Fifty years after Roberts’ MHD: Dynamos andplanetary flows today

Book of Abstracts

1

Magnetostrophy
Cheng-chin Wu & Paul H. Roberts
Maths Department, UCLA, Los Angeles
Abstract
A dynamo driven by motions unaffected by viscous forces is termed ‘magnetostrophic’, but cannot be found through today’s numerical simulations, which require substantial viscosity to stabilize solutions of the full MHD dynamo equations.
By using an alternative numerical technique, proposed by Taylor (1963, Proc. R.
Soc. Lond., vol. 274, p. 274), we recently obtained the first magnetostrophic dynamo
solutions ever derived (2015, GAFD, vol. 109, p. 84). These were axisymmetric andof mean-field type.
Taylor’s magnetostrophic theory ignores inertial forces. Consequently, there
are no torsional waves. We argued (2014, GAFD, vol. 108, p. 696) that torsional
waves would be restored if inertial forces were added to Taylor’s theory. In thispresentation, we compare mean-field solutions derived using the original Taylor
theory (‘OTT’) with those obtained from a modified Taylor theory (‘MTT’) in
which inertial forces are not ignored.

2

New Analysis of Magnetic Tornadoes
Arter Wayne
UKAEA, Culham Science Centre, Abingdon
Abstract
The recent work[1] showed how the equations of ideal, compressible magnetohydrodynamics (MHD) may be elegantly formulated in terms of Lie derivatives,
building on the work of Helmholtz, Walen and Arnold. The“linear” fields approach reduces ideal MHD to a low order set of non-linear
ordinary differential equations capable of further simplification, so has the potential
to enrich understanding of this difficult subject, which has application both tolaboratory and geophysical/astrophysical plasmas.
The just published work [2] extends the linear fields’ solution of compressible
nonlinear MHD to the case where the magnetic field depends on superlinear powersof position vector, usually but not always, expressed in Cartesian components. Implications of the resulting Lie-Taylor series expansion for physical applicability of
the Dolzhansky-Kirchhoff (D-K) “linear field” equations are found to be positive.
It is demonstrated how resistivity may be included in the D-K model. Arguments
are put forward that the D-K equations may be regarded as illustrating properties
of nonlinear MHD in the same sense that the Lorenz equations inform about theonset of convective turbulence. It is thereby suggested that the Lie-Taylor series approach may lead to valuable insights into MHD turbulence, especially fast timescaletransients and the role of plasmoids.
1. Arter, W. 2013 “Potential vorticity formulation of compressible magnetohydrodynamics. Phys. Rev. Lett. 110, 015004.” (doi:10.1103/PhysRevLett.110.015004)
2. Arter, W. 2017 “Beyond linear fields: the Lie–Taylor expansion”,Proc. R. Soc. A473,
20160525; http://dx.doi.org/10.1098/rspa.2016.0525

3

On the Nature of Diffusion-Free Heat Transfer Scalings in Dynamo Models
Jonathan Aurnou
UCLA, Los Angeles
Abstract
Many dynamo studies extrapolate numerical model results to planetary conditions by empirically constructing scaling laws. The seminal work of Christensen
and Aubert (2006) proposed a set of scaling laws that have been used throughout
the geoscience community. These scalings make use of specially-constructed parameters that are independent of fluid diffusivities, anticipating that large-scale turbulent processes will dominate the physics in planetary dynamo settings. With these‘diffusion-free’ parameterizations, the results of current numerical dynamo models
are argued to extrapolate directly to fully-turbulent planetary core systems; the effects of realistic fluid properties merit no further investigation. Here, we will discuss
the validity of diffusion-free heat transfer scaling arguments and their applicability
to planetary conditions. We do so by constructing synthetic heat transfer datasets
and examining their scaling properties alongside those proposed by Christensen and
Aubert (2006). Our results show that the diffusion-free parameters compress andstretch the heat transfer data, eliminating information and creating an artificialalignment of the data. Most significantly, diffusion-free heat transfer scalings arefound to be unrelated to bulk turbulence and are instead controlled by the onset
of non-magnetic rotating convection, itself determined by the viscous diffusivity of
the working fluid. Ultimately, our results, in conjunction with those of Stelzer andJackson (2013) and King and Buffett (2013), show that diffusion-free heat transfer
scalings are not validated by current-day numerical dynamo datasets and cannot
yet be extrapolated to planetary conditions.

