2.1 Pressure field

As a consequence of assumed parallel flow on the inflow and outflow, the pressure adopts hydrostatic balance over the section $BAE$ and $CD$ , and is given by

(2.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}=-(\unicode[STIX]{x1D70C}_{2}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(z))g,\end{eqnarray}$$

where the density of the fluid is written $\unicode[STIX]{x1D70C}(z)=\unicode[STIX]{x1D70C}_{2}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(z)$ and thus $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ is the excess density. Upstream we denote the depth of the dense fluid by $h_{a}$ , noting that for a gravity current $h_{a}$ vanishes. The pressure distribution on $CD$ is then given by

(2.7) $$\begin{eqnarray}p=\left\{\begin{array}{@{}ll@{}}p_{D}+\unicode[STIX]{x1D70C}_{2}g(H-z),\quad & h_{a}<z<H,\\ p_{D}+\unicode[STIX]{x1D70C}_{2}g(H-h_{a})+\unicode[STIX]{x1D70C}_{1}g(h_{a}-z),\quad & 0<z<h_{a},\end{array}\right.\end{eqnarray}$$

while the pressure distribution on $BE$ is given by

(2.8) $$\begin{eqnarray}p=p_{E}+\int _{0}^{H}(\unicode[STIX]{x1D70C}_{2}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C})g\;\text{d}z.\end{eqnarray}$$

In this integral (2.8), and throughout, $z$ -integrals are evaluated over the outflow boundary $BAE$ . It is then straightforward to evaluate the differences in pressure between the upper and lower boundaries at these inflow and outflow locations. These are given by

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle p_{C}-p_{D}=\unicode[STIX]{x1D70C}_{2}gH+(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})gh_{a}, & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle p_{B}-p_{E}=\unicode[STIX]{x1D70C}_{2}gH+\int _{0}^{H}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g\;\text{d}z. & \displaystyle\end{eqnarray}$$

The pressure differences on $BC$ and $DE$ may be expressed from (2.2), noting that $w=0$ on these boundaries. On $DE$ this yields

(2.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70C}_{2}u_{E}^{2}}{2}+p_{E}=\frac{\unicode[STIX]{x1D70C}_{2}U^{2}}{2}+p_{D}+\int _{D}^{E}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x,\end{eqnarray}$$

while on $BC$

(2.12) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70C}_{1}u_{B}^{2}}{2}+p_{B}+\int _{B}^{O}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x=\frac{\unicode[STIX]{x1D70C}_{C}U^{2}}{2}+p_{C}+\int _{C}^{O}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x,\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{C}$ is the density at $C$ (which is equal to $\unicode[STIX]{x1D70C}_{2}$ for the gravity current scenario and $\unicode[STIX]{x1D70C}_{1}$ for internal jumps). Also in the latter expression we have separated the dissipative effects into contributions from $OB$ and $OC$ .

We may now combine these expressions for the pressure differences over the boundaries of the control volume to satisfy the requirement that the pressure field is single-valued (2.5). Thus evaluating $(p_{C}-p_{D})+(p_{D}-p_{E})+(p_{E}-p_{B})+(p_{B}-p_{C})=0$ from (2.9)–(2.12), we deduce that

(2.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x1D70C}_{2}u_{E}^{2}}{2}-\frac{\unicode[STIX]{x1D70C}_{1}u_{B}^{2}}{2}-\frac{U^{2}}{2}(\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{C})=\int _{0}^{H}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g\;\text{d}z-(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})gh_{a}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\int _{D}^{E}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x-\int _{C}^{O}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x-\int _{O}^{B}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x.\end{eqnarray}$$

It is shown in appendix A that integration of the vorticity equation over the control volume leads to an identical condition, as it must, since they are derived from the same fundamental governing principles.

2.2 Global mass and momentum balances

Mass conservation is given by

(2.14) $$\begin{eqnarray}\int _{0}^{H}u\;\text{d}z=UH,\end{eqnarray}$$

while conservation of the dense solute is given by

(2.15) $$\begin{eqnarray}\int _{0}^{H}\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}\;u\;\text{d}z=(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})Uh_{a}.\end{eqnarray}$$

Since there are no $x$ -components of the stress tensor on the boundary of the control volume, the global balance of linear momentum along the $x$ -axis integrated over the domain is given by

(2.16) $$\begin{eqnarray}(p_{D}-p_{E})H+\unicode[STIX]{x1D70C}_{2}U^{2}(H-h_{a})+\unicode[STIX]{x1D70C}_{1}U^{2}h_{a}-\int _{0}^{H}\unicode[STIX]{x1D70C}u^{2}\;\text{d}z=\int _{0}^{H}z\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g\;\text{d}z-\frac{1}{2}(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2})gh_{a}^{2}.\end{eqnarray}$$

We emphasise that this balance, which is sometimes termed the ‘flow force balance’, does not explicitly include the effects of viscosity due to the assumed conditions on the boundary of the control volume. This does not imply that viscous effects are absent within the control volume. Indeed an inviscid analysis would eliminate coupling between the flowing layers and energy dissipation, assumptions that restrict the results to a very narrow range of parameter values. Some recent studies (Borden et al. Reference Borden, Meiburg and Constantinescu2012b , Borden & Meiburg Reference Borden and Meiburg2013a ,Reference Borden and Meiburg b , Konopliv et al. Reference Konopliv, Llewellyn-Smith, McElwaine and Meiburg2016) suggested that vorticity conservation for the control volume should be added to the formulation. However, as explained above and demonstrated in appendix A, this condition is equivalent to the requirement that the pressure field is single-valued and is embodied in (2.5) and (2.13).

The formulation above is the common basis for various models that predict the velocities of the front of gravity currents and of internal jumps, $U$ (and some related features, such as energy dissipation). For both flow problems, closure assumptions specify the outflow velocity and density profiles ( $u(z)$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(z)$ , respectively) and usually some additional assumptions are made about the energy dissipation or the source of vorticity. Then this framework yields an expression for the speed, $U$ , expressed dimensionlessly in terms of a Froude number, as a function of the relative depth of the dense fluid layer. Very often the outflow velocity is assumed to be piecewise constant in the layers with a discontinuity of velocity at the interface (i.e. a vortex sheet), but an alternative model pursued in this study features a velocity profile in which the velocity transitions between the layers over a finite thickness, denoted by $\unicode[STIX]{x1D702}$ , the magnitude of which emerges from the model. We term this latter class of models as ‘vortex-wake’ models, because the velocity field in the wake of the front or the jump is rotational. We note that although these approaches are related, the vortex-sheet model does not emerge as the limit of vanishing thickness of the vortex-wake model $(\unicode[STIX]{x1D702}\rightarrow 0)$ , because the thickness is part of the solution and not a free variable. The models coincide only in the special case for which $\unicode[STIX]{x1D702}=0$ is a solution and this corresponds to an energy conserving flow.

In what follows (§§ 3 and 4), we analyse the prediction from this framework for the flow in two configurations, namely the front of gravity currents and internal jumps. For each we show that a solution may be calculated on the basis of a vortex-sheet or vortex-wake idealisation for the motion. We note that these differing approaches invoke differing assumptions about the energy losses and/or diffusion of vorticity and they diverge somewhat from each other. The challenge is to reveal the conditions for agreement, because both models approximate the same physical problem. We further note that in the Boussinesq regime $((\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{1})/\unicode[STIX]{x1D70C}_{2}\ll 1)$ , the vortex-sheet problem for an internal jump coincides with the problem for a gravity current in the limit $h_{a}/H\rightarrow 0$ . In that case, the lower layer expands strongly, and hence $u_{B}/U\rightarrow 0$ (as for the gravity current problem). However, in the gravity current model, the stagnation pressure at $O$ is proportional to upper fluid density, $\unicode[STIX]{x1D70C}_{2}$ , and this makes a difference for the non-Boussinesq analysis. Therefore, in general, we need separate solutions for the gravity currents and internal jumps.

