Global MHD Instabilities of the Solar Tachocline

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Global MHD Instabilities of the Solar Tachocline

• Currently Active Collaborators (alphabetical)
• Paul Cally (Monash University HAO)
• Mausumi Dikpati (HAO)
• Peter Gilman (HAO)
• Mark Miesch (HAO)
• Aimee Norton (HAO)
• Matthias Rempel (HAO)
• Past Contributors (alphabetical)
• J. Boyd, P. Fox, D. Schecter

May 2004
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Motivations for Study of Global Instability of
Differential Rotation and Toroidal Fields in the
Solar 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

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Physical Setting of Solar Tachocline
Location and Extent
Straddles base of convection zone at r .713
R? Thickness lt 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??)
Physical Properties
Rotation Well constrained by helioseismic
inferences torsional oscillations? 1.3 year
oscillations in low latitudes?
Jets? Stratification Subadiabatic Overshoot
Radiative parts Sharp 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?
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Rotation Detail within Solar Tachocline
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Nonlinear 2D MHD Equations
Defining velocity magnetic filed respectively
asand using a modified pressure variable we
can write,
Continuity Equations
Equations of Motion
Induction Equations
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2D MHD Instability Reduction to Solvable System
Vorticity Equation
Classical Hydrodynamic Stability Problem
In which ? sin ? and
Boundary conditions ?, ? 0 at poles
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2D MHD Instability 2nd Order Equations for
Reference State Changes
For differential rotation (linear measure)
MaxwellStress
ReynoldsStress
For toroidal magnetic field (linear measure)
MixedStress
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Differential Rotation and Toroidal Field
Profiles Tested for Instability
Differential rotation (angular measure)
Toroidal field (angular measure)
With symmetric about the equator, and
anti-symmetric, unstable disturbances separate
also into two symmetries
Symmetric
Antisymmetric
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Barotropic Instability(sometimes also called
Inflection Point Instability)

• Barotropic pressure and density surfaces
  coincide in fluid (baroclinic when they dont)
• 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

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Barotropic Instability of Solar Differential
Rotation Measured by Helioseismic Data
(Charbonneau, Dikpati and Gilman, 1999)
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Properties of 2D MHD Instability of Differential
Rotation 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)
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Broad Toroidal Field Profiles Tested for Global
MHD Instability of Field and Differential Rotation
P
E
SP
NP
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Gaussian Type Banded Toroidal Field Profiles
Tested for Global MHD Instability of Field and
Differential Rotation
E
SP
NP
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Mechanisms of Global MHD Instability for Weak
Toroidal Fields (TF)
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Toroidal Ring Disturbance Patterns of
Longitudinal Wave Numbers m0, 1, 2
m 0
m 1
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Summary of Properties of 2D Instability of
Differential Rotation and Toroidal Field
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Critical or Singular Points in the Equations for
2D MHD Stability
Transformation of variables
Vorticity equation changes to
in which
So have singular points where one or both of
factors in S
.
or where the doppler shifted (angular) phase
vanish, i.e., at the poles, and where
velocity of the perturbation equals the local
(angular) Alfvén speed.
How many singular points there are depends on
profiles of .
of ordinary hydrodynamics is NOT a singular
Note that the usual critical point
there).
point here (H regular at such points, so
If let YS1/2 H, then k2
real if ci 0 complex if not
k2 is large in the neighborhood of singular
points defined above
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Example of Profile of Reynolds and Maxwell
Stresses of Unstable Disturbance of Longitudinal
Wave Number m1, in Relation to Alfvénic Singular
Points, of a Toroidal Band of 16 Width
(c) bw16
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Dominant Energy Flow in Unstable Solutions
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Energy Flow Diagram for Nonlinear 2D MHD System
with Forcing and Drag
(Dikpati, Cally and Gilman, 2004)
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Example of Clamshell Instability in Nonlinear
2D MHD System
(Cally, Dikpati and Gilman, 2003)
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Nonlinear Survey of Symmetric Tipping Mode in
Strong Bands
(Cally, Dikpati and Gilman 2003)
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Linear and Nonlinear Tip Angles
(Cally, Dikpati and Gilman, 2003)
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Nonlinear Tipping of Toroidal Fields in
Tachocline
Peak Toroidal Field 25 kG
Peak Toroidal Field 100 kG
(Cally, Dikpati and Gilman, 2003)
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Global MHD Instability with Kinetic (dk)
andMagnetic (dm) Drag
Banded TF
Broad TF
(Dikpati, Cally and Gilman, 2004)
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Evolution of Tip Angles of a1 Toroidal Bands for
Various Realizations with dk10dm, for Latitude
Placements of 30
(Dikpati, Cally and Gilman, 2004)
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Observation Evidence of Tipped Toroidal Ring?
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Tipped Toroidal Ring in Longitude-latitude
Coordinates Linear Solutions with Two Possible
Symmetries
(Cally, Dikpati and Gilman, 2003)
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Sparking 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)
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Schematic of Tipped Toroidal Ring in Sparking
Snake Model
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Schematic of Flux Emergence

