A sufficient condition for inviscid shear instability: Hurdle theorem and its application to alternating jets

Our formulation is based on multi-layer quasi-geostrophic systems (see section 5.3.2 of Vallis, 2017). 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 weather layer potential vorticity is written as

where $L_{d}$ is the first Rossby deformation length and $\Psi_{\rm deep}$ is the deep-layer streamfunction, also known as the dynamic topography. The ratio of the depth of the deep layer to the weather layer is taken to be very large, so that $\Psi_{\rm deep}$ may be treated as steady (Majda & Wang, 2005). We further assume for simplicity that the deep-layer circulation does not vary with longitude, $x$. Such variations can affect the phase-locking of long Rossby waves and thereby can play an indirect role, however the focus here is on unidirectional shear instability, which is relevant given the predominantly zonal nature of gas giant circulations. Under those assumptions, $\Psi_{\rm deep}$ 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 “$1\frac{1}{2}$ layer” model, but was recently re-branded as the “$1\frac{3}{4}$ 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).

In summary, the $1\frac{3}{4}$ layer model is written as conservation of quasi-geostrophic potential vorticity $\frac{DQ}{Dt}=0$ for (3) which now yields one nonlinear equation in one unknown, $\Psi(x,y,t)$. As previously noted, 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 (3) 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,

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

Note that in this model a Galilean transformation in $x$ is applied for both layers so that $U$, $U_{\rm deep}$ are replaced by $U-\alpha$, $U_{\rm deep}-\alpha$, respectively, implying that $Q^{\prime}$ is unchanged (note that this is not the case in the $1\frac{1}{2}$ layer model, where $U_{\rm deep}$ is related to bottom topography and hence not altered in the Galilean shift). The link (5) implies that 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. This is important in planetary science, as observing $U$ and $Q^{\prime}$ determines $U_{\rm deep}$, providing new insights into the nature of deep jets (Dowling, 1995b; Read et al., 2006, 2009a, 2009b). Furthermore, as we shall see shortly, if there is a neutrality hypothesis that can potentially be used to constraint the reciprocal Rossby Mach number, to be defined in (8), 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.

Given a wavenumber, $k\geq 0$, and deformation radius, $L_{d}>0$, equation (4) and the boundary conditions constitute an eigenvalue problem for the complex phase speed, $c=c_{r}+ic_{i}$. For fixed $L_{d}$, if there is a $k$ that yields $c_{i}>0$, the flow is unstable. It is easy to confirm that the complex conjugate of the eigenvalue is also an eigenvalue; however, when $|c_{i}|$ is small, only the mode with $c_{i}>0$ provides a good approximation to the viscous problem.

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. In the case of rotation with a non-zero planetary vorticity gradient, $\beta\neq 0$, the necessary and sufficient conditions for stability have also been discussed in the context of the analogous problem in plasma physics (e.g., Numata et al., 2007; Zhu et al., 2018). In the case of no rotation and no stratification, $\beta=0$ and $L_{d}\rightarrow\infty$, the PV gradient (5) simplifies to $Q^{\prime}=-U^{\prime\prime}$ and (4) reduces to the original Rayleigh equation, in which case the critical latitudes are inflection points of $U(y)$.

The reciprocal Rossby Mach number, $M^{-1}(y)$ emerges as being central to this work and there are various ways to motivate it. For example, in Section 1 we briefly commented on the physical interpretation by Dowling (2014, 2020). The following motivation is based on mathematical consideration applied to the second order ordinary differential equation (4). If equation (4) possesses a non-trivial solution that meets the boundary conditions, then in order for the solution to exhibit oscillatory behaviour at some point, the factor $Q^{\prime}/(U-c)$, needs to be sufficiently larger than the smallest eigenvalue, $\kappa_{0}^{2}$, of the eigenvalue problem formulated by the remaining terms with $k=0$:

with $\varphi(-L/2)=\varphi(L/2)=0$. This leads to the introduction of the reciprocal Rossby Mach number,

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. However, as we will explain later, in the Jupiter-style basic flows, there is only one optimal choice of $\alpha$, allowing us to remove the subscript. Note from (5) that $Q^{\prime}$ is invariant with respect to $\alpha$. Both $\kappa_{0}^{2}$ and its associated eigenfunction, $\varphi_{0}$, can be explicitly computed:

