Helical Turbulence Research Articles

The integral equation method for a steady kinematic dynamo problem

With only a few exceptions, the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method. However, the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics. The paper is intended to facilitate the use of integral equation solvers in dynamo theory. In concrete, the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter α. For the case of spherical geometry, three examples of the function α(r) with steady and oscillatory solutions are considered. A convergence rate proportional to the inverse squared of the number of grid points is achieved. Based on this method, a convergence accelerating strategy is developed and the convergence rate is improved remarkably. Typically, quite accurate results can be obtained with a few tens of grid points. In order to demonstrate its suitability for the treatment of dynamos in other than spherical domains, the method is also applied to α2 dynamos in rectangular boxes. The magnetic fields and the electric potentials for the first eigenvalues are visualized.

On the origin of activity in solar-type stars

The magnetic flux of solar coronal active regions is thought to originate in strong toroidal magnetic fields generated by a dynamo at the base of the convection zone. Once generated, this magnetic flux rises through the convection zone as discrete buoyant flux tubes, which may be formed into S2-shaped loops by their interaction with convective: cells and strong downdrafts. The loops are prevented from fragmentation by twist and curvature of their axes, which are writhed by the Coriolis effect and helical convective turbulence. These Σ-shaped loops emerge through the photosphere to form dipolar sunspot pairs and coronal active regions. These regions' free energy, relative magnetic helicity, and tendency to flare and erupt reflect the convection zone phenomena that dominate their journey to the surface, in which helical convective turbulence appears to play a primary role. Recent research leads me to suggest a new paradigm for activity in solar-type stars with deep-seated (tachocline) dynamos. In the present paradigm, dynamo models are expected to explain the distribution of activity in the H-R diagram, as reflected in mean chromospheric emission in lower main-sequence stars. In the new paradigm, dynamo action simply generates the flux that is necessary, but not sufficient, for such activity, and the amplitude of activity depends; most importantly on the kinetic helicity and turbulence of convection zone flows.

Understanding helical magnetic dynamo spectra with a non-linear four-scale theory

Recent MHD dynamo simulations for magnetic Prandtl number $>1$ demonstrate that when MHD turbulence is forced with sufficient kinetic helicity, the saturated magnetic energy spectrum evolves from having a single peak below the forcing scale to become doubly peaked with one peak at the system (=largest) scale and one at the forcing scale. The system scale field growth is well modeled by a recent nonlinear two-scale nonlinear helical dynamo theory in which the system and forcing scales carry magnetic helicity of opposite sign. But a two-scale theory cannot model the shift of the small-scale peak toward the forcing scale. Here I develop a four-scale helical dynamo theory which shows that the small-scale helical magnetic energy first saturates at very small scales, but then successively saturates at larger values at larger scales, eventually becoming dominated by the forcing scale. The transfer of the small scale peak to the forcing scale is completed by the end of the kinematic growth regime of the large scale field, and does not depend on magnetic Reynolds number $R_M$ for large $R_M$. The four-scale and two-scale theories subsequently evolve almost identically, and both show significant field growth on the system and forcing scales that is independent of $R_M$. In the present approach, the helical and nonhelical parts of the spectrum are largely decoupled. Implications for fractionally helical turbulence are discussed.

Origin of galactic and extragalactic magnetic fields

A variety of observations suggest that magnetic fields are present in all galaxies and galaxy clusters. These fields are characterized by a modest strength ${(10}^{\ensuremath{-}7}--{10}^{\ensuremath{-}5}\mathrm{G})$ and huge spatial scale $(\ensuremath{\lesssim}1\mathrm{Mpc}).$ It is generally assumed that magnetic fields in spiral galaxies arise from the combined action of differential rotation and helical turbulence, a process known as the \ensuremath{\alpha}\ensuremath{\omega} dynamo. However, fundamental questions concerning the nature of the dynamo as well as the origin of the seed fields necessary to prime it remain unclear. Moreover, the standard \ensuremath{\alpha}\ensuremath{\omega} dynamo does not explain the existence of magnetic fields in elliptical galaxies and clusters. The author summarizes what is known observationally about magnetic fields in galaxies, clusters, superclusters, and beyond. He then reviews the standard dynamo paradigm, the challenges that have been leveled against it, and several alternative scenarios. He concludes with a discussion of astrophysical and early-Universe candidates for seed fields.

