8. (W is scalar for displacement, T is scalar for traction)
Note that matrix does not depend on m

9. Algorithm for toroidal modes
Choose harmonic degree and frequency
Compute starting solution for (W,T)
Integrate equations to top of solid region
Is T(surface)=0? No: go change frequency and start again. Yes: we have a mode solution

10. T(surface) for harmonic degree 1

16. Radial and Spheroidal modes

25. Spheroidal modes

28. Minors
To simplify matters, we will consider the spheroidal mode equations in the Cowling approximation where we include all buoyancy terms but ignore perturbations to the gravitational potential

35. Spheroidal modes w/ self grav
(three times slower than for Cowling approx)

37. Red > 1%; green .1--1%; blue .01--.1%

38. Red>5; green 1--5; blue .1--1 microHz

39. Mode energy densities

42. Normalized radius
Dash=shear, solid=compressional energy density

43. (black dots are observed modes)

44. All modes for l=1

45. (normal normal modes)

46. hard to compute
ScS --not observed
(not-so-normal normal modes)