# Edge-wave phase-shifts versus normal-mode phase-tilts
in an Eady problem with a sloping boundary

J. Mak<EMAIL_ADDRESS>Department of Ocean Science, Hong Kong
University of Science and Technology, Hong Kong  Center for Ocean Research in
Hong Kong and Macau, Hong Kong University of Science and Technology, Hong Kong
National Oceanography Centre, Southampton, UK  N. Harnik  E. Heifetz  Porter
school of the Environment and Earth Sciences, Tel Aviv University, 69978,
Israel  G. Kumar<EMAIL_ADDRESS>Department of Ocean Science, Hong
Kong University of Science and Technology, Hong Kong  Center for Ocean
Research in Hong Kong and Macau, Hong Kong University of Science and
Technology, Hong Kong  Department of Mathematics, Faculty of Science and
Technology (IcfaiTech), ICFAI Foundation for Higher Education, Hyderabad
501203, Telangana, India  E. Q. Y. Ong  Climate Change Research Centre,
Australian Centre for Excellence in Antarctic Science, University of New South
Wales, Sydney, NSW, Australia  Australian Research Council Centre of
Excellence for Climate Extremes, University of New South Wales, Sydney, NSW,
Australia

###### Abstract

One mechanistic interpretation of baroclinic instability is that of mutual
constructive interference of Rossby edge-waves. While the two edge-waves and
their relative phase-shifts are invoked as part of the mechanistic
interpretation, for example in relation to suppression of baroclinic
instability over slopes, the phase-tilts of the related normal modes are often
presented instead. Here we highlight the differences between edge-wave phase-
shifts and normal-mode phase-tilts, in the context of an Eady problem modified
by the presence of a sloping boundary. The resulting problem remains tractable
analytically, and the interacting Rossby edge-waves can be asymmetric, in
contrast to the standard Eady case. We argue and present evidence that the
normal-mode phase-tilt is potentially a misleading quantity to use, and edge-
wave phase-shifts should be the ones that are mechanistically relevant. We
also provide a clarification for the mechanistic rationalization for
baroclinic instability in the presence of slopes (such as suppression of
growth rates) that is valid over all parameter space, in contrast to previous
attempts. We further present evidence that there is a strong correlation
between quantities diagnosed from the GEOMETRIC framework with the edge-wave
phase-shifts, but not the normal-mode phase-tilts. The result is noteworthy in
that the geometric framework makes no explicit reference to the edge-wave
structures in its construction, but the correlation suggests that in problems
where edge-wave structures are not so well-defined or readily available, the
GEOMETRIC framework should still capture mechanistic and dynamical
information. Some implications for parameterization of baroclinic instability
and relevant eddy-mean feedbacks are discussed. For completeness, we also
provide an explicit demonstration that the linear instability problem of the
present modified Eady problem is parity-time symmetric, and speculate on some
suggestive links between parity-time symmetry, shear instability, and the
edge-wave interaction mechanism.

##  I Introduction

Baroclinic dynamics and its turbulence are ubiquitous features in rotating
stratified systems, playing a key role in systems such as the Earth’s ocean
and atmosphere as well as other geophysical/astrophysical systems, for the
associated transport of buoyancy and impacts the overturning circulation
[e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Understanding the mechanisms and
conditions for instability, its transition to turbulence and its eventual
saturation is of interest in understanding and modeling of the evolution in
the relevant rotating stratified systems.

It is not too controversial to say the subject of baroclinic instability is
rather well-understood at least in the hydrodynamic regime, where the linear
instability phase of idealized models have analytical solutions [e.g., 11, 12,
13, 14], and general stability theorems may be derived [e.g., 15, 16, 17, 18].
The nonlinear phase can also be tackled, mostly by numerical means [e.g., 19,
20, 21, 22, 23, 24, 25, 26, 27, 28]. Often of interest in those cases are the
associated statistics such as meridional eddy buoyancy fluxes (baroclinic
instability usually leads to poleward eddy buoyancy flux in order for
reduction of available potential energy [e.g., 1]), which plays a role in the
eddy-mean interaction in the relevant rotating stratified systems, and informs
on the parameterization in numerical general circulation models. A link that
has been of particular interest is that of _quasi-linear control_ , i.e., to
what degree does the linear instability characteristics have an imprint on the
nonlinear dynamics. While one could argue that the processes that are being
parameterized are inherent manifestations of the nonlinear dynamics, and there
is no strong reason that the relevant linear analysis should play any role,
the fact remains that there does appear a relation between the two [e.g., 29,
30, 31, 32, 33]. A piece of work of relevance here is the GEOMETRIC framework
of [34, 35] (see also [36, 37]), which has highlighted a link between the
associated eddy fluxes in terms of geometric quantities associated with eddy
variance ellipses (such as anisotropy factors and angles) and the linear
instability properties. The scalings provided from the GEOMETRIC framework has
found particular skill in the parameterization of eddy buoyancy fluxes in
numerical ocean general circulation models [e.g., 38, 39, 40].

In the present case we are interested in the instability characteristics of
baroclinic instability in the presence of a slope, where ‘slope’ is broadly
interpreted to mean a slope as a physical boundary (e.g. topography in the
atmosphere and/or ocean), or motion in the presence of an impermeable surface
arising from the relevant fluid properties (e.g., adiabatic flow above/below a
sloping isentrope in planetary atmospheres, magnetic field effects in the
solar tachocline above the radiative zone). Analogous investigations of
classical baroclinic instability in the presence of weak slopes in the linear
and nonlinear regime exist, and is of particular relevance in the field of
oceanography. The presence of continental slopes is generally seen to suppress
eddy buoyancy fluxes over the slope regions [e.g., 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58], with consequences for the
material exchange between the shelf and open ocean environment. One possible
contributing factor for the observed suppression over slope regions is that
the linear instability is itself suppressed and/or less efficient over regions
with slopes [e.g., 41, 42, 43, 44, 45, 47, 53, 54]. In relation to the
GEOMETRIC parameterization, the work of [56] has found, by diagnoses, that a
tuning parameter $\alpha$ normally interpreted as a baroclinic eddy efficiency
for the feedback onto the mean state [e.g., 38] is suppressed over the slope
regions. A slope suppressed $\alpha$ has been found to lead to improvements in
idealized prognostic calculations [57], providing additional evidence that
there should be reduced eddy feedback over slopes. The links between the
observed suppression of this $\alpha$ parameter and possible links with linear
stability analysis are to be clarified, and is one of the goals of the present
work.

Why exactly are baroclinic instabilities suppressed over slopes? For this, we
note first that, in the absence of slopes, a kinematic/mechanistic
interpretation of baroclinic instability is normally given in terms of a pair
of counter-propagating Rossby edge-waves [e.g., 59, 60, 61] (although the
concept of instability arising from a constructive interference of edge-waves
appears to hold for general shear instabilities [e.g., 62, 63, 64, 65, 66]).
If the edge-waves are the building blocks of the instability, then knowing how
the edge-waves interact and form the instability should help with
parameterization efforts. The mechanism is illustrated in Fig. 1$a$ for the
classical Eady problem [12], and proceeds as follows:

  1. 1.

Rossby waves are supported on potential vorticity (PV) gradients, and in the
standard Eady set up, PV is only non-zero and localized at the upper and lower
boundaries, associated with buoyancy anomalies [e.g. 1], hence Rossby edge-
waves;

  2. 2.

the Rossby edge-waves carry PV anomalies, and the self-induced PV anomalies
have associated with it a velocity (via PV inversion, cf. [60]), such that the
edge-waves are _counter_ -propagating against the background mean flow;

  3. 3.

the waves interact with each other, and depending on the phase-shift of the
edge-waves, can lead to mutual amplification of the wave displacement (the
theoretical optimum being quarter of a wavelength or $\pi/2$ out of phase), as
well as some hindering/helping of the other wave’s propagation;

  4. 4.

the counter-propagation against the mean flow and the mutual interaction
between the waves can lead to a phase-locked configuration, and if the phase
shift is also conducive for displacement amplification, then we have (modal)
instability.

Figure 1: Physical schematic for the current modified Eady problem. ($a$)
Cross-section of the set up, showing the basic state linear flow, with Rossby
edge-waves drawn on at the vertical boundaries in an unstable configuration
for a case where the interaction is symmetric (cf., a standard Eady problem,
$\delta=0$ here); see text for a description of the counter-propagating Rossby
waves mechanism. ($b$) Head on view, showing the basic state linear flow going
in and coming out of the page, and the associated buoyancy profile via thermal
wind shear relation. The $\delta<0$ case (topography and basic state buoyancy
profile at opposite orientations) is illustrated, where the contribution from
the slope _reinforces_ the bottom PV gradient by allowing extra stretching of
a fluid column.

How is the counter-propagating Rossby wave mechanism modified by the presence
of slopes? In Fig. 1$b$ we show a case where the slope and the isopycnal
configuration are in opposite orientations (the $\delta<0$ regime later),
adapted from the work of [53]. In this set up, the vortex tube when moved in
the meridional direction $y$ is allowed to stretch more in the vertical
direction $z$, and so the vortex tube strengthens more than it would relative
to the flat case. Then it is perhaps clear what is going to happen is this
setting: the bottom edge-wave is modified because the slope (if small)
reinforces the value of the PV value at the bottom boundary, increasing the
bottom PV gradient, and thus changing the characteristic of the bottom edge-
wave. The work of [43] alludes to this but stops short of elaborating on the
details. The work of [53] goes slightly further by arguing that, in the
setting as in Fig. 1$b$, the bottom Rossby edge-wave would propagate _faster_
(because the associated PV gradient is enhanced), the associated doppler
shifted velocity $\overline{U}_{\rm bot}-c>0$ must be smaller in value (here
$\overline{U}$ denotes the basic state velocity, and suffix ‘bot’ and ‘top’
have the expected meanings). In order to maintain phase-locking, we need
$\overline{U}_{\rm top}-c$ to match the $\overline{U}_{\rm bot}-c>0$, and that
can only happen if $c$ _decreases_ in value, which for Rossby waves means
selecting a higher wavenumber, i.e. shorter waves, providing an explanation
for the observed instability bandwidth (see Fig 2$a$ in the $\delta<0$
regime).

We argue here the explanation provided by [53] is incomplete. A first issue is
that the argument solely based on phase-speed matching does not extend to the
converse case ($\delta>0$): the instability does not monotonically go to the
longest wavelength as the slope is increased in the other orientation. Second
is that the argument as is does not provide an explanation for the growth rate
behavior over parameter space (at least not explicitly). One of our aims of
this work is to provide a more complete argument and support it with further
analysis: essentially, _the mutual interaction matters_. We further clarify a
point regarding the usage of the term ‘phase-shift’. The work of [53] (and
also in standard textbooks such as [1]) invoke ‘phase-shifts’ when talking
about the instability mechanism associated with Rossby edge-waves in terms of
PV signatures, but demonstrate it with a phase-_tilt_ of the _streamfunction_
eigenfunction, where the phase-tilt coincides with that of the anticipated
optimum of $\pi/2$ (quarter of a wavelength; (see for example Fig. 1$d$ of
[53], and essentially Fig. 4$c$ in the present work). We provide arguments
that this can be misleading: it is the Rossby edge-wave basis in the PV
signature and their phase-shifts that are relevant for the mechanistic
interpretation, and those phase-shifts generically do not coincide with the
phase-_tilts_ diagnosed from the streamfunction eigenfunction. We show here
that while the diagnosed phase-tilt in the streamfunction _happens_ to be the
theoretical optimum $\pi/2$ for the most unstable mode, the associated PV
edge-wave phase-_shifts_ are essentially _never_ at $\pi/2$ over the whole
parameter space, the latter because, again, the _mutual interaction matters_.