4The Evolution of Magnetic-Coriolis Waves in Planetary Cores
Oliver Bardsley
University of Cambridge
Abstract
Motivated by dynamics in planetary cores, we consider the dispersion of waves
from localised sources in a rapidly-rotating fluid threaded by a mean magnetic field.
We focus on the evolution of waves packets launched as inertial-Alfvén waves,which resemble low-frequency inertial waves, in the sense that energy disperses axially on the fast timescale of the background rotation rate, though they also exhibit
slower propagation along magnetic field lines at the Alfvén velocity. When themagnetic field is uniform, inertial-Alfvén waves automatically focus energy radiation onto the rotation axis, ensuring that they dominate the dispersion pattern.
However, the situation changes when the mean magnetic field is non-uniform.
The wave packets are forced to evolve into a more general form of magnetic-Coriolis
wave, and hence refract off-axis. We show that the ultimate fate of any wave packet
is strongly dependent on the choice of launch location, i.e. the strength of themagnetic field at the source.

5

Juno at Jupiter: Initial Magnetic Field Results
Jeremy Bloxham
Harvard University, Cambridge, MA
Abstract
The Juno spacecraft entered orbit around Jupiter in July 2016. Every 53 days
Juno passes close to Jupiter, spending about 2 hours travelling from above thenorth pole, at a radial distance from the center of Jupiter of approximately 2 Rj, toperijove near the equator at a radial distance of approximately 1.06 Rj, and then to
above the south pole. Thus, with a nominal radius of the Jovian dynamo region of
0.85 Rj, Juno measures Jupiter’s magnetic field proportionately much closer to its
dynamo than is possible for any other planet.
By the end of October 2017, Juno will have completed eight passes separated byapproximately 45 degrees of longitude, Initial data show that Jupiter has a spatiallycomplex, intense magnetic field, with field strengths of almost 800,000 nT recorded
during its first pass. Using an alternative method of magnetic field analysis basedon an elastic net to model the field in physical space, the first pass revealed astrong pair of spots near the equator, suggestive of flux expulsion, and patches ofreverse flux at high latitudes. Although subsequent orbits have not measured quite
such strong fields as the first orbit, they nonetheless suggest considerable spatialcomplexity.
Jupiter and Saturn, the two gas giants in our solar system, have remarkably different magnetic fields. Understanding this difference is clearly a first order problem
in planetary magnetic fields.

6On effects of topography in rotating flows
Fabian Burmann
ETH Zürich
Abstract
Both, seismological studies and geodynamic arguments suggest that there is significant topography at the core mantle boundary (CMB). This leads to the question
whether the topography of the CMB could influence the flow in the Earth’s outercore. As a preliminary experiment, we investigate the effects of bottom topography
in the so-called Spin-Up, where motion of a contained fluid is created by a sudden increase of rotation rate. Experiments are performed in a cylindrical container
mounted on a rotating table and quantitative results are obtained with particle image velocimetry. Several horizontal length scales of topography (λ) are investigated,
ranging from cases where λ is much smaller then the lateral extend of the experiment (R) to cases where λ is a fraction of R. We find that there is an optimal λthat creates maximum dissipation of kinetic energy. Depending on the length scaleof the topography, kinetic energy is either dissipated in the boundary layer or in
the bulk of the fluid. Two different phases of fluid motion are present: a starting
flow in the from of solid rotation (phase I), which is later replaced by meso scalevortices on the length scale of bottom topography (phase II).

7

Balanced models for rapidly rotating convection-driven
magnetohydrodynamics
Michael Calkins
University of Colorado, Boulder
Abstract
Rotation is known to play a key role in the generation of planetary and stellar
magnetic fields. Rapidly rotating dynamos are characterized by a leading ordergeostrophic force balance between the Coriolis and pressure gradient forces; inertia, viscosity and the Lorentz force can all perturb this balance and yield quasigeostrophic (QG) magnetohydrodynamics. QG dynamos are routinely studied via
direct numerical simulation of the magnetohydrodynamic equations, but remaincomputationally intensive due to the range of spatiotemporal scales. However, simplified, QG dynamo models can be developed via asymptotic reduction of the governing equations. These models offer significantly reduced computational complexity
while simultaneously allowing for a simplified physical picture of the dynamics. A
summary of the theoretical properties and results from ongoing numerical simulations of these reduced QG dynamo and magnetoconvection models will be discussed.

8Geomagnetic polar minima do not arise from steady meridional circulation
Hao Cao
Harvard University, Cambridge MA
Abstract
Observations of the Earth’s magnetic field have revealed locally pronounced field
minima near each pole at the core-mantle boundary (CMB). The leading explanation for the existence of the polar magnetic minima has long been attributed
to the supposed large-scale overturning circulation of molten metal in the outer
core: fluid upwells within the inner core tangent cylinder toward the poles thendiverges toward lower latitudes when it reaches the CMB, where Coriolis effects
sweep the fluid into anticyclonic vortical flows. The diverging near-surface meridional circulation is believed to advectively draw magnetic flux away from the poles,
resulting in the low intensity or even reversed polar magnetic fields. However, theinter- connections between magnetic minima and meridional circulations have not
to date been ascertained quantitatively. Here we quantify the magnetic effects ofsteady, axisymmetric meridional circulation driven by Ekman and gyroscopic pumping. We numerically solve the axisymmetric magnetohydrodynamic equations for
Earth’s outer core under the magnetostrophic approximation, varying the amplitude of the thermal forcing, the imposed poloidal magnetic field amplitudes andgeometries, and the kinematic viscosity of the fluid. Extrapolated to core conditions, our results show that the change in polar magnetic field resulting from steady,large-scale meridional circulations in Earth’s outer core is less than 4% of the background field, significantly smaller than the 100% polar magnetic minima observedat the CMB. This suggests that the geomagnetic polar minima cannot be produced
solely by axisymmetric, steady meridional circulations, and must depend upon additional tangent cylinder dynamics, likely including non-axisymmetric, time-varying
processes.

