Nonlinear Rossby adjustment in a channel
Authors: HELFRICH K. R. , KUO ALLEN C. , PRATT L. J. 
Published: July 2002

The Rossby adjustment problem for a homogeneous fluid in a channel is solved for
large values of the initial depth discontinuity. We begin by analysing the classical
dam break problem in which the depth on one side of the discontinuity is zero. An
approximate solution for this case can be constructed by assuming semigeostrophic
dynamics and using the method of characteristics. This theory is supplemented by
numerical solutions to the full shallow water equations. The development of the flow
and the final, equilibrium volume transport are governed by the ratio of the Rossby
radius of deformation to the channel width, the only non-dimensional parameter.
After the dam is destroyed the rotating fluid spills down the dry section of the
channel forming a rarefying intrusion which, for northern hemisphere rotation, is
banked against the right-hand wall (facing downstream). As the channel width is
increased the speed of the leading edge (along the right-hand wall) exceeds the
intrusion speed for the non-rotating case, reaching the limiting value of 3.80 times the
linear Kelvin wave speed in the upstream basin. On the left side of the channel fluid
separates from the sidewall at a point whose speed decreases to zero as the channel
width approaches infinity. Numerical computations of the evolving flow show good
agreement with the semigeostrophic theory for widths less than about a deformation
radius. For larger widths cross-channel accelerations, absent in the semigeostrophic
approximation, reduce the agreement. The final equilibrium transport down the
channel is determined from the semigeostrophic theory and found to depart from the
non-rotating result for channels widths greater than about one deformation radius.
Rotation limits the transport to a constant maximum value for channel widths greater
than about four deformation radii.

The case in which the initial fluid depth downstream of the dam is non-zero is
then examined numerically. The leading rarefying intrusion is now replaced by a
Kelvin shock, or bore, whose speed is substantially less than the zero-depth intrusion
speed. The shock is either straight across the channel or attached only to the right-hand wall depending on the channel width and the additional parameter, the initial
depth difference. The shock speeds and amplitudes on the right-hand wall, for fixed
downstream depth, increase above the non-rotating values with increasing channel
width. However, rotation reduces the speed of a shock of given amplitude below the
non-rotating case. We also find evidence of resonant generation of Poincaré waves
by the bore. Shock characteristics are compared to theories of rotating shocks and
qualitative agreement is found except for the change in potential vorticity across the
shock, which is very sensitive to the model dissipation. Behind the leading shock
the flow evolves in much the same way as described by linear theory except for the
generation of strongly nonlinear transverse oscillations and rapid advection down the
right-hand channel wall of fluid originally upstream of the dam. Final steady-state
transports decrease from the zero upstream depth case as the initial depth difference
is decreased.