Nonlinear Fluctuation-Dissipation Theory to the Second Order

The second-order nonlinearresponse function is expressed in termsof a third-rank correlation in the dynamical variables, which will help the derivation of a nonlinear heat-e ux law. The second-order nonlinear term proved to have no ine uence on the absorption of the external energy by a system. A consequence of that fact is the linear growth or decay of the third-rank correlation tensor. In the case of e uids, the growth of the third-rank velocity correlation tensor will indicate a growth in helical turbulence, and the growth in helical turbulence can lead to the formation of cyclones.

Dissipation in helical turbulence

In helical turbulence a linear cascade of helicity accompanying the energy cascade has been suggested. Since energy and helicity have different dimensionality we suggest the existence of a characteristic inner scale, ξ=kH−1, for helicity dissipation in a regime of hydrodynamic fully developed turbulence and estimate it on dimensional grounds. This scale is always larger than the Kolmogorov scale, η=kE−1, and their ratio η/ξ vanishes in the high Reynolds number limit, so the flow will always be helicity free in the small scales.

A new look at the relationship between activity, dynamo number and Rossby number in late-type stars

ABSTRA C T The correlation between stellar activity, as measured by the indicator DRHK, and the Rossby number Ro in late-type stars is revisited in light of recent developments in solar dynamo theory. Different stellar interior models, based on both mixing-length theory and the full spectrum of turbulence, are used in order to see to what extent the correlation of activity with Rossby number is model dependent, or otherwise can be considered universal. Although we find some modest model dependence, we find that the correlation of activity with Rossby number is significantly better than with rotation period alone for all the models we consider. Dynamo theory suggests that activity should scale with the dynamo number. A current model of the solar dynamo, the so-called interface dynamo, proposes that the amplification of the toroidal magnetic field by differential rotation (the v-effect) and the production of the poloidal magnetic field from toroidal by helical turbulence (the a-effect) take place in different, adjacent layers near the base of the convection zone. A new scale analysis based on the interface dynamo shows that the appropriate dynamo number does not depend on the Rossby number alone, but also depends on an additional dimensionless factor related to the differential rotation. This leads to a new interpretation of the correlation between activity and Rossby number, which in turn leads to some conclusions about the magnitude of differential rotation in the dynamo layers of late-type main-sequence stars.

The structure of the global solar magnetic field excited by the turbulent dynamo mechanism

A mixing-length approximation is used to calculate Kλ for a Parker dynamo wave excited by the dynamo mechanism near the base of the solar convection zone (K is the wave number of the dynamo wave and λ the extent of the dynamo region). In a turbulent-dynamo model, this number characterizes the modes of the global magnetic field generated by a mechanism based on the joint action of the mean helical turbulence and solar differential rotation. Estimates are obtained for the helicity and radial angular-velocity gradient using the most recent helioseismological measurements at the growth phase of solar cycle 23. These estimates indicate that the dynamo mechanism most efficiently excites the fundamental antisymmetric (odd), dipole, mode of the poloidal field (Kλ≈−7) at low latitudes, while the conditions at latitudes above 50° are more favorable for the excitation of the lowest symmetric (even), quadrupole, mode (Kλ≈+8). The resulting north-south asymmetry of the poloidal field can explain the magnetic anomaly (“monopole” structure) of the polar fields observed near solar-cycle maxima. The effect of α quenching increases the calculated period of the dynamo-wave propagation from middle latitudes to the equator to about seven years, in rough agreement with the observed duration of the solar cycle.

Dynamo in helical MHD turbulence: quantum field theory approach

A quantum field model of helical MHD stochastically forced by gaussian hydrodynamic, magnetic and mixed noices is investigated. These helical noises lead to an exponential increase of magnetic fluctuations in the large scale range. Instabilities, which are produced in this process, are eliminated by spontaneous symmetry breaking mechanism accompanied by creation of the homogeneous stationary magnetic field.

Cascades in helical turbulence.

We suggest the existence of a characteristic inner scale xi for helicity dissipation in a regime of hydrodynamic fully developed turbulence and estimate it on dimensional grounds. This scale is always larger than the Kolmogorov scale eta and their ratio eta/xi vanishes in the high Reynolds number limit, so the flow will always be helicity free in the small scales. These ideas are illustrated in a shell model of turbulence.

