WASR 8000 · Week 7
WASR 8000 · Environmental Tracers in Hydrology · Fall 2026

The Gases That Do Nothing

An interactive primer on the five gases that never react — and therefore remember only physics: the temperature at the water table when rain became groundwater, the air that was trapped and squeezed into solution on the way down, and the helium that the rock has been adding ever since. Recharge thermometry, excess air and its three models, the inverse fit that lets the data choose among them, and what the records say about past climate and contamination histories. Builds on Weeks 3, 5 and 6; no tracer background needed.

Week 7 companion · read alongside the three readings ≈ 50–60 minutes 13 interactive demos · self-check quiz
1 · The opening question

Five degrees colder — or an artifact of a bubble?

In 1995 a group led by Martin Stute measured the noble gases dissolved in old groundwater from a semi-arid aquifer in tropical Brazil and reported that the last glacial maximum had been about 5 °C cooler there than today — a number that landed in the middle of an argument about how much the tropics cooled. Four years later two other noble-gas specialists argued that the cooling was an artifact: the "excess air" in the samples was fractionated, the fractionation had been modelled inadequately, and the temperature had absorbed the error. A year after that, a new model of excess air was applied to the same data and the 5 °C came back. Same samples, three verdicts. Where would you stand?

Kipfer and colleagues tell the story in two sentences: the Brazilian result "was challenged on grounds of insufficient modelling of the excess air fractionation by Ballentine and Hall (1999), but later confirmed by using the CE-model for excess air (Aeschbach-Hertig et al. 2000)." Other tropical records then fell into line, "suggesting that the tropics and subtropics … cooled rather uniformly by around 5°C during the LGM." The number survived — but only because the bubble finally got a physical description that could be tested.

That is Week 7 in one dispute. Noble gases are the tracers that do nothing chemically, which is exactly why their concentrations can be read as a record of physical conditions: the temperature of the water table when the water went under, the salinity and pressure of the air it last touched, and the air that was trapped in the pores and forced into solution. But two of those signals — temperature and trapped air — are tangled in the same four numbers, and every reading depends on a model of what the trapped bubble did. The craft of this week is untangling them, and then pressing on the model.

The three readings divide the labour: Kipfer, Aeschbach-Hertig, Peeters & Stute (2002) is the full review — components, solubilities, the three excess-air models, the inverse method, lakes, dating and paleoclimate; Aeschbach-Hertig, Peeters, Beyerle & Kipfer (1999) is the method paper that turned "correct for excess air, then read the temperature" into a least-squares fit with honest error bars (note: it predates the CE model, which arrived in 2000); White (2015, Ch. 12) supplies the geochemical backdrop — where the noble gases came from, why helium has three parents, and what the isotope ratios say about crust, mantle and air. This page is a map to all three, not a substitute for reading them.

2 · Why noble gases

A tracer that does nothing tells you the most

Every tracer so far in this course carried a chemical liability. Nitrate is eaten; CFC-11 is degraded under anoxia; carbon exchanges with the aquifer; even water's own isotopes exchange with the rock in the hot crust. Helium, neon, argon, krypton and xenon have full outer electron shells; helium's first ionization potential, 2373 kJ mol⁻¹, is the highest of any element, and although the potential falls steeply down the group (xenon's, 1170 kJ mol⁻¹, sits below hydrogen's), none of the five forms compounds or joins reactions in nature (the heavier ones, xenon in particular, can be adsorbed on mineral surfaces, but they are never chemically bound). Once dissolved, their concentrations can change only for physical reasons. That is the whole method. Kipfer et al. write the measured concentration of gas i as a sum of parts:

Ci,m = Ci,eq + Ci,ex + Ci,rad + Ci,terKipfer et al. eq. 5

The four parts, in the order the water acquires them

  • Equilibrium (eq) — what Henry's law dissolves from moist air at the water's temperature, salinity and atmospheric pressure. This is the thermometer, and the salinometer and altimeter.
  • Excess air (ex) — the surplus, ubiquitous in groundwater, from air bubbles trapped in the pores and pushed into solution by the pressure of the rising water table. Typically 10–50 % supersaturation. This is the confounder — and, read the right way, a rain gauge.
  • Radiogenic (rad) — produced by decay inside the aquifer: ⁴He from U and Th, ³He from tritium (the "tritiogenic" clock of Week 5), rarely ⁴⁰Ar from ⁴⁰K and ²¹Ne from (α,n) reactions.
  • Terrigenic (ter) — gases that flow in from outside the aquifer: crustal helium from deeper rock, or mantle helium with its distinctive ³He/⁴He.

