# Fully conservative hydraulic jumps and solibores in two-layer Boussinesq
fluids

Jānis Priede1,2 1Centre for Fluids and Complex Systems,
Coventry University, UK
2Department of Physics, University of Latvia, Riga, LV-1004, Latvia

###### Abstract

We consider a special type of hydraulic jumps (internal bores) which, in the
vertically bounded system of two immiscible fluids with slightly different
densities, conserve not only the mass and impulse but also the circulation and
energy. This is possible only at specific combinations of the upstream and
downstream states. Two such combinations are identified with arbitrary
upstream and downstream interface heights. The first has a cross symmetry
between the interface height and shear on both sides of the jump. This
symmetry, which is due to the invariance of the two-layer shallow-water system
with swapping the interface height and shear, ensures the automatic
conservation of the impulse and energy as well as the continuity of
characteristic velocities across the jump. The speed at which such jumps
propagate is uniquely defined by the conservation of the mass and circulation.
The other possibility is a marginally stable shear flow which can have fully
conservative jumps with discontinuous characteristic velocities. Both types of
conservative jumps are shown to represent a long-wave approximation to the so-
called solibores which appear as smooth permanent-shape solutions in a weakly
non-hydrostatic model. A new analytical solution for solibores is obtained and
found to agree very well with the previous DNS results for partial-depth lock
release flow. The finding that certain large-amplitude hydraulic jumps can be
fully conservative, while most are not such even in the inviscid
approximation, points toward the wave dispersion as a primary mechanism behind
the lossy nature of internal bores.

##  1 Introduction

Hydraulic jumps are steep variations in the height of the liquid surface which
can propagate at a nearly constant speed over relatively large distances. Such
step-like gravity waves are often referred to as bores. A well-known example
is that of tidal bores (Simpson, 1999). Hydraulic jumps can form also in
stably stratified fluids where they are known as internal bores (Baines,
1995). The latter occur in various geophysical flows, such as coastal oceans
(Scotti & Pineda, 2004) and the inversion layers in the atmosphere (Christie &
White, 1992).

One of the key characteristics of hydraulic jumps is their speed of
propagation. Determination of this speed theoretically is difficult because
the flow associated with bores tends to be very complex and often turbulent.
Notwithstanding this complexity, the speed at which a bore propagates can be
determined approximately by considering the conservation of relevant
quantities. In single fluid layers, the speed of propagation is determined by
the conservation of the mass and momentum fluxes across the front of the bore.
Application of the respective conservation laws integrally to a box enclosing
the jump constitutes the basis of the hydraulic approximation, also known as
the control volume method (Rayleigh, 1914).

The same front speed is defined also by the Rankine-Hugoniot jump conditions
for the hydrostatic shallow-water (SW) equations governing the conservation of
mass and momentum in single fluid layers (Whitham, 1974). The momentum,
however, is not the only dynamical quantity that can be conserved. In fact,
the hydrostatic SW equations admit an infinite number of locally conserved
quantities (Whitham, 1974, p. 459). However, only one such quantity can in
general be conserved besides the mass. For example, the jumps conserving
momentum do not in general conserve energy. The preference for momentum over
energy as a conserved quantity is motivated by physical considerations.
Namely, the loss of energy can be attributed to the viscous dissipation in
strongly turbulent bores. Turbulence can also enhance vorticity by stretching
and tilting vortices so increasing the total amount of enstrophy (Batchelor,
1967, Sec. 5.2, p. 270). This can affect the conservation of the circulation
flux across the jump. However, there is no analogous physical mechanism by
which turbulence or viscosity could affect the flux of momentum across the
jump.

The same considerations apply also to two-layer systems. However, there are a
few significant differences which concern two-layer systems bounded by a rigid
lid. The principal difference is a longitudinal pressure gradient. It appears
in this case due to the fixed total height and ensures that the volumetric
flux in one layer is equal but opposite to that in the other layer. This makes
the basic SW equations for separate layers non-local. As a result, the
longitudinal momentum is not in general conserved in such systems (Camassa et
al., 2012). But it does not mean that the two-layer hydrostatic SW equations
are inherently non-local as it is often thought (Fyhn et al., 2019). These
equations can be represented in a number of locally conservative forms which
are mathematically equivalent as long as the waves are smooth (Priede, 2023).
Firstly, there is a circulation conservation equation (Sandstrom & Quon,
1993). It yields the same speed of propagation as the vorticity front
condition obtained by Borden & Meiburg (2013) using the conventional control-
volume approach. Secondly, there is also a momentum-like quantity, called
pseudo-momentum or impulse (Benjamin, 1986), which is conserved locally in
two-layer systems bounded by a rigid lid. However, in contrast to the single-
layer case, the respective two-layer SW conservation equation is not uniquely
defined. It contains a free parameter $\alpha$ which defines the relative
contribution of each layer to the pressure at the interface (Priede, 2023).
This parameter, which is expected to depend on the density ratio, affects only
the hydraulic jumps but not continuous waves. The Rankine-Hugoniot jump
conditions resulting from the mass and impulse conservation equations with
$\alpha=1$ and $\alpha=-1$ are mathematically equivalent to the classical
front conditions for internal bores obtained by Wood & Simpson (1984) and
Klemp et al. (1997), respectively. The latter also includes the front
condition of Benjamin (1968) for gravity currents. The vorticity front
condition of Borden & Meiburg (2013) is in turn recovered in the limit
$|\alpha|\rightarrow\infty.$ With $\alpha=0,$which is suggested by symmetry
considerations for Boussinesq fluids, the predicted jump propagation
velocities agree well with the available numerical and experimental results
(Priede, 2023).

