2.1. Non-hydrostatic Boussinesq equations

We consider a rotating incompressible fluid with velocity $\boldsymbol{u}=(u,v,w)$ under the $f$ -plane approximation. With $z$ the vertical coordinate (normal to a reference geopotential surface), the fluid density ${\it\rho}$ and pressure $p$ are written as

(2.1) $$\begin{eqnarray}\displaystyle {\it\rho}(\boldsymbol{x},t) & = & \displaystyle {\it\rho}_{0}+\bar{{\it\rho}}(z)+{\it\rho}^{\prime }(\boldsymbol{x},t),\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle p(\boldsymbol{x},t) & = & \displaystyle \bar{p}(z)+p^{\prime }(\boldsymbol{x},t),\end{eqnarray}$$

${\it\rho}_{0}$

${\it\rho}_{0}+\bar{{\it\rho}}$

$\bar{p}$

(2.3) $$\begin{eqnarray}\frac{\text{d}\bar{p}}{\text{d}z}=-({\it\rho}_{0}+\bar{{\it\rho}})g,\end{eqnarray}$$

where both the gravitational and centrifugal force contribute to the constant $g$ (Salmon Reference Salmon1998). The perturbation buoyancy $b$ and the buoyancy frequency $N$ are then defined through

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle b=-g\,{\it\rho}^{\prime }/{\it\rho}_{0}, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle N^{2}=-\frac{g}{{\it\rho}_{0}}\frac{\text{d}\bar{{\it\rho}}}{\text{d}z}. & \displaystyle\end{eqnarray}$$

$N$

Making the Boussinesq approximation ( $\bar{{\it\rho}}$ , ${\it\rho}^{\prime }\ll {\it\rho}_{0}$ ), the equations of motion may be simplified to

(2.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\boldsymbol{u}}{\text{D}t}+f\,\hat{\boldsymbol{k}}\times \boldsymbol{u}=-{\it\rho}_{0}^{-1}\boldsymbol{{\rm\nabla}}p^{\prime }+b\,\hat{\boldsymbol{k}}, & \displaystyle\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}b}{\text{D}t}+N^{2}w=0, & \displaystyle\end{eqnarray}$$

(2.6c ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

$\text{D}/\text{D}t$

$\hat{\boldsymbol{k}}$

$z$

$f$

2.2. Potential vorticity and ageostrophic vorticity

Following Dritschel & Viúdez (Reference Dritschel and Viúdez2003), we reformulate (2.6) using a set of variables that makes explicit the leading-order hydrostatic–geostrophic balance at small Rossby number. To this end, let us define the dimensionless total PV $Q$ and PV anomaly $q$  as

(2.7) $$\begin{eqnarray}Q=1+q=\frac{1}{f}\left({\bf\omega}+f\hat{\boldsymbol{k}}\right)\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\left(\frac{b}{N^{2}}+z\right),\end{eqnarray}$$

where the vorticity ${\bf\omega}=({\bf\omega}_{h},{\it\zeta})=\boldsymbol{{\rm\nabla}}\times \boldsymbol{u}$ . The expression in the second bracket above is the isopycnal height. It is related to the density through $b/N^{2}+z=(1-{\it\rho}/{\it\rho}_{0})g/N^{2}$ . Hence, $Q$ is essentially the component of the absolute vorticity ${\bf\omega}+f\hat{\boldsymbol{k}}$ along the density gradient, normalised by $f$ . The basic state, ${\bf\omega}=b=0$ , then has $Q=1$ and $q=0$ .

A natural choice for the Rossby number is the PV anomaly $q$ itself, specifically its extreme value. Material conservation of PV implies that this Rossby number is constant. Its relation to the traditional Rossby number $U/f\,L$ may be seen by non-dimensionalising variables using QG scaling, i.e.  $(x,y)=L(\tilde{x},{\tilde{y}})$ , $z=H\tilde{z}$ , $t=(L/U)\tilde{t}$ , $(u,v)=U({\tilde{u}},\tilde{v})$ , $w=(U/f\,L)(H/L)U\tilde{w}$ and $b=(U/f\,L)N^{2}H\tilde{b}$ , where $L$ and $H$ are typical horizontal and vertical length scales respectively, and $U$ is a typical horizontal velocity scale. Then the PV may be written

(2.8) $$\begin{eqnarray}Q=1+\frac{U}{f\,L}\biggl(\frac{\partial \tilde{b}}{\partial \tilde{z}}+\tilde{{\it\zeta}}\biggr)+\biggl(\frac{U}{f\,L}\biggr)^{2}\tilde{{\bf\omega}}\boldsymbol{\cdot }\tilde{\boldsymbol{{\rm\nabla}}}\tilde{b}.\end{eqnarray}$$

The non-dimensional variables, with tildes, are of $O(1)$ . Hence, to leading order the PV anomaly $q$ is just the traditional Rossby number. The extreme value of $q$ is a better choice as it does not require ad hoc choices for $U$ and $L$ .

The PV alone, however, is not sufficient to describe the dynamics of the flow, since the original system (2.6) has three independent time derivatives, after taking into account incompressibility. Motivated by QG balance, two additional variables are chosen from the horizontal (vector) component of the dimensionless ageostrophic (and non-hydrostatic) vorticity

(2.9) $$\begin{eqnarray}\boldsymbol{A}\equiv \frac{{\bf\omega}}{f}+\frac{\boldsymbol{{\rm\nabla}}b}{f^{2}}.\end{eqnarray}$$

Notably, the horizontal component $\boldsymbol{A}_{h}=0$ for a flow in thermal wind balance (apart from often negligible contributions from horizontal derivatives of the vertical velocity). Therefore, $\boldsymbol{A}_{h}$ captures higher-order ageostrophic motions beyond the leading-order hydrostatic–geostrophic balance. These motions include inertia–gravity waves, but are typically dominated by higher-order balance effects (Dritschel & Viúdez Reference Dritschel and Viúdez2007).

2.3. Reformulation of the governing equations

Instead of $\{\boldsymbol{u},b,p\}$ , we shall use $\{q,\boldsymbol{A}_{h}\}$ as the prognostic variables to explicitly enforce material conservation of PV. This enables a more accurate representation of both the balanced PV-controlled dynamics as well as the residual imbalance (Dritschel & Viúdez Reference Dritschel and Viúdez2003, Reference Dritschel and Viúdez2007). The equations of motion for $q$ and $\boldsymbol{A}_{h}$ are

(2.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}q}{\text{D}t}=0, & \displaystyle\end{eqnarray}$$

(2.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}\boldsymbol{A}_{h}}{\text{D}t}+f\hat{\boldsymbol{k}}\times \boldsymbol{A}_{h}=\frac{1}{f}({\bf\omega}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\boldsymbol{u}_{h}+(1-{\it\sigma}^{-2})\boldsymbol{{\rm\nabla}}_{h}w-\frac{1}{f^{2}}(\boldsymbol{{\rm\nabla}}_{h}\boldsymbol{u})\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}b, & \displaystyle\end{eqnarray}$$

$\boldsymbol{u}_{h}$

$\boldsymbol{{\rm\nabla}}_{h}$

(2.11) $$\begin{eqnarray}{\it\sigma}\equiv \frac{f}{N}.\end{eqnarray}$$

We set ${\it\sigma}=0.1$ (a typical value for the oceans) for all the results reported here. The results are not sensitive to this parameter so long as ${\it\sigma}\lesssim 1$ (Dritschel & McKiver Reference Dritschel and McKiver2014).

