2.1. The steady-state flow pattern

Following Ungarish (Reference Ungarish2006), we start with the solution of a two-dimensional stratified steady flow field over a rigid free-slip bottom topography (that mimics the dense fluid) in a channel with an upper free-slip horizontal lid at $z=H$; see figure 1. The $\{x,z\}$ system is a frame of reference attached to the GC, the origin $O$ is the front stagnation point, the velocity components are $\{u,w\}$ and gravity acts in the $-z$ direction. We use dimensional variables unless stated otherwise. The idea is to explore the similarity between the flow configuration of a GC with an available result called Long's model (Long Reference Long1953, Reference Long1955) concerning the flow of a stratified fluid in a channel with an upper horizontal solid top and a prescribed bottom topography. As in the non-stratified case, the far-upstream velocity of the ambient is constant over the height of the channel, $0 \leqslant z \leqslant H$. This flow climbs the topography (current) at a relatively slow pace (compared with the horizontal propagation). The streamlines become horizontal again in the far-downstream region of the thinner channel, $h \leqslant z \leqslant H$. For an observer moving with the current (the bottom topography), the flow is steady.

In the non-stratified case, the downstream flow of the ambient fluid regains a uniform velocity, $UH/(H-h)$, and a pressure field that is linear with $z$. This facilitates the calculations and simplifies the results. However, when $S>0$, the linear density stratification in the upstream region must be compressed in a non-trivial manner into a thinner channel. The downstream flow field of the ambient fluid develops a quite complex $z$-dependent structure of velocity, density and pressure. The first task is to specify this flow field.

The geometry is like that of figure 1, but the coloured domain is a part of the solid bottom (the obstacle). To be specific, the obstacle (or topography) encountered by the unperturbed stratified fluid is defined by the bottom elevation function, $z=\chi (x)$. The value of $\chi$ is $0$ (i.e. no obstacle) for non-positive $x$. For $x>0$, the bottom is elevated to $\chi (x) >0$; and far downstream at the left, a parallel geometry is achieved again with $\chi (x) = h = \text {const.}>0$.

The far-upstream flow (at the right, $x \rightarrow - \infty$), where the bottom is flat, $z=\chi (x) = 0$, consists of parallel horizontal streamlines with constant velocity $U$ and a prescribed stable linearly changing density. Using the subscript $r$ (right) to denote this region, we write

(2.1a–c)\begin{equation} u_r(z) = U, \quad w_r(z) = 0, \quad \rho_{r}(z) = \rho_b - \widetilde{\Delta\rho} \times (z/ H), \end{equation}

where $\widetilde {\Delta \rho } = \rho _b - \rho _{o} = S(\rho _c - \rho _{o})$ in general, and $\widetilde {\Delta \rho } = \rho _c - \rho _{o}$ for the intrusion ($S=1$) case that is the topic of this paper.

Under the assumption of a two-dimensional steady Boussinesq inviscid flow, the analysis of Long (Reference Long1953) can be applied to reduce the set of governing Euler equations to a single partial differential equation (PDE) for the displacement of the streamline, $\delta (x,z)$, subject to the obvious $\delta = 0$ in the upstream right region, and $\delta =\chi (x)$ at the bottom. In the steady flow, streamlines and pathlines coincide, and hence the initial (upstream) density is conserved along these lines and allows the application of clear-cut conditions to the downstream domain for the flow over a simple bottom topography. Variants of the derivation method can be found in the literature; e.g. Shapiro (Reference Shapiro1992) includes the density variation by an Exner function, while Shivamoggi & Rollins (Reference Shivamoggi and Rollins2004) use a streamfunction for the modified velocity components $(\rho (z)/\rho _{o})^{{1/2}}\{u,w\}$. Mathematical manipulations produce a PDE for $\delta (x,z)$ (or the closely related streamfunction $\psi$), which, for some simple (but physically relevant) upstream conditions is a linear Helmholtz equation amenable to analytical solution. Here we use the analytical solution for the present configuration as reported in Baines (Reference Baines1995, chap. 5). The flow field is conveniently expressed with the aid of the perturbation (about the upstream flow) streamfunction, $-\partial \psi /\partial z =u'$, $\partial \psi /\partial x = w$, where $u=U+u'$. The result reads

