The creation of magnetic structures from velocity shear(Constraints for the solar dynamo)

Nic BrummellApplied Mathematics, University of California Santa Cruz

Geoff Vasil, Kelly Cline JILA/APS, University of Colorado

Lara Silvers, Mike ProctorDAMTP, University of Cambridge, UK

UCSD Feb 2009

We are Sun worshippers!Lots of fun things far away, but can we even understand our nearest and dearest star? Only 93.5 million miles away (150 mill km) => easy to observe! Run-of-the-mill, average star G2: 5000-6000 deg K 15 million degs K in core 2 million degs K in corona 4.5 billion years old 5.5 billion years to go! Mainly hydrogen Fusion in core When hydrogen runs out: burns helium -> red giant -> white dwarf 1.4 million km diameter (~ 100 x Earth)Looks pretty boring in white light, BUT .

The Sun

We are Sun worshippers!

Major puzzle: solar magnetic activity cycleSolar magnetic activity is VARIABLE but remarkably ORDERED.How does it work? A DYNAMO!But how does the dynamo work?

Observations: exteriorReally big stuff: Flares, Coronal holes,CMEs700+MmDays, monthsGiant cells: 200+ Mm10-20 daysSupergranulation: 30-50 Mm20 hoursMesogranulation: 7-10 Mm2 hoursGranulation: 1-2 Mm5 minsEven smaller stuff: Intergranular lanes,magnetic bright points, diffusionLook at sun in different ways,see different scales.TURBULENT!

Theories driven by observationsSolid body rotation in the coreInterface layer the TACHOCLINEDifferential rotation in the convection zoneThings we know surprisingly well -- solar interior structure (thanks to helioseismology)

Large-scale dynamo: TheoryExisting poloidal field is stretched by latitudinal (and/or radial) differential rotation into the toroidal direction an -like mechanismStrong toroidal field rises due to magnetic buoyancy and, under the influence of global rotation, twists into the poloidal direction an a-like mechanism

Strong toroidal field rises as structures?MAGNETIC BUOYANCY: the standard explanation

Magnetic field exerts a magnetic pressure (Lorenz force JxB can be split into pressure and tension ) Concentrated B contributes to the total pressure Isothermal pressure balance implies density lower in tubeOutside:Pt = Pg1Inside:Pg2 = Pt-Pm Pm ~ B2 => Pg2 < Pg1densityinside < densityoutsidePg = density*temperature

Magnetic buoyancy instabilities 101

Magnetic buoyancy instabilities 101

Magnetic buoyancy instabilities 101End result: Magnetic GRADIENTS that are important!(gradients in the direction of gravity)

Standard concept of the Omega effect

Large-scale dynamo: intuitive pictureGeneration/shredding of magnetic fieldTransport of magnetic field from CZ into tachocline -- TURBULENT TRANSPORT OF B ?W effect: Conversion of poloidal field to toroidal field -- DIFFERENTIAL ROTATION -- ORIGIN ?Formation of structures and magnetic buoyant rise -- MAGNETIC BUOYANCY / SHEAR ?a mechanism: Regeneration of poloidal field -- ROTATION, TURBULENT u, b CORRLNS??Recycling/breakdown of fieldEmergence of structuresSTUDY ELEMENTS OF THE DYNAMO

Heavily dependent on the tachoclineBoundary layer between the convection zone (differentially rotating) the radiative interior (solid body rotation)Layer (plus sub-layers?) of strong radial shear weaker latitudinal shearAlmost certainly threaded with magnetic field weak poloidal? strong toroidal?

We wish to examine, via fully nonlinear MHD simulations: the interaction of velocity shear and weak background (poloidal) magnetic fields. the production of strong (toroidal) magnetic fields by the shear magnetic structures produced any magnetic buoyancy instabilities of the magnetic configuration the nonlinear evolution of the system

So how does it work?

