2.1 The base state

As in Nguyen et al. (Reference Nguyen, Hua, Schopp and Carton2012) and part 1, we consider an axisymmetric pancake vortex with angular velocity

(2.1) $$\begin{eqnarray}{\it\Omega}(r,z)={\it\Omega}_{0}\text{e}^{-(r^{2}/R^{2}+z^{2}/{\it\Lambda}^{2})},\end{eqnarray}$$

where ( $r,{\it\theta},z$ ) are cylindrical coordinates with $z$ pointing upward, $R$ the radius, ${\it\Lambda}$ the typical half-thickness and ${\it\Omega}_{0}$ the maximum angular velocity. The radial and vertical inviscid momentum equations under the Boussinesq approximation for such steady base flow are

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle -r{\it\Omega}^{2}-fr{\it\Omega}=-\frac{1}{{\it\rho}_{0}}\frac{\partial p_{t}}{\partial r}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle 0=-\frac{1}{{\it\rho}_{0}}\frac{\partial p_{t}}{\partial z}-\frac{g}{{\it\rho}_{0}}{\it\rho}_{t}, & \displaystyle\end{eqnarray}$$

where $p_{t}$ and ${\it\rho}_{t}$ are the total pressure and density, respectively, $g$ is the gravity and ${\it\rho}_{0}$ a constant reference density. Combining (2.2) and (2.3) gives the thermal wind equation

(2.4) $$\begin{eqnarray}\frac{\partial {\it\rho}_{t}}{\partial r}=-\frac{{\it\rho}_{0}}{g}\frac{\partial }{\partial z}(r{\it\Omega}^{2}+fr{\it\Omega}),\end{eqnarray}$$

yielding

(2.5) $$\begin{eqnarray}{\it\rho}_{t}={\it\rho}_{0}+\bar{{\it\rho}}(z)+{\it\rho}_{b}(r,z),\end{eqnarray}$$

with $\bar{{\it\rho}}(z)=-N^{2}{\it\rho}_{0}z/g$ where $N=\sqrt{-g/{\it\rho}_{0}(\text{d}\bar{{\it\rho}}/\text{d}z)}$ is the Brunt–Väisälä frequency which is assumed constant and

(2.6) $$\begin{eqnarray}{\it\rho}_{b}(r,z)=-z\frac{{\it\rho}_{0}}{\text{g}}\left(\frac{R}{{\it\Lambda}}\right)^{2}\left({\it\Omega}+f\right){\it\Omega},\end{eqnarray}$$

is the density field associated with the vortex.

2.2 Linearized equations

We subject this vortex to infinitesimal perturbations of velocity $\boldsymbol{u}^{\prime }=[u_{r}^{\prime },u_{{\it\theta}}^{\prime },u_{z}^{\prime }]$ , pressure $p^{\prime }$ and density ${\it\rho}^{\prime }$ written as

(2.7) $$\begin{eqnarray}[u_{r}^{\prime },u_{{\it\theta}}^{\prime },u_{z}^{\prime },p^{\prime },{\it\rho}^{\prime }]=\left[u_{r}(r,z),u_{{\it\theta}}(r,z),u_{z}(r,z),{\it\rho}_{0}p(r,z),\frac{{\it\rho}_{0}}{g}{\it\rho}(r,z)\right]\text{e}^{-\text{i}{\it\omega}t+\text{i}m{\it\theta}}+\text{c.c}.,\end{eqnarray}$$

where $m$ is the azimuthal wavenumber and ${\it\omega}={\it\omega}_{r}+\text{i}{\it\omega}_{i}$ , with ${\it\omega}_{r}$ the frequency and ${\it\omega}_{i}$ the growth rate. The linearized Navier–Stokes equations under the Boussinesq approximation are

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}({\it\omega}-m{\it\Omega})u_{r}-(2{\it\Omega}+f)u_{{\it\theta}}=-\frac{\partial p}{\partial r}+{\it\nu}\left({\rm\nabla}^{2}u_{r}-\frac{1}{r^{2}}u_{r}-\frac{2}{r^{2}}\text{i}mu_{{\it\theta}}\right)\!, & \displaystyle\end{eqnarray}$$

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}({\it\omega}-m{\it\Omega})u_{{\it\theta}}+({\it\zeta}+f)u_{r}+\frac{\partial r{\it\Omega}}{\partial z}u_{z}=-\frac{\text{i}m}{r}p+{\it\nu}\left({\rm\nabla}^{2}u_{{\it\theta}}-\frac{1}{r^{2}}u_{{\it\theta}}+\frac{2}{r^{2}}\text{i}mu_{r}\right)\!, & \displaystyle\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}({\it\omega}-m{\it\Omega})u_{z}=-\frac{\partial p}{\partial z}-{\it\rho}+{\it\nu}{\rm\nabla}^{2}u_{z}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}({\it\omega}-m{\it\Omega}){\it\rho}+\frac{g}{{\it\rho}_{0}}\frac{\partial {\it\rho}_{b}}{\partial r}u_{r}+\frac{g}{{\it\rho}_{0}}\frac{\partial {\it\rho}_{b}}{\partial z}u_{z}=N^{2}u_{z}+{\it\kappa}{\rm\nabla}^{2}{\it\rho}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{r}\frac{\partial ru_{r}}{\partial r}+\frac{1}{r}\text{i}mu_{{\it\theta}}+\frac{\partial u_{z}}{\partial z}=0, & \displaystyle\end{eqnarray}$$

where ${\it\zeta}=(1/r)\partial r^{2}{\it\Omega}/\partial r$ is the vertical vorticity, ${\it\nu}$ the viscosity and ${\it\kappa}$ the diffusivity of the stratifying agent. The viscous and diffusive damping of the base state are neglected in (2.8)–(2.12) as classically done in stability analyses (Drazin & Reid Reference Drazin and Reid1981). This assumption is valid as long as the time taken by the perturbations to grow to finite amplitude is small compared to the viscous decay time of the base flow. The problem is governed by five non-dimensional numbers: aspect ratio ( ${\it\alpha}$ ), Froude number ( $F_{h}$ ), Rossby number ( $Ro$ ), Reynolds number ( $Re$ ), Schmidt number ( $Sc$ ), defined as follows:

(2.13a-e ) $$\begin{eqnarray}{\it\alpha}=\frac{{\it\Lambda}}{R},\quad F_{h}=\frac{{\it\Omega}_{0}}{N},\quad Ro=\frac{2{\it\Omega}_{0}}{f},\quad Re=\frac{{\it\Omega}_{0}R^{2}}{{\it\nu}},\quad Sc=\frac{{\it\nu}}{{\it\kappa}}.\end{eqnarray}$$

The Schmidt number is set to $Sc=1$ throughout the paper. In part 1, the effect of the Schmidt number has been studied and shown to affect only the axisymmetric mode of centrifugal instability. The equations (2.8)–(2.12) in the inviscid and non-diffusive limit are non-dimensionalized in appendix A and shown to reduce to the quasi-geostrophic equation when $Ro\ll 1$ and $F_{h}\ll 1$ whatever the aspect ratio ${\it\alpha}$ . For this reason, even if we consider here aspect ratios not as small as in the oceans, the quasi-geostrophic regime will be reached for sufficiently small Rossby and Froude numbers.

As explained in part 1, equations (2.8)–(2.12) are discretized with finite element methods using FreeFEM++ (Hecht Reference Hecht2012; Garnaud Reference Garnaud2012) in the domain $0\leqslant r\leqslant R_{\mathit{max}}$ and $-Z_{\mathit{max}}\leqslant z\leqslant Z_{\mathit{max}}$ . The size is taken as $R_{\mathit{max}}\geqslant 10R$ and $Z_{\mathit{max}}=5{\it\Lambda}$ . These sizes are slightly different compared to part 1 because some modes are more sensitive to radial confinement in the presence of background rotation. The mesh is adapted to the base flow so that the mesh is finer ( ${\sim}0.001R$ ) inside the vortex core than outside ( ${\sim}0.1R$ ). The boundary conditions at $r=0$ are $u_{r}=u_{{\it\theta}}=0$ for $m=0$ , $u_{z}=p={\it\rho}=0$ for $m=1$ and $\boldsymbol{u}=p={\it\rho}=0$ for $m\geqslant 2$ . At the other boundaries, $R=R_{\mathit{max}}$ and $z=\pm Z_{\mathit{max}}$ , all the perturbations are imposed to vanish: $\boldsymbol{u}=p={\it\rho}=0$ . The matrix version of (2.8)–(2.12) built by FreeFEM++ is then solved by means of an iterative Krylov–Schur scheme and a shift-invert method using the SLEPc and PETSc libraries (Hernandez, Roman & Vidal Reference Hernandez, Roman and Vidal2005; Garnaud Reference Garnaud2012; Garnaud et al. Reference Garnaud, Lesshafft, Schmid and Huerre2013; Balay et al. Reference Balay, Abhyankar, Adams, Brown, Brune, Buschelman, Eijkhout, Gropp, Kaushik and Knepley2014; Roman et al. Reference Roman, Campos, Romero and Tomas2015). More details can be found in part 1. The limit ${\it\alpha}\rightarrow \infty$ corresponding to a columnar vortex has been solved by means of a Chebyshev collocation spectral method (Antkowiak & Brancher Reference Antkowiak and Brancher2004).