9Saturn’s Internal Magnetic Field Revealed by Cassini Grand Finale
Hao Cao
Harvard University, Cambridge MA
Abstract
Saturn’s internal magnetic field has been puzzling since the first in-situ measurements during the Pioneer 11 Saturn flyby. Cassini magnetometer measurements
prior to the Grand Finale phase established 1) the highly axisymmetric nature of
Saturn’s internal magnetic field with a dipole tilt smaller than 0.06 degrees, 2) atleast an order of magnitude slower secular variation rate compared to that of thecurrent geomagnetic field, and 3) expulsion of magnetic fluxes from the equatorial
region towards high latitude.
The Cassini spacecraft entered the Grand Finale phase in April 2017, during
which time the spacecraft dived through the gap between Saturn’s atmosphere andthe inner edge of the D-ring 22 times before descending into the deep atmosphereof Saturn. The unprecedented proximity to Saturn (reaching ˜ 2500 km above thecloud deck) and the highly inclined nature of the Grand Finale orbits provided an
ideal opportunity to decode Saturn’s internal magnetic field. The fluxgate magnetometer onboard Cassini made precise vector measurements during the Grand
Finale phase. Magnetic signals from the interior of the planet and various magnetospheric currents were observed during the Grand Finale phase. Here we report themagnetometer measurements during the Cassini Grand Finale phase, new featuresof Saturn’s internal magnetic field revealed by these measurements (e.g., the highdegree magnetic moments, the level of axisymmetry beyond dipole), and implications for the deep interior of Saturn.

10

Strong Field Spherical Dynamos
Emmanuel Dormy
CNRS, Paris
Abstract
Some 40 years ago Paul Roberts introduced the notion of weak and strong field
branches for the Geodynamo. I will present numerical models recently obtained
which illustrate this conjectured bifurcation diagram. Direct numerical simulations
in a spherical shell can indeed exhibit the anticipated transition from a weak-field
dynamo branch, in which viscous effects set the dominant length scale, and thestrong-field branch, in which viscous and inertial effects are largely negligible.
In addition to the weak-dipolar branch and to the fluctuating-multipolar branch,widely discussed in the literature, a third regime can thus be identified.It corresponds to a strong-dipolar branch which appears to approach, in a numericallyaordable regime, the magnetostrophic limit relevant to the dynamics of the Earth’s
core. I will discuss the transitions between these branches and point to the relevanceto this strong-field dipolar branch to Geodynamo modelling.
I will finally argue that a distinguished limit needs to be sought to produce
numerical models relevant to the geodynamo, and that the usual approach of minimising the magnetic Prandtl number (ratio of the fluid kinematic viscosity to its
magnetic diffusivity) at a given Ekman number is misleading.

11Experiments and simulations of the magnetized spherical Couette problem
Ferran Garcia
Helmholtz-Zentrum Dresden-Rossendorf
Abstract
The magnetized spherical Couette system models experiments that are being
carried out at Helmholtz-Zentrum Dresden-Rossendorf (HZDR). A liquid metal isconfined within two differentially rotating spheres and immersed in a magnetic fieldparallel to the axis of rotation. Preliminary simulations of periodic and quasiperiodic
flows, arising at the first bifurcations, will be presented. The aim is to study in detailthe instabilities observed in the experiments and previous numerical studies. This
study will reveal how the flow patterns depend on control parameters, reproducing
thus different physical situations of the HZDR experiments.

