# Vertical convection regimes in a two-dimensional rectangular cavity: Prandtl
and aspect ratio dependance

Arman Khoubani1  Ashwin Vishnu Mohanan2  Pierre Augier1
Jan-Bert Flór1<EMAIL_ADDRESS>1Laboratoire des Écoulements
Géophysiques et Industriels, Université Grenoble Alpes, CNRS, Grenoble INP,
38000 Grenoble, France 2Swedish Meteorological and Hydrological Institute,
Norrköping, Sweden.

###### Abstract

Vertical convection is the fluid motion that is induced by the heating and
cooling of two opposed vertical boundaries of a rectangular cavity (see e.g
Wang et al., 2021). We consider the linear stability of the steady two-
dimensional flow reached at Rayleigh numbers of O($10^{8}$).

As a function of the Prandtl number, $Pr$, and the height-to-width aspect
ratio of the domain, $A$, the base flow of each case is computed numerically
and linear simulations are used to obtain the properties of the leading linear
instability mode. Flow regimes depend on the presence of a circulation in the
entire cavity, detachment of the thermal layer from the boundary or the corner
regions, and on the oscillation frequency relative to the natural frequency of
oscillation in the stably temperature-stratified interior, allowing for the
presence of internal waves or not. Accordingly the regime is called slow or
fast, respectively. Either the global circulation or internal waves in the
interior may couple the top and bottom buoyancy currents, while their absence
implies asymmetry in their perturbation amplitude.

Six flow regimes are found in the range of $0.1\leq Pr\leq 4$ and $0.5\leq
A\leq 2$. For $Pr\lessapprox 0.4$ and $A>1$ the base flow is driven by a large
circulation in the entire cavity. For $Pr\gtrapprox 0.7$ the thermal boundary
layers are thin and the instability is driven by the motion along the wall and
the detached boundary layer. A transition between these regimes is marked by a
dramatic change in oscillation frequency at $Pr=0.55\pm 0.15$ and $A<2$.

##  1 Introduction

Over the past six decades, vertical convection has attracted significant
interest due to its wide range of applications in industry, the environment as
well as in geophysics. Circulation patterns and instabilities that may arise
due to vertical heat transport along hot or cold isothermal boundaries are
relevant in view of the transport of heat (see e.g. Miroshnichenko & Sheremet,
2018). In the idealised case of a rectangular cavity, the typical flow
evolution is that, after turning on the heat forcing above its critical value
for convection, an upward motion arises at the heated boundary and a downward
motion at the cooled boundary, while stratification develops progressively in
the interior (Gill, 1966). When these motions reach the two horizontal
adiabatic boundaries, they turn into horizontal buoyancy currents. The flow
pattens in this cavity and related instabilities are determined by the
Rayleigh number, the Prandtl number, and the aspect ratio defined respectively
as

$Ra=\frac{g\beta\Delta TH^{3}}{\nu\kappa}\;\;Pr=\frac{\nu}{\kappa}\;\;\mbox{
and }\;\;A=\frac{H}{W},$ (1)

with $\beta$ the thermal expansion coefficient, $\nu$ the dynamic viscosity,
$\kappa$ the thermal diffusivity, and $H$ and $W$ the height and width of the
cavity, $\Delta T$ the horizontal temperature difference in the cavity, and
$g$ the gravitational constant. In view of the relatively low aspect ratio
considered ($A<4$), the Rayleigh number is based on the height $H$ of the
tank, as is most common (see e.g. Bejan, 2013), allowing also for comparison
with other results in the literature. In this study we consider the linear
instability of the steady circulating flow that is reached at intermediate
critical Rayleigh numbers O($10^{8}$) beyond the onset of the convective
instability for a range of Prandtl numbers. This steady circulating flow is
called later the base flow.

Applications vary with Prandtl number. Generally, higher Prandtl numbers apply
to geophysical flows with Prandtl numbers of 0.7 and 7 for air and water at
$20^{o}\;C$, respectively, and very high Prandtl numbers for the Earth mantle,
with magma viscosities somewhere around $10^{19}$ (e.g. Busse, 2006). For
seawater the Prandtl number ranges from $1<Pr<14$ as a function of temperature
and salinity. The lower Prandtl numbers apply to gases and liquid metals.
Atmospheric air has a Prandtl number in the range of $0.7<Pr<0.79$, Methane
gas in the range of $0.7<Pr<0.87$, whereas mixtures of liquid Helium may have
a Prandtl number between $0.2<Pr<0.6$ depending on its mixtures. Other
applications are semiconductor crystals with $Pr$ O($10^{-2}$) (see e.g.
Gelfgat et al., 1999), and nuclear engineering processes that are associated
with convective fluid motions of sodium, lead or alloys for cooling with
$Pr\approx 10^{-1}$ to $10^{-3}$ (see e.g. Grötzbach, 2013). Very small
Prandtl numbers apply to astrophysics, stellar and deep solar convection with
$Pr\approx 10^{-6}$ (see e.g. Guervilly et al., 2019; Pandey et al., 2021;
Garaud, 2021, etc.).

In the past, particular attention has been given to the flow transition to a
permanent oscillatory state that occurs in the corner regions of a cavity with
$A=1$ for $Pr=0.7$ and a Rayleigh number just above critical, i.e. $Ra\approx
10^{8}$. The thermal boundary layers detach and the presence of standing and
dissipative internal wave modes were observed in the interior (see Paolucci &
Chenoweth, 1989; Henkes & Hoogendoorn, 1990; Le Quéré & Behnia, 1998, and
references therein). Above the critical Rayleigh number, a shear instability
occurs in the vertical boundary layers, with a transition to chaos through
quasi periodicity (see e.g. Lappa, 2009). For larger Prandtl numbers
differences in behaviour occur since the thermal boundary layer is thinner
with a larger velocity gradient normal to the boundary, favouring shear
instability. Thus, an immediate transition to turbulence has been observed for
Prandtl numbers, $2.5\leq Pr\leq 7.0$, and a transition from steady to a
periodic state of the jet-like structure for the lower Prandtl number range
$0.25\leq Pr\leq 2.0$ (see Janssen & Henkes, 1995; Chenoweth & Paolucci,
1986).

For tall cavities with $1.0\leq A\leq 3.0$ ($Pr=0.7$) (see e.g. Xin & Le
Quéré, 2006, and references therein), the instability is determined by the
detachment of the boundary layer in the corner regions and the spatial
structure of normal modes that fill the cavity. The instability in the
boundary remains relatively small. The inclined flow structures in the
interior that were ascribed to internal waves (see Le Quéré & Behnia, 1998;
Xin & Le Quéré, 1995) are shown to be in fact part of the unstable mode (Xin &
Le Quéré, 2006). For cavities with $A\geq 3$, the traveling waves in the
vertical boundaries have about 10 times higher frequencies with the
instability in the vertical boundary layers being dominant. These wall mode
waves occur as Tollmien-Schlichting waves in the boundary layer for small
$Ra$, (Yahata, 1999; Xin & Le Quéré, 2006; Xin & Quéré, 2012). For smaller
values of $A$, internal waves in the interior dominate the instability
(Yahata, 1999). The instability mode is found to be either centrosymmetric or
anti-centrosymmetric, respectively. This instability is part of two Hopf
bifurcations that is encountered for increasing $Ra$ number (and fixed $Pr$
number), with consecutively the (anti-centrosymmetric or centrosymmetric)
internal wave modes, and for larger Rayleigh number the (anti-centrosymmetric
or centrosymmetric) wall modes (see Burroughs et al., 2004; Oteski et al.,
2015).

