Introduction
The firn layer is the upper 50–120 m of consolidating snow found in the
accumulation zones of ice sheets and glaciers. Within this perennial snowpack
a network of connected pores exists that facilitates the movement of air. The
firn layer is a mixed blessing. On the one hand it complicates the
interpretation of ice core records via the gas age–ice age difference
or Δage; , water-isotope diffusion
, the broadening of the gas age distribution
, diffusive isotopic fractionation of trace gases
, and non-atmospheric gas variations
originating from layered bubble trapping . On
the other hand, the firn provides a valuable archive of old air
, and the characteristics of firn air
movement give rise to additional signals that can be used as proxies for
local temperature change and
surface elevation , or as tools for ice core dating
. It is therefore clear that a complete
scientific understanding of firn air transport is critical to both the
correct interpretation of the ice core record, and for utilizing the unique
possibilities offered by the firn-derived proxies.
Commonly, firn air transport models include three mechanisms of air movement
. The first is downward
advection with the ice matrix, which operates along the full length of the
firn column. In a Eulerian frame of reference, the downward velocity of the
air (wair) is smaller than that of the ice itself (wice)
due to a back flux of air from pore compression . The
second mechanism is molecular diffusion in the open pores. As the tortuosity
of the pore space increases with depth, the effective diffusivity decreases;
at the so-called lock-in depth, molecular diffusion effectively ceases.
Molecular diffusion in the vertical direction leads to gravitational
enrichment in δ15N–N2 and other tracers. The third mechanism is
convective mixing, an umbrella term for several phenomena that vigorously
ventilate the upper few meters of the firn column
. Convective
mixing is commonly implemented in models as an eddy-type diffusivity that is
equal in magnitude for all gases, and does not enrich gravitationally
.
Diffusion effectively ceases at the lock-in depth, and consequently there is
no gravitational enrichment within the lock-in zone .
However, detailed studies at the NEEM site in northern Greenland and the
Megadunes site in Antarctica suggest continued vertical mixing within the
lock-in zone . Figure
shows the 14CO2 concentrations in NEEM firn air samples (black dots
with error bar), which reflect the atmospheric “bomb spike” in
Δ14CO2 caused by atmospheric nuclear weapons testing in the
1940s through 1960s. The two solid curves show model simulations using a firn
air transport model either with, or without vertical mixing in the lock-in
zone (LIZ). To correctly fit the firn air data, the models used in
require a LIZ mixing term of the order of
2×10-9 m2 s-1; at the time that study was published, the
nature of this mixing remained unclear, and a good fit to the data can be
obtained regardless of whether this diffusivity is implemented as
molecular-type, eddy-type, or a combination thereof. However, the remnant LIZ
mixing is unlikely to be caused by molecular diffusion because of (1) the
absence of LIZ gravitational enrichment, and (2) insights from percolation
theory that suggest that in a porous medium the gas diffusivity vanishes
abruptly at the percolation threshold (i.e., the lock-in depth) as the pore
connectivity decreases . Here we show that the LIZ
mixing can readily be explained by dispersion occurring in the deep firn that
is driven by air movement induced through synoptic-scale surface pressure
variations.
The 14CO2 bomb spike as observed and simulated in firn air
at NEEM, northern Greenland . Continued mixing within the
lock-in zone (LIZ) is needed to correctly simulate the smoothness of the
observations. Both simulations use the firn diffusivity values as presented
in ; the eddy-type LIZ diffusion was either turned on (red
curve) or off (blue curve). The case without LIZ mixing was not tuned
separately to the observations.
Steady viscous fluid flow through a disordered porous medium leads to
dispersion . On the
microscopic level, this process can be understood as a consequence of the
different flow paths available to tracer molecules. Figure
schematically compares fluid flow through a microscopically ordered (upper)
and disordered (lower) porous medium. In the upper panel all pathways are
equivalent, and tracer molecules transiting the porous medium each have the
same transit time and are not dispersed in passage. In the lower panel the
tracer molecules traveling through different sections of the pore space have
different transit times, and they are spatially dispersed as they transit the
medium. The different pathways not only have unequal lengths. According to
the Hagen–Poiseuille equation, the hydraulic conductance of a capillary (for
laminar flow) scales as ∝r4, with r the pore radius. Given that a
range of pore radii exist in natural firn, the r4 power law will
concentrate flow in the widest pathways, further broadening the distribution
of tracer transit times.