Only ⁴Herad, ³He (radiogenic and tritiogenic), occasionally ⁴⁰Arrad and very rarely ²¹Nerad are produced in amounts that show up in natural waters. Neon (mostly), argon, krypton and xenon in groundwater are therefore atmospheric gases only — which is why they can carry the recharge signal cleanly, and why helium needs special handling (Section 10).

Five gases, five different characters

Pick a gas. Solubility in water increases with atomic mass (White pp. 418–419; helium the least soluble, xenon the most — the reverse of their order in magma); diffusivity decreases with mass; atmospheric abundance is wildly uneven (argon is nearly 1 % of air, xenon 87 parts per billion). Those three facts set which gas responds to which process.

Keep the paradox in view all week: the noble gases are useful because they are useless to chemistry. A δ¹⁵N of nitrate had to be defended against three processes that could forge it; a xenon concentration has to be defended against only one — the trapped bubble — and against the arithmetic of separating it out.
3 · The equilibrium component

Henry's law is the thermometer

Water in contact with air dissolves each gas until the dissolved concentration is proportional to the gas's partial pressure. The proportionality constant depends on temperature (cold water holds more gas, and holds proportionally much more of the heavy gases) and on salinity (salt water holds less). The partial pressure depends on the gas's mixing ratio in dry air and on the total pressure minus the water-vapour pressure — which is how altitude enters. Kipfer et al. write all of it into one equilibrium concentration, Ci,eq(T, S, P), using the moist-air solubilities of Weiss (He, Ne, Ar, Kr) and Clever (Xe):

pi = zi · [ptot − ew(T)] ,   ptot(h) ≈ ptotsl · e−h/hatm ,   Ci,eq = Ci,eq(T, S, ptot)Kipfer et al. eqs. 3–4; Table 1B

Solubility explorer — five gases on fixed axes

Each panel is drawn in absolute units (cm³ STP per gram of water) on its own fixed axis, 0–30 °C. Move temperature and watch where the marker sits on each curve; add salt or altitude and watch the whole curve sink. The percentages in the tiles are the local slopes of the equilibrium curves — how much each gas's equilibrium concentration changes per degree at your temperature (the next section's Table 2 quotes the change of a sample's total concentration, excess air included, over a one-degree step, hence its slightly smaller numbers). Computed from the parameterizations in Kipfer et al.'s Table 1B; the values reproduce their Table 1 (S = 0.1 ‰, 1 atm) to within 0.3 %.

10.0 °C
0 ‰
0 m
Check against Kipfer et al. Table 1 (fresh water, 1 atm)
THe ×10⁻⁸Ne ×10⁻⁷Ar ×10⁻⁴Kr ×10⁻⁸Xe ×10⁻⁸

Table rows: Kipfer et al. (2002) Table 1, S = 0.1 ‰; page values computed at S = 0.1 ‰ for the comparison.

The precision that makes this a thermometer: a good laboratory measures each concentration to about 1 % and the elemental ratios to better than 0.6 %. Xenon's equilibrium concentration changes by about 3.5 % per degree at 10 °C, so a 1 % xenon measurement is, on its own, a 0.3 °C thermometer. Aeschbach-Hertig et al. (1999) equilibrated water with air in temperature-controlled rooms at 4, 15 and 30 °C and recovered the room temperatures to ±0.2 °C — the method's proof of concept.
The catch that sets up the rest of the week: the same four gases also respond to salinity, to pressure (altitude), and above all to trapped air. Four measurements, five unknowns. The next section shows what each unknown looks like across the five gases — the "fingerprint" that makes some pairs separable and others hopeless.
4 · Five knobs, five fingerprints

Why temperature and excess air can be told apart — and salinity and pressure cannot

Aeschbach-Hertig et al. built a synthetic sample (10 °C, fresh, 1 atm, with 3 × 10⁻³ cm³ STP g⁻¹ of dissolved air) and nudged one parameter at a time. The chart is their Table 2: how much each gas's concentration moves for a one-degree warming, a one-per-mil salting, a 0.01 atm rise in pressure, a small extra dose of air, and a small diffusive loss of the excess (the "R" of the partial re-equilibration model). Hover the bars.