For larger $Ra$, the flow becomes nonlinear with vortices detaching from the
boundary layer and penetrating into the stratified core. These penetrating
vortices excite internal waves with a frequency smaller than the Brunt-Väisäla
frequency thus perturbing the core fluid (Xin & Le Quéré, 1995). The
instability in a rectangular cavity can thus form in the corner region or in
the lateral boundary, and internal waves take part in the instability. Next to
the instability, for certain parameters, a large-scale circulation has been
also observed with the hot boundary layer motion ’connecting’ to the start of
the cold boundary layer motion. In the case of conducting horizontal
boundaries this gives rise to limit cycles (see Henkes & Hoogendoorn, 1990). A
comprehensive review of studies on vertical convection is given in a
historical perspective by Le Quéré (2022).

Though a range of Prandtl numbers and different aspect ratio have been
considered in the past, the critical Rayleigh number and the shape and
symmetry of the most unstable corresponding mode are known only for some
specific values of $Pr$ and $A$. There are no clear indications as to the
mechanism responsible for the instability, the presence of locally increasing
modes or global modes and their symmetry. In this research, in order to obtain
information about the shape of the most unstable mode, and to the critical
Rayleigh number, a numerical linear stability analysis is used. The
representation of the perturbations of amplitude, phase and vorticity of each
specific unstable mode allows investigation of the mechanism of instability
and the role of internal waves permitting different flow regimes to be
identified.

Apart from the reduced computational costs, the advantage of a two-dimensional
approach is that it is well-posed with a simple geometry and forcing, similar
to other flows for which the knowledge and understanding of regimes and the
transition between them is of fundamental interest. Some well known other
examples are Taylor-Couette flow or Rayleigh-Benard in thin gaps. More related
flows are the shear-driven cavity flow (see e.g. for the homogeneous case
Bengana et al. (2019), and for the stratified case Wu et al. (2018)), or the
self organised state of two-dimensional turbulence on a rectangular domain
interacting with vorticity generated at the slip-free boundaries (Konijnenberg
et al., 1998; Heijst et al., 2006). In the latter example, a large central
vortex interacts with the boundaries, implying aspects of symmetry with flow
phenomena analogous to large cell flows in vertical convection, but without
baroclinic effects.

In a three dimensional box with periodicity in the third direction (see Xin &
Quéré, 2012, for $Pr=0.7$), the instability starts for a 10 times smaller
Rayleigh number. The three dimensional effects are however modest, with low
frequency modes loosing their stability earlier than in the two-dimensional
case. Two dimensional simulations are found to be also satisfactory for larger
aspect ratio ($A=4$) and capture the general features of buoyancy-driven flow,
as long as it is not turbulent, i.e. up to $Ra\sim 10^{10}$ (Trias et al.,
2007). Also partial similarities with the two-dimensional counterpart have
been noticed in cubic cavities (see Gelfgat, 2017, 2020a, 2020b). This is
further discussed at the end of the conclusions.

In the next section, §2, the numerical code with the linear approach and the
decomposition into leading modes are discussed next to the nonlinear approach
and the diagnostics. In the subsequent section, §3, the results are presented
with the different base states, leading linear modes, and the different
observed regimes. In §4, the main conclusions are presented and further
discussed.

Figure 1: (a) Schematic of the problem with the walls kept at constant
temperatures 0.5 and -0.5 so that $\Delta T=1.0$. The initial condition for
the nonlinear simulations is $T_{0}=\Delta
T\times\left(x/x_{max}-0.5\right)+noise$ and $(u,v)=(0,0)$, and for the linear
simulations, $T_{p_{0}}=noise$ and $(u_{p},v_{p})=(0,0)$. (b) Location of the
interrogation points used to assess the state of the flow with the oscillation
frequency $\omega$ measured in the corner at $(x,z)=(0.1,0.1)$, for $A=1$,
$H=$(0,1) and $W=$(1,0), $N_{c}$ the stratification in the centre of the
cavity. The maximum speed in the buoyancy current is sketched by red and blue
curved lines. $L_{h}$ is the horizontal distance from the wall to the minimum
(in $z$) on the red line of the current at the top.

##  2 Numerical setup

###  2.1 Governing equations and linear stability approach

We consider a two-dimensional flow inside a rectangular cavity of aspect ratio
$A_{v}=H/W$ with cavity height $H$ and width $W$, adiabatic top and bottom
boundaries, and the two walls kept at a constant temperature with temperature
difference $\Delta T$ (see figure 1(a)). The scales of this problem are
$\Delta T$ for the temperature difference and for the length scales
$(x,y)\sim(\delta_{T},H)$ where $\delta_{T}=(\kappa t_{r})^{1/2}$ is the
thickness of the heated boundary layer, and $t_{r}$ is the reference time
defined below. Using the momentum equations, the friction term then scales
with the buoyancy term, i.e, $\nu V_{r}/(\delta_{T})^{2}\sim g\beta\Delta T$
which yields a characteristic speed $V_{r}=\left(\kappa/H\right)Ra^{1/2}$. The
time scale $t_{r}=H/V_{r}$ is obtained from the balance between the advective
term and diffusive term in the temperature equation. With these scales one
obtains for the dimensionless form of the continuity equation, Boussinesq
approximation of the Navier-Stokes equations, and the temperature equations,
respectively,

$\mathbf{\bnabla}\cdot\mathbf{v}=0,$ (2)

$\frac{\partial\mathbf{v}}{\partial
t}+\mathbf{v}\cdot\mathbf{\nabla\mathbf{v}}=-\mathbf{\nabla}P+\frac{Pr}{Ra^{1/2}}\mathbf{\nabla}^{2}\mathbf{v}+Pr\Theta\mathbf{e}_{z},$
(3)

$\frac{\partial\Theta}{\partial
t}+\mathbf{v}\cdot\mathbf{\nabla}\Theta=\frac{1}{Ra^{1/2}}\mathbf{\nabla}^{2}\Theta,$
(4)

Figure 2: Temperature signal of a nonlinear simulation at point (x, z) = (0.1,
0.1) for $A=1.0$, $Pr=0.71$ and $Ra=1.85\times 10^{8}$, with (a) the evolution
of the flow from $t=0$, and (b) the evolution to a steady state and subsequent
exponential growth in amplitude and saturation. The inset shows the
perturbation in log scale with the fit giving the growth rate (straight line).

with $\mathbf{v}$ the velocity vector, pressure $P$, and $\Theta$ the
dimensionless temperature $\left(T-T_{r}\right)/\Delta T$ with $T_{r}$ being
the average temperature (here $T_{r}=0$). As mentioned above, the control
parameters of this flow are the Rayleigh number, Prandtl number and aspect
ratio $A=H/W$ which depends on the chosen length $W$ since the heigh $H$ is
kept constant.

The no-slip condition is used for all boundaries. For the temperature, the
Dirichlet condition is used with $\Theta=0.5$ and $\Theta=-0.5$ on the two
lateral walls respectively, and for zero heat flux the Neumann condition at
the two horizontal boundaries.

Numerical simulations are performed with the spectral element code
Nek5000111https://nek5000.mcs.anl.gov using the two newly developed Python
packages Snek5000222https://snek5000.readthedocs.io and
Snek5000-cbox333https://github.com/snek5000/snek5000-cbox Mohanan et al.
(2023); Augier et al. (2019); Mohanan et al. (2019). The packages are
available online (see footnotes) and the data that we have produced here is
available as a Zenodo dataset444https://zenodo.org/record/7827872.

