1. Global Rotation
Rachel Howe, NSO

2. How deep is the tachocline, how wide is the shear?
With almost a whole cycle of observations from GONG and MDI, we can look at the stationary part of the rotation.
Forward-modeling exercises help to establish robustness of results.

3. The Questions How well can we measure the near-surface shear?
Is the slant in the convection-zone rotation real?
How well can we measure the shape of the tachocline?

4. The Data
106 overlapping 108-day sets from GONG, covering 10.5 years
0≤l≤150, coefficients up to a16
49 72-day sets from MDI, starting May 1996.
0≤l≤300, coefficients up to a36

5. Analysis Fit 11yr sinusoid to each coefficient
Use stationary part of fit
2dRLS and 2dOLA inversions
Convolve averaging kernels with test profiles to simulate inversions.

6. Analysis – the inversion problem
Kernel
Averaging Kernel
Aim of inversion is to go from a set of coefficients, each of which samples the rotation rate in a slightly different region, to a linear combination of those coefficients that gives a localized estimate of the rotation rate at a given location.
The averaging kernel is a linear combination of the data kernels that describes the way the region over which the rotation estimate is averaged.

7. Coefficients

8. Results: profiles
GONG
MDI
7,3 RLS
6,2 RLS
OLA

9. Results: profiles
GONG
MDI
6,2 RLS
7,3 RLS
OLA

10. Global Rotation Profile
Dashed lines inclined at 25 degrees to rotation axis.

11. Slope of Isorotation Contours
Dashed line shows slope of radial lines.
Gilman & Howe, 2003 (BB proceedings).

12. Results: Synthetic Profiles
GONG
MDI
7,3 RLS
6,2 RLS
OLA

13. Results: Synthetic Profiles
GONG
MDI
7,3 RLS
6,2 RLS
OLA

14. Results: Synthetic Profiles
GONG
MDI
7,3 RLS
6,2 RLS
OLA

15. Results: Synthetic Profiles
GONG
MDI
7,3 RLS
6,2 RLS
OLA

16. Discussion Slope in convection zone looks robust
Near-surface shear already resolved in single-set data
But reversal at high latitudes may be MDI(CA) artefact.
Some sensitivity to different tachcocline thickness.

17. Calibrating the Tachocline
Charbonneau et al (1999) used fits with error function model convolved with 1d averaging kernel
Tried this using OLA inversions, checking against synthetic profiles.

18. Fit Results (MDI OLA)

19. But …
We know rotation isn’t quite depth-independent in convection zone.
What does technique do with tilted-contour model and no thickness?

20. Aha! Fits to tilted-contour model Fits to real data
Fits to dashed-contour model

21. So …
Try fitting model with slope in convection zone taken into account, combined with error-function tachocline.

22. And we get this …

23. From which we conclude Deconvolution fitting does not work perfectly.
Forward modeling is only useful if the model is appropriate!
The tachocline could be quite thin, (0.03±0.04)Rsun once the slanted rotation contours are taken into account.
However, this still needs more study.