[Paper Review] Slow dynamos and decay of monopole magnetic fields in the early Universe

Garcia de Andrade|ArXiv.org|Jan 13, 2009

Solar and Space Plasma Dynamics2 references3 citations

TL;DR

This paper investigates slow dynamo action in monopole plasmas during the early Universe, showing that magnetic monopoles suppress fast dynamo growth and instead lead to a linearly increasing magnetic energy over time, $\delta E_M \approx t$, due to non-solenoidal magnetic fields and dissipative effects. The model uses Riemannian filament dynamics and Da Rios equations to link curvature, torsion, and monopole flux to dynamo efficiency.

ABSTRACT

Previously Liao and Shuryak [ extbf{Phys. Rev C (2008)}] have investigated electrical flux tubes in monopole plasmas, where magnetic fields are non-solenoidal in quark-QCD plasmas. In this paper slow dynamos in diffusive plasma [{ extbf{Phys. Plasmas extbf{15} (2008)}}] filaments (thin tubes) are obtained in the case of monopole plasmas. In the absence of diffusion the magnetic field decays in the Early Universe. The torsion is highly chaotic in dissipative large scale dynamos in the presence of magnetic monopoles. The magnetic field is given by the Heaviside step function in order to represent the non-uniform stretching of the dynamo filament. These results are obtained outside the junction condition. Stringent limits to the monopole flux were found by Lewis et al [ extbf{Phys Rev D (2000)}] by using the dispute between the dynamo action and monopole flux. Since magnetic monopoles flow dispute the dynamo action, it seems reasonable that their presence leads to a slow dynamo action in the best hypothesis or a decay of the magnetic field. Hindmarsh et al have computed the magnetic energy decay in the early universe as ${E}_{M}\approx{t^{-0.5}}$, while in our slow dynamo case linearization of the growth rate leads to a variation od magnetic energy of $δ{E}_{M}\approx{t}$, due to the presence of magnetic monopoles. Da Rios equations of vortex filaments are used to place constraints on the geometry of monopole plasma filaments.

Motivation & Objective

• To investigate the interplay between magnetic monopoles and dynamo action in the early Universe, particularly how monopole flux affects magnetic field evolution.
• To model the decay or growth of magnetic fields in monopole plasmas using a Riemannian filament model with non-uniform stretching.
• To determine whether the presence of magnetic monopoles leads to slow dynamo action or field decay, based on the violation of the solenoidal condition $\nabla \cdot \mathbf{B} = \rho_m$.
• To derive constraints on plasma filament curvature and torsion using the Da Rios equations in the context of monopole-driven dynamo systems.
• To compare the magnetic energy decay rate in monopole plasmas with known results such as $E_M \approx t^{-0.5}$ from Hindmarsh et al., showing a reversal in behavior due to monopole effects.

Proposed method

• Modeling the magnetic field as $\mathbf{B} = B_0(t) H_0(s - s_0) \mathbf{t}$, where $H_0$ is the Heaviside step function, to represent a localized field at the filament junction.
• Using the self-induction equation $\partial_t \mathbf{B} = \nabla \times (\mathbf{v} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}$ with a velocity field $\mathbf{v} = v_0 \delta(s - s_0) \mathbf{t}$ to simulate localized plasma flow.
• Introducing the monopole condition $\nabla \cdot \mathbf{B} = \rho_m$ to replace the solenoidal constraint, leading to a diffusion equation for monopole density: $\partial_t \rho_m = \eta \nabla^2 \rho_m$.
• Applying the Frenet-Serret formalism to describe filament dynamics, with curvature $\kappa$, torsion $\tau$, and tangent vector $\mathbf{t}$, and deriving evolution equations for $\kappa$ and $\tau$ under dynamo flow.
• Using the Da Rios equations for vortex filaments: $\kappa_t = -2\kappa_s \tau - \tau_s \kappa$ and $\tau_t = \left[-\frac{\kappa_{ss}}{\kappa} + \frac{\kappa^2}{2} - \tau^2\right]_s$, assuming stationarity to solve for $\tau(s)$ and $\kappa(s)$.
• Computing the magnetic energy integral $E_M = \frac{1}{8\pi} \int \mathbf{B}^2 dV$ and linearizing the exponential growth to obtain $\delta E_M \approx \eta \kappa^2 t$, linking energy growth to diffusion and curvature.

Experimental results

Research questions

• RQ1 How does the presence of magnetic monopoles alter the standard fast dynamo mechanism in early Universe plasmas?
• RQ2 What is the time evolution of magnetic energy in a monopole plasma under dissipative conditions, and how does it compare to the $t^{-0.5}$ decay observed by Hindmarsh et al.?
• RQ3 How do curvature and torsion of plasma filaments, governed by the Da Rios equations, constrain the growth or decay of magnetic fields in monopole-driven dynamos?
• RQ4 What role does the junction condition $s = s_0$ play in initiating or suppressing dynamo action, particularly through the Dirac delta function in velocity and field profiles?
• RQ5 Can the plasma flow velocity $v_0$ be expressed in terms of the magnetic diffusivity $\eta$ and filament radius, and what does this imply for large-scale vs. small-scale dynamo efficiency?

Key findings

• In the absence of diffusion ($\eta \to 0$), the dynamo growth rate $\gamma$ vanishes away from the junction, indicating that monopoles are necessary for any significant dynamo action.
• The magnetic energy variation is found to be $\delta E_M \approx \eta \kappa^2 t$, showing a linear growth in time due to the interplay between diffusion and filament curvature.
• The torsion of the filament is damped by monopole dissipation, with $\tau(s) = -1/(\eta s)$, indicating that higher dissipation leads to reduced torsional complexity.
• The curvature is given by $\kappa^2 = 1/(4\eta s^2)$, showing that curvature increases with decreasing diffusion or smaller scale structures.
• The plasma flow velocity is determined as $v_0 = -\text{Re}_m^2 / (4 s_0)$, indicating that large-scale dynamos (high $\text{Re}_m$) have faster flow speeds than small-scale ones.
• The model predicts a reversal in magnetic energy behavior: while standard models show decay as $E_M \approx t^{-0.5}$, the monopole-influenced system exhibits growth as $\delta E_M \approx t$, due to non-solenoidal field effects.