Percent change in each dissolved gas for a unit change in one parameter

Transcribed from Aeschbach-Hertig et al. (1999), Table 2. Base state: T = 10 °C, S = 0 ‰, P = 1 atm, A = 3 × 10⁻³ cm³ STP g⁻¹, R = 0.

Read the shapes, not the numbers

  • Temperature bites hardest on the heavy gases: xenon −3.35 % per °C, helium only −0.32 %. The fingerprint is a steep ramp from He to Xe.
  • Excess air is the mirror image: it fattens helium and neon (the least soluble gases, so a little air is a large fraction of what was dissolved) and barely touches xenon. A ramp the other way.
  • Pressure and salinity both ramp gently with mass — pressure because, with excess air present, it "is relatively more important for the heavy noble gases"; salinity because its effect "increase[s] with molar mass". The two ramps are so alike, and so like a combination of temperature and excess air, that "especially the effects of P and S are very similar".
  • Re-equilibration (diffusive loss of the excess) strips helium and neon most, because they diffuse fastest and because most of their excess was excess air to begin with.

The consequence

Two parameters with different fingerprints can be separated from one sample; two with the same fingerprint cannot. Temperature and excess air are opposite ramps — "readily identifiable", in Kipfer et al.'s words. Salinity and pressure are nearly the same gentle ramp — "hard to separate", and the errors explode if you try (Section 9's ladder: fitting T and P together costs ±0.62 °C; T, S and P together, ±7.6 °C). This is why a noble-gas study fixes salinity (fresh recharge, S ≈ 0) and pressure (from the altitude of the recharge area) before it fits temperature and air.

The ramp for excess air also explains a rule of thumb you will meet in every paper: excess air is measured by neon. Neon has no significant radiogenic or terrigenic source, so any neon above equilibrium is trapped air — and the relative neon excess, ΔNe, becomes the currency in which excess air is quoted.

5 · The bubble in the pore

Excess air: the ubiquitous surplus

Groundwater almost always holds more dissolved air than a lake would at the same temperature. Heaton and Vogel named it "excess air" (1979, 1981) after finding it in several South African aquifers of different lithology, hydrology and climate, and it has turned up in "virtually all noble gas studies of ground water" since. The mechanism is in the name of the zone where it forms: just above and below a rising water table, up to 10–20 % of the pore space is occupied by immobile bubbles of trapped soil air. The rising water loads them hydrostatically; pressure forces gas into solution; and because the bubbles cannot escape, the surplus stays. Excess air is quoted as the relative neon excess:

ΔNe = (CNe,m / CNe,eq − 1) × 100 %Kipfer et al. eq. 15

The arithmetic of a bubble

  • How much air? Unfractionated excess air of Ad cm³ STP per gram adds Ad·zi of each gas. A dose of 10⁻³ cm³ STP g⁻¹ raises neon by about 9 % at 10 °C and 10 % at 22 °C (Aeschbach-Hertig et al.: 1 % ΔNe per 10⁻⁴ cm³ STP g⁻¹, "exact for T = 22.4 °C, P = 1 atm, and S = 0 ‰"); Kipfer et al.'s rule of thumb is 10 % ΔNe per 10⁻³ cm³ STP g⁻¹, "requiring at least 0.05 atm overpressure".
  • How much pressure? To hold 2 × 10⁻³ cm³ STP g⁻¹ of air permanently in solution at 13 °C takes an excess pressure of about 0.1 atm — roughly a 1 m column of water on top of the bubble, or the surface tension of a bubble about 30 µm across (Kipfer et al., eq. 14 and the discussion after it).
  • Is that plausible? "Water table fluctuations of the order of 1 m probably occur in most recharge areas." Oxygen consumption in soil air could raise the other gases' partial pressures by up to 25 % if the O₂ were entirely removed, but it is usually not, so that route "is questionable" as a 0.1 atm source.
  • Where does the excess sit? In the light gases. The stack on the right shows a real sample from southern France read with unfractionated excess air — the neon is 29 % above equilibrium as published (27 % when recomputed here at sea level), the xenon barely 2 %.

