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# Necessary conditions for stability of alternating jets: Hurdle theorem

K. Deguchi1<EMAIL_ADDRESS>M. Hirota2  T. Dowling3  1School of
Mathematics, Monash University, VIC 3800, Australia 2Institute of Fluid
Science, Tohoku University, Sendai, 980-8577, Japan 3Department of Physics &
Astronomy, University of Louisville, KY 40299 USA

###### Abstract

We propose a simple method to identify unstable parameter regions in general
inviscid unidirectional shear flow stability problems. The theory is
applicable to a wide range of basic flows, including those that are non-
monotonic. We illustrate the method using a model of Jupiter’s alternating jet
streams based on the quasi-geostrophic equation. The main result is that the
flow is unstable if there is an interval in the flow domain for which the
reciprocal Rossby Mach number, calculated from the zonal flow and potential
vorticity, surpasses a certain threshold (or ‘hurdle’). The hurdle height
approaches unity when the domain size greatly exceeds the atmosphere’s
intrinsic deformation length, as it does on gas giants. In this case, the
Kelvin-Arnold sufficient condition of stability allows the instability to be
detected sharply, providing long-awaited theoretical support for the
hypothesis used to explain why Jupiter and Saturn jets are stably maintained
on the decadal timescale.

##  1 Introduction

The stability of inviscid, parallel shear flows has applications in
geophysical fluid dynamics, astrophysics, plasma physics, and engineering
thermofluid sciences. For the purely hydrodynamic cases, sufficient conditions
for stability can be traced back to the Rayleigh (1880) inflection-point
theorem, the refinement of which by Fjørtoft (1950) is one of the two
conditions derived using Arnold’s method (Arnol’d, 1966). The existence of two
distinct sufficient conditions were anticipated by Kelvin on energetic grounds
(Thomson, 1880), the same year as Rayleigh’s inflection point theorem. Today
they are called the Kelvin-Arnol’d $1^{\rm st}$ and $2^{\rm nd}$ shear-
stability theorems (KA-I and KA-II). KA-I corresponds to the Rayleigh-Fjørtoft
condition, while KA-II is relevant in planetary physics.

A breach of sufficient conditions for stability does not necessarily imply
instability. However, in applied fields, these conditions are sometimes
treated as if they were sharp stability criteria, contributing to potential
confusion. The reason for this may be that the known derived necessary
conditions for stability, i.e., conditions guaranteeing the existence of
instability, require rather complex assumptions about the basic flow $U(y)$.
Tollmien (1935) was the first to find a class of basic flows where the KA-I
condition becomes a sufficient and necessary condition for stability. Many
assumptions are required of $U$ in his class, e.g. $U$ is symmetric and
disappears on the wall, etc. Howard (1964) also showed that KA-I becomes a
sharp stability condition when the flow is considered in an unbounded domain
and the basic flow is assumed to be in the class $\mathcal{H}$ he defined.
Nevertheless the above results are only special cases and in general KA-I is
not sharp. In a bounded domain, it is in fact possible to construct
counterexamples that demonstrate that KA-I is not a necessary condition of
stability (see Tollmien, 1935; Drazin & Howard, 1966; Drazin & Reid, 1981).

The more general the basic flow considered, the more complex the argument
required to derive necessary conditions for stability. Using the Nyquist
method, Rosenbluth & Simon (1964) derived a sharp stability condition using an
integral quantity that can be calculated from the basic flow. The assumption
imposed by them on $U(y)$ is that it is monotonic and has only one inflection
point in the domain. The latter assumption was removed by Balmforth & Morrison
(1999), who also used the Nyquist method. Hirota et al. (2014) used the
variational method under almost the same assumptions and demonstrated that the
definiteness of the quadratic form introduced by Barston (1991) is a necessary
and sufficient condition for stability. In order to derive those conditions,
it is critical to assume the monotonicity of $U(y)$. The special nature of
this type of basic flow is that all neutral modes must possess only one
critical layer—where the phase speed of the mode matches $U(y)$—and moreover,
the position of this layer is fixed at one of the inflection points.

Demonstrating the existence of a neutral mode first and then perturbing the
wavenumber to find unstable modes has been a common approach since Tollmien
(1935) and Howard (1964). The assumptions employed by Howard (1964) were aimed
at enabling the use of Sturm-Liouville theory for neutral modes (see also
Morse & Feshbach, 1953), allowing the assertion of the existence of neutral
modes for non-monotonic $U$. For the bounded flows, Tung (1981) investigated
how much the assumptions about $U$ can be relaxed to demonstrate the presence
of unstable modes around neutral modes. While his analysis had some flaws,
later, Lin (2003) independently completed a mathematically rigorous theory.

One notable example where the non-monotonicity of the basic velocity field
becomes crucial is the alternating jet streams of Jupiter and Saturn. Their
evolution is governed by the behaviour of Rossby waves, which are analogous to
drift waves in plasma physics and arise when there is a gradient in the
potential vorticity (PV), a conserved fluid-dynamical quantity formed by the
combination of conservation of mass, momentum, and thermal energy (Vallis,
2017). Following standard practice, this article works within the quasi-
geostrophic framework, which admits Rossby waves but filters out sound waves
and buoyancy waves. The non-rotating, non-stratified KA-I result by Rayleigh
and Fjørtoft was followed by extensions to rotating, non-stratified flow (Kuo,
1949) and to rotating, stratified flow (Charney & Stern, 1962). The latter
result, widely known as the Charney-Stern stability criterion, established
that a sufficient condition for stability is the absence of a point where the
PV gradient changes sign, called a critical latitude (or critical line).
Physically, a critical latitude is where Rossby-wave propagation switches
direction relative to the flow; in other words, where PV eddies pivot.

Nondimensionalising the KA-I and KA-II conditions for the planetary atmosphere
problem uncovers the role played by the Rossby Mach number, $M$, which leads
to a physical interpretation of KA-II for which “supersonic” flows (in the
usual sense but with respect to Rossby waves rather than sound waves) are
stable; see Dowling (2014). The $M<0$ case is a second supersonic case that
has no analogy with sound waves; it corresponds to when Rossby waves travel
downstream relative to the flow. Getting KA-I and KA-II to concatenate
together via $M^{-1}<1$ is a compact way to look at the stability condition.
However, note that in general $M$ is a function of latitude, found from a
ratio of the zonal wind, $U(y)$, to the PV gradient, $Q^{\prime}(y)$, as we
clarify in the next section. Therefore, the above simple stability condition
can only be applied for basic flows belonging to a certain class where the
sign of $M^{-1}(y)$ does not change for all $y$. This class, in fact, closely
resembles the one defined by Howard (1964) and Lin (2003), and we base our
analysis on this ground.

Figure 1:  Example Jupiter profiles with respect to latitude of zonal wind,
$U$ (solid line), and potential vorticity gradient, $Q^{\prime}=dQ/dy=Q_{y}$
(dashed line), from the southern hemisphere, at 11 hPa pressure. A strong
correlation between $U$ and $Q_{y}$ is evident. Here, $dy=Rd\phi$, where $y$
is latitude in meters, $\phi$ is latitude in radians (planetographic), and
$R(\phi)$ is the local radius of curvature (see equation (5) in Dowling &
Ingersoll, 1989). These are a sample of the profiles produced by Read et al.
(2006) using the Cassini Composite Infrared Spectrometer (CIRS). Similar
correlations to this figure are found at other pressure levels.

Dowling (1993) discovered that Jupiter’s atmosphere has $U$ and $Q^{\prime}$
profiles strongly correlated. This analysis, based on Voyager observations of
the cloud-top vorticity field (Dowling & Ingersoll, 1989), was expanded for
Jupiter and Saturn by Read et al. (2006) and Read et al. (2009a, b), as
illustrated in figure 1. The correlation implies that the $M^{-1}(y)$ profile
is almost constant. If the profile is neutrally stable and the KA-II condition
is sharp, this constant must be unity. However, as already noted, the
stability condition based on the reciprocal Rossby Mach number is only a
sufficient condition for stability, and the significance of it regarding
necessary conditions for stability remains not well understood. Moreover, the
actual $M^{-1}(y)$ profile is not merely a constant, as can be seen from the
fact that the correlation in figure 1 is not perfect.

Particularly intriguing from a planetary atmospheric physics perspective is
thus when KA-II gives sharp stability boundaries. There are interesting
numerical results in this regard: Stamp & Dowling (1993) set up a sinusoidal
model basic flow such that $M$ becomes a constant, and numerically found that
the instability disappears at the KA-II stability boundary, $M=1$. Motivated
by these empirical results, our mathematical goals are twofold: (i) to derive
handy conditions that guarantee the presence of instability, and (ii) to
determine under what conditions KA-II achieves sharp stability boundaries. We
also attempt to emphasise the simplicity of the theoretical results, because
the pursuit of sharpness of the conditions often makes them difficult to use.
For example, from a practical standpoint, the condition proposed by Balmforth
& Morrison (1999) is not user-friendly due to the requirement of solving a
Fredholm integral equation. Also, to demonstrate the existence of instability
using the quadratic form in Hirota et al. (2014) or the Rayleigh quotient in
Lin (2003), it is necessary to find a convenient test function, but no useful
recipe is known for non-monotonic bounded basic flows.

The rest of the article is organised as follows. In Section 2, the quasi-
geostrophic equation is linearised around the basic flow to yield an
eigenvalue problem. The basic states are then classified in Section 3 to
clarify when the stability condition derived in Section 4 can be used. One of
the classes is identified as having Jupiter-style critical latitudes; it
yields an almost sharp stability condition that aligns with observations of
the jets on Jupiter and Saturn. Our strategy is to first extend the Rayleigh
quotient method developed by Howard (1964), Tung (1981), and Lin (2003) to the
quasi-geostrophic system and then use it to derive a simple necessary
condition for stability. In Section 5, the stability criteria are illustrated
with model profiles. Section 6 concludes with a summary of the theoretical
results and a discussion of their application to planetary physics problems.

##  2 Formulation of the problem

Our starting point is quasi-geostrophic conservation of potential vorticity on
a beta-plane (Vallis, 2017),

$\displaystyle\frac{DQ}{Dt}=Q_{t}-\Psi_{y}Q_{x}+\Psi_{x}Q_{y}=0\,,$ (1)

where $\Psi$ is the streamfunction for the predominantly horizontal flow and
the quasi-geostrophic potential vorticity is written as

$\displaystyle Q=f_{0}+\beta
y+\Psi_{xx}+\Psi_{yy}+\frac{f_{0}^{2}}{\rho}\left(\frac{\rho}{N^{2}}\Psi_{z}\right)_{z}.$
(2)

Here, $t$ is time and $x$, $y$, and $z$ are spatial coordinates in the zonal
(longitudinal), meridional (latitudinal), and vertical directions,
respectively; when these appear as a subscript, the meaning is partial
differentiation. The zonal and meridional wind components can be found as
$U=-\Psi_{y}$ and $V=\Psi_{x}$, respectively. In the quasi-geostrophic
framework, the static density and squared buoyancy frequency, $\rho$ and
$N^{2}$, are given functions of $z$.

The quasi-geostrophic equation can be derived by first applying the shallow
water approximation to the Navier-Stokes equations and then taking the limit
of strong stratification and rapid rotation. Conversely, the KA-I stability
condition can be extended from the quasi-geostrophic framework to the
primitive shallow-water framework (Ripa, 1983).

In (2), the terms $f_{0}+\beta y$ are the first two terms of the Taylor series
of the planetary vorticity, $f$, which is also called the Coriolis parameter.
The terms $\Psi_{xx}+\Psi_{yy}$ are the relative vorticity. The last term
involving two vertical derivatives is the stretching vorticity. If separation
of variables can be used in $z$, the stretching vorticity may be simplified
via the vertical eigenvalue problem

$\displaystyle\frac{f_{0}^{2}}{\rho}\left(\frac{\rho}{N^{2}}\varPhi_{z}\right)_{z}=-L_{d}^{-2}\varPhi\,,$
(3)

with suitable boundary conditions, in which case there is a different Rossby
deformation length, $L_{d}$, for each vertical mode. Usually it is the
smallest $L_{d}^{-2}$ (the first mode) that is of interest. Physically,
$L_{d}$ is the length scale that separates potential-energy dominated
structures, i.e., large-scale pressure highs and lows maintained by the
Coriolis effect, from kinetic-energy dominated structures, i.e., small-scale
pressure anomalies that get flattened by gravity. The physical role played by
$L_{d}$ is analogous to the Larmor radius in plasma flows (Hasegawa, 1985).

