Structuring eccentric-narrow planetary rings
(Abridged) A simple and general description of the dynamics of a narrow eccentric ring is presented.We view an eccentric ring which precesses uniformly at a slow rate as exhibiting a global $m=1$ mode originating from a standing wave superposed on an axisymmetric background.We adopt a continuum description using the language of fluid dynamics which gives equivalent results for the secular dynamics of thin rings as the the well known description using discrete elliptical streamlines formulated by Goldreich and Tremaine (1979). We use this to discuss the non linear mode interactions that appear through the excitation of higher $m$ modes due to the coupling of the $m=1$ mode with an external satellite potential, showing that they can lead to the excitation of the $m=1$ mode through a feedback process.Two conditions for the maintainance of a steady $m=1$ mode are obtained. One,being the condition for the normal mode pattern to precess uniformly requires a balance between the differential precession induced by the oblateness of the central planet,self-gravity and collisional effects and is the continuum form of that obtained from the $N$ streamline model of Goldreich and Tremaine (1979).The other condition is for the steady maintenance of the non-zero radial action of the ring on account of the normal mode.This requires a balance between input due to eccentric resonances due to external satellites and additional collisional damping associated with the presence of the $m=1$ mode We estimate that such a balance can occur in the $\epsilon-$ring of Uranus,given its currently observed physical and orbital parameters.