2. The model for the shape

We use a cylindrical system $r,{\it\theta},z$ which rotates with ${\it\Omega}\hat{\boldsymbol{z}}$ , while gravity $g$ acts in the $-z$ direction. The present work is restricted to vortices of constant density, ${\it\rho}_{c}$ , which are located at the level of neutral buoyancy, $z=0$ , in a fluid of linear stratification, ${\it\rho}_{a}(z)={\it\rho}_{c}(1-{\it\sigma}z)$ , where ${\it\sigma}$ is a positive constant. This means that $z=0$ is the plane of mirror-symmetry during the motion considered here, and that the buoyancy frequency $\mathscr{N}>0$ is a constant ( $\mathscr{N}^{2}={\it\sigma}g$ ). The flow is assumed axisymmetric and stable.

The volume of the vortex under consideration is $4{\rm\pi}\mathscr{V}$ (i.e.  $\mathscr{V}$ is the volume of the upper half of the vortex, per radian). We shall first consider (unless stated otherwise) a fixed, given $\mathscr{V}$ . At a later stage, we shall extend the analysis to $\mathscr{V}(t)$ due to sustained influx (injection).

For simplicity, we hereafter use balances for the upper half and per radian, unless stated otherwise.

The shape (thickness) $h(r,t)$ , for $r\in [0,R(t)]$ , of the vortex has been considered in numerous previous papers. $R(t)$ is the radius of the vortex. The recent papers of ALM12 and HML12 provide a review of the available results, and also present (and confirm) a more comprehensive formula.

We start with a short derivation of the shape. Evidently, the model which emerges in this work is an approximation for an ‘asymptotic’ range of parameters. For definiteness, we assume that $f/\!\mathscr{N}$ is of the order of unity, and again, that the flow is axisymmetric and stable. Essential to our derivation are the following additional restrictions/assumptions. The domain of ambient is much larger than the vortex, and hence its basic rotation and stratification are negligibly affected by the motion of the vortex. The density differences between the vortex and the environment are small (Boussinesq case), and the kinematic viscosity of the ambient and the vortex fluids is the same ${\it\nu}$ (this can be relaxed). The buoyancy force is dominated by gravity, $g\gg {\it\Omega}^{2}R$ . This implies that the pressure is hydrostatic in the $z$ direction. The dynamic flow is dominated by rotation (Coriolis) effects; this can be expressed as $E\ll 1$ , where the Ekman number of the vortex is

The justification of this definition of $E$ will emerge later. We also assume that the interface between the vortex and the ambient is sharp, and, again, the flow is axisymmetric and stable. These assumptions are relevant to systems in nature, and can be reproduced in the laboratory with moderate costs.

The motion starts at $t=0$ , for definiteness, with some compatible initial conditions that will be discussed later.

The main point on which we focus attention in this section is that the derivation of the interface shape formula, $z=h(r,t)$ for $0\leqslant r\leqslant R(t)$ (or rather a good approximation), can be decoupled from the angular momentum equation of the flow-field.

The first governing equation is volume conservation

where $\mathscr{V}$ is given.

The second equation is the height-averaged radial momentum balance

Here $u$ is the radial velocity (dimensional) and ${\it\omega}$ is the angular velocity (dimensionless, scaled with ${\it\Omega}$ ). This balance takes into account the hydrostatic pressure field of the vortex and ambient fluids, as well as pressure continuity at the interface; the result is that the pressure gradient $-[\partial p/({\it\rho}_{c}\partial r)]$ is replaced by the first term on the right-hand side of (2.3). The underlying assumption in this equation is that the viscous shear is negligible in both the axial and radial momentum balances; this is fulfilled when $E\ll 1$ . Various derivations of this equation are available in the literature (e.g. Ungarish Reference Ungarish2009 § 13.1 for $S=1$ ; ALM12 and HML12), and we shall not repeat it here.

