option is labelled the ‘transfer’ entropy production rate and includes all irreversible processes that transfer heat within the climate, radiative and material, but not those involved in the exchange of radiation with space. Estimates in three model climates put the material rate in the range 27-48mW/m2K, the transfer rate 67-76mW/m2K, and the planetary rate 1279-1312mW/m2K. The climate-relevance of each rate is probed by calculating their responses to climate changes in a simple radiative-convective model. An increased greenhouse effect causes a significant increase in the material and transfer entropy production rates but has no direct impact on the planetary rate. When the same surface temperature increase is forced by changing the albedo instead, the material and transfer entropy production rates increase less dramatically and the planetary rate also registers an increase. This is pertinent to solar radiation management as it demonstrates the difficulty of reversing greenhouse gas-mediated climate changes by albedo alterations. It is argued that the transfer perspective has particular significance in the climate system and warrants increased prominence.
... Investigations along this line are generally related to the second law of thermodynamics wherein entropy and entropy production are fundamental components, and thermodynamic extremal entropy production principles are often used to explain some collective behaviors of the complex Earth's system without knowing the details of the dynamics within the system. Such thermodynamic investigations have provided crucial insight into various processes of climatic importance in the past several decades [e.g., Paltridge, 1975Paltridge, , 1978Golitsyn and Mokhov, 1978;Nicolis and Nicolis, 1980;Grassl, 1981;Mobbs, 1982;Noda and Tokioka, 1983;Essex, 1984Essex, , 1987Wyant et al., 1988;Lesins, 1990;Peixoto et al., 1991;Stephens and O'Brien, 1993;Goody and Abdou, 1996;Goody, 2000;Ozawa et al., 2003;Paltridge et al., 2007;Pauluis et al., 2008;Wang et al., 2008;Lucarini et al., 2010;Wu and Liu, 2010]. However, the entropic aspects of climate theory have not yet been developed as well as those based on energy, momentum, and mass balances. ...
The study of the Earth's radiation entropy flux at the top of the atmosphere is reviewed with an emphasis on its estimation methods. Existing expressions for calculating radiation entropy flux scattered in different disciplines are surveyed, and their applicabilities are examined. It is found that the Earth's net radiation entropy flux estimated from these various expressions can differ substantially, more than the typical value of the entropy production rate associated with the atmospheric latent heat process. Comparison analysis shows that the commonly used expression of radiation entropy flux as the ratio of radiation energy flux to absolute temperature underestimates the Earth's radiation entropy flux by >30%. Theoretical analysis reveals that the large difference in the Earth's reflected solar radiation entropy flux among the different expressions arises mainly from the difference of the Earth's reflection properties (i.e., Lambertian or specular) assumed in these expressions. For the Earth system with typical shortwave albedo of 0.30 and longwave emissivity between 0.50 and 1.00, the Earth's net radiation entropy flux derived from the most accurate Planck's spectral expression ranges from 1.272 to 1.284 W m-2 K-1, amounting to the overall Earth's entropy production rate from 6.481 × 1014 to 6.547 × 1014 W K-1.
We propose an additional category of dimensionless groups based on the principle of {\it entropic similarity}, defined by ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. Since all processes involving work against friction, dissipation, diffusion, dispersion, mixing, separation, chemical reaction, gain of information or other irreversible changes are driven by (or must overcome) the second law of thermodynamics, it is appropriate to analyse these processes directly in terms of competing entropy-producing and transporting phenomena and the dominant entropic regime, rather than indirectly in terms of forces. In this study, we derive entropic groups for a wide variety of phenomena relevant to fluid mechanics, classified into diffusion and chemical reaction processes, dispersion mechanisms and wave phenomena. It is shown that many dimensionless groups traditionally derived by kinematic or dynamic similarity (including the Reynolds number) can also be recovered by entropic similarity -- albeit with a different entropic interpretation -- while a large number of new dimensionless groups are also identified. The analyses significantly expand the scope of dimensional analysis and similarity arguments for the resolution of new and existing problems in science and engineering.
