Flows in Rotating Channels:

K. R. Helfrich and L. J. Pratt

Department of Physical Oceanography
Woods Hole Oceanographic Institution
Woods Hole, MA 02543

1. Introduction

Nonlinear internal waves are common features of straits and are a consequence of the complicated interaction of time-dependent stratified flow with topographic features. These underlying flows are themselves highly nonlinear (typically hydraulically controlled) and exhibit significant spatial and temporal variation. Thus a full understanding of internal solitary wave propagation must include an understanding of the underlying flow which both generates and affects the evolution of the waves. In this short contribution we describe some results of two related problems which address the nonlinear dynamics of flow in rotating channels. The first is an extension of the dam break problem to the case of a rotating channel (Rossby adjustment in a channel). The second is the extension of Long's classic study of upstream influence and hydraulic adjustment over topography to flows in a rotating channel. In both examples we rely on solutions of the single layer shallow water equations and therefore do not model the nonhydrostatic effects which lead to solitary waves. However, the fully nonlinear shocks and hydraulic jumps which do arise are resolved and described.

2. Rossby Adjustment in a Channel

The setting for the dam break problem is a rectangular channel of width w. A cross-channel (y-direction) barrier separates two semi-infinite regions of homogeneous fluid of different depths. At time t = 0 the barrier is destroyed causing the upper fluid to intrude into the shallower fluid and the formation of a leading shock, or bore. The hydrostatic solution for no rotation is given by Stoker (1957). Gill (1976) examined the linear limit when rotation is included. A steady current is set up by Kelvin waves moving away from the barrier position. The Kelvin wave moving into the shallower layer is trapped to the right-hand wall (looking in the direction of the shallow layer) while the Kelvin wave moving into the deeper layer is trapped to the left wall. The steady flow approaches the section of the initial barrier along the left wall, crosses the channel at that section, and continues along the right wall.

When the initial depth difference is finite and rotation is included, the Kelvin wave moving into the shallow layer is replaced by a Kelvin bore. Because of the uncertainty of the proper shock-joining conditions and non-conservation of potential vorticity across the shock, an analytical solution to this problem has not been found. Thus we employ numerical methods to study the flow. Figure 1 shows a the depth field for a channel of nondimensional width w = 4 and an initial nondimensional depth in the shallow section d0 = 0.1, which is scaled by the initial depth in the deep region D. The horizontal dimensions are normalized by the deformation radius based on D. The flow is shown at t = 20. In this example the leading shock curves back upstream and its amplitude decays away from the right wall. Trailing the bore is a geostrophic boundary current. Near x = 0 there is a cross-channel geostrophic current due to the along channel gradient in depth.

Figure 1

In a narrower channel the shock structure is rather different. Figure 2 shows a case with w = 1 and d0 = 0.25. The leading bore is now nearly straight across the channel. The amplitude decays away from the right wall (y = -0.5), but increases near the left wall. Trailing the bore near the left wall is a packet of Poincaré waves. These waves appear to be generated by the resonant mechanism described by Tomasson and Melville (1992).

We find that the leading bore extends across the channel to attach at the left wall (within one deformation radius of its position on the right wall) when the channel width scaled by the deformation radius in the upstream region, w d0-1/2, is less than about 3. In all cases the shock is connected to the following geostrophic current by an ageostrophic boundary layer characterized by a strong transverse jet. Shock amplitudes, measured on the right wall, and shock speeds increase above the nonrotating solutions as w is increased for a given d0. However, the relation between bore speed and amplitude branches below the nonrotating relation as w (rotation) is increased. Finally, the potential vorticity jump across the shock is shown to depart significantly from the pseudo-inviscid estimates due to the viscous flux of vorticity, an effect that is at best crudely parameterized and poorly understood.

Figure 2

3. Long's Problem in a Rotating Channel

In Long's (1954, 1970) classic problem on upstream influence a topographic feature is rapidly grown into a nonrotating, hydrostatic, single layer flow. The response of the fluid, and in particular the development of upstream influence, can be determined by the location of the flow in the F - hm plane, where F is the Froude number of the initial flow and hm is the final height of the topography. Upstream influence occurs through the propagation of a shock, or bore, which causes a permanent change in the flow approaching the obstacle. At the crest of the obstacle the flow is critical and downstream a hydraulic jump back to subcritical flow may exist. In the weakly nonlinear and dispersive limit the upstream influence is achieved by the radiation of a train of solitary waves (e.g., Grimshaw and Smyth, 1986).

Using the semigeostrophic approximation and the assumption of uniform potential vorticity a theory is developed which gives the critical obstacle height above which upstream influence and hydraulic control at the topographic crest is achieved for rotating flow in a uniform channel. The critical height is a function of the initial flow Froude number (defined using the appropriate characteristic speeds in the rotating system) and the channel width relative to the deformation radius based on the upstream potential depth. The structure of the critical height curve is similar to the nonrotating version with increasing critical height as F increases above, or decreases below, one.

Figure 3

The theory is supplemented by numerical solutions of the full shallow water equations to explore the temporal development of the flow. The numerical solutions reveal numerous interesting features including upstream propagating shocks and separated rarefying intrusions, downstream hydraulic jumps (in both flow depth and width), flow separation and recirculation. Figure 3 shows one typical example for a channel of w = 2 and initial flow with F = 1. A symmetric topographic feature of height hm = 0.5 is grown into the flow between t = 0 and 2. A bore propagates upstream leaving behind a new subcritical upstream state which becomes supercritical as it flows over the topographic crest (x = 0). Just downstream of the topographic crest the flow separates from the left wall to form an expanding recirculation gyre. A small wedge of separated flow (zone of zero depth) remains over the topography. The numerical solutions show the semigeostrophic theory to give a generally good prediction of the critical obstacle height for upstream influence.

4. Summary

One goal of our work is to better understand the adjustment to hydraulically controlled flows in rotating channels. A consequence of the adjustment process, either in the Rossby adjustment problem or in the rotating version of Long's problem, is the generation of shocks, bores and jumps. In the context of internal solitary waves, the topic of this workshop, we repeat that wave generation often occurs due to flow interaction with topography and so any information on the dynamics of that interaction is useful in the question of solitary wave generation and propagation. Secondly, the shocks, bores and jumps we calculate are the fully nonlinear, hydrostatic limits of the (typically) weakly nonlinear solitary wave and undular bore solutions. As such, they may provide information on the initial, or boundary, conditions which result in solitary waves in the far field after enough time has passed for dispersive effects to become important.

References

Gill, A. E. 1976. Adjustment under gravity in a rotating channel. J. Fluid Mech. 77, 603-621.

Grimshaw, R. H. J. & N. Smyth 1986. Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429-464.

Long, R. R. 1954. Some aspects of the flow of stratified fluids. II. Experiments with a two-fluid system. Tellus 6, 97-115.

Long, R. R. 1970. Blocking effects in flow over obstacles. Tellus 22, 471-480.

Stoker, J. J. 1957. Water Waves. Interscience Publishers, 567 pp.

Tomasson, G. & W. K. Melville 1992. Geostrophic adjustment in a channel: nonlinear and dispersive effects. J. Fluid Mech. 241, 23-58.