1. Convection-driven dynamos in rotating spherical shells -
Scottish Universities
Environmental Research Centre
Convection-driven dynamos in
rotating spherical shells -
basic phenomenology
R.D. Simitev
School of Mathematics and Statistics
F.H. Busse
Institute of Physics
Good morning!
Thank you very much for the kind invitation to Zurich. I have enjoyed this experience a lot.
I would like tell you today of things I have learned from my numerical simulations of convection-driven dynamos in rotating spherical shells.
In particular, I would like to discuss two rather unusual properties of planetary dynamos:
(a) dynamo oscillations
(b) bistability and hysteresiss.
I believe these properties are are unusual because the geodynamo has been traditionally considered stationary in comparison to the solar dynamo which is oscillating regularly. Furthermore, I would like to propose that oscillations can serve a mechanism for reversals and excursions,
But , first let me give credit to my collaborator – Friedrich Busse from the Institute of Physics of the University of Bayreuth.
East Kilbride, 28 Oct

2. Motivation: Applications of spherical dynamos
Dynamos in
rotating spherical
shells
Solar and stellar
magnetism
Geomagnetism
Planetary magnetism
East Kilbride, 28 Oct
SOHO, EIT Consortium, ESA, NASA

3. Outline of the talk Mathematical formulation of the problem.
Numerical methods of solution.
Typical convection and dynamo features. Turbulence.
Overview of the basic effects controlling dynamo behaviour and types of dynamo solutions.
Oscillations of dipolar dynamos as a possible cause of geomagnetic excursions and reversals.
Bistability and hysteresis of fully nonlinear dypolar dynamos.
Conclusions.
Here is a short overview of my talk.
I will briefly state the mathematical problem we consider and outline the numerical methods we use for its solution.
East Kilbride, 28 Oct

4. Model remarks
Our main motivation has been to model planetary dynamos and the Geodynamo.
Accordingly, we have made a number of appropriate assumptions:
Boussinesq approximation – constant material properties; variation of density is only included in the gravity term.
Incompressible fluid – the velocity field is solenoidal.
Rapid rotation – we strive to increase the rotation rate as our main motivation has been to model the Geodynamo.
Relatively thick spherical shells - our main motivation has been to model the Geodynamo.
Direct numerical simulations – no assumptions for the eddy diffusivities.
Self-sustained magnetic field – we look for dynamo solutions and we do not impose external magnetic fields.
These may not always be appropriate in the Solar context but we believe they capture the basic physics of the dynamo process.
OUR APPROACH: systematic study of parameter dependencies and careful extrapolation to astrophyiscal objects – Earth, planets, stars.
Here is a short overview of my talk.
I will briefly state the mathematical problem we consider and outline the numerical methods we use for its solution.
East Kilbride, 28 Oct
Mathematical formulation of the problem
Numerical methods of solution
Overview of the basic effects controlling dynamo behaviour and types of dynamo solutions: a focus on low magnetic Prandtl number regime
Oscillations of dipolar dynamos as a possible cause of geomagnetic excursions and reversals
Bistability and hysteresis of fully nonlinear dypolar dynamos
Conclusions

5. Convective spherical shell dynamos
Model equations & parameters
Boussinesq approximation
Simitev & Busse, JFM, 2005
Basic state & scaling
Boundary Conditions
So here is the problem we are interested in.
We consider a spherical shell full of electrically conducting fluid and rotating whit a constant angular velocity about a fixed vertical axis.
We assume a basic static state with a linear temperature profile due to internal heat sources.
We assume the Boussinesq approximation. Then the deviations from the static state are governed by the Navier-Stokes equations with Coriolis and Lorentz forces, by the heat equation which describes energy inflow into the system and by the induction equation for the magnetic field.
When nondimensionalised using the following scales, 4 non-dimensional parameters appear – the Rayleigh number the Coriolis number the Prandtl and the magnetic Prandtl numbers.
We have used a variety of boundary conditions but in this talk I will focus on stress-free boundary conditions for velocity, insulation inner and outer region for the magnetic field and a fixed temperatures at the boundary.
Length scale:
Time scale:
Temp. scale:
Magn. flux density:
East Kilbride, 28 Oct

