Introduction
Clouds play a crucial role in the thermal balance and the
hydrologic cycle of the earth. Our understanding of the formation and
evolution of clouds, however, is far from making reliable predictions. The
difficulty is partially rooted in the turbulent nature of clouds due to the
wide range of scales involved in the process . In recent
years, there has been increasing awareness of the importance of turbulence on
clouds . The study of cloud–turbulence
interaction is difficult since both laboratory experiments and numerical
simulations cannot reproduce all the important physical and thermodynamical
parameters involved in real-world clouds. Observations in natural clouds are
therefore an irreplaceable aspect of the investigation. Most of these
observations are carried out with airborne campaigns that are limited in
resolution by the flying speed of the aircraft, even though significant
progress has been achieved in recent years by using slowly flying
instruments to improve spatial resolution .
Ground-based measurements at high-altitude research stations therefore play a
complementary role. In this work, we consider the Umweltforschungsstation
Schneefernerhaus (UFS) located just below the peak of Zugspitze in the German
Alps, at a height of 2650 m. Several short campaigns on cloud research
used UFS/Zugspitze as the measurement site . Here, by
using the meteorological data (wind speed, direction, temperature, humidity,
visibility, etc.) collected by the German Weather Service (DWD) from 2000 to
2012 and turbulence measurements recorded by multiple ultrasonic sensors
(sampled at 10 Hz) in 2010, we show that UFS is a well-suited station
for cloud–turbulence research. Note that our work is not concerned with
the climatology or mountain meteorology at Zugspitze, which are important
problems by themselves and require different methods of study, e.g., using
data from longer periods. Rather, by analyzing data recorded at UFS in the
same way as conventionally done for laboratory flows, we show that both the
turbulence and the cloud properties at UFS carry similar characteristics to
those in other well-studied turbulent flows and in airborne clouds. Our results
presented here can also serve as a benchmark characterization of the
turbulence and cloud physics conditions at UFS, which can be used for other
researchers who are interested in carrying out related studies at UFS to
evaluate the usefulness of the research station for their own investigation.
In this paper, we present the flow conditions and large-scale turbulence
at UFS. The small-scale turbulence will be characterized in an accompanying
paper . Here, the term “large scale” or “forcing
scale” corresponds to the range of the local peak in the energy spectrum and
is related to the scale at which the kinetic energy is supplied into the
turbulent motion; “small scale” or “dissipation scale” refers
to the range at which the viscous dissipation converts the kinetic energy
into heat. In-between lies the “inertial (sub)range”, which is the range of
scales where the kinetic energy is cascaded down to smaller and smaller
scales without significant loss. This nomenclature is conventional in the
fluid mechanics community (see, e.g., the
following textbooks: )
and is also widely adopted in other related communities, e.g., in
astrophysics and in atmospheric
science . In meteorology sometimes the entire range of
scales where turbulence dynamics are important is defined simply as
“microscale turbulence”, in contrast with the “macroscale turbulence” of the
large-scale synoptic flows . We will discuss in more
detail these different terminologies later in relation to the energy spectrum
measured at UFS.
Flow conditions at UFS
PDFs of the horizontal wind direction (left panel) and the wind
speed (right panel). In the plot of the horizontal wind direction, 0∘
corresponds to the wind coming from the north, and 90 and 270∘
correspond to the wind from the east and the west, respectively. The
condition for “clouds” is that the relative humidity be above 99 %.
Fraction of time during which the UFS is covered by clouds during any
month of the year. The red curve is the average cloud fraction, and the green
curves are the maximum and minimum monthly values over the period of data
analyzed. From the top to the bottom, the three panels are the total fraction
of cloud covering, the fraction of clouds from the east and the fraction of
clouds from the west.
Average fraction of time that clouds cover UFS at a given time of the
day (local time at UFS) in summer (top) and in winter (bottom).
Figure shows the local weather conditions at UFS,
including temperature, relative humidity, solar radiation and visibility,
using the hourly data from the Deutscher Wetterdienst (DWD, German Weather
Service) recorded between 2000 and 2012. The average temperature at UFS
during that period, -1.5 ∘C, was higher than the average
temperature at the peak of Zugspitze during the standard reference period.
