Probabilistic characterization of tidal mixing in a coastal embayment: a Markov Chain approach

Horizontal mixing by tidal streams and turbulent motions is modelled using a low-dimensional representation based on the theory of Markov Chains. Such a representation is expected to prove useful in the characterization of the bulk properties of mixing and exchange in coastal areas and thereby provide a basis for box, or reservoir, models of water quality and marine ecosystems. Irregular coastlines and complex bathymetry are common in many coastal environments and can cause highly structured tidal flows. Advective stirring by such flows, combined with the effect of turbulence, can give rise to complex mixing regimes and enhanced dispersion. The characterization of mixing proposed in this study is based on a discrete-time, finite-state Markov Chain model. We first provide a brief overview of Markov Chains and their use in modelling the ensemble effects of mixing in terms of the probability of a particle (or fluid parcel) making a transition from one region to another over a fixed number of tidal cycles. These ideas are then illustrated by examining tidal mixing in Passamaquoddy Bay, a tidally energetic coastal embayment close to the entrance of the Bay of Fundy off the east coast of Canada. The transition probabilities are estimated from the trajectories of order 105 particles calculated using tidal flow fields from a realistic numerical ocean model of the study region. Various quantities describing the tidal flushing, retention and exchange properties of Passamaquoddy Bay are determined from the Markov Chain model and shown to agree reasonably well with estimates based on the trajectories of the tracked particles.