6. Numerical Methods 3D non-linear problem: Linear problem:
Toroidal-poloidal representation
Spectral decomposition in spherical harmonics and Chebyshev polynomials
Scalar equations
Pseudo-spectral method. Time-stepping: Crank-Nicolson & Adams-Bashforth
For the numerical solution of the governing equations we use the familiar representation of the velocity and magn fileds into toroidal-poloidal scalars.
The scalar fields are then decomposed into spherical harmonics and Chebyshev polynomials.
Then we use a pseudo-spectral collocation method in which the equations are enforced at all grid points and the linear terms are computed in spectral space and the non-linear terms are computed in physical space and transformed to spectral space.
The integration in time involves a combination of a Crank-Nicholson step for the linear terms and an Adams-Bashforth step for the non-linear terms.
Resolution: radial=49, latitudinal=96, azimuthal=193.
Tilgner, IJNMF, 1999
Linear problem:
Galerkin spectral method for the linearised equations leading
to an eigenvalue problem for the critical parameters.
East Kilbride, 28 Oct

7. Typical properties of dynamo solutions.
Part I
Typical properties of dynamo solutions.
Turbulence
Spectra and spacial features
Temporal behaviour
Values of the magnetic Reynolds number
States of convection
Dynamo symmetries
East Kilbride, 28 Oct 2011
Ref: Simitev R.D. & Busse F.H., J. Fluid Mech., 532, 365, 2005.
Busse F.H. & Simitev R.D., Astron. Nachr., 326, 2005.
Bounds on the value of the magnetic Prandtl number for dynamo action
- Critical value of Rm,
- Turbulent flux expulsion.
Dynamo symmetry types as function of the magnetic Prandtl number.
Temporal and spacial structures of low Pm dynamos.
Energetics

8. Typical time dependence: turbulence
East Kilbride, 28 Oct 2011

9. Spectra and separation of scales
MD – red, FD - blue MD
East Kilbride, 28 Oct 2011

10. Typical magnetic Reynolds number for dynamo onset
East Kilbride, 28 Oct 2011

11. Finite-amplitude columnar convection
Simitev&Busse, NJP,2003
East Kilbride, 28 Oct

12. Dynamo symmetry types
Dynamo solutions exhibit symmetry because rapidly-rotating convection remains
equatorially-symmetric even in the turbulent regime.
Dipolar
Simitev, 2004
Quadrupolar
Hemispherical
East Kilbride, 28 Oct

13. Basic effects controlling dynamo behaviour
Part II
Basic effects controlling dynamo behaviour
Ref: Simitev R.D. & Busse F.H., J. Fluid Mech., 532, 365, 2005.
Busse F.H. & Simitev R.D., Astron. Nachr., 326, 2005.
Grote et al., 2003.
Bounds on the region of dynamo action
- Critical value of Rm,
- Turbulent flux expulsion.
Dynamo symmetry types as function of the magnetic Prandtl number.
Effect of magnetic field on convection.
East Kilbride, 28 Oct 2011

14. Bounds on the region for dynamo action
(R-Pm plane)
no dynamo
due to flux expulsion
Dynamo action is restricted by:
(a) vigour of convection - convection must be sufficiently vigorous to support dynamo action.
(b) magnetic field diffusivity – the magnetic diffusivity must be sufficiently low for the magnetic field to persist.
(c) flux expulsion - however, convection which is too vigorous can lead toexpulsion of magnetic field from small eddies.
no dynamo
due to low Rm
no dynamo
chaotic dipole
hemispherical
Simitev & Busse, JFM, 2005
East Kilbride, 28 Oct 2011

15. Decay of dynamo action due to flux expulsion
Note: With the increase of the value of Rayleigh number at all other parameter values fixed the magnetic energy components saturate and ultimately decrease due to flux expulsion and increasingly filamentary structure of the magn field.
Note: Ohmic dissipation continues to increase with R.
kinetic
magnetic Ex Mx Vx Ox
Busse & Simitev,2006
(B)
R= decay
dynamo
dynamo
decay
East Kilbride, 28 Oct