Component stack — after Kipfer et al. Fig. 4

Each bar is the measured concentration as a percentage of equilibrium at the recharge temperature (100 % line). Set the temperature, the excess air, the radiogenic ⁴He and the tritiogenic ³He, and watch which gases the bubble and the rock reach.

10.8 °C
3.0 ×10⁻³
+12 %
+45 %
equilibriumexcess airradiogenic ⁴Hetritiogenic ³He

The sample in Fig. 4 was "interpreted assuming unfractionated excess air" — the simplest of three possible stories about the bubble. Section 7 tells the other two, and Section 1's dispute is what happens when the story is wrong.
6 · Two gases, two unknowns

The neon–xenon diagram: reading temperature and air off one plot

Because neon answers to excess air and xenon to temperature, a plot of one against the other separates the two by geometry. Air-saturated water (ASW) traces a curve as temperature changes; adding plain air moves a sample off that curve along a straight line whose slope is the atmospheric Xe/Ne ratio. Read a sample back along the line to the curve and you have its recharge temperature; the length of the line is its excess air. Kipfer et al.'s Fig. 8 does exactly this for an alluvial aquifer in Switzerland: shallow winter samples land at 7.4 °C, shallow summer and autumn samples at 9.4 °C, deep samples at 8.7 °C.

Ne versus Xe, absolute units, fixed axes

ASW curve for 0–30 °C at your altitude; thin line = addition of unfractionated air to the sample; dashed = the same sample if its excess air were fractionated (CE model, F = 0.5). Hover the curve for temperatures.

8.7 °C
1.5 ×10⁻³
400 m

The three presets place samples at the three group temperatures of Fig. 8 with illustrative excess-air amounts; the figure's scatter along each line is the range of excess air among the wells. The mean temperatures from Ne and Xe alone (winter 7.4, summer 9.4, deep 8.7 °C) "lie close to the mean temperatures calculated from all noble gases (7.4, 9.2, and 8.6 °C, respectively)".

The disadvantage Kipfer et al. name is on the plot: to draw the line you must assume the slope — that is, the elemental composition of the excess air. Toggle the dashed line in your head: if the bubble was only partly dissolved (CE model), the excess is enriched in xenon relative to air, the line steepens, and reading the sample back along the atmospheric slope lands on the ASW curve at the wrong temperature — too cold, because the xenon the correction failed to remove reads as cold water. That is the Brazil dispute in one line. With argon and krypton measured as well, the assumption can be tested instead of made — Sections 7 and 9.
7 · Three stories about the same bubble

Unfractionated air, partial re-equilibration, closed-system equilibration

The excess is rarely plain air. "There is increasing evidence that excess air tends to be fractionated relative to atmospheric air, with an enrichment of the heavy gases," and any model of recharge conditions must say how. Kipfer et al.'s Table 2 lists the candidates. Each is a story about what a trapped bubble does; each has one or two parameters; and, disconcertingly, "in many cases all of them provide a reasonable fit to the measured concentrations of Ne, Ar, Kr, and Xe." Where they differ is in what they predict for helium, for the isotope ratios, and for the physical plausibility of their own parameters.

ModelWhat the bubble doesExcess of gas iParametersReference
UA · unfractionated excess airBubbles dissolve completely. The excess has the composition of air.Ad · ziAd: dissolved air (cm³ STP g⁻¹)Heaton & Vogel (1981)
PR · partial re-equilibrationBubbles dissolve completely, then part of the excess diffuses back out across the water table; light, fast gases leave first.Ad · zi · e−R·Di/DNeAd: initial dissolved air; R: degree of re-equilibrationStute et al. (1995b)
MR · multi-step partial re-equilibrationThe same, in n repeated dissolve-and-degas steps.Ad · zi · e−Ri (1 − e−nRi)/(1 − e−Ri)Ad, R per step; n stepsKipfer et al. (2002)
CE · closed-system equilibrationBubbles dissolve only partly; water and the remaining trapped air reach a new equilibrium at elevated pressure. No diffusion.(1 − F) Ae zi / (1 + F Ae zi/Ci,eq)Ae: entrapped air; F: reduction of its volume by dissolution and compressionAeschbach-Hertig et al. (2000)

The UA model is the special case R = 0 or F = 0 of the others, so "the problem of model choice essentially reduces to the PR / MR and CE models." The 1999 paper you are reading works with UA and PR; CE was published the following year and is the subject of Section 8.