###  2.1 The eigenvalue problem for planetary atmosphere

It is common practice in geophysical fluid dynamics to simplify the problem by
studying a multi-layer shallow-water model in which constant-density layers
are arranged in a (hydro)statically stable manner, with low density overlying
high density. A two-layer model of this type, with a fully dynamic weather
layer that overlies a layer containing a deep jet profile, $U_{\rm deep}$, was
first applied to Jupiter by Ingersoll & Cuong (1981). In this case, the
potential vorticity (2) is replaced by

$\displaystyle Q=f_{0}+\beta y+\Psi_{xx}+\Psi_{yy}-L_{d}^{-2}(\Psi-\Psi_{\rm
deep})\,,$ (4)

where $L_{d}$ is the first Rossby deformation length and $\Psi_{\rm deep}$ is
the deep-layer streamfunction, also known as the dynamic topography. Majda &
Wang (2005) proved that $\Psi_{\rm deep}$ may be treated as steady as the
ratio of the depth of the deep layer to the weather layer approaches infinity.
Thus, we assume that $\Psi_{\rm deep}$ and $U_{\rm deep}=-(\Psi_{\rm
deep})_{y}$ are functions of $y$ only. The weather layer corresponds to the
first-baroclinic-mode structure of the atmosphere of a gas giant, while the
barotropic structure of the gas-giant interior is modelled by $\Psi_{\rm
deep}$. This two-layer configuration has traditionally been called the
“$1$–$\nicefrac{{1}}{{2}}$ layer” model, but was recently re-branded as the
“$1$–$\nicefrac{{3}}{{4}}$ layer” model when $U_{\rm deep}$ has an alternating
jet profile rather than being still ($U_{\rm deep}=0$), to emphasise the
effect on the weather-layer dynamics that the corresponding undulations of
$\Psi_{\rm deep}$ have via stretching vorticity (Flierl et al., 2019).

Conservation of quasi-geostrophic potential vorticity for the
$1$–$\nicefrac{{3}}{{4}}$ layer model (4) yields one non-linear equation (1)
in one unknown, $\Psi(x,y,t)$. Gas giants are observed to be predominantly
zonal, hence for the linear stability analysis the basic state is assumed to
be only a function of latitude, $y$. The $x$ dimension on a gas giant is
periodic, hence the $x$ and $t$ dependencies are represented with a Fourier
component by replacing $\Psi(x,y,t)$ in (4) with
$\Psi(y)+\delta\,\psi(y)\exp[ik(x-ct)]$, where $k$ and $c$ are the zonal
wavenumber and phase speed, respectively, and $\delta$ sets the scale of the
amplitude. The $y$ dimension is assumed to span a channel of width $L$ centred
on the origin, $y\in(-L/2,L/2)$. When considering Jupiter’s atmosphere, we
assume that the channel is contained inside the northern or southern
extratropical domain (i.e., poleward of the equatorial jet), where quasi-
geostrophic theory holds. The meridional boundary conditions on the
perturbations are Dirichlet, $\psi(-L/2)=0$ and $\psi(L/2)=0$, unless
otherwise stated.

The problem is linearised by restricting to $|\delta|\ll 1$ and retaining only
$O(\delta)$ terms. These standard assumptions yield a linear ordinary
differential equation that governs the meridional structure of small-amplitude
perturbations,

$\psi^{\prime\prime}-(k^{2}+L_{d}^{-2})\psi+\frac{Q^{\prime}}{U-c}\psi=0\,,\qquad
y\in\Omega\,,$ (5)

where $\Omega=(-L/2,L/2)$ and a prime denotes ordinary differentiation with
respect to $y$. Three length scales exist for this problem: the channel width,
$L$, the deformation length, $L_{d}$, and the scale at which the basic flow
varies, $L_{U}$. In what follows, dimensional expressions are retained when
addressing physical phenomena, but in the mathematical case studies
expressions are implicitly non-dimensionalised using the length scale $L_{U}$
(i.e., $L$ and $L_{d}$ are $L/L_{U}$ and $L_{d}/L_{U}$ in the dimensional
form, respectively).

In the $1$–$\nicefrac{{3}}{{4}}$ layer model, $U(y)=-\Psi^{\prime}(y)$ and
$Q^{\prime}(y)$ are linked by

$\displaystyle
Q^{\prime}(y)=\beta-U^{\prime\prime}(y)+L_{d}^{-2}\big{(}U(y)-U_{\rm
deep}(y)\big{)}\,.$ (6)

Therefore two of the three basic flow profiles, $U(y)=-\Psi^{\prime}(y)$,
$U_{\rm deep}(y)$, and $Q^{\prime}(y)$ may be specified independently, after
which the third is fixed. The importance of (6) in planetary science arises
from the fact that observing $U$ and $Q^{\prime}$ determines $U_{\rm deep}$,
providing new insights into the nature of deep jets (Dowling, 1995; Read et
al., 2006, 2009a, 2009b). Furthermore, as we shall see shortly, if the Rossby
Mach number profile is given, then $U$ and $Q^{\prime}$ can be related. This
is useful as $U$ is easily observable, while precise measurements of
$Q^{\prime}$ require accurate temperature retrievals and thus pose a
relatively greater challenge. Physically, the gas giant zonal winds are
observed to be quite stable, hence in planetary physics, determining neutral
Rossby Mach number profiles has been a focus. In the wider context, it is
important to note that our mathematical analysis holds for the quasi-
geostrophic equation linearised around a broad range of $U(y)$ and
$Q^{\prime}(y)$ profiles.

Given a wavenumber, $k\geq 0$, and deformation radius, $L_{d}>0$, equation (5)
and the boundary condition constitute an eigenvalue problem for the complex
phase speed, $c=c_{r}+ic_{i}$. It is easy to confirm that the complex
conjugate of the eigenvalue is also an eigenvalue. Therefore if there is a $k$
that yields $c_{i}\neq 0$, the flow is unstable. We assume that $U$ and
$Q^{\prime}$ are $C^{1}$, so unless $(U-c)$ vanishes somewhere, the
eigenfunctions belong to the function space

$\displaystyle C_{0}^{2}=\\{f\in
C^{2}(\overline{\Omega})|f(-L/2)=f(L/2)=0\\},$ (7)

where $\overline{\Omega}=[-L/2,L/2]$.

Note that in the case of no rotation and no stratification, $\beta=0$ and
$L_{d}\rightarrow\infty$, such that the PV gradient (6) is
$Q^{\prime}=-U^{\prime\prime}$ and (5) reduces to the original Rayleigh
equation. In this case, the critical latitudes are inflection points of
$U(y)$. For the equation allowing $\beta\neq 0$, the necessary and sufficient
conditions for stability have also been discussed in the context of plasma
physics (e.g., Numata et al., 2007; Zhu et al., 2018).

###  2.2 The reciprocal Rossby Mach number

If the flow is unstable, physical intuition suggests that the size of the
driving term in (5), $\psi\,Q^{\prime}/(U-c)$, is sufficiently larger than
that of the diffusion effect. The latter may be estimated by the eigenvalue,
$\kappa^{2}$, of

$\displaystyle\varphi^{\prime\prime}-(k^{2}+L_{d}^{-2})\varphi=-\kappa^{2}\varphi,\qquad
y\in\Omega$ (8)

with $\varphi(-L/2)=\varphi(L/2)=0$. The most dangerous mode would have the
least diffusion effect, so set $k=0$ and seek the smallest eigenvalue,
$\kappa_{0}^{2}$. This is one way to motivate the introduction of the
reciprocal Rossby Mach number,

$\displaystyle
M_{\alpha}^{-1}(y)=\frac{1}{\kappa_{0}^{2}}\frac{Q^{\prime}}{U-\alpha}\,,$ (9)

where the subscript alpha explicitly shows the dependence on a reference-frame
shift of the zonal wind, $\alpha\in\mathbb{R}$. At this stage, $\alpha$ is
arbitrary, but we shall see that in the Jupiter-style basic flows, it is
uniquely determined, and hence the subscript can be removed. Note from (6)
that $Q^{\prime}$ is invariant with respect to $\alpha$. Both $\kappa_{0}^{2}$
and its associated eigenfunction, $\varphi_{0}$, can be explicitly computed:

$\displaystyle\kappa_{0}^{2}=\frac{\pi^{2}}{L^{2}}+L_{d}^{-2},\qquad\varphi_{0}=\cos\left(\frac{\pi}{L}y\right).$
(10)

The sufficient conditions for stability can be expressed in terms of
$M_{\alpha}^{-1}$ as follows.

###### Theorem 1

Suppose there exists $\alpha$ such that $M_{\alpha}^{-1}\leq 0$ for all
$y\in\Omega$, or that $0\leq M_{\alpha}^{-1}\leq 1$ for all $y\in\Omega$. Then
there is no unstable mode for any $k$.

Here, the former and latter conditions correspond to KA-I and KA-II,
respectively. Mathematically, the stability conditions can be used in the
equality cases, although they are often omitted in the physics literature
(Dowling, 2014). If $Q^{\prime}(y)$ does not change sign over $\Omega$, then
there are no critical latitudes and we can always choose a large enough or
small enough $\alpha$ to satisfy KA-I, which recovers the Charney-Stern
condition.

Theorem 1 can be shown by Arnol’d’s method; first find a Hamiltonian of the
dynamical system and show that it is a Lyapunov function under the
assumptions. The method has a wide range of applications and can also derive
nonlinear stability with respect to finite amplitude disturbances. However, if
we restrict ourselves to the linear eigenvalue problem, Theorem 1 can be
proved by a more straightforward approach, as shown in Appendix A.

The assertion of the theorem remains the same with Neumann or periodic
boundary conditions for (8), in which case (10) is replaced by

$\displaystyle\kappa_{0}^{2}=L_{d}^{-2},\qquad\varphi_{0}=1\,.$ (11)

Stamp & Dowling (1993) found a basic flow where KA-II may give a sharp
stability boundary imposing periodicity in the meridional direction. They used
the assumption $Q^{\prime}=aL_{d}^{-2}U$ so that $M^{-1}_{0}$ becomes a
constant $a$, together with a sinusoidal zonal wind profile $U=\cos(2\pi
y/L)$. The numerical calculations suggest that whenever $M_{0}^{-1}>1$,
unstable modes exist for some $k$. One of our tasks in Section 4 is to provide
a mathematical proof of this empirical result for a broad range of $U(y)$.

##  3 Classification of the basic flows

In Section 4 we shall find a necessary condition for stability. Hereafter we
assume the following properties for the basic flow:

  * •

$Q^{\prime}(y)$, $U(y)$ are $C^{1}$.

  * •

The PV gradient $Q^{\prime}(y)$ changes sign somewhere in $\Omega$.

  * •

The zeros of $Q^{\prime}(y)$, the critical latitudes, are simple.

  * •

$Q^{\prime\prime}(y_{l})\neq 0$ for all
$y_{l}\in\mathcal{Q}=\\{y\in\Omega|Q^{\prime}(y)=0\\}$.

Before proceeding to the next section, here we define two classes of the basic
flows. This is important as the strength of the stability conditions varies
between the classes. It is convenient to define the set of the critical
latitudes $\mathcal{Q}=\\{y\in\Omega|Q^{\prime}(y)=0\\}$ and the set of the
zonal wind nodes (zeros) in a suitable Galilean frame
$\mathcal{U}_{\alpha}=\\{y\in\Omega|U(y)=\alpha\\}$; note that the number of
elements of the latter set depends on $\alpha$.

###  3.1 Class (i)

Figure 2: Classification of basic flow profiles. For all cases, we consider
the PV gradient $Q^{\prime}(y)$ profile shown in panel (a). The two critical
latitudes are indicated by the horizontal red dashed lines. The $U(y)$ profile
shown in panel (b) is Class (i). The $U(y)$ profile shown in panel (c) is
Class (ii) but not Class (i). The $U(y)$ profile shown in panel (d) is not
Class (ii).

Let us assume the flow properties stated above. We say a basic flow belongs to
Class (i) if there exists an $\alpha\in\mathbb{R}$ such that
$M_{\alpha}^{-1}(y)$ becomes a continuous and one-signed function (i.e. non-
negative or non-positive for all $y\in\Omega$) in $\Omega$.

Class (i) typically occurs when the critical latitudes and the nodes of
$U-\alpha$ coincide, i.e. $\mathcal{Q}=\mathcal{U}_{\alpha}$. For example,
this is the case when $Q^{\prime}$ in figure 2a is paired with $U$ in figure
2b. On the other hand, the basic flow is not Class (i) when $U$ is replaced by
the profile shown in figure 2c or d.

Note that $U^{\prime}(y_{l})\neq 0$ for all $y_{l}\in\mathcal{Q}$, because
otherwise $M_{\alpha}^{-1}$ becomes singular, in view of the last assumption
made above. It is easy to check that $M_{\alpha}^{-1}\neq 0$ for all
$y\in\Omega$. Moreover, we can show that the $\alpha\in\mathbb{R}$ that makes
$M^{-1}_{\alpha}$ continuous is uniquely determined, and $\alpha$ must be
chosen from the range of the zonal flow

$\displaystyle\mathcal{R}=\left(\min_{y\in\overline{\Omega}}U,\max_{y\in\overline{\Omega}}U\right).$
(12)

The proof is as follows. If there exists $y_{l}\in\mathcal{Q}$ that does not
belong to $\mathcal{U}_{\alpha}$, the function $M^{-1}_{\alpha}$ cannot be
continuous. If there exists $y_{j}\in\mathcal{U}_{\alpha}$ that does not
belong to $\mathcal{Q}$, the function $M^{-1}_{\alpha}$ must change sign
there; note that $Q^{\prime\prime}(y_{l})\neq 0$ by assumption. There should
be at least one critical latitude by assumption, so $\alpha\in\mathcal{R}$.

