# Regime transitions in stratified shear flows: the link between horizontal
and inclined ducts

M. Duran-Matute1<EMAIL_ADDRESS>S.J. Kaptein1  H.J.H. Clercx1  1 Fluid
Dynamics Laboratory and J.M. Burgers Centre, Department of Applied Physics,
Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The
Netherlands * Currently at: Flanders Hydraulics Research, Berchemlei 115, 2140
Antwerp, Belgium

###### Abstract

We present the analytical solution for the two-dimensional velocity and
density fields within the _high-advection/low-diffusion_ (HALD) approximation
for laminar stratified inclined duct (SID) flows. The hallmark of the HALD
approximation is that advection is negligible in the momentum equations but
dominates in the transport equation. The analytical solution is valid for
laminar flows in the limit of long ducts and for high Schmidt number values.
Under such conditions, the flow depends on a single non-dimensional parameter
$\Upsilon=\mbox{{Re}}_{g}\Gamma/K$ with $\mbox{{Re}}_{g}$ the gravitational
Reynolds number, $\Gamma$ the aspect ratio of the duct, and $K$ a geometrical
parameter depending on the tilt of the duct. The analytical solution in the
HALD approximation is validated against results from laboratory experiments,
and allows us to gain new insight into the dynamics and properties of SID
flows. Most importantly, constant values of $\Upsilon$ mark the transitions
between different flow regimes for both horizontal and inclined ducts.

###### keywords:

##  1 Introduction

The understanding and improved modelling of stratified turbulent flows is of
the utmost importance in several environmental situations. A particular flow
configuration that attracts a lot of attention is the stratified shear flow.
The interest in this flow is twofold. On the one hand, it is reminiscent of
several flows occurring in the environment such as the estuarine gravitational
circulation (Geyer & MacCready, 2014), the lock exchange flow (Ottolenghi et
al., 2016; Härtel et al., 2000), and the exchange flow occurring in ocean
straits (Gregg et al., 1999). On the other hand, it presents rich dynamics
encompassing the emergence of instabilities (Kaminski et al., 2014;
Ducimetière et al., 2021), Holmboe waves (Salehipour et al., 2016; Lefauve et
al., 2018), and stratified turbulence (Salehipour et al., 2018; Smith et al.,
2021; Lefauve & Linden, 2020a, 2022a, 2022b).

The pre-eminent experimental setup is the stratified inclined duct (SID)
(Macagno & Rouse, 1961; Meyer & Linden, 2014; Lefauve et al., 2019a; Lefauve &
Linden, 2020a, 2022a, 2022b), where two large tanks with fluid with different
densities are linked by an inclined, long duct (see figure 1). In recent
years, there has been vast progress in the understanding of the flow in SID
experiments driven by improved measurement capabilities that allow for
simultaneous detailed measurements of the three-dimensional (3D) density and
velocity fields (Partridge et al., 2019). A central research line has been the
transitions between flow regimes: from laminar to the emergence of interfacial
waves, to intermittently turbulent, and to fully turbulent (Macagno & Rouse,
1961; Meyer & Linden, 2014; Lefauve et al., 2019a; Lefauve & Linden, 2020a).
Although these different regimes have been observed for 60 years, explaining
them over a wide range of parameter values and determining the functional
dependence of the regime transition on the governing parameters remains a
challenge. In fact, one of the unanswered questions is “How to explain flow
regime transitions in horizontal ducts or ducts inclined at a slightly
negative angle?” (Lefauve et al., 2019a).

Lefauve et al. (2019a) distinguished between two situations: _lazy_ and
_forced_ flows. To explain this distinction, it is necessary to define the
internal angle of the duct $\alpha=\arctan(H/L)$, where $H$ is the height of
the duct and $L$ its length. _Lazy_ flows are defined as those occurring when
the inclination angle $\theta$ of the duct satisfies
$\alpha\gg\theta>-\alpha$, and _forced_ flows as those occurring when
$\theta>\alpha$. Between lazy and forced flows, a smooth transition occurs.
The term _forced_ refers to the increased importance of the gravitational
forcing due to the duct’s tilt. Meyer & Linden (2014) and Lefauve et al.
(2019a) have proposed two different conditions for the regime transitions in
_forced_ flows showing good agreement with experimental data. However, these
conditions are unreliable in the transition between lazy and forced flows, and
completely unsuitable for lazy flows.

The current paper proposes an explanation for the regime transitions in SID
experiments that spans both _lazy_ and _forced_ flows (i.e. encompassing
slightly negative inclinations, horizontal ducts, and positive inclinations)
with a Schmidt number value much larger than unity. This explanation is made
possible by extending the recent description by Kaptein et al. (2020) of the
_high-advection/low-diffusion_ (HALD) approximation for laminar flows in
horizontal ducts. In this approximation, advection can be neglected from the
momentum equation while it dominates in the density transport equation.
Recently, Kaptein (2021) proposed new curves denoting the regime transitions
based on a generalized Reynolds number using an empirical parametrisation.
Although, these curves show an overall better agreement than previously
proposed options, the fact that they are based on an empirical derivation
impedes a thorough physical explanation.
To provide such an explanation and propose transition curves based on physical
principles, we derive, in the current paper, a two-dimensional (2D) analytical
solution for the flow in the HALD approximation for both horizontal and
slightly inclined ducts. This analytical solution provides several new
insights into SID flows. In particular, it provides the non-dimensional
parameter governing the regime transition for SID flows for horizontal and
inclined ducts. The new understanding of SID flows should allow to execute new
targeted experimental campaigns and to further exploit the results of SID
experiments for the understanding of large scale environmental exchange flows.

##  2 Description of the system and background

Figure 1: Schematic representation of the side view of a stratified inclined
duct (SID) experimental setup. The duct of length $L$ and height $H$ is
inclined at an angle $\theta$ with respect to the horizontal. The duct
connects two large tanks: one with water with density
$\rho=\bar{\rho}+\Delta\rho/2$ and the other with density
$\rho=\bar{\rho}-\Delta\rho/2$. The internal angle of the duct is
$\alpha=\arctan(H/L)$. The along-duct coordinate is $x$ and the coordinate
perpendicular to the bottom and the top of the duct is $z$. The origin $O$ of
the coordinate system is located at the center of the duct.

The SID setup, mentioned earlier and sketched in figure 1, consists of two
tanks with fluid at densities $\bar{\rho}\pm\Delta\rho/2$ (due to differences
in, for example, salt concentration or temperature), joined by a duct. The
duct has length $L$ and height $H$, and it is inclined at an angle $\theta$
with respect to the horizontal. The fluid is considered to have uniform and
constant viscosity $\nu$. It is convenient to define the buoyancy velocity
scale $U_{g}\equiv\sqrt{g^{\prime}H}$, where $g^{\prime}\equiv
g\Delta\rho/\bar{\rho}$ with $g$ the gravitational acceleration. Besides the
angle of the duct $\theta$, the system can be described by three non-
dimensional parameters: the aspect ratio of the duct
$\Gamma\equiv\tan\alpha=H/L$, the gravitational Reynolds number

$\mbox{{Re}}_{g}\equiv\frac{HU_{g}}{2\nu}=\frac{H\sqrt{g^{\prime}H}}{2\nu},$
(1)

and the Schmidt number $Sc\equiv\nu/\kappa$ with $\kappa$ the diffusivity of
salt (or heat, in which case the Schmidt number is referred to as the Prandtl
number). We consider long ducts: $\Gamma\ll 1$. For ducts with finite width
$W$, we must introduce an additional parameter $B=W/H$.

