The AirCore is a long coiled tube that acts as a “tape recorder” of the composition of air as it is slowly filled or flushed. When launched by balloon with one end of the tube open and the other closed, the initial fill air flows out during ascent as the outside air pressure drops. During descent atmospheric air flows back in. I describe how we can associate the position of an air parcel in the tube with the altitude it came from by modeling the dynamics of the fill process. The conditions that need to be satisfied for the model to be accurate are derived. The extent of the mixing of air parcels that enter at different times is calculated so that we know how many independent samples are in the tube upon landing and later when the AirCore is analyzed.

When the AirCore is filling with atmospheric air coming in through the open end, the newly sampled air pushes the air that is already in the tube deeper into the tube while compressing it. This mode of sampling is entirely passive, relying on the pressure continuing to increase as the altitude becomes lower during descent. The AirCore could also be flushed by a pump without any need for pressure changes of the outside air that is being
sampled. I conceived the idea of the AirCore in the late 1990s after we had
found

With Jim Smith and Michael Hahn, two members of our group in those days, we
verified that there is very little mixing along the length of the tube by
pushing slugs of air from two different reference air cylinders, alternating
between high and low CO

We realized that AirCore technology could become extremely useful for the validation of satellite retrievals of column-averaged mole fractions of greenhouse gases. The measurements of a gas sample captured by the AirCore are calibrated, but care has to be taken, as with all air samples in containers, that no artifacts are introduced by the container or by gas-handling procedures. In contrast, remote sensing estimates of greenhouse gases can in principle never be calibrated. Metrology, the science of measurement, defines what a calibration is. Using a measurement standard, one presents the measurement method with a known value, under controlled conditions, so that the measurement indication is related to a quantity value (paraphrased from Bureau International de Poids et Mesures, 2008). In the case of greenhouse gases in the atmosphere, the conditions cannot be controlled. In addition, we realized that the regular deployment of AirCores could be a cost-effective way to monitor and study an evolving atmospheric circulation as climate change progresses, as proposed by Fred Moore (Moore et al., 2014).

Developments of the AirCore by various groups have been described in other papers, for example by Wagenhäuser et al. (2021) and Membrive et al. (2017). However, there has been neither a comprehensive treatment of fill dynamics nor a detailed look at mixing, hence this paper.

Diffusive mixing over large distances is exceedingly slow, but there is
another use of diffusion. Flow inside the tube is laminar, which has maximum
speed in the center and zero speed at the wall. With velocities that differ
from zero to some finite value, why does laminar flow not “smear out” our
tape recorder signal by mixing air parcels that came in at different times?
Again, molecular diffusion comes to the rescue. Using the square root
relationship above, if the inner radius of the tube is 0.3 cm, it takes a
CO

The AirCore collects a continuous sample. Instead of valves, distance in the tube is used to keep separated the air that has been sampled from different pressure altitudes. The number of independent samples (the inverse of vertical resolution) in the tube decreases as the time between collection and measurement becomes longer. The measurement or “readout” of the vertical profile is carried out by attaching an analytical instrument to one end of the tube and a cylinder with air of a well-known composition to the other end. The latter pushes the sampled air slowly through the analyzer. The procedure, as well as various tests of mixing, has been described by Anna Karion of GML (Karion et al., 2010).

How do we accurately associate position in the tube with the geometric altitude or pressure altitude that the sample at that position came from? It is the first question we address in this paper. The filling does not occur uniformly as a function of pressure altitude. The second question is how far the mixing of adjacent air parcels extends as a result of molecular diffusion and secondarily as a result of the flow itself. It will be addressed in Sect. 6. I wrote the first version of the fill dynamics calculation to make the association of altitude with position in 2005, called rocketfall.pro, coded in Interactive Data Language (IDL). Undergraduate students in the engineering department at the University of Colorado were getting ready to put an AirCore on a NASA rocket, and I was worried about there not being enough time to passively collect air from the stratosphere as the rocket was falling at supersonic speeds. There have been several successive versions of the algorithm since then. The significantly improved IDL version of July 2021 is described in this paper.