The importance of Class (i) can be seen from the fact that it is the case that
Theorem 1 may be able to show stability, when there is a critical latitude.
The KA-I and KA-II stability conditions can be combined; the flow is stable if

$\displaystyle\max_{y\in\overline{\Omega}}M^{-1}\leq 1.$ (13)

Here the reciprocal Rossby Mach number is simply written as $M^{-1}$, as the
choice of $\alpha$ is trivial. Note if $M_{\alpha}^{-1}(y)$ changes sign (i.e.
the basic flow is not Class (i)) mathematically the condition (13) cannot
guarantee stability.

Figure 3:  Comparison of Jupiter profiles of $U$ (solid) and $Q_{y}L_{d}^{2}$
(dashed), using the same data as in figure 1. The deformation length is set
here to $L_{d}=c_{\text{gw}}/|f_{0}|=1750\,\rm km$, using the gravity wave
speed, $c_{\text{gw}}=454\,\rm m\,s^{-1}$, inferred from the Comet Shoemaker-
Levy 9 impact analysis (Hammel et al., 1995) and the Coriolis parameter,
$f_{0}$, evaluated at the G-fragment impact site, latitude
$\phi_{0}=-47.5^{\circ}$ (planetographic); the figure is centred on
$\phi_{0}$. Since $L/L_{d}\gg 1$ on Jupiter, it follows that
$\kappa_{0}^{2}\approx L_{d}^{-2}$, such that the similarity between the two
profiles implies $M^{-1}\approx 1$ with $\alpha\approx 0$. This, and further
evidence discussed in the text, support the identification of Class (i) as the
Jupiter-style class.

Class (i) is an extension of class $\mathcal{H}$ in Howard (1964), and class
$\mathcal{K}^{+}$ in Lin (2003) to the quasi-geostrophic equation, to which
the “Jupiter-style shear flow” (Dowling, 2020; Afanasyev & Dowling, 2022)
belongs. As briefly commented in section 1, observations support that $M^{-1}$
is around unity in Jupiter’s and Saturn’s atmosphere (Dowling, 1993; Read et
al., 2006, 2009a, 2009b). This configuration can be inferred from the most
accurate available observational data, as depicted in figure 3.

To understand why such a mean flow is achieved, it is necessary to consider
the nonlinear evolution of the disturbances and their mutual interaction with
the mean flow. It is shown in Dowling (1993) that when Jupiter’s observed
basic flow is made the initial condition for a local shallow water model, it
will rapidly evolve to become marginally stable. In addition, slowly moving,
planetary-scale thermal features have been regularly observed on Jupiter
(Fisher et al., 2016, and references therein). These lines of evidence suggest
that a gas giant’s alternating jets tend to evolve until the longest Rossby
waves are coherent across them, becoming stationary with respect to deep-
seated, large-scale pressure anomalies; this has been called a “princess and
the pea” phenomenon (Stamp & Dowling, 1993).

The basic flows that are not Class (i) may also be physically important. For
example, $M_{\alpha}^{-1}$ changes sign on a seasonal basis in Earth’s
atmosphere (Du et al., 2015). For this reason we next consider a class larger
than Class (i).

###  3.2 Class (ii)

Figure 4: The PV gradient profile $Q^{\prime}(y)$ defines the sets
$\Omega_{+}=\\{y\in\Omega|Q^{\prime}(y)>0\\}$,
$\Omega_{-}=\\{y\in\Omega|Q^{\prime}(y)<0\\}$. For the zonal wind profiles
$U(y)$ shown in panels (b) and (c), the set
$\mathcal{D}=\mathcal{R}\backslash(\\{U(y)|y\in\Omega_{+}\\}\cup\\{U(y)|y\in\Omega_{-}\\})$
has two and one points, respectively. Therefore both cases are Class (ii).

Again assume the flow properties listed at the beginning of Section 3. We say
a basic flow belongs to Class (ii) if there exists an $\alpha\in\mathcal{R}$
satisfying the following conditions, called Class (ii) conditions.

  * •

$M_{\alpha}^{-1}(y)$ is continuous.

  * •

$M_{\alpha}^{-1}(y_{j})$, where $y_{j}\in\mathcal{U}_{\alpha}$, are one-
signed, and at least one of them is non-zero.

Note that if a basic flow is Class (i), then it is Class (ii). However, for
Class (ii) basic flows, in general there can be more than one $\alpha$ that
make $M_{\alpha}^{-1}$ continuous. In figures 2a,c, for $\alpha=0$ the lower
critical latitude cancels the singularity and $M_{\alpha}^{-1}$ changes sign
at the upper critical latitude. Alternatively, an appropriately negative
$\alpha$ may be chosen such that $U-\alpha$ vanishes at the upper critical
latitude.

In order to see the physical implication of the second condition in the bullet
points above, let us consider the neutral solutions, setting $c_{i}=0$. The
key is that they can potentially become singular at the critical layers, the
points at which $U(y)$ coincides with the phase speed $c_{r}$. Mathematically
the set of the critical layers,
$\mathcal{U}_{c_{r}}=\\{y\in\Omega|U(y)=c_{r}\\}$, are regular singular points
of (5) when $c_{i}=0$. The appropriate tool for analysing the behaviour of
solutions around such singularities is Frobenius’ method, from which it can be
shown that $\psi$ is continuous but $\psi^{\prime}$ may be discontinuous at
the critical layers (Lin, 1955). A necessary and sufficient condition for no
jumps to exist at a critical layer is that either $Q^{\prime}=0$ or $\psi=0$,
or both, occur there. Here, following Drazin & Reid (1981), we briefly explain
the nature of the the critical layers without going into the details of
Frobenius’ method.

Because of the singularities, neutral solutions must be understood as the
vanishing $c_{i}$ limit in unstable solutions. Let us multiply (5) by
$\psi^{*}$ and take the imaginary part, keeping finite $c_{i}$:

$\displaystyle\psi^{*}\psi^{\prime\prime}-\psi\psi^{*}\,{}^{\prime\prime}+\frac{2ic_{i}Q^{\prime}}{(U-c_{r})^{2}+c_{i}^{2}}|\psi|^{2}=0\,.$
(14)

Integrate (14) across a critical layer, $y=y_{c}$ (i.e., $U(y_{c})=c_{r}$),
and take the limit $c_{i}\rightarrow 0+$,

$\displaystyle[\psi^{*}\psi^{\prime}-\psi\psi^{*}\,{}^{\prime}]^{y_{c}+}_{y_{c}-}$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\rightarrow
0+}\lim_{c_{i}\rightarrow
0+}\int^{y_{c}+\epsilon}_{y_{c}-\epsilon}\frac{-2ic_{i}Q^{\prime}}{(U-c_{r})^{2}+c_{i}^{2}}|\psi|^{2}dy$
(15)

An intuitive way to show the limit is as follows. If $\epsilon>0$ is
sufficiently small, we may be able to assume that $Q^{\prime}|\psi|^{2}$ is
almost a constant, and that $U-c_{r}$ is approximately
$U_{c}^{\prime}(y-y_{c})$, where $U^{\prime}_{c}=U^{\prime}(y_{c})$. Then the
right hand side of (15) can be explicitly worked out by using the well-known
formula that can be found by differentiating the arctangent function:

$\displaystyle-2i(Q^{\prime}|\psi|^{2})|_{y=y_{c}}\lim_{\epsilon\rightarrow
0+}\lim_{c_{i}\rightarrow
0+}\frac{2}{U_{c}^{\prime}}\arctan\left(\frac{\epsilon
U_{c}^{\prime}}{c_{i}}\right)=-2i\pi\left.\left(\frac{Q^{\prime}|\psi|^{2}}{|U^{\prime}|}\right)\right|_{y=y_{c}}.$
(16)

The above result suggests that if multiple critical layers appear as
$\mathcal{U}_{c_{r}}=\\{y_{1},y_{2},\dots,y_{N}\\}$, integrating (14) over
$\Omega$ yields

$\displaystyle
0=\sum_{j=1}^{N}\left.\left(\frac{Q^{\prime}|\psi|^{2}}{|U^{\prime}|}\right)\right|_{y=y_{j}}.$
(17)

If $Q^{\prime}=0$ or $\psi=0$ happen at all the critical layers, there are no
jumps at all, and hence the neutral solution $\psi$ is real and $C^{2}_{0}$.
Here, it is convenient to define terminology to distinguish between neutral-
mode types:

Pathological mode

no critical layers (empty $\mathcal{U}_{c_{r}}$) or $\psi$ vanishes at all
critical layers;

Singular mode

eigenfunction is not $C^{2}_{0}$;

Regular mode

eigenfunction is not pathological and is $C^{2}_{0}$.

Our strategy in Section 4 is similar to the two-step procedure in Howard
(1964) and Lin (2003): first prove the conditions that guarantee the existence
of a neutral solution, and then establish the existence of unstable modes in
its neighbourhood in parameter space. A regular mode must exist for these
processes to take place, and guaranteeing this essentially corresponds to the
second Class (ii) condition – for the singular modes we cannot use the
standard Sturm-Liouville theory, and for the pathological modes we have
difficulty in showing neighbourhood instability.

To check if a basic flow belongs to Class (ii), it is useful to define the
sets $\\{U(y)|y\in\Omega_{+}\\}$ and $\\{U(y)|y\in\Omega_{-}\\}$, decomposing
$\Omega$ into the three parts, $\Omega_{+}=\\{y\in\Omega|Q^{\prime}(y)>0\\}$,
$\Omega_{-}=\\{y\in\Omega|Q^{\prime}(y)<0\\}$, and $\mathcal{Q}$. Then if the
set

$\displaystyle\mathcal{D}=\mathcal{R}\backslash(\\{U(y)|y\in\Omega_{+}\\}\cup\\{U(y)|y\in\Omega_{-}\\})$
(18)

is non-empty, we can select $\alpha$ from this set to show that the basic flow
is Class (ii). If $y_{j}\in\Omega$, $j=1,2,\dots$ satisfies $U(y_{j})=\alpha$,
then $Q^{\prime}(y_{j})=0$ for all $j$ because neither $Q^{\prime}(y_{j})>0$
nor $Q^{\prime}(y_{j})<0$ happen. Thus $M^{-1}_{\alpha}(y)$ is a continuous
function. Moreover, for sufficiently small $\epsilon>0$, either
$\alpha+\epsilon\in\\{U(y)|y\in\Omega_{+}\\}$ or
$\alpha+\epsilon\in\\{U(y)|y\in\Omega_{-}\\}$ occurs. In the former case it
can be checked that $M^{-1}|_{y=y_{j}}>0$ for all $j$ and in the latter case
that $M^{-1}|_{y=y_{j}}<0$ for all $j$.

To visually check whether the basic flow is Class (ii), first find
$\Omega_{+}$ and $\Omega_{-}$ using $Q^{\prime}(y)$ and then illustrate
$\\{U(y)|y\in\Omega_{+}\\}$ and $\\{U(y)|y\in\Omega_{-}\\}$. For example, with
$U(y)$ shown in figure 4b and c, the number of elements in $\mathcal{D}$ is
one and two, respectively. The condition that the point at which
$\\{U(y)|y\in\Omega_{+}\\}$ and $\\{U(y)|y\in\Omega_{-}\\}$ are separated be
in the range of $U(y)$ is quite loose, such that a wide array of basic flows
belongs to Class (ii). In particular, if $U(y)$ is monotonic and there is at
least one critical latitude, $\mathcal{D}$ should be non-empty. On the other
hand, for the basic flow $U(y)$ shown in figure 2d, $\mathcal{D}$ is empty,
and in fact the basic flow cannot be Class (ii).