Three major simplifications are introduced next: the inertial acceleration term on the left-hand side is negligible, $z=h(r,t)$ is continuous, and ${\it\omega}$ is a function of $t$ (but not of $r$ ). These assumptions are, typically, justifiable when $E\ll 1$ , and after the formation (initial adjustment) of the vortex. The informal justification of the first two simplifications is what can be called ‘common knowledge’ in the branches of rotating fluids and gravity currents: Greenspan (Reference Greenspan1968) demonstrates that in axisymmetric bodies of rotating fluid with small $E$ , significant radial motion, $O({\it\Omega}{\it\omega}r)$ can be sustained only in thin viscous layers; and Ungarish (Reference Ungarish2009) shows that Coriolis effects stop the radial propagation of a typical intrusion after about a $1/{\it\Omega}$ time interval from release, and that this is accompanied by a sharp (but continuous) decrease of $h$ to $0$ at the outer radius $r=R$ of the vortex. The third simplification is supported by consistency with observations.

The power of these simplifications is that (2.3) (with zero left-hand side) can be integrated subject to the boundary condition $h=0$ at the outer radius $r=R$ . This yields the height of the vortex

where $R=R(t)$ is the radius of the vortex, ${\it\xi}$ is the reduced radial coordinate,

and

The aspect ratio is

Substitution of (2.4) into (2.2) yields the radius

We note that (2.4)–(2.8) are exactly the model of ALM12 and HML12, for a vortex of constant density, upon the change of notation that these papers use the Rossby number $Ro={\it\omega}/2$ . We prefer here the use of the variable ${\it\omega}$ because of the direct connection with the analysis of the spin-up process. Moreover, the definition of $Ro$ varies in the literature, and this may be confusing. We reiterate that our ${\it\omega}$ is the dimensionless angular velocity (scaled with ${\it\Omega}$ ) with respect to the rotating frame.

We emphasize that these equations are for a general, as yet unspecified, ${\it\omega}(t)$ . Physical considerations indicate that $-1\leqslant {\it\omega}<0$ . The lower bound is counter-rotation with respect to the ambient fluid in the rotating frame, i.e. fixed in the absolute (laboratory) frame. The upper bound is co-rotation with the ambient fluid.

ALM12 and HML12 pointed out that in realistic circumstances $|{\it\omega}|$ decreases with $t$ . Typically, the initial value of ${\it\omega}$ is close to $-1$ (see appendix A). We therefore define the ‘standard’ or ‘reference’ situation as

These values are the maximum of ${\it\alpha}(t)$ and minimum of $R(t)$ . Also, we note in passing that the ‘standard’ case is quite robust. If we take ${\it\omega}=-0.7$ instead of $-1$ , ${\it\phi}$ decreases by $9\,\%$ ; therefore, ${\it\alpha}$ and $R$ change by only $-4.6\,\%$ and $1.6\,\%$ , respectively.

ALM12 and HML12 provide experimental and Navier–Stokes simulation evidence that (2.7) is a reliable and accurate relationship for the aspect ratio of vortices in a wide range of circumstances. The most convincing component of this evidence is the observation that (2.7) is valid over very long periods of time (hundreds of revolutions of the system): in each tested system, the measured ${\it\omega}$ decays significantly, and this is accompanied by the decrease of the measured aspect ratio ${\it\alpha}$ in agreement with the theoretical correlation.

However, (2.7) is a descriptive rather than a predictive model. Accurate as it is, this is still only one equation for two variables, ${\it\alpha}(t)$ and ${\it\omega}(t)$ . In general, we do not know the value of ${\it\omega}(t)$ needed for the calculation of ${\it\alpha}$ (except for the initial situation when ${\it\omega}\approx -1$ ). We emphasize that the values of ${\it\omega}(t)$ used by ALM12 and HML12 in their verification of (2.7) were obtained by laborious experiments or three-dimensional Navier–Stokes simulations, not by an asymptotic model. Indeed, (2.7) is a quasi-static result. It does not provide any information concerning the dynamic mechanisms that affect the possible change of ${\it\omega}$ with time. With hindsight, this is not surprising, because the derivation of (2.7), as discussed above, uses only momentum balances in $z$ and $r$ directions, plus volume conservation. We did not use any conservation law for ${\it\omega}$ . We conclude that the upgrade from the descriptive to the predictive model requires an additional equation: the angular momentum balance.

