Q2. How many times can the L2 operator be used to reduce the order of the!-?
the order of the !-derivatives can be reduced down to first order by using Eq. 10 as many times as needed, the L2 operator introduced being again well adapted to the spherical harmonic expansion.
Q3. What is the effect of the singularity in spheroidal geometry?
In the spheroidal geometry, the singularity is reported to the focus circle, but the effects of this singular circle seem to be rather limited in the present linear approach where only m=1 terms are included.
Q4. What is the eigen mode of rotation for an inviscid flow?
In spheroidal symmetry, expanding the scalar fields of any vector field u in spherical harmonics and integrating their angular parts leads to the existence of only two contributions, which can be written, in the cartesian basis, as:!" pv1 = #6 $% st1(µ)( ) ˆ ex + & st1(µ)( ) ˆ ey[ ]0µc ' sinh(2µ)dµ" pv2 = 2 6#5 & sp2(µ)( ) ˆ ex + % sp2(µ)( ) ˆ ey[ ]0µc ' dµ()* *+* *(24)It turns out that this pseudo-vorticity 'pv, apart a constant prefactor, is nothing else but the projection of the vector field u on the two orthogonal spinover modes!u 211and!u 21 1whichcorrespond to the inertial vortical flows around!ˆ e x and!ˆ e y , respectively:!"pv = #coth(2µc ) (u211 * $u)spheroid% ˆ ex + (u21 1 * $u)spheroid% ˆ ey &' ( () * + +(25)The evolution of the system will then be followed by investigating the time-variation of this vector 'pv as well as its dependence on viscosity and eccentricity.
Q5. What is the spherical Mie representation of a vector field?
Within this system, an extension- 3 -of the spherical Mie representation of a vector field [19] is first considered: a solenoidal vector field (here the velocity field) is decomposed into the sum of two unique vectors which are themselves derived from two scalar fields, namely the sphero-poloidal and the spherotoroidal ones.
Q6. What is the oblate spheroidal coordinate system?
In the present work, equation of motion of a rotating, viscous, incompressible fluidwithin a spheroidal container is directly treated in real space by using an appropriate set of coordinates, namely the oblate spheroidal coordinate system.
Q7. What is the eigenfrequency of the spinover mode?
2: Calculated variation of the decay factor , (left scale) and eigenfrequency 'd (right scale) of the spinover mode for Ekman numbers Ek ranging from 10 -9 to 10-4 and eccentricities e = 0.02 (left hand side) and 0.266 (right hand side); lines are least square fits; horizontal dotted lines are the theoretical limits for Ek = 0.
Q8. what is the radial structure of the viscous boundary layer?
4: Radial structure of the viscous boundary layer, seen through the scaling of the first four sphero-toroidal radial components for e = 0.266; successive curves correspond to Ek = 7. 10 -5 to 7. 10-9 (factor 10 between two successive values); the scaled radial coordinate is (µc - µ).(Ek) -1/2 (µc = 2 for e = 0.266).- 26 -REFERENCES[1]
Q9. What was the spheroidal geometry of the rigid boundary of the Earth?
The spheroidal geometry of the rigid boundary of the latter was explicitely considered, giving rise to specific effects on the Earth's nutation compared to the case of a spherical container, but viscosity was neglected.
Q10. Why is the inner core not considered in this study?
It is worth noting that the inner core, which is often included in spherical symmetry in order to avoid a singularity at the origin, is not considered here.
Q11. what is the eigen mode of rotation for an inviscid flow?
It is the eigen mode of rotation for an inviscid flow, which is excited when a fluid is in rapid rotation around the z-axis and its rotation axis is suddenly tilted by a slight quantity [4].
Q12. What is the spherical symmetry of the vector field?
As for the spherical symmetry, this Mie-like representation allows one to work with only two scalar fields instead of the three velocity components, once the pressure has been eliminated from the equations.
Q13. What is the spherical harmonics expansion for stlm?
For the azimutal part, only m = 1 terms are needed here (see Sect. 4), since the (m = -1) terms can be deduced through the general relations:!st l"m (µ ) = (-1) m stlm (µ)( )*sp l"m (µ ) = (-1) m splm (µ)( )*# $ % & %(18)The corresponding matrices consist in a penta-diagonal matrix associated with theradial grid, the elements of which are band-matrices of order lmax, arising from the coupling- 12 -of scalar field components having different l values.
Q14. What is the simplest way to extend the spherical coordinate system?
At last, realistic geophysical applications would require to include an inner boundary with a possibly- 17 -different eccentricity, which would demand to extend the present coordinate system but would simplify the inner boundary condition.
Q15. What is the way to study rotating bodies?
Using a more realistic coordinate system appears however highly desirable, because natural bodies or cavities are often ellipsoidal rather than spherical.