Abstract:
An extension to the energy-conserving theory of gravity currents in
rectangular rotating channels is presented, in which an upstream potential
vorticity boundary condition in the current is applied. It is assumed that the fluid
is inviscid; that the Boussinesq approximation applies; that the fundamental
properties of momentum ,energy, volume flux and potential vorticity are
conserved between upstream and downstream locations; and that the flow is
dissipation less. The upstream potential vorticity in the current is set through the
introduction of a new parameter δ, that defines the ratio of the reference depth of
the current to the ambient fluid. Flow types are established as a function δ and
the rotation rate, and a fourth flow geometry is identified in addition to the three
previously identified for rotating gravity currents. Detailed solutions are
obtained for three cases δ =0.5, 1.0 and 1.5, where δ <1 is relevant to currents
originating from a shallow source and δ >1 to currents where the source region
is deeper than the downstream depth, for example where a deep ocean flow
encounters a plateau. The governing equations and solutions for each case are
derived, quantifying the flow in terms of the depth, width and front speed. Cross
stream velocity profiles are provided for both the ambient fluid and the current.
These predict the evolution of a complex circulation within the current as the
rotation rate is varied. The ambient fluid exhibits similar trends to those
predicted by the energy conserving theory, with the Froude number tending to
√2 at the right-hand wall at high rotation rates. The introduction of the potential
vorticity boundary condition into the energy-conserving theory does not appear
to have a substantial effect on the main flow parameters (such as current speed
and width); however it does provide an insight into the complex dynamics of the
flow within the current.