In appendix B, it is shown that the equations (2.8)–(2.12) in the quasi-geostrophic and inviscid limits ( $Ro\rightarrow 0,F_{h}\rightarrow 0,Re=\infty$ ) can be solved by separation of variables and a shooting method for $Bu=Ro^{2}{\it\alpha}^{2}/(4F_{h}^{2})=1$ . Hence, this particular case has been used as a validation test for the code based on FreeFEM++ and SLEPc. The eigenvalues obtained by the two numerical methods are in good agreement (see appendix B).

3.1 Azimuthal wavenumber $m=0$

For $m=0$ , it has been found in part 1 that only a centrifugal instability exists in a stratified fluid when the buoyancy Reynolds number $ReF_{h}^{2}$ is sufficiently high. This remains true when the Rossby number is varied. Figure 1 shows two examples of spectra for $Ro=20$ and $Ro=-10$ together with the one for $Ro=\infty$ , all for the same set of parameters: ${\it\alpha}=0.5,F_{h}=0.5$ and $Re=10\,000$ . The unstable modes are shown by symbols and are labelled $(m,i)$ where $i$ is the mode number. For each point, there exist actually two modes with different symmetry with respect to the mid-plane $z=0$ : anti-symmetric (○) and symmetric (

$Ro$

(3.4) $$\begin{eqnarray}{\it\Phi}=\left.\frac{1}{r^{3}}\frac{\partial }{\partial r}\left(r^{4}\left({\it\Omega}+\frac{f}{2}\right)^{2}\right)\right|_{{\it\rho}_{t}},\end{eqnarray}$$

Figure 3.

Maximum growth rate of centrifugal instability for $m=0$ as a function of $Ro$ for a pancake vortex for ${\it\alpha}=0.5$  (

) and a columnar vortex (

) for $F_{h}=0.5$ and $Re=10\,000$ . The dashed grey line (

) shows the maximum of the asymptotic growth rate (3.5) and the solid line (——) shows the upper limit for the growth rate of centrifugal instability for $m=0$ : $\sqrt{-\text{min}({\it\phi})}$ . The vertical shaded region is the region unstable to gravitational instability. The upper $x$ -axis indicates the corresponding value of square root of the Burger number $\sqrt{Bu}$ .

is negative. The radial derivative in (3.4) is taken along constant density surfaces as specified by the generalisation of the Rayleigh criterion to baroclinic vortices (Solberg Reference Solberg1936; Eliassen & Kleinschmidt Reference Eliassen, Kleinschmidt and Bartels1957). The minimum of ${\it\Phi}$ is located on the symmetry plane $z=0$ whatever $Ro$ but the region of negative ${\it\Phi}$ has a bean shape for $Ro>\exp (2)=7.39$ and a croissant shape (extending to the axis at $z=\sqrt{\ln (-Ro)}$ ) for $Ro<-1$ . The number of oscillations of the modes along the vertical increases with the mode number but there is always a single oscillation along the radial direction for the parameters of figure 1. The variation of the growth rate of the most unstable mode as a function of $Ro$ for ${\it\alpha}=0.5,F_{h}=0.5$ and $Re=10\,000$ is summarized in figure 3 (dashed line with circles). Centrifugal instability is stabilized for small Rossby number in the range $-3.5<Ro<17$ . A similar evolution of the maximum growth rate is observed for a columnar vortex (grey continuous line). As shown in part 1, the growth rate of the centrifugal instability can be predicted using the asymptotic formula of Billant & Gallaire (Reference Billant and Gallaire2005) for large axial wavenumber $k$ for a columnar vortex with the addition of the leading viscous term as in Lazar et al. (Reference Lazar, Stegner and Heifetz2013b ):

(3.5) $$\begin{eqnarray}{\it\omega}={\it\omega}^{(0)}-\frac{{\it\omega}^{(1)}N}{k}-\text{i}{\it\nu}k^{2},\end{eqnarray}$$

where

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\omega}^{(0)}=m{\it\Omega}(r_{0})+\text{i}\sqrt{-{\it\phi}(r_{0})}, & \displaystyle\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\omega}^{(1)}=\frac{(2n+1)\text{i}}{2\sqrt{2}}\sqrt{\frac{{\it\phi}^{\prime \prime }(r_{0})-2m^{2}{\it\Omega}^{\prime }(r_{0})^{2}+2\text{i}m\sqrt{-{\it\phi}(r_{0})}{\it\Omega}^{\prime \prime }(r_{0})}{-{\it\phi}(r_{0})}}\sqrt{1-\frac{{\it\phi}(r_{0})}{N^{2}}}, & \displaystyle\end{eqnarray}$$

with $n$ a non-negative integer, ${\it\phi}=(2{\it\Omega}+f)({\it\zeta}+f)$ and $r_{0}$ is given by

(3.8) $$\begin{eqnarray}\displaystyle {\it\phi}^{\prime }(r_{0})=-2\text{i}m{\it\Omega}^{\prime }(r_{0})\sqrt{-{\it\phi}(r_{0})}. & & \displaystyle\end{eqnarray}$$

The maximum growth rate predicted by (3.5) for $F_{h}=0.5$ and $Re=10\,000$ is shown by the dashed grey line in figure 3. It is close to the maximum growth rate for both columnar and pancake vortices. We can remark that the maximum growth rate for $F_{h}=0.5$ and $Re=10\,000$ is approximately three times smaller than the theoretical upper limit for the growth rate of a centrifugal instability which is $\sqrt{-\min ({\it\phi})}$ for $m=0$ (solid line in figure 3) and which is attained in the limit $k\rightarrow \infty$ and ${\it\nu}\rightarrow 0$ . Note that $\min ({\it\phi})=\min ({\it\Phi})$ since ${\it\Phi}$ is minimum on the symmetry plane $z=0$ .

3.2 Azimuthal wavenumber $m=1$

A spectrum for $m=1$ for a large Rossby number $Ro=20$ and ${\it\alpha}=0.5$ , $F_{h}=0.5$ and $Re=10\,000$ is displayed in figure 4(b). The spectrum of a columnar vortex for the same parameter is also shown by a continuous grey line for comparison. These spectra are qualitatively similar to those for $Ro=\infty$ (shown in figure 4(a) for reference) in which three types of modes have been identified in part 1. From this basis, we can easily classify the different modes in figure 4(b). The series of modes ((1, 1)–(1, 4), (1, 6)–(1, 8)) correspond to centrifugal instability. The eigenmodes differ again by the number of oscillations along the vertical (compare modes (1, 1) and (1, 3) in figure 5) but they have all a single oscillation along the radial direction. They are localized near the region where ${\it\Phi}$ is negative and they tend to be aligned along the base isopycnals (figure 5 a,b). The mode (1, 5) in figure 4(b) corresponds to a mixed Gent–McWilliams–centrifugal instability since it exhibits both the characteristics of centrifugal and Gent–McWilliams instabilities (figure 5 c). The latter instability, which is also called the internal instability, bends the vortex and is due to the presence of a critical layer where ${\it\Omega}={\it\omega}_{r}$ and where the vertical vorticity radial gradient is positive (Gent & McWilliams Reference Gent and McWilliams1986; Smyth & McWilliams Reference Smyth and McWilliams1998; Yim & Billant Reference Yim and Billant2015). The mode (1, 9) is almost neutral with zero frequency for large $Re$ . Its radial velocity perturbation (figure 5 d) is almost identical to the angular velocity of the base flow, implying that it displaces the vortex without deforming it. As shown in part 1, this mode corresponds to the displacement mode originating from the translational invariance. In § 4.1, it will be shown that the instability of the displacement mode is due to viscous effects.

Figure 4.

Growth rate ( ${\it\omega}_{i}/{\it\Omega}_{0}$ ) and frequency ( ${\it\omega}_{r}/{\it\Omega}_{0}$ ) spectra for $m=1$ for different Rossby numbers $Ro$ : (a) $Ro=\infty$ , (b) $Ro=20$ , (c) $Ro=-10$ (d) $Ro=5$ , (e) $Ro=2$ and (f) $Ro=1.43$ for $F_{h}=0.5$ and $Re=10\,000$ . Discrete symbols (○: for symmetric and 

 for anti-symmetric modes) correspond to pancake vortices for ${\it\alpha}=0.5$ and thick continuous lines (

) correspond to columnar vortices ( ${\it\alpha}=\infty$ ).

Figure 5.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of centrifugal instability (CI) (a) mode (1, 1) and (b) mode (1, 3), (c) mixed Gent–McWilliams–centrifugal instability (GMW-CI) (1, 5) and (d) the displacement mode (DM) (1, 9) for $Ro=20$ (figure 4 b) ( $m=1,{\it\alpha}=0.5,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The dashed line indicates the contour where the Rayleigh discriminant ${\it\Phi}$ vanishes.

