Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids

Abstract

Rotating fluids support waves. These inertial waves propagate obliquely through the fluid, with an angle that is fixed with respect to the rotation axis. Upon reflection, their wavelength is unchanged only when the wall obeys the local reflectional symmetry, that is, when it is either parallel or perpendicular to the rotation axis. For internal gravity waves in a density-stratified fluid, sloping boundaries thus break the symmetry of ray paths, in a two-dimensional container, predicting their focusing upon attractors: particular paths onto which the wave rays, and hence the energy, converge, and to which the wave energy returns after a small number of refections. Laboratory observations, presented here, show that, despite the intrinsic three-dimensionality of inertial waves, attractors still occur. The intensified wave energy on the attractor encourages centrifugal instabilities, leading to a mean flow. Evidence of this comes from dye spreading, observed to develop most rapidly over the location where the attractor reflects from the sloping wall, being the place where focusing and instabilities occur. This mean flow, resulting from the mixing of angular momentum, accompanying the intensification of the wave field at that location, has geophysical implications, because the ocean, atmosphere and Earth's liquid outer core can be regarded as asymmetrically contained. The relevance of wave focusing in a rotating, spherical shell, the modifications due to the addition of radial stratification, and its implications for observed equatorial current patterns and inertial oscillations are discussed. The well-known universality of oceanic, gravito-inertial wave spectra might reflect complementary, divergent (chaotic) wave-ray behaviour, which occurs in containers obeying the reflectional symmetry, but in which symmetry is broken in the horizontal plane. Periodic orbits still exist, but now repell.

Rapidly rotating cylinder flow with an oscillating sidewall

“…Therefore, much can be predicted by ray tracing the characteristics from the origin of the inertial waves and successively reflecting the rays on the container walls. Phenomena such as wave focusing and wave attractors have been analyzed using ray tracing [51][52][53][54]. In our problem, the wave beams originate at the corners where the end walls and the sidewall meet.…”

Mean flow generation by Görtler vortices in a rotating annulus with librating side walls

“…Bretherton and Turner (1968) proposed the same argument and explained how the presence of a preferred direction, arising from the Coriolis force, induces an anisotropic mixing, resulting in a mean radial flux of angular momentum which homogenizes fluid's angular momentum (see also Manton, 1973). Thompson (1979) argued that when inertial wave amplitude is large, centrifugal instability gives rise to a strong local mixing which produces a stable vortex spreading along the axis of rotation (see also Maas, 2001). More recently, Kloosterziel et al (2007) investigated development of inertial instability in initially barotropic vortices in a uniformly rotating and stratified fluid.…”

Discrete and continuous hamiltonian systems for wave modelling

“…Later, Maas [62] presented a semianalytical structural solution in a rectangular parallelepiped with straight walls. Due to their symmetrical shape all three containers do not have any net focussing which inertial waves are otherwise prone to develop [61]. The axial spheroid has a symmetric structure, and thus compensates every reflected focussing wave with a reflected defocussing wave.…”

“…In the case of an axial cylinder or a rectangular parallelepiped, the walls are either parallel or perpendicular to the rotation axis, therefore such walls possess a local reflectional symmetry. A simple tilt of one of the walls immediately results in symmetry breaking and hence in wave focussing and defocussing, such that due to dominance of the former, wave attractors may appear (e.g., [61]). …”

“…Inertial waves were identified in a rotating axial cylinder [34,73,65,67,53], in a slightly tilted free surface cylinder [102], in a sphere [4], in a tilted spheroid (tilt of its axis of rotation with respect to the axis of the cavity) [64,106], in a (truncated) cone [12], in a rectangular parallelepiped [14,54,15] and in a trapezoid [61,69].…”