12Simulations of dynamo action from a precession driven flow in a cylindricalcontainer
Andre Giesecke
Helmholtz-Zentrum Dresden-Rossendorf
Abstract
Since many years precession is regarded as an alternative flow driving mechanism that may account, e.g., for remarkable features of the ancient lunar magnetic
field or as a complementary power source for the geodynamo. Precessional forcing
is also of great interest from the experimental point of view because it represents
a natural forcing mechanism that allows an efficient driving of conducting fluid
flows on the laboratory scale without making use of propellers or pumps. Within
the project DRESDYN (DREsden Sodium facility for DYNamo and thermohydraulic studies) a dynamo experiment is under development at Helmholtz-Zentrum
Dresden-Rossendorf in which a precession driven flow of liquid sodium with a magnetic Reynolds number of up to Rm=700 will be used to drive dynamo action.
Our present study addresses preparative numerical simulations and flow measurements at a small model experiment running with water. In dependence of precession
ratio and Reynolds number the resulting hydrodynamic flow patterns and amplitudes provide the essential ingredients for kinematic dynamo models that are used
to estimate whether the particular flow is able to drive a dynamo. In the strongly
non-linear regime the flow essentially consists of standing inertial waves. Most remarkable feature is the resonant-like occurrence of a stationary axisymmetric mode
which emerges around a precession ratio Ωp /Ωc = 0.1. Kinematic dynamo models
applying the time-averaged flow from the hydrodynamic simulations exhibit dynamo
action at a critical magnetic Reynolds number of Rmc =430 which is well within therange that will be achieved in the planned sodium experiment.

13

Behaviors and transitions along the path to magnetostrophic convection
Alexander Grannan
UCLA, Los Angeles
Abstract
Turbulent convection constrained by Coriolis and Lorentz forces is important for
generating planetary magnetic fields. Theoretical predictions suggest that the convective regime for planetary dynamos is a unique magnetostrophic state where Coriolis and Lorentz forces are the leading order balance. However, many experimental
studies have focused only on rotating and non-rotating turbulent convection andrelatively few laboratory experiments are capable of investigating turbulent magnetoconvection and the turbulent magnetostrophic regime. In this work, we perform
one laboratory experiment in a cylinder at a fixed heat flux using the liquid metal
gallium in order to investigate, sequentially: Rayleigh-Benard convection without
any imposed constraints, magnetoconvection with a Lorentz constraint imposed byvertical magnetic field, rotating convection with a Coriolis constraint imposed by rotation, and finally the magnetostrophic convective regime. Using an array of internaland external temperature probes, we show that each regime along the path to magnetostrophic convection is unique. The behaviors and transitions in the dominantmodes of convection as well as their fundamental frequencies and wavenumbers areinvestigated.

14

Dipole Oscillations and Stability of the Geodynamo
David Gubbins
University of Leeds
Abstract
Paleointensity studies have shown Earth’s dipole moment continually fluctuates
by about 40% of the mean during intervals of “stable” polarity: the dynamo neverreaches stable equilibrium. Strong oscillations can arise from extreme sensitivity of
the critical Ra at very low Ekman number. I have explored this with a series ofnumerical dynamo experiments in which only Ra, the Roberts number (q), and initial conditions, vary. Three regimes of dynamo action result: weak-field (Ra>RaD)
and strong-field (Ra>RaS), depending on whether the initial field is weak or strong,and subcritical (Ra<RaD), when a weak initial field gives no dynamo but a strong
initial condition does. At Ra> RaS the weak field energy increases exponentiallyto a critical point where it accelerates even faster to join the strong field branch.
Exponential growth corresponds to a kinematic dynamo with little change to thefluid flow; the faster growth is accompanied by an increase in length scale measured by energy/dissipation. Increasing Ra further produces only a small increasein magnetic energy. The strong-field and subcritical dynamos have strong oscillations that decrease in size as Ra increases initially, then increase again at larger Ra
as expected; they are associated with changes in length scale, as in the growth fromweak- to strong-field dynamo action.

15

Large-scale flows and magnetic fields produced by rotating convection in aquasi-geostrophic model in spherical geometryCéline Guervilly
Newcastle University
Abstract
We present numerical simulations of rotating Boussinesq convection at low Rossby
numbers using a quasi-geostrophic (QG) model in spherical geometry. For low
Prandtl numbers (Pr∼O(0.1)) and low Ekman numbers (Ek<1e-6), the convection
occurs on a “strong” branch, characterised by Reynolds numbers greater than 1000,
that can extend well below the linear onset of convection (Ra∼0.6Racrit for Ek=1e-8
and Pr=0.01). These QG results hold in the 3D system for similar parameters asshown by the numerical study of Kaplan et al (2017 PRL 119). In this talk, we
discuss the characteristics of the QG flows on the strong branch, such as the increaseof the convective scale with the thermal forcing and the formation of intense zonal
flows consisting of persistent multiple jets. These zonal flows strongly affect the convective heat transport, leading to the formation of a mean temperature staircase.
We also show preliminary results of dynamos generated by the quasi-geostrophic
flows at low magnetic Prandtl numbers.