Figure 6.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of (a) centrifugal instability $(1,1)$ and (b) Gent–McWilliams instability (GMWI) (1, 5) for $Ro=-10$ (figure 4 c) ( $m=1,{\it\alpha}=0.5,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The dashed line indicates the contour where the Rayleigh discriminant ${\it\Phi}$ vanishes.

A similar spectrum is observed for $Ro=-10$ (figure 4 c) except that there is a clear separation between centrifugal and Gent–McWilliams instabilities. This is similar to the columnar vortex case (grey solid line) where Gent–McWilliams instability and centrifugal instability correspond to two distinct growth rate peaks for negative Rossby numbers in contrast to positive Rossby numbers (Yim & Billant Reference Yim and Billant2015). The structure of centrifugal instabilities (figure 6 a) is similar to those for $Ro=20$ (figure 5 a,b). Gent–McWilliams instability (figure 6 b) is localized in the vortex core and corresponds more clearly to a bending of the vortex as a whole than for $Ro=20$ (figure 5 c) where the mixed Gent–McWilliams–centrifugal mode tends to concentrate at the top and bottom of the pancake vortex.

When the absolute value of the Rossby number is further decreased, the centrifugal instability disappears since ${\it\Phi}>0$ everywhere when $-1\leqslant Ro\leqslant 7.39$ and only the Gent–McWilliams instability remains, as exemplified in figure 4(d) for $Ro=5$ . In this case, there are three unstable Gent–McWilliams modes and the first two are displayed in figure 7. The first one (figure 7 a) is similar to the one previously shown in figure 6(b) for $Ro=-10$ while the second one exhibits one more oscillation along the vertical and is thus symmetric (figure 7 b). There is still also the displacement mode ((1, 4) in figure 4 d) with a very weak frequency and growth rate. All these modes are close to the spectrum of a columnar vortex shown by the grey solid line (figure 4 d). The maximum frequency of the unstable branch of the columnar vortex is ${\it\omega}_{r}={\it\Omega}_{0}\text{e}^{-2}=0.135{\it\Omega}_{0}$ . This corresponds to the maximum frequency for which the gradient of the vertical vorticity ${\it\zeta}^{\prime }(r_{c})$ is positive at the critical radius $r_{c}$ where ${\it\Omega}(r_{c})={\it\omega}_{r}$ .

Figure 7.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of Gent–McWilliams instability (GMWI): (a) mode $(1,1)$ and (b) mode $(1,2)$ for $Ro=5$ (figure 4 d) ( $m=1,{\it\alpha}=0.5,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ .

Figure 8.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of the mixed baroclinic–Gent–McWilliams instability (BGMWI): mode $(1,1)$ for $Ro=2$ (figure 4 e) ( $m=1,{\it\alpha}=0.5,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ .

Figure 9.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of the (a) mode $(1,1)$ , (b) mode $(1,2)$ , (c) mode $(1,3)$ and (d) mode $(1,6)$ for $Ro=1.43$ (figure 4 f) ( $m=1,{\it\alpha}=0.5,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The double dotted dashed line (

) shows where the isopycnal potential vorticity gradient (3.9) changes sign.

So far, all the unstable modes observed for a pancake vortex derive from those for a columnar vortex. However, when the Rossby number is further decreased to $Ro=2$ (figure 4 e) and then to $Ro=1.43$ (figure 4 f), all the unstable modes for pancake vortices have no counterparts in columnar vortices except the displacement mode near the origin ((1, 4) in figure 4(e) and (1, 7) in figure 4 f). For $Ro=2$ (figure 4 e), the most unstable mode (1, 1) has a frequency larger than $0.135{\it\Omega}_{0}$ but is still close to the spectra of the columnar vortex. The structure of this mode (figure 8 a) is similar to Gent–McWilliams instability for $Ro=5$ (figure 7 a) but it tends to be distorted near $r=R$ . In § 5, we will show that the energy source of this mode is the potential energy of the base flow instead of the kinetic energy. For this reason, we shall call it mixed baroclinic–Gent–McWilliams instability. Two other modes start also to be observed near the frequency ${\it\omega}_{r}=0.25{\it\Omega}_{0}$ (figure 4 e) but they are weakly unstable. In contrast, for $Ro=1.43$ (figure 4 f), there are many modes around this frequency and their growth rate is much larger. The leading mode (1,1) corresponds in fact to baroclinic–Gent–McWilliams mode whose frequency and growth rate have increased until merging with the series of modes aligned near the frequency ${\it\omega}_{r}=0.25{\it\Omega}_{0}$ . A selection of these modes are depicted in figure 9. The first mode (1,1) shows some similarities with the one for $Ro=2$ (figure 8) but it is clearly more concentrated in the vortex core $r/R_{0}\leqslant 0.5$ and is maximum at a slightly higher vertical level $z/{\it\Lambda}=\pm 0.7$ . The next modes are similar but exhibit more radial oscillations (figure 9 b–d) with still two vertical oscillations, i.e. one per half-vertical plane. Similar modes have been observed in stratified non-rotating fluids (part 1) when the isopycnal deformations are sufficiently strong and have been attributed to baroclinic instability (BI). The isopycnals are indeed more and more distorted when the Rossby number decreases for given aspect ratio and Froude number. The isopycnals even overturn, i.e. $\max (\partial {\it\rho}_{t}/\partial z)>0$ , when $Ro\leqslant 1.3$ for ${\it\alpha}=0.5$ and $F_{h}=0.5$ . In quasi-geostrophic fluids, a necessary condition for baroclinic instability is the sign change of the potential vorticity gradient along the isopycnals (Charney & Stern Reference Charney and Stern1962; Eliassen Reference Eliassen1983; Hoskins, McIntyre & Robertson Reference Hoskins, McIntyre and Robertson1985; Ménesguen, McWilliams & Molemaker Reference Ménesguen, McWilliams and Molemaker2012b ):

(3.9) $$\begin{eqnarray}\left.{\displaystyle \frac{\partial {\it\Pi}}{\partial r}}\right|_{{\it\rho}_{t}}={\displaystyle \frac{\partial {\it\Pi}}{\partial r}}-{\displaystyle \frac{\partial {\it\Pi}}{\partial z}}{\displaystyle \frac{{\displaystyle \frac{\partial {\it\rho}_{t}}{\partial r}}}{{\displaystyle \frac{\partial {\it\rho}_{t}}{\partial z}}}}=0,\end{eqnarray}$$

where ${\it\Pi}=({\it\zeta}+f)\partial {\it\rho}_{t}/\partial z-r\partial {\it\Omega}/\partial z\partial {\it\rho}_{t}/\partial r$ is the potential vorticity of the base flow. The double dotted dashed line in figure 9 shows where $\partial {\it\Pi}/\partial r|_{{\it\rho}_{t}}=0$ . As can be seen, the modes develop in the vicinity of this line suggesting that they are due to baroclinic instability. In this respect, the leading mode (1,1) (figure 9 a) is not different from the following modes (figure 9 b–d) and could be equally classified as baroclinic. However, we will continue to call it baroclinic–Gent–McWilliams mode since it derives continuously from Gent–McWilliams instability as $Ro$ increases. A detailed study of these baroclinic modes will be conducted in § 8.

Figure 10.

(a) Maximum growth rates and (b) corresponding frequencies of the different types of instability as a function of $Ro$ for $m=1$ , ${\it\alpha}=0.5,F_{h}=0.5$ and $Re=10\,000$ : centrifugal instability 

, displacement mode 

, Gent–McWilliams instability 

, mixed Gent–McWilliams–centrifugal instability 

, mixed baroclinic–Gent–McWilliams instability 

 and baroclinic instability 

. The upper $x$ axis indicates the corresponding value of the square root of the Burger number $\sqrt{Bu}$ . The shaded region is gravitationally unstable.

The effect of the Rossby number on the growth rate and frequency of the most unstable modes of each instability for $m=1$ is summarized in figure 10. As already seen for $m=0$ , centrifugal instability (dashed line with circles) is stabilized for small negative $Ro\gtrsim -2$ . For positive Rossby number, centrifugal instability transforms continuously into the Gent–McWilliams instability (dashed line with filled triangle) when $Ro\lesssim 7$ (Yim & Billant Reference Yim and Billant2015). The growth rate of the mixed Gent–McWilliams–centrifugal instability (dashed line with open triangle) is almost constant for large Rossby number. For negative Rossby number, the Gent–McWilliams instability stabilizes around $Ro\sim -2$ as for the centrifugal instability. For positive Rossby number, the growth rate of the Gent–McWilliams instability decreases as the Rossby number decreases and becomes minimum for $Ro\simeq 2$ corresponding to a Burger number $Bu=({\it\alpha}Ro/2F_{h})^{2}\equiv (N{\it\Lambda}/fR)^{2}$ of approximately unity, as indicated in the upper $x$ axis in figure 10. Below this Rossby number, the frequency (figure 10 b) increases beyond the cutoff frequency ${\it\omega}_{r}=0.135{\it\Omega}_{0}$ above which the Gent–McWilliams instability no longer exists in the case of a columnar vortex (Yim & Billant Reference Yim and Billant2015). Simultaneously, the growth rate again increases and the instability becomes a mixed baroclinic–Gent–McWilliams instability (dashed line with star). For smaller Rossby number, the instability merges with the pure baroclinic instability (dashed line with diamond) with a growth rate increasing dramatically as $Ro$ decreases. However, the frequency saturates at ${\it\omega}_{r}\simeq 0.25{\it\Omega}_{0}$ . The growth rate of the displacement mode (dashed line with square) remains very low for any Rossby number.

