Equilibrium State of Magnetized Cloud

The concentrated random magnetic field in a magnetized cloud can be initialized by setting up the magnetic field distribution, usually a linear combination of orthogonal wave functions (like sin and cos series). In order to study for instance, the instability of such a magnetized cloud under anisotropic thermal diffusion and self gravity, one has to obtain an equilibrium. This equilibrium is required so that with a strong field filling the cloud, the cloud will not be torn apart or develop filaments inside, which will be recognized as an effect of instability. Unfortunately, unlike the uniform field case where the magnetic pressure is always constant, the field configuration being used in the magnetized cloud study has an often spherical symmetric distribution. The field is also usually cut off around the clump edge so that one cloud's field will not affect another cloud. These properties suggest that the force introduced by the field is usually non zero throughout the cloud.

There are several ways to obtain the equilibrium state of a magnetized cloud. The easiest way is to set up a magnetic field distribution with random frequency spectrum inside the cloud and find out the magnetic pressure at each point. Then set up the cloud with constant density but varying temperature so that the thermal pressure at each point exactly balances the magnetic pressure. Outside the cloud, the magnetic field is cut off. So the thermal pressure equals the total pressure. The next few animations show such a magnetized cloud with various magnetic beta.

beta = 10

beta = 4

As we can see from the above animations, with high beta, this approach works OK in that there is no significant changes in the density and field distributions. But as beta goes to 4, we can see the density varies as well as the field strength. The beta = 1 case shows even greater change. The reason is that the Lorentz force is not uniform given the non uniform distribution of the field so that the divergence of the magnetic tensor does not act as gradient of a scalar pressure. Since our primary interest is in those low beta cases, we will want something more sophisticated.

Another approach is to set up a cloud as above, and then let it evolve for a period of time to let the field relax to some steady state. With this method, we can achieve magnetized cloud with any desired field spectrum since the only thing we need to do is throw in the field and let it treat itself. The down side of this method though, is that (1) for large simulations with multiple magnetized clump, it can take very long to reach a force free state and we have no time how long it will take. (2) it does not work with low beta. as shown in the following two animations with beta = 1 and beta = 0.2, the density imbalance grows to more than 20%.

beta = 1

beta = 0.2

Force Free Symmetric Magnetic Field

In order to achieve an equilibrium of magnetized cloud, especially in the case of very low beta, we may want to construct an initial field which has the two properties: (1) the Lorentz force is zero at every point. (2) the field has a certain kind of symmetry to fit the cloud (cylindrical symmetry for the 2.5D cloud, spherical symmetry for the 3D cloud) (3) the field should have a controllable spectrum (4) the field should be weak far from the symmetry center.

Surprisingly, all these conditions can be achieved at once, at least to some extent. Starting from the Lorentz equation:

we know that

Since we are treating source free situation, we have:

which implies


So we end up with the following equation:


The force free magnetic field can be categorized by different alpha functions (alpha can not be arbitrary though), the simplest category might be those with alpha = constant. This set automatically satisfies equation (1). To solve it, take the curl of equation (2). We end up with Helmholtz equation of B with alpha being the wave number:

Consider the scalar version of the wave equation


Notice that not all solutions of the wave equation would automatically satisfy the force free equation.

It can be shown that the three independent solution to the B equation can be written in terms of psi:


It is easy to show that B_p and B_t represents a certain type of poloidal and toroidal field.

There is a property:

But we know that


So the force free equation is true for the pair:

The conclusion is that given any poloidal field, there is a toroidal field that can exactly cancel the Lorentz field induced by the former, making the whole field force free.
The solution is then written as:

We then observe that the solution field is the curl of a vector, where in spherical coordinates, l is the unit vector along radial directions.:

Procedure Finding Force Free Magnetized Cloud

(1) Finding the general solution to the scalar wave equation:

In cylindrical coordinates, the eigenfunctions are (suppose 2.5D so that everything is uniform along z direction):

In spherical coordinates, the eigenfunctions are:

where brackets stands for any linear combination.
The Bessel functions of the first and second kind can be found by calling the function in fortran portable library (add the "USE IFPORT" statement in your module).
Since there is no intrinsic spherical Bessel or Legendre functions in fortran library, I have written some calculation routines into the CommonFunctions module.
One of them is Gamma function, since it can be obtained quickly by doing a simple iteration. Another is the associated Legendre polynomial.
The spherical Bessel function of the first kind can be found by utilizing a fairly simple finite series of Gamma functions. But the spherical Bessel functions of the second kind seem to require a much lengthy computation (partly because the series method is not useful since the first and second kind Bessel functions are independent). I left it out since we do not require the completeness of the linear space. The point here is that someone may have to fill in this hole in the future.

(2) Finding the vector potential:

(3) Cut off the potential at cloud edge. In order to maintain the force free property, the cloud is treated so that the density radius is smaller than the field radius. So that the field will extend outside the cloud, and then cut off. This field extension should be not too large to affect other clouds.

(4) Find the field and test the force free property for use.

A simple solution to the 2.5D cylindrical solution is given by:

It is easy to prove that this solution satisfies the force free equation by using the derivative relations of Bessel functions.

The magnetic energy of this set up is plotted below. No cut off is applied here to zero out the field energy outside the cloud, so we can see the field energy similar to point diffraction patterns extending into the outer area. We can see that the setup has nice centralized field energy inside the cloud while the field does not exert any force on the cloud material. Another nice feature about this setup is that the field has no radial component so that it will be an equilibrium even if there is an anisotropic conduction along field lines.

The stability with self gravity and thermal diffusion of such a cloud might be interesting. Since we can put in a Bonner Ebert cloud which is also force free. The cloud would be unstable to radial pokes because of anisotropic thermal instabilities. As we know, the thermal instability will reorient the field and in this case, the force free property can not be maintained, I guess…

Random Force Free Magnetized Cloud

Since the force free field can be add up and maintain the force free property, we can achieve broad spectrum by adding in a large number of spherical force free fields with various frequencies.
Given an initial field, we can also decompose it into the combination of eigenfunctions and then find out a cancelling field to obtain a force free state.