Since the speed of propagation predicted by the SW model depends on $\alpha,$
circulation like energy is not in general conserved by the jumps that conserve
the mass and impulse. This is because the related conservation laws are
mutually equivalent only for smooth (differentiable) solutions (Whitham,
1974). There are, however, a few exceptions that correspond to the jumps whose
speed of propagation does not depend on $\alpha.$ Such jumps, which conserve
all four quantities, are the focus of the present paper. We show that these
fully conservative jumps represent a long-wave approximation to the solibores
which are permanent-shape smooth solutions appearing in the weakly non-
hydrostatic approximation (Esler & Pearce, 2011). We also obtain an exact
analytical solution for such solibores and find a good agreement with the
numerical results of Khodkar et al. (2017) for the partial-height lock-
exchange flow in a two-layer system with a small density contrast.

The paper is organized as follows. In section 2, we formulate the problem and
present the basic equations including the generalized SW momentum equation.
Jump conditions are introduced and analysed in section 3. In section 4, we
show that the characteristics of fully conservative jumps are the same as
those of solibores and obtain a relatively simple analytical solution for the
latter. The paper is concluded with section 5 which contains a summary and
discussion of the main results including a comparison with numerical results.

##  2 Two-layer SW model

Figure 1: Sketch of the problem showing a horizontal channel of a constant
height $H$ bounded by two parallel solid walls and filled with two inviscid
immiscible fluids with constant densities $\rho^{+}$ and $\rho^{-},$ where
$h^{+}=\zeta(x,t)$ and $h^{-}=H-h^{+}$are the depths of the bottom and top
layers, respectively.

Consider a horizontal channel of a constant height $H$ which is bounded by two
parallel solid walls and filled with two inviscid immiscible fluids with
constant densities $\rho^{+}$ and $\rho^{-}$ as shown in Fig. 1. The fluids
are subject to a downward gravity force with the free fall acceleration $g.$
The interface separating the fluids is located at the height $z=\zeta(x,t),$
which is equal to the depth of the bottom layer $h^{+}$ and varies with the
horizontal position $x$ and the time $t.$ The velocity $\boldsymbol{u}^{\pm}$
and the pressure $p^{\pm}$ in each layer are governed by the Euler equation

$\partial_{t}\boldsymbol{u}+\boldsymbol{u}\cdot\boldsymbol{\nabla}\boldsymbol{u}=-\rho^{-1}\boldsymbol{\nabla}p+\boldsymbol{g}$
(1)

and the incompressibility constraint
$\boldsymbol{\nabla}\cdot\boldsymbol{u}=0.$ Henceforth, for the sake of
brevity, we drop $\pm$ indices wherever analogous expressions apply to both
layers. At the interface $z=\zeta(x,t)$, we have the continuity of pressure,
$[p]\equiv p^{+}-p^{-}=0,$ and the kinematic condition

$w=\frac{d\zeta}{dt}=\zeta_{t}+u\zeta_{x},$ (2)

where $\frac{d}{dt}$ denotes the material derivative, $u$ and $w$ are the $x$
and $z$ components of velocity, and the subscripts $t$ and $x$ stand for the
corresponding partial derivatives.

To make the paper self-contained, below we shall present a brief derivation of
the basic shallow water equations. Firstly, integrating the incompressibility
constraint over the depth of each layer and using (2), we obtain

$h_{t}+(h\bar{u})_{x}=0,$ (3)

where the overbar denotes the depth average. Secondly, doing the same for the
horizontal $(x)$ component of (1), we have

$(h\bar{u})_{t}+(h\overline{u^{2}})_{x}=-\rho^{-1}h\overline{p}_{x}.$ (4)

Pressure is obtained by integrating the vertical $(z)$ component of (1) as
follows

$p(x,z,t)=\mathit{\Pi}(x,t)+\rho\int_{\zeta}^{z}(w_{t}+\boldsymbol{u}\cdot\boldsymbol{\nabla}w-g)dz,$
(5)

where $\mathit{\Pi}(x,t)=\left.p^{\pm}(x,z,t)\right|_{z=\zeta}$ is the
distribution of pressure along the interface. Thirdly, averaging the
$x$-component of the gradient of the pressure (5) over the depth of each
layer, after a few rearrangements, we obtain

$\overline{p}_{x}=\left(\mathit{\Pi}+\rho
g\zeta+\rho\overline{(z-z_{0})(w_{t}+\boldsymbol{u}\cdot\boldsymbol{\nabla}w)}\right)_{x},$
(6)

which defines the RHS of (4) with $z_{0}=0$ and $z_{0}=H$ for the bottom and
top layers, respectively.