On the macroscopic level, dispersion can be described as a diffusive process
with a diffusivity Ddisp with units of m2 s-1
. Dispersion is much stronger in the longitudinal
(along-flow) direction than in the transverse direction. This distinction is
important in many applications (for example a point-source contamination in
an aquifer subject to groundwater flow), but in firn air applications the
source (the atmosphere) is laterally uniform, and only the longitudinal
dispersion matters. Fluid flow at a macroscopic mean velocity v results in
dispersion of a magnitude
Ddisp=αL×v,
where αL is the so-called longitudinal dispersivity, which is
a property of the medium with units of m. In the case of firn air dispersion
it is important to point out that Ddisp does not lead to
gravitational enrichment, as it originates in macroscopic air movement that
does not discriminate between N2 isotopologues. There are two sources of
macroscopic air movement in deep firn: the aforementioned back flux due to
pore compaction, and flow driven by synoptic-scale surface pressure
variations. Here we will show that the latter is several orders of magnitude
larger, and hence dominates the dispersive mixing. The upper few meters of
the firn are furthermore subject to wind-driven air flow
e.g.; the dispersion caused by this flow is minimal
owing to the low dispersivity of near-surface firn, and overwhelmed by
diffusion and convective mixing.
In this work we shall use published gas chromatography experiments by
, which can be used to derive both the diffusivity and
the dispersivity of firn samples. correctly inferred
that molecular diffusion is by far the dominant process in mixing firn air,
and that study therefore focused on the first-order process of diffusive
transport. We want to emphasize that Jakob Schwander fully realized the
potential of these measurements to estimate dispersive gas mixing in firn
also, as he wrote: “This dependence of D on the gas-flow velocity can be
used to estimate how air flow in the firn caused by atmospheric pressure
variations contributes to the gas mixing rate” . Since
1988 there has been enormous progress in observing and modeling firn air
movement, and presently it is pertinent to also investigate the second-order
effect of firn dispersion. Still, the present work should be considered a
footnote to the seminal achievement of , and the many
deep and early insights in firn air movement it contains. The reader is
strongly encouraged to (re-)read the original work.
Schematic depiction of dispersion. Fluid movement through a
microscopically homogeneous, low-tortuosity porous medium (upper panel) is
non-dispersive; fluid movement through a microscopically disordered porous
medium (lower panel) disperses tracer molecules. Macroscopic fluid flow is
from left to right.
Air pressure dynamics in the firn column
Mathematical description
Here we present a mathematical description of air pressure dynamics in polar
firn, aimed towards understanding firn air movement in deep firn in response
to surface pressure variations.
In hydrostatic equilibrium the firn air pressure p(z) increases with depth
z following the barometric equation:
p(z)=p0exp-MgRTz,
with p0 the surface pressure, M the molar mass of air, g the
gravitational acceleration, R the gas constant, and T the absolute
temperature. Deviations from this hydrostatic balance result in viscous air
flow, with a velocity given by Darcy's law:
v=-kμdpdz-MgRTp,
with k the permeability of the firn and μ the (temperature- and
pressure-dependent) dynamic viscosity of air.
The continuity equation for macroscopic air movement within the firn is
ddtsopρ=-ddzsopρv,
with ρ the fluid density and sop the firn open porosity.
Assuming static, isothermal firn properties (i.e.,
dsop/dt=0 and dT/dz=0), and using
the ideal gas law then gives
dpdt=-1sopddzsoppv.
Propagation of pressure variations into the firn at WAIS Divide.
(a) Pressure response at indicated depths to a unit-step surface
pressure increase of 1 mbar at time t=0. (b) Low-pass filter
characteristics of the firn column to surface pressure variability.
(c) Surface pressure history of year 2010 at the WAIS Divide site
from AWS measurements (orange) and from the ERA-Interim reanalysis (green).
(d) power spectral density (PSD) of the time series shown
in (c).
Substituting Darcy's law into the continuity equation yields
dpdt=1μsop[k*pd2pdz2+k*dpdz+dk*dzp-2k*MgRTpdpdz-dk*dzMgRTp2],
where k*=k⋅sop.
Equation () is a nonlinear differential equation that is
difficult to solve numerically due to the various terms that scale as
O(p2), a complication that arises from the compressibility of
air, which makes the fluid density ρ a function of pressure. Equation
() can be simplified substantially by making the following
approximation. In the continuity equation Eq. () the time-varying
pressure p(t) is substituted by the mean annual pressure p‾ to
get
dpdt=-1sopddzsopp‾v,
which in turn simplifies Eq. () to
dpdt=p‾μsop[k*d2pdz2+dk*dz-k*MgRTdpdz-dk*dzMgRTp].
Equation () is a linear second-order partial differential
equation that can be solved easily using finite difference methods.
The approximation in Eq. () makes the fluid density independent
of the pressure; this changes the firn air mass flux from J=ρv=pMv/RT
to J‾=ρ‾v=p‾Mv/RT. By doing so an error
is introduced because p(t) can differ from p‾. This error,
however, is rather small given that the atmospheric pressure p never
deviates by more than 5 % from the mean annual pressure p‾.