2.3 Dimensionless variables and governing equations

At this point it is convenient to introduce dimensionless variables. We scale lengths with respect to the channel height $H$ , velocities with respect to oncoming speed $U$ and densities with respect to the density of the less dense fluid, $\unicode[STIX]{x1D70C}_{2}$ . There are five dimensionless quantities. The Froude number, $Fr$ , and excess density ratio, $S$ , are given by

(2.17a,b ) $$\begin{eqnarray}Fr^{2}=\frac{U^{2}}{g^{\prime }h}\quad \text{and}\quad S=\frac{\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{2}}{\unicode[STIX]{x1D70C}_{2}},\end{eqnarray}$$

where the reduced gravity $g^{\prime }=Sg$ and we define $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}(z)/\unicode[STIX]{x1D70C}_{2}=S\;r(z)$ . There are also two dimensionless parameters that measure the dissipation along the channel boundaries. On the streamlines $DE$ and $BC$ , noting the independence of the velocity field to the streamwise coordinate at the outflow and inflow, we write

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{2}g^{\prime }h\unicode[STIX]{x1D6FF}_{t}\equiv -\int _{D}^{E}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x=-\int _{D}^{E}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}z^{2}}\;\text{d}x\equiv -\int _{D}^{E}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}{\unicode[STIX]{x2202}z}\;\text{d}x, & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{2}g^{\prime }h\unicode[STIX]{x1D6FF}_{b}\equiv -\int _{C}^{B}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u\;\text{d}x=-\int _{C}^{B}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}z^{2}}\;\text{d}x\equiv -\int _{C}^{B}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}{\unicode[STIX]{x2202}z}\;\text{d}x, & \displaystyle\end{eqnarray}$$

where in (2.18) the viscosity corresponds to that of the less dense fluid, while in (2.19) it corresponds to that of the fluid along $BC$ (see figure 1). The dimensionless parameters, $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ measure the head loss on the streamlines $DE$ and $BC$ , respectively. Equivalently they measure the diffusion of vorticity from the boundaries into the flow. Additionally the parameter $R=\unicode[STIX]{x1D70C}_{C}/\unicode[STIX]{x1D70C}_{2}$ , which equals $1$ for gravity currents and $1+S$ for internal jumps. Henceforth all variables will be assumed to be dimensionless; in particular the scaled downstream interface height varies between zero and unity $(0\leqslant h\leqslant 1)$ .

The balances over the control volume have yielded four independent expressions that must be satisfied by the motion. These are conservation of mass (2.14) and dense species (2.15), the consistency of the pressure distribution to ensure that it is single-valued (2.13) and the balance of streamwise momentum (2.16), which we simplify by substituting (2.11) and (2.18) for the pressure difference $p_{D}-p_{E}$ . In dimensionless form these are given by

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{1}u\;\text{d}z=1, & \displaystyle\end{eqnarray}$$

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{1}ru\;\text{d}z=h_{a}, & \displaystyle\end{eqnarray}$$

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{Fr^{2}}{2}(u_{E}^{2}-(1+S)u_{B}^{2}-(1-R))=\frac{1}{h}\left(\int _{0}^{1}r\;\text{d}z-h_{a}\right)-\unicode[STIX]{x1D6FF}_{t}+\unicode[STIX]{x1D6FF}_{b}, & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}\left(\frac{u_{E}^{2}}{2}-\int _{0}^{1}(1+Sr)u^{2}\;\text{d}z+\frac{1}{2}+Sh_{a}\right)+\unicode[STIX]{x1D6FF}_{t}=\frac{1}{h}\left(\int _{0}^{1}zr\;\text{d}z-\frac{h_{a}^{2}}{2}\right). & \displaystyle\end{eqnarray}$$

Finally we comment that in §§ 3 and 4, we will compute the rate of dissipation within the flow, denoted by $\dot{{\mathcal{D}}}$ and rendered dimensionless with respect to $\unicode[STIX]{x1D70C}_{2}(g^{\prime }h)^{3/2}h$ .

3.1 Vortex-sheet model $(\unicode[STIX]{x1D702}=0)$

The classical approach specifies a constant velocity $u_{E}$ over $h<z\leqslant 1$ . This implies that $\unicode[STIX]{x1D719}=0$ for $z>h$ and corresponds to a sharp velocity transition at the interface $z=h$ , a vortex sheet, and thus $\unicode[STIX]{x1D702}=0$ . The typical members of this group of models are the work of Benjamin (Reference Benjamin1968) and the recent circulation-based solutions (Borden & Meiburg Reference Borden and Meiburg2013a , Konopliv et al. Reference Konopliv, Llewellyn-Smith, McElwaine and Meiburg2016).

With $\unicode[STIX]{x1D702}=0$ , the previous results for mass conservation (3.4), pressure compatibility (3.3) and momentum balance (3.5) (already reduced to (3.9)–(3.10)) can be expressed as follows:

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}=2(1-h)^{2}(1+\unicode[STIX]{x1D6FF}_{b}-\unicode[STIX]{x1D6FF}_{t}), & \displaystyle\end{eqnarray}$$

(3.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}_{t}+\frac{h^{2}}{1-h^{2}}\unicode[STIX]{x1D6FF}_{b}=\frac{1}{2}h\frac{1-2h}{1-h^{2}}. & \displaystyle\end{eqnarray}$$

An important outcome (see (3.7)) is that the dimensionless head loss is constant, and equal to $\unicode[STIX]{x1D6FF}_{t}$ , on all the streamlines in the less dense ambient (except for the stagnation line $CO$ ). In this setting with $\unicode[STIX]{x1D702}=0$ , the dimensionless dissipation is $\dot{{\mathcal{D}}}=Fr\unicode[STIX]{x1D6FF}_{t}/h$ (see (3.6)).

The dilemma is the choice of $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ . As observed above, the attempt to impose $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}=0$ is compatible with (3.12)–(3.13) only at the points $h=0$ and $1/2$ . At any other value of $h$ , at least one of the $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ must be non-zero. Furthermore the attempt to set $\unicode[STIX]{x1D6FF}_{t}=0$ fails, because in this case (3.13) yields the unacceptable $\unicode[STIX]{x1D6FF}_{b}\sim 1/h$ for $h\ll 1$ .

Figure 2.

Schematic of the expected vorticity $\unicode[STIX]{x1D714}$ distribution in a vortex-sheet model of the flow around the front of gravity currents. In this ‘classical’ model, the vorticity vanishes $(\unicode[STIX]{x1D714}=0)$ on the boundaries of the control volume ( $ABCDE$ ) due to free-slip conditions and the assumptions of parallel in- and outflow, except for at the vortex sheet. Eddies are depicted to show vorticity generation at the interface between the fluids of different densities. Above the head of the current, we expect significant upward diffusion and advection of vorticity because of $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z<0$ and the vertical component of the velocity field is non-vanishing. We expect insignificant vorticity ( $\unicode[STIX]{x1D714}\approx 0$ ) in the domain above $CO$ , because the inflow is uniform. This scenario demonstrates that under the ‘classical’ model it is expected that the diffusion of vorticity from the boundary $OC$ is much smaller than the diffusion of vorticity from $DE$ . Thus $\unicode[STIX]{x1D6FF}_{t}\gg |\unicode[STIX]{x1D6FF}_{b}|\approx 0$ in general, and this supports Benjamin’s model for the front speed with $\unicode[STIX]{x1D6FF}_{b}=0$ in (3.14). Note that a positive $\unicode[STIX]{x1D6FF}_{b}$ (as predicted by the circulation-based $Fr$ ) requires $\unicode[STIX]{x1D714}<0$ above $OC$ .