##  4 A handy necessary condition for stability: the hurdle theorem

###  4.1 A sufficient condition for existence of a neutral mode

Let us assume that for some $\alpha\in\mathbb{R}$ the reciprocal Rossby Mach
number $M^{-1}_{\alpha}(y)$ becomes continuous. Write $\lambda=-k^{2}$. Then
from (5) the neutral modes with the phase speed $c=\alpha$ must satisfy

$\displaystyle-\psi^{\prime\prime}+(L_{d}^{-2}-\kappa_{0}^{2}M_{\alpha}^{-1})\psi=\lambda\psi,\qquad
y\in(-L/2,L/2),$ (19)

and the Dirichlet boundary conditions. This is a regular Sturm-Liouville
problem with eigenvalue $\lambda$. It is well-known that all eigenvalues are
real, and they can be ordered as $\lambda_{0}<\lambda_{1}<\lambda_{2}<\dots$,
where $\lambda_{n}\rightarrow\infty$ as $n\rightarrow\infty$. By integration
by parts, it is easy to check that the associated eigenfunctions,
$\psi_{0},\psi_{1},\psi_{2},\dots$, are real and satisfy the equation

$\displaystyle
R_{\alpha}(\psi_{n})\equiv\frac{\int_{\Omega}\\{|\psi_{n}^{\prime}|^{2}+(L_{d}^{-2}-\kappa_{0}^{2}M_{\alpha}^{-1})|\psi_{n}|^{2}\\}dy}{\int_{\Omega}|\psi_{n}|^{2}dy}=\lambda_{n}.$
(20)

Here the Rayleigh quotient, $R_{\alpha}$, depends on $\alpha$. It should also
be noted from Sturm’s oscillation theorem that the $n$th eigenfunction
$\psi_{n}$ has $n$ zeros in the interval $(-L/2,L/2)$, a useful fact to be
used in later sections.

The minimum eigenvalue $\lambda_{0}$ can be found by the optimisation problem

$\displaystyle\lambda_{0}=\min_{\phi\in C^{2}_{0}}R_{\alpha}(\phi),$ (21)

where the unique minimiser is the zeroth eigenfunction $\phi=\psi_{0}$. Note
that there is no problem to extend the search space to

$\displaystyle
H_{0}^{1}=\left\\{\phi\left|\int_{\Omega}|\phi|^{2}dy<\infty,\int_{\Omega}|\phi^{\prime}|^{2}dy<\infty,\phi(-L/2)=\phi(L/2)=0\right.\right\\}$
(22)

essentially because the eigenfunctions of the Sturm-Liouville problem are a
complete orthonormal basis of $H_{0}^{1}$. Here, following the usual notation
in mathematics, $H$ implies that the space is a Hilbert space, the superscript
$1$ implies that the square integrability of the first derivative, and the
subscript $0$ implies that the Dirichlet boundary conditions are satisfied.

A sufficient condition for existence of a neutral mode is then summarised as
follows.

###### Theorem 2

Suppose there is $\alpha\in\mathbb{R}$ that makes $M^{-1}_{\alpha}$
continuous. If there exists $g(y)\in H^{1}_{0}$ such that $R_{\alpha}(g)\leq
0$, there is a neutral mode with the phase speed $c_{r}=\alpha$.

If the assumption of the theorem is met, a neutral mode can be found because
$\lambda_{0}=\min_{\phi\in H_{0}^{1}}R(\phi)\leq R(g)\leq 0$ implies that the
minimiser of (21) is the neutral eigenfunction having the wavenumber
$k=k_{0}\equiv\sqrt{-\lambda_{0}}\geq 0$. For this mode, the critical layers
(the points $y_{j}$ at which $U(y_{j})=c_{r}$ is satisfied) appear on the
critical latitudes (i.e. $\mathcal{Q}$). For Class (i) basic flows, the
critical layers and critical latitudes are the same so that
$\mathcal{Q}=\mathcal{U}_{c_{r}}$ for regular modes, and there are no singular
modes. However, in general, the position at which $Q^{\prime}$ disappears may
not be a critical layer of a regular mode (see figure 4b, for example).
Furthermore, the critical layers of a singular mode may not belong to
$\mathcal{Q}$.

In the next section, we shall show the following theorem:

###### Theorem 3

Suppose the basic flow is Class (ii), and fix $\alpha$ so that the Class (ii)
conditions are satisfied. If there exists $g(y)\in H^{1}_{0}$ such that
$R_{\alpha}(g)<0$, the flow is unstable. Furthermore, if the basic flow is
Class (i), a necessary and sufficient condition for stability is that no such
$g(y)\in H^{1}_{0}$ exists.

The second half of the theorem is somewhat similar to that derived by Howard
(1964) and Lin (2003) for shear flows, but the first half is entirely new. We
also remark that in the case of the Rayleigh equation, it can be proved that
the phase speed of the neutral solution is in the range of $U(y)$, denoted
$\mathcal{R}$, by Howard’s semicircle theorem, and that there are no
pathological modes when $k>0$. Those facts are used, for example, in proofs by
Howard (1964), Balmforth & Morrison (1999), and Hirota et al. (2014), but they
do not in general apply to stability problems more complex than the Rayleigh
equation. This is essentially the reason why Tung (1981) struggled to
incorporate the effect of non-zero $\beta$ into the theory, but as we will see
in the next section, the solution is, in fact, simple.

###  4.2 Existence of an unstable mode

Here we show Theorem 3. The proof is rather technical and readers who are not
interested in the mathematical details may skip this subsection without losing
the thread of the discussion. We first note that upon writing $\gamma\equiv-
L_{d}^{-2}-k^{2}=\lambda-L_{d}^{-2}$, (5) becomes

$\displaystyle\psi^{\prime\prime}+\left(\frac{Q^{\prime}}{U-c}+\gamma\right)\psi=0.$
(23)

The corresponding dispersion relation, $F(c,\gamma)=0$, can be formulated by
two linearly independent solutions of (23). They are analytic functions in the
upper or lower half complex $c$-plane, and behave regularly with respect to
$\gamma$, and so does $F(c,\gamma)$. A neutral solution is obtained when
$c_{i}$ is brought close to zero in the dispersion relation. From the symmetry
of the inviscid problem, neutral solutions always exist as pairs, i.e. the
$c_{i}=0+$ mode and the $c_{i}=0-$ mode.

Let us suppose that the assumptions of Theorem 3 are satisfied. From Theorem 2
we know that there is a neutral mode $\psi_{0}$ with wavenumber
$k_{0}=\sqrt{-\lambda_{0}}>0$ and phase speed $\alpha$. This mode satisfies

$\displaystyle\mathcal{L}\psi_{0}\equiv\psi_{0}^{\prime\prime}+\left(\frac{Q^{\prime}}{U-\alpha}+\gamma_{0}\right)\psi_{0}=0.$
(24)

Here, we define the linear operator $\mathcal{L}$ for later use.

In view of the argument at the beginning of this section, we can compute the
coefficients of the Taylor expansion of $c(\gamma)$ around the neutral mode in
the upper half complex plane. Writing
$\gamma_{0}\equiv\lambda_{0}-L_{d}^{-2}$, around the neutral mode, the
following expansion holds,

$\displaystyle
c_{i}(\gamma)=\left.\frac{dc_{i}}{d\lambda}\right|_{0}(\gamma-\gamma_{0})+O((\gamma-\gamma_{0})^{2})\,.$
(25)

Here and hereafter, the symbol ‘$|_{0}$’ attached to a quantity means that it
is evaluated at the $c_{i}=0+$ neutral mode. The expression (25) is valid as
long as $c_{i}$ does not become negative. Our goal below is to show
$\left.\frac{dc_{i}}{d\gamma}\right|_{0}\neq 0$; then (25) implies that an
unstable mode can be found by slightly varying $\gamma$ from the neutral
value.

Differentiate (23) by $\gamma$,

$\displaystyle\psi_{\gamma}^{\prime\prime}+\left(\frac{Q^{\prime}}{U-c}+\gamma\right)\psi_{\gamma}+\left(1+\left(\frac{Q^{\prime}}{U-c}\right)_{\gamma}\right)\psi=0\,,$
(26)

where the subscript $\gamma$ denotes the differentiation. Evaluate this
equation at the neutral parameter,

$\displaystyle\mathcal{L}\psi_{\gamma}|_{0}+\left(1+\left.\left(\frac{Q^{\prime}}{U-c}\right)_{\gamma}\right|_{0}\right)\psi_{0}=0\,.$
(27)

Now, combine (24) and (27) to obtain

$\displaystyle
0=\int_{\Omega}(\psi_{0}\mathcal{L}\psi_{\gamma}|_{0}-\psi_{\gamma}|_{0}\mathcal{L}\psi_{0})dy+\int_{\Omega}\left(1+\left.\left(\frac{Q^{\prime}}{U-c}\right)_{\gamma}\right|_{0}\right)\psi_{0}^{2}dy\,.$
(28)

The first integral vanishes after integration by parts. Therefore

$\displaystyle\left.\frac{dc}{d\gamma}\right|_{0}=-\frac{\int_{\Omega}\psi_{0}^{2}dy}{K}=-\frac{(K_{r}-iK_{i})\int_{\Omega}\psi_{0}^{2}dy}{K_{r}^{2}+K_{i}^{2}}\,,$
(29)

where

$\displaystyle K=K_{r}+iK_{i}=\lim_{c_{i}\rightarrow
0+}\int_{\Omega}\frac{Q^{\prime}\psi_{0}^{2}}{(U-\alpha-ic_{i})^{2}}dy\,.$
(30)

The limit can be worked out as

$\displaystyle
K=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Omega}\frac{Q^{\prime}\psi_{0}^{2}}{(U-\alpha)^{2}}dy+i\pi\kappa_{0}^{2}\sum_{j=1}^{N}\left.\left(\frac{M_{\alpha}^{-1}\psi_{0}^{2}}{|U^{\prime}|}\right)\right|_{y=y_{j}}\,,$
(31)

noting that the integrand of (30) becomes singular at
$\mathcal{U}_{\alpha}=\\{y_{1},y_{2},\dots,y_{N}\\}$. The dashed integral
represents the Cauchy principle value integral.

Here, since $\alpha\in\mathcal{R}$, the set $\mathcal{U}_{\alpha}$ is non-
empty. The terms in the summation in (31) cannot cancel out because all the
$M_{\alpha}^{-1}(y_{j})$ are one-signed from the second Class (ii) condition.
Also $U^{\prime}(y_{j})\neq 0$ for all $j$, as otherwise $M_{\alpha}^{-1}(y)$
cannot be continuous. Moreover, $\psi_{0}$ does not vanish in $\Omega$ because
it is the least oscillatory eigenmode, as remarked just below (20). From (31)
$K_{i}$ is non-zero, and therefore we can conclude
$\left.\frac{dc_{i}}{d\gamma}\right|_{0}\neq 0$ using (29).

We still need to show the latter half of Theorem 3 for Class (i) basic flows.
It is sufficient to consider the case $M_{\alpha}^{-1}(y)>0$ for all $y$, as
otherwise the flow must be stable from Theorem 1. Suppose
$R_{\alpha}(\phi)\geq 0$ for all $\phi(y)\in H^{1}_{0}$. In this case, if an
unstable mode $\psi$ exists, the following integral must be fulfilled:

$\displaystyle\int_{\Omega}|\psi^{\prime}|^{2}+(L_{d}^{2}-\kappa_{0}^{2}M_{\alpha}^{-1})|\psi|^{2}dy\geq
0.$ (32)

However, for any unstable mode we can deduce (53), which yields the
contradicting result

$\displaystyle\kappa_{0}^{2}\int_{\Omega}M_{\alpha}^{-1}|\psi|^{2}dy>\int_{\Omega}\\{|\psi^{\prime}|^{2}+L_{d}^{-2}|\psi|^{2}\\}dy$
(33)

when $M_{\alpha}^{-1}(y)>0$ for all $y$. Thus, the flow must be stable.

###  4.3 Simple integral-based stability conditions

If a function $g\in H^{1}_{0}$ that makes the Rayleigh quotient negative is
found, the existence of instability is guaranteed by Theorem 3. The remaining
question for deriving simple necessary conditions for stability is how to
choose $g$. A good test function is $g=\varphi_{0}$, the function introduced
in (10). Using (20), the condition $R_{\alpha}(\varphi_{0})<0$ becomes

$\displaystyle
1<\frac{2}{L}\int^{\frac{L}{2}}_{-\frac{L}{2}}M_{\alpha}^{-1}\cos^{2}\left(\frac{\pi
y}{L}\right)dy\,.$ (34)

Note that this result depends on the boundary conditions. For example, if
periodic boundary conditions are used, the eigenvalue problem (8) produces
(11), and thus the condition (34) must be replaced by

$\displaystyle
1<\frac{1}{L}\int^{\frac{L}{2}}_{-\frac{L}{2}}M_{\alpha}^{-1}dy\,.$ (35)

This explains the numerical results of Stamp & Dowling (1993), because the
basic flow is clearly Class (i), and when $M_{0}^{-1}$ is a constant,
according to (35), an unstable mode should appear whenever $M_{0}^{-1}>1$. The
stability condition (13), now reduced to $M_{0}^{-1}\leq 1$, and the KA-II
stability boundary, $M_{0}^{-1}=1$, is indeed sharp. However, if the constant
nature of the reciprocal Mach Rossby number is not perfect, as in Jupiter’s
atmosphere, the discussion here does not guarantee sharpness.

Figure 5: An example of unstable $M^{-1}_{\alpha}(y)$ profile. Observe that
the blue curve overcomes the hurdle $h$.