The hourly average temperature at UFS varies between -10 and
+6 ∘C throughout the year. The average relative humidity is
also nearly the same throughout the year, but with much larger fluctuations
in wintertime. As a check, we analyzed the 1 min data between 2008 and
2012, also recorded by DWD. The results are the same as those from the hourly
data over the same period. In the presentation below, the term “DWD data”
refers to the recording over the longer period, i.e., the hourly data between
2000 and 2012.
We now turn to the flow and turbulence conditions at UFS, which is the main
focus of this paper. To see the possible relation between clouds and the
flow conditions, we define the “cloud” events as when the relative
humidity is above 99 %. We varied this condition to when the relative
humidity is above 95 %, and to include other conditions such as requiring the
temperature to be above 0 ∘C or the visibility to be below 200 m.
The statistics obtained are virtually the same. Figure
shows the probability density functions (PDFs) of the horizontal wind
direction and the wind speed under both the no-clouds and clouds conditions.
Note that we use the word “wind” to merely refer to air flow. Due to
the local topography, the winds at UFS are primarily in the east–west
direction. This dominance of a preferred wind direction is very convenient
for fixed instruments, such as a hot-wire and particle size analyzer (see also
the accompanying paper, ). The variations of the wind
direction and the PDF of wind speed are different for wind from the west and
from the east, which is most likely due to the different topography on the
two sides. Winds from the west have to pass over the mountain ridge before
reaching UFS and hence are generally more intense and spread over wider
angles. The east winds usually flow along the valley and are mostly free from
the effect of the mountain. In comparison, the flow conditions (wind
direction and wind speed) are almost independent of whether the wind is
carrying clouds or not.
As a way to quantify the difference between the wind from the east and
from the west, we fitted the wind speed PDFs shown in Fig. as Weibull distributions. The shape parameters of
the fitted Weibull distribution are 1.50, 1.48, 1.33 and 1.33 for
wind from the east with clouds, wind from the east without clouds, wind from
the west with clouds and wind from the west without clouds, respectively. As
expected the shape parameter depends only on whether the wind is from the
east or from the west, and it is not sensitive to whether there are clouds or
not. In all cases the shape parameter of the fitted Weibull distribution is
smaller than 2, which signals a wide distribution of wind speed. The shape
parameter for wind from the east is slightly larger, which is consistent with
the local topology; i.e., wind from the east is coming from the valley, with
less influence by the mountain, while wind from the west is coming over the
ridge, from the wind hole and moves along the mountain before reaching the
measurement site.
As we are interested in the events of clouds covering UFS, we checked the
time fraction of cloud covering at a given month of the year.
Figure shows that it is more likely to observe
clouds in the “summer” (from April to September) than in the “winter”
(from October to March). This is especially true for clouds from the east
because these clouds are almost exclusively formed from the convection rising
from the valley, which correlates with solar radiation. Clouds from the west
can be carried by the dominating westerlies, as stated in Sect. , and therefore do not show a simple dependence on
the season. On the other hand, there is still a pronounced peak around July.
Overall, the probability that the UFS is covered in clouds is more than 25 %
in the summer, with a peak of nearly 30 % in July. In the summer, the
probability of having clouds from the east and the west is approximately 10
and 15 %, respectively. Therefore, the UFS offers a good possibility to
compare these two types of clouds and the associated turbulence.
To evaluate the chance of measuring clouds, we also checked the cloud
fraction during the time of the day in summer and in winter, as shown in Fig. . In the summertime the clouds are
most likely to occur during the late afternoon and later in the night, which
reflects the fact that the summer clouds are usually formed from convection
originated in the local valley. During the wintertime the probability of
cloud occurrence is independent of the time of day, suggesting that cloud
cover is more associated with synoptic-scale weather.
The ultrasonic wind sensors installed on the roof of UFS. The five
ultrasonic sensors are labeled with numbers from 0 to 4. The top sensor
(number 0) is approximately 6 m above the roof.
Large-scale turbulence
To measure large-scale turbulence, we installed five ultrasonic sensors on a
mast located on the roof of the round tower of the UFS
(Fig. ). The ultrasonic sensors are manufactured by Thies
Clima (Göttingen, Germany), and each of them measures the full wind velocity
vector in three dimensions at a sampling frequency of 10 Hz. The five
sensors are arranged in a configuration that forms two tetrahedrons sharing
one face (Fig. ). The analysis shown here is mainly from the
measurements by the top sensor, which is approximately 6 m above the
roof and 20 m away from the mountain. The wind measured by this sensor
is least influenced by the mast itself and other sensors.