16. Types of dynamos in the parameter space
No dynamo
Regular and chaotic non-oscillatory dipolar dynamos (at large Pm/P
and not far above dynamo onset)
Oscillatory dipolar dynamos (at values of R larger than those of
non-oscillatory dipoles)
Hemispherical dynamos – always oscillatory
Quadrupolar dynamos – always oscillatory
We have performed a large number of simulations and in the parameter space and the following picture emerges.
Above dynamo onset four different types of dynamos become preferred depending on the value of the magnetic Prandtl number.
At large values of Pm dipolar dynamos are preferred while at small values of Pm quadrupolar dynamos are preferred. In between one finds mixed and hemispherical dynamos.
The quadrupolar dynamos always oscillate while the dipolar dynamos near dynamo onset are non-oscillatory.
As the Rayleigh number is increase the dipolar dynamos also become perturbed by higher multipole components and start to oscillate.
East Kilbride, 28 Oct

17. Bounds on the value of Pm for dynamo action
(P-Pm plane)
dipolar
hemispherical
quadrupolar
mixed
Simitev & Busse, JFM, 2005
decaying
Note: The minimal value of Pm decreases as P decreases.
(As a rough rule the value of the critical Pm is of the same order as the ordinary Prandtl number)
East Kilbride, 28 Oct

18. Effect of self-sustained magnetic field on convection
Simitev, 2004
There is little evidence that a generated magnetic field plays a role similar to externally imposed field and counteracts the Coriolis force.
Rather, the main effect of a generated field is to inhibit differental rotation and thereby increase amplitude of convection and its heat transport.
East Kilbride, 28 Oct

19. Oscillations of dipolar dynamos as a possible
Part III
Oscillations of dipolar dynamos as a possible
cause of geomagnetic excursions and reversals
Ref: Busse F.H. & Simitev R.D., Phys. Earth Planet. Inter., 168, 237, 2008.
Busse F.H. & Simitev R.D., Geophys. Astrophys. Fluid Dyn., 100, 2006.
Examples of linear oscillations
Parker's plane layer theory of dynamo wave
Non-linear oscillations
Mechanism of excursions and reversals
East Kilbride, 28 Oct

20. Non-oscillatory dynamos
exist if the dipolar component is strongly dominant,
have large ratio of Pm/P, so that quadrupolar components are not strong,
are not too turbulent for otherwise higher harmonics will enter A B C A
Here are snapshots of three different dipolar dynamos.
At the bottom of the slide you may see the time-series of the energy density on the case in the middle. The time seried demonstrate a chaotic time dependence but the structure of the dipolar field changes very little with time and remains basically as shown in the snapshots. B C B
East Kilbride, 28 Oct

21. Example of a quadrupolar oscillation
One period
Mean meridional filedlines
of constant (left),
(right)
and radial magn. field.
Here is for example an oscillation in a quadrupolar dynamo. The meridional plots cover a full period of the oscillation and you can see that the solution oscillates in an almost perfectly periodic fashion.
At the bottom you may see the values of the dipolar and quadrupolar coefficients of the poloidal field in the middle of the shell. The dipolar component is negligible and the quadrupolar oscillates very regularly.
Time series of toroidal and
poloidal magn.
coefficients.
East Kilbride, 28 Oct

22. An example of a dipolar oscillation
Notes:
Half-period of oscillation (column-by-column)
East Kilbride, 28 Oct

23. An example of a dipolar oscillation
Notes:
Half-period of oscillation (column-by-column)
East Kilbride, 28 Oct

24. Effect of oscillations on convection
Mean meridional magn. fieldlines (clockwise)
Energy densities
Finally, I would like to mention that the dynamo oscillations have an effect on convection as well.
The right half of the slide shows energy densities in this case. You may observe that the regular hemispherical oscillations affect the differential rotation. the oscillations in the differential rotation lead to regular oscillations in all other convection components.
East Kilbride, 28 Oct

