eurm10 msam10  LABEL:firstpage–LABEL:lastpage

#  LaTeX 2ε Input Guide for Authors

Cheng-Nian Xiao<EMAIL_ADDRESS>Inanc Senocak  Department of Mechanical
Engineering and Materials Science,
University of Pittsburgh, 3700 O’Hara St, Pittsburgh, PA 15261, USA

(10 August 2001 and in revised form 10 August 2004; 2001)

#  Instabilities of longitudinal vortex rolls in katabatic Prandtl slope flows

Cheng-Nian Xiao<EMAIL_ADDRESS>Inanc Senocak  Department of Mechanical
Engineering and Materials Science,
University of Pittsburgh, 3700 O’Hara St, Pittsburgh, PA 15261, USA

(10 August 2001 and in revised form 10 August 2004; 2001)

###### Abstract

Stationary counter-rotating longitudinal vortex pairs emerge from one-
dimensional Prandtl slope flows under katabatic as well as anabatic conditions
due to a linear instability when the imposed surface heat flux magnitude is
sufficiently strong relative to the stable ambient stratification. For
anabatic flows, these vortices have already been identified to exhibit an
unique topology that bears a striking resemblance to speaker-wires since they
stay coherent as a single unit without the presence of another vortex pair.
Under katabatic conditions and at a constant Prandtl number, we find that the
longitudinal vortices emerging at a range of different slope angles possess
the similar topology as their anabatic counterparts. We determine the
existence of both fundamental and subharmonic secondary instabilities
depending on the slope angle for the most likely transverse base flow
wavelength. Our results indicate that the most dominant instability shifts
from a fundamental to subharmonic mode with increasing slope angle. At shallow
slopes, this dynamic contrast with the speaker-wire vortices in anabatic slope
flows at the same angle which for which the subharmonic instability is clearly
dominant. These modes are responsible for the bending and movement of single
or multiple speaker-wire vortices, which may merge or reconnect to lead to
dynamically more unstable states, eventually leading to transition towards
turbulence. We demonstrate that at sufficiently steep slopes, the dynamics of
these vortex pairs are dominated by long-wave reconnections or two-dimensional
mergers between adjacent pairs.

###### doi:

S002211200100456X

††volume: 000

##  1 Introduction

Prandtl’s model for katabatic and anabatic slope flows is a popular
abstraction used to understand the fundamental characteristics of stably
stratified flows in cold weather conditions over non-flat terrains, such as
nocturnal winds over hills or katabatic flows over the (Ant-)arctic ice sheets
(Prandtl, 1942, 1952; Gutman, 1983; Zardi & Whiteman, 2013). The model can be
solved exactly for laminar conditions to obtain buoyancy and velocity
solutions that are sinusoidal profiles damped exponentially with increasing
height (Shapiro & Fedorovich, 2004; Fedorovich & Shapiro, 2009). The profiles
demonstrate a dominant near-surface along-slope jet which is topped by a weak
reverse flow that decays with growing height, as shown in Fig. 1.

In previous investigations, the stability of Prandtl’s slope flows and
travelling waves along a vertical wall has been studied by Gill & Davey
(1969); McBain et al. (2007); Maryada et al. (2022). We have analyzed the
linear stability of katabatic (Xiao & Senocak, 2019) and anabatic (Xiao &
Senocak, 2020) Prandtl slope flows subject to constant surface heat fluxes for
inclination angles $2^{\circ}<\alpha<70^{\circ}$ and found the dominating mode
to be a stationary longitudinal roll instability at shallow slopes and a
traveling wave instability at slope angles larger than $10^{\circ}$ (anabatic)
or $70^{\circ}$(katabatic). While the nonlinear travelling wave solutions
arising from the corresponding linear mode of the Prandtl slope flows over a
vertical wall have been analyzed by McBain et al. (2007); Maryada et al.
(2022), we instead directed our focus to an investigation of the dynamics of
longitudinal rolls arising from the primary longitudinal roll instability
under anabatic conditions over a shallow slope with $\alpha=3^{\circ}$ subject
to constant surface heat flux (Xiao & Senocak, 2022b), which is a novel vortex
configuration that has not been studied before. Our prior studies have
demonstrated that the vortex instabilities of stably stratified flows are
determined by the slope angle $\alpha$, the transverse vortex spacing $b_{y}$,
as well as the dimensionless stratification perturbation parameter $\Pi_{s}$,
which can also be interpreted as the surface heat flux normalized by the
background stratification. In this present study, we intend to focus on the
dynamics of stationary vortices in katabatic slope flows. In contrast to
anabatic slope flows which appear only fleetingly during morning transition
before the positive surface heat flux overpowers the surrounding stable
stratification, katabatic conditions can be stable and long-lasting as
manifested in nocturnal boundary layers or above the (Ant-)arctic ice sheet.
Another key difference is the ability of katabatic conditions to support
stationary longitudinal vortices emerging from a primary instability at very
steep slopes up to $65^{\circ}$ (Xiao & Senocak, 2019), whereas anabatic
conditions do not admit this for slopes larger than $10^{\circ}$(Xiao &
Senocak, 2020).

Numerous experiments under simpler configurations, i.e. without stratification
or surface inclination, have confirmed the emergence and persistence of highly
organised two-dimensional (2-D) vortex structures in the course of a
developing shear layer downstream of a splitter plate(Winant & Browand, 1974;
Brown & Roshko, 1974). These vortices have been noticed to arise under a
multitude of different flow conditions, and appear to be remarkably robust
with respect to external disturbances (Wygnanski et al., 1979). It is evident
that any progress in the understanding of turbulent transition will need to
accommodate the dynamics and interaction of these large vortex structures. In
a seminal work on vortex dynamics, Crow (1970) has studied the linear
stability for parallel counter-rotating vortex pairs suspended in quiescent
air with neutral stratification and discovered that sinusoidal bending of each
individual vortex arises due to a symmetric mode now named after him, which
then results to vortex merger and ring formation at precisely those locations
where the the neighboring vortices approach each other due to bending. This
work has been extended by Zheng & Hardin (2017) to include the analysis of
anti-symmetric modes in vortex pairs, whereas Crouch (1997) studied the
dynamics of two unsteadily tumbling vortex pairs.

For more complicated flow configurations which involve the simultaneous
presence of heat transfer, ambient stratification and non-flat surfaces, the
scenario outlined above is expected to be insufficient to describe the entire
flow physics. However, despite the fact that many real-life phenomena such as
geophysical flows contain some or all of these complicating factors, there has
been very few attempts to properly comprehend vortex instabilities in these
challenging situations.

The role of stratification on Stuart vorticies has been studied by Miyazaki &
Fukumoto (1992) who showed that elliptic instabilities causing anti-symmetric
distortion of vortex cores are suppressed by the presence of stable
stratification. However, Le Dizès & Billant (2009) has found that stable
stratification can also destabilize a single vertical columnar vortex as a
result of resonance with internal gravity waves, which is a so-called
radiative instability. Stable stratification can also give rise to a vortex
instability in rotating flows called the ”zig-zag” instability, first
identified and investigated by Billant & Chomaz (2000); Billant (2010). Its
formation is due to additional self induction as well as mutual induction
between vortices as a result of stratification and rotation that are aligned
with the main vorticity. Its main effect is symmetric bending of individual
vortices in co-rotating pairs and anti-symmetric bending in counter-rotating
pairs.

Most of the current and ongoing studies on vortex instabilities such as those
listed above have implicitly assumed that the main merger dynamics depend on
no more than two vortices, which can be either counter-rotating or co-
rotating. A notable departure from this tradition is our recent work on the
instability of speaker-wire vortices arising in anabatic Prandtl slope flows
with surface heating Xiao & Senocak (2022b), which required at least two
counter-rotating pairs with four vortices to enable vortex mergers or
reconnections. This is due to the fact that all fundamental modes are anti-
symmetric such that only subharmonic instabilities can move vortices from
adjacent pairs closer toward each other. It has been found that the spacing
between adjacent vortex pairs is a main factor for determining the strength of
the most dominant modes, which is directly correlated to the likelihood of
vortex mergers. An intuitive interpretation of the results presented in Xiao &
Senocak (2022b) is that the further the initial spacing of the base vortices
is from that of the most stable configuration, the stronger is the subharmonic
instability of the vortex configuration which would cause vortex mergers or
the formation of new vortices in order to attain the most optimal vortex
spacing. We have also observed in Xiao & Senocak (2022b) that pure mergers,
i.e. the process of fusing two neighboring vortices into a single one without
any bending, can be a dominant dynamics due to the fact that the two-
dimensional vortex instability is almost as strong as the 3D mode with the
maximal possible growth rate.