We mentioned above that the dimensionless head losses along the boundary streamlines, $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ , result from flow-field features inside the control volume, and we recall the connection between $\unicode[STIX]{x1D6FF}_{t}$ , $\unicode[STIX]{x1D6FF}_{b}$ and vorticity fluxes at the top and bottom boundary (see (2.18)–(2.19)). Figure 2 shows the expected vorticity distribution. The top and bottom boundaries are free-slip with $\unicode[STIX]{x1D714}=0$ , the upstream flow is irrotational with $\unicode[STIX]{x1D714}=0$ , the dense fluid is motionless with $\unicode[STIX]{x1D714}=0$ . We cannot identify a mechanism that can justify the presence of a significant non-zero $\unicode[STIX]{x1D714}$ above the $CO$ line, and this implies $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z=0$ on $CO$ , and hence $\unicode[STIX]{x1D6FF}_{b}=0$ . On the other hand, $\unicode[STIX]{x1D714}>0$ is expected above the interface, a gradient $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z<0$ can be justified below the $DE$ boundary, and hence $\unicode[STIX]{x1D6FF}_{t}>0$ is the only physically consistent representation.

These observations yield simple guidelines that constrain the closure of the vortex-sheet models of gravity currents. There is no head loss on streamlines prior to contraction/expansion, i.e. upstream of $O$ $(\unicode[STIX]{x1D6FF}_{b}=0)$ and the head loss in the contracting layer must be positive, in general $(\unicode[STIX]{x1D6FF}_{t}>0)$ . The head loss terms reduce $Fr$ (see (3.12)), and hence the ideal $Fr_{I}$ is the upper bound of this result, as expected.

These conditions $\unicode[STIX]{x1D6FF}_{t}>0,\unicode[STIX]{x1D6FF}_{b}=0$ lead directly to the solution due to Benjamin (Reference Benjamin1968):

(3.14a,b ) $$\begin{eqnarray}Fr^{2}=Fr_{B}^{2}\equiv \frac{(2-h)(1-h)}{1+h};\quad \unicode[STIX]{x1D6FF}_{t}=\frac{1}{2}h\frac{1-2h}{1-h^{2}}.\end{eqnarray}$$

Note that our discussion of feasible values of $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ was not based upon energetic considerations. Rather, we have framed the discussion in terms of vorticity. Moreover, by the same logic we infer that $\unicode[STIX]{x1D6FF}_{t}<0$ is not feasible, and hence the range of validity of (3.14) is $h\in (0,1/2]$ .

A key weakness in this argument based upon vorticity is the reliance on the presence of a significant, non-vanishing gradient $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z$ in the entire contracted/accelerated fluid, which generates head loss in all the streamlines, including the top one. There is little convincing physical justification for the presence of significant $\unicode[STIX]{x1D714}$ far away from the interface at high Reynolds number. However, this must be accepted as a mathematical simplification needed for the classical model. Since having adopted the control volume analysis, we have no means for setting the details of the flow inside the control volume. Our decision to set a vortex sheet $(\unicode[STIX]{x1D702}=0)$ at the outflux boundary can be sustained only if the head loss $\unicode[STIX]{x1D6FF}_{t}$ is homogeneous in the fluid above the interface. The less dense fluid contracts in a domain close to the interface in the control volume and there is no doubt that $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z<0$ there. The best we can do with the vortex-sheet model is to demand consistency of the sign and magnitude between the physical gradient of vorticity ( $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z$ ) and the modelled head loss, $\unicode[STIX]{x1D6FF}_{t}$ .

A pseudo-inviscid solution for the Froude number, $Fr$ , has been suggested by the circulation-based analysis in Borden & Meiburg (Reference Borden and Meiburg2013a ) and Konopliv et al. (Reference Konopliv, Llewellyn-Smith, McElwaine and Meiburg2016). These papers argue that the viscous contribution to the vorticity balance (3.12) vanishes, but do allow for the effects of head loss and dissipation in the momentum and energy balances (which implies $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}>0$ ). Mathematically, this yields $Fr(h)=Fr_{I}\equiv \sqrt{2}(1-h)$ (see (3.11)) and (from (3.13))

(3.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}={\textstyle \frac{1}{2}}h(1-2h).\end{eqnarray}$$

We consult again figure 2. The positive $\unicode[STIX]{x1D6FF}_{t}$ requires a negative $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z$ in the domain above the current, and this is consistent with the expected distribution of vorticity, as for Benjamin’s solution discussed above. On the other hand, the positive $\unicode[STIX]{x1D6FF}_{b}$ requires $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z<0$ at the $CO$ boundary, which needs a significant negative $\unicode[STIX]{x1D714}$ in the upstream domain, to the right of the stagnation point $O$ (see figure 2). In our opinion, there is no simple mechanism that can justify a negative $\unicode[STIX]{x1D714}$ above the $OC$ line. The incoming flow field is irrotational and diffusion from the interface is not expected to supply negative $\unicode[STIX]{x1D714}$ either. Consequently, the solution based on the $\unicode[STIX]{x1D6FF}_{b}=\unicode[STIX]{x1D6FF}_{t}>0$ scenario appears not to be physically acceptable. We emphasise that this conclusion is based on vorticity, not energy, arguments. Both the Benjamin and circulation-based results yield predictions for the Froude number, $Fr$ , that are quantitatively close, and both find $\dot{{\mathcal{D}}}>0$ in the same range of validity $(0<h\leqslant 1/2)$ . The novelty of the interpretation developed in this paper is that energy-dissipation considerations are less incisive than vorticity-distribution arguments in the selection of top/bottom head loss closures for jump models.

We note in passing that there is ambiguity in the circulation-based derivation concerning the status of the $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}>0$ component. Borden & Meiburg (Reference Borden and Meiburg2013a ) and Konopliv et al. (Reference Konopliv, Llewellyn-Smith, McElwaine and Meiburg2016) derive $Fr$ from the inviscid form of (3.12) and then substitute this $Fr$ into the momentum balance and obtain (3.15), apparently as a consequence of their result. These separate calculations obscure the distinction between the assumptions that underpin the idealisation and the predictions that it affords. In our opinion, the use of the inviscid form of (3.12) is valid only in a steady-state situation in which the momentum equation (2.23) is also satisfied and hence only under the condition $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}$ . We showed above that the governing system reduces to (3.12) and (3.13), which provides two equations for the three unknowns $Fr$ , $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ . Hence any solution requires an additional expression linking the variables. In any case, the observations that $\unicode[STIX]{x1D6FF}_{t}>0$ in general, and that $\unicode[STIX]{x1D6FF}_{b}>0$ appears to be physically unacceptable, indicate that imposing $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}$ may not be a good modelling assumption, although it does coincide with the inviscid result $Fr_{I}$ and thus forms an upper bound for $Fr$ .

Figure 3.

Schematic of the vortex-wake model for flow around the front of a gravity current. Vorticity is only non-vanishing within the layer of thickness $\unicode[STIX]{x1D702}$ where the velocity transitions from $u=0$ to $u=u_{E}$ . Outside this transition layer the vorticity vanishes, and thus there is no head loss on the boundaries $BC$ and $DE$ (and so $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}=0$ ).

