Dynamos driven by Helical Waves: scaling laws for numerical dynamos and for the planets

We derive scaling relationships for planetary dynamos based on a balance between energy production and Joule dissipation, and between the curl of the buoyancy and Coriolis forces. These scaling relationships are deduced for the particular case of dynamos driven by helical waves, but are shown to have a much broader applicability. They are consistent with the evidence of the numerical dynamos, yielding predictions consistent with published empirical scaling laws and also with the observed transition from dipolar to multipolar dynamos. A direct comparison with the observational evidence for the planets is hampered by the fact that we do not know what sets the smallest scale of the motion in the planets. Nevertheless, we use our scaling relationships to show that the traditional assumption that the Elsasser number is of order unity is inconsistent with the observation that the gas-giant dynamos are dipolar dynamos, as is the more recent suggestion that the strength of the dipole is independent of rotation rate and controlled by the buoyancy flux alone. On the other hand, we show that the observational data is consistent with the hypothesis that a dipolar dynamo saturates at the lowest permissible magnetic energy compatible with a given buoyancy flux

Inertial-Alfvén waves as columnar helices in planetary cores

We consider a rapidly rotating, Boussinesq fluid stirred by buoyant anomalies. In such a system it is known that, in the absence of a magnetic field, inertial waves whose wave vectors lie normal to the rotation axis play a key role in establishing quasi-geostrophic motion. In particular, buoyant anomalies radiate low-frequency inertial wave packets which disperse along the rotation axis, leading to axially elongated columnar vortices. Here we focus on the influence of an ambient magnetic field on this process, motivated by the dynamics of planetary cores. We find that, once again, the waves responsible for establishing quasi-geostrophic structures have wave vectors normal to the rotation axis; however, these are not conventional inertial waves, but rather hybrid 'inertial-Alfvén waves'. Their frequency equals that of an Alfvén wave but their axial group velocity is half that of the equivalent inertial wave. They have maximal kinetic, magnetic and cross-helicity, carry magnetic and kinetic energy in equal amounts, and are particularly potent in establishing columnar, helical vortices through the spontaneous emission of axially elongated wave packets. Although our hybrid inertial-Alfvén waves have been overlooked in dynamo literature to date, we speculate that they in fact play a central role in planetary dynamos.Engineering and Physical Sciences Research Counci

On the spatial segregation of helicity by inertial waves in dynamo simulations and planetary cores

The distribution of kinetic helicity in a dipolar planetary dynamo is central to the success of that dynamo. Motivated by the helicity distributions observed in numerical simulations of the Earth’s dynamo, we consider the relationship between the kinetic helicity, h=\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{u}, and the buoyancy field that acts as a source of helicity, where $\boldsymbol{u}$ is velocity. We show that, in the absence of a magnetic field, helicity evolves in accordance with the equation \unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{F}+S_{h}, where the flux, $\boldsymbol{F}$, represents the transport of helicity by inertial waves, and the helicity source, $S_{h}$, involves the product of the buoyancy and the velocity fields. In the numerical simulations it is observed that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south, a feature which the authors had previously attributed to the transport of helicity by waves (Davidson &amp; Ranjan, Geophys. J. Intl, vol. 202, 2015, pp. 1646–1662). It is also observed that there is a strong spatial correlation between the distribution of $h$ and of $S_{h}$, with $S_{h}$ also predominantly negative in the north and positive in the south. This correlation tentatively suggests that it is the in situ generation of helicity by buoyancy that establishes the distribution of $h$ outside the tangent cylinder, rather than the dispersal of helicity by waves, as had been previously argued by the authors. However, although $h$ and $S_{h}$ are strongly correlated, there is no such correlation between \unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t and $S_{h}$, as might be expected if the distribution of $h$ were established by an in situ generation mechanism. We explain these various observations by showing that inertial waves interact with the buoyancy field in such a way as to induce a source $S_{h}$ which has the same sign as the helicity in the local wave flux, and that the sign of $h$ is simply determined by the direction of that flux. We conclude that the observed distributions of $h$ and $S_{h}$ outside the tangent cylinder are consistent with the transport of helicity by waves.</jats:p

On the helicity characteristics and induced emf of magnetic-Coriolis wave packets