Note that this 5 % error is much smaller than other errors such as the
uncertainty in the firn permeability k. Moreover, in the next section it is
shown that the pressure response time of the firn is orders of magnitude
faster than the synoptic-scale pressure variations that drive barometric
pumping in the deep firn, rendering a 5 % error irrelevant. We also
neglect the small viscosity changes due to temperature and pressure.
Propagating pressure anomalies into the firn
In the numerical solutions presented here, Eq. () is solved
using a Crank–Nicolson finite difference method, which employs implicit time
stepping. At the surface boundary the pressure is set to equal the
atmospheric pressure forcing; at the lower boundary the pressure gradient is
set to zero. The sop parameterization of is
used as well as the k parameterization of .
In a first experiment we force the model with an atmospheric pressure p(t)=p‾+u(t), where u(t) denotes a 1 mbar magnitude unit step (or
Heaviside) function at time t=0. Firn properties of the West Antarctic Ice
Sheet (WAIS) Divide site in West Antarctica are used.
Equation () is solved at 0.1 m depth resolution and at
time increment Δt=1×10-10 year = 3.2×10-3 s. The propagation of the
pressure anomaly into the firn is shown in Fig. a
at four depth levels; the depths z=67 m and z=76 m are selected because
they are the lock-in depth and deepest firn air sampling depth, respectively.
Note that the pressure variations at depth are amplified relative to the
surface forcing due to the hydrostatic effect (Eq. ). Let
t1/2(z) be the time required at depth z for the pressure
increase to reach half the final amplitude. The depth profile of
t1/2(z) is shown in Fig. a (note the
logarithmic scale). It is clear that pressure fluctuations propagate
relatively fast through the firn column; at z=20 m depth the
response time t1/2=20 s, and at the lock-in depth
t1/2=230 s. This fast response need not be a
surprise, given that in free air pressure variations propagate at the speed of sound; in firn, the
propagation speed is limited by the finite permeability of the medium. The
largest increase in t1/2(z) is seen within the lock-in zone,
where firn permeability becomes vanishingly small as sop approaches
zero.
The response curves of Fig. a can be used via
Fourier transform to derive the low-pass filter characteristics of the firn
to pressure variations; this is shown in Fig. b.
For a surface pressure oscillation at any given frequency, the transfer
function shows by how much the amplitude of that signal is attenuated in the
firn (so a transfer function value of 1 means that the pressure oscillation
is transmitted at full amplitude). The model shows that a surface pressure
oscillation with a period of 1 h or longer will have a nearly unattenuated
response at the LID (yellow curve); pressure variations with a period of
1 min or shorter (such as wind pumping events driven by wind gusts over
surface topography) are completely dampened before they reach the LID. These
conclusions depend on the assumption that the permeability measured on
centimeter-scale samples is also valid on the macroscopic scale of the entire firn
column. The ultimate test of these model results would be an in situ pressure
gauge buried in the deep firn near the firn–ice transition, but to our
knowledge such measurements have only been performed in the upper few meters
of snowpacks e.g..
The firn transfer function of Fig. b can be
compared to the spectral density of pressure variability at the site.
Figure c shows hourly pressure data from an
automated weather station (AWS) located at the WAIS Divide site
, as well as 6-hourly data from the ERA-Interim reanalysis
; Fig. d shows the power spectral
densities of both time series. From analyzing the firn transfer functions, it
is clear that most of the synoptic-scale pressure variability observed in
these records will be expressed at full magnitude all the way down to the
firn–ice transition.
Firn air pressure response at the WAIS Divide site.
(a) Response time t1/2(z) to a unit-step surface
forcing. (b) The 2σ air displacement Δz as a function
of depth (blue), and the mean absolute air velocity due to barometric pumping
(red), both forced by the 2010 surface pressure time series
(Fig. c).
Barometric pumping
The pressure variations described above are associated with net macroscopic
air movement in the firn column, a phenomenon known as barometric or
atmospheric pumping . Whenever surface pressure increases,
firn air will move downwards as the underlying air parcels are being
compressed (the same number of molecules occupy a smaller volume at higher
p), resulting in atmospheric air entering the upper firn column. Also, vice
versa, when surface pressure decreases, the firn air moves upwards in the
column with the upper air parcels being expelled into the free atmosphere.
One can think of the entire firn column as “breathing” in and out in
response to high- and low-pressure systems, respectively.
The 2σ barometrically driven vertical displacement of air parcels
(Δz) at the WAIS Divide is plotted in blue in Fig. b. Near
the surface, barometric pumping displaces air vertically of the order of
1 m, which presumably contributes to the establishment of a
well-mixed zone (or “convective zone”). Note that the curve shows the
2σ vertical displacement; the peak-to-peak displacement is much larger
and of the order of 3.5 m near the surface. Figure b
furthermore shows the mean (absolute) velocity |v| in red. Both curves of
course closely resemble each other, given that the vertical velocity is the
first derivative of the air displacement. We plotted the actual air
velocities in the pores themselves, i.e., the air velocity averaged over the
pore cross section. To obtain the velocity per unit of bulk sample
cross section, one has to multiply the values by sop.
