Appendix A: Minimum entropy production rate

General equation for the entropy production rate is given by (Landau & Lifshits 1987):

 

$$% \frac{\partial S}{\partial t}\,= \int \frac{\kappa ({\rm grad}\... ...\right)^2\, {\rm d}V\,+ \int \frac{\xi}{T}\,({\rm div}\, {\bf v})^2\, {\rm d}V.$$ (A.1)

Here $\partial S/\partial t$ is the rate of the total entropy production in the system, the velocity v and temperature T are functions of the coordinate (x,y,z), the volume element is denoted by dV, the thermal conductivity, dynamic viscosity, and the second viscosity coefficients - by $\kappa$, $\eta$, and $\xi$, respectively.

The terms included in this equation account for only hydro- and thermodynamic processes since we assume the ionization balance in each point of the region which means that the radiative heating and cooling do not contribute to the entropy production. The second viscosity equals zero for dilute monoatomic gases (Chapman & Cowling 1970) so we omit it from further consideration.

We also assume that heat conductivity $\kappa$ and viscosity $\eta$ are dominated by turbulence, and hence the Prandtl number is about unity (Monin & Yaglom 1975):

$$% {\rm Pr} = \frac{\eta\, C_{\rm p}}{\kappa} \simeq 1,$$ (A.2)

where $C_{\rm p}$ is a specific heat capacity at constant pressure. The relation (A.2) is valid for both the ionized and neutral gas.

From observations, only one component v_x (along the sightline) of the velocity vector v is known. Therefore we are compelled to neglect in (A.1) all terms including derivatives other than $\partial v_x/\partial x$. The second right hand term in (A.1) can be re-written in the form:

 

$$% \int \frac{\eta}{2T}\,\left(\frac{\partial v_i}{\partial x_k}\,... ...frac{4}{3}\int \frac{\eta}{2T}\,\left({\rm div}\, {\vec v}\right)^2\, {\rm d}V.$$ (A.3)

After all these assumptions, we obtain the following simplified 1D form for the entropy production rate

 

$$% \frac{\partial \tilde{S}}{\partial t}\,= \int \kappa \left[ \fr... ...tilde{T}}\, \left(\frac{{\rm d}\tilde{v}_x}{{\rm d}x}\right)^2\right] {\rm d}x,$$ (A.4)

where $\tilde{v}_x = v_x/\sigma_{\rm v}$, $\tilde{T} = T/T_0$, with $\sigma_{\rm v}$ and T₀ being scales (characteristic values) for velocity and temperature inside the considered region, ${\cal A}$ is a constant of about 1, and $\tilde{S}$ is the entropy per unit area.

For a monatomic ideal gas $C_{\rm p} = 5nR/2$ (n is the number of gram-moles and R is the universal gas constant), and if the gas is fully ionized then $\kappa \propto T^{5/2}$ (e.g. Lang 1999).

Given the values of $\kappa$, Pr, T₀, $C_{\rm p}$, and $\sigma_{\rm v}$, we can rearrange the computational points $\{ v_1, \ldots, v_k \}$ and $\{ n_1, \ldots, n_k \}$ in such a manner, that minimum of (A.4) will be achieved. Minimization of (A.4) was carried out by means of combinatorial simulated annealing technique (Press et al. 1992). The obtained configuration of the density and velocity distributions should have the least dissipation and therefore will exist longer than all others.