Long-lived “bubbles” of wildfire smoke or volcanic aerosol have recently been observed in the stratosphere, co-located with ozone, carbon monoxide, and water vapour anomalies. These bubbles often survive for several weeks, during which time they ascend through vertical distances of 15 km or more. Meteorological analysis data suggest that this aerosol is contained within strong, persistent anticyclonic vortices. Absorption of solar radiation by the aerosol is hypothesised to drive the ascent of the bubbles, but the dynamics of how this heating gives rise to a single-sign anticyclonic vorticity anomaly have thus far been unclear. We present a description of heating-driven stratospheric vortices, based on an axisymmetric balanced model. The simplest version of this model includes a specified localised heating moving upwards at fixed velocity and produces a steadily translating solution with a single-signed anticyclonic vortex co-located with the heating, with corresponding temperature anomalies forming a vertical dipole, matching observations. A more complex version includes the two-way interaction between a heating tracer, representing the aerosol, and the dynamics. An evolving tracer provides heating which drives a secondary circulation, and this in turn transports the tracer. Through this two-way interaction an initial distribution of tracer drives a circulation and forms a self-lofting tracer-filled anticyclonic vortex. Scaling arguments show that upward velocity is proportional to heating magnitude, but the magnitude of peak quasigeostrophic vorticity is

Smoke from wildfires in Australia in 2020 has been observed to enter the stratosphere and, unexpectedly, to form long-lived coherent anomalies that persist for several weeks and that ascend through distances of several kilometres

This apparent co-location of tracers with coherent vortices is consistent with a physical interpretation where strong vortices effectively act to isolate tracers (which may be chemical species, small particles such as wildfire smoke or volcanic aerosol (henceforth collectively referred to as “aerosol”), etc) from their environment, thus maintaining large anomalous concentrations that would otherwise be reduced through the effects of mixing. The idea of vortex isolation in the stratosphere has previously been much discussed in the context of the winter polar vortex, for example, the isolation of regions of low ozone concentration in the austral spring lower stratosphere following chemical destruction

Much of our physical understanding of vortex isolation originates in the studies of two-dimensional turbulence, where it has been observed that flow self-organises into strong, relatively long-lived coherent vortices of different signs

Therefore, in one sense, the existence of a persistent aerosol (i.e. tracer) anomaly within coherent vortices, as recently observed, is expected from the above physical description. However much of the previous discussion of persistent coherent vortices, both in the atmosphere and the ocean, has emphasised the material conservation of potential vorticity (PV) and focused on whether the vortices can survive external perturbation. In the case of the recently observed smoke-containing stratospheric vortices, it has been noted that

The previously cited papers on aerosol-filled vortices

This single-signed PV anomaly is quite different to the balanced dynamical response to specified localised (stationary) heating expected from simple theory for axisymmetric flow, which has been applied, for example, to extratropical anticyclones and cyclones in the troposphere and lower stratosphere

Accordingly, key specific questions which motivate further study of the dynamics of smoke- or aerosol-filled vortices are the following:

How does an isolated anticyclonic vortex emerge as a response to heating and why is the anticyclonic vortex apparently centred at the same level as the heating rather than above it?

What determines the rate of rise of the tracer anomaly and accompanying anticyclonic vortex?

What determines the strength of the vortex and the corresponding temperature anomaly?

Once aerosol is injected into the stratosphere, what is the mechanism for its organisation into long-lived ascending heating-driven vortex structures and under what conditions is this organisation likely to take place?

A non-standard aspect of the dynamics of the observed stratospheric vortices is that the heating field is determined by the aerosol which co-evolves with the dynamical fields. A simple representation is that the aerosol is a tracer field transported by the flow, and the heating is simply proportional to the tracer concentration. Solution of the Eliassen problem for a localised heating shows that, alongside the response in PV, there is also a secondary circulation response, which is upward motion in the centre of the heating region

In Sect. 3, explicit numerical solutions are presented for the evolution from initial conditions of a distribution of smoke that is localised in the horizontal and vertical. The highly simplified case of specified ascent of the smoke (and hence the heating) is considered first, followed by the fully interactive case, where the aerosol drives a secondary circulation through its heating effect and is transported by that circulation. Given the inability of the axisymmetric model to explicitly represent deformation by the large-scale flow and the consequent vortex stripping, a simple adjustment of the smoke to represent this effect within the axisymmetric formulation is also presented and discussed. Further aspects explored include the effects of thermal damping and non-Boussinesq effects. A summary of our key findings is provided at the end of Sect.