Another source of macroscopic air movement in deep firn is the gradual
closure of the pore space by the densification process, which leads to an
upward air flow relative to the firn matrix . The
velocity of this (accumulation-rate-dependent) back flux is of the order of
10-9 to 10-8 m s-1, and is clearly negligible in
magnitude compared to the barometrically driven flow.
Dispersive mixing
Experimental dispersivity of polar firn samples
Here we revisit the published firn diffusion experiments by
to estimate the dispersivity of polar firn. In these
experiments, cylindrical firn samples (30 mm diameter, 50 mm
length) from Siple station, Antarctica, were placed in a -20 ∘C
chamber inside a gas chromatograph (GC) and flushed through with a pure N2
carrier gas at a controlled superficial flow velocity ranging from 0.175 to
0.789 mm s-1. A small amount of CO2 or O2 is
injected into the N2 carrier-gas stream before the firn sample, and the
eluting peak is measured using a thermal conductivity detector. The effective
firn diffusivity (as a function of flow speed) is reconstructed from the
peak width in the chromatogram using standard GC theory. More procedural
details are given in the original publication .
Firn dispersivity. (a) Effective diffusivity of CO2 as a
function of flow speed as measured by on firn samples
from Siple station. (b) Experimental firn longitudinal (i.e.,
along-flow) dispersivity αL as a function of sample open
porosity (white dots) with best fit (solid blue line); the shaded area
indicates the estimated 95 % confidence interval of the fit.
(c) Dispersive mixing at NEEM, using the dispersivity envelope
(95 % confidence) from (b), multiplied by the mean absolute firn
air velocity induced by barometric pumping at that site. The dashed black
(red) line indicates the optimal level of dispersive mixing needed in the
firn air transport model to fit the NEEM (WAIS Divide) firn air observations
.
The results at four representative sampling depths are shown in
Fig. a, in each case at five different flow speeds. The
solid lines give the linear least-squares fit to the diffusivity data
(second-order terms are small, and neglected here). The (extrapolated) intercept with
the y axis (zero flow speed) gives the molecular diffusivity of the sample.
The data show that firn diffusivity decreases with depth as the tortuosity of
the pore space increases; this result has since been confirmed many times
both via direct measurements and via inverse
methods .
The slope of the fit represents αL, as per
Eq. (), with the longitudinal (or along-flow) orientation
being vertical in the natural firn setting. Near the surface (z=2 m) the reconstructed diffusivity is
independent of the flow speed, suggesting αL=0 m. This
is due to the homogeneous and open pore geometry of shallow firn, in which
the traveled path length and flow speed is identical for all pore clusters
that contribute to the flow, and hence tracer molecules traveling through
various parts of the pore space do not get dispersed. For deeper firn samples
we see an increase in the slope of the fit, and therefore an increase in
αL. This increased dispersivity is caused by heterogeneity of
the pore geometry, by which the different pore clusters have increasingly
disparate path lengths, flow speeds, and cul-de-sacs, which serve to disperse
tracer molecules traveling through various parts of the pore space.
The firn dispersivity data are plotted in Fig. b as a
function of sop (white dots), where we used the average value of
the CO2 and O2 experiments in . The data suggest
an exponential dependence on sop of the form
αL(sop)=a⋅exp(b⋅sop).
Propagating the 95 % confidence intervals on the αL values
estimated in the slope fitting, the following values for the fitting
parameters are suggested: a=1.26±0.40m and b=-25.7±2.2. Unfortunately there are no experimental data within the lock-in zone,
which makes the exponential extrapolation of Eq. () uncertain.
Using a Monte Carlo scheme, we construct an uncertainty envelope by fitting
functions of the form a⋅exp(b⋅sop+c⋅sop2), while forcing the curve to intercept the y axis at an
αL(sop=0) value randomly selected from the interval
0.3 to 3 m, and randomly perturbing the αL data
within their 95 % confidence intervals; this envelope is indicated as the
blue shading in Fig. b.
Dispersive mixing driven by barometric pumping
Multiplying the experimental αL values by the estimated mean
absolute velocity profile allows for a theoretical estimate of dispersive
mixing in the firn column, as is shown in Fig. c for NEEM
as the blue curve with the uncertainty envelope. Note that the pore velocities of
(Fig. b) are first multiplied by sop to convert them
into superficial velocities (average velocity in an open tube with the same
diameter as the sample) as used by . The NEEM site is
used here because it was the venue for an intensive study of firn air
processes by eight different research groups and had a very clear signal of
continued mixing within the lock-in zone (Fig. ). Six
different firn air transport models were applied to two separate boreholes at
the NEEM site, which together suggest a lock-in zone diffusivity in the range
of ∼2×10-9 to 7×10-9 m2 s-1
; the theoretical estimate of dispersive mixing derived
here corresponds very well with this range.