3. Angular momentum considerations

The angular velocity ${\it\omega}$ of a contained volume of fluid can increase with time in a process that is usually referred to as ‘spin-up’. The typical driving force is the shear due to the fact that the domain under consideration is embedded in a rigid container (of height $2H$ ) which rotates with a larger angular velocity ${\it\omega}_{b}$ . Viscous boundary layers appear at the boundary, and sustain a torque that is proportional to ${\it\omega}_{b}-{\it\omega}(t)$ and continuously reduces the difference to zero. During the spin-up, particles (or rather shells) of fluid change their ${\it\omega}$ due to radial displacement and change of height (the potential-vorticity effect). (For connection with our problem, suppose that the gravity acts parallel with the axis of rotation and ${\it\omega}_{b}=0$ .)

When $E$ is small, this process is well understood, and amenable to accurate modelling, for a homogeneous fluid in a solid container (Greenspan Reference Greenspan1968). The dominant viscous effect is concentrated in Ekman boundary layers of thickness $({\it\nu}/{\it\Omega})^{1/2}$ (in dimensionless form, $E^{1/2}$ ; $E={\it\nu}/({\it\Omega}H^{2})$ ). These layers form quickly (in about one revolution of the system) and are subsequently quasi-steady; they induce a weak ‘secondary flow’ (or circulation) inside the container that causes, roughly, the decay of the initial ${\it\omega}$ on the time scale $E^{-1/2}/{\it\Omega}$ . Some residual discrepancies decay on the diffusion time scale $E^{-1}/{\it\Omega}$ . Even this simple scenario requires some care and adaptations in practical circumstances: the height of the container may vary with radius; if an open boundary is present the spin-up time increases; side-wall boundary layers also contribute. The accuracy of the spin-up solution also depends on the accuracy of the initial condition ${\it\omega}(0)$ ; in practical circumstances, this may also contain uncertainties.

Significant modifications occur when the spun-up domain is embedded in another fluid (rather than in a solid container), and gravity buoyancy is significant. First, at the interface between the fluids there is a double Ekman layer effect: while the ambient shears and pumps the embedded fluid in one direction, this fluid shears and pumps the ambient into the opposite direction; therefore, the effective Ekman shear layer is thicker (by roughly 2). Second, the shape of the fluid domain under consideration varies during the process; moreover, there is coupling between the shape and ${\it\omega}$ . Third, when the embedding fluid is linearly stratified, the buoyancy hinders the motion (circulation) of the particles activated by the the Ekman layer pumping, and consequently this effect may be ‘suppressed’ even before a significant reduction of $|{\it\omega}|$ is attained. These features were analysed in the papers of Walin (Reference Walin1969), Flor et al. (Reference Flor, Ungarish and Bush2002), Ungarish & Mang (Reference Ungarish and Mang2003) and the references therein.

The vortex under present investigation contains the spin-up components in their more complicated form. The formulation and solution of the flow-field that matches in detail all the effects involved is a formidable task; moreover, the expected outcome will be too complicated for useful insights and quick computation. We shall therefore not pursue this approach here. For progress, we shall attempt only an approximate global solution for the entire vortex, as follows.

Consider the angular momentum balance of the vortex

where ${\it\Gamma}$ is the angular momentum of the volume inside the interface, and $M$ is the shear torque on the envelope. The system is sketched in figure 2. Again, we make the budget for only the upper-half of the vortex, per radian.

Figure 2.

Schematic description of the spin-up flow. The interface of the vortex (upper half) is shown by the thick (black) curve. The domain between the thin lines is the ‘boundary layer’ ${\it\delta}$ in which transition of ${\it\omega}$ occurs. The density is ${\it\rho}_{c}$ (constant) inside the vortex, and ${\it\rho}_{c}(1-{\it\sigma}z)$ in the ambient fluid. $\mathscr{N}^{2}={\it\sigma}g$ .