The fields $\boldsymbol{u}$ and $b$ can be recovered from $\{q,\boldsymbol{A}_{h}\}$ as follows. Expressing $\boldsymbol{A}$ in terms of a potential ${\bf\varphi}$ ,

(2.12) $$\begin{eqnarray}\boldsymbol{A}={\rm\nabla}^{2}{\bf\varphi},\end{eqnarray}$$

we have from (2.9),

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}=-f\,\boldsymbol{{\rm\nabla}}\times {\bf\varphi}, & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle b=f^{2}\,\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\varphi}. & \displaystyle\end{eqnarray}$$

${\bf\varphi}_{h}$

${\it\phi}$

${\bf\varphi}$

$\{q,\boldsymbol{A}_{h}\}$

(2.15) $$\begin{eqnarray}\displaystyle & \boldsymbol{A}_{h}={\rm\nabla}^{2}{\bf\varphi}_{h}, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle q=\left({\rm\nabla}_{h}^{2}+{\it\sigma}^{2}\frac{\partial ^{2}}{\partial z^{2}}\right){\it\phi}-(1-{\it\sigma}^{2})\boldsymbol{{\rm\nabla}}_{h}\boldsymbol{\cdot }\frac{\partial {\bf\varphi}_{h}}{\partial z}+{\it\sigma}^{2}\boldsymbol{{\rm\nabla}}(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\varphi})\boldsymbol{\cdot }[{\rm\nabla}^{2}{\bf\varphi}-\boldsymbol{{\rm\nabla}}(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\varphi})].\qquad & \displaystyle\end{eqnarray}$$

${\bf\varphi}_{h}$

${\it\phi}$

(2.17) $$\begin{eqnarray}{\it\sigma}^{2}({\it\phi}_{zz}{\rm\nabla}_{h}^{2}{\it\phi}-{\it\phi}_{xz}^{2}-{\it\phi}_{yz}^{2})+B_{1}{\rm\nabla}_{h}^{2}{\it\phi}+B_{2}{\it\phi}_{zz}+C_{1}{\it\phi}_{xz}+C_{2}{\it\phi}_{yz}+D=0,\end{eqnarray}$$

with the coefficients $B_{1}$ , $B_{2}$ , $C_{1}$ , $C_{2}$ and $D$ depending on ${\it\sigma}$ , ${\bf\varphi}_{h}$ and $q$ .

2.4. The contour-advective semi-Lagrangian algorithm and near-balanced initialisation

We solve the prognostic equations (2.10) numerically in a triply periodic box of horizontal side lengths $2{\rm\pi}L{\it\sigma}^{-1}\times 2{\rm\pi}L{\it\sigma}^{-1}$ and height $2{\rm\pi}L$ . In the following, we scale length and time such that $L=1$ and $N=2{\rm\pi}$ . Hence, the buoyancy period is 1 and (given ${\it\sigma}=0.1$ ) the inertial period $T_{\mathit{ip}}=2{\rm\pi}/f=10$ .

The PV anomaly $q$ is represented by contours on isopycnal surfaces. The evolution of $q$ , governed by (2.10a ), is carried out by solving $\text{d}x/\text{d}t=u(\boldsymbol{x},t)$ and $\text{d}y/\text{d}t=v(\boldsymbol{x},t)$ on these surfaces, using the contour-advective semi-Lagrangian (CASL) algorithm (Dritschel & Ambaum Reference Dritschel and Ambaum1997). The ageostrophic field $\boldsymbol{A}_{h}$ is discretised on a three-dimensional grid and (2.10b ) is solved spectrally with explicit leap-frog time-stepping, together with very weak hyperviscosity and the high-order Fourier filtering scheme described in Hou & Li (Reference Hou and Li2007). A detailed description of the numerical method can be found in the appendix of Dritschel & Viúdez (Reference Dritschel and Viúdez2003). For the present work, we use 512 isopycnal surfaces and a grid size of $128^{3}$ , apart from a few exceptional cases at double resolution noted below. The fixed time step ranges from 0.025 to 0.1 depending on $q$ . For all other CASL parameters such as the surgical scale, we use the standardised values given in the bottom part of table 1 in Fontane & Dritschel (Reference Fontane and Dritschel2009).

The initial conditions are prepared so that, at time $t=0$ , an ellipsoidal volume of constant PV anomaly $q_{0}$ in a near-balanced state is located at the domain centre. This ensures that the initial emission of IGWs by adjustment is minimal. We achieve this by starting from a rest state at $t=-T_{\mathit{init}}$ , then smoothly ramp up $q$ inside the ellipsoid from 0 to $q_{0}$ according to the formula

(2.18) $$\begin{eqnarray}q(t)=q_{0}\sin ^{2}\left[\frac{{\rm\pi}}{2}\left(1+\frac{t}{T_{\mathit{init}}}\right)\right]\end{eqnarray}$$

while integrating the system forward over the initialisation period $-T_{\mathit{init}}<t<0$ . We take $T_{\mathit{init}}=20|q_{0}|T_{\mathit{ip}}$ to ensure minimal excitation of IGWs. Note that both the PV contours and the ageostrophic field $\boldsymbol{A}_{h}$ evolve during the initialisation period. We start at $t=-T_{\mathit{init}}$ with an upright ellipsoid whose shape is specified by the two horizontal semi-axis lengths $a_{0}$ and $b_{0}$ ( $a_{0}<b_{0}$ ) and the vertical semi-axis length $c_{0}$ . We define the initial horizontal aspect ratio ${\it\lambda}_{0}$ and initial vertical aspect ratio ${\it\mu}_{0}$ at $t=-T_{\mathit{init}}$ as follows:

(2.19a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\lambda}_{0}=\frac{a_{0}}{b_{0}}, & \displaystyle\end{eqnarray}$$

(2.19b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\mu}_{0}=\frac{c_{0}}{{\it\sigma}\sqrt{a_{0}b_{0}}}. & \displaystyle\end{eqnarray}$$

Throughout any given simulation, the linear dimensions of the vortex in the $x$ -, $y$ - and $z$ -directions remain less than one-third of the respective domain edge lengths, so that the effects of periodicity are small. The simulations run until the vortex reaches an ‘equilibrium’ state, by which we mean that the various vortex properties become approximately statistically stationary. The total simulated time ranges from $300T_{\mathit{ip}}$ to $2000T_{\mathit{ip}}$ . With all other parameters held fixed, each simulation is characterised by ${\it\lambda}_{0}$ , ${\it\mu}_{0}$ and the Rossby number $q_{0}$ . Our principal data set consists of 308 cases spanning a large range of aspect ratios and includes both cyclones and anticyclones with $|q_{0}|\leqslant 0.5$ . We have also performed a number of simulations at larger $|q_{0}|$ and higher resolution; these results will be included in some of the analyses presented in the following sections.

3.1. Evolution and long-term fate

Depending on the initial conditions ( $q_{0}$ , ${\it\lambda}_{0}$ and ${\it\mu}_{0}$ ), the vortex displays a rich variety of evolutionary behaviours and reaches different equilibrium configurations at large times. We now give an account of the various phenomena observed in the broad range of numerical experiments conducted.

