Dynamics of wake vortices
Wake vortices are detected using the λ2 field, which is defined as
the second eigenvalue of the symmetric tensor S2+Ω2 with S and Ω being the
symmetric and the antisymmetric parts of the velocity gradient tensor,
respectively . The vortex cores are
characterized by negative values of λ2 with lower values
corresponding to stronger vortices. The initial minimum value of λ2
is found in the flow region inside the wake vortices, λ2,0=minx,y,z{λ2(x,y,z;0)}=-49.67s-2, which is much
smaller than the value corresponding to the vortex tubes of atmospheric
turbulence, λ2atm=-0.001s-2. The evolution
of λ2 iso-surfaces reported in Fig. illustrates the
mechanism of vortex instability for the reference case. It is known from
stability analysis that counter-rotating vortices are prone to both long-wave
(Crow) instability with wavelength λlw=8.6b and
short-wave (elliptical) instability with wavelength λsw=0.37b . For the
present study, these values are λLW=405m and
λSW=17.4m, respectively. The vortical structures
identifying the wake vortices and the background turbulence are characterized
by the values λ2=λ2,0/10 and λ2=λ2,0/400 000≈λ2atm/10, respectively. The figure
shows that turbulence triggers perturbations of the wake vortex system that
develop until the two vortex tubes collide and break up. The largest flow
structure corresponds to the size of the axial domain and can then be
identified with λLW. Starting at t=2.0min, the
vortex tubes start to get closer and eventually collide at t=2.2min at two axial locations, x=250m and x=350m. In the other simulations, a single collision between the
vortex tubes is observed, see Fig. . It is also interesting
to observe the emergence of another flow structure with wavelength of about
25 m starting at t=1min, which can be identified with a
short-wave instability. Although this has a higher growth rate
than the long-wave instability, in the
present case it is not sufficiently strong to overwhelm it and break the
vortices before the latter develops. Elliptical instability in combination
with Crow instability have been studied theoretically
e.g.,, experimentally
e.g.,, and numerically
e.g.,. Their emergence depends on
characteristics of the vortex system (vortex core radius and spacing) and on
the initial perturbation of the vortex system. Typical perturbations include
sinusoidal waves corresponding to the excited instability modes
,
broadband white noise, and spectral perturbations satisfying model spectra of
isotropic or anisotropic turbulence. The latter approach has been used in the
wake vortex literature and particularly in the studies that attempted to
analyze the interaction of the wake with the background turbulence
and more recently by who focused on the
identification of vortex flow topology, and by who
analyzed passive scalar transport in addition to the fine-scale structures of
the wake. The main difference with the present study is that here the
background turbulence is sustained, yet the three-dimensional snapshots shown in these
studies as well as the various pictures of wake instabilities reported in the
literature , , and
corroborate the structures visualized in
Fig. . Isometric views of iso-surfaces λ2=λ2,0/10 and λ2=λ2,0/1000 are drawn in
Fig. for cases 1, 2, 3, and 6 taken immediately before and
after the vortex tubes collide. The snapshots are similar for the three
cases except for the double collision appearing in case 2, which ensures the
reproducibility of these flow structures by the present LES. In general, the
vortex breakup always starts in the lowermost part of the wake because the
wake descent is inversely proportional to the vortex separation (which goes
to 0 at the collision point).
Iso-contours λ2=λ2,0/10 (black) and
λ2=λ2,0/1000 (colored) taken immediately before
(left panels) and after (right panels) the vortex tubes collide. Black
iso-contours represent the wake vortices, while colored
iso-contours represent the secondary vortices.
Time histories of vortex length ratios xv/Lx and λ2/λ2,0.
for cases 1, 2, 3, and 6. The horizontal dashed lines for each case indicate the
threshold values λ2=λ2,0/10 that is used to discriminate the
wake vortex tubes from the background turbulence. The two vertical dashed lines
correspond to the times selected in the
iso-contours of Fig. (i.e., immediately before and after the vortex breakup) and
define the lifespans of the wake vortices.
