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In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behaviour is different from that of an
$\unicode[STIX]{x1D6FC}^{2}$
dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time dependent. The growing mean field can be modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. (J. Fluid Mech., vol. 783, 2015, pp. 23–45), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15 % supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.
This paper investigates the motion of three-dimensional ideal magnetohydrodynamics with incompressible flows. The governing equation is performed at steady state, with the magnetic field parallel to the plasma flow. The equations of stationary equilibrium are derived and described mathematically in Cartesian space. Two approaches for derivation of general three-dimensional solutions for Alfvénic and non-Alfvénic flows at constant and variable fluid densities are constructed. The general vector and scalar potentials of the velocity field are used to derive general formulas of general three-dimensional solutions for Alfvénic and non-Alfvénic flows. To verify the general results we have obtained, some examples are presented. An application that may be of interest for coronal loops and solar prominences is presented.
Boundary layers in space and astrophysical plasmas are the location of complex dynamics where different mechanisms coexist and compete, eventually leading to plasma mixing. In this work, we present fully kinetic particle-in-cell simulations of different boundary layers characterized by the following main ingredients: a velocity shear, a density gradient and a magnetic gradient localized at the same position. In particular, the presence of a density gradient drives the development of the lower-hybrid drift instability (LHDI), which competes with the Kelvin–Helmholtz instability (KHI) in the development of the boundary layer. Depending on the density gradient, the LHDI can even dominate the dynamics of the layer. Because these two instabilities grow on different spatial and temporal scales, when the LHDI develops faster than the KHI an inverse cascade is generated, at least in two dimensions. This inverse cascade, starting at the LHDI kinetic scales, generates structures at scale lengths at which the KHI would typically develop. When that is the case, those structures can suppress the KHI itself because they significantly affect the underlying velocity shear gradient. We conclude that, depending on the density gradient, the velocity jump and the width of the boundary layer, the LHDI in its nonlinear phase can become the primary instability for plasma mixing. These numerical simulations show that the LHDI is likely to be a dominant process at the magnetopause of Mercury. These results are expected to be of direct impact to the interpretation of the forthcoming BepiColombo observations.
Beam-driven instabilities are considered in a pulsar plasma assuming that both the background plasma and the beam are relativistic Jüttner distributions. In the rest frame of the background, the only waves that can satisfy the resonance condition are in a tiny range of slightly subluminal phase speeds. The growth rate for the kinetic (or maser) version of the weak-beam instability is much smaller than has been estimated for a relativistically streaming Gaussian distribution, and the reasons for this are discussed. The growth rate for the reactive version of the weak-beam instability is treated in a conventional way. We compare the results with exact calculations, and find that the approximate solutions are not consistent with the exact results. We conclude that, for plausible parameters, there is no reactive version of the instability. The growth rate in the pulsar frame is smaller than that in the rest frame of the background plasma by a factor
$2\unicode[STIX]{x1D6FE}_{\text{s}}$
, where
$\unicode[STIX]{x1D6FE}_{\text{s}}=10^{2}{-}10^{3}$
is the Lorentz factor of the bulk motion of the background plasma, placing a further constraint on effective wave growth. Based on these results, we argue that beam-driven wave growth probably plays no role in pulsar radio emission.
We present a Vlasov–DArwin numerical code (ViDA) specifically designed to address plasma physics problems, where small-scale high accuracy is requested even during the nonlinear regime to guarantee a clean description of the plasma dynamics at fine spatial scales. The algorithm provides a low-noise description of proton and electron kinetic dynamics, by splitting in time the multi-advection Vlasov equation in phase space. Maxwell equations for the electric and magnetic fields are reorganized according to the Darwin approximation to remove light waves. Several numerical tests show that ViDA successfully reproduces the propagation of linear and nonlinear waves and captures the physics of magnetic reconnection. We also discuss preliminary tests of the parallelization algorithm efficiency, performed at CINECA on the Marconi-KNL cluster. ViDA will allow the running of Eulerian simulations of a non-relativistic fully kinetic collisionless plasma and it is expected to provide relevant insights into important problems of plasma astrophysics such as, for instance, the development of the turbulent cascade at electron scales and the structure and dynamics of electron-scale magnetic reconnection, such as the electron diffusion region.
