The dipole–multipole transition in rapidly rotating dynamos is investigated through the analysis of forced magnetohydrodynamic waves in an unstably stratified fluid. The focus of this study is on the inertia-free limit applicable to planetary cores, where the Rossby number is small not only on the core depth but also on the length scale of columnar convection. By progressively increasing the buoyant forcing in a linear magnetoconvection model, the slow magnetic–Archimedean–Coriolis (MAC) waves are significantly attenuated so that their kinetic helicity decreases to zero; the fast MAC wave helicity, on the other hand, is practically unaffected. In turn, polarity reversals in low-inertia spherical dynamos are shown to occur when the slow MAC waves disappear under strong forcing. Two dynamically similar regimes are identified – the suppression of slow waves in a strongly forced dynamo and the excitation of slow waves in a moderately forced dynamo starting from a small seed field. While the former regime results in polarity reversals, the latter regime produces the axial dipole from a chaotic multipolar state. For either polarity transition, a local Rayleigh number based on the mean wavenumber of the energy-containing scales bears the same linear relationship with the square of the peak magnetic field measured at the transition. The self-similarity of the dipole–multipole transition can place a constraint on the Rayleigh number for polarity reversals in the Earth.

In systems where the standard $\alpha$ effect is inoperative, one often explains the existence of mean magnetic fields by invoking the ‘incoherent $\alpha$ effect’, which appeals to fluctuations of the mean kinetic helicity at a mesoscale. Most previous studies, while considering fluctuations in the mean kinetic helicity, treated the mean turbulent kinetic energy at the mesoscale as a constant, despite the fact that both these quantities involve second-order velocity correlations. The mean turbulent kinetic energy affects the mean magnetic field through both turbulent diffusion and turbulent diamagnetism. In this work, we use a double-averaging procedure to analytically show that fluctuations of the mean turbulent kinetic energy at the mesoscale (giving rise to $\eta$-fluctuations at the mesoscale, where the scalar $\eta$ is the turbulent diffusivity) can lead to the growth of a large-scale magnetic field even when the kinetic helicity is zero pointwise. Constraints on the operation of such a dynamo are expressed in terms of dynamo numbers that depend on the correlation length, correlation time and strength of these fluctuations. In the white-noise limit, we find that these fluctuations reduce the overall turbulent diffusion, while also contributing a drift term which does not affect the growth of the field. We also study the effects of non-zero correlation time and anisotropy. Turbulent diamagnetism, which arises due to inhomogeneities in the turbulent kinetic energy, leads to growing mean-field solutions even when the $\eta$-fluctuations are statistically isotropic.

We derive a mathematical model of an electromagnetic (EM) field generated by tsunamigenic seabed deformation over an ocean of constant depth. We solve the governing Maxwell equations for the EM field, coupled with a potential flow model of Cauchy–Poisson type for the transient fluid motion forced by seabed deformation. Our new model advances previous studies, where simplified formulae without direct forcing were assumed for the wave field. Using complex integration and large-time asymptotics, we obtain a novel analytical solution for the magnetic field propagating at large distance from the seabed deformation in two dimensions. We show that this magnetic field is made of two terms, one proportional to an Airy function, and thus propagating similarly to the surface gravity wave, and one proportional to a Scorer function, which exhibits a phase lag with respect to the surface gravity wave. Such a phase lag explains the time difference between the arrival of the EM field and the surface gravity wave generated by seabed deformation, which were observed in recent measurements and numerical results. Finally, we discuss the opportunity to detect EM fields as precursors of surface gravity waves in tsunami early warning systems. We introduce a novel non-dimensional parameter to identify the propagation regime of the magnetic field, i.e. self-induction versus diffusion-dominated. We show that tsunami early warning via EM field is possible for diffusion-dominated regimes when the water depth is less than 2 km. Our findings provide a rigorous analytical explanation of existing observations and numerical results.