In order to measure the lifespan of wake vortices, their intensities are
evaluated as a function of time: the minimum value of λ2 of each
vortex is measured along the longitudinal axis. The condition λ2<λ2,0/10 is used to discriminate wake vortices from secondary
vortices. Noticing on Fig. that this condition may not be
reached on some parts of the longitudinal axis, we define the vortex length
xv as the length of the longitudinal axis range covering all
axial sections where this condition is verified. The length ratio
xv/Lx and the average vortex intensity λ¯2 are
plotted in Fig. for each vortex and for cases 1,2,3, and
6. The magnitude of λ¯2 quickly decreases at the beginning before
its rate of variation stabilizes at a constant value of -0.1|λ2,0|
per minute. The ratio xv/Lx is initially equal to 1, meaning
that coherent vortices can be detectable over all the axial domain. After a
couple of minutes, the magnitude of λ¯2 decreases abruptly and
falls below -|λ2,0|/10 within a few seconds. In the meantime,
xv decreases and reaches 0 within the same time span. The
abrupt slope change is the mark of the vortex breakup as shown by snapshots
of Fig. , which are taken at the times indicated by the
vertical dashed lines in Fig. . The wake lifespan
tb is defined by the abrupt slope change and values are given in
Table . The results obtained for all cases studied are
compared in Fig. to the analytical estimation of
and contrasted with a limited set of measurements
and numerical data found in the literature
. In spite of the large
scatter of data (even with similar turbulence level), the numerical values of
tb are in the range of experimental values. All collected data
show a similar trend: weaker relative turbulence leads to longer lifespan (as
observed for example by ). Similar
trends were recently reported using the normalized Brunt–Väisälä
frequency N*=Nt0 and the vortex sinking distance
zb=w0×tb instead of the vortex lifespan
tb
,
and Fig. shows a general good agreement with those data.
Lifespans of wake vortices as a function of relative turbulence
intensity. The solid line is the analytical estimation by
, the dashed line outlines lifespans
obtained by numerical simulations of
, and black crosses are
in situ measurements .
Simulations: case 1, case 2,
case 3, and case 6.
Non-dimensional maximum descent zb/b of wake vortices as a function
of non-dimensional Brunt–Väisälä frequency N×t0 for the present simulations and a
collection of experimental and numerical data
(adapted from ). Simulations: case 1, case 2,
case 3, and case 6. Note that zb/b=w0×tb/b=tb/t0
with t0=2πb02/Γ. In all cases considered in this study, t0≃25s and N×t0≃0.3 (see Table).
In order to evaluate the vertical displacement of the wake vortices, vertical
profiles λ̃2(z,t)=minx,y{λ2(x,y,z;t)} are
taken at different times and shown in Fig. . They descend
roughly 100 m until t=1min, in agreement with previous
observational studies . The minimum values of
λ2 are coherent with values found in Fig. and
indicate the region containing the vortices. The vertical spread of this
region increases at a rate of 20mmin-1, which is due to the
development of instabilities discussed in Fig. . We observed in
Fig. that the vertical displacement increases in the regions
where the vortex separation goes to 0, this phenomenon is observed again
in Fig. as the vertical spreading has accelerated at t=2min in cases 1, 3, and 6. Besides, the minimum value of
λ̃2 has increased towards 0 in cases 1 and 3 in accordance with
Fig. . At the end of the vortex regime, the primary wake is
found between 200 and 300m below flight level.
Vertical profiles of λ̃2/λ2,0 that is used
to track the wake vortices as they move downwards and
break up. The four profiles shown for each case correspond to
t=0.5min,
t=1min,
t=1.5min,
t=2min.
Contrail microphysics
Snapshots of ice crystals spatial distribution for cases 1,
2, and 3. Crystals are colored with diameters. The figure shows
the development of the secondary wake at t=1.5min and 3min
(center and right panels) and the formation of puffs at t=3min (right panels).
It can be observed that the size of crystals slightly increases during the instability process of the
vortex regime (t≤1.5min), whereas it increases considerably in the dissipation regime (t=3min)
with larger crystals found in the secondary wake.