The helicity constraint in turbulent dynamos with shear

ABSTRA C T The evolution of magnetic fields is studied using simulations of forced helical turbulence with strong imposed shear. After some initial exponential growth, the magnetic field develops a large-scale travelling wave pattern. The resulting field structure possesses magnetic helicity, which is conserved in a periodic box by the ideal magnetohydrodynamics equations and can hence only change on a resistive time-scale. This strongly constrains the growth time of the large-scale magnetic field, but less strongly constrains the length of the cycle period. Comparing this with the case without shear, the time-scale for large-scale field amplification is shortened by a factor Q, which depends on the relative importance of shear and helical turbulence, and which also controls the ratio of toroidal to poloidal field. The results of the simulations can be reproduced qualitatively and quantitatively with a mean-field aV-dynamo model with alpha-effect and turbulent magnetic diffusivity coefficients that are less strongly quenched than in the corresponding a 2 -dynamo.

Numerical Simulation of Evolution of Helical-Vortex Instability in a Fluid Layer Heated from Below

Large-scale helical-vortex instability is simulated numerically for laminar convection regimes. The simulation is based on the nonlinear Boussinesq equations supplemented by an external force of special form, whose structure is identical to the tensor structure of the generative term in the equation for the hydrodynamic alpha-effect. Introducing the external force makes it possible to model the average effect produced by small-scale helical turbulence: generation of a positive feedback between the toroidal and poloidal components of the velocity vector field. The structure and energetics of helical-vortex flows developing in a convectively unstable medium are analyzed. The parameters of helical and non-helical convection regimes are compared. Certain novel effects related to spiral feedback are discussed: generation of a toroidal velocity field in structures with the poloidal circulation typical of ordinary convection; enlargement of the horizontal scale of supercritical motions as a result of the merging of vortex cells, accompanied by a significant increase in the circulation strength in the larger helical vortices formed; qualitative changes in the energetics.

Effects of Helicity and System Rotation on Decaying Homogeneous Turbulence.

To consider the nonreflection property of the turbulence statistics in a rotating system, a new method has been proposed to introduce turbulence helicity into an isotropic turbulent field for direct numerical simulation (DNS). Effects of the helicity and the rotation on turbulence statistics and vortex structure are investigated by theoretical analysis and DNS of the homogeneous decaying turbulence. Although DNS results show that both rotation and the helicity inhibit energy decay, theoretical analysis reveals the occurrence of very different mechanisms for the rotation and the helicity for producing the inhibition effect. The helicity directly decreases the energy transfer while the rotation suppresses the energy transfer due to the so-called scrambling effect. The other effect of system rotation is that rotation elongates the vortex structure along the rotation axis, however, the presence of the helicity appears to weaken this tendency.

Mean Field Theory Interpretation of Solar Polarity Reversal

A mechanism of the polarity reversal of the solar magnetic field is explored on the basis of the mean field or turbulent dynamo theory. In the low-latitude region of the convective zone, the toroidal magnetic field, which is the origin of sunspots, is generated by the rotational motion of fluids, with the turbulent cross helicity as the intermediary. This field generates the poloidal field of dipole type through the alpha or turbulent helicity effect. The latter, in turn, contributes to the annihilation of the turbulent cross helicity, resulting in the decay of the toroidal magnetic field. This process indicates less room for the occurrence of the fully developed poloidal field in the low-latitude region and paves the way for the polarity reversal through the change of the sign of the turbulent cross helicity. A simple model mimicking the periodic polarity reversal is presented, and the relationship of the reversal period to the ratio of the poloidal to toroidal fields is given. The meridional-flow velocity at the solar surface is estimated, giving a result consistent with observations.