(2.2)$$\begin{gather} \psi(x,z)=U \times \chi(x) \, \frac{\sin[\beta(1-z/H)]} {\sin[\beta(1-\chi(x)/H)]} = U \times \delta (x,z), \end{gather}$$

(2.3)$$\begin{gather}\rho(x,z)= \rho_b + (\rho_{b}-\rho_{o}) \left [-\frac{z}{H} + \frac{\chi(x)}{H}\frac{\sin[\beta(1-z/H)]} {\sin [\beta (1-\chi(x)/H)]} \right ], \end{gather}$$

where

(2.4)\begin{equation} \beta = \frac{[(\rho_{b}/\rho_{o}-1) g H]^{{1/2}}}{U} = \frac{(S g' H)^{{1/2}}}{U} = \frac{\mathcal{N} H}{U}. \end{equation}

We combine this result with the presumed steady-state bottom GC. We replace the solid bottom obstacle with a stationary fluid of density $\rho _{c}$ in the domain $(x \geqslant 0,\ 0 \leqslant z \leqslant \chi (x))$. The $\{x,z\}$ system is now a frame of reference attached to the GC, and the origin $O$ is the front stagnation point. We assume that, like in the non-stratified case, under certain conditions the structure of the parallel horizontal far-upstream and downstream flow regions of the ambient, given by (2.2) and (2.3), is preserved. In this system, the right upstream flow, where $\chi (x) = 0$, is unchanged and given by (2.1a–c). In the left (subscript $l$) region, where $\chi (x) = h$, the parallel horizontal flow satisfies $w_l=0$ and

(2.5)$$\begin{gather} u_l(z) = \left \{\begin{array}{ll} 0 & ( 0 \leqslant z < h), \\ U \left \{ 1 + \dfrac{ a}{{1-a}}\dfrac {{{\gamma}}}{\sin{{\gamma}}}\cos \left [ \dfrac {{{\gamma}}}{{1-a}}\left(1-\dfrac{z}{H}\right) \right ] \right \} & ( h \leqslant z \leqslant H), \end{array}\right. \end{gather}$$

(2.6)$$\begin{gather} \rho_l(z) = \left \{ \begin{array}{ll} \rho_{c} & ( 0 \leqslant z < h), \\ \rho_r(z) + \widetilde{\Delta\rho} \dfrac{a}{\sin {{\gamma}}} \sin \left[\dfrac{{{\gamma}}}{{1-a}}\left(1-\dfrac{z}{H}\right) \right] & ( h \leqslant z \leqslant H), \end{array} \right. \end{gather}$$

and

(2.7)$$\begin{gather} \delta_l(z) = H \frac {a}{\sin {{\gamma}}} \sin \left[\frac {{{\gamma}}}{{1-a}}\left(1-\frac{z}{H}\right) \right] \quad (h \leqslant z \leqslant H), \end{gather}$$

where

(2.8)\begin{equation} {{\gamma}} = ({1-a}) \sqrt{S}/(\sqrt{{a}} \hat{U}) , \end{equation}

and, again, $\widetilde {\Delta \rho } = \rho _b - \rho _{o}$ in general, and $\widetilde {\Delta \rho } = \rho _c - \rho _{o}$ for the intrusion ($S=1$) case.

We can verify by substitution that $\delta _l(z = H) =0$ (the horizontal top streamline) and $\delta _l(z = h) = h$ (the streamline displaced from the bottom on the right to the interface with the current on the left). We also note that this flow satisfies the free-slip boundary condition $(\partial u_{l}/\partial z) =0$ at $z=H$. The substitution ${{\gamma }} =0$ yields undefined $0/0$ terms in the solution, and hence the non-stratified flow is calculated as the $S \to 0$ limit.

The application of these results for GCs with small and moderate values of $S$ can be found in Ungarish (Reference Ungarish2006) and White & Helfrich (Reference White and Helfrich2008).