While we show that the quantities diagnosed from the streamfunction
eigenfunctions directly bear little resemblance to the quantities associated
with the PV edge-wave basis, we report here that applying the GEOMETRIC
framework to the modified Eady problem and performing analysis in terms of
appropriate combinations of the normal-mode eigenfunctions, the resulting
quantities, in particular the vertical eddy tilt, _do_ correspond well to the
quantities diagnosed from the PV edge-wave basis, even though the GEOMETRIC
framework makes no explicit reference to the edge-waves. Within the GEOMETRIC
framework and in the context of linear theory, the suppression of the
instability and eddy efficiency as invoked in parameterization of baroclinic
eddies is attributed principally to changes in the _buoyancy anisotropy_ of
the eddies. The link between GEOMETRIC and PV edge-waves may be of further
interest: in cases where the edge-wave basis are not well-defined, it may in
fact be useful to perform an analysis in the GEOMETRIC framework instead.

In Sec. II we formulate and provide an overview of the instability
characteristics of the modified Eady problem, and make precise our arguments
on why we think the existing descriptions of the instability mechanism in the
modified Eady problem is incomplete. We additionally highlight that the
modified Eady problem here is in fact parity-time ($\mathcal{PT}$) symmetric
[e.g., 67, 68, 69, 70], which has consequences for the solution spectrum, but
defer the expanded details and related discussion to Appendix A. In Sec. III
we provide a rephrasing of the modified Eady problem in terms of an edge-wave
basis [cf., 71, 61], where the presence of the slope (related to a topographic
PV signature) manifests as an independent adjustable parameter that controls
the degree of asymmetry between upper and lower edge-waves (the standard Eady
problem being the case with symmetric interaction). We provide an internally
consistent physical rationalization of the instability mechanism, with
explicit references to the phase-shift and asymmetry in the wave amplitude
ratios that is valid over all parameter space. We further clarify the issue of
‘phase-shifts’, by comparing results from the edge-wave basis and a standard
analysis of the streamfunction eigenfunction. In Sec. IV we demonstrate links
between the quantities of interest from the GEOMETRIC framework and the edge-
wave analysis, highlighting a link between the eddy angles with edge-wave
phase-shifts [cf., 72], as well as providing an analysis for what contributes
to the suppression to the eddy efficiency parameter $\alpha$ that is
prescribed in parameterizations. We summarize our results in Sec. V, and
discuss some implications of our results for parameterization of baroclinic
processes.

##  II Overview of the modified Eady problem

The physical set up is as illustrated in Fig. 1, for the Northern Hemisphere
with Coriolis parameter $f_{0}>0$. We start with the quasi-geostrophic (QG)
equations [e.g. 1] formulated on a $f$-plane with potential vorticity (PV)
advection in the interior, and QG buoyancy advection on the vertical surfaces,
i.e.,

$\frac{\mathrm{D}q}{\mathrm{D}t}=0,\quad
z\in(H,-H_{b}),\qquad\qquad\frac{\mathrm{D}b}{\mathrm{D}t}=0,\quad
z=H,-H_{b},$ (1)

where the domain of interest is between $z=H$ and $z=-H_{b}(y)$, $H_{b}(y)$
represents the bottom slope, $\mathrm{D}/\mathrm{D}t=\partial/\partial
t+\boldsymbol{u}\cdot\nabla$ is the material derivative, $\boldsymbol{u}$ is
the geostrophic velocity with associated streamfunction $\psi$, $\nabla$ the
horizontal gradient operator, and $(x,y,z)$ denotes the zonal (East-West),
meridional (North-South) and vertical co-ordinate. The PV $q$ and QG buoyancy
$b$ are defined as

$q=\nabla^{2}\psi+\frac{\partial}{\partial z}\frac{f_{0}}{N_{0}^{2}}b,\qquad
b=f_{0}\frac{\partial\psi}{\partial z},$

where $N_{0}^{2}=\partial\overline{B}/\partial z$ is the buoyancy frequency
associated with the prescribed background stratification, where $\overline{B}$
is the basic state buoyancy profile to be prescribed with the basic state
velocity $\overline{U}$. Contributions from the small slope will arise through
the buoyancy equation in the advective term via $w=\boldsymbol{u}\cdot\nabla
H_{b}$, arising from the no-normal flow condition on the bottom boundary.

We make an assumption that $\partial H_{b}(y)/\partial y$ is small (more
precisely, that $(N_{0}/f_{0})\partial H_{b}/\partial y$ is of order Rossby
number [e.g., 43, 53]), and that $\partial H_{b}(y)/\partial y$ can be
approximated by a small constant contribution only in the boundary condition
at $z=-H$. Then, linearizing against the basic state
$\boldsymbol{u}=\overline{U}\boldsymbol{e}_{x}=\Lambda z$ (and so
$\overline{B}=-f_{0}\Lambda y$ by thermal wind shear relation), the governing
linear equations are

$\displaystyle\left(\frac{\partial}{\partial t}+\Lambda
z\frac{\partial}{\partial
x}\right)\left(\nabla^{2}\psi+\frac{f_{0}^{2}}{N_{0}^{2}}\frac{\partial^{2}\psi}{\partial
z^{2}}\right)=0,\qquad$ $\displaystyle z\in(-H,H),$ (2a)
$\displaystyle\left(\frac{\partial}{\partial t}+\Lambda
H\frac{\partial}{\partial x}\right)\frac{\partial\psi}{\partial
z}-\Lambda\frac{\partial\psi}{\partial x}=0,\qquad$ $\displaystyle z=H,$ (2b)
$\displaystyle\left(\frac{\partial}{\partial t}-\Lambda
H\frac{\partial}{\partial x}\right)\frac{\partial\psi}{\partial
z}-\left(\Lambda-\frac{N_{0}^{2}}{f_{0}}\frac{\partial H_{b}}{\partial
y}\right)\frac{\partial\psi}{\partial x}=0,\qquad$ $\displaystyle z=-H.$ (2c)

Non-dimensionalizing by the horizontal length-scale $L$, vertical length-scale
$H$, and time-scale $T=L/U=L/(\Lambda H)$, we have

$\displaystyle\left(\frac{\partial}{\partial t}+z\frac{\partial}{\partial
x}\right)\left(\nabla^{2}\psi+F^{2}\frac{\partial^{2}\psi}{\partial
z^{2}}\right)=0,\qquad$ $\displaystyle z\in(-1,1),$ (3a)
$\displaystyle\left(\frac{\partial}{\partial t}+\frac{\partial}{\partial
x}\right)\frac{\partial\psi}{\partial z}-\frac{\partial\psi}{\partial
x}=0,\qquad$ $\displaystyle z=1,$ (3b)
$\displaystyle\left(\frac{\partial}{\partial t}-\frac{\partial}{\partial
x}\right)\frac{\partial\psi}{\partial
z}-(1-\delta)\frac{\partial\psi}{\partial x}=0,\qquad$ $\displaystyle z=-1,$
(3c)

where $F^{2}=(f_{0}L/NH)^{2}$ and is related to the inverse of the Burger
number. A key non-dimensional parameter in the present system is (in terms of
dimensional variables)

$\delta=\frac{\partial H_{b}}{\partial
y}\left(\frac{N_{0}^{2}}{f_{0}\Lambda}\right)=\left.\frac{\partial
H_{b}}{\partial y}\right/\frac{-\partial\overline{B}/\partial
y}{\partial\overline{B}/\partial z}=\frac{\partial H_{b}/\partial y}{s},$ (4)

i.e. the parameter $\delta$ relates to the orientation of intersection between
the background isopycnal slopes $s$ with the bottom slope (a $\delta<0$ case
is illustrated in Fig. 1$b$); this parameter is related to the $\alpha_{T}$
parameter in [47]. With that, the case $\delta<0$ and $\delta>0$ are sometimes
known as retrograde or prograde configurations, although we will not be using
that terminology here. One could relate the $\delta$ parameter here to a
topographic $\beta$ term, but we refer to reader to the work of [43] for that
since we do not invoke that term in this work.

With appropriate horizontal boundary choices and conditions (periodic in zonal
$x$ direction, periodic or appropriate no-normal flow boundary conditions in
meridional $y$ direction), we consider solutions of the form

$\psi(x,y,z,t)=\tilde{\psi}(z)\exp[\mathrm{i}(kx-\omega t)]g(y),$ (5)

where $\tilde{\psi}$ is a vertical structure function in the streamfunction,
$g(y)$ is an appropriate eigenfunction of the Laplacian operator so that
$\partial^{2}g/\partial y^{2}=-l^{2}g$ (e.g., combinations of $\sin ly$ and
$\cos ly$ as appropriate), $\mathrm{i}=\sqrt{-1}$, $(k,l)$ are the zonal and
meridional wavenumbers, $\omega=kc=k(c_{r}+\mathrm{i}c_{i})$ is the angular
frequency, $c$ is the (complex) phase-speed; we have modal instability if
$c_{i}>0$. The modified Eady problem has zero PV signature in the interior, so
the vertical structure function satisfies

$\tilde{\psi}(z)=a\cosh\mu z+b\sinh\mu z,\qquad\mu^{2}=F^{2}(k^{2}+l^{2}).$
(6)

The constants $a$ and $b$ are fixed by the compatibility condition resulting
from the vertical boundary conditions. Making the shorthand $C=\cosh\mu$ and
$S=\sinh\mu$, after some algebraic manipulation, the dispersion relation is
given by

$0=c^{2}+\frac{\delta}{2\mu}\left(\frac{C}{S}+\frac{S}{C}\right)c+\frac{\delta^{2}}{4\mu^{2}}-\left(\frac{1-\delta/2}{\mu}-\frac{C}{S}\right)\left(\frac{1-\delta/2}{\mu}-\frac{S}{C}\right).$
(7)

Computing for $c$ analytically or numerically, and denoting the solutions of
the plus and minus branch as $c^{\pm}$ for ease of discussion, plots of
$c^{\pm}_{r}$ and growth rate of the instability $kc_{i}^{\pm}$ as a function
of wavenumber $k$ and $\delta$ can be constructed, and a sample of this for
the case $l=0$ is shown in Fig. 2; we will focus mostly on the $l=0$ case in
this work since the gravest meridional mode seems to be associated with the
largest growth rates at fixed $\delta$. Note that for our computations shown
for the remainder of this work that we take $F=1$ for simplicity.
Additionally, compared to more standard non-dimensional formulations [e.g., 1,
43, 47, 53], our values of $k$ and $c$ are smaller and larger by a factor of 2
respectively, but $kc_{i}$ remain the same.