Figure 11.

(a) Maximum growth rate and (b) corresponding frequency of centrifugal instability (

) for columnar (thick continuous lines) and pancake ( ${\it\alpha}=0.5$ ) (

) vortices as a function of $Ro$ for $m=1,F_{h}=0.5$ and $Re=10\,000$ . The dashed grey line shows the maximum of the asymptotic growth rate (3.5). The dotted line indicates the Gent–McWilliams instability.

Similar results have been obtained by Nguyen et al. (Reference Nguyen, Hua, Schopp and Carton2012) in quasi-geostrophic fluids except that the centrifugal instability is absent since $Ro\ll 1$ . Thus, they have reported for $m=1$ only two modes that they distinguish by their symmetry: a symmetric and an anti-symmetric mode. When the Burger number is of order unity or larger, the symmetric mode corresponds to the displacement mode herein and the anti-symmetric mode is similar to the Gent–McWilliams most unstable mode. In contrast, when $Bu$ is small, the symmetric and anti-symmetric modes correspond to the leading baroclinic modes herein (BGMWI or BI depending on $Bu$ ). The growth rate of the anti-symmetric mode is minimum for $Bu=1$ as in figure 10 when the transition between the Gent–McWilliams and baroclinic–Gent–McWilliams instabilities occur.

Finally, figure 11 shows that the maximum growth rate and corresponding frequency of the centrifugal instability for pancake (dashed line with circle) and columnar (grey continuous line) vortices are close. The maximum growth rate and associated frequency predicted by (3.5) is also represented by a dashed line. It underestimates the observed growth rate of both pancake and columnar vortices for positive Rossby number. This discrepancy comes from the smooth transition between Gent–McWilliams and centrifugal instabilities as $Ro$ varies.

Figure 12.

Growth rate ( ${\it\omega}_{i}/{\it\Omega}_{0}$ ) and frequency ( ${\it\omega}_{r}/{\it\Omega}_{0}$ ) spectra for $m=2$ for different Rossby numbers $Ro$ : (a) $Ro=\infty$ , (b) $Ro=20$ , (c) $Ro=0.8$ and (d) $Ro=0.25$ for $F_{h}=0.5$ and $Re=10\,000$ . Discrete symbols (○: for symmetric and 

for anti-symmetric modes) correspond to pancake vortices for ${\it\alpha}=1.2$ and thick continuous lines (

) correspond to columnar vortices ( ${\it\alpha}=\infty$ ).

In summary, for $m=1$ , the baroclinic–Gent–McWilliams instability dominates when $Ro$ is small and the centrifugal instability takes over when $Ro$ is large. In between, there exists a Gent–McWilliams instability for positive $Ro$ . For negative $Ro$ , the centrifugal instability is always dominant. Although only one particular set of parameters ( ${\it\alpha}=0.5,F_{h}=0.5$ and $Re=10\,000$ ) has been presented, we shall see in § 4 that this picture is typical of other parameter combinations in the strongly stratified regime with moderate and strong rotation.

3.3 Azimuthal wavenumber $m=2$

Finally, we present the effect of the Rossby number on the stability of the azimuthal wavenumber $m=2$ for the set of parameters: ${\it\alpha}=1.2,F_{h}=0.5$ and $Re=10\,000$ . Note that the aspect ratio is changed compared to $m=0$ and $m=1$ in order to show more typical examples of spectra. In the strongly stratified non-rotating case (part 1), two instabilities have been found: a centrifugal instability for sufficiently large buoyancy Reynolds number $ReF_{h}^{2}$ and a shear instability when $F_{h}/{\it\alpha}\leqslant 0.5$ . These correspond to the modes (2, 1)–(2, 2) and (2, 3), respectively in figure 12(a). When the Rossby number is decreased, a scenario similar to the one described for $m=1$ occurs. Centrifugal instability becomes less dominant for moderate Rossby number (see figure 12(b) for $Ro=20$ ) and disappears for small Rossby number (see figure 12(c) for $Ro=0.8$ ) since ${\it\Phi}>0$ for $-1<Ro<7.39$ . An example of the structure of a centrifugal mode for $Ro=20$ is depicted in figure 13(b). In contrast, a baroclinic instability appears for small Rossby number (modes (2, 1)–(2, 4) in figure 12(d) for $Ro=0.25$ ). As for $m=1$ (figure 9), the baroclinic modes differ by the number of oscillations in the radial direction and by their symmetry with respect to the mid-plane $z=0$ (not shown).

Figure 13.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of (a) shear instability (SI) $(2,1)$ and (b) centrifugal instability (CI) $(2,2)$ for $Ro=20$ (figure 12 b) ( $m=2,{\it\alpha}=1.2,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The dash dotted line in (a) shows the inflection radius $r_{I}$ where ${\it\zeta}^{\prime }(r_{I})=0$ . The dashed line in (b) indicates the contour where the Rayleigh discriminant ${\it\Phi}$ vanishes.

Figure 14.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of the mixed baroclinic–shear instability (BSI) (a) mode $(2,1)$ for $Ro=0.8$ (figure 12 c) and (b) mode $(2,5)$ for $Ro=0.25$ (figure 12 d) ( $m=2,{\it\alpha}=1.2,F_{h}=0.5,Re=10\,000$ ). The quasi-horizontal lines are isopycnals. The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The dash dotted line shows the inflection radius $r_{I}$ where ${\it\zeta}^{\prime }(r_{I})=0$ . The solid line in (b) show the critical layer where ${\it\omega}_{r}=m{\it\Omega}(r,z)$ .

Figure 15.

(a) Maximum growth rates and (b) corresponding frequencies of the different types of instabilities as a function of $Ro$ for $m=2,{\it\alpha}=1.2$ , $F_{h}=0.5$ and $Re=10\,000$ : centrifugal instability 

, shear instability 

, baroclinic–shear instability 

 and baroclinic instability 

. The upper $x$ axis indicates the corresponding value of the square root of the Burger number $\sqrt{Bu}$ . The shaded region is gravitationally unstable.

For all the Rossby numbers presented in figure 12, the shear instability remains present around the same frequency in the vicinity of the shear instability branch of the columnar vortex (grey continuous line near ${\it\omega}_{r}/{\it\Omega}_{0}=0.26$ ). However, its growth rate for the pancake vortex varies with $Ro$ while the maximum growth rate for the columnar vortex is independent of the Rossby number since the dominant mode is two-dimensional. The shape of the shear instability mode also varies somewhat with the Rossby number (see figure 13(a) for $Ro=20$ , figure 14 for $Ro=0.8$ and $Ro=0.25$ ). The mode for $Ro=20$ is almost identical to the one found for $Ro=\infty$ (part 1). The mode for $Ro=0.25$ (figure 14 b) is similar but extends vertically around $r/R=r_{I}/R=1.4$ where $r_{I}$ is the inflection point such that $\partial {\it\zeta}(r,z)/\partial r=0$ at $z=0$ . A similar mode has been found by Nguyen et al. (Reference Nguyen, Hua, Schopp and Carton2012) in quasi-geostrophic fluids. They argued that this mode originates from a baroclinic instability induced by the critical layer where ${\it\omega}_{r}=m{\it\Omega}$ . As can be seen in figure 14(b), the mode tends indeed to be distorted along the critical layer (shown by a solid line). However, figure 12 shows that this mode derives continuously from the shear instability. In addition, we will show in § 5 that the energy source of this instability is the potential energy of the base flow instead of the kinetic energy. For this reason, we will call here this instability the mixed baroclinic–shear instability (BSI).

Figure 15 outlines the effect of the Rossby number on the maximum growth rate and associated frequency of each instability. Centrifugal instability (dashed line with open circle) is stabilized for $-3\leqslant Ro\leqslant 17$ and is stronger for negative Rossby number than for positive ones as already observed for $m=0$ and $m=1$ . Strikingly, the opposite behaviour is observed for shear instability (dashed line with filled circle): it tends to be enhanced for moderate positive Rossby numbers and attenuated for finite negative Rossby numbers. However, the growth rate of the shear instability decreases to zero as $Ro$ decreases from $Ro=7$ to $Ro=1$ . Below $Ro=1$ (which corresponds to $Bu=1$ as indicated in the upper $x$ axis) it starts to increase again but the instability is then of the mixed type: baroclinic–shear instability. Such a growth rate minimum for $Bu\simeq 1$ is consistent with the results of Nguyen et al. (Reference Nguyen, Hua, Schopp and Carton2012). This mixed instability exists down to $Ro=-0.6$ while the classical shear instability reappears for $Ro\simeq -23$ . The baroclinic instability co-exists with the mixed baroclinic-shear instability for small Rossby numbers in the range $0\leqslant Ro\leqslant 0.4$ . Its growth rate becomes very high as soon as the Rossby number threshold is crossed but its frequency remains almost constant ${\it\omega}_{r}\simeq 0.5{\it\Omega}_{0}$ (figure 15 b).