Model lab — what each story does to the excess

Left: the elemental pattern of the excess relative to air, normalised to neon (a value of 1 means "the same proportion as in air"; UA is flat at 1 by definition). Right: the He/Ne ratio of the excess, Lex, against the model's fractionation parameter — a reproduction of Kipfer et al.'s Fig. 7 at 10 °C, with your setting marked. The atmospheric ratio Lair = 0.288 is the ceiling for every model; the ratio in air-saturated water, Leq, is the floor for CE but not for PR.

3.0 ×10⁻³

Excess pattern relative to air (Ne = 1)

Lex = (He/Ne)ex against F (CE) or R (PR) — after Fig. 7

Why PR fell out of favour. To explain strongly fractionated samples, PR needs enormous initial excesses that are then almost all lost again: Stute et al.'s Brazilian aquifer required initial neon excesses of about 300 %, "corresponding to Ad-values of up to 3 × 10⁻² cm³ STP g⁻¹. Complete dissolution of such amounts of air would require excess pressures of more than 1.5 atm, or — if hydrostatic pressure dominates — water table fluctuations of more than 15 m, which seem rather unrealistic." Aeschbach-Hertig et al.'s own Botswana samples needed 290 % (25 × 10⁻³ cm³ STP g⁻¹) degassed by a factor of five; their July samples from the Töss valley needed 100 % and a loss of more than 80 %. PR also predicts diffusive fractionation of neon isotopes — a lowered ²⁰Ne/²²Ne — which "has not been observed in field studies."
Why CE won, provisionally. It explains the enrichment of heavy gases without any diffusion, with parameters that mean something: Ae ≈ the trapped air-to-water volume ratio, and the pressure on the bubble. Analysis of ²⁰Ne/²²Ne "provides the best option to distinguish between the models and so far appears to favor the CE-model." But read the hedge that follows: "Field and laboratory studies under various conditions are needed to understand which model provides the best approximation of reality under which conditions." That sentence is your Block 2 discussant's opening.
8 · The closed-system equilibration model

A bubble that dissolves only as far as the pressure lets it

The CE model's idea is simple enough to draw. Start with air-saturated water and a finite volume of trapped air per gram of water, Ae. Raise the water table; the bubble is now under a total pressure Ptot greater than atmospheric. Gas dissolves until the water is in a new equilibrium with what is left of the bubble — in a closed system, with nothing escaping. The excess of each gas is then:

CCEi,ex = (1 − F) Ae zi / (1 + F Ae zi/Ci,eq) ,   F = v/q ,   v = Vg/Vg0 ,   q = (Ptot − ew)/(P − ew)Kipfer et al. eqs. 27–28

v is the fraction of the trapped gas left in the bubble, q the pressure on it relative to the atmosphere; "any pair of the parameters Ae, F, q, and v fully determines the amount and composition of excess air, the most intuitive choice being Ae (≈ air / water volume ratio) and q (≈ pressure exerted on the entrapped air)." The three are coupled by one physical requirement: the partial pressures in what is left of the bubble must add up to Ptot. Given Ae, q and the temperature, that requirement fixes F — which is what the lab below solves, using the solubilities of N₂, O₂ and Ar, the gases that make up the bubble.

CE lab — set the bubble and the pressure, read the excess

10.0 °C
0.030
1.20

Excess of each gas relative to its equilibrium concentration

Fixed 0–200 % axis. Coloured = the CE result; grey = unfractionated air with the same neon excess (what a UA reading of this sample would assume for the other gases).

