Force-free fields: non-uniqueness and non-existence’

Illustrating the conceptual difficulties of force-free extrapolation.

Nonlinear force-free equilibria satisfy $\mathit{j}=\alpha \mathit{B}$ with $\alpha$ a non-trivial function of position. The fact that $\alpha$ has to be constant along each magnetic field line makes the system “integrable”, limiting the topological complexity of these fields. However, because the governing equation $(\mathrm{\nabla }\times \mathit{B})\times \mathit{B}=\mathbf{0}$ is nonlinear, when computing force-free extrapolations it is difficult to know what boundary conditions will give a well-posed problem.

In this post, I have collected two analytical “counter-examples” to show explicitly that the solutions can be non-unique and even non-existent for magnetic field boundary conditions such as solar magnetograms. These facts are well-known but I can’t find anywhere in the literature where they are explicitly demonstrated. For simplicity, we will use Cartesian coordinates where the solar photosphere is represented by the infinite plane $y=0$ and the corona where we want to find a force-free field is the half-space $y>0.$

Example of non-uniqueness

The nicest example I have found originates in Low (1977) and is discussed on p352 of Schindler’s book. It is a solution of the (pressureless) Grad-Shafranov equation for a magnetic field $\mathit{B}=\mathrm{\nabla }\times (A(x,y){\mathit{e}}_z)+B_z(A){\mathit{e}}_z,$ where the out-of-plane component is chosen to be $B_z(A)=\sqrt{\lambda }{\mathrm{e}}^{-A}.$ This turns Grad-Shafranov into the Liouville equation ${\mathrm{\nabla }}^{2}A=\lambda {\mathrm{e}}^{-2A},$ so that solutions can be constructed analytically. We consider the particular pair of solutions $$A=\ln (1+x^2+y^2-2h_{\pm }y),h_{\pm }=\pm \sqrt{1-\lambda /4},$$ where $\lambda <4.$ These two solutions have different topology, corresponding to an arcade for $h_{-}$ and a “flux rope” for $h_{+},$ respectively. (When $\lambda =0$ the arcade solution becomes a potential arcade and the flux rope solution becomes a line current singularity surrounded by current free field.) Here is an illustration of the two solutions:

From my point of view here, the interesting thing is to look at the photospheric boundary conditions of these two solutions. The magnetic field on $y=0$ is $$\mathit{B}(x,0)=\frac{-2h_{\pm }}{1+x^2}{\mathit{e}}_x-\frac{2x}{1+x^2}{\mathit{e}}_y+\frac{\sqrt{\lambda }}{1+x^2}{\mathit{e}}_z,$$ showing that they both have the same vertical field $B_y(x,0).$ Incidentally, this vertical field doesn’t depend on $\lambda ,$ so you can generate a whole sequence of fields with the same normal-component on the boundary, ranging from potential with $\lambda =0$ to force-free.

Notice that the $h_{+}$ and $h_{-}$ fields for the same $\lambda$ also share the same $B_z(x,0)$ distribution, though their $B_x(x,0)$ components differ in sign. So far, I haven’t found an explicit example of two force-free fields that have exactly the same $\mathit{B}$ in the photosphere for all three components – i.e., the same vector magnetogram. To me, whether this is possible is still an open question.

Example of non-existence

It is known that not all vector magnetograms $\mathit{B}$ are compatible with a force-free extrapolation (for any $\alpha$) – for example, this is mentioned in the Living Reviews article by Wiegelmann and Sakurai. This is important when working with observed vector magnetogram data, meaning that they must be pre-processed for compatibility with the force-free assumption. Here, I will show how to prove this non-existence explicitly. The idea is to use the Virial Theorem (originally introduced in the context of MHD by Chandrasekhar, I think).

