Large-Scale Inhomogeneous Thermodynamics
--And applications for atmospheric energetics
Yong Zhu
Chapter 1 Introduction
In the study of atmospheric thermodynamics, an air parcel, if it is sufficiently small, may be assumed as a classical thermodynamic system (such as a small piece of ideal gas), of which the thermodynamic equilibrium state is uniform. Variations in the thermodynamic state may be illustrated by the state equation and the first law of thermodynamics (Dufour and van Mieghem, 1975; Iribarne and Godson, 1981). When the atmosphere is considered as a single thermodynamic system, additional assumptions and relationships are applied, since the atmosphere manifests the particular features different from those of a classical thermodynamic system. These features are summarized briefly in the following.
Firstly, we never see a uniform state of the atmosphere without parcel motions attained by molecular diffusions. The steady states viewed from the datasets over meteorological scales are inhomogeneous, owing to the gravitation and rotation of the Earth and the heat exchanges between the atmosphere and surroundings. Thus, we may not assume a uniform state to represent an equilibrium state of the atmosphere in the study of atmospheric dynamics and thermodynamics. The zeroth law of classical thermodynamics cannot be applied for the inhomogeneous thermodynamic system. The thermodynamic processes, such as energy conversions, must be studied between inhomogeneous equilibrium states of the atmosphere. Unlike in the classical thermodynamic systems, the inhomogeneous equilibrium state attained with conservation of mass and energy may not be unique. The weather predictions are actually looking for one of the equilibrium states according to the initial conditions and physical processes provided.
Secondly, the atmosphere in the gravitational field possesses geopotential energy as well as internal energy and kinetic energy. The energy conservation law of the whole atmosphere is different from the first law of classical thermodynamics or the energy conservation law of ideal gases, as it includes also the kinetic energy equation derived from the atmospheric momentum equation (Brunt, 1944; Starr, 1951; Hess, 1959). According to this law, the mechanic work and parcel kinetic energy may be converted from the geopotential energy and internal energy through the processes inside the atmosphere. Thus, the atmosphere reduces its height of gravity center or increases its static stability when producing kinetic energy in the dry processes. For a provided initial state, the atmosphere possesses multiple barotropic reference states with different static stabilities and energy partitions. A real reference state attained by energy conversion depends on the process as well as initial state.
Thirdly, the development of vertical motions in a fluid is produced by the buoyancy force doing mechanic work for fluid parcels. The buoyancy force on air parcels is zero in the statically stable atmosphere, as the downward gravitational force is balanced by the upward pressure gradient force. Thus, the transfer of geopotential energy and internal energy into kinetic energy may take place only in the statically unstable atmosphere if it is barotropic. In other words, the energy conversions in the atmosphere are restricted by the static stability, which depends on not only the intensity but also the direction of vertical gradient of potential temperature. While, the thermodynamic entropy is independent of the gradient direction. Therefore, the ability of energy conversion in the atmosphere may not be measured by the classical thermodynamic entropy, which depends on magnitude of the gradient only.
Fourthly, the disorderliness of atmosphere depends not only on the distribution of molecules at different microscopic energy levels, but also on the distribution of parcels with different potential temperatures. While, the classical thermodynamic entropy is calculated by integrating the entropy for each element of volume or mass, and so depends on the thermodynamic state of each element instead of the element distribution. In the reversible adiabatic processes with conservation of parcel potential temperature, the classical thermodynamic entropy is conserved, but the thermodynamic disorderliness of atmosphere may nevertheless be increased irreversibly by turbulent parcel motions. Thus, a reversible process predicted by the second law of thermodynamics may not be really reversible in the inhomogeneous fluid system. The thermodynamic irreversibility related to the turbulent diffusions may be independent of that related to molecular diffusions. Also, the extremal state with maximum thermodynamic entropy attained through parcel motions are different from the isothermal state attained by molecular diffusions. As the isothermal heat-death state is seldom observed and is not interested for the study of energy conversion in the atmosphere, some particular physical relationship should be used to filter out the dead reference state.
Fifthly, the conversion of geopotential energy in the atmosphere is irreversible in isolation with increasing the static stability or the vertical gradients of air density and potential temperature. These reductions in the vertical disorderliness resulting from parcel motions are different from the changes of horizontal disorderliness. When the statically unstable atmosphere becomes stable by adiabatic parcel convection, the thermodynamic entropy may not change but the process is irreversible. Thus, there are thermodynamically and geopotentially irreversible processes in the atmosphere related to the molecular or turbulent diffusion and geopotential energy conversion respectively. These processes may be independent of each other, and so the thermodynamic entropy is no longer a perfect discriminate parameter for the variation tendency of the atmosphere. The adiabatic parcel exchanges in the vertical direction cannot happen in the statically stable atmosphere, though the thermodynamic entropy is conserved. Also, the statically stable atmosphere cannot reduce its static stability by parcel motions, thought the entropy increases in the process. Thus, we need another discriminate parameter together with the thermodynamic entropy to indicate the variation tendency and irreversibility of the geophysical and thermodynamic system.
Sixthly, an isolated small piece of gas may reach the uniform equilibrium state with maximum thermodynamic entropy through molecular diffusions. While, the major processes on the meteorological scales are related to parcel motions driven by external Newtonian forces in the atmosphere. Under the effect of the gravitation, the atmosphere tends to lower its gravity center, or increase its vertical disorderliness in mass density and potential temperature. In other words, the energy conversions occur in an unstable atmosphere drive the geophysical fluid system to the equilibrium state possessing maximum kinetic energy, minimum geopotential energy and thermodynamic entropy, compared with other equilibrium states. The isothermal heat-death atmosphere with maximum thermodynamic entropy may not be observed. Thus, there is the principle of minimum thermodynamic entropy exchange which can be applied to depict the climatological mean fields (Paltridge, 1975, 1978). This principle makes the studies on the maximum available potential energy or kinetic energy generation practically meaningful. However, owing to some other dynamic equilibrium constraints, such as the geostrophic balance and thermal wind balance, the climatological mean state of the open atmosphere is different from the extremal state of minimum thermodynamic entropy. It possesses negative entropy sources, and so the energy conversions in the atmosphere are always possible.
These major differences discussed above make the atmosphere a particular thermodynamic system in the gravitational field, which is different from the classical thermodynamic systems studied by the classical thermodynamics. Also, it is necessary to create a new thermodynamics to study the thermodynamic processes in the large-scale compressible geophysical fluid. The atmospheric energetics is an application of the new thermodynamics. To introduce the new thermodynamics, we review simply in the next chapter two basic systems studied in the classical physics.
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