Figure 2: General instability characteristics for the case of $l=0$. ($a$)
Growth rate as a function of the non-dimensional zonal wavenumber $k$ and
$\delta$ parameter, with darker shadings denoting higher values, sample
contours of growth rates, and the green dashed line denoting
$k_{\max}(\delta)$ where the growth rate is maximized. Also shown are the
growth rates $kc^{+}_{i}$ and phase-speed $c_{r}^{\pm}$ for ($b,c$)
$\delta=+0.5$, ($d,e$) the standard Eady problem $\delta=0.0$, and ($f,g$)
$\delta=-0.5$. The shaded regions in panels $b$-$g$ denote the regions where
there the growth rates are non-zero.

We make an observation that Eq. (3) is invariant under the transformation

$\mathcal{P}:(x,y,z)\mapsto(-x,-y,-z),\qquad\mathcal{T}:(t,\psi)\mapsto(-t,-\psi),$
(8)

so the system is parity-time ($\mathcal{PT}$) symmetric, where parity refers
to a flipping or mirroring of space, while time symmetry refers to reversal of
time [67, 68, 69, 70]. Note that we regard $\delta$ as a given and prescribed
parameter of the system, so $\delta$ does not transform under $\mathcal{P}$.
The fact that the equations are $\mathcal{PT}$ symmetric implies the solution
spectrum has certain properties (e.g., Fig. 2$b$-$g$), and has very suggestive
links for shear instability as well as the edge-wave interpretation for shear
instabilities. The details are somewhat of a digression, and we refer the
reader to Appendix A for details.

There are a few features that may be observed from Fig. 2:

  1. 1.

when $\delta=0$ and there is instability, $c^{\pm}_{r}=0$;

  2. 2.

when $\delta\neq 0$ and there is instability, $c^{+}_{r}=c^{-}_{r}$, and if
$\delta>0$, $c^{\pm}_{r}<0$ (and vice-versa for $\delta<0$);

  3. 3.

there is no instability for $\delta\geq 1$, but instability seems to persist
for $\delta<0$;

  4. 4.

relative to the standard $\delta=0$ case, the most unstable wavelength $k_{\rm
max}$ decreases somewhat for $\delta>0$, but persistently increases for
$\delta<0$, with the unstable bandwidth shifting to larger wavenumbers.

A test of the wave interaction interpretation would be whether we are able to
rationalize the changes in the instability characteristics via changes to the
properties of these edge-waves that is valid over the whole parameter space.

The first point simply arises from our choice of problem formulation, where
our domain goes from $z=\pm 1$ and our velocity profile is anti-symmetric
about $z=0$ (cf. $c^{\pm}_{r}=(\overline{U}_{\rm top}+\overline{U}_{\rm
bot})/2=0.5$ with the more standard formulations and choice of non-
dimensionalization in other works [e.g., 1, 43, 47, 53]). The second point
relates to the first point, and together with the third point can be
rationalized as follows. Noting first that, with counter-propagation of Rossby
edge-waves against the mean flow, the top wave intrinsically propagates to the
left, while the bottom wave would propagate to the right. Noting that
$\partial H_{b}/\partial y\sim\delta$, the presence of the slope is to alter
the background PV gradient on which the the edge-waves are propagating (cf.
Fig. 1$b$, via PV stretching), and in the present setting, $\delta>0$
_counter_ acts while $\delta<0$ _reinforces_ the PV gradient provided by the
background state; mathematically this is through the $(1-\delta)$ term in Eq.
(3). For $\delta>0$, the bottom wave is _weakened_ and propagates _slower_ for
a fixed wavenumber, and one might expect the instability has more of the
characteristic of the top wave propagating _left_ , resulting in an unstable
mode with $c^{\pm}_{r}<c^{\pm}_{r}(\delta=0)=0$. On the other hand, for
$\delta<0$, the bottom wave is _strengthened_ and propagates _faster_ for a
fixed wavenumber, the bottom wave dominates and resulting in an unstable mode
with $c^{\pm}_{r}>c^{\pm}_{r}(\delta=0)=0$. These observations can be verified
with standard formulations, where we have instead
$c^{\pm}_{r}\lessgtr(\overline{U}_{\max}+\overline{U}_{\min})/2=c^{\pm}_{r}(\delta=0)=0$.
The third point is also consistent with the mechanistic picture: there is no
instability for $\delta\geq 1$, because counter-propagation is then no longer
possible, but for $\delta<0$ counter-propagation appears to always be
possible, and could in principle persist at increasing wavenumbers, albeit
over a decreasing bandwidth. Some of these points were already noted by the
work of [47, 53].

The fourth point however is not covered by the explanation given in [47, 53]
as such (although it was neither work’s main focus). Those works argue that
for phase-locking the edge-wave phase-speeds must match, and therefore look
for conditions where the phase-speeds match. While that argument functions
well for the $\delta<0$ regime, it (1) fails for the opposite case of
$\delta>0$, where the most unstable wavenumber do not uniformly go to longer
waves or smaller wavenumbers (green dashed line of Fig. 2$a$), and (2) does
not explain the changes in the strength of instability over parameter space.
The reason we will argue for is simply that _mutual edge-wave interaction
matters_ , is part of the solution and central to the counter-propagating
Rossby wave mechanism, and cannot simply be ignored, as is done when
considering simply phase-speeds can match. The associated mutual interaction
leads to extra helping/hindering of the wave propagation that affects phase-
locking, and strength of interaction affects the growth rates.

##  III Clarifying the instability mechanism in terms of Rossby edge-waves

###  III.1 edge-wave formulation in phase-amplitude variables

To quantify the impact of slopes and its modification to the background PV on
the wave interaction mechanism, here we consider expressing the problems
explicitly in terms of Rossby or PV edge-waves and its interaction, rather
than in the streamfunction eigenfunction. The streamfunction eigenfunction
$\tilde{\psi}(z)$ as it stands is in general a tilted structure in space that
could be regarded as a superposition of edge-wave structures, and the problem
here is in the definition of an appropriate basis of PV edge-wave functions.
While there is a general approach for constructing the PV edge-waves in terms
of wave activity variables such as pseudomomentum and pseudoenergy [e.g., 73,
18, 61, 74, 66], we do not need that amount of complexity here since the edge-
wave locations are well defined for the present problem. For elucidation
purposes we will derive the structure and governing equations explicitly.

Consider expressing the streamfunction eigenfunction $\tilde{\psi}(z)$ in
terms of a linear superposition of untilted structures focused at the top and
bottom boundary (subscript $T$ and $B$ respectively) such that

$\tilde{\psi}=\tilde{\psi}_{T}+\tilde{\psi}_{B},\qquad\tilde{q}=\tilde{q}_{T}+\tilde{q}_{B},$
(9)

where tilde denotes functions that are $z$ only. Assuming modal solutions as
in (5), $\tilde{q}$ and $\tilde{\psi}$ are related via

$\tilde{q}=-\mu^{2}\tilde{\psi}+\frac{\partial^{2}\tilde{\psi}}{\partial
z^{2}},$ (10)

subject to the boundary conditions that

$\left.\frac{\partial\tilde{\psi}_{T}}{\partial
z}\right|_{z=-1}=0,\qquad\left.\frac{\partial\tilde{\psi}_{B}}{\partial
z}\right|_{z=+1}=0.$ (11)

Denoting $\hat{\delta}$ to be the Dirac $\delta$-distribution, if we take
(abusing mathematical rigor somewhat)

$\tilde{q}_{B}=\hat{q}_{B}(t)\hat{\delta}(z+1),\qquad\tilde{q}_{T}=\hat{q}_{T}(t)\hat{\delta}(z-1),$
(12)

i.e. PV anomaly of an edge-wave is non-zero only at the associated locations,
then either by manually constructing a solution [cf. 71], or by noting that we
are in effect looking for the Green’s function associated with the one-
dimensional Helmholtz operator (in Eq. 10) subject to homogeneous Neumann
conditions (in Eq. 11), for which solutions are documented (e.g., online
Green’s function libraries, with appropriate changes of variable), or
otherwise, the relevant solutions are

$\tilde{\psi}_{B}=-\hat{q}_{B}\frac{\cosh\mu(1-z)}{\mu\sinh
2\mu},\qquad\tilde{\psi}_{T}=-\hat{q}_{T}\frac{\cosh\mu(1+z)}{\mu\sinh 2\mu}.$
(13)

Note that, with (13),

$\tilde{b}=\frac{\partial\tilde{\psi}}{\partial
z}=\begin{cases}-\hat{q}_{T},&z=+1,\\\ +\hat{q}_{B},&z=-1,\end{cases}$ (14)

demonstrating the explicit relation between buoyancy and PV anomalies, and
that the top edge-wave induces no PV anomaly at the location of the other
edge-wave.

Taking $\hat{q}_{T}=T\mathrm{e}^{\mathrm{i}\epsilon_{T}}$ and
$\hat{q}_{B}=B\mathrm{e}^{\mathrm{i}\epsilon_{B}}$, substituting (13) into the
governing equations (3$b,c$) and considering the real and imaginary parts lead
to

$\displaystyle\frac{1}{T}\frac{\partial T}{\partial t}$
$\displaystyle=+\frac{k}{\mu\sinh 2\mu}\frac{B}{T}\sin\Delta\epsilon,$ (15a)
$\displaystyle\frac{1}{B}\frac{\partial B}{\partial t}$
$\displaystyle=-\frac{k(1-\delta)}{\mu\sinh
2\mu}\frac{T}{B}\sin\Delta\epsilon,$ (15b)
$\displaystyle-\frac{1}{k}\frac{\partial\epsilon_{T}}{\partial t}$
$\displaystyle=+\left[1-\frac{1}{\mu\sinh 2\mu}\left(\cosh
2\mu+\frac{B}{T}\cos\Delta\epsilon\right)\right],$ (15c)
$\displaystyle-\frac{1}{k}\frac{\partial\epsilon_{B}}{\partial t}$
$\displaystyle=-\left[1-\frac{(1-\delta)}{\mu\sinh 2\mu}\left(\cosh
2\mu-\frac{T}{B}\cos\Delta\epsilon\right)\right],$ (15d)

where we define $\Delta\epsilon=\epsilon_{T}-\epsilon_{B}$ as the phase-shift
of the edge-wave in terms of PV signature; $\Delta\epsilon>0$ means the top
wave has a PV signature that is _lagging behind_ the bottom wave PV signature
(cf. Fig. 1$a$). The set of equations are cast in a form that is more similar
to Eq. (14) of [62] for the dimensional formulation of the Rayleigh shear
profile problem, but is equivalent to Eq. (7) of [71], who consider the phase-
shift in terms of the buoyancy variable instead. Note that since
$\tilde{\psi}\sim-\tilde{q}$, the phase-shift applies also to the
streamfunction; contrast this to $\tilde{v}=\mathrm{i}k\tilde{\psi}$, which
would be shifted by $\pi/2$, and $\tilde{b}$ which would be shifted instead by
$\pi$. Here, $k/(\mu\sinh 2\mu)$ plays the role of the vertical interaction
(cf. $\mathrm{e}^{-k}$ in [62] for the Rayleigh profile in the barotropic
setting). Taking the amplitudes $T$ and $B$ as positive without loss of
generality, we note that we need $\Delta\epsilon\in(0,\pi)$ for growth of
edge-waves, which is consistent with what we know about baroclinic
instability: an unstable mode has the PV, streamfunction and meridional flow
patterns leaning _against_ the shear (top signal lagging bottom signal; see
Fig. 4 for example), while the buoyancy pattern leans _into_ the shear
corresponding to a shift by $\pi$ [e.g., 1].

