First, consider the case of a constant size particle, which is assumed to sediment vertically with a downward speed vsed > 0. The evolution of the particle's position X(t), Z(t) is then given by the following: dXdt=Ucos⁡(kX+mZ-ωt+ϕ),dZdt=Wcos⁡(kX+mZ-ωt+ϕ+δϕ)-vsed, where U and W are the amplitude of the wave in horizontal and vertical wind respectively, ω is the angular frequency, k is the horizontal wavenumber, m is the vertical wavenumber, and ϕ is the wave phase. Note that we consider a gravity wave in the midfrequency range (f ≪ ω ≪ N with f the local Coriolis frequency and N the Brunt–Väisälä frequency), so that the polarization relations e.g., impose that the wave horizontal wind perturbation is aligned with the horizontal wavenumber. The 2-D plane x–z is chosen along the direction of propagation of the wave (and is not necessarily zonal). Near the equator (f → 0), these formulas also describe equatorial Kelvin waves and x–z is then a vertical–zonal plane. For both types of waves, the polarization relations also give the following: W=-kmUandδϕ=0.

Then, there is an analytical formula for the vertical trajectory Z(t) of the particle with initial position X0, Z0: Z(t)=Wω+mvsedsin⁡ω+mvsedtcos⁡ϕ0+sin⁡ϕ01-cos⁡ω+mvsedt-vsedt+Z0, with ϕ0 = kX0 + mZ0 + ϕ. This analytical solution highlights the difference with a no-wave case, which would simply give Znowave(t) = -vsedt + Z0, but also with a Lagrangian air parcel, whose vertical position is given by the following: Zparcel(t)=Wωsin⁡(ωt)cos⁡ϕ0+[1-cos⁡(ωt)]sin⁡ϕ0+Z0.

As gravity waves in the upper troposphere mostly propagate from lower levels, their vertical group velocity is positive, which implies that their vertical phase speed cϕz = ωm is negative . In the following, we take the convention ω > 0 so that a negative vertical phase speed is associated with m < 0. Depending on the ratio between cϕz and vsed, different cases may arise. If |cϕz| ≫ vsed, the particles follow the air parcels and oscillate vertically. The particles are close to perfectly Lagrangian, and there is no net effect of the wave. If |cϕz| ≪ vsed, the particles travel in a stationary wave field, i.e., through positive and negative wave phases, whose contributions cancel out in a long enough temporal average. The interesting interaction appears when cϕz and vsed are of the same order: then, as ω + mvsed < ω and ω + mvsed > mvsed, the variations seen by the particles have longer periods than those seen by air parcels or by particles that would fall through a stationary wave field with the same vertical structure. This is due to the fact that the particles travel in the same direction as the wave phase. In particular, when cϕz ≃ -vsed (for t ≪ 1ω+mvsed), one has the following: Z(t)≃Wcos⁡ϕ0-vsedt+Z0, and thus the wave can bring a significant contribution to the displacement of the crystal, a contribution which does not cancel out after one wave period. Such a configuration could also significantly modify the lifetime of the ice crystals, which could stay longer or shorter times in saturated or supersaturated regions than if only sedimentation were moving them. Since we only consider nonbreaking waves, it should be noted, however, that the vertical wind contribution cannot overcome sedimentation in the long-term because stability requirements impose that W < cϕz ≃ vsed (see Appendix ). Furthermore, Eq. () implies a significant contribution of the wave to the motion of an individual ice crystal, but not necessarily an average effect on the ice crystal population. Indeed, if there is no preferential location of the crystals in the wave phases ϕ0 then some crystals are accelerated but others are slowed down.

For illustration, we show the effect of a monochromatic wave on the vertical motion of falling particles in a special configuration in Fig. . We assume that the particles fall at a constant speed of 2 cm s-1 (equivalent to a ≃ 20 µm diameter spherical ice particle at T = 190 K and P ≃ 120 hPa). The wave is chosen with a period T = 2πω ≃ 1 day, a shallow vertical wavelength λz = 2πm = 4 km. The chosen squared Brunt–Väisälä frequency is N‾2 = 2 × 10-4 rad2 s-2, i.e., intermediate between the stratosphere (where N‾2 = 4 × 10-4 rad2 s-2 typically but can be as large as 1 × 10-3 rad2 s2) and the troposphere (where typically N‾2 = 1 × 10-4 rad2 s-2 but can be smaller near the bottom of the TTL), as expected for the transition region of the TTL. The horizontal wavelength is prescribed from the dispersion relation: λx = 2πk = 2πN‾mω ≃ 800 km. The Eulerian temperature perturbation has an amplitude AT = 1 K, so that the vertical velocity amplitude is equal to W=gN‾2ωATT‾≃2cms-1, with g = 9.81 m s-2, T‾ = 185 K, and U = |m|kW ≃ 3.7 m s-1. Overall, the chosen wave characteristics are similar to those of equatorial waves commonly observed in radiosondes e.g.,. Although the only requirement for a significant effect on the particle's speed is that the fall speed of the particle is close to the vertical phase speed of the wave, the integrated effect on the particle's displacement will be larger for low-frequency waves, such as the one chosen in this example.