When the characteristic horizontal length scale $L$ is much larger than the
height $H:$ $H/L=\epsilon\ll 1,$ the exact depth-averaged equations obtained
above can be simplified using the shallow-water approximation. In this case,
the incompressibility constraint implies $w/u=O(\epsilon)$ and (6)
correspondingly reduces to

$\overline{p}_{x}=(\mathit{\Pi}+\rho g\zeta)_{x}+O(\epsilon^{2}),$ (7)

where the leading-order term is purely hydrostatic and $O(\epsilon^{2})$
represents a small dynamical pressure correction due to the vertical velocity
$w.$ Besides that the flow in both layers is considered to be irrotational:
$\boldsymbol{\omega}=\boldsymbol{\nabla}\times\boldsymbol{u}=0.$ According to
the inviscid vorticity equation
$\frac{d\boldsymbol{\omega}}{dt}=(\boldsymbol{\omega}\cdot\boldsymbol{\nabla})\boldsymbol{u},$
this property is preserved by (1). In the leading-order approximation, the
irrotationality constraint reduces to $\partial_{z}u^{(0)}=0$. It means that
the horizontal velocity can be written as

$u=\bar{u}+\tilde{u},$ (8)

where the deviation from the average $\bar{u}$ implied by (7) is
$\tilde{u}=O(\epsilon^{2}).$ Consequently, in the second term of (4), we can
substitute $\overline{u^{2}}=\bar{u}^{2}+O(\epsilon^{4}).$ Finally, using (3)
and ignoring the $O(\epsilon^{2})$ dynamical pressure correction, (4) can be
written as

$\rho\left(\bar{u}_{t}+\tfrac{1}{2}\text{$\bar{u}^{2}$}_{x}+g\zeta_{x}\right)=-\mathit{\Pi}_{x}.$
(9)

This equation along with (3) constitute the basic set of SW equations in the
leading-order (hydrostatic) approximation.

The vertical velocity, which gives rise to a non-hydrostatic pressure
perturbation, follows from the incompressibility constraint and (8) as

$w(z)=-\int_{z_{0}}^{z}u_{x}dz=-(z-z_{0})\bar{u}_{x}+O(\epsilon^{2}),$ (10)

where $z_{0}$ is defined as in (6) to satisfy the impermeability conditions
$w(0)=w(H)=0.$ Substituting this into (6), after a few transformations, we
obtain the well-known result for the first-order pressure correction (Green &
Naghdi, 1976; Liska et al., 1995; Choi & Camassa, 1999)

$h\overline{p}_{x}^{(1)}=-\tfrac{1}{3}\rho\left(h^{3}(D_{t}\bar{u}_{x}-\text{$\bar{u}_{x}$}^{2}\right)_{x}+O(\epsilon^{4})=\tfrac{1}{3}\rho\left(h^{2}D_{t}^{2}h\right)_{x}+O(\epsilon^{4}),$
(11)

where $D_{t}\equiv\partial_{t}+\bar{u}\partial_{x}$ and
$\bar{u}_{x}=-h^{-1}D_{t}h.$ The latter relation follows from (3) and ensures
that (2) is satisfied by (10) up to $O(\epsilon^{2}).$

The system of four SW equations (9) and (3) contains five unknowns: $u^{\pm}$,
$h^{\pm}$ and $\mathit{\Pi},$ and is completed by adding the fixed height
constraint $\\{h\\}\equiv h^{+}+h^{-}=H.$ Henceforth, we simplify the notation
by omitting the bar over $u$ and using the curly brackets to denote the sum of
the enclosed quantities.

Two more unknowns are eliminated as follows. Firstly, adding the mass
conservation equations for each layer together and using $\\{h\\}_{t}\equiv
0,$ we obtain $\\{uh\\}=\Phi(t),$ which is the total flow rate. The channel is
assumed to be laterally closed, which means $\Phi\equiv 0,$ and, thus,
$u^{-}h^{-}=-u^{+}h^{+}.$ Secondly, the pressure gradient $\mathit{\Pi}_{x}$
can be eliminated by subtracting the two equations (9) one from another. This
leaves only two unknowns, $U\equiv u^{+}h^{+}$ and $h=h^{+},$ and two
equations, which can be written in a locally conservative form as

$\displaystyle\left(\left\\{\rho/h\right\\}U\right)_{t}+\left(\tfrac{1}{2}\left[\rho/h^{2}\right]U^{2}+g\left[\rho\right]h\right)_{x}$
$\displaystyle=-\left[\overline{p}_{x}^{(1)}\right],$ (12) $\displaystyle
h_{t}+U_{x}$ $\displaystyle=0.$ (13)

where the square brackets denote the difference of the enclosed quantities
between the bottom and top layers: $\left[f\right]\equiv f^{+}-f^{-}.$ The
pressure correction on the RHS of (12), which is defined by (11), can be cast
in the following locally conservative form

$\overline{p}_{x}^{(1)}=\frac{\rho}{3}\left(\left(hD_{t}^{2}h+\tfrac{1}{2}(D_{t}h)^{2}-h_{t}D_{t}h\right)_{x}+\left(h_{x}D_{t}h\right)_{t}\right).$
(14)

In the hydrostatic approximation, which will be considered first, this
dynamical pressure correction is irrelevant.

In the following, the density difference is assumed to be small. According to
the Boussinesq approximation, this difference is important only for the
gravity of fluids, which drives the flow. For the inertia, this difference is
ignored, which simplifies the problem significantly. A further simplification
is achieved by using the total height $H$ and the characteristic gravity wave
speed $C=\sqrt{2Hg[\rho]/\\{\rho\\}}$ as the length and velocity scales,
respectively, and $H/C$ as the time scale.

