On the force of vertical winds in the upper atmosphere
OOn the force of vertical winds in the upper atmosphere
Arjun Berera ∗ and Daniel J. Brener † The Higgs Centre for Theoretical Physics,James Clerk Maxwell Building,Peter Guthrie Tait Road, EH9 3FD, Edinburgh. (Dated: August 10, 2020)For many decades vertical winds have been observed at high altitudes of the Earth’s atmosphere,in the mesosphere and thermosphere layers. These observations have been used with a simple onedimensional model to make estimates of possible altitude climbs by biologically sized particles deeperinto the thermosphere. A particle transport mechanism is suggested from the literature on auroralarcs, indicating that an altitude of 120 km could be reached by a nanometer sized particle whichis higher than the measured 77 km limit on the biosphere. Vertical wind observations in the uppermesophere and lower thermosphere are challenging and so we suggest that particles could reachaltitudes greater than 120 km, depending on the magnitude of the vertical wind. Applications ofthe larger vertical winds in the upper atmosphere to astrobiology and climate science are explored.
I. INTRODUCTION
Vertical winds up to 100 m/s have been observed in theupper mesosphere and thermosphere layers of the Earth’satmosphere for many decades. These vertical winds areobserved to be sustained on the order of minutes to anhour. High altitude measurements of vertical winds havelimited temporal, spatial, and vertical coverage. Groundbased measurement techniques using optical imaging arerestricted to narrow altitude ranges but have a greatertemporal coverage. In situ measurements, using rocketsto disperse trackable tracer gases, are temporally limited,but are able to sample larger altitude ranges. Reviews ofvertical winds in the thermosphere are given by Smith in1998 [1] and Larsen and Meriwether in 2012 [2].Due to the Earth’s geomagnetic field, particles in thesolar wind are channeled such that they penetrate theatmosphere in the polar regions. The charged particlescollide with the neutral air, transferring their energy andmomentum. The transfer of momentum and subsequentheating at these altitudes is understood to cause some ofthe largest observed vertical winds. The mean verticalwind in the polar regions at these altitudes is typicallyin order of a few tens of m/s. However, upward verticalwinds of up to 150 m/s have been observed [3] [4] [5]at around 240 km. At polar regions where the auroraoccur, known as the auroral zone, vertical winds havebeen observed with magnitudes around 50 m/s [6] [7][8] [9] [10] [11] [12] [13] [14] [15]. A couple of studieshave reported vertical winds >
100 m/s [3] [4] [16]. Mostof these observations are of vertical winds at altitudesaround 250 km, but some are as low as 90 km (e.g. [15])and they are usually seen during geomagnetic storms.Little theoretical progress has been made in under-standing the large vertical wind observations [2] [17] [18]. ∗ [email protected] † [email protected] Deviations from the hydrostatic balance have been stud-ied (e.g. [19], [20] [21]), but they do not last long enoughto create vertical winds sustained over several hours [2].Fully non-hydrostatical models (e.g. the Global Iono-sphere Thermosphere Model [22]) are unable to repro-duce the large vertical winds because they typically runat too low a spatial and temporal resolution [18] [23].Various measurements have shown there are particlesof a radius around the size of a micron and in reportedconcentrations of approximately 1 particle cm − in thestratosphere. [24] [25] [26] [27] [28]. Additional studiesfound that these particles include bacteria [29] [26] [30].The highest altitude that biological particles have beenfound is 77 km [31]. These were fungal spores, with aradius within an order of magnitude of a micron. Bacte-ria have been found as high as 41 km [26] [30]. However,these are likely to be underestimates, as the studies werevery limited due to the technical difficulty of growingcultures of the bacteria and fungi captured in sealed cap-sules from rockets sent to these altitudes. More recently,cosmic dust samples from the surface of the InternationalSpace Station (ISS) were found to have DNA from sev-eral kinds of bacteria which were genetically similar tothose found in the Barents and Kara seas’ coastal zones[32]. The investigators hypothesised that the wild landand marine bacteria DNA could transfer from the loweratmosphere into the ionosphere-thermosphere using theascending branch of the global electric circuit or the bac-teria found may have had a space origin.A mechanism known as gravito-photophoresis, arisingfrom irradiation of particles