The present study aims to extend our previous analysis to investigate the
dynamics of the longitudinal rolls that emerge as the steady configuration of
a linear instability under katabatic conditions at shallow slope angles, i.e.
subject to surface cooling. Several distinctions of vortex instabilities under
the Prandtl’s model sets our work apart from the other aforementioned studies
on vortex instabilities such as Crow’s instability, elliptic instability, zig-
zag instability, and secondary convection rolls. Most significantly, Prandtl’s
model includes the following key components missing in other configurations:
Firstly, a constant ambient stratification that is positioned at an oblique
angle to the longitudinal rolls aligned with the main flow direction and
secondly, a solid surface wall containing its own boundary layer as part of
the Prandtl base flow. Unsurprisingly, the combination of these complicating
factors renders a theoretical approach highly difficult and, to the best of
our understanding, is not present in the published literature besides our own
earlier related work under anabatic conditions. As demonstrated in Xiao &
Senocak (2022b), the dominant vortex instability in anabatic Prandtl slope
flows involves two counter-rotating vortex pairs, also termed speaker-wire
vortices, thus four individual vortices in total. Following the analysis in
Xiao & Senocak (2022b), we apply a modal linear bi-global stability analysis
to identify different instabilities which can arise from the base flow
vortices under katabatic conditions. We will further determine how these
instabilities are controlled by external conditions as well as their role in
the subsequent transition of slope flows to more unstable configurations.

##  2 Speaker-wire vortices in katabatic slope flows

###  2.1 Governing equations and characteristic flow scales

(a)

(b)

(c)

Figure 1:  Base flow profiles for slope flows under Prandtl’s model: a) Sketch
of slope geometry and the rotated coordinate system with 1-D Prandtl slope
flow profile; b) Longitudinal vortex pairs (speaker-wire vortices) arising as
instability from the 1-D laminar anabatic or katabatic Prandtl slope flow,
with definition of coordinate axes, vortex width $a_{y}$, and spacing $b_{y}$
which is equivalent to the transverse wavelength $\lambda_{y}$ of the primary
longitudinal roll instability; c) cross-slope $yz$ plane with cross-sections
of three speaker-wire vortex pairs, indicating of the vortex width $a_{y}$ and
vortex spacing $b_{y}$.

Let us consider the idealised Prandtl slope flow configuration as shown in
figure 1(a), where $\alpha$ is the slope angle, and gravity $\mathbf{g}$ acts
downward in the vertical direction. A constant uniform negative buoyancy flux
$B_{S}$ is imposed at the surface to achieve katabatic conditions. We consider
a rotated Cartesian coordinate system whose $x$ axis is aligned with the
planar inclined surface. The direction normal to the slope surface is
represented by the $z$ axis, whereas the cross-flow transverse direction is
aligned with the $y$ coordinate, as shown in Figs. 1(a)-1(b). Let $u$ be the
along-slope (longitudinal), $v$ be the cross-slope (transverse), and $w$ be
the slope-normal velocity components, such that $\mathbf{u}=u_{i}=[u,v,w]$ is
the velocity vector. The unit gravity vector in the rotated coordinate system
is then given by $g_{i}=(g_{1},g_{2},g_{3})=[\sin\alpha,0,\cos\alpha]$.

The potential temperature, buoyancy, and the Brunt-Väisälä frequency are
denoted by $\theta,b,N$, respectively, where $N$ is related to the ambient
potential temperature as
$N=\sqrt{\frac{g}{\Theta_{r}}\frac{\partial\Theta_{e}}{\partial z^{\prime}}}$.
The buoyancy is defined as a perturbation potential temperature as
$b=g(\Theta-\Theta_{e})/\Theta_{r}$, where $\Theta_{r}$ is a reference
potential temperature and $\Theta_{e}$ is the environmental (ambient)
potential temperature. The kinematic viscosity and thermal diffusivity of the
fluid are denoted by $\nu,\beta$, respectively, and they are assumed to be
constant. The transport equations for momentum with a Boussinesq approximation
and perturbaton buoyancy fields are written as follows:

$\displaystyle\frac{\partial\mathbf{u}}{\partial
t}+\nabla\cdot(\mathbf{u}\otimes\mathbf{u})=$
$\displaystyle~{}-\frac{1}{\rho}\nabla p+bg\
\mathbf{n}_{\alpha}+\nu\Delta\mathbf{u},$ (1) $\displaystyle\frac{\partial
b}{\partial t}+\nabla\cdot(b\mathbf{u})=$ $\displaystyle~{}\beta\Delta
b-N^{2}\ (\mathbf{n}_{\alpha}\cdot\mathbf{u}).$ (2)

where $\mathbf{n}_{\alpha}=(\sin\alpha,0,\cos\alpha)$ is the slope-normal unit
vector. The conservation of mass principle is imposed by a divergence-free
velocity field

$\displaystyle\nabla\cdot\mathbf{u}=0.$ (3)

At the surface $z=0$, a negative buoyancy flux $B_{s}$ is imposed to generate
anabatic flow conditions against a constant stable ambient stratified
environment quantified with $N^{2}$.

For the one-dimensional flow problem, Shapiro & Fedorovich (2004) extended the
exact solution of Prandtl (1942) to include a constant buoyancy flux at the
surface instead of a constant temperature surface condition and introduced the
following characteristic flow scales Fedorovich & Shapiro (2009):

$\displaystyle l_{0}=$ $\displaystyle~{}\text{Pr}^{\nicefrac{{-1}}{{4}}}\
\nu^{\nicefrac{{1}}{{2}}}N^{\nicefrac{{-1}}{{2}}}\sin^{\nicefrac{{-1}}{{2}}}\alpha,$
(4) $\displaystyle u_{0}=$ $\displaystyle~{}\text{Pr}^{\nicefrac{{1}}{{4}}}\
\nu^{\nicefrac{{-1}}{{2}}}N^{\nicefrac{{-3}}{{2}}}B_{s}\sin^{\nicefrac{{-1}}{{2}}}\alpha,$
(5) $\displaystyle b_{0}=$ $\displaystyle~{}\text{Pr}^{\nicefrac{{3}}{{4}}}\
\nu^{\nicefrac{{-1}}{{2}}}N^{\nicefrac{{-1}}{{2}}}B_{s}\sin^{\nicefrac{{-1}}{{2}}}\alpha,$
(6)

where $\text{Pr}={\nu}/{\beta}$ is the Prandtl number. A time scale
$t_{0}:=l_{0}/|u_{0}|=\sqrt{\nu\beta}~{}N|B_{s}|^{-1}$, and a shear scale
$S_{0}:=|u_{0}|/l_{0}=\sqrt{Pr/\nu}N^{-1}|B_{s}|$ can also be defined from the
above scales. We observe from (4)-(6) that the length scale characterizing the
laminar boundary layer thickness is independent of the surface flux $B_{s}$,
whereas the magnitude of both the reference velocity and buoyancy scale varies
linearly with $B_{s}$. Since only the magnitude of the surface buoyancy flux
appears in the flow scale definitions, they are the same for both katabatic
and anabatic conditions at the same surface buoyancy flux magnitude but with
opposite signs. Subsequently, these characteristic scales will be applied to
normalize all flow equations and quantities presented herein. Specifically,
the dimensionless stratification perturbation number $\Pi_{s}$ as introduced
in Xiao & Senocak (2019), can be regarded as the imposed surface buoyancy flux
$B_{s}$ normalised by the background stratification scale $\beta N^{2}$. This
unique parameter is determined from the given external flow parameters as
follows:

$\displaystyle\Pi_{s}\equiv\frac{|B_{s}|}{\beta N^{2}}.$ (7)

It should be mentioned that the $\Pi_{s}$ is not restricted to Prandtl slope
flows; it is a necessary additional parameter whenever there exists multiple
independent stratification mechanisms. In Xiao & Senocak (2022a), we
demonstrated the significance of $\Pi_{s}$ as an independent dimensionless
parameter in stable open channels flows that are stratified by the
simultaneous action of surface cooling as well as a static ambient
stratification.