16

Waves in spherical dynamo simulations and their implications for planetary
dynamos
Kumiko Hori
Kobe University/University of Leeds
Abstract
Magnetohydrodynamic waves excited in deep interiors of rapidly rotating planets
can produce secular time-variations of the magnetic fields. In strong-field dynamos,
in which the inertial and viscous forces are small compared to the magnetostrophic
forces (i.e. Coriolis, pressure, Lorentz, and buoyancy forces), unique wave motions
can occur in both axisymmetric and nonaxisymmetric modes. Examples include
torsional Alfven waves and magnetic Rossby waves. To explore the relevance of thesewaves and their dynamics, we use DNS of spherical dynamos driven by Boussinesq
or anelastic convection. The linear theory gives good results for observed wave
speeds, although nonlinear Lorentz terms were found to influence the waveforms.
Waves in data, combined with theory, could allow us to infer physical properties
within the Earth’s core or Jupiter’s metallic hydrogen region.

17

On the unique flow morphologies in rotating convection of liquid metals
Susanne Horn
UCLA, Los Angeles
Abstract
Rotating convection is one of the underlying fundamental processes occurring in
the Earth’s outer liquid metal core.
The low Prandtl number (Pr) that characterises liquid metals also implies that there
are inherently different instability mechanisms present compared to moderate- or
high-Pr fluids.
As a consequence, a distinct type of convection develops that is formed of large-scale
oscillatory inertial modes.
To study the associated unique flow morphologies, we have conducted direct numerical simulations of rotating convection in a cylinder filled with a fluid with Pr=0.025
and rotating with Ekman numbers as low as 5x10-6 . The Rayleigh number (Ra)range was chosen in such a way that we could investigate the purely oscillatory
regime close to onset, the mixed regime of oscillatory and wall modes, and alsobroad band turbulence. The obtained flow fields were analysed using the Dynamic
Mode Decomposition (DMD). This technique allows to extract and identify the
structures that govern the dynamics of the system as well as their corresponding
frequencies.
We find that different types of oscillatory modes coexist, they can precess prograde
and retrograde and also occur in the form of quasi-axisymmetric torsional modes.
For higher Ra, they additionally couple and non-linearly interact with wall modes.
Leveraging on these results we are in the progress of designing a dynamic observer,
with the objective to connect our results to laboratory experiments. This will enable
us to draw a complete picture of the multi-modal flow in low-Pr convection.

18

Torsional waves in a rotating spherical shell: transmission and reflection in
the Earth’s outer core
Dominique Jault
University Grenoble Alpes / CNRS
Abstract
I will discuss 1D models and solutions for torsional Alfvén waves in a sphericalshell. Transmission at the tangent cylinder and reflection at the equator have notbeen fully elucidated yet. There are interesting analogies with waves in a canal of
abruptly varying depth and with motions of a string with frictional support at one
end. The solution in the equatorial region in the presence of a magnetic and viscousboundary layer deserves special care.

19

Rotating magnetic waves in stably stratified layers
Chris Jones
University of Leeds
Abstract
Waves in a thin layer on a rotating sphere are studied. The effect of a toroidal
magnetic field is considered, using the shallow water ideal MHD equations. The work
is motivated by suggestions that there is a stably stratified layer below the Earth’s
core mantle boundary, and the existence of stable layers in stellar tachoclines.
Magnetic instabilities can occur when the field is strong. Waves can be divided
into magneto-inertial gravity waves, magneto-Kelvin waves, fast and slow magnetic
Rossby waves. The properties of these waves will be discussed.

20

Can libration maintain Enceladus’s ocean?
Rich Kerswell
University of Cambridge
Abstract
Current consensus is that Enceladus (the sixth largest moon of Saturn) has a
global subsurface ocean of predominantly water. Given the cold temperatures inthis part of the solar system (the surface temperature of Enceladus could be as
low as -200C), it is an ongoing challenge to explain why this ocean doesn’t freeze.
I’ll briefly discuss how to assess whether (pure) libration of the ice shell across theocean and terrestial core could be important.

21

Decadal polar motion and electromagnetic core-mantle coupling:
implications from geodynamo simulations
Weijia Kuang
NASA Goddard Space Flight Center
Abstract
It has long been observed that Earth’s rotation axis orientation, called polar
motion if defined in the terrestrial reference frame, varies on a wide range of timescales. It has also long been speculated that exchange of equatorial angular momentum between the Earth’s fluid outer core and the solid mantle could be a major
contributor to the observed decadal polar motion. However, none of the major
coupling mechanisms, including the EM coupling across the core-mantle boundary(CMB), is found to be responsible for the angular momentum exchange.
The EM torques evaluated from geomagnetic observations are partial, and could
be extremely underestimates of the true EM coupling torque because, except a fraction due to advection of the poloidal field by the core flow, other contributions fromthe outer core to the toroidal field in the mantle has been largely ignored. Using geodynamo simulations we have found that the missing parts can be ordersof magnitude stronger than those retained in the previous studies, leading to amuch stronger EM coupling torque which, as suggested by our simulation results,
could potentially yield an order ∼ 5 mas polar motion on time scales of 20 to 60
years. Should the EM coupling be responsible for the observed decadal polar motion, geodetic measurements could then be used to infer toroidal magnetic field andthe convective flow beneath the CMB.

