## Abstract

A hierarchy of inhomogeneous, nonstationary stochastic models of material transport is formulated, and its properties are described. The transport models from the hierarchy sequence provide progressively more skillful simulations of the subgrid-scale transport by mesoscale eddies, which are typically not resolved in coarse-grid representations of the ocean circulation. The stochastic transport models yield random motion of individual passive particles, and the probability density function of the particle population can be interpreted as the concentration of a passive tracer.

Performance of the models is evaluated by (a) estimating their parameters from Eulerian and Lagrangian statistics of a fluid-dynamic reference solution, (b) solving for the transport, and (c) comparing the stochastic and fluid-dynamic transports. The reference solution represents midlatitude oceanic gyres, and it is found by solving steadily forced, quasigeostrophic equations of motion at large Reynolds number. The gyres are characterized by abundant coherent structures, such as swift, meandering currents, strong vortices, eddies, and planetary waves. The common, nondiffusive spreading of material (i.e., single-particle dispersion that is a nonlinear function of time) is induced by all these structures on intermediate times and by inhomogeneity and lateral boundaries on longer times. The higher-order members of the hierarchy are developed specially for simulating nondiffusive transports by turbulence in the presence of organized fluid patterns.

The simplest, but least skillful, member of the hierarchy is the commonly used diffusion model. In terms of the random particle motion, the diffusion is equivalent to the random walk (Markov-0) process for particle positions. The higher-order members of the hierarchy are the Markov-1 (a.k.a. Langevin or random acceleration), Markov-2, and Markov-3 models, which are jointly Markovian for particle position and its time derivatives. Each model in the hierarchy incorporates all features of the models below it. The Markov-1 model simulates short-time ballistic behavior associated with exponentially decaying Lagrangian velocity correlations, but on large times it is overly dispersive because it does not account for trapping of material by the coherent structures. The Markov-2 model brings in the capability to simulate intermediate-time, subdiffusive (slow) spreading associated with such trappings and with both decaying and oscillating Lagrangian velocity correlations. The Markov-3 model is also capable of simulating intermediate-time, superdiffusive (fast) spreading associated with sustained particle drifts combined with the trapping phenomenon and with the related asymmetry of the decaying and oscillating Lagrangian velocity correlations.

*Corresponding author address:* Pavel S. Berloff, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, CA 90095-1567. Email: pavel@atmos.ucla.edu.