Figure  shows the initial positions of the particles (red dots) and their positions after one wave period (blue dots), so that all meteorological fields have the same value as at the beginning and the air parcels have returned to their initial positions. However, the particles have descended in altitude due to sedimentation. Without the wave, the particles would just fall to the green positions. Due to the presence of the wave, advection by the vertical and horizontal winds significantly disperses the particles vertically relative to the no-wave case. Although we did not select the wave characteristics other than the intrinsic frequency to obtain it, a significant impact can be expected. We also checked that the monochromatic wave was stable (see Appendix ).

Consistent with Eq. (), Fig.  shows that if the particles are of constant size with vsed ≃ cϕz, there is a significant impact on the motion of individual particles. However, if the particles are initialized in all phases of the wave, there is no effect on the mean downward transport of the particles' population. The main impact is to disperse the particles vertically with some particles falling more slowly but others falling more rapidly when the wave is present. Which of the two (increased or suppressed fall of the particles) will prevail in a realistic setting (i.e., including growth and sublimation)?

We now present numerical simulations of the full Eq. (). The setup is the one presented in Sect.  (constant background relative humidity and idealized wave), but without the additional simplifications in Sect. . The full equations are solved, and the dependency of the microphysical parameters on the varying background temperature, pressure and crystal size are included. We still assume that there are few ice crystals, i.e., there is no water consumption. The goal is to extend the analysis of the simplified system, to check whether the expected patterns of cloud occurrence in preferred wave phases appear and to illustrate the impact of wave advection on transport. For that purpose, a population of ice crystals with radius 5 µm is initialized in all phases of the idealized wave. They then grow or sublimate depending on the environment's relative humidity. Sublimated ice crystals are irremediably lost, while the others continue their trajectories down to the bottom of the domain.

Figure  displays the results of a simulation at RHic = 85 %, integrated during 1 day. The ice crystals are initialized with a radius of 5 µm around 17 km (16.5 to 17.5 km) in regularly gridded horizontal and vertical positions spanning all the phases of the wave. The different colors correspond to simulations accounting for wave advection or not, and will be discussed later.

Consistent with the analysis of the simplified system in Sect.  (see Fig. ), three types of ice crystal behaviors are evident in Fig.  (blue dots):

Ice crystals initialized in (highly) subsaturated regions (RHi < 70 %, black in Fig. ) quickly sublimate.

The qualitative behavior between the full ice crystal trajectories and the simplified system are thus similar. Now, is the remaining ice crystal population sensitive to the background relative humidity?

In Fig. , the results of the previous simulation (RHic = 85 %) are compared with the dry case (RHic = 63 %) after 1 day of integration, i.e., half a wave period. The end position of the crystals are shown by the blue dots. The main characteristics described in Sect.  can again be noticed, with the remaining ice crystals (about 5 % of the initialized crystals in both cases) preferentially encountered where the relative humidity is near 100 % and in the ∂T′/∂z < 0 (∂RHi/∂z > 0) phase of the wave. We also note that, in the dry case (Fig. b), there are remaining crystals after half a wave period even though those were not initialized in the region of perpetual oscillations expected from the theoretical analysis (their radius of 5 µm being too small, see Fig. c). They are, however, sufficiently close to this point to remain near the elliptic point a significant amount of time. Thus, due to the presence of the wave, those ice crystals can survive longer and remain in the TTL even though the large-scale state is subsaturated.

When the background relative humidity RHic is increased even further (e.g., RHic ≥ 100 %), the impact is similar with the remaining ice crystals near the elliptic point, which will be located at a different position in the phase such that RHi(Ψf) = 100 %. However, it should be noted that, for the ice crystals away from the elliptic point, the background relative humidity changes the relative proportion of ice crystals that sublimate versus those that quickly fall out of the TTL through the lower boundary of the domain.

In Sect. , we emphasized that ice crystals may encounter significant wave advection when they are more frequent in specific wave phases. The feedback between sedimentation and growth tends to localize the remaining ice crystals preferentially in the specific wave phase where the elliptic point is located. Hence, we can expect a mean impact on the downward velocity of the ice crystals.