Thermal diffusivity k=k(z) ( not k(r,T:x,y,z) ) :Ck(layer1)/Ck(layer2)=(m2+1)/(m1+1)Stiffness, S = (m2-mad)/(mad-m1)e.g. Brummell, Clune & Toomre, 2002 Full compressible MHD (poloidal/toroidal) DNS Cartesian Pseudospectral / finite-difference Semi-implicit HIGHER Rq, Re, Pe, LOWER Pr than global sims (resolves from diffusive scale UP)Local simulations of elements of the dynamo

So how do we solve these?Blue Gene/LFastest machine in the world!~ 213,000 cpus596 Tflops peak, 478 sustained

The mega effectInteraction between shear (gradients) in the azimuthal velocity and the poloidal magnetic field produces strong toroidal magnetic field. TWO POSSIBILITIES:(the usual)(the not-so-usual)

Good reasons for the not-so-usual But really interested in gradients for magnetic buoyancy:t BT = BP + , = r*sin() Differentiate MDI observational results:

Therefore reasons: Radial shear is stronger Need radial gradients for magnetic buoyancy Some radial component likely Horizontal scale selection - better option?

Case (a): Latitudinal shear+(a) Latitudinal shear:BqdVf/dq=BfBfqflatitudelongitude

Mimic some properties of the tachocline : Use a convectively stable layer Force* a velocity shear in both the vertical (z) and one horizontal (y) direction. e.g. U(y,z) = f(z) cos(2 p y/ym) where f(z) is a polynomial function chosen to confine the shear to a particular layer between zu and zl (and to be sufficiently continuous) Shear flow is hydrodynamically stableThen add an initial magnetic field: B0 = (0, By , 0) with By = 1 Model: Localised latitudinal velocity shear* Add term in the equations that induces desired flow in absence of magnetic effects

Induction of strong toroidal field by shear

Increasing Rm: Magnetic buoyancy instabilityA more interesting movie! Instability Cyclic activity Two out-of-phase sequences of identical but oppositely-directed magnetic structures. Broken the y-reflectional symmetry

Latitudinal shear: conclusionsFirst demonstration that buoyantly-rising strong (toroidal) magnetic structures can be spontaneously generated by the action of shear on a background (poloidal) magnetic field.

BUT Radial gradients stronger => stronger toroidal field?Currently the structures have scale of the latitudinal shear = BIG (too big)

SO

Try radial gradients?

So consider Case (b): Radial shearOne might expect:Yes! but turns out that this is much harder to make work than you might think!

Radial shear: Magnetic buoyancyMagnetic buoyancy instability: volume rendering of abs(B)

768^3200,000 cpu hours yikes!

Measured Taylor microscale Reynolds number ~ 30Pm=0.625

Radial shear: Looks good, but problems BxzTimeNot much flux transport.Inefficient

Radial shear: ProblemsThe forcing required to make this work is HUGE! Target velocity of forcing is about Mach 15! Yikes!If you run the same case without magnetic field, keeping the same large forcing, get some serious Kelvin-Helmholtz turbulence!

WHY SUCH A LARGE FORCING?RE-CAST THIS QUESTION

Is the tachocline this strongly maintained? Is it maintained by external processes independent of tachocline dynamics? (USUAL POV)Or must maintaining processes take into account internal dynamics? (UNUSUAL POV) Must the tachocline be STRONGLY MAINTAINED against the tachocline dynamics?In other words: is the shear we oberve the target shear of a a particular forcing or a modified shear returned at the end of a complex nonlinear process?

Two simple 1-D analytic models to try and answer this question:Completely unmaintained shearWeakly maintained shearIf these dont work in some sense, then answer is shear is strongly-maintained.

We evaluate under what conditions a magnetic buoyancy instability can manifest itself

Stability conditionCondition for instability (Newcomb 1961):around a mechanical equiilibrium can be recast aswhere N is the Brunt-Vaisalla frequencyNote that NONE of this is really valid in our case, of course! : No mechanical equilibrium : background state evolves! Shear flow on topBUT look to see if this is even CLOSE to being satisfied as an indication of LIKELIHOOD of magnetic buoyancy Thermodynamics: Adiabatic => N = N0, const. (Reasonable but well come back to this)

Model 1:No maintenance of the shear (slow rundown) things happen fast so can ignore complex nonlinear back reactions on the shear maintainence of shear independent of tachocline dynamicsExamine time-dependent buildup of the toroidal field:No forcing, linearisation Induction eqn: stretching production of tor field by shear on pol fieldMomentum eqn: linear back reaction on the shearEasy: assume background density const (but dont have to)Initial conditions: Bx=0 and a given profile u=U0(z) => solutions=> Alfven wave dynamics

Alfven wave dynamicsBx(t,z)U(t,z)Analytical: constant wave speedInitial condition: U0 = tachocline-like jump,Magnitude U0, width z

Full 3D simulations: Failures :(We have plenty of examples of the types of dynamics we describe

Forcing not large enough:System heads to diffusively-balanced state plus Alfven waves.