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

If $Q^{\prime}(y)$ does not change sign over $\Omega$, i.e. there are no PV extrema, we can always choose a large enough or small enough $\alpha$ to satisfy KA-I, which recovers the Charney-Stern condition (KA-I is also known as a sufficient condition for the absence of over-reflections of Rossby waves; see Lindzen & Tung, 1978). In other words, the interesting case occurs when there is at least one PV extremum. We assume the following properties for both Class (i) and Class (ii) basic flows:

• •

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

• •

  The zeros of $Q^{\prime}(y)$, the PV extrema, are isolated.

• •

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

We say a basic flow belongs to Class (i) if there exists an $\alpha\in\mathcal{R}$ such that the following additional condition is satisfied.

• •

  $M_{\alpha}^{-1}(y)$ is continuous and one-signed in $\Omega$.

Here $\mathcal{R}$ is the range of the zonal flow

In the definition, ‘a function is one-signed’ means that it is non-negative or non-positive for all $y\in\Omega$. Of course, from Theorem 1, the non-positive cases are stable.

Class (i) occurs when the PV extrema and the zeros of $U-\alpha$ coincide, i.e. $Y_{Q}=Y_{U,\alpha}$. 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.

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 PV extremum. The KA-I and KA-II stability conditions can be combined; the flow is stable if

Here the reciprocal Rossby Mach number is simply written as $M^{-1}$, as the choice of $\alpha$ is trivial (see Theorem 3 in Section 6.1). Note if $M_{\alpha}^{-1}(y)$ changes sign (i.e. the basic flow is not Class (i)), mathematically the condition (11) cannot guarantee stability.

Class (i) is the geophysically interesting case to which the “Jupiter-style” shear flow belongs, assuming a suitable $U_{\rm{deep}}$ (Dowling, 2020; Afanasyev & Dowling, 2022). As mentioned 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. The link between Class (i) and Jupiter’s atmosphere is further discussed in Section 5.

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). Our main result, Theorem 2, which we call the hurdle theorem, can show the existence of instability for a wider range of basic flows than Class (i).

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 in $\Omega$.

• •

  $M_{\alpha}^{-1}(y_{j})$ is one-signed for all $y_{j}\in Y_{U,\alpha}$.

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 PV extremum cancels the singularity and $M_{\alpha}^{-1}$ changes sign at the upper PV extremum. Alternatively, an appropriately negative $\alpha$ may be chosen such that $U-\alpha$ vanishes at the upper PV extremum.

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 $Y_{Q}$. Then if the set

is non-empty, we may be able to select $\alpha$ from this set. Furthermore, in the vicinity of the chosen point, there should be no points belonging to

to satisfy the second Class (ii) condition.

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 in the range of $U(y)$ is not particularly restrictive, 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 PV extremum, $\mathcal{D}$ 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).

Our main result is the following ‘hurdle theorem’, which provides a sufficient condition for instability (and the contrapositive necessary condition for stability). This theorem is proved in Section 6:

In short, if the reciprocal Rossby Mach number profile surpasses the hurdle $h$, then the flow is unstable, as illustrated in figure 5. As can be seen from the proof in Section 6, this result is actually independent of the conditions applied at the boundaries (though it depends on $L$).

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

This is the contrapositive of (11), implying that the KA-II stability condition is almost sharp if the basic flow is Class (i) and $L_{d}$ is small (recall that for Class (ii) the stability condition (11) is not valid).

Consider the quasi-geostrophic problem (4). In setting up a model basic flow, we assume that the PV gradient profile has the form

where $a,b>0$ are two free 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

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 zero (i.e. $0\in\mathcal{R}$), the existence of a PV extremum is guaranteed. Then, since both $a$ and $b$ are positive, the function (18) 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

are considered. Here, lengths are implicitly scaled by $L_{U}$. The profile (19) may be regarded as a model for Jupiter’s alternating jets (with $L_{U}$ being the jet peak-to-peak length scale), while (20) is a simpler case where there is only one PV extremum.

