Abstracts - KTH Mechanics

Scattering of linear Rossby waves by abrupt topography
Gareth W. Owen ∗ ,I.DavidAbrahams † and Andrew J. Willmott ∗
Planetary Rossby waves propagate across the world’s ocean basins. They play a
vital rôle in the long term distribution of ocean vorticity and to the oceans’ rapid
adjustment to atmospheric and climatic variability. Knowledge of the interaction
between these waves and topography is crucial to a full understanding of this process,
and hence to the transportation of energy, mixing and ocean circulation. Previously,
the interaction of baroclinic Rossby waves with abrupt topography was modelled using
barotropic or simple layered models. Here a more realistic model is employed, with a
continuously stratified fluid of buoyancy frequency N(z). The topographic variation
consists of a ‘top-hat’ ridge of infinitesimal or finite width.
The quasi-geostrophic streamfunction, ψ, satisfies the linear Rossby wave equation
∂
∂t
∂ 2 ψ
∂x2 + ∂2ψ ∂
+
∂y2 ∂z
f 2
N 2
∂ψ
∂z
+ β ∂ψ
∂x =0.
The vertical modes satisfy a Sturm-Liouville eigenvalue problem, which predicts a
qualitatively different modal structure than that of layered models. Applying a mode
matching technique results in a singular algebraic system of equations. This is solved
using a novel method in order to determine the scattering coefficients. In the case
where this algebraic method breaks down we employ matched asymptotic expansions.
We consider two ideal stratifications: a constant buoyancy frequency 1 and a buoyancy
frequency that decays exponentially with depth 2 . It is found that the scattered
field depends crucially upon the stratification. When the density variation is confined
to a thin thermocline a large amount of the incident wave energy is reflected by
a small ridge. Appreciable energy conversion between the propagating modes takes
place in this case and the scattered field differs markedly from the two-layer solution.
∗ Mathematics, Keele University, Keele, Staffs. ST5 5BG, UK.
† Mathematics, University of Manchester, Manchester M13 9PL, UK.
1 Owen, Abrahams, Willmott and Hughes, J. Fluid Mech. 465, 131 (2002).
2 Owen, Willmott, Abrahams and Mansley, Geophys. and Astrophys. Fluid Dyn. 99, 219 (2005).
Figure 1: Satellite altimetry data reveals strong Rossby wave activity in regions close
to ocean ridges. Phase shifts are consistent with conversion between baroclinic modes.
147