###  4.4 A simple stability condition using a hurdle

The above conditions require us to examine the profile of $M^{-1}_{\alpha}(y)$
across the entire domain, $\Omega$. This may not be convenient in terms of
application to planetary atmospheres, for example, the quasi-geostrophic
approximation may not be accurate near the equatorward boundary. Fortunately,
it turns out we can show the existence of instability by just checking a local
part of the basic flow. Consider a comparison problem in a subinterval
$[y_{1},y_{2}]\subset\Omega$ with some $\widetilde{M}^{-1}(y)$ profile.

$\displaystyle\widetilde{\psi}^{\prime\prime}-(\widetilde{k}^{2}+L_{d}^{-2})\widetilde{\psi}+\kappa_{0}^{2}\widetilde{M}^{-1}\widetilde{\psi}=0,\qquad
y\in(y_{1},y_{2})\,.$ (36)

We impose the Dirichlet condition,
$\widetilde{\psi}(y_{1})=\widetilde{\psi}(y_{2})=0$. If this problem has a
neutral mode, $\widetilde{\psi}_{0}(y)$ say, and
$\widetilde{M}^{-1}<M_{\alpha}^{-1}$ on $[y_{1},y_{2}]$, the original problem
(19) must be unstable. The reason is as follows. We set the test function as

$g(y)=\begin{cases}\widetilde{\psi}_{0}(y)\,,&\text{if $y\in[y_{1},y_{2}]$}\\\
0\,,&\text{otherwise}\end{cases}$ (37)

using the neutral mode. Then the Rayleigh quotient can be computed as

$\displaystyle R(g)$ $\displaystyle=$
$\displaystyle\frac{\int_{y_{1}}^{y_{2}}|\widetilde{\psi}_{0}^{\prime}|^{2}+(L_{d}^{-2}-\kappa_{0}^{2}M_{\alpha}^{-1})|\widetilde{\psi}_{0}|^{2}dy}{\int_{y_{1}}^{y_{2}}|\widetilde{\psi}_{0}|^{2}dy}$
(38) $\displaystyle<$
$\displaystyle\frac{\int_{y_{1}}^{y_{2}}|\widetilde{\psi}_{0}^{\prime}|^{2}+(L_{d}^{-2}-\kappa_{0}^{2}\widetilde{M}^{-1})|\widetilde{\psi}_{0}|^{2}dy}{\int_{y_{1}}^{y_{2}}|\widetilde{\psi}_{0}|^{2}dy}=-\widetilde{k}^{2}\leq
0\,,$

where $\widetilde{k}$ is the wavenumber of the neutral mode
$\widetilde{\psi}_{0}(y)$. A particularly convenient comparison problem is
when $\widetilde{M}^{-1}$ is a constant, since the problem can be solved
analytically using trigonometric functions. It is easy to check that a neutral
mode with a wavenumber $\widetilde{k}$ can be found when

$\displaystyle\widetilde{M}^{-1}=\frac{\frac{\pi^{2}}{(y_{2}-y_{1})^{2}}+(\widetilde{k}^{2}+L_{d}^{-2})}{\kappa_{0}^{2}}\,.$
(39)

We want to make $\widetilde{M}^{-1}$ as small as possible to produce
instability criteria as sharp as possible, so we let $\widetilde{k}\rightarrow
0$. Then the condition $\widetilde{M}^{-1}<M_{\alpha}^{-1}$ on $[y_{1},y_{2}]$
yields the main result of this article:

###### Theorem 4

Suppose the basic flow is Class (ii). Fix an $\alpha$ so that the Class (ii)
conditions are satisfied. The flow is unstable if there is an interval
$[y_{1},y_{2}]\subset\Omega$ over which

$\displaystyle
h\equiv\frac{\frac{\pi^{2}}{(y_{2}-y_{1})^{2}}+L_{d}^{-2}}{\frac{\pi^{2}}{L^{2}}+L_{d}^{-2}}<M_{\alpha}^{-1}$
(40)

is satisfied.

In short, if the reciprocal Rossby Mach number profile surpasses the hurdle
$h$ as shown in figure 5, the flow is unstable. As can be seen from the proof,
this result is actually independent of the boundary conditions. In Section 5,
we stress-test this ‘hurdle theorem’ with model basic flow profiles.

Note that the height of the hurdle, $h$, is always higher than unity, and
depends on the width of the hurdle. However, if
$L_{d}^{-2}\gg\pi^{2}/(y_{2}-y_{1})^{2}$, the hurdle height is almost 1. Thus,
in this limit the condition for the existence of instability asymptotes to

$\displaystyle\max_{y\in\overline{\Omega}}M_{\alpha}^{-1}>1.$ (41)

This is the contrapositive of (13), implying that the KA-II stability
condition is almost sharp if the basic flow is Class (i) and $L_{d}$ is small.

On Jupiter, the jet wavelength is $L_{U}\approx 2\pi L_{d}$, as can be seen
from figure 1 using the fact that $1^{\circ}$ latitude is approximately
$1200\,\rm km$; a similar relationship holds for Saturn. On the other hand,
the length scale of the variation of $M^{-1}$ seems slower. For example, in
figure 1, the nature of the correlation differs between the latitude ranges of
$-30^{\circ}$ to $-20^{\circ}$ and $-50^{\circ}$ to $-30^{\circ}$. Even if the
variation of $M^{-1}$ is considered to be about $L_{U}$, the sharpness (41)
reasonably holds, because using $y_{2}-y_{1}\approx 2\pi L_{d}$, the hurdle
height is $h\approx 1.25$. Note that we can assume $L\gg L_{d}$ for planetary
atmosphere problems.

##  5 Analysis of model profiles

In the first two subsections below, we deal with a model that generalises the
Stamp & Dowling (1993) basic flow. In the parameter range treated in section
5.1, the basic flow belongs to Class (i), whereas in section 5.2, an example
is analysed that showcases the typical phenomena that arise when the basic
flow belongs to Class (ii) and not to Class (i). In section 5.3, we clarify
where our analysis stands in the long history of shear flow research using the
Rayleigh equation. Our conditions have broader applicability compared to
existing ones, however, there may be cases where existing conditions are more
suitable when they can be applied. To demonstrate this, we purposefully choose
a basic flow for which the condition by Tollmien (1935) is applicable.

###  5.1 The quasi-geostrophic case: Class (i)

(a) (b)

Figure 6:  (a) The eigenmodes of (46) for $L=16,L_{d}=1,a=1.5,b=3.5$. There
are four neutral modes for this parameter set. The solid curves are the
theoretical results assuming large $L$ ($k\approx
1.225,0.6707,0.5494,0.2476$). The symbols are the numerical results ($k\approx
1.269,0.7178,0.6016,0.3564$). The red curve and the filled circles is the
localised mode. The green curves and the open circles are the oscillatory
modes. (b) The neutral $k$ for $L_{d}=1,a=1.5,b=3.5$. The solid curves are the
theoretical results, and the larger $L$, the better the agreement with the
numerical results, shown by the symbols. In the limit $L\rightarrow\infty$,
the oscillatory modes (green curves) form a continuous spectrum in the
interval $k\in(0,\sqrt{0.5})$. There is only one localised mode (red curve) at
$k=\sqrt{1.5}$.

(a) (b)

(c) (d)

Figure 7: (a) The theoretical neutral curve (47) for $L_{d}=1$. (b) Comparison
of the theoretical neutral curve (green) and the numerical results (black) for
$L_{d}=1$. The three numerical neutral curves are computed using $L=10,12,16$.
(c) Comparison of the numerical neutral curve (black) and the unstable
parameter region by Theorem 4 (red shaded). $L=16,L_{d}=1$. The $b$–$a$ plane
is used to make it easier to see the relationship with figure 10. (d) The
theoretical curve (47) for $L_{d}^{-1}=1,2,4,8$. The blue shaded region is
stable from Theorem 1.

Consider the quasi-geostrophic problem (5). In setting up a model basic flow,
we assume that the PV gradient profile has the form

$\displaystyle
Q^{\prime}(y)=\kappa^{2}_{0}\\{a+(b-a)\,\text{sech}^{2}(y)\\}U(y)\,,$ (42)

where $a,b>0$ are parameters. The advantage of using this designed basic flow
is that, regardless of $U(y)$, the reciprocal Rossby Mach number with the
choice $\alpha=0$ becomes

$\displaystyle M_{0}^{-1}(y)=a+(b-a)\,\text{sech}^{2}(y)\,.$ (43)

The profile (42) is an extension of the model for Jupiter’s atmosphere
considered by Stamp & Dowling (1993) to the case where $M_{0}^{-1}(y)$ is not
a constant. Indeed, the case $a=b$ corresponds to the PV profile considered in
Stamp & Dowling (1993). When $b>a$, the $M_{0}^{-1}$ profile has a hump of
height $b$ centred at $y=0$, and likewise a dip when $b<a$.

If $U(y)$ has a node (i.e. $0\in\mathcal{R}$), the existence of a critical
latitude is guaranteed. Then, since both $a$ and $b$ are positive, the
function (43) is one-signed for all $y$, implying that the model basic flow
belongs to Class (i), i.e. “Jupiter-style”. The two different zonal wind
profiles

$\displaystyle U(y)=\sin(2\pi y),$ (44) $\displaystyle U(y)=\tanh(y),$ (45)

are considered. Here, lengths are implicitly scaled by $L_{U}$. The profile
(44) may be regarded as a model for Jupiter’s alternating jets (with $L_{U}$
being the jet peak-to-peak length scale), while (45) is a simpler case where
there is only one critical latitude.

Recall that for Class (i) the phase speed of the regular neutral modes is
uniquely determined, and for our basic flow it is $0$. Thus, setting $c=0$ in
(5), the equation satisfied by the neutral solutions can be found as

$\displaystyle\psi^{\prime\prime}-(k^{2}+L_{d}^{-2}-\kappa_{0}^{2}M^{-1})\psi=0\,,$
(46)

simplifying the notation as $M^{-1}=M_{0}^{-1}$. This equation and the
Dirichlet boundary condition constitute an eigenvalue problem with
$\lambda=-k^{2}$ as the eigenvalue. Considering the model flow is emulating
the atmosphere of Jupiter, contemplating the case of large $L$ is natural. As
we shall see shortly, when $L$ is large, it is possible to study (46)
analytically.

(a)

(b)

Figure 8: The black crosses are numerical solutions of the eigenvalue problem
(5) with the PV gradient profile (42) and the parameters
$L=16,L_{d}=1,a=1.5,b=3.5$. The magenta squares are the neutral modes computed
in figure 6a. The dotted curves indicate the upper bound of $c_{i}$ by (68)
with (71). The panels on the right depict contour plots of the streamfunction
$\psi(y)e^{ikx}+\psi^{*}(y)e^{-ikx}$ at the wavenumber that maximises the
growth rate $kc_{i}$. The sign of the contours is distinguished by red and
blue. (a) $U=\sin(2\pi y)$, (b) $U=\tanh(y)$.

To see the behaviour of a typical neutral mode, let us choose the parameter
$L_{d}^{-2}(b-a)=2$ that makes the analysis simple. First, we note that
$\psi=\text{sech}y$ approximately satisfies (46) when
$k=\sqrt{1+L_{d}^{-2}(a-1)}$, because $\kappa_{0}^{2}\approx L_{d}^{-2}$. This
mode decays exponentially for large $|y|$ as shown by the red curve in figure
6a. Such localised modes do not interact with the boundary, and they are
robust to changes in $L$. However, there are also oscillatory modes that do
not show exponential behaviour near the boundaries; see the green curves in
figure 6a. Those curves can also be found theoretically. Let
$\omega=\sqrt{L_{d}^{-2}(a-1)-k^{2}}$ be real. Then $\psi=\omega\cos(\omega
y)-(\text{tanh}y)\sin(\omega y)$ and $\psi=\omega\sin(\omega
y)+(\text{tanh}y)\cos(\omega y)$ satisfy (46). If $L$ is large, the former and
latter solutions approximately satisfy the boundary conditions when
$\tan(\omega L/2)=\omega$ and $\tan(\omega L/2)=-\omega^{-1}$, respectively.
There are many $\omega$ that meet these requirements, and whenever
$k=\sqrt{L_{d}^{-2}(a-1)-\omega^{2}}$ is real, (46) has a neutral mode that
oscillates near the boundaries. The number of allowed values of $k$ increases
with increasing $L$, as shown in figure 6b, and in the limit of
$L\rightarrow\infty$ the eigenvalues of the oscillatory modes form a
continuous spectrum occupying $k\in(0,L_{d}^{-1}\sqrt{a-1})$.