22

Cascade to large-scale vortices in spherical rotating convection
Yufeng Lin
University of Cambridge
Abstract
Large-scale coherent structures have been observed recently in numerical simulations of rapidly rotating convection in Cartesian boxes. However, similar structures
have not yet been reported in the spherical geometry, which is more relevant togeophysical applications. In this talk, we will report numerical evidence of suchlarge-scale coherent structures in a rapidly rotating whole sphere driven by internal
heating. When the Rayleigh number is well above the critical value and a stress-freeboundary condition is used for the velocity field, small-scale convective flows merge
into large-scale structures, leading to a long-lived, axial-invariant cyclone aroundthe rotation axis. We will discuss the formation of such large-scale cyclones and theheat transport associated with these large-scale structures.

23

Taylor state dynamos found by optimal control: axisymmetric examples
Phil Livermore
University of Leeds
Abstract
In 1963, Taylor promulgated an inertia-free and viscous-free model as the asymptotic limit of Earth’s dynamo. In this theoretical limit, the velocity and the magnetic
field organize themselves in a special manner such that the Lorentz torque acting
on a geostrophic cylinder has to tend to zero as the inertial force and the viscous
force tend to zero. We propose a new approach for solving this system based upon
the concept of optimal control theory. In our method, the Lorentz torque is treatedas the target function to be minimized at each time step and we seek the optimal
geostrophic flow such that the Lorentz torque remains arbitrarily small at the future time step. We demonstrate the success of this approach on illustrative models,
namely 2D axisymmetric systems driven by prescribed α- and αω-effects. In thisposter we will describe a number of dynamo solutions in the Taylor state.

24

Optimisation of Kinematic Dynamo in a sphere
Jiawen Luo
ETH Zürich
Abstract
In this study, we use a variational optimization approach to optimize laminar
kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic
Reynolds number for dynamo action. Magnetic boundary condition is chosen to beeither pseudo-vacuum or perfect-conductor boundaries. The spectra corresponding
to two magnetic boundary conditions are identical by changing u into -u (Favier andProctor 2013). A no-slip boundary for flow field gives a critical magnetic Reynolds
number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimalpairs (u, BT ) are found possessing certain rotation symmetries (anti-symmetries)
and optimal flows share some common features. u localizes in a small region where
it spirals up with very large velocity and vorticity, which is close to a Beltrami flowin the sense that u and curl u is nearly
R parallelRin that region. Moreover, calculationof different helicities indicates that A.u and B.u might be important values thatare expected to be small at the optima, with A the vector potential of B. A cheap
way to achieve this is through symmetry. Comparisons with a recently obtained
result by Chen Long using insulating magnetic boundary are made. We also discuss
the energy requirement for dynamo action following (Proctor 2015).

25

Multiscale numerical simulations of magnetoconvection at low magnetic
Prandtl and Rossby numbers
Stefano Maffei
University of Colorado, Boulder
Abstract
The dynamics of the Earth’s outer core is characterized by low values of the
Rossby, Ekman and magnetic Prandtl numbers. These values indicate the large
spectra of temporal and spatial scales that need to be accounted for in realistic
numerical simulations of the system. Current direct numerical simulation (DNS) are
not capable of reaching this extreme regime, suggesting that a new class of models
is required to account for the rich dynamics expected in the natural system. Here
we present results from a quasi-geostrophic, multiscale model based on the scale
separation implied by the low Rossby number typical of rapidly rotating systems.
In particular, we investigate a plane layer geometry where convection is driven by an
imposed temperature gradient and the hydrodynamic equations are modified by thepresence of a large scale magnetic field. Analytical investigation shows that at valuesof thermal and magnetic Prandtl numbers relevant for liquid metals, the energetic
requirements for the onset of convection is not significantly altered even in thepresence of strong magnetic fields. Results from strongly forced nonlinear numericalsimulations show the presence of an inverse cascade, typical of 2-D turbulence, whenno or weak magnetic field is applied. For higher values of the magnetic field theinverse cascade is quenched.

26

Direction of the wave propagation and shock normal variation detection into
earth magnetic ramp region in sunward earth radial distance 10.5 earth
Mangetopouse transition region
Jivraj Pipaliya
University of Sheffield
Abstract
One of the key aspects of the in-situ measurements of collision less shocks is to
identify direction of the wave propagation variation with shock normal in Quasi
perpendicular shock front earth magnetic ramp transition region. It is argued that
direction of the wave propagation and shock normal variation occurs at the earthmagnetic ramp co-related to angle variation between shock normal and upstream
magnetic field

27

A Possible Origin of the Earth’s Dipole Magnetic Field
John Shebalin
George Mason University
Abstract
The Earth’s quasi-steady dipole magnetic field may well be due to MHD turbulence within the outer core. Here, the theoretical and computational results thatlead to this hypothesis will be discussed, and new numerical results will be presented.
A comparison of IGRF spectra with spectra from new forced, dissipative Fourier
method numerical simulations will be presented. This will require a discussion of
the relationship between spherical harmonic degree and turbulence wavenumber,which are generally not equivalent. These results also include a demonstration thatenergetic, quasi-stationary, largest-scale modes exist and are due to intrinsic properties of MHD turbulence. These quasi-stationary Fourier modes are analogous to
the dipole term in a spherical harmonic expansion.