25. Linear Oscillations: Parker dynamo waves
Axisymmetric field:
Following Parker's (1955) plane layer analysis of dynamo waves:
Using a linear wave solution ansatz:
we can obtain an expression for the growth rate
Assuming pseudo isotropic turbulence the alpha-coefficient is related to the helicity
I hope to have demonstrated on the previous examples that dynamo oscillations appear very frequently throughout the parameter regime.
To understand them in terms of a theoretical model we follow the dynamo wave analysis of Parker. In particular we consider a plane layer kinematic dynamo with a time dependent axisymmetric field.
We use the following representation if the velocity and magn fields in axisymmetric and poloidal terms. The axisymmetric and the polidal parts of the induction equation are ... In the poloidal equation we include an alpla effect term as suggested by Parker.
The equations are linear and the can be solved with a linear wave ansatz. from this we can easily obtain a dispersion relation and solve it to find the period of dynamo oscillations.
We assume that the alpha effect included in the poloidal equation is due to psedo-isotropic fluid flow turbulence and so the alpha-effect coefficient is related to the helicity of the flow by this expression.
Period:
East Kilbride, 28 Oct

26. Period of oscillations: model vs. numerics
On this slide have plotted the expression for the period obtained by the linear analysis on the previous slide in comparison with the periods determined from our numerical solutions. The theoretical values are plotted with empty symbols and the numerical ones with solid symbols,
The theoretical model and the full numerical solutions agree quite well for throughout various regions of the parameter space.
East Kilbride, 28 Oct

27. Non-linear dynamo oscillations
Mean meridional magn. fieldlines (clockwise)
Energy densities
Busse & Simitev, PEPI, 2008
Not all dynamo oscillations occur in a linear fashion. Here is an example in which the magnetic field an convection interact strongly.
Sometimes oscillations are caused by a temporary shift of the mean toroidal flux from a dipolar to a quadrupolar symmetry.
In this particular example this even leads to an aperiodic magnetic field reversal as demonstrated in the bottom panel of the time series.
East Kilbride, 28 Oct

28. Reversals cased by toroidal flux oscillations
Oscillating FD
dynamo
Busse & Simitev, PEPI, 2008
East Kilbride, 28 Oct 2011

29. Reversals cased by toroidal flux oscillations
Oscillating FD
dynamo
Busse & Simitev, PEPI, 2008
East Kilbride, 28 Oct 2011

30. Some similarities with geomagnetic observations
Reversed magnetic field appears first at low latitudes as also observed by Clement (Nature, 2004).
Average duration of a reversal event is ~20000yrs - roughly consistent with observations.
For each reversal we observe several excursion events.
Amplitude of the field increases more sharply after a reversal than than it decays before the reversal.
Busse & Simitev, PEPI, 2008
Other similarities with geomagnetic observations are that ...
East Kilbride, 28 Oct

31. Bistability and hysteresis of dipolar dynamos
Part IV
Bistability and hysteresis of dipolar dynamos
generated by chaotic convection
in rotating spherical shells
Ref: Simitev, R., Busse F.H., Europhysics Letters, 85, 19001, 2009.
Two types of dipolar dynamos generated by chaotic convection at identical external parameter values
The transition between Mean Dipolar (MD) and Fluctuating Dipolar (FD) dynamos and the hysteresis phenomenon
Contrasting properties of Mean Dipolar (MD) and Fluctuating Dipolar (FD) dynamos
Oscillations of Fluctuating Dipolar (FD) dynamos and reversals
East Kilbride, 28 Oct

32. Two types of dipolar dynamos generated by chaotic convection
Energy densities
Fully chaotic (large-scale turbulent) regime.
Two chaotic attractors for the same parameter values.
Essential qualitative difference: contribution of the
mean poloidal dipolar energy
black mean poloidal
green......fluctuating poloidal
red mean toroidal
blue fluctuating toroidal
All case that will be demonstrated in this talk are in a fully turbulent state.
Typical examples of the variation of the energy components are shown on this silde. To characterize convection and dynamo solutions ion more detail we split the total magnetic and kinetic energies to poloidal and toroidal components. The poloidal and toroidal components both have mean and fluctuating parts. Furthermore each energy component may be classified with according to its symmetry with respect to the equatorial plane. The kinetic energy is ... the magnetic is dipolar and quadrupolar.
Apart from the obvious quantitative differences, in the above plot, an essential qualitative difference may be noticed.
In the left panel, the black line which is the mean poloidal dipolar energy is
is the dominant magn component while in the right panel it is relatively insignificant.
East Kilbride, 28 Oct