For $L_{d}^{-2}(b-a)\neq 2$, the theoretical analysis is a bit more
complicated (Appendix C). However, the final results are neat and clean as
summarised in figure 7a, which shows the parameter plane with $b-1$ and $a-1$
as abscissa and ordinate. Clearly the third quadrant must be stable as labeled
because of the KA stability condition (13). Moreover, if $L$ is large, the
first and second quadrants are unstable; the easiest way to see this is to
note that the right hand side of (34) is almost $a$, because the integral is
largely unaffected by what happens in the hump or dip. The analysis in
Appendix C also shows that even if $a$ is slightly above 1, many oscillatory
modes will appear when $L$ is large. The stability of the flow is in fact non-
trivial only in the fourth quadrant of figure 7a, but somewhat surprisingly,
the stability in this region can be found analytically (see Appendix C). In
summary, the theoretical neutral curve for $L\gg 1$ can be described as

$a=\begin{cases}1\,,&\text{if $b\in(0,1)$}\\\ 1-L_{d}^{2}(b-1)^{2}\,,&\text{if
$b\geq 1\qquad.$}\end{cases}$ (47)

This neutral curve is determined by the zeroth mode, $\psi_{0}$, and as seen
in Section 4, it must delimit the stable and unstable parameter regions, as
shown in figure 7a. In the fourth quadrant, only localised modes appear, and
this is the reason why the neutral curve is so simple. Applying the shooting
or Chebyshev collocation method to (46), we can calculate the neutral curve
for finite $L$. The computational results indeed approach the theoretical
result (47) as $L$ is increased, as shown in figure 7b.

Without resorting to eigenvalue computations, Theorem 4 can give a rough idea
of the behaviour of neutral curves. In figure 7c is shown the unstable region
that can be found by setting the various hurdles for $M^{-1}(y)$. The
numerically calculated neutral curve should be sandwiched between the KA
stable region and the unstable region by the hurdle theory. The latter region
has a piecewise smooth boundary because the behaviour of hurdles in each
quadrant is different. The entire first quadrant is unstable, as can be found
by considering the full width hurdle of height unity. In the second quadrant
the hurdle giving the best results sits between one of the boundaries and the
dip. In the third quadrant, when $b$ is large enough the hump will be able to
hurdle over. For this to be seen, the value of $b$ must be at least larger
than 6 for $L_{d}=1$, meaning that the hurdle estimation does not give sharp
results. However, as noted just below (41), the smaller is $L_{d}$, the more
accurate is the hurdle at detecting instability. In line with this
expectation, the theoretical neutral curve approaches the KA stability
boundary with decreasing $L_{d}$ (figure 7d).

The regular neutral modes do not depend on $U(y)$, but the unstable modes
stemming from them do. Figure 8 shows the eigenvalue $c$ of (5) calculated for
the same parameters as in figure 6a. For the oscillatory zonal flow (44), all
instabilities persist down to $k=0$ (figure 8a). However, for the monotonic
zonal flow (45), the first and second neutral modes, as well as the third and
fourth neutral modes, are connected as seen in figure 8b, resulting in a
qualitatively completely different diagram. For both cases, the growth rate
$kc_{i}$ is well below the analytically derived upper bound (see Appendix D)
shown by the dotted curve. The streamfunction at the maximum growth rate is
indicated by arrows. The fastest growing mode inherits the properties of the
localised neutral mode and forms a strong vortex in the centre of the region.
The second and subsequent modes spread over the whole domain, like the
oscillatory neutral modes.

Figure 9:  Stability analysis in the fourth quadrant of figure 7a, with
$U=\sin(2\pi y)$. Red plus symbols: $L=16,L_{d}=1,a=0.5,b=2$. Green cross
symbols: $L=16,L_{d}=1/8,a=0.5,b=1.2$. In both cases, the most unstable
eigenfunctions are indicated by arrows.

Of greatest interest from a planetary physics perspective is the fourth
quadrant of figure 7a. When the basic flow $U=\sin(2\pi y)$ is used, the
spatial scale of the variations in $M^{-1}$ is about twice as much as those in
$U$, making it an ideal set up. Figure 9 shows the growth rates computed for
the two different parameter sets, $(L,L_{d},a,b)=(16,1,0.5,2)$ and
$(L,L_{d},a,b)=(16,1/8,0.5,1.2)$. The latter setting is close to the situation
found in Jupiter’s atmosphere, where as mentioned just below (4.23),
$L_{U}\approx 2\pi L_{d}$, such that $L_{d}\approx 1/2\pi$ in the non-
dimensional model context. In the chosen base flow, there exist many critical
latitudes (i.e., critical layers), but since $a=0.5$, most of the critical
latitudes are supersonic ($M>1$). For both eigenfunctions shown in figure 9,
one can observe that disturbances are localised near the centre of the domain,
i.e., where the critical latitudes are subsonic ($M<1$). This is not a
phenomenon specific to the current model, but is expected to be the case for
any $M^{-1}$ profile. This expectation arises from the fact that, according to
Sturm’s oscillation theorem, neutral modes exhibit oscillatory behaviour only
when $M^{-1}$ well exceeds unity, and it is at the critical latitude/layer
that vortices form in unstable modes. When $L_{d}$ is small, as soon as the
hump of $M^{-1}$ exceeds 1, an unstable mode occurs, as noted at the end of
section 4.4. This is why the eigenfunction for the $L_{d}=1/8$ case shown in
figure 9 has smaller vortices.

When $L_{d}$ is small, the middle critical latitude in the model is nearly
sonic, when the flow is nearly neutral. This situation is consistent with the
observations by Read et al. (2006, 2009a) where it is revealed that Jupiter
and Saturn each have at least a dozen stable (supersonic) or marginally stable
(sonic) critical latitudes. The alternating jets of those planets persist on a
decadal timescale (Porco et al., 2003), and as first spotted by Dowling &
Ingersoll (1989) and later confirmed by Marcus & Shetty (2011), Jupiter’s
Great Red Spot indeed straddles a critical latitude.

Conversely, if the method used here allows us to estimate $L_{d}$ from the
$M^{-1}$ profile under the assumption of neutrality, it has significance in
planetary physics. At least in the context of $1$–$\nicefrac{{3}}{{4}}$ layer
dynamics, we can calculate $U_{\rm deep}$ from equation (2.6) using a
numerically obtained neutral $L_{d}$. This has been historically important for
probing the deep jets on Jupiter (Dowling, 1995), but additional
considerations for Jupiter in combination with Juno gravity inversions of
interior circulations, and for Saturn in combination with ring-wave seismology
(i.e., kronoseismology), are warranted.

###  5.2 The quasi-geostrophic case: Class (ii)

Figure 10:  The stability of the basic flow $U=\tanh(y)$ with the parameters
$L=16,L_{d}=1$. The black curve is the numerically obtained neutral curve. The
red shaded area is the unstable parameter region found by Theorem 4.

(a) (b)

Figure 11: (a) Reciprocal Mach number profile for $L_{d}=1,L=16,a=0.5,b=-0.5$.
The basic flow $U=\tanh(y)$ is used. (b) The numerically obtained neutral
modes at the same parameters. They have the phase speed $c_{r}\approx 0.7071$.
Figure 12: The same plots as figure 8 but for $L_{d}=1,L=16,a=0.5,b=-0.5$. The
basic flow $U=\tanh(y)$ is used. The neutral solutions, indicated by the
magenta squares, are identical to those shown in figure 11b.

Interesting things happen when considering negative $b$ in (42). There is no
longer any guarantee that the basic flow is Class (i), as the sign of the
$M^{-1}_{0}$ profile (43) may change. The neutral curve does depend on the
choice of $U$, because the phase speed of neutral modes is not always zero.
Here we mainly consider the second zonal wind profile $U(y)=\tanh(y)$, fixing
$L=16,L_{d}=1$. When $a$ and $b$ are positive, the neutral curve is obtained
as shown in figure 7c. What happens if the neutral curve is extended to the
region where $b$ is negative? For example, consider the situation when $a$ is
smaller than 1 and $b$ is negative. Then clearly $M_{0}^{-1}$ will be
everywhere smaller than 1, so one might think that the existence of
instability cannot be said by Theorem 4. However, the results actually look
like figure 10. The emergence of the unexpected unstable region is due to the
modes with non-zero phase speed.

It is easy to see that the critical latitudes are zeros of $U(y)$ and
$\\{a+(b-a)\,\text{sech}^{2}(y)\\}$. For example, consider the case
$a=0.5,b=-0.5$. The zeros of $\\{a+(b-a)\,\text{sech}^{2}(y)\\}$ are at $\pm
0.8814$, and thus the critical latitudes are
$\mathcal{Q}=\\{-0.8814,0,0.8814\\}$. One of these three points must coincide
with the zero of the $U(y)-\alpha$, for $M^{-1}_{\alpha}$ to be continuous.
Thus there are regular neutral modes with $c_{r}=0$ and $c_{r}\approx\pm
U(0.8814)\approx\pm 0.7071$. If $\alpha=0$ is chosen, the $M^{-1}_{\alpha}$
profile is everywhere less than unity as shown by the red curve in figure 11a.
Hence, the flow is stable for steady modes. However, when $\alpha=0.7071$ is
used, $M^{-1}_{\alpha}$ exceeds 1 significantly, as indicated by the blue
curve in 11b.

Travelling wave type neutral modes with a phase speed of 0.7071 indeed appear
for the parameter choice $a=0.5,b=-0.5$ (figure 11b). They are clearly regular
modes. For $\alpha=0.7071$, Class (ii) conditions are satisfied. This follows
from the monotonicity of $U(y)$; see the comments at the end of section 3.2.
As would be expected from these facts, unstable modes appear when $k$ is
reduced from the neutral values (figure 12). The eigenfunctions with the
largest growth rates are similar to the neutral mode, with vortices
concentrated in regions where $M^{-1}_{\alpha}$ exceeds 1.

The fact that the neutral curve for $b<0$ depends on $U$ suggests that the
situation is much more complicated when the basic flow is not of Class (i).
For example, if the basic flow $U(y)=\sin(2\pi y)$ is used, the only reference
shift $\alpha$ that would make $M^{-1}_{\alpha}$ continuous is 0, when
$a>0,b<0$, and $L$ is large. However, this does not mean that considering only
steady modes is sufficient. The reason for this is that when
$\\{U(y)|y\in\Omega_{-}\\}\cap\\{U(y)|y\in\Omega_{+}\\}$ is non-empty, there
may be a singular neutral mode (observe that the summation in (17) may
cancel). Such singular modes are beyond the scope of this paper and there is
no need to consider them as long as we assume that Jupiter’s atmosphere is of
Class (i).

###  5.3 The Rayleigh equation case: internally heated flow

Figure 13: The stability of the Rayleigh equation with the model flow profile
(48). The blue regions are KA stable because there are no inflection points in
$U(y)$. The red region is unstable according to Theorem 4.

Here we consider the Rayleigh equation case, i.e.,
$Q^{\prime}=-U^{\prime\prime}$ and $L_{d}^{-2}=0$ in (5). The basic flow we
choose is

$\displaystyle U(y)=\frac{y^{4}-6y^{2}+5}{12}+a(1-y^{2})\,,$ (48)

which appears when internally heated fluid flows through a vertically oriented
channel; see Nagata & Generalis (2002). The walls are set at $y=-1$ and 1,
hence $L=2$. The parameter $a$ is the ratio of the Reynolds number to the
Grashof number. Based on the Orr-Sommerfeld numerical calculations, Uhlmann &
Nagata (2006) argued that for $O(1)$ wavenumbers, instability appears when
$a\in(-5/12,0)\approx(-0.4166,0)$, because of the existence of inflection
points and reverse flow. Note that unstable modes also exists at small
wavenumber parameter regions due to a viscous instability mechanism (i.e.,
Tollmien–Schlichting waves), but that is not the current focus.

The basic flow (48) has two inflection points in $\Omega=(-1,1)$ when
$a\in(-1/2,0)$, $y_{c}=\pm\sqrt{2a+1}$. For other values of $a$, there are no
inflection points and the flow must be stable according to KA-I; see figure
13. Using $\alpha=U(y_{c})=-a(5a+2)/3$, the reciprocal Rossby Mach number can
be obtained as

$\displaystyle
M^{-1}_{\alpha}(y)=\frac{L^{2}}{\pi^{2}}\frac{-U^{\prime\prime}}{U-\alpha}=\frac{1}{\pi^{2}}\frac{48}{10a+5-y^{2}}\,.$
(49)

This is continuous in $\Omega=(-1,1)$ when $a>-2/5$, so we conclude that the
flow is Class (i) if $a\in(-2/5,0)$. In the remaining parameter range
$a\in(-1/2,-2/5]$, there are no $\alpha\in\mathcal{R}$ for which
$M^{-1}_{\alpha}(y)$ is continuous, and hence the basic flow is not Class (ii)
(and thus not Class (i)).