In a rapidly rotating Boussinesq fluid, buoyant anomalies radiate low-frequency inertial wave packets which disperse along the rotation axis. The wave packets lead to axially elongated vortices, which propagate negative (positive) kinetic helicity upwards (downwards) with respect to the rotation vector. The kinetic helicity carried by the inertial wave packets is near-maximal, relative to the velocity and vorticity fields. In classical mean-field theory, kinetic helicity is often associated with the α-effect, which is thought to be an important ingredient for planetary dynamos. The modification of inertial wave packets in the presence of a transverse uniform magnetic field is investigated here, motivated by small-scale dynamics in planetary cores, where a large-scale magnetic field affects fluid motions. We study numerically the dispersion of wave packets from an isolated buoyant source and from a random layer of buoyant anomalies, while varying the Lehnert number Le – the ratio of the frequencies of Alfvén and inertial waves. We find that for Le < 0.1, the vortices are columnar and continue to segregate kinetic helicity so that it is negative (positive) above (below) the buoyant source. Importantly, the wave packets induce an α-effect, which remains strong and coherent for Earth-like values of the Lehnert number (Le < 0.1). The interaction of wave packets emitted by multiple neighbouring buoyant sources results in an α-effect that is stronger than the α-effect induced by wave packets launched from an isolated buoyant source, and we provide an analytical explanation for this. The coherence of the α-effect induced by the wave packets, for Earth-like values of the Lehnert number, lends support to the α2 dynamo model driven by helical waves

A physical conjecture for the dipolar-multipolar dynamo transition

In numerical simulations of planetary dynamos there is an abrupt transition in the dynamics of both the velocity and magnetic fields at a ‘local’ Rossby number of 0.1. For smaller Rossby numbers there are helical columnar structures aligned with the rotation axis, which efficiently maintain a dipolar field. However, when the thermal forcing is increased, these columns break down and the field becomes multi-polar. Similarly, in rotating turbulence experiments and simulations there is a sharp transition at a Rossby number of ${\sim}0.4$. Again, helical axial columnar structures are found for lower Rossby numbers, and there is strong evidence that these columns are created by inertial waves, at least on short time scales. We perform direct numerical simulations of the flow induced by a layer of buoyant anomalies subject to strong rotation, inspired by the equatorially biased heat flux in convective planetary dynamos. We assess the role of inertial waves in generating columnar structures. At high rotation rates (or weak forcing) we find columnar flow structures that segregate helicity either side of the buoyant layer, whose axial length scale increases linearly, as predicted by the theory of low-frequency inertial waves. As the rotation rate is weakened and the magnitude of the buoyant perturbations is increased, we identify a portion of the flow which is more strongly three-dimensional. We show that the flow in this region is turbulent, and has a Rossby number above a critical value $Ro^{crit}\sim 0.4$, consistent with previous findings in rotating turbulence. We suggest that the discrepancy between the transition value found here (and in rotating turbulence experiments), and that seen in the numerical dynamos ($Ro^{crit}\sim 0.1$), is a result of a different choice of the length scale used to define the local $Ro$. We show that when a proxy for the flow length scale perpendicular to the rotation axis is used in this definition, the numerical dynamo transition lies at $Ro^{crit}\sim 0.5$. Based on this we hypothesise that inertial waves, continually launched by buoyant anomalies, sustain the columnar structures in dynamo simulations, and that the transition documented in these simulations is due to the inability of inertial waves to propagate for Ro>Ro^{crit}.The Leverhulme Trust U

Are planetary dynamos driven by helical waves?

In most numerical simulations of the Earth’s core the dynamo is located outside the tangent cylinder and, in a zero-order sense, takes the form of a classical α2 dynamo. Such a dynamo usually requires a distribution of helicity, h , which is asymmetric about the equator and in the simulations it is observed that, outside the tangent cylinder, the helicity is predominantly negative in the north and positive in the south. If we are to extrapolate the results of these simulations to the planets, we must understand how this asymmetry in helicity is established and ask if the same mechanism is likely to operate in a planet. In some of the early numerical dynamos, which were too viscous by a factor of at least 109 , as measured by the Ekman number, the asymmetric helicity distribution was attributed to Ekman pumping. However, Ekman pumping plays much less of a role in more recent, and less viscous, numerical dynamos, and almost certainly plays no significant role in the core of a planet. So the question remains: what establishes the asymmetric helicity distribution in the simulations and is this mechanism likely to carry over to planetary cores? In this paper we review the evidence that planetary dynamos, and their numerical analogues, might be maintained by helical waves, especially inertial waves, excited in and around the equatorial regions. This cartoon arises from the observation that there tends to be a statistical bias in the buoyancy flux towards the equatorial regions, and so waves are preferentially excited there. Moreover, upward (downward) propagating inertial waves carry negative (positive) helicity, which leads naturally to a segregation in h .The authors thank the Leverhulme Trust for their generous support through the grant RPG-2015-195/RG77943