by sunlight, has been shownto elevate micron scale particles to altitudes of approx-imately 83 km [33]. This mechanism has been little in-vestigated with only a handful of papers examining itseffects on the upper atmosphere (e.g. [33] [34]).The existence of noctilucent clouds at altitudes of 80- 100km provides evidence that small dust-like particlescan be found there [35]. These clouds are most often ob-served closer to the poles or latitudes greater than 50 de-grees, and so are also known as polar mesospheric clouds. a r X i v : . [ phy s i c s . a o - ph ] A ug A definite source of the condensation nuclei would bedust from meteors and passing comets. These are knownto release large quantities of dust into the upper atmo-sphere. However, it is also possible that the dust particlesare terrestrial in origin. Volcanic eruptions are terres-trial events capable of significant upward thrust, project-ing ash into the stratosphere [36] [37]. Modelling of the1883 Krakatao eruption would indicate that dust fromthe volcanic ash cloud diffused up to around 85 km [38][39]. This idea is supported by observations of noctilu-cent clouds that appeared at the time of the eruption[40]. A recent study found that 50 - 100 nm sized parti-cles could be projected in volcanic eruptions to the uppermesosphere, and 10 nm to more than 120 km [41].There is a growing interest in the upper atmospherefrom the astrobiology community, but there has been noconcerted research effort placed on the mesosphere andthermosphere [42] [43]. In the last decade, work has beendone to extend global numerical weather prediction mod-els into the thermosphere (e.g. the Met Office UnifiedModel [17]) and whole atmosphere modelling studies areconsidering the role of the mesosphere and thermospherein the Earth’s climate [44] [45] [46].The issues of modelling the vertical winds motivated usto take an unconventional, yet simpler approach to mak-ing some first estimates of the maximum altitude pos-sible for a large heavy particle to be projected to fromthe middle atmosphere. The purpose of this paper isto illustrate the approximate strength of these extremewinds which has implications for transporting particleslarger than molecules higher than experiments have yetmeasured.By considering the possibility of the existence of theselarger particles at altitudes higher than previously con-sidered we open up other interesting applications in otherfields such as astrobiology and climate science which wediscuss in the penultimate section.
II. THEORYA. 1-Dimensional model
The particles we are interested in are larger and denserthan the surrounding air. Therefore any transport ofsuch particles is largely dependant on their inertial fea-tures. In describing how a particle is carried by the wind,two forces shall be considered. The first is the weight ofthe particle due to its mass. The second is the force car-ried by the momentum of the vertical wind, distributedover the surface of the particle.We wish to make first order estimates of these forceson a particle, so we will model the particle as a disk.This means that all the upward vertical wind hits theparticle at the same time. No significant part of theparticle feels more wind against itself. This assumptionis critical as it reduces the complexity of the problemto only one dimension. It is also helpful as it mirrors the most abundant measurements of vertical winds inthe upper atmosphere as instantaneous winds at discretealtitude points.Let the particle be a disc of radius r and thickness h .As illustrated in figure 1, and let it sit with its majoraxis perpendicular to the vertical wind direction. So thedensity of such a particle, ρ p can be written as ρ p = mπr h , (1)where m is the mass, r the radius and h the thickness ofthe particle. Consider a vertical wind blowing upwardswith velocity, w which impacts on the circular disc sur-face. Then the mass of the air which will impact on thebase surface of the particle in some finite time t is, M air = ρπr wt, (2)where ρ is the density of the air in a cylindrical volumeelement. FIG. 1. Particle as a disc, orientated such that the windimpacts on its major axis, pushing it upwards.
The standard model for the drag force on a particle ofmass m moving at a velocity v , is given by m dvdt = −
12 Γ ρπr v , (3)where Γ is the drag coefficient. If we now substitute ourexpression for the particles mass (equation 1) we find, dvdt = −
12 Γ ρρ p h v . (4)At an altitude of 85 km the air density, ρ ∼ − kgm − using Γ = 1, which is typical for a rough body and aparticle velocity, v = 10 ms − , the term (cid:12)(cid:12) dvdt (cid:12)(cid:12) becomes ∼ − h . If the drag is to be small, say ∼ .