3.2 Vortex-wake model

Our new approach is to close the integral model of mass and momentum balances using the conditions $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}=0$ on the top and bottom boundaries, but to allow the vortex layer to have non-vanishing thickness (see figure 3). This is in accord with our expectation that the ambient flow is mostly irrotational, therefore $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z=0$ on and near the $z=0,H$ boundaries, and, according to (2.18)–(2.19), $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}=0$ . On the other hand, the viscous and rotational effects exist in a layer close to the interface, and this justifies the finite thickness of the transition region in the velocity field (i.e. there is a vortex layer of thickness $\unicode[STIX]{x1D702}$ instead of the vortex sheet as assumed in the classical models).

The governing equations are (3.2)–(3.5) with $\unicode[STIX]{x1D6FF}_{t}=\unicode[STIX]{x1D6FF}_{b}=0$ , and the dissipation is given by (3.6). We immediately deduce from (3.3) that $Fr\;u_{E}=\sqrt{2}$ and then from (3.4) and (3.5) that

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}=2\left(1-\int _{0}^{1}f\;\text{d}z\right)^{2}, & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}=\frac{h\left(1-\displaystyle \int _{0}^{1}f\;\text{d}z\right)^{2}}{2\displaystyle \int _{0}^{1}f\;\text{d}z+\left(\displaystyle \int _{0}^{1}f\;\text{d}z\right)^{2}-2\displaystyle \int _{0}^{1}f^{2}\;\text{d}z}. & \displaystyle\end{eqnarray}$$

We substitute for $f(z)$ from (3.2) and eliminate $Fr^{2}$ to find that the thickness $\unicode[STIX]{x1D702}$ satisfies the following quadratic equation:

(3.18) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{2}\left(\int _{0}^{1}\unicode[STIX]{x1D719}(s)\;\text{d}s\right)^{2}+2\unicode[STIX]{x1D702}\left((1+h)\int _{0}^{1}\unicode[STIX]{x1D719}(s)\;\text{d}s-\int _{0}^{1}\unicode[STIX]{x1D719}(s)^{2}\;\text{d}s\right)+\frac{h(2h-1)}{2}=0.\end{eqnarray}$$

It is noteworthy that this quadratic has real roots for $0\leqslant h\leqslant 1/2$ , which is the domain of validity. Also from (3.6), the dimensionless dissipation is given by

(3.19) $$\begin{eqnarray}\dot{{\mathcal{D}}}=\frac{\sqrt{2}\unicode[STIX]{x1D702}}{h}\int _{0}^{1}(2\unicode[STIX]{x1D719}-\unicode[STIX]{x1D719}^{2})(1-\unicode[STIX]{x1D719})\;\text{d}s.\end{eqnarray}$$

If we now impose a linear variation of the velocity field within the wake $\unicode[STIX]{x1D719}(s)=1-s$ , we find the wake thickness, Froude number and dissipation are given by

(3.20) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{L}\equiv \frac{2}{3}\left(\sqrt{1+\frac{21}{2}h}-1-3h\right),\quad Fr=Fr_{L}\equiv \frac{\sqrt{2}}{3}\left(4-\sqrt{1+\frac{21}{2}h}\right)\\ \quad \text{and}\quad \displaystyle \dot{{\mathcal{D}}}=\dot{{\mathcal{D}}}_{L}\equiv {\displaystyle \frac{\sqrt{2}\unicode[STIX]{x1D702}_{L}}{4h}},\end{array}\right\}\quad & & \displaystyle\end{eqnarray}$$

where the subscript $L$ refers to assumption of a linear variation within the wake. These dependencies are plotted in figures 4–6, where we observe that dissipation vanishes when $h=1/2$ and reaches a maximum when $h\rightarrow 0$ . Notably, when $h\ll 1$ we find that

(3.21a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{L}=\frac{3h}{2}+\cdots \,,\quad Fr_{L}=\sqrt{2}+\cdots \quad \text{and}\quad \dot{{\mathcal{D}}}_{L}=\frac{3\sqrt{2}}{8}+\cdots \,.\end{eqnarray}$$

Furthermore, when $|1/2-h|\ll 1$ we find that

(3.22a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{L}=\frac{3}{5}\left(\frac{1}{2}-h\right)+\cdots \,,\quad Fr_{L}=\frac{\sqrt{2}}{2}+\cdots \quad \text{and}\quad \dot{{\mathcal{D}}}_{L}=\frac{3}{5}\left(\frac{1}{2}-h\right)+\cdots \,.\end{eqnarray}$$

Figure 4.

The Froude number, $Fr$ , as a function of dimensionless gravity current depth, $h$ , for various models: Benjamin, $Fr_{B}$ ; circulation, $Fr_{C}$ ; vortex-wake linear, $Fr_{L}$ ; vortex-wake exponential, $Fr_{E}$ .

Conversely if the velocity variation is exponential with $\unicode[STIX]{x1D719}(s)=\exp (-\unicode[STIX]{x1D6FC}s)$ , then using $\exp (-1/\unicode[STIX]{x1D702})\ll 1$ , the wake thickness and Froude number are given by

(3.23) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{E}\equiv \frac{\unicode[STIX]{x1D6FC}}{2}\left(\sqrt{1+6h}-1-2h\right),\quad Fr=Fr_{E}\equiv \frac{1}{\sqrt{2}}\left(3-\sqrt{1+6h}\right)\\ \quad \text{and}\quad \dot{{\mathcal{D}}}=\dot{{\mathcal{D}}}_{E}\equiv {\displaystyle \frac{5\sqrt{2}\unicode[STIX]{x1D702}_{E}}{6\unicode[STIX]{x1D6FC}h}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where the subscript $E$ refers to the exponentially varying velocity. It is interesting to note that neither the Froude number, $Fr_{E}$ , nor the dissipation $\dot{{\mathcal{D}}}_{E}$ depend upon the value of $\unicode[STIX]{x1D6FC}$ . Furthermore, if we choose $\unicode[STIX]{x1D6FC}=3$ , which corresponds to the velocity field being within 5 % of its asymptotic value at $z=h+\unicode[STIX]{x1D702}$ , then $\unicode[STIX]{x1D702}_{L}=\unicode[STIX]{x1D702}_{E}=3h/2+\cdots \,$ when $h\ll 1$ .

Figure 5.

The dissipation, $\dot{{\mathcal{D}}}$ , as a function of dimensionless gravity current depth, $h$ , for various models: Benjamin, $\dot{{\mathcal{D}}}_{B}$ ; circulation, $\dot{{\mathcal{D}}}_{C}$ ; vortex-wake linear, $\dot{{\mathcal{D}}}_{L}$ ; vortex-wake exponential, $\dot{{\mathcal{D}}}_{E}$ .

Figure 6.

The thickness of the vortex layer, $\unicode[STIX]{x1D702}$ , and the relative thickness of the layer, $\unicode[STIX]{x1D702}/h$ , as functions of the dimensionless depth of the gravity current for the linear, $\unicode[STIX]{x1D702}_{L}$ , and exponential (with $\unicode[STIX]{x1D6FC}=3$ ),  $\unicode[STIX]{x1D702}_{E}$ , models of the vortex wake.

Both of these dependencies are plotted in figures 4–6. We note that the predicted Froude numbers from these vortex-wake formulations are very close to those derived by Benjamin (Reference Benjamin1968) and Borden & Meiburg (Reference Borden and Meiburg2013a ) – and the vortex-wake predictions precisely coincide with the inviscid result (3.11) (and with the vortex-sheet models) when $h=0$ and $h=1/2$ , The largest discrepancy with the Benjamin result is $4\,\%$ for the linear profile, and only $2\,\%$ for the exponential profile. This is a remarkable agreement. The discrepancies with the circulation-based model of Borden & Meiburg (Reference Borden and Meiburg2013a ) are a little larger (see figure 4).

