global mhd instabilities of the solar tachocline

High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)

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Global MHD Instabilities of the Solar Tachocline

Currently Active Collaborators Currently Active Collaborators (alphabetical):(alphabetical):Paul Cally (Monash University & HAO)

Mausumi Dikpati (HAO)Peter Gilman (HAO)Mark Miesch (HAO)Aimee Norton (HAO)

Matthias Rempel (HAO)

Past Contributors Past Contributors (alphabetical):(alphabetical):J. Boyd, P. Fox, D. Schecter

May 2004May 2004

2Peter Gilman November Peter Gilman November 20042004

Motivations for Study of Global Instability of Differential Motivations for Study of Global Instability of Differential Rotation and Toroidal Fields in the Rotation and Toroidal Fields in the

Solar TachoclineSolar Tachocline

• May produce latitudinal angular momentum transport that keeps tachocline thin and couples to an angular momentum cycle with the convection zone

• Can generate global magnetic patterns that can imprint on the convection zone and photosphere above

• Can contribute to the physics of the solar dynamo through generation of kinetic and current helicity

• Can produce preferred longitudes for emergence of active regions


Physical Setting of Solar TachoclinePhysical Setting of Solar Tachocline

Location and Extent

Physical Properties

Straddles “base” of convection zone at r = .713 R

Thickness < 0.05 R, may be as thin as .02 R - .03 R

Shape may depart from spherical. Prolate? Thicker at high latitudes?

Convection zone base change from oxygen abundance? (To slightly below .713?)

Rotation Well constrained by helioseismic inferences; torsional oscillations?; 1.3 year oscillations in low latitudes? Jets?

Stratification Subadiabatic; Overshoot & Radiative partsSharp or smooth transition?

Magnetic Field Strong (~100kG inferred from theory for trajectories of rising tubes) Tipped toroidal fields?Broad or narrow in latitude? Stored in overshoot and/or radiative part?


2 40 0 2 4sin sins r s r s r

Rotation Detail within Solar TachoclineRotation Detail within Solar Tachocline


Nonlinear 2D MHD EquationsNonlinear 2D MHD Equations

Defining velocity & magnetic filed respectively asand using a modified pressure variable we can write,

,ˆˆ ,ˆˆ baBvuV

,ρp

Continuity Equations:

,0cos

,0cos

ba

vu

Equations of Motion:

cos

cos2cos

1 22u

vvvu

t

u

,coscoscos

1

a

bb

cos

cos2

22u

vuvu

t

v

,coscos

aba

Induction Equations:

.0)(cos

1

,0

vaubt

b

vaubt

a


2D MHD Instability: Reduction to 2D MHD Instability: Reduction to Solvable SystemSolvable System

cos

1 ;

cos

1 ,

b,

φ

χavu

irctim iccceau ;~, ,cos ,cos )(

00

In which = sin and

(Legendre Operator) 2

22

11

mdd

dd

L

Vorticity Equation:

0)1( )1()( 202

2

02

02

2

0

d

dL

d

dLc

Classical Hydrodynamic Stability Problem

“Boundary” conditions: , χ = 0 at poles

Induction Equation: 00 )( c


2D MHD Instability: 22D MHD Instability: 2ndnd Order Equations Order Equations for Reference State Changesfor Reference State Changes

22

1cos ( ' ' ' ')

cos

ua b u v

t

( ' ' ' ')a

u b at

For differential rotation (linear measure):

For toroidal magnetic field (linear measure):

“Mixed”Stress

MaxwellStress

ReynoldsStress


Differential Rotation and Toroidal Field Differential Rotation and Toroidal Field Profiles Tested for InstabilityProfiles Tested for Instability

Differential rotation (angular measure): 2 4

0 0 2 4s s s

0 2 4sin (0); rotation of equator; 0 0.3 (surface value)s s s

Toroidal field (angular measure) 0

With symmetric about the equator, and anti-symmetric, unstable disturbances separate also into two symmetries:

0 0

Symmetric:Symmetric:

AntisymmetricAntisymmetric: :

symmetric, antisymmetric antisymmetric, symmetric

1/ 23

0 ( , have either sign); node at aa b a b b 4( 0.1 10 gauss)a B

0 gaussian profiles of arbitrary amplitude, width, and latitude of peak


Barotropic InstabilityBarotropic Instability(sometimes also called Inflection Point Instability)(sometimes also called Inflection Point Instability)

• Barotropic: pressure and density surfaces coincide in fluid (baroclinic when they don’t)

• Instability originally discovered by Rayleigh, put in atmospheric setting by H.L. Kuo

• As meteorologists use it, instability is of axisymmetric zonal flow, a function of latitude only, to 2D (long. – lat.) wavelike disturbances

• Disturbances grow by extracting kinetic energy from the flow, by Reynolds stresses that transport angular momentum away from the local maximum in zonal flow

• Necessary condition for instability: gradient of total vorticity of zonal flow changes sign – hence “inflection point”


Barotropic Instability of Solar Differential Rotation Barotropic Instability of Solar Differential Rotation Measured by Helioseismic DataMeasured by Helioseismic Data

(Charbonneau, Dikpati and Gilman, 1999)


Properties of 2D MHD Instability of Differential Properties of 2D MHD Instability of Differential Rotation and Toroidal Magnetic FieldRotation and Toroidal Magnetic Field

ToroidalMagnetic Field

DifferentialRotation

Angular momentum transport toward the poles primarily by the Maxwell Stress (perturbations field lines tilt

upstream away from equator)

Magnetic flux transport away from the peak toroidal field by the Mixed Stress (phase

difference in longitude between perturbation velocities & magnetic fields)


Broad Toroidal Field Profiles Tested for Global MHD Broad Toroidal Field Profiles Tested for Global MHD Instability of Field and Differential RotationInstability of Field and Differential Rotation

E P

SP NP


Gaussian Type Banded Toroidal Field Profiles Gaussian Type Banded Toroidal Field Profiles Tested for Global MHD Instability of Field Tested for Global MHD Instability of Field

and Differential Rotationand Differential Rotation

ESP NP


Mechanisms of Global MHD Instability for Mechanisms of Global MHD Instability for Weak Toroidal Fields (TF)Weak Toroidal Fields (TF)


Toroidal Ring Disturbance Patterns of Toroidal Ring Disturbance Patterns of Longitudinal Wave Numbers m=0, 1, 2Longitudinal Wave Numbers m=0, 1, 2

• Toroidal ring shrinks• Fluid in ring spins up

m = 0

• Toroidal ring deforms, creating Maxwell Stress• Fluid flow inside ring deforms but does not spin up

m = 2

m = 1

• Toroidal ring tips but remains same circumference; creates Maxwell stress

• Fluid in ring keeps same speed but flow tips


Summary of Properties of 2D Instability of Summary of Properties of 2D Instability of Differential Rotation and Toroidal FieldDifferential Rotation and Toroidal Field

Property Without Toroidal Field With Toroidal Field

Unstable?Unstable for differential rotation >~20% with term (Watson result ~29% with

no term)

Unstable for almost all differential rotation and toroidal fields

Growth Rate Determined by shear magnitude efold ~ few months

Determined by shear magnitude and field strength and profile shortest efolds ~

few months

Phase Velocities Between minimum and maximum rotation rate

Between min and max rotation rate for broad fields; for narrow fields acquires

rotation rate at latitude of peak field

Semi-circle theorem, bounding growth rates and phase

velocitiesYes Yes

Unstable longitude wave numbers m

m = 1 only for broad fieldsm up to at least 6 for narrow profiles

Energy source Differential Rotation Differential rotation for weak fields, toroidal field for strong or narrow fields

Changes in reference state predicted High Latitude Jets Sharp changes in differential rotation

and toroidal field

Disturbance symmetries about equator Both unstable

Both unstable, with velocities and magnetic field paired with opposite

symmetry; symmetry switching may occur

1 3m

44


Transformation of variables: 0 c H

Vorticity equation changes to

d H

d S

dS

d

dH

d

mc c

d

d

S

SH

2

2 2

2

2

0 21 1

12

1

21

0

in which S c 1 20

202 So have singular points where one or both of factors in S

0 0 c

How many singular points there are depends on profiles of .0 and 0

Note that the usual critical point 0 0 c of ordinary hydrodynamics is NOT a singular

0 there).