Tests have been conducted for $Pr=0.71$ and $A=1$ and results for the growth
rate and oscillation frequency have been validated with respect to former
studies in the literature. The resolution employed (see table I Appendix A)
allows for the study of the motion in the thin boundary layers near the walls,
and accuracy in growth rate and oscillation frequency. Figure 2 shows a
typical temperature signal measured in the corner of the cavity (see figure
1(b)) for the nonlinear simulation. The large difference in temperature
between figures 2(a) and (b) reveals the transient to the base state ($t<250$)
and the subsequent growth and saturation of the instability ($t>250$),
respectively. From $t$=0 to 250, the motions along the boundaries and the
stratification in the interior develop. The base state is a steady state with
minimum amplitude of oscillations (at approximately $t\approx 500$), with
motions in thermal boundary layers in the presence of a stratified interior.
From $t=500$ onwards, a linear instability leads to the exponential growth of
the amplitude of the oscillation up till about $t=1375$, after which it
saturates and small nonlinear oscillations in amplitude develop. The
oscillation frequency shows a perfect exponential growth in the range of about
$500<t<1400$ (see the inset in figure 2(b)).

For each set of control parameters $Pr$ and $A$, nonlinear simulations are
performed to obtain a first approximation of the critical Rayleigh number
$Ra_{c}$. Nonlinear simulations reach a steady state at Rayleigh number of
O($\sim 10^{8}$) but still smaller than the critical Rayleigh number, i.e.
$Ra<Ra_{c}$, while for slightly higher $Ra$ numbers, the flow starts to
oscillate at $t\approx 500$. Three nonlinear simulations are performed for
three values of $Ra$ slightly larger than the estimated $Ra_{c}$ using the
Selective Frequency Damping (SFD) method of Åkervik et al. (2006), which give
us three steady base states. This method has also been used recently for
computing the base state of the flow that is induced when tilting a cavity
containing a stably stratified fluid (see Grayer et al., 2020).

Subsequently, the linear stability of these steady base flows is considered.
Using perturbed variables $\Theta=\Theta_{b}+{\theta}^{\prime}$,
$\mathbf{V}=\mathbf{V}_{b}+{\mathbf{v}}^{\prime}$, $P=P_{b}+{p}^{\prime}$
where subscript $b$ represents the base state and superscript ′ the
perturbation, and neglecting second order terms, we obtain linearised
perturbation equations of the form

$\frac{\partial{\mathbf{v}}^{\prime}}{\partial
t}+\mathbf{V_{b}}\cdot\mathbf{\nabla{\mathbf{v}}^{\prime}}+{\mathbf{v}}^{\prime}\cdot\mathbf{\nabla\mathbf{V_{b}}}=-\mathbf{\nabla}{p}^{\prime}+\frac{Pr}{Ra^{1/2}}\mathbf{\nabla}^{2}{\mathbf{v}}^{\prime}+Pr{\theta}^{\prime}\mathbf{e}_{z},$
(5)

$\frac{\partial{\theta}^{\prime}}{\partial
t}+\mathbf{V_{b}}\cdot\mathbf{\nabla}{\theta}^{\prime}+{\mathbf{v}}^{\prime}\cdot\mathbf{\nabla}\Theta_{b}=\frac{1}{Ra^{1/2}}\mathbf{\nabla}^{2}{\theta}^{\prime}.$
(6)

Linear simulations are run for the three steady base states obtained from the
SFD method, and corresponding to three unstable $Ra$ values. A small amount of
noise of the order of $10^{-6}$ is added so that, due to the linear
instability, an exponential growth of the leading mode is observed in the
whole cavity. This noise is random and therefore not necessarily symmetric.
Then, the growth rate is determined for each linear simulation, thus providing
eventually three different growth rates. Using a linear interpolation of these
growth rates, the critical Rayleigh number $Ra_{c}$ is extrapolated from the
value for zero growth-rate. The base flow and the perturbation analysed in the
next section are obtained from the nonlinear and linear simulations for the
Rayleigh number just above this critical Rayleigh number.

###  2.2 Decomposition of the leading linear mode

In view of former observations of this instability (see e.g. Xin & Le Quéré,
2006), the different variables can be decomposed during the oscillating
exponential growth into

$\theta^{\prime}\left(x,z,t\right)=A_{\theta}(x,z)\cos\left(\omega
t+\Phi_{\theta}\left(x,z\right)\right)e^{\sigma_{r}t},$ (7)

$u^{\prime}\left(x,z,t\right)=A_{u}\left(x,z\right)\cos\left(\omega
t+\Phi_{u}\left(x,z\right)\right)e^{\sigma_{r}t},$ (8)

and,

$w^{\prime}\left(x,z,t\right)=A_{w}\left(x,z\right)\cos\left(\omega
t+\Phi_{w}\left(x,z\right)\right)e^{\sigma_{r}t},$ (9)

where $A(x,z)$, $\omega$, $\Phi(x,z)$, and $\sigma_{r}$ are amplitude,
frequency, phase, and growth rate of the field variables, respectively. The
growth rate $\sigma$ and the oscillation frequency $\omega$ are computed by an
algorithm based on Hilbert transforms. The amplitude fields are then obtained
by taking the time maximum of the perturbation variables divided by
$e^{\sigma_{r}t}$. Finally, the phase fields are obtained with one curve fit
per grid point and variable.

##  3 Numerical results

###  3.1 The steady base flow and diagnostics

Figure 3: The base flow states for $A=2.0$ and different $Pr$: (a) $Pr=0.35$,
(b) $Pr=0.53$, (c) $Pr=0.71$, (d) $Pr=1.4$, (e) $Pr=2.0$ and (f) $Pr=2.8$.
Stream lines are supplied with arrows indicating flow direction, and color
represents the temperature.

Figure 4: The base flow state for $Pr=0.35$ and different $A$: (a) $A=0.5$,
(b) $A=1.0$, (c) $A=1.5$ and (d) $A=2.0$. Lines show the stream lines with the
arrows the flow direction, and color represents the temperature.

Figure 3(a-f) shows the steady base flow for different Prandtl numbers and
constant aspect ratio $A=2.0$. From figures 3(a-f) one notices that for larger
Prandtl numbers the buoyancy currents detach and meander along the horizontal
boundaries, the number of meanders depending on the $Pr$ number. Clear
experimental and numerical support for this meandering is shown by Xu et al.
(2008). The wave length of this meandering buoyancy current changes
significantly with $Pr$, while its presence is limited by the horizontal
extend $W$ of the cavity represented by $A$ (note $H$ is kept constant).

Figure 5: (a) Meander length scale $L_{h}/H$ against Prandtl number $Pr$, with
the dashed lines representing the aspect ratios $A=H/W$, and (b) $N_{c}$
Brunt-Väisälä frequency scaled with the characteristic time $t_{r}$ at the
center of cavity as a function of $Pr$ number. For comparison the slope of
$\sqrt{Pr}$ predicted by 10 is shown by the dashed line.

When the width of the cavity is smaller than this wave length, the head of the
buoyancy current joins the start of the cold boundary layer, and vice versa
near the bottom boundary, such that the two currents reinforce each others
inertia leading to a large scale, fast circulation along the boundaries (see
figure 3(a)). With increasing $Pr$ number, ($Pr<0.7$ in figure 3), the large-
scale circulation decreases in strength and the buoyancy current detaches from
the horizontal boundary, i.e. $L_{h}<W$, and it meanders locally. For larger
$Pr$ numbers, ($Pr>1$, see figures 3(d,e,f)) a horizontal exchange flow
establishes in the interior between the two thermal boundary layers, The
boundary layers are thinner for these higher $Pr$ numbers. For a constant $Pr$
it depends on the aspect ratio whether there are cell patterns (as in figure
3(a)) or rather horizontal exchanges between the thermal boundary layers (as
in figure 3(f)) (see Xin & Le Quéré, 2006).

