Chiral dynamos sourced by chiral magnetic waves

October 21, 2020

Contents

1 Introduction 1

2 Physical background and methods 1 2.1 Chiral magnetic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3 Generation of μ_{5} via the chiral separation effect and subsequent chiral dynamo 2

3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1.1 System of equations and analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1.2 Conservation of total chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1.3 Budget equation of total energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 Exploring the generation of μ_{5} spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Chiral dynamos and at different frequencies of CMWs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.4 Generation of μ_{5} by chiral magnetic waves at different Reynolds numbers . . . . . . . . . . . . . . . . 4

4 Conclusion 5

1 Introduction

Central question: Can chiral magnetic waves generate enough μ_{5} to self-consistently drive chiral dynamos?

Plan for this paper:

• Analysis of the influence of the initial μ_{5} power spectrum on the chiral dynamo instability (Section ??). How is the chiral dynamo affected if μ_{5} is not constant, but has an extended power spectrum? To answer this question, we perform DNS without the chiral separation effect (CSE).

• In Section 3.2 we explicitly include the CSE and consider scenarios without an initial chiral asymmetry. However, a generation of μ_{5} results from gradients in μ. In most of our tested scenarios, a μ_{5} is generated with an extended power spectrum. We will identify initial conditions that generate μ_{5} on individual modes only.

• In Section ?? we focus on scenarios where the μ_{5} power spectrum is generated on a single wavenumber (This can be achieved if a weak external magnetic field is applied and the initial energy spectra of the magnetic field and μ are located in single modes.). We will analyze the exponential increase of the magnetic field strength over many orders of magnitude and the subsequent steady state phase.

2 Physical background and methods

2.1 Chiral magnetic effect

2.2 Numerical setup

We use the Pencil Code^{1} to solve equations (??)–(??) in a three-dimensional periodic domain of size L^{3} = (2π)^{3} with a resolution of up to 224^{3}. This code employs a third-order accurate time-stepping method of [? ] and sixth-order explicit finite differences in space [? ? ]. The smallest wavenumber covered in the numerical domain is k_{1} = 2π/L = 1 which we use for normalization of length scales. All velocities are normalized to the sound speed c_{s} = 1 and further the mean fluid density to \overline{ρ} = 1.

1http://pencil-code.nordita.org/

Further, the magnetic Prandtl number is 1, i.e. the magnetic diffusivity equals the viscosity. Time is normalized by the diffusion time t_{η} = (ηk_{1}^{2})^{−1}. An overview of all runs presented in this paper is given in Tab. ?? and Tab. 1 in Appendix 4.

3 Generation of μ_{5} via the chiral separation effect and subsequent chiral dynamo

In this section we are extending the system of equations (??)–(??) plus (??) to include also the chiral separation effect (CSE). We then consider scenarios in which there is no initial chiral asymmetry but μ_{5} can be generated by gradients in μ via the CSE term in the evolution equation of μ_{5}. The goal is to explore whether a v_{5} or even an α_{5} dynamo can be driven in such a scenario.

3.1 Theoretical background

3.1.1 System of equations and analytical solutions

The CSE couples the chiral chemical potential (μ_{5}) with the chemical potential (μ). The respective coupling constants are C_{5} and C_{μ}. In this section we consider the following set of equations:

\frac{∂B}{∂t} = ∇ × [U × B − η (∇ × B − μ_{5}B)]  (1)
ρ\frac{DU}{Dt} = (∇ × B) × B − ∇p + ∇·(2νρS)  (2)
\frac{Dρ}{Dt} = −ρ ∇ · U  (3)
\frac{Dμ_{5}}{Dt} = −D_{5} ∇^{4}μ_{5} + λ η [B·(∇ × B) − μ_{5}B^{2}] − C_{5}(B·∇)μ,  (4)
\frac{Dμ}{Dt} = −D_{μ} ∇^{4}μ − C_{μ}(B·∇)μ_{5}  (5)

The coupling between μ_{5} and μ leads to the interesting phenomenon of chiral magnetic waves [CMWs; see ? ]. When considering the coupled linearized equations (4) and (5), the frequency of CMWs is found to be

