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Axel Brandenburg
Publications and source records attributed to Axel Brandenburg.
Nonlinear current helicity fluxes in turbulent dynamos and alpha quenching.
Large scale dynamos produce small scale current helicity as a waste product that quenches the large scale dynamo process (alpha effect). This quenching can be catastrophic (i.e., intensify with magnetic Reynolds number) unless one has fluxes of small scale magnetic (or current) helicity out of the system. We derive the form of helicity fluxes in turbulent dynamos, taking also into account the nonlinear effects of Lorentz forces due to fluctuating fields. We confirm the form of an earlier derived magnetic helicity flux term, and also show that it is not renormalized by the small scale magnetic field, just like turbulent diffusion. Additional nonlinear fluxes are identified, which are driven by the anisotropic and antisymmetric parts of the magnetic correlations. These could provide further ways for turbulent dynamos to transport out small scale magnetic helicity, so as to avoid catastrophic quenching.
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Delayed correlation between turbulent energy injection and dissipation.
The dimensionless kinetic energy dissipation rate C(epsilon) is estimated from numerical simulations of statistically stationary isotropic box turbulence that is slightly compressible. The Taylor microscale Reynolds number (Re(lambda)) range is 20< or approximately equal to Re(lambda) < or approximately equal to 220 and the statistical stationarity is achieved with a random phase forcing method. The strong Re(lambda) dependence of C(epsilon) abates when Re(lambda) approximately 100 after which C(epsilon) slowly approaches approximately 0.5, a value slightly different from previously reported simulations but in good agreement with experimental results. If C(epsilon) is estimated at a specific time step from the time series of the quantities involved it is necessary to account for the time lag between energy injection and energy dissipation. Also, the resulting value can differ from the ensemble averaged value by up to +/-30%. This may explain the spread in results from previously published estimates of C(epsilon).
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Suppression of small scale dynamo action by an imposed magnetic field.
Nonhelical hydromagnetic turbulence with an externally imposed magnetic field is investigated using direct numerical simulations. It is shown that the imposed magnetic field lowers the spectral magnetic energy in the inertial range. This is explained by a suppression of the small scale dynamo. At large scales, however, the spectral magnetic energy increases with increasing imposed field strength for moderately strong fields, and decreases only slightly for even stronger fields. The presence of Alfve n waves is explicitly confirmed by monitoring the evolution of magnetic field and velocity at one point. The frequency omega agrees with v(A) k(1) , where v(A) is the Alfve n speed and k(1) is the smallest wave number in the box.
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Inertial range scaling in numerical turbulence with hyperviscosity.
Numerical turbulence with hyperviscosity is studied and compared with direct simulations using ordinary viscosity and data from wind tunnel experiments. It is shown that the inertial range scaling is similar in all three cases. Furthermore, the bottleneck effect is approximately equally broad (about one order of magnitude) in these cases and only its height is increased in the hyperviscous case-presumably as a consequence of the steeper decent of the spectrum in the hyperviscous subrange. The mean normalized dissipation rate is found to be in agreement with both wind tunnel experiments and direct simulations. The structure function exponents agree with the She-Leveque model. Decaying turbulence with hyperviscosity still gives the usual t(-1.25) decay law for the kinetic energy, and also the bottleneck effect is still present and about equally strong.
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Simulations of nonhelical hydromagnetic turbulence.
Nonhelical hydromagnetic forced turbulence is investigated using large scale simulations on up to 256 processors and 1024(3) mesh points. The magnetic Prandtl number is varied between 1/8 and 30, although in most cases it is unity. When the magnetic Reynolds number is based on the inverse forcing wave number, the critical value for dynamo action is shown to be around 35 for magnetic Prandtl number of unity. For small magnetic Prandtl numbers we find the critical magnetic Reynolds number to increase with decreasing magnetic Prandtl number. The Kazantsev k(3/2) spectrum for magnetic energy is confirmed for the kinematic regime, i.e., when nonlinear effects are still unimportant and when the magnetic Prandtl number is unity. In the nonlinear regime, the energy budget converges for large Reynolds numbers (around 1000) such that for our parameters about 70% is in kinetic energy and about 30% is in magnetic energy. The energy dissipation rates are converged to 30% viscous dissipation and 70% resistive dissipation. Second-order structure functions of the Elsasser variables give evidence for a k(-5/3) spectrum. Nevertheless, the three-dimensional spectrum is close to k(-3/2), but we argue that this is due to the bottleneck effect. The bottleneck effect is shown to be equally strong both for magnetic and nonmagnetic turbulence, but it is far weaker in one-dimensional spectra that are normally studied in laboratory turbulence. Structure function exponents for other orders are well described by the She-Leveque formula, but the velocity field is significantly less intermittent and the magnetic field is more intermittent than the Elsasser variables.
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Magnetic helicity evolution in a periodic domain with imposed field.
In helical hydromagnetic turbulence with an imposed magnetic field (which is constant in space and time) the magnetic helicity of the field within a periodic domain is no longer an invariant of the ideal equations. Alternatively, there is a generalized magnetic helicity that is an invariant of the ideal equations. It is shown that this quantity is not gauge invariant and that it can therefore not be used in practice. Instead, the evolution equation of the magnetic helicity of the field describing the deviation from the imposed field is shown to be a useful tool. It is demonstrated that this tool can determine steady state quenching of the alpha-effect. A simple three-scale model is derived to describe the evolution of the magnetic helicity and to predict its sign as a function of the imposed field strength. The results of the model agree favorably with simulations.