The second law of thermodynamics implies a relationship between the net entropy export by Earth and its internal irreversible entropy production. The application of this constraint for the purpose of understanding Earth’s climate is reviewed. Both radiative processes and material processes are responsible for irreversible entropy production in the climate system. With a focus on material processes, an entropy budget for the climate system is derived that accounts for the multiphase nature of the hydrological cycle. The entropy budget facilitates a heat-engine perspective of atmospheric circulations that has been used to propose theories for convective updraft velocities, tropical cyclone intensity, and the atmospheric meridional heat transport. Such theories can be successful, however, only if they properly account for the irreversible entropy production associated with water in all its phases in the atmosphere. Irreversibility associated with such moist processes is particularly important in the context of global climate change, for which the concentration of water vapor in the atmosphere is expected to increase, and recent developments toward understanding the response of the atmospheric heat engine to climate change are discussed. Finally, the application of variational approaches to the climate and geophysical flows is reviewed, including the use of equilibrium statistical mechanics to predict the behavior of long-lived coherent structures and the controversial maximum entropy production principle.
Motivated by the circumstantial evidence of the pervasive nature of the
`general coldness' of climate model simulations, a theoretical analysis
is made of the model response expected from the presence of both
physical and aphysical sources of entropy under the joint conditions
that the net flux of energy through the upper and lower boundaries of
the atmosphere and the isentropic temporally, areally integrated entropy
source must vanish. These joint conditions are essential for a simulated
global climate state to be without drift. The application of these
conditions in the presence of positive definite aphysical entropy
sources leads to the conclusion that the model-simulated climate state
will be characterized by a general coldness, in particular in the upper
polar troposphere and lower tropical troposphere as observed in 104 out
of 105 possible outcomes from 35 different simulations by 14 climate
models.In assessing the magnitude of this effect, a 10°C bias in
mean temperature corresponds with a relatively small error of 4% in the
mean heat addition of an isentropic layer. This correspondence reveals
the extreme sensitivity of a climate model's temperature response to
aphysical entropy sources introduced by spurious numerical
dispersion/diffusion, Gibbs oscillations, parameterizations, and other
factors. In accurately simulating hydrologic and chemical processes,
this difficulty is compounded in the sense that the saturation specific
humidity doubles for each additional 10°C increase in temperature
and the inherent strong dependence of both processes on temperature,
pressure, and amount of water substances-vapor, liquid, and ice. A
strategy that makes climate simulations tractable is that numerical
trade-offs occur among the various parameterizations of the components
of heat addition. These trade-offs allow models to be tuned to simulate
a reasonable state achievable for given resolution, numerics, and
parameterizations. Oreskes et al. label this step `calibration' and
suggest that in such situations, `empirical adequacy is forced.'The
results of this analysis in combination with Carathéordary's
statement of the Second Law reveals in the strict sense that the
presence of positive definite aphysical sources of entropy in a climate
model precludes the simulation of unbiased distributions of the heat
addition and temperature. Since in the strict sense the true state
cannot be simulated, several questions follow: are reasonable states of
global and regional climate change simulated for the right reasons; just
what are reasonable states; and how are the right reasons to be
determined in view of the trade-offs among the several components of
parameterized heat addition?
[1] Despite best efforts, there still is no physical theory for climate based on the physics of the laboratory regime (i.e., fluid mechanics, radiative transfer, classical thermodynamics, etc.). This paper builds from previous discussions on how laboratory-regime assumptions may lock our current theoretical efforts into the laboratory regime and how we might get around this problem. Using ultralong time photographic exposures (known as solargraphs) for inspiration, it draws into question classical thinking about intensive thermodynamic variables for theoretical climate purposes while using the fact that physical flows remain well defined even for climate regimes. These flows are characterized here as “generalized wind.” A simple example based on radiative energy and entropy transfer illustrates how these generalized wind fields can partially replace what is lost in moving away from laboratory-regime physics. These winds are shown to carry the dynamics in a modified form of radiation-like fluid dynamics that, together with radiation, might be possible to close in climate regimes.