To show that more precisely, we have also performed simulations without the wave advection. The crystals just fall, seeing the relative humidity perturbations created by the wave, but not the wave-induced wind perturbations. The end position of the ice crystals for the different simulations are represented in Figs.  and  by the green dots. For the low relative humidity case (RHic = 63 %), the differences in the positions of the remaining ice crystals between the simulations with and without the wave-induced wind is limited, which is expected since the phase where the elliptic point is located in that case is characterized by very small vertical wind anomalies Wcos⁡Ψe ≪ W. Yet, more crystals survive and do no sublimate in the full wind simulation than in the no-wind one (∼ 5 % vs. 1 %). In the high relative humidity setup, the differences between the full simulations and the no wind simulations already noticed for low RHi are enhanced: the downward sedimentation velocity of the crystals has been slowed down by almost a factor of 2, due to the crystals remaining in the phase Wcos⁡Ψe ≃ W. This is in striking contrast with the simulation for which the wave wind was not accounted for.

The differences seen between these 2 simulations suggest that wave advection slows down the descent of the ice crystals. Indeed, when the full wave is appropriately accounted for, the crystals' downward motion at the elliptic point happens at a (negative) vertical speed: wcfull=vsedfull+Wcos⁡Ψe=cϕz+Wcos⁡Ψe, whereas if the advection by the wave-induced wind is neglected, the crystals fall more rapidly and their downward vertical speed becomes larger (more negative), specifically wcnohor=wcnowind=cϕz<wcfull(<0). This is due to the fact that the vertical wind anomaly w′ = Wcos⁡Ψe is positive at the elliptic point. Thus, the wave advection can significantly slow down the fall of ice crystals (in our example, by a factor of nearly 2). It is interesting to note that this effect does not only come from the wave-induced vertical wind but also from the horizontal wind disturbance, as detailed in Appendix . The contribution of the wave to both wind components is thus central to entirely apprehend its impacts on ice crystals.

Overall, the experiments described above show that advection by the wave-induced wind has an impact on the sedimentation–growth of ice crystals and can significantly diminish the sedimentation mass flux, which suggests a mean impact of wave advection on the average dehydration efficiency of ice crystals. The downward water mass flux needed to close the water budget of the TTL may then be significantly affected by the waves. However, those simulations remain idealized: for instance, the described wave-driven localization is tied to the initialization of the crystals in all phases of the wave, in particular in the phase where RHi ≃ 100 %. Ice nucleation at low TTL temperatures is only active for RHi ≃ 120 % (heterogeneous nucleation) to 160 % (homogeneous nucleation) so that small-scale gravity waves superimposed to the large-scale wave would be required to bring all the air to the supersaturation threshold for nucleation. Given the number of additional assumptions and the complexity that would be required to represent them in a more complete setting, we leave those investigations in future work. Rather, to investigate the relevance of this potential effect to the atmosphere, we turn to aircraft observations in the TTL from ATTREX. In both 2013 and 2014 the Global Hawk carried a fast cloud droplet probe (FCDP) which was used to measure the size distribution and concentration of ice crystals from 1 to 50 µm diameter, and measured at about 1 Hz (180 m horizontal resolution). Temperature and pressure were provided at 1 Hz by the NASA Ames Meteorological Measurement System (MMS). Finally, of interest for this study, a microwave temperature profiler provided estimates of the local temperature lapse rate with a resolution of about 3 km along the flight track.

To evaluate the impact of the wave vertical wind on the ice mass flux, it would seem natural to use the standard altitude coordinate. In that case, the average ice mass vertical flux Ficez through a constant-altitude surface, is expressed as follows: Ficez=w′qice‾-Fsed, where qice is the ice mass concentration and Fsed the sedimentation flux, given by Fsed=∫0+∞m(D)N(D)vsed(D)dD, with m(D) the mass of an ice crystal of maximum dimension D, N(D) the number density distribution and vsed(D) the sedimentation speed. Assuming spherical ice crystals, D is the diameter and m(D) = ρiceπ6D3.

In practice, however, the method suggested by Eq. () turns out to be difficult to apply to observations, since vertical velocities are dominated by high-frequency waves whose mean impact cancels out (w′qice‾ = 0) but which add noise to the observational estimate. To overcome these issues, we use isentropic coordinates in which reversible motions are filtered out. This acknowledges that efficient, irreversible dehydration only occurs when ice crystals cross isentropic surfaces. Switching to isentropic coordinates θ on the vertical, the ice crystal vertical speed becomes vθ=vsed∂θ∂z. The wave advection impact is then hidden in the wave-induced stability fluctuations ∂θ′∂z: ∂θ∂z(x,y,z,t)=∂θ‾∂z(z)︸background+∂θ′∂z(x,y,z,t)︸wave, and this can be readily computed from observations of vertical temperature profiles and ice crystals' size, such as those from ATTREX campaign used by . The results, presented in Appendix , show generally smaller ∂θ′∂z within clouds and are consistent with , who showed that within clouds lower ∂T′∂z was found.