1. (a) Quasi-stable: a quasi-stable vortex deforms extremely slowly and remains close to the initial configuration during the whole simulation. Typically, the fractional change in the semi-axis lengths over $100T_{\mathit{ip}}$ is less than 1 %. A quasi-stable vortex often has ${\it\lambda}_{0}\geqslant 0.5$ .

2. (b) Shape alteration: an unstable vortex usually evolves and restablises into a new equilibrium state by changing its shape. We classify the eventual vortex shape as roughly ellipsoidal if $\langle {\it\varrho}\rangle >0.95$ and non-ellipsoidal if $\langle {\it\varrho}\rangle \leqslant 0.95$ . The evolution process often involves the shedding of vortex filaments (though there are exceptions, see § 3.2). We find that the different scenarios for the nonlinear evolution of a QG vortex described in Dritschel et al. (Reference Dritschel, Scott and Reinaud2005) also apply to our non-hydrostatic Boussinesq system. Figure 2(a) shows an example of filaments being spun off and forming rings near the top and bottom of a prolate vortex ( ${\it\mu}_{0}>1$ ).

3. (c) Binary system: it is not uncommon for ejected filaments to roll up into small satellite vortices which orbit around the main vortex, e.g. see figure 2(b). However, in a few rare cases involving a strongly horizontally elongated vortex, the vortex develops a dumbbell shape before breaking up into a roughly ellipsoidal primary vortex and a smaller secondary vortex. The two vortices revolve around each other forming a binary system. Figure 2(c) shows an example of an anticyclonic binary system. In the analysis presented below, we focus on the primary vortex only. The formation of a binary system has not been observed in QG dynamics.

4. (d) Shape oscillation: as a vortex becomes unstable, instead of converging towards an equilibrium shape, its boundary may start to morph away from and then back to its initial shape periodically. There are also cases where the vortex evolves away from its initial shape and oscillates between two new shapes. Furthermore, it is possible for a vortex to exhibit transient damped shape oscillations as it evolves towards an equilibrium configuration. Two modes of shape oscillation have been observed; they are illustrated in figure 2(d). Figure 2(e) shows a vortex undergoing transient shape oscillation.

5. (e) Tumbling: we find that in some cases where ${\it\mu}_{0}$ is close to unity, the vortex starts to tumble, irrespective of any subsequent development in the vortex shape. The notion of a vertical axis, and hence the definition of the semi-axis lengths, becomes ambiguous for a tumbling ellipsoid. In such cases, after the vortex has reached a statistical steady state, we define $a(t)$ , $b(t)$ and $c(t)$ such that they have the same order in length as $a_{0}$ , $b_{0}$ and $c_{0}$ .

Figure 2.

Examples of vortex evolution at different initial horizontal ( ${\it\lambda}_{0}$ ) and vertical ( ${\it\mu}_{0}$ ) aspect ratios. (a) A prolate vortex ( $q_{0}=0.5$ , ${\it\lambda}_{0}=0.6$ , ${\it\mu}_{0}=2.4$ ) becomes unstable at $t\approx 1500$ , spinning off filaments that form rings near the top and bottom of the vortex. Time series of the semi-axis lengths $a$ and $b$ of its equivalent oriented ellipsoid shows that at the time it becomes unstable, the vortex acquires a near-circular cross-section ( $\langle {\it\lambda}\rangle =0.87$ ). The final equilibrium shape is roughly ellipsoidal as ${\it\varrho}>0.95$ at large $t$ .(b) PV fragments come off one end of a long oblate vortex ( $q_{0}=0.5$ , ${\it\lambda}_{0}=0.3$ , ${\it\mu}_{0}=0.4$ ) as it rotates. Two major fragments form satellite vortices orbiting around the main vortex while the finer ones are extended into thin filaments. (c) A binary system formed from a highly horizontally elongated vortex ( $q_{0}=-0.25$ , ${\it\lambda}_{0}=0.2$ , ${\it\mu}_{0}=0.6$ ). (d) Two modes of vortex shape oscillation: top and bottom of the vortex moving (i) in phase ( $q_{0}=0.25$ , ${\it\lambda}_{0}=0.5$ , ${\it\mu}_{0}=1.4$ ) or (ii) out of phase ( $q_{0}=-0.5$ , ${\it\lambda}_{0}=0.3$ , ${\it\mu}_{0}=1.0$ ). (e) A vortex ( $q_{0}=0.25$ , ${\it\lambda}_{0}=0.6$ , ${\it\mu}_{0}=2.4$ ) oscillates between two different shapes after initial strong filament shedding. The oscillation fades over time as the vortex tends to an equilibrium non-ellipsoidal shape ( $\langle {\it\varrho}\rangle \approx 0.91$ ). This is reflected in the damped oscillation of ${\it\varrho}(t)$ starting at $t\approx 1800$ .

Figure 3.

Phase diagrams showing the equilibrium configuration of a vortex at large time, for varying aspect ratios ${\it\lambda}_{0}$ and ${\it\mu}_{0}$ and for different PV anomalies or Rossby numbers $q_{0}$ : (a) QG; (b) $q_{0}=0.25$ ; (c) $q_{0}=-0.25$ ; (d) $q_{0}=0.5$ ; (e) $q_{0}=-0.5$ . The symbols denote the shape of the vortex, as follows: ○, little change from the initial shape at $t=0$ ; ♢, roughly ellipsoidal ( $\langle {\it\varrho}\rangle >0.95$ ); ▵, non-ellipsoidal ( $\langle {\it\varrho}\rangle \leqslant 0.95$ ); 

, binary system. The quasi-stable cases are shown with filled circles. A symbol inscribed with another indicates that the vortex undergoes shape oscillation, while a dashed inner symbol means that the oscillation is transient. A hatched symbol shows that the orientation of the vortex is varying due to tumbling (slanted lines) or shape oscillation (vertical lines). For $q_{0}=-0.5$ , inside the lower right region bounded by the dashed line are cases in which the vortex deforms smoothly without filament shedding.

3.2. Stability and phase diagram

In § 3.1 above, we described a rich variety of possible behaviours of a vortex and illustrated them in a few examples. We now present results from our full principal data set consisting of 308 simulations. We consider four different values of Rossby number: $q_{0}=\pm 0.25$ and $\pm 0.5$ . For the aspect ratios, ${\it\lambda}_{0}$ varies from 0.2 to 0.8 in steps of 0.1 and ${\it\mu}_{0}$ from 0.4 to 2.4 in steps of 0.2. Figure 3(b)–(e) summarises the equilibrium configuration for all cases. For completeness, we also performed simulations of the QG system (Dritschel et al. Reference Dritschel, Scott and Reinaud2005) representing the limit of zero Rossby number. The results are shown in figure 3(a).

