Abstract
General characteristics of entropy production in a fluid system are investigated from a thermodynamic viewpoint. A basic expression for entropy production due to irreversible transport of heat or momentum is formulated together with balance equations of energy and momentum in a fluid system. It is shown that entropy production always decreases with time when the system is of a pure diffusion type without advection of heat or momentum. The minimum entropy production (MinEP) property is thus intrinsic to a pure diffusion-type system. However, this MinEP property disappears when the system is subject to advection of heat or momentum due to dynamic motion. When the rate of advection exceeds the rate of diffusion of heat or momentum, entropy production tends to increase over time. The maximum entropy production (MaxEP), suggested as a selection principle for steady states of nonlinear non-equilibrium systems, can therefore be understood as a characteristic feature of systems with dynamic instability. The observed mean state of vertical convection of the atmosphere is consistent with the condition for MaxEP presented in this study.
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Notes
- 1.
- 2.
Assuming linearity, \( {\boldsymbol{\Uppi}}:\nabla {\mathbf{v}} = \, [ 2\mu (\nabla {\mathbf{v}})^{\text{s}} - \left( { 2/ 3} \right)\mu (\nabla \cdot {\mathbf{v}}){\boldsymbol{\delta}} \left]: \right[(\nabla {\mathbf{v}})^{\text{s}} \, + \, \, (\nabla {\mathbf{v}} )^{\text{a}} ] \, = { 2}\mu (\nabla {\mathbf{v}})^{\text{s}} :(\nabla {\mathbf{v}})^{\text{s}} - \left( { 2/ 3} \right)\mu (\nabla \cdot {\mathbf{v}})^{ 2},\) with δ denoting the unit tensor, and T s and T a denoting symmetric and asymmetric parts of a tensor T. Then, \( \partial ({\boldsymbol{\Uppi}}:\nabla {\mathbf{v}})/\partial t = { 2}[ 2\mu (\nabla {\mathbf{v}})^{\text{s}} - \left( { 2/ 3} \right)\mu (\nabla \cdot {\mathbf{v}}){\boldsymbol{\delta}}\left]: \right[\nabla (\partial {\mathbf{v}}/\partial {\text{t}})]^{\text{s}} = 2{\boldsymbol{\Uppi}}:\nabla (\partial {\mathbf{v}}/\partial {t}) \).
- 3.
In a general case, the forth term in the right-hand side of Eq. (6.12) should be expressed as a sum of the viscosity μ and the second viscosity λ. Using Stokes’ relation (λ = −2μ/3), μ + λ = μ/3.
- 4.
There are a few exceptions. Laminar (or non-turbulent) flow can be realized even with advection of momentum, e.g., in a converging nozzle. However, the flow direction is not parallel in this case, and it may not be regarded as “laminar” in the strict sense of the word.
- 5.
The exact correspondence between the time evolution of a system and the probability of states requires an additional assumption, which is related to profound and not yet fully solved problems of the ergodic hypothesis.
- 6.
There is also generation of mechanical energy by volume expansion of the air at the surface. The rate of energy conversion is about 2 W m−2, which thereafter dissipates into heat in the atmosphere. We have included this rate in the convective energy transport considered here.
Abbreviations
- A :
-
Surface of a system or the Earth (m2)
- c v :
-
Specific heat at constant volume (J K−1 kg−1)
- e :
-
Unit vector (–)
- F c :
-
Convective heat flux density (sensible and latent heat) (J m−2 s−1)
- F r :
-
Radiation flux density (J m−2 s−1)
- F LW :
-
Longwave radiation flux density (J m−2 s−1)
- F SW :
-
Shortwave radiation flux density (J m−2 s−1)
- J i :
-
Diffusive flux density of i-th component
- J h :
-
Diffusive flux density of heat (J m−2 s−1)
- J m :
-
Diffusive flux density of momentum (kg m−2 s−1)
- k :
-
Thermal conductivity (J m−1 K−1 s−1)
- L h :
-
Kinetic coefficient for heat diffusion (J m−1 K s−1)
- n :
-
Unit vector normal to system’s surface (–)
- p :
-
Pressure (Pa)
- t :
-
Time (s)
- T :
-
Temperature (K)
- T e :
-
Effective radiation temperature at the top of the atmosphere (K)
- T r :
-
Effective radiation temperature (K)
- T s :
-
Surface temperature (K)
- T sun :
-
Emission temperature of the sun (K)
- V :
-
Volume of a system (m3)
- v :
-
Velocity (m s−1)
- X i :
-
Gradient of intensive variable for i-th diffusive flux
- δ :
-
Unit tensor (–)
- \( \kappa \) :
-
Thermal diffusivity (m2 s−1)
- \( \lambda \) :
-
Second viscosity (kg m−1 s−1)
- \( \mu \) :
-
Viscosity (kg s−1 m−1)
- \( \nu \) :
-
Kinematic viscosity (m2 s−1)
- \( {\boldsymbol{\Uppi}} \) :
-
Viscous stress tensor (Pa)
- \( \rho \) :
-
Density (kg m−3)
- σB :
-
Stefan–Boltzmann constant ≈ 5.67 × 10−8 (J m−2 K−4 s−1)
- \( \dot{\sigma } \) :
-
Rate of entropy production (J K−1 s−1)
- \( \dot{\sigma }_{\text{conv}} \) :
-
Rate of entropy production due to convective heat flux (J K−1 s−1)
- \( \dot{\sigma }_{\text{h}} \) :
-
Rate of entropy production due to heat diffusion (J K−1 s−1)
- \( \dot{\sigma }_{\text{m}} \) :
-
Rate of entropy production due to momentum diffusion (J K−1 s−1)
- \( \dot{\sigma }_{\text{rad}} \) :
-
Rate of entropy production due to absorption of radiation (J K−1 s−1)
- \( \dot{\sigma }_{\text{tot}} \) :
-