There is thus a systematic relation between clouds and anomalies of ∂θ′∂z. This relation suggests that there is an impact of wave advection on the ice flux. To quantify this more precisely, we use a relevant quantity for dehydration in isentropic coordinates, the cross-isentropic vertical mass flux of total water FH2Oθ: FH2Oθ=qH2Oθ˙∂θ∂z-Fsed=1∂θ∂zqH2Oθ˙-Fsed∂θ‾∂z-Fsed∂θ′∂z, with qH2O the total water mass concentration. In the first part of the equality, the term proportional to the diabatic heating rate θ˙ corresponds to air masses crossing isentropes and transporting their water vapor and water condensates. The term Fsed corresponds to the cross-isentropic flux due to sedimentation. We have here neglected eddy diffusive fluxes. Permanent dehydration due to cloud formation will be brought by this irreversible cross isentropic water mass flux FH2Oθ. The second part of the equation splits the sedimentation term into two terms, in order to emphasize the modulation introduced by the waves through their stability impact. Strictly speaking, the impact is actually more on the radiatively driven mass flux than on the sedimentation flux. However, we apply this splitting for the purpose of illustration, since a direct estimate would require further information such as the radiative heating rates within the clouds observed during the campaign. Figure  represents the sedimentation flux for the whole ATTREX campaign, as well as for the eastern and western Pacific flights separately. It can be clearly seen that the downward flux due to sedimentation is stronger over the cold, convective western Pacific than over the eastern Pacific. Furthermore, the wave advection impact is represented by the black curves for the whole campaign. The full curve corresponds to the term Fsed∂θ′∂z/∂θ∂z, which represents the wave advection impact on sedimentation. In the upper TTL, it is on average about 10 % of the mean downward flux. This influence is not very large, but still significant. The wave advection impact might hence be worth taking into account.

The frequency distribution of RHi in the upper troposphere shows two robust characteristics: (1) the common occurrence of 100 % relative humidity within clouds and (2) the high clear-sky supersaturations that get more frequent with increasing altitudes and decreasing temperatures in the TTL e.g.,. While those observations have both received explanations, the analysis presented above suggests that a new mechanism (wave-driven localization) may also contribute.

First, regarding (1), the explanation usually invoked is that the presence of ice crystals damps relative humidity variations towards saturation, by absorbing or releasing water molecules depending on the relative humidity. Using a theoretical parcel model framework, explain this by the relaxation towards a quasi-equilibrium supersaturation state, close to RHi = 100 %. In their stochastic parcel model driven by colored noise temperature variability, also found a damping of the initial mean supersaturation by the ice crystals and a stabilization of the relative humidity with small fluctuations around 100 %. In both studies, large excursions away from RHi = 100 % were prevented by the stabilizing effect brought by the presence of the ice crystals.

In the simulations presented above (Fig. ), the average relative humidity in clouds is also close to 100 %, but for a different reason. Indeed, contrary to the traditional explanations, the feedback of the cloud particles on the ambient relative humidity is not included in our simulations. Hence, ice crystals cannot regulate the vapor field towards RHi ≃ 100 %. What happens, on the contrary, is that only the ice crystals initially located near RHi ≃ 100 % remain and are constrained to follow the saturated regions due to the wave-driven localization.

Regarding point (2), it is thought that the increased occurrence of high supersaturations encountered in clear skies at low temperatures is due to the increase with decreasing temperature of the supersaturation threshold required for nucleation. These higher clear-sky supersaturations are often observed in the very cold TTL, and the absence of clouds in such conditions limits the efficiency of dehydration . In our simulations (see for example Fig. ), we also note that the amplitude of the RHi oscillations increases with altitude (whereas the temperature amplitude remains constant). The reason for these higher supersaturations at low temperatures in our simulations is the nonlinearity of the Clausius–Clapeyron relation with respect to temperature, combined with constant gravity wave temperature amplitude through the TTL. This may be seen from Eq. () (which is derived from the Clausius–Clapeyron equation): wave-induced RHi oscillations have their amplitude ΔRHi equal to ΔRHi=RHic(Z‾)βG(T‾)Zwave. In our simulations, βG(T‾) is the only factor that does depend on altitude (through the background state temperature gradient T‾(z)). Given that βG(T‾) increases with decreasing temperature (see Eq. ) and the other factors are constant, it follows that high supersaturation (and subsaturation) are more common (and larger) at low temperatures.