We use a fluid dynamics model and a subset of the flight data, namely the
time, pressure and temperature of outside air and the temperature of the
tube, as input data. The starting point is Poiseuille's equation for steady-state laminar flow in a tube with a circular cross section:

The viscosity (

The

The first one is that inertial effects, i.e., accelerations, die out very
rapidly. Suppose we suddenly set the pressure gradient that is driving the
flow to zero. What is the timescale for the flow to die down? We can
estimate the time it takes for the flow to adjust by using Eq. (1). The
average speed of the flow is

Next we assume that the temperature of the gas is the same as that of the
wall. How rapidly does the temperature of the gas equilibrate with the wall
of the tube? The heat capacity of a volume of air is

Is the flow always laminar as Eq. (1) assumes? The Reynolds number is

The tube is wound up in a coil with a typical diameter of 20 to 30 cm. As the
flow goes around the coil there will be a centrifugal force away from the
center of the coil. The centrifugal force is greatest where the flow has the
maximum velocity, 2

If the tube is elliptical (as a result of bending, for example) instead of
circular, we can use a good approximation for the change in flow resistance.
Following Lekner (2019), Eq. (1) can be written for volume flow as (

We assumed the ideal gas law. Non-ideality is often described by the virial
expansion relating pressure and density,

When the mean free path increases at lower pressures, there could be
“wall slip”, non-zero velocity at the wall which can be modeled as an
effective decrease in viscosity increasing the volume flow. Berg (2005)
gives an approximate expression for the factor by which the flow increases,

When

The above expressions for viscosity,

For the diffusivity of trace gases in air as a function of temperature and
pressure, we use the empirical equation presented by Massman (1998),

In Figs. 1–4 the flight is shown of a small diameter (

Descent velocity (negative) and rate of fill air outflow followed by air sample inflow during the flight of GMD008.

In Fig. 2 the outside air temperature first cools while in the troposphere,
then becomes nearly constant in the tropopause and starts increasing again
higher into the stratosphere. GMD008 was well insulated but still partially
followed the outside temperatures with a delay. In the right panel the total
amount of air in the tube is plotted relative to how much it would be if it
had the same pressure and temperature everywhere in the tube as the outside
air. The vertical line shows the ratio equaling 1 if they were the same. During ascent
in the troposphere (up to about 10 km), the air in the tube is warmer and
thus less dense than outside air. In the tropopause the tube continues to
cool so that the “deficit” becomes smaller, but at higher altitudes,
around

Flight of GMD008.

In Fig. 3 the fill rate is plotted (moles per hectopascal of ambient pressure gain)
divided by the final fill (moles of air) at valve closure. At sea level the
final pressure is close to 1013 hPa so that the average fraction of the final
fill amount per hectopascal will be approximately 0.001. The uptick upon
landing (very close to the

Flight of GMD008. The vertical line at

Flight of GMD008. The turnaround at high altitude. Inflow velocity and pressure inside the AirCore from the moment the ascent stops and descent begins. Times are in seconds after start descent.

Comparison of start of fill process for two AirCore configurations.

Let us look now at an AirCore with larger diameters (Fig. 5). This one had
26 m of

Flight of AC01 in Oklahoma.

Flight of AC01 in Oklahoma. Compare with Fig. 3.

If one wants to sample still higher into the stratosphere, the diameter of
the first 10 to 20 m at the open end needs to be widened further than 6 mm
diameter (Table 1). All of this is consistent with Fig. (7), where we also see that at the start of the descent the outflow velocity inside the tube drops by a factor of

Flight of AC01 in Oklahoma, showing inflow velocity and pressure gradients. Compare with Fig. 4. Note the much smaller fill delay than in Fig. 4. The pressure drop across the two valves and dryer is visible here along the

In these calculations I have experimented with another strategy to fill the AirCore. One could launch it with both valves open, but the one in the back is closed as soon as the descent starts. That would decrease the amount of fill air that remains in the back. However, the difference from having the back valve closed during the entire flight is negligible.

So far the treatment of valves and the dryer has been missing from this description. As a first approximation we could treat the valves as short pieces of tubing with a reasonably “average” inner diameter and length such that their internal volume is correct. This does not provide enough flow resistance when we compare it to differential-pressure measurements made during flights between the closed end of the AirCore and the outside ambient air (Fig. 8).