The rate of dissipation is shown in figure 5. Both the classical ( $\unicode[STIX]{x1D702}=0$ ) and vortex-wake ( $\unicode[STIX]{x1D6FF}_{b}=\unicode[STIX]{x1D6FF}_{t}=0$ ) models yield the same qualitative behaviour: a positive dissipation $\dot{{\mathcal{D}}}$ which vanishes when $h=1/2$ and which is maximised when $h=0$ . Quantitatively, the new model is slightly less dissipative. The wake model of Klemp et al. (Reference Klemp, Rotunno and Skamarock1994) led to broadly similar results. However their approach did not explicitly calculate the thickness of the wake, $\unicode[STIX]{x1D702}$ , and was based upon a linearisation that neglected the term of order $\unicode[STIX]{x1D702}\unicode[STIX]{x1D719}^{2}$ . We note that had this approximation been made in (3.16)–(3.18), then we would have recovered their result for the Froude number, $Fr$ .

In figure 6 we show the thickness of the velocity-transition layer for both the linear and exponential variations ((3.20) and (3.23), respectively), where for the latter we choose $\unicode[STIX]{x1D6FC}=3$ , so that $\unicode[STIX]{x1D719}(1)<0.05$ (i.e. the layer thickness corresponds to the position where the velocity field is within 5 % of its asymptotic value). The interesting feature is that $\unicode[STIX]{x1D702}/h$ is finite and non-zero for relatively thin currents $h\ll 1$ ; the wakes are also relatively thick in this regime. Additionally, the wake thickness vanishes for $h=1/2$ and at this value $Fr=Fr_{I}=1/\sqrt{2}$ .

4.1 Vortex-sheet model $(\unicode[STIX]{x1D702}=0)$

In a vortex-sheet model of the flow downstream from the internal jump, both the density and velocity fields are discontinuous at the interface $z=h$ , and in terms of the expression for the velocity field given in (4.2), we impose $\unicode[STIX]{x1D713}(s)=s/|s|$ . Then from (4.3) and (4.4) we deduce

(4.9a,b ) $$\begin{eqnarray}u_{B}=\frac{h_{a}}{h}\quad \text{and}\quad u_{E}=\frac{1-h_{a}}{1-h}.\end{eqnarray}$$

Substituting these expressions into (4.5) and (4.6) we find two coupled algebraic equations for three variables, $Fr$ , $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ , given by

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{Fr^{2}}{2}\left(\left(\frac{1-h_{a}}{1-h}\right)^{2}-(1+S)\frac{h_{a}^{2}}{h^{2}}+S\right)=1-\frac{h_{a}}{h}+\unicode[STIX]{x1D6FF}_{b}-\unicode[STIX]{x1D6FF}_{t}, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}\left(\frac{1}{2}\left(\frac{1-h_{a}}{1-h}\right)^{2}-\frac{(1-h_{a})^{2}}{(1-h)}-(1+S)\frac{h_{a}^{2}}{h}+\frac{1}{2}+Sh_{a}\right)+\unicode[STIX]{x1D6FF}_{t}=\frac{h^{2}-h_{a}^{2}}{2h}. & \displaystyle\end{eqnarray}$$

Since the velocity field on the outflow is piecewise constant, a Bernoulli balance yields that head loss on each streamline in the upper fluid is $\unicode[STIX]{x1D6FF}_{t}$ , while it is $\unicode[STIX]{x1D6FF}_{b}$ in the lower fluid. We may thus evaluate the dimensionless dissipation in the flow, which can be rewritten as

(4.12) $$\begin{eqnarray}\dot{{\mathcal{D}}}=\frac{Fr}{h}(\unicode[STIX]{x1D6FF}_{t}(1-h_{a})+\unicode[STIX]{x1D6FF}_{b}h_{a}).\end{eqnarray}$$

We may readily identify this expression for the dissipation as the sum from each layer. Physical considerations demand that the total dissipation must be positive semi-definite $\dot{{\mathcal{D}}}\geqslant 0$ , but importantly this criterion does not imply that both layers must be dissipative.

To complete the model, we must invoke a closure assumption because our systems is reduced to three variables $(Fr,\unicode[STIX]{x1D6FF}_{t},\unicode[STIX]{x1D6FF}_{b})$ and two equations (4.10) and (4.11). This situation is entirely analogous to the problem for gravity current fronts – and in the context of internal jumps, different investigators have adopted different strategies (see Wood & Simpson Reference Wood and Simpson1984; Klemp et al. Reference Klemp, Rotunno and Skamarock1997; Li & Cummins Reference Li and Cummins1998; Borden & Meiburg Reference Borden and Meiburg2013b ; Baines Reference Baines2016). We find that it is convenient to write $\unicode[STIX]{x1D6FF}_{b}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF}_{t}$ and then eliminating $\unicode[STIX]{x1D6FF}_{t}$ between (4.10) and (4.11), we find that

(4.13) $$\begin{eqnarray}Fr^{2}=\frac{(h-h_{a})(1+\textstyle \frac{1}{2}(\unicode[STIX]{x1D706}-1)(h+h_{a}))}{h(q_{1}+(\unicode[STIX]{x1D706}-1)q_{2})},\end{eqnarray}$$

where

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle q_{1}=\frac{1}{2}\left(\frac{(1-h_{a})^{2}}{(1-h)^{2}}-(1+S)\frac{h_{a}^{2}}{h^{2}}+S\right), & \displaystyle\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle q_{2}=\frac{1}{2}\frac{(1-h_{a})^{2}}{(1-h)^{2}}-\frac{(1-h_{a})^{2}}{1-h}-(1+S)\frac{h_{a}^{2}}{h}+\frac{1}{2}+Sh_{a}. & \displaystyle\end{eqnarray}$$

We may then recover different formulae by substituting different values of $\unicode[STIX]{x1D706}$ (cf. Li & Cummins Reference Li and Cummins1998). Wood & Simpson (Reference Wood and Simpson1984) assumed that there was no dissipation in the contracting layer and so $\unicode[STIX]{x1D6FF}_{t}=0$ ( $\unicode[STIX]{x1D706}\rightarrow \infty$ ). In contrast Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) assumed that there was no dissipation in the expanding layer, expressed as $\unicode[STIX]{x1D6FF}_{b}=0$ and so $\unicode[STIX]{x1D706}=0$ . Finally, and most recently, Borden & Meiburg (Reference Borden and Meiburg2013b ) and Baines (Reference Baines2016) assumed a form of vorticity conservation that is equivalent to assuming that the dissipation per unit mass along the bounding streamlines $DE$ and $BC$ is equal ( $\unicode[STIX]{x1D6FF}_{b}=(1+S)\unicode[STIX]{x1D6FF}_{t}$ ). We also note that that if $\unicode[STIX]{x1D706}=1$ then the deduced Froude number from (4.13) is equal to the ideal, ‘inviscid’ result (4.7), although the dissipation is not necessarily vanishing in this case. It is evident therefore that all previous expressions for the speed of the bore require, or imply, a closure assumption.