Figure 4 shows flows for $Pr=0.35$ and varying aspect ratio $A$. When the
width of the cavity is large (see figure 4(a) for $A=0.5$), the buoyancy
current meanders along the horizontal boundary. With increasing aspect ratio
$A$, the meander length-scale of the buoyancy current becomes smaller than the
width of the cavity, and there is an increasing tendency for the formation of
cell circulation. Two circulation cells appear for $A=2$ in figure 4(d).

In all cases, a stable density gradient $-\beta dT/dz$ is present in the
interior. The buoyancy (Brunt-Väisäilä) frequency scaled with time
$t_{r}=H^{2}/(\kappa{Ra}^{1/2})$ is given by

$(N_{c}t_{r})^{2}=g\beta\frac{dT}{dz}\frac{H^{4}}{\kappa^{2}}\frac{1}{Ra}=Pr\frac{H}{\Delta
T}\frac{dT}{dz},$ (10)

showing that the Prandtl number, height of the reservoir and density gradient
are relevant for the scaled buoyancy frequency, with slightly stronger
stratifications for shallow cavities (small $A$) and weaker stratifications
for high and narrow cavities (large $A$).

In the simulations, the buoyancy frequency $N_{c}$ is determined by the
density gradient in a small region around the centre of the tank. The meander
length-scale of the buoyancy current, $L_{h}$, is defined as the horizontal
distance from the wall where the line that describes the maximum speed (see
Figure 1(b)) has a minimum in $z$. $L_{h}$ has been determined from the flow
near the top. In the perturbed state, the instability causes oscillations in
the temperature with frequency, $\omega$. Its amplitude increases due to the
linear instability, and though present in the entire tank, they are mainly
visible in the corner regions where the thermal motion along the wall is
blocked by the horizontal boundary (see e.g. Le Quéré & Behnia, 1998; Xin & Le
Quéré, 2006, and references therein). Thus, to analyse this flow, the
frequency of the oscillation frequency of the instability mode, $\omega$, the
density stratification in the interior, $N_{c}$, and the wave length of the
meandering buoyancy current, $L_{h}$, are measured at the locations shown in
Figure 1(b).

When the temperature oscillations have a higher frequency than the buoyancy
frequency, i.e. $\omega>N_{c}$, internal waves cannot propagate in the
interior and are evanescent. In contrast, when $\omega<N_{c}$, the buoyancy
currents near top and bottom boundaries may couple due to the internal waves
that propagate into the interior. Even though a larger value of $N$ may exist
near the top and bottom boundaries, the value $N_{c}$ in the centre of the
tank is taken as reference value since it is more relevant for the coupling of
top and bottom regions. As mentioned, flows with $\omega/N_{c}>1$ are called
fast, and flows with $\omega/N_{c}<1$ slow since allowing for the propagation
of internal gravity waves, and, in most cases, for the coupling between the
two buoyancy currents. In the absence of internal waves, top and bottom
instability generally start to grow independently with different perturbation
amplitude, resulting in asymmetry. This is referred to as amplitude asymmetry,
or shortly asymmetry and is further detailed below.

For the base states, figure 5(a) shows the (scaled) wave length of the
meandering buoyancy current $L_{h}/H$ against $Pr$ for different aspect-ratio
of the cavity. For small $Pr$ there is no detachment and $L_{h}$ is larger
than the width of the cavity, i.e. $L_{h}=1/A$. In this case, the horizontal
buoyancy currents and boundary layers at the side walls reinforce each other
leading to a large cell circulation. For larger $Pr$ numbers, the boundary
layers are thin and the aspect ratio $A$ has no influence on the value of
$L_{h}/H$. The effects obtained for decreasing $Pr$ can also be obtained for
larger aspect ratio. Boundary layers grow with distance and become thicker and
have more inertia, leading to relatively larger values of $L_{h}$.

Figure 5(b) shows that the stratification in the interior increases less for
$Pr>0.4$ than for $Pr<0.4$ since larger gradients form near the horizontal
boundaries, and a weaker stratification forms in the interior. The aspect
ratio $A$ has an influence on the internal stratification only for smaller
Prandtl numbers for which the large cell circulation dominates.

The asymmetry in the value of the amplitude between the top and bottom
currents is considered in particular since its relation with the presence of
internal waves is novel and provides new insights. For both currents the
growth rates are the same, but due to an asymmetry in the initial noise the
amplitudes of the perturbations may be different. Writing out the equations
for the temperature, we have then,

$\theta^{\prime}(\mathbf{x},t)=\left(C_{+}A_{+}(\mathbf{x})+C_{-}A_{-}(\mathbf{x})\right)\cos(\omega
t+\phi(\mathbf{x}))\,e^{\sigma t},$

with the ratio $C_{+}/C_{-}$ depending on the initial noise. Thus, in contrast
to the coupled case with top and bottom currents having the same amplitude and
internal waves being part of the same global mode, we have in the uncoupled
case two local regions that are growing independently.

The symmetry conditions are imposed by the boundary conditions, so that for a
solution $\Phi$ with

$\Phi(\mathbf{x})=\begin{pmatrix}\Theta_{b}(\mathbf{x})\\\
\mathbf{V_{b}}(\mathbf{x})\\\ P_{b}(\mathbf{x})\\\
\xi_{b}(\mathbf{x})\end{pmatrix}$

with $\xi_{b}=\nabla\times\mathbf{V_{b}}$ the vorticity in the basic state,
the equations are invariant for the transformations

$R\,\Phi(\mathbf{x})=\begin{pmatrix}-\Theta_{b}(\mathbf{-x})\\\
-\mathbf{V_{b}}(\mathbf{-x})\\\ P_{b}(\mathbf{-x})\\\
\xi_{b}(\mathbf{-x}).\end{pmatrix}$

In case there is symmetry, the solution can be either ”centro-symmetric”, i.e.
$R\,\Phi(\mathbf{x})=\Phi(\mathbf{x})$, or ”anti centro-symmetric”, i.e.
$R\,\Phi(\mathbf{x})=-\Phi(\mathbf{x})$ (see Burroughs et al., 2004). Counter
intuitively, for centro-symmetry, the velocity phase must be opposite to that
in the other corner whereas the vorticity must be equal, and vice versa for
anti centro-symmetry. This centro-symmetry was tested considering the point
reflected value with respect to the centre of the tank. Thus, the
nondimensional temperature in the top half of the cavity
$\theta^{\prime}_{\text{top}}$ is compared with its flipped counterpart in the
bottom half of the cavity $\theta_{\text{bot}}^{{}^{\prime}\text{flip}}$. The
parameters for centro -symmetry and anti centro-symmetry then become,
respectively,

$I_{C}=\frac{\langle\left|\theta^{\prime}_{\text{top}}-\theta_{\text{bot}}^{{}^{\prime}\text{flip}}\right|\rangle}{\sqrt{\langle\theta^{\prime
2}\rangle}},$
$I_{A}=\frac{\langle\left|\theta^{\prime}_{\text{top}}+\theta_{\text{bot}}^{{}^{\prime}\text{flip}}\right|\rangle}{\sqrt{\langle\theta^{\prime
2}\rangle}},$

with the brackets standing for the average over the domain, and the enumerator
representing the rms-value of $\theta^{\prime}$. When the perturbation is
centro-symmetric or anti-centro-symmetric, either $I_{C}$ or $I_{A}$,
respectively, is small and never zero because of numerical noise. Both values
are large when there is no such symmetry. To identify centro-symmetry, the
difference of the reciprocals of $I_{C}$ and $I_{A}$

$I_{dif}=\frac{1}{I_{C}}-\frac{1}{I_{A}},$ (11)

is used, with anti centro-symmetry for $I_{dif}>0$, i.e. the temperature
perturbations in the top and bottom halves of the cavity have the same sign,
and centro-symmetry for $I_{dif}<0$, i.e. opposite temperature perturbations
in the top and bottom halves of the cavity. When $I_{dif}\simeq 0$, there is
no symmetry, i.e. the top and bottom currents are asymmetric in amplitude. The
expression (11) therefore provides simultaneously information about centro
symmetry and amplitude symmetry.