###  2.2 Steady speaker-wire vortices as longitudinal rolls

The governing flow equations (1)-(3) can be solved on the 2-D cross-slope $yz$
plane to arrive at the steady longitudinal rolls that arise as a saturated
linear instability for slope angles less than $70^{\circ}$ when $\Pi_{s}$ is
sufficiently large Xiao & Senocak (2020). To create the steady vortices that
serve as base flows for the secondary stability analysis, the initial flow
field is set to be the laminar Prandtl flow profile superposed with a weak
sinusoidal disturbance varying along the transverse $y$ direction. For the
simulated cross-slope $yz$ plane, the width is chosen to be an integer
multiple of the targeted transverse spacing $b_{y}=\lambda_{y}$ of the vortex
pairs, and the the total height of the domain is chosen to be at least
$50l_{0}$. Each length scale $l_{0}$ along both the vertical and transverse
direction is resolved by at least two mesh points. The evolution of the flow
field is simulated until steady state is reached, which happens within a
suitable range of $\Pi_{s}$ values and results in stationary vortices that are
the saturation of the growing linear instability mode. For the spacing of the
base vortices, we choose the transverse wavelength $\lambda_{y}$ of the most
dominant longitudinal roll instability, i.e. the primary mode with the largest
growth rate for the one-dimensional Prandtl flow profile. The dependence of
$\lambda_{y}$ as a function of the stratification parameter $\Pi_{s}$ for
different slope angles is shown in Fig. 2, which indicates a clear decrease of
the transverse wavelength with increasing value of $\Pi_{s}$ and slope angle
$\alpha$. Further details about the structure of the vortices that serve as
base flow for the secondary stability analysis are provided along with the
stability results below.

Figure 2: Transverse wavelength of the most dominant linear instabilities for
laminar 1-D katabatic Prandtl slope flows at different slope angles $\alpha$
plotted against $\Pi_{s}$.

##  3 Linear secondary instability analysis of speaker-wire vortices

Let $(U,V,W,B)$ be the 2-D flow profile of the steady longitudinal rolls as
computed from Eqn. (1)-(2), and assuming that modal disturbances to this base
flow are of the form

$\mathbf{q}(x,y,z,t)=\left[\hat{u}(y,z),\hat{v}(y,z),\hat{w}(y,z),\hat{p}(y,z),\hat{b}(y,z)\right]\exp({ik_{x}x+\omega
t}),$

then the equations governing the evolution of the flow disturbances
approximated to first the order have the following form:

$\displaystyle ik_{x}\hat{u}+\frac{\partial\hat{v}}{\partial
y}+\frac{\partial\hat{w}}{\partial z}$ $\displaystyle=0,$ (8)
$\displaystyle\omega\hat{u}+iUk_{x}\hat{u}+\frac{\partial U}{\partial
y}\hat{v}+\frac{\partial U}{\partial z}\hat{w}$
$\displaystyle+\frac{\partial\hat{u}}{\partial
y}V+\frac{\partial\hat{u}}{\partial z}W$
$\displaystyle=-ik_{x}\hat{p}-\frac{Pr}{\Pi_{s}}\sin\alpha\left(-k_{x}^{2}\hat{u}+\frac{\partial^{2}\hat{u}}{\partial
y^{2}}+\frac{\partial^{2}\hat{u}}{\partial z^{2}}+\hat{b}\right),$ (9)
$\displaystyle\omega\hat{v}+iUk_{x}\hat{v}+\frac{\partial V}{\partial
y}\hat{v}$ $\displaystyle+\frac{\partial V}{\partial
z}\hat{w}+\frac{\partial\hat{v}}{\partial y}V+\frac{\partial\hat{v}}{\partial
z}W$ $\displaystyle=-\frac{\hat{p}}{\partial
y}-\frac{Pr}{\Pi_{s}}\sin\alpha\left(-k_{x}^{2}\hat{v}+\frac{\partial^{2}\hat{v}}{\partial
y^{2}}+\frac{\partial^{2}\hat{v}}{\partial z^{2}}\right),$ (10)
$\displaystyle\omega\hat{w}+iUk_{x}\hat{w}+\frac{\partial W}{\partial
y}\hat{v}$ $\displaystyle+\frac{\partial W}{\partial
z}\hat{w}+\frac{\partial\hat{w}}{\partial y}V+$
$\displaystyle\frac{\partial\hat{w}}{\partial
z}W=-\frac{\partial\hat{p}}{\partial z}$
$\displaystyle-\frac{Pr}{\Pi_{s}}\sin\alpha\left(-k_{x}^{2}\hat{w}+\frac{\partial^{2}\hat{w}}{\partial
y^{2}}+\frac{\partial^{2}\hat{w}}{\partial z^{2}}+\hat{b}\cot\alpha\right),$
(11) $\displaystyle\omega\hat{b}+iUk_{x}\hat{b}+\frac{\partial B}{\partial
y}\hat{v}$ $\displaystyle+\frac{\partial B}{\partial
z}\hat{w}+\frac{\partial\hat{b}}{\partial y}V+\frac{\partial\hat{b}}{\partial
z}W$
$\displaystyle=-\frac{\sin\alpha}{\Pi_{s}}\left(-k_{x}^{2}\hat{b}+\frac{\partial^{2}\hat{b}}{\partial
y^{2}}+\frac{\partial^{2}\hat{b}}{\partial
z^{2}}-(\hat{u}+\hat{w}\cot\alpha)\right),$ (12)

where $\hat{u},\hat{v},\hat{w},\hat{p},\hat{b}$ describe the shape of the flow
disturbance along the slope normal and transverse directions, normalised by
the flow scales given in (4)-(6). The normalised base flow field describing
the steady vortices is denoted by $(U,V,W,B)$, and the slope angles studied
here are $\alpha=3^{\circ},12^{\circ},22^{\circ},30^{\circ}$.

The linearised equations for bi-global stability analysis as shown above can
be written as a generalised eigenvalue problem in the following way:

$\displaystyle A(k_{x})\mathbf{\hat{q}}(y,z)=\omega
B(k_{x})\mathbf{\hat{q}}(y,z),$ (13)

The shape of the complex disturbance vector

$\mathbf{\hat{q}}(y,z)=[\hat{u}(y,z),\hat{v}(y,z),\hat{w}(y,z),\hat{p}(y,z),\hat{b}(y,z)]^{T}$

varies in the slope-normal (z) and transverse direction (y), where
$(\hat{u},\hat{v},\hat{w})$ are the along-slope, cross-slope (transverse) and
slope-normal disturbance velocity components. As a bi-global stability
analysis, the slope-normal and transverse dimensions are fully resolved and
the disturbance variation along the streamwise direction is approximated by
only one Fourier mode with wave number $k_{x}$. When $k_{x}$ is zero, then the
corresponding mode is 2-D without any streamwise variation, whereas a positive
$k_{x}$ implies a full 3-D disturbance. The appropriate boundary conditions
for this problem are no slip for disturbance velocities at $z=0$ and free-slip
at $z\rightarrow\infty$, whereas for the buoyancy disturbance,
$\partial\hat{b}/\partial z|_{z=0}=\partial\hat{b}/\partial
z|_{z\rightarrow\infty}=0$ are imposed. The slope-normal derivative of
pressure disturbance $\hat{p}$ is also set to zero at both $z=0$ and
$z\rightarrow\infty$. On both transverse boundaries, periodic conditions are
imposed for all variables. The generalised eigenvalue problem (13) is
discretised via spectral elements in the transverse plane, which is available
in Nektar++ Cantwell et al. (2015). For a base flow containing two full
transverse spatial periods, i.e. two speaker-wire vortex pairs, the cross-
slope dimension $L_{y}=2\lambda_{y}$ is chosen for the simulation domain; 2
degrees of freedom are used to resolve a unit length scale $l_{0}$ in the
cross-slope and vertical directions, and the resulting generalised eigenvalue
problem are solved with the modified Arnoldi algorithm as implemented in
Nektar++. Linear stability of the problem is associated with the real part of
the eigenvalues $\omega$, where $\Re\\{\omega\\}>0$ represents a positive
exponential growth for the corresponding eigenmode, thus an unstable mode. The
imaginary part of $\omega$ is the temporal oscillation frequency for the
corresponding eigenmode, and $\Im\\{\omega\\}=0$ represents a stationary mode.