Figure shows the spectrum of the horizontal wind velocity
in the east–west direction (solid line), measured from the 1 min DWD
data between January 2008 and July 2012 (circles), with our sonic sensor at
10 Hz between September 2010 and September 2011 (crosses). The sonic
sensor spectrum shown was obtained by averaging the spectra from records of
every 2 h and hence only extends between 0.5 h-1 and 5 Hz.
The two spectra agree very well in the frequency range covered by both data
records, i.e., between 0.5 h-1 and 0.5 min-1. The
peak at f=1.15×10-5 Hz in the spectrum corresponds to the
diurnal forcing. There is a noticeable “knee” at a frequency range of about 1 h-1, which can be made clearly visible from the compensated spectrum f⋅E(f) shown in the bottom panel of Fig. . This
is the so-called “spectral gap” and is known to exist in spectra measured
in many atmospheric flows, under vastly different geographic
conditions . The spectral gap is commonly considered as the
separation between the timescales of the synoptic flow and the local
“microscale turbulence” that bears more universal features as assumed
by the Kolmorogov hypotheses . In detailed turbulence studies such
as we are concerned with here, it is customary to divide this range into
large scales, small scales and the “inertial range” in between. The
large scales are where the energy is supplied into the turbulent motion,
which may be loosely related to the peak after the spectral gap in the
compensated spectrum, i.e., at timescales of roughly 1 to 10 min. The
small scales are at much faster timescales or much smaller length scales
that are dominated by viscous dissipation, which cannot be resolved from the
sonic sensor measurements.
To access the turbulence properties at faster timescales, we added the
spectrum from a hot-wire anemometer measurement sampled at 1 kHz, which was
conducted in August 2009 (triangles). The hot-wire spectrum overlaps well
with the spectrum from the sonic sensor in the range of frequency between
10-2 and 10-1 Hz. The deviation of the sonic sensor spectrum is
most likely due to its limited resolution in wind velocity measurement. The
composited spectrum from all three spectra shows that, for flows at UFS, the
spectrum at 10-2⪅f⪅102 Hz indeed follows the
Kolmogorov spectrum Euu(f)∝f-5/3, which is shown by the
dash-dotted lines in both panels of Fig. . Note that it
gives fEuu(f)∝f-2/3 in the compensated plot. As suggested
by Fig. , in our following analysis of the turbulence, we
divide our records to segments that are no longer than 10 min.
The composite spectrum of the east–west horizontal wind
velocity, including data from the 1 min DWD recording between January
2008 and July 2012 (circles), measured from our sonic sensor sampling at
10 Hz between September 2010 and September 2011 (crosses), and from a
one-component hot-wire anemometry sampled at 1 kHz in August 2009
(triangles). The thick solid line is a smoothed composite spectrum using all
three spectra mentioned above. Note that, due to the limited resolution, the
10 Hz sonic sensor data deviate from the hot-wire data at f≳100 Hz and are not used beyond that. The Kolmogorov Euu(f)∝f-5/3
spectrum was shown for comparison (dash-dotted line). Note that it gives fEuu(f)∝f-2/3 in the compensated plot in the bottom panel. The
vertical dotted line marks the frequency corresponding to a 1 h period. The
inset of the top panel shows the local slope dlnE/dlnf of the
smoothed spectrum, from which the scaling range can be identified. The
bottom panel shows the compensated spectrum f⋅Euu(f), in which the
so-called “spectral gap” at f≈1 h-1 is clearly
visible.