Ion motion in a collisionless shock front is affected by macroscopic large-scale weakly varying and microscopic small-scale fast varying magnetic and electric fields. With the increase of the Mach number the role of the microscopic field is expected to become progressively more important. Using a combination of hybrid simulations and test particle analysis, we show that in moderately supercritical shocks macroscopic fields play the main role in ion motion across the shock. Pressure balance across the shock is only weakly broken and non-stationarity is related to the deviations from the total pressure from the constant value.
We present three-dimensional direct numerical simulations and an analytic model of reflection-driven magnetohydrodynamic (MHD) turbulence in the solar wind. Our simulations describe transverse, non-compressive MHD fluctuations within a narrow magnetic flux tube that extends from the photosphere, through the chromosphere and corona and out to a heliocentric distance
$r$
of 21 solar radii
$(R_{\odot })$
. We launch outward-propagating ‘
$\boldsymbol{z}^{+}$
fluctuations’ into the simulation domain by imposing a randomly evolving photospheric velocity field. As these fluctuations propagate away from the Sun, they undergo partial reflection, producing inward-propagating ‘
$\boldsymbol{z}^{-}$
fluctuations’. Counter-propagating fluctuations subsequently interact, causing fluctuation energy to cascade to small scales and dissipate. Our analytic model incorporates dynamic alignment, allows for strongly or weakly turbulent nonlinear interactions and divides the
$\boldsymbol{z}^{+}$
fluctuations into two populations with different characteristic radial correlation lengths. The inertial-range power spectra of
$\boldsymbol{z}^{+}$
and
$\boldsymbol{z}^{-}$
fluctuations in our simulations evolve toward a
$k_{\bot }^{-3/2}$
scaling at
$r>10R_{\odot }$
, where
$k_{\bot }$
is the wave-vector component perpendicular to the background magnetic field. In two of our simulations, the
$\boldsymbol{z}^{+}$
power spectra are much flatter between the coronal base and
$r\simeq 4R_{\odot }$
. We argue that these spectral scalings are caused by: (i) high-pass filtering in the upper chromosphere; (ii) the anomalous coherence of inertial-range
$\boldsymbol{z}^{-}$
fluctuations in a reference frame propagating outwards with the
$\boldsymbol{z}^{+}$
fluctuations; and (iii) the change in the sign of the radial derivative of the Alfvén speed at
$r=r_{\text{m}}\simeq 1.7R_{\odot }$
, which disrupts this anomalous coherence between
$r=r_{\text{m}}$
and
$r\simeq 2r_{\text{m}}$
. At
$r>1.3R_{\odot }$
, the turbulent heating rate in our simulations is comparable to the turbulent heating rate in a previously developed solar-wind model that agreed with a number of observational constraints, consistent with the hypothesis that MHD turbulence accounts for much of the heating of the fast solar wind.
These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental contexts disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) small- and large-scale magnetohydrodynamic (MHD) dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small- and large-scale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.
We present an ideal two-fluid wave mode analysis for a pair plasma, extending an earlier study for cold conditions to the warm pair plasma case. Starting from the completely symmetrized means for writing the governing linearized equations in the pair fluid rest frame, we discuss the governing dispersion relation containing all six pairs of forward and backward propagating modes, which are conveniently labelled as S, A, F, M, O and X. These relate to the slow (S), Alfvén (A) and fast (F) magnetohydrodynamic waves, include a modified (M) electrostatic mode, as well as the electromagnetic O and X branches. In the dispersion relation, only two parameters appear, which define the pair plasma magnetization
$E^{2}\in [0,\infty ]$
and the squared pair plasma sound speed
$v^{2}$
, measured in units of the light speed
$c$
. The description is valid also in the highly relativistic regime, where either a high magnetization and/or a relativistic temperature (hence sound speed) is reached. We recover the exact relativistic single-fluid magnetohydrodynamic expressions for the S, A and F families in the low wavenumber–frequency regime, which can be obtained for any choice of the equation of state. We argue that, as in a cold pair plasma, purely parallel or purely perpendicular propagation with respect to the magnetic field vector
$\boldsymbol{B}$
is special, and near-parallel or near-perpendicular orientations demonstrate avoided crossings of branches at computable wavenumbers and frequencies. The complete six-mode phase and group diagram views are provided as well, visually demonstrating the intricate anisotropies in all wave modes, as well as their transformations. Analytic expressions for all six wave group speeds at both small and large wavenumbers complement the analysis.
