Trajectory encounter number as a diagnostic of mixing potential in fluid flows 1

6 Fluid parcels can exchange water properties when coming in contact with each other, leading to 7 mixing. The trajectory encounter number, which quantifies the number of fluid parcel trajectories 8 that pass close to a reference trajectory over a finite time interval, is introduced as a measure of 9 the mixing potential of a flow. Regions characterized by low encounter numbers, such as cores 10 of coherent eddies, have low mixing potential, whereas turbulent or chaotic regions characterized 11 by large encounter numbers have high mixing potential. The encounter number diagnostic was 12 used to characterize mixing potential in 3 flows of increasing complexity: the Duffing Oscillator, 13 the Bickley Jet, and the altimetry-based velocity in the Gulf Stream Extension region. An 14 additional example was presented where the encounter number was combined with the u-star15 approach of Pratt et al., 2016 to characterize the mixing potential for a specific tracer distribution 16 in the Bickley Jet flow. Analytical relationships were derived connecting encounter number to 17 diffusivity for purely-diffusive flows, and to shear and strain rates for linear shear and linear 18 strain flows, respectively. It is shown that in a diffusive regime the encounter number grows as a 19 square-root of time, whereas in a linear shear and strain flows the encounter number is 20 proportional to time. 21 I. Encounter number 22 a. main idea 23 Mixing is an irreversible exchange of properties between different water masses. This process is 24 important for maintaining the oceanic large-scale stratification and general circulation, and it 25 plays a key role in the redistribution of bio-geo-chemical tracers throughout the world oceans. 26 Mixing occurs between different water masses when they come in direct contact with each other. 27 Thus, mixing potential of the flow, i.e., the opportunity for mixing to occur, is generally 28 enhanced in regions where water parcels meet or encounter many other water parcels, such as, 29 for example, within a chaotic zone –a region of the flow that is in a state of chaotic advection. 30 There, the separation between initially nearby water parcels grows exponentially in time and, in 31 the infinite time limit, each water parcel encounters all the other water parcels within the same 32 zone. Similarly, high encounter numbers will exist in turbulent regions. In contrast, mixing 33 potential is expected to be smaller in regions where water parcels do not experience many 34 encounters with other water parcels and remain close to their initial neighbors as the flow 35 evolves. This would be the case, for example, for a water parcel that is located inside a coherent 36 Nonlin. Processes Geophys. Discuss., doi:10.5194/npg-2016-72, 2016 Manuscript under review for journal Nonlin. Processes Geophys. Published: 7 December 2016 c © Author(s) 2016. CC-BY 3.0 License.