Global MHD Instabilities of the Solar Tachocline

1. 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

2. 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

3. 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 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?

4. Rotation Detail within Solar Tachocline

5. 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:

6. 2D MHD Instability: Reduction to Solvable System (Legendre Operator) Induction Equation: Classical Hydrodynamic Stability Problem Vorticity Equation: In which  = sin  and “Boundary” conditions:, χ= 0 at poles

7. 2D MHD Instability: 2nd Order Equations for Reference State Changes For differential rotation (linear measure): MaxwellStress ReynoldsStress For toroidal magnetic field (linear measure): “Mixed”Stress

8. 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:

9. 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”

10. Barotropic Instability of Solar Differential Rotation Measured by Helioseismic Data (Charbonneau, Dikpati and Gilman, 1999)

11. Properties of 2D MHD Instability of Differential Rotation and Toroidal Magnetic Field ToroidalMagnetic Field DifferentialRotation Magnetic flux transport away from the peak toroidal field by the Mixed Stress (phase difference in longitude between perturbation velocities & magnetic fields) Angular momentum transport toward the poles primarily by the Maxwell Stress (perturbations field lines tilt upstream away from equator)

12. Broad Toroidal Field Profiles Tested for Global MHD Instability of Field and Differential Rotation P E SP NP

13. Gaussian Type Banded Toroidal Field Profiles Tested for Global MHD Instability of Field and Differential Rotation SP E NP

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

15. Toroidal Ring Disturbance Patterns of Longitudinal Wave Numbers m=0, 1, 2 • Toroidal ring tips but remains same circumference; creates Maxwell stress • Fluid in ring keeps same speed but flow tips • Toroidal ring shrinks • Fluid in ring spins up m = 2 • Toroidal ring deforms, creating Maxwell Stress • Fluid flow inside ring deforms but does not spin up m = 0 m = 1

16. Summary of Properties of 2D Instability of Differential Rotation and Toroidal Field

17. Critical or Singular Points in the Equations for 2D MHD Stability and 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 Y=S1/2 H, then : k2 real if ci =0; complex if not k2 is large in the neighborhood of singular points defined above

18. 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°

19. Dominant Energy Flow in Unstable Solutions

20. Energy Flow Diagram for Nonlinear 2D MHD System with Forcing and Drag (Dikpati, Cally and Gilman, 2004)

21. Example of “Clamshell” Instability in Nonlinear 2D MHD System (Cally, Dikpati and Gilman, 2003)

22. Nonlinear Survey of Symmetric Tipping Mode in Strong Bands (Cally, Dikpati and Gilman 2003)

23. Linear and Nonlinear Tip Angles (Cally, Dikpati and Gilman, 2003)

24. Nonlinear Tipping of Toroidal Fields in Tachocline Peak Toroidal Field 25 kG Peak Toroidal Field 100 kG (Cally, Dikpati and Gilman, 2003)

25. Global MHD Instability with Kinetic (dk) andMagnetic (dm) Drag Banded TF Broad TF (Dikpati, Cally and Gilman, 2004)

26. 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)

27. Observation Evidence of Tipped Toroidal Ring?

28. Tipped Toroidal Ring in Longitude-latitude Coordinates Linear Solutions with Two Possible Symmetries (Cally, Dikpati and Gilman, 2003)

29. “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)

30. Schematic of Tipped Toroidal Ring in “Sparking Snake” Model

31. 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)

32. Histogram of Sunspot Pair Angles

33. Global Instabilities of Solar Tachocline Dynamo Potential Assume Differential Rotation from Helioseismology

34. 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)

35. 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 ωc rotation of solar interior G ~ 10-1 for Overshoot Tachocline G ~ 102 for Radiative Tachocline (Dikpati, Gilman and Rempel, 2003)

36. Relationship among Effective Gravity G Subadiabatic Stratification and Undisturbed Shell Thickness H (Dikpati, Gilman and Rempel, 2003)

37. Shallow Water Equations of Motion and Mass Continuity

38. Shallow Water Induction and Flux Continuity Equations

39. Singular Points For cases of solar interest: Sr , Sm = 0 are important, Sg = 0 is not Occur at latitudes where: hσ 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

40. 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

41. MHD Shallow Water Equilibriumfor Banded Toroidal Fields Overshoot Layer (G=0.1) (Dikpati, Gilman and Rempel, 2003)

42. 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

43. Stability Diagrams for HD Shallow Water System G Differential Rotation G r/Ro (Dikpati and Gilman, 2001)

44. Growth Rates for Unstable ModesFor Broad Toroidal Field a = 1.0 s4 / s0 = 0 m = 1, S m = 1, A a= 0.5 s4 / s0 = 0 m = 1, S m = 1, A a = 0.1 s4 / s0 = 0 m = 1, S m = 1, A a= 0.2 s4 / s0 = 0 m = 1, S m = 1, A (Gilman and Dikpati, 2002)

45. Growth Rates of Unstable Modes for Broad Toroidal Fields Overshoot Layer Radiative Layer a a (Gilman and Dikpati, 2002)

46. Domains of Unstable Toroidal Field Bands Overshoot Layer Radiative Layer (Dikpati, Gilman and Rempel, 2003)

47. 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)

48. 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

49. 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

50. 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.