Basic Equations of Ideal One-Fluid Magnetohydrodynamics: (Part V) The Energy Equations and Summary

SOURCE FOR CONTENT: Priest E., Magnetohydrodynamics of the Sun, 2014. Ch. 2. Cambridge University Press.

 

The final subset of equations deals with the energy equations. My undergraduate research did not take into account the thermodynamics of conducting fluid in order to keep the math relatively simple. However, in order to understand MHD one must take into account these considerations. Therefore, there are three essential equations that are indicative of the energy equations:

I. Heat Equation:

We may write this equation in terms of the entropy S as

\displaystyle \rho T \bigg\{\frac{\partial S}{\partial t}+(\nabla \cdot \textbf{v})S\bigg\}=-\mathcal{L}, (1)

where \mathcal{L} represents the net effect of energy sinks and sources and is called the energy loss function. For simplicity, one typically writes the form of the heat equation to be

\displaystyle \frac{\rho^{\gamma}}{\gamma -1}\frac{d}{dt}\bigg\{\frac{P}{\rho^{\gamma}}\bigg\}=-\mathcal{L}. (2)

2. Conduction

For this equation one considers the explicit form of the energy loss function as being

\displaystyle \mathcal{L}=\nabla \cdot \textbf{q}+L_{r}-\frac{J^{2}}{\sigma}-F_{H}, (3)

where \textbf{q} represents heat flux by particle conduction, L_{r} is the net radiation, J^{2}/\sigma is the Ohmic dissipation, and F_{H} represents external heating sources, if any exist.  The term \textbf{q} is given by

\textbf{q}=-\kappa \nabla T, (4)

where \kappa is the thermal conduction tensor.

3. Radiation

The equation for radiation can be written as a variation of the diffusion equation for temperature

\displaystyle \frac{DT}{Dt}=\kappa \nabla^{2}T (5)

where \kappa here denotes the thermal diffusivity given by

\displaystyle \kappa = \frac{\kappa_{r}}{\rho c_{P}}. (6)

We may write the final form of the energy equation as

\displaystyle \frac{\rho^{\gamma}}{\gamma-1}\frac{d}{dt}\bigg\{\frac{P}{\rho^{\gamma}}\bigg\}=-\nabla \cdot \textbf{q}-L_{r}+J^{2}/\sigma+F_{H}, (7)

where \textbf{q} is given by Eq.(4).

As far as my undergraduate research is concerned, I am including these equations to be complete.

 

So to summarize the series so far, I have derived most of the basic equations of ideal one-fluid model of magnetohydrodynamics. The equations are

\displaystyle \frac{\partial \textbf{B}}{\partial t}=(\textbf{v}\times \textbf{B})+\lambda \nabla^{2}\textbf{B}, (A)

\displaystyle \frac{\partial \textbf{v}}{\partial t}+(\nabla \cdot \textbf{v})\textbf{v}=-\frac{1}{\rho}\nabla\bigg\{P+\frac{B^{2}}{2\mu_{0}}\bigg\}+\frac{(\nabla \cdot \textbf{B})\textbf{B}}{\mu_{0}\rho}, (B)

\displaystyle \frac{\partial \rho}{\partial t}+(\nabla \cdot \rho\textbf{v})=0, (C)

\displaystyle \frac{\partial \Omega}{\partial t}+(\nabla \cdot \textbf{v})\Omega = (\nabla \cdot \Omega)\textbf{v}+\nu \nabla^{2}\Omega, (D)

\displaystyle P = \frac{k_{B}}{m}\rho T = \frac{\tilde{R}}{\tilde{\mu}}\rho T, (E) (Ideal Gas Law)

and

\displaystyle \frac{\rho^{\gamma}}{\gamma-1}\frac{d}{dt}\bigg\{\frac{P}{\rho^{\gamma}}\bigg\}=-\nabla \cdot \textbf{q}-L_{r}+J^{2}/\sigma +F_{H}.  (F)

We also have the following ancillary equations

\displaystyle (\nabla \cdot \textbf{B})=0, (G.1)

since we haven’t found evidence of the existence of magnetic monopoles. We also have that

\displaystyle \nabla \times \textbf{B}=\mu_{0}\textbf{J}, (G.2)

where we are assuming that the plasma velocity v << c (i.e. non-relativistic). Finally for incompressible flows we know that (\nabla \cdot \textbf{v})=0 corresponding to isopycnal flows.

 

In the next post, I will discuss some of the consequences of these equations and some elementary theorems involving conservation of magnetic flux and magnetic field line topology.

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