A thermodynamic minimum principle valid for photon radiation is shown to hold for arbitrary geometries. It is successfully extended to neutrinos, in the zero mass and chemical potential case, following a parallel development of photon and neutrino statistics. This minimum principle stems more from that of Planck than that of classical Onsager-Prigogine irreversible thermodynamics. Its extension from bosons to fermions suggests that it may have a still wider validity.
This chapter explores the classical concepts of entropy and the processes that change it in relation to the Lorenz energy cycle and climate. Entropy concepts are fundamental to the study of large-scale exchange, monsoons, and the thermodynamic efficiency of differential heating in maintaining atmospheric circulation. Entropy concepts are essential for understanding reversible and irreversible thermodynamic processes. The study of entropy and its flux is also important for understanding radiative and chemical processes in relation to the earth system. Certain underlying concepts of the second law and entropy with the Lorenz energy cycle and the climate state are linked in the chapter. The understanding of the climate state as a thermally forced circulation is incomplete without including entropy and its exchange. Certain key contributions, both past and present, are reviewed in bringing to the forefront theimportance of entropy and its balance in modeling the climate state., In-depth relations linking the classical Carnot cycle with the Lorenz energy cycle are set forth. The chapter also emphasizes on the relevance of mean entropy sources on the intensity of winter and summer time circulations.
The flux of entropy carried from the planet by radiation was estimated, from Earth Radiation Budget Experiment data, by Stephens and O'Brien. the present paper provides the numerical context for Stephens and O'Brien's results by deriving: (1) the maximum possible flux of entropy, subject only to the constraint that the planet be in energy balance with incoming solar radiation; (2) the latitude profiles of albedo and brightness temperature that maximize the entropy flux for a planet without horizontal transport of energy; (3) the entropy flux for a planet with such efficient transport that the brightness temperature is uniform from equator to pole. In each case, the entropy fluxes are computed as functions of the planetary albedo. the entropy fluxes in models (2) and (3) are found to be almost indistinguishable numerically. the result is extended to a family of latitude profiles for albedo and brightness temperature which interpolate between the dynamical extremes represented by models (2) and (3), and it is shown that all members of the family have essentially the same entropy-flux. This suggests that measurements of the planetary entropy-flux derived from satellite radiances may not enable us to distinguish easily between different atmospheric and oceanic circulations. the entropy flux for planet earth reported by Stephens and O'Brien is observed to lie close to the maximum.
This paper introduces an approximate method for estimating the global distributions of the entropy fluxes flowing through the upper boundary of the climate system, and provides an estimate of the entropy exchange between the Earth and space and the entropy production of the planet. The entropy fluxes are those associated with the exchange of radiation between the Earth and space and a specific method is described to derive these entropy fluxes from available measurements of the earth's radiation budget. Entropy fluxes calculated from the Earth Radiation Budget Experiment measurements show how the long-wave entropy flux densities dominate the total entropy fluxes at all latitudes compared with the entropy flux densities associated with reflected sunlight. It appears that the entropy production of the planet is both constant for the 36 months of data considered and very near its maximum possible value. The mean value of this production is 0.68×1015W K-1 and the amplitude of the annual cycle is approximately 1 to 2% of this value, which is within the uncertainty of our estimate. -from Authors
Recently, there have been attempts at computing the various terms in the steady entropy balance of the climate system. Differing numbers have been published. The global atmospheric entropy production rate by absorption and emission of long-wave radiation is one contribution among many others. This paper is mainly devoted to the calculation of just that contribution for the simple model of a gray atmosphere. The inescapable conclusion is that it is substantial, out-weighing, with the unlikely exception of phase changes, all other contributions not explicitly considered here. The results are compared with those in accounts of a similar nature.