2.1.1. Switch to intrusion

Here we focus attention on the special case of the symmetric intrusion which corresponds to $\rho _b = \rho _c$ and $S=1$. Therefore, in the subsequent analysis, $\rho _b$ ‘disappears’ (unless specified otherwise) and we keep in mind that the density at the bottom, $z=0$, of the ambient fluid is $\rho _c$. For the convenience of the reader, we repeat some definitions using $S=1$. Now $g' = \mathcal {N}^{2} H$ and ${{\gamma }} = (1-a)/(\sqrt {a} \hat {U})$. Again, the speed of propagation of the intrusion is $U$. The pertinent scaled speed of propagation, referred to as the Froude number of the intrusion, is

(2.9)\begin{equation} \hat{U} = Fr = \frac{U}{(g' h)^{{1/2}}} = \frac{U}{\mathcal{N} H \sqrt{a}}. \end{equation}

We note that different reference speeds like (1) $\mathcal {N} h$ and (2) $2 \mathcal {N} H$ are relevant and have also been used in the literature. The transformation of the present $Fr$ is achieved by multiplication with the factor $1/\sqrt {a}$ for scaling (1) and $\sqrt {a}/2$ for scaling (2). (The scaling (2) is used in figure 4 and the value is denoted $Fr_2$.) Because of the various reference speeds that make sense, it is not possible to refer to an accepted value of $Fr$ for intrusions that is of the order of unity over a significant range of $a$.

Typical profiles of density and $u$ in the downstream (left) flow of a symmetric intrusion, as predicted by (2.5) and (2.6), are shown in figure 2.

Figure 2.

Typical profiles of density and velocity in the downstream (figure 1) flow for intrusion $\rho _b = \rho _c$, predicted by (2.5) and (2.6). Note the sharp change of $u(z)$ in the vortex sheet and the free slip $\partial u/ \partial z =0$ at the top. The values of ${{\gamma }}$ and $a$ illustrated here do not satisfy the CV balances.

In our analysis, the subscripts A–E indicate the value at the corresponding point on the boundary of the CV sketched in figure 1; and $\textrm {{A}}^{-}$ ($\textrm {{A}}^{+}$) refer to point A just below (respectively above) the intrusion–ambient interface.

2.2. Pressure field

The pressure is continuous and a single-valued function. In the horizontal flow $w_r=0$ and $w_l =0$ domains, the steady-state $z$-momentum equations reduce to the hydrostatic balance $\partial p/\partial z = -\rho (z) g$, where $p$ is the pressure, and hence we obtain the following.

1. (1) For the CD boundary:

   (2.10)\begin{equation} p_r(z) = p_C - g \int_0^{z} \rho_r(z')\,\textrm{d}z' \quad \text{or} \quad p_r(z) = p_D + g \int_z^{H} \rho_r(z')\,\textrm{d}z' . \end{equation}

2. (2) For the BE boundary:

   (2.11)\begin{equation} p_l(z) = p_B - g \int_0^{z} \rho_l(z')\,\textrm{d}z' \quad \text{or} \quad p_l(z) = p_E + g \int_z^{H} \rho_l(z')\,\textrm{d}z'. \end{equation}

Since $\rho _r(z)$ and $\rho _l(z)$ are known, see (2.1a–c) and (2.6), we can express the pressures on the vertical boundaries of the CV analytically. However, there are constants of integration $p_B$, $p_C$, $p_D$ and $p_E$ that require further specification. One of them (say $p_D$) can be set to zero; for the others, we employ the dynamic connection along horizontal streamlines (Bernoulli equation) and pressure-loop continuity.

On the streamlines along the horizontal boundaries, the ambient fluid may display head loss. There is convincing evidence from previous investigations that the ideal Bernoulli equation (with no head loss) leads to some contradictions in the subsequent analysis. (A short discussion of the need for and justification of the head loss effect is given in Appendix B.) To incorporate this effect into the Bernoulli equation, we introduce the dimensionless $\varDelta _b$ and $\varDelta _t$ (in general, functions of $a$) for the bottom and top lines, and hence the modified Bernoulli equation yields

(2.12a,b)\begin{equation} p_B = p_O = p_C + \tfrac{1}{2} \rho_{c} U^{2} - \rho_{o} g'h \varDelta_b, \quad p_E = p_D + \tfrac{1}{2} \rho_{o} (U^{2} - u_E^{2}) - \rho_{o} g'h \varDelta_t \end{equation}

(the first equation, (2.12a), uses the $u_B = u_O =0$ stagnation-point condition).

Note that the head loss $\varDelta _t$ defined at the top implies that the same head loss is present on all the streamlines of the ambient on the AE boundary. The $\varDelta _t$ value affects $p_E$ along the top streamline $z=H$, and then this head loss is carried down from point E to $z< H$ by the hydrostatic pressure (second equation of (2.11)), without changing the profiles of $u_l(z)$ and $\rho _l(z)$. In other words, $\varDelta _t$ represents a loss of pressure (i.e. energy) in the entire domain of downstream ambient, not only on the top streamline, which is consistent with Long's solution. Formally, Long's flow is derived from inviscid Euler equations that are expected to conserve energy (zero head loss), but since the solution is an integral of a PDE, it can accommodate some constants. This implies that some internal friction in the transition flow from left to right may be assumed, under the condition that it produces the same head loss on all the streamlines. Therefore, the simulation of Khodkar, Allam & Meiburg (Reference Khodkar, Allam and Meiburg2018) based on the Navier–Stokes (not Euler) equations with free-slip boundary conditions, at a large $Re$, is a relevant test for the present $Fr$ result. Moreover, since the head loss is the same for the entire ambient flow, the rate of energy dissipation $\dot {\mathcal {D}}$ of the CV is simply $\rho _{o} U H g' h \varDelta _t$. Therefore, the $\varDelta _t=0$ situation is called energy-conserving, and results with $\varDelta _t \geqslant 0$ are energetically valid.

In the present formulation, $\varDelta _b$ does not contribute to the energy dissipation because it is confined to the stagnation line. This renders the sign of $\varDelta _b$ inconclusive. A decrease of the stagnation pressure ($\varDelta _b > 0$) makes sense, but a small increase cannot be excluded. In any case, it is clear that: (a) both $\varDelta _{b}$ and $\varDelta _{t}$ are expected to be small, otherwise the CV solution is dominated by the details of the internal flow, which were supposed to be excluded from the analysis; and (b)  there is no reason to expect equality of $\varDelta _b$ and $\varDelta _t$, not in magnitude and not even in sign. In any case, at least one of the $\varDelta _{b}$ and $\varDelta _{t}$ must be a non-zero function of $a$ for a non-trivial $Fr(a)$ result.

In the two-fluid flow field, an equilibrium (steady state) can be maintained only for certain values of $U$ (i.e. $Fr(a)$) determined by CV balances and other physical consideration. The necessary balances of volume and mass continuity are satisfied by the flow field (2.5)–(2.6), while the pressure is given by (2.10)–(2.12a,b). The other requirements are considered below.

2.3. Momentum balance

Following Benjamin, we consider the momentum balance for the rectangular CV. The assumptions of steady state and vanishing $x$-component viscous stress (on the boundaries and in the horizontal flow domains) impose the flow-force balance

(2.13)\begin{equation} \int^{H}_0 (\rho_lu_l^{2} - \rho_ru_r^{2}) \,\textrm{d} z = \rho_{o} \left [ \int_h^{H} u^{2}_l(z) \,\textrm{d}z - H U^{2} \right ] = \int_0^{H} [ p_r(z) - p_l(z) ] \,\textrm{d}z. \end{equation}

Here the Boussinesq simplification, $\rho _l \approx \rho _r \approx \rho _{o}$, was applied to the convection terms.

The evaluation of the integral of the dynamic $u_l^{2}(z)$ term upon use of (2.5) is straightforward. The pressure terms are also known, as explained above, from (2.10)–(2.12a,b). However, in the calculation of the pressure integral of (2.13), we can use either the first or the second of equations (2.12a,b). In the first case, the flow-force balance (2.13) is reduced to an equation for ${{\gamma }}(a)$ that contains the parameter $\varDelta _b$; in the second case, we obtain an equation for ${{\gamma }}(a)$ that contains the parameter $\varDelta _t$. The equations are not the same, which means that $\varDelta _b$ and $\varDelta _t$ are not independent, and, in particular, at least one of the $\varDelta _b$ and $\varDelta _t$ must be non-zero. It is convenient to focus on the following two cases.

1. (1) We use (2.12a) to connect the pressures of the left and right sides. We obtain

   (2.14)\begin{align} &\frac{{\hat{U}}^{2}}{2( {1-a})}[ 1 + a -2a^{2}+ a^{2} ( {{\gamma}} ^{2} + ( {{\gamma}} \cot{{\gamma}} )^{2} + {{\gamma}} \cot {{\gamma}} ) ] \nonumber\\ &\quad = \frac{1}{2}a - \frac{1}{3} a^{2} + ({1-a})^{2} \frac{1- {{\gamma}} \cot {{\gamma}} }{ {{\gamma}} ^{2}} + \varDelta_b. \end{align}
   Recall that ${\hat {U}}^{2} = (1-a)^{2} /(a {{\gamma }} ^{2})$. Next, we follow the classical approach of Benjamin: assume $\varDelta _b$ = 0. We obtain, after some algebra, the flow-force balance in the form
   (2.15)\begin{equation} f({{\gamma}}) = {1-a} + a (2-a) {{\gamma}} \cot {{\gamma}} + (a {{\gamma}} \cot {{\gamma}})^{2} - \frac{1}{3} a^{2} {{\gamma}} ^{2} \frac{a}{{1-a}} = 0. \end{equation}
   The root(s) of this equation, for given $a$, provide the desired solution $Fr(a)= \hat {U} = (1-a)/(a^{{1/2}} {{\gamma }})$ for the case $\varDelta _b =0$. This is called model B (for Benjamin).

2. (2) We use (2.12b) to connect the pressures of the left and right sides. We obtain

   (2.16)\begin{align} &\frac{{\hat{U}}^{2}}{2({1-a})} [ 1 + a -2a^{2} + a^{2} ( {{\gamma}} ^{2} + ( {{\gamma}} \cot{{\gamma}} )^{2} + {{\gamma}} \cot {{\gamma}} ) ] \nonumber\\ &\quad - \frac{{\hat{U}}^{2}}{2} \left[ 1 + \frac{ a}{{1-a}}\frac {{{\gamma}}}{\sin {{\gamma}}} \right ]^{2} = \varPsi({{\gamma}}) + \varDelta_t, \end{align}
   where
   (2.17)\begin{equation} \varPsi({{\gamma}}) ={-} \frac{1}{3} a^{2} + ({1-a})^{2} \left( 1 - \frac{{{\gamma}}}{ \sin {{\gamma}} }\right) \frac{1}{{{\gamma}} ^{2}} - (1-a) \frac{a }{{{\gamma}} \sin {{\gamma}}} ( 1 - \cos {{\gamma}} ) . \end{equation}
   Again, recall that ${\hat {U}}^{2} = (1-a)^{2} /(a {{\gamma }} ^{2})$. Next, if we assume $\varDelta _t = 0$, we obtain
   (2.18)\begin{align} f_1({{\gamma}}) = (1-a) \left[ 2 (1-a) + a {{\gamma}} \cot {{\gamma}} - 2 \frac{{{\gamma}}}{\sin {{\gamma}}} \right] - \left(\frac{a {{\gamma}}}{\sin {{\gamma}}} \right)^{2} - 2{{\gamma}}^{2} \varPsi ({{\gamma}}) = 0. \end{align}
   The root(s) of this equation, for given $a$, provide the desired solution $Fr(a)= \hat {U} = (1-a)/(a^{{1/2}} {{\gamma }})$ for the case $\varDelta _t =0$. This is called model EC (for energy-conserving).

To finish the solution, we recall that both (2.14) and (2.16) must be satisfied. Thus, if one of the $\varDelta _b$ and $\varDelta _t$ is set zero to calculate ${{\gamma }}(a)$, the other equation should be used to calculate the other non-zero $\varDelta$.

2.4. Dissipation and stability

In the physical context of the intrusion problem, the solution ${{\gamma }}(a)$ that satisfies volume, mass and momentum balances of the CV (e.g. roots of (2.15) or (2.18)) must be subjected to some further requirements. Only real-valued positive roots of $f({{\gamma }})$ or $f_1({{\gamma }})$ are of interest, and several branches ${{\gamma }}_1 < {{\gamma }}_2 < {{\gamma }}_3 < \cdots$ must be taken into account. An inspection reveals that $f({{\gamma }})$ and $f_1({{\gamma }})$ are singular, ${\sim }(\sin {{\gamma }})^{-2}$ for ${{\gamma }} = k {\rm \pi}, k=1,2, \ldots,$ and hence the vicinity of these points is excluded from the search for roots. In general, for a given $a$, several real-valued distinct roots, ${{\gamma }}_1 < {{\gamma }}_2 < \cdots$ were found. However, after further tests (see below), it turns out that (with few exceptions) only the first root is relevant.

In general, the flow dissipates energy. As mentioned above, Long's solution admits a constant head loss $\varDelta _t$ for all the streamlines from right to left, except for the stagnation line CO with a possibly different $\varDelta _b$. For a given result $Fr(a)$ (i.e.  a known combination of $a$ and ${{\gamma }}$), the corresponding $\varDelta _b$ and $\varDelta _t$ must be calculated (from (2.14) or (2.16), as relevant) and inspected for sign and magnitude, as discussed later.

For some values of ${{\gamma }}$, the flow field on the left, (2.5)–(2.6), displays negative $u$ (referred to as a kinematically invalid solution) and unstable $\partial \rho /\partial z >0$. We observe that these effects both depend on the magnitude of the coefficient

(2.19)\begin{equation} \theta = \left \{ \begin{array}{ll} 0 & (0 < {{\gamma}} \leqslant {\rm \pi}/{2}), \\ \dfrac{a}{{1-a}} {{\gamma}} |{\cot{{\gamma}}}| & ({\rm \pi}/{2} < {{\gamma}} < {\rm \pi}), \\ \dfrac{a}{{1-a}} \dfrac{{{\gamma}}}{|{\sin{{\gamma}}}| } & ( {\rm \pi}< {{\gamma}}). \end{array}\right. \end{equation}

For a kinematically valid and stable flow for $S \in (0,1]$, the condition $\theta \leqslant 1$ is required, and therefore we shall reject solutions that produce $\theta > 1$. The domains of stability as a ${{\gamma }}/{\rm \pi}$–$a$ diagram are shown in figure 3.

Figure 3.

Stability diagram $\gamma /{\rm \pi}$–$a$. The solid black line is $\theta =1$, and above this line is the domain of instability (also marked by the large red cross) where $\theta >1$.

The flow with $U = V = \mathcal {N} H/{\rm \pi}$ is called critical, and the situation $U>V$ is defined as supercritical. Substitution into (2.8) (with $S=1$) of $U = V = \mathcal {N} H/{\rm \pi}$ shows that the line $a =1 - {{\gamma }}/{\rm \pi}$ separates the two domains, as shown by the dashed blue line in figure 3. This diagram derived from (2.8) is a harbinger of peculiar results: a significant part of the stable domain is occupied by fast (supercritical or close-by) intrusions. The subsequent analysis confirms this anticipation.

We note that figure 3 has also been used for GCs with small and moderate $S$ by Ungarish (Reference Ungarish2006). However, when $S$ is small (weak stratification) a fast (compared to $V$ of the internal wave) propagation is a common occurrence, because the difference $\rho _{c}-\rho _b$ enhances the driving force, in contrast to the case of intrusion ($\rho _b= \rho _c$).

2.5. Vorticity considerations

The relevant vorticity component is $\omega = \partial u/ \partial z - \partial w / \partial x$. Consider the CV: one has $\omega = 0$ on the horizontal boundaries $z = 0, H$, on the vertical inflow boundary CD (the constant horizontal $U$ condition) and on the $\textrm {{BA}}^{-}$ boundary (the stagnant current). On the outflow boundary $\textrm {{A}}^{+}\textrm {E}$, the ambient fluid carries a significant vorticity, $\omega = \partial u/\partial z$, which can be calculated from (2.5). In addition, vorticity may be present in a vortex sheet at $z = h$ (compressed into point A on the BE boundary). We note that the flux of vorticity along this boundary from point B (at $z = 0$ where $u_B = 0$) to some other point M (at $0 < z \leqslant H$) is given by

(2.20)\begin{equation} \int_B^{M} \omega u \,\textrm{d}z = \int_B^{M} u \frac{\partial u}{\partial z}\,\textrm{d}z = \tfrac{1}{2} u_M^{2}. \end{equation}

Equation (2.20) incorporates the vortex sheet as point M moves from $\textrm {{A}}^{-}$ to $\textrm {{A}}^{+}$.