In figure 16, the maximum growth rate and associated frequency of the centrifugal instability for $m=2$ for a pancake vortex is further compared to the one for a columnar vortex. As already seen for $m=0$ and $m=1$ , they are close and in reasonable agreement with the asymptotic formula (3.5).

Figure 16.

(a) Maximum growth rates and (b) corresponding frequencies of centrifugal instability for columnar (thick continuous lines,

) and pancake ( ${\it\alpha}=1.2$ ) (symbolled lines,

) vortices for $m=2,F_{h}=0.5$ and $Re=10\,000$ . The dashed grey line (

) shows the maximum of the asymptotic growth rate (3.5).

4.1 Effect of the Reynolds number

Figure 17 shows the effect of the Reynolds number on the maximum growth rate and associated frequency for ${\it\alpha}=0.5,F_{h}=0.3$ and $Ro=1.25$ . For these parameters, only the Gent–McWilliams instability and displacement mode are unstable for $m=1$ while only the baroclinic–shear instability is unstable for $m=2$ . The growth rates of the Gent–McWilliams instability (dashed line with triangles) and the baroclinic–shear instability (dashed line with stars) asymptote to a constant value for large $Re$ and decreases to zero for $Re\simeq 1000$ and $Re\simeq 2000$ , respectively. Surprisingly, the growth rate of the displacement mode (dashed line with squares) first increases when the Reynolds number decreases. Thereby, it is most unstable for a finite Reynolds number $Re\simeq 500$ . The instability of the displacement mode is therefore of viscous origin. A similar viscous instability of the displacement mode of a columnar vortex exists in stratified-rotating fluids (Billant Reference Billant2010; Riedinger, Le Dizès & Meunier Reference Riedinger, Le Dizès and Meunier2010). Further analyses would be necessary to explain its detailed mechanism. In summary, for the given parameters ${\it\alpha}=0.5,F_{h}=0.5$ and $Ro=1.25$ , the $m=1$ displacement instability is dominant for $Re<2000$ while the $m=1$ Gent–McWilliams instability is dominant for higher $Re$ .

Figure 17.

(a) Maximum growth rates and (b) corresponding frequencies as a function of the Reynolds number $Re$ for the Gent–McWilliams instability (

), displacement mode (

) for $m=1$ and the baroclinic–shear instability (

) for $m=2$ for ${\it\alpha}=0.5$ , $F_{h}=0.3$ and $Ro=1.25$ .

4.2 Effect of the Froude number

The Froude number is now varied for $Re=10\,000$ still for ${\it\alpha}=0.5$ and $Ro=1.25$ (figure 18). The displacement mode (dashed line with squares) keeps a very low growth rate and frequency independently of the Froude number. In contrast, the growth rate of the Gent–McWilliams instability (dashed line with triangles) fluctuates with $F_{h}$ : it exhibits two successive maxima as the Froude number increases before increasing widely when it transforms into the mixed baroclinic–Gent–McWilliams instability. The structure of the mode for some selected Froude numbers is displayed in figure 19. For $F_{h}=0.05$ , the mode exhibits five oscillations along the vertical and is symmetric while for $F_{h}=0.2$ , it has only two oscillations occupying the whole pancake vortex and is anti-symmetric as seen before (see figure 7 a). This change of structure explains why there is a slight frequency jump in figure 18(b) (dashed line with triangles) and two growth rate maxima (figure 18 a). For $F_{h}=0.4$ (figure 19 c), the mode is of mixed type: baroclinic–Gent–McWilliams with a frequency above the cutoff ${\it\omega}_{r}=0.135{\it\Omega}_{0}$ of the Gent–McWilliams instability and slightly below the frequency ${\it\omega}_{r}=0.25{\it\Omega}_{0}$ of the baroclinic instability (figure 18 b).

Figure 18.

(a) Maximum growth rates and (b) corresponding frequencies as a function of the Froude number $F_{h}$ for ${\it\alpha}=0.5$ , $Ro=1.25$ and $Re=10\,000$ of the different instabilities: displacement mode (

); Gent–McWilliams instability (

); baroclinic-Gent–McWilliams instability (

); baroclinic instability (

) for $m=1$ ; shear instability (

); baroclinic-shear instability (

) and baroclinic instability for $m=2$ (

). The shaded area indicates the gravitationally unstable region.

Figure 19.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of the most unstable mode for different Froude numbers: Gent–McWilliams instability for (a)  $F_{h}=0.05$ and (b) $F_{h}=0.2$ and (c) mixed baroclinic–Gent–McWilliams instability for $F_{h}=0.4$ for $m=1,{\it\alpha}=0.5,Ro=1.25$ and $Re=10\,000$ . The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ .

Figure 20.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of the most unstable modes for different Froude numbers: shear instability for (a) $F_{h}=0.05$ and mixed baroclinic–shear instability for (b) $F_{h}=0.3$ and (c) $F_{h}=0.45$ for $m=2$ , ${\it\alpha}=0.5,Ro=1.25$ and $Re=10\,000$ . The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The solid line (—) is the critical layer ( $r_{c},z_{c}$ ) where $m{\it\Omega}(r_{c},z_{c})={\it\omega}_{r}$ .

Figure 21.

(a) Maximum growth rates and (b) corresponding frequencies as a function of aspect ratio ${\it\alpha}$ for $F_{h}=0.3$ , $Ro=1.25$ and $Re=10\,000$ of the different instabilities: displacement mode (

); Gent–McWilliams instability (

); baroclinic–Gent–McWilliams instability (

); baroclinic instability (

) for $m=1$ ; shear instability (

); baroclinic–shear instability (

) and baroclinic instability for $m=2$  (

). The shaded area indicates the gravitationally unstable region.

Shear instability for $m=2$ (dashed line with filled circles in figure 18) is most unstable as $F_{h}\rightarrow 0$ and stabilizes for $F_{h}\simeq 0.1$ . However, it becomes unstable again for $F_{h}\geqslant 0.3$ but under the mixed form of a shear–baroclinic instability (dashed line with stars). When $F_{h}\geqslant 0.4$ , the baroclinic instability for $m=2$ (dashed line with crosses) becomes quickly strongly unstable as $F_{h}$ increases (with a growth rate higher than the upper limit of figure 18 a) but with a constant frequency ${\it\omega}_{r}\simeq 0.5{\it\Omega}_{0}$ . Examples of modes corresponding to these three types of instability for $m=2$ are shown in figure 20. For small $F_{h}$ , the shear instability (figure 20 a) is strongly localized near $z=0$ where the radial shear is maximum. In contrast, baroclinic–shear instability (figure 20 b,c) occupies the whole pancake vortex as already seen in § 3.3.

4.3 Effect of the aspect ratio

We now vary the aspect ratio keeping the other parameters to the same values as before: $F_{h}=0.3,Ro=1.25$ and $Re=10\,000$ (figure 21). For $m=1$ , the displacement mode (dotted line with squares) remains only marginally unstable while the growth rate of the Gent–McWilliams instability (dashed line with triangles) is large and varies non-monotonically with the aspect ratio. It exhibits two maxima, for ${\it\alpha}\simeq 1$ and for large ${\it\alpha}$ . For very small ${\it\alpha}$ , the growth rate rises again but the instability is then under the form of the mixed baroclinic–Gent–McWilliams instability. The full spectra for three aspect ratios are displayed in figure 22 along with the spectra for ${\it\alpha}=\infty$ , i.e. for a columnar vortex (solid line). For small aspect ratio (figure 22 a), there exist only two unstable modes: the displacement mode near the origin and the baroclinic–Gent–McWilliams instability near ${\it\omega}=0.2{\it\Omega}_{0}$ whose eigenmode is shown in figure 23(a). This markedly differs from the columnar case (solid line). In contrast, for larger aspect ratios ( ${\it\alpha}=3$ , figure 22 b), many modes are present within the frequency range of the columnar vortex. They are however less unstable than for ${\it\alpha}=0.38$ . At even larger aspect ratio ${\it\alpha}=10$ (figure 22 c) the spectrum becomes denser with a bell shape resembling the spectra of the columnar vortex. However, the maximum growth rate for the columnar vortex is still higher. The most unstable eigenmode for ${\it\alpha}=10$ (figure 23 c) presents a vertical wavelength ${\it\lambda}\simeq 5.23R$ close to the most amplified wavelength of the columnar vortex ${\it\lambda}\simeq 4.5R$ . A longer typical wavelength ${\it\lambda}\simeq 6.16R$ is observed for ${\it\alpha}=3$ (figure 23 b).

Figure 22.

Growth rate ( ${\it\omega}_{i}/{\it\Omega}_{0}$ ) and frequency ( ${\it\omega}_{r}/{\it\Omega}_{0}$ ) spectra for $m=1$ for different aspect ratios: (a) ${\it\alpha}=0.38$ , (b) ${\it\alpha}=3$ and (c) ${\it\alpha}=10$ for $F_{h}=0.3,Ro=1.25$ and $Re=10\,000$ . The solid line (——) indicates the spectrum for the columnar vortex for the same $F_{h},Ro$ and $Re$ .