###  3.1 Instabilities of katabatic speaker-wire vortices at different slope
angles

To investigate the secondary linear instability of speaker-wire vortices,
eigenvalues with the highest maximal real values for a range of streamwise
wave numbers $k_{x}$ are computed for different stratification perturbation
parameters $\Pi_{s}$ of the base flow at different slope angles $\alpha$. The
transverse wavelength $\lambda_{y}$ of the base flow vortices was chosen to be
the most dangerous wavelength at the given $\Pi_{s}$ as determined from linear
stability analysis of the 1D Prandtl flow profile, which is shown in Fig. 2.
Thus in contrast to the study for vortices under anabatic conditions (Xiao &
Senocak, 2022b), the focus here is on the effects of the slope angle instead
of the vortex spacing. Throughout our analysis, we also assume a constant
Prandtl number $Pr=0.71$ for all the cases.

We observe from Fig. 2 that the slope inclination $\alpha$ exerts a
significant effect on the optimal transverse wavelength of the base vortices
when all other flow parameters are held constant. Hence, we expect that the
secondary instabilities arising from these base flow vortices will also be
qualitatively different depending on $\alpha$. At a given slope angle and
Prandtl number, three independent parameters determine the growth rates and
oscillation frequencies of the secondary instability, which are the
stratification parameter $\Pi_{s}$, the streamwise instability wave number
$k_{x}=2\pi/\lambda_{x}$ and the transverse base flow wave number
$k_{y}=2\pi/\lambda_{y}$, where $\lambda_{y}=b_{y}$ equals the spacing of the
base speaker-wire vortex. In the following, we will separately present and
discuss the results of the stability analysis for base speaker-wire vortex at
four different slope angles $\alpha$, going from $\alpha=3^{\circ}$ to
$\alpha=30^{\circ}$. The base flow used for modal analysis of each case
consists of two speaker-wire vortices arising from the primary linear
instability mode, i.e. the transverse domain size is twice the wavelength of
the primary vortex instability as described in Xiao & Senocak (2020). This
choice of domain size ensures that potential subharmonic eigenmodes with twice
the transverse wavelength of the base vortices can be picked up.

####  3.1.1 Shallow slope: $\alpha=3^{\circ}$

At the shallow slope $\alpha=3^{\circ}$, it turns out that the transverse
spatial period of all the secondary instability modes equals the transverse
wavelength $\lambda_{y}$ of the base vortices which is also the spacing
$b_{y}$ between the core of adjacent vorte pairs, hence they are all
fundamental modes. The shape of this mode on the transverse plane is
visualized via the contour plot for the streamwise vorticity $\omega_{x}$ as
shown in Fig. 3. Thus, this observation is in contrast with the anabatic slope
flow at the same angle $\alpha=3^{\circ}$ where the most dominant instability
is a subharmonic mode Xiao & Senocak (2022b).

Figure 4(a) shows the growth rates of the most unstable secondary modes as a
function of streamwise wavenumbers $k_{x}$ within the range $[0,0.15]$ for
four different values of stratification parameter $\Pi_{s}=1.6,1.7,1.8,1.9$.
The growth rate of any mode at any streamwise wavenumber $k_{x}$ grows with
increasing $\Pi_{s}$, which implies a stronger surface cooling for the
katabatic slope flow. 2-D modes have zero longitudinal wave number, i.e.
$k_{x}=0$, and thus only vary on the transverse $yz$ plane but are constant
along the streamwise direction.

For the 3-D modes, which have non-zero wave numbers $k_{x}>0$, we observe from
figure 4(a) that at large wave numbers $k_{x}>0.15$, the growth rates tend to
increase with decreasing $k_{x}$ until reaching a maximal value exceeding the
2-D growth rate ($k_{x}=0$) at a optimal wave number $k_{x}\approx 0.07$, and
from that point on they decrease with decreasing wavenumber to reach the 2-D
growth rate at $k_{x}=0$. In contrast to both the subharmonic and fundamental
eigenmodes of speaker-wire vortices under anabatic conditions at the same
slope angle of $\alpha=3^{\circ}$ (Xiao & Senocak, 2022b), the most dominant
fundamental modes at katabatic conditions are clearly 3-D since the growth
rate of the mode at the optimal non-zero wave number is an order of magnitude
larger than that of the zero-wavenumber 2-D mode.

All 3-D fundamental modes with positive longitudinal wavenumbers $k_{x}>0$ are
oscillatory whose frequency increases monotonically with growing $k_{x}$ or
decreasing wavelength, as shown in figure 4(b). The 2-D mode with $k_{x}=0$ is
stationary, i.e. with zero imaginary part of its eigenvalue. At small
streamwise wavenumbers $k_{x}$, the oscillation frequencies of the 3-D mode
decrease to zero with decreasing $k_{x}$ to converge to the stationary 2-D
mode at $k_{x}=0$. As figure 4(b) shows, within the small wave number range,
the normalised frequencies of all three cases shown here appear to obey a
simple linear dispersion relation given by $\Im(\omega)=\eta\cdot k_{x}$,
where $\eta\approx 0.025$ is empirically determined to fit all three curves.
The accuracy of this fit shows that the group speed of the long-wave
fundamental vortex instabilities given by
$c=\frac{\partial\Im(\omega)}{\partial k_{x}}=\eta$ is nearly constant and
equals $\eta=0.025$. However, this simple relation no longer holds for modes
with wave numbers $k_{x}>0.05$ whose oscillation frequencies increase faster
than linear with growing wave numbers.

Existence of both 2-D and 3-D fundamental vortex instabilities has also been
identified in the study of co-rotating Stuart vortex arrays by Pierrehumbert &
Widnall (1982) or as instabilities in a shear layer by Corcos & Sherman
(1984), where they are shown to be responsible for the displacement or bending
of neighboring vortices. In contrast to the anabatic slope flow at a slope
angle of $\alpha=3^{\circ}$ (Xiao & Senocak, 2022b), there are no subharmonic
modes in the katabatic slope flow with angle $\alpha=3^{\circ}$ . As will be
shown later, the fundamental modes for this case as well as at larger slope
angles are all anti-msymmetric. This suggest that at the initial onset of
instabilities, the neighboring vortex tubes will bend and move synchronously
in parallel fashion. This is confirmed by direct numerical simulations via
solving the full nonlinear governing equations, as shown in the animation
available as supplementary movie 1. From the animation, the consistent growth
of the anti-symmetric fundamental mode which eventually generates novel
structures on top of the original vortices can be seen.

Figure 3:  Visualization of streamwise vorticity magnitude for the fundamental
mode along with base vortices on cross-slope $yz$ vertical plane in katabatic
flow at $\alpha=3^{\circ}$ and $\Pi_{s}=1.6$ with streamwise wave number
$k_{x}=0.07$. Thick contour lines represent the base flow, and thin lines are
from the instability mode. Contours are drawn at 8% of maximal magnitude in
both cases. Only half of the total transverse domain is shown, equalling the
transverse wavelength $\lambda_{y}$ of the base vortices.

(a)

(b)

Figure 4: Fundamental modes of katabatic speaker-wire vortices at slope angle
$\alpha=3^{\circ}$ for different values of $\Pi_{s}$ as a function of the
longitudinal wave number $k_{x}$: Growth rate $\Re(\omega)$ (a) and
oscillation frequency $\Im(\omega)$ (b). The linear fit for the dispersion
relation between oscillation frequency and streamwise wave number is also
shown in (b).

####  3.1.2  Moderately steep slopes $\alpha=12^{\circ}$

At a slope angle of $\alpha=12^{\circ}$, both fundamental and subharmonic
secondary instabilities are found to emerge. The transverse wavelength of
fundamental mode equals the spacing $b_{y}$ between adjacent base flow
speaker-wire vortices, whereas the subharmonic mode have transverse
wavelengths twice as large. Hence the additional appearance of the subharmonic
mode is a consequence of the increasing slope angle compared to the shallow
slope at $\alpha=3^{\circ}$.

