The entropic energy of geophysical fluid systems

Abstract

The main objective of this work is to develop an integral theory for a wide class of fluid systems that not only includes the concepts of balance of mass, momentum, and energy, but also of entropy and the second law of thermodynamics. This theory provides an explicit relationship between the integrals of kinetic energy and of entropy in terms of a new energy form, the entropic energy.

Linearly viscous fluids whose entropy functions are strictly concave are defined as equilibrium regular fluids. A fluid system consists of an equilibrium regular fluid of fixed mass M, whose body force is a consequence of a steady, positive, monotonic potential field, and whose bounds satisfy certain specified conditions. Geophysical fluid systems are a non-degenerate set of fluid systems.

In geophysical fluid systems there exists a unique state for given values of system mass and system total energy E, called the associated equilibrium state, that is motionless, hydrostatic, and isothermal (Theorem 2.1). The product of the temperature T0 of this state and the difference S0 – S in system entropies between it and a natural state is equal to the entropic energy N, the sum of the system kinetic energy K and a static energy form T₀. The entropic energy is a positive measure, resembling an L₂-norm, of the deviation of the natural state from its associated equilibrium state, and thus the associated equilibrium state is thermodynamically stable in Gibbs sense (Theorem 2.2).

If a geophysical fluid system is in isolation (E = 0, S ≥ 0) entropic energy is bounded by its value at the time isolation occurs and its rate of change can never become positive for as long as the system remains in isolation (Theorem 2.3). When a system is energetically isolated (E = 0), the rate of change of circulation intensity (as measured by ≥) is controlled only by the rate of entropy production, and necessary conditions for an increase or maintenance of intensity in certain physically-realistic systems can be formulated solely in terms of the average temperatures in regions of heating and cooling (Theorem 3.3).

Finally, suppose the sum of the convergences of radiation and heat fluxes vanishes somewhere in the domain of a geophysical fluid system. Then, whether in energetic or complete isolation or neither, the system must contain accelerations or motions at a given instant if its rate of change of entropic energy is non-zero at that instant (Theorem 3.1). Moreover, the system must be in motion on part of an interval of time if its entropic energy is constantly changing on that interval (Theorem 3.2).