The dispersivity αL and the pumping velocity |v| have
opposite trends with sop. However, because αL scales
more strongly with sop, it ends up dominating the behavior of
Ddisp, which generally increases as sop gets smaller –
the exception to this pattern is the lower half of the lock-in zone
(sop<0.05), where dispersive mixing decreases again as the air
movement approaches zero. Dispersive mixing occurs throughout the firn
column; however, within the diffusive zone it is overwhelmed by molecular
diffusion, which is about 2–3 orders of magnitude larger. Molecular
diffusion effectively ceases at the lock-in depth, and therefore dispersion
dominates only in the lock-in depth.
Dispersive mixing in firn air transport modeling
In this section we use the theoretical estimates of dispersive mixing
strength in a firn air model to investigate whether it is consistent with the
measured trace gas concentrations in air samples extracted from the pore
space. We use the NEEM and WAIS Divide
sites, which are among the most well-characterized firn air sites in the
literature. At both locations we use ERA-Interim 6-hourly surface pressure
data at the nearest grid point (0.75∘ × 0.75∘
resolution grid) for the calendar years 2010 and 2011 to calculate the
barometrically driven air velocity and dispersive mixing strength. The
theoretical Ddisp values at WAIS Divide are about 15 % higher than those at NEEM (the latter
are shown as the blue enveloped curve in Fig. c). We
assume that dispersive mixing at all times was equal to the average of the
2010–2011 period.
Next, using the CIC firn air model , we calibrate the
molecular diffusivity profile at both sites using well-established methods
,
where dispersive mixing strength is set to Ddisp=γ⋅Ddisp0, with Ddisp0 the theoretical dispersive mixing
strength, and γ a scaling factor that is changed in the calibration
routine in order to optimize the model fit to the firn air data. The
Ddisp profiles that optimize the fit are shown in
Fig. c as dashed lines. The fit to the firn air data is
optimized at NEEM and WAIS Divide when we set dispersive mixing in the model to around 50 and 58 %
of the theoretical estimate, respectively.
The firn air data indicate that WAIS Divide has more dispersive mixing than NEEM, as also predicted by our
theoretical calculations. This should thus be considered a robust result.
While the theoretical estimates are of the correct order of magnitude, they
appear to overestimate the dispersion suggested by observed trace gas
concentrations. There may be several causes for this mismatch. First, to fit
the same tracer data, different firn air transport models require slightly
different diffusivity profiles , and some of the mismatch
could be an artifact of the firn air model used. Second, at the low
sop values of the lock-in zone (where dispersion dominates the
transport) our αL parameterization relies on extrapolating the
observational estimates (Fig. b), and
Eq. () may thus overestimate the true firn dispersivity.
Third, the data by give the dispersion under steady
flow conditions, whereas barometric pumping drives an air flow that is
variable in time; the analogy from electronics would be direct current (DC)
and alternating current (AC), respectively. Conceivably the AC dispersivity
of firn is lower than the DC dispersivity we derived from Jakob Schwander's
data, for example because in an alternating flow a portion of the air will
retrace its path back to its original position (reducing the dispersion),
which does not occur in direct flow.
Gravitational enrichment at Law Dome
A site with strong barometric pumping is the Law Dome site in coastal eastern
Antarctica, where firn air has been sampled at the high-accumulation DE08
site 1.2 m year-1 ice equivalent; , and the DSSW20K site 0.16 m year-1 ice
equivalent; . We calculate dispersion at DE08
to be about 65 % stronger than it is at NEEM.
Model simulations and firn air data for the Law Dome
DSSW20K (a–d) and DE08 (e) sites, with model
scenarios color-coded. The first scenario (CTRL, blue) is a control run with
no convective mixing or dispersion to show gravitational equilibrium. The
second scenario (+CZ + Ddisp, red) includes convective mixing
(following with H=2 m and Deddy,0=1×10-5 m2 s-1) and dispersion to optimize the fit to
all tracers except δ15N. The third scenario (150×Ddisp, yellow) uses strongly enhanced dispersive mixing to optimize
the fit to δ15N data. The fourth scenario (layered Ddisp,
green) uses layered dispersion and diffusivity to optimize the fit to all
tracer data. Model details and atmospheric trace gas forcings are as in
. Spatial model resolution is Δz=0.1 m
for scenarios 1–3, and Δz=0.01 m for scenario 4.
(a) DSSW20K methane mixing ratio; (b) DSSW20K 14CO2
mixing ratio; (c) DSSW20K δ15N–N2;
(d) DSSW20K 86Kr excess; (e) DE08
δ15N–N2.
