The Sun’s internal rotation

Michael Thompson

University of Sheffield

michael.thompson@sheffield.ac.uk

Global-mode seismology

Measure mode properties ω; A, Γ; line-shapesEigenfunctions / spherical harmonics

Frequencies ωnlm(t) depend on conditions in solar interior determining wave propagation

ωnlm – degeneracy lifted by rotation and by structural asphericities and magnetic fields

Inversion provides maps such as of c and ρ and rotation and wave-speed asphericities

North-south averagesSnapshots at successive times: typically 2-month averages

Spherical harmonics

Data from space – in particular from three instruments on the SOHO sateliteSOI-MDIGOLFVIRGO

Data from ground-basednetworks, in particularGONG, BiSON, LOWL GONG network sites

Radial Stratification

Sound speed profile

Temperature

/12 Tc

(Sound speed is in units of hundreds of km/s)

Helioseismic determination of location of base of convection zone at 0.713 +/- 0.003 R (Christensen-Dalsgaard et al. 1991). No temporal or latitudinal variation in this location (Basu & Antia 2001).

Note that this measures the extent of the essentially adiabaticallystratified region. Sun’s transition smoother than in models.

Christensen-Dalsgaard (2004)

Sun minus model

New model(Z=0.015)

Old model(Z=0.020)

Old ‘standard’ Model S (Christensen-Dalsgaard et al. 1996)left little room for overshoot – near-perfect match to depth ofSun’s convection zone. But newly determined abundancesgive a model with 1-2% shallower convection zone. (See Basu & Antia 2004; Bahcall et al. 2004 submitted; C-D 2004)

Mean internal rotation

Rotation rate inferred from MDI data using two different analysis techniques

Radial cuts through inferred rotation profile of the solar interior (at latitudes indicated)

Dashed lines are at 25 degrees to polar axis – a better match to the isorotation contours than radial lines (or Taylor columns!)

Rotation rate in the deeper interior

(LOWL + BiSON data)Fast core strongly ruled out (unless very small). Possibly slightly slow core rotation. But inversionsconsistent with uniform rotation in deep interior.

The Tachocline

Artificial data Solar data

Charbonneau et al. (1999)

Kosovichev (1996) – fit to early data, characterizing the tachocline transition with the function

0.5{ 1 + erf [2 ( r - r0 ) / w ] }

Obtained width w = (0.09+/-0.05)R and central locationr0 = (0.692+/-0.005)R, i.e. just beneath base of convection zone.

More recent analyses have typically yielded substantially smaller width, e.g. Charbonneau et al. (1999):width w = (0.039+/-0.013)RPosition r0 = (0.693+/-0.002)R at equator.Prolate with location different by (0.024+/-0.004)R at 60 degrees.

Extent of prolateness confirmed by Basu & Antia (2001), whofurther found no temporal variation in the tachocline properties.

Near-surface shear layer

Near-surface rotation rate from inversion of f-mode splittings

Mean shear gradient as function of latitude

Logarithmic derivative

Logarithmic derivative

Gradient using only l<250 f modes

Conclusion: Rotational gradient negative (logarithmic derivative approx. -1) at low-/mid-latitudes. Vanishes just above 50 degrees latitude.Small or positive gradient in shallow layer at high latitude.

High-latitude jet (Schou et al. 1998)Evidence weak – could be real buttime-dependent.

Jet at 75 degreesapparent in RLSinversion of first144-days of MDI data.

Mean Angular Velocity in 3-D ModelsMean Angular Velocity in 3-D Models

REASONABLEREASONABLECONTACT WITHCONTACT WITHHELIOSEISMICHELIOSEISMICDEDUCTIONSDEDUCTIONS

Case ABCase ABBrun & ToomreBrun & Toomre

Courtesy J. Toomre

Angular Momentum BalanceAngular Momentum Balance

REYNOLDS STRESSESREYNOLDS STRESSES (R) transport angular momentum TOWARD equator, unlike (R) transport angular momentum TOWARD equator, unlike MERIDIONAL CIRCULATIONMERIDIONAL CIRCULATION (MC) SLOW POLES from weak MC at high latitudes (MC) SLOW POLES from weak MC at high latitudes

