Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

By Michel Rieutord, B. Georgeot, L. Valdettaro. Journal of Fluid Mechanics, 2000

168 citations recorded by the source corpus.

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.

Subfields: Molecular Biology, Oceanography, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/63510560f62426ecb692155e81a0f73754d47d3d

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Citations in the corpus

Resonance locking as the source of rapid tidal migration in the Jupiter and Saturn moon systems (2016, background): “…mode frequencies, or they could correspond to frequencies at which inertial waves are focused onto attractors to create enhanced tidal dissipation (Rieutord et al. 2001; Ogilvie & Lin 2004; Papaloizou & Ivanov 2005; Ogilvie & Lin 2007; Ivanov & Papaloizou 2007; Goodman & Lackner 2009; Rieutord &…”
Tidal Dissipation in Stars and Giant Planets (2014, background): “In a spherical shell, however, for most frequencies the rays are focused towards a small number of limit cycles known as wave attractors (Maas & Lam 1995; Rieutord et al. 2001), which exist in certain bands of frequency, depending on the radius ratio of the shell.”
INTERNAL GRAVITY WAVES: From Instabilities to Turbulence (2002, unknown): “They tend to an inviscid fractal solution in the limit of small viscosity, as calculated by Rieutord et al. (2001).”
Tidal Dissipation in Rotating Giant Planets (2003, methodology): “Principal Method Our principal method of solution makes use of a Chebyshev pseudospectral approach (e.g., Boyd 2001) as used by, e.g., Rieutord et al. (2001) in their study of free inertial waves in an incompressible fluid contained in a spherical annulus.”
Geophysical and astrophysical fluid dynamics beyond the traditional approximation (2008, background): “But, because of their fine structure, most of the physically relevant dissipation was found to take place near the singularities [Rieutord et al., 2001].”