Critical or Singular Points in the Equations Critical or Singular Points in the Equations for 2D MHD Stabilityfor 2D MHD Stability

point here (H regular at such points, so

.

vanish, i.e., at the poles, and where or where the doppler shifted (angular) phase

velocity of the perturbation equals the local (angular) Alfvén speed.

If let Y=S1/2 H, then : k2 real if ci =0; complex if not 2

22

d Yk Y=0

d

k2 is large in the neighborhood of singular points defined above


Example of Profile of Reynolds and Maxwell Stresses of Unstable Example of Profile of Reynolds and Maxwell Stresses of Unstable Disturbance of Longitudinal Wave Number m=1, in Relation to Disturbance of Longitudinal Wave Number m=1, in Relation to

AlfvAlfvénic Singular Points, of a Toroidal Band of nic Singular Points, of a Toroidal Band of 16° Width16° Width

(c) bw=16°


Dominant Energy Flow in Unstable SolutionsDominant Energy Flow in Unstable Solutions

Low, broad, toroidal field :a

' ' K M K K

High or narrow toroidal field :a

' 'M K M


Energy Flow Diagram for Nonlinear 2D MHD Energy Flow Diagram for Nonlinear 2D MHD System with Forcing and DragSystem with Forcing and Drag

(Dikpati, Cally and Gilman, 2004)


Example of “Clamshell” Instability in Nonlinear 2D Example of “Clamshell” Instability in Nonlinear 2D MHD SystemMHD System

(Cally, Dikpati and Gilman, 2003)


Nonlinear Survey of Symmetric Nonlinear Survey of Symmetric Tipping Mode in Strong BandsTipping Mode in Strong Bands

(Cally, Dikpati and Gilman 2003)


Linear and Nonlinear Tip AnglesLinear and Nonlinear Tip Angles



Nonlinear Tipping of Toroidal Fields Nonlinear Tipping of Toroidal Fields in Tachoclinein Tachocline


Peak Toroidal Field 25 kG Peak Toroidal Field 100 kG


Global MHD Instability with Kinetic (Global MHD Instability with Kinetic (ddkk) and) andMagnetic (Magnetic (ddmm) Drag) Drag

o a


Broad TF Banded TF


Evolution of Tip Angles of Evolution of Tip Angles of aa=1 Toroidal Bands for =1 Toroidal Bands for Various Realizations with dk=10dm, for Latitude Various Realizations with dk=10dm, for Latitude

Placements of 30°Placements of 30°



Observation Evidence of Tipped Observation Evidence of Tipped Toroidal Ring?Toroidal Ring?


Tipped Toroidal Ring in Longitude-latitude Coordinates Tipped Toroidal Ring in Longitude-latitude Coordinates Linear Solutions with Two Possible SymmetriesLinear Solutions with Two Possible Symmetries



““Sparking Snake” ModelSparking Snake” Model

• Imagine snake on interior spherical surface

• Sends out ‘sparks’ given specific trajectories to outer spherical surface

• Assign snake geometry & dynamics

• Analyze results to determine if an observer could decipher the underlying geometry

(Gilman & Norton)


Schematic of Tipped Toroidal Ring in “Sparking Schematic of Tipped Toroidal Ring in “Sparking Snake” ModelSnake” Model


Schematic of Flux EmergenceSchematic of Flux Emergence

(Norton and Gilman, 2004)

• Important that we discriminate between a spread in latitudes from flux emergence and one from tipped toroidal field

• Schematic illustrating flux trajectory variations dependent upon field strength of source toroidal ring

• Ellipses represent contours of toroidal field strength

• Strongest flux ropes rise radially, weaker rise non-radially


Histogram of Sunspot Pair AnglesHistogram of Sunspot Pair Angles


Global Instabilities of Solar TachoclineAssume Differential Rotation from Helioseismology

Hydrostatic Models Result

2D HD Stable

2D MHD Unstable for wide range of toroidal fields

“Shallow Water” HD Overshoot part Unstable

Radiative part Stable

Shallow Water” MHD Both Parts Unstable

SW HD Instabilities suppressed for broad peak fields 10 kG

Multi-layer SW HD, MHD Expect Instability

3D HD, MHD Expect Instability; unstable for MHD when DR, TF independent of radius

3D Nonhydrostatic HD, MHD

More modes of Instability

Magnetic buoyancy enters

Dyn

am

o P

ote

nti

al


What is MHD Shallow Water System?What is MHD Shallow Water System?

• Spherical Shell of fluid with outer boundary that can deform

• Upper boundary a material surface

• Horizontal flow, fields in shell are independent of radius

• Vertical flow, field linear functions of radius, zero at inner boundary

• Magnetohydrostatic radial force balance

• Horizontal gradient of total pressure is proportional to the horizontal gradient of shell thickness

• Horizontal divergence of magnetic flux in a radial column is zero

(Gilman, 2000)


Effective Gravity Parameter (G)Effective Gravity Parameter (G)

in which:

2

2

1

2t ad

t c p

g HG

r H

gt gravity at tachocline depth

fractional departure from adiabatic temperature gradientH thickness of tachocline “shell”Hp pressure scale height

rt solar radius at tachocline depth

ωc rotation of solar interior

ad

G ~ 10-1 for Overshoot TachoclineG ~ 102 for Radiative Tachocline

(Dikpati, Gilman and Rempel, 2003)


Relationship among Effective Gravity G Relationship among Effective Gravity G Subadiabatic Stratification and Subadiabatic Stratification and

Undisturbed Shell Thickness HUndisturbed Shell Thickness Had



Shallow Water Equations of Shallow Water Equations of Motion and Mass ContinuityMotion and Mass Continuity

2 21 1

coscos cos cos 2

u h v v u vG u

t

2 21

cos ,cos cos 2

b b a ba

2 2

coscos 2

v h u v u vG u

t

2 2

cos ,cos 2

a b a ba

1 11 1 1 cos ,

cos cosh h u h v

t


Shallow Water Induction and Shallow Water Induction and Flux Continuity EquationsFlux Continuity Equations

cos cos ,cos cos

a a u u aub va v b

t

1cos cos ,

cos cos cos

b b u v aub va v b

t

1 11 1 cos 0.

cos cosh a h b


Singular PointsSingular Points

hσ is departure of shell thickness from uniform thickness

For cases of solar interest:Sr , Sm = 0 are important, Sg = 0 is not

• Singular points define places of rapid phase shifts with latitude in unstable modes• Therefore much of disturbance structure, as well as energy conversion processes,

determined in this neighborhood• Play major role in interpreting instability as a form of resonance

Occur at latitudes where:

2 2 0m o r oS c 0r o rS c

22 21 (1 )g o r o oS c G h


Equilibrium in MHD Shallow Water SystemEquilibrium in MHD Shallow Water System

• Balance between hydrostatic pressure gradient and magnetic curvature where toroidal field is strong

• Balance between magnetic curvature stress and coriolis force curvature with prograde jet inside toroidal field band

• Actual solar case may be in between

In general, a balance among three latitudinal forces, including hydrostatic pressure gradient, magnetic curvature stress, and

coriolis forces

Important Limiting Cases:


MHD Shallow Water EquilibriumMHD Shallow Water Equilibriumfor Banded Toroidal Fieldsfor Banded Toroidal Fields


Overshoot Layer (G=0.1)


Schematic of Possible Modes of Instability in Schematic of Possible Modes of Instability in MHD “Shallow Water” ShellMHD “Shallow Water” Shell

• h increases poleward• Toroidal ring shrinks• Fluid in ring spins up

m = 0

• h redistributes but no net poleward rise• Toroidal ring deforms, creating Maxwell Stress• Fluid flow inside ring deforms but does not spin up

m = 2

m = 1

• h redistributed but no net rise• Toroidal ring tips but remains

same circumference• Fluid in ring keeps same speed

but flow tips


Stability Diagrams for Stability Diagrams for HD Shallow Water SystemHD Shallow Water System

(Dikpati and Gilman, 2001)

ad

G r/Ro

G

Dif

fere

ntia

l Rot

atio

n


Growth Rates for Unstable ModesGrowth Rates for Unstable ModesFor Broad Toroidal FieldFor Broad Toroidal Field