Global thermodynamics is the study of the relationships between the total mass, energy, and entropy of the atmosphere. The main result is an explicit, exact relation between the entropy, the kinetic energy, and the static entropic energy, a new form of static energy. The stability of atmospheric motions and the maintenance of the circulation are investigated with the total entropic energy (the sum of the kinetic and static entropic energies). The dynamical significance of Gibbs' concept of thermal equilibrium is that the total entropic energy decreases monotonically to zero in isolated atmospheres as a consequence of the Second Law. Furthermore, the generation of entropic energy is related specifically to the destruction of entropy by heating at high temperature relative to cooling at low temperature. An energy budget for zonal and eddy forms of entropic energy is constructed, and illustrates the catalytic effect for the general circulation of this destruction of entropy. Finally, the argument is reversed and a stability postulate is applied to the total entropic energy to infer the existence of an entropy function. The concept of entropic energy represents a new integral of the equations of atmospheric motion that combines the previous energy and entropy integrals into one statement, giving a mathematical version of the relations between entropy and motion. DOI: 10.1111/j.2153-3490.1973.tb01599.x
Thesis (Ph.D.)--Pennsylvania State University. Microfilm (positive).
An extremal principle is formulated and applied to a one-dimensional
energy balance model. The principle is based on the maximization of the
generation of available potential energy, and reduces to a maximization
of the correlation between heating and temperature deviation. The
maximization is subject to a constraint of global energy balance, and
makes no reference to a dynamical transport mechanism. The maximizing
temperature distribution has a larger meridional gradient than the
observed temperature distribution. Our results suggest the general
circulation operates at nearly its maximum efficiency.
An extremum principle is developed for radiative transfer in a gray
atmosphere by using a purely thermal example from Planck's (1913) work
on heat radiation. Entropy is accounted for, as is Prigogine's (1947,
1967) theorem describing equilibrium as a thermodynamic state of minimal
entropy. Attention is given to cases where thermodynamic equilibrium is
not present. An isothermal black slab heated on one side is considered.
The extremum is shown to be an unconstrained steady state and a minimum.
The steady state is also demonstrated not to be equilibrium and a
countering force on the slab is modeled. The discussion is extended to
local thermodynamic equilibrium and to absorption by two isothermal gray
slabs. Energy balance conditions for the latter are calculated in order
to obtain a generalized extremum. Constraints on the equilibrium are
found using irreversible thermodynamics. The problem of a gray
atmosphere is then approached in terms of a continuum of isothermal
slabs.
The rate of increase of the entropy of a simple viscous fluid, capable
of conducting heat, is investigated in greater detail than has hitherto
been customary. It is shown that, if Kelvin's hypothesis concerning the
absolute temperature be adopted, and the usual law of viscosity be
assumed, the requirements of the second law are satisfied.
It is suggested that worthwhile results may be obtained by the use of a
suitable form of the second law of thermodynamics for the calculation of
the efficiency of the earth's surface and atmosphere, considered as the
heat engine that is driving the large-scale atmospheric circulation. A
relation is obtained connecting the rate at which the kinetic energy of
the general circulation is dissipated by turbulence with the integral
over the atmosphere of the net rate of absorption of radiation divided
by the temperature at which it is absorbed. This is done on the
apparently plausible assumption that any other terms in the atmosphere's
entropy budget are relatively small. The method used is described, and
its application is indicated with the aid of a sample calculation using
illustrative data. It appears that the engine driving the atmospheric
circulation is, from one point of view, more efficient than might be
supposed.
Attempts to link the theory of irreversible thermodynamics to the study
of climate have utilized an entity which has been identified as the
entropy production rate. However, this entity does not properly account
for irreversibility due to the interaction of the radiation field with
matter, that is, changes in the entropy of the radiation field, which
cannot in general be properly described by the thermodynamic relations,
are not accounted for. Calculations show that the entity used in these
attempts deviates substantially from the correct rate of entropy
production, both in terms of magnitude and sensitivity.
