Unresolved Mach-number dependence of the turbulent dynamo: growth-rate and saturation asymptotics beyond Mach 20 and a predictive theory for supersonic, compressible turbulence

The turbulent (fluctuation) dynamo amplifies weak seed magnetic fields by chaotic stretching in electrically conducting turbulence. Federrath et al. (Phys. Rev. Lett. 107, 114504, 2011) simulated driven, isothermal magnetohydrodynamic turbulence from the subsonic to the supersonic regime and measured how the kinematic growth rate $\Gamma$ and the saturation ratio $(E_m/E_k)_{\rm sat}$ of magnetic to kinetic energy depend on the rms Mach number $\mathcal{M}$, over a grid spanning $\mathcal{M} \approx 0.02$-$20$, and on whether the forcing is solenoidal or compressive. They found amplification rates and saturation levels decreasing strongly at the transition from subsonic to supersonic flow, and reduced dynamo efficiency under compressive driving, but their high-$\mathcal{M}$ fit imposed an empirical $\Gamma \propto \mathcal{M}^{1/3}$ scaling (in eddy-turnover units) that was only calibrated up to $\mathcal{M} \approx 20$ and was not claimed to be the true asymptote. The following remain open: the true asymptotic behavior of $\Gamma(\mathcal{M})$ and $(E_m/E_k)_{\rm sat}(\mathcal{M})$ for $\mathcal{M} \gg 20$, and a predictive dynamo theory covering small magnetic Prandtl number ($\mathrm{Pm} \lesssim 1$) and shock-dominated (Burgers-type) compressible turbulence.

Background

The turbulent or fluctuation dynamo is the exponential amplification of a weak seed magnetic field by random stretching and folding of field lines in electrically conducting turbulence; it is the leading mechanism for magnetizing the interstellar and intracluster plasma. Its kinematic phase is characterized by a growth rate $\Gamma$ of magnetic energy, and its subsequent nonlinear saturation by the ratio $(E_m/E_k)_{\rm sat}$ of saturated magnetic to turbulent kinetic energy. Following Federrath et al. (2011), $\Gamma$ is measured in units of the inverse eddy-turnover time $t_{\rm ed}^{-1}$, where $t_{\rm ed} = L/(2\mathcal{M}c_s)$ is defined at the largest scales $L/2$ of a periodic simulation domain of size $L$ and $c_s$ is the sound speed; because $t_{\rm ed}$ itself depends on $\mathcal{M}$, the $\mathcal{M}$-dependence of $\Gamma$ is normalization-dependent, and the scaling $\Gamma \propto \mathcal{M}^{1/3}$ in units of $t_{\rm ed}^{-1}$ corresponds to $\Gamma \propto \mathcal{M}^{4/3}c_s/L$ in $\mathcal{M}$-independent physical units. All growth rates in this statement use the $t_{\rm ed}^{-1}$ convention unless stated otherwise. The rms Mach number $\mathcal{M}$ (ratio of flow speed to sound speed) measures compressibility: $\mathcal{M} \gtrsim 1$ flows are supersonic and contain shocks. Forcing of the turbulence can be solenoidal (divergence-free, vorticity-dominated), compressive (curl-free, dominated by shocks and density fluctuations), or a mixture; astrophysical driving lies between the solenoidal and compressive extremes. The magnetic Prandtl number $\mathrm{Pm}$ is the ratio of viscous to magnetic diffusivity, the Reynolds number $\mathrm{Re}$ measures the ratio of inertial to viscous forces, and the magnetic Reynolds number $\mathrm{Rm} = \mathrm{Pm}\,\mathrm{Re}$ must exceed a critical value $\mathrm{Rm}_{\rm crit}$ for dynamo action. Classical analytic dynamo theory uses the Kazantsev closure, a model in which the velocity is a prescribed Gaussian random field, delta-correlated in time, with a power-law spectrum; its predictions depend on the assumed spectral slope, e.g. Kolmogorov (incompressible, slope exponent 1/3) versus Burgers (shock-dominated, slope exponent 1/2) turbulence. Federrath, Chabrier, Schober, Banerjee, Klessen and Schleicher (Phys. Rev. Lett. 107, 114504, 2011) ran a grid of simulations with solenoidal and compressive forcing over $\mathcal{M} \approx 0.02$-$20$, most in ideal MHD, with four representative non-ideal runs demonstrating convergence at $\mathrm{Pm} \approx 2$ and $\mathrm{Re} \approx 1500$. They found that $\Gamma$ and $(E_m/E_k)_{\rm sat}$ decrease strongly at the transition from subsonic to supersonic flow — a drop of the growth rate at $\mathcal{M} \approx 1$ is noticeable for both forcing types — and that solenoidal forcing gives higher growth rates and saturation levels than compressive forcing; over the supersonic range the fitted growth rate follows an imposed $\Gamma \propto \mathcal{M}^{1/3}$ tail in good agreement with the models up to $\mathcal{M} \approx 20$, while the saturation level continues to decrease with increasing $\mathcal{M}$. Under purely compressive forcing, vorticity (which the dynamo feeds on) is generated only indirectly, through the viscous term $2\nu\,\nabla\times(\mathbf{S}\nabla\ln\rho)$ involving the strain tensor $\mathbf{S}$ acting on density gradients, and in oblique, colliding shocks, where vorticity generation starts to dominate over the destruction of coherent vortical motions as $\mathcal{M}$ is increased further. Their fits used an empirical function whose high-$\mathcal{M}$ behavior ($\Gamma \propto \mathcal{M}^{1/3}$ in units of $t_{\rm ed}^{-1}$) was fixed by hand, and they explicitly cautioned that the fits do not necessarily reflect the true asymptotic behavior, noting that even higher $\mathcal{M}$ has to be investigated to see whether $\Gamma \propto \mathcal{M}^{1/3}$ holds in that limit. They also stressed that existing analytic predictions, derived for incompressible Kolmogorov turbulence, do not capture the observed $\mathcal{M}$ and forcing dependence, and called for an extension of dynamo theory to small $\mathrm{Pm}$ and from Kolmogorov to Burgers-type, shock-dominated turbulence.

Problem Statement

Federrath et al. (2011) established empirically, and left explicitly unresolved, how turbulent dynamo action depends on compressibility in supersonic turbulence. The problem is to obtain a predictive account of the fluctuation dynamo in driven supersonic turbulence, comprising two jointly supporting parts: (1) determine the true asymptotic behavior at large Mach number of the kinematic growth rate $\Gamma(\mathcal{M})$ and the saturation ratio $(E_m/E_k)_{\rm sat}(\mathcal{M})$, in particular testing whether the empirically imposed scaling $\Gamma \propto \mathcal{M}^{1/3}$ (in units of $t_{\rm ed}^{-1}$) holds for $\mathcal{M}$ substantially larger than the $\mathcal{M} \approx 20$ maximum of the original simulations, for both solenoidal and compressive forcing; and (2) extend dynamo theory beyond the incompressible Kolmogorov assumptions to small magnetic Prandtl number $\mathrm{Pm} \lesssim 1$ and to compressible, shock-dominated (Burgers-type) turbulence, so that the observed dependence of dynamo growth and saturation on $\mathcal{M}$ and on the solenoidal-versus-compressive forcing type is predicted rather than fitted. Parts (1) and (2) are linked by one scientific objective: a closed, predictive description of supersonic turbulent dynamo action in which simulation measurements of $\Gamma$ and $(E_m/E_k)_{\rm sat}$ over $\mathcal{M}$ and forcing type, including the untested $\mathcal{M} \gg 20$ limit, are reproduced by theory rather than fitted after the fact.

References

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