While there are four independent variables, the equations depend only on the
amplitude ratios and the phase difference, and could be considered a two-
dimensional dynamical system. Following the notation of [62], we define the
amplitude ratio as $\tan\gamma=T/B$. Noting then various trigonometric
identities such as

$\frac{B^{2}-T^{2}}{B^{2}+T^{2}}=\cos
2\gamma,\qquad\frac{B^{2}+T^{2}}{2BT}=\frac{1}{2\sin 2\gamma},$

(15) takes the form

$\displaystyle\frac{\partial\gamma}{\partial t}$
$\displaystyle=\frac{k}{\mu\sinh 2\mu}\sin\Delta\epsilon(\cos
2\gamma+\delta\sin^{2}\gamma),$ (16a)
$\displaystyle\frac{\partial\Delta\epsilon}{\partial t}$
$\displaystyle=\frac{2k}{\mu\sinh
2\mu}\left[\left(1-\frac{\delta}{2}\right)\cosh 2\mu-\mu\sinh
2\mu+\left(\frac{1}{\sin
2\gamma}-\frac{\delta}{2}\tan\gamma\right)\cos\Delta\epsilon\right].$ (16b)

The set of equations should be compared with Eq. (15) of [62], noting the
difference in the interaction function ($k/(\mu\sinh 2\mu$) vs.
$\mathrm{e}^{-k}$), arising from the differing physics between the systems
being considered, encapsulated in the different Green’s functions of the
associated system.

A dynamical system could be described in terms of phase portraits [e.g., 75],
and phase portraits associated with the most unstable wavenumber $k$ (with
$l=0$) for sample choices of $\delta$ are shown in Fig. 3. Note that the
stable and unstable equilibrium points of the dynamical system (repellers and
attractors) are associated with the unstable and stable normal-modes
respectively in the unstable bandwidth [cf. 62, 61]. As the stability
boundaries are passed there is a bifurcation, and the equilibrium points
become centers located at $\gamma=0,\pi/2$, and $\Delta\epsilon=\pm\pi$ (not
shown), associated with neutral and freely propagating edge-waves with no
change in amplitude or phase. The non-equilibrium points have been argued to
correspond to non-modal growth [e.g., 71, 62], and the phase portraits
indicate the regime transition in terms of edge-waves as the non-modal
instabilities develop, but we leave this for the interested reader to pursue.

Figure 3: Phase portrait and the fixed points of the dynamical system (stable
and unstable mode marked on as a circle and cross respectively), for three
choices of $\delta$, at wavenumber ($k=k_{\max}(\delta)$, $l=0$).

Focusing on the unstable modes (the attractors), we note that the associated
PV phase-shifts $\Delta\epsilon$ is _not_ $\pi/2$, even for the $\delta=0$
case, which seems to contradict what is generally documented about the Eady
problem having a phase-shift of $\pi/2$ [e.g., 1]. There is in fact no
discrepancy: taking the $\delta=0$ case as an example (i.e., standard Eady
problem), we compute the values of $\gamma$ and $\Delta\epsilon$ associated
with the stable equilibrium point and construct the edge-wave couplets as well
as their sum, and these are shown in Fig. 4 (recalling
$\hat{q}=B\mathrm{e}^{\mathrm{i}\epsilon_{B}}$, we take the reference to be
$B=1$ and $\epsilon_{B}=0$) . While $\Delta\epsilon\neq\pi/2$, their
combination does lead to a phase-_tilt_ in the _streamfunction eigenfunction_
$\Delta\epsilon_{\rm eigen}=\pi/2$, and the reconstructed solution can be
shown to coincide exactly with the one obtained from the more standard normal-
mode analysis. A non-optimal shift is realized simply because the mutual
interaction from the interaction function (which is wavenumber dependent) is
also part of the solution, and provides extra hindering of the wave
propagation required for phase-locking [e.g., 62, 61, 72, 65]. Considerations
based simply on phase-shift (or phase-tilt) and/or phase-speed matching is
incomplete, because it is ignoring the interaction component.

Figure 4: Edge-waves components and the combination demonstrated for the most
unstable mode of the standard Eady problem ($\delta=0$). ($a,b$) Associated
edge-wave structure $\psi_{T,B}$ (referenced by $B=1$ and $\epsilon_{B}=0$)
given in (13) of the attracting fixed point in Fig. 3$b$, with
$\Delta\epsilon\neq\pi/2$. ($c$) The reconstructed normal-mode
$\psi=\psi_{T}+\psi_{B}$ that has a phase-tilt $\Delta\epsilon_{\rm
eigen}=\pi/2$.

The statement in the above paragraph holds true for different values of $k$
and $\delta$ (not shown). For synchronized growth of edge-waves associated
with stable equilibrium points of (16), we have $\Delta\epsilon\in(0,\pi)$ and
$\gamma\in(0,\pi/2)$, so that the growth rate can be inferred from (15), given
by

$\sigma=\left|\frac{k}{\mu\sinh
2\mu}\frac{1}{\tan\gamma}\sin\Delta\epsilon\right|=\left|\frac{k(1-\delta)}{\mu\sinh
2\mu}\tan\gamma\sin\Delta\epsilon\right|,$ (17)

which numerically coincides with that of Fig. 2$a$ (not shown). The phase-
speeds of the normal-modes $c_{r}$ can also in principle be reconstructed in
principle from the edge-wave basis via consideration of the self- and induced-
propagation by the edge-waves [cf., 61, 66]. The edge-wave formulation here
encompasses the standard formulation of the modified Eady problem, which is
not surprising given it is really a reformulation of the same problem, but
expressing it in a basis that allows for a mechanistic interpretation to be
drawn from.

We show in Fig. 5 the amplitude ratios $\gamma$ and edge-wave phase-shift
$\Delta\epsilon$ of equilibrium points of (16) associated with unstable modes.
Starting first with the amplitude ratio $\gamma$, it is clear that $\gamma$
only depend on $\delta$. The analytic expression for $\gamma$ can be obtained
by noting that, with synchronized growth and growth rate given by (17), we
must have (with appropriate normalization and/or shifts in the phase)

$\left|\frac{1}{\tan\gamma}\right|=\left|\frac{B}{T}\right|=\sqrt{1-\delta},$
(18)

which is $k$ independent. Notice that: (1) $|B/T|$ is ill-defined for
$\delta>1$, corresponding to the case where there is no instability (since
there is no counter-propagation possible for the bottom wave as the bottom
background PV gradient has switched signs); (2) $|B|=0$ for $\delta=1$ for
physically sound solutions (coinciding with vanishing PV gradient at the
bottom); (3) $|T|>|B|$ for $\delta\in(0,1)$, i.e. weaker bottom wave; (4)
$|T|=|B|$ for $\delta=0$, and there is no asymmetry in the standard Eady case;
(5) $|B|>|T|$ for $\delta<0$, and there is _always_ instability possible for
$\delta<0$. The observations are consistent with our physical expectations
highlighted in the previous sections.

Figure 5: ($a$) Normalized amplitude ratio $\gamma$ in multiples of $\pi$ as a
function of $k$ and $\delta$, and ($b$) the same quantity along the line of
$k_{\max}(\delta)$ (red line, with $\gamma_{\rm eigen}$ as faint green line);
note $0.25$ here denotes equal amplitude at top and bottom. ($c$) The
normalized edge-wave phase-shift $\Delta\epsilon$ in multiples of $\pi$ as a
function of $k$ and $\delta$, and ($d$) the same quantity along the line of
$k_{\max}(\delta)$ (red line); note $+0.5$ here denotes that the top edge-wave
_lags behind_ the bottom edge-wave by a quarter wavelength. The line of
$k_{\max}(\delta)$ has been marked on panel ($a,c$) as the green dashed line.

The behavior of $\Delta\epsilon$ in Fig. 5$c$ is in line with kinematic
arguments from edge-waves that help or hinder each other’s propagation, with
implications for phase-locking. We generally need $\Delta\epsilon\in(0,\pi)$
for constructive interference. At fixed $\delta$, long waves propagate faster
(since these are Rossby edge-waves), and the edge-waves need to _hinder_ each
other to maintain phase-locking, which for instability requires
$\Delta\epsilon\in(\pi/2,\pi)$. The converse holds for shorter waves,
requiring $\Delta\epsilon\in(0,\pi/2)$ [e.g., 61].

###  III.2 Physical rationalization of the interacting edge-wave process

Our interpretation for the role of topographic PV on the Counter-propagation
Rossby Wave mechanism is then as follows. Starting with the $\delta<0$ case,
the bottom wave is strengthened and propagate faster, but also leads to a
stronger induced velocity at the top wave. The strong interaction leads to a
stronger _hindering_ effect for fixed wavenumber, which will in general lead
to a sub-optimal phase-shift configuration. One way to drive the configuration
towards a more optimal configuration would be to go towards _shorter_ waves,
which offsets the increased interaction introduced by a stronger bottom wave
from the fact that $\delta<0$ (since the interaction function goes like
$k/(\mu\sinh 2\mu)$ for the present Eady system). This also then explains the
reduction in the growth rate as we shift towards shorter waves (Fig. 2$a$):
while the efficiency could be changed via the phase-shift, the interaction
strength is _decreased_. It would seem that, in the $\delta<0$ setting, it is
_always_ possible to compensate for the increase in interaction from
$\delta\to-\infty$ by reducing the interaction function via increases in $k$,
albeit over an increasingly narrow bandwidth of wavenumbers. The phase-speeds
are positive (Fig. 2$g$) because the bottom edge-wave propagating to the right
(or eastwards) dominates over the top edge-wave.

For the $1>\delta>0$ regime, the bottom wave is weakened (from (18)) and leads
to a weaker hindering effect in general. What this means is that the top wave
is now propagating too fast, and this effect would have to be offset by
increasing the interaction function via decreasing in the wavenumber, i.e.,
going to longer wavelengths. However, unlike the $\delta<0$ case, reduction in
$k$ leads an increase in edge-wave propagation since we are dealing with
Rossby edge-waves, and beyond a certain point it is simply not possible for
the bottom wave’s induced velocity and the background flow to hold the top
wave into a phase-locked position, and instability is no longer possible. As
$\delta\nearrow 1$, the PV gradient vanishes, counter-propagation is no longer
possible, and no phase-locking can be achieved. Note that $\delta\geq 1$
coincides with the non-satisfaction of the Charney–Stern condition that it is
necessary for the background PV gradient to change sign in the domain in order
for instability, which had previously been interpreted as a condition required
for counter-propagation [e.g., 61, 74, 76]. The phase-speeds are negative
(Fig. 2$c$) because the top edge-wave propagating to the left (or wastwards)
dominates over the bottom edge-wave.

