Most commonly, eddy-induced transport is parameterised according to down-gradient diffusion. That is, transport is assumed to be a Gaussian process with no memory. The eddy diffusivity constant K quantifies the efficiency of tracer transport. This constant is frequently taken to be isotropic and homogeneous. However, both numerical experiments and observational data have suggested that this in fact is not the case. For example, float trajectories and estimates made from satellite altimetry in the North Atlantic show spreading to be predominately zonal except in the vicinity of the gulf stream where the preferred direction of spread is aligned with the jet core (Rypina et al. 2012). Furthermore, numerical experiments in a wind-driven 2 layer double-gyre model is shown to produce regions of super and sub diffusivity (Berloff et al. 2002a), that is regions where Lagrangian particles spread at a rate that is either faster or slower than that of a diffusive process.
It is still poorly understood exactly what is behind these observed behaviours of oceanic transport. Berloff et al. argue that multiple alternating jets, which are eddy-driven and are therefore not simulated by non-eddy resolving models, may be a key factor in influencing tracer spreading. The shear dispersion in these jets causes a significant difference between zonal and meridional tracer transport (Berloff et al. 2011).
Strong mean advection can be a driving factor of transport, such as in the interior of the Gulf Stream, where its boundaries also act as a barrier to transport. However, where the eddies are much stronger than the mean flow, mean advection cannot account for the anisotropy alone. Kamenkovich et al. (2015) implemented a 3-layer quasi geostrophic numerical model, with a wind stress forcing in the top layer, and bottom friction in the bottom layer, and deployed neutrally buoyant Lagrangian particles. This also revealed that the dispersion rates were strongly anisotropic and transport was also non-diffusive. The flow was decomposed into three components: time-mean advection, large-scale zonal transients and the remainder of the eddy-field to assist in examining the effects of each component on the anisotropic transport. They found that zonal transients are the most important effect in explaining the largely zonal anisotropic diffusivity.
O' Dwyer et al. (2000) argues that the distribution of potential vorticity plays a significant role in float dispersion in the North Atlantic, in particular, near the Gulf Stream where the potential vorticity is nearly uniform, the dispersion is strongly isotropic. It appears that floats spread preferentially along PV contours.
A simple dynamical model simulating a meandering jet will be used as the test case. Different parameter regimes will be explored which will determine the strength and chaotic nature of the jet, such as how well-defined the meanders are. Much like the gulf stream, the meanders create jet rings; regions of circular motion, much like a vortex. Lagrangian particles will be advected using this velocity field. Various Lagrangian statistics can be employed to study the transport and mixing properties, such as single and two particle dispersion, Lyapunov exponents and Lagrangian time correlations. These statistics will be used to determine the how leaky the jet is the transport, but also how the jet rings behave as barriers to transport. Furthermore the anisotropic and non-diffusive spreading will be studied in the frame of possible explanations as stated above.
Possible approaches to address these limitations can be explored and applied to this model. One possible approach is to consider Markov Chains of higher order. A zeroth order Markov model is simply a Gaussian process, i.e diffusion. However, if we introduce white noise at higher orders, such as to the velocity field and acceleration, the particles can then retain memory and exhibit sub and super diffusive behaviours (Berloff et al. 2002b)
The analysis of the simple test case will hopefully be the ground work in deriving a new more suitable approach to parameterising eddy-induced transport in OGCMs.
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