We now study quantitatively the turbulent flows at UFS. We present here
large-scale flow measurements. Analysis of small-scale turbulence is reported
in an accompanying paper . We analyzed in detail the
continuous recording of wind velocities by the top sensor between September
and December 2010. From these recordings, we select the “steady” events that
are defined as segments with periods between 1 and 2 min, during which
the fluctuation of wind around its mean is less than 25 % of the mean. For
these segments, we then used Taylor's frozen turbulence hypothesis to obtain
spatial correlations and structure functions from the time series of velocity
data. The longitudinal and transverse velocity auto-correlations, f(r,t) and
g(r,t), are defined as
f(r,t)=u1(x+e1r,t)u1(x,t)u12 ,g(r,t)=u2,3(x+e1r,t)u2,3(x,t)u2,32,
where the subscripts 1, 2 and 3 refer to vector components and
e1 is an unit vector in the 1 direction. Component 1 is the mean
flow direction of the segment, which could vary from segment to segment,
e.g., from east to west, and the other two components are in the directions
perpendicular to the mean velocity. The longitudinal and transverse integral
scales, LLL and LNN, are
LLL(t)=L11(t)≡∫0∞f(r,t)dr,LNN(t)=L22,33(t)≡∫0∞g(r,t)dr.
Since an integration up to infinity is practically not possible, we determine
the integral length scales by integrating correlations up to the first
zero-crossing. The integral length scales correspond roughly to the sizes of
the biggest eddies in the flow and therefore vary significantly in
environmental flows. Figure show the distribution of the
measured integral length scales. The most probable values of the longitudinal
and transverse length scales are LLL≈9 m and LNN=(L22+L33)/2≈4.5 m, which agree surprisingly well with the relation of
LLL=2LNN for homogeneous and isotropic turbulence
e.g.,. We note that these values are also consistent with
the measurement height of 6 m and are not necessarily representative of the
largest eddies in the atmospheric boundary layer, which could possess much
bigger scales .
On the other hand, the averages of the instantaneous length scale ratio is
LLLLNN=1.51, which indicates
that the large-scale wind conditions at the measurement site are in fact
anisotropic. This large-scale anisotropy may be explained by wind shear
induced by surrounding structures. A 6 m tall lidar tower located in the
northwest of the measurement site blocks the flow from that direction.
Because of the presence of the lidar tower, wind coming from the west
experiences a strong shear at the measurement site. In contrast, wind coming
from the east is less influenced because no structures are located on the
east side of the measurement site. Indeed, the average length scale ratio
LLLLNN is 1.79 for east winds and
1.26 for west winds, which also suggests that the east wind is in general
more isotropic.
PDFs of the integral length scales measured using Taylor's
hypothesis. Note that L11 corresponds to the longitudinal integral scale
LLL and L22 and L33 correspond to the transverse integral
scale LNN. The most probable value of LLL is 9 m, and the most
probable values of the transverse scales are L22≈4 m and L33≈5 m.
The second-order (top panel) and the third-order (bottom panel)
velocity structure functions obtained from the ultrasonic sensor data. The
dashed lines in the top panel are fits to Eqs. () and (),
which give the values of ε. These values are then used in
Eq. () to plot the two dashed lines in the bottom panel.
Energy dissipation rate per unit mass, ε, is the most important
quantity to characterize turbulence. For cloud studies, the turbulence energy
dissipation rate determines the other parameters of cloud droplets such as
the Stokes number and the settling parameter . We estimate the
energy dissipation rate from the longitudinal and transverse second-order
velocity structure functions by fitting the inertial range scaling:
DLL=C2εr2/3
and
DNN=43C2εr2/3
with a value of C2=2.1 e.g.,. As shown in the top panel
of Fig. , the measured structure functions
DLL and DNN indeed show a scaling region consistent with
Eqs. () and (), where we fit for ε. We then
check these values against the measured third-order structure function, which
satisfies an exact scaling law
DLLL=-45εr.
As shown in the bottom panel of Fig. , the
measured values of DLLL are within a factor of 2 from the lines
-45εr with the energy dissipation rates ε
estimated from the second-order velocity structure functions DLL and
DNN. It is known that the scaling of the third-order structure function,
Eq. (), is more sensitive to Reynolds number, noise and the
inhomogeneity of the flow . We therefore take the averages
of the values obtained using Eqs. () and () as the
measured ε. The energy dissipation rates determined in this way
are in the range of 10-4 to 10-2 m2 s-3, which are comparable
with previously reported values of atmospheric turbulence
measurements and are also typical for turbulence in
clouds .
PDFs of the Taylor-scale Reynolds number Rλ. The most
probable value is Rλ≈3000.
