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
High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)
The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research
under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.
May 2004
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
Peter Gilman
2
November 2004
Physical Setting of Solar Tachocline
Location and Extent
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?)
Physical Properties
Rotation
Stratification
Magnetic Field
Peter Gilman
Well constrained by helioseismic inferences;
torsional oscillations?; 1.3 year oscillations in
low latitudes? Jets?
Subadiabatic; Overshoot & Radiative parts
Sharp or smooth transition?
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?
3
November 2004
Rotation Detail within Solar Tachocline
0  s0  r   s2  r  sin 2   s4  r  sin 4 
Peter Gilman
4
November 2004
Nonlinear 2D MHD Equations


ˆ
ˆ
Defining velocity & magnetic filed respectively as V  u  v , B  aˆ  bˆ,
and using a modified pressure variable   p ρ, we can write,
Continuity Equations:
Equations
of Motion:
u

v cos    0,

 
a

b cos    0,

 
u
1   u 2  v 2 
v  v







u
cos


 cos    
t cos   
2


1 
b  b 






a
cos

,
cos   cos    

v
  u 2  v 2 
u  v







u
cos


 cos    
t  
2



a  b







a
cos

,
 cos    

Induction Equations:
Peter Gilman
a 
ub  va   0,

t 
b
1 

(ub  va )  0.
t cos  
5
November 2004
2D MHD Instability: Reduction to
Solvable System
u  

1 
χ
1 



, v 
; a  , b 

cos  
φ
cos  
u   0 cos  , a   0 cos  ,  ,  ~ eim( ct ) ; c  cr  ici
Vorticity Equation:
Classical Hydrodynamic Stability Problem
d2




d2
2
( 0  c) L 
 0 (1   )   0 L  
 0 (1   2 )  0
d 2
d 2
In which  = sin  and
d
L
d


m2


2 d

 1   d   
2

 1 
Induction Equation:
“Boundary” conditions:
Peter Gilman
 (Legendre Operator)
(0  c)    0
, χ = 0 at poles
6
November 2004
2D MHD Instability: 2nd Order Equations
for Reference State Changes
For differential rotation (linear measure):
u
1 
2

cos
 (a ' b '  u ' v ')
2
 t cos   

Maxwell
Stress

Reynolds
Stress
For toroidal magnetic field (linear measure):
a 

(u ' b '  ' a ')
 t 
“Mixed”
Stress
Peter Gilman
7
November 2004
Differential Rotation and Toroidal Field
Profiles Tested for Instability
Differential rotation (angular measure): 0  s0  s2   s4 
  sin (0); s0  rotation of equator; 0  s2  s4  0.3 (surface value)
2
Toroidal field (angular measure)  0
 0  a  b 3 (a, b have either sign); node at    a b
(a  0.1  B  104 gauss)
 0  gaussian profiles of arbitrary amplitude, width,
and latitude of peak

4

1/ 2
With 0 symmetric about the equator, and  0 anti-symmetric, unstable
disturbances separate also into two symmetries:
Symmetric:  symmetric,  antisymmetric
Antisymmetric:  antisymmetric,  symmetric
Peter Gilman
8
November 2004
Barotropic 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”
Peter Gilman
9
November 2004
Barotropic Instability of Solar Differential
Rotation Measured by Helioseismic Data
(Charbonneau, Dikpati and Gilman, 1999)
Peter Gilman
10
November 2004
Properties of 2D MHD Instability of Differential
Rotation and Toroidal Magnetic Field
Toroidal
Magnetic Field
Differential
Rotation
Angular momentum transport toward
the poles primarily by the Maxwell
Stress (perturbations field lines tilt
upstream away from equator)
Peter Gilman
Magnetic flux transport away from the peak
toroidal field by the Mixed Stress (phase
difference in longitude between perturbation
velocities & magnetic fields)
11
November 2004
Broad Toroidal Field Profiles Tested for Global MHD
Instability of Field and Differential Rotation
Peter Gilman
E
P
SP
NP
12
November 2004
Gaussian Type Banded Toroidal Field Profiles
Tested for Global MHD Instability of Field
and Differential Rotation
SP
Peter Gilman
E
NP
13
November 2004
Mechanisms of Global MHD Instability for
Weak Toroidal Fields (TF)
Peter Gilman
14
November 2004
Toroidal Ring Disturbance Patterns of
Longitudinal Wave Numbers m=0, 1, 2
m=0
m=1
• Toroidal ring tips but remains same circumference;
• Toroidal ring shrinks
• Fluid in ring spins up
•
creates Maxwell stress
Fluid in ring keeps same speed but flow tips
m=2
• Toroidal ring deforms, creating Maxwell Stress
• Fluid flow inside ring deforms but does not spin up
Peter Gilman
15
November 2004
Summary of Properties of 2D Instability of
Differential Rotation and Toroidal Field
Property
Without Toroidal Field
With Toroidal Field
Unstable?
Unstable for differential rotation >~20%
with  4 term (Watson result ~29% with
no  4 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
velocities
Yes
Yes
Unstable longitude wave
numbers m
1 m  3
m = 1 only for broad fields
m 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
Both unstable
Both unstable, with velocities and
magnetic field paired with opposite
symmetry; symmetry switching
may occur
Disturbance symmetries about
equator
Peter Gilman
16
November 2004
Critical or Singular Points in the Equations
for 2D MHD Stability
Transformation of variables:
  0  c H
d
S 