In the hydrostatic Boussinesq approximation, (12) and (13) take a remarkably
symmetric form (Milewski & Tabak, 2015):

$\displaystyle\vartheta_{t}+\tfrac{1}{2}(\eta(1-\vartheta^{2}))_{x}$
$\displaystyle=0,$ (15)
$\displaystyle\eta_{t}+\tfrac{1}{2}(\vartheta(1-\eta^{2}))_{x}$
$\displaystyle=0,$ (16)

where $\eta=[h]$ and $\vartheta=[u]$ are the dimensionless depth and velocity
differentials between the top and bottom layers. Subsequently, the former is
referred to as the interface height and the latter as the shear velocity. Note
that this basic system of two-layer SW equations is invariant to swapping
$\eta$ and $\vartheta.$ The same symmetry holds also for the momentum and
energy equations

$\displaystyle(\eta\vartheta)_{t}+\tfrac{1}{4}(\eta^{2}+\vartheta^{2}-3\eta^{2}\vartheta^{2})_{x}$
$\displaystyle=0,$ (17)
$\displaystyle(\eta^{2}+\vartheta^{2}-\eta^{2}\vartheta^{2})_{t}+(\eta\vartheta(1-\eta^{2})(1-\vartheta^{2}))_{x}$
$\displaystyle=0,$ (18)

which are obtained by multiplying (15) with $\eta$ and $\eta^{2}\vartheta,$
respectively, and then using (16) to convert the resulting expression into
locally conservative form. It is important to note that this
$\eta\leftrightarrow\vartheta$ symmetry is limited to the Boussinesq
approximation. Also, note that the conserved quantity $\eta\vartheta$ in (17)
represents a pseudo-momentum (impulse) (Priede, 2023). An infinite sequence of
further conservation laws can be constructed in a similar way (Milewski &
Tabak, 2015). The generalized momentum equation can formally be obtained by
multiplying (15) with an arbitrary constant $\alpha$ and adding to (17):

$((\eta+\alpha)\vartheta)_{t}+\tfrac{1}{4}(\eta^{2}+\vartheta^{2}-3\eta^{2}\vartheta^{2}+2\alpha\eta(1-\vartheta^{2}))_{x}=0.$
(19)

For a more detailed derivation of equations (17)–(19), we refer to Priede
(2023). It has to be stressed that equations (15) and (17)–(19) are mutually
equivalent and can be transformed one into another using (16) only if
$\vartheta$ and $\eta$ are differentiable. This, however, is not the case
across hydraulic jumps which will be considered in the next section. As the
problem is governed by two equations, only two corresponding jump conditions
can in general be satisfied.

The constant $\alpha$ in (19), which defines the relative contribution of each
layer to the pressure gradient along the interface, is supposed to depend only
on the ratio of densities. For nearly equal densities, which is the case
covered by the Boussinesq approximation, $\alpha\approx 0$ can be expected.
This corresponds to both layers affecting the pressure at the interface with
equal weight coefficients. Note that with $\alpha=0$ (19) reduces to the basic
momentum equation (17) so restoring the $\eta\leftrightarrow\vartheta$
symmetry of the Boussinesq approximation. This symmetry is recovered also in
the limit |$\alpha|\rightarrow\infty,$ in which (19) reduces to the
circulation conservation equation (15).

Two-layer SW equations for Boussinesq fluids can also be written in the
canonical form

$R_{t}^{\pm}-\lambda^{\pm}R_{x}^{\pm}=0,$ (20)

where

$R^{\pm}=-\eta\vartheta\pm\sqrt{(1-\eta^{2})(1-\vartheta^{2})}$ (21)

are the Riemann invariants and

$\lambda^{\pm}=\tfrac{3}{4}R^{\pm}+\tfrac{1}{4}R^{\mp}=-\eta\vartheta\pm\tfrac{1}{2}\sqrt{(1-\eta^{2})(1-\vartheta^{2})}$
(22)

are the associated characteristic velocities (Long, 1956; Cavanie, 1969;
Ovsyannikov, 1979; Sandstrom & Quon, 1993; Baines, 1995; Chumakova et al.,
2009).

For the interface confined between the top and bottom boundaries, which
corresponds to $\eta^{2}\leq 1,$ the characteristic velocities (22) are real
and, thus, the equations are of hyperbolic type if $\vartheta^{2}\leq 1.$ This
hyperbolicity constraint on the shear velocity ensures the absence of the
long-wave Kelvin-Helmholtz instability which would otherwise disrupt the
interface (Milewski et al., 2004). It has to be noted that this instability is
different from the usual short-wave Kelvin-Helmholtz which does not appear in
the hydrostatic SW approximation (Esler & Pearce, 2011).

##  3 Hydraulic jumps

Figure 2: A jump with the upstream interface height $\eta_{-}=\eta$ and the
shear velocity $\vartheta_{-}=\vartheta$ propagating at the speed $\dot{\xi}$
into a still fluid ahead $(\vartheta_{+}=0)$ with the interface located at the
height $\eta_{+}=\eta_{0}.$

Consider a discontinuity in $\eta$ and $\vartheta$ at the point $x=\xi(t)$
across which the respective variables jump by
$\left\llbracket\eta\right\rrbracket\equiv\eta_{+}-\eta_{-}$ and
$\left\llbracket\vartheta\right\rrbracket\equiv\vartheta_{+}-\vartheta_{-}.$
Here the plus and minus subscripts denote the corresponding quantities at the
front and behind the jump. The double-square brackets stand for the
differential of the enclosed quantity across the jump. Integrating (16) and
(19) across the jump, which is equivalent to substituting the spatial
derivative $f_{x}$ with $\left\llbracket f\right\rrbracket$ and the time
derivative $f_{t}$ with $-\dot{\xi}\left\llbracket f\right\rrbracket$
(Whitham, 1974), the jump propagation velocity can be expressed, respectively,
as (Priede, 2023)

