We investigate the asymptotic properties of inertial modes confined in a spherical
shell when viscosity tends to zero. We first consider the mapping made by the
characteristics of the hyperbolic equation (Poincaré's equation) satisfied by inviscid
solutions. Characteristics are straight lines in a meridional section of the shell, and the
mapping shows that, generically, these lines converge towards a periodic orbit which
acts like an attractor (the associated Lyapunov exponent is always negative or zero).
We show that these attractors exist in bands of frequencies the size of which decreases
with the number of reflection points of the attractor. At the bounding frequencies the
associated Lyapunov exponent is generically either zero or minus infinity. We further
show that for a given frequency the number of coexisting attractors is finite.We then examine the relation between this characteristic path and eigensolutions of
the inviscid problem and show that in a purely two-dimensional problem, convergence
towards an attractor means that the associated velocity field is not square-integrable.
We give arguments which generalize this result to three dimensions. Then, using
a sphere immersed in a fluid filling the whole space, we study the critical latitude
singularity and show that the velocity field diverges as 1/√d, d being the distance to
the characteristic grazing the inner sphere.We then consider the viscous problem and show how viscosity transforms singularities
into internal shear layers which in general reveal an attractor expected at the
eigenfrequency of the mode. Investigating the structure of these shear layers, we find
that they are nested layers, the thinnest and most internal layer scaling with E1/3, E
being the Ekman number; for this latter layer, we give its analytical form and show
its similarity to vertical 1/3-shear layers of steady flows. Using an inertial wave packet
travelling around an attractor, we give a lower bound on the thickness of shear layers
and show how eigenfrequencies can be computed in principle. Finally, we show that
as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense
in [0, 2Ω], contrary to the case of the full sphere (Ω is the angular velocity of the
system).Hence, our geometrical approach opens the possibility of describing the eigenmodes
and eigenvalues for astrophysical/geophysical Ekman numbers
(10−10–10−20), which are out of reach numerically,
and this for a wide class of containers.
Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum
