2. Topology, geometry and vorticity: a few examples from the literature

The conceptual framework of the present investigation is historically well-known, and lies at the very heart of the foundations of Euler’s hydrodynamics. In this section we briefly present two examples of the subtle interplay between the topology of a fluid’s boundary and vorticity generation, by focusing on the similarities that make a unifying viewpoint possible. While these examples are not directly related to the core of our investigation (as they differ, respectively, in the setting and in the boundary conditions for the velocity), they can help provide some motivations and a common stream of ideas.

As mentioned in the Introduction, Klein’s ‘Kaffeelöffel’ gedanken experiment (Klein Reference Klein1910) provides one of the most intriguing example of such a class of phenomena. Let us briefly discuss it following Saffman (Reference Saffman1992). Consider a plate of width $2\,L$ moving through an irrotational, homogeneous and incompressible Euler fluid with velocity $U$ normal to the plate. In the lab frame with horizontal coordinates $x$ , taken in the direction of the motion, and $y$ , along the plate, the appropriate complex velocity potential ${\it\Phi}$ is

(2.1) $$\begin{eqnarray}{\it\Phi}({\it\varsigma})=\,U\cdot ({\it\varsigma}-\sqrt{L^{2}+{\it\varsigma}^{2}}),\quad {\it\varsigma}=x+\text{i}\,y.\end{eqnarray}$$

This yields an effective vortex sheet bound to the plate, with vorticity

(2.2) $$\begin{eqnarray}{\it\omega}(x,y)=2U\,{\it\delta}(x)\,\frac{y}{\sqrt{L^{2}-y^{2}}}\end{eqnarray}$$

$(x=0,y\in [-L,L])$ , with ${\it\delta}(x)$ the Dirac delta-function. With the plate in place, no streamline crossing it can be drawn, hence the circulation around any closed contour in the fluid domain vanishes. When the plate is instantaneously removed (or, as suitable in a thought-experiment, is dissolved into the fluid) the vortex sheet (2.2) is transferred to the fluid and becomes free to evolve with time. After the plate removal, one can envisage closed contours drawn in the fluid domain crossing the segment $(x=0,y\in [-L,L])$ . Along such contours the circulation may not vanish, in general. Thus the generation of non-trivial circulation can be ascribed to the different topologies of the fluid domain, which is simply connected after the plate has melted away, while it is topologically non-trivial when the plate is present. From a local viewpoint, the origin of vorticity is the jump across the plate of the $y$ -component of the velocity $v=\pm U\,y/\sqrt{L^{2}-y^{2}}$ for $x\rightarrow 0^{\pm }$ (which diverges at the tips $y=\pm L$ of the plate).

Recent experimental investigations by Wibawa et al. (Reference Wibawa, Steele, Dahl, Rival, Weymouth and Triantafyllou2012) demonstrate this effect by means of a surface-piercing foil, towed at constant speed and suddenly pulled vertically out of the fluid (which, incidentally, closely resembles the ‘oar immersed in an infinite fluid’ schematics of Klein’s Reference Klein1910 original paper). Of course, viscosity is always present in the real experiment. However, when the foil is lifted the transfer of boundary layer vorticity into the fluid takes place suddenly, in contrast to the steady release always present in the wake, thereby locally mimicking the mechanism of Klein’s idealized experiment. This demonstrates the possibility of vorticity generation due to variations of the fluid domain geometry even in real (albeit low-viscosity) fluids.

Figure 1.

(a,b) Schematic of the initial evolution of the vortex sheet (a) and interface (b) for the Kaffeelöffel and dam-break thought-experiments. (c,d) Level sets of Kaffeelöffel instantaneous streamfunction (c) and dam-break pressure (d) at $t=0^{+}$ in the lab frame. Parameters are $L=1$ , $U=1$ (c), and ${\it\rho}_{1}=0.5$ , ${\it\rho}_{2}=2$ (d). Only the horizontal strips spanning $-1\leqslant y\leqslant 1$ and $0\leqslant z\leqslant 1$ , respectively for plate and gate, are depicted, showing the discontinuous derivatives of the isolines of streamfunction (c) and pressure (d) across the vortex sheet and density interface, respectively.

Some mathematical similarities exist between the Kaffeelöffel thought-experiment and a particular realization of the asymptotically static, layered fluids which are the subject of our study. Consider the ‘internal’ dam-break problem for a two-fluid system confined in a channel (see Camassa et al. Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013, for details). This setup is defined as an incompressible Euler fluid, initially at rest and under gravity, of density ${\it\rho}_{2}$ (resp. ${\it\rho}_{1}$ ) filling the right side $x>0$ (resp. left side $x<0$ ) of a horizontal channel between horizontal rigid boundaries located at heights $z=0,1$ (with ${\it\rho}_{2}>{\it\rho}_{1}$ , say). Such a configuration can be achieved with a gate at $x=0$ initially separating the two fluids, which is then instantaneously removed or dissolved at time $t=0$ . Immediately after the gate removal, the initial horizontal component of the pressure gradient as $x\rightarrow 0^{\pm }$ can be computed explicitly,

(2.3a−c ) $$\begin{eqnarray}\displaystyle \frac{\partial p}{\partial x}(0^{\pm },z)={\it\Upsilon}_{\pm }\log \left(\frac{1+\cos ({\rm\pi}\,z)}{1-\cos ({\rm\pi}\,z)}\right),\quad {\it\Upsilon}_{-}\equiv \frac{g\,{\it\rho}_{1}\,({\it\rho}_{2}-{\it\rho}_{1})}{{\rm\pi}\,({\it\rho}_{2}+{\it\rho}_{1})},\quad {\it\Upsilon}_{+}\equiv \frac{g\,{\it\rho}_{2}\,({\it\rho}_{2}-{\it\rho}_{1})}{{\rm\pi}\,({\it\rho}_{2}+{\it\rho}_{1})}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Thus, at time $t=0^{+}$ the pressure gradient jumps across the interface at $x=0$ between the two fluids, and diverges at the points ( $z=0,1$ ) where the gate was in contact with the rigid horizontal boundaries. These points, after the gate removal, become the intersections of the interface between the two density layers with the confining boundaries of the fluid. At the same time, and consequently, the fluid acceleration $-\boldsymbol{{\rm\nabla}}p/{\it\rho}$ has a continuous horizontal component but suffers a jump, proportional to the density difference, for the vertical component. Thus, the acceleration in the dam-break problem mirrors the continuity properties of the velocity of the Kaffeelöffel case, with continuous normal and discontinuous tangential components in both problems. The pressure for the dam-break case could also be viewed as the counterpart of the streamfunction in the Kaffeelöffel vortex sheet generation: both pressure and streamfunction, respectively for the dam-break and the Kaffeelöffel case, are continuous but suffer jumps in their horizontal gradient components. Moreover, the mathematical singularities in both setups play a similar role, at least initially, in the evolution of vortex sheets. The shear that develops as a consequence of the jump in the vertical component of the pressure gradient for the dam-break (recall that the pressure is continuous at $x=0$ ) is ‘pulled’ horizontally to the left and right, respectively, for $z<1/2$ and $z>1/2$ , by the pressure forces exerted on the fluid. These are strongest at the intersection points where the pressure gradient diverges at $z=0$ and $z=1$ . The result is an S-shaped vortex sheet at the early times of evolution following the gate removal (see Camassa et al. Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013, figure 16, for a direct numerical simulation), not unlike the way the initial vortex sheet in the Kaffeelöffel experiment extends (and bends) in response to the singular pressure forces at the plate tips once this is removed. This similarity is illustrated in figure 1. (We remark that the pressure/streamfunction analogy between the dam-break and Kaffeelöffel thought-experiments does not quite hold for times before the removal of the objects, since the pressure jumps across the dam before its removal, while the streamfunction maintains its continuity. Also note that, unlike the dam break setup, the fluid domain for the Kaffeelöffel experiment is the whole plane $(x,y)$ .)

Consequences of the interplay between vorticity balance laws and topological changes of isopycnals in layered flows might also be familiar under a different guise, which we will now briefly sketch in another idealized example. Consider the setup of a gravity current in a two-layer incompressible Euler fluid studied by von Kármán (Reference von Kármán1940) and Benjamin (Reference Benjamin1968), depicted in figure 2, that can be described as follows. In a two-dimensional domain confined between two rigid, horizontal plates, a flat interface at infinity separates the two fluids to the right of the domain. There, the fluid in the upper layer moves to the left with constant horizontal velocity, feeding the advancement from right to left of a point at finite range where the interface contacts the upper plate. For this solution of the Euler equations, the only location of the interface at right-infinity compatible with total horizontal momentum and energy conservation is exactly at the middle of the channel, regardless of the density difference between the two fluids (Benjamin Reference Benjamin1968). Accordingly, the gravity-current front moves from right to left at a constant speed selected by this depth of the interface. If $h$ still denotes the vertical distance between the plates and $h_{2}^{\prime }$ is the thickness of the lower-layer fluid to the far right of the domain, one has

(2.4) $$\begin{eqnarray}h_{2}^{\prime }=h/2,\end{eqnarray}$$

while the asymptotic velocities to the right of the lower and upper fluids are, respectively, $u_{2}^{\prime }=c_{0}/2$ and $u_{1}^{\prime }=-u_{2}^{\prime }$ , where $c_{0}\equiv \sqrt{g\,h\,(1-{\it\rho}_{r})}$ is the critical long-wave speed, with the definition of density ratio ${\it\rho}_{r}\equiv {\it\rho}_{1}/{\it\rho}_{2}$ . In contrast, suppose that the interface never contacts the upper plate at finite range but may approach it asymptotically towards infinity to the left. For this setup, a solution of the Euler equations compatible with momentum conservation also exists (see e.g. Choi & Camassa Reference Choi and Camassa1999), and can be interpreted as a limiting case of a so-called conjugate-state solution. In this case, the thickness of the lower-layer fluid to the far right is

(2.5) $$\begin{eqnarray}h_{2}^{\prime }=\frac{h}{\sqrt{{\it\rho}_{r}}+1},\end{eqnarray}$$

while the asymptotic velocities are, respectively, $u_{2}^{\prime }=c_{0}h_{2}\,/h-c$ and $u_{1}^{\prime }=-u_{2}^{\prime }/\sqrt{{\it\rho}_{r}}$ , where $c=c_{0}/(1+\sqrt{{\it\rho}_{r}})$ is the velocity of the front, and $h_{2}$ is the thickness of the lower fluid at the far left of the channel.

Figure 2.

Schematic of gravity-current (solid) and conjugate-state (dash) propagation in a two-layer fluid. A lab-frame observer’s test section is highlighted in the region between $-L$ and $L$ . Sketches refer to times where the front of either solution with respect to the test section is: (a) approaching from far to the right, (b) moving through, (c) travelling out to the far left, leaving a steady horizontal vortex sheet behind.

In a test section of width $2L$ , as either the gravity current or the conjugate-state front moves from the far right to far left (see figure 2), an observer would see the total fluid vorticity increase from zero to a constant value at sufficiently large times. This is because the fluid in the test window is initially at rest, and after the front has moved through, the stratified fluid remains in steady motion with a vortex sheet determined by the velocity jump across the interface, for either the gravity-current or the conjugate-state solutions. As can be seen from the expression of the fluid velocities, at large times the total vorticity in the test section is the same for both gravity currents and the limiting case of conjugate states, that is

(2.6) $$\begin{eqnarray}{\it\Gamma}_{w}=2L\sqrt{g\,h\,(1-{\it\rho}_{r})},\end{eqnarray}$$

but, interestingly, the effect of the intersection with the upper wall of the interface between the two fluids causes the abrupt difference between its asymptotic height at large times seen by comparing (2.4) for the gravity current solution versus (2.5) for the conjugate state, respectively, with ${\it\rho}_{r}<1$ . Thus, the difference between measurable quantities such as the time-asymptotic location of the interface between the two fluids in the test section can ultimately be attributed to the effect of the topological change of a density isoline.

3. Two-dimensional systems: a unifying perspective on momentum and vorticity conservation laws

We begin by recalling some results obtained in Camassa et al. (Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2012, Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013) concerning motion in a two-dimensional fluid domain (a channel) of the form ${\it\Omega}\equiv \mathbb{R}\times [z_{-},z_{+}]$ . If $(x,z)$ and $(u,w)$ denote the Cartesian components of coordinates and velocity field, respectively, the total horizontal fluid momentum is

(3.1) $$\begin{eqnarray}{\it\Pi}\equiv \int _{{\it\Omega}}{\it\rho}u\,\text{d}x\,\text{d}z.\end{eqnarray}$$

It can be shown that ${\it\Pi}$ is not a constant of the motion, since its time-derivative is

(3.2) $$\begin{eqnarray}\dot{{\it\Pi}}=-h\langle p\rangle _{{\it\Delta}},\end{eqnarray}$$

where $h=z_{+}-z_{-}$ is the channel height and the layer-average $\langle p\rangle _{{\it\Delta}}$ of the pressure jump at the channel end-points is

(3.3) $$\begin{eqnarray}\langle p\rangle _{{\it\Delta}}\equiv \frac{1}{h}\int _{z_{-}}^{z_{+}}(p(+\infty ,z)-p(-\infty ,z))\,\text{d}z.\end{eqnarray}$$

Moreover, it turns out that $\langle p\rangle _{{\it\Delta}}=p(+\infty ,\bar{z})-p(-\infty ,\bar{z})$ for any $\bar{z}\in [z_{-},z_{+}]$ , and that this pressure difference is in general non-zero, as already remarked in Benjamin (Reference Benjamin1986) and further substantiated in Camassa et al. (Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2012, Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013).

In two dimensions the local vorticity ${\it\omega}=w_{x}-u_{z}$ is a scalar. The total vorticity ${\it\Gamma}$ is

(3.4) $$\begin{eqnarray}{\it\Gamma}\equiv \int _{{\it\Omega}}{\it\omega}\,\text{d}x\,\text{d}z=\int _{\mathbb{R}}(u^{-}-u^{+})\,\text{d}x,\end{eqnarray}$$

where $f^{\pm }(x,t)\equiv f(x,z_{\pm },t)$ denote, hereafter, the boundary values of the quantity $f(x,z,t)$ . By setting $\lim _{|x|\rightarrow \infty }{\it\rho}^{+}(x)\equiv {\it\rho}_{1}$ , $\lim _{|x|\rightarrow \infty }{\it\rho}^{-}(x)\equiv {\it\rho}_{2}$ , and ${\it\rho}_{_{{\it\Delta}}}\equiv {\it\rho}_{2}-{\it\rho}_{1}$ , and by using the horizontal component of the Euler equations, the time-derivative of the total vorticity turns out to be

(3.5) $$\begin{eqnarray}\dot{{\it\Gamma}}=\frac{{\it\rho}_{_{{\it\Delta}}}\langle p\rangle _{{\it\Delta}}}{{\it\rho}_{1}{\it\rho}_{2}}+\int _{\mathbb{R}}\left(\frac{{\it\rho}_{x}^{+}}{({\it\rho}^{+})^{2}}p^{+}-\frac{{\it\rho}_{x}^{-}}{({\it\rho}^{-})^{2}}p^{-}\right)\text{d}x.\end{eqnarray}$$

Equations (3.2) and (3.5) suggest the definition of the quantity

(3.6) $$\begin{eqnarray}\mathscr{G}\equiv {\it\Gamma}+\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}h}{\it\Pi},\end{eqnarray}$$

whose time-evolution is

(3.7) $$\begin{eqnarray}\dot{\mathscr{G}}=\int _{\mathbb{R}}\left(\frac{{\it\rho}_{x}^{+}}{({\it\rho}^{+})^{2}}\,p^{+}-\frac{{\it\rho}_{x}^{-}}{({\it\rho}^{-})^{2}}\,p^{-}\right)\text{d}x.\end{eqnarray}$$

If the density is constant on the upper and lower lids (i.e. if ${\it\rho}^{+}(x)={\it\rho}_{1}$ and ${\it\rho}^{-}(x)={\it\rho}_{2}$ for all $x$ ), then $\mathscr{G}$ is a conserved quantity, related to those classified in Benjamin (Reference Benjamin1986) by

(3.8) $$\begin{eqnarray}\mathscr{G}=\frac{{\it\rho}_{_{{\it\Delta}}}}{h{\it\rho}_{1}{\it\rho}_{2}}\mathscr{I}+\frac{z_{+}{\it\rho}_{1}-z_{-}{\it\rho}_{2}}{h{\it\rho}_{1}{\it\rho}_{2}}\mathscr{C},\end{eqnarray}$$

where

(3.9) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \mathscr{C}\equiv \int _{{\it\Omega}}{\it\sigma}~\text{d}x\,\text{d}z=\frac{{\it\Pi}^{-}}{h}-\frac{{\it\Pi}^{+}}{h},\\ \displaystyle \mathscr{I}\equiv \int _{{\it\Omega}}z{\it\sigma}~\text{d}x\,\text{d}z={\it\Pi}-\frac{z_{+}}{h}{\it\Pi}^{+}+\frac{z_{-}}{h}{\it\Pi}^{-},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with ${\it\sigma}\equiv ({\it\rho}w)_{x}-({\it\rho}u)_{z}$ and ${\it\Pi}^{\pm }\equiv h\int _{\mathbb{R}}{\it\rho}^{\pm }u^{\pm }\,\text{d}x$ . Because of property (3.7), the quantity $\mathscr{G}$ acts as an indicator of the topological features of the pycnocline surfaces with respect to the fluid domain boundaries. We will henceforth refer to $\mathscr{G}$ as the ‘topological vorticity’.

As pointed out in Camassa et al. (Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2014), whenever the density at the lids fails to be constant, the determination of the motion invariants (and of the Hamiltonian representation of the system) is substantially altered; in particular, boundary values of all the fields have to be carefully considered. The upshot of such a topological change is that the quantities (3.9) are no longer conserved, since their balance laws are

(3.10) $$\begin{eqnarray}\dot{\mathscr{C}}=\mathscr{J}^{+}-\mathscr{J}^{-}\end{eqnarray}$$

and

(3.11) $$\begin{eqnarray}\dot{\mathscr{I}}=z_{+}\mathscr{J}^{+}-z_{-}\mathscr{J}^{-},\end{eqnarray}$$

where $\mathscr{J}^{\pm }$ are given by

(3.12) $$\begin{eqnarray}\mathscr{J}^{\pm }\equiv \frac{1}{2}\int _{-\infty }^{+\infty }{\it\rho}_{\,\,x}^{\pm }(u^{\pm })^{2}\,\text{d}x.\end{eqnarray}$$

The same lack of conservation happens for the topological vorticity $\mathscr{G}$ (see (3.7)). We remark that, if the boundary density is not a constant, relation (3.8) should also be modified to

(3.13) $$\begin{eqnarray}\mathscr{G}+\frac{1}{{\it\rho}_{1}}\int _{\mathbb{R}}({\it\rho}_{1}-{\it\rho}^{+})\,u^{+}\text{d}x-\frac{1}{{\it\rho}_{2}}\int _{\mathbb{R}}({\it\rho}_{2}-{\it\rho}^{-})\,u^{-}\,\text{d}x=\frac{{\it\rho}_{_{{\it\Delta}}}}{h{\it\rho}_{1}{\it\rho}_{2}}\mathscr{I}+\frac{z_{+}{\it\rho}_{1}-z_{-}{\it\rho}_{2}}{h{\it\rho}_{1}{\it\rho}_{2}}\mathscr{C}.\end{eqnarray}$$

These results become particularly transparent in the simplified case of a sharp two-layer stratification, depicted in figure 3. In this case ${\it\rho}^{+}$ (resp. ${\it\rho}^{-}$ ) can be non-constant only if the lower (resp. upper) fluid touches the upper (resp. lower) boundary. Suppose this happens for $x$ in some suitable intervals of the real line; then we can write

(3.14a,b ) $$\begin{eqnarray}\frac{1}{{\it\rho}^{+}}=\frac{1}{{\it\rho}_{1}}-\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{i}{\it\chi}_{[{\it\alpha}_{i}^{+},{\it\beta}_{i}^{+}]},\quad \frac{1}{{\it\rho}^{-}}=\frac{1}{{\it\rho}_{2}}+\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{j}{\it\chi}_{[{\it\alpha}_{j}^{-},{\it\beta}_{j}^{-}]},\end{eqnarray}$$

where the characteristic function ${\it\chi}$ is ${\it\chi}_{[a,b]}(x)=1$ if $x\in [a,b]$ and zero otherwise. Thus, spatial derivatives give rise to Dirac-deltas at the density jumps:

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{1}{{\it\rho}^{+}}\right)_{x}=\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{i}({\it\delta}(x-{\it\beta}_{i}^{+})-{\it\delta}(x-{\it\alpha}_{i}^{+})), & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{1}{{\it\rho}^{-}}\right)_{x}=-\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{j}({\it\delta}(x-{\it\beta}_{j}^{-})-{\it\delta}(x-{\it\alpha}_{j}^{-})). & \displaystyle\end{eqnarray}$$

${\it\Gamma}$

(3.17) $$\begin{eqnarray}\displaystyle \dot{{\it\Gamma}} & = & \displaystyle \frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\left(\langle p\rangle _{{\it\Delta}}-\mathop{\sum }_{i}(p^{+}({\it\beta}_{i}^{+})-p^{+}({\it\alpha}_{i}^{+}))-\mathop{\sum }_{j}(p^{-}({\it\beta}_{j}^{-})-p^{-}({\it\alpha}_{j}^{-}))\right)\nonumber\\ \displaystyle & = & \displaystyle \frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\left(\langle p\rangle _{{\it\Delta}}-\mathop{\sum }_{i}P_{{\it\Delta},i}^{+}-\mathop{\sum }_{j}P_{{\it\Delta},j}^{-}\right),\end{eqnarray}$$

where $P_{{\it\Delta},i}^{+}\equiv p^{+}({\it\beta}_{i}^{+})-p^{+}({\it\alpha}_{i}^{+})$ and $P_{{\it\Delta},j}^{-}\equiv p^{-}({\it\beta}_{j}^{-})-p^{-}({\it\alpha}_{j}^{-})$ are the local boundary pressure differences. These local differences need not vanish in general (an instance of such non-trivial contributions is illustrated in Camassa et al. Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013). They are related to the variation of the amount of boundary momentum contained in the intervals $[{\it\alpha}_{i}^{\pm },{\it\beta}_{i}^{\pm }]$ by

(3.18) $$\begin{eqnarray}\displaystyle P_{{\it\Delta},i}^{+} & = & \displaystyle \int _{{\it\alpha}_{i}^{+}}^{{\it\beta}_{i}^{+}}p_{x}^{+}\,\text{d}x=-\int _{{\it\alpha}_{i}^{+}}^{{\it\beta}_{i}^{+}}{\it\rho}_{2}(u_{t}^{+}+u^{+}u_{x}^{+})\,\text{d}x\nonumber\\ \displaystyle & = & \displaystyle -\dot{{\it\Pi}}_{i}^{+}+{\it\rho}_{2}(u^{+}({\it\beta}_{i}^{+})\dot{{\it\beta}}_{i}^{+}-u^{+}({\it\alpha}_{i}^{+})\dot{{\it\alpha}}_{i}^{+})-\left.{\textstyle \frac{1}{2}}{\it\rho}_{2}(u^{+})^{2}\right|_{{\it\alpha}_{i}^{+}}^{{\it\beta}_{i}^{+}},\end{eqnarray}$$

where ${\it\Pi}_{i}^{+}\equiv h\int _{{\it\alpha}_{i}^{+}}^{{\it\beta}_{i}^{+}}{\it\rho}_{2}u^{+}\,\text{d}x$ . Similarly, we obtain

(3.19) $$\begin{eqnarray}P_{{\it\Delta},j}^{-}=-\dot{{\it\Pi}}_{j}^{-}+{\it\rho}_{1}(u^{-}({\it\beta}_{j}^{-})\dot{{\it\beta}}_{j}^{-}-u^{-}({\it\alpha}_{j}^{-})\dot{{\it\alpha}}_{j}^{-})-\left.{\textstyle \frac{1}{2}}{\it\rho}_{1}(u^{-})^{2}\right|_{{\it\alpha}_{j}^{-}}^{{\it\beta}_{j}^{-}},\end{eqnarray}$$

where ${\it\Pi}_{j}^{-}\equiv h\int _{{\it\alpha}_{j}^{-}}^{{\it\beta}_{j}^{-}}{\it\rho}_{1}u^{-}\,\text{d}x$ . By using the relations

(3.20a,b ) $$\begin{eqnarray}\dot{{\it\beta}}_{i}^{\pm }=u^{\pm }({\it\beta}_{i}^{\pm }),\quad \dot{{\it\alpha}}_{i}^{\pm }=u^{\pm }({\it\alpha}_{i}^{\pm }),\end{eqnarray}$$

we can write the local pressure differences as

(3.21a,b ) $$\begin{eqnarray}P_{{\it\Delta},i}^{+}=-\dot{{\it\Pi}}_{i}^{+}+\left.{\textstyle \frac{1}{2}}{\it\rho}_{2}(u^{+})^{2}\right|_{{\it\alpha}_{i}^{+}}^{{\it\beta}_{i}^{+}},\quad P_{{\it\Delta},j}^{-}=-\dot{{\it\Pi}}_{j}^{-}+\left.{\textstyle \frac{1}{2}}{\it\rho}_{1}(u^{-})^{2}\right|_{{\it\alpha}_{j}^{-}}^{{\it\beta}_{j}^{-}}.\end{eqnarray}$$

Figure 3.

Two-dimensional setup: typical interface configuration contributing to total vorticity evolution through boundary terms.

The second and the third terms in (3.17) can be considered as topological contributions to the time-variation of ${\it\Gamma}$ . As shown in Camassa et al. (Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013), if the initial velocities are zero these contributions are proportional to ${\it\rho}_{_{{\it\Delta}}}^{2}$ , and can therefore dominate the first term which is proportional to ${\it\rho}_{_{{\it\Delta}}}^{3}$ . This can be given the following interpretation. Consider, for simplicity, a sequence of zero-velocity initial fluid configurations in which the lower fluid is arranged with ‘bumps’ whose height grows towards the maximal height $z_{+}$ of the upper bounding plate. As long as a sliver of lighter fluid between the upper lid and the top of a bump of heavier fluid is present the initial variation of total vorticity is given by $\dot{{\it\Gamma}}={\it\rho}_{_{{\it\Delta}}}\,\langle p\rangle _{{\it\Delta}}/({\it\rho}_{1}{\it\rho}_{2})$ . This quantity varies smoothly with the bump heights. On the other hand, when one or more slivers of upper fluid between bumps and the upper plate disappear, jumps can occur in the time-variation of ${\it\Gamma}$ , since this is now given by all the terms in (3.17). Such abrupt changes should be expected to happen for sequences of interface profiles limiting to profiles that intersect either lid, such as those depicted in figure 3.

4. The three-dimensional case: momentum and vorticity balance

Paradoxical results about momentum conservation in stratified, incompressible Euler fluids in unbounded domains emerge due to the interplay between incompressibility and stratification (Benjamin Reference Benjamin1986; Camassa et al. Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2012, Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013). The paradoxical nature of these results can be traced back to the effects of incompressibility in homogeneous-density fluids. The classical computation of the total momentum integral over unbounded domains (Saffman Reference Saffman1992; Falkovich Reference Falkovich2011) with localized force distributions shows that the total momentum, and its variation in time, depend conditionally on the limit to infinity of finite volume integrals. While for homogeneous fluids in the two-dimensional case of a channel the limit of infinite volumes does not require handling conditionally convergent integrals, the same setup for inhomogeneous fluids is more delicate, as already pointed out in Benjamin (Reference Benjamin1986) and further discussed in Camassa et al. (Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2012, Reference Camassa, Chen, Falqui, Ortenzi and Pedroni2013). To a larger extent, three-dimensional settings for stratified fluids require addressing the issue of conditional convergence of volume and surface integrals. This should perhaps be expected in the case of localized density distributions away from hydrostatic equilibrium, as these can be viewed as impulsive force sources, thus leading to similar issues as for their three-dimensional homogeneous counterpart. In what follows, unless otherwise specified, we will always use bounded control volumes ${\it\Omega}$ and take their unbounded limits to handle the case of infinite channels. Depending on the situation, we will explicitly indicate when such volumes can be taken as material objects, whether by using impermeable walls or by letting them evolve with the fluid motion according to the Lagrangian perspective.

Next, we first briefly review the three-dimensional analogue of the horizontal momentum evolution equation (3.2), and then move on to examine the total vorticity equation (3.5) in three-dimensional settings.

4.1. Three-dimensional momentum and impulse evolution

For a stratified fluid under gravity confined between two rigid infinite horizontal planes the time-derivative of the total horizontal momentum may depend on the shape of the lateral boundary in the limit to infinite domains. For a stratified fluid in three dimensions, this can be seen by suitably reinterpreting the link between total momentum

(4.1) $$\begin{eqnarray}{\it\bf\Pi}=\int _{{\it\Omega}}{\rm\pi}\,\text{d}V\equiv \int _{{\it\Omega}}{\it\rho}\boldsymbol{u}\,\text{d}V,\end{eqnarray}$$

and hydrodynamical impulse (in analogy with the two-dimensional case as in Benjamin Reference Benjamin1986),

(4.2) $$\begin{eqnarray}\boldsymbol{I}\equiv \frac{1}{2}\int _{{\it\Omega}}\boldsymbol{x}\times {\bf\sigma}\,\text{d}V\,\quad \text{with }{\bf\sigma}\equiv \boldsymbol{{\rm\nabla}}\times ({\it\rho}\boldsymbol{u}),\end{eqnarray}$$

by replacing the velocity vector $\boldsymbol{u}$ used in Saffman (Reference Saffman1992) and Falkovich (Reference Falkovich2011) with the momentum vector ${\rm\pi}={\it\rho}\boldsymbol{u}$ . Using standard formulae of vector calculus we can write the relation between momentum ${\it\bf\Pi}$ and impulse $\boldsymbol{I}$ as

(4.3) $$\begin{eqnarray}{\it\bf\Pi}=\boldsymbol{I}+\frac{1}{2}\int _{\partial {\it\Omega}}({\it\rho}\boldsymbol{u}(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{x})-\boldsymbol{n}({\it\rho}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{x}))\text{d}S.\end{eqnarray}$$

Here, following general definitions, see e.g. Saffman (Reference Saffman1992) or Majda & Bertozzi (Reference Majda and Bertozzi2002), we have restored the factor $1/2$ in the definition of the impulse, so that its first component is $\mathscr{I}/2$ with respect to the definition of the two-dimensional case in the previous section. The last integral in (4.3) is zero for sufficiently fast-decaying velocities at infinity (for instance, if ${\it\Omega}$ is a sphere, for velocities decaying faster than $r^{-3}$ ), but may very well be non-zero in a number of physical solutions of the Euler equations (for homogeneous fluids, numerous examples can be found in Lamb Reference Lamb1932). Moreover, as soon as the velocity field has a non-vanishing dipole moment (and thus the integral on the right-hand side of (4.3) is only conditionally convergent), the surface integral depends on the geometry of the boundary $\partial {\it\Omega}$ of the fluid volume ${\it\Omega}$ . In the special case of a ‘cylindrical’ shape ${\it\Omega}=W\times [z_{-},z_{+}]$ as in figure 4 one gets

(4.4) $$\begin{eqnarray}\boldsymbol{I}_{\bot }={\it\bf\Pi}_{\bot }-\frac{z_{+}}{2h}{\it\bf\Pi}_{\bot }^{+}+\frac{z_{-}}{2h}{\it\bf\Pi}_{\bot }^{-}-\frac{1}{2}\int _{\partial {\it\Omega}_{\bot }}((\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{x}){\it\rho}\boldsymbol{u}_{\bot }-({\it\rho}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{x})\boldsymbol{n})\,\text{d}S,\end{eqnarray}$$

where the subscript $\bot$ denotes horizontal components of vectors, $\partial {\it\Omega}_{\bot }$ is the lateral boundary and, in analogy with the notation of § 3, we define

(4.5) $$\begin{eqnarray}{\it\bf\Pi}_{\bot }^{\pm }=h\int _{W}{\rm\pi}_{\bot }(x,y,z_{\pm })\,\text{d}x\,\text{d}y.\end{eqnarray}$$

When the density along the plates $z=z_{\pm }$ is constant, and for zero-velocity initial conditions, the time-derivative of the horizontal impulse $\boldsymbol{I}_{\bot }$ can be shown to vanish at time $t=0$ (with a suitable modification of the arguments presented in e.g. Saffman Reference Saffman1992, for the case of a homogeneous fluid), so that (4.4) yields

(4.6) $$\begin{eqnarray}\dot{{\it\bf\Pi}}_{\bot }=\frac{z_{+}}{2h}\,\dot{{\it\bf\Pi}}_{\bot }^{+}-\frac{z_{-}}{2h}\dot{{\it\bf\Pi}}_{\bot }^{-}+\frac{1}{2}\int _{\partial {\it\Omega}_{\bot }}((\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{x})\,\boldsymbol{{\rm\nabla}}_{\bot }p-(\boldsymbol{{\rm\nabla}}p\boldsymbol{\cdot }\boldsymbol{x})\boldsymbol{n})\,\text{d}S.\end{eqnarray}$$

The last term makes apparent how the geometry of the lateral vertical boundary $\partial {\it\Omega}_{\bot }$ explicitly enters the time-variation of the total momentum. Notice that the pressure gradient plays, in the time-variation formula, exactly the same role as played by the (horizontal) momentum ${\it\rho}\boldsymbol{u}_{\bot }$ in (4.3). We may expect non-trivial contributions when the dipole moment of the pressure gradient is non-zero, so that the integral on the right-hand side of (4.6) is not absolutely convergent. Some explicit computations of such geometry-dependent contributions are reported in appendix A.

Figure 4.

Schematics of cylindrical fluid domains for impulse and momentum time-variations.

We end this section with a remark: while for constant density along the plates in two-dimensional settings the horizontal impulse is always conserved (Benjamin Reference Benjamin1986) as this corresponds to the Noether’s symmetry of horizontal translational invariance, in three dimensions this no longer holds (in general) for stratified fluid between infinite horizontal plates. In particular, (4.6) does not hold beyond initial times, since the term $\dot{\boldsymbol{I}}_{\bot }$ is not zero in general. Conservation of horizontal impulse is recovered, for instance, when ${\bf\omega}\boldsymbol{\cdot }\boldsymbol{n}=0$ along the plates. The influence of this condition on the vorticity evolution is discussed next.

4.2. Three-dimensional vorticity

The evolution equation for the vorticity in three dimensions is given by

(4.7) $$\begin{eqnarray}{\bf\omega}_{t}+(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}){\bf\omega}-{\bf\omega}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}+\boldsymbol{{\rm\nabla}}\times \left(\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}}\right)=0.\end{eqnarray}$$

In the absence of vortex sheets, this implies that the time-variation of the total fluid vorticity is

(4.8) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}=-\int _{\partial {\it\Omega}}(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}){\bf\omega}\,\text{d}S+\int _{\partial {\it\Omega}}({\bf\omega}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{u}\,\text{d}S-\int _{{\it\Omega}}\boldsymbol{{\rm\nabla}}\times \left(\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}}\right)\text{d}V.\end{eqnarray}$$

With either a material interpretation or with wall boundary conditions for the control volume ${\it\Omega}$ , the first transport term on the right-hand side of this equation would be absent. This property would be preserved in time for the case of a homogeneous fluid, as a consequence of Kelvin circulation invariance. The second and third terms on the right-hand side of expression (4.8) can be naturally classified as the kinematic and baroclinic terms, denoted respectively by $\dot{{\it\bf\Gamma}}_{kin}$ and $\dot{{\it\bf\Gamma}}_{bar}$ . With a material surface defined by ${\bf\omega}\boldsymbol{\cdot }\boldsymbol{n}=0$ at some instant in time, the second (kinematic) term would also be absent from the right-hand side of (4.8). While in general both the kinetic and baroclinic terms contribute to the variation of total vorticity (as already remarked by Poincaré Reference Poincaré1893), we shall henceforth concentrate on the latter to show how the topology of velocity and density discontinuities play a fundamental role in the variation of the total vorticity.

The consequences of non-trivial stratifications, giving rise to the baroclinic term in (4.8), are the focus of this section. First, we study the more intuitive case of a sharp two-layer stratification in a cylindrically shaped domain ${\it\Omega}=W\times [z_{-},z_{+}]$ . In this setting, as we shall see, the topological aspects emerge in a particularly transparent way.

4.2.1. The two-layer case

In the two-layer case the density, and possibly the tangent component of the velocity, are discontinuous at the interface $z={\it\eta}(x,y,t)$ between the two fluids. In particular, we have

(4.9) $$\begin{eqnarray}{\it\rho}={\it\rho}_{2}-{\it\rho}_{{\it\Delta}}H(z-{\it\eta}(x,y,t)),\end{eqnarray}$$

where we are maintaining the notation of § 3, i.e.  ${\it\rho}_{2}$ (resp.  ${\it\rho}_{1}$ ) is the density of the lower (resp. upper) fluid, ${\it\rho}_{_{{\it\Delta}}}={\it\rho}_{2}-{\it\rho}_{1}$ , and $H$ is the Heaviside function.

To compute the baroclinic contribution of the total vorticity evolution

(4.10) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=-\int _{{\it\Omega}}\boldsymbol{{\rm\nabla}}\times \left(\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}}\right)\text{d}V,\end{eqnarray}$$

we need to carefully interpret the curl in the distributional sense owing to the fact that the density has a jump along the interface ${\it\Sigma}$ between the two fluids. This leads to

(4.11) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=-\int _{{\it\Sigma}}\boldsymbol{n}_{1}\times \left(\frac{\boldsymbol{{\rm\nabla}}p_{1}}{{\it\rho}_{1}}\right)\text{d}S-\int _{{\it\Sigma}}\boldsymbol{n}_{2}\times \left(\frac{\boldsymbol{{\rm\nabla}}p_{2}}{{\it\rho}_{2}}\right)\text{d}S,\end{eqnarray}$$

where $p_{1}$ (resp. $p_{2}$ ) is the pressure in the upper (resp. lower) layer, and $\boldsymbol{n}_{1}$ (resp. $\boldsymbol{n}_{2}$ ) is the unit normal to ${\it\Sigma}$ pointing to the upper (resp. lower) layer. By standard theorems of vector calculus (see e.g. Smirnov Reference Smirnov1964, p. 322) we obtain

(4.12) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\frac{1}{{\it\rho}_{1}}\int _{\partial {\it\Sigma}}p_{1}\boldsymbol{t}\,\text{d}l-\frac{1}{{\it\rho}_{2}}\int _{\partial {\it\Sigma}}p_{2}\boldsymbol{t}\,\text{d}l,\end{eqnarray}$$

where the tangent vector $\boldsymbol{t}$ is oriented according to the rule that it lead to counterclockwise circulation along $\partial {\it\Sigma}$ with respect to the normal $\boldsymbol{n}_{2}$ (see e.g. figure 5). Continuity of the pressure yields the final formula

(4.13) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\int _{\partial {\it\Sigma}}p\boldsymbol{t}\,\text{d}l.\end{eqnarray}$$

If we denote by ${\it\gamma}_{{\it\alpha}}$ the connected components of the interface ${\it\Sigma}$ , we obviously have

(4.14) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{{\it\alpha}}\int _{{\it\gamma}_{{\it\alpha}}}p\boldsymbol{t}_{{\it\alpha}}\,\text{d}l_{{\it\alpha}},\end{eqnarray}$$

where the tangent vector is oriented according to the rule that ‘a – long-armed and short-legged – walker upright in the direction of the outward normal to $\partial {\it\Omega}$ , and moving along the loop ${\it\gamma}_{{\it\alpha}}$ according to the orientation set by  $\boldsymbol{t}_{{\it\alpha}}$ , would keep the left-hand immersed in the heavier fluid’.

Figure 5.

A simple case of a two-layer configuration with a topologically non-trivial interface.

In most applications of (4.14), the pycnocline will intersect the lateral boundary $\partial {\it\Omega}_{\bot }$ along a closed loop ${\it\gamma}_{_{\infty }}$ , and it is convenient to single out the corresponding term in (4.14) so that it reads

(4.15) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\int _{{\it\gamma}_{_{\infty }}}p\boldsymbol{t}_{\infty }\,\text{d}l_{\infty }+\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{{\it\beta}}\int _{{\it\gamma}_{{\it\beta}}}p\boldsymbol{t}_{{\it\beta}}\,\text{d}l_{{\it\beta}}.\end{eqnarray}$$

It is natural to denote the second term as the ‘topological contribution’ to $\dot{{\it\bf\Gamma}}_{bar}$ , as it keeps track of the topology of the pycnocline deformations of this surface, as illustrated in figure 6.

Figure 6.

Two-layer system: sketch of possible interface topologies. Each interface ‘hole’ contributes a term to the evolution of total vorticity. The shape of the hole boundary, and not the contact angle, is the only interface feature affecting the baroclinic term of vorticity evolution.

Two remarks are in order. First, note that this formula does not carry any information on the way the interface between the two fluids intersects the boundary of the domain ${\it\Omega}$ . In particular, angles of intersection do not play a role in the final expression (4.14). Second, the sign choice in (4.14) agrees with the two-dimensional case (3.17), taking into account that ${\it\Gamma}=-{\it\bf\Gamma}\boldsymbol{\cdot }\boldsymbol{j}$ . To see this, consider the setup depicted in figure 7: both top and bottom circuits between the two fluids end up being oriented counterclockwise for an upright observer viewing the channel from above.

Figure 7.

Two-layer system: orientations of top and bottom circuits of the interface intersections with horizontal boundaries.

Let us now define the three-dimensional analogue of the topological vorticity $\mathscr{G}$ introduced in § 3 (for a specific class of fluid domains ${\it\Omega}$ and interface configurations as graphs over the horizontal variables $z={\it\eta}(x,y,t)$ ). This vector topological vorticity is given by

(4.16) $$\begin{eqnarray}\mathbf{\mathscr{G}}\equiv {\it\bf\Gamma}+\frac{{\it\rho}_{_{{\it\Delta}}}}{h{\it\rho}_{1}{\it\rho}_{2}}{\it\bf\Pi}\times \boldsymbol{k},\end{eqnarray}$$

and its time-evolution can be computed as follows. As can be readily shown, the time-variation of the total horizontal momentum of the fluid filling ${\it\Omega}$ is given by

(4.17) $$\begin{eqnarray}\dot{{\it\bf\Pi}}_{\bot }=\int _{{\it\Omega}}\frac{\partial }{\partial t}({\it\rho}\boldsymbol{u}_{\bot })\,\text{d}V=-\int _{\partial {\it\Omega}_{\bot }}{\it\rho}\boldsymbol{u}_{\bot }\,(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n})\,\text{d}S-\int _{\partial {\it\Omega}_{\bot }}p\boldsymbol{n}\,\text{d}S.\end{eqnarray}$$

Let us suppose that the base $W$ of the domain ${\it\Omega}$ is sufficiently large that the hydrostatic condition on pressure can be applied asymptotically at $\partial {\it\Omega}_{\bot }$ ,

(4.18) $$\begin{eqnarray}\left.p_{j}|_{\partial {\it\Omega}_{\bot }}\sim -{\it\rho}_{j}g(z-{\it\eta})\right|_{\partial {\it\Omega}_{\bot }}+\,P|_{{\it\gamma}_{_{\infty }}},\quad j=1,2,\end{eqnarray}$$

where $P$ is the interfacial pressure. (That the pressure $p$ can be asymptotic to a non-constant value due to the directional dependence of $P$ , even under the limiting hydrostatic assumption, might not be immediately clear; a few explicit examples will be provided in appendix A.) Further, we assume that horizontal velocity goes to zero sufficiently fast (for a dipole term it would be $O(|\boldsymbol{x}_{\bot }|^{-3})$ ) so that the transport term in (4.17) can be neglected, and that the interface between the two fluids be asymptotically horizontal, as appropriate for hydrostatic equilibria. This means that ${\it\eta}$ is a constant ( ${\it\eta}=z_{0}$ , say) on the lateral vertical boundary of ${\it\Omega}$ , so that the intersection curve ${\it\gamma}_{_{\infty }}$ is horizontal, i.e.  $\boldsymbol{t}_{\infty }\boldsymbol{\cdot }\boldsymbol{k}=0$ . We obtain

(4.19) $$\begin{eqnarray}\dot{{\it\bf\Pi}}_{\bot }=-\int _{\partial {\it\Omega}_{\bot }}(P-{\it\rho}g(z-{\it\eta}))\boldsymbol{n}\,\text{d}S=-\int _{\partial {\it\Omega}_{\bot }}P\boldsymbol{n}\,\text{d}S.\end{eqnarray}$$

Since $\boldsymbol{t}_{\infty }=\boldsymbol{n}\times \boldsymbol{k}$ , taking the cross-product with $\boldsymbol{k}$ we get

(4.20) $$\begin{eqnarray}\dot{{\it\bf\Pi}}\times \boldsymbol{k}=\dot{{\it\bf\Pi}}_{\bot }\times \boldsymbol{k}=-\int _{\partial {\it\Omega}_{\bot }}P\boldsymbol{t}_{\infty }\,\text{d}S=-h\int _{{\it\gamma}_{_{\infty }}}P\boldsymbol{t}_{\infty }\,\text{d}l,\end{eqnarray}$$

where $l$ is the arc length parameter of ${\it\gamma}_{_{\infty }}$ . Similarly to the two-dimensional case, when the density is constant along the upper and lower lids (so that the pycnocline does not intersect either of the bounding channel plates) it follows from the baroclinic vorticity variation defined by (4.14) that the evolution of $\mathbf{\mathscr{G}}$ is simply

(4.21) $$\begin{eqnarray}\dot{\mathbf{\mathscr{G}}}=\dot{{\it\bf\Gamma}}_{kin}.\end{eqnarray}$$

For the special surfaces $\partial {\it\Omega}$ defined by ${\bf\omega}\boldsymbol{\cdot }\boldsymbol{n}=0$ , which in the channel case can imply ${\bf\omega}\boldsymbol{\cdot }\boldsymbol{k}=0$ along the plates, the right-hand side of this relation would vanish for the velocity class we are considering (decaying at infinity as $O(1/|\boldsymbol{x}_{\bot }|^{3})$ ), so that $\mathbf{\mathscr{G}}$ turns out to be a constant of the motion.

Conversely, let us now consider topologically non-trivial situations, where the pycnocline intersects the upper or lower lids along some closed curves, and hence coincides with the rigid plates in the regions internal to these curves. In this case, (4.21) becomes

(4.22) $$\begin{eqnarray}\dot{\mathbf{\mathscr{G}}}=\dot{{\it\bf\Gamma}}_{kin}+\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\mathop{\sum }_{{\it\beta}}\int _{{\it\gamma}_{{\it\beta}}}p\boldsymbol{t}_{{\it\beta}}\,\text{d}l_{{\it\beta}},\end{eqnarray}$$

where the sum is taken over the top and bottom intersection curves (thus excluding the limiting ${\it\gamma}_{_{\infty }}$ defined above along the lateral boundary of ${\it\Omega}$ ), i.e. the curves defined by ${\it\eta}(x,y)=z_{+}$ and ${\it\eta}(x,y)=z_{-}$ . Note that the vertical component of $\dot{\mathbf{\mathscr{G}}}$ is not affected by the contributions from the contours ${\it\gamma}_{{\it\beta}}$ , unlike its horizontal components.

Equation (4.22) conveys the topological contribution to the balance law for $\mathbf{\mathscr{G}}$ due to non-trivial pycnocline configurations generating non-simply connected interfaces between the two fluids.

4.2.2. Continuous stratifications

The topological effects studied above for two-layer configurations extend directly to the more physically relevant case of continuous stratifications, whenever the density fails to be constant along the bounding horizontal lids. The baroclinic term of the ${\it\bf\Gamma}$ evolution is now

(4.23) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=-\int _{{\it\Omega}}\boldsymbol{{\rm\nabla}}\times \left(\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}}\right)\text{d}V=\int _{\partial {\it\Omega}}p\boldsymbol{n}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S.\end{eqnarray}$$

Therefore, in the cylindrical geometry we are considering, $\dot{{\it\bf\Gamma}}_{bar}$ can be split into three different components,

(4.24) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\int _{\partial {\it\Omega}_{\bot }}p\boldsymbol{n}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S-\int _{W}p\boldsymbol{k}\times \boldsymbol{{\rm\nabla}}\left.\left(\frac{1}{{\it\rho}}\right)\right|_{z_{-}}\text{d}S+\int _{W}p\boldsymbol{k}\times \boldsymbol{{\rm\nabla}}\left.\left(\frac{1}{{\it\rho}}\right)\right|_{z_{+}}\text{d}S,\end{eqnarray}$$

where the second and third terms on the right-hand side are evaluated at the top and bottom plates $z=z_{\pm }$ . Note that these two terms vanish when ${\it\rho}$ is constant along the plates. For most applications, all isopycnals can be assumed to intersect the lateral boundary $\partial {\it\Omega}_{\bot }$ along closed curves as the domain $W$ limits to infinity, since the fluid will be assumed to be in isotropic hydrostatic equilibrium at infinity. Thus, the terms in (4.24) are the counterpart for continuously stratified fluids of those in (4.15); in particular, the first term can be viewed as the continuous version of the first term (4.15), while the second and third terms constitute the topological contribution from density isopycnals intersecting the bounding plates at $z=z_{\pm }$ .

Despite their similarity with the layered case, continuous stratifications do require some care in handling the integrals at the top and lower lids in (4.24) whenever the density varies along the lids. In fact, let us assume that some isopycnal surfaces have non-trivial intersections with the upper (lower) lid. Typically, such intersections would foliate the lid into a family of closed curves, which can be viewed as ‘holes’ in the isopycnal surface itself. (Not all such curves are necessarily smooth and/or simple, as ‘separatrix’-like configurations can be envisioned to happen.)

Under these circumstances, the topological terms in (4.24) can be computed as follows. Define

(4.25) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}^{\pm }\equiv \pm \int _{W}p\boldsymbol{k}\times \boldsymbol{{\rm\nabla}}\left.\left(\frac{1}{{\it\rho}}\right)\right|_{z_{\pm }}\text{d}S=\pm \int _{W}\frac{p^{\pm }}{({\it\rho}^{\pm })^{2}}\boldsymbol{T}_{\pm }\,\text{d}x\,\text{d}y,\end{eqnarray}$$

where

(4.26) $$\begin{eqnarray}\boldsymbol{T}_{\pm }\equiv \boldsymbol{{\rm\nabla}}{\it\rho}^{\pm }\times \boldsymbol{k}=(({\it\rho}^{\pm })_{y},-({\it\rho}^{\pm })_{x},0)\,\end{eqnarray}$$

is a tangent vector to the curves of constant density on the lids. For simplicity, we focus on the upper lid $z=z_{+}$ , and assume a non-constant smooth distribution ${\it\rho}(x,y,z_{+})\equiv {\it\rho}^{+}(x,y)$ , with ${\it\rho}^{+}(x,y)={\it\rho}_{1}$ on the boundary of $W$ . Let us also assume that ${\it\rho}^{+}$ has only one critical point in $W$ , with critical value ${\it\rho}_{max}>{\it\rho}_{1}$ . Then all curves of constant density on the upper plate are closed and bounded. By a change of variables from Cartesian to isopycnal coordinates $(x,y)\mapsto ({\it\rho}^{+},l)$ , such that $l$ is the arc length parameter on such curves, we get, by taking into account that the Jacobian ${\it\rho}_{x}l_{y}-{\it\rho}_{y}l_{x}$ equals $\sqrt{{\it\rho}_{x}^{2}+{\it\rho}_{y}^{2}}$ ,

(4.27) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}^{+}=\int _{W}\frac{p^{+}}{({\it\rho}^{+})^{2}}\boldsymbol{T}_{+}\,\text{d}x\,\text{d}y=\int _{{\it\rho}_{1}}^{{\it\rho}_{max}}\left(\oint \frac{p^{+}}{({\it\rho}^{+})^{2}}\boldsymbol{t}_{+}\,\text{d}l\right)\text{d}{\it\rho}^{+},\end{eqnarray}$$

where $\boldsymbol{t}_{+}=(x_{l},y_{l})$ is the normalized tangent vector. If the function ${\it\rho}^{+}$ has more than one critical point, as in figure 8, every critical point will provide a similar contribution to the time-variation of the total vorticity. The same argument applies to the lower lid.

Figure 8.

A typical density configuration at the upper boundary of the fluid. A similar configuration can exist for the lower boundary with asymptotic density ${\it\rho}_{2}$ .

