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Hydraulic Control of Flows with Nonuniform Potential Vorticity

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  • 1 Wood Hole Oceanographic Institution, Woods Hole, MA 02543
  • | 2 Scripps Institution of Oceanography, La Jolla, CA 92093
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Abstract

The hydraulics of flow contained in a channel and having nonuniform potential vorticity is considered from a general standpoint. The channel cross section is rectangular and the potential vorticity is assumed to be prescribed in terms of the streamfunction. We show that the general computational problem can be expressed in two traditional forms, the first of which consists of an algebraic relation between the channel geometry and a single dependent flow variable and the second of which consists of a pair of quasi-linear differential equations relating the geometry to two dependent flow variables. From these forms we derive a general “branch condition” indicating a merger of different solutions having the same flow rate and energy and show that this condition implies that the flow is critical with respect to a certain long wave. It is shown that critical flow can occur only at the sill in a channel of constant width (with one exception) at a point of width extremum in a flat bottom channel. We also discuss the situation in which the fluid becomes detached from one of sidewalls.

An example is given in which the potential vorticity is a linear function of the streamfunction and the rotation rate is zero, a case which can be solved analytically. When the potential vorticity gradient points downstream, allowing propagation of potential vorticity waves against the flow, multiple pairs of steady states are possible, each having a unique modal structure. Critical control of the higher-mode solutions is primarily over vorticity, rather than depth. Flow reversals arise in some situations, possible invalidating the prescription of potential vorticity.

Abstract

The hydraulics of flow contained in a channel and having nonuniform potential vorticity is considered from a general standpoint. The channel cross section is rectangular and the potential vorticity is assumed to be prescribed in terms of the streamfunction. We show that the general computational problem can be expressed in two traditional forms, the first of which consists of an algebraic relation between the channel geometry and a single dependent flow variable and the second of which consists of a pair of quasi-linear differential equations relating the geometry to two dependent flow variables. From these forms we derive a general “branch condition” indicating a merger of different solutions having the same flow rate and energy and show that this condition implies that the flow is critical with respect to a certain long wave. It is shown that critical flow can occur only at the sill in a channel of constant width (with one exception) at a point of width extremum in a flat bottom channel. We also discuss the situation in which the fluid becomes detached from one of sidewalls.

An example is given in which the potential vorticity is a linear function of the streamfunction and the rotation rate is zero, a case which can be solved analytically. When the potential vorticity gradient points downstream, allowing propagation of potential vorticity waves against the flow, multiple pairs of steady states are possible, each having a unique modal structure. Critical control of the higher-mode solutions is primarily over vorticity, rather than depth. Flow reversals arise in some situations, possible invalidating the prescription of potential vorticity.

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