FEATURED IMAGE CREDIT: U.R. Christensen from the Nature article: “Earth Science: A Sheet-Metal Dynamo”
The image shows the overall distortion of magnetic field lines inside the core and its’ effect on the magnetic field outside.
This post shall continue to derive the principal equations of ideal one-fluid magnetohydrodynamics. Here I shall derive the continuity equation and the vorticity equation. Furthermore, I shall show that the Boussinesq approximation results in a zero-valued divergence. I have consulted the following works while researching this topic:
Davidson, P.A., 2001. An Introduction to Magnetohydrodynamics. 3-4,6.
Murphy, N., 2016. Ideal Magnetohydrodynamics. Lecture presentation. Harvard-Smithsonian Center for Astrophysics.
Consider a fluid element through which fluid passes. The mass of this element can be represented as the volume integral of the material density :
Take the first order time derivative of the mass, we arrive at
We now make a slight notation change; let the triple integral be represented as
Now, the mass flux through a surface element is . Thus, the integral of the mass flux is
We may write this as
Now, we invoke Gauss’s theorem (also known as the Divergence Theorem) of the form
Since the integral cannot be zero, the integrand must be. Therefore, we get the continuity equation:
Recall the following equation from a previous post (specifically Eq. (6) of that post)
Now we define the concept of vorticity. Conceptually, this refers to the rotation of the fluid within its velocity field. Mathematically, we define the vorticity to be the curl of the fluid velocity:
Now recall, the vector identity
which upon rearrangement is
Using Eq.(10.2) with Eq.(8) we get
Recall our definition of vorticity. Upon taking the curl of Eq.(11) we arrive at a variation of the induction equation (see post)
Next, we invoke another vector identity
Using Eq. (12.2) in Eq. (12.1) yields the vorticity equation
Returning to the continuity equation:
I will now show that the flow does not diverge. In other words, there are no sources in the fluid velocity field. The Boussinesq approximation’s main assertion is that of isopycnal flow (i.e. flow of constant density). Therefore, let . Substitution into the continuity equation yields the following
The temporal derivative is a derivative of a non-zero constant which is itself zero. This just leaves . Now, since , this then means that
Hence, the divergence of the fluid’s velocity field is zero.