In Sect. 4, a different problem is considered in which the smoke is initially confined to a horizontally homogeneous layer. The geometry is assumed to be two-dimensional and periodic in the horizontal rather than axisymmetric. A linear stability problem is solved to demonstrate that this configuration is unstable as a result of the coupling between the smoke and the dynamics. Numerical solutions, under the QG approximation, can follow the evolution out of the linear regime and show how this coupling leads to self-organisation of the flow to give a discrete set of rising smoke plumes. Section 5 summarises the results and discusses the implications for the formation and evolution of smoke-driven vortices in the real atmosphere. It is argued that, whilst the axisymmetric or two-dimensional models have fundamental limitations, the conclusions obtained from these models can be combined with knowledge of two-dimensional or QG vortex dynamics from much previous work to give useful insights into the behaviour of the aerosol-driven vortices in the real 3-D atmosphere.

To describe the dynamics resulting from an aerosol-like tracer that generates diabatic heating and consequently anomalies in vorticity, temperature, and velocity, we consider an axisymmetric framework on an

As is standard, we use Eq. (

It is convenient to combine Eqs. (

The principles underlying the behaviour allowed by these equations are well-known. A first important implication is that the flow in the (

The condition for the PDE for

A significant simplification is to make the quasigeostrophic approximation, which requires small Rossby number

To the dynamical equations presented above we add the evolution equation for a heating tracer, with concentration

The dynamical equations and the aerosol tracer equation are coupled by allowing the heating

The rest of this section will now focus on the equations resulting from the quasigeostrophic approximation. We briefly discuss the practicalities of solving the non-quasigeostrophic balanced vortex formulation in Sect.

There is a corresponding equation for the velocities (

Equations (

In the simulations to be reported below, we solve initial value problems starting from a configuration in which the flow is at rest (i.e. there is no initial PV anomaly), and the initial distribution of the tracer

Parameter choices are as follows and will be retained in further sections unless otherwise stated. The Coriolis frequency is calculated at 45° N, i.e.

On solving the non-quasigeostrophic balanced model equations, it is generally found that there is catastrophic breakdown of the numerical solution at some time and simple approaches such as reducing time step do not resolve this problem. Overall, the early time evolution of the tracer (and hence the heating rate) in the quasigeostrophic equations matches the evolution of the non-quasigeostrophic equations: the tracer profile right before the non-QG numerical solution breaks down is plotted in Fig.

Solutions of the Eliassen problem, Eq. (1), and quasigeostrophic problem, Eq. (4), with no thermal relaxation.

Examination of the minimum potential vorticity as this “breakdown time” is approached shows the PV approaching zero (roughly as a linear function of time), suggesting that non-ellipticity is the cause of the breakdown (Fig.

Loss of ellipticity is a familiar problem when solving the non-quasigeostrophic axisymmetric (or two-dimensional) vortex models, e.g. in studies of tropical cyclones. The physical interpretation is often that the axisymmetric flow becomes unstable to symmetric instability or inertial instability

Given the ad hoc nature of these adjustment procedures, and some arguments in the tropical cyclone literature that different adjustments can give quite different outcomes for the evolution

This section explores solutions of the axisymmetric quasigeostrophic formulation starting from the Gaussian initial condition Eq. (

The dynamics of the system above, without the coupling of tracer to heating, have been much studied

Schematic diagram of

The structure shown in Fig.

We hypothesise that this apparent difference can be resolved by incorporating the effect of the previously noted upwelling on the aerosol tracer and its associated heating, which will be displaced upward. The relevant problem to consider is not the response to a stationary localised heating but the response to an ascending localised heating. As the heating arrives in a region it will provide an anticyclonic PV forcing, and as it leaves it will provide a cancelling cyclonic PV forcing, suggesting that the response will be an anticyclonic PV anomaly moving with the heating.