UBx

Amplitude of BxBx grows but saturates.Really is just superposition of two waves, and only grows when they are interfering.

Bx grows whilst interference is in shear zone, then halts in the constant parts of U0i.e. Bx grows until i.e. grows for Alfven crossing time tA across half layer.

Two phases of dynamics:Growth (actually interfering waves) for t < tABx grows linearly like and u ~ U0Wave propagation for t > tA with

Can Bx get big enough for magnetic buoyancy instability?Linear growth up to a saturation max value

Model 1: Instability?Estimate gradients of Bx from this amplitude simply as (Bx(max)-0)/z/2Differentiate

Plug into mag buoy instability condition (adiabatic version)

or Richardson number of forced flow, Ri < z/HSmall!!Need highly shear unstable original flow for possibility of mag buoy instabilities!!Rule of thumb: Ri < 1/4 => turbulent shearTachocline Ri ~ 1000?

Maybe NO maintenance is not enough! Try some weak maintenance? => Model 2

Model 2: Integrate, combine and figure out some bounds => estimate of max amplitude of toroidal fieldWeak maintenance of the shearExamine time-independent state weakly forced stateInductionMomentumPm = magnetic Prandtl numberMagnetic energy bounded by the kinetic energy of the the forcing (but with Pm factor)Stability? Ri/Pm < z/H or Ri < Pm . z/H EVEN WORSE!

Conclusions:UNDER THE ASSUMPTIONS-- WEAK OR NO MAINTENANCE OF SHEAR=> SHEAR INDEPENDENT OF TACHOCLINE DYNAMICSthen to get magnetic buoyancy instabilities NEED-- large forcing (large = hydrodynamically unstable)-- or high PmWe have confirmed these simple ideas with massive numerical simulations! (buoyancy found at small Pm for large forcing, or at large Pm for small forcing) Ultimate conclusion:The maintenance of the tachocline needs to know about tachocline dynamics.In other words, we have to do a more self-consistent problem.Pm=4000U0=1.2

Radial shear: Problem avoidanceTo avoid these theoretical restrictions, need:Large forcing Large magnetic Prandtl numberBoth work:Seen alreadyLarge Pm

Pm = 4000

Radial shear: Thoughts Action of radial shear on radial field -- conclusions: For magnetic buoyancy to occur, must stretch the field fast enough that Alfven waves cannot radiate the magnetic energy away.Simple models, confrmed by numerical experiments, show that this requires a very strong initial/target shear -- one that is probably hydrodynamically unstable (or a high Pm which is not valid for the Sun).The models assumed that the tachocline dynamics and the processes that maintained the tachocline itself were independent. At this stage we concluded that this must not be true and a more holistic, more complicated model of tachocline dynamics was needed.We had pondered one quirky caveat in this work however, and had only speculated on its effect. That was the possibility of a DOUBLE-DIFFUSIVE INSTABILITY.

Double-diffusive instabilities 101In the presence of TWO components that affect the density, can have an instability even if the overall density stratification is convectively stable.e.g. ocean water: warm salty water overlying cooler fresh water

IF the diffusivity of the stabilising component is much LARGER than that of the stabilising component, can have instability:Parcel perturbed upwards should return downwards due to temperature deficit but this diffuses away quickly, and so continues upwards due to relative freshness.warmsaltycoolfreshOverall density stratification stableThermal stratification stabilisingSalinity stratification destabilising

Double-diffusive magnetic buoyancyIn the earlier work , we assumed that the thermodynamics adjusted adiabatically.If we assumed that the thermal diffusivity was sufficiently large, then we could re-do the work assuming that the adjustment was isothermal.Now we have to include the diffusivities via the ratio (Note: not Pm!)This is equivalent to looking for double-diffusive instabilities in this system. Adiabatic Isothermal

Clearly, if is small, then this inequality is more likely satisfied.