For $a\in(-2/5,0)$ we can use Theorem 4 to show that the flow is unstable when
$a\in(-\frac{2}{5},\frac{24}{\pi^{2}}-\frac{1}{2})\approx(-0.4,-0.01366)$.
This result can be found by simply setting $y_{2}=1,y_{1}=-1$; in this case
the height of the hurdle is 1 according to (40). Note that the stability
condition (13) is satisfied when $a$ is greater than
$\frac{24}{\pi^{2}}-\frac{2}{5}\approx 0.08634$, but it is not useful because
the flow is not Class (i) there, due to the absence of critical latitudes
(inflection points).

If $a\notin(-1/2,-1/3)$, the basic flow belongs to the class considered in
Tollmien (1935), and therefore the KA-I condition provides a sharp stability
boundary at $a=0$. This result more accurately pinpoints the stability
boundary than (13). However, in terms of applicability, the advantage is
reversed, since $(-1/2,-1/3)\subset(-1/2,-2/5]$ (see figure 13). In fact,
Tollmien’s theory is equivalent to taking $\psi=U(y)$ as a test function,
which is only possible for very special class of basic flows. Numerical
computations using the shooting or Chebyshev collocation method reveal that
the flow is unstable when $a\in(-0.5,0)$; see Appendix B.

##  6 Conclusion

Rayleigh’s inflection-point theorem of 1880 ushered in a century-long series
of sufficient conditions for the stability of inviscid, parallel shear flows.
Yet, well into the $21^{\rm st}$ century, necessary conditions have remained
elusive for applied researchers. We have developed a simple methodology that
allows for the detection of inviscid instability, inspired by empirical
observations in the stability of Jupiter’s alternative jets. All that is
needed to operate our method is a comparison of $h$ defined in (4.22) with the
profile of the reciprocal Rossby Mach number, $M_{\alpha}^{-1}(y)$, which can
be easily calculated from the basic flow (see section 2.2). The true stability
boundary in the parameter space should be found between the unstable regions
detected by Theorem 4 and the stable regions deduced by the KA criteria.

Our method relies solely on the fundamental tools of Sturm-Liouville theory,
but exhibits a high level of applicability beyond Rayleigh’s equation
framework. Similar stability conditions are likely to be found for various
inviscid shear-flow stability problems in science and engineering, including
hypersonic, stratified, magneto-hydrodynamic, and/or non-Newtonian flows. To
the best of our knowledge, the possibility of such a general and practical
stability theory has not been previously pointed out.

As noted in section 1, in order for the necessary conditions of stability to
apply, the basic flow must belong to a class with certain favourable
properties. We assume there is at least one critical latitude in the domain
and identify two basic-flow classes of interest, for which
$M^{-1}_{\alpha}(y)$ is continuous in some reference zonal wind shift,
$\alpha$. From the perspective of the general inviscid stability problem, one
of the novelties of our work is the extension of the applicability of
stability conditions through the introduction of new classifications. As
illustrated by Theorem 3, our necessary conditions apply to Class (ii), which
covers almost all monotonic flows and a wide range of non-monotonic flows. If
the basic flow belongs to this class, it can be asserted that there exists the
least oscillatory regular neutral mode (section 4.1) and that instability
occurs when the parameters are slightly altered (section 4.3). Theorem 3
requires a test function, but the stability condition can be simplified using
a kind of comparison principle (section 4.5). A convenient method to determine
whether the basic flow belongs to Class (ii) is by plotting sets
$\\{U(y)|y\in\Omega_{+}\\}$ and $\\{U(y)|y\in\Omega_{-}\\}$ (see figure 4),
and checking for the existence of a point separating them.

In general, Class (ii) admits more than one reference-frame shift, $\alpha$
(see section 5.3). However, for a subclass of it, referred to as Class (i),
$\alpha$ is uniquely determined. For the latter class, $M^{-1}(y)$, which can
now be safely unsubscripted, does not change sign, as identified with Jupiter
and Saturn. An important feature of Class (i) is that the critical latitudes
must coincide with the critical layers, where the zonal wind speed matches
with the shift $\alpha$. The regular neutral modes must have the unique phase
speed $\alpha$, which explains why Read et al. (2009b) observed that long-
wavelength Rossby waves in Saturn have the same $10^{\rm h}34^{\rm m}$
rotation period.

Our results also offer new insights into the theoretical understanding of
pattern formation in planetary atmospheres. The KA stability theorems were re-
expressed by Dowling (2014) in terms the Rossby Mach number, revealing a key
attribute that supersonic critical latitudes are stable. However, this left
the natural question of how subsonic a critical latitude must be before
becoming unstable, which is answered precisely by Theorem 4, and illustrated
with detailed model analyses in section 5.1.

In the case where $M^{-1}$ is a constant, the sonic condition, $M^{-1}=1$,
gives the stability boundary. This result holds true regardless of the
boundary conditions because there exists a regular neutral mode that can be
expressed as a trigonometric function. Therefore, the conjecture of Stamp &
Dowling (1993) is mathematically confirmed to be correct. The correlations of
the zonal flow $U$ and the PV gradient $Q^{\prime}$ seen in figure 1 thus
suggests that the Jupiter’s alternate jets are formed to be linearly neutral.
However, a closer look reveals that $M^{-1}$ is not perfectly constant, and
unsteady dynamics may occur at neutral or subsonic critical latitudes (see
Read et al., 2006, 2009a).

In section 5.1 we extended the Stamp & Dowling (1993) model to study the case
where $M^{-1}$ is not constant. The $\rm sech^{2}$ bump profile considered for
$M^{-1}(y)$ (see (5.2)) is interesting from the perspective of planetary
physics because the behaviour of neutral curves can be analysed analytically
to some extent. There exist qualitatively distinct neutral states, oscillatory
and localised modes. In the context of Jupiter’s atmosphere, only the latter
appears, and the KA-II condition becomes almost sharp. This implies that in
atmospheres sustained over long periods, the $M^{-1}$ profile cannot become
too subsonic, aligning with observational facts. Numerical experiments (e.g.,
Dowling, 1993) suggest that vortices appear at subsonic latitudes, as
theoretically expected for nearly neutral unstable modes. This is also
consistent with typical observations. For instance, in figure 1, the $Q_{y}$
curve sits above $U$ in the latitude range of -30 to -20 degrees. In this
latitude range, where $M^{-1}$ is likely greater than 1, features like the
Great Red Spot and Oval BA are observed. Further to this point, Dowling (2020)
pointed out that Saturn has two different, persistently wavy jets sandwiched
between straight jets in its northern hemisphere, the Ribbon and the Polar
Hexagon, and offered the hypothesis that these are examples of ‘subsonic’
regions sandwiched between ‘sonic’ or ‘supersonic’ regions. Although the
temperatures fields necessary to empirically determine the stretching
vorticity, and hence the PV field, at and below Saturn’s cloud tops are
difficult to obtain with the data in hand, the morphology of these wavy jets
is well established, which suggests that an analysis couched in terms of
$M^{-1}(y)$ along the lines outlined here would be instructive.

As a useful exercise, one can plot $M^{-1}(y)$ profile using the data shown in
figure 1 and $L_{d}$ estimated in figure 3, to demonstrate that the presence
of a hurdle that aligns with the range of latitudes $-30$ to $-20$ degrees,
and that at other latitudes, $M^{-1}(y)$ is characterised by several humps
that peak close to unity. However, such arguments inevitably entail a number
of caveats. Remote-sensing observations are accompanied by noise and the
analysis can be sensitive to the uncertainties; for example, the deformation
length enters as the square. We must also take into account the extent to
which model assumptions correspond to reality, like those in the beta-plane
$1$–$\nicefrac{{3}}{{4}}$ quasi-geostrophic model employed here. The value of
$L_{d}$ may vary with latitude and we do not have good estimates in hand
outside the vicinity of the latitude rage shown in figure 3. Moreover, when
substituting the figure 1 data into (2.9), it is evident that, regardless of
how we choose the zonal wind shift, $M^{-1}_{\alpha}(y)$ cannot be made into a
continuous function.

The assumption that Jupiter’s atmosphere is Class (i) is only inferred from
observations, and the actual data are not even Class (ii). Therefore, it
raises the question of how robust the results of this paper are under more
general conditions. To address this, a numerical experiment is performed. We
spline interpolate $U$ and $Q_{y}$ in the latitude range shown in figure 3,
corresponding to $L\approx 10860\,\rm km$, and numerically solve the
eigenvalue problem (5). The stability boundary is found to be $L_{d}\approx
1950\,\rm km$, with the most dangerous mode, $k=0$, travelling with a phase
speed $c_{r}\approx 6\,\rm ms^{-1}$. Plotting $(U-c_{r})$ and
$\kappa_{0}^{-2}Q^{\prime}$ with $\kappa_{0}^{-1}\approx 1700\,\rm km$
obtained through (2.10) gives the same level of correlation as in figure 3.
Changing the artificially set boundary conditions produces somewhat different
results, but $\kappa_{0}^{-1}$ always takes a value similar to the observed
value of $L_{d}$. It is therefore evident that in practice, the results of
this paper are applicable to profiles that do not strictly fall into Class
(ii).

Whether simple stability criteria can be obtained for basic flows that do not
belong to Class (ii) is one of the open questions in inviscid stability
theory. A central issue here is determining under what conditions singular
neutral modes must occur. Such modes are observed in various fluid dynamics
problems; for instance, in the crossflow instability on aircraft wings, where
they play a crucial role in the transition process. The discussion in section
3.2 shows that at least the requirement that the intersection of sets
$\\{U(y)|y\in\Omega_{+}\\}$ and $\\{U(y)|y\in\Omega_{-}\\}$ be non-empty is
necessary. However, such conditions do not immediately imply the existence of
neutral modes. Due to the presence of jumps at singular points,
$\lambda=-k^{2}$ is no longer obtained as an eigenvalue of a self-adjoint
operator, and the optimisation of a quadratic form carried out in this paper
cannot be used to demonstrate the presence of neutral modes.

Another interesting open problem is extension of the theory to basic flows
that vary spatially in a more complicated manner than considered here. For
example, as alluded to above, we have only considered the
$1$–$\nicefrac{{3}}{{4}}$ layer quasi-geostrophic model, but actual Jupiter
and Saturn atmospheres have continuous vertical-structure. In other shear flow
studies, despite the change of the basic flow in two directions being the
prevalent configuration in almost all realistic flows, the Rayleigh-Fjørtoft
condition, which only holds in idealised situations, is often employed to
explain the origin of instability. A typical example is the Kelvin-Helmholtz
instability observed in flows over a riblet (e.g. García-Mayoral & Jiménez,
2011). Uhlmann & Nagata (2006) studied inviscid instability in duct flows
using second derivatives of the basic flow along the steepest direction of the
flow field. Curiously, they found that the presence of inflection points
calculated in this manner is effective in detecting instability. This is
further evidence that the method developed in this paper could be generalised.
A difficulty with this extension is that the necessary conditions for the
existence of a neutral mode are not known, although a sufficient condition has
recently been found in the case of a single critical layer (Deguchi, 2019).

Acknowledgement. The data for figures 1 and 3 were kindly provided by P.L.
Read.

Declaration of interests. The authors report no conflict of interest.

##  Appendix A Sufficient conditions of stability

Consider the integral of $\psi^{*}\times$(5) over $\Omega=(-L/2,L/2)$.
Integration by parts yields

$\displaystyle\int_{\Omega}\frac{Q^{\prime}(U-c^{*})}{|U-c|^{2}}|\psi|^{2}dy=\int_{\Omega}\\{|\psi^{\prime}|^{2}+(k^{2}+L_{d}^{-2})|\psi|^{2}\\}dy\,,$
(50)

whose real and imaginary parts are

$\displaystyle\int_{\Omega}\frac{Q^{\prime}(U-c_{r})}{|U-c|^{2}}|\psi|^{2}dy=\int_{\Omega}\\{|\psi^{\prime}|^{2}+(k^{2}+L_{d}^{-2})|\psi|^{2}\\}dy,~{}~{}~{}c_{i}\int_{\Omega}\frac{Q^{\prime}}{|U-c|^{2}}|\psi|^{2}dy=0\,,$

respectively. When we assume $c_{i}\neq 0$, the above two equations can be
combined into

$\displaystyle\int_{\Omega}\frac{Q^{\prime}(U-w)}{|U-c|^{2}}|\psi|^{2}dy=\int_{\Omega}\\{|\psi^{\prime}|^{2}+(k^{2}+L_{d}^{-2})|\psi|^{2}\\}dy\,,$
(51)

where $w\in\mathbb{R}$ is arbitrary. If there exists $w$ such that
$Q^{\prime}/(U-w)\leq 0$ for all $y$, (51) cannot be satisfied, meaning that
the flow is stable. In other words, the stability is guaranteed if there
exists $w\in\mathbb{R}$ such that $M_{w}^{-1}\leq 0$ for all $y\in\Omega$
(i.e. KA-I). The above derivation is essentially the standard argument for
shear flows due to Rayleigh and Fjørtoft.