Figure 23.

Real part of the radial velocity perturbation Re( $u_{r}$ ) of the most unstable mode for different aspect ratios: mixed baroclinic–Gent–McWilliams instability (a) ${\it\alpha}=0.38$ and Gent–McWilliams instability for (b) ${\it\alpha}=3$ and (c) ${\it\alpha}=10$ for $F_{h}=0.3,Ro=1.25$ and $Re=10\,000$ . The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ .

As seen in figure 21(a), the shear instability (dashed line with filled circles) is only unstable for large aspect ratio ${\it\alpha}\geqslant 2$ . However, for small ${\it\alpha}$ , it reappears in the form of the mixed baroclinic–shear instability (dashed line with stars). The pure baroclinic instability for $m=2$ (dashed line with cross) also arises for very small aspect ratio.

Figure 24.

Maximum growth rates as a function of vertical Froude number $F_{h}/{\it\alpha}$ for $Ro=1.25$ , $Re=10\,000$ , (a) $m=1$ and (b) $m=2$ . The upper $x$ axis indicates the corresponding value of $\sqrt{Bu}$ . The filled coloured symbols show the results when $F_{h}$ is varied for ${\it\alpha}=0.5$ , open symbols when ${\it\alpha}$ is varied for $F_{h}=0.3$ , $Re=10\,000$ and black filled symbols when ${\it\alpha}$ is varied for $F_{h}=0.05$ and $Re=360\,000$ . The shaded area indicates the gravitationally unstable region. Different symbols are used for each instability: Gent–McWilliams (

), baroclinic–Gent–McWilliams (

), shear (

), baroclinic–shear (

) and baroclinic (

) instabilities.

4.4 Scaling in terms of the vertical Froude number

The two previous sections have revealed different growth rate variations as a function of the Froude number $F_{h}$ and aspect ratio ${\it\alpha}$ . However, figure 24 shows that these growth rate variations are actually almost identical when represented as a function of the vertical Froude number $F_{h}/{\it\alpha}$ . The only deviation from this self-similarity is for small $F_{h}/{\it\alpha}$ : when $F_{h}/{\it\alpha}<0.1$ , the growth rate of the Gent–McWilliams instability grows with decreasing $F_{h}/{\it\alpha}$ when ${\it\alpha}$ is varied and $F_{h}$ kept constant, whereas it decreases when ${\it\alpha}$ is fixed and $F_{h}$ is varied. A small discrepancy is also observed for $m=2$ when $F_{h}/{\it\alpha}$ is small. These differences come from viscous effects due to vertical shear since they scale like $1/\mathscr{R}$ where $\mathscr{R}=ReF_{h}^{2}$ is the buoyancy Reynolds number, because the typical vertical scale $L_{v}$ scales like $L_{v}\sim F_{h}R$ . Thus, viscous effects increase when the Froude number decreases keeping the Reynolds number constant. To prove this, the results for a smaller Froude number $F_{h}=0.05$ but higher Reynolds number $Re=360\,000$ are also displayed in figure 24 (black filled symbols). This Reynolds number has been chosen so that the buoyancy Reynolds number $\mathscr{R}=900$ is the same as for $F_{h}=0.3$ and $Re=10\,000$ . We see that the growth rate variations are almost identical for these two parameter sets, confirming that the vertical Froude number scaling holds if the buoyancy Reynolds number is sufficiently large.

If the Rossby number is also varied but kept small, the growth rate curves for different $Ro$ collapse when represented as a function of $2F_{h}/({\it\alpha}Ro)=1/\sqrt{Bu}$ in agreement with the quasi-geostrophic theory (figure 25). We note that the transition from the Gent–McWilliams instability to the baroclinic–Gent–McWilliams instability occurs near $Bu\simeq 1$ as in figure 24(a) (see the upper $x$ axis) and in a quasi-geostrophic fluid (Nguyen et al. Reference Nguyen, Hua, Schopp and Carton2012). However, the growth rate curves are no longer self-similar when $2F_{h}/({\it\alpha}Ro)$ approaches the threshold (3.2) for gravitational instability. Indeed, this threshold depends on $Ro$ and is thus different for $Ro=0.2$ and $Ro=0.5$ . The transition from shear instability to a baroclinic–shear instability also occurs around $Bu=1$ (figure 25 b).

Figure 25.

Maximum growth rates as a function of $2F_{h}/({\it\alpha}Ro)$ for $Ro=0.5$ (open symbols) and $Ro=0.2$ (filled symbols) for $F_{h}=0.3$ , $Re=10\,000$ (a) $m=1$ and (b) $m=2$ . The upper $x$ axis indicates the corresponding value of $\sqrt{Bu}$ . Different symbols are used for each instability: Gent–McWilliams (

), baroclinic–Gent–McWilliams (

), shear (

), baroclinic–shear (

) and baroclinic (

) instabilities. The vertical dashed and dashed-dotted lines show the thresholds for gravitational instability for $Ro=0.5$ and $Ro=0.2$ , respectively.

8.1 A typical example

Figure 32 shows the spectra for ${\it\alpha}=0.5,F_{h}=0.3,Ro=0.4$ and $Re=10\,000$ for different azimuthal wavenumbers $m$ . These control parameters are just below the threshold of gravitational instability which is $F_{h}=0.32$ for ${\it\alpha}=0.5$ and $Ro=0.4$ . Among all azimuthal wavenumbers, $m=2$ is the most unstable. For each $m$ , the most unstable mode is anti-symmetric (black symbol) and the second most unstable mode is symmetric (grey symbol). Nevertheless, the growth rate difference between anti-symmetric and symmetric modes becomes very small as $m$ increases. We will show that these modes are due to baroclinic instability. However, the displacement mode is also observed for $m=1$ near the origin and the baroclinic–shear instability for $m=2$ is located near ${\it\omega}_{r}/{\it\Omega}_{0}=0.18$ with a small growth rate. Note also that the leading mode for $m=1$ is the baroclinic–Gent–McWilliams instability which derives continuously from the Gent–McWilliams instability. For higher azimuthal wavenumbers, $m\geqslant 3$ , only the baroclinic instability exists. No instability has been found for $m=0$ and $m\geqslant 6$ . The characteristic frequency of each azimuthal wavenumber is proportional to $m$ : ${\it\omega}_{r}/{\it\Omega}_{0}\simeq 0.25m$ , i.e. the azimuthal phase velocity is constant. In fact, this corresponds to the angular velocity of the base flow ${\it\Omega}(r_{b},z_{b})\simeq 0.25$ at the point $r_{b}=0,z_{b}=1.17{\it\Lambda}$ where the vertical density gradient $\partial {\it\rho}_{t}/\partial z$ is maximum for ${\it\alpha}=0.5,F_{h}=0.3$ and $Ro=0.4$ .

Figure 32.

Growth rate ( ${\it\omega}_{i}/{\it\Omega}_{0}$ ) and frequency ( ${\it\omega}_{r}/{\it\Omega}_{0}$ ) spectra for different azimuthal wavenumbers: $m=1$  (

);  $m=2$  (

);  $m=3$  (

);  $m=4$  (

);  $m=5$  (

) for ${\it\alpha}=0.5$ , $F_{h}=0.3,Ro=0.4$ , and $Re=10\,000$ . Grey and black symbols indicate symmetric and anti-symmetric modes, respectively.

Figure 33 shows the first three anti-symmetric eigenmodes for $m=3$ . As already shown for $m=1$ (figure 9), the number of oscillations in the radial direction increases with the mode number while the vertical structure remains the same. Figure 34(a) shows the growth rate as a function of the radial wavenumber $l=2{\rm\pi}/{\it\lambda}_{r}$ where the radial wavelength ${\it\lambda}_{r}$ is estimated as twice the distance between two successive extrema, as illustrated in figure 33(b). For each azimuthal wavenumber, the growth rate decreases monotonically as $l$ increases. As already visible in figure 32, the maximum growth rate exhibits a bell-shaped curve as a function of $m$ (figure 34 b). These wavenumber properties are very reminiscent of baroclinic instability in parallel shear flows (Vallis Reference Vallis2006).

Figure 33.

Real part of the radial velocity perturbation Re $(u_{r})$ for the first three anti-symmetric baroclinic eigenmodes for $m=3$ : (a) $(3,1)$ , (b) $(3,3)$ and (c) $(3,5)$ for ${\it\alpha}=0.5$ , $F_{h}=0.3$ , $Ro=0.4$ and $Re=10\,000$ . The dotted line indicates the extension of the base flow by showing the contour where ${\it\Omega}=0.1{\it\Omega}_{0}$ . The double dotted dashed line (

) shows where the isopycnal potential vorticity gradient (3.9) changes sign.

Figure 34.

(a) Growth rate ( ${\it\omega}_{i}/{\it\Omega}_{0}$ ) as a function of the radial wavenumber $l$ for different azimuthal wavenumbers:  $m=1$  (

);  $m=2$  (

);  $m=3$  (

);  $m=4$  (

);  $m=5$  (

) and (b) maximum growth rate as a function of the azimuthal wavenumber $m$ for ${\it\alpha}=0.5$ , $F_{h}=0.3,Ro=0.4,$ and $Re=10\,000$ .

