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.

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.

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.

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.

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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 .