###  III.3 Analysis in terms of the instability normal-mode

Here we provide an analogous analysis to demonstrate how different the results
are if the instability streamfunction eigenfunction from (5) is utilized
instead. Focusing on unstable modes, given a value of $c$, we can obtain the
coefficients $a$ and $b$ for the vertical structure function $\tilde{\psi}(z)$
from (6). Given $\tilde{\psi}(z)=\tilde{\psi}_{r}+\mathrm{i}\tilde{\psi}_{i}$,
we can compute for a (normalized) amplitude and phase of the eigenfunction via
[e.g., 1, 53]

$|\tilde{\psi}(z)|^{2}=\tilde{\psi}_{r}^{2}(z)+\tilde{\psi}_{i}^{2}(z),\qquad\epsilon(z)=\arctan\frac{\tilde{\psi}_{i}(z)}{\tilde{\psi}_{r}(z)}.$
(19)

Analogous to our edge-wave analysis in Sec. III, we introduce the quantities

$\tan\gamma_{\rm
eigen}=\frac{|\tilde{\psi}(z=1)|}{|\tilde{\psi}(z=-1)|},\qquad\Delta\epsilon_{\rm
eigen}=\epsilon(z=1)-\epsilon(z=-1)$ (20)

as a measure of the amplitude ratio and phase-_tilt_ (rather than phase-shift)
between the streamfunction eigenfunction at the top and bottom of the domain
respectively. Fig. 6 shows the amplitude ratio $\gamma_{\rm eigen}$ and phase-
tilt $\Delta\epsilon_{\rm eigen}$ as measured through the eigenfunction
$\tilde{\psi}(z)$.

Figure 6: Analogue of Fig. 5 but for quantities diagnosed from the normal-mode
directly. ($a$) Normalized amplitude ratio $\gamma_{\rm eigen}$ in units of
$\pi$ as a function of $k$ and $\delta$, and ($b$) the same quantity along the
line of $k_{\max}(\delta)$ (green line, with $\gamma$ from Fig. 5$b$ as faint
red line); note $0.25$ here denotes equal amplitude at top and bottom. ($c$)
The normalized phase-tilt in the streamfunction $\Delta\epsilon_{\rm eigen}$
in units of $\pi$ as a function of $k$ and $\delta$, and ($d$) the same
quantity along the line of $k_{\max}(\delta)$ (green line, with
$\Delta\epsilon$ from Fig. 5$d$ as faint red line); note $+0.5$ here denotes
that the top edge-wave _lags behind_ the bottom edge-wave by a quarter
wavelength. The line of $k_{\max}(\delta)$ marked on as green has been marked
on panel ($a,c$) as the green dashed line.

Starting first with the amplitude ratio, we note that the standard Eady case
with $\delta=0$ has $\gamma_{\rm eigen}=\pi/4$ throughout the unstable
bandwidth, i.e. the top and bottom of the normal-mode have equal amplitude,
and in fact the normal-mode is symmetric about $z=0$ for $\delta=0$ (not
shown; [cf., 77]). For $\delta>0$, $\gamma_{\rm eigen}>\pi/4$, i.e. the
normal-mode amplitude at the bottom is smaller than that at the top, and
conversely for $\delta<0$, consistent with results from physical expectations
and edge-wave analysis. However, compared to the equivalent graph in terms of
edge-waves in Fig. 6$a,b$, there is a wavenumber dependence in $\gamma_{\rm
eigen}$, when our physical argument would suggest that the amplitude ratio
should only depend on $\delta$. The observation of $\gamma_{\rm eigen}$ is
consistent if we remember that diagnosed quantities between two separate edge-
wave structures do not necessarily need to be the same as that diagnosed from
their combinations.

For the phase-tilt in the streamfunction, in the standard Eady case with
$\delta=0$, the maximally growing mode has $\Delta\epsilon_{\rm eigen}=\pi/2$,
where the top signal lags behind the bottom signal by quarter of a wavelength.
As the wavenumber increases, the tilt in the normal-mode reduces, while the
opposite is true as the wavenumber decreases. For $\delta\neq 0$, the behavior
of $\Delta\epsilon_{\rm eigen}$ in the streamfunction is asymmetric with the
sign of $\delta$. Notably, the most unstable wavenumber for $\delta<0$ has
$\Delta\epsilon_{\rm eigen}=\pi/2$, indicating the most unstable mode is still
able to access a standard Eady-like configuration in the normal-mode, albeit
with decreasing growth rate. However, while a phase-shift of $\pi/2$ would be
the optimum configuration for constructive interference in the edge-wave
interaction framework [e.g., 60, 71, 61], we should note that the phase-shift
$\Delta\epsilon$ need not coincide with the phase-tilt $\Delta\epsilon_{\rm
eigen}$ (cf. Fig. 6$c,d$ and Fig. 5$c,d$). As was demonstrated in Fig. 4,
$\Delta\epsilon$ is generally not at the expected optimal because one needs to
take into account of the interaction function.

One could then wonder whether there is in fact any specific meaning to the
value of the phase-tilt $\Delta\epsilon_{\rm eigen}$. It is certainly true
that the phase-tilt $\Delta\epsilon_{\rm eigen}$ (as well as the PV edge-wave
phase-shift) should relate to an _energetic_ interpretation of the instability
[e.g., §6.7.2 of 1], in relation to the fact that perturbations of meridional
velocity $v^{\prime}$ and buoyancy $b^{\prime}$ should be overall positively
correlated, so that the zonally averaged meridional buoyancy flux
$\overline{v^{\prime}b^{\prime}}$ is poleward, leading to a decrease in
available potential energy. While the statement about the energetics is true,
the problematic aspect is linking that to the raw value of
$\Delta\epsilon_{\rm eigen}$. Since in the linear instability analysis we
should only be talking about rates and efficiencies, we might suspect that
$\Delta\epsilon_{\rm eigen}=\pi/2$ would correspond to maximum efficiency in
$\overline{v^{\prime}b^{\prime}}$ in reducing available potential energy, i.e.
maximum linear correlation. A straightforward linear regression analysis for
the correlation of $v^{\prime}$ and $b^{\prime}$ over the whole spatial domain
(not shown) indicates this is simply not true: $v^{\prime}$ and $b^{\prime}$
are maximally correlated in the linear sense at small $k$, and its dependence
as a function of $k$ and fixed $\delta$ bears no resemblance to
$\Delta\epsilon_{\rm eigen}$ or the growth rate plots (the scatter plots of
$v^{\prime}$ against $b^{\prime}$ becomes increasingly ‘circular’ with
increasing $k$). The behavior of $\overline{v^{\prime}b^{\prime}}$ itself
(making the choice of normalizing the eigenfunction $\tilde{\psi}(z)$ to have
unit magnitude) also bears no resemblance to $\Delta\epsilon_{\rm eigen}$ or
the growth rate: for fixed $\delta$, maximum $\overline{v^{\prime}b^{\prime}}$
occurs at a $k$ larger than the wavenumber at which there is maximum growth
(not shown). So while $\Delta\epsilon_{\rm eigen}$ would in some way be
related to the energetic as well as the kinematic/mechanistic view of the
instability problem, it may perhaps be simpler to not attribute too much
meaning to $\Delta\epsilon_{\rm eigen}$. The fact that $\Delta\epsilon_{\rm
eigen}=\pi/2$ for the most unstable mode is curious and is perhaps worthy of
further exploration, but we argue here that if we are invoking the edge-wave
interaction mechanism, it is perhaps quantitatively misleading or not entirely
appropriate to provide evidence in terms of something that is not in edge-wave
form.

##  IV Analysis in the GEOMETRIC framework

The previous section highlights subtleties when using the streamfunction
eigenfunction with a mechanistic explanation, and argues that it is the edge-
wave basis that are more dynamically relevant. Does that mean the instability
eigenfunctions have little utility relative to the edge-wave basis? We provide
a processing of the eigenfunctions in terms of the GEOMETRIC framework of [34,
35] (see also [36, 37]) that considers geometric quantities such as anisotropy
factors and angles of eddy variance ellipses, and turns out to have
mechanistic links with baroclinic instability (cf. the barotropic case,
considered in [72]). The quantitative links between energetics and mechanistic
interpretations are demonstrated here for a case where both the edge-wave
basis and eddy fluxes are well-defined. The strong correlation of the
geometric quantities with that diagnosed from the edge-wave framework provides
a suggestion that in cases where the edge-wave basis is less well-defined
(e.g., the linear Charney–Green problem, or data from the nonlinear evolution
of baroclinic instability), the GEOMETRIC framework may still be utilized and
has energetic and dynamical relevance.

As a recap to the work of [36, 34, 35], in the QG limit, it is known that the
eddy forcing on to the mean state is determined by the object [e.g., 34]

$\boldsymbol{\mathsf{E}}=\begin{pmatrix}-M+P&N&0\\\ N&M+P&0\\\
-S&R&0\end{pmatrix},$ (21)

with

$\displaystyle M=\frac{1}{2}\overline{v^{\prime 2}-u^{\prime
2}}=-\gamma_{m}E\cos 2\phi_{m}\cos^{2}\lambda,\qquad
N=\overline{u^{\prime}v^{\prime}}=\gamma_{m}E\sin 2\phi_{m}\cos^{2}\lambda,$
(22) $\displaystyle P=\frac{1}{2N_{0}}\overline{b^{\prime
2}}=E\sin^{2}\lambda,\qquad
R=\frac{f_{0}}{N_{0}^{2}}\overline{u^{\prime}b^{\prime}}=\gamma_{b}\frac{f_{0}}{N_{0}}E\cos\phi_{b}\sin
2\lambda,\qquad
S=\frac{f_{0}}{N_{0}^{2}}\overline{v^{\prime}b^{\prime}}=\gamma_{b}\frac{f_{0}}{N_{0}}E\sin\phi_{b}\sin
2\lambda,$

where $M,N$ denote the eddy momentum fluxes (related to the Reynolds
stresses), $R,S$ denote the eddy buoyancy fluxes (related to the form
stresses), $P$ is the eddy potential energy, and $E=P+K$ is the total eddy
energy, with the eddy kinetic energy $K$ defined in the usual way. In the
framework, $E$ becomes the only dimensional variable, which is arbitrary up to
a multiplicative constant for the linear instability problem. By contrast, the
non-dimensional geometric quantities related to the eddy variance ellipses are
independent of the arbitrary mulitplicative constant, and are given by

$\displaystyle\gamma_{m}=\frac{\sqrt{M^{2}+N^{2}}}{K},\qquad\gamma_{b}=\frac{N_{0}}{2f_{0}}\sqrt{\frac{R^{2}+S^{2}}{KP}},$
(23) $\displaystyle\sin
2\phi_{m}=\frac{N}{\sqrt{M^{2}+N^{2}}},\qquad\sin\phi_{b}=\frac{S}{\sqrt{R^{2}+S^{2}}},$
$\displaystyle\frac{K}{E}=\cos^{2}\lambda,\quad\frac{P}{E}=\sin^{2}\lambda,\quad\tan^{2}\lambda=\frac{P}{K},$

where the overbar denotes a mean operator (zonal average for the present
work), $\gamma_{m,b}$ are the momentum and buoyancy anisotropy parameters,
$\phi_{m,b}$ are angle parameters related to the eddy momentum and buoyancy
ellipses, while $\lambda$ is an angle relating to the eddy energy partition.
Note that there is a degeneracy in the angle parameters, where for example we
could define $\phi_{b}$ in terms of $\cos\phi_{b}=R/\sqrt{R^{2}+S^{2}}$ as in
[34].

