Two-dimensional stability of a wake vortex dipole in a stratified atmosphere.
Authors: Saulgeot Pierre , Brion Vincent , Bonne Nicolas , Dormy Emmanuel , Jacquin Laurent 
Published: March 2024

A recent study (Lee et al., 2021) has shown that contrails are the main contributor to aviation-related radiative forcing. However, the same study shows that this contribution is highly imprecise due to numerous uncertainties. Among the most important are the numerous contingencies regarding the vertical and horizontal extent of ice plumes, as well as their altitude, which may differ from the flight level of the emitting aircraft, rising to hundreds of meters. This uncertainty is largely due to its interaction with the aircraft&#8217;s dynamic wake, which, very soon after the aircraft&#8217;s passage, is reduced to two counter-rotating vortices known as wingtip vortices.These two vortices descend by induction into the atmosphere, driving the plumes to lower altitudes. However, these dynamics are influenced by atmospheric stratification, as shown in Spalart (1996). In most cases, the two wake vortices continue their descent, but certain dynamic structures are created in their vicinity by the baroclinic torque due to buoyancy, and rise to flight altitude. The wake then splits into two parts: one descending into the atmosphere and the other rising back up to, or slightly above, flight altitude. A long, rising column of fluid joins the two wakes. The plume initially trapped around the two vortices can then evolve in three different ways. Either the plume remains with the vortices well below the flight altitude, or it rises to this altitude or even higher, entrained in the secondary wake, or it is distributed between the two wakes and the column uniting them, as shown by Saulgeot et al. (2023).Among the parameters influencing these dynamics is the relationship between atmospheric stratification, quantified by the Brunt-V&#228;is&#228;l&#228; frequency N, and the characteristic time &#964;0 of the vortex dipole&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#964;0 = b0/ W0where the natural motion of the vortices is a descent at constant speed W0 caused by mutual induction. This is the reference time scale, and the initial vortex separation b0 is the reference distance. In this scale framework, the effective stratification of the vortex flow is measured by the inverse of the Froude number&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160; Fr&#8722;1 = N&#964;0.&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160;&#160; The intermediate vorticity column plays a fundamental role in the upwelling of the plume: it is the only link between the primary and secondary wakes and can therefore influence both the latter and the plume. At the end of the two-dimensional phase of wake evolution, before the onset of the Crow instability, this column can destabilize, isolating the two parts of the wake and preventing the plume from rising. This can be thought of as thermal plume jet instabilities. These are of two types: sinusoidal and varicose. In most cases, the two instabilities follow one another (see figure 1): the varicose instability appears first, then the sinusoidal instability takes over due to a higher growth rate. Nevertheless, the appearance of one or the other can be observed independently.Figure 1: Vorticity field for Fr&#8722;1 =0.6. (a) t =7.3&#964;0; (b) t =7.4&#964;0; (c) t =7.5&#964;0; (d) t =7.6&#964;0; (e) t =7.7&#964;0;&#160;(f) t =7.8&#964;0; (g) t =7.9&#964;0.