The shapes of the subharmonic and fundamental modes on the transverse plane
are visualized together with the base flow vortices via the contour plot for
the streamwise vorticity $\omega_{x}$ as shown in Figs. 5(a) and 5(b). It can
be seen that compared to the $3^{\circ}$ case as shown in Fig. 3, the base
vortices are around half as wide along the transverse direction, but also
clearly taller in the vertical. As per definition, the subharmonic modes have
a transverse wavelength twice as large as their fundamental counterparts.

Figure 6(a) presents the growth rates of the most unstable fundamental modes
as a function of streamwise wavenumbers $k_{x}$ within the range $[0,0.15]$
for four different values of normalized surface heat flux
$\Pi_{s}=2.6,2.7,2.8,2.9$. The growth rates of the strongest subharmonic modes
for streamwise wavenumbers $k_{x}$ within $[0,0.03]$ are shown in Fig. 6(b).
The smaller wavenumber range for the strongest subharmonic modes displayed
here implies that their streamwise wavelengths are longer compared to the
fundamental modes. For comparison, the growth rate for the strongest
subharmonic mode at $\Pi_{s}=2.6$ is also displayed together with all the
fundamental modes Fig. 6(a).

Compared to the shallow slope with $\alpha=3^{\circ}$, larger $\Pi_{s}$ values
are needed to sustain the base vortices that arise as the primary roll
instability from the 1D Prandtl slope flow profile (Xiao & Senocak, 2019). It
is evident from Fig. 6(a) that a higher value of $\Pi_{s}$, implying a larger
normalized surface heat flux magnitude, increases the growth rate of all
fundamental modes at any given wave number. However, the same $\Pi_{s}$
dependence is not true for the subharmonic modes. As Fig. 6(b) shows, the
growth rate for subharmonic modes attains its maximum at $\Pi_{s}=2.7$ and is
smaller for larger as well as smaller $\Pi_{s}$ values.

(a)

(b)

Figure 5:  Visualization of streamwise vorticity magnitude for the fundamental
mode on cross-slope $yz$ vertical plane in katabatic flow at
$\alpha=12^{\circ}$ and $\Pi_{s}=2.6$: (a) fundamental mode with $k_{x}=0.1$;
(b) subharmonic mode with $k_{x}=0.015$. Thick contour lines represent the
base flow, and thin lines are from the instability mode. Contours are drawn at
8% of maximal magnitude in both cases.

(a)

(b)

(c)

(d)

Figure 6:  Comparison between fundamental and subharmonic modes of katabatic
speaker-wire vortices at slope angle $\alpha=12^{\circ}$ for different values
of $\Pi_{s}$ as a function of the longitudinal wave number $k_{x}$ : (a)-(b)
Growth rates $\Re(\omega)$; (c)-(d) oscillation frequency $\Im(\omega)$.
Fundamental modes are shown on the left(a,c), and subharmonic modes are shown
on the right(b,d). The linear fit for the relations between oscillation
frequency and streamwise wave number of the subharmonic mode is also shown in
(d).

From figure 6(a), we observe that for modes with sufficiently large wave
numbers $k_{x}>0.01$, the fundamental mode is stronger than its subharmonic
counterpart. The fundamental modes attain the strongest growth at an optimal
non-zero wave number $k_{x}$ between 0.08 and 0.1, and they become
monotonically weaker with decreasing wave number below that value. On the
other hand, as we observe from figure 6(b), the subharmonic modes gain
strength with decreasing wave number such that the most dominant mode is
purely 2-D at $k_{x}=0$. Hence, for very large streamwise wavelengths with
$k_{x}<0.01$, fundamental modes are weaker than their subharmonic
counterparts.

From figure 6(c), we observe that all fundamental modes with positive
streamwise wavenumbers $k_{x}>0$ are oscillatory. For $k_{x}=0$, the
fundamental modes are all stationary, and their frequencies as a function of
$k_{x}$ attains one local maximum in the low-wavenumber range at $k_{x}\approx
0.004$ and begin to increase with growing wavenumbers for $k_{x}>0.012$, as
shown in figure 6(c). However, at higher wavenumbers beyond $k_{x}=0.1$, this
trend no longer holds, as the fundamental mode frequencies either stagnate
(for $\Pi_{s}=2.8,2.9$) or even slightly decrease (for $\Pi_{s}=2.6,2.7$) with
increasing $k_{x}$. On the other hand, figure 6(d) shows that the frequency of
subharmonic modes increases monotonically with growing wavenumber. At
$k_{x}=0$, the 2-D subharmonic mode can be seen to be stationary. As Fig. 6(d)
shows, for wavenumber values less than $k_{x}=0.01$, the normalised
frequencies for all four $\Pi_{s}$ values fit a linear dispersion relation
where the group velocity given by $\eta=\frac{\partial\Im(\omega)}{\partial
k_{x}}$ increases with growing $\Pi_{s}$ from $\eta=0.011$ at $\Pi_{s}=2.6$ to
$\eta=0.05$ for $\Pi_{s}=2.9$. This differs from the behavior of subharmonic
modes in anabatic slope flows where the group velocity $\eta$ remains the same
for different $\Pi_{s}$ values. Fig. 6(d) shows that at larger wavenumbers,
the linear dispersion relation no longer holds because the oscillation
frequencies increase at a faster rate than linear rate with growing
wavenumbers for $k_{x}>0.01$.

The above-mentioned properties of fundamental and subharmonic modes have
significant impact on the dynamics of the base vortices. Since the strongest
subharmonic mode is 2-D, we can expect fusion between neighboring vortex pairs
instead of wavy reconnections that result in vortex rings like in the Crow
instability found in Crow (1970). The fundamental modes which dominate at
shorter wavelengths are antisymmetric and will bend each individual vortex
parallel to its adjacent neighbors. This co-existence of both dynamics can be
seen from animations obtained from direct numerical integration of the full
nonlinear governing equations as shown in the supplementary movie 2. The
animation displays the initial presence of the anti-symmetric fundamental mode
which gradually gives way to the subharmonic mode dynamics which moves
neighboring vortex pairs closer for mergers.

####  3.1.3  Steep slopes $\alpha=22^{\circ}$

For even steeper slopes with an angle of $\alpha=22^{\circ}$, subharmonic and
fundamental secondary instabilities again manifest themselves. The typical
shapes of these two different kinds of modes on the transverse plane are
visualized alongside the base flow vortices via the contour plot for the
streamwise vorticity $\omega_{x}$ as shown in Figs. 7(a) and 7(b). Compared to
the previous two cases at smaller angles as shown in Fig. 3 and Figs.
5(a)-5(b), the base vortices are narrower along the transverse direction, but
also slightly taller in the vertical direction.

(a)

(b)

Figure 7:  Visualization of streamwise vorticity magnitude for the fundamental
mode on cross-slope $yz$ vertical plane in katabatic flow at
$\alpha=22^{\circ}$ and $\Pi_{s}=3.9$: (a) fundamental mode with $k_{x}=0.13$;
(b) subharmonic mode with $k_{x}=0.015$. Subharmonic modes have widths twice
as large as fundamental ones. Thick contour lines represent the base flow, and
thin lines are from the instability mode. Contours are drawn at 8% of maximal
magnitude in both cases.

The growth rates of the most unstable fundamental modes as a function of
streamwise wavenumbers $k_{x}$ within the range $[0,0.2]$ are shown for four
different values of normalized surface heat flux $\Pi_{s}=3.9,4.1,4.3,4.5$ in
figure 8(a). Again, a higher $\Pi_{s}$ value than at the lower slope angles is
needed to trigger and maintain the primary roll instability for the base
vortices. The growth rates of the strongest subharmonic modes for streamwise
wavenumbers $k_{x}$ within $[0,0.04]$ are shown in figure 8(b). It can be seen
that a higher value of $\Pi_{s}$ strengthens all instability modes at any
given wave number. By comparing the growth rate plots shown in Figs. 8(a) and
8(b) side by side, We observe that the fundamental modes and subharamonic
modes are of a similar strength across all wave numbers. The main distinction
between the two mode types is the fact that the strongest fundamental mode are
3-D and achieve the highest growth at an optimal non-zero wave number $k_{x}$
between 0.02 for $\Pi_{s}=3.9$ and 0.11 for $\Pi_{s}=4.5$, displaying a clear
trend of increasing streamwise wavelength with decreasing $\Pi_{s}$ value. On
the other hand, the subharmonic modes become monotically stronger with
decreasing wave number such that the most dominant mode is purely 2-D at
$k_{x}=0$, same as in the previously studied slope angles.