• 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

(Norton and Gilman, 2004)
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Histogram of Sunspot Pair Angles
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Global Instabilities of Solar Tachocline
Assume Differential Rotation from Helioseismology
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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)
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Effective Gravity Parameter (G)
in which
gt gravity at tachocline depth fractional
departure from adiabatic temperature
gradient H thickness of tachocline
shell Hp pressure scale height rt solar radius
at tachocline depth ?c rotation of solar interior
G 10-1 for Overshoot Tachocline G 102 for
Radiative Tachocline
(Dikpati, Gilman and Rempel, 2003)
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Relationship among Effective Gravity G
Subadiabatic Stratification and
Undisturbed Shell Thickness H
(Dikpati, Gilman and Rempel, 2003)
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Shallow Water Equations of Motion and Mass
Continuity
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Shallow Water Induction and Flux Continuity
Equations
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Singular Points
Occur at latitudes where
hs is departure of shell thickness from uniform
thickness

• 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

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Equilibrium in MHD Shallow Water System
In general, a balance among three latitudinal
forces, including hydrostatic pressure gradient,
magnetic curvature stress, and coriolis forces
Important Limiting Cases

• 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

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MHD Shallow Water Equilibriumfor Banded Toroidal
Fields
Overshoot Layer (G0.1)
(Dikpati, Gilman and Rempel, 2003)
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Schematic of Possible Modes of Instability in
MHD Shallow Water Shell
m 0
m 1

• h redistributed but no net rise
• Toroidal ring tips but remains same circumference
• Fluid in ring keeps same speed but flow tips

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

m 2

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

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Stability Diagrams for HD Shallow Water System
(Dikpati and Gilman, 2001)
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Growth Rates for Unstable ModesFor Broad
Toroidal Field
(Gilman and Dikpati, 2002)
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Growth Rates of Unstable Modes for Broad
Toroidal Fields
Overshoot Layer
Radiative Layer
a
a
(Gilman and Dikpati, 2002)
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Domains of Unstable Toroidal Field Bands
Overshoot Layer
Radiative Layer
(Dikpati, Gilman and Rempel, 2003)
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Global 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)
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Growth Rates For 3D Global MHD Instability
No Boundary Conditions Top and Bottom
Pressure 0 TopVertical Velocity 0 Bottom
0.1 yr
Vertical Velocity 0 Top and Bottom
1 yr
n
0.1 yr
0.1 yr
1 yr
1 yr
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Summary 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 m1
  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

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Two distinct possible sources of jets

• 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)
• 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.

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Jet balancing magnetic curvature stress
If 2nd term is not too big, then
jet-like toroidal flow
core rotation rate
solar-like differential rotation
jet parameter
toroidal field
e0 no jet e1 full jet
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Jet amplitudes for various toroidal field bands
and their latitude locations
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2D MHD Instability 2nd Order Equations for
Reference State Changes
For differential rotation (linear measure)
MaxwellStress
ReynoldsStress
For toroidal magnetic field (linear measure)
MixedStress
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Jet amplitudes from nonlinear hydrodynamic
calculations
Dikpati 2004 (in preparation)
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Jet amplitudes in 2D MHD nonlinear calculations
Results are for a 10-degree toroidal band with
100 kG peak field placed at 40-degree latitude
Start with an initial 30 jet System stabilizes
with a 20 jet
Start with no jet, system
stabilizes with a 20 jet
(Cally, Dikpati Gilman, 2004)
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Conditions under which hydrodynamic instability
can occur and produce a high-latitude jet, when a
100 kG toroidal field band is present
Narrow bands and low band latitudes

• band of width lt latitude
• band of width lt latitude
• band of width lt latitude