Chaotic advection in a steady, three-dimensional, Ekman-driven eddy

Abstract

We investigate and quantify stirring due to chaotic advection within a steady, three-dimensional, Ekman-driven, rotating cylinder flow. The flow field has vertical overturning and horizontal swirling motion, and is an idealization of motion observed in some ocean eddies. The flow is characterized by strong background rotation, and we explore variations in Ekman and Rossby numbers, E and R o , over ranges appropriate for the ocean mesoscale and submesoscale. A high-resolution spectral element model is used in conjunction with linear analytical theory, weakly nonlinear resonance analysis and a kinematic model in order to map out the barriers, manifolds, resonance layers and other objects that provide a template for chaotic stirring. As expected, chaos arises when a radially symmetric background state is perturbed by a symmetrybreaking disturbance. In the background state, each trajectory lives on a torus and some of the latter survive the perturbation and act as barriers to chaotic transport, a result consistent with an extension of the KAM theorem for three-dimensional, volume-preserving flow. For shallow eddies, where E is O(1), the flow is dominated by thin resonant layers sandwiched between KAM-type barriers, and the stirring rate is weak. On the other hand, eddies with moderately small E experience thicker resonant layers, wider-spread chaos and much more rapid stirring. This trend reverses for sufficiently small E, corresponding to deep eddies, where the vertical rigidity imposed by strong rotation limits the stirring. The bulk stirring rate, estimated from a passive tracer release, confirms the non-monotonic variation in stirring rate with E. This result is shown to be consistent with linear Ekman layer theory in conjunction with a resonant width calculation and the Taylor-Proudman theorem. The theory is able to roughly predict the value of E at which stirring is maximum. For large disturbances, the stirring rate becomes monotonic over the range of Ekman numbers explored. We also explore variation in the eddy aspect ratio.

Lagrangian chaos in confined two-dimensional oscillatory convection

“…Other interesting generalisations to three dimensions are perturbed axisymmetric geometries in which the two-dimensional set-up is recovered approximatively in each meridional section. When the unperturbed flow possesses a component along the invariant azimuthal direction θ, the Lagrangian system has a Hamiltonian structure, and the occurrence of KAM tori as transport barriers was demonstrated in a Poincaré section θ = cst (Fountain et al 2000;Pratta et al 2014). Steady states in a cubic geometry have been investigated by Tric, Labrosse & Betrouni (2000).…”

A Multiwavenumber Theory for Eddy Diffusivities and Its Application to the Southeast Pacific (DIMES) Region

“…This concept dates back to Bretherton (1966) and Green (1970) and can be interpreted as follows: If eddies are stationary relative to the mean flow, eddies have sufficient time to stir and mix the same tracers, which also move with the mean flow; on the other hand, if the eddies propagate relative to the mean flow, eddies do not mix the same tracers, and mixing is suppressed (Klocker and Abernathey 2014). This idea has also recently been applied to studies about chaotic advection (Pratt et al 2014).…”

Three-Dimensional Chaotic Advection by Mixed Layer Baroclinic Instabilities

“…Analysis of other oceanic flows in which vertical velocities might play an important role, such as coastal upwelling regions, in which vertical velocities are one order of magnitude smaller than the horizontal velocities, reveals a still dominating effect of vertical shear (Bettencourt et al 2012). It would be interesting to extend the analysis here proposed to other kind of flows, such as idealized flows (e.g., Pratt et al 2013;Rypina et al 2015) Langmuir turbulence (e.g., Van Roekel et al 2012), in which vertical velocities are comparable to the horizontal velocities and the emerging turbulence is no longer quasi two dimensional.…”