28

Baroclinically-driven flows and dynamo action in rotating spherical fluid
shellsRadostin Simitev
University of Glasgow
Abstract
The dynamics of stably stratified stellar radiative zones is of considerable interest due to the availability of increasingly detailed observations of Solar and stellarinteriors. We reports non-axisymmetric and time-dependent simulations of flows ofanelastic fluids driven by baroclinic torques in stably stratified rotating sphericalshells - a system serving as an elemental model of a stellar radiative zone. With
increasing baroclinicity a sequence of bifurcations from simpler to more complex
flows is found in which some of the available symmetries of the problem are broken
subsequently. The poloidal component of the flow grows relative to the dominanttoroidal component with increasing baroclinicity. The possibility of magnetic field
generation thus arises and this paper proceeds to provide some indications for selfsustained dynamo action in baroclinically-driven flows. We speculate that magnetic
fields in stably stratified stellar interiors are thus not necessarily of fossil origin as
it is often assumed.

29

The Equatorial Ekman Layer
Andrew Soward
Newcastle University
Abstract
The steady incompressible viscous flow in the wide gap between spheres rotating about a common axis at slightly different rates (small Ekman number E) has
a long and celebrated history. The problem has particular application to the geodynamo and other planetary dynamos, where the moving electrically conducting
fluid responsible for the dynamo is confined in such a geometry. A comprehensive
asymptotic study, in the limit E  1, was undertaken by Stewartson (J. Fluid
Mech. 1966, vol. 26, pp. 131-144). In addition to the inner and outer boundary
E 1/2 Ekman layers, he identified a set of nested shear layers on the inner spheretangent cylinder: inside E 2/7 , outside E 1/4 . They embed an E 1/3 shear layer, whichmerges with the inner sphere Ekman layer to form the E 2/5 Equatorial Ekman layerof axial length E 1/5 . We are unaware of any analytic solution of this E 2/5 layer. We
present a numerical solution, which adopts non-local integral boundary condition
to account for the far field behaviour as the E 2/5 and E 1/3 layers separate and become distinct. We compare our higher order extension of Stewartson’s asymptotic
similarity solution with our numerics.

30

Evolution of MHD waves in a rotating fluid
Binod Sreenivasan
Indian Institute of Science, Bangalore
Abstract
Turbulence in the Earth’s core is thought to be driven by blobs of light elements
released from the inner core boundary. The long-time evolution of a flow disturbance subject to background rotation and a magnetic field provides some insight
into the character of small-scale turbulence in the core. In this study, two regimes
are analysed in the inviscid limit, that of strong rotation where the inertial wave
frequency is much higher than the Alfvén wave frequency, and that of weak rotation
where the Alfvén wave frequency is dominant. In either regime, the evolution consists of a damped wave-dominated phase followed by a diffusion-dominated phase.
For strong rotation, the laws of energy decay in the damped wave phase are obtained by considering the decay of the fast and slow Magneto-Coriolis (MC) waves
individually. The diffusion-dominated phase obeys the decay laws in the well-known
quasi-static approximation. The wave–diffusion transition time scale indicates thatthe wave phase of decay is very long, so that small-scale turbulence in the core ischaracterised by perpetual damped wave motions. Interestingly, the induced magnetic field is far more efficient than the velocity in supporting slow MC waves. In theregime of weak rotation, the fast and slow MC wave solutions merge and tend to thenon-dispersive damped Alfvén wave solution. Here the decay laws in non-rotating
MHD turbulence are recovered. The present study suggests that a mean intensityof ∼10 mT or higher is plausible for the toroidal magnetic field within the Earth’s
core.

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Dynamos, instabilities, inverse problems: Paul Roberts’ legacy forexperimental MHD
Frank Stefani
Helmholtz-Zentrum Dresden-Rossendorf
Abstract
Retracing Paul Roberts’ footsteps, I survey the recent experimental activities
related to dynamo action, magnetically triggered flow instabilities, Alfvén waves,and magnetic flow tomography. I’m certain he will be most pleased by those developments that have superseded some of the pessimistic prognoses made in his seminal
1967 book: ”Since processes of self-excitation are out of the question...” (p. 172),
is just a case in point...