Meyer & Linden (2014) proposed an empirical condition for the transition
between different regimes by defining the Grashof number as
$\mbox{{Gr}}=2\sin\theta\mbox{{Re}}_{g}^{2}/\Gamma$ which quantifies the ratio
of the buoyancy force to the viscous force. They proposed the critical value
$\mbox{{Gr}}=4\times 10^{7}$ for the transition between the intermittently
turbulent and the turbulent regimes showing good agreement with experimental
results. Lefauve et al. (2019a) proposed that the transitions between
different regimes for a SID setup with a given $\Gamma$-value occur at
constant $\theta\Rey_{g}$-values. Lefauve & Linden (2020a) compared their
proposed transitions against several experimental data sets including those of
Meyer & Linden (2014). They remarked particularly good agreement with
experiments for _forced_ flows (i.e. $\theta>\alpha$) when, in addition,
$\Gamma\mbox{{Re}}_{g}\lesssim 50$. However, the comparison was inconclusive
for other values of $\theta$ and $\Gamma\Rey_{g}$.

Both previously mentioned conditions [those proposed by Meyer & Linden (2014)
and Lefauve et al. (2019a)] have a crucial shortcoming: they are not valid for
$\theta\leq 0$ because the proposed governing parameters (Gr and
$\theta\mbox{{Re}}_{g}$) are then equal or smaller than zero. This would mean
that for $\theta\leq 0$, the flow does not transition away from laminar, which
is in disagreement with experimental results. Moreover, it is well known that
the Grashof number defined as $\mbox{{Gr}}=\mbox{{Re}}_{g}^{2}$ is the
governing parameter in the case of a horizontal duct (see e.g. Härtel et al.,
2000; Hogg et al., 2001). Hence, the definition of Gr proposed by Meyer &
Linden (2014) is inconsistent with the relevant definition for horizontal
ducts. For these reasons, there is still the need to find a physical
explanation and a generalized governing parameter that is valid for positively
inclined, horizontal, and negatively inclined ducts.

To find such a generalized governing parameter, we take horizontal ducts as
our starting point. In such ducts, diffusion dominates and
$U/U_{g}\propto\mbox{{Re}}_{g}\Gamma$ (with $U$ denoting the typical magnitude
of the velocity) for $\mbox{{Re}}_{g}\Gamma\ll(180/\mbox{{Sc}})^{1/2}$ using
our definitions [or $\mbox{{Re}}_{g}\Gamma\ll 12(5/\mbox{{Sc}})^{1/2}$ if $H$,
instead of $H/2$ is taken as the typical length scales as done by Kaptein et
al. (2020)]. This approximation is known as the _viscous advective-diffusive
(VAD) solution_ (Cormack et al., 1974; Hogg et al., 2001), the _hydrostatic-
viscous balance_ (Lefauve & Linden, 2020a) or _the diffusion-dominated regime_
(Kaptein et al., 2020). In the opposite case of
$\mbox{{Re}}_{g}\Gamma\gg(180/\mbox{{Sc}})^{1/2}$, two distinct approximations
arise depending on Sc. For $\mbox{{Sc}}\sim 1$, the flow tends to the
hydraulic limit, in which $U/U_{g}\propto 1$ (Hogg et al., 2001). This is the
theoretical limit for a steady, inviscid, irrotational, hydrostatic flow in
which the two layers are of equal thickness all along the duct (Gu & Lawrence,
2005; Lefauve & Linden, 2020a).

However, for $\mbox{{Sc}}\gg 1$, the novel HALD approximation by Kaptein et
al. (2020) predicts $U/U_{g}\propto\mbox{{Re}}_{g}\Gamma$, similarly to the
VAD solution but with a different proportionality constant. This linear growth
of velocity is, most probably, only possible because regime transitions
towards a turbulent flow were not considered. This suggests that the HALD
approximation should hold between the VAD solution and the waves flow regime
(eventually leading to turbulence). In other words, the upper bound for the
HALD approximation should correspond to the transition from the laminar regime
to the waves regime. It is this hypothesis that we will endeavour to verify in
this paper.

##  3 Analytical description of the HALD approximation

In this section, we focus on the HALD approximation and derive its
consequences for regime transitions in inclined ducts. First, in Section 3.1,
we explain the origin of the HALD approximation using scaling analysis for a
long duct. In Section 3.2, we present the analytical solution for the two-
dimensional velocity and density fields in the HALD approximation to then
discuss, in Section 3.3, its implications for the regime transitions.

###  3.1 Scaling analysis

Although scaling analysis for SID flow has been previously done by Lefauve et
al. (2019a), the HALD approximation was not considered because simultaneous
analysis of the Navier-Stokes equation and the transport equation was not
carried out. We perform here this simultaneous analysis, and in addition, we
take a slightly different approach in the non-dimensionalization that allows
us to simplify the equations by assuming a long duct ($\Gamma\ll 1$).

We consider the 2D, steady flow in a $(x,z)$ cross-section of an infinitely
wide duct, where $x$ is the along-channel coordinate and $z$ the coordinate
going from the bottom to the top of the duct. The flow velocity is
$\boldsymbol{v}=(u,0,w)$. The flow is described by the continuity and the
steady Navier-Stokes equations with the Boussinesq approximation for an
incompressible fluid:

$\displaystyle\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}$
$\displaystyle=$ $\displaystyle 0,$ (2) $\displaystyle u\frac{\partial
u}{\partial x}+w\frac{\partial u}{\partial z}$ $\displaystyle=$
$\displaystyle-\frac{1}{\bar{\rho}}\frac{\partial p}{\partial
x}+\nu\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial
z^{2}}\right)+g\frac{\rho^{\prime}}{\bar{\rho}}\sin\theta,$ (3) $\displaystyle
u\frac{\partial w}{\partial x}+w\frac{\partial w}{\partial z}$
$\displaystyle=$ $\displaystyle-\frac{1}{\bar{\rho}}\frac{\partial p}{\partial
z}+\nu\left(\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial
z^{2}}\right)-g\frac{\rho^{\prime}}{\bar{\rho}}\cos\theta,$ (4)

with $p$ the pressure and $\rho^{\prime}$ the variable part of the total
density. The reference density $\bar{\rho}$ is related to the total density
$\rho$ and the varying part $\rho^{\prime}$ by
$\rho=\bar{\rho}+\rho^{\prime}$, with $\bar{\rho}\gg\rho^{\prime}$. A linear
equation of state relates the density $\rho$ to, for example, salt
concentration or fluid temperature. In this way, the density $\rho$ is also
governed by a steady transport equation:

$u\frac{\partial\rho}{\partial x}+w\frac{\partial\rho}{\partial
z}=\kappa\left(\frac{\partial^{2}\rho}{\partial
x^{2}}+\frac{\partial^{2}\rho}{\partial z^{2}}\right).$ (5)

We define the non-dimensional variables denoted by a tilde such that

$u=U\tilde{u},\>w=U\Gamma\tilde{w},\>x=\dfrac{L}{2}\tilde{x},\>z=\dfrac{H}{2}\tilde{z},\>\rho^{\prime}=\Delta\rho\,\tilde{\rho}^{\prime},\>p=\dfrac{2\bar{\rho}U\nu
L}{H^{2}}\tilde{p}.$ (6)