Global thermodynamics is the study of the relationships between the total mass, energy, and entropy of the atmosphere. The main result is an explicit, exact relation between the entropy, the kinetic energy, and the static entropic energy, a new form of static energy. The stability of atmospheric motions and the maintenance of the circulation are investigated with the total entropic energy (the sum of the kinetic and static entropic energies). The dynamical significance of Gibbs' concept of thermal equilibrium is that the total entropic energy decreases monotonically to zero in isolated atmospheres as a consequence of the Second Law. Furthermore, the generation of entropic energy is related specifically to the destruction of entropy by heating at high temperature relative to cooling at low temperature. An energy budget for zonal and eddy forms of entropic energy is constructed, and illustrates the catalytic effect for the general circulation of this destruction of entropy. Finally, the argument is reversed and a stability postulate is applied to the total entropic energy to infer the existence of an entropy function. The concept of entropic energy represents a new integral of the equations of atmospheric motion that combines the previous energy and entropy integrals into one statement, giving a mathematical version of the relations between entropy and motion.
It can be proved for certain small-scale convective heat transfer processes that the preferred steady-state mode is one of maximum entropy production. The constraint is more or less equivalent to one of maximum kinetic energy dissipation or of maximum convective heat transport. Evidence is accumulating that the same constraint may apply on the much larger scale of the earth-atmosphere system. The concept is accepted as the basis of a purely thermodynamic model of the mean annual global climate. The model allows a prieri calculation of the broad-scale geographic distributions of cloud, surface temperature, horizontal energy fluxes in the ocean and in the atmosphere, net radiant energy inputs, etc. The agreement with observation strongly supports the basic concept. It suggests also that, to the extent allowed by the degrees of freedom in the dynamics, the partition of atmospheric and oceanic energy flow is determined by a requirement to equalize the local dissipations in the two media.
The entropy balance associated with a Budyko-Sellers climatic model is developed. It is shown that different regimes, associated with decreasing, as well as increasing values of entropy production (which measures the rate of dissipation in the system) in the course of time are possible. This immediately poses the problem of stability of steady states of the climatic system. An explicit criterion of climatic stability is thus derived, which is expressed in terms of thermodynamic quantities related to excess entropy production. The results are illustrated on simple cases involving diffusive energy transport. A comparison with Paltridge's minimum entropy exchange principle is also attempted.
It is found that the mean meridional distribution of temperature, cloud cover and meridional energy flux can be predicted with extraordinary accuracy by application of a simple minimum principle to a multi-box model of the globe which contains no direct specification of the system dynamics. The minimized quantity is related to the global net rate of production of entropy. It is the sum over all latitude zones of the ratio of net radiant energy input to the effective emission temperature of the zone. The result suggests that global dynamics is something of a passive variable which alters so as to satisfy a condition akin to minimum energy dissipation.
Even though irreversible thermodynamics is meant to be a universal theory for nonequilibrium processes, that theory has not taken nonequilibrium radiation processes, which can normally be ignored, into account; thus it is primarily a theory for matter. When the simple bilinear entropy production rate of irreversible thermodynamics is modified, to account for the entropy production due to radiation while preserving local equilibrium for matter, the simple bilinear form of that rate, on which the current theory depends, is lost. Despite substantial compromises, generalized definitions for forces and fluxes cannot restore that simple homogeneous bilinear form, nor do they succeed in generating even generalized bilinearity in the entropy production rates for some problems concerning radiation. The possibility of a broader nonequilibrium theory is suggested by radiation problems, characterized by an example due to Planck, where those generalizations fail and the entropy production rate is a minimum in the steady state.
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 1947. Vita. Includes bibliographical references (leaves 106-111). by Herbert B. Callen. Ph.D.
Entropy and climate, " in: Man's Impact on Climate
- H Fortak
Fortak, H., " Entropy and climate, " in: Man's Impact on Climate (W. Bach, J. Pankrath and W. Kellog, ed.), Elsevier, Amsterdam (1979).
On the treatment of radiation in the entropy budget of the earth-atmosphere system, " in: New Perspectives in Climate Modelling
- U Callies
- F Herbert
Callies, U. and Herbert, F., " On the treatment of radiation in the entropy budget of the earth-atmosphere system, " in: New Perspectives in Climate Modelling (A. L. Berger and C. Nicolis, eds.), Elsevier, Amsterdam (1984).