$\displaystyle\dot{\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\left\llbracket\vartheta(1-\eta^{2})\right\rrbracket}{\left\llbracket\eta\right\rrbracket},$
(23) $\displaystyle\dot{\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\left\llbracket\eta^{2}+\vartheta^{2}-3\eta^{2}\vartheta^{2}+2\alpha(1-\vartheta^{2})\right\rrbracket}{\left\llbracket(\eta+\alpha)\vartheta\right\rrbracket}.$
(24)

For a jump to be feasible, it has to satisfy the hyperbolicity constraint
$\vartheta_{\pm}^{2}\leq 1$ as well as the energy constraint. The latter
follows from the integration of (18) across the jump and defines the
associated energy production rate (Priede, 2023)

$\dot{\varepsilon}=\left\llbracket\eta\vartheta(1-\eta^{2})(1-\vartheta^{2})\right\rrbracket-\dot{\xi}\left\llbracket\eta^{2}+\vartheta^{2}-\eta^{2}\vartheta^{2}\right\rrbracket\leq
0,$ (25)

which cannot be positive as the energy can only be dissipated or dispersed but
not generated by the jump.

Applying (23) and (24) to a jump with the upstream interface height
$\eta_{-}=\eta$ propagating into a quiescent fluid $(\vartheta_{+}=0)$ with
the interface located at the height $\eta_{+}=\eta_{0},$ as shown in Fig. 2,
after a few rearrangements, we obtain:

$\displaystyle\vartheta_{-}^{\pm}$
$\displaystyle=\pm\frac{(\eta_{0}-\eta)(\eta_{0}+\eta+2\alpha)^{1/2}}{((1-\eta^{2})(\eta_{0}-\eta)+2(\eta+\alpha)(1-\eta_{0}\eta))^{1/2}},$
(26) $\displaystyle\dot{\xi}^{\pm}$
$\displaystyle=-\vartheta_{-}^{\pm}\frac{1-\eta^{2}}{2(\eta_{0}-\eta)},$ (27)

where $\vartheta_{-}^{\pm}$ is the upstream shear velocity and the plus and
minus signs refer to the opposite directions of propagation, i.e.,
$\dot{\xi}^{+}=-\dot{\xi}^{-},$ which are both permitted by the mass and
momentum balance conditions. Because the energy balance (25) changes sign with
the direction of propagation, only one direction of propagation is in general
permitted for each jump. This does not apply to the jumps which satisfy (25)
exactly. Such energy-conserving jumps, which can in principle propagate
downstream as well as upstream, will be considered in the following.

A distinctive feature of these jumps is the invariance of their velocity of
propagation (27) with $\alpha$ (Priede, 2023). It implies that these jumps
conserve also circulation. If this is the case, the $\alpha$ terms in the
generalized momentum equation (19) cancel out so making the associated jump
condition independent of $\alpha.$ For this to happen, the shear velocity (26)
at $\alpha=0$ has to be the same as that at $\alpha\rightarrow\infty.$ This
requirement results in $\eta(\eta-\eta_{0})=0,$ which has two solutions:
$\eta=\eta_{0}$ and $\eta=0.$ The first one is irrelevant as it corresponds to
a uniform state with a constant interface height $\eta_{0}.$ The second one
describes a jump from $\eta_{0}$ to the channel mid-height $\eta=0.$ The
corresponding upstream shear velocity and the speed of propagation following
from (26) and (27) are $\vartheta_{-}^{\pm}=\pm\eta_{0}$ and
$\dot{\xi}^{\pm}=\mp\frac{1}{2}.$

Owing to the symmetry of this jump $\vartheta_{\pm}=\eta_{\mp},$ the impulse
and energy are conserved automatically, i.e., independently of the velocity of
propagation. Besides that the same symmetry makes both characteristic
velocities (22) continuous across the jump:
$\left\llbracket\lambda^{\pm}\right\rrbracket=0.$ It means that the positive,
as well as negative characteristics, emanating upstream from the jump are
parallel to the corresponding characteristics emanating downstream. Likewise
continuous are also the Riemann invariants (21).

The above properties are shared by a broader class of fully conservative jumps
which can be sustained by a non-zero downstream shear $(\vartheta_{+}\not=0).$
The continuity condition $\left\llbracket\lambda^{\pm}\right\rrbracket=0$,
which after a few rearrangements can be rewritten as
$\left\llbracket(\eta\pm\vartheta)^{2}\right\rrbracket=0,$ yields two pairs of
possible solutions:

$\displaystyle(\eta_{+},\vartheta_{+})$
$\displaystyle=\pm(\eta_{-},\vartheta_{-}),$ (28)
$\displaystyle(\eta_{+},\vartheta_{+})$
$\displaystyle=\pm(\vartheta_{-},\eta_{-}).$ (29)