For Class (i) the phase speed of the neutral modes is uniquely determined (Section 6, Theorem 3), and for our basic flows it is $0$. Thus, setting $c=0$ in (4), the equation satisfied by the neutral solutions can be found as

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 that the model flows are meant to emulate the atmosphere of Jupiter, contemplating the case of large $L$ is natural.

When $L$ is large, it is possible to study (21) analytically. 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 (21) 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 evanescent 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 (21). 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, (21) 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 B). 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 labelled because of the KA stability condition (11). 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 (40) is almost $a$, because the integral is largely unaffected by what happens in the hump or dip. The analysis just below (B2) 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 (B4)). In summary, the theoretical neutral curve for $L\gg 1$ can be described as

This neutral curve delimits 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 (21), we can calculate the neutral curve for finite $L$. The computational results indeed approach the theoretical result (22) as $L$ is increased, as shown in figure 7b.

One of the novel features of Theorem 2 is that, without resorting to eigenvalue computations, applying a simple hurdle yields a useful estimate of the behaviour of neutral curves. In figure 7c, the unstable region is depicted, which can be identified 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 identified 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 (15), the smaller $L_{d}$ becomes, 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 neutral modes do not depend on $U(y)$, but the unstable modes stemming from them do. Figure 8 shows the eigenvalue $c$ of (4) calculated for the same parameters as in figure 6a. For the oscillatory zonal flow (19), all instabilities persist down to $k=0$ (figure 8a). However, for the monotonic zonal flow (20), 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 shown by the dotted curve (see supplementary material where Pedlosky’s bound is tightened using the approach by Deguchi, 2021). 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.

Of particular interest from a planetary physics perspective is the fourth quadrant of figure 7a. 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 somewhat close to the situation found in Jupiter’s atmosphere, as we shall see in Section 5. In the chosen base flow $U=\sin(2\pi y)$, there exist many critical latitudes, and in Section 6 it is shown that they must coincides with the PV extrema for Class (i). 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 ($M^{-1}>1$). Since $a=0.5$, other critical latitudes are supersonic ($M^{-1}<1$). The occurrence of vortices at subsonic latitudes is not a phenomenon specific to the current model. Mathematically, 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. 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 3.3. This is why the eigenfunction for the $L_{d}=1/8$ case shown in figure 9 has smaller vortices.

Interesting things happen when considering negative $b$ in (17). There is no longer any guarantee that the basic flow is Class (i), as the sign of the $M^{-1}_{0}$ profile (18) 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; the results look like figure 10. The emergence of the unstable region is due to the modes with non-zero phase speed, because clearly $M_{0}^{-1}$ will be everywhere smaller than 1.

It is easy to see that the PV extrema 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 PV extrema are $Y_{Q}=\{-0.8814,0,0.8814\}$. 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 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). For $\alpha=0.7071$, Class (ii) conditions are satisfied. This follows from the monotonicity of $U(y)$; see the comments below (13). 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 (see the remark below (26)). Such singular modes are beyond the scope of this paper, and in fact there is no need to consider them in the planetary context as long as we assume that Jupiter’s atmosphere is of Class (i).

We first note that the following theorem holds.

This theorem can be shown as follows. The possibility of $M^{-1}_{\alpha}$ vanishing only occurs at $y_{l}\in Y_{Q}$. For Class (i), $U^{\prime}(y_{l})\neq 0$ for all $y_{l}\in Y_{Q}$, because otherwise $M_{\alpha}^{-1}$ becomes singular, in view of the assumption that $Q^{\prime\prime}(y_{l})\neq 0$ for all $y_{l}\in Y_{Q}$. L’Hôpital’s rule now suggests that $M_{\alpha}^{-1}(y_{l})=Q^{\prime\prime}(y_{l})/U^{\prime}(y_{l})\neq 0$. We can also show that the $\alpha\in\mathbb{R}$ that makes $M^{-1}_{\alpha}$ continuous and one-signed is uniquely determined. Suppose there are two possible values of $\alpha$, $\alpha_{1}$ and $\alpha_{2}$, say. Since $M_{\alpha_{1}}^{-1}$ and $M_{\alpha_{2}}^{-1}$ are continuous functions that do not vanish in $\Omega$, $(M_{\alpha_{1}}-M_{\alpha_{2}})$ is a continuous function. However, this implies that $\frac{\alpha_{2}-\alpha_{1}}{Q^{\prime}}$ is a continuous function, which is not possible unless $\alpha_{1}=\alpha_{2}$.

