Part 1 : solar dynamo models [Paul] Part 2 : Fluctuations ... · ISSI Dynamo tutorial 1 5 From...

ISSI Dynamo tutorial 1 1

Dynamo tutorial

Part 1 : solar dynamo models [Paul]

Part 2 : Fluctuations and intermittency[Dario]

Part 3 : From dynamo to interplanetary magnetic field [Paul]


Dynamo tutorialPart 1 : solar dynamo models

The solar cycle magnetic fieldGlobal 3D MHD simulations 2D axisymmetric solar dynamo models Mean-field electrodynamics and models Babcock-Leighton modelsModels based on HD/MHD instabilities


The solar magnetic cycle


Maxwell’s equations


From Maxwell to MHD (1)

Step 1: Drop displacement current to revert to the originalform of Ampère’s Law:

Step 2: Write down Ohm’s Law in a frame co-moving withthe fluid:

Step 3: Non-relativistic transformation back to the laboratoryframe of reference:


From Maxwell to MHD (2)Step 4: Combine with Ampère’s Law to express the electric

field as:

Step 5: Substitute into Faraday’s Law to get the justly famousmagnetohydrodynamical induction equation:

where we defined the magnetic diffusivity as:


The MHD equations


Global 3D MHD simulations


Simulation frameworkSimulate anelastic convection in thickrotating fluid shell, convectivelyunstable in upper two thirds, usingEULAG-MHD in « ideal » mode.

This type of ILES simulation is oftenrefered to as « minimally diffusive »;

This allows to reach a maximallyturbulent state on a given mesh;

A flow or magnetic structure candevelop gradients reaching the meshscale and remain nonlinearly stable;

Simulations can be run in strongly turbulent regimes over very long times.


Kinetic and magnetic energies

(120 s.d.=10 yr)


Magnetic cycles (1)

Large-scale organisation of the magnetic field takes place primarilyat and immediately below the base of the convecting fluid layers

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sont requis pour visionner cette image.


Magnetic cycles (1)Zonally-averaged Bphi at r/R =0.718

Zonally-averaged Bphi at -58o latitude


Successes and problems

KiloGauss-strength large-scale magnetic fields, antisymmetric aboutequator and undergoing regular polarity reversals on decadal timescales.

Cycle period four times too long, and strong fields concentratedat mid-latitudes, rather than low latitudes.

Reasonably solar-like internal differential rotation, and solar-likecyclic torsional oscillations (correct amplitude and phasing).

Internal magnetic field dominated by toroidal component andstrongly concentrated immediately beneath core-envelope interface.

Well-defined dipole moment, well-aligned with rotation axis,but oscillating in phase with internal toroidal component.

On long timescales, tendency for hemispheric decoupling, and/ortransitions to non-axisymmetric oscillatory modes.


REALITY CHECK

The numerical treatment of unresolved scales influences a lot

the global dynamo behavior !


Axisymmetric+kinematic formulation of solar dynamo models


Model setup Solve MHD induction equation in spherical polar coordinatesfor large-scale (~R), axisymmetric magnetic field in a sphereof electrically conducting fluid:

Evolving under the influence of a steady, axisymmetriclarge-scale flow:

Match solutions to potential field in r > R .


Kinematic axisymmetric dynamo


Differential rotation

Fast

Slo

w


Shearing by axisymmetricdifferential rotation


Meridional circulation


REALITY CHECK

Many independent global MHD simulations suggest that magnetic

backreaction on large-scale flows is an important (perhaps the dominant)

amplitude-limiting mechanism


Kinematic axisymmetric dynamos


Poloidal source terms

1. Turbulent alpha-effect

2. Active region decay (Babcock-Leighton mechanism)

3. Helical hydrodynamical instabilities

4. Magnetohydrodynamical instabilities(flux tubes, Spruit-Tayler)


REALITY CHECK

We are forcing fundamentally non-axisymmetric processes into an axisymmetric model formulation


Mean-field electrodynamicsand dynamo models


The basic idea[ Parker, E.N., ApJ, 122, 293 (1955) ]

Cyclonic convective updraft/downdraftsacting on a pre-existing toroidal magneticfield will twist the fieldlines into poloidalplanes (in the high Rm regime)