(a)

(b)

(c)

(d)

Figure 8:  Comparison between fundamental and subharmonic modes of katabatic
speaker-wire vortices at slope angle $\alpha=22^{\circ}$ for different values
of $\Pi_{s}$ as a function of the longitudinal wave number $k_{x}$ : (a)-(b)
Growth rates $\Re(\omega)$; (c)-(d) oscillation frequency $\Im(\omega)$.
Fundamental modes are shown on the left(a,c), and subharmonic modes are shown
on the right(b,d). The linear fit for the dispersion relation of the
subharmonic mode is also shown in (d).

All the fundamental modes with non-zero streamwise wavenumbers studied in this
case are oscillatory. However, the two-dimensional mode with $k_{x}=0$ is
stationary. As shown in figure 8(c), within the very small wavenumber range
$0<k_{x}<0.01$, the frequencies for modes with different $\Pi_{s}$ values
attain a local maximum at $k_{x}\approx 0.005$. In the wavenumber range
$0.01<k_{x}<0.08$, the fundamental mode frequencies increase with growing
wavenumber. However, at higher wavenumbers $k_{x}>0.1$, this trend no longer
holds, and the frequencies even slightly decrease with increasing $k_{x}$. In
contrast, figure 8(d) indicates that the frequency of subharmonic modes is a
monotonically growing function of their wavenumber $k_{x}$. All 2-D
subharmonic modes ($k_{x}=0$) are stationary, and for the low wavenumber
range, the frequencies for all four $\Pi_{s}$ values can be described via a
linear dispersion relation given by $\Im(\omega)=\eta\cdot k_{x}$. In contrast
to the previous case at smaller slope angle, however, the group velocity
$\eta$ decreases with growing $\Pi_{s}$ from $\eta=0.045$ at $\Pi_{s}=3.9$ to
$\eta=0.007$ for $\Pi_{s}=4.5$. At higher wavenumbers beyond $k_{x}=0.03$, the
oscillation frequencies of the subharmonic modes start to grow slower than the
previous linear rate with increasing wavenumbers.

From these computed properties of fundamental and subharmonic secondary
instabilities, we can derive helpful inferences about the dynamics of vortices
at the steep slope angle of $\alpha=22^{\circ}$. Since the strongest
subharmonic mode dominate at lower wave numbers, we can expect long-wave
reconnections or pure mergers between neighboring vortex pairs involving four
vortices. The fundamental modes which are stronger at shorter wavelengths will
bend each individual vortex parallel to its adjacent neighbors. In the full
nonlinear vortex evolution however, it turns out that the the subharmonic
modes appear to have a stronger impact on the vortex dynamics than predicted
from linear stability analysis, as vortex mergers and reconnections seem to be
the dominant feature of such flows with little signature of the wavy bending
signature of the fundamental modes. This is very similar to the dynamics at
higher slope angles as described below and shown in the animation contained in
supplementary movie 3.

####  3.1.4  Very steep slopes $\alpha=30^{\circ}$

The subharmonic and fundamental instabilities of speaker-wire vortices at the
steepest slope in this study with $\alpha=30^{\circ}$ are visualized together
with the base vortices via the contour plot for the streamwise vorticity
$\omega_{x}$ on the transverse plane in Figs. 9(a) and 9(b). When compared to
the previous cases at smaller angles as shown in Fig. 3, Figs. 5(a)-5(b) and
Figs. 7(a)-7(b), it can be seen that the base flow vortices continue to shrink
along the transverse direction and grow in the vertical with increasing slope
angle. As a consequence, the transverse wavelengths of the fundamental and
subharmonic modes are also clearly less than that of the modes at lower slope
angles.

(a)

(b)

Figure 9:  Visualization of streamwise vorticity magnitude for the fundamental
mode on cross-slope $yz$ vertical plane in katabatic flow at
$\alpha=30^{\circ}$ and $\Pi_{s}=6.1$: (a) fundamental mode with $k_{x}=0.06$;
(b) subharmonic mode with $k_{x}=0.03$. Thick contour lines represent the base
flow, and thin lines are from the instability mode. Contours are drawn at 8%
of maximal magnitude in both cases.

The growth rates of the most unstable fundamental modes as a function of
streamwise wavenumbers $k_{x}$ within the range $[0,0.25]$ for four different
values of normalized surface heat flux $\Pi_{s}=5.5,5.7,5.9,6.1$ are shown in
figure 10(a). The growth rates of the strongest subharmonic modes for
streamwise wavenumbers $k_{x}$ within $[0,0.1]$ are shown in figure 10(b).
Similar like in the previous cases, the strongest subharmonic instabilities
have larger streamwise wavelengths than their fundamental counterparts. For
sufficiently small wavenumbers, i.e. $0<k_{x}<0.06$, it can be seen that the
subharmonic modes are clearly stronger than the fundamental ones, with the
distinctively dominating 2-D subharmonic mode at $k_{x}=0$. However, since the
growth rate of subharmonic modes decays rapidly with growing wavenumber
whereas the growth rates of fundamental modes are far less sensitive to
wavenumber variation, at sufficiently large wavenumbers $k_{x}>0.1$, only the
fundamental modes have positive growth rates.

(a)

(b)

(c)

(d)

Figure 10:  Comparison between fundamental and subharmonic modes of katabatic
speaker-wire vortices at slope angle $\alpha=30^{\circ}$ for different values
of $\Pi_{s}$ as a function of the longitudinal wave number $k_{x}$ : (a)-(b)
Growth rates $\Re(\omega)$; (c)-(d) oscillation frequency $\Im(\omega)$.
Fundamental modes are shown on the left(a,c), and subharmonic modes are shown
on the right(b,d). The linear fit for the dispersion relation of the
subharmonic mode is also shown in (d).

All fundamental modes with non-zero longitudinal wavenumbers $k_{x}>0$ are
oscillatory; their frequencies are doubly-parabolic functions of $k_{x}$ with
one local maximum in the low-wavenumber range at $k_{x}\approx 0.01$ and a
global maximum at $k_{x}\approx 0.15$, as shown in figure 10(c). On the other
hand, figure 10(d) shows that the frequencies of subharmonic modes have a less
complicated relationship to their wavenumbers. All 2-D subharmonic
modes($k_{x}=0$) are stationary, and for wavenumber values less than
$k_{x}=0.04$, the normalised frequencies for all four $\Pi_{s}$ values seem to
fit a linear dispersion relation given by $\Im(\omega)=\eta\cdot k_{x}$, where
the group velocity is $\eta=0.054$ for $\Pi_{s}=5.5$ and decreases with
growing $\Pi_{s}$ to $\eta=0.032$ at $\Pi_{s}=6.1$. However, at larger
wavenumbers $k_{x}>0.04$, this linear relation no longer holds true. For
$\Pi_{s}=5.5$, the frequency increases at a smaller rate with growing $k_{x}$
whereas for the cases with larger $\Pi_{s}$ values, the frequency starts to
decrease with growing wavenumber when $k_{x}>0.06$.

From these properties of fundamental and subharmonic vortex instabilities at
the very steep slope, we may infer that 2-D and long-wave subharmonic modes
are the dominant driver of vortex dynamics. They would manifest themselves as
pure mergers or long-wave reconnections between neighboring vortex pairs. Even
though the linear stability results may suggest that the fundamental modes to
be visible at shorter wavelengths, in reality, they are overshadowed by the
effect of the far stronger long-wave subharmonic modes. This can be seen from
animations obtained from direct numerical integration of the full nonlinear
governing equations in which the dominating dynamics are vortex mergers and
long-wave instabilities, as shown in the animation available as supplementary
movie 3.

###  3.2  Vortex dynamics due to fundamental and subharmonic modes

The role of fundamental instabilities (i.e. modes which have the same
transverse spatial period as the base flow) on the dynamics of an array of
longitudinal rolls has been extensively documented in previous studies. As an
example, they have also been identified as instabilities of Rayleigh-Benard
convection rolls which aim at distorting the structure and spacing of the
rolls to bring them closer to the optimal wavelength Clever & Busse (1974);
Pierrehumbert & Widnall (1982). Other well-known representatives for such
modes include the Crow instabilities which are responsible for the bending and
reconnection of a vortex pair suspended in quiescent air Crow (1970).