The first solution corresponding to the plus sign in (28) is continuous and so
irrelevant. The other three are symmetric jumps which automatically satisfy
the energy balance condition (25) as well as that of the impulse (24) with
$\alpha=0.$ The second solution corresponding to the minus sign in (28), which
represents a centrally symmetric jump, conserves the mass and circulation only
if $\vartheta_{+}=\pm\eta_{+}$ and $\dot{\xi}=\pm\frac{1}{2}(1-\eta_{+}^{2}).$
This is just a particular case of a more general solution that follows from
the mass and circulation conservation laws for the second pair of jumps (29).
In this case, we obtain the speed of propagation which can be written in terms
of the upstream and downstream interface heights as

$\dot{\xi}^{\pm}=\pm\frac{1}{2}(1+\eta_{+}\eta_{-}).$ (30)

The corresponding shear velocities according to (29) are
$(\vartheta_{+},\vartheta_{-})=\mp(\eta_{-},\eta_{+}).$ Note that the previous
two solutions with $\eta_{+}=0$ and $\eta_{+}=-\eta_{-}$ are particular cases
of (30).

In marginally hyperbolic shear flows with $\vartheta_{+}=\vartheta_{-}=\mp 1,$
which is a very specific case, fully conservative jumps with discontinuous
characteristic velocities are possible. Such jumps propagate at the speed
$\dot{\xi}^{\pm}=\pm\frac{1}{2}(\eta_{+}+\eta_{-})$ which ensures the
conservation of impulse and mass, whilst the energy and circulation are
conserved automatically.

In the next section, we show that these fully conservative hydraulic jumps
represent a long-wave approximation to the so-called solibores which appear as
permanent-shape solutions in the weakly non-hydrostatic approximation
described by (11) (Esler & Pearce, 2011).

##  4 Weakly non-hydrostatic analytical solution for solibores

Let us now turn to a weakly non-hydrostatic approximation, which is defined by
the dynamical pressure correction (14) in the circulation conservation
equation (12), and search for permanent-shape waves travelling at the speed
$c.$ Such waves are stationary in the co-moving frame of reference where they
vary depending only on $x^{\prime}=x-ct.$ With the time derivative written as
$\partial_{t}\equiv-c\partial_{x},$ the last two terms in (14) cancel out, and
correspondingly (12) and (13) take the form

$\displaystyle\left(\tfrac{1}{2}\eta(1-\vartheta^{2})-c\vartheta\right)_{x}$
$\displaystyle=$
$\displaystyle-\tfrac{1}{3}\left[hD_{t}^{2}h+\tfrac{1}{2}(D_{t}h)^{2}\right]_{x},$
(31) $\displaystyle\left(\tfrac{1}{2}\vartheta(1-\eta^{2})-c\eta\right)_{x}$
$\displaystyle=$ $\displaystyle 0,$ (32)

where $h$ stands for $h^{\pm}=\frac{1}{2}(1\pm\eta)$ and $D_{t}$ for
$D_{t}^{\pm}\equiv(u^{\pm}-c)\partial_{x}$ with the plus and minus indices
referring to the top and bottom layers. Now (31) and (32) can be integrated
once to obtain

$\displaystyle\tfrac{1}{2}\eta(1-\vartheta^{2})-c(\vartheta-\vartheta_{0})$
$\displaystyle=$
$\displaystyle-\tfrac{1}{3}\left[hD_{t}^{2}h+\tfrac{1}{2}(D_{t}h)^{2}\right],$
(33) $\displaystyle\tfrac{1}{2}\vartheta(1-\eta^{2})-c(\eta-\eta_{0})$
$\displaystyle=$ $\displaystyle 0,$ (34)

where $\vartheta_{0}$ and $\eta_{0}$ are the constants of integration. Note
that if the fluid far up- or downstream is at rest $(\vartheta=0),$ then
according to (34) $\eta_{0}$ is equal to the respective interface height;
$c\vartheta_{0}=A$ is the flux of circulation which is one of the conserved
quantities.

Using the identity $hD_{t}^{\pm}\equiv-h_{0}^{\pm}c\partial_{x},$ which
follows from $D_{t}^{\pm}\equiv(u^{\pm}-c)\partial_{x}$ and (34) when written
in terms of the original variables as $h^{\pm}(u^{\pm}-c)=-h_{0}^{\pm}c$, the
RHS of (33) can be transformed as follows

$-\frac{c^{2}}{3}\left[h_{0}^{2}\left(\left(h_{x}/h\right)_{x}+\tfrac{1}{2}\left(h_{x}/h\right)^{2}\right)\right]=-\frac{c^{2}}{3}\left\\{(h_{0}h_{x})^{2}/h\right\\}_{x}/\text{$\eta_{x}$}.$

Then multiplying (33) with $\eta_{x}$ and using (32) to substitute
$c\eta_{x},$ after a few rearrangements, the resulting equation can be
integrated once more to obtain

$A\eta+B-\tfrac{1}{4}(1-\eta^{2})(1+\vartheta^{2})=-\frac{c^{2}}{12}\frac{1-2\eta_{0}\eta+\eta_{0}^{2}}{1-\eta^{2}}\text{$\eta_{x}$}^{2},$

where $B$ is a constant of integration which represents another conserved
quantity, the flux of impulse. Finally, using (34) to eliminate $\vartheta,$
we obtain