To derive the Virial Theorem, we start by writing $\mathit{j}\times \mathit{B}$ as the divergence of the Maxwell stress tensor, $$\mathit{j}\times \mathit{B}=\mathrm{\nabla }\cdot (\frac{B^2}{2{\mu }_0}\mathbb{I}-\frac{\mathit{B}\mathit{B}}{{\mu }_0}).$$ Setting this equal to zero, dotting with the position vector $\mathit{r},$ and integrating, leads after some algebra to $${\int }_V\frac{B^2}{2{\mu }_0}\mathrm{d}V={\oint }_{\partial V}(\frac{B^2}{2{\mu }_0}\mathit{r}-\frac{\mathit{r}\cdot \mathit{B}}{{\mu }_0}\mathit{B})\cdot \mathrm{d}\mathit{S}(†),$$ where $V$ is the coronal volume and $V $is the photospheric boundary (we assume that the solution decays at infinity).

The neat thing about the Virial Theorem $(†)$ is that the energy of any force-free field is given purely by a boundary integral, if you know all three components of $\mathit{B}$ on the boundary. To prove that a force-free field cannot exist for some particular boundary map $\mathit{B}(x,0,z)$ on $y=0$ (we consider the Cartesian case for simplicity), note that the lowest-energy magnetic field (that decays at infinity) is the potential extrapolation that matches the given normal component $B_y(x,0,z).$ Call the energy of this potential field $E_p.$ If we can find a distribution of $B_x(x,0,z)$ and $B_z(x,0,z)$ where $(†)$ gives a lower energy than $E_p,$ then we have a contradiction meaning that such a field cannot exist.

Initially, I tried to use the Low (1977) solution above to construct an example of this. But I think the potential field solution in that case actually has unbounded energy, so it’s a bit of a dodgy example. A better one is to take a submerged 3D dipole, $$\mathit{B}=\frac{3\mathit{r}(\mathit{m}\times \mathit{r})-r^2\mathit{m}}{r^5},\mathit{m}={\mathit{e}}_x,\mathit{r}=x{\mathit{e}}_x+(y+1){\mathit{e}}_y+z{\mathit{e}}_z.$$ On the photosphere, this potential field has $$B_x=\frac{2x^2-z^2-1}{(x^2+z^2+1{)}^{5/2}},B_y=\frac{3x}{(x^2+z^2+1{)}^{5/2}},B_z=\frac{3xz}{(x^2+z^2+1{)}^{5/2}}.$$ Using $(†),$ with the help of Wolfram Alpha, I calculate the energy in the half-space $y>0$ to be $$E_p=\frac{\pi }{8{\mu }_0}.$$ Now if I simply change the sign of $B_z(x,0,z),$ I get a different vector map whose energy according to $(†)$ is $$E=-\frac{\pi }{16{\mu }_0}<E_p.$$ This indeed contradicts the fact that the potential field is the minimum-energy field with that distribution of $B_y(x,0,z),$ so I conclude that this vector map permits no force-free extrapolation. In fact, I didn’t need to bother with the potential field here, as a negative energy is always impossible!

Edit

Jean-Jacques Aly subsequently pointed out to me a simpler proof of non-existence that doesn’t use the Virial Theorem. The subject is discussed in his 1989 Solar Physics paper. His argument goes as follows. A force-free field satisfies $\mathrm{\nabla }\cdot (\frac{B^2}{2{\mu }_0}\mathbb{I}-\frac{\mathit{B}\mathit{B}}{{\mu }_0})=\mathbf{0},$ so integrating over $V,$ using the Divergence Theorem, and taking the $x$-component shows that any force-free field must satisfy the integral relation $${\int }_{y=0}{B}_x{B}_y\mathrm{d}S=0.$$ It is clear that you can readily find examples of fields for which this is violated. Jean-Jacques suggests considering the alternative boundary condition ${\mathit{B}}^{\mathbf{\prime }}(y=0)=\mathit{B}(y=0)+k{B}_y(y=0){\mathit{e}}_x,$ where $k$ is some constant. Then $${\int }_{y=0}{B}_x^{\prime }{B}_y^{\prime }\mathrm{d}S=k{\int }_{y=0}{B}_y^2\mathrm{d}S,$$ so ${\mathit{B}}^{\prime }$ has no force-free extrapolation for any $k>0.$ In fact, since $k$ can be made arbitrarily small, we can jump from existence to non-existence by an arbitrarily small change in the boundary conditions!

This argument also readily extends to the spherical case if we take the $\varphi$-component of the integral relation to get $${\int }_S{B}_r{B}_{\varphi }\mathrm{d}S=0$$ on the spherical surface.