The collective effect of many suchevents is the production of an electricalcurrent flowing parallel to the backgroundtoroidal magnetic field; such a currentsystem contributes to the production ofa poloidal magnetic component


Scale separation Separate flow and magnetic field into large-scale, « laminar » component, and a small-scale, « turbulent » component:

Assume now that a good separation of scales exists betweenthese two components, so that it becomes possible to definean averaging operator:

such that:

This is NOT a linearization! No assumption is made at this stagewith regards to the relative magnitudes of flow and field components


The turbulent EMF (1) Substitute into MHD induction equation and apply averaging operator:

with : TURBULENT ELECTROMOTIVE FORCE !

Still no approximation !!

with :

Now subtract averaged induction equation from original inductionequation to obtain evolution equation for b :


The turbulent EMF (2) Now, the whole point of the mean-field approach is NOT to have to dealexplicitly with the small scales; since the PDE for b is linear, with theterm acting as a source; therefore there must exit alinear relationship between b and B, and also between B and ;We develop the mean emf as

Where the various tensorial coefficients can be a function of ,of the statistical properties of u, on the magnetic diffusivity, butNOT of .

Specifying these closure relationships is the crux of the mean-field approach


The alpha-effect (1) Consider the first term in our EMF development:

If u is an isotropic random field, there can be no preferred directionin space, and the alpha-tensor must also be isotropic:

This leads to:

The mean turbulent EMF is parallel to the mean magnetic field!

This is called the « alpha-effect »


The alpha-effect (2) Computing the alpha-tensor requires a knowledge of the statisticalproperties of the turbulent flow, more precisely the cross-correlationbetween velocity components; under the assumption that b << B,if the turbulence is only mildly anisotropic and inhomogeneous, theso-called Second-Order Correlation Approximation leads to

The alpha-effect is proportional to the fluid helicity!

If the mild-anisotropy is provided by rotation, and the inhomogeneityby stratification, then we have

where is the correlation time for the turbulence.


Turbulent diffusivity Turn now to the second term in our EMF development:

In cases where u is isotropic, we have , and thus:

The mathematical form of this expression suggests that canbe interpreted as a turbulent diffusivity of the large-scale field.for homogeneous, isotropic turbulence with correlation time ,it can be shown that

This result is expected to hold also in mildly anisotropic, mildly inhomogeneous turbulence. In general,


REALITY CHECK

All commonly-used formulations for the alpha and beta-tensors apply to

physical regimes that are far removed from solar interior

conditions


Scalings and dynamo numbers

Three dimensionless groupingshave materialized:

Length scale: solar/stellar radius:Time scale: turbulent diffusion time: 0


The mean-field zoo

The alpha-effect is the source of both poloidal and toroidalmagnetic components; works without a large-scale flow!planetary dynamos are believed to be of this kind.

Both the alpha-effect and differential rotation shear contribute totoroidal field production; stellar dynamos could be of this kind ifdifferential rotation is weak, and/or if dynamo action takes placein a very thin layer.

Differential rotation shear is the source of the toroidal component;the alpha-effect is the source of only the poloidal component. the solar dynamo is believed to be of this kind.


Linear alpha-Omega solutions (1)

Choice of alpha:

Solve the axisymmetric kinematic mean-field alpha-Omega dynamoequation in a differentially rotating sphere of electrically conductingfluid, embedded in vacuum; in spherical polar coordinates:



The growth rate, frequency,and eigenmode morphologyare completely determinedby the product of the twodynamo numbers



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Positive alpha-effect Negative alpha-effect


Linear alpha-Omega solutions (4)Time-latitude « butterfly » diagram

Equivalent in axisymmetric numerical model: constant-rcut at r/R=0.7, versus latitude (vertical) and time (horizontal)

[ http://solarscience.msfc.nasa.gov/images/bfly.gif ]

http://solarscience.msfc.nasa.gov/images/bfly.gif


Nonlinear models: alpha-quenching (1)

We expect that the Lorentz force should opposethe cyclonic motions giving rise to the alpha-effect;

We also expect this to become important when themagnetic energy becomes comparable to the kinetic energy of the turbulent fluid motions, i.e.:

This motivates the following ad hoc expressionfor « alpha-quenching »:



The magnetic diffusivity is the primary determinant of the cycle period







Magnetic fields concentrated at too high latitude;Try instead a latitudinal dependency for alpha:


REALITY CHECK

Alpha-quenching is a ridiculously oversimplistic representation of the

complex nonlinear interactions between a flow and a magnetic field


Alpha-Omega dynamos with meridional circulation (1)



Equatorward propagation of the deep toroidal field is now due to advection by the meridional flow, not « dynamo waves » effect.