A remarkable property of both Law Dome sites is that LIZ gravitational
δ15N enrichment is much less than would be expected based on
diffusive equilibrium. For DSSW20K and DE08 this is shown in
Fig. c and e, respectively; note that the shallowest
δ15N data may be impacted by thermal diffusion due to temperature
seasonality . Here we will focus on the DSSW20K site,
for which more data are available. At DSSW20K gravitational enrichment in
δ15N appears to stop ∼7 m above the actual lock-in depth.
Here we explore the possibility that this anomaly is due to dispersive mixing
in the deep firn. First, we calibrate the CIC firn model to the DSSW20K site
using firn air data of CH4, CO2, SF6, Δ14CO2, CFC-11,
CFC-12, and CFC-113 using established methods
. Following our findings at the NEEM and WAIS Divide
sites, we set the deep firn dispersion to equal 55 % of our theoretical
estimate based on Law Dome surface pressure time series from the ERA-Interim
reanalysis. Next, we explore four instructive modeling scenarios that are
color-coded in Fig. .
In the first scenario (blue curves), we have eliminated both near-surface
convection and deep-firn dispersion to show the δ15N gravitational
signal in the absence of macroscopic mixing (Fig. c). It is
clear that the δ15N data in the lock-in zone are
depleted about 40 per meg relative to gravitational equilibrium. In the second scenario (red
curves) we add the convective and dispersive mixing, and find an improved fit
to all tracers except δ15N. This second scenario is equivalent to
best current modeling practices, and comparable to methods used above at the
NEEM and WAIS Divide sites. We must conclude that our best practices cannot
account for the anomalous deep-firn δ15N signal seen at DSSW20K
(Fig. c) and DE08 (not shown).
In the third scenario (yellow curves), we attempt to fit the δ15N
data simply by increasing the magnitude of Ddisp. We have to
increase dispersion 150-fold to simulate δ15N correctly; however
doing so seriously compromises the model fit to all other tracers
(Fig. a–b), showing this approach to be invalid.
Diffusion and dispersion in layered firn. (a) Schematic of
conceptual model where the horizontal blue lines depict dense firn strata
with limited vertical connectivity. (b) Layering as implemented in
the firn air diffusion model; the zoom is on four annual layers just above
the lock-in depth.
In all modeling scenarios so far we have assumed that molecular diffusion and
dispersion are both one-dimensional processes that vary smoothly with depth
and that occur independently without interactions between them. In reality, the pore space
is a three-dimensional network that is strongly impacted by density layering
. Diffusion and dispersion
have generally opposite relationships to density. Thus, near the lock-in zone
of a firn with strong density layering, we should expect to see alternating
bands of high diffusivity (associated with low-density strata) and bands with
high dispersion (associated with high-density strata). The high-density
strata have a larger fraction of closed bubbles, and therefore relatively few
vertically connective pathways; these pathways will channel the
barometrically induced flow, thereby becoming focal points of dispersive
mixing. This situation is depicted schematically in Fig. a. In
this conceptual model, dispersion dominates transport in the vertical
direction, which leads to a strongly reduced gravitational enrichment.
However, in the transverse directions molecular diffusion still dominates,
particularly in the low-density strata. In a layered firn it is thus
conceivable that a mixed zone exists between the
molecular-diffusion-dominated diffusive zone and the dispersion-dominated
lock-in zone, in which gravitational enrichment is very weak, yet molecular
diffusion is still active.
Next we attempt to capture the dynamics of such a layered firn in our
one-dimensional model. In our fourth modeling scenario (green curves), we use
an idealized layered firn model, with alternating annual bands of diffusive
and dispersive mixing; details are shown in Fig. . The
dispersion has been set to perfectly compensate the reduced molecular
diffusion, and therefore the total mixing is comparable to that of scenario
2. We let the compensation be perfect for CO2, which means it is imperfect
for other trace gases. We find that using such a layered approach can
simulate both the regular tracers and δ15N correctly
(Fig. a–c). These modeling results confirm that
δ15N enrichment can cease before the lock-in depth is reached. The
common practice of defining the lock-in as the depth where δ15N
enrichment stops e.g., may thus be invalid at sites
that have both strong layering and large barometric variations. Instead, the
lock-in depth should be identified with transient tracers such as CO2 and
CH4, using their sharp inflection point. However, our fourth modeling
scenario is still flawed as it attempts to represent what is fundamentally a
three-dimensional process into a one-dimensional model. Furthermore our choice of perfectly
compensating variations in dispersive and diffusive mixing is of course
questionable – it is included here as an illustrative example only. To gain
a meaningful representation of these processes it may be necessary to move to
firn air transport models of two or more dimensions.
Discussion
Synoptic-scale barometric pumping strength for Greenland and
Antarctica using 6-hourly ERA-Interim reanalysis surface pressure values for
the period 1 January 2010 through 31 December 2011. The color scale gives the
root mean square of the pressure rate of change dp/dt in
units of mbar day-1. Key ice core drilling locations are indicated.
