Chaotic transport by Rossby waves in shear flow

Abstract

Transport and mixing properties of Rossby waves in shear flow are studied using tools from Hamiltonian chaos theory. The destruction of barriers to transport is studied analytically, by using the resonance overlap criterion and the concept of separatrix reconnection, and numerically by using PoincarC sections. Attention is restricted to the case of symmetric velocity profiles with a single maximum; the Bickley jet with velocity profile sech' is considered in detail. Motivated by linear stability analysis and experimental results, a simple Hamiltonian model is proposed to study transport by waves in these shear flows. Chaotic transport, both for the general case and for the sech* profile, is investigated. The resonance overlap criterion and the concept of separatrix reconnection are used to obtain an estimate for the destruction of barriers to transport and the notion of banded chaos is introduced to characterize the transport that typically occurs in symmetric shear flows. Comparison between the analytical estimates for barrier destruction and the numerical results is given. The role of potential vorticity conservation in chaotic transport is discussed. An area preserving map, termed standard nontwist map, is obtained from the Hamiltonian model. It is shown that the map reproduces the transport properties and the separatrix reconnection observed in the Hamiltonian model. The conclusions reached are used to explain experimental results on transport and mixing by Rossby waves in rotating fluids.

Transport properties in nontwist area-preserving maps

“…Nonmonotonicity occurs when the flow shear reverses sign along some shearless curve, as is the case for zonal flows of geophysical and atmospheric interest. 4,7 One example is the Bickley jet for which the velocity profiles are symmetric with a single maximum.…”

“…Such waves play an important role in zonal sheared flows that occur in both oceans and atmospheres, like the Gulf stream and the jet current. 7 Due to the nonintegrable nature of the combined Hamiltonian, chaotic behavior occurs for the advected particle trajectories, even though the fluid velocity field need not be chaotic at all.…”

Controlling the Unsteady Analogue of Saddle Stagnation Points

“…Their stable and unstable manifolds separate the flow regime into regions of fluid which do not mix. Analysis of saddle stagnation points is well established in fluid applications ranging from groundwater modeling [29,46], macro-and micromixing devices [17,2,6,50,42,21], and oceanographic flows [48,3,36,11,9,40]. The analogous entity in unsteady flows is that of a hyperbolic trajectory, a specific type of time-varying fluid parcel trajectory which possesses time-varying stable and unstable manifolds, whose locations govern fluid transport; cf.…”

Renormalization and destruction of 1/γ2 tori in the standard nontwist map

“…Refs. 3,4), in celestial mechanics, [5] fluid dynamics [1] and atomic physics. [6] It has been shown [7,8] that nontwist regions appear generically in area-preserving maps that have a tripling bifurcation of an elliptic fixed point.…”