2
c


c




0
d 1   2  
d 2 H 1 dS dH  1 
m2 
H  0
Vorticity equation changes to


2 

d 2 S d d 1   2  1   2 
S





2
2
2
S  1    0  c   0 . So have singular points where one or both of factors in S
in which




vanish, i.e., at the poles, and where  0  c    0 or where the doppler shifted (angular) phase
velocity of the perturbation equals the local (angular) Alfvén speed.
How many singular points there are depends on profiles of
0
.
and
0
Note that the usual critical point  0  c  0 of ordinary hydrodynamics is NOT a singular
point here (H regular at such points, so   0 there).
If let Y=S1/2 H, then
d2Y
 k 2 Y=0 : k2 real if ci =0; complex if not
2
d
k2 is large in the neighborhood of singular points defined above
Peter Gilman
17
November 2004
Example of Profile of Reynolds and Maxwell Stresses of
Unstable Disturbance of Longitudinal Wave Number m=1, in
Relation to Alfvénic Singular Points, of a Toroidal Band of
16° Width
(c) bw=16°
Peter Gilman
18
November 2004
Dominant Energy Flow in Unstable Solutions
Low, broad, toroidal field a :
K  M '  K'  K
High or narrow toroidal field a :
M  K'  M '
Peter Gilman
19
November 2004
Energy Flow Diagram for Nonlinear 2D MHD
System with Forcing and Drag
(Dikpati, Cally and Gilman, 2004)
Peter Gilman
20
November 2004
Example of “Clamshell” Instability in
Nonlinear 2D MHD System
(Cally, Dikpati and Gilman, 2003)
Peter Gilman
21
November 2004
Nonlinear Survey of Symmetric
Tipping Mode in Strong Bands
(Cally, Dikpati and Gilman 2003)
Peter Gilman
22
November 2004
Linear and Nonlinear Tip Angles
(Cally, Dikpati and Gilman, 2003)
Peter Gilman
23
November 2004
Nonlinear Tipping of Toroidal Fields
in Tachocline
Peak Toroidal Field 25 kG
Peak Toroidal Field 100 kG
(Cally, Dikpati and Gilman, 2003)
Peter Gilman
24
November 2004
Global MHD Instability with Kinetic (dk) and
Magnetic (dm) Drag
Broad TF o  a 
Banded TF
(Dikpati, Cally and Gilman, 2004)
Peter Gilman
25
November 2004
Evolution of Tip Angles of a=1 Toroidal Bands
for Various Realizations with dk=10dm, for
Latitude Placements of 30°
(Dikpati, Cally and Gilman, 2004)
Peter Gilman
26
November 2004
Observation Evidence of Tipped
Toroidal Ring?
Peter Gilman
27
November 2004
Tipped Toroidal Ring in Longitude-latitude Coordinates
Linear Solutions with Two Possible Symmetries
(Cally, Dikpati and Gilman, 2003)
Peter Gilman
28
November 2004
“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)
Peter Gilman
29
November 2004
Schematic of Tipped Toroidal Ring in
“Sparking Snake” Model
Peter Gilman
30
November 2004
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)
Peter Gilman
31
November 2004
Histogram of Sunspot Pair Angles
Peter Gilman
32
November 2004
Global Instabilities of Solar Tachocline
Dynamo Potential
Assume Differential Rotation from Helioseismology
Hydrostatic Models
Result
2D HD
2D MHD
Stable
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
Peter Gilman
Magnetic buoyancy enters
33
November 2004
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)
Peter Gilman
34
November 2004
Effective Gravity Parameter (G)
2
1 gt   ad H
G
2  rtc 2 H p
in which:
gt
   ad
H
Hp
rt
ωc
gravity at tachocline depth
fractional departure from adiabatic temperature gradient
thickness of tachocline “shell”
pressure scale height
solar radius at tachocline depth
rotation of solar interior
G ~ 10-1 for Overshoot Tachocline
G ~ 102 for Radiative Tachocline
(Dikpati, Gilman and Rempel, 2003)
Peter Gilman
35
November 2004
Relationship among Effective Gravity G
Subadiabatic Stratification    ad and
Undisturbed Shell Thickness H
(Dikpati, Gilman and Rempel, 2003)
Peter Gilman
36
November 2004
Shallow Water Equations of
Motion and Mass Continuity