One idea relating to parameterization of eddy fluxes is that the non-
dimensional geometric parameters might be more universal and related to
dynamics and/or instability characteristics, so are perhaps be easier to
parameterize. For example, it is known that in barotropic/horizontal shear
instabilities, $\phi_{m}$ directly relates to the tilt angle of the eddy,
where if the eddy tilts into the shear we have instability, while if eddy
tilts with the shear we would have the converse, with eddies fluxing momentum
back into the mean state [e.g., 36, 34, 37, 78, 72]. In the present modified
Eady problem, it can be demonstrated that it is the eddy buoyancy rather than
momentum fluxes that are non-trivial, consistent with the present set up
leading to a pure baroclinic instability. The parameters of interest are then

$\tan 2\phi_{t}=\gamma_{b}\tan 2\lambda,\quad\gamma_{t}=\frac{\cos
2\lambda}{\cos 2\phi_{t}},$ (24)

where $\phi_{t}$ and $\gamma_{t}$ are the angle and anisotropy parameter of a
vertical eddy in physical space [36, 34, 79]. From this, we note that we can
define a non-dimensional parameter $\alpha$ where

$\alpha=\gamma_{b}\sin\phi_{b}\sin 2\lambda=\gamma_{t}\sin\phi_{b}\sin
2\phi_{t}.$ (25)

The $\alpha$ parameter is a combination of geometric parameters that closely
relates to the Eady growth rate, is bounded in magnitude by unity in the QG
limit [34], and is one of the tuning parameters that is at present prescribed
in parameterizations of baroclinic restratification effects [e.g., 39]. An
interest here is on the dependence of $\phi_{t}$ and $\alpha$ on $\delta$, and
what is the dominant contribution to the variation of $\alpha$, with the
possibility to aid/inform our parameterization efforts for baroclinic eddies
and its feedback onto the mean state in theoretical and/or numerical models
[e.g., 80, 39].

For the case of $l=0$, i.e. no meridional variation, $u^{\prime}=0$, and so
$R=N=0$ while $M^{2}=K$, and so

$\gamma_{m}=1,\quad\phi_{m}=0,\quad\phi_{b}=\pm\frac{\pi}{2},\quad\alpha=\pm\gamma_{b}\sin
2\lambda=\gamma_{t}\sin 2\phi_{t}.$ (26)

For the present set up it is the sign of $\alpha$ that distinguishes whether
we have instability or not, since $\phi_{b}$ is defined in term of the zonal
mean meridional advection of eddy buoyancy fluxes $S$, which is the principal
interest for baroclinic instability ($S>0$ is poleward flux of buoyancy, and
so $\sin\phi_{b}$ and $\alpha$ are both positive).

Figure 7: Vertically averaged ($a$)
$\overline{\alpha}^{z}=\overline{\gamma_{b}\sin 2\lambda}^{z}$ ($c$) buoyancy
anisotropy factor $\overline{\gamma_{b}}^{z}$ and ($e$) vertical tile angle
$\overline{\phi_{t}}^{z}$ (in multiples of $\pi$) as a function of $k$ and
$\delta$, and ($b,d,f$) the respective quantities along the line of
$k_{\max}(\delta)$. The line of $k_{\max}(\delta)$ marked on as green has been
marked on panel ($a,c,e$) as the green dashed line.

Fig. 7 shows the value of the vertically averaged $\alpha$, buoyancy
anisotropy $\gamma_{b}$ and the vertical tilt $\phi_{t}$ over the unstable
region in parameter space. It may be seen that $\alpha$ correlates strongly
with the growth rate (Fig. 2$a$), in line with the analysis of [34]. Although
the values of $\alpha$ reported here are around an order of magnitude larger
than what are used in parameterizations for numerical ocean models [e.g., 39],
we note that the diagnosed value here is for the linear instability analysis,
whilst the values used in parameterizations are applied as averages over both
energetic and quiescent regions and to mimic the feedback during the nonlinear
phase. The sensitivity of $\alpha$ to $\delta$ is of most relevance,
suggesting it should be $\alpha$ that is decreased in value in the presence of
slopes, particularly in the $\delta<0$ scenario (where topographic and
isentropic/isopycnal slopes are opposite in orientation), consistent with a
previous diagnostic result in the nonlinear regime [56].

The dominant contribution to the variation of $\alpha$ (in both the average
and pointwise sense) is principally through the buoyancy anisotropy parameter
$\gamma_{b}$ (see Fig 7$a$ and $c$). The eddy energy partition angle parameter
$\lambda$ plays a secondary role, since $\sin 2\lambda$ is diagnosed to be
close to 1 in value in the pointwise sense (not shown). The presence of a
sloping boundary would naturally be expected to force an anisotropy, and the
feature of $\alpha$ correlating well with $\gamma_{b}$ and not $\sin 2\lambda$
seems to be consistent with diagnoses from numerical simulations in a global
ocean circulation model in the nonlinear regime [80], and diagnoses in the
nonlinear regime of an idealized baroclinic flow over a topographic slope
(personal communication with Huaiyu Wei and Yan Wang). The decreasing values
of $\gamma_{b}$ for fixed $\delta$ as $k$ is increased corresponds to a
statement in the previous section that the scatter plot of $v^{\prime}$
against $b^{\prime}$ becomes increasing ‘circular’ in the same limit [cf. 34].
If we consider instead $\alpha$ in terms of $\phi_{t}$ and $\gamma_{t}$, then
we see here that $\sin 2\phi_{t}$ would not correlate well with $\alpha$, and
neither would $\gamma_{t}$ (not shown). So while there is a flexibility for
the form of $\alpha$ used, it would seem that, in the linear regime at least,
it is $\gamma_{b}$ that is more relevant, and the presence of the slopes
modifies the anisotropy of the state.

The vertical eddy angle $\phi_{t}$ has the behavior that $\phi_{t}\searrow 0$
as $k$ is increased for fixed $\delta$ (Fig. 7$e$). One interpretation of the
tilt angle could be that it is related to $\Delta\epsilon$ of two interacting
edge-waves [cf., 72, for the horizontal case], having an optimal phase shift
leading to instability ($\phi_{t}=\pi/4$ might be expected to be analogous to
the optimum phase-shift of $\Delta\epsilon=\pi/2$; cf. [72] for the case where
the velocity shear is purely in the horizontal). Indeed, Fig. 7$e,f$ showing
$\phi_{t}$ over parameter space resembles that of $\Delta\epsilon$ over
parameter space, as shown in Fig. 5$c,d$. In that sense, even though
$\phi_{t}$ is defined in terms of $S\sim\overline{v^{\prime}b^{\prime}}$
(through the definition of $\gamma_{b}$) and makes no reference to edge-wave
structures whatsoever, there are apparent mechanistic links of $\phi_{t}$ with
$\Delta\epsilon$. In that regard, $\phi_{t}$ could perhaps serve as a possible
proxy for edge-wave phase-shifts that is easier to diagnose in cases where the
definition of edge-wave structures becomes more ambiguous (e.g., dynamics in a
nonlinear setting). Again, the phase-tilt $\Delta\epsilon_{\rm eigen}$ (Fig.
6$c$) bears little resemblance to $\phi_{t}$ (Fig. 7$e$) nor $\Delta\epsilon$
(Fig. 5$e$) over parameter space.

The diagnosed results above are presented as vertically averaged quantities,
which does not demonstrate the asymmetry introduced with $\delta$. Examination
of the full vertical profiles does in fact show the quantities to be
increasingly concentrated towards the lower boundary when $\delta<0$ (and
vice-versa), consistent with the known behavior of $\gamma$ in (18) and Fig.
5$a$, the lower edge-wave being the increasingly dominant contribution (not
shown). We close this section with a note that the above results should be
interpreted with the caveat that the diagnoses are in the linear and nonlinear
regime respectively, and the two are not necessarily directly comparable
(e.g., it is not obvious that the instability has to leave a strong imprint on
the nonlinear eddy fluxes locally).

##  V Closing remarks

###  V.1 Conclusions

The present work aims to clarify and point out some links between several
concepts in baroclinic instability, such as the underlying mathematical
symmetries of the governing system, a mechanistic interpretation of the shear
instability problem [e.g., 59, 60, 61, 66, 81], and geometric parameters of
the eddy variance ellipses [e.g., 36, 34, 72]. We considered the Eady problem
in the quasi-geostrophic system as a working example, modified to include a
weak linear bottom slope. Making an assumption about the magnitude of slopes,
the presence of the slope only affects the dynamics via a bottom boundary
contribution to potential vorticity (PV), and the standard analysis leads to
closed form solutions, as is known in the literature [e.g., 41, 42, 43, 44,
45, 53]. The resulting modified Eady system was claimed here to be parity-time
($\mathcal{PT}$) symmetric [cf. 68, 69, 70, 81], with consequences for the the
solution spectrum; we refer the reader to Appendix A for the full details and
other more speculative links.

To clarify aspects of the edge-wave mechanism and the modification by the
presence of a slope, we perform an edge-wave interaction analysis [cf., 71,
61, 66]. For the present system, where edge-wave locations are well-defined,
the edge-wave analysis is really a rephrasing of the standard instability
problem in a different choice of basis, and the standard modal instability
problem is effectively one of finding fixed points of a two-dimensional
dynamical system for the amplitude ratio and edge-wave phase-shift.
Physically, we expect that for $\delta<0$, where topographic slope and
isentrope/isopycnals have horizontal gradients of opposite signs, more
vertical stretching is allowed (cf. Fig. 1$b$), and thus is adds to the
background PV gradient at the bottom. The bottom wave is then stronger, and
its characteristic should be more apparent in the overall normal-modes. For
$\delta<0$, $c_{r}^{\pm}>0=c_{r}^{\pm}(\delta=0)$ with our choice of basic
state set up, since the bottom wave counter-propagates to the right (or east).
The opposite is true for $\delta>0$. The asymmetry is supported by the
resulting edge-wave analysis, where the amplitude ratios are simply functions
of $\delta$ but not of the wavenumber (Fig. 5$a$); the dependence on $\delta$
only should be expected from the PV point of view, but such a dependence is
not in fact seen in the analysis of the tilted streamfunction eigenfunction
(Fig. 6$a$).