The hypothesised behaviour can be illustrated by explicit calculation, (i) in a model in which the heating is simply specified as ascending at a given rate (Sect.

On the basis of the above qualitative discussion, before considering the full problem in which the tracer evolves according to Eq. (

The initial distribution of the tracer

Tracer-filled vortex of radius 500 km, consistent with 1000 km vortex diameter detected from the Australian wildfires

However another prediction of this calculation is that, as a result of the early-time evolution, a positive PV anomaly is left below the initial region of heating (Fig.

Further information on the response to a specified upward-moving heating with zero thermal damping is given in Fig.

Evolution of temperature and QGPV for steadily upward-moving heating, including influence of thermal damping:

Motivated by the numerical solutions, we consider a steadily translating solution. We define a new coordinate

The form of Eq. (

In this idealised problem

Thermal damping, through radiative transfer, is expected to modify the response to the forcing of PV by the heating and hence to modify other dynamical quantities including temperature. This can be investigated by including non-zero values of

The results shown in Fig.

For this simple experiment where the tracer is moved upwards at constant speed, the thermal damping has two major effects. The first is that, as can be seen in the right-hand panels, it dissipates the low-level cyclonic PV anomaly, which is not being maintained by any forcing. As described by

To consider the extent to which the thermal damping affects the dynamics, and in particular the size of the potential vorticity anomaly, it is necessary to consider the size of the second term on the right-hand side of Eq. (

To assess the effect of thermal damping on the vertical velocity, first note from the argument above that if

Explicit calculation indeed shows that, for a fixed

There are two key conclusions from the above discussion. The first is that the magnitude of the quasigeostrophic PV anomaly, in this steadily translating case, is independent of

We now solve the full quasigeostrophic equations where the tracer equation is solved explicitly and determines the heating. The secondary circulation is solved to find

To probe how the vortex evolution depends on the initial aspect ratio of the tracer bubble, we explore three different initial conditions in Fig.

Evolution of a tracer-filled vortex, where the tracer generates heating and its own secondary circulation. No radiative damping has been included (

In all three cases, the maximum tracer abundances occur at the top of the tracer structure, which is consistent with detected aerosol bubbles following the Australian and Canadian wildfires

The precise relation between the tracer distribution (hence the heating distribution) and the vertical velocity distribution is expressed by Eq. (

We now turn our attention to the shape of the potential vorticity anomalies, which are forced by the heating and are therefore determined by the time history of the tracer. As expected from the simulations with specified upward motion of the heating, the dominant features are an anticyclonic potential vorticity anomaly moving upward with the tracer and a cyclonic potential vorticity anomaly left at a fixed level below and essentially determined by the initial distribution of the tracer. The fact that for the shallower initial conditions there is a central portion of the tracer distribution that is ascending more rapidly and an outer part that is ascending less rapidly implies similar geometry for the anticyclonic potential vorticity anomaly. The central part of the potential vorticity anomaly is largely forced by the central part of the tracer distribution and ascends with it. The outer part of the potential vorticity anomaly is largely forced by the outer part of the tracer distribution and ascends with it, more slowly than the central part. This effect is not seen for the deep initial condition because the entire tracer distribution moves upwards together.

Full quasigeostrophic model integrated from the standard initial condition for 50 d with thermal damping for

We now consider the role of thermal damping in the fully coupled model. While the conclusions from simple scaling arguments we put forth in Sect. 3.2 that arose from the imposed ascent case would be expected to hold in certain situations, the behaviour seen in the fully coupled model is more likely to describe strong aerosol-filled vortices. Tracer, vertical velocity, and potential vorticity profiles after 50 d are shown in Fig.