Centro-symmetry is generally better solved with different methods that provide
a continuous distribution for the critical Rayleigh number and the centro-
symmetry as a function of Prandtl number (see e.g. Lyubimova et al., 2009, and
references therein). Since this was not the aim of the present investigation,
the centro symmetry below is provided for completeness, and the details are
left for future study.

Below, the six observed regimes for increasing $Pr$ are discussed for a single
aspect ratio $A=1$. For the regimes shown in figures 6 the field of the base
flow is shown next to the fields of the perturbations predicted by the linear
mode decomposition, with (b) amplitude, (c) phase, and (d) vorticity. The
perturbation of the vorticity field (d) reveals the spatio-temporal nature of
the disturbing wave packets. The phase map (c) shows the distribution of the
length scales in the field and can be considered as a signature of the
internal waves. In addition, two time steps show the base flow perturbed with
the linear perturbation (see figure 7(a,b)).

Movies of the unsteady states are provided as supplementary material. The
effect of varying aspect ratio $A$, along with the measured parameters
$\omega/N_{c}$ and the regime diagram in the space set by $A$ and $Pr$ are
discussed below.

###  3.2 Six unsteady regimes

Figure 6: Flow maps for $A=1.0$ and different different values of Prandtl
number $Pr$ and oscillation frequency $\omega/N_{c}$, (case I-VI) with (a)
base state (b) temperature amplitude $A_{\theta’}$ (c) phase map
$\Phi_{\theta’}$, and (d) vorticity perturbation $\xi’$. Supplementary movies
of the regimes I Fast Circulation cells (FCc) (Movie 1), II Slow Circulation
cells (SCc) (Movie 2), III Fast Corner flow (FCo) (Movie 3), IV Slow Corner
flow (SCo) (Movie 4),V Fast Corner with Evanescent waves (FCoE) (Movie 5) and
VI Fast instabilities at the Wall (FW) (Movie 6) are available on line.

Figure 7: The base flows corresponding to regimes shown in figure 6 plus the
perturbation at (a) $t=0$ and (b) $t=0.25T$. Arrows on streamlines indicate
flow direction and color represents temperature. Case VI is not shown since
differences with case V are very small. For all cases I-VI, the same movies as
referred to in figures 6, are represented on line.

The identification of the different regimes, shown in figure 6, is based on
the detachment of the buoyancy current from the top and bottom boundary and/or
the location of the instability, the existence of a large scale circulation,
and whether the scaled oscillation frequency is slow ($\omega/N_{c}\leq 1$) or
fast ($\omega/N_{c}>1$) allowing for presence of internal waves (or not). With
increasing Prandtl number, the plumes detach earlier from the horizontal
boundaries, and the presence of internal wave motions in the interior changes
accordingly (see figures 6 I-VI); Figures 7 I-V show the transition from a
convective regime with circulation cells moving locally through the tank to
flows confined to boundary currents with horizontal exchange between them.
Below we present each flow regime.

In case I ($Pr=0.1$) (see figure 6), there is no detachment of the buoyancy
current from the horizontal boundaries and the hot buoyancy current joins the
cold thermal boundary layer resulting in a large cell circulation (see e.g.
Xin & Le Quéré, 2006), thus coupling the top and bottom motions. The phase
plot (figure 6(c) I) shows a central rotary motion. In the perturbed flow (see
figures 7(a,b) I) smaller cells move in the interior with the large cell
circulation showing that convective motions dominate. The oscillation
frequency of this mode is larger than the buoyancy frequency,
$\omega/N_{c}=2.26$, and internal waves can therefore not propagate. This flow
regime is referred to as the Fast Cell Circulation (FCc).

In case II ($Pr=0.35$) (see figure 6 II), the buoyancy current detaches from
the boundary and meanders with a wavelength of about half the cavity width
(figure 6(a) II). (Note that for the case $Pr=0.1$ shown above an increase in
width of the cavity would also result in a detachment of the buoyancy
current). This detachment causes the radiation of internal waves into the
interior (see phase plots in figure 6(c) II) that have a dominant vertical
mode with $\omega/N_{c}$ close to 1.0, leading to quasi vertical iso-phase
lines. Vorticity perturbations (figure 6(d) II) show a maximum shear in the
detached buoyancy current. The perturbed flow (figure 7(a,b) II) consists of
two large cell structures with a dominant vertical transport, and large
oscillations in the density profile revealing the presence of internal waves.
In contrast to the FCc case above, energy of the detached buoyancy currents is
dispersed into internal wave motions such that the oscillation frequency is
relatively small or ’slow’. This regime is referred to as Slow Circulation
Cells (SCc).

In case III, $Pr=0.53$, (see figures 6 III) the flow pattern changes from a
convective flow to a horizontal exchange flow with a shift in direction at
mid-height (figure 6(a) III). The buoyancy currents detach from the horizontal
boundary close to the thermal wall. Oscillations are fast, $\omega/N_{c}>3$,
and internal waves cannot propagate through the stratified interior, and the
phase plot (figure 6(c) III) shows very different length scales in the
buoyancy currents compared to those in the interior. In the absence of
internal waves and a large scale circulation, the top and bottom motions are
decoupled. Thus, the independent growth of the unstable regions leads to
different perturbation amplitudes implying amplitude asymmetry (see figure
6(b) III). Because of the fast oscillations and the localisation of the
instability in the corner regions this flow is referred to as the Fast Corner
flow (FCo).

In case IV, $Pr=0.71$, (see figures 6 IV) we recover the case studied in
detail in former studies (see e.g. Paolucci, 1990; Le Quéré & Behnia, 1998;
Xin & Le Quéré, 2006, and references therein), and more recently by Grayer et
al. (2020). Here $\omega/N_{c}\approx 0.5$ allowing for internal waves in the
interior. These internal waves and the unstable corner regions are part of the
same global instability mode (Xin & Le Quéré, 2006), as the phase plot in
(figure 6(c) IV) clearly shows. The amplitude of the internal waves in the
interior is relatively weak compared to the amplitude in the corner regions.
The perturbed base flow (figures 7(a,b) IV) shows next to the oscillating
corner regions, recirculating regions in the interior that are slightly
flattened by the internal buoyancy stratification. The absolute value of the
perturbation amplitude of the top and bottom current are identical or very
close, which is referred to as symmetric in amplitude (not to confuse with
centro-symmetry). The vorticity perturbations (figures 6(d) IV) have opposite
signs at the top and bottom corner, revealing that this mode is anti centro-
symmetric (see the definition in §3.1), which is the mode that appears for the
lowest Rayleigh number in agreement with Burroughs et al. (2004); Oteski et
al. (2015). The centro-symmetric mode appears for a slightly higher Rayleigh
number.