Numerical simulations of the geodynamo (and other planetary dynamos) have made significant progress in recent years. As computing power has advanced, some new models claim to be ever more appropriate for understanding Earth's core dynamics. One measure of the success of such models is the ability to replicate the expected balance between forces operating within the core; Coriolis and Lorentz forces are predicted to be most important. The picture is complicated for an incompressible flow by the existence of the pressure gradient force which renders the gradient parts of all other forces dynamically unimportant. This can confuse the situation, especially when the scale dependence of forces is considered. In this work we investigate force balances through the alternative approach of eliminating gradient parts of each force to form ‘solenoidal force balances’. We perform a length-scale-dependent analysis for several spherical simulations and find that removal of gradient parts offers an alternative picture of the force balance compared with looking at traditional forces alone. Solenoidal force balances provide some agreement with the results of previous studies but also significant differences. They offer a cleaner overall picture of the dynamics and introduce differences at smaller scales. This has implications for geodynamo models purporting to have reached Earth-like regimes: in order to achieve a meaningful comparison of forces, only the solenoidal part of forces should be considered.

Dynamos driven by rotating convection in the plane layer geometry are investigated numerically for a range of Ekman number ($E$), magnetic Prandtl number ($Pm$) and Rayleigh number ($Ra$). The primary purpose of the investigation is to compare results of the simulations with previously developed asymptotic theory that is applicable in the limit of rapid rotation. We find that all of the simulations are in the quasi-geostrophic regime in which the Coriolis and pressure gradient forces are approximately balanced at leading order, whereas all other forces, including the Lorentz force, act as perturbations. Agreement between simulation output and asymptotic scalings for the energetics, flow speeds, magnetic field amplitude and length scales is found. The transition from large-scale dynamos to small-scale dynamos is well described by the magnetic Reynolds number based on the small convective length scale, $\widetilde {Rm}$, with large-scale dynamos preferred when $\widetilde {Rm} \lesssim O(1)$. The magnitude of the large-scale magnetic field is observed to saturate and become approximately constant with increasing Rayleigh number. Energy spectra show that all length scales present in the flow field and the small-scale magnetic field are consistent with a scaling of $E^{1/3}$, even in the turbulent regime. For a fixed value of $E$, we find that the viscous dissipation length scale is approximately constant over a broad range of $Ra$; the ohmic dissipation length scale is approximately constant within the large-scale dynamo regime, but transitions to a $\widetilde {Rm}^{-1/2}$ scaling in the small-scale dynamo regime.

In the limit of large magnetic Reynolds numbers, it is shown that a smooth differential rotation can lead to fast dynamo action, provided that the electrical conductivity or magnetic permeability is anisotropic. If the shear is infinite, for example between two rotating solid bodies, the anisotropic dynamo becomes furious, meaning that the magnetic growth rate increases toward infinity with an increasing magnetic Reynolds number.

Nonlinear Hall-magnetohydrodynamic dynamos associated with coherent structures in subcritical shear flows are investigated by using unstable invariant solutions. The dynamo solution found has a relatively simple structure, but it captures the features of the typical nonlinear structures seen in simulations, such as current sheets. As is well known, the Hall effect destroys the symmetry of the magnetohydrodynamic equations and thus modifies the structure of the current sheet and mean field of the solution. Depending on the strength of the Hall effect, the generation of the magnetic field changes in a complex manner. However, a too strong Hall effect always acts to suppress the magnetic field generation. The hydrodynamic/magnetic Reynolds number dependence of the critical ion skin depth at which the dynamos start to feel the Hall effect is of interest from an astrophysical point of view. An important consequence of the matched asymptotic expansion analysis of the solution is that the higher the Reynolds number, the smaller the Hall current affects the flow. We also briefly discuss how the above results for a relatively simple shear flow can be extended to more general flows such as infinite homogeneous shear flows and boundary layer flows. The analysis of the latter flows suggests that interestingly a strong induction of the generated magnetic field might occur when there is a background shear layer.