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Self-similar scaling in decaying numerical turbulence.
Decaying turbulence is studied numerically using as initial condition a random flow whose shell-integrated energy spectrum increases with wave number k like k(q). Alternatively, initial conditions are generated from a driven turbulence simulation by simply stopping the driving. It is known that the dependence of the decaying energy spectrum on wave number, time, and viscosity can be collapsed onto a unique scaling function that depends only on two parameters. This is confirmed using three-dimensional simulations and the dependence of the scaling function on its two arguments is determined.
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Bottleneck effect in three-dimensional turbulence simulations.
At numerical resolutions around 512(3) and above, three-dimensional energy spectra from turbulence simulations begin to show noticeably shallower spectra than k(-5/3) near the dissipation wave number ("bottleneck effect"). This effect is shown to be significantly weaker in one-dimensional spectra such as those obtained in wind tunnel turbulence. The difference can be understood in terms of the transformation between the one-dimensional and three-dimensional energy spectra under the assumption that the turbulent velocity field is isotropic. Transversal and longitudinal energy spectra are similar and can both accurately be computed from the full three-dimensional spectra. Second-order structure functions are less susceptible to the bottleneck effect and may be better suited for inferring the scaling exponent from numerical simulation data.
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Contributions to the theory of a two-scale homogeneous dynamo experiment.
The principle of the two-scale dynamo experiment at the Forschungszentrum Karlsruhe is closely related to that of the Roberts dynamo working with a simple fluid flow which is, with respect to proper Cartesian coordinates x, y, and z, periodic in x and y and independent of z. A modified Roberts dynamo problem is considered with a flow more similar to that in the experimental device. Solutions are calculated numerically, and on this basis an estimate of the excitation condition of the experimental dynamo is given. The modified Roberts dynamo problem is also considered in the framework of the mean-field dynamo theory, in which the crucial induction effect of the fluid motion is an anisotropic alpha effect. Numerical results are given for the dependence of the mean-field coefficients on the fluid flow rates. The excitation condition of the dynamo is also discussed within this framework. The behavior of the dynamo in the nonlinear regime, i.e., with backreaction of the magnetic field on the fluid flow, depends on the effect of the Lorentz force on the flow rates. The quantities determining this effect are calculated numerically. The results for the mean-field coefficients and the quantities describing the backreaction provide corrections to earlier results, which were obtained under simplifying assumptions.
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Solitons in the noisy Burgers equation.
We investigate numerically the coupled diffusion-advective type field equations originating from the canonical phase space approach to the noisy Burgers equation or the equivalent Kardar-Parisi-Zhang equation in one spatial dimension. The equations support stable right hand and left hand solitons and in the low viscosity limit a long-lived soliton pair excitation. We find that two identical pair excitations scatter transparently subject to a size-dependent phase shift and that identical solitons scatter on a static soliton transparently without a phase shift. The soliton pair excitation and the scattering configurations are interpreted in terms of growing step and nucleation events in the interface growth profile. Finally, we show that growing steps perform an anomalous random walk with dynamic exponent z=3/2.
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Nonlinear states of the screw dynamo.
The self-excitation of magnetic field by a spiral Couette flow between two coaxial cylinders is considered. We solve numerically the fully nonlinear, three-dimensional magnetohydrodynamic (MHD) equations for magnetic Prandtl numbers P(m) (ratio of kinematic viscosity to magnetic diffusivity) between 0.14 and 10 and kinematic and magnetic Reynolds numbers up to about 2000. In the initial stage of exponential field growth (kinematic dynamo regime), we find that the dynamo switches from one distinct regime to another as the radial width delta(r)(B) of the magnetic field distribution becomes smaller than the separation of the field maximum from the flow boundary. The saturation of magnetic field growth is due to a reduction in the velocity shear resulting mainly from the azimuthally and axially averaged part of the Lorentz force, which agrees with an asymptotic result for the limit of P(m)<<1. In the parameter regime considered, the magnetic energy decreases with kinematic Reynolds number as Re-0.84, which is approximately as predicted by the nonlinear asymptotic theory (approximately Re(-1)). However, when the velocity field is maintained by a volume force (rather than by viscous stress) the dependence of magnetic energy on the kinematic Reynolds number is much weaker.
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Effect of hyperdiffusivity on turbulent dynamos with helicity.
In numerical studies of turbulence, hyperviscosity is often used as a tool to extend the inertial subrange and to reduce the dissipative subrange. By analogy, hyperdiffusivity (or hyperresistivity) is sometimes used in magnetohydrodynamics. The underlying assumption is that only the small scales are affected by this manipulation. In the present paper, possible side effects on the evolution of the large-scale magnetic field are investigated. It is found that for turbulent flows with helicity, hyperdiffusivity causes the dynamo-generated magnetic field to saturate at a higher level than normal diffusivity. This result is successfully interpreted in terms of magnetic helicity conservation, which also predicts that full saturation is reached only after a time comparable to the large-scale magnetic (hyper)diffusion time.
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