More explicitly,

where we used the assumption ${\it\omega}={\it\omega}(t)$ and the results (2.4), (2.8). Then, after some algebra (see appendix C) and recalling that here $\mathscr{V}$ is constant, we obtain

where

Since ${\it\omega}<0$ , this term is always positive; its influence on the spin-up may be significant, as discussed later. Equations (3.3)–(3.4) reveal the complexity of the spin-up: the rate of change of angular momentum is distributed among several components, and not a linear function of $\text{d}{\it\omega}/\text{d}t$ .

For the torque we write

The fraction in the integral stands for the shear, while ${\it\delta}$ is the thickness of the transition layer from the domain with ${\it\omega}<0$ (inside the vortex) to the unperturbed ambient fluid where ${\it\omega}=0$ . While ${\it\omega}$ is negative, the torque is positive, and this drives the spin-up process. Since ${\it\omega}={\it\omega}(t)$ we rewrite (3.5) as

Here $\bar{{\it\delta}}=\bar{{\it\delta}}(t)$ is the weighted-average thickness of the angular momentum transfer layer, defined by

where ${\it\xi}=r/R$ . This definition is convenient because, when ${\it\delta}$ is $r$ -independent, it returns $\bar{{\it\delta}}={\it\delta}$ .

To close the model we must specify the effective thickness of the shear layer, $\bar{{\it\delta}}(t)$ . This is the most challenging component of our model. To our knowledge, no direct investigation of this effect has been performed, and we must rely on pieces of theoretical results gained for related spin-up problems (Walin Reference Walin1969; Flor et al. Reference Flor, Ungarish and Bush2002; Ungarish & Mang Reference Ungarish and Mang2003). We consider the spin-up process over a long time period. Evidently, the first stage is dominated by the double Ekman layer of thickness ${\it\delta}_{1}=2({\it\nu}/{\it\Omega})^{1/2}$ .

However, the stratification of the ambient fluid tends to suppress the upper part of the Ekman layer. The result is that on the time scale $t_{s}$ the thickness of the transition layer of ambient fluid near the centre increases to ${\it\delta}_{0s}$ , where

Subsequently, the layer at the centre thickens further by diffusion according to $[{\it\nu}(t-t_{s})]^{1/2}$ . The thicker layer prevails over half of the radius (approximately), then decreases to ${\it\delta}_{1}$ at $r=R$ . Based on analytical considerations and numerical solutions (Walin Reference Walin1969; Flor et al. Reference Flor, Ungarish and Bush2002; Ungarish & Mang Reference Ungarish and Mang2003), we therefore suggest that the boundary layer of the vortex can be approximated by the functions

where

to which we add the diffusion contribution for $t>t_{s}$ when relevant. To improve the accuracy, we use the instantaneous $R(t)$ instead of $R(0)$ for the calculation of ${\it\delta}_{0s}$ in the last equation.

With (3.9) the integral in (3.7) can be calculated analytically, and we thus obtain $\bar{{\it\delta}}(t)$ . The details are presented in appendix B.

We admit that the calculation of the shear term is a bold approximation, and that we have neglected the curvature of the interface. We are presently unable to estimate the approximation errors. The justifications for the use of this approximation are as follows. (i) In the present state of the art, such a closure is necessary for progress, and it will be quite straightforward to replace it with a better one when available. (ii) The predictions of the resulting model seem to be qualitatively correct and quantitatively in reasonable agreement when compared to the available data acquired by laboratory experiments and Navier–Stokes simulations.

Finally, we substitute (3.3) and (3.6) into the angular momentum balance (3.1). It is convenient to use the dimensionless time variable $({\it\Omega}t)$ . We thus obtain the spin-up equation

Here $\mathscr{V}$ is constant. $J(t)$ and $R(t)$ are explicit functions of ${\it\omega}(t)$ via (2.6)–(2.8) and (3.4), while $\bar{{\it\delta}}(t)$ is also explicitly calculated as specified above.