Of course, our simplifications and the above considerations are not strictly applicable to the Earth cirrus clouds, which behave in a more complicated manner. For instance, in the TTL, clouds characterized by RHi ≃ 100 % generally have large numbers of ice crystals , which is consistent with the vapor-quenching explanation. Furthermore, even for TTL cirrus with low ice crystal number, the assumption made here of negligible water consumption is unrealistic. Damping of the super and subsaturation by the moving ice crystals will tend to broaden the equilibrium regions of RHi ≃ 100 %. Hence, we do not claim that the quenching does not occur and lead to RHi ≃ 100 %, but it is worthwhile to note that another mechanism can lead to the same relation. Our idealized examples may thus point to a process overlooked up to now.

The relevance of wave-driven localization to cirrus clouds' evolution and dehydration efficiency depends on their microphysical properties: size distribution and crystal shape. Those points are explained below.

First, regarding the size distribution, it is important to note that our considerations are relevant for crystals whose sedimentation speed vsed is close to cϕz, which limits the usefulness to rather small ice crystals (a few tens of microns for the maximum dimension). However, the sedimentation flux Fsed depends on the fourth- or fifth-order moment of the size distribution depending on the regime in which the crystals fall, and is hence more tied to the large ice crystals. The wave impact might hence be negligible if the crystals that dominate Fsed are large compared to those that are affected by the wave-driven localization. To investigate whether this is the case, we show in Fig.  the mean size distribution of the number, mass and mass flux of ice crystals from the ATTREX FCDP, and 2DS (2-D stereo probe) cloud probes during the 2014 field campaign. The mass and sedimentation flux are computed assuming spherical particles for the FCDP bins in which, for the 2DS, the maximum dimension, area and mass measured are directly used. The sedimentation speed needed for the mass-flux computation follows the work of with the measured maximum dimension, surface area and mass used as inputs to the formulas. Small ice crystals with sizes of a few tens of microns diameter or less dominate the number distribution, with a peak around 10 µm corresponding to fall speeds around a few millimeters per second. However, larger ice crystals become more important when the mass distribution is considered, and even more for the mass flux distribution. Nevertheless, the observations suggest that in the TTL, at levels higher than θ = 360 K, the mass flux is still dominated by crystals with maximum dimension smaller than 50 µm. In that case, the wave–sedimentation interaction described above will have an influence on the dehydration efficiency, since vsed is not much larger than |cϕz|.

Second, regarding crystal shape, we have been assuming so far that the ice crystals all have spherical shapes. Although this is mostly the case in observations of crystals dimensions below 65 µm at the largest, larger crystals are clearly aspherical , which can strongly diminish their sedimentation speed : for instance, the sedimentation speed for hexagonal plates might be diminished by 40 % relative to the spherical case . This is especially important since those largest, aspherical ice crystals dominate Fsed. Hence, the sedimentation flux may be more affected by the microphysical characteristics of the crystals (their shape, etc.) than by the wave advection effect. Nevertheless, ATTREX observations do suggest that this effect is present. This might be related to the fact that, at high altitude, a large portion of the mass flux Fsed is associated with crystals below 65 µm maximum dimension (Fig. ).

Our calculations emphasize the importance of an accurate representation of the waves for cirrus modeling. In particular, the simulations presented in Sect.  show that in the presence of the wave, ice crystals can survive in regions where the background environment is subsaturated. Climate models lack the vertical resolution to represent fine-vertical-scale equatorial waves, so that the processes described in this paper are in a large part absent. The situation is less critical for operational weather prediction models, but still found that large disagreements could arise between the winds represented in those products and the observed ones.

The role of equatorial and gravity waves has long been acknowledged in Lagrangian cirrus cloud models, which generally include a parameterization of wave fluctuations e.g.,. Most of the time, however, only temperature fluctuations are accounted for, and the wave advection is ignored. The studies by and by are exceptions, and the particle-following approach remains rarely used with most Lagrangian models only following air parcels. In models purely following air parcels e.g.,, a removal time is prescribed for the falling ice and the wave-driven localization is entirely absent. Column models in isentropic coordinates, such as the one of and , partly include this effect, but they neglect horizontal wind vertical shear which can modify the wave impact on the crystals'motion (see Sect. ). By diminishing the average downward speed of the ice crystals, wave advection first limits the dehydration of the TTL, and then increases its average cloudiness by keeping surviving crystals within it. Neglecting that effect could add significant uncertainties in our understanding of the water budget of the TTL and lower stratosphere.