Here, we define $U=2Q/H$ with $Q$ the volume flow rate through the duct in one
direction. We do not set, _a priori_ , $U_{g}$ as the typical velocity scale
because we do not know the relationship between $U$ (or $Q$) and $U_{g}$. In
fact, determining this relationship is a crucial step in deriving the
analytical solution in the HALD approximation.
Using the non-dimensional variables defined in (6), (2)–(5) can be written as

$\displaystyle\frac{\partial\tilde{u}}{\partial\tilde{x}}+\frac{\partial\tilde{w}}{\partial\tilde{z}}$
$\displaystyle=$ $\displaystyle 0,$ (7)
$\displaystyle\Rey\Gamma\left(\tilde{u}\frac{\partial\tilde{u}}{\partial\tilde{x}}+\tilde{w}\frac{\partial\tilde{u}}{\partial\tilde{z}}\right)$
$\displaystyle=$
$\displaystyle-\frac{\partial\tilde{p}}{\partial\tilde{x}}+\Gamma^{2}\frac{\partial^{2}\tilde{u}}{\partial\tilde{x}^{2}}+\frac{\partial^{2}\tilde{u}}{\partial\tilde{z}^{2}}+\dfrac{\Rey}{2\mbox{{Fr}}^{2}}\tilde{\rho}^{\prime}\sin\theta,$
(8)
$\displaystyle\Rey\Gamma^{3}\left(\tilde{u}\frac{\partial\tilde{w}}{\partial\tilde{x}}+\tilde{w}\frac{\partial\tilde{w}}{\partial\tilde{z}}\right)$
$\displaystyle=$
$\displaystyle-\frac{\partial\tilde{p}}{\partial\tilde{z}}+\Gamma^{4}\frac{\partial^{2}\tilde{w}}{\partial\tilde{x}^{2}}+\Gamma^{2}\frac{\partial^{2}\tilde{w}}{\partial\tilde{z}^{2}}-\dfrac{\Rey\Gamma}{2\mbox{{Fr}}^{2}}\tilde{\rho}^{\prime}\cos\theta,$
(9)
$\displaystyle\Rey\,\mbox{{Sc}}\,\Gamma\left(\tilde{u}\frac{\partial\tilde{\rho}^{\prime}}{\partial\tilde{x}}+\tilde{w}\frac{\partial\tilde{\rho}^{\prime}}{\partial\tilde{z}}\right)$
$\displaystyle=$
$\displaystyle\Gamma^{2}\frac{\partial^{2}\tilde{\rho}^{\prime}}{\partial\tilde{x}^{2}}+\frac{\partial^{2}\tilde{\rho}^{\prime}}{\partial\tilde{z}^{2}},$
(10)

where the Reynolds number is defined as $\Rey=UH/(2\nu)$ and the Froude number
as $\mbox{{Fr}}=U/U_{g}$. In this way,
$\Rey/\mbox{{Fr}}^{2}=\Rey_{g}/\mbox{{Fr}}$. Notice that $\Rey\Gamma$ gives,
in (8), the relative importance of inertia compared to vertical viscous
momentum diffusion.

We now make use of the fact that $\Gamma\ll 1$ to describe the main
characteristics and balances in the HALD approximation. It can immediately be
seen, as expected for long ducts, that horizontal viscous diffusion can be
neglected from (8) and (9). Furthermore, we consider small $\Rey$-values
($\Rey\ll\Gamma^{-1}$) such that the non-linear terms in (8) and (9) are also
neglected. Vertical viscous diffusion in (9), which is of order $\Gamma^{2}$
can also be immediately neglected with respect to the pressure gradient and
gravity forces. This means that (9) yields, at zeroth order, the hydrostatic
balance:

$\frac{\partial\tilde{p}}{\partial\tilde{z}}=-\dfrac{\Rey\Gamma}{2\mbox{{Fr}}^{2}}\tilde{\rho}^{\prime}\cos\theta,$
(11)

which holds provided that
$\Rey\Gamma/\mbox{{Fr}}^{2}=\mbox{{Re}}_{g}\Gamma/\mbox{{Fr}}\sim\textit{O}(1)$,
in which case, $\mbox{{Fr}}=U/U_{g}\propto\Rey_{g}\Gamma$ in agreement with
the numerical results of Kaptein et al. (2020).

We now turn our attention to (8), where the remaining terms yield

$\frac{\partial^{2}\tilde{u}}{\partial\tilde{z}^{2}}=\frac{\partial\tilde{p}}{\partial\tilde{x}}-\dfrac{\Rey}{2\mbox{{Fr}}^{2}}\tilde{\rho}^{\prime}\sin\theta.$
(12)

Since $\Rey\Gamma/\mbox{{Fr}}^{2}\sim\textit{O}(1)$ [i.e.
$\Rey/\mbox{{Fr}}^{2}\sim\textit{O}(\Gamma^{-1})$], the last term on the right
hand side is only retained for the case
$\Gamma^{-1}\sin\theta\sim\textit{O}(1)$. As $\sin\theta\to 0$, the last term
is neglected yielding the governing equation for a horizontal duct.

Finally, the $z$-component of the velocity, $\tilde{w}$, is given by the
continuity equation

$\frac{\partial\tilde{u}}{\partial\tilde{x}}+\frac{\partial\tilde{w}}{\partial\tilde{z}}=0.$
(13)

Notice that neglecting the non-linear terms in (8) because of a small
$\Rey$-value ($\Rey\ll\Gamma^{-1}$ ) does not guarantee that advection can
also be neglected in the transport equation (10). Instead, there are three
distinct possibilities depending on the value of the Péclet number
$\mbox{{Pe}}=\Rey\mbox{{Sc}}$. $i$) If $\mbox{{Pe}}\ll\Gamma^{-1}$, advection
is negligible, and due to the dominance of diffusion and the no-flux condition
at the top and bottom, isopycnals tend to be vertical; $ii$) for
$\mbox{{Pe}}\sim\textit{O}(\Gamma^{-1})$, the governing equations for the VAD
solution are recovered provided that $\Rey\ll\Gamma^{-1}$ and $\tilde{w}=0$;
$iii$) for $\mbox{{Pe}}\gg\Gamma^{-1}$, advection dominates over diffusion and
both terms on the right hand side of (10) can be neglected yielding at leading
order:

$\tilde{u}\frac{\partial\tilde{\rho}^{\prime}}{\partial\tilde{x}}+\tilde{w}\frac{\partial\tilde{\rho}^{\prime}}{\partial\tilde{z}}=0.$
(14)

This last case gives the simplified transport equation in the HALD
approximation.

In short, the HALD approximation is characterized by a hydrostatic balance in
the vertical, while in the horizontal, the flow is driven by a pressure
gradient force, and if the angle $\theta$ is large enough, by gravity in the
along-duct direction. Finally, we have the hallmark of the HALD approximation:
advection is negligible in the momentum equations due to a small $\Rey$-value
but dominates the transport equation due to a large Pe-value (i.e. large Sc-
value). Hence, this approximation is relevant, for example, for water flows
where the density difference is caused by a difference in salt concentration
($\mbox{{Sc}}\approx 700$) and not by a difference in temperature ($Sc\approx
7$).