The effect of a fundamental instability on vortices at a slope angle of
$12^{\circ}$, which is representative for fundamental modes at other angles as
well, is visualised in Fig. 11(a). As can be seen, this 3-D fundamental mode
causes sinusoidal bending and distortion of each speaker-wire vortex. It is
worth noticing that the fundamental modes for speaker-wire vortices are anti-
symmetric, i.e. they can only bend both vortices within a pair along the same
direction, as shown in Fig. 11(a).

Like their fundamental counterparts, 2-D and 3-D subharmonic vortex
instabilities which have twice the transverse wavelength as the base flow also
play a prominent role in shaping the vortex dynamics, such as in co-rotating
Stuart vortex arrays Pierrehumbert & Widnall (1982) or as secondary
instabilities in a shear layer by Corcos & Sherman (1984), where they are
shown to be responsible for the merging of neighboring vortices. As visualised
in Fig. 11(b), the 3-D secondary subharmonic modes of the speaker-wire
vortices in katabatic slope flows bend adjacent speaker-wire vortices in
opposite directions as to facilitate their reconnection in the 3-D case and
merger for the 2-D mode. In a recent investigation, the presence of long-wave
vortex reconnections associated with the low-frequency content in flow spectra
has been observed in the DNS of steep katabatic slope flows with angle
$\alpha>20^{\circ}$ (Henao-Garcia et al., 2023).

We observe from Fig. 11(a) and 11(b) that both the fundamental and subharmonic
modes are anti-symmetric within a single counter-rotating pair, i.e. they bend
the two sister rolls of the same pair along the same direction, thus
preventing a vortex reconnection or merger of neighboring vortices within the
same pair in contrast to what is observed for a single vortex pair with Crow
instability (Crow, 1970). This means that a single vortex pair (i.e. a
speaker-wire vortex) can remain in its basic pair structure even after the
initial onset of instabilities, thus justifying their designation as a
coherent vortex structure. Similar vortex structures which remains stable and
coherent over long wavelengths have been observed in the Langmuir vortices on
the surface of seas and oceans as described in Craik & Leibovich (1976).

As outlined in the previous discussions, the properties of secondary modes for
steady speaker-wire vortices are heavily influenced by the size of the slope
angle $\alpha$. This in turn translates into different vortex dynamics at
different slope angles. For the four different slope angles
$\alpha=3^{\circ},12^{\circ},22^{\circ},30^{\circ}$ studied here, our analysis
has shown that with increasing values of $\alpha$, the subharmonic modes
become more dominant relative to their fundamental counterparts. At shallow
slopes with $\alpha=3^{\circ}$, only fundamental modes are supported, whereas
at steep slopes with $\alpha>30^{\circ}$, the subharmonic modes have the
largest growth rates and are clearly stronger than their fundamental
counterparts in the low wavenumber range. When $\alpha$ takes on an
intermediate value such as between $10^{\circ}$ and $20^{\circ}$, both
fundamental and subharmonic modes have similar growth rates at stratification
parameter $\Pi_{s}$ not far above the linear stability threshold. To track the
further evolution of the vortex dynamics beyond the initial destabilization by
the instability modes, we have carried out a series 3-D numerical simulations
based on the full nonlinear set of equations (1)-((3), initializing the flow
field with an array of at least 4 base vortex pairs upon which a fundamental
or subharmonic mode with $10\%$ of the base flow strength has been added. The
simulation domain and boundary conditions for cross-slope $yz$ plane are the
same as described in section 2.2, while 32 Fourier modes have been used to
resolve the along-slope $x$ direction, which is sufficient for our purpose to
study the higher order vortex dynamics before the onset of full turbulence.
The animated simulation in supplementary movie 1 shows the initial growth of
the fundamental mode in longitudinal rolls at the slope angle
$\alpha=3^{\circ}$, which later leads to novel structures on top of the
original base vortices. Supplementary movie 2 shows the initial vortex
dynamics at $\alpha=12^{\circ}$ due to to the anti-symmetric fundamental mode,
which later makes way for a long-wave subharmonic mode that merges two
adjacent vortex pairs. The animation shown in supplementary movie 3
illustrates how the subharmonic mode at $\alpha=30^{\circ}$ causes vortex
reconnections and mergers between rolls from two adjacent pairs, without any
apparent signature from a short-wave fundamental mode such as sinuosoidal
bending of vortices.

(a)

(b)

Figure 11:  Distortion of longitudinal speaker-wire vortices in katabatic
flows due to instabilities: Effect of (a) Fundamental mode at
$\alpha=12^{\circ},\Pi_{s}=2.9$ and (b) subharmonic mode at
$\alpha=30^{\circ},\Pi_{s}=6.1$. Visualization of streamwise vorticity
contours at 8% of maximal magnitude for base vortices added with the
corresponding instability modes.

###  3.3 Comparison of secondary vortex modes under katabatic and anabatic
conditions

In our earlier work (Xiao & Senocak, 2020), we have established that in
anabatic slope flows, the stationary roll mode is only the dominant primary
instability at slope angle values less than 10 degrees whereas at steeper
slopes $\alpha>10^{\circ}$, the 2-D travelling wave mode would replace it as
the stronger instability. On the other hand, under katabatic slope flow
conditions, the stationary longitudinal rolls can emerge naturally as a result
of the strongest instability mode over a wide range of slope angles up to 70
degrees. Thus, a comparison between the dynamics of longitudinal rolls under
these two conditions is only possible for a narrow range of shallow slopes
less than $10^{\circ}$. In this work, the slope angle $\alpha=3^{\circ}$ is
chosen as the common reference point to contrast anabatic and katabatic slope
flows, which is a realistic slope angle that can be observed in actual
terrains. For further comparison purposes, we also introduce vortices formed
under katabatic conditions at the steep slope angle $\alpha=30^{\circ}$.

To illustrate the most significant differences between katabatic and anabatic
slope flow conditions, we show the optimal transverse wavelengths of the
primary roll instability for the one-dimensional Prandtl flow profile at
different slope angles as a function of $\Pi_{s}$ in Fig. 12(a). The
aforementioned trend of decreasing vortex spacing with increasing slope angle
for katabatic flows is clearly evident, and for both katabatic as well as
anabatic conditions, the optimal vortex width decreases with increasing
$\Pi_{s}$ value.

At the shallow slope with angle $\alpha=3^{\circ}$ where steady vortices can
form under both katabatic and anabatic conditions, we observe from Fig. 12(a)
that for the same value of $\Pi_{s}=1.9$ but with opposite signs of surface
heat flux, the transverse wavelength $\lambda_{y}$ of the most dominant vortex
in the katabatic case is an order of magnitude larger than under anabatic
conditions. This observation is visualized in Fig. 12(b), which displays the
streamwise vorticity contours of both anabatic and katabatic vortices at
$\alpha=3^{\circ}$, and it is obvious that the vortices formed under katabatic
conditions are multiple times wider and taller. Fig. 12(b) also shows that
only at a far steeper slope angle of $\alpha=30^{\circ}$ do the katabatic
vortices assume a similar size as their anabatic counterparts at
$\alpha=3^{\circ}$.

