Trajectory encounter volume as a diagnostic of mixing potential in fluid flows

Abstract

Abstract. Fluid parcels can exchange water properties when coming into contact with each other, leading to mixing. The trajectory encounter mass and a related simplified quantity, the encounter volume, are introduced as a measure of the mixing potential of a flow. The encounter volume quantifies the volume of fluid that passes close to a reference trajectory over a finite time interval. Regions characterized by a low encounter volume, such as the cores of coherent eddies, have a low mixing potential, whereas turbulent or chaotic regions characterized by a large encounter volume have a high mixing potential. The encounter volume diagnostic is used to characterize the mixing potential in three flows of increasing complexity: the Duffing oscillator, the Bickley jet and the altimetry-based velocity in the Gulf Stream extension region. An additional example is presented in which the encounter volume is combined with the u * approach of Pratt et al. (2016) to characterize the mixing potential for a specific tracer distribution in the Bickley jet flow. Analytical relationships are derived that connect the encounter volume to the shear and strain rates for linear shear and linear strain flows, respectively. It is shown that in both flows the encounter volume is proportional to time.

Connection between encounter volume and diffusivity in geophysical flows

“…In that reference frame, the reference particle will always stay at the origin, while other particles will still be involved in a random walk motion, but with a diffusivity twice that in the stationary frame, κ moving =2κ stationary (Rypina and Pratt, 2017). The problem of finding the encounter number is then reduced to counting the number of randomly walking particles (with diffusivity κ moving ) that come within radius R of the origin in the moving frame.…”

“…Rypina and Pratt (2017) discuss how the encounter volume can be used to identify Lagrangian coherent structures (LCS) such as stable and unstable manifolds of hyperbolic trajectories and regions foliated by the KAM-like tori surrounding elliptic trajectories in realistic geophysical flows. A detailed comparison between the encounter volume method and some other Lagrangian methods of LCS identification, as well as the dependences on parameters, t 0 , T , R, and on grid spacing (or on the number of trajectories, K), and the relative advantages of different techniques, was given in Rypina and Pratt (2017). The interested reader is referred to that earlier paper for details.…”

“…Here the indicator function I is 1 if true and 0 if false, and K is the total number of particles. As in Rypina and Pratt (2017), we define encounter volume based on the number of encounters with different trajectories, not the total number of encounter events (see the schematic diagram of trajectory encounters in Fig. 1).…”

“…In order to formally define the encounter volume V , Rypina and Pratt (2017) subdivide the entire fluid into infinitesimal fluid elements with volumes dV i , and define the encounter volume for each fluid element to be the total volume of fluid that passes within a radius R of it over a finite time interval t 0 < t < t 0 + T , i.e., V (x 0 ; t 0 , T , R) = lim dV i →0 i dV i .…”

“…Rypina and Pratt (2017) introduce a trajectory encounter volume, V , the volume of fluid that comes in contact with the reference fluid parcel over a finite time interval. The increase in V over time is one measure of the mixing potential of the element, "mixing" being the irreversible exchange of properties between different water parcels.…”