In case V, $Pr=2.8$, the thermal boundary layer thickness is thin and the
perturbation maxima are limited to very small corner regions (figures 6(a-d)
V). The oscillation frequency is again increased (figure 6(c) V). Heat is
diffused relatively slowly for this $Pr$-number so that the temperature
gradients near the top and bottom boundaries are larger than in the interior.
The scaled oscillation frequency near the boundaries is $\omega/N\approx 0.8$
while in the interior $\omega/N\approx 1$, implying close to evanescent waves
in the interior as can be inferred from their almost vertical propagation
direction (figures 6(c) V). The coupling between the top and bottom currents
is therefore also weak, and there is amplitude symmetry only for some aspect
ratio. In the interior, there is a smooth exchange flow (figures 7(a,b) V).
This regime is referred to as Fast Corner flow with Evanescent internal waves
(FCoE).

In case VI, $Pr=4$, a simulation for a single aspect ratio has been conducted,
showing that the oscillation frequency of the instability (here
$\omega/N_{c}\approx 1.8$) increases further with $Pr$ (see (see figures
6(a-d) VI). Internal waves emitted by the buoyancy currents near the top and
bottom are evanescent and cannot propagate further into the interior. The two
buoyancy currents have again different perturbation amplitudes and are
therefore asymmetric in amplitude (figure 6(b) VI). The scales of motion in
the boundary layer, top and bottom buoyancy currents as well as the interior
are indeed very different (see figure 6(c) VI), showing two independent
unstable regions. As will be shown below this is a shear instability located
at the wall. This regime is referred to as the Fast instability at the Wall
(FW).

Figure 8: Base flows showing close ups of the streamline -temperature fields
of the corner regions for $A=1$ with maximum amplitude (a-d) for Prandtl
numbers from 0.53 to 4.0, and (e-h) the corresponding amplitude fields. Note
that zooms and aspect ratios in (a,b; e,f) and (c,d; g,h) are chosen
differently for visualisation purposes.

####  3.2.1 Corner and boundary layer flows

When zooming in the corner regions (figure 8) substantial flow changes can be
noticed for different $Pr$. For $Pr=0.53$, the detachment and position of
$L_{h}$ (see figure 1b)) is small. Rolling vortex billows emerging from shear
instability move to the corner (see supplementary movies) leading a maximum of
the instability in the boundary layer near the corner (see figure 8(a,e)). For
$Pr=0.71$, the instability has its maxima in the corner in the detached
buoyancy current, where the main mixing occurs (see figures 8(b,f)). For
$Pr=2.8$, the buoyancy current hardly detaches, and the flow is characterised
by the presence of a downward jet close to the wall (at $0.98$ in figure
8(c,g)). The instability is located inside the detached buoyancy current and
close to the boundary (figure 8(g)).

For $Pr=4$, the boundary layer detaches right in the corner region where it
also results in a downward return flow (figure 8(d)). The temperature gradient
has its maximum very near to the wall. Outside the boundary layer, the strong
shear between the upward boundary layer motion and the downward return flow
with rotary motions similar as that for Kelvin-Helmholtz type instabilities
(see figures 8(d,h), and supplementary material), suggests a shear instability
(see also Janssen & Henkes, 1995; Xu et al., 2008). In the interior, there is
a horizontal exchange flow from right to left above mid depth, and below rom
left to right (see figures 6 with $Pr\geq 0.53$). This exchange flow becomes
dominant with increasing $Pr$ number.

The phase plots in figures 6(c)V-VI, ($Pr=2.8$ and $Pr=4.0$) show a large
difference in scale between the boundary layers and the interior. Internal
waves in the interior emerge from the buoyancy currents at the top and bottom
boundaries, and are related to up- and down-ward motions of entrainment and
detrainment, as shown in figures 8(c,d and g,h). Comparing these latter plots
showing the perturbation amplitude (figures 8(g,h)), one notices that the
downward flow for $Pr=4.0$ goes to mid-depth, and for $Pr=2.8$ to only about
half that distance. In contrast ro the case $Pr=2.8$, for $Pr=4$ the
instability is located at the boundary where it has its maximum amplitude.

###  3.3 Instabilities and Regime diagram

Figure 9: (a) Maximum normalised oscillation frequency ($\omega/N_{c}$) and
(b) regime diagram in space set by $Pr$ number and aspect ratio $A$, with the
color indicating the logarithmic value of the normalised oscillation frequency
$\omega/N_{c}$. Regime names are with F for Fast when $\omega/N_{c}>1$ and S
for slow when $\omega/N_{c}<1$, $\blacksquare$ Fast Circulation Cells (FCc); +
Slow Circulation Cells; $\blacktriangle$ Fast Corner flow (FCo); $\bigstar$
Slow Corner flow (SCo); + Fast Corner Evanescent waves (FCoE), and
$\blacklozenge$ Fast instability at the wall (FW). (c) Diagram showing the
symmetry index $I_{dif}$ (11), with the extremes (blue and red) corresponding
to symmetric cases (anti centro-symmetric – letter A – and centro-symmetric –
letter C –, respectively), and yellow for amplitude asymmetry and no coupling
between top and bottom regions.

Figure 9(a) presents the normalised oscillation frequency of the leading
linear mode with respect to $Pr$ for the four different values of $A$. Here
$N_{c}$ varies with $Pr$ (see figure 5) but the main variation is in $\omega$.
The sharp jump between at $Pr\approx 0.55\pm 0.15$ separates two different
regimes, one for small $Pr$ and one for large $Pr$, which can be deduced from
scaling arguments (see Gill, 1966). From the stationary heat equation one can
deduce that the velocity along the boundary scales as $\kappa/\ell$ when
$\ell$ is the length scale of the boundary layer. When introducing this
scaling into the vorticity equation, with $\xi=\nabla\times{\bf v_{b}}$ we
obtain for the convective term ${\bf
v_{b}}\cdot\nabla\xi\sim\kappa^{2}/\ell^{3}$ and the diffusive term
$\nu\nabla^{2}\xi\sim\nu\kappa/\ell^{3}$. The ratio of the diffusive term over
the convective term is equal to $Pr=\nu/\kappa$.

For small Prandtl number the instability is thus convectively driven with
large cell circulations (see also figure 7). In figure 9(a) ($Pr\lessapprox
0.5$) the flow is characterised by a cell circulation that is affected by the
cavity aspect-ratio $A$. For small cavity widths (large $A$), the buoyancy
current remains attached to the horizontal boundary, and reinforces the motion
in the thermal boundary layer at the wall, resulting in a fast motion with a
high frequency of oscillation (see e.g. red squares in figure 5(a)). For large
cavity widths (small $A$), the buoyancy currents detach from the horizontal
boundaries, and radiate internal waves into the interior. The cell circulation
is therefore weakened, resulting in a relatively slow motion with a low
frequency of oscillation (see e.g. blue squares figure 5(a)), and thus a
smaller $Pr$-number is needed to increase $\omega/N_{c}$.

For large Prandtl numbers the diffusion term is larger than the convective
term from which it was concluded that the instability is buoyancy driven (see
McBain et al., 2007). But the detrainment and entrainment motions and the
related return flows that are due to these horizontal boundaries cannot be
neglected. Janssen & Henkes (1995) (for different $Pr$ numbers, 0.25, 0.71 and
2), and Yahata (1999) (for $Pr=0.71$) suggested this to be a shear instability
with a change in instability from shear driven for $A>3.65$, to internal wave
driven for $A<3.41$. For the present range in $Pr$ numbers, Figure 9(a) shows
strong variations around $Pr=0.55\pm 0.15$ before the general trend to thinner
boundaries and larger $\omega/N_{c}$ values for $Pr\gtrapprox 1$.

