Phenomenology of Turbulent Dynamo Growth and Saturation

Stepanov, Rodion; Plunian, Franck

doi:10.1086/587795

Cited by 15 publications

(21 citation statements)

References 22 publications

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“…With hydrodynamic turbulence intermittency 2 , the She & Leveque (1994) phenomenology provides the value ζ 1 = 1/9+2(1−(2/3) 1/3 ) ≈ 0.364 and then the small-scale dynamo phenomenology gives β = β ≈ 0.466, in agreement with the numerical result of Stepanov & Plunian (2008). In our simulations, the values we obtain for ζ 1 (shown in Table 1 and computed from ten independent runs with ν = η = 10 −10 ) are closer to the She & Leveque 1994 value than to onethird (no intermittency), although both are within 3-σ error bars.…”

Section: Resultssupporting

confidence: 81%

“…The α exponent in T m controls the strength of long-range (in Fourier space) nonlinear interactions: following Stepanov & Plunian (2008), we use α = −1 for strong nonlocal interactions, and α = −5/2 for weak nonlocal interactions; α = −∞ would correspond to no nonlocal interactions, i.e., to a local model.…”

Section: Model Equations and Numerical Set Upmentioning

confidence: 99%

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Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Buchlin

2011

A&A

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Context. Direct numerical simulations of plasmas have shown that the dynamo effect is efficient even at low Prandtl numbers, i.e., the critical magnetic Reynolds number Rm c that is necessary for a dynamo to be efficient becomes smaller than the hydrodynamic Reynolds number Re when Re → ∞. Aims. We test the conjecture that Rm c tends to a finite value when Re → ∞, and we study the behavior of the dynamo growth factor γ at very low and high magnetic Prandtl numbers. Methods. We use local and nonlocal shell models of magnetohydrodynamic (MHD) turbulence with parameters covering a much wider range of Reynolds numbers than direct numerical simulations, that is of astrophysical relevance. Results. We confirm that Rm c tends to a finite value when Re → ∞. As Rm → ∞, the limit to the dynamo growth factor γ in the kinematic regime follows Re β , and, similarly, the limit for Re → ∞ of γ behaves like Rm β , with β ≈ β ≈ 0.4. Conclusions. Our comparison with a phenomenology based on an intermittent small-scale turbulent dynamo, together with the differences between the growth rates in the different local and nonlocal models, indicate that nonlocal terms contribute weakly to the dynamo effect.

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Section: Resultssupporting

confidence: 81%

Section: Model Equations and Numerical Set Upmentioning

confidence: 99%

Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Buchlin

2011

A&A

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“…For large Pm, E b (k)/E u (k) grows monotonically with k, but it decreases with k for small Pm due to the nature of energy spectra described in the last two sections. Our results are in general agreement with the results of Stepanov and Plunian [35]. Note that the inertial range in our simulation is broader because our viscosity and the magnetic diffusivity are smaller than those of Stepanov and Plunian [35].…”

Section: Dynamo With Large Prandtl Numberssupporting

confidence: 91%

“…In the shell model of Frick et al [33], dB n /dt does not involve U n at all, while the shell model of Plunian and Stepanov [34] includes nonlocal terms. In the shell model of Stepanov and Plunian [17], Lessinnes et al [18], and Stepanov and Plunian [35], the structures of N n [B, B] and N n [U, U ] are the same, and so are those of N n [U, B] and N n [B, U ]. As a result, these models do not yield correct magnetic-to-magnetic energy transfers (B2B) due to N n [U, B], as well as magnetic-to-velocity (B2U ) and velocityto-magnetic (U 2B) transfers arising due to N n [B, B] and N n [B, U ].…”

Section: Description Of Our Mhd Shell Modelmentioning

confidence: 99%

Dynamos at extreme magnetic Prandtl numbers: insights from shell models

Verma

Kumar

2016

Journal of Turbulence

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We present an MHD shell model suitable for computation of various energy fluxes of magnetohydrodynamic turbulence for very small and very large magnetic Prandtl numbers Pm; such computations are inaccessible to direct numerical simulations. For small Pm, we observe that both kinetic and magnetic energy spectra scale as k −5/3 in the inertial range, but the dissipative magnetic energy scales as k −11/3 exp(−k/kη). Here, the kinetic energy at large length scale feeds the large-scale magnetic field that cascades to small-scale magnetic field, which gets dissipated by Joule heating. The large-Pm dynamo has a similar behaviour except that the dissipative kinetic energy scales as k −13/3 . For this case, the large-scale velocity field transfers energy to the large-scale magnetic field, which gets transferred to small-scale velocity and magnetic fields; the energy of the small-scale magnetic field also gets transferred to the small-scale velocity field, and the energy thus accumulated is dissipated by the viscous force.

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“…It is worth noting that our shell model considers local interactions between shells, namely, each shell interacts only with the first two neighbor shells on each side. At large Pm it is known that non-local interactions give an important contribution in the magnetic energy spectrum at small-scales [42][43][44]. This contribution, due to non-local interactions, was estimated by Plunian and Stepanov using a non-local shell model [43].…”

Section: Dynamical Runsmentioning

confidence: 99%

Finite-time singularities and flow regularization in a hydromagnetic shell model at extreme magnetic Prandtl numbers

Nigro

Carbone

2015

New J. Phys.

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Conventional surveys on the existence of singularities in fluid systems for vanishing dissipation have hitherto tried to infer some insight by searching for spatial features developing in asymptotic regimes. This approach has not yet produced a conclusive answer. One of the difficulties preventing us from getting a definitive answer is the limitations of direct numerical simulations which do not yet have a high enough resolution so far as to properly describe spatial fine structures in asymptotic regimes. In this paper, instead of searching for spatial details, we suggest seeking a principle, that would be able to discriminate between singular or not-singular behavior, among the integral and purely dynamical properties of a fluid system. We investigate the singularities developed by a hydromagnetic shell model during the magnetohydrodynamic turbulent cascade. Our results show that when the viscosity is equal to the magnetic diffusivity (unit magnetic Prandtl number) singularities appear in a finite time. A complex behavior is observed at extreme magnetic Prandtl numbers. In particular, the singularities persist in the limit of vanishing viscosity, while a complete regularization is observed in the limit of vanishing diffusivity. This dynamics is related to differences between the magnetic and the kinetic energy cascades towards small scales. Finally a comparison between the three-dimensional and the two-dimensional cases leads to conjecture that the existence of singularities may be related to the conservation of different ideal invariants.

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Phenomenology of Turbulent Dynamo Growth and Saturation

Cited by 15 publications

References 22 publications

Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Dynamos at extreme magnetic Prandtl numbers: insights from shell models

Finite-time singularities and flow regularization in a hydromagnetic shell model at extreme magnetic Prandtl numbers

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