RR

MCMC

RR

MCMCVV

VV

totaltotal totaltotalMERIDIONALMERIDIONALCIRCULATIONCIRCULATION

Poleward flowPoleward flowat equatorat equator

Courtesy J. Toomre

Temporal variations of the Sun’s internal rotation

1986 2000

Torsional oscillations (Howard & LaBonte 1980)

Evolution of surface velocity field (upper panel) and surface magnetic field (lower panel)

Ulrich 2001

Torsional oscillations of whole convection zone

Differencing rotation inversions relative to solar minimum (1996) at successive 72-day epochs reveals the torsional oscillations at low and high latitudes through much of theconvection zone.

It is also revealing to stack fewer results, so the evolution of the whole convection zone can be seen. Here we difference rotation inversions relative to solar minimum at successive 1-year epochs.

Variation of rotation at fixed depth (3 different inversions)

GONGRLS

MDIRLS

MDIOLA

0.99 R 0.95 R 0.90 R 0.84 R

0.99 R

0.95 R

0.90 R

0.84 R

Equator 30o 60o 75o

Systematic variations – consistent between analyses

Howeet al.(2004)

Variation of rotation at fixed depth (3 different inversions)

30o 45o15oEquator

GONG

MDI

MDI

0.8 R

0.9 R

Flows inferred with ring-diagram analysis (10 Mm deep)

Haber et al.

1999

1997

Longitudinal averages of flows from ring-diagram analysis

Zonal flowsat 1 Mm and 7 Mm depth(note torsional oscillation)

Meridional flowsMostly poleward but with transientcounter-cell in N hemisphere

Comparison of near-surface torsional oscillation flow from global analysis (contours) and ring analysis(coloured shading) – symmetrized.

SSW and Active Complex 9393

7 Mm7 Mm

16 Mm16 Mm

Flows near and beneath active region Apr 2001

Phase of variations in the CZ

(A) Amplitude and ( B) phase of 11-year harmonic fit

(C) 22-yr extrapolation as function of latitude; with residuals also shown

(D) 22-yr extrapolation as function of depth; with residuals also shown

Comparison of phase of variations with isorotation contours(nb different colour scheme)

Constant-phase surfaces tend to align with contours of constant rotation.

Summary: Variability in the convection zone

Komm et al. 2003

Angular momentum variations

Aug 2002May 1996

Evolving of zonal flows

Vorontsov et al. 2002

Torsional oscillations – what is relation to magnetic field?

Whole convection zone involvedStrong evolution at high latitude

Physical understanding of the phase propagation of perturbations through the convection zone?

Role of polar open field-lines?

Variability in and near the tachocline

Rotation residuals show a quasi-periodic 1.3-yr variation at low latitudes (Howe et al. 2000).

Boberg et al. 2002

Solar mean magnetic field

1975 2000

Wavelet analysis of the Sun’s mean photospheric magnetic field: prominent periods are the rotation period and its 2nd harmonic, and the 1.3/1.4-yr period

0 11 22

Toroidal magnetic field strength Angular velocity variationsTime (yrs)

Lat

itud

e

Equator

Time (yrs)

0 11 22 D

ep

th

Surface

CZ base

The evolution of the angular velocity and the magnetic field arecoupled.

Results of Covas et al. (2000, 2001) model (spatio-temporal fragmentation):

Summary: variability in and near tachocline

Howe et al. 2000

1996 2002

Variations in Ω ( r , θ ; t )1.3-yr variations in inferred rotation rate at low latitudes above and beneath tachocline

Signature of dynamo field evolution?Radiative interior also involved in solar cycle?

Link between tachocline and 1.3/1.4-yr variations in • solar wind, • aurorae,• solar mean magnetic field ?

Some questions

• What further constraints on tachocline / c.z. base would be useful?

• Is comparison of variability in c.z. between Sun and numerical models worthwhile?

• What determines the phase propagation of variations in the c.z.?

• What is the nature of the torsional oscillation? What causes the high-latitude variability?