u
1 h
v  v 
1   u2  v2 
 G


 u cos  



t
cos  cos   
 cos   2 
b

cos
 b 

1   a 2  b2 
     a cos   cos   2  ,




   u2  v2 
v
h
u  v 
 G


 u cos   


t
 cos   
   2 
a

cos
 b 
   a 2  b2 
     a cos     2  ,





1 
1 
1  h  u  
1  h  v cos   ,
1  h   
t
cos 
cos 
Peter Gilman
37
November 2004
Shallow Water Induction and
Flux Continuity Equations
a 
a

 ub  va  
t 
cos 
 u 

u

v
cos




  
 cos 


b
1 
b

 ub  va  
t
cos  
cos 
 a 


b
cos



  
,


 u 

v

v
cos




  
 cos 


 a 


b
cos



  
,


1 
1 
1  h  a  
 1  h  b cos    0.
cos  
cos  
Peter Gilman
38
November 2004
Singular Points
Occur at latitudes where:
Sr  o  cr  0
2
Sm  o  cr   o2  0
2
S g  1   2 o  cr   o2   G (1  ho )


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
Peter Gilman
39
November 2004
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
Peter Gilman
40
November 2004
MHD Shallow Water Equilibrium
for Banded Toroidal Fields
Overshoot Layer (G=0.1)
(Dikpati, Gilman and Rempel, 2003)
Peter Gilman
41
November 2004
Schematic of Possible Modes of Instability in
MHD “Shallow Water” Shell
m=0
• h increases poleward
• Toroidal ring shrinks
• Fluid in ring spins up
m=1
• h redistributed but no net rise
• Toroidal ring tips but remains
m=2
•
same circumference
Fluid in ring keeps same speed
but flow tips
• h redistributes but no net poleward rise
• Toroidal ring deforms, creating Maxwell Stress
• Fluid flow inside ring deforms but does not spin up
Peter Gilman
42
November 2004
Stability Diagrams for
HD Shallow Water System
G
Differential Rotation
   ad
G
r/Ro
(Dikpati and Gilman, 2001)
Peter Gilman
43
November 2004
Growth Rates for Unstable Modes
For Broad Toroidal Field
a = 0.5
s4 / s 0 = 0
m = 1, S
m = 1, A
a = 1.0
s4 / s 0 = 0
m = 1, S
m = 1, A
a = 0.2
s4 / s 0 = 0
m = 1, S
m = 1, A
a = 0.1
s4 / s0 = 0
m = 1, S
m = 1, A
(Gilman and Dikpati, 2002)
Peter Gilman
44
November 2004
Growth Rates of Unstable Modes
for Broad Toroidal Fields
Overshoot Layer
Radiative Layer
a
a
(Gilman and Dikpati, 2002)
Peter Gilman
45
November 2004
Domains of Unstable Toroidal Field Bands
Overshoot Layer
Radiative Layer
(Dikpati, Gilman and Rempel, 2003)
Peter Gilman
46
November 2004
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)
Peter Gilman
47
November 2004
Growth Rates For 3D Global MHD Instability
No Boundary Conditions
Top and Bottom
Pressure = 0 Top
Vertical Velocity = 0 Bottom
0.1 yr
Vertical Velocity = 0
Top and Bottom
1 yr
n=
0.1 yr
0.1 yr
1 yr
Peter Gilman
1 yr
48
November 2004
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 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
Peter Gilman
49
November 2004
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.
Peter Gilman
50
November 2004
Jet balancing magnetic curvature stress
If 2nd term is not too big, then
 j : jet-like toroidal flow
c : core rotation rate
 s : solar-like differential rotation
0

1   
2 1/ 2
: jet parameter
: toroidal field
Peter Gilman
ε=0
ε=1
51
no jet
full jet
November 2004
Jet amplitudes for various toroidal field bands and
their latitude locations
Peter Gilman
52
November 2004
2D MHD Instability: 2nd Order Equations
for Reference State Changes
For differential rotation (linear measure):
u
1 
2

cos
 (a ' b '  u ' v ')
2
 t cos   

Maxwell
Stress

Reynolds
Stress
For toroidal magnetic field (linear measure):
a 

(u ' b '  ' a ')
 t 
“Mixed”
Stress
Peter Gilman
53
November 2004
Jet amplitudes from nonlinear hydrodynamic calculations
Dikpati 2004 (in preparation)
Peter Gilman
54
November 2004
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)
Peter Gilman
55
November 2004
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 10  width < 10  latitude
 band of 5 width < 30 latitude
 band of 2 width < 50 latitude
Peter Gilman
56
November 2004