Pressure difference (

In panel (a) we calculate that during the descent the air enters the tube too
easily so that the altitudes assigned to the air sample in the stratosphere
would be biased high. We could decrease the chosen average inner diameter
(not a well-defined value) of the valves (panel b), optimized so that the
difference between calculated and measured

The flow inside a valve can be complicated, with sharp corners, turbulence,
sudden acceleration through a flow restriction with its associated heating
and cooling of the gas, etc. The industry has introduced flow coefficients
(

For a high-pressure drop (

In Fig. 8c we optimized both

Figure 9 shows a potential test procedure for determining

A potential test procedure to determine

The fill dynamics calculation has produced time series of air density,
pressure and temperature, and flow velocity everywhere in the tube as a
function of time, from the start of the fill process, which begins a varying
amount of time after the AirCore has started its descent, to the time of
valve closure. We divide the final amount of air in the tube at closure into
400–500 equal mass packets. Starting from 400 we increase the number, which
shrinks the size of each packet, until the remaining fill air in the back of
the tube comprises an exact integer number of packets. For each mass packet,
after it has entered the tube, we follow it through the tube, as it is pushed
toward the back while being compressed by packets entering later. The time
steps are defined by when a new packet has fully entered, and they are
longer at the start of the fill. The molecular diffusivity

For an AirCore with (an almost) uniform diameter, we get mixing as in Fig. 10a.
Close to the open end at position 0 m, there is very little mixing because
the time to mix was short. Near the closed end at 93 m the spread of mixing
deviates from what is seen in the first approximately

Root-mean-square diffusive mixing when the valve at position 0 is closed.

For an AirCore with two sections of different diameters, we see an interesting
effect (Fig. 10b). The air that comes in at high altitudes and ends up in
the back of the tube has to go through the

In Eq. (7)

Two additional cases of mixing upon valve closure.

In Fig. 11a when the tube had descended to 850 mbar, the atmospheric pressure
data were changed to simulate an updraft (lowering outside pressure)
followed by a downdraft. The most recent seven mass packets were lost from the
tube during the updraft and replaced by new air during the downdraft (above-average rate of increase of outside pressure). As a result, the air sample
that just escaped from being lost is now adjacent to the replacement air,
creating the jump in rms mixing because it has been

We will now express the amount of spreading (in both directions – twice the
rms distance) of each equal-mass “packet” of air as a fraction of the
total mass of air in the tube, assuming that the temperature inside the tube
has become uniform. If that fraction were 0.01 everywhere in the tube, there
would be slightly less than 100 independent samples in the AirCore. It would be slightly
less because the remaining fill air in the back takes up space. Figure 12
shows a more realistic situation. Each sample takes up the same volume,
separated by the blue vertical lines, producing vertical boxes. If there is
almost no mixing, as in the case of the last sample that entered the
AirCore, the sample almost completely fills the first volume (or box in Fig. 12a), which is indicated by the value of 1.0 on the

In Fig. 12b we plot the situation near the closed end. As in Fig. 12a, the mixing of only every fifth air packet is plotted, here ending with the first that came in at the highest altitude, centered approximately at 0.991. The remaining fill air in this case has the mass of four packets, and the curves of fill air and of the total air sample (sum of all packets) cross over at exactly the point where the fourth box from the right starts.

Mixing at a closed end. The AirCore is to the right of the 0 cm point.

How we
calculate mixing at a closed end (at

Let us assume that after the valve has been closed there has been a delay of 0.5 h before analysis starts. Therefore, additional diffusion has taken
place, as shown in Fig. 14 for the case

Mixing after 30 min of storage, for AirCore

Often the AirCore is analyzed significantly later than 30 min after valve
closure, and the measurement process itself may take 0.5 h. In Fig. 15 the state of mixing 4 h after valve closure has been calculated, and two AirCore configurations are compared. The spreading width of air packets near the closed end is nearly twice as large for the

Mixing after 4 h of storage.

The mixing calculated above allows for a realistic and precise estimate of
the altitude resolution of the full air sample, both when the AirCore is
analyzed in the field promptly after landing or hours or even days later.
When the air is slowly pushed through an analyzer, we obtain a
quasi-continuous curve for the mole fraction of the gases of interest as a
function of fractional cumulative mass in the tube which is linked to flight
data such as pressure altitude, geometric altitude, latitude, and longitude as calculated from the filling dynamics. We define the information
content as the number of independent air samples that are inside the tube
or the number of degrees of freedom (DoF). Longer wait times before analysis
decrease DoF. For example, 0.5 h after landing DoF is potentially 154
for the Trainou flight, while after another delay of 4 h, DoF has
dropped to 67. This is “potential DoF” because it could be decreased
further by additional mixing in the measurement cell or by successive
analyzer cells measuring different gas species. In the section above we
chose more than 400 equal-mass packets to calculate mixing. This was done to
prevent a possibly low numerical resolution of the mixing calculation which
would unnecessarily create a low bias in DoF estimates. Ideally, the
measurement process could be modeled in a way similar to the fill and mixing
calculation above, convolving the packets leaving the AirCore with a pulse
response of the measurement cell. The response could be measured separately
by introducing a sharp spike just before the cell and recording how it is
mixed and flushed out. This would be similar to the spiking method described
by Wagenhäuser et al. (2021). In the worst case the measurement cell would
be perfectly mixed, giving rise to exponential flushing. In that case, after
one cell volume has entered from the AirCore into the measurement cell, the
latter still contains a fraction

