Helicity and Magnetic Helicity

SOURCE FOR CONTENT: Davidson, P.A., 2001, An Introduction to Magnetohydrodynamics. 3.  The purpose of this post is to introduce the concepts of helicity as an integral invariant and its magnetic analog. More specifically, I will be showing that is invariant and thus correlates to the conservation of vortex line topology using the approach given in […]

“Proof” of Alfven’s Theorem of Flux Freezing

SOURCE FOR CONTENT: Choudhuri, A.R., 2010. Astrophysics for Physicists. Ch. 8.  In the previous post we saw the consequences of different regimes of the magnetic Reynolds’ number under which either diffusion or advection of the magnetic field dominates. In this post, I shall be doing a “proof” of Alven’s Theorem of Flux Freezing. (I hesitate to call […]

Deriving the Bessel Function of the First Kind for Zeroth Order

NOTE: I verified the solution using the following text: Boyce, W. and DiPrima, R. Elementary Differential Equations.  In this post, I shall be deriving the Bessel function of the first kind for the zeroth order Bessel differential equation. Bessel’s equation is encountered when solving differential equations in cylindrical coordinates and is of the form where describes […]

Basics of Tensor Calculus and General Relativity-Vectors and Introduction to Tensors (Part II-Continuation of Vectors)

SOURCE FOR CONTENT: Neuenschwander, D.E., 2015. Tensor Calculus for Physics. Johns Hopkins University Press. Ch.1 In the preceding post of this series, we saw how we may define a vector in the traditional sense. There is another formulation which is the focus of this post. One becomes familiar with this formulation typically in a second course […]

Consequences and some Elementary Theorems of the Ideal One-Fluid Magnetohydrodynamic Equations

SOURCE FOR CONTENT: Priest, E. Magnetohydrodynamics of the Sun, 2014. Cambridge University Press. Ch.2.; Davidson, P.A., 2001. An Introduction to Magnetohydrodynamics. Ch.4.  We have seen how to derive the induction equation from Maxwell’s equations assuming no charge and assuming that the plasma velocity is non-relativistic. Thus, we have the induction equation as being Many texts […]

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 […]