Crystals appear more mixed in the strong atmospheric turbulence case
(top panels).
Vertical profiles of normalized number ñ(z)/ntot (left panels) and mass M̃i(z)/Mv,0 (right panels) of ice crystals.
The four profiles shown for each case correspond to
t=0.5 min,
t=1 min,
t=1.5 min,
t=2 min.
As explained in Sect. , vapor deposits on ice crystals at a
rate that depends on the local ice supersaturation. This process is
represented in Fig. , which shows the spatial
distribution of ice crystals colored with diameter dp for
different levels of atmospheric turbulence. In the early stages of the vortex
regime (until t=0.85min), the crystals are distributed in the
primary wake with diameters ranging between 2 and
6 µm. The vortex cores are visible and resemble pictures of
vortex tubes see for example. Note that
regions of low crystal density (i.e., far from the cores) contain larger
crystals because, for given ambient supersaturation, the same amount of
ambient water vapor has to be shared among less crystals. At the end of the
vortex regime (t=1.5min), the vortex tubes are still visible and
the secondary wake has started to form. Crystals remain relatively small in
the primary wake while they grow mostly in the secondary wake where they are
exposed to a supersaturated environment. The presence of atmospheric
turbulence folds the structure of the secondary wake and the peripheral
regions of the primary wake. At t=3min, the primary wake is
rearranged into a puff as a consequence of the turbulent dissipation that
results from the breakup (named induced turbulence hereafter). The figure
shows that turbulence increases the mixing of the contrail with ambient air
and favors the development of Crow instability. The vertical extension
reduces with the turbulence intensity because the vortices tend to
destabilize and break up earlier in the case of strong turbulence compared to
weak turbulence. This is further confirmed by the vertical profiles of ice
crystal number and mass reported in Fig. taken at the
same time as in Fig. .
Normalized contrail volume per unit flight distance. Note the increased mixing following the
vortex breakup where the exhaust material is released into the atmosphere.
Contrail diffusion is analyzed in Fig. that
shows the evolution of the contrail volume per unit flight distance
Vp normalized by the initial volume Vp,0.
The contrail volume is computed as the sum of volume of the grid cells
containing at least one crystal and defined over a regular mesh with
1 m of resolution. Up to t=1min the volume
expansion is similar for the three turbulence levels. At approximately
t=tb the volume expands faster with stronger
turbulence, as also observed in snapshots of
Fig. . It is interesting to note that the
expansion rate is the same for all cases at the end of the simulation,
even for case 6 (T0=215K). This means that the wake
turbulence is the main contributor to contrail diffusion in the
dissipation regime (even though atmospheric turbulence is expected to
become predominant as wake turbulence dissipates).
Ice mass per unit flight distance normalized by the mass of
emitted water vapor.
Adiabatic compression reduces Mi at the
end of the vortex regime, and this is particularly effective when the
atmosphere is weakly supersaturated (case 4). The breakup of
the vortices causes Mi to increase at a rate that
depends on atmospheric temperature, saturation, and, to a much lesser extent,
turbulence.
Vertical profiles of normalized mass M̃i(z)/Mv,0. Left panel: profiles at
different wakes ages in the dissipation regime for case 2. Right panel: comparison at t=4.5min between case 2 (s0=1.3) and case 4 (s0=1.1).
Figure shows the evolution of ice mass per meter of flight
Mi normalized by the mass of vapor emitted by the engines
Mv,0=15gm-1. The mass Mi is obtained
using Eq. ():
Mi=1Lz∑pncmp=1Lz∑pncπ6dp3ρice.