We first investigate the relation between the stability and the geometry of the vortex. Figure 3 shows that, generally, a small vertical aspect ratio and a near-circular horizontal cross-section are favourable conditions for stability. In figure 3, the circles occupy a region in the lower right of the ${\it\lambda}_{0}{-}{\it\mu}_{0}$ phase plane. Because the circles indicate that the vortex maintains or oscillates about its initial shape, and in anticipation of the results in § 3.3 below, we loosely refer to this region as the ‘stability region’. As we move away from the stability region by decreasing ${\it\lambda}_{0}$ or increasing ${\it\mu}_{0}$ , we first encounter cases of unstable vortices which evolve into a roughly ellipsoidal equilibrium shape. This is then followed by the non-ellipsoidal cases as we move further from the stability region. This pattern is especially conspicuous for the QG system. For $q_{0}=-0.5$ , no stability region is found within the parameter ranges considered. However, in the region below the dashed line in figure 3(e), the vortex deforms noticeably over several hundred inertial periods in a smooth manner, without shedding filaments. Thus, the evolution in these cases is not unlike that of a quasi-stable vortex except that it happens over a much shorter time scale.

We now turn to the effects of the Rossby number $q_{0}$ on the stability of the vortex. From figure 3, we see that as $q_{0}$ decreases, the stability region diminishes while its boundary shifts downwards and towards the right. Therefore we conclude that, for fixed aspect ratios ${\it\lambda}_{0}$ and ${\it\mu}_{0}$ , increasing $q_{0}$ enhances stability. As a consequence, a cyclone ( $q_{0}>0$ ) tends to be more stable than an anticyclone ( $q_{0}<0$ ) of the same shape and $|q_{0}|$ .

In most cases, the vortex rotates with a fixed orientation about the vertical axis (the vortex may be tilted). However, the orientation of the vortex varies if it is tumbling or undergoing the out-of-phase-type shape oscillation depicted in figure 2(d). An interesting observation from our simulations is that a vortex with a vertical aspect ratio ${\it\mu}_{0}$ around unity is most susceptible to orientation variation (hatched symbols in figure 3).

3.3. Aspect ratios of the late-time equilibria

In § 3.2 above, we have seen that the stability of a vortex depends crucially on its aspect ratios ${\it\lambda}$ and ${\it\mu}$ . In the present section, we look into how ${\it\lambda}$ and ${\it\mu}$ change as an unstable vortex evolves into an equilibrium configuration. The discussion surrounding (3.1) points out that the equivalent oriented ellipsoid effectively captures the vortex shape. Hence, we can reliably estimate the aspect ratios ${\it\lambda}(t)$ and ${\it\mu}(t)$ of the vortex at time $t$ from $a(t)$ , $b(t)$ and $c(t)$ using expressions analogous to (2.19). The aspect ratios of the equilibrium vortex shape are then represented by the time-averaged values $\langle {\it\lambda}\rangle$ and $\langle {\it\mu}\rangle$ over the statistically steady state.

Figure 4 illustrates the change in ${\it\lambda}$ and ${\it\mu}$ between the initial shape at $t=0$ and the equilibrium shape at large $t$ . Note that $\{{\it\lambda}_{0},{\it\mu}_{0}\}\equiv \{{\it\lambda}(-T_{\mathit{init}}),{\it\mu}(-T_{\mathit{init}})\}$ and there is a small difference between $\{{\it\lambda}(0),{\it\mu}(0)\}$ and $\{{\it\lambda}_{0},{\it\mu}_{0}\}$ because the vortex is allowed to evolve during the initialisation period. A prominent feature of figure 4 is that ${\it\lambda}$ and ${\it\mu}$ migrate towards and cluster around the stability boundary (except for the binary systems which evolve deep into the stability region, see § 3.2 for a definition of the ‘stability region’). The deformation of cyclones is mostly achieved with only a minor change in ${\it\mu}$ , while a few of the anticyclones show relatively large changes in ${\it\mu}$ . In the majority of cases, the vortex eventually acquires a near-circular horizontal cross-section with $\langle {\it\lambda}\rangle >0.8$ . The time series in figure 2(a) shows an example of $a$ and $b$ approaching each other.

Figure 4.

Changes in the horizontal aspect ratio ${\it\lambda}$ and the vertical aspect ratio ${\it\mu}$ of the vortex. Crosses are values at the end of the initialisation period $t=0$ . Solid circles are time-averaged values $\langle {\it\lambda}\rangle$ and $\langle {\it\mu}\rangle$ over the statistically steady state at large time. (a) $q_{0}=0.25$ ; (b) $q_{0}=-0.25$ ; (c) $q_{0}=0.5$ ; (d) $q_{0}=-0.5$ .

We emphasise that, with the exception of the quasi-stable cases, the final equilibrium shape of the vortex is not a perfect ellipsoid, see figure 1. Thus, the clustering of $\langle {\it\lambda}\rangle$ and $\langle {\it\mu}\rangle$ seen in figure 4 suggests that the stability of the vortex depends primarily on the aspect ratios – gross measures of the vortex shape – not on the detailed vortex shape. The pattern displayed in figure 4 has interesting implications for the vortex rotation rate, as elaborated in § 4.

4.1. Vortex rotation rate estimated by a higher-order balanced model

A number of balanced models have been developed to extend the QG approximation (see e.g. Warn et al. Reference Warn, Bokhove, Shepherd and Vallis1995, Muraki, Snyder & Rotunno Reference Muraki, Snyder and Rotunno1999 and references therein). Here, we adopt the nonlinear quasi-geostrophic (NQG) balanced model of McKiver & Dritschel (Reference McKiver and Dritschel2008), see appendix C. The NQG balanced model uses a Rossby-number expansion to second order, $O(q_{0}^{2})$ , except that it retains the exact definition of PV. This greatly improves the estimate of the underlying balanced flow, as demonstrated explicitly in McKiver & Dritschel (Reference McKiver and Dritschel2008). The resulting nonlinear PV inversion relations allow one to reconstruct the velocity and buoyancy fields to $O(q_{0}^{2})$ from a given instantaneous PV field.

In figure 6(b), we plot $|\langle {\it\Omega}\rangle |$ versus $q_{0}$ for several cases having similar aspect ratios $\langle {\it\lambda}\rangle$ and $\langle {\it\mu}\rangle$ . Corresponding to each of these cases, we construct the velocity field from an ellipsoid of uniform PV with the same $\langle {\it\lambda}\rangle$ , $\langle {\it\mu}\rangle$ and $q_{0}$ by NQG PV inversion. The rotation rates of such NQG-balanced ellipsoids are then calculated (see appendix B). We find close agreement, typically within 3 %, between the theoretical NQG rotation rate and the data across all $q_{0}$ . Figure 6(b) also shows the rotation rate of an NQG ellipsoid with ${\it\lambda}=0.8$ and ${\it\mu}=1.0$ as a function of $q_{0}$ . These results clearly reveal the emergence of asymmetry in the cyclonic and anticyclonic rotation rate as soon as finite Rossby number effects are taken into account.

5.1. Definitions and method of analysis

The control exerted by PV on the dynamics of vortices in rotating stably stratified geophysical flows can be, for all practical purpose, nearly complete, with IGWs playing a negligible role. This state of affairs is known as ‘balance’. Balance is favoured when the magnitude of the Rossby number $|q_{0}|\lesssim 1$ and the frequency ratio $f/N\lesssim 1$ , even in strongly turbulent flows (Dritschel & Viúdez Reference Dritschel and Viúdez2007; Dritschel & McKiver Reference Dritschel and McKiver2014), at least for carefully prepared initial conditions.

