In this paper, we describe the nonlinear outcome of spiral shocks in protoplanetary disks. Spiral shocks, for most protoplanetary disk conditions, create a loss of vertical force balance in the post-shock region and result in rapid expansion of the gas perpendicular to the disk midplane. This expansion has characteristics similar to hydraulic jumps, which occur in incompressible fluids. We present a theory to describe the behavior of these hybrids between shocks and hydraulic jumps (shock bores) and then compare the theory to three-dimensional hydrodynamics simulations. We discuss the fully three-dimensional shock structures that shock bores produce and discuss possible consequences for disk mixing, turbulence, and evolution of solids.

39 pages, 18 figures, 1 table. Edited to match as closely as possible the ApJ proofs, which resulted in the correction of several typos. In addition, section 5.3 was slightly altered because an error in an analysis tool was discovered; the differences between the entropy gradient method and the Schwarzschild criterion method are minor. Figure 18 now only includes what was Figure18a