tokens: ). Despite this wide range of applications, the nonlinear dynamics of eccentric discs remains poorly understood. In the Solar system, several of the rings of Saturn and Uranus are observed to be elliptical, and a theory for narrow eccentric rings (with small eccentricities but arbitrary eccentricity gradients) composed of weakly collisional particles has been developed (Borderies, Goldreich Tremaine 1983 ; Chiang Goldreich 2000 ). For small eccentricities, eccentric discs can be described as slowly precessing one-armed density waves (or shocks, e.g. Okazaki 1991 ; Papaloizou, Savonije Henrichs 1992 ; Lee Goodman 1999 ; Goodchild Ogilvie 2006 ; Ogilvie 2008 ; Saini, Gulati Sridhar 2009 ). A particular solution for a global uniformly eccentric (and apsidally aligned) Keplerian disc has also been studied by Statler ( 2001 ). A general secular theory for three-dimensional eccentric discs with arbitrary eccentricities and eccentricity gradients has been developed (Ogilvie 2001 ), and we aim to explore (and test using numerical simulations) some of its implications in this work. Secular theory describes dynamical processes that occur on time-scales that are much longer than the orbital time-scale. In the more familiar context of a system of multiple planets around a star, the Keplerian orbit of each planet is subject to perturbing forces that are proportional to the planet-to-star mass ratio; secular theory accounts for the long-term behaviour of eccentricity and inclination resulting from the orbit-averaged perturbing forces. For a thin gaseous disc around a star, however, pressure provides the most important departure from Keplerian orbital motion, being of second order in the aspect ratio H / R of the disc. Other subdominant forces, such as those due to the self-gravity of the disc (Tremaine 2001 ) and viscous or turbulent stresses (Ogilvie 2001 ), can also be important. In this situation, secular theory describes the long-term evolution of eccentricity due to gas pressure (and other forces) in thin discs, but neglects contributions that are of fourth order or higher in H / R . In recent work, we developed a local model of an eccentric disc (Ogilvie Barker 2014 ), which is similar to the shearing box that is commonly used to study the dynamics of circular astrophysical discs (Goldreich Lynden-Bell 1965 ; Hawley, Gammie Balbus 1995 ). We used this model to analyse the vertical oscillatory flows that are driven by the variation in the vertical gravity around a non-circular orbit (Ogilvie Barker 2014 ), and subsequently studied the local linear stability of these discs and their vertical flows (Barker Ogilvie 2014 ). We found that eccentric discs are generically unstable (in three dimensions), being subject to a small-scale instability in which inertial waves are driven by a parametric resonance. This instability is expected to lead to wave activity or turbulence, and to damping of the disc eccentricity and eccentricity gradients (Papaloizou 2005b
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