We now have a closed predictive model. The background system (input parameters ${\it\Omega}=f/2,\mathscr{N}\!/f$ and ${\it\nu}$ ), the volume of the vortex $\mathscr{V}$ (dimensional), and the initial ${\it\omega}(0)$ are given. The solution of (3.11), combined with (2.6)–(2.8), provides ${\it\omega},R$ , and, most importantly, ${\it\alpha}$ for the subsequent motion. Theoretically, the integration of (3.11) can be carried out ‘to infinity’, but there are indications that in real systems instabilities become dominant when $|{\it\omega}|<0.01$ , and hence we think that ${\it\omega}=-0.01$ is a plausible end point for the prediction of our model.

4. Slow sustained injection

Suppose that the vortex is produced and sustained by injection (influx) from a source about the axis at rate $q$ , which is active for many rotations of the system. We assume that the injection is ‘slow’, so that our model can be used within some small extensions. This implies that a relatively small volume is added per $1/{\it\Omega}$ , or $q{\it\Omega}\ll \mathscr{V}_{0}=qt_{0}$ , where $t_{0}$ is the time at which we start using the model. The result $t_{0}\gg 1/{\it\Omega}$ is quite trivial, and, surprisingly, independent of $q$ . (A more stringent criterion could involve the Ekman layer circulation speed, but the result is too cumbersome for useful insights.)

We therefore assume that a vortex which satisfies (2.4)–(2.8) exists for $t\geqslant t_{0}=3/{\it\Omega}$ , with initial ${\it\omega}(0)$ . The volume is $qt$ (per half-vortex per radian). We wish to predict the subsequent ${\it\omega}(t)$ and ${\it\alpha}(t)$ .

The extension of the model is as follows. First, we replace the constant $\mathscr{V}$ with the time-dependent counterpart $\mathscr{V}(t)=qt$ in (2.2), (2.8) and (3.2). The results (2.4)–(2.8) carry over.

The presence of $\mathscr{V}(t)$ in (3.2) contributes a significant dynamic effect on the spin-up process via a modification of $\text{d}{\it\Gamma}/\text{d}t$ for (3.1). After some algebra (see appendix C) and arrangement, the spin-up equation reads

Here $\bar{{\it\delta}}$ is calculated as before, using the appropriate $R(t)$ .

The last term on the right-hand side of (4.1) is the dynamic contribution of the injection to the angular momentum variation. It reproduces the fact that the particles of injected fluid that propagate to larger radii are decelerated by Coriolis effects. This is actually a ‘spin-down’ contribution to the entire vortex, or a ‘sink’ that absorbs part of the external torque. This spin-down effect is strong when (i)  ${\it\omega}$ is close to zero, and (ii)  ${\it\Omega}t$ is not large. (We note in passing that this term essentially explains why ‘rapid injection’ creates ${\it\omega}(0)\approx -1$ : when the contribution of the first term is small, ${\it\omega}$ tends to $-1$ in less than one $1/{\it\Omega}$ .)

However, in a long spin-up process, the effect of the spin-down (last) term is not so dramatic. The first reason is the obvious trade-off between the terms on the right-hand side of (4.1). The spin-up (first and positive) term becomes dominant as the second (negative) term drives ${\it\omega}$ to $-1$ . This keeps a balance during which ${\it\omega}$ is spun up. The second reason is the fairly rapid decay of the spin-down term like $({\it\Omega}t)^{-1}$ . The $R^{2}/\mathscr{V}(t)$ factor in the first term also decays with time, but more slowly, ${\sim}({\it\Omega}t)^{-1/3}$ . The hindering effect of $J$ is like in the fixed-volume case.

