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Non-linear evolution of the parametric instability of inertial waves inherentto eccentric discs is studied by way of a new local numerical model. Modecoupling of tidal deformation with the disc eccentricity is known to produceexponentially growing eccentricities at certain mean-motion resonances.However, the details of an efficient saturation mechanism balancing this growthstill are not fully understood. This paper develops a local numerical model foran eccentric quasi-axisymmetric shearing box which generalises the often-usedcartesian shearing box model. The numerical method is an overall second orderwell-balanced finite volume method which maintains the stratified andoscillatory steady-state solution by construction. This implementation isemployed to study the non-linear outcome of the parametric instability ineccentric discs with vertical structure. Stratification is found to constrainthe perturbation energy near the mid-plane and localise the effective region ofinertial wave breaking that sources turbulence. A saturated marginally sonicturbulent state results from the non-linear breaking of inertial waves and issubsequently unstable to large-scale axisymmetric zonal flow structures. Thisresulting limit-cycle behaviour reduces access to the eccentric energy sourceand prevents substantial transport of angular momentum radially through thedisc. Still, the saturation of this parametric instability of inertial waves isshown to damp eccentricity on a time-scale of a thousand orbital periods. Itmay thus be a promising mechanism for intermittently regaining balance with theexponential growth of eccentricity from the eccentric Lindblad resonances andmay also help explain the occurrence of "bursty" dynamics such as the superhumpphenomenon.

Non-linear hydrodynamic instability and turbulence in eccentric astrophysical discs with vertical structure