To find KA-II, we rewrite the left hand side of (51) by introducing an
$\alpha\in\mathbb{R}$ and setting $w=2c_{r}-\alpha$:

$\displaystyle\int_{\Omega}\frac{(U-w)(U-\alpha)}{(U-c_{r})^{2}+c_{i}^{2}}\frac{Q^{\prime}}{U-\alpha}|\psi|^{2}dy=\int_{\Omega}\frac{(U-c_{r})^{2}-(c_{r}-\alpha)^{2}}{(U-c_{r})^{2}+c_{i}^{2}}\frac{Q^{\prime}}{U-\alpha}|\psi|^{2}dy\,.~{}~{}~{}~{}$
(52)

Using Poincare’s inequality
$\frac{L^{2}}{\pi^{2}}\int_{\Omega}|\phi^{\prime}|^{2}dy\geq\int_{\Omega}|\phi|^{2}dy$,
from (51) we can deduce

$\displaystyle\kappa_{0}^{2}\int_{\Omega}ZM_{\alpha}^{-1}|\psi|^{2}dy=\int_{\Omega}\\{|\psi^{\prime}|^{2}+(k^{2}+L_{d}^{-2})|\psi|^{2}\\}dy\geq(\kappa_{0}^{2}+k^{2})\int_{\Omega}|\psi|^{2}dy,~{}~{}~{}~{}$
(53)

where
$Z(y)=\frac{(U-c_{r})^{2}-(c_{r}-\alpha)^{2}}{(U-c_{r})^{2}+c_{i}^{2}}<1$ for
all $y$. If $M_{\alpha}^{-1}\in[0,1]$ for all $y$ the inequality (53) cannot
be satisfied, so the flow must be stable.

##  Appendix B The stability of (48) around $a=-0.5$

Figure 14: The green curve is the solution of the asymptotic problem (55). The
red filled circles are the solution of the Rayleigh equation with the basic
flow (48). The growth rate $c_{i}$ computed for various $k$ for $a=-0.45$, and
then the data are rescaled as $c_{0i}=c_{i}/\epsilon^{4}$, $k_{0}=k/\epsilon$,
using $\epsilon=\sqrt{0.5-0.45}\approx 0.2236$.

When the parameter $a$ in figure 13 is slightly larger than -0.5, there is an
instability showing asymptotic behaviour. Let us consider
$\epsilon=\sqrt{a+\frac{1}{2}}$ as a small parameter. Then we write
$y=\epsilon Y$, $k=\epsilon^{-1}k_{0}$ and

$\displaystyle c=-\frac{1}{12}+\epsilon^{2}+\epsilon^{4}c_{0}+\cdots$ (54)

in the Rayleigh equation. Noting
$U-c=\epsilon^{4}(\frac{Y^{4}}{12}-Y^{2}-c_{0})+\cdots$ and
$U^{\prime\prime}=\epsilon^{2}(Y^{2}-2)+\cdots$, the leading order equation
can be found as

$\left(\frac{Y^{4}}{12}-Y^{2}-c_{0}\right)(\psi_{YY}-k_{0}^{2}\psi)-(Y^{2}-2)\psi=0.$
(55)

Here we must impose $\psi\rightarrow 0$ as $|Y|\rightarrow\infty$. This
eigenvalue problem can be solved by the shooting method to find the eigenvalue
$c_{0}=c_{0r}+ic_{0i}$ for fixed $k_{0}$. The asymptotic growth rate $c_{0i}$,
shown in figure 14, captures the behaviour of the full solution very well even
when $\epsilon$ is moderately small. Since this instability appears around
singular neutral modes, it is outside the application of the sufficient
conditions of stability available so far.

##  Appendix C Analysis of (46) for $L\gg 1$

Writing

$\displaystyle\nu=\frac{\sqrt{1+4\kappa_{0}^{2}(b-a)}-1}{2},\quad\mu=\sqrt{k^{2}+L_{d}^{-2}-\kappa_{0}^{2}a}\,,$
(56)

the general solution of (46) can be written as

$\displaystyle\psi(y)=C_{1}P^{\mu}_{\nu}(\text{tanh}y)+C_{2}Q^{\mu}_{\nu}(\text{tanh}y)$
(57)

using arbitrary constants $C_{1}$ and $C_{2}$. Here $P^{\mu}_{\nu}$ and
$Q^{\mu}_{\nu}$ are the associated Legendre functions of the first and second
kind, respectively. The constant $\mu$ determines the behaviour of the
solution at large $|y|$. If $\mu$ is real, the behaviour is exponential, while
if $\mu$ is imaginary, the behaviour is oscillatory (see Bielski (2013), for
example). Now we assume that $L$ is large. Since $\kappa_{0}^{2}\approx
L_{d}^{-2}$ under this assumption, the constant $\mu$ can become imaginary
only when $a>1$. This is the case where the oscillatory modes may appear.

If $\mu$ and $\nu$ are integers satisfying $0\leq\mu\leq\nu$, the neutral mode
can be obtained by using the Legendre polynomials. The localised mode
$\psi=\text{sech}y$ found in section 5.2 is the special case $\nu=\mu=1$. The
parabolic part of the neutral curve (47) can be found by generalising this
mode, because from Sturm’s oscillation theory, the mode without nodes, i.e.
$\psi_{0}$, is usually the most dangerous. This mode can be written explicitly
as $\psi_{0}(y)=(\text{sech}y)^{\nu}$, because it satisfies (46) when
$\nu=\mu$. For the solution to decay as $|y|\rightarrow\infty$, it is
necessary for $\nu$ to be greater than 0, a condition that is indeed satisfied
in the forth quadrant of figure 6a. The condition $\nu=\mu$ can be written as

$\displaystyle
k^{2}=\frac{1}{2}\left(1+2L_{d}^{-2}(b-1)-\sqrt{1+2L_{d}^{-2}(b-a)}\right)$
(58)

using (56). Thus the neutral mode $\psi_{0}$ exists when the right hand side
of (58) is non-negative. After some algebra, this condition can be simplified
to

$\displaystyle a\geq 1-L^{-2}_{d}(b-1)^{2}\,.$ (59)

##  Appendix D Eigenvalue bounds

Figure 15: The points are eigenvalues of (5) in the complex plane, obtained at
various $k$. The blue solid curve is the inner envelope bound. The magenta
dashed curve is the Pedlosky semicircle bound. Top panel: The same set up as
figure 12. Bottom panel:
$L=16,L_{d}=2,U=\tanh(y),Q^{\prime}=0.2-U^{\prime\prime}$.

Here we derive the bound of the unstable eigenvalues $c$ in the complex plane
following the inner envelope theory by Deguchi (2021). The first step is to
write $\eta=\psi/(U-c)$ and $B=Q^{\prime}+U^{\prime\prime}$ in (5),

$\displaystyle(U-c)(2U^{\prime}\eta^{\prime}+B\eta)+(U-c)^{2}\\{\eta^{\prime\prime}-(k^{2}+L_{d}^{-2})\eta\\}=0\,.$
(60)

Then multiplying $\eta^{*}$ and integrating over $\Omega$, we get

$\displaystyle\int_{\Omega}(U-c)^{2}Wdy=\int_{\Omega}(U-c)B|\eta|^{2}dy\,,$
(61)

where $W\equiv|\eta^{\prime}|^{2}+(k^{2}+L_{d}^{-2})|\eta|^{2}$. The real and
imaginary parts of (61) are

$\displaystyle\int_{\Omega}\\{(U-c_{r})^{2}-c_{i}^{2}\\}Wdy=\int_{\Omega}(U-c_{r})B|\eta|^{2}dy\,,$
(62) $\displaystyle-
c_{i}\int_{\Omega}2(U-c_{r})Wdy=-c_{i}\int_{\Omega}B|\eta|^{2}dy\,.$ (63)

Using the identity

$\displaystyle\int_{\Omega}\\{(U-r_{c})^{2}-(c_{r}-r_{c})^{2}-c_{i}^{2}\\}Wdy$
$\displaystyle=\int_{\Omega}\\{(U-c_{r})^{2}+2(U-c_{r})(c_{r}-r_{c})-c_{i}^{2}\\}Wdy$
(64)

that holds for any $r_{c}\in\mathbb{R}$ to (62),

$\displaystyle\int_{\Omega}\\{(U-r_{c})^{2}-(c_{r}-r_{c})^{2}-c_{i}^{2}\\}Wdy$
$\displaystyle=\int_{\Omega}(U-c_{r})B|\eta|^{2}dy+(c_{r}-r_{c})\int_{\Omega}2(U-c_{r})Wdy\,.$
(65)

The right hand side can be simplified using (63),

$\displaystyle\int_{\Omega}\\{(U-r_{c})^{2}-(c_{r}-r_{c})^{2}-c_{i}^{2}\\}Wdy=\int_{\Omega}(U-r_{c})B|\eta|^{2}dy\,.$
(66)

This integral equation implies that if we can find $R(r_{c})>0$ such that

$\displaystyle\int_{\Omega}(U-r_{c})^{2}Wdy-\int_{\Omega}(U-r_{c})B|\eta|^{2}dy\leq
R^{2}\int_{\Omega}Wdy\,,$ (67)

then we have the semicircle bound

$\displaystyle(c_{r}-r_{c})^{2}+c_{i}^{2}\leq R^{2}(r_{c})\,.$ (68)

The radius $R$ depends on the centre of the semicircle $r_{c}$. The tightest
possible bound can therefore be established by plotting families of
semicircles on the complex plane with different $r_{c}$ and taking their inner
envelope.

To run this algorithm we still need to find $R(r_{c})$ satisfying the
inequality (67). We decompose $\overline{\Omega}=[-1,1]$ into two parts,
$\Omega_{1}=\\{y\in\overline{\Omega}|(r_{c}-U)B\geq 0\\}$ and
$\Omega_{2}=\\{y\in\overline{\Omega}|(U-r_{c})B>0\\}$. Then using

$\displaystyle\int_{\Omega}Wdy\geq(\kappa^{2}_{0}+k^{2})\int_{\Omega}|\eta|^{2}dy\,,$
(69)

which can be deduced by Poincare’s inequality, the left hand side of (67) can
be estimated as

$\displaystyle\int_{\Omega}(U-r_{c})^{2}Wdy-\int_{\Omega_{1}}(U-r_{c})B|\eta|^{2}dy-\int_{\Omega_{2}}(U-r_{c})B|\eta|^{2}dy$
$\displaystyle\leq\left\\{\max_{\overline{\Omega}}(U-r_{c})^{2}+\frac{1}{\kappa_{0}^{2}+k^{2}}\max_{\Omega_{1}}(r_{c}-U)B\right\\}\int_{\Omega}Wdy\,.$
(70)

Therefore the radius of the inner envelope theory bound, appeared in (68), can
be found as

$\displaystyle
R(r_{c})=\sqrt{\max_{\overline{\Omega}}(U-r_{c})^{2}+\frac{1}{\kappa_{0}^{2}+k^{2}}\max\left(\max_{\overline{\Omega}}(r_{c}-U)B,0\right)}\,.$
(71)

Let us consider the special case $r_{c}=(U_{\text{max}}+U_{\text{min}})/2$,
where $U_{\text{max}}=\max_{\overline{\Omega}}U$ and
$U_{\text{min}}=\min_{\overline{\Omega}}U$. Then we can show that the right
hand side of (71) is larger than or equal to the Howard semicircle radius
$(U_{\text{max}}-U_{\text{min}})/2$ (Howard, 1961). However (71) is smaller
than or equal to the well-known Pedlosky semicircle radius (Pedlosky, 1964),

$\displaystyle\sqrt{\left(\frac{U_{\text{max}}+U_{\text{min}}}{2}\right)^{2}+\frac{1}{\kappa_{0}^{2}+k^{2}}\left(\frac{U_{\text{max}}+U_{\text{min}}}{2}\right)\max_{\Omega}|B|}\,,$
(72)

generalised for non-constant $B$ and finite $L_{d}^{-1}$.

Figure 15 top panel shows the comparison of the eigenvalues obtained in 12 and
the bounds obtained above with $k=0$. The magenta dashed curve is the Pedlosky
semicircle bound, which can be found by (68) with the centre $r_{c}=0$ and the
radius (72). The blue solid curve is the inner envelope bound. In this
example, the inner envelope of $(c_{r}-r_{c})^{2}+c_{i}^{2}=R^{2}(r_{c})$ is
merely a circle centred at the origin, but its radius is smaller than that of
Pedlosky’s. Figure 15 bottom panel is another example. The zonal flow is also
$U=\tanh(y)$, but $B$ is set to a constant of 0.2. In this case the inner
envelope bound is not a circle and part of it overlaps the Pedlosky bound.
Finally, we note that if $(\frac{U_{\text{max}}+U_{\text{min}}}{2}-U)B$ is
negative for all $y\in\Omega$, then the inner envelope bound matches to
Howard’s semicircle. This occurs, for example, in the situation shown in
figure 8b.

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