As described in § 2.4, we initialise each flow by artificially ramping up the PV anomaly over many inertial periods while otherwise solving the full equations of motion. This results in an ‘initial’ state nearly void of IGWs when $|q_{0}|\lesssim 1$ and $f/N\lesssim 1$ . To explore the possible spontaneous emission of IGWs at later times and to estimate their contribution to the dynamics, we need to quantify balance and the residual imbalance or IGW activity. One approach to diagnose the balanced component from a given PV field is to invert the ‘balance conditions’ obtained from a Rossby number expansion of the governing equations. Such balance conditions define an instantaneous state of balance. The traditional QG balance and the higher-order NQG balance in § 4.1 fall into this category. Here, we take another approach, called optimal potential vorticity (OPV) balance (Viúdez & Dritschel Reference Viúdez and Dritschel2004), which does not enforce any specific balanced conditions but instead seeks the ‘minimally imbalanced’ state through a numerical iterative procedure.

Given the PV distribution at some chosen diagnostic time $t$ , the OPV procedure carries out a series of artificial forward and backward integrations of the equations of motion, over a specified time period ${\it\tau}$ . Each integration involves ramping up or ramping down the PV anomaly. The objective is to arrive at the original given PV distribution from a different one in a state of rest at time $t-{\it\tau}$ , after ramping up the PV anomaly and otherwise solving the full equations of motion. The resulting fields are called ‘balanced’, even though they may contain small residual imbalance (Dritschel & Viúdez Reference Dritschel and Viúdez2007; Viúdez Reference Viúdez2007). Note, the balance depends only on the PV distribution provided and the ramp period ${\it\tau}$ . The choice of ${\it\tau}$ is strongly affected by the amount of distortion of the PV distribution occurring between $t-{\it\tau}$ (the base distribution) and $t$ (the given target distribution) during the ramp. If ${\it\tau}$ is too long, the base distribution is too distorted to ever converge to the target one. A compromise must be made. Following previous studies (Dritschel & Viúdez Reference Dritschel and Viúdez2007; McKiver & Dritschel Reference McKiver and Dritschel2008) and after conducting further tests here, we took ${\it\tau}$ to be four inertial periods. This is sufficiently long not to excite significant IGW activity, though not long enough to avoid all of it. In this sense, it is better to view OPV balance as a procedure which seeks the minimally imbalanced state, subject to the constraint that the base PV distribution can actually evolve into the target PV distribution. The technical details of OPV balance are summarised in appendix D.

Given the OPV balanced flow ${\bf\varphi}_{\mathit{bal}}$ at a chosen diagnostic time $t$ , the imbalanced flow (or ‘imbalance’) is defined simply by subtraction from the full flow,

(5.1) $$\begin{eqnarray}{\bf\varphi}_{\mathit{imb}}={\bf\varphi}-{\bf\varphi}_{\mathit{bal}}.\end{eqnarray}$$

The balanced flow is further divided into a QG balanced part ${\bf\varphi}_{\mathit{QG}}=(0,0,{\it\phi}_{\mathit{QG}})$ and the remaining ‘ageostrophic’ balanced part

(5.2) $$\begin{eqnarray}{\bf\varphi}_{\mathit{AGb}}={\bf\varphi}_{\mathit{bal}}-{\bf\varphi}_{\mathit{QG}}.\end{eqnarray}$$

Here, ${\it\phi}_{\mathit{QG}}$ is defined in the standard way, inverting the QG linear operator on the PV anomaly $q$ (see (C 3) in appendix C) on flat isopycnal surfaces, not on the actual displaced surfaces (the displacement is proportional to $q_{0}$ , and the effect of this displacement is $O(q_{0}^{2})$ on the balanced flow and therefore an ageostrophic effect).

We next focus on two cases which serve to illustrate how the nature of both the balanced flow and the imbalance vary with the vortex shape and the Rossby number $q_{0}$ . In each case, we examine a time at or near the end of the simulation when the vortex is in a near-equilibrium state. As discussed below, the imbalance is often largest during a period of instability and rapid change, though it never contributes significantly to the total energy. We finish by examining the fractional energy associated with the balanced and imbalanced flow components as a function of the Rossby number $q_{0}$ .

5.2. Oblate weakly eccentric vortices

We first look at an oblate cyclone and anticyclone at $|q_{0}|=0.25$ with ${\it\lambda}_{0}=0.8$ and ${\it\mu}_{0}=0.4$ . These vortices are quasi-stable: in the late-time equilibrium state $\langle {\it\lambda}\rangle \approx$ ${\it\lambda}(t=0)=0.81$ in both cases, and $\langle {\it\mu}\rangle \approx {\it\mu}(t=0)=0.35$ and 0.46 for the cyclone and anticyclone respectively.

Figure 8(a–c) shows the balanced and imbalanced components of the scaled vertical vorticity ${\it\zeta}/f$ for the cyclone while figure 8(d–f) shows the analogous fields for the anticyclone. First of all, the QG balanced fields are entirely as expected: positive vorticity inside the vortex for the cyclone and negative for the anticyclone. Just outside, the vorticity switches sign. Again, this is consistent since, inside the cyclone, vortex lines are more bunched together, while outside they must be more spread out in order to relax to the undisturbed state with absolute vorticity $f\hat{\boldsymbol{k}}$ far from the vortex. The anticyclone is exactly the opposite. On the other hand, the ageostrophic balanced fields do not preserve the antisymmetry between cyclones and anticyclones. In both cases, ${\it\zeta}_{\mathit{AGb}}$ is negative inside the vortex, and weakly positive just outside of it. This is because ${\it\zeta}_{\mathit{AGb}}$ depends quadratically, to leading order, on the QG balanced fields (see appendix C or, explicitly, (2.23) of McKiver & Dritschel (Reference McKiver and Dritschel2008)).

Figure 8.

A vertical $y{-}z$ cross-section at $x=-{\it\sigma}^{-1}{\rm\pi}/32$ of the scaled vertical vorticity ${\it\zeta}/f$ for a cyclonic vortex or ‘cyclone’ (a–c) and an anticyclonic vortex or ‘anticyclone’ (d–f) with $q_{0}=\pm 0.25$ respectively. The initial aspect ratios are ${\it\lambda}_{0}=0.8$ and ${\it\mu}_{0}=0.4$ in both cases. The QG balanced part ${\it\zeta}_{\mathit{QG}}/f$ is shown in (a) and (d), the ageostrophic balanced part ${\it\zeta}_{\mathit{AGb}}/f$ in (b) and (e), the imbalanced part ${\it\zeta}_{\mathit{imb}}/f$ in (c) and (f). The fields are depicted at $t=480T_{\mathit{ip}}$ . The bold black contour indicates the vortex edge, while the thin solid and thin dashed contours in (a,b,d,e) denote positive and negative field values, respectively. Note: the plotted domain is squeezed horizontally by the factor ${\it\sigma}=f/N=0.1$ . A nonlinear colour scale is used for the imbalanced fields to emphasise the weaker wave structure outside the vortex. The actual maximum of $|{\it\zeta}_{\mathit{imb}}/f|$ is approximately $8\times 10^{-4}$ .

