Basic Equations of Ideal One-Fluid Magnetohydrodynamics (Part II)

Continuing with the derivation of the ideal one-fluid MHD equations, the next equation governs the motion of a parcel of fluid (in this case plasma). This momentum equation stems from the Navier-Stokes’ equation. The derivation of this equation will be reserved for a future post. However, the solution of this equation will not be attempted. (Incidentally, the proof of the existence and uniqueness of solutions to the Navier-Stokes’ equation is one of the Millennium problems described by the Clay Institute of Mathematics.)

The purpose of this post is to derive the momentum equation from the Navier-Stokes’ equation.

The Navier-Stokes’ equation has the form

$\frac{\partial \textbf{v}}{\partial t} + (\nabla \cdot \textbf{v})\textbf{v} = \textbf{F} - \frac{1}{\rho}\nabla P+\nu \nabla^{2}\textbf{v}, (1)$

where $\textbf{F}$ represents a source of external forces, $\textbf{v}$ is the velocity field, $\nabla P$ is the pressure gradient, $\rho$ is the material density, $\nu$ is kinematic viscosity, and $\nabla^{2}\textbf{v}$ is the laplacian of the velocity field. More specifically, it is a consequence of the viscous stress tensor whose components can cause the parcel of fluid to experience stresses and strains.

Defining the magnetic force per unit volume as

$\textbf{F}=\frac{\textbf{J}\times \textbf{B}}{\rho}, (2)$

where we recall that $\textbf{J}$ is the current density defined by Ohm’s law in a previous post, and also recall from Basic Equations of Ideal One-Fluid Magnetohydrodynamics (Part I) the equation

$\nabla \times \textbf{B}=\mu_{0}\textbf{J}, (3)$

if we solve for the current density $\textbf{J}$ , we get

$\textbf{J}=\frac{1}{\mu_{0}\rho}[(\nabla \times \textbf{B})\times \textbf{B}], (4)$

where $\mu_{0}$ is the permeability of free space. Now we invoke the vector identity

$[(\nabla \times \textbf{B})\times \textbf{B}]=(\nabla \cdot \textbf{B})\textbf{B}-\nabla \bigg\{\frac{B^{2}}{2}\bigg\}. (5)$

At this point, we assume that we are dealing with laminar flows in which case the term $\nu=0.$ The Navier-Stokes’ equation becomes the Euler equation at this point. (Despite great understanding of classical mechanics, one phenomena for which we cannot account is turbulence and sources of friction, so this assumption is made out of necessity as well as simplicity. For processes in which turbulence cannot be neglected the best we can do in this regard is to parameterize turbulence in numerical models.) Using the vector identity as well as making use of our assumption of laminar flows, ideal one-fluid momentum equation is

$\frac{\partial \textbf{v}}{\partial t}+ (\nabla \cdot \textbf{v})\textbf{v}=-\frac{1}{\rho}\nabla \bigg\{P + \frac{B^{2}}{2\mu_{0}}\bigg\}+\frac{(\nabla \cdot \textbf{B})\textbf{B}}{\mu_{0}\rho}, (6)$

where the additive term to the pressure $\frac{B^{2}}{2\mu_{0}}$ is the magnetic pressure exerted on magnetic field lines and the additional term $\frac{(\nabla \cdot \textbf{B})\textbf{B}}{\mu_{0}\rho}$ is the magnetic tension acting along the magnetic field lines.

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