Rate of helicity production by solar rotation

In recent years, solar observers have discovered a striking pattern in the distribution of coronal magnetic structures: northern hemisphere structures tend to have negative magnetic helicity, while structures in the south tend to have positive magnetic helicity. This hemispheric dependence extends from photospheric observations to in situ measurements of magnetic clouds in the solar wind. Understanding the source of the hemispheric sign dependence, as well as its implications for solar and space physics has become known as the solar chirality problem. Rotation of open fields creates the Parker spiral which carries outward 1047 Mx2 of magnetic helicity (in each hemisphere) during a solar cycle. In addition, rough estimates suggest that each hemisphere sheds on the order 1045 Mx2 in coronal mass ejections each cycle. Both the α effect (arising from helical turbulence) and the Ω effect (arising from differential rotation) should contribute to the hemispheric chirality. We show that the Ω effect contribution can be captured in a surface integral, even though the helicity itself is stored deep in the convection zone. We then evaluate this surface integral using solar magnetogram data and differential rotation curves. Throughout the 22 year cycle studied (1976–1998) the helicity production in the interior by differential rotation had the correct sign compared to observations of coronal structures ‐ negative in the north and positive in the south. The net helicity flow into each hemisphere over this cycle was approximately 4 × 1046 Mx2. For comparison, we estimate the α effect contribution; this may well be as high or higher than the differential rotation contribution. The subsurface helicity can be transported to the corona with buoyant rising flux tubes. Evidently only a small fraction of the subsurface helicity escapes to the surface to supply coronal mass ejections.

Anomalous scaling in a shell model of helical turbulence

In a helical flow there is a subrange of the inertial range in which there is a cascade of both energy and helicity. In this range the scaling exponents associated with the cascade of helicity can be defined. These scaling exponents are calculated from a simulation of the GOY shell model. The scaling exponents for even moments are associated with the scaling of the symmetric part of the probability density functions while the odd moments are associated with the anti-symmetric part of the probability density functions.

Fast magnetic and turbulent-wave dynamos in electron magnetohydrodynamics

The influence of inertia on the spontaneous amplification of large-scale perturbations by electron magnetohydrodynamic (EMHD) turbulence is studied in a 212-dimensional (212-D) model. It is shown that electron inertia results in the modification of α-like effects, which are due to the helicity of the turbulence. Under some conditions no instability of the mean field due to helicity of turbulence will occur. Electron inertia can also change the sign of turbulent viscosity in the equation of the third component of the mean field, making it negative. The possibility is discussed of the generation of 3-D whistlers by 2-D EMHD turbulence due to either helicity or to anisotropy of the turbulence. Unlike the case of 2-D large-scale perturbations, turbulent resistivity is anisotropic and leads to the destabilization of whistler modes, independently of the sign of the resistivity coefficient.

Peculiarities of turbulence in a flow with vortices

In the atmospheric boundary layer there exist groups of roll vortices contributing considerably to the momentum transfer, motion properties, etc. There are data pointing to an important role of such vortices in the formation of such a dangerous extreme weather event (EWE) as a tornado. It is known that helical turbulence mode is the cause of the similarity of atmospheric and magnetohydrodynamic turbulence. To clarify the peculiarities of EWE origination, experimental studies of MHD turbulence were carried out under laboratory conditions.

On spontaneous amplification of large-scale perturbations by two-dimensional electron magnetohydrodynamic turbulence

The influence of small-scale turbulence on large-scale perturbations is studied in the model of 2-D electron magnetohydrodynamics (EMHD), in which the magnetic field has all three components. It is shown that in addition to positive turbulent viscosity and an analogue of α- effect caused by helicity of turbulence, which are typical to 3-D EMHD turbulence, such 2-D turbulence results in turbulent resistivity, which is caused by anisotropy of turbulence and can be negative.

Modification of turbulent helical/nonhelical flows with small-scale energy input

Modification of helical/nonhelical turbulence by adding small-scale turbulence is investigated with the aid of direct numerical simulations of isotropic turbulence. Detailed characteristics of the energy and helicity spectral transfer were examined. It is found that nonlocal energy cascade through isosceles triads toward the wave-number range of the energy input is markedly enhanced, thus the turbulent intensity is decreased faster than a flow field with no energy input. The optimum wave number for turbulence reduction depends only on the characteristic wave number of the energy-containing range, once the magnitude of the energy input is given. A model equation for isosceles triads derived from the EDQNM approximation indicates that the optimum wave number increases with increasing energy ratio between the energy input and the parent turbulence. For helical turbulence, the sign of helicity is crucial for the helicity and energy transfer as well as the decay rate. When the helicity of small-scale energy input has the same sign as that of the parent turbulence, enhancement of the energy cascade is notedly deteriorated. The mechanism of suppressing interaction between the parent helical turbulence and the small-scale energy input is also explained with the aid of the EDQNM approximation.