Turning to the imbalanced vertical vorticity ${\it\zeta}_{\mathit{imb}}$ (figure 8 c, f), we see much smaller-scale features, strongest at the edge or inside the vortex. These features resemble wave crests running principally diagonally, as one would expect of IGWs ‘forced’ at sub-inertial frequencies (note the St Andrews cross at the centre of the cyclone). The forcing is provided by the rotating distorted vortex, and we see here evidence of a continuous spontaneous emission of IGWs by the balanced vortical flow. Some wave trapping is evident in the anticyclone, but the amplitude of the IGWs is closely comparable for both vortices. The amplitudes of the waves are exceedingly weak; imbalanced field values are $10^{2}{-}10^{3}$ times smaller than balanced field values at this Rossby number, $|q_{0}|=0.25$ . This remains true even at $|q_{0}|=0.5$ (see § 5.4 for a discussion of the Rossby number scaling).

Analogous results for the anomalous ‘static stability’,

(5.3) $$\begin{eqnarray}{\it\gamma}=\frac{1}{N^{2}}\frac{\partial b}{\partial z},\end{eqnarray}$$

are presented in figure 9 (a–c for the cyclone and d–f for the anticyclone). Note that ${\it\gamma}$ is the negative of the vertical derivative of the isopycnal displacement. The local squared buoyancy frequency equals $N^{2}(1+{\it\gamma})$ , so ${\it\gamma}<-1$ indicates static instability (overturning isopycnals). The structure of ${\it\gamma}_{\mathit{QG}}$ in the vortex interior implies increased stability ( ${\it\gamma}>0$ ) for cyclones, and decreased stability ( ${\it\gamma}<0$ ) for anticyclones. Note that ${\it\gamma}_{\mathit{QG}}$ has the same sign as ${\it\zeta}_{\mathit{QG}}/f$ in the vortex interior, but the opposite sign outside. The balanced ageostrophic component ${\it\gamma}_{\mathit{AGb}}$ (at second order in Rossby number) has almost no effect on stability within the vortex. Only at the vortex edge is ${\it\gamma}$ reduced, but it is increased above and below it, for both the cyclone and the anticyclone.

Figure 9.

As in figure 8, but for the anomalous ‘static stability’ ${\it\gamma}=b_{z}/N^{2}$ , with the QG balanced part ${\it\gamma}_{\mathit{QG}}$ in (a) and (d), the ageostrophic balanced part ${\it\gamma}_{\mathit{AGb}}$ in (b) and (e), and the imbalanced part ${\it\gamma}_{\mathit{imb}}$ in (c) and (f). The top row (a–c) is for the cyclone and the bottom row (d–f) is for the anticyclone. A nonlinear colour scale is used for the imbalanced fields. The maximum $|{\it\gamma}_{\mathit{imb}}|$ is approximately $6\times 10^{-4}$ .

The imbalanced field ${\it\gamma}_{\mathit{imb}}$ again displays evidence of IGW activity, more pronounced above and below the anticyclone (which is less oblate than the cyclone at this time). The pattern of $-{\it\gamma}_{\mathit{imb}}$ is remarkably similar to ${\it\zeta}_{\mathit{imb}}/f$ . In fact, for plane-wave solutions of the linearised equations about a state of rest, these two fields exactly cancel. The sum ${\it\zeta}_{\mathit{imb}}/f+{\it\gamma}_{\mathit{imb}}$ is indeed much smaller outside the vortex than either field in isolation, as seen in figure 10 displaying one-dimensional cross-sections from the two fields of the cyclone. This confirms that the imbalanced fields mainly consist of IGWs.

Figure 10.

One-dimensional cross-sections along $y$ of ${\it\zeta}_{\mathit{imb}}/f$ (solid) and $-{\it\gamma}_{\mathit{imb}}$ (dashed) for the oblate cyclone shown in figure 8. The vertical dot-dashed lines indicate the location of the vortex edge. (a) A transect through the vortex at $(x,z)=(-{\it\sigma}^{-1}{\rm\pi}/32,0)$ (insets show regions having much weaker fields). (b) A transect below the vortex at $(x,z)=(-{\it\sigma}^{-1}{\rm\pi}/32,-{\rm\pi}/2)$ . The relation ${\it\zeta}_{\mathit{imb}}/f+{\it\gamma}_{\mathit{imb}}=0$ satisfied by plane-wave IGWs holds well, verifying that the imbalanced fields mainly consist of IGWs.

The vertical velocity $w$ contains only a balanced ageostrophic component $w_{\mathit{AGb}}$ and the imbalanced one $w_{\mathit{imb}}$ . These are shown in figure 11 (for the cyclone) and figure 12 (for the anticyclone). The balanced component $w_{\mathit{AGb}}$ displays a three-dimensional octopolar structure, typical of a rotating ellipsoid around which isopycnals rise and fall antisymmetrically about the midplane $z=0$ as the vortex rotates. In $w$ , the cyclone exhibits a significantly higher degree of balance, with the magnitude of $w_{\mathit{imb}}$ being nearly 10 times smaller than $w_{\mathit{AGb}}$ . For the anticyclone, by contrast, $w_{\mathit{AGb}}$ and $w_{\mathit{imb}}$ are comparable (note that the colour scales for the imbalanced fields in the figures are nonlinear), and it is evident that OPV balance struggles to clearly define balance. In both flows, waves appear to be trapped inside the vortex, especially in the horizontal cross-sections. Wave crests run approximately perpendicular to the flow at the vortex edge, but become radial approaching the centre of the vortex. While the imbalance is particularly visible here, it nonetheless plays an insignificant role in the flow evolution. This echoes previous findings when care is taken to carefully balance the initial conditions (Dritschel & Viúdez Reference Dritschel and Viúdez2007; McKiver & Dritschel Reference McKiver and Dritschel2008).

Figure 11.

The ageostrophic balanced part $w_{\mathit{AGb}}$ (a–c) and the imbalanced part $w_{\mathit{imb}}$ (d–f) of the vertical velocity associated with a cyclonic oblate vortex ( $q_{0}=0.25$ , ${\it\lambda}_{0}=0.8$ and ${\it\mu}_{0}=0.4$ ). (a) and (d) show, respectively, three-dimensional surfaces at $w_{\mathit{AGb}}=\pm 5\times 10^{-6}$ and $w_{\mathit{imb}}=\pm 5\times 10^{-7}$ (positive values are in red); (b) and (e) are vertical $y{-}z$ cross-sections at $x=-{\it\sigma}^{-1}{\rm\pi}/32$ ; (c) and (f) are horizontal $x{-}y$ cross-sections at $z=-{\rm\pi}/32$ . A nonlinear colour scale is used in the $w_{\mathit{imb}}$ cross-section plots. The maximum $|w_{\mathit{imb}}|$ is approximately $5\times 10^{-6}$ .

Figure 12.

As in figure 11, but for an anticyclonic vortex with $q_{0}=-0.25$ . Surfaces at $w_{\mathit{AGb}}=\pm 5\times 10^{-6}$ and $w_{\mathit{imb}}=\pm 2\times 10^{-6}$ are shown in (a) and (d) respectively. A nonlinear colour scale is used in the $w_{\mathit{imb}}$ cross-section plots. The maximum $|w_{\mathit{imb}}|$ is approximately $2\times 10^{-5}$ .