###  3.2 Analytical solution in the HALD approximation

To facilitate the comparison with previous work (e.g. Lefauve et al., 2019a;
Kaptein et al., 2020), we make the velocity components and the pressure
dimensionless with $U_{g}$ (denoted with hats) such that

$\hat{u}=\mbox{{Fr}}\,\tilde{u},\quad\hat{w}=\mbox{{Fr}}\,\tilde{w},\quad\hat{p}=\mbox{{Fr}}\,\tilde{p}.$
(15)

The governing equations (11)–(14), known as the HALD approximation, are then
rewritten as

$\displaystyle\frac{\partial\hat{p}}{\partial\tilde{z}}$ $\displaystyle=$
$\displaystyle-\frac{\Rey_{g}\Gamma}{2}\tilde{\rho}^{\prime}\cos\theta,$ (16)
$\displaystyle\frac{\partial^{2}\hat{u}}{\partial\tilde{z}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\partial\hat{p}}{\partial\tilde{x}}-\frac{\Rey_{g}}{2}\tilde{\rho}^{\prime}\sin\theta,$
(17)
$\displaystyle\frac{\partial\hat{u}}{\partial\tilde{x}}+\frac{\partial\hat{w}}{\partial\tilde{z}}$
$\displaystyle=$ $\displaystyle 0,$ (18)
$\displaystyle\hat{u}\frac{\partial\tilde{\rho}^{\prime}}{\partial\tilde{x}}+\hat{w}\frac{\partial\tilde{\rho}^{\prime}}{\partial\tilde{z}}$
$\displaystyle=$ $\displaystyle 0.$ (19)

For simplicity in the notation, we drop the tildes and hats from now on.

To find an expression for the along-channel velocity $u$ and the density
$\rho^{\prime}$ in the HALD approximation, we begin with the transport
equation (19). For the solution to this equation, we propose

$\displaystyle\rho^{\prime}(x,z)=\frac{1}{2}-\mathcal{H}(z-\eta(x)),$ (20)

where $\mathcal{H}(z)$ is the Heaviside function defined as:
$\mathcal{H}(z>0)=1$, $\mathcal{H}(z<0)=0$, and $\mathcal{H}(z=0)=1/2$. The
physical meaning of this solution is that the density gets organized in two
layers with a sharp interface located at $z=\eta(x)$ with $\eta(0)=0$ due to
the definition of the coordinate system. Notice that

$\displaystyle\frac{\partial\rho^{\prime}}{\partial z}$ $\displaystyle=$
$\displaystyle-\delta(z-\eta(x)),$ (21)
$\displaystyle\frac{\partial\rho^{\prime}}{\partial x}$ $\displaystyle=$
$\displaystyle\eta^{\prime}(x)\delta(z-\eta(x))=-\eta^{\prime}(x)\frac{\partial\rho^{\prime}}{\partial
z}$ (22)

with $\delta$ the Dirac delta function. Substituting these two expressions
into (19) yields

$(u\eta^{\prime}(x)-w)\delta(z-\eta(x))=0.$ (23)

Here, it can be seen that indeed (20) is a solution to (19) since
$\delta(z-\eta(x))\neq 0$ only for $z=\eta(x)$ (i.e. at the interface), where
$u=w=0$ due to symmetry. Formally, the solution is not defined at the
interface, so it is convenient to approach the problem by treating each layer
separately, with the interface as a boundary between them. The proposed
solution further satisfies $\partial^{2}\rho^{\prime}/\partial z^{2}=0$ in
each of the layers. We use the index $\zeta=\pm 1$ with $\zeta=+1$ referring
to the top layer and $\zeta=-1$ to the bottom layer. Using this notation,
$\rho^{\prime}_{\zeta}=-\zeta/2$.

We now turn our attention to (16). Integrating this equation with respect to
$z$ yields an expression for the pressure:

$p_{\zeta}(x,z)=\Rey_{g}\Gamma\dfrac{\zeta}{4}z\cos\theta+\gamma_{\zeta}(x)$
(24)

with $\gamma_{\zeta}(x)$ integration constants, which are functions of $x$ to
be determined using the boundary conditions. At $z=1$, the pressure is an
unknown function of $x$, but it is convenient to define, without loss of
generality,

$p_{+1}(x,z=1)=\mbox{{Re}}_{g}\Gamma[f(x)-\eta(x)\cos\theta]/4$ (25)

with $f(x)$ an unknown function. Using (24) and (25), one obtains an
expression for $\gamma_{+1}(x)$ as a function of $\eta(x)$ and $f(x)$, and
applying continuity of pressure at $z=\eta(x)$ yields an expression for
$\gamma_{-1}(x)$. The pressure as a function of $x$ and $z$ is then given by

$p_{\zeta}(x,z)=\zeta\dfrac{\Rey_{g}\Gamma}{4}[\cos\theta(z-\eta(x)-\zeta)+\zeta
f(x)]$ (26)

so that the horizontal pressure gradient is

$\dfrac{\partial p_{\zeta}}{\partial
x}=\zeta\dfrac{\Rey_{g}\Gamma}{4}\left(\cos\theta\dfrac{d\eta(x)}{dx}-\zeta\dfrac{df(x)}{dx}\right).$
(27)

Substituting (27) into (17) yields

$\frac{\partial^{2}u_{\zeta}}{\partial
z^{2}}=-\zeta\dfrac{\Rey_{g}\Gamma}{4}F_{\zeta}(x)$ (28)

with

$F_{\zeta}(x)=\cos\theta\dfrac{d\eta(x)}{dx}-\zeta\dfrac{df(x)}{dx}-\dfrac{\sin\theta}{\Gamma}.$
(29)

Integrating twice with respect to $z$ and applying the boundary conditions
$u(z=\pm 1)=u(z=\eta(x))=0$ gives the following expression for $u_{\zeta}$:

$u_{\zeta}(x,z)=-\zeta\dfrac{\Rey_{g}\Gamma}{8}F_{\zeta}(x)(z-\zeta)(z-\eta(x)).$
(30)

The condition that

$\int_{-1}^{1}u(x,z)dz=0,$ (31)

by integrating $u_{-1}(x,z)$ from $z=-1$ to $z=\eta(x)$ and $u_{+1}(x,z)$ from
$z=\eta(x)$ to $z=1$, yields

$\dfrac{df(x)}{dx}=-\dfrac{\eta(x)[3+\eta^{2}(x)]}{1+3\eta^{2}(x)}\left(\cos\theta\dfrac{d\eta(x)}{dx}-\dfrac{\sin\theta}{\Gamma}\right),$
(32)

and $F_{\zeta}(x)$ can be rewritten as

$F_{\zeta}(x)=\dfrac{[1+\zeta\eta(x)]^{3}}{1+3\eta^{2}(x)}\cos\theta\left(\dfrac{d\eta(x)}{dx}-\dfrac{\tan\theta}{\Gamma}\right).$
(33)

Replacing this expression for $F_{\zeta}$ into (28) clearly shows that the
flow has two drivers: a baroclinic pressure gradient due to a sloping
interface between the two layers with homogeneous density and gravity due to
the tilt of the duct.

The only missing part of the solution is to determine the shape of the
interface $\eta(x)$. For this, we consider the flow rate through each of the
layers yielding

$-\int_{\eta(x)}^{1}u_{+1}(x,z)dz=\mbox{{Fr}}=\int_{-1}^{\eta(x)}u_{-1}(x,z)dz,$
(34)

where we used the definition $\mbox{{Fr}}=U/U_{g}$ with $U=2Q/H$. In this way,
we obtain an autonomous differential equation for $\eta(x)$:

$\dfrac{d\eta(x)}{dx}=-\dfrac{48}{K\cos\theta}\dfrac{1+3\eta^{2}(x)}{[1-\eta^{2}(x)]^{3}}+\dfrac{\tan\theta}{\Gamma},$
(35)

where $K=\mbox{{Re}}_{g}\Gamma/\mbox{{Fr}}$ within the HALD approximation.

