Accretion discs with non-zero central torque

We present analytical and numerical solutions for accretion discs subject to a non-zero central torque. We express this in terms of a single parameter, $f$, which is the ratio of outward viscous flux of angular momentum from the inner boundary to the inward advected flux of angular momentum there. The standard "accretion" disc, where the central boundary condition is zero-torque, is represented by $f=0$. A "decretion" disc, where the radial velocity at the inner boundary is zero, is represented by $f\rightarrow\infty$. For $f > 0$ a torque is applied to the disc at the inner boundary, which feeds both angular momentum and energy into the disc. This can arise, for example, in the case of a circumbinary disc where resonances transfer energy and angular momentum from the binary to the disc orbits, or where the disc is around a rotating magnetic star which can allow the disc orbits to be accelerated outwards at the magnetospheric radius. We present steady-state solutions to the disc structure as a function of $f$, and for arbitrary kinematic viscosity $\nu$. For time-dependent discs, we solve the equations using a Green's function approach for the specific case of $\nu \propto R$ and provide an example numerical solution to the equations for the case of $\nu \propto R^{3/2}$. We find that for values of $f\lesssim 0.1$ the disc solutions closely resemble "accretion" discs. For values of $f \gtrsim 10$ the solutions initially resemble "decretion" discs, but at sufficiently late times exhibit the properties of "accretion" discs. We discuss the application of this theory to different astrophysical systems, and in particular the values of the $f$ parameter that are expected in different cases.