The magnetohydrodynamic (MHD) turbulence appears in engineering laboratory flows and is a common phenomenon in natural systems, e.g. stellar and planetary interiors and atmospheres and the interstellar medium. The applications in engineering are particularly interesting due to the recent advancement of tokamak devices, reaching very high plasma temperatures, thus giving hope for the production of thermonuclear fusion power. In the case of astrophysical applications, perhaps the main feature of the MHD turbulence is its ability to generate and sustain large-scale and small-scale magnetic fields. However, a crucial effect of the MHD turbulence is also the enhancement of large-scale diffusion via interactions of small-scale pulsations, i.e. the generation of the so-called turbulent viscosity and turbulent magnetic diffusivity, which typically exceed by orders of magnitude their molecular counterparts. The enhanced resistivity plays an important role in the turbulent dynamo process. Estimates of the turbulent electromotive force (EMF), including the so-called $\alpha$-effect responsible for amplification of the magnetic energy and the turbulent magnetic diffusion are desired. Here, we apply the renormalization group technique to extract the final expression for the turbulent EMF from the fully nonlinear dynamical equations (Navier–Stokes, induction equation). The simplified renormalized set of dynamical equations, including the equations for the means and fluctuations, is derived and the effective turbulent coefficients such as the viscosity, resistivity, the $\alpha$-coefficient and the Lorentz-force coefficients are explicitly calculated. The results are also used to demonstrate the influence of magnetic fields on energy and helicity spectra of strongly turbulent flows, in particular the magnetic energy spectrum.

The evolution of a buoyancy disturbance in a stratified incompressible fluid permeated by a uniform vertical magnetic field is investigated. Two regimes are considered in the absence of background rotation – that of strong stratification, where the internal gravity wave frequency $\omega _A$ is much higher in magnitude than the magnetic (Alfvén) wave frequency $\omega _M$, and that of strong magnetic field, where $\omega _M$ is dominant. For small but finite magnetic diffusion, perturbations that initially lie in the strong-field regime are shown to cross over to the regime of strong stratification, so that small-scale motions may exist as damped internal gravity waves at large times. The induced magnetic field propagates as damped Alfvén waves for a much longer time than the velocity before undergoing the above transition. With strong rotation, the unstably stratified system that satisfies the inequality $|\omega _C| > |\omega _M| \gg |\omega _A| \gg |\omega _\eta |$, where $\omega _C$ is the inertial wave frequency and $\omega _\eta$ is the diffusion frequency, is of relevance to convection-driven dynamos. Here, a parameter space with $|\omega _M/\omega _C| \sim 0.1$ is found wherein the flow intensity of the slow magnetic-Archimedean-Coriolis (MAC) waves is of the same order of magnitude as that of the fast MAC waves. Slow wave motions at horizontal length scales much smaller than the width of the fluid layer can therefore generate substantial helicity in rapidly rotating dynamos. The excitation of slow MAC waves at scales of $\sim$10 km in the Earth's core may play a crucial role in the generation of the axial dipole field.

An informal introduction is provided to a range of topics in fluid dynamics having a topological character. These topics include flows with boundary singularities, Lagrangian chaos, frozen-in fields, magnetohydrodynamic analogies, fast- and slow-dynamo mechanisms, magnetic relaxation, minimum-energy states, knotted flux tubes, vortex reconnection and the finite-time singularity problem. The paper concludes with a number of open questions concerning the above topics.