Using the measured values of ε and the rms velocity fluctuation
u, we estimate the Taylor microscale Reynolds number for each segment as
Rλ=15u4εν,
where ν is the kinematic viscosity of air. This definition assumes that
the turbulence is homogeneous and isotropic e.g., and that
the largest length scale can be estimated as L∼u3/ε, which
we discuss next. The PDF of Rλ obtained using Eq. ()
is shown in Fig. . The maximum Reynolds number measured is
Rλ∼O(104), and the most probable value is Rλ≈3000.
With the values of ε, L and u, we check the normalized
energy dissipation rate
Cε=εLu3.
It has been found that the normalized energy dissipation rate Cε
is a constant of approximately 0.5 for a wide range of Reynolds numbers,
including both laboratory flows and flows in the
atmosphere . This observation has also been
called the “zeroth law of turbulence”, as Kolmogorov's hypotheses assume
that the mean energy dissipation rate is independent of the viscosity at high
Reynolds numbers .
Normalized energy dissipation rates conditioned on wind direction
(top) and cloudiness (bottom).
Figure shows the current measurements
of Cε as a function of Reynolds number. In agreement with
earlier measurements, a value of about 0.5 was found. It can also be seen
that the value of Cε depends weakly on the wind direction, which
may be attributed to the different flow conditions of the east and the west
wind. As described before, winds from the west are subject to stronger shear
than winds from the east. Previous data from turbulence in homogeneous shear
flows showed that, in shear flows, ε estimated from isotropic
relations is smaller than the true energy dissipation
rates . Therefore, for wind from the west, the measured
Cε could be smaller than the real values. On the other hand, the
effect of clouds on the normalized energy dissipation rate Cε is
not very clear. There might be a slight increase in Cε
associated with the occurrence of clouds, but better converged data are
needed before drawing any conclusions.
Finally, to further investigate the deviation of the turbulent flow at UFS
from the ideal isotropic conditions, we show the measured events on the
so-called “Lumley triangle”, which is the realizable region on the plane
spanned by the two non-trivial invariants of the deviatoric part of the
Reynolds stress tensor . The top curved side of
the “triangle” represents two-component turbulence; i.e., one component of
the fluctuating velocity vanishes, e.g., 〈u12〉=0, or is
much smaller than the other two components. Both of the two straight sides of
the triangle represent axisymmetric turbulence, i.e., two components of the
fluctuations are the same, e.g., 〈u22〉=〈u32〉. The left side corresponds to the state of the third component being
smaller than the other two, e.g., 〈u22〉=〈u32〉>〈u12〉, which may be termed as “axisymmetric
turbulence with one smaller eigenvalue” ; while the right side
corresponds to the opposite state: i.e., the third component is larger than
the other two, e.g., 〈u12〉>〈u22〉=〈u32〉, which can be similarly termed as “axisymmetric
turbulence with one larger eigenvalue” . When mapped onto that
plane, any realistic turbulent flow must lie within the Lumley triangle. The
origin on that map represents the isotropic flows. The closer to the origin,
the more isotropic the flow is. As shown in Fig. ,
the flows at UFS (for winds from the east) are not strictly isotropic. In
fact, the turbulence is close to being axisymmetric, with both “one smaller
eigenvalue” and “one larger eigenvalue” cases. As the Reynolds number
increases, however, there is a trend for the flow to become more isotropic.
We also compared the flows at UFS with two widely used laboratory turbulent
flows, i.e, the von Kármán swirling flow between two counter-rotating
disks and the propeller-driven turbulent flow
within an icosahedra, named the Lagrangian exploration module, or
LEM . The turbulent flows at UFS are less isotropic than the
LEM flow, which is designed to achieve high homogeneity and
isotropy . On the other hand, the degree of isotropy of the UFS
flows is comparable and, in many cases, even better than the von Kármán
flow, especially as the Reynolds number increases. For winds from the west,
the range of anisotropy is approximately the same as that of the east winds,
but no clear change with Reynolds number can be observed. This is most likely
due to the effect of shear as discussed before.
Mapping of the eastward turbulent wind flows at UFS on the Lumley
triangle, in comparison with two laboratory flows: the von Kármán flow
and the LEM flow. Here η and ξ are the two non-trivial invariants of
the Reynolds stress tensor . The symbols
representing the UFS flows are color-coded with the Reynolds number
Rλ, whose range is indicated by the colorbar.