The analogue of the topological vorticity vector $\mathbf{\mathscr{G}}$ for continuous stratifications can be defined with an identical expression in terms of the vorticity ${\it\bf\Gamma}$ and the total momentum ${\it\bf\Pi}$ to that of the two-layer case (4.16). Perhaps surprisingly, the time-variations of $\mathbf{\mathscr{G}}$ reduce to the same expressions as those for the two-layer case under natural assumptions. In fact, under asymptotic isotropic hydrostatic equilibrium at infinity, ${\it\rho}(\boldsymbol{x},t)\rightarrow {\it\rho}_{\infty }(z)$ , and for velocities whose asymptotic decay guarantees the vanishing of the first term on the right-hand side of (4.17) (e.g. in a cylindrical geometry, $\boldsymbol{u}_{\bot }$ decaying faster than $|\boldsymbol{x}_{\bot }|^{-1/2}$ would suffice), the time-derivative of the total momentum yields

(4.28) $$\begin{eqnarray}\dot{{\it\bf\Pi}}\times \boldsymbol{k}=-\int _{\partial {\it\Omega}_{\bot }}p\boldsymbol{n}\times \boldsymbol{k}\,\text{d}S=-\int _{\partial {\it\Omega}_{\bot }}p\boldsymbol{t}_{\infty }\,\text{d}S=-\int _{\partial {\it\Omega}_{\bot }}(P(l)+r(z))\boldsymbol{t}_{\infty }\,\text{d}S.\end{eqnarray}$$

Here, for the last term we have used the asymptotic property of the pressure (see appendix A for the case of zero initial velocities) which implies

(4.29) $$\begin{eqnarray}p(\boldsymbol{x},t)\sim P(l,t)-g\int ^{z}{\it\rho}_{\infty }(s)\,\text{d}s\equiv P(l,t)+r(z)\quad \text{as}~\boldsymbol{x}_{\bot }\rightarrow \infty .\end{eqnarray}$$

In (4.28) the second term vanishes, as

(4.30) $$\begin{eqnarray}\int _{\partial {\it\Omega}_{\bot }}r(z)\boldsymbol{t}_{\infty }\,\text{d}S=\left(\int _{z_{-}}^{z_{+}}r(z)\,\text{d}z\right)\left(\oint _{\partial W}\boldsymbol{t}_{\infty }\,\text{d}l\right)=0,\end{eqnarray}$$

since the integral of the unit tangent vector along its loop vanishes. Hence,

(4.31) $$\begin{eqnarray}\dot{{\it\bf\Pi}}\times \boldsymbol{k}=-\oint _{\partial W}P(l)\boldsymbol{t}_{\infty }\,\text{d}l\,\int _{z_{-}}^{z_{+}}\text{d}z=-h\oint _{\partial W}P(l)\boldsymbol{t}_{\infty }\,\text{d}l.\end{eqnarray}$$

This expression matches its two-layer counterpart (4.20), upon observing that for the present configuration ${\it\gamma}_{\infty }$ is the rigid translation of the boundary of the plate domain $\partial W$ .

When the density is constant along the upper and lower plates (namely, ${\it\rho}(x,y,z_{+})={\it\rho}_{1}$ and ${\it\rho}(x,y,z_{-})={\it\rho}_{2}$ ) and the fluid is asymptotically in isotropic equilibrium at infinity, performing similar manipulations with the baroclinic vorticity time-variation, equation (4.10) yields

(4.32) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\oint _{\partial W}P(l)\boldsymbol{t}_{\infty }\,\text{d}l\,\int _{z_{-}}^{z_{+}}\left(\frac{1}{{\it\rho}}\right)_{z}\text{d}z\,=\frac{{\it\rho}_{_{{\it\Delta}}}}{{\it\rho}_{1}{\it\rho}_{2}}\oint _{\partial W}P(l)\boldsymbol{t}_{\infty }\,\text{d}l.\end{eqnarray}$$

Here again we have used the null property of the integral of the tangent for cancelling the term

(4.33) $$\begin{eqnarray}\left(\oint _{\partial W}\boldsymbol{t}_{\infty }\,\text{d}l\right)\left(\int _{z_{-}}^{z_{+}}r(z)\left(\frac{1}{{\it\rho}}\right)_{z}\text{d}z\right)\end{eqnarray}$$

which arises in the computation. The time-derivative $\dot{{\it\bf\Gamma}}_{bar}$ is thus

(4.34) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=-\frac{{\it\rho}_{_{{\it\Delta}}}}{h{\it\rho}_{1}{\it\rho}_{2}}\dot{{\it\bf\Pi}}\times \boldsymbol{k},\end{eqnarray}$$

just as in the two-layer case. Hence,

(4.35) $$\begin{eqnarray}\dot{\mathbf{\mathscr{G}}}=\dot{{\it\bf\Gamma}}_{kin}\end{eqnarray}$$

still holds for the continuously stratified case, under the assumptions we have made. As before, for the non-trivial topology of isopycnals outlined above, the evolution of the topological vorticity $\mathbf{\mathscr{G}}$ acquires the topological contribution to the total vorticity (4.27), since the evolution of the momentum ${\it\bf\Pi}$ with continuous stratification remains unaffected by the topology.

5. The case of a free surface

Finally, we briefly discuss how the previous results are modified by removing the constraint of a rigid lid and replacing it with a free surface at $z={\it\eta}(x,y,t)$ , with constant pressure $p=p_{a}$ . Even if pycnoclines can intersect the free boundary, there are now substantial dynamical differences between the topological contributions to the vorticity time-evolution with respect to rigid horizontal boundaries.

For the free-boundary case the interpretation of the control volume

(5.1) $$\begin{eqnarray}{\it\Omega}=\{(x,y,z)\mid (x,y)\in W,~z_{-}<z<{\it\eta}(x,y,t)\}\end{eqnarray}$$

is naturally Lagrangian. Similarly to rigidly bounded fluids, the time-derivative of the total vorticity can still be split into its kinematic and baroclinic components, with the latter defined in (4.23) as

(5.2) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=-\int _{{\it\Omega}}\boldsymbol{{\rm\nabla}}\times \left(\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}}\right)\text{d}V=\int _{\partial {\it\Omega}}p\boldsymbol{n}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S.\end{eqnarray}$$

Therefore

(5.3) $$\begin{eqnarray}\dot{{\it\bf\Gamma}}_{bar}=\int _{\partial {\it\Omega}_{\bot }}p\boldsymbol{n}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S+\int _{z={\it\eta}}p\boldsymbol{n}_{{\it\eta}}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S-\int _{z=z_{-}}p\boldsymbol{k}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S,\end{eqnarray}$$

where the free-surface integral must be identically zero, since clearly the arbitrary constant in the definition of pressure cannot contribute to the vorticity time-variation. This can be seen directly by the same change of variables as used in (4.27) from Cartesian to isopycnal coordinates. In the cylindrical domain of the previous section we have applied this change of variables at the fixed upper lid, while now it must be applied to the time-varying free surface of the fluid. Denoting by ${\it\rho}^{+}$ the fluid density at the free surface, the interface baroclinic contribution to the vorticity evolution is

(5.4) $$\begin{eqnarray}\int _{z={\it\eta}}p\boldsymbol{n}_{{\it\eta}}\times \boldsymbol{{\rm\nabla}}\left(\frac{1}{{\it\rho}}\right)\text{d}S=-p_{a}\int _{{\it\rho}_{1}}^{{\it\rho}_{max}}\frac{1}{({\it\rho}^{+})^{2}}\left(\oint \boldsymbol{t}_{+}\,\text{d}l\right)\text{d}{\it\rho}^{+}=0,\end{eqnarray}$$

where the last equality holds because the circulation of the tangent vector along a closed curve is zero. The first term in (5.3) is the non-topological contribution already encountered in the rigid-lid case of the previous section, and vanishes for fluids asymptotically in hydrostatic equilibrium. However, unlike the rigid-lid case, the constant-pressure boundary condition kills the topological contribution from the top, leaving only the isopycnal topology at the bottom to give a possibly non-zero contribution. The free-surface contribution to vorticity is therefore purely kinematical, due to the fact that the normal velocity to the free surface may not be identically zero.