ΔNe against trapped air at your q — the pressure, not the reservoir, sets the scale

What the lab shows: with a small bubble the water swallows it whole and the result is UA — the excess grows in proportion to Ae. Past a threshold (about 2 × 10⁻³ cm³ STP g⁻¹ per 0.1 atm of overpressure at 13 °C — eq. 14), the bubble survives, the excess is fractionated toward the heavy gases, and the neon excess stops depending on how much air was trapped: it peaks near 2(q − 1) just past the threshold and declines slowly toward q − 1 as the reservoir grows, so from there on q, not Ae, sets the scale. That is Kipfer et al.'s central claim about excess air: its size "is limited not by the available air reservoir Ae, but by the pressure acting on that reservoir (q)", so ΔNe "is essentially a measure of the pressure on the entrapped air."
Typical numbers, and what they imply. Six aquifers gave Ae of 0.02–0.04 cm³ STP g⁻¹ — "a few percent of the pore space were occupied by entrapped air during infiltration" — with q ≈ 1.2 for three temperate aquifers and ≈ 1.5 for three semi-arid ones. Read as hydrostatic load, those q values mean water-table amplitudes of about 2 and 5 m, "probably at the upper limit of what can be expected." Notice the ratio: Ae ≈ 10⁻² but the air that ends up dissolved (the UA-equivalent Ad) is ≈ 10⁻³. "It seems that typically more air is entrapped than can be dissolved at the prevailing pressure."

Press hard here — the assumptions the CE model carries

The course plan asks Block 2 to lean on the excess-air models. Here is where the weight goes; each item is drawn from the readings' own caveats.

9 · Letting the data choose

Four numbers, three unknowns, one statistic

The old way was iterative: subtract a guessed amount of atmospheric excess air from each gas, compute a temperature from each corrected concentration, and adjust the guess until the four temperatures agree. It works, and the scatter of the four temperatures is a rough error bar. Aeschbach-Hertig et al. (1999) replaced it with an inverse fit: treat T, S, P and the excess-air parameters as unknowns, fix the ones you know (fresh recharge, S ≈ 0; altitude gives P), and find the rest by minimising the weighted misfit

χ² = Σi (Cimeas − Cimod)² / σi²Kipfer et al. eq. 32; Aeschbach-Hertig et al. eq. 12

The weights are the measurement errors, which buys three things: every gas counts in proportion to how well it was measured; the covariance of the fit gives honest parameter errors (and their correlations); and the minimum χ² itself is a test of the model. With n measured gases and m fitted parameters, a correct model gives χ² ≈ ν = n − m on average. If the probability of a χ² that large is below a cut-off — "pc = 0.01 proved to be appropriate", in Kipfer et al.'s later practice; the 1999 paper itself rejected at p < 0.05 — the model is rejected for that sample. Apply the same model to N samples and the summed χ² tests it against the whole data set, "with a much larger number of degrees of freedom": a model can fail a data set while passing every sample in it.

χ² lab — synthetic samples with a hidden truth

Choose the world that made the sample (its true temperature and excess-air story are hidden until you fit), add 1 % measurement noise to Ne, Ar, Kr and Xe, and fit each of the three models. Watch χ², the degrees of freedom, the p-value, the verdict at pc = 0.01 — and what the wrong model does to the temperature. New noise draws give the sampling spread; Aeschbach-Hertig et al.'s Monte Carlo of 1,000 such draws returned T = 10.007 ± 0.216 °C and A = (2.995 ± 0.147) × 10⁻³ for a UA world at 10 °C.

The sample: measured concentrations relative to ASW at the fitted temperature

Bars: measured ÷ equilibrium at the currently fitted T (100 % = equilibrium). Error bars: ±1 % (1σ). Hollow markers: the fitted model's prediction.

Pick a model and fit.
Fit history for this sample
ModelT (°C)Excess airχ²νpVerdict

The uncertainty ladder — what each extra unknown costs

Kipfer et al.'s Table 3: the 1σ error of the fitted parameters for a synthetic sample at 10 °C, S = 0, 1 atm, Ad = 3 × 10⁻³, with 1 % errors on all four concentrations, as the set of free parameters grows. Choose which error to display.

Temperature alone: ±0.19 °C. Temperature with unfractionated air: ±0.21 — excess air costs almost nothing when its fingerprint is so different. Fractionated air (T, Ae, F or T, Ad, R) enlarges it "by a factor of two to three". Temperature with pressure: ±0.62; with salinity: ±0.82; and all three together, ±7.6 °C — "it is practically impossible to simultaneously fit T, S, and P."
The sentence that licenses paleoclimate: "each individual parameter of the equilibrium component can be well determined even in the presence of fractionated excess air. This fact provides the basis for the application of noble gas concentrations in ground water as indicators of paleotemperature." The price is that you must know the altitude and the freshness of the recharge — or find them another way.
10 · Helium

The gas with three parents

Neon through xenon in groundwater are atmospheric; helium is not. Its ⁴He has an atmospheric part (equilibrium plus excess air), a terrigenic part from the rock (crustal or mantle), and its ³He has all of those plus the decay product of tritium — the Week 5 clock. Separating them is bookkeeping with three equations (Kipfer et al. eqs. 29a–c), and the bookkeeping hinges on two assumptions: the He/Ne ratio of the excess air, Lex, and the ³He/⁴He ratio of the terrigenic helium, Rter.