Equation (35) can be further used to rewrite $F_{\zeta}(x)$ in (33) yielding

$F_{\zeta}(x)=-\dfrac{48}{K}\dfrac{1}{[1-\zeta\eta(x)]^{3}},$ (36)

such that (28) is now written as

$\frac{\partial^{2}u_{\zeta}}{\partial
z^{2}}=\dfrac{24\mbox{{Fr}}}{[1-\zeta\eta(x)]^{3}}\rho^{\prime},$ (37)

where we have decided to write explicitly $\rho^{\prime}$ (instead of
$\rho^{\prime}=\rho^{\prime}_{\zeta}=-\zeta/2$) because this form of the
equation will be useful for further analysis. Finally, the along-duct velocity
given by (30) can be written as

$u_{\zeta}(x,z)=\dfrac{6\zeta\mbox{{Fr}}}{[1-\zeta\eta(x)]^{3}}(z-\zeta)(z-\eta(x)),$
(38)

where we can notice that the velocity profile at $x=0$ is self-similar:

$u_{\zeta}(0,z)=6\zeta\mbox{{Fr}}(z-\zeta)z.$ (39)

To reach the complete solution, it is still necessary to determine the value
for $K$ (equivalent to obtaining the value of Fr for given values of
$\mbox{{Re}}_{g}$ and $\Gamma$), to also obtain $\eta(x)$ by solving (35). A
similar problem was previously encountered for horizontal ducts by Gu &
Lawrence (2005) and for inclined ducts by Lefauve et al. (2018) who found an
autonomous equation for the slope of the interface using internal hydraulics.
In those cases, an additional condition was proposed using maximum exchange
flow theory. This theory states that the maximum flow rate possible is such
that the flow becomes critical at the edges of the channel (i.e. that the
composite Froude number becomes unity at $x=\pm 1$); see e.g. Dalziel (1991);
Gu & Lawrence (2005) for details. In such a case, the flow is said to be
hydraulically controlled. However, enforcing the composite Froude number to be
equal to unity at the edges is inconsistent with the assumptions used to
define the HALD approximation, in particular, that
$\Rey/\mbox{{Fr}}^{2}\sim\Rey_{g}/\mbox{{Fr}}\sim\textit{O}(\Gamma^{-1})$ as
obtained from the hydrostatic balance (11). In other words, the laminar flow
in the HALD approximation is not hydraulically controlled.
We propose a different condition inspired in the results by Kaptein et al.
(2020) for the simulation used to exemplify the HALD approximation
($\mbox{{Re}}_{g}=1000$, $\Gamma=1/60$, $\mbox{{Sc}}=300$, $\theta=0^{\circ}$)
and shown in figure 2. In this figure, we can see that the current inside the
duct turns sharply when reaching the end of the duct, with the light fluid
turning upwards at $x=-1$ and the dense fluid turning downwards at $x=1$. The
currents at the edges of the duct must have a dimensionless thickness in the
$x$-direction smaller than $\Gamma$, meaning that it is thin with respect to
the length of the duct. We propose then to impose $\eta(\pm 1)=\mp 1$ such
that $d\eta/dx|_{x=\pm 1}=\mp\infty$, where we are then assuming an error of
order $\Gamma$. Notice that, even though the flow is not hydraulically
controlled, the flow at the edges of the duct remains crucial in determining
the slope of the interface, and hence, the volume flow rate.

Figure 2: Density field from the numerical simulation by Kaptein et al. (2020)
for $\mbox{{Re}}_{g}=1000$, $\Gamma=1/60$, $\mbox{{Sc}}=300$, and $\theta=0$.
The white dashed lines represent the limits of the duct ($x=\pm 1$). The black
dash line represents the interface given by (41) with $K=131$.

Since the case of horizontal ducts can be treated analytically, we focus first
on this case for which (35) simplifies to

$\dfrac{d\eta(x)}{dx}=-\dfrac{48}{K}\dfrac{1+3\eta(x)^{2}}{[1-\eta(x)^{2}]^{3}},$
(40)

yielding

$x=-\dfrac{K}{19440}[320\sqrt{3}\arctan(\sqrt{3}\eta(x))-555\eta(x)+150\eta(x)^{3}-27\eta(x)^{5}].$
(41)

Taking $\eta(\pm 1)=\mp 1$ yields $K\approx 131$ and
$d\eta/dx|_{x=0}=-48/K\approx-0.366$ which is close to the slope of $-1/3$
obtained empirically by Kaptein et al. (2020) and shows good agreement with
the density field shown in figure 2.

To find the value of $K$ for the inclined ducts, which depends on $\theta$ and
$\Gamma$, we follow a similar approach as for horizontal ducts, but we do the
calculations numerically. The value of $K$ is obtained by solving (35) while
enforcing $\eta(\pm 1)=\mp 1$. Strictly speaking, we obtain the value of
$K\cos\theta$, but since $\theta\ll 1$, $K\cos\theta\approx K$. The solution
to (35) for three different values of $\theta$ is shown in figure 3. It can be
seen that several properties of the shape of the interface expected from
previous work (see e.g. Gu & Lawrence, 2005; Lefauve et al., 2019a; Kaptein et
al., 2020) and observed in figure 2 are reproduced by the analytical solution.
For example, the solution naturally yields a slope of the interface that is
constant to a good approximation over a large portion of the duct around
$x=0$, and that it bends up or down as it approaches the edges of the duct.
Furthermore, the slope of the interface at $x=0$ given by

$S=\left.\dfrac{d\eta(x)}{dx}\right|_{x=0}=-\dfrac{48}{K\cos\theta}+\dfrac{\tan\theta}{\Gamma}.$
(42)

decreases for increasing values of $\theta$. The values of $K\cos\theta$ and
$S$ as a function of $\Gamma^{-1}\tan\theta$ are shown in figure 4.

The fact that it is possible to obtain the value of $S$ as a function of
$\Gamma^{-1}\tan\theta$ is a critical difference with respect to the
parametrization used by Kaptein (2021) to determine the regime transition
curves. In that work, it was assumed that the slope would vary linearly for
$\Gamma^{-1}\tan\theta<1$ with $S=-1/3$ for $\Gamma^{-1}\tan\theta=0$ and
$S=0$ for $\Gamma^{-1}\tan\theta\geq 1$. Although certain features and trends
of the regime transition curves might be reproduced using this assumptions,
differences are also expected when the value of $S$ is shown in figure 4b) is
considered.

Figure 3: Shape of the interface for three different values of
$\Gamma^{-1}\tan\theta$ as obtained from solving (35).

Figure 4: a) Value of $K\cos\theta\approx K$ and b) value of $S$, the slope of
the interface at $x=0$, as a function of $\Gamma^{-1}\tan\theta$. The value of
$K\cos\theta$ and $S$ are such that the solution to (35) satisfies $\eta(\pm
1)=\mp 1$.