01 ms − , thatwould give h > − m which is certainly possible. At150 km the air density, ρ ∼ − kgm − which using thesame procedure gives h > − m for the drag to be small,which is also not an unrealistic size of particle to find.The force due to the vertical wind is essentially thesame as that due to the drag force, but acting in theopposite direction, so the particle can be considered tobe moving with the flow of the wind after some finite time t , to allow the wind to have sufficient force (drag) on theparticle. Hence air resistance, against the direction ofthe particles motion, shall be henceforth neglected.Let the velocity of the vertical wind, (cid:126)w be upwards,perpendicular to the Earth’s surface (ˆ k ) so that (cid:126)w = w ˆ k . Assume that the vertical wind is constant with bothaltitude and time. Let the velocity of the particle movingupwards with the wind be (cid:126)v = v ˆ k . Hence the velocity ofthe wind relative to the particle is ( w − v )ˆ k . Since all themotion is restricted to one dimension, the ˆ k unit vectorsshall be dropped from now on, with upwards defined aspositive.From equation 2, the mass of the air impacting on thelower surface of the particle in some time t is M air = ρπr ( w − v ) t, (5)which gives the momentum transferred by the wind tothe particle, p = ρπr ( w − v ) t. (6)Hence the force on the particle due to the wind is givenby F wind-particle = dpdt (7)= ρπr ( w − v ) . (8)The equation for the two forces, the weight and verticalwind momentum, is therefore m dvdt = − mg + ρπr ( w − v ) . (9)As the relative velocity force term in the above equa-tion is squared, the force due to the vertical wind cannotchange sign. So in the regime where the vertical windblows downwards, meaning (cid:126)w < (cid:126)v , the equation at themoment wrongly suggests the particle will still move up-wards. Put simply, taking the limit where (cid:126)w → −∞ , theacceleration of the particle will remain in the positive ˆ k upward direction. To give the force the correct sign, theHeaviside unit step function, H is introduced as H ( w ) = (cid:40) , if w > − , if w < dvdt = − g + H ( w − v ) ρ ( z, t ) ρ p h ( w ( z, t ) − v ( t )) . (10)Physically this equation says that if the force due to thevertical upward wind is strong enough, it can overcomethe force due to the particles weight, and cause the par-ticle to accelerate upwards. To further check this equa-tion’s stability the equation is solved by separation toyield, t = 1 g (cid:90) v ( t )0 dv (cid:48) b ( w − v (cid:48) ) − t the par-ticle reaches a velocity v ( t ) having been at rest initiallyand b = Hρ ( z ) ghρ p . This integral has an analytical solution, v = w − (cid:114) gk (cid:32) (cid:113) kg w − (cid:113) kg w +1 (cid:33) exp( − t √ kg )1 − (cid:32) (cid:113) kg w − (cid:113) kg w +1 (cid:33) exp( − t √ kg ) , (12)where k = H ( w − v ) ρ ( z,t ) ρ p h . It is assumed that t is suffi-ciently small such that the air density ρ ( z ) and verticalwind speed w remain constant. In the limit that t → ∞ ,we obtain the correct steady state solution, as found inequation 14. When t = 0, we find v = 0, just as we hadhoped. This result is included for completeness.The Heaviside function is introduced to maintain somestability when working with real observations which cansometimes have cases where w <
0, but really this equa-tion is only valid for the cases where v < w and w > t to reach the steady state. Thisis when the wind just balances the weight of the particle,and at this point the particles velocity will approach, butgenerally not reach, the speed of the wind. To see this,consider the limiting cases of the parameter k when thewind is blowing upwards such that H ( w − v ) is positivein equation 12. When the parameter k is maximised theparticles velocity is closer to that of the wind. Due tothe disk geometry chosen, k is maximised when the par-ticle thickness h is minimised with respect to the densityof the particle ρ p . In other words the optimum shape isthat of a pancake where the particles mass is distributedover as large a surface area as possible. Additionally, thehigher the air density, the stronger the force of the ver-tical wind, hence the smaller the difference between thespeed of the particle and the speed of the wind.The steady state solution for equation 10, is dvdt = 0.This corresponds to, − g + k ( w ( z, t ) − v ( t )) = 0 , where k = H ( w − v ) ρ ( z, t ) ρ p h , (13)giving in the steady state, v = w − (cid:114) gk . (14)The negative square-root is selected as v < w is required,and for v > (cid:112) gk < w . This gives theminimum wind needed to get the particle to reach thesteady state, w ≥ (cid:18) gρ p hρ ( z ) (cid:19) . (15)This condition provides a means for understanding themagnitudes of the vertical wind needed to propel it up-wards and shall be referred to as the threshold velocity.We will examine three different particles to illustrate theforces of vertical winds in the literature and make furtherhypotheses. The first particle we will call our standardparticle , with general dimension of a nanometer (heightand diameter) and standard dust