33. Regions and transition
Ratio of fluctuating to mean poloidal magn energy
Two types of dipolar dynamos
Mean Dipolar (MD)
Fluctuating Dipolar (FD)
MD and FD dynamos correspond to rather different chaotic attractors in a fully chaotioc system
The transition between them is not gradual but is an abrupt jump as a critical parameter value is surpassed.
The nature of the transition is complicated.
Therefore, I will prefer to represent the transition between the tow states as shown on this slide.
Let me now discuss in more detail the nature of this transition.
East Kilbride, 28 Oct

34. Bistability and hysteresis in the MD <==> FD transition
Bistability and hysteresis in the ratio of fluctuating poloidal to mean poloidal magn energy
(a)
(b)
(c)
in all cases:
The coexistence is
not an isolated phenomenon
but can be traced with
variation of the parameters.
PMD= PFD = 0.5
σMD = σFD = 1
The coexistence is not an isolated phenomenon limited to the case shown.
In fact we have ....
East Kilbride, 28 Oct

35. (period of relaxation oscillations, clockwise)
The hysteresis is a purely magnetic effect
After magnetic field is suppressed both MD and FD dynamos equilibrate to statistically identical convective states
(period of relaxation oscillations, clockwise)
The hysteresis loops are purely magnetic phenomena. This may be easily demonstrated if we switch off the magnetic field in any of the dynamo solutions.
For instance, performing that in the case discussed earlier we observe that one the magnetic field is removed the non-magnetic convection rapidly equilibrates to the same convective state.
In this case this happens to be relaxation oscillations of thermal convection which is one of the well-known coherent states of turbulent rotationg convection..
Magnetic field is artificially suppressed, i.e. non-magnetic convection
East Kilbride, 28 Oct

36. A property comparison of MD and FD dynamos
(Spatial structures)
Snapshots of
spatial structures
The stronger magnetic field of
MD dynamos counteracts differential
rotation (diff rotation, meridional streamlines)
Both MD and FD dynamos appear dipolar from the outside (radial magn field at r=ro+1.3, Earth's surface)
FD dynamos have a somewhat more irregular and small-scale internal structure
(Bphi and meridional fieldlines)
East Kilbride, 28 Oct

37. (Temporal variations)
A property comparison of MD and FD dynamos
(Temporal variations)
Mean Dipolar (MD) dynamos are non-oscillatory.
Fluctuating Dipolar (FD) dynamos are oscillatory.
Half-period of oscillation in a FD dynamo (row-by-row)
Notes:
East Kilbride, 28 Oct

38. Conclusion
We have described typical properties of self-consistent spherical dynamos in rotating spherical shells.
Dynamo oscillations are typical temporal behaviour,
Oscillations may lead to aperiodic reversals similar to geomagnetic reversals.
Co-existence of chaotic attractors.
In conclusion I hope to have been able to demonstrate that dynamo oscillations both of quadrupolar and dipolar dynamos are found very frequently in the parameter regions which are currently accessible to numerical simulations.
Some of the oscillations can be understood in terms of linear dynamo waves while others are not so easy to interpret. However they might hold the key to understanding reversals and other interesting dynamo behaviour.
Thank you for your attention!
East Kilbride, 28 Oct
Simitev, R., Busse, F.H., Prandtl number dependence of convection driven dynamos in rotating spherical fluid shells, J. Fluid Mech., 532, 365 – 388, 2005.
Busse, F.H., Simitev, R., Parameter dependences of convection driven dynamos in rotating spherical fluid shells, GAFD, 100(4-5), pp , 2006.
Tilgner, A., Spectral methods for the simulation of incompressible flows in spherical shells, Int. J. Numer. Meth. Fluids, 30, pp , 1999.
Parker, E.N., Hydromagnetic dynamo models, Astrophys. J., 122, pp , 1955.
Busse, F.H., Simitev, R., Toroidal flux oscillations as possible causes of geomagnetic excursions and reversals, Phys. Earth Planet. Inter., DOI: /j.pepi