ω_{CMW} = [C_{5} C_{μ}(k_{x}B_{0})^{2} − \frac{1}{4} (λη B_{0}^{2} − (D_{5} + D_{μ})k^{2})^{2}]^{1/2} ,  (6)

where the external magnetic field B_{0} is along x axis. The damping rate of these waves is

γ_{CMW} = −\frac{1}{2} [λη B_{0}^{2} + (D_{5} + D_{μ})k^{2}] ,  (7)

In the case of the hyper-diffusion (D_{5} + D_{μ})k^{2} should be replaced by (D_{5} + D_{μ})k^{4}. In the following, we are considering the following initial conditions where μ_{5}(t = 0) = 0 and μ(t = 0) is either in form of Gaussian noise or sin waves in x direction. The initial magnetic field is weak and form of Gaussian noise or Beltrami fields. In some of the DNS in this section we additionally apply an external magnetic field with B_{ex} = (B_{ex,x}, 0, 0) and B_{ex,x} = 10^{−6}. The there is no initial velocity field. Let us analyze the system of equations (1)–(5) with such initial conditions. In Phase 1, a chiral asymmetry is generated via the last term in Eq. (4). For times less than the periode of a CMW, we can approximate the evolution of μ_{5} as

μ_{5}(t) ≈ C_{5}(B·∇)μ t.  (8)

In all our DNS the external field will be in x direction. When we then consider μ(t = 0) = μ_{0}sin(k_{x}x) we find

μ_{5}(t) ≈ C_{5}B_{ex,x}k_{x}μ(t) t.  (9)

Note that μ is a function of time due to the dissipation term in Eq. (5). We define the start of Phase 2, when μ_{5} exceeds 1 and a v_{5} amplifies the magnetic field exponentially with the growth rate (??). Note that during this phase, μ_{5} continues to grow according to Eq. (9). In the beginning of Phase 3 the generation of μ_{5} stops. This is the case, once the terms λ η [B·(∇ × B) − μ_{5}B^{2}] and C_{5}(B·∇)μ become comparable. It is important to note that the generation term of μ_{5} is determined by the external large-scale magnetic field, while the destruction term is determined mostly by the rms magnetic fields that is much larger than B_{ex,x} during phase 3. An equilibrium state is then reached when

λ η μ_{5}B^{2} ≈ C_{5}B_{ex,x}k_{x}μ.  (10)

Hence for a given set of parameters, a finial value of the production efficiency

\frac{μ_{5}B^{2}}{μ} ≈ \frac{C_{5}B_{ex,x}k_{x}}{λ η}  (11)

can be reached. We can also introduce the magnetic field strength B_{eq} that corresponds to the terms λ η [B·(∇ × B) − μ_{5}B^{2}] and C_{5}(B·∇)μ being comparable:

B_{eq} ≡ (\frac{C_{5}B_{ex,x}k_{x}}{λ η} )^{1/2} .  (12)

[Run B1]  [Run B2]  [Run B3]  [Run B4]  [Run B5]

Figure 1: Generation of μ_{5} by the CSE using different initial conditions in 2D DNS. In the top row, solid lines show positive quantities and dashed lines shows the absolute value of a quantity with negative sign. 1st column: The initial μ and B are Gaussion noise. 2nd column: The initial μ is Gaussion noise, the initial B is a Beltrami field on k = 1. 3rd column: The initial μ is a sin wave field on k = 6, the initial B is Gaussion noise. 4th column: The initial μ is a sin wave field on k = 6, B is a Beltrami field on k = 1. 5th column: Same as 4th column, but with weak external magnetic field.

3.1.2 Conservation of total chirality

We use the induction equation and the equation for the vector potential:

\frac{∂B}{∂t} = −∇ × E,  (13)
\frac{∂A}{∂t} = −E + ∇Φ,  (14)

where B = ∇ × A and Φ is the electrostatic potential. Multiplying Equation (13) by A and Equation (14) by B, and adding them, we obtain an evolution equation for the magnetic helicity density, A·B:

\frac{∂A·B}{∂t} + ∇· (E × A + B Φ) = −2E·B.  (15)