The conditions are examined in which stationary hydraulic jumps may occur in a continuously stratified layer of fluid of finite thickness moving over a horizontal boundary at z = 0 and beneath a deep static layer of uniform density. The velocity and density in the flowing layer are modified by turbulent mixing in the transition region. Entrainment of fluid from the overlying static layer is possible. Results are expressed in terms of a Froude number, Fr, characterizing the flow upstream of a transition. A Froude number, Fr*, is found that must be exceeded if conditions for the conservation of volume, mass and momentum fluxes across a hydraulic transition are satisfied. The condition Fr > Fr* is satisfied if the kinetic energy (KE) per unit area is greater than the potential energy per unit area, or if ∫0h [u2(z) − z2N2(z)]dz > 0, in a flow of speed u(z), in a layer of thickness h, with buoyancy frequency N(z). In the particular case (referred to as an ‘η profile’) of a flow with velocity and density that are constant if z ≤ ηh, decrease linearly if ηh ≤ z ≤ h, and in which u(z) = 0 and density is constant when z ≥ h, long linear internal waves can propagate upstream, ahead of a stationary hydraulic jump, for Fr in a range Fr* < Fr < Frc; here Frc is the largest Fr at which long waves, and wave energy, can propagate upstream in a flow with specified η. (Profiles other than the η profile exhibit similar properties.) It is concluded that whilst, in general, Fr > Fr* is a necessary condition for a hydraulic jump to occur, a more stringent condition may apply in cases where Fr* < Frc, i.e. that Fr > Frc.
Physical constraints are imposed on the form of the flow downstream of the hydraulic jump or transition that relate, for example, to its static and dynamic stability and its stability against a further hydraulic jump. A further condition is imposed that relates the rate of dissipation of turbulent KE within a transition to the loss in energy flux of the flow in passing through the transition from the upstream side to the downstream. The constraints restrict solutions for the downstream flow to those in which the flux of energy carried downstream by internal waves is negligible and in which dissipation of energy occurs within the transition region.
Although the problem is formulated in general terms, particular examples are given for η profiles, specifically when η = 0 and 0.4. The jump amplitude, the entrainment rate, the loss of energy flux and the shape of the velocity and density profiles in the flow downstream of a transition are determined when Fr > Fr* (and extending to those with Fr > Frc) in a number of extreme conditions: when the loss of energy flux in transitions is maximized, when the entrainment is maximized, when the jump amplitude is least and when loss of energy flux is maximized subject to the entrainment into the transitions being made zero. The ratio of the layer thickness downstream and upstream of a transition, the jump amplitude, is typically at least 1.4 when jumps are just possible (i.e. when Fr ~ Frc). The amplitude, entrainment and non-dimensionalized loss in energy flux increase with Fr in each of the extreme conditions, and the maximum loss in energy flux is close to that when the entrainment is greatest. The magnitude of the advective and diffusive fluxes across isopycnal surfaces, i.e. the diapycnal fluxes characterizing turbulent mixing in the transition region, also increase with Fr. Results are compared to those in which the velocity and density profiles downstream of the transition are similar to those upstream, and comparisons are also made with equivalent two-layer profiles and with a cosine-shaped profile with continuous gradients of velocity and density. If Fr is larger than a certain value (about 7 and > Frc, if η = 0.4), no solutions for flows downstream of a transition are found if there is no entrainment, implying that fluid must be entrained if a transition is to occur in flows with large Fr. Although the extreme conditions of loss of energy flux, jump amplitude or entrainment provide limits that it must satisfy, in general no unique downstream flow is found for a given flow upstream of a jump.