In general, it is not straightforward to compare the time-dependent behaviour of the fixed-volume and slow-injection vortices. Both are expected to display ${\it\omega}\approx -1$ in the initial stage (with ${\it\alpha}\mathscr{N}\!/f\approx 0.5$ ), and both will eventually be spun up to small $|{\it\omega}|$ (accompanied by ${\it\alpha}\mathscr{N}\!/f\approx 0.5(2|{\it\omega}|)^{1/2}$ ). The details, however, depend on the values of $q$ (for the injection case) and $\mathscr{V}$ (of the fixed vortex). We could not find a simple similarity rule between these systems.

Another immediate extension of the model is to the case of slow sustained injection over a finite period of time, $t\geqslant t_{i}$ , after which a vortex of fixed volume exists. In this case we integrate (4.1) as follows: with both right-hand side terms up to $t_{i}$ ; then, for $t>t_{i}$ , with the second term on the right-hand side turned off (set to zero). For $t>t_{i}$ the fixed-volume vortex behaviour is recovered; however, the ‘initial conditions’ are, typically, different from the standard fixed-volume case. Technically, our model can now imitate the vortex of fixed $\mathscr{V}$ created by rapid injection: take $q=\mathscr{V}{\it\Omega}/n$ , and turn off the source at $t_{i}=n/{\it\Omega}$ , where $n=4$ or $5$ . For definiteness, take ${\it\omega}(0)=-0.5$ ; tests show that an adjustment towards $-1$ occurs. We keep in mind, however, that the results in this initial stage lack physical accuracy. The underlying assumptions concerning the hydrostatic and radial-momentum balances cannot be justified during this $t_{i}$ . These assumption gain validity around $t_{i}$ , and it is therefore justified to consider the solution for the subsequent time.

Finally, we note that the assumption of constant-rate injection can be relaxed. For a prescribed general smoothly increasing $\mathscr{V}(t)$ , the only change in the previous model will be to replace $1/({\it\Omega}t)$ in (4.1) with $[1/({\it\Omega}\mathscr{V}(t))](\text{d}\mathscr{V}/\text{d}t)$ .

Figure 3.

Comparison with fixed-volume experiment of ALM12: ${\it\alpha}\mathscr{N}\!/f$ as a function of $t/T$ (number of revolutions, where $T=2{\rm\pi}/{\it\Omega}$ ). The thick solid line shows the model prediction, the symbols joined by the thin line are experimental data, $f=6.8~\text{s}^{-1},\mathscr{N}=1.6~\text{s}^{-1}$ . The model uses $f/\!\mathscr{N}=4.25$ , $E=5.1\times 10^{-5}$ , ${\it\omega}(0)=-0.95$ .

5. Comparisons

Given the bold approximations and simplifications of our model, confirmation by comparison with experimental and Navier–Stokes simulations is needed. Unfortunately, most of the pertinent available data are concerned with the short time interval of formation and adjustment (up to about 20 revolutions, say). In other cases, clear-cut comparisons are not possible because the reported data were obtained with initial conditions incompatible with the model, or use implicit scalings, or were contaminated by instabilities. For example, in the experiments of Grant et al. (Reference Grant, Sundermeyer and Hebert2011), the vortex (lens) was created by an oscillating grid (at the middle of the tank). The volume of the resulting vortex, and the value of the initial ${\it\omega}(0)$ , which are needed in our model, cannot be well estimated. All we can say is that the observations are consistent with our model if we use some plausible estimates of $\mathscr{V}$ and ${\it\omega}(0)$ .

We have therefore restricted our comparison to four cases. In these comparisons the time is rescaled for the number of revolutions $t/T$ , where $T=2{\rm\pi}/{\it\Omega}$ . The experimental data concern water, and hence for the corresponding $E$ we use ${\it\nu}=0.01~\text{cm}^{2}~\text{s}^{-1}$ .