###  3.3 Implications of the HALD approximation for the regime transition

The derivation of the analytical solution in the HALD approximation yielded
that $K\cos\theta$ is a constant for a given value of $\Gamma^{-1}\tan\theta$
to solve (35) while enforcing $\eta(\pm 1)=\mp 1$. For the upcoming discussion
and an easier comparison with the work by Lefauve et al. (2019a), it is
convenient to use the fact that, for a long duct with small tilt angle
($\theta,\Gamma\ll 1$), $K\cos\theta\approx K$ and
$\Gamma^{-1}\tan\theta\approx\theta/\alpha$. In such a case, the solution in
the HALD approximation yields that $K=\Rey_{g}\Gamma/\mbox{{Fr}}$ must be a
constant for a given value of $\theta/\alpha$. Both $\Rey_{g}$ and $\Gamma$
are control parameters of the problem, while Fr is a response parameter
equivalent to the non-dimensional volume flow rate. Hence, if the value of
$\Rey_{g}\Gamma$ is increased while keeping $\theta/\alpha$ fixed, the value
of Fr should also increase to keep the value of $K$ constant.
It is convenient to define the parameter

$\Upsilon=\frac{\Rey_{g}\Gamma}{K}.$ (43)

Notice that $\Upsilon$ is a control parameter since $\mbox{{Re}}_{g}$ and
$\Gamma$ are directly imposed for a given experiment, while $K$ can be
interpreted as a geometrical parameter imposed indirectly through the choice
of the value of $\theta/\alpha$. The limit in the validity of the HALD
approximation, for a given value of $\theta/\alpha$, should then be given by a
critical value $\Upsilon=\Upsilon_{c}$. Within the HALD approximation (i.e.
for $\Upsilon\leq\Upsilon_{c}$), $\Upsilon=\mbox{{Fr}}$, but outside the HALD
approximation (i.e. for $\Upsilon>\Upsilon_{c}$ ), $\Upsilon>\mbox{{Fr}}$. In
this way, it is expected that a transition (a qualitative change in the flow)
must occur when $\Upsilon=\Upsilon_{c}$. Moreover, we can use the fact that,
in the HALD approximation, $\mbox{{Fr}}=\Upsilon=\mbox{{Re}}_{g}\Gamma/K$ and
the hydrostatic balance holds to show that (8) can be rewritten as

$\Upsilon^{2}K\left(u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial
z}\right)=\Gamma^{2}\frac{\partial^{2}u}{\partial
x^{2}}+\frac{\partial^{2}u}{\partial
z^{2}}+\dfrac{K}{2}\left(\dfrac{\sin\theta}{\Gamma}\rho^{\prime}-\cos\theta\int_{-1}^{z}\rho^{\prime}dz^{\prime}\right),$
(44)

with $z^{\prime}$ a dummy variable. From this equation, it is clear that
$\Upsilon^{2}$ determines the relative magnitude of the non-linear terms.

To derive the leading balances defining the HALD approximation in Section 3.1,
we used the severe condition that
$\Rey=\mbox{{Re}}_{g}\mbox{{Fr}}\ll\Gamma^{-1}$ (equivalent to
$\mbox{{Fr}}\mbox{{Re}}_{g}\Gamma\ll 1$) to neglect the non-linear terms in
the horizontal momentum balance. However,
$\mbox{{Fr}}\mbox{{Re}}_{g}\Gamma\approx 0.6$ for the simulation used by
Kaptein et al. (2020) to illustrate the HALD regime and shown in figure 2, and
where no waves are observed. Hence, this simulation already suggests that the
range of validity of the HALD approximation might extends beyond this strict
condition. In the following sections, we study the range of validity of the
HALD approximation and the consequence for regime transitions in four
different experimental setups.

##  4 Experimental verification

Results from laboratory experiments are used to verify the theoretical results
derived in the previous section. We have two main aims: $i$) show that the
solution in the HALD approximation described in Section 3.2 does exist and is
observed experimentally, $ii$) show that curves of $\Upsilon=\text{constant}$
define the transition between different regimes.

###  4.1 Description of the data

We use the experimental data sets by Meyer & Linden (2014), Lefauve et al.
(2019a) and Lefauve & Linden (2020a). These data sets have been discussed and
made available online (https://doi.org/10.17863/CAM.48821,
https://doi.org/10.17863/CAM.41410) by Lefauve et al. (2019b) and Lefauve &
Linden (2020a, b). Four ducts with different dimensions were used. We will
refer to them as mSID, tSID, LSID and HSID in accordance with Lefauve & Linden
(2020a). Table 1 summarizes the characteristics of all the setups. For each
setup, the inclination angle of the duct $\theta$ and the gravitational
Reynolds number $\Rey_{g}$ were varied independently. The fluid used was a
salt (NaCl) solution ($Sc\approx 700$), and the value of $\Rey_{g}$ was varied
by changing the salt concentration in each of the tanks. In total, we use 738
data points to study the regime transitions.

name |  $H$ (mm) | Cross-section | $\Gamma^{-1}$ | $B$ | $Sc$ |  $\theta$ (deg.) | $\mbox{{Re}}_{g}$
---|---|---|---|---|---|---|---
mSID | 45 |  | 30 | 1 | 700 | [-1,6] | [300, 6 000]
LSID | 100 |  | 30 | 1 | 700 | [-1,4] | [2 000, 20 000]
HSID | 100 |  | 15 | 1 | 700 | [0,4] | [1 000, 20 000]
tSID | 90 |  | 15 | 1/4 | 700 | [-1,3] | [3 000, 15 000]
Table 1: Characteristics of the experiments used in this paper. Four duct
geometries [abbreviated mSID (m for mini), LSID (L for large), HSID (H for
half), tSID (t for tall)] are used (Lefauve & Linden, 2020a). We list the
values of the dimensionless numbers describing each duct geometry ($\Gamma$
and $B$), the value of $Sc$ for salt in water, and the ranges of $\theta$ and
$\mbox{{Re}}_{g}$ explored.

Meyer & Linden (2014) distinguished four different regimes: laminar
($\mathsf{L}$), Holmboe waves ($\mathsf{H}$), intermittently turbulent
($\mathsf{I}$), and turbulent ($\mathsf{T}$). Lefauve & Linden (2020a)
introduced a ‘waves’ ($\mathsf{W}$) regime where waves other than Holmboe
waves were observed. The different regimes were mostly identified by
shadowgraph observations over a subsection of the duct, following the
qualitative description of each regime by Meyer & Linden (2014). A schematic
of the shadowgraph set-up was presented by Lefauve et al. (2018). The observed
regime and the mass flow rate as a function of the governing parameters are
provided by Lefauve & Linden (2020b). A complication for the comparison with
the experimental data is that, as pointed out by Lefauve & Linden (2020a),
there is a surprising difference between the results from the LSID and the
mSID setups: the regions in the $(\theta,\mbox{{Re}}_{g})$-plane where the
different regimes occur do not coincide even if the values of all the
dimensionless parameters are identical.

To show that the proposed solution in the HALD approximation is observed
experimentally, we analyse the along-duct velocity and the density fields for
two experiments in the mSID duct (provided by Lefauve et al., 2019b). The
first experiment falls within the $\mathsf{L}$ regime with $\theta=2^{\circ}$
and $\mbox{{Re}}_{g}=398$, while the second falls within the $\mathsf{H}$
regime with $\theta=1^{\circ}$ and $\mbox{{Re}}_{g}=1455$. A detailed
description of the experiments and the methodology is given by Lefauve et al.
(2019a). The second experiment, which falls within the $\mathsf{H}$ regime, is
not expected to match with the analytical derivation. However, it is the only
experiment in the dataset for a _lazy_ flow for which the velocity and density
field are available, and we still expect to see certain similarities to the
analytical solution in the HALD approximation.