As has been noted in the previous section, neither the cyclonic potential vorticity anomaly nor its associated dipole temperature signal has been emphasised in the literature, though there is some indication of such an anomaly in the modelling study by

The details of the coupled tracer–dynamics structure discussed in the preceding section are significantly different from those that have typically been observed, such as the persistence of anomalies at lower levels, the cyclonic PV anomaly, and the trailing tracer features and accompanying PV anomalies that are particularly prominent for the standard and shallow initial conditions. That said, it is also the case that the regions of substantial tracer concentration extend well below the anticyclonic PV anomaly in the central part of the upward-moving structure, and this applies even in the case of the deep initial condition. These differences can be attributed to our model being axisymmetric and hence not capturing 3D effects such as vortex stripping (referred to here as generally resulting from a combination of horizontal and vertical shear), which allow the vorticity distribution to have a very direct effect on, for example, tracer dispersion. These missing effects can be incorporated in a very simple ad hoc way in the axisymmetric model by incorporating an adjustment to the tracer field whereby any tracer lying in regions where the PV or vorticity anomaly is less than a critical threshold is instantaneously removed. The justification is that in reality tracers outside of coherent vortices will be rapidly mixed. To focus on the dynamics of the persistent anticyclonic vortices, rather than on their initial formation, the adjustment is applied at an intermediate time, when, within the interactive tracer–dynamics model, the regions of anticyclonic PV are already significantly displaced from the region where the tracer was initially concentrated and, furthermore, the tracer is retained only in anticyclonic regions. The simulations reported in the previous section were repeated and the adjustment applied only after 14 d, but it was applied continuously after that time. The criterion for retaining (or removing) the tracer could be varied, and in the illustrative cases to be shown it was chosen on the basis of PV being less than (or greater than) the value

Figure

Vertical slice of

The abrupt adjustment of the tracer field at 14 d is of course unrealistic. Furthermore, the fact that the coincidence between the tracer field and the anticyclonic vorticity is greater with the adjustment than without it is a direct consequence of the adjustment and therefore by itself not very significant. The important point that this calculation illustrates is that the coherent upward-propagating tracer–vortex structure is robust to the inclusion of the adjustment, giving greater confidence that the mechanisms described here are viable in the real atmosphere. Careful comparison of Fig.

Thus far, by using a very large value for

Full quasigeostrophic model with vortex stripping adjustment,

The variation of

A further general point about non-Boussinesq effects suggested by these solutions is that the decreasing density tends to offset the effect on ascent rates of tracer leakage from the upward-moving bubble. This may be an important effect in the real atmosphere where the tracer bubbles may ascend around 15 km, equivalent to a factor of 10 reduction in density. Said differently, if the tracer bubble was not leaking, the non-Boussinesq effects would act to increase the volume of the bubble and hence the ascent rate via the diffusive argument made above. This increased bubble volume and speedup of vortex ascent is not detected in the observed cases on record thus far, suggesting that the tracer bubbles leak material, consistent with existing observational evidence

We now present Fig.

Schematic summarising the physical processes in our theoretical model for aerosol-filled vortices. For reference, panel

As we submitted this paper we became aware that an independent paper on the dynamics of heating-driven vortices had been submitted for publication elsewhere

The key similarities between the two studies are, first, the PV-based description of the dynamics and, second, the introduction of an active tracer representing sunlight-absorbing aerosols whose mixing ratio determines the short-wave heating rate, such that

One significant difference between the two studies is in the initial condition. We consider an initial condition where there is a specified tracer anomaly and the flow is at rest. P2024, on the other hand, consider an initial condition with and without an anticyclone present, together with the tracer anomaly, and report that the coherence and ascent of the tracer bubble are enhanced with the initial anticyclone. We will discuss this point further in Sect. 5.

The important methodological differences between P2024 and our study are as follows. P2024 use both a simplified axisymmetric model and a full 3-D non-hydrostatic numerical model. In their axisymmetric model formulation, P2024 adapt coordinate transformations previously applied in tropical cyclone modelling approaches

The strong advantages of the P2024 approach are that the use of the numerical model avoids the technical difficulties of integrating balanced equations when PV becomes near-zero and that the 1-D model gives clear insight into some of the mechanisms that are operating in the numerical model. Therefore, P2024 are able to provide a self-consistent prediction that PV values become very small in the anticyclone. Our scaling arguments and model simulations have similarly predicted that PV values in the anticyclone are small, following from

A particular shortcoming of QG dynamics is that the equations neglect vertical advection of PV, and the evolution of the QGPV Eq. (