A numerical investigation of convection-driven dynamos is carried out in the plane layer geometry. Dynamos with different magnetic Prandtl numbers $Pm$ are simulated over a broad range of the Rayleigh number $Ra$. The heat transport, as characterized by the Nusselt number $Nu$, shows an initial departure from the heat transport scaling of non-magnetic Rayleigh–Bénard convection (RBC) as the magnetic field grows in magnitude; as $Ra$ is increased further, the data suggest that $Nu$ grows approximately as $Ra^{2/7}$, but with a smaller prefactor in comparison with RBC. Viscous ($\epsilon _u$) and ohmic ($\epsilon _B$) dissipation contribute approximately equally to $Nu$ at the highest $Ra$ investigated; both ohmic and viscous dissipation approach a Reynolds-number-dependent scaling of the form $Re^a$, where $a \approx 2.8$. The ratio of magnetic to kinetic energy approaches a $Pm$-dependent constant as $Ra$ is increased, with the constant value increasing with $Pm$. The ohmic dissipation length scale depends on $Ra$ in such a way that it is always smaller, and decreases more rapidly with increasing $Ra$, than the viscous dissipation length scale for all investigated values of $Pm$.

The generation of a magnetic field in an electrically conducting fluid generally involves the complicated nonlinear interaction of flow turbulence, rotation and field. This dynamo process is of great importance in geophysics, planetary science and astrophysics, since magnetic fields are known to play a key role in the dynamics of these systems. This paper gives an introduction to dynamo theory for the fluid dynamicist. It proceeds by laying the groundwork, introducing the equations and techniques that are at the heart of dynamo theory, before presenting some simple dynamo solutions. The problems currently exercising dynamo theorists are then introduced, along with the attempts to make progress. The paper concludes with the argument that progress in dynamo theory will be made in the future by utilising and advancing some of the current breakthroughs in neutral fluid turbulence such as those in transition, self-sustaining processes, turbulence/mean-flow interaction, statistical and data-driven methods and maintenance and loss of balance.

Nonlinear three-dimensional dynamo equilibrium solutions of viscous-resistive magneto-hydrodynamic equations are continued to formally infinite magnetic and hydrodynamic Reynolds numbers. The external driving mechanism of the dynamo is a uniform shear, which constitutes the base laminar flow and cannot support any kinematic dynamo. Nevertheless, an efficient subcritical nonlinear instability mechanism is found to be able to generate large-scale coherent structures known as streaks, for both velocity and magnetic fields. A finite amount of magnetic field generation is identified at the self-consistent asymptotic limit of the nonlinear solutions, thereby confirming the existence of an effective nonlinear dynamo action at astronomically large Reynolds numbers.

We study dynamo action in rotating, plane layer Boussinesq convection in the absence of inertia. This allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. We have identified three families of solutions, defined in terms of what is the dominant contribution to the velocity. In weak field dynamos the dominant contribution is the thermal component, in super strong field dynamos the dominant contribution is magnetic and in strong field dynamos the two components are comparable. For each of these solutions we investigate the force balance in the momentum equation to determine the relative importance of the viscous, buoyancy, Coriolis and magnetic forces. We do this by extracting the solenoidal part of the individual terms in the momentum equation, thereby removing their pressure contributions. This is numerically preferable to the more common practice of taking the curl of the momentum equation, which introduces an extra derivative. We find that, irrespective of the type of dynamo solution, the dynamics is controlled by the horizontal forces (in projection). Furthermore, in the progression from weak to strong to super strong dynamos, we find that the viscous forces in the thermal equation become negligible, thereby leading to a balance between buoyancy and Coriolis forces. On the other hand, no corresponding trend is observed in the magnetic part of the momentum equation: the viscous stresses always remain significant. This can be attributed to the different degrees of smoothness of the Coriolis and Lorentz forces, the latter having contributions from strong, filamentary structures. We discuss how our findings relate to dynamo solutions in which viscosity plays no role whatsoever – so-called Taylor states.

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (J. Fluid Mech., vol. 868, 2019, pp. 176–211). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous resistive MHD equations in plane Couette flow have been utilised. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll–streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in the companion paper. When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of ‘self-sustained shear-driven dynamos’ at the zero external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in the companion paper. Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions.