$\text{$\eta_{x}$}^{2}+P(\eta)=0,$ (35)

where

$P(\eta)=-\frac{3}{c^{2}}\frac{(1-\eta^{2})(1-\eta^{2}-4(A\eta+B))+4c^{2}(\eta-\eta_{0})^{2}}{1-\eta^{2}+(\eta-\eta_{0})^{2}}$
(36)

is analogous to potential energy when $\text{$\eta_{x}$}^{2}$ is interpreted
as kinetic energy with $x$ representing the time. Thus (35) can be thought of
as describing the conservation of total energy for a body performing periodic
oscillations in the potential well defined by (36). The oscillations occur
between the points at which the velocity-like quantity $\eta_{x}$ changes
sign. This happens at the turning points $\eta_{x}=0$ which correspond to two
extremes of $\eta.$ According to (35), these points are the zeros of the
numerator of $P(\eta).$ As the latter is a quartic function of $\eta$, it can
have in general four roots. For a finite-amplitude solution to be possible,
all roots have to be real (Esler & Pearce, 2011). If two roots coincide but
the other two are distinct, the oscillation period becomes infinite resulting
in the so-called solitary wave. If also the other two roots coincide so that
there are two distinct double roots, we end up with only a half-solitary wave
which is termed the solibore (Esler & Pearce, 2011).

In the following, we focus on the last case and determine the unknown
constants in (36) by factorizing the numerator of $P(\eta)$ as follows

$(1-\eta^{2})(1-\eta^{2}-4(A\eta+B))+4c^{2}(\eta-\eta_{0})^{2}=(\eta-\eta_{+})^{2}(\eta-\eta_{-})^{2},$
(37)

where $\eta_{+}$ and $\eta_{-}$ are two double roots of the quartic defining
respectively the downtream and upstream interface heights. Comparing the
coefficients at the same powers of $\eta$ on both sides of (37) and
eliminating $A$ and $B$, we have

$4c^{2}\left(1\pm\eta_{0}\right)^{2}=\left(\eta_{+}+\eta_{-}\pm(1+\eta_{+}\eta_{-})\right)^{2},$
(38)

which represents a system of two equations corresponding to the same choice of
sign on both sides. There are two solutions for $c$ and $\eta_{0}$ possible
depending on the combination of signs when taking the square root of both
sides of (38). The first solution

$c=\pm\tfrac{1}{2}(1+\eta_{+}\eta_{-}),\quad\eta_{0}=\frac{\eta_{+}+\eta_{-}}{1+\eta_{+}\eta_{-}},$

with $(\vartheta_{+},\vartheta_{-})=\mp(\eta_{-},\eta_{+})$ following from
(34) corresponds to the conservative jumps with continuous characteristic
velocities found in the previous section. The second solution

$c=\pm\tfrac{1}{2}(\eta_{+}+\eta_{-}),\quad\eta_{0}=\frac{1+\eta_{+}\eta_{-}}{\eta_{+}+\eta_{-}},$

with $\vartheta_{+}=\vartheta_{-}=\mp 1$ resulting from (34) corresponds to
the conservative jumps with discontinuous characteristic velocities. This type
of solibores seems to be of little practical relevance because of the very
specific shear values required for their existence.

Using the factorization of the numerator of $P(\eta)$ defined by (37), we can
integrate (35) analytically as follows

$\displaystyle x(\eta)$
$\displaystyle=\frac{c}{\sqrt{3}}\int\frac{q(\eta)\,\mathrm{d}\eta}{(\eta-\eta_{+})(\eta-\eta_{-})}$
$\displaystyle=\frac{2c}{\sqrt{3}(\eta_{+}-\eta_{-})}\left[q(\eta_{+})\text{arctanh}\left(\frac{q(\eta)}{q(\eta_{+})}\right)-q(\eta_{-})\text{arctanh}\left(\frac{q(\eta_{-})}{q(\eta)}\right)\right],$
(39)

where $q(\eta)=\sqrt{1-\eta^{2}+(\eta-\eta_{0})^{2}}$ and $\eta_{+}$ and
$\eta_{-}$ are ordered so that $q(\eta_{+})\geq q(\eta_{-}).$ In the symmetric
continuous case $\eta_{+}=-\eta_{-},$ when $\eta_{0}=0$ and
$c=\pm(1-\eta_{+}^{2})/2,$ (39) can be written explicitly as
$\eta(x)=\eta_{+}\tanh\left(\frac{2\sqrt{3}x}{\eta_{+}-\eta_{+}^{-1}}\right).$

Figure 3: Solibores advancing into a quiescent downstream state with various
interface heights $\eta_{0}<0.$ The dots show the approximate solution of
Benjamin (1968) for energy-conserving gravity current.

For a quiescent downstream state, when $\vartheta_{+}=\eta_{-}=0,$
$\eta_{0}=\eta_{+}$ and $c=\pm\frac{1}{2},$ which is practically the most
relevant, (39) is plotted in Fig. 3a for various interface heights
$\eta_{0}<0.$ Note that (39) is invariant with the symmetry transformation
$\eta\rightarrow-\eta.$ Therefore, the solutions with $\eta_{0}>0$ are mirror
symmetric images of those with $\eta_{0}<0$ shown in Fig. 3. It is important
to note that the solution for $\eta_{0}=-1$ corresponds to the limit of an
infinitesimally shallow bottom layer. As seen, the solibore shape in this
limiting case is very similar to that of the gravity current found by Benjamin
(1968). The difference is mostly at the bottom which is approached
tangentially by the bore whereas the gravity current approaches it at the
angle of $\pi/3$ (von Kármán, 1940). This exact inviscid solution, which
contains a singularity at the contact point where the velocity varies as $\sim
r^{-1/2}$ with the distance $r,$ is outside the scope of the SW approximation.