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An Asymptotic Model of Rapidly Rotating Convective Dynamos
Steven Tobias
University of Leeds
Abstract
Magnetic field generation in planets is modified by the presence of strong rotation. In such planets, the primary balance is controlled by the rotation rate anddissipative processes, with the small magnetic Prandtl number being key. Rotation and pressure always play an important role and magnetic field may enter into
the leading order balance, though the relative importance of geostrophic motions
to magnetostrophic motions is uncertain (and presumably scale-dependent). HereI shall present an asymptotic model of rotating convective dynamos and magnetoconvection where the leading order balance is that of geostrophy. The magnetic
energy in such dynamos is asymptotically larger than the kinetic energy. I shall describe the derivation of these model, some numerical integrations and some possible
generalisations.

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Anisotropic particle diffusion in field-guided magnetohydrodynamic
turbulence
Yue-Kin Tsang
University of Leeds
Abstract
Magnetohydrodynamic (MHD) turbulence with a background magnetic field isrelevant to many astrophysical processes such as solar wind transport. We investigate how a mean guided-field alters the Lagrangian transport of tracer particles in
MHD turbulence. We find that the mean-squared-displacement grows linearly withtime, indicating a diffusive behavior. As the strength of the guided-field increases,
the diffusion becomes anisotropic with larger diffusivity in the field-parallel direction. Interestingly, associated with such transition are changes in the behavior of
the Lagrangian velocity correlation function. The Lagrangian velocity decorrelation
time exhibits a power-law scaling with the root-mean-squared velocity. The scaling
exponent shifts from -1 to -2 as anisotropic diffusion develops in the system, signalling a switching from the hydrodynamic-like regime to a MHD turbulence regimeof interacting Alfven waves.

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Liquid metal convection in a horizontal magnetic field
Tobias Vogt
Dresden
Abstract
Magnetohydrodynamic Rayleigh-Bénard convection was studied experimentallyand numerically using a liquid metal inside a flat box with square horizontal crosssection and aspect ratio five. The magnetic field in our experiments is applied inhorizontal direction perpendicular to the temperature gradient. A sufficiently strong
magnetic field causes a quasi-two-dimensional convection pattern in a way that theconvection rolls are oriented along the magnetic field lines. We used variable fieldstrength in this study in order to investigate the transition from a disordered toquasi-two-dimensional convection. We present systematic flow measurements thatwere performed by means of ultrasound Doppler velocimetry. We observed several
different flow patterns depending on the balance between Lorentz force and buoyancy. The velocity measurements give an insight into the dynamics of the primaryconvection rolls, the secondary flow induced by Ekman pumping and reveal theexistence of small vortices that develop around the convection rolls. The measurements were accompanied by direct numerical simulations in order to illustrate thetime dependent, three dimensional velocity field. The overall agreement between
experiment and numerical results is very good.

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Advective core surface flows deduced from geomagnetic secular variation
data
Kathy Whaler
University of Edinburgh
Abstract
In 1965, Paul Roberts (with Scott) pointed out the ambiguity in deducing advecting core surface flows from the geomagnetic secular variation, and began discussions about ways to reduce the uncertainty. This seminal paper set the scene formuch subsequent research into the subject, including further work on defining theambiguity by Backus (1968) who also identified how to establish that the data areconsistent with a purely advective. A number of authors subsequently have madeassumptions that allow the ambiguity to be reduced, and provide methods to test
for consistency with the assumption. This has provided information on the dynamical regimes of the core, and informed geodynamo investigations. Flow models havealso been validated by comparing their predictions of decadal timescale changes inlength of day with the observations. Here, we present core surface flow models
covering the period from the start of the CHAMP satellite era to the present day,using data from the Swarm satellite constellation, and filling in the period between
the two missions by geomagnetic observatory data. We use annual differences ofmonthly mean data to define the secular variation, either from observatories or an
updated ’virtual observatories’ (VO) method applied to satellite data. Our models
share many of the features identified in other studies, such as the planetary gyre,and flow accelerations at the times of geomagnetic impulses or ’jerks’. The models
fit the data to within their uncertainties, but the residuals are correlated, possiblyreflecting incomplete removal of external fields.

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Asymptotic theory for equatorially antisymmetric convection in rotating
fluid spheres
Keke Zhang
University of Exeter
Abstract
This paper is concerned with the classical problem of convective instabilities in
rapidly rotating,self-gravitating, internally heated Boussinesq fluid spheres studied
by P. H. Roberts (1968).We present a new asymptotic solution describing convectiondriven torsional oscillation – whose flow velocity and pressure are fully analyticaland in closed form.The asymptotic solution is in the form of equatorially antisymmetric torsional oscillation, in quantitative agreement with the result of thenumerical simulation, and physically preferred in a special range of small Prandtl
number for rapidly rotating fluid spheres with the stress-free boundary condition.

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