Whilst accepting that the calculations of our study are subject to the limitations of QG theory whereas P2024's are not, there are ingredients of the problem that might allow the QG model we have presented to be more useful than might appear. The first is that, the scaling arguments we lay out in this study match estimates from observations of, for example, relative vorticity, and with suitable modification of parameters can be extended beyond quasigeostrophy

Thus far, we have explored the dynamics of a vortex evolving from an initial condition of a localised bubble of heating tracer as described by an initial condition Eq. (

To study this problem without requiring a full three-dimensional numerical simulation, it is helpful to assume that the configuration is two-dimensional, depending only on the two Cartesian coordinates

Adopting the QG framework, the required modified form of Eq. (

The tracer abundance,

We now consider the configuration where the tracer abundance (and hence the heating) is a function of

First consider the case where there is no aerosol, i.e.

Next, consider the case where the heating has a constant vertical gradient, i.e.

The scenario of constant

The problem is simplified a little by rewriting Eq. (

The behaviour can be illustrated by considering a Gaussian vertical profile,

Figure

The structures simulated here have some resemblance to those seen in simulations of the maintenance of tropopause-level cirrus clouds by the circulation, forced by cloud radiative heating via the interaction of cloud dynamics, microphysics, and radiation

Having established that there is unstable growth of small disturbances to a horizontally homogeneous tracer layer, the behaviour when the disturbances reach finite amplitude may be investigated by solving the complete quasigeostrophic system Eq. (13), without the linearisation assumption in the Cartesian form of the tracer equation (Eq.

The early stages of the evolution seen in Fig.

Solutions of the full quasigeostrophic equations in Eq. (13) in Cartesian coordinates with periodic boundary conditions, showing the tracer distribution at

These two-dimensional results notwithstanding, as we have previously noted, the assumption of two-dimensionality, whether in Cartesian geometry or as axisymmetry, misses potentially important mechanisms such as vortex isolation, aerosol bubbles pinching off ascending plumes, and coalescence of plumes. A more complete study would require analysis of the full 3-D problem.

Following penetration of wildfire smoke or of volcanic aerosol into the stratosphere, recent studies have detected evidence of aerosol-filled anticyclonic vortices that persist for several weeks and ascend for large distances, typically 10–20 km. Aerosols are known to be effective absorbers of radiation, and their presence in large concentrations will therefore give substantial heating effects at the location of these aerosol-filled vortices. Various important details of the observed dynamical structures require further explanation, such as the fact that a single-signed anticyclonic potential vorticity anomaly is co-located with a localised heating and the ascent of the vortex across isentropic surfaces, which cannot be explained by material conservation of potential vorticity.

In this paper we have considered a simplified dynamical description of these vortices, starting with an assumption of axisymmetry together with hydrostatic and gradient wind balance, which leads to the classical Eliassen problem for the response of a vortex in a rotating stratified fluid to applied heating. The novel ingredient here is that the heating is determined by a tracer, representing sunlight-absorbing aerosol, which is transported upward by the secondary circulation, which is itself part of the Eliassen response to the tracer heating. There is therefore a two-way coupling between the evolution of the tracer and the circulation. In reality the observed aerosol-driven vortices are contained within a larger-scale three-dimensional stratospheric flow, which is likely to have a strong deforming effect. Hence, the assumption of axisymmetry has several limitations, which are discussed in more detail below. In particular, an axisymmetric theory cannot account for the isolation of tracers within strong vortices, which is a well-known phenomenon in geophysical fluid dynamics and is likely to be a major part of an overall description. A further simplification in most of our explicit calculations is that we use the QG form of the Eliassen problem rather than the full non-QG form. This may limit the applicability of some of the detailed predictions, though not necessarily the broader quantitative predictions such as scaling estimates.

In Sect. 1 we highlighted four key specific questions which motivate further study of the dynamics of aerosol-filled vortices:

How does an isolated anticyclonic vortex emerge as a response to heating and why is the anticyclonic vortex apparently centred at the same level as the heating rather than above it?

What determines the rate of rise of the tracer anomaly and accompanying anticyclonic vortex?

What determines the strength of the vortex and the corresponding temperature anomaly?

Once aerosol is injected into the stratosphere, what is the mechanism for its organisation into long-lived ascending heating-driven vortex structures and under what conditions is this organisation likely to take place?