Models with meridional circulation (2)


Models with meridional circulation (3)


REALITY CHECK

Both observations and numerical simulations indicate that the solar meridional flow is spatiotemporally far more complex than the steady, single-cell pattern used in the vast

majority of models


Models based onthe Babcock-Leighton mechanism


Sunspot as as emerging toroidal flux ropes

Hale et al. 1919, ApJ, 49, 153

Parker 1955, ApJ, 121, 491

Latitude

Tilt


1952 « magnetograph »


2001 magnetogram


Active region decay (1)

Synoptic magnetogram courtesy D. Hathaway, NASA/MSFC[ http://solarscience.msfc.nasa.gov/images/magbfly.gif ]Toroidal flux emerging in active regions in one cycle: ~10^17 Wb

Peak polar cap flux: ~10^14 Wb


(2b)


Active region decay (3)Zonal means


REALITY CHECK

Is the Babcock-Leighton effect anessential part of the dynamo loop,

or a mere surface side effect of sunspot decay ?


Babcock-Leighton dynamo model (1)


Babcock-Leighton dynamo model (2)A Babcock-Leighton source term for the axisymmetricdynamo equations:

Concentrated in surface layers

Non-local in BPeaking at mid-latitudes

The source term operatesonly in a finite range of toroidal field strengths.





The turnover time of the meridional flowis the primary determinant of the cycle period


REALITY CHECK

Can the solar polar field really be kinematically subducted by the

meridional flow, like assumed by all Babcock-Leighton models ?


Hinode


Babcock-Leighton versus alpha-effectThere are serious potential problems with the operation of thealpha-effect at high field strength; not so with the B-L mechanism

The B-L mechanism operates only in a finite range of field strength;potentially problematic in the presence of large cycle amplitude fluctuations.

Both models can produce tolerably solar-like toroidal field butterflydiagrams, and yield the proper phase relationship between surfacepoloidal and deep toroial components (with circulation included inthe mean-field model)

The alpha-effect (or something analogous) appears unavoidablein stratified, rotating turbulence.

A decadal period arises « naturally » in B-L models; in mean-fieldmodels, it requires tuning the value of the turbulent magnetic diffusivity


Models based onHD and/or MHD instabilities


Buoyant rise of toroidal flux ropes[ Caligari et al. 1995, ApJ, 441, 886 ]

Destabilization and buoyant rise of thin toroidal magnetic flux tubesstored immediately below the core-envelope interface (overshoot)

Non-axisymmetric modes of loworder (m=1 or m=2) are mosteasily destabilized;

Conservation of angular momentumgenerates a flow along the axisof the rising loop;

The Coriolis force acting on theflow in the legs of the loop impartsa twist that shows up as an E-Wtilt upon emergence through the surface.


Instability of toroidal flux tubesstored in the tachocline

Dia

gram

cou

rtesy

A. F

erriz

Mas

Toroidal flux ropes storedIn the overshoot region ofthe tachocline are sensitiveto a buoyantly-driven undulatory instability withlong growth rate; these« helical waves » providea mean EMF parallel tothe tube axis, I.e., a formof alpha-effect.

Dynamo regions Emergence region


REALITY CHECK

We currently do not have a « concensus dynamo model » for the

basic solar cycle


Coming up after recess:

Dynamo tutorialPart 2 : Fluctuations and intermittency

Part 1 : solar dynamo models [Paul] Part 2 : Fluctuations ... · ISSI Dynamo tutorial 1 5 From...

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Transcript of Part 1 : solar dynamo models [Paul] Part 2 : Fluctuations ... · ISSI Dynamo tutorial 1 5 From...