Previous investigations invested considerable effort into the question of which of these many variants is preferable. These studies do not provide a theoretical criterion for the choice of $\unicode[STIX]{x1D706}$ , and actually there is no reason why $\unicode[STIX]{x1D706}$ should be a constant. The main comparison has been between measurement of the speed of the jump, $U$ , in laboratory experiments and Navier–Stokes simulations, and the model predictions, along with qualitative arguments concerning the dissipation. In general, the conclusion is that the model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) performs better than that of Wood & Simpson (Reference Wood and Simpson1984) (although the predicted $U$ often exceeds the data). The circulation-based models of $U$ are close to, but slightly larger than, the Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) results. However there are difficulties in drawing firm conclusions from comparisons with laboratory experiments and simulation data, because they are usually restricted to a relatively narrow range of flow parameters (e.g. the flows are often Boussinesq and with equal viscosities). The assumptions of the modelled flow (e.g. free-slip boundaries, laminar steady state, Schmidt number $Sc\rightarrow \infty$ ) do not conform precisely to either laboratory experiments or computational simulations. Moreover, there is no theoretical interpretation or criterion to underpin the modelling and closure assumptions.

Figure 7.

Schematic of the vorticity, $\unicode[STIX]{x1D714}$ , distribution in a two-layer jump under the classical model of a vortex sheet at $A$ . The vorticity vanishes ( $\unicode[STIX]{x1D714}=0$ ) at the boundaries of the control volume, except for at the vortex sheet. The eddies indicate vorticity generation and note that $\unicode[STIX]{x1D714}>0$ in the lower (expanding) layer. This implies that $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z>0$ , and $\unicode[STIX]{x1D6FF}_{b}<0$ . This scenario demonstrates that under the ‘classical’ model of a vortex sheet that it is inconsistent to set $\unicode[STIX]{x1D6FF}_{b}=(1+S)\unicode[STIX]{x1D6FF}_{t}$ since they must be opposite signed and furthermore because the total dissipation must be positive, there must be dissipation in the upper contracting layer (thus supporting the model of Klemp et al. (Reference Klemp, Rotunno and Skamarock1997)).

We suggest a more self-contained approach. We exploit again the connection between the boundary head losses, $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ , and the boundary gradients of vorticity, $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z$ (see (2.18) and (2.19)). The expected vorticity distribution is shown in figure 7. The lower layer expands, the upper contracts, and hence $u_{2}>u_{1}$ and $\unicode[STIX]{x1D714}>0$ is produced at the interface. In view of the $\unicode[STIX]{x1D714}=0$ boundary conditions at $z=0$ and $1$ , $(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z)>0$ in the region with fluid of density $\unicode[STIX]{x1D70C}_{1}$ and $(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}z)<0$ in the region with fluid of density $\unicode[STIX]{x1D70C}_{2}$ . Thus according to (2.18)–(2.19), we deduce that $\unicode[STIX]{x1D6FF}_{t}>0$ and $\unicode[STIX]{x1D6FF}_{b}<0$ . In other words, the upper layer is dissipative, while the lower one gains energy: see (4.12). Immediately this implies that the model due to Wood & Simpson (Reference Wood and Simpson1984) is problematic: by imposing $\unicode[STIX]{x1D6FF}_{t}=0$ and $\unicode[STIX]{x1D6FF}_{b}>0$ , it prevents the dissipation effect from the proper layer, and enforces it on the wrong layer. The model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) with $\unicode[STIX]{x1D6FF}_{t}>0$ and $\unicode[STIX]{x1D6FF}_{b}=0$ , on the other hand, is consistent with the vorticity field in the upper (major) layer, while the needed negative $\unicode[STIX]{x1D6FF}_{b}$ is approximated by assuming that it vanishes. The circulation-based model with $\unicode[STIX]{x1D6FF}_{t}>0,\unicode[STIX]{x1D6FF}_{b}>0$ is consistent with the vorticity field in the upper layer, but imposes the wrong sign in the lower layer. On balance, therefore, the model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) appears to be the most consistent compromise. We emphasise that we used no arguments concerning the energetics of the flow in our assessment of the models with different $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ , in contrast with the widely accepted suggestion of Li & Cummins (Reference Li and Cummins1998).

The expressions of head loss on streamlines (2.18)–(2.19) reveal that $\unicode[STIX]{x1D6FF}_{t}$ and $\unicode[STIX]{x1D6FF}_{b}$ can be interpreted as distributed viscous drag forces per unit area, which arise due to shear at the interface downstream of the internal jump. The contracting upper layer moves faster than the expanding lower layer, and hence a positive drag is expected in this domain, $\unicode[STIX]{x1D6FF}_{t}>0$ . By the same argument, $\unicode[STIX]{x1D6FF}_{b}<0$ – and therefore the inferred signs of these terms are in agreement with the argument developed above. Moreover, in vorticity balance (A 2), the term $\unicode[STIX]{x1D6FF}_{b}-\unicode[STIX]{x1D6FF}_{t}$ can be interpreted as the torque of the distributed drag forces, and we expect it to be negative (clockwise).

Together, these observations yield a simple guiding principle for the closures for classical (vortex-sheet) models of internal jumps. In general, the head loss in the contracting layer must be positive, and in the expanding layer negative. From (4.10), we see that the head loss terms $\unicode[STIX]{x1D6FF}_{t}>0$ and $\unicode[STIX]{x1D6FF}_{b}<0$ lead to a reduction in predicted speed of the jump, $Fr$ and the inviscid expression, $Fr_{I}$ , is the upper bound of this result, as expected. We note again that the situation $\unicode[STIX]{x1D6FF}_{b}=\unicode[STIX]{x1D6FF}_{t}$ , which also reproduces $Fr_{I}$ , contradicts the observation that $\unicode[STIX]{x1D6FF}_{b}>0$ and therefore lacks physical justification.

The validity of the models is traditionally restricted by requiring the total dissipation $\dot{{\mathcal{D}}}\geqslant 0$ . We comment that this is not a sharp criterion in general, because some models employ non-physical $\unicode[STIX]{x1D6FF}_{b}>0$ , and this distorts the value of $\dot{{\mathcal{D}}}$ . This is in particular problematic for the model due to Wood & Simpson (Reference Wood and Simpson1984) and circulation-based models (Borden & Meiburg Reference Borden and Meiburg2013a ; Baines Reference Baines2016). The model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) imposes vanishing dissipation in the lower layer ( $\unicode[STIX]{x1D6FF}_{b}=0$ and $\dot{{\mathcal{D}}_{1}}=0$ ) and thus demands that the upper layer alone is dissipative (so that $\dot{{\mathcal{D}}}>0$ ), whereas the discussion above indicates that $\unicode[STIX]{x1D6FF}_{b}<0$ is consistent with the anticipated distribution of vorticity.

We conclude that considerations of the vorticity distribution are sharper than the energy-dissipation arguments (although the latter are still of value). For example, these considerations predict that the model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) is preferable to the other closures – and we know that it is in broad agreement with the data. However, the new insights are still mostly qualitative. We argue with confidence that the ratio $\unicode[STIX]{x1D6FF}_{b}/\unicode[STIX]{x1D6FF}_{t}<0$ should be negative and finite. A model with $\unicode[STIX]{x1D6FF}_{b}/\unicode[STIX]{x1D6FF}_{t}<0$ will be an improved substitute to the Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) model ( $\unicode[STIX]{x1D6FF}_{b}=0$ ), but we do not have a reliable estimate for the optimal value of this ratio.

4.2 The vortex-wake model

Figure 8.

Schematic of the vorticity $\unicode[STIX]{x1D714}$ distribution in an internal jump with a finite, non-vanishing thickness transition layer over which the velocity varies from $u_{B}$ to $u_{E}$ . Vorticity is non-vanishing within this layer, but vanishes everywhere else within the domain.

In this section we examine a different closure, namely the existence of a finite thickness layer through which the velocity varies from $u_{B}$ to $u_{E}$ (see figure 8). Within this layer there is non-vanishing vorticity generated by the baroclinic torque associated with the expansion of the lower, relatively dense layer. We examine the dynamics integrated over a control volume enclosing the jump on the assumption that streamlines of the motion have become parallel with the channel boundaries at the segment $BE$ and the dissipative effects at the boundary vanish, so that $\unicode[STIX]{x1D6FF}_{b}=\unicode[STIX]{x1D6FF}_{t}=0$ .