8.2 Parametric study

Figure 35 outlines the effect of the Rossby number on the maximum growth rate and corresponding frequency of the baroclinic instability for each azimuthal wavenumber for ${\it\alpha}=0.5,F_{h}=0.3$ and $Re=10\,000$ . When $Ro$ increases, the growth rate decreases faster as $m$ increases. Hence, $m=3$ is the most unstable azimuthal wavenumber when $Ro$ is close to the threshold for the gravitational instability whereas $m=1$ becomes the most unstable for $Ro\geqslant 0.5$ . In between, $m=2$ is the most unstable. In contrast, the frequency of each azimuthal wavenumber is independent of the Rossby number (figure 35 b). Similarly, figure 36 shows the effect of the Froude number for ${\it\alpha}=0.5,Ro=0.4$ and $Re=10\,000$ . The frequency is again independent of the Froude number (figure 36 b) whereas the maximum growth rate (figure 36 a) increases with $F_{h}/{\it\alpha}$ at a rate increasing with $m$ .

Figure 35.

(a) Maximum growth rate and (b) corresponding frequency of the baroclinic instability (dark solid lines) as a function of $Ro$ for different azimuthal wavenumbers: $m=1$  (

);  $m=2$  (

);  $m=3$  (

);  $m=4$  (

);  $m=5$  (

) for ${\it\alpha}=0.5,F_{h}=0.3$ and $Re=10\,000$ . The shaded area indicates the gravitationally unstable region.

Figure 36.

(a) Maximum growth rate and (b) corresponding frequency of the baroclinic instability (dark solid lines) as a function of $F_{h}/{\it\alpha}$ for different azimuthal wavenumbers: $m=1$  (

);  $m=2$  (

);  $m=3$  (

);  $m=4$  (

);  $m=5$  (

) for ${\it\alpha}=0.5,Ro=0.4,$ and $Re=10\,000$ . The shaded area indicates the gravitationally unstable region.

8.3 A simple analytical model

In part 1, a model consisting of a bounded vortex with an angular velocity varying only in the vertical direction has been considered and shown to account qualitatively for the characteristics of baroclinic instability in stratified non-rotating fluids. Such a model takes into account the main features of the base flow in the core of the pancake vortex where the baroclinic instability develops. Its stability can be solved analytically when the vertical variations are weak. Here, this model is extended to take into account a background rotation. The base angular velocity of the vortex is assumed to be

(8.1) $$\begin{eqnarray}{\it\Omega}=\tilde{{\it\Omega}}_{0}-\tilde{{\it\Omega}}_{1}z,\end{eqnarray}$$

where $\tilde{{\it\Omega}}_{0}$ and $\tilde{{\it\Omega}}_{1}$ are constants. From the thermal wind relation (2.4), the base density is obtained as

(8.2) $$\begin{eqnarray}{\it\rho}_{b}=\frac{{\it\rho}_{0}}{g}\left[\tilde{{\it\Omega}}_{0}+\frac{f}{2}-\tilde{{\it\Omega}}_{1}z\right]\tilde{{\it\Omega}}_{1}r^{2}.\end{eqnarray}$$

As in part 1, we consider that the base flow is contained in a rigid cylinder of radius $R$ and height $H$ between $z=-H/2$ and $z=H/2$ . By assuming that the vertical variations are weak, i.e. $\tilde{{\it\Omega}}_{1}H\ll |\tilde{{\it\Omega}}_{0}+f/2|$ , the equations (2.8)–(2.12) in the inviscid limit can be reduced at leading order in $\tilde{{\it\Omega}}_{1}$ to

(8.3) $$\begin{eqnarray}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p}{\partial r}\right)-\left[\frac{m^{2}}{r^{2}}+C^{2}\frac{\partial ^{2}}{\partial z^{2}}\right]p=0+O(\tilde{{\it\Omega}}_{1}^{2}),\end{eqnarray}$$

where $C=2|\tilde{{\it\Omega}}_{0}+f/2|/N$ . Note that the hypothesis $\tilde{{\it\Omega}}_{1}H\ll |\tilde{{\it\Omega}}_{0}+f/2|$ is not valid around $\tilde{Ro}\equiv 2\tilde{{\it\Omega}}_{0}/f=-1$ . Here, the dimensionless numbers are denoted with a tilde in order to distinguish them from the ones defined in (2.13). The Coriolis parameter $f$ enters the problem only through the constant $C$ in (8.3). The general solution of (8.3) is

(8.4) $$\begin{eqnarray}p=\text{J}_{m}(Ckr)\left(A\cosh kz+B\sinh kz\right),\end{eqnarray}$$

where $\text{J}_{m}$ is the Bessel function of order $m$ of the first kind and $A$ and $B$ are constants. Imposing the boundary conditions $u_{z}(z=\pm H/2)=0$ and $u_{r}(r=R)=0$ yields two relations similar to those for the classical Eady problem (Vallis Reference Vallis2006)

(8.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\omega}=m\tilde{{\it\Omega}}_{0}+\frac{m\tilde{{\it\Omega}}_{1}}{k}\sqrt{\left(1-\frac{kH}{2\tanh (kH/2)}\right)\left(1-\frac{kH\tanh (kH/2)}{2}\right)}, & \displaystyle\end{eqnarray}$$

(8.6) $$\begin{eqnarray}\displaystyle & \displaystyle CkR={\it\mu}_{m,n}, & \displaystyle\end{eqnarray}$$

where ${\it\mu}_{m,n}$ is the $n$ th root of J $_{m}$ . Combining the condition for instability $kH<2.4$ and the fact that ${\it\mu}_{m,n}>{\it\mu}_{1,1}=3.83$ , yields the instability condition

(8.7) $$\begin{eqnarray}\frac{\tilde{F_{h}}}{\tilde{{\it\alpha}}}\left|1+\frac{1}{\tilde{Ro}}\right|>0.8,\end{eqnarray}$$

where $\tilde{{\it\alpha}}=H/R$ and $\tilde{F_{h}}=\tilde{{\it\Omega}}_{0}/N$ . This condition simply states that the Burger number $\tilde{Bu}=\tilde{{\it\alpha}}^{2}N^{2}/(f+2\tilde{{\it\Omega}}_{0})^{2}$ based on the absolute angular velocity $\tilde{{\it\Omega}}_{0}+f/2$ should be below a threshold $\tilde{Bu}<0.4$ for instability. When $\tilde{Ro}\ll 1$ , equation (8.7) recovers the classical condition for baroclinic instability (Eady Reference Eady1949; Saunders Reference Saunders1973; Hide & Mason Reference Hide and Mason1975). We can also derive scaling laws for the maximum growth rate and the most amplified wavenumber (8.5). For any $m$ , the growth rate is maximum for the first root of the Bessel function $n=1$ . An asymptotic expansion of this root for large $m$ is ${\it\mu}_{m,1}=m+1.856m^{1/3}+O(m^{-1/3})$ (Abramowitz & Stegun Reference Abramowitz and Stegun1972). Taking only the leading order of this expansion, i.e. ${\it\mu}_{m,1}\sim m$ , (8.6) implies $m/k=CR$ . The growth rate is thus maximum when the term inside the square root in (8.5) is minimum, i.e. when $kH=1.6$ . The most amplified azimuthal wavenumber is therefore

(8.8) $$\begin{eqnarray}m_{\mathit{max}}\simeq 3.2\frac{\tilde{F_{h}}}{\tilde{{\it\alpha}}}\left|1+\frac{1}{\tilde{Ro}}\right|,\end{eqnarray}$$

and the maximum growth rate is

(8.9) $$\begin{eqnarray}{\it\omega}_{imax}\simeq 0.6\tilde{{\it\Omega}}_{1}R\frac{\tilde{F_{h}}}{\tilde{{\it\alpha}}}\left|1+\frac{1}{\tilde{Ro}}\right|,\end{eqnarray}$$

for large $m$ . These scaling laws are tested in figure 37 for baroclinic instability of the pancake vortex by assuming that $F_{h}/{\it\alpha}|1+1/Ro|$ is equivalent to $\tilde{F_{h}}/\tilde{{\it\alpha}}|1+1/\tilde{Ro}|$ . Different combinations of $F_{h}$ and $Ro$ are shown for ${\it\alpha}=0.5$ and $Re=10\,000$ . As seen in figure 37(a), the maximum growth rates align along a straight line when represented as a function of $F_{h}/{\it\alpha}|1+1/Ro|$ . Baroclinic instability only occurs for $F_{h}/{\it\alpha}|1+1/Ro|\geqslant 1.46$ in qualitative agreement with (8.7). Note that the leftmost point (star) in figure 37(a), which is slightly away from the other points, is for a negative Rossby number $Ro=-5$ . Similarly, figure 37(b) shows that the most amplified azimuthal wavenumber increases approximately linearly with $F_{h}/{\it\alpha}|1+1/Ro|$ in agreement with (8.8).