At the beginning of the vortex regime, the emitted water vapor
Mv,0 has been completely deposed on ice crystals. The ambient
vapor entrained in the plume during the first 10 s (see, e.g.,
) also contribute to the overall ice mass so that
initially Mi>Mv,0. When ambient air is subsaturated
(case 5), the ice mass decreases exponentially, consistently with
observations of non-persistent contrails. When ambient air is supersaturated
(all but case 5), the ice mass reaches a maximum at t=25s. This
maximum indicates that all available vapor (exhaust vapor plus atmospheric
vapor trapped in the wake) has been converted into ice so that s=1 inside
the contrail. As vapor density in the atmosphere scales with s0, this
maximum is higher when the atmosphere is more supersaturated. In cases 2, 3,
4, and 6, the ice mass decreases from t=1min to the end of the
vortex regime. This decrease indicates the sublimation of crystals, explained
by the process of adiabatic compression occurring in the primary wake
. However, the mixing of the
secondary wake with ambient air partly compensates sublimation so that the
ice mass reduction is less pronounced when increasing the turbulence
intensity, and even completely compensates sublimation in case 1. In the
dissipation regime, the ice mass grows again as the consequence of induced
turbulence. At t=4min, Mi reaches
2Mv,0 when s0=1.10 (case 4), 4.5Mv,0 when
T0=215K (case 6), and 8.5Mv,0 when s0=1.30,
T0=218K, regardless of the turbulence level. The less-pronounced increase of ice mass for lower ambient temperature can be
understood by observing that ambient saturation is kept constant between
cases 2 and 6. Hence, the density of water vapor ρv=sρvs,i(T) decreases when decreasing T (since
ρvs,i is a monotonic function of
temperature). In case 6, the mass of vapor entraining in the contrail and
condensing into ice is then reduced compared to case 2 as shown in
Fig. .
The vertical structure of the contrail is further analyzed in
Fig. in the dissipation regime. While the “curtain”
connecting the primary and secondary wakes forms as soon as the vortex pair
start the descent, ice tends to substantially accumulate in the secondary
wake after the breakup until, at t=4.5min, the two peaks are
approximately the same for case 2 (s0=1.3). For case 4 (s0=1.1) most of
ice crystals in the primary wake sublimated so that only the peak in the
secondary wake is apparent at the end of the dissipation regime. (Note these
results closely resemble those obtained recently by
).
Normalized number of activated particles (fraction of surviving
crystals). Adiabatic compression is
strong enough in weakly supersaturated atmosphere (case 4) to
completely sublimate 30% of crystals, but it is not
able to evaporate any crystals in strongly supersaturated
atmosphere.
The number of particles surviving the adiabatic compression is
an important parameter to consider when evaluating the global and climate impact
of contrail
e.g.,.
Figure shows the fraction of surviving
crystals, and their values at the end of the simulation are summarized in
Table for all considered cases. These data indicate that
almost all crystals survive when s0=1.30 and 75% survive
when s0=1.10, which is higher than the results obtained by
and recently by
and . As
discussed in Sect. , this can be caused by the different
microphysical setup (cf. inclusion of Kelvin effect) and/or the mixing
efficiency predicted by the two models, especially in the peripheral region
of the primary wake in Fig. .
Ratio of deposed mass. In the vortex regime, the contrail
is close to equilibrium (Mi≈Me) only
when the atmosphere is strongly supersaturated. In the dissipation regime,
the strong mixing between the contrail and ambient air causes a depart
from the equilibrium state (Mi<Me).
It is interesting to evaluate the mass of ice that would be formed by
a model that would enforce equilibrium between ice and vapor phase at
each time step. This kind of models may be attractive as they are less
computationally expensive. The equilibrium ice mass
Me is defined by
Me=Mi+Mv,awithMv,a=∫Vp(ρv-ρvs,i)dVp
and represents the ice mass of ice if all available vapor were
instantaneously deposed onto crystals (note that Mv,a is not the
available mass of vapor in the “true” situation).