The regime diagram in Figure 9(b) shows the oscillation frequency $\omega/N$
represented by the colour as a function of the $Pr$ number and aspect ratio
$A$, with the symbols indicating the different regimes discussed above. For
$Pr\lessapprox 0.5$ the above reasoning with a dominating cell-driven motion
is refrained in this diagram. But for $Pr>0.5$, not less than 4 regimes appear
due to the changing influences of internal waves (as indicated in figure 9(b)
for $Pr>0.5$) and decreasing thickness of the boundary layer at the wall with
$Pr$. In regime FCo there are no waves and no coupling; in SCo for the
majority of cases there are waves and the top and bottom region couple. For
even larger $Pr$ numbers internal waves weaken, and the thermal boundary
layers are thinner (FCoE). For $Pr=4.0$ (regime FW) there are no waves and no
coupling. In contrast to the regime FCo, the instability occurs at the
boundary. Since for larger $Pr$, the lateral boundary layers will be even
closer to the walls, one may speculate that this regime will not further
change. Simulations were tested for aspect ratio, $A=6$ and $Pr=0.7$ as in Xin
& Le Quéré (2006) showing again the FCc-regime. In view of the mentioned
increase of the boundary layer thickness with height, and the increasing
inertia of the buoyancy current for taller cavities and thus larger $L_{h}$,
the regime FCo most likely disappears for larger aspect ratio $A$.

Figure 9(c) shows the amplitude asymmetry according to the definition of
equation (11) as a function of $Pr$ and $A$. In the FCo-regime due to the
absence of internal waves and cell circulation the flow is asymmetric in
amplitude. For larger $Pr$ (regime SCo) waves are present, and generally
(except for $A=1$, $Pr=1.4$) couple top and bottom regions. Not all flows with
internal waves are found to be symmetric in amplitude. Wave patterns in the
interior vary considerably depending on aspect ratio of the cavity and the
location of the detachment of the buoyancy currents, and some do not allow for
coupling. This may explain the isolated points of asymmetry in amplitude
(orange, green and yellow) for regimes with internal waves (SCc, SCo in figure
9(b)). In the regime FCoE ($Pr=2.8$) internal waves are close to evanescent,
so that the coupling is generally weak in this regime, causing asymmetry in
the amplitude for most cases, and complete asymmetry in the regime FW
($Pr=4$). Regime FCc is symmetric in amplitude due to the coupling by the cell
circulation, and SCc due to internal waves except for one case ($A=1$ and
$Pr=0.2$). For $Pr\gtrapprox 0.5$ the cell circulation was found to increase
in strength for larger aspect ratio $A>1$, and appeared to cause amplitude
symmetry also in the regime FCo, thus showing some overlap between the regimes
(dashed line in figures 9(b,c)). This suggest that for higher aspect ratio
$A\geq 2$ and $Pr<0.6$ the flow is also dominated by convectively driven
cells.

Figure 9(c) also shows anti centro- and centro-symmetry (letters A and C,
respectively). These results are coherent with former results (see e.g.
Burroughs et al. (2004); Oteski et al. (2015); Xin & Le Quéré (2006)).
However, no systematic variation with the regime diagram and amplitude
asymmetry was found in this context. It is expected that more pertinent
results could be found with a continuous parameter variation method (see e.g.
Lyubimova et al., 2009). This is left open for further research.

Figures 9(a,b) suggest that there is a transition in instability in the regime
$Pr=0.55\pm 0.15$. The increase of both, the shear and the density gradient
between the detached buoyancy currents and the motion in the boundary layer
with $Pr$, make it hard, however, to determine when exactly the instability is
buoyancy or shear driven. A quantitative comparison is needed to determine the
precise nature of the instability for each $Pr$ number. This is undertaken in
a separate study.

Outside the space of this regime diagram, for aspect ratio greater than 3 or
4, the boundary layer will become unstable with, for large $Ra$, the
detachment of vortices disturbing the internal stratification and generating
internal waves. (Xin & Le Quéré, 1995). In the limit of very large $Pr$ and
large $A$, the onset of local cells was found in the core, the number of the
cells being set by the scale of the instability in the boundary layer (see
Daniels, 1985, 1987, and references therein)

Figures 10(a) and (b) show the variation of the critical Rayleigh number and
Reynolds number as a function of $Pr$, respectively. There is a clear
transition between the two regimes, with $Ra_{c}\sim Pr^{2}$ for $Pr<1$ and
$Ra_{c}\sim Pr^{5}$ for $Pr>1$. In between, (i.e. for $0.5<Pr\lessapprox 2$),
there is an intermediate region where internal waves are present and play a
role for the dynamics. In both figures (10 a and b), for $Pr>1$ the aspect
ratio does not affect the results. Finally, it is noteworthy that the Reynolds
number defined as $Re_{c}=Ra_{c}^{0.5}/Pr$ could serve as an appropriate
critical value for the onset of instability since it varies just between
$(2\pm 1)10^{4}$ for $Pr<1$, and for $Pr>1$, $Re_{c}=10^{4}Pr^{2}$, whereas
the Rayleigh number varies over 6 decades (see figure 10(b)). This Reynolds
number compares well with a Reynolds number based on the maximum velocity in
the cavity and the appropriate typical length scale.

Figure 10: Critical Rayleigh number as a function of Prandtl number and (b)
the Reynolds number derived from the critical Rayleigh number and Prandtl
numbet agains Prandtl number for different aspect ratio $A$ (see legend). The
dashed lines represent the power laws discussed in the text.

##  4 Conclusions and discussion

The present investigation shows that there is a large variation in flow
regimes depending on the Prandtl number and aspect ratio as is represented in
the regime diagram of figure 9(b). The role of the detachment in the corner
regions and the internal waves on the dynamics is investigated.

The regimes depend on the variation in the detachment of the buoyancy current
with Prandtl number and its effect on the circulation inside the cavity, and
on the other hand, the presence of internal waves. When there is no detachment
(for $Pr<0.5$) and the buoyancy current is limited by the horizontal extend of
the cavity, a cell circulation develops. The instability mode is global, and
top and bottom motions have amplitude symmetry, i.e. they have same (absolute)
perturbation amplitude.

When there is detachment of the buoyancy current, there is a local circulation
that is determined by the dynamics of the buoyancy current. The coupling
between top and bottom motions, and therewith their symmetry in amplitude then
depends on the presence of internal waves. Generally, in the absence of
internal waves and large-scale circulation, the perturbation amplitudes in the
corner region differ and the flow is asymmetric. However, some exceptions
occur, and not all internal wave patterns allow for this coupling.

The critical Rayleigh number depends on the Prandtl number, with two regimes,
one for $Pr<1$ with some variation due to aspect ratio and $Ra_{c}\sim Pr^{2}$
, and one for $1<Pr<4$ where the aspect ratio does not affect the results and
$Ra_{c}\sim Pr^{5}$. As mentioned above, this can equally be expressed in
terms of the Reynolds number. For our extreme value of $Pr=4$, Janssen &
Henkes (1995) find a critical Rayleigh number $Ra_{c}=2.5\cdot 10^{10}$, while
for $Pr=10$ Wang et al. (2021) find a critical Rayleigh number of
$Ra_{c}=6\cdot 10^{10}$, showing a downward tendency. This subject is open to
further research. On the other hand, for very small Prandtl numbers, Gelfgat
et al. (1999) found critical $Ra$ numbers of order O($10^{-4}$) so that on
this side, a further decrease in critical Rayleigh number can be expected with
higher critical Rayleigh numbers for the smaller aspect ratio $A$.