Turbulence is commonly observed in nearly collisionless heliospheric plasmas, including the solar wind and corona and the Earth’s magnetosphere. Understanding the collisionless mechanisms responsible for the energy transfer from the turbulent fluctuations to the particles is a frontier in kinetic turbulence research. Collisionless energy transfer from the turbulence to the particles can take place reversibly, resulting in non-thermal energy in the particle velocity distribution functions (VDFs) before eventual collisional thermalization is realized. Exploiting the information contained in the fluctuations in the VDFs is valuable. Here we apply a recently developed method based on VDFs, the field–particle correlation technique, to a
$\unicode[STIX]{x1D6FD}=1$
, solar-wind-like, low-frequency Alfvénic turbulence simulation with well-resolved phase space to identify the field–particle energy transfer in velocity space. The field–particle correlations reveal that the energy transfer, mediated by the parallel electric field, results in significant structuring of the VDF in the direction parallel to the magnetic field. Fourier modes representing the length scales between the ion and electron gyroradii show that energy transfer is resonant in nature, localized in velocity space to the Landau resonances for each Fourier mode. The energy transfer closely follows the Landau resonant velocities with varying perpendicular wavenumber
$k_{\bot }$
and plasma
$\unicode[STIX]{x1D6FD}$
. This resonant signature, consistent with Landau damping, is observed in all diagnosed Fourier modes that cover the dissipation range of the simulation.
Wave dispersion in a pulsar plasma is discussed emphasizing the relevance of different inertial frames, notably the plasma rest frame
${\mathcal{K}}$
and the pulsar frame
${\mathcal{K}}^{\prime }$
in which the plasma is streaming with speed
$\unicode[STIX]{x1D6FD}_{\text{s}}$
. The effect of a Lorentz transformation on both subluminal,
$|z|<1$
, and superluminal,
$|z|>1$
, waves is discussed. It is argued that the preferred choice for a relativistically streaming distribution should be a Lorentz-transformed Jüttner distribution; such a distribution is compared with other choices including a relativistically streaming Gaussian distribution. A Lorentz transformation of the dielectric tensor is written down, and used to derive an explicit relation between the relativistic plasma dispersion functions in
${\mathcal{K}}$
and
${\mathcal{K}}^{\prime }$
. It is shown that the dispersion equation can be written in an invariant form, implying a one-to-one correspondence between wave modes in any two inertial frames. Although there are only three modes in the plasma rest frame, it is possible for backward-propagating or negative-frequency solutions in
${\mathcal{K}}$
to transform into additional forward-propagating, positive-frequency solutions in
${\mathcal{K}}^{\prime }$
that may be regarded as additional modes.
A modified Vlasov equation is obtained by developing a covariant statistical mechanics for a system of electrons without considering the effects of the ions and including the Landau–Lifshitz equation of motion. General dispersion relations for the transverse and longitudinal modes for any temperature are expressed. The results are similar to those found by Hakim & Mangeney (Phys. Fluids, vol. 14, 1971, pp. 2751–2781) for both the modified Vlasov equation and the dispersion relations. However, for the longitudinal mode, unlike the development of Hakim and Mangeney, correct expansions are done in order to give a numerical approach to obtain the longitudinal relativistic dispersion relations for any value of the wavenumber. Accordingly, new loop solutions, with turning points, crossing the super-luminous region and the super-thermal region are found. Although the expressions for the Landau damping and the damping due to the radiation reaction force coincide with the Hakim and Mangeney results for some particular cases, in general they are different. A Landau anti-damping appears in the second branch of the loop in a small region between the cutoff point and the intersection with the super-thermal line. The analysis of this effect leads us to a kind of wave pulse. We will call them bipolar waves. The treatment contains the relativistic interactions between all the electrons in the system with retarded effects. This explain the differences with Zhang’s recent work (Phys. Plasmas, vol. 20, 2013, 092112–092132). It is shown that for low densities, the cutoff of the wave is due to the dispersion relations and not due to the radiation reaction force damping. While for both high densities and temperatures, the damping due to the radiation reaction force is important.
We study the damping of collisionless Alfvénic turbulence in a strongly magnetised plasma by two mechanisms: stochastic heating (whose efficiency depends on the local turbulence amplitude
$\unicode[STIX]{x1D6FF}z_{\unicode[STIX]{x1D706}}$
) and linear Landau damping (whose efficiency is independent of
$\unicode[STIX]{x1D6FF}z_{\unicode[STIX]{x1D706}}$
), describing in detail how they affect and are affected by intermittency. The overall efficiency of linear Landau damping is not affected by intermittency in critically balanced turbulence, while stochastic heating is much more efficient in the presence of intermittent turbulence. Moreover, stochastic heating leads to a drop in the scale-dependent kurtosis over a narrow range of scales around the ion gyroscale.
Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle
$\unicode[STIX]{x1D703}=0$
) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation
$|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$
, where
$z=\unicode[STIX]{x1D714}/k_{\Vert }c$
is the phase speed and
$\unicode[STIX]{x1D6FD}_{\text{A}}c$
is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function,
$z^{2}W(z)$
, which is negative for
$|z|<z_{0}$
and has a sharp maximum at
$|z|=z_{\text{m}}$
, with
$1-z_{\text{m}}<1-z_{0}\ll 1$
. We give numerical estimates for the maximum of
$z^{2}W(z)$
and for
$z_{\text{m}}$
and
$z_{0}$
for a one-dimensional Jüttner distribution. The L and A modes reconnect, for
$z_{\text{A}}>z_{0}$
, to form the O and Alfvén modes for oblique propagation (
$\unicode[STIX]{x1D703}\neq 0$
). For
$z_{\text{A}}<z_{0}$
the Alfvén and O mode curves reconnect forming a new mode that exists only for
$\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$
. The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around
$\unicode[STIX]{x1D703}\approx 0$
. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.
I analyse and numerically evaluate the radiation field generated by an experimentally realized embodiment of an electric polarization current whose rotating distribution pattern moves with linear speeds exceeding the speed of light in vacuum. I find that the flux density of the resulting emission (i) has a dominant value and is linearly polarized within a sharply delineated radiation beam whose orientation and polar width are determined by the range of values of the linear speeds of the rotating source distribution, and (ii) decays with the distance
$d$
from the source as
$d^{-\unicode[STIX]{x1D6FC}}$
in which the value of
$\unicode[STIX]{x1D6FC}$
lies between
$1$
and
$2$
(instead of being equal to
$2$
as in a conventional radiation) across the beam. In that the rate at which boundaries of the retarded distribution of such a source change with time depends on its duration monotonically, this is an intrinsically transient emission process: temporal rate of change of the energy density of the radiation generated by it has a time-averaged value that is negative (instead of being zero as in a conventional radiation) at points where the envelopes of the wave fronts emanating from the constituent volume elements of the source distribution are cusped. The difference in the fluxes of power across any two spheres centred on the source is in this case balanced by the change with time of the energy contained inside the shell bounded by those spheres. These results are relevant not only to long-range transmitters in communications technology but also to astrophysical objects containing rapidly rotating neutron stars (such as pulsars) and to the interpretation of the energetics of the multi-wavelength emissions from sources that lie at cosmological distances (such as radio and gamma-ray bursts). The analysis presented in this paper is self-contained and supersedes my earlier works on this problem.
It is shown that in low-beta, weakly collisional plasmas, such as the solar corona, some instances of the solar wind, the aurora, inner regions of accretion discs, their coronae and some laboratory plasmas, Alfvénic fluctuations produce no ion heating within the gyrokinetic approximation, i.e. as long as their amplitudes (at the Larmor scale) are small and their frequencies stay below the ion-Larmor frequency (even though their spatial scales can be above or below the ion Larmor scale). Thus, all low-frequency ion heating in such plasmas is due to compressive fluctuations (‘slow modes’): density perturbations and non-Maxwellian perturbations of the ion distribution function. Because these fluctuations energetically decouple from the Alfvénic ones already in the inertial range, the above conclusion means that the energy partition between ions and electrons in low-beta plasmas is decided at the outer scale, where turbulence is launched, and can be determined from magnetohydrodynamic (MHD) models of the relevant astrophysical systems. Any additional ion heating must come from non-gyrokinetic mechanisms such as cyclotron heating or the stochastic heating owing to distortions of ions’ Larmor orbits. An exception to these conclusions occurs in the Hall limit, i.e. when the ratio of the ion to electron temperatures is as low as the ion beta (equivalently, the electron beta is order unity). In this regime, slow modes couple to Alfvénic ones well above the Larmor scale (viz., at the ion inertial or ion sound scale), so the Alfvénic and compressive cascades join and then separate again into two cascades of fluctuations that linearly resemble kinetic Alfvén and ion-cyclotron waves, with the former heating electrons and the latter ions. The two cascades are shown to decouple, scalings for them are derived and it is argued physically that the two species will be heated by them at approximately equal rates.