Figure shows the ratio of deposed mass
Mi/Me. At t=25s, the ratio is close to
1 consistently with the equilibrium state suggested earlier in this
section. The competition between ice sublimation due to the adiabatic
compression and ice deposition by mixing of the secondary wake is seen again
here at the end of the vortex regime. Turbulence favors mixing and
entrainment of supersaturated ambient air into the plume. This in turn
increases the ice deposition rate, which scales with the local
supersaturation or Yv(xp) (the amount of vapor
available at particle position). When turbulence is strong (case 1)
deposition is stronger than sublimation as Mi/Me≲1, whereas when turbulence is weak (case 3) sublimation is stronger
than deposition as Mi/Me≳1. When
supersaturation is reduced to s0=1.10 (case 4), sublimation is much
stronger than deposition and the equilibrium mass Me approaches
0 (while Mi/Me diverges so the assumption of
equilibrium is not valid for s0=1.10 ). In the dissipation regime,
the ratio Mi/Me reduces due to the mixing produced by
the induced turbulence and reaches a constant value of 0.7 when s0=1.30 (regardless turbulence intensity or background temperature) and 0.45
when s0=1.10. This result can be explained with an estimation of the
rate of change of vapor in the contrail. On the one hand, the quantity of vapor
is increased by mixing at a rate of Qm=s0ρvs,i(T0)dVp/dt, neglecting temperature variations in the contrail vicinity.
On the other hand, the quantity of vapor is decreased by deposition at a rate
of Qd=ndmp/dt=Cn(s-1)ρvs,i(T0), where C depends on the mean
particles radius and can be assumed constant between 1 and 2 min as the ice
mass (see Fig. ). The temperature dependence of
G(Knp)Dv,p in Eq. () can
reasonably be neglected in this context. Figure
shows that theses rates are balanced at the end of the simulation. When T0
is reduced, both Qm and Qd are decreased by the same
amount, which does not change the balance (although the time needed to reach
this balance increases). When s0 is reduced by 15% from 1.30
to 1.10, Qm is reduced by 15% while Qd
is reduced (through n) by 25%, and the balance is then changed
towards higher amounts of vapor in the contrail. A potential drawback in the
definition of Mv,a in Eq. () is that it depends on
the definition of Vp, which may not be suitable to evaluate the
non-equilibrium in the actual contrail. To that end, we calculated the mean
saturation ratio s‾ by means of an ensemble average
s(xp) over all ice particles. Its evolution is shown in
Fig. . In the middle of the vortex phase between 1 and
2 min, thermodynamic conditions are very close to equilibrium (relative
humidity is slightly less than 100%) because of the sublimation
due to adiabatic heating balancing the deposition due to the entrainment of
fresh ambient vapor (similar levels of relative humidity were observed for
example by , their Fig. 8). In the dissipation
phase humidity is greater than 100% as the ice crystals
originally trapped in the vortices are fully exposed to ambient vapor
although it does not exceed 105% for the conditions of this
study.
Mean saturation ratio computed as an ensemble average over all ice particles. The contrails is slightly subsaturated during the initial vortex descent between 1 and 2 min and supersaturated in the dissipation phase after 2 min.
Mean optical thickness. The horizontal black line
represent the visibility criterion δ¯>0.03
. The mean optical thickness decreases
during the vortex regime and stabilizes during the dissipation
regime. Its evolution depends mainly on atmospheric temperature,
saturation, and, to a lesser extent, turbulence. The symbols on
the right of the figure shows the values of the optical thickness for different aircraft
and atmospheric situations measured in the CONCERT campaign
as reported by .
The optical thickness of the contrail δ¯ is shown in
Fig. . It is evaluated by first
computing the optical thickness δ(Sxy) over each
column Sxy=[x,x+Δx[×[y,y+Δy[×Lz and averaging δ over the regions where
δ(Sxy)>0. The sunlight is assimilated to a monochromatic wave of
wavelength λ0=550nm, the refractive index of
water is μ0=1.31, and the extinction coefficient Q is
approximated by the anomalous diffraction theory
:
Q(ρ)=2-4ρsinρ+4ρ2(1-cosρ),
where ρ≡2π(μ0-1)dp/λ0.
Over a column Sxy, the
optical thickness is computed as follows
δ(Sxy)=1ΔxΔy∑xp∈Sxyncπ4dp2Q(ρ).