Some of the regimes represented in figure 9(b) have been observed in former
investigations. For air-filled cavities of aspect ratio of 6 and 7 e.g. the
FCc regime was also found in Xin & Le Quéré (2006), whereas the SCo regime is
well studied in e.g. Le Quéré & Behnia (1998) and Oteski et al. (2015), but no
detailed information has been found about the instabilities of the FCoE- and
FW-regimes. Most remarkable is the FCo-regime with its drastic increase in
oscillation frequency of a factor of 10 for Prandtl numbers in the range of
$Pr=0.55\pm 0.15$. This regime has not been shown before. It most likely
disappears for larger aspect ratio, with for $A\leq 2$ and approximately
$Pr<0.55$ the flow being dominated by convectively driven cells.

For $0.4\leq Pr\leq 0.7$, the shear and the temperature gradient between the
corner region and upward motion near the wall increase, and the corner region
oscillates with internal waves in the interior. There is uncertainty about the
origin of the instability mechanism, and the roles of shear and internal
waves. For $Pr>2.8$ the region of instability changes again and occurs between
the downward motion in the detached corner flow and the upward motion at the
wall at some distance from the boundary. Also here the temperature gradient is
large. Therefore, in order to determine the origin of the instability, a
comparison of the individual terms in the momentum equations, as well as plots
for Rayleigh criterion, centrifugal instability, and Richardson number is
needed. Since this is a rather elaborate effort, it will be presented
elsewhere.

This investigation is limited to a single mode, obtained for the lowest
critical $Ra$-number. When increasing $Ra$, bifurcations with other modes may
appear, as shown by Oteski et al. (2015) for $Pr=0.7$ for air, and in a three
dimensional box by (e.g. Gelfgat, 2017). In view of former results obtained
with DNS (see Trias et al., 2007), one may nevertheless expect that the
present results will provide also a good guide line for the three dimensional
case, as long as the Rayleigh number is small ($Ra<10^{10}$). Preliminary
tests in the present research have shown that three-dimensional instabilities
are absent as long as the cavity depth is about 10% of the cavity-height (i.e.
$0.1\times H$). Therefore we expect that the present regimes have their
footprint also in a quasi three-dimensional environment.

[Supplementary data]Supplementary material and movies are available at
https://doi.org/10.1017/jfm.2019…

[Acknowledgments] The authors thank the anonymous referees for their helpful
comments. Further, they acknowledge Olivier De-Marchi, Gabriel Moreau and
Cyrille Bonamy of the LEGI’s informatics team for their support. A CC-BY
public copyright license has been applied by the authors to the present
document and will be applied to all subsequent versions up to the Author
Accepted Manuscript arising from this submission, in accordance with the
grant’s open access conditions.

[Funding] This project was funded by the project LEFE/IMAGO-2019 contract
COSTRIO. AK acknowledges the finance of his PhD thesis from the school STEP of
the University Grenoble Alpes. Part of this work was performed using resources
provided by CINES under GENCI allocation number A0120107567.

[Declaration of Interests]The authors report no conflict of interest.

[Data availability statement.] The data that support the findings of this
study are openly available in Zenodo at https://zenodo.org/records/7827872,
reference number 7827872.

[Author ORCID’s]
A. Khoubani https://orcid.org/0000-0002-0295-5308;
A. V. Mohanan https://orcid.org/0000-0002-2979-6327;
P. Augier https://orcid.org/0000-0001-9481-4459;
J.-B. Flór https://orcid.org/0000-0002-7114-2263.

## Appendix A

$A$ | $Pr$ | $Ra(10^{7})$ | $nx$ | $ny$ |  | $A$ | $Pr$ | $Ra(10^{7})$ | $nx$ | $ny$
---|---|---|---|---|---|---|---|---|---|---
0.5 | 0.1 | 0.369 | 80 | 40 |  | 1 | 0.1 | 0.1406 | 40 | 40
0.5 | 0.2 | 2.2 | 80 | 40 |  | 1 | 0.2 | 2.605 | 40 | 40
0.5 | 0.35 | 12.8 | 80 | 40 |  | 1 | 0.35 | 4.939 | 40 | 40
0.5 | 0.44 | 16.5 | 88 | 44 |  | 1 | 0.44 | 10.962 | 44 | 44
0.5 | 0.53 | 24 | 88 | 44 |  | 1 | 0.53 | 14.82 | 44 | 44
0.5 | 0.62 | 24.1 | 88 | 44 |  | 1 | 0.62 | 13.26 | 44 | 44
0.5 | 0.69 | 19.85 | 88 | 44 |  | 1 | 0.69 | 20.40 | 44 | 44
0.5 | 0.71 | 23.75 | 88 | 44 |  | 1 | 0.71 | 18.35 | 44 | 44
0.5 | 1 | 12.13 | 96 | 48 |  | 1 | 1 | 12.88 | 48 | 48
0.5 | 1.4 | 28.15 | 104 | 52 |  | 1 | 1.4 | 35.15 | 52 | 52
0.5 | 2 | 310 | 160 | 80 |  | 1 | 2 | 2000 | 80 | 80
0.5 | 2.8 | 2880 | 200 | 100 |  | 1 | 2.8 | 2800 | 100 | 100
|  |  |  |  |  | 1 | 4 | 11950 | 100 | 100

$A$ | $Pr$ | $Ra(10^{7})$ | $nx$ | $ny$ |  | $A$ | $Pr$ | $Ra(10^{7})$ | $nx$ | $ny$
---|---|---|---|---|---|---|---|---|---|---
1.5 | 0.1 | 0.213 | 27 | 40 |  | 2 | 0.1 | 0.594200 | 20 | 40
1.5 | 0.2 | 0.739 | 27 | 40 |  | 2 | 0.2 | 0.6408 | 20 | 40
1.5 | 0.35 | 6.19 | 27 | 40 |  | 2 | 0.35 | 2.6788 | 20 | 40
1.5 | 0.44 | 13.2 | 33 | 50 |  | 2 | 0.44 | 6.7 | 28 | 56
1.5 | 0.53 | 13.18 | 33 | 50 |  | 2 | 0.53 | 7.95 | 25 | 50
1.5 | 0.62 | 12.135 | 33 | 50 |  | 2 | 0.62 | 12.63 | 25 | 50
1.5 | 0.69 | 18.26 | 33 | 50 |  | 2 | 0.69 | 17 | 25 | 50
1.5 | 0.71 | 16.325 | 33 | 50 |  | 2 | 0.71 | 15.87 | 25 | 50
1.5 | 1 | 158 | 37 | 56 |  | 2 | 1 | 14.125 | 28 | 56
1.5 | 1.4 | 34.1 | 40 | 60 |  | 2 | 1.4 | 34.15 | 30 | 60
1.5 | 2 | 197.2 | 53 | 80 |  | 2 | 2 | 181.25 | 40 | 80
1.5 | 2.8 | 2600 | 80 | 120 |  | 2 | 2.8 | 2500 | 60 | 120
Table 1: Input parameters for the linear simulations. $nx$ and $ny$ are number
of elements in x and y directions, respectively, and the polynomial order $O$
is for all cases 7 so that the number of grid points is $(O+1)^{2}\times
nx\times ny$.

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