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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): - PowerPoint PPT Presentation
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High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)
The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Researchunder sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.
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
3Peter Gilman November Peter Gilman November 20042004
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?
4Peter Gilman November Peter Gilman November 20042004
2 40 0 2 4sin sins r s r s r
Rotation Detail within Solar TachoclineRotation Detail within Solar Tachocline
5Peter Gilman November Peter Gilman November 20042004
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
6Peter Gilman November Peter Gilman November 20042004
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
7Peter Gilman November Peter Gilman November 20042004
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
8Peter Gilman November Peter Gilman November 20042004
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
9Peter Gilman November Peter Gilman November 20042004
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”
10Peter Gilman November Peter Gilman November 20042004
Barotropic Instability of Solar Differential Rotation Barotropic Instability of Solar Differential Rotation Measured by Helioseismic DataMeasured by Helioseismic Data
(Charbonneau, Dikpati and Gilman, 1999)
11Peter Gilman November Peter Gilman November 20042004
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)
12Peter Gilman November Peter Gilman November 20042004
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
13Peter Gilman November Peter Gilman November 20042004
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
14Peter Gilman November Peter Gilman November 20042004
Mechanisms of Global MHD Instability for Mechanisms of Global MHD Instability for Weak Toroidal Fields (TF)Weak Toroidal Fields (TF)
15Peter Gilman November Peter Gilman November 20042004
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
16Peter Gilman November Peter Gilman November 20042004
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
17Peter Gilman November Peter Gilman November 20042004
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
18Peter Gilman November Peter Gilman November 20042004
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°
19Peter Gilman November Peter Gilman November 20042004
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
20Peter Gilman November Peter Gilman November 20042004
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)
21Peter Gilman November Peter Gilman November 20042004
Example of “Clamshell” Instability in Nonlinear 2D Example of “Clamshell” Instability in Nonlinear 2D MHD SystemMHD System
(Cally, Dikpati and Gilman, 2003)
22Peter Gilman November Peter Gilman November 20042004
Nonlinear Survey of Symmetric Nonlinear Survey of Symmetric Tipping Mode in Strong BandsTipping Mode in Strong Bands
(Cally, Dikpati and Gilman 2003)
23Peter Gilman November Peter Gilman November 20042004
Linear and Nonlinear Tip AnglesLinear and Nonlinear Tip Angles
(Cally, Dikpati and Gilman, 2003)
24Peter Gilman November Peter Gilman November 20042004
Nonlinear Tipping of Toroidal Fields Nonlinear Tipping of Toroidal Fields in Tachoclinein Tachocline
(Cally, Dikpati and Gilman, 2003)
Peak Toroidal Field 25 kG Peak Toroidal Field 100 kG
25Peter Gilman November Peter Gilman November 20042004
Global MHD Instability with Kinetic (Global MHD Instability with Kinetic (ddkk) and) andMagnetic (Magnetic (ddmm) Drag) Drag
o a
(Dikpati, Cally and Gilman, 2004)
Broad TF Banded TF
26Peter Gilman November Peter Gilman November 20042004
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°
(Dikpati, Cally and Gilman, 2004)
27Peter Gilman November Peter Gilman November 20042004
Observation Evidence of Tipped Observation Evidence of Tipped Toroidal Ring?Toroidal Ring?
28Peter Gilman November Peter Gilman November 20042004
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
(Cally, Dikpati and Gilman, 2003)
29Peter Gilman November Peter Gilman November 20042004
““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)
30Peter Gilman November Peter Gilman November 20042004
Schematic of Tipped Toroidal Ring in “Sparking Schematic of Tipped Toroidal Ring in “Sparking Snake” ModelSnake” Model
31Peter Gilman November Peter Gilman November 20042004
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
32Peter Gilman November Peter Gilman November 20042004
Histogram of Sunspot Pair AnglesHistogram of Sunspot Pair Angles
33Peter Gilman November Peter Gilman November 20042004
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
34Peter Gilman November Peter Gilman November 20042004
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)
35Peter Gilman November Peter Gilman November 20042004
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)
36Peter Gilman November Peter Gilman November 20042004
Relationship among Effective Gravity G Relationship among Effective Gravity G Subadiabatic Stratification and Subadiabatic Stratification and
Undisturbed Shell Thickness HUndisturbed Shell Thickness Had
(Dikpati, Gilman and Rempel, 2003)
37Peter Gilman November Peter Gilman November 20042004
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
38Peter Gilman November Peter Gilman November 20042004
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
39Peter Gilman November Peter Gilman November 20042004
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
40Peter Gilman November Peter Gilman November 20042004
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:
41Peter Gilman November Peter Gilman November 20042004
MHD Shallow Water EquilibriumMHD Shallow Water Equilibriumfor Banded Toroidal Fieldsfor Banded Toroidal Fields
(Dikpati, Gilman and Rempel, 2003)
Overshoot Layer (G=0.1)
42Peter Gilman November Peter Gilman November 20042004
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
43Peter Gilman November Peter Gilman November 20042004
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
44Peter Gilman November Peter Gilman November 20042004
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
45Peter Gilman November Peter Gilman November 20042004
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
46Peter Gilman November Peter Gilman November 20042004
Domains of Unstable Toroidal Field BandsDomains of Unstable Toroidal Field BandsOvershoot Layer
(Dikpati, Gilman and Rempel, 2003)
Radiative Layer
47Peter Gilman November Peter Gilman November 20042004
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)
48Peter Gilman November Peter Gilman November 20042004
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
49Peter Gilman November Peter Gilman November 20042004
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
50Peter Gilman November Peter Gilman November 20042004
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.
51Peter Gilman November Peter Gilman November 20042004
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
52Peter Gilman November Peter Gilman November 20042004
Jet amplitudes for various toroidal field bands and their
latitude locations
53Peter Gilman November Peter Gilman November 20042004
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
54Peter Gilman November Peter Gilman November 20042004
Jet amplitudes from nonlinear hydrodynamic calculations
Dikpati 2004 (in preparation)
55Peter Gilman November Peter Gilman November 20042004
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
56Peter Gilman November Peter Gilman November 20042004
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
band of width < latitude
band of width < latitude
10 10
5 30
502
Narrow bands and low band latitudes