A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow

Wind forcing of the ocean generates a spectrum of inertia–gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) are highly energetic and play a significant role in the slow, large-scale dynamics of the ocean. To analyse this role, we develop a new model of the non-dissipative interactions between NIWs and balanced motion. The model is derived using the generalised-Lagrangian-mean (GLM) framework (specifically, the ‘glm’ variant of Soward & Roberts, J. Fluid Mech., vol. 661, 2010, pp. 45–72), taking advantage of the time-scale separation between the two types of motion to average over the short NIW period. We combine Salmon’s (J. Fluid Mech., vol. 719, 2013, pp. 165–182) variational formulation of GLM with Whitham averaging to obtain a system of equations governing the joint evolution of NIWs and mean flow. Assuming that the mean flow is geostrophically balanced reduces this system to a simple model coupling Young & Ben Jelloul’s (J. Mar. Res., vol. 55, 1997, pp. 735–766) equation for NIWs with a modified quasi-geostrophic (QG) equation. In this coupled model, the mean flow affects the NIWs through advection and refraction; conversely, the NIWs affect the mean flow by modifying the potential-vorticity (PV) inversion – the relation between advected PV and advecting mean velocity – through a quadratic wave term, consistent with the GLM results of Bühler & McIntyre (J. Fluid Mech., vol. 354, 1998, pp. 301–343). The coupled model is Hamiltonian and its conservation laws, for wave action and energy in particular, prove illuminating: on their basis, we identify a new interaction mechanism whereby NIWs forced at large scales extract energy from the balanced flow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. A rough estimate suggests that this mechanism could provide a significant sink of energy for mesoscale motion and play a part in the global energetics of the ocean. Idealised two-dimensional models are derived and simulated numerically to gain insight into NIW–mean-flow interaction processes. A simulation of a one-dimensional barotropic jet demonstrates how NIWs forced by wind slow down the jet as they propagate into the ocean interior. A simulation assuming plane travelling NIWs in the vertical shows how a vortex dipole is deflected by NIWs, illustrating the irreversible nature of the interactions. In both simulations energy is transferred from the mean flow to the NIWs.

Dynamic Potential Vorticity Initialization and the Diagnosis of Mesoscale Motion

A new method for diagnosing the balanced three-dimensional velocity from a given density field in mesoscale oceanic flows is described. The method is referred to as dynamic potential vorticity initialization (PVI) and is based on the idea of letting the inertia–gravity waves produced by the initially imbalanced mass density and velocity fields develop and evolve in time while the balanced components of these fields adjust during the diagnostic period to a prescribed initial potential vorticity (PV) field. Technically this is achieved first by calculating the prescribed PV field from given density and geostrophic velocity fields; then the PV anomaly is multiplied by a simple time-dependent ramp function, initially zero but tending to unity over the diagnostic period. In this way, the PV anomaly builds up to the prescribed anomaly. During this time, the full three-dimensional primitive equations—except for the PV equation—are integrated for several inertial periods. At the end of the diagnostic period the density and velocity fields are found to adjust to the prescribed PV field and the approximate balanced vortical motion is obtained. This adjustment involves the generation and propagation of fast, small-amplitude inertia–gravity waves, which appear to have negligible impact on the final near-balanced motion. Several practical applications of this method are illustrated. The highly nonlinear, complex breakup of baroclinically unstable currents into eddies, fronts, and filamentary structures is examined. The capability of the method to generate the balanced three-dimensional motion is measured by analyzing the ageostrophic horizontal and vertical velocity—the latter is the velocity component most sensitive to initialization, and one for which a quasigeostrophic diagnostic solution is available for comparison purposes. The authors find that the diagnosed fields are closer to the actual fields than are either the geostrophic or the quasigeostrophic approximations. Dynamic PV initialization thus appears to be a promising way of improving the diagnosis of balanced mesoscale motions.

Planetary waves in a stratified ocean of variable depth. Part 1. Two-layer model

Linear Rossby waves in a two-layer ocean with a corrugated bottom relief (the isobaths are straight parallel lines) are investigated. The case of a rough bottom relief (the wave scale L is much greater than the bottom relief scale Lb) is studied analytically by the method of multiple scales. A special numerical technique is developed to investigate the waves over a periodic bottom relief for arbitrary relationships between L and Lb.There are three types of modes in the two-layer case: barotropic, topographic, and baroclinic. The structure and frequencies of the modes depend substantially on the ratio Δ = (Δh/h2)/(L/a) measuring the relative strength of the topography and β-effect. Here Δh/h2 is the typical relative height of topographic inhomogeneity and a is the Earth's radius. The topographic and barotropic mode frequencies depend weakly on the stratification for small and large Δ and increase monotonically with increasing Δ. Both these modes become close to pure topographic modes for Δ>1.The dependence of the baroclinic mode on Δ is more non-trivial. The frequency of this mode is of the order of f0L2i/aL (Li is the internal Rossby scale) irrespective of the magnitude of Δ. At the same time the spatial structure of the mode depends strongly on Δ. With increasing Δ the relative magnitude of motion in the lower layer decreases. For Δ>1 the motion in the mode is confined mainly to the upper layer and is very weak in the lower one. A similar concentration of mesoscale motion in an upper layer over an abrupt bottom topography has been observed in the real ocean many times.Another important physical effect is the so-called ‘screening’. It implies that for Lb<Li the small-scale component of the wave with scale Lb is confined to the lower layer, whereas in the upper layer the scale of the motion L is always greater than or of the order of, Li. In other words, the stratification prevents the ingress of motion with scale smaller than the internal Rossby scale into the main thermocline.