Double-diffusive instabilityTwo simulations differing only in the thermal diffusivity and therefore Fixed magnetic and viscous diffusivity (=> Prandtl number changes)Lower is unstable!

Double-diffusive instabilityFluctuations are induced in B and w

Double-diffusive instabilityAdded complexity:Since this is not a bifurcation about a static equilibrium but rather one about a dynamically-evolving background state, there is a condition on the growth rate as well as just a Rayleigh number criterion (Ratot~RaT-RaB)Growth rate must be faster than the evolution of the background for instability to manifest.Background evolves on the Alfven timescale of the background field, so now dynamics depends on background field strength!Instability occurs weaker and higher up.

We are still working on the nonlinear evolution of this instability

ConclusionsSolar boundary layer -- the tachocline -- essential for solar dynamics.Dynamics of the tachocline very mysterious.In terms of generating buoyant magnetic structures:TYPE 1: Latitudinal shear can produce buoyant structures but of the wrong scales.TYPE 2: Radial shear struggles to produce buoyant structures at solar parameters by regular magnetic buoyancy instabilities.This is due to the fact that a system laced radially with magnetic field can radiate away magnetic energy through Alfven waves. This will occur always be the case because it is unlikely that you can have Type 1 alone (no poloidal field)Differences between Type 1 and Type 2? Layers vs finite structures? etc Is the tachocline shear strongly maintained against internal processes?TYPE 3: Double-diffusive instead of regular magnetic buoyancy instabilities CAN occur at solar parameter. It remains to be seen if this is a viable mechanism of magnetic transport.Are any of these mechanisms strong enough to transport magnetic field through a convection zone?

The End

Latitudinal shear AND radial fieldBy=1, Bz=0 (no vertical field)By=1, Bz=1 (vertical field)Very similar dynamics except for tilted geometry: Equilibria are equilibria. Instabilities still occur.So why do layers behave differently?

Layer dynamicsProbably due to the fact that layers CANNOT have the following sort of behaviour:No horizontal pressure gradients: no poloidal flows driven in layers.Wide but finite layers? Ends rise first, waves in middle?

Wide but finite layers8x wider shear than previous latitudinal shear caseShows some signs of what we expect: Ends rise first Middle spreads by wavesMore work to come

Another escape route?A completely different type of instability: DOUBLE-DIFFUSIVE INSTABILITIESFor the work we presented so far, arguments were made under an adiabatic parcel argument in general.Adiabatic => thermally-sealed.Opposite extremely: isothermal => infinitely leaky (in heat)

High Pm, low forcingPm=4000U0=1.2

Radial shearWhy was such a large forcing necessary?Initial forcing of shear: r dt uo ~ FInitial induction of Bx: dt Bx0 ~ Bz dz u0Back-reaction on shear: r dtu1 ~ dx(p + a Bx0Bx0) + Bz dz Bx0 + m dzz u0 + F

Question: Does magnetic buoyancy act before magnetic tension destroys the shear?

magnetic tensionmagnetic pressure

Radial shearAnswer: Only if the forcing is LARGE r dtu1 ~ dx(p + a Bx0Bx0) + Bz dz Bx0 + m dzz u0 + F

Usually balance diffusion and forcing so that in the absence of magnetic fields, a steady target velocity is produced. r dtu1 ~ dx(p + a Bx0Bx0) + Bz dz Bx0 + m dzz u0 + F

Here, must also balance magnetic tension with a larger forcing otherwise tension acts back on the shear before the magnetic pressure becomes significant dynamics are dominated by Alfven wavesXXdt Bxn ~ Bz dz un r dtun ~ Bz dz Bxn + F

Radial shear: ProblemsWant to work out effectively whether it is possible to amplify the horizontal field Bx fast enough to go buoyantly unstable before Alfven waves radiate away the magnetic energy.i.e. you have an Alfven crossing time of the layer to get Bx as big as possible! Theory:Unforced case (rundown problem from an initial velocity shear profile):

For instability, we need (ballpark based on Newcomb condition): dPM/dz ~ P*N0(z)2/g magnetic pressure, PM = Bx(z)2/2 ; N0(z) = buoyancy frequency of the initial background profileThe best that unforced shear can do after roughly one Alfven time is Bx ~ Bz tA zu Afven time, tA ~ z/vA Therefore, for instability, require z (zu(z))2 ~ N0(z)2 HP z/HP ~ Ri i.e. for z small, Ri must be small => SHEAR FLOW IS (HYDRODYNAMICALLY) UNSTABLE!