Implications for firn air and ice core studies
There are probably three factors that contribute to the magnitude of
dispersive mixing at any given site.
Magnitude of barometric variability. Figure
shows maps for Greenland and Antarctica of the time-averaged root mean square
of the surface pressure change dp/dt expressed in
mbar day-1; this is a good proxy for the barometric pumping power
available at a given site, as the air flow velocity scales with the rate of
pressure change. The maps suggest that barometric pumping is strongest near
the ice sheet margins, and weakest in the interior. The Dome A ice core
drilling site at Kunlun station has the smallest barometric pumping of all
sites, whereas coastal cores such as Law Dome and James Ross Island (JRI)
have strong barometric pumping.
Firn column thickness. The amount of barometrically driven
air flow at any given depth depends on the total size of the air reservoir
below that depth (which is why air flow decreases with depth in
Fig. ). All other things being equal, a thicker firn column will
have higher air flow velocities and hence more dispersive mixing than a thin
column.
Layering. As argued in Sect. , firn density
layering can enhance dispersive mixing by increasing medium heterogeneity,
specifically through the formation of high-density, high-dispersivity layers.
Melt layers and ice lenses may similarly act as focal points for dispersive
mixing.
The NEEM and WAIS Divide sites have comparable firn thickness and density
layering, and therefore the stronger barometric variability at the WAIS
Divide site results in stronger dispersive mixing at that site
(Fig. ). The Law Dome DE08 and DSSW20K sites experience
the same barometric variations, yet DE08 has a thicker firn column and
DSSW20K has more (annually spaced) high-density layers. It is therefore not clear a
priori which of these two Law Dome sites has stronger dispersive
mixing, and unfortunately the available firn air data are of insufficient
resolution to establish this unambiguously.
Dispersive mixing influences the ice core record in several ways, the most
important of which is via the broadening of the gas age distribution. A
comparative firn model study at the NEEM site showed that the low-diffusion
lock-in zone environment contributes more to the broadening of the final age
distribution than the diffusive zone does . The weak
barometric variability in interior Antarctica
(Fig. ) may be part of the reason why gas records
from central East Antarctic ice cores such as EPICA Dome C have surprisingly little smoothing, as evidenced by the fact that
abrupt methane transitions during e.g., the Younger Dryas period and
Dansgaard–Oeschger cycle are well preserved .
Dispersive mixing potentially has implications for the use of δ15N
as a proxy for past firn column thickness, which is an additional constraint
on past Δage
.
The cited studies all assume δ15N reflects the thickness of the
diffusive zone, which is then used to estimate the lock-in depth by adding an
estimated convective zone thickness. As we showed in Sect. ,
under circumstances of strong layering and intense barometric pumping,
δ15N may underestimate the lock-in depth where Δage is fixed
– this effect is likely to be important at coastal coring sites such as Law
Dome, James Ross Island, Berkner Island, and Roosevelt Island where
barometric variability is strong, as well as at sites influenced by melt
layers. It has been suggested that glacial firn has more pronounced layering
than present-day firn . However, climate models
participating in PMIP2 (Paleoclimate Modeling Intercomparison Project
Phase 2) disagree on the sign and magnitude of the change in cyclonic
activity around Antarctica between the Preindustrial and Last Glacial Maximum
. It is therefore conceivable, but highly uncertain, that
dispersive mixing was stronger during glacial times. Several studies have
noted a δ15N model–data mismatch in central Antarctica during
glacial climate conditions, with densification models simulating a thickening
of the firn column (relative to present) and δ15N data suggesting a
thinning . We speculate that enhanced glacial
dispersive mixing could contribute to the observed low gravitational
enrichment in δ15N. Overall, more multi-tracer, high-resolution
firn air studies at sites with strong barometric variability will be needed
to better understand the influence of barometric pumping on gravitational
enrichment and the δ15N proxy.
86Kr excess as a potential proxy of past synoptic activity
The degree of isotopic gravitational enrichment of any given gas species in
the firn depends on the relative strength of molecular diffusion, which acts
to drive isotopic enrichment towards gravitational equilibrium, and
macroscopic transport processes (convection, advection and dispersion), which
act to erase the enrichment. Slow-diffusing gases such as krypton (Kr) and
xenon (Xe) will therefore always be less isotopically enriched than
fast-diffusing gases such as N2 and argon (Ar). This effect was first
observed by , who studied deep air convection (a form of
macroscopic transport) at the Antarctic Megadunes site
.
Here we define 86Kr excess as
86Krexcess = δ86Kr / 82Kr|tc-δ40Ar / 36Ar|tc,
where the subscript “tc” (“thermally corrected”) denotes that the
isotopic ratios have been corrected for thermal fractionation either by using
a thermal model or by paired δ15N–δ40Ar data
. Due to the different molecular
diffusivities of N2 and Kr, they are in a different state of gravitational
disequilibrium, which makes 86Krexcess a measure of the
aggregate strength of non-diffusive transport processes in the firn column.
