Go to previous page Go up Go to next page

13.3 The r-mode instability

In order to further demonstrate the usefulness of the canonical energy, let us prove the instability of the inertial r-modes. For a general inertial mode we have (cf. [75Jump To The Next Citation Point] for a discussion of the single fluid problem using notation which closely resembles the one we adopt here)
i i ˙i 2 v ∼ δv ∼ ξ ∼ Ω and δΦ ∼ δn ∼ Ω . (257 )
If we also assume axial-led modes, like the r-modes, then we have δvr ∼ Ω2 and the continuity equation leads to
∇i δvi ∼ Ω3 → ∇iξi ∼ Ω2. (258 )

Under these assumptions we find that Ec becomes (to order Ω2)

1 ∫ [ 2 || i ||2 i∗ j ] Ec ≈ -- ρ |∂tξ| − |v ∇i ξ| + ξ ξ ∇i∇j (Φ + ˜μ) dV. (259 ) 2
We can rewrite the last term using the equation governing the axisymmetric equilibrium. Keeping only terms of order Ω2 we have
1 ξi∗ξj∇i ∇j (Φ + ˜μ ) ≈ -Ω2 ξi∗ξj∇i ∇j(r2 sin2 θ). (260 ) 2
A bit more work then leads to
1 [ | |2 ] --Ω2ξi∗ξj∇i∇j (r2sin2θ) = Ω2r2 cos2θ ||ξ θ|| + sin2 θ|ξϕ|2 (261 ) 2
and
{ [ ]} || i ||2 2 2 2 2 [ θ ϕ∗ ϕ θ∗] 2 2 || θ||2 2 ϕ 2 |v ∇iξj| = Ω m |ξ| − 2imr sinθ cosθ ξ ξ − ξ ξ + r cos θ|ξ | + sin θ |ξ | , (262 )
which means that the canonical energy can be written in the form
∫ { [ ]} Ec ≈ − 1- ρ (m Ω − ω )(m Ω + ω)|ξ|2 − 2im Ω2r2 sin θcos θ ξθξϕ∗ − ξϕξθ∗ dV (263 ) 2
for an axial-led mode.

Introducing the axial stream function U we have

iU ξθ = − -------∂ϕYlmeiωt, (264 ) r2sinθ ϕ --iU--- m iωt ξ = r2sin θ∂θYl e , (265 )
where m m Yl = Yl (θ,ϕ ) are the standard spherical harmonics. This leads to
2 |U|2[ 1 m 2 m 2] |ξ| = --2- --2--|∂ϕYl | + |∂θY l | (266 ) r sin θ
and
[ ] 1 cosθ ir2sinθ cosθ ξθξϕ∗ − ξϕ ξθ∗ = -2-----m |U |2[Yml ∂θYlm∗ + Ylm∗∂θYlm]. (267 ) r sinθ

After performing the angular integrals, we find that

{ } l(l +-1) 2m2-Ω2-- ∫ 2 Ec = − 2 (m Ω − ω)(m Ω + ω ) − l(l + 1) ρ|U | dr. (268 )
Combining this with the r-mode frequency [75]
[ ] 2 ω = m Ω 1 − -------- (269 ) l(l + 1)
we see that Ec < 0 for all l > 1 r-modes, i.e. they are all unstable. The l = m = 1 r-mode is a special case, leading to E = 0 c.
  Go to previous page Go up Go to next page