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!

Moon_Astronomy_Club

(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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