Figure 37.

(a) Maximum growth rate and (b) most amplified azimuthal wavenumber $m$ of the baroclinic instability for different combinations of $Ro$ and $F_{h}$ : $Ro=0.4$ , $F_{h}$ varies

, $F_{h}=0.3$ , $Ro$ varies

, $Ro=1$ , $F_{h}=0.42$

, $Ro=5$ , $F_{h}=0.67$

, $Ro=10$ , $F_{h}=0.7$

, $Ro=-5$ , $F_{h}=0.8$

and $Ro=\infty$ , $F_{h}$ varies 

. Other parameters are fixed to ${\it\alpha}=0.5$ , $Re=10\,000$ . The dotted line in (a) is a fit.

Figure 38 displays a map of the domain of existence of the baroclinic instability in the parameter space ( $F_{h}/{\it\alpha}$ , $Ro$ ). The dashed line represents the threshold $F_{h}/{\it\alpha}|1+1/Ro|=1.46$ deduced from figure 37(a). The shaded region is gravitationally unstable. The circle symbols indicate the parameters for which the baroclinic instability has been observed for $m=2$ , $Re=10\,000$ and ${\it\alpha}=0.5$ while the cross symbols correspond to the parameters stable to the baroclinic instability for $m=2$ . We can see that there is a good agreement between the threshold $F_{h}/{\it\alpha}|1+1/Ro|=1.46$ and the numerical results for positive Rossby number. However, the threshold departs from the numerical results for negative Rossby numbers around $Ro\simeq -5$ . This is most likely due to the assumption of slow vertical variation $\tilde{{\it\Omega}}_{1}H\ll \tilde{{\it\Omega}}_{0}+f/2$ used to derive the theoretical scaling laws. This hypothesis indeed breaks down around $\tilde{Ro}=-1$ and this may affect a large range of negative Rossby numbers $Ro$ . Furthermore, the location where the density gradient is maximum actually varies with the Rossby number, implying that $\tilde{{\it\Omega}}_{0}$ also varies with $Ro$ . These variations could be taken into account in a refined analysis.

Figure 38.

Domain of existence of the baroclinic instability for $m=2$ as a function of $F_{h}/{\it\alpha}$ and $Ro$ for ${\it\alpha}=0.5$ and $Re=10\,000$ : circles (

) indicate the parameters unstable to baroclinic instability, crosses ( $\times$ ) are for the stable case. The shaded area indicates the region unstable to gravitational instability and the dashed line (– – –) corresponds to the threshold $F_{h}/{\it\alpha}|1+1/Ro|=1.46$ .

Figure 39.

Domains of existence of the different instabilities for the azimuthal wavenumbers (a,b) $m=0$ , (c,d) $m=1$ and (e,f) $m=2$ as a function of $Ro$ and $F_{h}/{\it\alpha}$ (a,c,e) and $Bu=Ro^{2}{\it\alpha}^{2}/(4F_{h}^{2})$ (b,d,f) for $Re=10\,000$ and various aspect ratio ${\it\alpha}$ . The symbols indicate the different instabilities for each set of parameters investigated: in (a–f): centrifugal (CI)

; baroclinic (BI)

; and stable ( $\times$ ). In (c,d): Gent–McWilliams (GMWI) ▴; baroclinic–Gent–McWilliams (BGMWI) $+$ . In (e,f): shear (SI) ▴ and baroclinic–shear (BSI) $+$ instabilities. In the white region in (d), only the displacement mode (DM) is unstable for finite $Re$ . The shaded area indicates the gravitationally unstable region (GI), the thick solid line shows the threshold for centrifugal instability for ${\it\alpha}=0.5$ derived from (3.5), the dotted line is the threshold $F_{h}/{\it\alpha}<c(Ro)/{\rm\pi}$ for shear instability and the dashed line is the threshold for baroclinic instability: $F_{h}/{\it\alpha}|1+1/Ro|=1.46$ . The dashed dotted lines in (c,d) and (e,f) show the empirical thresholds for baroclinic–Gent–McWilliams instability ( $Bu=0.7Ro+0.1$ ) and baroclinic–shear instability ( $Bu=1.3/(-Ro+2.26)$ for $|Ro|<2$ and $Bu=2.5Ro$ for $Ro>2$ ) instabilities, respectively.

9.1 Parameter space ( $F_{h}/{\it\alpha}$ , $Ro$ )

Figure 39(a,c,e) summarize the different instabilities observed for each azimuthal wavenumber $m=0,1$ and 2 in the parameter space ( $F_{h}/{\it\alpha}$ , $Ro$ ) for $Re=10\,000$ and various aspect ratios. The symbols indicate the instability type while the lines indicate the different semi-theoretical thresholds that have been derived throughout the paper. These conditions are generally in good agreement with the numerical results for all the parameters investigated. In summary, a centrifugal instability exists for $m=0,1,2$ for sufficiently high Rossby and Froude numbers. The solid lines correspond to the thresholds for centrifugal instability for ${\it\alpha}=0.5$ that can be obtained from the asymptotic formula (3.5). This formula shows that the aspect ratio of the vortex has no effect and there is a stabilization at low Froude numbers because the buoyancy Reynolds number $\mathscr{R}=ReF_{h}^{2}$ , which controls viscous effects, decreases. Shear instability for $m=2$ is present below a critical vertical Froude number $F_{h}/{\it\alpha}$ depending on the Rossby number $F_{h}/{\it\alpha}<c(Ro)/{\rm\pi}$ , where $c(Ro)$ is defined in (6.1) (dotted line in figure 39 e). However, for higher $F_{h}/{\it\alpha}$ , it reappears under the mixed form of a baroclinic–shear instability when the Rossby number is not too large (dashed dotted line in figure 39(e) $Ro\lesssim 10F_{h}^{2}/{\it\alpha}^{2}$ for $Ro>0$ and $Ro\gtrsim -1.1F_{h}/{\it\alpha}$ for $Ro<0$ ). The Gent–McWilliams instability for $m=1$ (triangles in figure 39 c) is observed over wide ranges of $Ro$ and $F_{h}/{\it\alpha}$ except when it transforms to the baroclinic–Gent–McWilliams instability (squares in figure 39 c) for small Rossby numbers such that $F_{h}/{\it\alpha}>|Ro|/(2\sqrt{0.7Ro+0.1})$ (dashed dotted lines in figure 39 c). A baroclinic instability occurs when $F_{h}/{\it\alpha}|1+1/Ro|>1.46$ (dashed lines in figure 39 c,e), i.e. only in a small band close to the threshold for the gravitational instability (shaded region).

9.2 Parameter space ( $Bu$ , $Ro$ )

Figures 39(b,d,f) display the same instability maps but focused on the region $|Ro|<2$ which pertains to most mesoscale oceanic vortices. In addition, the $x$ -axis is now the Burger number $Bu=Ro^{2}{\it\alpha}^{2}/(4F_{h}^{2})$ instead of the vertical Froude number $F_{h}/{\it\alpha}$ . Indeed, the Burger number is more appropriate to describe the region of small Rossby number since it is the only non-dimensional parameter in the quasi-geostrophic limit. Only the range $0\leqslant Bu\leqslant 2$ is displayed as in Nguyen et al. (Reference Nguyen, Hua, Schopp and Carton2012). In these ranges, we see that centrifugal instability for ${\it\alpha}=0.5$ exists only in the bottom left corner for sufficiently negative $Ro$ and small $Bu$ . The Gent–McWilliams instability (triangles in figure 39 d) occurs when $Bu\gtrsim 1$ . In contrast, the baroclinic–Gent–McWilliams instability exists mostly for positive $Ro$ in a band adjacent to the domain of baroclinic instability. The threshold can be fitted approximately by $Bu\simeq 0.7Ro+0.1$ (dashed dotted line in figure 39 d). Hence, there is an intermediate range of Burger number where only the displacement mode is unstable for $m=1$ for finite Reynolds number. For $m=2$ (figure 39 f), the domain of existence of the baroclinic–shear instability is also contiguous to the domain unstable to baroclinic instability. The upper Burger number limit is given by $Bu\simeq 1.3/(-Ro+2.26)$ . Just above this threshold, $m=2$ perturbations are stable since shear instability starts to be active only when $Bu\gtrsim 7$ . These results are in general consistent with those in the quasi-geostrophic limit ( $Ro\rightarrow 0$ ) in continuously stratified fluids (Nguyen et al. Reference Nguyen, Hua, Schopp and Carton2012) or in two-layer fluids (Ikeda Reference Ikeda1981; Flierl Reference Flierl1988; Helfrich & Send Reference Helfrich and Send1988; Benilov Reference Benilov2003). In particular, Flierl (Reference Flierl1988) reports for piecewise profiles that the $m=1$ mode is unstable for $Bu>1$ while higher azimuthal modes, $m\geqslant 2$ , are unstable to shear instability for $Bu>O(1)$ for sufficiently steep vorticity profiles and to baroclinic instability for $Bu<O(1)$ (see for example his figure 10 b). A neutrally stable region around $Bu=O(1)$ is also observed for moderately steep vorticity profiles as in figure 39.