Rational large Reynolds number matched asymptotic expansions of three-dimensional nonlinear magneto-hydrodynamic (MHD) states are the concern of this contribution. The nonlinear MHD states, assumed to be predominantly driven by a unidirectional shear, can be sustained without any linear instability of the base flow and hence are responsible for subcritical transition to turbulence. Two classes of nonlinear MHD states are found. The first class of nonlinear states emerged out of a nice combination of the purely hydrodynamic vortex/wave interaction theory by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and the resonant absorption theories on Alfvén waves, developed in the solar physics community (e.g. Sakurai et al. Solar Phys., vol. 133, 1991, pp. 227–245; Goossens et al. Solar Phys., vol. 157, 1995, pp. 75–102). Similar to the hydrodynamic theory, the mechanism of the MHD states can be explained by the successive interaction of the roll, streak and wave fields, which are now defined both for the hydrodynamic and magnetic fields. The derivation of this ‘vortex/Alfvén wave interaction’ state is rather straightforward as the scalings for both of the hydrodynamic and magnetic fields are identical. It turns out that the leading-order magnetic field of the asymptotic states appears only when a small external magnetic field is present. However, it does not mean that purely shear-driven dynamos are not possible. In fact, the second class of ‘self-sustained shear-driven dynamo theory’ shows a magnetic generation that is slightly smaller in size in the absence of any external field. Despite its small size, the magnetic field causes the novel feedback mechanism in the velocity field through resonant absorption, wherein the magnetic wave becomes more strongly amplified than the hydrodynamic counterpart.

We present analytical examples of fluid dynamos that saturate through the action of the Coriolis and inertial terms of the Navier–Stokes equation. The flow is driven by a body force and is subject to global rotation and uniform sweeping velocity. The model can be studied down to arbitrarily low viscosity and naturally leads to the strong-field scaling regime for the magnetic energy produced above threshold: the magnetic energy is proportional to the global rotation rate and independent of the viscosity $\unicode[STIX]{x1D708}$. Depending on the relative orientations of global rotation and large-scale sweeping, the dynamo bifurcation is either supercritical or subcritical. In the supercritical case, the magnetic energy follows the scaling law for supercritical strong-field dynamos predicted on dimensional grounds by Pétrélis & Fauve (Eur. Phys. J. B, vol. 22, 2001, pp. 271–276). In the subcritical case, the system jumps to a finite-amplitude dynamo branch. The magnetic energy obeys a magneto-geostrophic scaling law (Roberts & Soward, Annu. Rev. Fluid Mech., vol. 4, 1972, pp. 117–154), with a turbulent Elsasser number of the order of unity, where the magnetic diffusivity of the standard Elsasser number appears to be replaced by an eddy diffusivity. In the absence of global rotation, the dynamo bifurcation is subcritical and the saturated magnetic energy obeys the equipartition scaling regime. We consider both the vicinity of the dynamo threshold and the limit of large distance from threshold to put these various scaling behaviours on firm analytical ground.

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported byWu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.

We present a variational optimization method that can identify the most efficient kinematic dynamo in a sphere, where efficiency is based on the value of a magnetic Reynolds number that uses enstrophy to characterize the inductive effects of the fluid flow. In this large-scale optimization, we restrict the flow to be steady and incompressible, and the boundary of the sphere to be no-slip and electrically insulating. We impose these boundary conditions using a Galerkin method in terms of specifically designed vector field bases. We solve iteratively for the flow field and the accompanying magnetic eigenfunction in order to find the minimal critical magnetic Reynolds number $Rm_{c,min}$ for the onset of a dynamo. Although nonlinear, this iteration procedure converges to a single solution and there is no evidence that this is not a global optimum. We find that $Rm_{c,min}=64.45$ is at least three times lower than that of any published example of a spherical kinematic dynamo generated by steady flows, and our optimal dynamo clearly operates above the theoretical lower bounds for dynamo action. The corresponding optimal flow has a spatially localized helical structure in the centre of the sphere, and the dominant components are invariant under rotation by $\unicode[STIX]{x03C0}$.