5.3. Prolate vortices

We next compare the fields associated with two highly prolate vortices ( ${\it\lambda}_{0}=0.7$ , ${\it\mu}_{0}=2.4$ and $q_{0}=\pm 0.25$ ) with the oblate vortices examined just above. These prolate vortices are unstable and undergo mild filament shedding at the top and bottom. At equilibrium, $\{\langle {\it\lambda}\rangle ,\langle {\it\mu}\rangle \}=\{0.95,2.21\}$ and $\{0.91,2.29\}$ for the cyclone and anticyclone respectively.

Figure 13 shows components of vertical vorticity, ${\it\zeta}_{\mathit{QG}}$ , ${\it\zeta}_{\mathit{AGb}}$ and ${\it\zeta}_{\mathit{imb}}$ (normalised by $f$ ), for a cyclone (a–c) and an anticyclone (d–f). Comparing with the oblate vortices in figure 8, we observe broadly similar features, especially in the balanced components. The ageostrophic component again decreases ${\it\zeta}$ in the vortex core in both cases, but the decrease is less here due to the more barotropic character of the vortices. The imbalance ${\it\zeta}_{\mathit{imb}}$ is also more confined to the vortex interior. In all cases, one sees a characteristic St Andrews cross pattern associated with IGWs emanating from the rotating non-axisymmetric vortex. Note, however, that these waves are exceptionally weak, carrying vorticity nearly $10^{3}$ times smaller than the balanced component.

Figure 13.

A vertical $y{-}z$ cross-section at $x=-{\it\sigma}^{-1}{\rm\pi}/32$ of the scaled vertical vorticity ${\it\zeta}/f$ for a cyclone (a–c) and an anticyclone (d–f) with $q_{0}=\pm 0.25$ respectively. The initial aspect ratios are ${\it\lambda}_{0}=0.7$ and ${\it\mu}_{0}=2.4$ in both cases. The layout is the same as in figure 8 with the QG balanced part ${\it\zeta}_{\mathit{QG}}/f$ in (a) and (d), the ageostrophic balanced part ${\it\zeta}_{\mathit{AGb}}/f$ in (b) and (e), and the imbalanced part ${\it\zeta}_{\mathit{imb}}/f$ in (c) and (f). The fields are depicted at $t=1200T_{\mathit{ip}}$ . A nonlinear colour scale emphasising the weak wave pattern outside the vortex is used for the imbalanced fields. The actual maximum of $|{\it\zeta}_{\mathit{imb}}/f|$ is approximately $8\times 10^{-4}$ .

Regarding the anomalous static stability ${\it\gamma}=b_{z}/N^{2}$ in figure 14, the balanced components are broadly similar to those shown for the oblate vortices in figure 9: QG balance enhances the stability of cyclones and reduces the stability of anticyclones. The ageostrophic component ${\it\gamma}_{\mathit{AGb}}$ is insignificant within the vortex and has a minor effect on stability. Regarding the imbalance, ${\it\gamma}_{\mathit{imb}}$ has a similar structure to ${\it\zeta}_{\mathit{imb}}/f$ , as expected for pure IGWs, and again the imbalance contributes negligibly to the dynamics ( ${\it\gamma}_{\mathit{imb}}$ is $10^{2}$ times smaller than ${\it\gamma}_{\mathit{QG}}$ ).

Figure 14.

As in figure 13, but for the anomalous static stability ${\it\gamma}=b_{z}/N^{2}$ , with the QG balanced part ${\it\gamma}_{\mathit{QG}}$ in (a) and (d), the ageostrophic balanced part ${\it\gamma}_{\mathit{AGb}}$ in (b) and (e), and the imbalanced part ${\it\gamma}_{\mathit{imb}}$ in (c) and (f). The top row (a–c) is for the cyclone and the bottom row (d–f) is for the anticyclone. A nonlinear colour scale is used for the imbalanced fields. The maximum $|{\it\gamma}_{\mathit{imb}}|$ is approximately $2\times 10^{-3}$ .

The vertical velocity components $w_{\mathit{AGb}}$ and $w_{\mathit{imb}}$ , shown in figure 15 (for the cyclone) and in figure 16 (for the anticyclone), are overall significantly different from their oblate-vortex counterparts in figures 11 and 12. First, the ageostrophic component is less well organised and more concentrated in the interior of prolate vortices, perhaps due to fact that vertical motions are less constrained. The pattern is much more complex, especially for the cyclonic state, where no octopolar pattern is evident. The imbalance $w_{\mathit{imb}}$ displays the characteristic St Andrews cross pattern in the $y{-}z$ cross-section, but in the $x{-}y$ cross-section wave crests are seen to be predominantly radial, switching abruptly to azimuthal just outside the vortex boundary. This structure is not seen clearly for the oblate vortices examined, and is probably due to the greater horizontal distortion of the vortex. Finally, as a fraction of $w_{\mathit{AGb}}$ , the imbalance $w_{\mathit{imb}}$ is appreciably weaker for these prolate vortices compared to their oblate counterparts. Here, $w_{\mathit{imb}}$ is 20–30 times smaller than $w_{\mathit{AGb}}$ . Even in vertical velocity, these flows exhibit a strong degree of balance.

Figure 15.

The ageostrophic balanced part $w_{\mathit{AGb}}$ (a–c) and the imbalanced part $w_{\mathit{imb}}$ (d–f) of the vertical velocity associated with a cyclonic prolate vortex ( $q_{0}=0.25$ , ${\it\lambda}_{0}=0.7$ and ${\it\mu}_{0}=2.4$ ). (a) and (d) show, respectively, three-dimensional surfaces at $w_{\mathit{AGb}}=\pm 1\times 10^{-5}$ and $w_{\mathit{imb}}=\pm 6\times 10^{-7}$ (positive values are in red); (b) and (e) are vertical $y{-}z$ cross-sections at $x=-{\it\sigma}^{-1}{\rm\pi}/32$ ; (c) and (f) are horizontal $x{-}y$ cross-sections at $z=-{\rm\pi}/32$ . A nonlinear colour scale is used in the $w_{\mathit{imb}}$ cross-section plots. The maximum $|w_{\mathit{imb}}|$ is approximately $4\times 10^{-6}$ .

Figure 16.

As in figure 15, but for an anticyclonic vortex with $q_{0}=-0.25$ . Surfaces at $w_{\mathit{AGb}}=\pm 1\times 10^{-5}$ and $w_{\mathit{imb}}=\pm 1.3\times 10^{-6}$ are shown in (a) and (d) respectively. A nonlinear colour scale is used in the $w_{\mathit{imb}}$ cross-section plots. The maximum $|w_{\mathit{imb}}|$ is approximately $7\times 10^{-6}$ .