density 1 g / cm [47],giving it a mass of ∼ × − kg.In figure 2, the condition equation 15 for our standardparticle and two biologically defined ones is presented.We used the NRLMSISE-00 model (see [48] and [49]) byimplementing the Python module fluids and its class atmosphere.ATMOSPHERE NRLMSISE00 , to give the aver-age air density, ρ ( z ) as a function of altitude and themethod from the U.S. Standard Atmosphere 1976 is usedto determine the variation of gravity with altitude [50].The change in the vertical velocity required for such aparticle to reach steady state is exponential and thenlinear. This corresponds to the change in the air densityprofile between the mesopause and thermosphere. FIG. 2. Threshold velocity equation 15 for three differ-ent particles. Standard dust particle of density 1000 kgm − ,height and radius of a nanometer with a mass of ∼ × − kg(green). Virus sized particle of density 196 kgm − , thickness109 nm (H1N1 virus from [51]) (blue). Small bacteria or bac-teria organelle sized particle of density 2000 kgm − , height of40 nm, radius ∼ µ m and mass of 10 − kg (orange). In our model, after some short transient period, the ve-locity of the particle will become close to that of the wind,which is the threshold velocity. We now use steady statesolution, equation 14, to make some simple estimates forthe vertical distance a particle could be carried upwardsin the vertical winds reported by different observationalstudies.Measurements of ion velocities can be considered as aproxy for neutral winds below altitudes of around 105km. In [15], they used this technique to measure ver-tical winds in Greenland on two different nights at analtitude of about 103 km. As found in most other verti-cal wind data, the wind displayed oscillatory behaviour
FIG. 3. Time series of vertical ion velocities (neutral verticalwind proxy) at 103 km on September 5 (top) and 12 (bottom)2003. The dotted lines indicate 20 m/s. From [15]. where it switched between upward and downward. Thevertical winds range in magnitude between 10 - 50 m/sand on both nights there are long periods of consistentupward wind. If we consider a wind of 50 m/s for oursmall bacteria or bacteria organelle sized particle (den-sity 2000 kgm − , height of 40 nm, radius ∼ µ m, massof 10 − kg) and use the U.S Standard Atmosphere valuefor the air density at 100 km of 5 . × − kgm − [52],we find that the upward velocity of the particle is 13 m / s(note there will be a difference from the orange curvein Figure 2 as that uses a slightly different air density).Since the wind typically grows to a maximum over around20-30 minutes lets suppose then the particle on averagehas a vertical velocity of ∼ / s. This would mean thatthe particle would climb around 8 . / s will take 50 minutes to move from 140 to 240km (these are the rough altitudes at which they corre-lated the vertical winds). However the horizontal windsover this altitude range have magnitudes in the range 200- 500 m/s. They hypothesise that when the horizontalwind is weaker, there may be higher correlation in thevertical wind in the horizontal and or between these twodifferent altitudes.If we take the case of a lower magnitude horizontalwind, 200 m/s then over the 20 minutes for a particleto climb 8.4 km it will have moved ∼
240 km in the hor-izontal which is within the maximum correlation scaleobserved. If the horizontal wind magnitude is just 50m/s more then we reach this maximum observed 300 kmcorrelation length. However, it could be argued that thecorrelation measurements are representative of the largerauroral arc which would extend much further than 300km. Observations of the horizontal winds and modellingdo indicate that auroral arcs could form coherent neu-tral winds in the E emission airglow band in both thehorizontal and vertical. More observational studies arerequired to determine these coherent properties.There are many examples in the literature where wehave sustained periods of vertical winds in the lower ther-mosphere which may or may not be related to auroralarcs. The mechanisms for many of these large verticalwinds remain unknown [2]. Finally let us take these es-timates to the extreme by considering a particle withdimensions like that of the H1N1 virus which has massof around 0.8 fg and diameter of 109 nm, giving it a muchlower density than bacteria of ∼
196 kgm − [51]. Notethat for our disk model we assume that the diameter isequal to the height of the disk. The threshold velocityprofile for such a particle is plotted in 2 as the blue curve.Then consider the case where this particle is caught ina vertical wind of 50 m/s along a long auroral arc. Ifwe assume that the air density the particle experiencesbetween 110 - 120 km is constant at ∼ × − kgm − as it moves with the updraft, then the particle will movesupwards with a relative velocity of ∼ III. MOTIVATION