(Gilman and Dikpati, 2002)

a = 1.0s4 / s0 = 0m = 1, Sm = 1, A

a = 0.5s4 / s0 = 0m = 1, Sm = 1, A

a = 0.1s4 / s0 = 0m = 1, Sm = 1, A

a = 0.2s4 / s0 = 0m = 1, Sm = 1, A


Growth Rates of Unstable Modes Growth Rates of Unstable Modes for Broad Toroidal Fieldsfor Broad Toroidal Fields

Overshoot Layer Radiative Layer

(Gilman and Dikpati, 2002)

a a


Domains of Unstable Toroidal Field BandsDomains of Unstable Toroidal Field BandsOvershoot Layer


Radiative Layer


Global MHD Instability of Tachocline in 3DGlobal MHD Instability of Tachocline in 3D

• General problem of instability from latitudinal and radial gradients of rotation and toroidal field is non separable. (much bigger calculation therefore required)

• Special case of 3D disturbances on DR and TR that are functions of latitude only.

• There are strong mathematical similarities to 2D and SW cases, depending on boundary conditions chosen.

• Has eigen functions with multiple nodes in vertical; representable by sines and cosines with wave number n.

• For strong TF, must take account of magnetically generated departures from Boussinesq gas equation of state.

• High n modes should be substantially damped by vertical diffusion or wave processes in tachocline

(Gilman, 2000)


Growth Rates For 3D Global MHD InstabilityNo Boundary Conditions Top and Bottom

n =

Vertical Velocity = 0 Top and Bottom

Pressure = 0 TopVertical Velocity = 0 Bottom

1 yr

1 yr1 yr

0.1 yr

0.1 yr

0.1 yr


Summary of Global MHD Instability ResultsSummary of Global MHD Instability Results

• Combinations of differential rotation and toroidal field likely to be present in the solar tachocline, are likely to be unstable to global disturbances of longitudinal wave number m=1 and sometimes higher

• The instability is primarily 2D, but likely to persist in 3D as well

• Instability can lead to a significant “tipping” of the toroidal field away from coinciding with latitude circles, which might be responsible for some aspects of patterns of sunspot location

• In 3D, the instability is likely to be an important component of the global solar dynamo, as a producer of poloidal from toroidal fields, and as a source of m 0 surface magnetic patterns


Two distinct possible sources of jets

1. Prograde jet to balance magnetic curvature stress associated with toroidal field band

(at mid latitudes, 100 kG TF would require 200 m/s prograde jet

if Coriolis force completely balances curvature stress)

2. Global HD or MHD instability extracts angular momentum from low latitudes and deposits it in narrow band at higher latitudes

So if we can find jets from helioseismic analysis, it could be evidence for 1 and/or 2 above.


Jet balancing magnetic curvature stress

ε=0 no jet ε=1 full jet

j

cs

2/120 1

: jet-like toroidal flow

: core rotation rate

: solar-like differential rotation: jet parameter

: toroidal field

If 2nd term is not too big, then


Jet amplitudes for various toroidal field bands and their

latitude locations


2D MHD Instability: 22D MHD Instability: 2ndnd Order Equations Order Equations for Reference State Changesfor Reference State Changes

22

1cos ( ' ' ' ')

cos

ua b u v

t

( ' ' ' ')a

u b at

For differential rotation (linear measure):

For toroidal magnetic field (linear measure):

“Mixed”Stress

MaxwellStress

ReynoldsStress


Jet amplitudes from nonlinear hydrodynamic calculations

Dikpati 2004 (in preparation)


Jet amplitudes in 2D MHD nonlinear calculations

Start with an initial ~30% jetSystem stabilizes with a ~20% jet

Start with no jet, system stabilizes with a ~20% jet

(Cally, Dikpati & Gilman, 2004)

Results are for a 10-degree toroidal band with 100 kG peak field placed at 40-degree latitude


Conditions under which hydrodynamic instability can occur and produce a high-latitude jet, when a 100

kG toroidal field band is present

band of width < latitude



10 10

5 30

502

Narrow bands and low band latitudes

global mhd instabilities of the solar tachocline

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