In Fig. , when ambient air is not
saturated (case 5), δ¯ decreases exponentially and, by 2.5 min, it falls
below the threshold δ¯<0.03 used by
as a visibility criterion.
Although the visibility of a contrail does depend not only on the optical depth but on
many other parameters (angle between observation and sun, contrast against background,
aerosols between observer and contrail, etc.), the simplistic treatment used here is in line
with those employed in previous numerical simulations of contrails .
When ambient air
is supersaturated, δ¯ increases by 25% of the
initial value by the time the contrail reaches the equilibrium state.
Afterwards, δ¯ decreases by the end of the vortex regime,
which is due to the microphysical processes mentioned above combined
with the dilution of the contrail that reduces the number density of
ice crystals and thus δ¯ in every cases. In case 4 (s0=1.10), the more pronounced sublimation results in a
stronger reduction of δ¯. In the early stages of the
dissipation regime, the mean optical thickness has larger fluctuations
that are possibly due to the induced wake turbulence and to the form of the
extinction coefficient that is an oscillating function of the argument (in particular
the ice crystal diameter dp).
As the contrail expands and diffuses, these oscillations are damped
and the optical thickness attains a value of around 0.06 when s0=1.10 (case 4), 0.18 when T0=215K
(case 6), and 0.22 when s0=1.30, T0=218K, regardless of the turbulence level. Mean contrail optical thickness
of the order of 0.2 to 0.3 were reported by and
for the CONCERT campaign and similar aircraft.
also reported qualitatively similar
results with the EULAG-LCM LES model and
the CoCiP model , i.e., 0.05<δ¯<0.1 in weakly supersaturated atmospheres and 0.1<δ¯<1 in strongly supersaturated atmospheres. Optical
thickness is lower when s0 or T0 is reduced as they both reduce
density of water in the atmosphere.
Crystal diameter distributions at 5 s,
1 min, 3 min for the three levels of saturation
and measurements from (bottom).
Figure compares the particle size distributions with
those obtained by in situ measurements .
Note that a computed size distribution represents an average over the whole
contrail whereas measurements are done locally. This may lead to
significantly different values where the contrail is highly inhomogeneous
such as in the peripheral regions of the contrail with low densities and
large crystals. Nevertheless, Fig. shows many common
properties between simulations and measurements: the distributions have
log-normal shapes which broadens and reduces in density as time advances.
When s0=1.30 (case 2) the distribution readily shifts towards larger
sizes, whereas when s0=1.10 (case 4) the distribution does not shift. At
t=3min, “sublimation tail” appears in the size distribution
containing small ice crystals (with diameter less than 0.1 µm) that
are prone to sublimate as the result of adiabatic compression. When s0=0.95 the sublimation tail appears as soon as t=1min, and the
distribution decreases more quickly and shifts towards smaller sizes, showing
that the contrail sublimates as a whole. Measurements have smaller sizes and
larger densities compared to simulations. This difference may be due not only to the
different conditions encountered in the measurements and those used in the
present computations but also to the uncertainty in the number of
nucleation sites as the same mass of water has to be shared on a different
number of crystals. These data were obtained from different campaigns or
independent flight measurements where the ambient conditions and the
characteristics of the aircraft generating the contrail could be
substantially different from those considered in this study. In more recent
campaigns, carried out time-averaged
measurements to reduce effects of contrail heterogeneity and obtained much
broader distributions. Compared to the present results, the number of large
crystals (dp>2µm) is equivalent, but they measured
an important number of crystals smaller (dp<2µm)
than the present model. This could be due to the limited number of numerical
particles and should lead to an underestimation of δ¯. However, the
coefficients of the summation in Eq. () scale with
dp2, which mitigates the effect of missing the smallest
particles on the mean optical thickness. Despite these differences,
Fig. shows a reasonable agreement in terms of order of
magnitude and shape of the crystal diameter distribution.
Top panel: normalized number of activated particles
(fraction of surviving crystals). Bottom panel: mean optical
thickness. Plain lines indicates simulations with the Kelvin
effect activated.