arXiv subjects

Hongsong Chou

Publications and source records attributed to Hongsong Chou.

5 recordsLinked to original sources

arXiv2001

The Dependence of Dynamo $α$-Effect on Reynolds Numbers, Magnetic Prandtl Number, and the Statistics of MHD Turbulence

We generalize the derivation of dynamo coefficient $α$ of Field et al (1999) to include the following two aspects: first, the de-correlation times of velocity field and magnetic field are different; second, the magnetic Prandtl number can be arbitrary. We find that the contributions of velocity field and magnetic field to the $α$ effect are not equal, but affected by their different statistical properties. In the limit of large kinetic Reynolds number and large magnetic Reynolds number, $α$-coefficient may not be small if the de-correlation times of velocity field and magnetic field are shorter than the eddy turn-over time of the MHD turbulence. We also show that under certain circumstances, for example if the kinetic helicity and current helicity are comparable, $α$ depends insensitively on magnetic Prandtl number, while if either the kinetic helicity or the current helicity is dominated by the other one, a different magnetic Prandtl number will significantly change the dynamo $α$ effect.

astro-ph↗

arXiv2001

Conservation of Magnetic Helicity and Its Constraint on $α$-Effect of Dynamo Theory

Dynamical studies of MHD turbulence on the one hand, and arguments based upon magnetic helicity on the other, have yielded seemingly contradictory estimates for the $α$ parameter in turbulent dynamo theory. Here we show, with direct numerical simulation of three-dimensional magnetohydrodynamic turbulence with a mean magnetic field, $\OB$, that the constraint on the dynamo $α$-effect set by the magnetic helicity is time-dependent. A time-scale $t_c$ is introduced such that for $t - < \b \cdot \nabla \times \b >)$; for $t>t_c$, the classical result of the $α$-coefficient given by the Mean-Field Electrodynamics is reduced by a factor of $1/({R_m |\OB|^2/v_{rms}^2})$, as argued by Gruzinov & Diamond, Seehafer and Cattaneo & Hughes. Here, $R_m$ is the magnetic Reynolds number, $v_{rms}$ the rms velocity of the turbulence, $τ_{cor}$ the correlation time of the turbulence, and $\overline B$ is in velocity unit. The applicability of and connection between different models of dynamo theory are also discussed.

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arXiv2001

Numerical Analysis of Magnetic Field Amplification by Turbulence

We apply a Fourier spectral numerical method to 3D incompressible MHD turbulence with a magnetic Prandtl number $Pr \geq 1$. We examine the processes by which an initially weak, large-scale seed magnetic field and an initially weak, small-scale, impulse-like seed magnetic field are amplified. We find that in both cases the magnetic energy spectrum grows at all scales. The growth rates at different amplification stages are analyzed. For a large-scale seed magnetic field, the magnetic energy density grows as $\sim t^2$ for the first few turbulence eddy turnover times, followed by a dynamic growth stage, where nonlinear interactions between different scales of the turbulence contribute to an exponential growth rate that is largely determined by the turbulence eddy turnover time. For a seed magnetic field that is initially set up at a small scale in the turbulence, during the kinematic development stage, the growth rate of magnetic energy is $\propto 1/τ_{max}$, where $τ_{max}$ is the eddy turnover time of the smallest eddies of the turbulence. The kinematic growth stage is followed by a dynamic growth stage, where nonlinearity plays important role. During such dynamic growth stage, the growth rate of total magnetic energy is determined by both the magnetic energy amplification within the turbulence inertial range and that within the turbulence dissipation range.

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arXiv2000

A Note on Power-Laws of Internet Topology

The three Power-Laws proposed by Faloutsos et al(1999) are important discoveries among many recent works on finding hidden rules in the seemingly chaotic Internet topology. In this note, we want to point out that the first two laws discovered by Faloutsos et al(1999, hereafter, {\it Faloutsos' Power Laws}) are in fact equivalent. That is, as long as any one of them is true, the other can be derived from it, and {\it vice versa}. Although these two laws are equivalent, they provide different ways to measure the exponents of their corresponding power law relations. We also show that these two measures will give equivalent results, but with different error bars. We argue that for nodes of not very large out-degree($\leq 32$ in our simulation), the first Faloutsos' Power Law is superior to the second one in giving a better estimate of the exponent, while for nodes of very large out-degree($> 32$) the power law relation may not be present, at least for the relation between the frequency of out-degree and node out-degree.

cs.NI↗

arXiv1998

Nonlinear Alpha Effect in Dynamo Theory

We extend the standard two-scale theory of the turbulent dynamo coefficient $α$ to include the nonlinear back reaction of the mean field $\bar B$ on the turbulence. We calculate the turbulent emf as a power series in $\bar B$, assuming that the base state of the turbulence ($\bar B=0$) is isotropic, and, for simplicity, that the magnetic diffusivity equals the kinematic viscosity. The power series converges for all $\bar B$, and for the special case that the spectrum of the turbulence is sharply peaked in $k$, our result is proportional to a tabulated function of the magnetic Reynolds number $R_M$ and the ratio $β$ of $\bar B$ (in velocity units) to the rms turbulent velocity $v_0$. For $β\to 0$ (linear regime) we recover the results of Steenbeck et al. (1966) as modified by Pouquet et al. (1976). For $R_M\gg 1$, the usual astrophysical case, $α$ starts to decrease at $β\sim 1$, dropping like $β^{-2}$ as $β\to \infty$. Hence for large $R_M$, $α$ saturates at $\bar B\sim v_0$, as estimated by Kraichnan (1979), rather than at $\bar B\sim R^{-1/2}_Mv_0$, as inferred by Cattaneo and Hughes (1996) from their numerical simulations at $R_M$=100. We plan to carry out simulations with various values of $R_M$ to investigate the discrepency.

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