I∗ Q[cm3s−1] f [s−1] g′[cm s−2] H0[cm] D[cm]
0.33 75 1 70.9 2 5
1.23 62 1.5 14.9 2 5
4.08 97 1.5 2.3 2 5
Table 2.5Experimental parameters for the experiments shown in figure 2.15.
For intermediate values ofI∗, the vortex has a smaller surface area, but has a shape that more closely resembles a circle. The circulation within the vortex is much stronger and the vortex is in general much larger than the width of the current. Finally, for high values of I∗, the vortex becomes unstable as it sheds smaller vortices that are then carried along with the propagating boundary current. The large disturbance to the boundary current along the bottom edge of the tank in the third image of figure 2.15 is an example of a smaller vortex that has been shed from the main outflow bulge. The vortex can become indistinguishable from the boundary current for the highest values ofI∗.
The parameter I∗ is similar to the one used in the work of TL, I = f Q1/5
2g′3/5, which was defined as the aspect ratio of the current (depth divided by width). With the introduction of changing source conditions in our experiments, the source aspect ratioAs=H0/Drepresents a second dimensionless parameter that we may expect to play a role in determining the flow dynamics. The parameterI∗, as defined above, combines both dimensionless parameters with a specific choice of curve in the two-dimensional parameter space. The reasoning behind this choice of curve is based on the work of TL. They found three distinct flow regimes which occurred at low (I ∼0.05), intermediate (I∼0.30) and high (I ∼2.76) values ofI.
Their classification was based on the relative size of the outflow vortex and the boundary current. Using the scalings for the current depthh0z and the current widthw0zused by TL the parametersIandI∗are related by
I∗=I(2D
H0)4/5= ( h0D
w0H0)4/5= (Ac
As)4/5, (2.3)
whereAc=h0z/w0zis the current aspect ratio. Therefore, the parameter Iis related to the current aspect ratio byI= (Ac/2)4/5and thus we have chosen the same scaling for the souce aspect ratio of(As/2)4/5when definingI∗.
For the three source openings used in the experiments, the source aspect ratio takes only two values atAs=0.4 andAs=0.8 (see figure 2.6). Therefore, since the value only
2.6 Flow classification 39
Figure 2.16Example curves in the two-dimensional paramter space defined by the current aspect ratioAcand the source aspect ratioAs.
varies by a factor of two the choice of a specific curve when definingI∗is not unreasonable.
Were the source aspect ratio allowed to vary over a larger range in future experiments, the relationship between the two aspect ratios of the current and the source would need to be investigated further. Figure 2.16 plots the specific curve used when defining I∗ alongside possible alternatives based on the source aspect ratio changing by a factor of two.
The parameterI∗is effectively telling us how the flow changes from the initial conditions at the source. Initially the flow knows the fixed source conditions with depthH0and width D, but will quickly adjust to the geostrophic depthh0zshortly after leaving the source and will no longer be constrained by the source conditions. In general, for a small value ofI∗we expect a shallow and wide current and for large values ofI∗the current will be much deeper and narrower.
A qualitative analysis of all of the experiments reveals the approximate boundaries of the values of I∗ between each regime. For those experiments in the regime of lowI∗ the values range from 0.15≤I∗≤0.50. Those in the second regime of intermediateI∗ have values in the range 0.39≤I∗≤1.73. Finally, the experiments in the third regime of highI∗ have values in the range 1.48≤I∗≤5.61. There is a slight degree of overlap between the values seen in the three regimes which is due to the qualitative nature of their description.
We estimate the transition from an elongated to a contracted vortex for 0.39≤I∗≤0.50
and the transition to instability for 1.37≤I∗≤1.73. The value of the parameterI∗will be used throughout this thesis to refer to the experiments being considered. The experimental data are grouped into low, intermediate and highI∗regimes and the properties of the flow investigated for each of the regimes.
Chapter 3
Steady state model for rotating gravity currents
3.1 Problem description
When estuarine water is discharged into the coastal zone, gravity-driven surface flows can be established as a result of the density difference between the river water and the ocean. These flows are often of a sufficiently large scale that the current dynamics are affected by the Coriolis force arising from the rotation of the earth. This causes the discharged freshwater to be confined near to the coast, where it will propagate as a coastal gravity current. In this chapter we introduce a theoretical model to describe the properties of a rotating gravity current in a steady state.
We have previously discussed experiments by TL, which examined rotating gravity currents in the laboratory. Here we present the key results of their geostrophic model for a steady state current under the assumption of zero PV. They find scalings for the maximum current depthh0z, maximum current widthw0z, and constant current velocityu0z in terms of the experimental parameters, which are
h0z= (2f Q
g′ )1/2, (3.1)
w0z= (8g′Q
f3 )1/4, (3.2)
u0z= 3
29/4(f g′Q)1/4. (3.3)
They also obtain an equation for the current profile, hz(y) where y is the across-current coordinate, given by
hz=h0z− f2
2g′y2. (3.4)
We seek to build on the work of TL by constructing a steady state model for the boundary current in the case of a finite potential vorticity source. The motivation for the introduction of a finite potential vorticity source arises from the fact that real outflows, such as rivers, have a finite value of PV. As discussed in chapter 2, the experimental setup is based on that of TL, but with a modified source structure that better represents the natural environment.
The freshwater is injected horizontally in the experiments, as seen in nature, in contrast to the vertical source used by TL. The introduction of a horizontal source of finite depth introduces a finite potential vorticity into the flow and the effects of this will be the focus of our investigation as we seek to answer the first key question of this thesis: what is the effect of finite PV on the flow?
An alternative approach to modelling a buoyant outflow in a rotating environment is presented by Pichevin and Nof (1997), hereafter PN. The main focus of their work was to show that the growth of the bulge next to the source is a result of the imbalance of momentum in the system and the slow offshore migration of the bulge is required to bal-ance the momentum-flux lost in the current downstream. The key part of the work of PN which we are interested in here is the use of an upstream Bernoulli condition to fix the constant velocity of the boundary current along the wall. Their model relies on knowing the value of the upstream BernoulliB∗PN=1+Frup/2, forFrup the upstream Froude num-ber, and they carry out numerical simulations of the flow for different values of this parameter.
Alongside our extension of the TL model to incorporate a finite PV source, we also present an alternative steady state model for the current that allows for a finite current ve-locity along the boundary wall. The use of a Bernoulli condition, as in the work of PN, is one way in which this velocity can be fixed. The motivation for the inclusion of this second model can be seen in the experimental results in chapter 4 where the PIV data show that the velocity close to the wall,∼0.1 cm, is non-zero and can be as large as the peak velocity in the current in some instances. We first introduce a steady state model for a finite PV source with a general wall velocity, before then considering the specific cases of a zero and a finite value fixed by an upstream Bernoulli condition at the source. The models will then be compared with one another and their zero PV counterparts to determine the effect that finite