Class (i) is an extension of class $\mathcal{H}$ in Howard (1964), and class $\mathcal{K}^{+}$ in Lin (2003) to the quasi-geostrophic equation. However, such strong restrictions for basic flows are not necessary to derive the hurdle theorem, as remarked in Section 3.

Let us consider the neutral solutions, setting $c_{i}=0$. The key to classify neutral modes is to note that they can potentially become singular at the critical latitudes, the points at which $U(y)$ coincides with the phase speed $c_{r}$. Mathematically, the set of the critical latitudes, $Y_{U,c_{r}}=\{y\in\Omega|U(y)=c_{r}\}$, are regular singular points of (4) 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 latitudes (Lin, 1955). A necessary and sufficient condition for no jumps to exist at a critical latitude 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 critical latitudes 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 (4) by $\psi^{*}$ and take the imaginary part, keeping finite $c_{i}$:

Integrate (23) across a critical latitude, $y=y_{c}$ (i.e., $U(y_{c})=c_{r}$), and take the limit $c_{i}\rightarrow 0+$,

An intuitive way to evaluate the right hand side 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 (24) can be explicitly worked out by using the well-known formula that can be found by differentiating the arctangent function:

The above argument is consistent with the viscous problem when the singularity of the inviscid solution is regularised by viscosity (Lin, 1955). The regularisation by inertia is also possible (Haberman (1972); Robinson (1974)) – although such a situation is relevant to nonlinear equilibrium states (e.g. Deguchi & Walton (2018)), here we only consider the linear problem.

The above result suggests that if multiple critical latitudes appear as $Y_{U,c_{r}}=\{y_{1},y_{2},\dots,y_{N}\}$, integrating (23) over $\Omega$ yields

If $Q^{\prime}=0$ or $\psi=0$ happen at all the critical latitudes, 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

empty $Y_{U,c_{r}}$ or $\psi$ vanishes at all critical latitudes;

Regular mode

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

Singular mode

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

In standard textbooks like Drazin & Reid (1981) and in previous research, there has been no distinction made between pathological modes and regular modes. We shall show in Section 6.3 that when the first Class (ii) condition is satisfied and the reciprocal Rossby Mach number surpasses a hurdle, neutral solutions exist with some choices of $k$, and at least one of them must be a regular mode. Moreover, if the second Class (ii) condition holds, unstable modes must exist around the (non-pathological, least oscillatory) regular neutral mode (Section 6.4).

If the intersection of sets $\{U(y)|y\in\Omega_{+}\}$ and $\{U(y)|y\in\Omega_{-}\}$ is non-empty, the summation in (26) may cancel, and hence a singular mode may exist (thus for Class (i) there are no singular modes). For the singular modes we cannot use the standard Sturm-Liouville theory to be used in sections 6.3 and 6.4, and for the pathological modes we have difficulty in showing neighbourhood instability.

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 (4) the neutral modes with the phase speed $c=\alpha$ must satisfy

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

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 below. In figure 6a, the red curve is $\psi_{0}$, and the other curves may be $\psi_{1},\psi_{2},\psi_{3}$. Likewise the modes shown in figure 11b are the zeroth and first modes.

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

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

by a density argument. 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.

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 (29) is the neutral eigenfunction having the wavenumber $k=k_{0}\equiv\sqrt{-\lambda_{0}}\geq 0$. This neutral mode is $\psi_{0}$ introduced earlier. The neutral curves shown in figures 7a, 10, 13 are indeed determined by $\psi_{0}$, and crucially, this mode must be a regular mode.

Since $c_{r}=\alpha$ for $\psi_{0}$, the critical latitudes (the points $y_{j}$ at which $U(y_{j})=c_{r}$ is satisfied) must appear on the PV extrema (i.e. $Y_{U,c_{r}}\subset Y_{Q}$). However, for Class (i) basic flows, the stronger result $Y_{Q}=Y_{U,c_{r}}$ can be shown for $\psi_{0}$, because $Y_{Q}=Y_{U,\alpha}$ as remarked just below (10), and there is only one choice of $\alpha$ from Theorem 3.