Magnetic helicity flux gives information about the topology of a magnetic field passing through a boundary. In solar physics applications, this boundary is the photosphere and magnetic helicity flux has become an important quantity in analysing magnetic fields emerging into the solar atmosphere. In this work we investigate the evolution of magnetic helicity flux in magnetohydrodynamic (MHD) simulations of solar flux emergence. We consider emerging magnetic fields with different topologies and investigate how the magnetic helicity flux patterns correspond to the dynamics of emergence. To investigate how the helicity input is connected to the emergence process, we consider two forms of the helicity flux. The first is the standard form giving topological information weighted by magnetic flux. The second form represents the net winding and can be interpreted as the standard helicity flux less the magnetic flux. Both quantities provide important and distinct information about the structure of the emerging field and these quantities differ significantly for mixed sign helicity fields. A novel aspect of this study is that we account for the varying morphology of the photosphere due to the motion of the dense plasma lifted into the chromosphere. Our results will prove useful for the interpretation of magnetic helicity flux maps in solar observations.
In a magnetized, collisionless plasma, the magnetic moment of the constituent particles is an adiabatic invariant. An increase in the magnetic-field strength in such a plasma thus leads to an increase in the thermal pressure perpendicular to the field lines. Above a
$\unicode[STIX]{x1D6FD}$
-dependent threshold (where
$\unicode[STIX]{x1D6FD}$
is the ratio of thermal to magnetic pressure), this pressure anisotropy drives the mirror instability, producing strong distortions in the field lines on ion-Larmor scales. The impact of this instability on magnetic reconnection is investigated using a simple analytical model for the formation of a current sheet (CS) and the associated production of pressure anisotropy. The difficulty in maintaining an isotropic, Maxwellian particle distribution during the formation and subsequent thinning of a CS in a collisionless plasma, coupled with the low threshold for the mirror instability in a high-
$\unicode[STIX]{x1D6FD}$
plasma, imply that the geometry of reconnecting magnetic fields can differ radically from the standard Harris-sheet profile often used in simulations of collisionless reconnection. As a result, depending on the rate of CS formation and the initial CS thickness, tearing modes whose growth rates and wavenumbers are boosted by this difference may disrupt the mirror-infested CS before standard tearing modes can develop. A quantitative theory is developed to illustrate this process, which may find application in the tearing-mediated disruption of kinetic magnetorotational ‘channel’ modes.
We present a complete analysis of all wave modes in a cold pair plasma, significantly extending standard textbook treatments. Instead of identifying the maximal number of two propagating waves at fixed frequency
$\unicode[STIX]{x1D714}$
, we introduce a unique labelling of all 5 mode pairs described by the general dispersion relation
$\unicode[STIX]{x1D714}(k)$
, starting from their natural ordering at small wavenumber
$k$
. There, the 5 pairs start off as Alfvén (A), fast magnetosonic (F), modified electrostatic (M) and electromagnetic O and X branches, and each
$\unicode[STIX]{x1D714}(k)$
branch smoothly connects to large wavenumber resonances or limits. For cold pair plasmas, these 5 branches show avoided crossings, which become true crossings at exactly parallel or perpendicular orientation. Only for those orientations, we find a changed connectivity between small and large wavenumber behaviour. Analysing phase and group diagrams for all 5 wave modes, distinctly different from the Clemmow–Mullaly–Allis representation, reveals the true anisotropy of the A, M and O branches.
Global magnetohydrodynamic (MHD) instabilities are investigated in a computationally tractable two-dimensional model of the solar tachocline. The model’s differential rotation yields stability in the absence of a magnetic field, but if a magnetic field is present, a joint instability is observed. We analyse the nonlinear development of the instability via fully nonlinear direct numerical simulation, the generalized quasi-linear approximation (GQL) and direct statistical simulation (DSS) based upon low-order expansion in equal-time cumulants. As the magnetic diffusivity is decreased, the nonlinear development of the instability becomes more complicated until eventually a set of parameters is identified that produces a previously unidentified long-term cycle in which energy is transformed from kinetic energy to magnetic energy and back. We find that the periodic transitions, which mimic some aspects of solar variability – for example, the quasiperiodic seasonal exchange of energy between toroidal field and waves or eddies – are unable to be reproduced when eddy-scattering processes are excluded from the model.