The phase-shift of the edge-waves $\Delta\epsilon$ associated with the most
unstable mode is not necessarily at the theoretical optimum of $\pi/2$
(actually slightly larger), and generically differs from the phase-_tilt_ in
the unstable streamfunction $\Delta\epsilon_{\rm eigen}$, which does seem to
be at $\pi/2$ for $\delta\leq 0$. We argue that $\Delta\epsilon$ rather than
$\Delta\epsilon_{\rm eigen}$ should be the quantity of interest if a
kinematic/mechanistic interpretation is to be invoked. The unstable normal-
mode is a linear combination of the edge-wave structures (e.g. Fig. 4), and
the phase-shifts in the untilted edge-waves structures do not have to
correspond to the phase-tilts in the tilted streamfunction eigenfunctions. We
are of the opinion that references to $\Delta\epsilon_{\rm eigen}$ should
generally be avoided (an exception perhaps for the Phillips problem [13],
where the entries of the normal-modes are defined as per-layer quantities and
could be argued to already be in edge-wave form). If some reference is to be
made to the energetics of the instability, the geometric parameters associated
with eddy variance ellipses [e.g., 60, 34] such as buoyancy anisotropy
$\gamma_{b}$ serve as better measures of the correlation for the meridional
velocity and buoyancy perturbations $v^{\prime}$ and $b^{\prime}$. On the
other hand, the vertical eddy tilt $\phi_{t}$, although defined with no
reference to the edge-wave structures themselves, display characteristics of
the edge-wave phase-shift $\Delta\epsilon$. The realized eddy efficiency
parameter $\alpha$ is shown to correlate strongly with the growth rate, in
line with the definition given in [34]. It is found here that the dominant
contribution to $\alpha$ comes from $\gamma_{b}$, and so decreases in $\alpha$
with variations in $\delta$ arises the decrease in $\gamma_{b}$, consistent
somewhat with previous diagnoses from a nonlinear realization of baroclinic
dynamics in a numerical ocean model [82].

Previous works have argued that the changes in instability characteristics
arises from changes in the edge-wave interaction, but only considers the need
for phase-speed matching [53]. We argue that view is incomplete as the
argument does not extend to all of parameter space, and does not explain all
the instability characteristics. Here we clarify the mechanistic explanation.
For $\delta<0$, the bottom wave increases in strength, and not only does it
intrinsically propagate faster (to the right or east), but leads to a larger
action-at-a-distance. The top wave experiences a larger _hindering_ , which
can be compensated by reducing the vertical interaction function via moving to
larger wavenumbers (which also reduces the intrinsic propagation speed of both
edge-waves), and the weaker interaction function leads to weaker growth rates.
It seems that it is always possible for such a compensation to be achieved in
the $\delta<0$ case, albeit over a decreasing bandwidth. For $\delta>0$, the
opposite is true, except the reduction in interaction from the weakened bottom
wave cannot be arbitrarily increased by moving to smaller wavenumbers, since
that increases the intrinsic propagation speed of both edge waves. In this
case the top edge-wave will become too fast for the given background flow and
wave interaction for phase-locking to be possible. The instability bandwidth
decreases in width and bounded away from the small wavenumber as $\delta$ is
increased towards 1. At $\delta\geq 1$ there is no instability whatsoever, as
counter-propagation of the bottom wave is no longer possible since the PV
gradient vanishes or reverses sign.

Although the present work focuses largely on the zero meridional wavenumber
case $l=0$, the results and observations apply to the $l\neq 0$ case. As a
demonstration, the instability characteristics of $l=0.5$ with the meridional
structure function $g(y)=\sin(ly)$ is shown in Fig. 8. Here the instabilities
generally possess weaker growth rates, with a change in the domain of
instability in the $\delta>0$ region, but otherwise the general behavior of
the growth rates and phase-speed $c_{r}^{\pm}$ are similar to what has been
reported (since the system is still $\mathcal{PT}$ symmetric). The physical
rationalization in terms of edge waves is still applicable. The reduction in
growth rates can be attributed to weakened interaction from $l\neq 0$, since
that increases the value of $\mu=\mbox{Bu}\sqrt{k^{2}+l^{2}}$. The expanded
domain of instability for $\delta>0$ can be rationalized as edge-waves having
smaller intrinsic propagation speed in general (because of the increased
$\mu$), and so the top edge-wave can be held in a phase-locking configuration
for the given basic flow, and the weakened interaction associated with weaker
bottom edge-wave can in fact be compensated by increasing the interaction via
decreasing the zonal wavenumber $k$. Edge-wave amplitude ratios $\gamma$ is as
given by (18), and the edge-wave phase-shift $\Delta\epsilon$ has similar
behavior to that observed in Fig. 5$c$ (not shown). While the computation of
the geometric parameters is slightly more complex as the relevant geometric
variables are now two-dimensional, the conclusions and interpretations are
essentially the same, except that the dominant contribution to $\alpha$ now
comes from $\gamma_{b}\sin\phi_{b}$ (since $R$ is no longer trivial and so
$\phi_{b}\neq\pm\pi/2$), but the energy partition contribution $\sin 2\lambda$
is still close to 1 and largely constant over the domain. In particular, the
vertical eddy tilt $\phi_{t}$ follows the phase-shift of the edge-waves more
so than the normal-mode phase-tilts (in terms of spatial distribution, average
values down the center line, or domain-averaged values; not shown).

Figure 8: Sample instability characteristics for $l=0.5$, with the non-trivial
meridional structure function $g(y)=\sin(y/2)$. ($a$) Growth rate as a
function of the non-dimensional wavenumber $k$ and $\delta$ parameter, with
darker shadings denoting higher values (strength of shading is the same used
in Fig. 2), sample contours of growth rates, and the green dashed line
denoting $k_{\max}(\delta)$ where the growth rate is maximized. Also shown are
the growth rates $kc^{+}_{i}$ and phase-speed $c_{r}^{\pm}$ for ($b,c$)
$\delta=+0.5$, ($d,e$) the standard Eady problem $\delta=0.0$, and ($f,g$)
$\delta=-0.5$. The shaded regions in panels $b$-$g$ denote the regions where
there the growth rates are non-zero.

###  V.2 Discussions and outlooks

The present work explores the behavior of the $\alpha$ parameter from a linear
instability point of view, to supplement parameterization efforts of
particular relevance to ocean modeling. We find that there is indeed a
suppression of $\alpha$ in the presence of slopes, and the dominant
contribution comes principally from changes in the buoyancy anisotropy
parameter (as well as the eddy buoyancy angle where it is present). The result
here would be consistent with previous observations that buoyancy fluxes are
suppressed over sloped regions [e.g., 43, 44, 46, 48, 49, 50, 51, 52, 53, 55],
with previous diagnostic results but in the nonlinear regime in a realistic
global ocean model [80], and is consistent with a parameterization that
$\alpha$ be suppressed over slopes in a way that is dependent on the slope
Burger number [56], related to the $\delta$ utilized in the present work. This
work thus provides a consistency rationalization for the proposal of [56],
which was empirical in nature. While the present work is for the linear
regime, previous works have suggested that the linear instability
characteristics can be useful in informing parameterizations that are invoked
for the nonlinear regime [e.g., 29, 30, 31, 32, 33].

The stabilization of baroclinic instability in the presence of small slopes
(related to the suppression of $\alpha$) is rationalized in the PV point of
view, where the presence of a slope modifies the background PV gradient and
leads to modifications in the interaction of the Rossby edge-waves. The
rationalization is given in Sec. III.2, but is also summarized in the
schematic given in Fig. 9. The point we clarify here is that it is the
asymmetry in the wave amplitudes that lead to changes in the mutual
interaction, that in turn modifies the phase-locking configuration, and the
end result dictates a change in the phase-shift. Arguments based solely on
phase-shifts and phase-speed matching is missing a key link in the
interaction, and does not fully explain the instability characteristics over
the whole parameter space. Further, we highlight that, from a mechanistic
point of view, it is the untilted edge-wave basis and its phase-shift that is
of relevance, and the phase-tilt in the tilted normal-modes is generally to be
avoided as it can be misleading. The physical rationalization applies
generically to baroclinic instability over slopes; sample analysis on the
analogous Phillips problem [cf., 45] shows similar results and interpretations
to here (not shown). It would further be interesting to see how some of the
analysis and points of view (edge-waves and GEOMETRIC framework) here carry
over to the case of transient / optimal growth [e.g., 83], but we leave this
for a follow up study.

Figure 9: Pictorial schematic for the change in instability characteristics
over parameter space in terms of interacting Rossby edge-waves. ($a$) For the
$\delta<0$ case, the bottom PV gradient is stronger, leader to a stronger
bottom edge-wave that strengthens the magnitude of interaction and edge-wave
propagation speed (represented by longer arrows), but this effect can be
compensated by going to shorter waves (represented here by the thinner
arrows). ($b$) The standard Eady case $\delta=0$, where the interaction is
symmetric. ($c$) For $1>\delta>0$, the bottom PV gradient weakens, leading to
a weaker bottom edge-wave and a weaker interaction (represented by the shorter
arrows), but this can be offset somewhat by going to longer waves (represented
by the thicker arrows). ($d$) For $\delta>1$, the PV gradient switches sign
and the bottom edge-wave no longer counter-propagates against the background
mean flow, no phase-locking configuration is possible, and there is no
instability.

It was found that the vertical tilt angle $\phi_{t}$ (rather than the buoyancy
angle $\phi_{b}$) possesses qualitative similarities to the edge-wave phase
shifts, even though the former is defined with no reference to the edge-waves
whatsoever. The observation is in line with the results in [72], who find that
in the barotropic instability of the Rayleigh sheet that there is a relation
between the horizontal tilt angles $\phi_{m}$ (the momentum flux angles) and
the edge-wave phase-shift. Then perhaps a parameterization of baroclinic
instability could be based on the geometric parameters such as $\phi_{t}$,
which informs the spatial structure of the eddy fluxes as well as an
efficiency via $\alpha$, with a realized magnitude based on a parameterized
eddy energy [cf. 34, 39]. For diagnostic purposes, in cases where the edge-
waves are not so well-defined (e.g., nonlinear cases, although see next
paragraph), diagnoses of $\phi_{t}$ could be done instead as a proxy for the
phase-shift, since the geometric parameters considered here applies
generically to cases where eddy fluxes can be diagnosed [34].

