### 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. [75] for a discussion of the single fluid problem
using notation which closely resembles the one we adopt here)
If we also assume axial-led modes, like the r-modes, then we have and the continuity equation
leads to
Under these assumptions we find that becomes (to order )

We can rewrite the last term using the equation governing the axisymmetric equilibrium. Keeping only
terms of order we have
A bit more work then leads to
and
which means that the canonical energy can be written in the form
for an axial-led mode.
Introducing the axial stream function we have

where are the standard spherical harmonics. This leads to
and
After performing the angular integrals, we find that

Combining this with the r-mode frequency [75]
we see that for all r-modes, i.e. they are all unstable. The r-mode is a special
case, leading to .