Regarding question (i), the axisymmetric model provides a clear explanation. An upward-moving localised heating field provides an upward-moving dipolar potential vorticity forcing, anticyclonic above and cyclonic below. The effect of the ascent is to give an anticyclonic potential vorticity anomaly moving upward with the heating and leaving behind a stationary cyclonic anomaly just below the initial location of the heating. The upward movement of the anticyclonic PV anomaly is not a result of material conservation of potential vorticity but results instead from the heating-induced forcing. We demonstrated this first by omitting the tracer–dynamical coupling and simply specifying upward motion of the heating field, both as the solution to an initial value problem and as an analytical steadily translating solution of the dynamical equations. We then included the tracer–dynamical coupling and showed the evolution of an initial tracer, and hence heating field, led to ascent of the top part of the tracer distribution with an accompanying anticyclonic vortex.

With respect to question (ii), for our self-lofting scaling estimates to hold, a minimum heating rate

For question (iii), the scaling analysis presented in this study suggested that the quasigeostrophic potential vorticity magnitude (and relative vorticity magnitude) in the tracer-filled vortex would be

Turning now to question (iv), as noted above, the axisymmetric model had some success in demonstrating the emergence of a detached, ascending tracer-filled vortex for suitable conditions on the initial tracer distribution. So, it could be the case that the details of injection (which we do not address), which sets up the initial tracer distribution, are key to the emergence of the tracer-filled vortices. We also investigated another possibility: that the coupling between tracer and heating played an active role in the initial stages of vortex emergence. We considered the stability of an initially horizontal layer of tracer, which is arguably the most general initial profile that can be considered. The configuration was shown to be unstable, as a result of self-reinforcement between heating and ascent at levels where the tracer concentration was decreasing with height. Furthermore, as the disturbances reached finite amplitude, they resulted in the break-up of the tracer layer and the formation of rising structures of tracer plumes. Over time, these plumes penetrated increasingly deep into the stratosphere and their horizontal length scale appeared to increase. The model in this case was two-dimensional which, as for the axisymmetric model, implied absence of any vortex isolation or vortex stripping effects. Inclusion of such effects would require a three-dimensional calculation. Nonetheless, what seems likely from our results is that tracer-filled vortices emerge as a result of a combination of various effects, which include the two-way interaction between tracer and dynamics as demonstrated in the axisymmetric and two-dimensional models, the geometry of the tracer injection into the stratosphere, and the effects of the background stratospheric flow.

As has been noted previously, evidence from studies of vortex isolation and vortex stripping suggests that a vortex is more likely to remain coherent and to isolate tracer within it if the vorticity magnitude is sufficiently large relative to external shear and strain rates. The conclusions above are that the typical relative vorticity of an ascending anticyclonic vortex, once it has formed, is

As we have noted previously, as we submitted this paper for publication, we became aware of an independent study on the dynamics of diabatically driven stratospheric anticyclones that had been submitted elsewhere

The models presented in this paper are intentionally simplified, and we conclude by discussing how these simplifications might be relaxed. One key choice was that the heating was proportional to the tracer concentration. This was based on the assumption that the heating arises primarily from short-wave absorption by wildfire aerosols, such as black carbon, sulfate aerosols, etc. Calculations by

As has been noted at various points in the paper, a simplification made here is the neglect of 3-D effects, principally the competing effects of vortex isolation and the distortion of vortices by external strain and shear, leading to stripping away of the outer layers of vortices and eventually to vortex destruction. For example, the role of vertical shear has been noted in breaking apart

Equation

In this case the variation of

In

In this case the vertical velocity

Rewriting Eq. (

Numerical solutions of the governing equations for the various dynamical models can be obtained using the numerical methods and parameter values provided in this study's main text.

No data sets were used in this article.

KS and PHH designed research, performed research, analysed solutions, and wrote the paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank the one anonymous reviewer and Bernard Legras for reviewing this study, as well as Thomas Birner for serving as editor.

Kasturi Shah was supported by a postdoctoral fellowship from the James S. McDonnell Foundation (grant no. 2021-3208).

This paper was edited by Thomas Birner and reviewed by Bernard Legras and one anonymous referee.