Nem = Neeq + Neex  ·  ⁴Hem = ⁴Heeq + Lex·Neex + ⁴Heter  ·  ³Hem = Req·⁴Heeq + Rex·Lex·Neex + Rter·⁴Heter + ³Hetrieqs. 29a–c

Helium separator

A sample at the chosen temperature, with the neon excess, total ⁴He and ³He/⁴He you set. Change the two assumptions and watch ⁴Heter, ³Hetri and the ³H–³He age move.

10.0 °C
20 %
1.22
1.16
10 TU

Where the ⁴He and ³He come from — measured totals split into parts

equilibriumexcess airterrigenictritiogenic
The negative-helium diagnostic. "In all previous studies, only He and Ne were measured and the assumption Lex = Lair was applied. Because Lair is actually only the upper limit of Lex, this approach tends to overestimate the atmospheric and thus to underestimate the non-atmospheric He components. A clear sign that this approach is not always appropriate is the common occurrence of negative values for ⁴Heter or even ³Hetri." In the Cape Cod sewage-plume study, 55 of 91 samples came out with negative terrigenic helium. The physical requirement ⁴Heter ≥ 0 sets an upper limit on Lex for each sample; most of those limits were above 0.25, "well within the range predicted by the CE-model."
The ratio ladder (White, Ch. 12). ³He/⁴He is quoted as R/Ra, relative to air's 1.384 × 10⁻⁶. Crustal rocks: 0.01–0.1 (the radiogenic production ratio is only 0.4–1 × 10⁻⁸). Mid-ocean-ridge basalt: 8.8 ± 2.5. Mantle-derived rocks generally, ocean-island basalts included: 5–50. The Solar System's primordial value: about 120; the solar wind: about 334–336. Air is a mixture of what leaks from crust and mantle, and helium is "the only element for which the Earth is not a closed system" — it bleeds to space with a residence time of only 10⁶–10⁷ years. For hydrology the ladder is a diagnostic: a terrigenic component near 0.015 Ra (the 2 × 10⁻⁸ of the chip) is crustal and barely perturbs the tritiogenic budget; one near 7 Ra is mantle helium, and the ³He it brings can swamp the tritium clock.
11 · The non-atmospheric gases as clocks

What accumulates, and what decays

Once the atmospheric parts are separated, what remains is time. Tritiogenic ³He grows as tritium decays, from the moment the water is cut off from the air; radiogenic ⁴He accumulates from the rock for as long as the water stays underground; and five radioactive noble-gas isotopes, made in the atmosphere or by industry, decay on schedules from days to hundreds of thousands of years. Between them they cover, in Kipfer et al.'s summary, "months to about 50 years" (³H–³He, ⁸⁵Kr, with ²²²Rn for the first 20 days), "thousands to millions of years" (⁴He, ⁴⁰Ar, ⁸¹Kr), and the awkward 100–1,000 years in between (³⁹Ar).

Half-lives and windows from Kipfer et al. (2002). Sample volumes are the ones they quote for the rare isotopes: about 500 L of water for ³⁹Ar and about 15,000 L for ⁸¹Kr, whose atmospheric abundances are near 10⁻¹⁵ of their element; ⁸⁵Kr/Kr is near 10⁻¹¹.

The ³H–³He clock, from isolation onward

Kipfer et al.'s Fig. 9, computed: while the water is open to the air, its tritiogenic ³He escapes and stays at zero; once isolated, ³Hetri grows exactly as ³H decays. Their ratio is a unique function of the time since isolation — the ³H–³He age — and, unlike either concentration alone, needs no input history.