The velocity is assumed to transition between $z=h-\unicode[STIX]{x1D702}$ and $z=h+\unicode[STIX]{x1D702}$ on the assumption that the viscosities of the two fluids are comparable (and the extension to differing viscosities and the resultant asymmetric velocity fields is tackled below). It will also be assumed, and later confirmed, that $\unicode[STIX]{x1D702}<h$ and $\unicode[STIX]{x1D702}<1-h$ , so that the transition layer is separate from the channel boundaries. Here we model a linear variation within the layer, so that $\unicode[STIX]{x1D713}(s)=s$ for $|s|<1$ and $\unicode[STIX]{x1D713}(s)=s/|s|$ for $|s|>1$ , and in appendix B we analyse an alternative that yields almost identical results.

The flow is modelled using (2.20)–(2.23), for which we need the following integrals:

(4.16a,b ) $$\begin{eqnarray}\int _{0}^{1}f\;\text{d}z=1-2h,\quad \int _{0}^{h}f\;\text{d}z=-h+\frac{1}{2}\unicode[STIX]{x1D702},\end{eqnarray}$$

(4.17a,b ) $$\begin{eqnarray}\int _{0}^{1}f^{2}\;\text{d}z=1-\frac{4}{3}\unicode[STIX]{x1D702},\quad \int _{0}^{h}f^{2}\;\text{d}z=h-\frac{2}{3}\unicode[STIX]{x1D702}.\end{eqnarray}$$

Conservation of mass and solute then yields

(4.18a,b ) $$\begin{eqnarray}u_{E}(1-h)+u_{B}h=1\quad \text{and}\quad u_{B}(h-\textstyle \frac{1}{4}\unicode[STIX]{x1D702})+\textstyle \frac{1}{4}u_{E}\unicode[STIX]{x1D702}=h_{a}.\end{eqnarray}$$

The requirement of a single-valued pressure field given by

(4.19) $$\begin{eqnarray}Fr^{2}(u_{E}^{2}-(1+S)u_{B}^{2}+S)=\frac{2(h-h_{a})}{h}.\end{eqnarray}$$

Finally the balance of streamwise momentum leads to

(4.20) $$\begin{eqnarray}\displaystyle & & \displaystyle Fr^{2}\left[u_{E}^{2}\left(h-\frac{1}{2}+\frac{\unicode[STIX]{x1D702}}{3}-\frac{S\unicode[STIX]{x1D702}}{12}\right)-u_{B}^{2}\left(h(1+S)-\frac{\unicode[STIX]{x1D702}}{3}-\frac{5S\unicode[STIX]{x1D702}}{12}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.-\,u_{E}u_{B}\left(\frac{2}{3}+\frac{S}{3}\right)\unicode[STIX]{x1D702}+\frac{1}{2}+Sh_{a}\right]=\frac{h^{2}-h_{a}^{2}}{2h}.\end{eqnarray}$$

It is possible to solve these algebraic equations to find that the dimensionless speed of the internal jump, expressed as $Fr$ , satisfies

(4.21) $$\begin{eqnarray}Fr^{2}=\frac{1}{4h}\frac{[4h(1-h)-\unicode[STIX]{x1D702}]^{2}}{[(Sh-(S+1))\unicode[STIX]{x1D702}+2(S(1-h)^{2}.(h+h_{a})-2hh_{a}+h+h_{a})]}.\end{eqnarray}$$

Note that this expression reduces to the inviscid result $Fr=Fr_{I}$ (see (4.7)) when $\unicode[STIX]{x1D702}=0$ . The thickness of the transition layer is determined by the quadratic equation

(4.22) $$\begin{eqnarray}\displaystyle & & \displaystyle 3S\unicode[STIX]{x1D702}^{2}-2[3Sh(h-3h_{a})+6Sh_{a}+(S+5)(h_{a}-h)]\unicode[STIX]{x1D702}\nonumber\\ \displaystyle & & \displaystyle \quad -\,12(h_{a}-h)^{2}(S(1-h)^{2}-2h+1)=0.\end{eqnarray}$$

It is noteworthy that this expression has real solutions when $h_{a}<h<(S+1-\sqrt{1+S})/S$ and that $\unicode[STIX]{x1D702}$ vanishes as the largest and smallest values of $h$ in this range, yielding the ‘inviscid’ solution for $Fr=Fr_{I}$ (4.7). Furthermore, for Boussinesq jumps $(S=0)$ , we immediately deduce that

(4.23a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D702}=\frac{6}{5}(h-h_{a})(1-2h)\quad \text{and}\quad Fr^{2}=\frac{(4h^{2}-7h-3h_{a}(1-2h))^{2}}{20h(3h^{2}+h+4h_{a}(1-2h))}.\end{eqnarray}$$

Dissipation is a by-product of this calculation. By substituting the assumed forms of the velocity we may evaluate the dimensionless dissipation in the lower and upper layers, denoted by $\dot{{\mathcal{D}}}_{1}$ and $\dot{{\mathcal{D}}}_{2}$ , respectively, and given by

(4.24a,b ) $$\begin{eqnarray}\dot{{\mathcal{D}}}_{1}=-\frac{Fr^{3}\unicode[STIX]{x1D702}(1+S)}{64h}(u_{E}-u_{B})(u_{E}+3u_{B})^{2}\quad \text{and}\quad \dot{{\mathcal{D}}}_{2}=\frac{Fr^{3}\unicode[STIX]{x1D702}}{64h}(u_{E}-u_{B})(3u_{E}+u_{B})^{2}.\end{eqnarray}$$

Thus we deduce that since $u_{B}<u_{E}$ there is in general some energy gain in the lower layer since $\dot{{\mathcal{D}}}_{1}<0$ . In the upper layer there is dissipation $(\dot{{\mathcal{D}}}_{2}>0)$ and in general for Boussinesq systems for which $u_{B}\ll u_{E}$ , $\dot{{\mathcal{D}}}_{2}\gg |\dot{{\mathcal{D}}}_{1}|$ . We reiterate that this result is an intrinsic outcome of the vortex-wake solution and furthermore we note that the flow is only non-dissipative when $\unicode[STIX]{x1D702}=0$ , which occurs at $h=(S+1-\sqrt{1+S})/S$ , in addition to the trivial cases $h=h_{a}$ (and $u_{B}=u_{E}$ ).

Figure 9.

Properties of an internal jump as functions of the dimensionless downstream thickness of the jump, $h$ , for various dimensionless upstream thicknesses: (a) the scaled thickness of the vortex-wake, $\unicode[STIX]{x1D702}/h$ ; (b) the Froude number of the jump, $Fr$ ; (c) the dissipation in each layer, $\dot{{\mathcal{D}}}_{1}({<}0)$ and $\dot{{\mathcal{D}}}_{2}$ (dashed lines) and the total dissipation $\dot{{\mathcal{D}}}_{1}+\dot{{\mathcal{D}}}_{2}$ (solid lines). In (b) and (c), the prediction from the vortex-sheet model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1994) are also plotted (dot-dash lines). In (b) the data points are from the simulations reported by Borden et al. (Reference Borden, Meiburg and Constantinescu2012b ) for $h_{a}=0.2$ $(\circ )$ , $h_{a}=0.1$ (▫) and $h_{a}=0.05~(\times )$ .

Figure 10.