6. Concluding remarks

We considered the time-dependent motion of an axisymmetric anticyclonic vortex in a large ambient stratified fluid rotating with constant ${\it\Omega}$ . We focused attention on the long-time behaviour (many revolutions of the system) that follow the initial adjustment (2–3 revolutions say). We developed a closed simple model for the prediction of the aspect ratio ${\it\alpha}$ (and actually the shape) and internal angular velocity ${\it\omega}$ as functions of time $t$ . (In our model ${\it\omega}$ is scaled with ${\it\Omega}$ ; the literature sometimes uses the Rossby number $Ro={\it\omega}/2$ .) This model is an extension of the model of ALM12 and HML12, which derived the connection between ${\it\alpha}$ and ${\it\omega}$ . We added the balance of angular momentum, and analysed the spin-up process. The solution is obtained numerically by the integration of one ODE, with insignificant computational effort. The model can be applied to cases of both fixed-volume and injection-sustained vortices.

The model elucidates, qualitatively and quantitatively, the salient features of the vortex on the long-time scale. The continuous ‘decay’ of the aspect ratio of the vortex over many revolutions of the system can be attributed to the spin-up effect. Both the fixed-volume and injection-sustained vortices start with ${\it\omega}\approx -1$ , typically. The model provides useful insights into the spin-up mechanism, as follows. For the fixed-volume vortex, the shear torque decays with time because the thickness of the boundary layer, $\bar{{\it\delta}}$ , increases. It starts as a double Ekman layer (between two fluids) but it quite quickly expands due to stratification effects, and later by viscous diffusion. This prolongs the spin-up somewhat beyond the classical $E^{-1/2}/{\it\Omega}$ time interval. In this case ${\it\omega}$ decays continuously towards $0$ , and ${\it\alpha}$ decreases (the vortex expands), but the notable changes of ${\it\alpha}$ occur, typically, over many revolutions of the system. Moreover, when $|{\it\omega}|$ becomes small, the momentum of angular inertia of the vortex increases like $(1+(1/3)|{\it\omega}|^{-1})$ ; this further hinders the spin-up, and prolongs the process. For the sustained-injection vortex, the injected volume tends to spin down the vortex, and counteracts the effect of the shear torque. However, after some time interval (say 10 revolutions) the vortex is sufficiently large to render the spin-down effect of injection marginal. Then, the spin-up is roughly like in the fixed-volume case. In particular, the hindrance mechanism of the increasing momentum of angular inertia like $(1+(1/3)|{\it\omega}|^{-1})$ is relevant when $|{\it\omega}|$ is small.

We admit that our spin-up equation contains bold simplification. We ignored the variation of ${\it\omega}$ with $r$ , and the effect of the curvature of the interface on the torque. We also discarded the fact that the spin-up of the vortex and its elongation are bound to modify the stratification of the ambient in the vicinity of the vortex. To assess the importance of these effects it is necessary to perform special experiments and Navier–Stokes computations which must be left for future work. Our model assumes a sharp interface between the vortex and the ambient; however, in realistic circumstances the turbulence and component diffusion may blur the boundary and invalidate the predictions.

Comparisons of the prediction of the model with experimental and Navier–Stokes simulation data were performed for four cases. In three cases the agreement is good. In one case, the model predicts a much faster spin-up than the observed one; we could not find a clear-cut explanation for this discrepancy.

We think that this model is a useful contribution to the investigation of the vortex in a rotating stratified ambient, and a valuable extension to the model of ALM12 and HML12. The present model is restricted to an anticyclonic vortex ( ${\it\omega}<0$ ) of constant density (i.e. the internal stratification $\mathscr{N}_{c}=0$ ). The study of ALM12 and HML12 also considered (i) cases with internal stratification, $\mathscr{N}_{c}>0$ , and (ii) the cyclonic vortex with ${\it\omega}>0$ . We are presently unable to extend the spin-up considerations to these cases, because (i) there are indications that the spin-up circulation affects $\mathscr{N}_{c}$ , but a mathematical approximation for this behaviour is still unavailable, to our knowledge, and (ii) HML12 shows that ${\it\omega}>0$ requires $\mathscr{N}_{c}>0$ . Extension of the model to these important regimes must be left for future work.