###  4.2 Experimental confirmation of the HALD approximation

Figure 5: Time-averaged density field in the $(x,z)$-plane at $y=0$ for two
experiments. a) Experiment within the $\mathsf{L}$ regime ($\theta=2^{\circ}$,
$\mbox{{Re}}_{g}=398$). b) Experiment within the $\mathsf{H}$ regime
($\theta=1^{\circ}$, $\mbox{{Re}}_{g}=1455$). The black dashed line represents
the line $z=Sx$ with $S=-0.20$ in a) and $S=-0.25$ in b) as given by (42).

Figure 5 shows the experimentally obtained time-averaged density fields for
two experiments and the location of the interface as predicted by the HALD
approximation. For the experiment within the $\mathsf{L}$ regime, the HALD
approximation yields $K\approx 39$, $S\approx-0.20$ and $\Upsilon\approx
0.34$, while for the experiment within the $\mathsf{H}$ regime, it yields
$K\approx 62$, $S\approx-0.25$ and $\Upsilon\approx 0.78$. For both
experiments, the analytical solution for the inclination of the interface
agrees well with the experiments within the measurement area.

Figure 6: The terms of the horizontal momentum balance in the HALD
approximation (37) as a function of $z$ for $y=0$ and $x\approx-0.4$ for the
same two experiments as in figure 5. The term on the left hand side of (37) is
plotted with a black solid line and the term on the right hand side with a
black dashed line. The dashed grey line represents both sides of the equation
from the analytic solution in the HALD approximation which assumes an
infinitely sharp interface and $\mbox{{Fr}}=\Upsilon$.

As a second step, we investigate if (37) is indeed the leading order balance.
Figure 6 shows with black lines both the left and right hand side of (37) as a
function of $z$ for $y=0$ and $x\approx-0.4$ from the two experiments in the
$\mathsf{L}$ and $\mathsf{H}$ regimes. As can be seen, both terms overlap well
for both experiments, with the experiment in the $\mathsf{H}$ regime shows
larger deviations, particularly, close to the interface. Nonetheless, we can
conclude that (37) is indeed the main balance in both experiments even if we
are comparing a cross-section of a 3D flow with a 2D analytical model.

For comparison, figure 6 also shows with dashed grey lines the profile of
either term of (37) assuming an infinitely thin interface and taking
$\mbox{{Fr}}=\Upsilon$ (i.e. assuming that the HALD approximation is perfectly
satisfied). The interface in the experiments is not infinitely thin due to
small, yet finite, diffusion in the vertical direction which is fully
neglected in the HALD approximation. However, one must be careful in making
interpretations about the thickness of the interface in the $\mathsf{H}$
regime because the time averaging makes it seem artificially thicker. Still,
assuming that the density is organized in two homogeneous layers is a good
approximation for both experiments. Moreover, we observe that the theory
overpredicts the volume flow rate (i.e. $\Upsilon>\mbox{{Fr}}$), as
illustrated by the difference between the grey dashed line and the black lines
in figure 6. Computing the value of Fr at this location in the flow, yields
$\mbox{{Fr}}=0.27$ for the experiment in the $\mathsf{L}$ regime and
$\mbox{{Fr}}=0.34$ for the experiment in the $\mathsf{H}$ regime. Although a
difference between $Fr$ and $\Upsilon$ would be expected for the experiment in
the $\mathsf{H}$ regime since the non-linear terms are certainly non-
negligible, it is difficult to understand from only these two experiments the
reason behind this difference. For a more detailed analysis, we use the mass
flux per unit width

$Q_{m}=\int_{-1}^{1}|\rho^{\prime}(z)u(z)|dz$ (45)

which was computed for several experiments. The value of $Q_{m}$ is commonly
used to compare (and thus characterize) the flow, with $Q_{m}=1/2$
corresponding to the hydraulic limit (Hogg et al., 2001; Lefauve et al.,
2019a), while $Q_{m}=\Upsilon/2$ in the HALD approximation.

The value of $Q_{m}$ as a function $\Upsilon/2$ for the experiments in the
mSID setup are shown in figure 7. The values of $Q_{m}$ can then be compared
to the expected values in the HALD approximation ($Q_{m}=\Upsilon/2$) and to
the hydraulic limit ($Q_{m}=1/2$). For $\Upsilon/2\lesssim 1/2$, the HALD
approximation gives a good estimate for the value of $Q_{m}$. For $\theta\geq
1.0^{\circ}$, $Q_{m}\approx\Upsilon/2$ up to $\Upsilon/2=1/2$. Then, a sharp
transition to the hydraulic limit occurs, and $Q_{m}\approx 1/2$ for
$\Upsilon/2\geq 1/2$. For $\theta=0.5^{\circ}$, $Q_{m}$ seems to saturate
reaching a maximum value $Q_{m}\approx 0.4<1/2$. The results for
$\theta=0^{\circ}$ seem to suggest that $Q_{m}$ will also not reach the value
$1/2$ for a horizontal duct. The impossibility of reaching $Q_{m}=1/2$ is in
agreement with frictional hydraulic theory which states that frictional
effects reduce the maximum possible value of $Q_{m}$ (Gu & Lawrence, 2005). It
is only for $\theta\geq 1.0^{\circ}$ that the additional forcing by gravity in
the along-duct direction reduces the importance of frictional effects and
makes the direct transition from the HALD to the hydraulic limit ($Q_{m}=1/2$)
possible.

Figure 7: Mass flux per unit width $Q_{m}$ as a function of the parameter
$\Upsilon/2$ for the experiments in the mSID setup. The color denotes the
value of the angle $\theta$. The solid black lines represent
$Q_{m}=\Upsilon/2$ (the expected value in the HALD approximation) and
$Q_{m}=1/2$ (the expected value in the hydraulic limit).

Finally, we compare the shape of the velocity profiles. Figure 8 shows the
normalized vertical profiles of the along-duct component of the velocity at
$y=0$ and $x\approx-0.4$ for the experiments in the $\mathsf{L}$ and
$\mathsf{H}$ regimes. It was already pointed out that the HALD approximation
overestimates the magnitude of the velocity (i.e. $\Upsilon>\mbox{{Fr}}$) for
these two experiments. Hence, we focus here only on the shape of the velocity
profile. For the experiment in the $\mathsf{L}$ regime, the flow is stationary
so the instantaneous and time-averaged velocity profiles coincide.
Furthermore, both profiles show an overall good agreement with the analytical
profile given by (38). On the other hand, the instantaneous velocity profiles
for the experiment in the $\mathsf{H}$ regime show, as expected, some temporal
variability. Nonetheless, the shape of the time-averaged profile agrees well
with the analytical profile given by (38). The largest discrepancy in the
shape can be observed at around $z\approx-0.5$.
In general, the flow in the $\mathsf{L}$ regime is very well described by the
HALD approximation, while this is less so for the experiment in the
$\mathsf{H}$ regime. This observation further supports the idea that the
transition from the $\mathsf{L}$ regime to the $\mathsf{H}$ or $\mathsf{W}$
regimes corresponds to limit of validity of the HALD approximation. Notice
that $\mbox{{Fr}}\mbox{{Re}}_{g}\Gamma\approx 3.6$ for the experiment in the
$\mathsf{L}$ regime, showing that the HALD approximation still gives a good
representation of SID flows beyond the restrictive condition
$\mbox{{Fr}}\mbox{{Re}}_{g}\Gamma\ll 1$ used to neglect the non-linear terms
in (8).