There are several reasons for using 86Kr and 40Ar rather than e.g.,
136Xe and 15N. First, Kr is over 10 times as abundant in the
atmosphere as Xe and therefore more easily measured. Second, δ40Ar
is more precise in terms of gravitational signal-to-noise ratio, and less
affected by thermal diffusion than δ15N. Third, δ40Ar
and δ86Kr can both be measured on air extracted from the same
physical ice sample after removal of O2, N2, and other reactive gases
via gettering . Because δ86Kr is always
less gravitationally enriched than δ40Ar, 86Kr excess is
always negative.
Figure d shows 86Kr excess for all four modeling
scenarios discussed in Sect. . In scenario 1 (blue curves)
the only macroscopic transport mechanism is advection, which results in a
86Krexcess of -5 per meg. Adding convection and modest
dispersion (scenario 2, red curve) increases 86Krexcess in
magnitude to -10.2 per meg, reflecting the increased degree of
gravitational disequilibrium in the firn column. Both scenarios with strong
dispersion (scenarios 3 and 4, yellow and green curves, respectively) show a
further increase in 86Krexcess magnitude to 16.5–18 per meg.
Measurements on the WAIS Divide ice core show
86Krexcess values of around -35 per meg during the late
Holocene (analytical precision is better than 20 per meg). However, older
sections of the core show 86Krexcess values as low as -90
per meg , suggesting periods of greatly enhanced
gravitational disequilibrium in the firn column (data not shown). One
tantalizing interpretation could be that these very negative
86Krexcess values represent periods of enhanced synoptic
activity in (West) Antarctica, driving strong dispersive mixing.
We propose here that 86Krexcess may act as a proxy for past
synoptic-scale pressure variability (or storminess/cyclone density). Before
this interpretation can be accepted, however, additional work is needed. The
large spatial variability in barometric variability over Antarctica
(Fig. ) provides a valuable opportunity to verify
whether ice core 86Krexcess indeed scales with local
barometric variability. Additional work is needed to reliably correct
86Krexcess for the influence of advection and convection in
the firn column , preferably through detailed studies of
86Krexcess in modern-day firn. If corroborated,
86Krexcess could hold important clues about past changes to
the large-scale atmospheric circulation, particularly when combined with
reconstructions of past wind conditions from surface roughness
. In Greenland, changes in synoptic activity are
presumably linked to the buildup of the Laurentide ice sheet and its
orographic influence on the storm tracks . In Antarctica,
synoptic activity may be linked to e.g., meridional movement of the
eddy-driven jet , atmospheric teleconnections to the
tropical Pacific , or the position and depth of the Amundsen
Sea low .
Summary and conclusions
In this work we show that surface pressure variability on synoptic timescales
drives macroscopic air movement in the deep firn, which in turn leads to
dispersion of trace gases in the firn open porosity. The work resolves an
outstanding question regarding the nature of lock-in zone mixing deduced from
detailed firn air experiments at the north Greenland NEEM site .
We present a mathematical description of the propagation of pressure
anomalies in polar firn. We find that pressure variations on the timescale of
order 1 h or slower are propagated to the firn–ice transition at full
amplitude; variations on shorter timescales are attenuated. Net
barometrically driven air movement is on the centimeter scale in the deep
firn, and on the meter scale in the upper firn; mean velocities are of the order of 10-6 m s-1. The precise values of the air
displacement and velocity depend primarily on the firn thickness and the
barometric variability at the site.
We use published firn sample gas chromatography experiments to estimate the
dispersivity αL of firn, which is the proportionality constant
in the relationship between superficial gas velocity and the apparent
dispersion strength (Eq. ). We find that
αL scales exponentially with the open porosity sop,
with αL≈0.1 m at the lock-in depth. Combining
simulated air velocities and firn dispersivity, we calculate dispersive mixing
in the deep firn to be of the order of 10-9 m2 s-1,
with precise values again depending on firn thickness and the barometric
variability at the site.
We apply these theoretical estimates of dispersion in a firn air transport
model, and find that they overestimate the amount of lock-in zone
dispersivity needed to optimize the fit to firn air trace gas measurements;
this mismatch may be due to the fact that our firn dispersivity
parameterization is based on steady-flow conditions, whereas barometric
pumping induces a time-variable flow. We suggest that strong dispersive
mixing at Law Dome, Antarctica, in combination with firn layering, may halt
gravitational enrichment in δ15N before the lock-in zone is
reached.
The dispersive mixing discussed here increases scientific understanding of
firn air transport, and has direct implications for the modeling thereof. The
ice core record is impacted by dispersive mixing primarily through the
widening of the gas age distribution. We propose that 86Kr excess may be
an ice core proxy for past synoptic activity, which is linked to the
large-scale atmospheric circulation.