Evolution of the magnetic field in spatially inhomogeneous axion structures

Abstract

We study the time evolution of magnetic fields in various configurations of spatially inhomogeneous pseudoscalar fields that are a coherent superposition of axions. For such systems, we derive a new induction equation for the magnetic field, which takes this inhomogeneity into account. Based on this equation, we study the evolution of a pair of Chern–Simons waves interacting with a linearly decreasing pseudoscalar field. The nonzero gradient of the pseudoscalar field leads to the mixing of these waves. We then consider the problem in a compact domain in the case where the initial Chern–Simons wave is mirror symmetric. The pseudoscalar field inhomogeneity then leads to an effective change in the $\alpha$ dynamo parameter. Thus, the influence of a spatially inhomogeneous pseudoscalar field on the magnetic field evolution bears a strong dependence on the system geometry.

Appendix Kelvin transformation

We suppose that the Kelvin transformation is a stereographic projection of the sphere $S^3$ outside the indicated point $pt$ onto the Euclidean space $\mathbb{R}^3$. This transformation is conformal, i.e., the angle between two vectors remains unchanged. In [21], the Kelvin transformation was used to construct the MHD soliton in Euclidean space with regular conditions at infinity.

It is noteworthy that the Kelvin transformation $T\colon S^3 \setminus \{pt\} \to \mathbb{R}^3$ does not change the equations. In the case where the selected point $pt$ is the north pole on the sphere ($\psi=\theta=0$), the magnetic mode $\mathbf{B}_\mathrm{A}$ transforms into the (generalized) toroidal mode $\mathbb{B}_\mathrm{A}$; and the magnetic mode $\mathbf{B}_\mathrm {B}$, into the (generalized) poloidal field $\mathbb{B}_\mathrm{B}$. We calculate only the transformation of magnetic modes.

The Kelvin transformation $T$ is defined as

$$ (\phi,\theta,\psi) \mapsto (r=\cot(\theta),\Psi,\phi),$$

(A.1)

where, in the image, the spherical system of coordinates with the latitude $\Psi$ is used and the longitude $\phi$ is taken into account. The image of the coordinate $\psi$ is calculated explicitly. It is independent of the coordinate $\phi$ of the image. The absolute value of the gradient of $\cot(\theta)$ determines the modulus of the conformal transformation $T$. This scaling coefficient, which is a function in the image, is denoted by $K$.

The Kelvin transformation maps the magnetic field $\mathbf{B}$ ($\mathbf{B}=\mathbf{B}_\mathrm {A}$ or $\mathbf{B}=\mathbf{B}_\mathrm {B}$), which is a divergence-free field on the initial sphere, into the magnetic field $\mathbb{B}= T_{\ast}(\mathbf{B})/K^3$ in Euclidean space, where $K$ is the scaling coefficient defined above. This function behaves asymptotically as $\propto r^2$, where $T_{\ast}$ is the translation of the vector via the differential of $T$. The vector field $\mathbb{B}$ is a magnetic field, because the flux of $\mathbb{B}$ that passes through the surface $T(\Sigma) \subset \mathbb{R}^3$ is equal to the flux of $\mathbf{B}$ through the surface $\Sigma \subset S^3$.

It can be seen that $\mathbb{B}_\mathrm{B} = T_{\ast}(\mathbf{B}_\mathrm{B})K^{-3}$ varies with the longitude $\phi$ and has the form of a toroidal magnetic mode. The mode $\mathbb{B}_\mathrm{A}=T_{\ast}(\mathbf{B}_\mathrm{A})K^{-3}$ is directed perpendicular to the coordinates $(r,\Psi)$ and is a poloidal one. The scaling coefficient $K$ is related to the form of the magnetic volume $K^3 \mathbf{d} x$ in $\mathbb{R}^3$ and has the asymptotic behavior $\propto r^6$ along the radius. Both fields $\mathbb{B}_\mathrm{A}$ and $\mathbb{B}_\mathrm{B}$ share the asymptotic behavior $\propto r^{-4}$.

The operator $\operatorname{curl}$ in $\mathbb{R}^3$ can be rewritten in the form

$$ \operatorname{curl}(K^{-2}T_{\ast}(F)) = K^{-3}T_{\ast}(\operatorname{curl}_\mathrm{S3}(F)),$$

(A.2)

where the operator $\operatorname{curl}_\mathrm{S3}$ and the vector field $F$ lie at the origin of coordinates of $S^3$. In particular, at infinity, where $K=K(r) \to +\infty$, the magnetic modes are almost potential and do not produce currents.

The magnetic helicity is preserved by the transformations $T$. The helicity is calculated as an improper integral of the form

$$ \chi_{\mathbb{B}} =\int_{\mathbb{R}^3} (\mathbb{A} \cdot \mathbb{B})\,\mathbf{d} x = \int_{\mathbb{R}^3} (K^{-2}T_{\ast}(\mathbf{A}),K^{-3}T_{\ast}(\mathbf{B}))\, \mathbf{d} x = \int_{S^3} ((\mathbf{A} \cdot \mathbf{B}))\, \mathbf{d} V = \chi_{\mathbf{B}},$$

(A.3)

where $\operatorname{curl}_\mathrm{S3}(\mathbf{A})=\mathbf{B}$. Here, the formulas $K^3\mathbf{d} V = \mathbf{d} x$ and $K^2((\dots,\dots)) = (\dots, \dots)$ relating the volume forms and the scalar products in the original image and the image are used.

Arnold’s inequality

Arnold’s inequality [19, Theorem 1.5, Ch. III] for the magnetic field on $S^3$ is written as

$$ \int_{S^3} (\mathbf{B} \cdot \mathbf{B})\, \mathbf{d} V \ge\frac{1}{2R}\int_{S^3} (\mathbf{A} \cdot \mathbf{B})\, \mathbf{d} V = \frac{1}{2R}\chi_{\mathbf{B}},$$

(A.4)

where $1/2R$ is the quantity that is inverse to the smallest eigenvalue of the operator $\operatorname{curl}$; $R$ is the sphere radius, which is the scaling coefficient; and $\chi_\mathbf{B}$ is the magnetic helicity.

Arnold’s inequality is generalized to the case of an unbounded conducting domain with the variable magnetic permeability $K(r)$ as

$$ \int_{\mathbb{R}^3} (\mathbb{B})^2 K^3\, \mathbf{d} x = \int_{S^3} (\mathbf{B})^2 K^{-1}\, \mathbf{d} V \ge \max_{S^3}(K^{-1}) \int_{S^3} (\mathbf{B})^2\, \mathbf{d} V \ge \frac{1}{2R}\chi_{\mathbf{B}},$$

(A.5)

where $r$ is the distance to the origin and $R$ is the scale of the magnetic permeability inhomogeneity.