Properties of an internal jump as functions of the dimensionless downstream thickness of the jump, $h$ , for various dimensionless upstream thicknesses: (a) the scaled thickness of the vortex-wake, $\unicode[STIX]{x1D702}/h$ ; (b) the dimensionless speed of the jump, $V$ ; (c) the total dissipation $\dot{{\mathcal{D}}}$ (solid lines). In (b) and (c), the prediction from the vortex-sheet model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1994) is also plotted (dot-dash lines). In (b) the data points are from the simulations reported by Borden et al. (Reference Borden, Meiburg and Constantinescu2012b ) for $h/h_{a}=1.3~(\times )$ and $h/h_{a}=1.3$ (▫).

We illustrate our results in figures 9 and 10 for the Boussinesq regime $(S=0)$ . In figure 9, we plot the scaled thickness, $\unicode[STIX]{x1D702}/h$ , the Froude number of the jump, $Fr$ , and dissipation, $\dot{{\mathcal{D}}}_{1}$ and $\dot{{\mathcal{D}}}_{2}$ , as functions of the dimensionless downstream thickness of the jump, $h$ , along with the predictions from the vortex-sheet model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) (for which $\unicode[STIX]{x1D706}=0$ in (4.13)) and with results from Navier–Stokes simulations due to Borden et al. (Reference Borden, Meiburg and Constantinescu2012b ). We find that in general the predictions of the bore speed from the vortex-wake model are somewhat slower than the vortex-sheet model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) and are in general in quite good agreement with the data from the numerical simulations. (Other vortex-sheet models lead to higher predicted velocities and are not plotted in figure 9.) The predicted dissipation is slightly higher in the vortex-wake model than in the vortex-sheet model and the lower layer always exhibits a negative dissipation $\dot{{\mathcal{D}}}_{1}<0$ for the parameters studied here, although it is of significantly smaller magnitude than $\dot{{\mathcal{D}}}_{2}$ .

From figure 9, we also note that $\unicode[STIX]{x1D702}/h$ exhibits a maximum for $h_{a}<h<1$ . In the Boussinesq regime this is easy to evaluate (see (4.23)) – it occurs at $h=(h_{a}/2)^{1/2}$ and at this height of downstream jump, $\unicode[STIX]{x1D702}/h=6(1-{2h_{a}}^{1/2})^{2}/5$ . Thus the model of a symmetric wake becomes invalid when $\unicode[STIX]{x1D702}>h$ , which in the Boussinesq regime occurs when $h_{a}<(11-2\sqrt{30})/12=0.00374$ . For greater density differences between the layers, this threshold value of $h_{a}$ increases, but as discussed below such flows are modelled better by an asymmetric wake (see § 4.3).

The predictions are also plotted as functions of the dimensionless depth of the upstream fluid, $h_{a}$ , in figure 10 for $h/h_{a}=1.3$ and $1.9$ , and the results are also compared with numerical simulations of the Navier–Stokes equations (Borden et al. Reference Borden, Meiburg and Constantinescu2012b ) in the Boussinesq regime $(S=0)$ . In this figure we examine the dependence of the dimensional bore speed scaled by $(\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gh_{a}/\unicode[STIX]{x1D70C}_{1})^{1/2}$ and in terms of the dependent variables used here, this dimensionless speed $V=Fr((1+S)h/h_{a})^{1/2}$ . We again note the utility of the vortex-wake model in that it is able to reproduce the simulation data more accurately than the vortex-sheet model due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997).

4.3 Asymmetric wakes

An interesting flexibility of the vortex-wake model is that the layer over which the velocity transitions between $u_{B}$ and $u_{E}$ need not be of equal thickness above and below the elevation $z=h$ , in particular when the fluids differ strongly in viscosities and densities. This effect may be included by writing $u=(u_{B}+u_{E})/2+(u_{E}-u_{B})f(z)/2$ and demanding a piecewise model for $f(z)$ in $z>h$ and $z<h$ . Denoting the thickness of the transition zones in the lower and upper layers by $\unicode[STIX]{x1D702}_{1}$ and $\unicode[STIX]{x1D702}_{2}$ , respectively, we impose $f(-\unicode[STIX]{x1D702}_{1})=-1$ and $f(\unicode[STIX]{x1D702}_{2})=1$ , while for continuity of velocity and shear stress, we enforce

(4.25a,b ) $$\begin{eqnarray}f(h^{-})=f(h^{+})\quad \text{and}\quad \unicode[STIX]{x1D707}_{1}f^{\prime }(h^{-})=\unicode[STIX]{x1D707}_{2}f^{\prime }(h^{+}).\end{eqnarray}$$

If we assume that there is linear dependence upon position within the transition layer, we find that

(4.26) $$\begin{eqnarray}f=\frac{1}{1+\unicode[STIX]{x1D6EC}}\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D6EC}-1+2{\displaystyle \frac{(z-h)}{\unicode[STIX]{x1D702}_{1}}},\quad & h-\unicode[STIX]{x1D702}_{1}<z<h,\\ \unicode[STIX]{x1D6EC}-1+2{\displaystyle \frac{(z-h)}{\unicode[STIX]{x1D702}_{2}}},\quad & h<z<h+\unicode[STIX]{x1D702}_{2},\end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D702}_{1}\unicode[STIX]{x1D707}_{2})/(\unicode[STIX]{x1D702}_{2}\unicode[STIX]{x1D707}_{1})$ . To close the problem then, we require an expression in addition to the conservation expressions derived above.

We proceed by arguing that in the steady laminar state, the thickness of the transition layer at a downstream distance, $x$ , is proportional to $[\unicode[STIX]{x1D707}x/(\unicode[STIX]{x1D70C}U)]^{1/2}$ . Thus the ratio of the thickness $\unicode[STIX]{x1D702}_{1}/\unicode[STIX]{x1D702}_{2}=[(\unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D70C}_{2})/(\unicode[STIX]{x1D707}_{2}\unicode[STIX]{x1D70C}_{1})]^{1/2}$ . We shall test our formulation against recent simulations of the Navier–Stokes equations by Ungarish, Borden & Meiburg (Reference Ungarish, Borden and Meiburg2014) that examined non-Boussinesq jumps when the dynamic viscosities of the fluids are equal. Thus we write $\unicode[STIX]{x1D702}_{1}/\unicode[STIX]{x1D702}_{2}=(1+S)^{1/2}$ and we analyse the conservation equations to deduce the thickness of the lower layer $\unicode[STIX]{x1D702}_{1}$ and the dimensionless speeds of the jump, $Fr$ . Results are plotted in figure 11, along with results from simulations (Ungarish et al. Reference Ungarish, Borden and Meiburg2014) and from the vortex-sheet model of Klemp et al. (Reference Klemp, Rotunno and Skamarock1997). We see that the scaled velocity, $V$ , decreases with increasing $\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1}$ and that the vortex-wake model captures the dependency quite accurately. Strongly non-Boussinesq models (with $S>3$ , say) may require a complicated analysis because of their tendency to adopt single layer dynamics, as pointed out by Ungarish et al. (Reference Ungarish, Borden and Meiburg2014), and because of undular wave production, which is not captured by this steady analysis (Borden et al. Reference Borden, Koblitz and Meiburg2012a ).

Figure 11.

The rescaled speed of an internal jump, $V$ , as a function of the density ratio, $(1+S)^{-1}$ , for various ratios of the downstream to upstream thicknesses of the lower layer ( $h/h_{1}=2.3,1.85,1.35$ ). Also plotted are the predictions due to Klemp et al. (Reference Klemp, Rotunno and Skamarock1997) (dot-dash line) and the results from Navier–Stokes simulations by Ungarish et al. (Reference Ungarish, Borden and Meiburg2014).