For the present linear problem the edge-waves are well-defined, and Sec. III
provides a manual construction of the edge-wave basis (e.g., where one part of
the edge-wave couplet has an untilted structure and no PV signal at the
location of the other edge-wave). There are problems where edge wave
structures are harder to define (e.g., instability in the presence of
planetary $\beta$ [11, 14, 72], the Rayleigh sheet problem in the presence of
a magnetic field [65]), or cases where one might naturally expect linear
theory to play no longer play a dominant role (e.g., during a nonlinear
evolution). However, there are still ways to define an edge-wave basis, for
example making use of orthogonality in the wave activity quantities such as
pseudomomentum and pseudoenergy [e.g., 73, 18, 61, 66]. It is possible to
extend the present analysis to some of the other models, such as the
Charney–Green model, which has been pointed to possibly be more relevant than
the Eady model in certain oceanic settings [e.g., 43, 46, 50, 52], problems
with smooth profiles and/or profiles containing multiple PV gradients (e.g.
jet profiles), and mixed barotropic-baroclinic problems [e.g., 84, 85]. In
particular, if one believes that there is some sort of quasi-linear control on
the linear instability on the nonlinear evolution, then perhaps the procedure
will be illuminating for an analogous analysis for simulations of the
nonlinear phase of the shear instabilities [e.g., 86]. These are left as
possible future investigations to be pursued.

###### Acknowledgements.

This work was supported by the the RGC General Research Fund 11308021 and the
Center for Ocean Research in Hong Kong and Macau, a joint research center
between the Laoshan Laboratory and Hong Kong University of Science and
Technology. The Python Jupyter notebook used to generate and analyze data for
the problem may be found at
https://github.com/julianmak/julianmak.github.io/blob/master/files/Eady/Eady_analysis.ipynb.
GK was supported by a post-doctoral funding through the Center for Ocean
Research in Hong Kong and Macau. Some of the work was initiated while EQYO was
a visiting student at the Hong Kong University of Science and Technology. EQYO
is supported by the Australian Government Research Training Program 678
Scholarship (RTP) and the Australian Research Council Special 676 Research
Initiative, Australian Centre for Excellence in Antarctic Science (ARC project
number 677 SR200100008). JM would like to thank Liiyung Yeow and the
hospitality of the staff at the Edison Hotel at Penang, Malaysia that lead to
the discussion of $\mathcal{PT}$-symmetry in classical systems and the writing
of the first draft of the present article. Authorship is alphabetical after
the first author. Resources, Supervision, Project administration, Funding
acquisition: JM. Conceptualization, Visualization: JM, NH, EH. Methodology:
JM, EH, GK. Software: JM, GK. Formal analysis: JM, NH, EH, GK. Writing -
Original Draft: JM, NH, EH. Validation, Writing - Review & Editing: everyone.

##  Appendix A Parity-Time $\mathcal{PT}$ symmetry of the modified Eady
problem

Note that in the text we made the observation that Eq. (3) is invariant under
the transformation

$\mathcal{P}:(x,y,z)\mapsto(-x,-y,-z),\qquad\mathcal{T}:(t,\psi)\mapsto(-t,-\psi).$
(27)

More formally, an operator $\mathcal{H}$ governing a system is Parity-Time
($\mathcal{PT}$) symmetric if it satisfies

$(\mathcal{PT})\mathcal{H}^{*}(\mathcal{PT})^{-1}=\mathcal{H},$ (28)

where $\mathcal{H}^{*}$ denotes the complex conjugate (rather than the
Hermitian conjugate $\mathcal{H}^{\dagger}$, which for a matrix representation
involves a transpose on top of the element-wise complex conjugate).

$\mathcal{PT}$ symmetry is a concept that originates from quantum physics
[e.g., 67], but has found recent interest in classical fluid systems also
[e.g., 68, 69, 70]. Here we show explicitly that the present modified Eady
problem is $\mathcal{PT}$ symmetric, which leads to certain features in the
solution spectrum that was previously highlighted in Sec. II. The ideas and
tools were previous given in the work of [70] for the Phillips-like problem
(two layer QG with uniform flow in each layer, cf. [13, 1]). The following
exposition is largely given for completeness, but also serves to highlight
very suggestive links between $\mathcal{PT}$ symmetry, shear instability and
the edge-wave formalism, possibly enabling tools to be borrowed from quantum
physics to further our understanding of classical fluid systems (e.g.
nonlinear shear instability), or providing classical analogues with well-
understood physics to complement the mathematical analysis of quantum systems.

Note that we can write the linearized system (3) in the form

$\frac{\partial}{\partial t}\mathcal{L}\varphi=\mathcal{M}\varphi,$ (29)

which is a generalized eigenvalue problem for the relevant operators
$\mathcal{L}$ and $\mathcal{M}$ acting on an eigenfunction $\varphi$. If we
are taking modal solutions as in (5), then $\mathcal{L}$ has an explicit
representation whose inverse that can in principle be computed for, and we can
define $M=\mathcal{L}^{-1}\mathcal{M}$ where

$c\phi=M\phi,$ (30)

with the eigenvector $\phi=(a,b)$ in this case. For the system here, it can be
shown that (using again the notation $C=\cosh\mu$ and $S=\sinh\mu$)

$L=\mu\begin{pmatrix}-S&C\\\ S&C\end{pmatrix},\quad L^{-1}=\frac{1}{2\mu
CS}\begin{pmatrix}-C&C\\\ S&S\end{pmatrix},\quad
M=\frac{-1}{SC}\begin{pmatrix}\cfrac{\delta}{2\mu}C^{2}&\left(1-\cfrac{\delta}{2}\right)\cfrac{CS}{\mu}-C^{2}\\\
\left(1-\cfrac{\delta}{2}\right)\cfrac{CS}{\mu}-S^{2}&\cfrac{\delta}{2\mu}S^{2}\end{pmatrix},$
(31)

where $L$ denotes the representation of $\mathcal{L}$ when modal solutions (5)
are taken.

In the present work, $M$ is already real, $\mathcal{PT}$ happens to be the
negative identity in the matrix representation relevant for the present system
[70], so $M$ is $\mathcal{PT}$-symmetric. Another way to see that $M$ is
$\mathcal{PT}$-symmetric is to note that, for $P$ and $T$ denoting the matrix
representations of $\mathcal{P}$ and $\mathcal{T}$, $\mu$ is invariant under
$P:(k,l)\mapsto(-k,-l)$ (interpreting $\mu=\mbox{Bu}|k|$ if $l=0$), while $T$
introduces a minus sign through its action on the eigenvector $\phi$, but this
is done twice, so since $M$ is real, the above equality holds.

A particular consequence for $M$ being $\mathcal{PT}$ symmetric is that the
entries $M$ are all real (which we have manually demonstrated here in (31),
but is in fact a general feature). Further, the dispersion relation (30), for
a two component system, satisfies

$c^{2}-\mbox{Tr}(M)+\mbox{Det}(M)=0,\qquad
c=\frac{1}{2}\left(\mbox{Tr}(M)\pm\sqrt{\mbox{Tr}(M)^{2}-4\mbox{Det}(M)}\right),$
(32)

where the trace and determinant of $M$ are given by

$M=\begin{pmatrix}a_{1}&a_{2}\\\
a_{3}&a_{4}\end{pmatrix},\qquad\mbox{Tr}(M)=a_{1}+a_{3},\qquad\mbox{Det}(M)=a_{1}a_{4}-a_{2}a_{3}.$

The entries of $M$ are given in (31), and the resulting dispersion relation
and be shown to coincide exactly with that given (7) obtained from standard
means. Further, the eigenfunction of the system (defined up to some arbitrary
constant) can be written as

$\phi=\begin{pmatrix}a\\\ b\end{pmatrix}=\begin{pmatrix}a_{2}\\\
-a_{1}+\cfrac{1}{2}\left(\mbox{Tr}(M)\pm\sqrt{\mbox{Tr}(M)^{2}-4\mbox{Det}(M)}\right)\end{pmatrix},$
(33)

and the eigenfunction $\phi$ is $\mathcal{PT}$-symmetric if $\phi$ is real.
However, for the eigenvector of an unstable mode in the present system,
$\mbox{Tr}(M)^{2}-4\mbox{Det}(M)<0$, and the resulting eigenvector becomes
complex and ceases being an eigenfunction of the $\mathcal{PT}$ operator
(because there is an extra minus introduced under a complex conjugate of the
eigenvector), and we have what is termed _spontaneous breaking of
$\mathcal{PT}$-symmetry_. The boundary between the region with and without
spontaneous $\mathcal{PT}$-symmetry breaking are called _exceptional points_ ,
and these correspond precisely to the locations of marginal stability. The
collision of eigenvalues $c^{\pm}$ on the real axis into complex conjugates is
related to what are known as _Krein collisions_ [e.g., 67, 69].

It is perhaps easy to see that, using the same framework of [70], other shear
flow instability problems should also be $\mathcal{PT}$ symmetric, such as
Phillips-like problems [cf., 45, 70], Charney–Green-like problems [cf. 11, 14,
1], the standard Rayleigh sheet problem [81] in hydrodynamics, the Rayleigh
sheet problem in magnetohydrodynamics with a uniform background magnetic field
[e.g., 65], and more general shear flow problems presumably. Additionally,
given the link between shear instabilities and its physical interpretation as
a pair of interacting edge-waves, one is left to wonder on the exact links
between $\mathcal{PT}$ symmetry and interacting edge waves. Suggestive links
include that spontaneous breaking of $\mathcal{PT}$ symmetry seems to
correspond exactly to when phase-locking occurs [cf., 70]. A Krein collision
of the eigenvalue occurs at exception points requires opposite signed Krein
signatures, reminiscent of the requirement that shear instabilities require
modes of opposite signed wave activity to collide (as positive/negative energy
modes [87] or pseudomomentum [e.g., 73]), which is sometimes interpreted as a
necessary (but not sufficient) condition for instability is the need for
counter-propagation of edge-waves [e.g., 61].

While an explicit link to the edge-wave system with $\mathcal{PT}$ symmetry
has been highlighted in the recent work of [81], and low-dimensional edge-wave
systems (low-dimensional in the sense of dynamical systems) could be
considered as a rephrasing of certain shear instability problem [e.g., 61],
the links with the general shear instability problem for general shear flows
remain to be explored (since generic shear flow instability problems should be
regarded an infinite dimensional dynamical system). For example, what is the
analogue of the Krein signature of the modes of the governing operator in the
more standard fluid dynamics context? (Likely something related to
pseudomomentum or pseudoenergy?) How are the standard shear flow linear
stability conditions related to the properties of the governing operator, and
in terms of interacting edge-waves? (Related to proofs of a purely real
spectrum in $\mathcal{PT}$ symmetric systems [e.g., 88, 89]?) Are there
conditions beyond the standard necessary but not sufficient conditions of
shear instability derivable from the $\mathcal{PT}$ symmetry property, using
techniques drawn from quantum physics? Most references of $\mathcal{PT}$
symmetry in relation to shear instability systems so far is on the linear
problem, but are there nonlinear analogues of the stability conditions such as
the Arnol’d conditions [e.g., 18] derivable from a similar approach? Such
links could in principle provide a mechanistic/physical interpretation to a
theoretical properties related to $\mathcal{PT}$ symmetry, and techniques in
relation to analyzing $\mathcal{PT}$ symmetric systems from quantum physics
could provide new approaches for improving our understanding of classical
fluid/plasma systems. The role of $\mathcal{PT}$ symmetry for fluid/plasma
systems remains to be fully explored, and research in this direction is a
subject of future work.

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