Formation of eyes in large-scale cyclonic vortices

We present numerical simulations of steady, laminar, axisymmetric convection of a Boussinesq fluid in a shallow, rotating, cylindrical domain. The flow is driven by an imposed vertical heat flux and shaped by the background rotation of the domain. The geometry is inspired by that of tropical cyclones and the global flow pattern consists of a shallow, swirling vortex combined with a poloidal flow in the r-z plane which is predominantly inward near the bottom boundary and outward along the upper surface. Our numerical experiments confirm that, as suggested by Oruba et al 2017, an eye forms at the centre of the vortex which is reminiscent of that seen in a tropical cyclone and is characterised by a local reversal in the direction of the poloidal flow. We establish scaling laws for the flow and map out the conditions under which an eye will, or will not, form. We show that, to leading order, the velocity scales with V=(\alpha g \beta)^{1/2}H, where g is gravity, \alpha the expansion coefficient, \beta the background temperature gradient, and H is the depth of the domain. We also show that the two most important parameters controlling the flow are Re=VH/\nu and Ro=V/\Omega H, where \Omega is the background rotation rate and \nu the viscosity. The Prandtl number and aspect ratio also play an important, if secondary, role. Finally, and most importantly, we establish the criteria required for eye formation. These consist of a lower bound on Re, upper and lower bounds on Ro, and an upper bound on Ekman number.Comment: 18 pages, 14 figures, 1 tabl

Dynamics of a trapped vortex in rotating convection

We consider axisymmetric rotating convection in a cylindrical domain, focussing on the eye that can form at the centre of a cyclonic vortex. Upon increasing the thermal forcing we observe that the the system undergoes a Hopf bifurcation from a state with a steady eye to one in which the eye oscillates. For an aspect ratio, Ekman number, and Prandtl number of 0.1 we find that the critical Reynolds number at which this transition occurs is 398. We examine the nature of the oscillations and propose that the behaviour results from an inertial wave trapped in the eye, with the frequencies falling within the expected range for inertial waves, and the oscillations displaying clear similarities to a standing inertial wave in a cylinder. We also examine the effect of Ekman number on the oscillation, finding that there is upper limit beyond which oscillations do not occur.Cambridge Philosophical Societ

Scale locality of the energy cascade using real space quantities

The classical energy cascade in turbulence as described by Richardson and Kolmogorov is predominantly a conjecture relying on the locality of interactions between scales of turbulence. This picture is generally accepted and assumes that energy and enstrophy transfers occur between neighbouring scales of turbulence and that vortex stretching plays a major role in the dynamics of this energy cascade. Direct numerical simulation data for Re_λ ranging from 37 to 1131 is used to gather evidence for the cascade by investigating the energy and enstrophy fluxes between scales and the interplay between vorticity at one scale and strain at an adjacent scale. This is achieved by using a bandpass filter to educe the turbulent structures at various length scales allowing one to determine the fluxes between these scales and to interrogate the role of non-local (in physical-space) vortex stretching. It is shown that the structures of a length scale L mostly transfer their energy to structures of size 0.3L and that most of the enstrophy flux goes from structures of scale L to 0.3L. Furthermore, vortical structures of a length scale L_ω are stretched mostly by straining structures of size 3 to 5L_ω and the stretching by eddies of sizes larger than 10L_ω is negligible. The stretching is dominated by the most extensive principal strain rate of the straining structures. These observations are found to be independent of Re_λ for the range investigated in this study. These results provide strong evidence for the classical view of an energy cascade transferring energy from large to small scales through a hierarchy of steps, each step consisting of the stretching of vortices by somewhat larger structures.Qualcomm European Research Studentship Fund in Technolog

The dispersion of magnetic-Coriolis waves in planetary cores

We consider the dispersion of waves in a rapidly rotating, Boussinesq fluid which is threaded by a magnetic field and stirred by slowly gravitating buoyant blobs. Motivated by dynamics in the core of the Earth, we focus on the evolution of inertial-Alfven wave packets radiated from the buoyant anomalies. These waves resemble conventional low-frequency inertial waves, in the sense that energy disperses on the fast timescale of the background rotation rate, though they also exhibit slower Alfven-like propagation along magnetic field lines. When the magnetic field is uniform, inertial-Alfven waves automatically focus energy radiation onto the rotation axis, a property they share with conventional low-frequency inertial waves in the hydrodynamic case, and which ensures that they dominate the dispersion pattern. However, the situation changes significantly when the magnetic field, B, is non-uniform. In particular, we show any non-uniformity of B causes inertial-Alfven waves to evolve into a more general form of magnetic-Coriolis (MC) wave, and that these waves refract, dispersing somewhat offaxis. Moreover, if inertial-Alfven waves are launched near the equator they can be confined to low latitudes by a critical layer at which the axial group velocity drops to zero. Given that the magnetic field in a planetary core is inevitably non-uniform, we conclude that quasi-geostrophy is most likely achieved through a combination of weakly modified inertial waves and a form of slightly off-axis MC wave in which the inertial and Alfven frequencies are comparable