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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.

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

The field in which the interaction of electrically conducting fluids (i.e. plasmas) and magnetic fields are studied is called magnetohydrodynamics (MHD). As an undergraduate, I investigated the MHD processes occurring in the core of Jupiter. The project had two components: calculation of the magnetic field by coding via MATLAB and investigating the magnetohydrodynamics of the metallic hydrogen in the jovian interior.

The purpose of this post is to derive the the induction equation of MHD from first principles as well as to describe one of my undergraduate research topics.

We start with Maxwell’s equations of electrodynamics:

$\nabla \cdot \textbf{E}=\frac{\rho}{\epsilon_{0}}, (1)$

$\nabla \cdot \textbf{B}=0, (2)$

$\nabla \times \textbf{E}=-\frac{\partial \textbf{B}}{\partial t}, (3)$

$\nabla \times \textbf{B}=\mu_{0}\textbf{J}+\mu_{0}\epsilon_{0}\frac{\partial \textbf{E}}{\partial t}. (4)$

In the above equations, $\textbf{E}$ is the electric field, $\textbf{B}$ is the magnetic field, $\rho$ is the charge density, $\epsilon_{0}$ and $\mu_{0}$ are the permittivity and permeability of free space, respectively, and $\nabla$ is the del differential operator. In the context of ideal MHD, we assume that the displacement current (the second term in Eq.(4)) becomes negligible since the velocities considered are believed to be non-relativistic. Therefore, Eq.(4) becomes

$\nabla \times \textbf{B}=\mu_{0}\textbf{J}. (5)$

A few ancillary expressions are required at this point, namely: Ohm’s law $\textbf{J}=\sigma \textbf{E}$ and the Lorentz force $\textbf{F}=q(\textbf{E}+\textbf{v}\times\textbf{B})$ . Combining these two gives

$\textbf{J}=\sigma(\textbf{E}+\textbf{v}\times\textbf{B}). (6)$

Solving for $\textbf{J}$ in Eq.(5) and substitution into Eq.(6) and then solving for the electric field $\textbf{E}$ yields

$\textbf{E}=\frac{1}{\mu_{0}\sigma}(\nabla \times \textbf{B})-(\textbf{v}\times\textbf{B}), (7)$

Taking the curl of the electric field above, while noting the identity $\nabla \times (\nabla \times \textbf{A})=\nabla(\nabla \cdot \textbf{A})-\nabla^{2}\textbf{A}$ we get the induction equation

$\frac{\partial \textbf{B}}{\partial t}= (\textbf{v}\times \textbf{B})+\lambda \nabla^{2}\textbf{B}, (8)$

where $\lambda$ is the magnetic diffusivity.

The induction equation is the first of the basic equations of ideal one-fluid MHD.

An Introduction…

A bit of background about myself: Since my sophomore year of high school I have been interested in astronomy, physics, and mathematics. I received my Bachelor’s degree in Earth and Planetary Sciences concentrating in astronomy/astrophysics from Western Connecticut State University. My coursework and independent readings ultimately led to minors in mathematics and physics.

My intention for this blog is to serve as a reference in astrophysics and related topics to myself as well as others. I aim to share my own research interests and consider selected problems that have fascinated me. I also hope to communicate recent news in the fields of physics and astronomy and discuss the implications of discoveries made.

DISCLAIMER: I am by no means an expert, and as such the posts that I create are of my opinion and my own logic. I may be wrong sometimes, and I hope that the people who see this (assuming that anyone sees this) will respect that.

That being said… Enjoy!

(ABOVE: An image of the moon taken with a lunar and planetary imaging camera mounted to a Newtonian 130mm reflecting telescope.)

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