Ungarish & Hogg (Reference Ungarish and Hogg2018) pointed out the connection between the vorticity balance and the pressure continuity over the boundary of the CV. We proceed as follows. Starting at point D, we calculate the pressure at point E, first in the clockwise sense and then in the opposite direction, using the balances (2.10)–(2.12a,b). The equality of the two $p_E$ results (the pressure is single-valued) yields

(2.21)\begin{equation} \frac{1}{2} u_E^{2} = g' h \left [ \frac{1}{2} a + \frac{1-a}{{{\gamma}} \sin {{\gamma}}} (1 - \cos {{\gamma}}) \right ] + g' h (\varDelta_b - \varDelta_t) . \end{equation}

The same procedure for point A in the ambient (i.e. $\textrm {{A}}^{+}$) gives

(2.22)\begin{equation} \tfrac{1}{2}u_A^{2} = g' h (\varDelta_b - \varDelta_t). \end{equation}

Evidently, (2.21) and (2.22) are vorticity balances. The term on the left-hand side is the vorticity outflux according to (2.20). The right-hand side in these equations expresses the effects that support and control the vorticity outflow: the term in the square brackets is the baroclinic torque, and the next term expresses the vorticity contribution of the head loss mechanism. The contribution of the head loss can be interpreted as a torque, because $\varDelta _{b}$ and $\varDelta _{t}$ affect the pressure distribution on the horizontal boundaries of the CV. However, this contribution can also be interpreted as diffusion of vorticity on these boundaries because of the connection pointed out by equation (B1) and associated discussion. For this reason, the $\varDelta _b - \varDelta _t = 0$ case is regarded as based on ‘conservation’ of vorticity or circulation.

There is a significant difference from the system of GCs in the homogeneous ambient, whose entire vorticity outflux is performed in the vortex sheet from the stationary current to the ambient, which moves with $z$-constant velocity. Therefore, in the homogeneous case, there is no difference between points $\textrm {{A}}^{+}$ and E; see Appendix A.

In the intrusion system, the vorticity flux at point $\textrm {{A}}^{+}$ (just above $z=h$ in the ambient) lacks baroclinic torque. This is because there is no density difference between the bottom streamline and the core of the intrusion. The vorticity at $\textrm {{A}}^{+}$ is controlled by the dissipative term $(\varDelta _b - \varDelta _t)$. This has interesting consequences. Using (2.5) for $z/H = h/H = a$, we rewrite (2.22) as

(2.23)\begin{equation} U^{2} [ 1 + {a}{{\gamma}} \cot {{\gamma}} /(1-a) ]^{2} = 2g' h (\varDelta_b - \varDelta_t). \end{equation}

We conclude that the standard Benjamin-type assumption, $\varDelta _b = 0$, is problematic in general, because it requires a negative $\varDelta _t$. If we also impose $\varDelta _t =0$, (2.23) requires $a {{\gamma }} \cot {{\gamma }} = - (1-a)$. Substitution into the flow-force balance (2.15) shows that the only solution is the trivial ${{\gamma }} = 0$. As anticipated, the solution with $\varDelta _t=\varDelta _b =0$ is irrelevant.

Here it is useful to go back to the $S<1$ case, to elucidate why the vorticity balance at point A is not in conflict with the $\varDelta _b = 0$ assumption. The counterpart of (2.22) is

(2.24)\begin{equation} \tfrac{1}{2} u_A^{2} = g' h \left [ (1-S) \right ] + g' h (\varDelta_b - \varDelta_t). \end{equation}

A baroclinic torque $\propto (1-S)$ is present. It is evident that, for small and moderate values of $S$, Benjamin's postulate $\varDelta _b=0$ can coexist well with a positive and small $\varDelta _t$. Moreover, assuming $0<\varDelta _t <0.1$, we estimate that for $S<0.9$ Benjamin's model will work along the usual pattern. This explains why the GC solutions of Ungarish (Reference Ungarish2006) and White & Helfrich (Reference White and Helfrich2008) did not encounter the $\varDelta _t<0$ difficulties of the intrusion.

Thus, contrary to the homogeneous-ambient GC, for which $\varDelta _b=0$ is the standard and successful assumption introduced by Benjamin, in the present intrusion case, $\varDelta _b =0$ is problematic. The $Fr$ models must be reconsidered, as done next.