(a)

(b)

Figure 12: Comparison of the most dominant primary instabilities of Prandtl
slope flows under katabatic and anabatic conditions leading to longitudinal
rolls, at different slope angles $\alpha$ and as a function of $\Pi_{s}$: (a)
transverse wavelength of the most dominant mode, with magenta asterisks
indicating cases for which steady vortices have been obtained through
numerical simulations; (b) transverse cross-sections of longitudinal speaker-
wire vortices arising from 1-D Prandtl flow profile for katabatic and anabatic
slope flows at different angles for the three cases marked by asterisks in
(a). Reference length scale is chosen to be the value at $\alpha=3^{\circ}$ as
given in equation (4). Streamwise vorticity contour at 12% of respective
maximal magnitude is visualized. In (b), thick black lines represent katabatic
flow at $\alpha=3^{\circ},\Pi_{s}=1.9$; medium blue lines represent katabatic
flow at $\alpha=30^{\circ},\Pi_{s}=6.1$, and thin red lines represent anabatic
flow with $\alpha=3^{\circ},\Pi_{s}=1.9$;

The growth rates of secondary vortex instabilities under katabatic and
anabatic conditions are displayed in Fig. 13(a). For the same slope angle of
$3^{\circ}$ and stratification parameter $\Pi_{s}=1.9$, there are no
instability modes for the base vortices at their most preferred width under
anabatic conditions, whereas the katabatic configuration is subject to a
fundamental secondary instability. Hence, a higher value of $\Pi_{s}=2.3$ is
chosen for the anabatic slope flow in order to compare its vortex instability
against the katabatic conditions at $\Pi_{s}=1.9$. As described previously,
there exists no subharmonic mode for katabatic vortices at $\alpha=3^{\circ}$,
whereas the anabatic vortices are susceptible to both fundamental and
subharmonic instabilities. Fig. 13(a) shows that even though the katabatic
configuration has a lower $\Pi_{s}$ value, its secondary instability exhibits
an up to four times larger growth rate than the strongest modes for anabatic
vortices. Another key distinction is that the most dominant mode in the
katabatic case is clearly 3-D since its highest growth rate at the non-zero
optimal streamwise wavenumber is clearly larger than the growth rate of the
2-D mode with zero wavenumber. There is no such behavior in the anabatic case
where for both fundamental and subharmonic modes, the largest growth rates are
attained at the lowest wavenumbers, and the growth rates remain relatively
constant for low wavenumbers. It is interesting to notice that for katabatic
vortices at $\alpha=30^{\circ}$ which are of a similar size as the anabatic
vortices at the shallow $\alpha=3^{\circ}$, the growth rates of the
fundamental and subharmonic modes at the low wavenumber range are comparable
to those under anabatic conditions, which indicates that the size and shape of
the base vortices rather than the slope angle directly exerts a major
influence on secondary mode growth for long-wave modes.

The oscillation frequencies of secondary vortex instabilities under katabatic
and anabatic conditions are plotted in figure 13(b). It is evident that the
secondary modes under anabatic conditions have clearly higher frequencies at
all wavenumbers than those of modes for katabatic vortices. A major
distinction is that while the frequencies of both subharmonic and fundamental
modes in the anabatic case fit a linear dispersion relation over the entire
range of streamwise wavenumber $0<k_{x}<0.08$, no such simple dependency
exists for the frequencies of secondary modes under katabatic conditions which
can increase at different rates or even decrease with growing $k_{x}$.

(a)

(b)

Figure 13: Comparison of instability modes for the longitudinal base vortices
under katabatic and anabatic conditions in Prandtl slope flows, at different
slope angles $\alpha$ and as a function of streamwise wave number $k_{x}$: (a)
growth rate; (b) oscillation frequency (inset shows detailed view for
katabatic conditions). The values of $\Pi_{s}$ are the same for the
corresponding vortices as in the previous figure.

##  4 Conclusions

We carried out a bi-global linear stability analysis to gain insights into the
dynamics of longitudinal rolls, a.k.a. speaker-wire vortices, that emerge due
to a primary instability from the one-dimensional katabatic slope flows over a
range of slope angles. The katabatic conditions are the analogue to the
anabatic slope flow conditions, whose effects on vortex dynamics are described
in our prior work (Xiao & Senocak, 2022b).

Our base flow configuration under katabatic conditions is uniquely different
than other well-known counter-rotating vortex pairs (Crow, 1970; Billant,
2010; Hattori et al., 2021; Busse & Clever, 1979; Clever & Busse, 1974). The
katabatic Prandtl slope flow includes an independent background stratification
that is at an angle to the cooled solid slope surface. Another unique feature
of our configuration is the fact that the stationary longitudinal rolls
serving as base flow have three non-zero velocity components even though the
flow field is still 2-D, i.e. it only varies along the vertical and transverse
dimensions. As a result, it is not too surprising that the instability
dynamics of speaker-wire vortices investigated in the current work are
distinct from the hitherto known vortex-pair instabilities.

Similar to our earlier investigation of speaker-wire vortices under anabatic
conditions, we have established that the counter-rotating vortex pair form a
coherent stable unit (i.e. a speaker-wire vortex) that can only lose their
individual rolls to mergers or reconnections in the presence of another
speaker-wire vortex. Our results for secondary instabilities of speaker-wire
vortices have shown that the slope angle $\alpha$ as well as the dimensionless
stratification perturbation parameter $\Pi_{s}$, which can be interpreted as a
normalized surface heat flux or the strength of the surface thermal forcing
relative to the ambient stratification, play a major role in determining which
modes are the most significant in destabilizing vortex rolls formed in
katabatic slope flows. The major distinction between vortex instabilities in
the anabatic and katabatic configurations is the fact that in the latter case,
stationary vortices can emerge from the 1D Prandtl base flow profile for all
slope angles less than around $70^{\circ}$, whereas this is restricted only to
non-steep slopes less than $10^{\circ}$ for anabatic slope flows.

For slope angles in the range of $3^{\circ}-30^{\circ}$ at which the 1D
katabatic Prandtl slope flow profile is naturally susceptible to the
stationary roll mode as primary instability, the most dominant vortex
instability shifts from the fundamental mode towards the subharmonic mode with
longer steamwise wavelengths with increasing slope steepness. For a shallow
slope with $\alpha=3^{\circ}$, the only unstable mode is the fundamental
instability, no matter how strong the imposed surface heat flux is. This
contrasts with vortices in anabatic slope flows at the same angle for which
the subharmonic instability is dominant. For larger slope angles, the
subharmonic mode begins to emerge as well. At $\alpha=12^{\circ}$, subharmonic
mode is still clearly weaker than its fundamental counterpart except at the
smallest wave numbers. For even steeper slopes with $\alpha=22^{\circ}$, both
subharmonic modes and fundamental modes are of comparable strength over a
broad range of wave numbers. At the steepest slope angle $\alpha=30^{\circ}$
investigated in this work, the subharmonic mode is clearly more dominant than
its fundamental counterpart except at the largest wavenumbers. The main
difference between the two mode types is that the fundamental instability is a
3-D oscillatory mode whose growth rate at the optimal nonzero streamwise
wavenumber is many times larger than the growth rate of the 2-D zero wave
number mode. In contrast, the subharmonic mode attains its strongest growth at
zero streamwise wavenumber where its oscillation frequency is also zero, hence
it is a 2-D stationary mode. This mode exhibits its two-dimensional nature by
directly merging two entire rolls from two adjacent but separate speaker-wire
vortices without any bending in the streamwise direction.

The 3-D fundamental mode, with a transverse wavelength equivalent to the
vortex separation of the base speaker-wire vortices, is anti-symmetric and
bends all rolls in all speaker-wire vortices along the same direction, thus
the distances between them remain the same as in the original base
configuration before the onset of instability. Due to the absence of any
symmetric instabilities, the fundamental mode cannot bend the two vortices
within a pair towards each other. However, after sufficiently long time, the
long-wave or two-dimensional subharmonic instability will eventually manifest
itself to move two neighboring speaker-wire vortices from different pairs
towards each other, resulting in the merger between two adjacent rolls which
are not from the same speaker-wire vortex as in the base configuration as
described above.

Our results demonstrate that the only possible vortex merger or reconnection
dynamics under Prandtl’s katbatic slope flow model is triggered by a long-wave
subharmonic mode which requires four vortex rolls in two speaker-wire
vortices. In contrast, one single speaker-wire vortex is able to maintain its
two-roll structure even after the onset of the fundamental instabilities. The
dependence of vortex dynamics on the slope angle is an extension of our
earlier study for vortices in anabatic slope flows which focused on the effect
of vortex width and separation (Xiao & Senocak, 2022b). Hence, the vortex
dynamics under katabatic conditions investigated in this work are expected to
contribute toward a better understanding of turbulent transition in stably
stratified boundary layers on non-flat surfaces.

Supplementary movies. 3 Supplementary movies have been attached.

Acknowledgments: This material is based upon work supported by the National
Science Foundation under Grant No. (1936445). Research was sponsored in part
by the University of Pittsburgh, Center for Research Computing through the
computing resources provided.

Declaration of Interests: The authors report no conflict of interest.

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