The AirCore can consist of one or more sections of different lengths, each
with a different inner diameter. For example, GML has flown AirCores with a
wider bore at the open end and a narrow bore at the closed end, in order to
get a better vertical resolution for the stratosphere. The sections can be
divided into a number of smaller segments when Eq. (2) is discretized for
a numerical solution (Fig. 16):

Coordinate system in the AirCore. The coordinate along the length of the tube is

The first term

On the right hand side we have defined the pressure

This is a tridiagonal matrix equation,

If the tube is open at

If both sides are open, each with a different defined constant pressure, then after initially transient, the flow settles to steady-state flow corresponding to Poiseuille's equation.

This describes the core algorithm, of which there are two versions, called tubeflowstep3.pro and tubeflowstep3Cv.pro. They have been programmed in Interactive Data Language (IDL). These algorithms have the flexibility to accommodate segments of the tube that have different lengths as well as diameters, flows in both directions, one or two valves open, a temperature gradient along the tube with its corresponding viscosity gradient, and variable time steps.

Another routine, called analyzefill_Gaus_ict.pro, reads the lengths and diameters of tube
sections; valves and dryer; and the relevant flight data, namely outside air
pressure and temperature and the temperature of the AirCore at different
locations along the tube, all as a function of time. If

The
analyzefill_Gaus_ict.pro program also reads altitude, latitude, and longitude, but they are not needed for the flow dynamics calculation per se; analyzefill_Gaus_ict.pro also sets up the coordinate system and initializes variables. By calling tubeflowstep3.pro at every time step or tubeflowstep3Cv.pro if

Although developed simultaneously with analyzefill_Gaus_ict.pro for the passively filled AirCore, the tubeflowstep3Cv program can also be used to model flow when the AirCore is actively filled with a pump and some form of flow and pressure control. In that case a program equivalent to analyzefill_Gaus_ict.pro would need to be developed.

Importantly, the code in analyzefill_Gaus_ict.pro also produces diagnostic graphics showing how the fill proceeded. In fact, all figures in this paper have been produced by analyzefill_Gaus_ict.pro except for Figs. 9 and 13.

Laboratory measurements of the flow properties of valves, as expressed in the flow coefficient

The precision of the sample mixing estimates could be improved by laboratory measurements of the pulse response of analyzers, especially when an AirCore is analyzed quickly in the field because very little mixing has yet occurred for the air that came in last.

In addition to measuring the pressure inside the tube during a flight at the closed end, one could consider measuring the pressure inside at a place closely behind the valve(s) plus dryer at the open end. It does not need to be done routinely, but it would give a history of the total pressure drop across the valve and dryer only.

In cases where people want to fly AirCores without a dryer, it could be helpful to study wall effects. Water vapor tends to adhere tightly to many surfaces, and as anyone experienced with vacuums knows, it can take a long time to pry it off the walls. One possible experiment would be to inject a short pulse of wet air at one end of a dry tube and register what comes out at the other end. How much stays behind, and for how long? How does that affect other species? In general, wall effects could make the AirCore into a (very poor) gas chromatograph if gases have sufficiently different adsorption/desorption properties.

The main flight analysis program and subroutines in Interactive Data Language (IDL) are available at

AirCore flight data from GML are available at

The contact author has declared that there are no competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I thank Anna Karion, Huilin Chen, Colm Sweeney, Tim Newberger, Jack Higgs, Sonja Wolter, and Bianca Baier of GML for making our lab's AirCore program blossom and Bianca Baier for providing flight data of the Trainou flight used in this paper. Especially the controlled return on a glider is a very promising improvement over the return by parachute.

This research has been supported by the National Oceanic and Atmospheric Administration (NOAA “base” funding).

This paper was edited by Thomas Röckmann and reviewed by Julien Moyé and one anonymous referee.