Figure 8: Normalized velocity profiles of the along-duct component as a
function of $z$ for $y=0$ and $x\approx-0.4$ for the same two experiments as
in figures 5 and 6. The grey lines represent the instantaneous velocities, the
black dashed line the time-averaged velocity, and the red line represents the
velocity in the HALD approximation given by (38).

###  4.3 Regime transitions

Figure 9: Location of the different regimes in the ($\Rey_{g}$,$\theta$) plane
for the four different setups: mSID, LSID, HSID, and tSID. The different
symbols represent the different regimes laminar ($\mathsf{L}$), Holmboe waves
($\mathsf{H}$), other waves ($\mathsf{W}$), intermittently turbulent
($\mathsf{I}$) and turbulent ($\mathsf{T}$). The solid lines represent curves
of constant $\Upsilon$ with the value indicated along the line. The dashed
line represents $\theta/\alpha=1$. The dotted lines are examples of the
transition between regimes for _forced_ flows given by
$\theta\mbox{{Re}}_{g}=\text{const.}$ as proposed by Lefauve et al. (2019a).

Figure 9 shows the location of the different regimes in the parameter space
$(\theta,\mbox{{Re}}_{g})$ for all experimental setups with superimposed
curves of constant $\Upsilon$. It can be clearly seen that these curves
correspond to the transitions between the different regimes through all values
of $\theta/\alpha$. For $\theta/\alpha\gtrsim 1$, curves given by
$\Upsilon=\text{const.}$ are equivalent to curves given by
$\theta\mbox{{Re}}_{g}=\text{const.}$ for a given duct. This is in agreement
with the transitions derived for _forced_ flows by Lefauve et al. (2019a)
using an energetics budget approach and that Lefauve & Linden (2020b) derived
using a two-layer frictional hydraulics model. However, the agreement of the
curves of constant $\Upsilon$-value based on the HALD approximation extends to
_lazy_ flows ($\theta/\alpha<1$) including $\theta/\alpha\leq 0$ where the
curves given by $\theta\mbox{{Re}}_{g}=\text{const.}$ do not make physical
sense.

Although curves of constant $\Upsilon$ match the transitions for all ducts,
the particular values of $\Upsilon$ associated with the transitions differ. In
the mSID setup, the transitions between successive regimes are given by
$\Upsilon\approx 0.5,$ 1, and 2. For the LSID setup, the values of $\Upsilon$
marking these transitions are a factor 2 larger, for the HSID a factor 3
larger, and for the tSID a factor 5 larger, approximately. In spite of the
differences in these values, there is a remarkable similarity: there is always
a factor 2 between the values of $\Upsilon$ marking successive regime
transitions. This observation further supports the fact that $\Upsilon$ is the
parameter controlling the transitions between the different regimes. To
explain why constant values of $\Upsilon$ define these transitions, we return
to (44) where we noted that the relative magnitude of the non-linear terms is
given by $\Upsilon^{2}$ in the HALD approximation. However, the experimental
data suggest that this property extends beyond this approximation, and hence,
beyond the L regime.

The difference in the transition values for the tSID setup with respect to the
other setups is not surprising since there is additional friction from the
side walls due to the small width of the duct. Additional friction for a given
forcing results in a smaller value of Fr (i.e. smaller magnitude of the non-
linear terms). On the other hand, the difference between the mSID and LSID
(both with $\Gamma=1/30$), and the HSID (with $\Gamma=1/15$) needs further
explanation since the HALD approximation accounts for having different
$\Gamma$ values. As already mentioned, the difference between the LSID and
mSID was already found to be particularly surprising by Lefauve & Linden
(2020a) since both have the same values for $\Gamma$, $B$, and Sc.

We believe that the key to understand further the regime transitions lies at
the edges of the duct. There are three main reasons for this: $i$) the flow in
the HALD approximation is largely determined by the flow at the edges of the
duct (i.e. the analytical solution depends on the boundary condition at $x=\pm
1$); $ii$) the HALD approximation formally breaks down immediately outside the
duct; $iii$) instabilities first emerge at the edges of the duct where the
non-linear terms are larger since both $u$ and $\partial u/\partial x$ grow as
$|x|\to 1$, which can be deduced from conservation of mass and the shape of
the interface. Hence, further knowledge of the flow and the experimental
setups at the edges of the duct is needed to clarify these points.

##  5 Conclusions

In the current paper, we present an analytical solution for the flow in
stratified inclined duct (SID) experiments within the _high-advection/low-
diffusion_ (HALD) approximation introduced by Kaptein et al. (2020) for
horizontal ducts. This approximation is derived for long ducts (i.e. with
aspect ratio $\Gamma=H/L\ll 1$), small-inclination angles $\theta$ (including
$\theta=0$ and slightly negative angles), small Reynolds number values
($\Rey\ll\Gamma^{-1}$), and large Péclet number values
($Pe=Re\,Sc\gg\Gamma^{-1}$), or in other words, large Schmidt number values
$[\mbox{{Sc}}\gg(\Rey\Gamma)^{-1}]$. Under these conditions, the flow gets
organized in two homogeneous layers and depends only on one non-dimensional
control parameter $\Upsilon=\mbox{{Re}}_{g}\Gamma/K$, where $\mbox{{Re}}_{g}$
is the gravitational Reynolds number and $K$ is a parameter that depends
exclusively on the ratio $\Gamma^{-1}\tan\theta\approx\theta/\alpha$ with
$\alpha$ the internal angle of the duct. Comparison with results from
laboratory experiments confirm the existence of the HALD approximation,
demonstrate the suitability of the analytical solution to describe the flow,
and show that the range of validity extends beyond the restrictive conditions
used to derive it.

The main gain from the derivation of the HALD approximation and the analytical
solution is that they provide new insights into the nature and dynamics of
flows in SID experiments. The HALD approximation presents the relevant
balances in the laminar regime, and it clearly exposes the two forcing
mechanisms: a baroclinic pressure gradient due to the inclined interface and
gravity in the along-duct direction. For $\theta/\alpha\gtrsim 1$, the
inclination of the interface might be neglected, but for
$\theta/\alpha\lesssim 1$, it is a crucial or even the main component of the
forcing despite the fact the slope of the interface is small ($\approx
48\Gamma/131$ for $\theta=0$). Furthermore, within the HALD approximation, the
control parameter $\Upsilon$ is equal to the Froude number Fr, which
represents the non-dimensional volume flux, or in other words, the magnitude
of the along-channel velocity.

As the value of $\Upsilon$ is increased, the HALD approximation breaks down
when a critical $\Upsilon$-value is reached, giving rise tot he emergence of
waves. Most importantly, even outside the HALD approximation, the curves given
by $\Upsilon=\text{const.}$ keep describing the regime transitions observed
experimentally for horizontal and slightly inclined ducts because
$\Upsilon^{2}$ gives the magnitude of the non-linear parameters. The
particular values of $\Upsilon$ at which transitions occur are setup
dependent. Our results put an end to a 60 year-old quest for the curves
describing the regime transitions in SID flows.

## Acknowledgements

This research was funded by STW now NWO/TTW (the Netherlands) through the
project “Sustainable engineering of the Rhine region of freshwater influence”
(no. 12682). The authors would like to acknowledge A. Lefauve and P.F. Linden
for making the experimental data available.

## Declaration of interests

The authors report no conflict of interest.

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