5.4. Energetics and Rossby number scaling

We next quantify the importance of the balanced and imbalanced parts of the flow using the energy norm

(5.4) $$\begin{eqnarray}E_{i}={\textstyle \frac{1}{2}}\langle |\boldsymbol{u}_{i}|^{2}+b_{i}^{2}/N^{2}\rangle ,\end{eqnarray}$$

where $i=\mathit{QG}$ , $\mathit{AGb}$ or $\mathit{imb}$ , and in this section $\langle \cdot \rangle$ denotes a domain average. We consider two cases, each having the same initial aspect ratios ${\it\lambda}_{0}=0.2$ and ${\it\mu}_{0}=2.4$ , but opposite PV, $q_{0}=\pm 0.5$ . Also, the simulations are carried out at double the standard resolution in each direction (i.e. at $256^{3}$ with subgrid PV kept to scales 16 times finer), and using a time step ${\rm\Delta}t=0.075$ . These prolate and initially highly eccentric vortices quickly become unstable and exhibit strong filament shedding between $t\approx 200T_{\mathit{ip}}$ and $t\approx 1500T_{\mathit{ip}}$ before finally settling down with $\{\langle {\it\lambda}\rangle ,\langle {\it\mu}\rangle \}=\{0.90,2.29\}$ and $\{0.88,1.73\}$ for the cyclone and anticyclone respectively.

Figure 17 shows the time evolution of the energy ratios $E_{\mathit{QG}}/E_{\mathit{tot}}$ , $E_{\mathit{AGb}}/E_{\mathit{tot}}$ and $E_{\mathit{imb}}/E_{\mathit{tot}}$ where $E_{\mathit{tot}}$ is the total energy, found from (5.4) using the full fields $\boldsymbol{u}$ and $b$ . Notably, $E_{\mathit{QG}}>E_{\mathit{tot}}$ and generally increases in time for the cyclone; just the opposite is true for the anticyclone. Hence, QG balance overestimates the energy of cyclones and underestimates that of anticyclones (this has been found to be true generally in all cases examined). We also find that oblate cyclones have larger $E_{\mathit{QG}}/E_{\mathit{tot}}\ ({>}1)$ than prolate ones, while oblate anticyclones have smaller $E_{\mathit{QG}}/E_{\mathit{tot}}\ ({<}1)$ than prolate ones. In other words, increasing vortex oblateness increases the difference $|E_{\mathit{QG}}/E_{\mathit{tot}}-1|$ .

Figure 17.

Time evolution of the energy norms $E_{\mathit{QG}}$ , $E_{\mathit{AGb}}$ and $E_{\mathit{imb}}$ (defined in (5.4)) for a cyclone (a–c) and an anticyclone (d–f) at $q_{0}=\pm 0.5$ respectively. Both vortices have the same initial aspect ratios, ${\it\lambda}_{0}=0.2$ and ${\it\mu}_{0}=2.4$ . The insets show the total energy $E_{tot}$ versus time.

Turning to the ageostrophic balanced part $E_{\mathit{AGb}}$ , this is seen to be just 1–2 % of $E_{\mathit{tot}}$ even at these moderately high Rossby numbers. $E_{\mathit{AGb}}/E_{\mathit{tot}}$ generally grows in time and appears to level off at late times. Note that two estimates of $E_{\mathit{AGb}}$ are provided in the figure, one obtained using OPV balance and the other using NQG balance. Both give closely comparable measures of the small ageostrophic balanced part. Note, simple scaling estimates suggest $E_{\mathit{AGb}}/E_{\mathit{tot}}\sim q_{0}^{2}$ (see below), which overestimates the observed energy ratio by more than an order of magnitude if one naively assumes an $O(1)$ prefactor in the scaling.

Finally, regarding the imbalanced part $E_{\mathit{imb}}$ , first of all it is much smaller than even $E_{\mathit{AGb}}$ , with energy ratios of the order $10^{-5}$ or smaller. $E_{\mathit{imb}}/E_{\mathit{tot}}$ is largest when the vortex undergoes deformation at early times and decays to very small values at late times. Here, one can see a substantial difference between the OPV and NQG estimates of imbalance. Generally, the OPV estimate is about an order of magnitude smaller except during the early period of evolution. In this early period, the flow is rapidly changing due to instability, and OPV struggles to clearly define balance. Indeed, for such time-varying flows, no perfect definition of balance exists; all one can be sure of is that NQG balance provides an overestimate of the imbalance, since presumably one can construct more accurate balance methods for any specific flow situation (cf. Mohebalhojeh & Dritschel Reference Mohebalhojeh and Dritschel2001). OPV balance generally underestimates the imbalance for slowly evolving flows since inevitably some of the imbalance is attributed to the balanced flow, by construction (see appendix D). It is less clear what happens for rapidly evolving flows, but the results of figure 17 indicate that OPV balance overestimates the imbalance in this situation. The upshot is that we can bound the actual level of imbalance below the NQG estimate and above the OPV one, provided the OPV estimate is lower than the NQG one.

We next examine how the QG, balanced ageostrophic and imbalanced energies scale with the PV-based Rossby number $q_{0}$ . We consider a single vortex shape with ${\it\lambda}_{0}=0.3$ and ${\it\mu}_{0}=1.6$ for many different values of $q_{0}$ , both positive and negative. We compute $E_{\mathit{QG}}$ , $E_{\mathit{AGb}}$ and $E_{\mathit{imb}}$ at or near the final time in each simulation, when the vortex has reached an equilibrium form. The vortex shape at equilibrium depends on $q_{0}$ with $0.68<\langle {\it\lambda}\rangle <0.95$ and $1.26<\langle {\it\mu}\rangle <1.59$ .

In QG theory, the balanced fields are proportional to $q_{0}$ , and hence one would expect $E_{\mathit{QG}}$ to scale with $q_{0}^{2}$ . The balanced ageostrophic fields are of higher order, and according to the scaling presented in appendix C, one would expect $E_{\mathit{AGb}}\sim q_{0}^{4}$ . For the imbalanced fields, the scaling is less clear. What we find is shown in figure 18, plotting the values for $E_{\mathit{QG}}$ , $E_{\mathit{AGb}}$ and $E_{\mathit{imb}}$ as a function of $|q_{0}|$ up to $|q_{0}|=0.75$ , using circles for cyclonic states ( $q_{0}>0$ ) and crosses for anticyclonic ones ( $q_{0}<0$ ). The QG scaling $E_{\mathit{QG}}\sim q_{0}^{2}$ is robust at least up to $|q_{0}|=0.5$ , and furthermore $E_{\mathit{AGb}}\sim q_{0}^{4}$ over all Rossby numbers considered, with consistently higher values for anticyclones (typically 2–3 times larger). We find that this is generally true for all the vortex shapes in our principal data set ( $q_{0}=\pm 0.25$ , $\pm 0.5$ ), implying that, for the same Rossby number magnitude $|q_{0}|$ , cyclones are closer to QG balance. The imbalance also exhibits a quartic scaling, $E_{\mathit{imb}}\sim q_{0}^{4}$ , albeit with an amplitude $10^{4}{-}10^{5}$ times smaller than $E_{\mathit{AGb}}$ . There is no clear trend that anticyclones exhibit more imbalance than cyclones except for $|q_{0}|\gtrsim 0.5$ . However, we cannot draw a general conclusion based on just one vortex shape (and it is too costly to examine many different vortex shapes in this way).

Figure 18.

Log–log plot of the energy norms for the QG, ageostrophic balanced and imbalanced fields, defined in (5.4), versus the PV-based Rossby number $q_{0}$ . Circles represent cyclonic states and crosses anticyclonic states. The initial aspect ratios of the vortex are ${\it\lambda}_{0}=0.3$ and ${\it\mu}_{0}=1.6$ .