Total rate of entropy production in the atmosphere (J K−1 s−1)
- stat:
-
Static state with no motion
- lam:
-
Laminar flow state
- adv:
-
State with advection
References
Ziegler, H.: Zwei Extremalprinzipien der irreversiblen Thermodynamik. Ing. Arch. 30, 410–416 (1961)
Paltridge, G.W.: Global dynamics and climate—a system of minimum entropy exchange. Q. J. Roy. Meteorol. Soc. 101, 475–484 (1975)
Paltridge, G.W.: The steady-state format of global climate. Q. J. Roy. Meteorol. Soc. 104, 927–945 (1978)
Schneider, E.D., Kay, J.J.: Life as a manifestation of the second law of thermodynamics. Math. Comput. Model. 19, 25–48 (1994)
Ozawa, H., Shimokawa, S., Sakuma, H.: Thermodynamics of fluid turbulence: a unified approach to the maximum transport properties. Phys. Rev. E 64, 026303 (2001)
Lorenz, R.D., Lunine, J.I., Withers, P.G., McKay, C.P.: Titan, Mars and Earth: entropy production by latitudinal heat transport. Geophys. Res. Lett. 28, 415–418 (2001)
Shimokawa, S., Ozawa, H.: On the thermodynamics of the oceanic general circulation: irreversible transition to a state with higher rate of entropy production. Q. J. Roy. Meteorol. Soc. 128, 2115–2128 (2002)
Shimokawa, S., Ozawa, H.: Thermodynamics of irreversible transitions in the oceanic general circulation. Geophys. Res. Lett. 34, L12606 (2007). doi:10.1029/2007GL030208
Hill, A.: Entropy production as the selection rule between different growth morphologies. Nature 348, 426–428 (1990)
Nohguchi, Y., Ozawa, H.: On the vortex formation at the moving front of lightweight granular particles. Physica D 238, 20–26 (2009)
Lorenz, E.N.: Generation of available potential energy and the intensity of the general circulation. In: Pfeffer, R.L. (ed.) Dynamics of Climate, pp. 86–92. Pergamon, Oxford (1960)
Ozawa, H., Ohmura, A., Lorenz, R.D., Pujol, T.: The second law of thermodynamics and the global climate system: a review of the maximum entropy production principle. Rev. Geophys. 41, 1018 (2003). doi:10.1029/2002RG000113
Dewar, R.: Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states. J. Phys. A 36, 631–641 (2003)
Niven, R.K.: Steady state of a dissipative flow-controlled system and the maximum entropy production principle. Phys. Rev. E 80, 021113 (2009)
Prigogine, I.: Modération et transformations irréversibles des systémes ouverts. Bulletin de la Classe des Sciences. Academie Royale de Belgique 31, 600–606 (1945)
Glansdorff, P., Prigogine, I.: Sur les propriétés différentielles de la production d’entropie. Physica 20, 773–780 (1954)
De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North-Holland, Amsterdam (1962)
Gyarmati, I.: Non-equilibrium Thermodynamics. Field Theory and Variational Principles. Springer, Berlin (1970). [First published in Hungarian, 1967]
Prigogine, I.: From Being to Becoming. Time and Complexity in the Physical Sciences. Freeman, San Fransisco (1980)
Sawada, Y.: A thermodynamic variational principle in nonlinear non-equilibrium phenomena. Prog. Theor. Phys. 66, 68–76 (1981)
Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Pergamon Press, Oxford (1987). [First published in Russian (1944)]
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)
Helmholtz, H.: Zur Theorie der stationären Ströme in reibenden Flüssigkeiten. Verhandlungen des naturhistorisch-medicinischen Vereins zu Heidelberg 5, 1–7 (1869)
Rayleigh, Lord.: On the motion of a viscous fluid. Phil. Mag. 26, 776–786 (1913)
Suzuki, M., Sawada, Y.: Relative stabilities of metastable states of convecting charged-fluid systems by computer simulation. Phys. Rev. A 27, 478–489 (1983)
Kawazura, Y., Yoshida, Z.: Entropy production rate in a flux-driven self-organizing system. Phys. Rev. E 82, 066403 (2010)
Malkus, W.V.R.: Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539 (1956)
Busse, F.H.: Bounds for turbulent shear flow. J. Fluid Mech. 41, 219–240 (1970)
Malkus, W.V.R.: Borders of disorder: in turbulent channel flow. J. Fluid Mech. 489, 185–198 (2003)
Paulus Jr, D.M., Gaggioli, R.A.: Some observations of entropy extrema in fluid flow. Energy 29, 2487–2500 (2004)
Niven, R.K.: Simultaneous extrema in the entropy production for steady-state fluid flow in parallel pipes. J. Non-Equilib. Thermodyn. 35, 347–378 (2010)
Ozawa, H., Ohmura, A.: Thermodynamics of a global-mean state of the atmosphere —a state of maximum entropy increase. J. Clim. 10, 441–445 (1997)
Heisenberg, W.: Nonlinear problems in physics. Phys. Today 20(5), 27–33 (1967)
Acknowledgments
The authors wish to express their cordial thanks to the organizers of the MaxEP 2011 Workshop in Canberra where the authors’ interest in this subject has been stimulated. Valuable comments from two anonymous reviewers are also gratefully acknowledged.
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Ozawa, H., Shimokawa, S. (2014). The Time Evolution of Entropy Production in Nonlinear Dynamic Systems. In: Dewar, R., Lineweaver, C., Niven, R., Regenauer-Lieb, K. (eds) Beyond the Second Law. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40154-1_6
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