SELF-SIMILAR GRAVITY CURRENTS WITH VARIABLE INFLOW REVISITED - PLANE CURRENTS

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of th e intruding fluid varies in time according to a power law of the type t(alpha). The resistance of the ambient fluid is taken into account by a boundary condition of the von Karman type, that depends on a parame ter beta that is a function of the density ratio of the fluids. The fl ow is characterized by beta,alpha and the Froude number F-0 near the s ource. We find four kinds of self-similar solutions: subcritical conti nuous solutions (Type I), continuous solutions with a supercritical-su bcritical transition (Type II), discontinuous solutions (Type III) tha t have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The curren t is always subcritical near the front, but near the source it is subc ritical (F-0 < 1) for Type I currents, and supercritical (F-0 > 1) for Types II, III, and IV. Type I solutions have already been found by ot her authors, but Type II, III, and IV currents are novel. We find the intervals of parameters for which these solutions exist, and discuss t heir properties. For constant-volume currents one obtains Type I solut ions for any beta that, when beta > 2, have a 'dry' region near the or igin. For steady inflow one finds Type I currents for 0 < beta < infin ity and Type II and III currents for any beta, if F-0 is sufficiently large.