Inertial Waves in a Rotating Spherical Shell with Homogeneous Boundary Conditions

We find the analytical form of inertial waves in an incompressible, rotating fluid constrained by concentric inner and outer spherical surfaces with homogeneous boundary conditions on the normal components of velocity and vorticity. These fields are represented by Galerkin expansions whose basis consists of toroidal and poloidal vector functions, i.e., products and curls of products of spherical Bessel functions and vector spherical harmonics. These vector basis functions also satisfy the Helmholtz equation and this has the benefit of providing each basis function with a well-defined wavenumber. Eigenmodes and associated eigenfrequencies are determined for both the ideal and dissipative cases. These eigenmodes are formed from linear combinations of the Galerkin expansion basis functions. The system is truncated to numerically study inertial wave structure, varying the number of eigenmodes. The largest system considered in detail is a 25 eigenmode system and a graphical depiction is presented of the five lowest dissipation eigenmodes, all of which are non-oscillatory. These results may be useful in understanding data produced by numerical simulations of fluid and magnetofluid turbulence in a spherical shell that use a Galerkin, toroidal–poloidal basis as well as qualitative features of liquids confined by a spherical shell.

Vortex cluster arising from an axisymmetric inertial wave attractor

We present an experimental and numerical study of the nonlinear dynamics of an inertial wave attractor in an axisymmetric geometrical setting. The rotating ring-shaped fluid domain is delimited by two vertical coaxial cylinders, a conical bottom and a horizontal wave generator at the top: the vertical cross-section is a trapezium, while the horizontal cross-section is a ring. Forcing is introduced via axisymmetric low-amplitude volume-conserving oscillatory motion of the upper lid. The experiment shows an important result: at sufficiently strong forcing and long time scale, a saturated fully nonlinear regime develops as a consequence of an energy transfer draining energy towards a slow two-dimensional manifold represented by a regular polygonal system of axially oriented cyclonic vortices undergoing a slow prograde motion around the inner cylinder. We explore the long-term nonlinear behaviour of the system by performing a series of numerical simulations for a set of fixed forcing amplitudes. This study shows a rich variety of dynamical regimes, including a linear behaviour, a triadic resonance instability, a progressive frequency enrichment reminiscent of weak inertial wave turbulence and the generation of a slow manifold in the form of a polygonal vortex cluster confirming the experimental observation. This vortex cluster is discussed in detail, and we show that it stems from the summation and merging of wave-like components of the vorticity field. The nature of these wave components, the possibility of their detection under general conditions and the ultimate fate of the vortex clusters at even longer time scale remain to be explored.

Near-inertial wave critical layers over sloping bathymetry

This study describes a specific type of critical layer for near-inertial waves (NIWs) that forms when isopycnals run parallel to sloping bathymetry. Upon entering this slantwise critical layer, the group velocity of the waves decreases to zero and the NIWs become trapped and amplified, which can enhance mixing. A realistic simulation of anticyclonic eddies on the Texas-Louisiana shelf reveals that such critical layers can form where the eddies impinge onto the sloping bottom. Velocity shear bands in the simulation indicate that windforced NIWs are radiated downward from the surface in the eddies, bend upward near the bottom, and enter critical layers over the continental shelf, resulting in inertially-modulated enhanced mixing. Idealized simulations designed to capture this flow reproduce the wave propagation and enhanced mixing. The link between the enhanced mixing and wave trapping in the slantwise critical layer is made using ray-tracing and an analysis of the waves’ energetics in the idealized simulations. An ensemble of simulations is performed spanning the relevant parameter space that demonstrates that the strength of the mixing is correlated with the degree to which NIWs are trapped in the critical layers. While the application here is for a shallow coastal setting, the mechanisms could be active in the open ocean as well where isopycnals align with bathymetry.

Idealized Numerical Modelling of Internal Waves Through Density Staircases

<p>The Arctic Ocean contains a warm layer originating from the Atlantic Ocean below the pycnocline which has a thermohaline staircase structure that inhibits vertical mixing. If this heat were to rise to the surface, the rate of sea ice loss would increase dramatically. Wind stress and ice floes generate internal waves which can cause vertical mixing. As the ice cover in the Arctic continues to decline, it will be important to predict how these changing internal waves propagate through such stratification profiles. Here, we investigate how density staircases enhance or limit downward near-inertial wave propagation. We use direct numerical simulations to solve the Boussinesq equations of motion using spectral methods. We simulate the propagation of internal waves through a vertically stratified fluid which includes one or more steps (i.e., mixed layers). We find that we reproduce the results of laboratory experiments showing transmission and reflection of internal waves from one or two mixed layers. We then extend our parameter regime to simulate the propagation of internal waves through a more realistic stratification profile tending toward that of the Arctic pycnocline.</p>

Near-inertial wave modulated turbulence in a Kuroshio anticyclonic eddy

<p>Storm-generated near-inertial waves are a significant source for deep-ocean mixing. However, their energy pathways beyond wind generation and equatorward propagation as low modes are still elusive. Previous studies suggest that the bulk of inertial wind power is lost in the nearfield of storm forcing, but there is little observational evidence to confirm this.</p><p>Finescale horizontal velocity, temperature, salinity and microscale temperature profiles to 500-m depth were collected in the Kuroshio-Oyashio Confluence east of Japan during the storm-seasons of 2016 and 2017 with chi-augmented EM-APEX floats. Temporal sampling was at 1-h resolution during storms, sufficient to resolve near-inertial motions. Turbulent dissipation rates  and diapycnal diffusivities K were inferred from microscale temperature-gradient spectra.  Several floats were trapped near the velocity maximum of anticyclonic eddies.  Mesoscale eddies are known to trap and amplify near-inertial waves and to modulate near-inertial wave distribution and dissipation.</p><p>Near-inertial energy-fluxes within the eddy are mostly inward and downward. Signatures of a critical layer, e.g., increasing vertical wavenumbers, shear, and turbulence are present at the depth where the eddy vorticity approaches the surface value, and strong vertical mean shears and vorticity-gradients occur. Turbulence is reduced by a factor of 10 above 180-m depth, despite elevated near-inertial energy, and enhanced between 200 and 255 m. Three mechanisms for the generation of enhanced turbulence are hypothesized: i) local and remotely forced near-inertial waves superimposing to create shear-unstable layers, ii) kinematic superposition of eddy and near-inertial shear that generates patches of turbulence at inertial periods, iii) a near-inertial critical layer due to dynamic wave/mean interaction. Ray tracing simulations will be performed to examine whether vertical vorticity gradients and/or Doppler shifting are responsible for the presence of a critical layer.</p>