Radial shear: ProblemsTheory:Forced case (forcing set up to balance dffusion in absence of B):

There are some steady solutions:BzzBx + zzu -zzu0 = 0Bzzu + zzBx = 0 Combining these givesQ Bx - zzBx = (Bz/) zu0 Q = (Bz)2/In terms of the velocity jump U = |zu0| dz and the magnetic Prandtl number M = / there is a sharp bound:sup(PM) M 0 U2/8Buoyancy needs:z/HP ~ Ri/MEven worse for solar values: since M is small, Ri must be even smaller.

Radial shear: Thoughts Action of radial shear on radial field -- conclusions: For magnetic buoyancy to occur, must stretch the field fast enough that Alfven waves cannot radiate the magnetic energy away.Simple models, confrmed by numerical experiments, show that this requires a very strong initial/target shear -- one that is probably hydrodynamically unstable.In the case where magnetic buoyancy is induced, simulations show that the presence of magnetic field suppresses the hydrodynamic instability, so that the resultant MHD velocity shear is reasonable. Adding the magnetic field back into the hydrodynamic simulation seems to return the system to the same MHD state.In the case where magnetic buoyancy is not induced, a diffusive balance results.These results are true for forced AND slow rundown (very low diffusivity) cases.Two big questions remain:Even when buoyancy does occur, it is inefficient at transporting magnetic flux vertically. Why?The latitudinal shear problem worked just fine. Would adding vertical field to this problem ruin things?

Radial shear: Problem avoidanceTo avoid these theoretical restrictions, need:Large forcing Large magnetic Prandtl numberBoth work:Seen alreadyLarge Pm

Pm = 4000

Latitudinal shear AND radial fieldBy=1, Bz=0 (no vertical field)By=1, Bz=1 (vertical field)Very similar dynamics except for tilted geometry: Equilibria are equilibria. Instabilities still occur.So why do layers behave differently?

Layer dynamicsProbably due to the fact that layers CANNOT have the following sort of behaviour:No horizontal pressure gradients: no poloidal flows driven in layers.Wide but finite layers? Ends rise first, waves in middle?

Wide but finite layers8x wider shear than previous latitudinal shear caseShows some signs of what we expect: Ends rise first Middle spreads by wavesMore work to come

ConclusionsSolar boundary layer -- the tachocline -- essential for solar dynamics.Dynamics of the tachocline very mysterious (Why is it so thin? Is it turbulent? -- Much more to come from Steve Tobias later).In terms of generating buoyant magnetic structures:Radial shear acting on radial field is much less efficient than latitudinal shear acting on latitudinal fieldThis is due to the fact that a system laced radially with magnetic field can radiate away magnetic energy through Alfven waves.Horizontal field => waves confined to structures => energy not radiated away?Furthermore, the dynamics of layers are very different from the dynamics of tubesNo poloidal flowsLayers => tethering of structures by magnetic tensionWhat is the role of waves in the tachocline?(Important for angular momentum transport too)

The End

Latitudinal shear with radial field By=1, Bx=0 (old)By=1, Bx=1 (new)Very similar dynamics except for tilted geometry: Equilibria are equilibria. Instabilities still occur.So why do layers behave differently?

Latest: 1D shear (Grad student: Geoff Vasil)Structures have twist

Radial shear: Magnetic buoyancyHigh enough forcing: strong enough magnetic layer created such that magnetic buoyancy can act.However, things are much more complicated and turbulent than previous case

Alfven wave dynamicsBx(t,z)U(t,z)Toy Calculation:with density stratificationAnalytical:constant wave speedInitial condition: U0 = tachocline-like jump, jump U0, width z:

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