The purpose of this paper is to show using some simpleestimates that particles larger than the air molecules, canbe lifted in the upper atmosphere, raising the possibilityfor biological particles to projected to higher altitudesthan presently have been investigated. The estimates wepresent are just crude conjecture to an extent. However,it is all that can be estimated given the lack of observa-tions and today’s insufficient model capabilities availablewithout resorting to model forcing. Nonetheless, this re-sult has importance; we find that there is the possibil-ity of these larger particles being carried from the uppermesosphere into the thermosphere. Our simple equationfor the steady state velocity of a particle blown upwardsby the wind can be used by others to determine order ofmagnitude effects of vertical winds.By showing that it is possible for large, heavy par-ticles to reach these high altitudes, simply by verticalwind transport, interesting possibilities and questions areopened up. For example, it has been suggested that ifbiological particles can be found at a minimum altitudeof 150 km, then hypervelocity space dust, which contin-uously impacts the atmosphere, has enough momentumto facilitate the planetary escape of such particles [60]. Itwas this work that also suggested vertical winds as a pos-sible mechanism to facilitate the upward climb, and ourestimates here using reported observations of large verti-cal winds shows that it is conceivable for such particlesto be projected from near the highest measured altitudesin the mesosphere up to 120 km, which is 30 km off theminimum altitude given in the panspermia theory.The highest altitude that biological particles have beenmeasured is 77 km [31]. Using specially adapted meteo-rological rockets Imshenetsky et al. found bacterial andfungal organisms in the mesosphere between 48 and 77km. To our knowledge no further studies like this havebeen conducted since to push this biosphere boundaryfurther. The recent analysis of swabs taken from theexterior of the ISS, which has an altitude of 400 km,does strongly indicate that particles of a biological ori-gin (whole bacteria or DNA fragments) can reach deepinto the thermosphere [32]. Our estimates support thehypothesis that these results are from the Earth’s at-mosphere. Most likely in the form of DNA or organellefragments due to the size/mass constraints as the highersuch particles can be projected into the thermosphere,the more they will be effected by the ascending branchof the global electric circuit as [32] suggested.So we would suggest that biological particles can befound at altitudes higher than 77 km, especially consid-ering the long history of large vertical winds measured inthe upper mesosphere and thermosphere that occur dur-ing geomagnetic storms. The results of this paper con-tribute to the small but building body of evidence thatthe upper atmosphere should be of interest to the astrobi-ology community and that further experiment campaignsare needed.Future work could consider the atmosphere of Mars,for which this vertical winds mechanism might be moresuitable. The weight of a particle on Mars is 38% less than Earth and it was recently reported that dust stormson Mars project dust up to around 80 km [61] If biolog-ical particles can be found at these high altitudes thenthey could be sampled and studied by satellites or probeswithout having to land on the planets surface, using amethodology similar to that of [32].Our results are complementary to the ideas associatedwith gravito-photophoresis, which has applications forclimate science, as it has been suggested that particlescould be engineered to reflect sunlight and be propelledby photophoretic forces [62]. Future work could examinecombinations of thermospheric vertical winds and propul-sion by photophoretic force to determine the possibleeffects of geoengineering in the upper atmosphere. Wethink it is important to understand the possible impactsof any proposed geoengineering solutions on the upperthermosphere. This region has much more diverse chem-ical processes catalysed by the stronger solar radiationand ionospheric phenomena. So even if an engineeredparticle was to reach these high altitudes for a brief pe-riod that could be long enough to create unexpected freeradicals which over gradual build up could have unfortu-nate consequences for atmospheric composition.
IV. CONCLUSION
The observations of large vertical winds reported fordecades in the upper mesosphere and thermosphere havebeen used as the basis for developing a one dimensionalvertical transport model for particles larger and heavierthan the air itself. In the context of other observationsand modelling studies of auroral arcs, our estimates in-dicate that such particles could indeed be transporteddeeper into the thermosphere than the previously mea-sured and considered value of 77 km. We argue the casethat a nanometer sized particle could climb to an alti-tude of 120 km via vertical winds generated along au-roral arcs. We call for further field campaigns to de-termine better the horizontal distribution of these largevertical winds and particularly between 90 - 150 km, aswell as for modelling groups to consider examining three-dimensional Lagrangian coherent structures in the ther-mosphere.
ACKNOWLEDGMENT
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