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Inertia-gravity waves beyond the inertial latitude. Part 1. Inviscid singular focusing

Published online by Cambridge University Press:  15 October 2010

VICTOR I. SHRIRA  and
WILLIAM A. TOWNSEND
VICTOR I. SHRIRA*
Affiliation:
Department of Mathematics, EPSAM, Keele University, Keele ST5 5BG, UK
WILLIAM A. TOWNSEND
Affiliation:
Department of Mathematics, EPSAM, Keele University, Keele ST5 5BG, UK
*
†Email address for correspondence: v.i.shrira@keele.ac.uk
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Abstract

The paper is concerned with analytical study of inertia-gravity waves in rotating density-stratified ideal fluid confined in a spherical shell. It primarily aims at clarifying the possible role of these motions in deep ocean mixing. Recently, it was found that on the ‘non-traditional’ β-plane inertia-gravity internal waves can propagate polewards beyond their inertial latitude, where the wave frequency equals the local Coriolis parameter, by turning into subinertial modes trapped in the narrowing waveguides around the local minima of buoyancy frequency N. The behaviour of characteristics was established: wave horizontal and vertical scales decrease as the wave advances polewards and tend to zero at a latitude corresponding to an attractor of characteristics. However, the basic questions about wave evolution, its quantitative description and the possibility of its reflection from the critical latitude remain open. The present work addresses these issues by studying the linear inviscid evolution of finite bandwidth wavepackets on the ‘non-traditional’ β-plane past the inertial latitude for generic oceanic stratification. Beyond the inertial latitude, the wave field is confined in narrowing waveguides of three distinct generic types around different local minima of the buoyancy frequency. In the oceanic context, the widest is adjacent to the flat bottom, the thinnest is the upper mixed layer, and the middle one is located between the seasonal and main thermocline. We find explicit asymptotic solutions describing the wave field in the WKB approximation. As a byproduct, the conservation of wave action principle is explicitly formulated for all types of internal waves on the ‘non-traditional’ β-plane. The wave velocities and vertical shear tend to infinity and become singular at the attractor latitude or its vicinity for both monochromatic and finite bandwidth packets. We call this phenomenon singular focusing. These WKB solutions are shown to remain valid up to singularity for the bottom and mid-ocean waveguides. The main conclusion is that even in the inviscid setting the wave evolution towards smaller and smaller horizontal and vertical scales is irreversible: there is no reflection. For situations typical of deep ocean, a simultaneous increase in wave amplitude and decrease of vertical scale causes a sharp increase of vertical shear, which may lead to wave breaking and increased mixing.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 478 - 509
Copyright
Copyright © Cambridge University Press 2010

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