##  5 Summary and discussion

In the present paper, we considered a special type of hydraulic jumps which,
in the two-layer Boussinesq system bounded by a rigid lid, can conserve not
only the mass and impulse, as usual, but also the circulation and energy. This
is possible only at certain combinations of the upstream and downstream states
for which the speed of propagation becomes invariant with the free parameter
specifying the contribution of each layer to the interfacial pressure
distribution in the generalised SW momentum conservation law.

Jumps propagating into a quiescent fluid are fully conservative only if the
interface upstream is located at the mid-height of the channel. The velocity
of propagation of such jumps is independent of their height and equal to
$\dot{\xi}=0.5$ in the usual dimensionless units. These fully conservative
jumps have a special symmetry with the upstream shear being equal to the
downstream interface height and vice versa: $\vartheta_{\pm}=\eta_{\mp}.$ This
symmetry ensures the automatic conservation of the impulse and energy as well
as the continuity of both characteristic velocities across the jump. Namely,
the positive and negative characteristics emanating on one side of the jump
are parallel to the corresponding characteristics emanating on the other side.
This apparently allows the jump to propagate without expanding or contracting.

Using this property we identified a broader class of fully conservative
symmetric jumps which propagate in shear flows with the velocity
$\dot{\xi}=\pm\frac{1}{2}(1+\eta_{+}\eta_{-}).$ The $\pm$ sign here reflects
the fact that, in the inviscid approximation, these jumps can theoretically
propagate in either direction. In real fluids, which are viscous, physical
considerations suggest that they have to propagate in the direction that
increases the shear. For example, a jump leaving a viscous fluid right behind
it at rest would be unphysical.

We also found that if the shear upstream and downstream is marginally
hyperbolic, i.e. $\vartheta_{-}=\vartheta_{+}=\mp 1,$ another type of
conservative jumps are possible. These jumps are in general asymmetric, have
discontinuous characteristic velocities and propagate at the speed
$\dot{\xi}=\pm\frac{1}{2}(\eta_{+}+\eta_{-}).$ Because of the very specific
shear values required for the existence of this type of jumps, they appear of
little practical relevance.

It is important to note that the fully conservative jumps considered in this
study satisfy four conditions but constrain only three out of five jump
parameters. Namely, such jumps still have two degrees of freedom – the
upstream and downstream interface heights. This is obviously due to the
$\eta\leftrightarrow\vartheta$ symmetry which is an exclusive feature of the
Boussinesq approximation enabling the automatic conservation of two
quantities.

Finally, the characteristics of fully conservative jumps were shown to be the
same as those of the so-called solibores which appear as permanent-shape
solutions in weakly non-hydrostatic approximation. An exact analytical
solution for solibores was obtained.

(a)

(b)

Figure 4: Comparison of the analytical solution (39) with the DNS results of
Khodkar et al. (2017) for partial lock exchange flows with the lock height
$h_{-}=0.9$ (a) and $h_{-}=0.8$ (b). The dashed lines show the analytical
solution and the dots show the approximate solution for the energy-conserving
gravity current obtained by Benjamin (1968). The DNS results shown in the
background are from figures 9(e,f) of Khodkar et al. (2017).

Note that two such fully conservative jumps feature in the partial lock
exchange flow when the lock is taller than the channel mid-height (Politis &
Priede, 2022). These jumps represent the front of a gravity current, which
propagates downstream along the bottom, and a bore, which propagates upstream
in the upper half of the channel. Analytical solution (39) can be seen in
figure 4 to agree well with the two-dimensional DNS results of Khodkar et al.
(2017) when the $x$-axis is scaled by a factor of $\sqrt{2}$, which is
apparently due to a different internal length scale used in the numerical code
for the $x$-coordinate. The necessity of this rescaling becomes obvious when
the numerical results of Khodkar et al. (2017) are compared with the inviscid
solution of Benjamin (1968) for the energy-conserving gravity current. This
solution can be used as a benchmark because it is known to provide a good
approximation to both experimental and numerical results (Härtel et al.,
2000). Once the rescaled numerical results match the benchmark solution for
gravity current, they agree also with the solution for solibore obtained in
this study. The agreement is very good for both heights of the lock
considered. As discussed above, the inviscid solution obtained by Benjamin
(1968) is very close to the respective solibore solution except in the
vicinity of the contact point. It is interesting to note that both cases are
equivalent in the SW approximation but treated somewhat differently in the
conventional control volume approach. This, however, does not affect the
predicted speed propagation which is the same for gravity currents (Benjamin,
1968) and internal bores with a vanishingly shallow downstream bottom layer
(Klemp et al., 1997).

The finding that certain large-amplitude hydraulic jumps can be fully
conservative while most are not such even in the inviscid approximation,
points toward the wave dispersion as a primary mechanism behind the lossy
nature of internal bores. Namely, it is the absence of dispersion in solibores
that makes the corresponding jumps fully conservative while all other jumps
are in general dispersive (Esler & Pearce, 2011). Such dispersive jumps may be
amenable to Whitham’s modulation theory (Whitham, 1965; El & Hoefer, 2016)
which could resolve the non-uniqueness in their speed of propagation emerging
in the hydrostatic SW approximation.

###### Acknowledgements.

Declaration of Interests. None.

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