τ = (1/λ) · ln(1 + ³Hetri/³H) ,   λ = 0.05626 yr⁻¹ (t½ = 4500 d = 12.32 yr) ,   1 cm³ STP g⁻¹ = 4.019 × 10¹⁴ TUeq. 33
21.0 yr
³H (normalised to 1 at isolation)³Hetri³Hetri/³H (lower panel)

Two cautions from the text. For τ ≪ 1/λ the age is linear in the ratio (eq. 34) but "not of ³H", so a mixture's apparent age "is always biased towards the component with the higher ³H concentration." And in the unsaturated zone the ³He escapes: the clock starts at the water table, not at the ground surface — which is why deep unsaturated zones lag the input functions of ⁸⁵Kr, CFCs and SF₆ as well.

The ⁴He accumulation age — and its two rate constants

If ⁴He accumulates at a constant rate JHe, the residence time is simply τ = ⁴Herad/JHe (eq. 35). "The problem is to determine the accumulation rate." In situ production from the aquifer's own uranium and thorium (eq. 36, Andrews & Lee 1979) is calculable; a flux from the whole crust below (Torgersen & Clarke 1985) is not — and the two can differ by orders of magnitude. Set both and compare the ages.

JHe = ΛHerw) (CUPU + CThPTh) (1 − Θ)/Θ ,  PU = 1.19 × 10⁻¹³, PTh = 2.88 × 10⁻¹⁴ cm³ STP µg⁻¹ yr⁻¹eq. 36
3.0 µg g⁻¹
10.0 µg g⁻¹
0.20
100 m
3.2 ×10⁻⁷

The point of the two rate constants is Week 10's whole argument in miniature. In situ production alone gave ⁴He ages "significantly larger than ¹⁴C ages" in the first studies, which is how the whole-crust flux hypothesis was born; that hypothesis "is far from being universally accepted"; and so, "from a practical point of view … this uncertainty has practically prevented the application of ⁴He accumulation as a quantitative dating tool" — unless it is calibrated against ¹⁴C or ⁸¹Kr, or unless in situ production can be shown to dominate, as at the Aquia aquifer in Section 12.
12 · Reading the ice age in a well

From a noble-gas temperature to a climate

The equilibrium component records the temperature at which the water last exchanged with air. For groundwater that is not the rain's temperature but the ground's: percolating water "equilibrates continuously with ground air until it reaches the capillary fringe and the quasi-saturated zone", so the noble-gas temperature (NGT) is, after the excess-air correction, the mean ground temperature at the water table. From there to a climate statement is a chain of local relationships — and the chain, not the thermometer, carries most of the uncertainty.

Four gasesNe, Ar, Kr, Xe ± 1 % NGTfit with an excess-air model Water-table temperature≈ NGT, unless the table is 1–2 m deep, recharge is very high, or the table is below ~30 m Ground (soil) temperaturewithin ~1 °C Mean annual air temperatureground is typically 1 ± 1 °C warmer

The instruction that follows in the text: convert NGT to air temperature "based on local relationships between ground and air temperature or calibrated locally by analysing young ground water that infiltrated under known climate conditions." Mixing in the aquifer and along long well screens smooths the record, so "the noble gas method cannot be used to study short-term climate fluctuations, but it is well-suited to derive quantitative estimates of mean temperatures during major climate states" — above all the difference between the last glacial maximum (about 21 kyr ago) and the Holocene.

Three records

Charts are redrawn from Kipfer et al.'s Figs. 21–23 at reading precision (about ±0.3 °C, ±1 kyr, ±10 % ΔNe); the numbers quoted in the panels are the authors' own. Where a panel is marked schematic, only the trend is shown.

How cold, where. Northern-latitude records that appear complete give a maximum LGM cooling near 9 °C (Stute & Deák 1989; the Aquia record); records with a gap — no recharge under permafrost or ice between about 10 and 20 kyr — may miss the coldest interval, so their 5–7 °C "may not represent the maximum cooling." Tropical and subtropical records converge on about 5 °C: Brazil, Namibia (also 5.5 °C by the N₂/Ar method at Uitenhage and 6 °C from a Cango Caves speleothem), Niger, Nigeria, Oman.
The chronology problem. The NGT record needs a time axis, and that axis is Week 3's problem. Bunter and Stampriet are dated by ¹⁴C; Aquia's carbonate chemistry made ¹⁴C unreliable, so its axis is a ⁴He accumulation age "based on the assumption of pure in situ production" (Section 11). Conversely, "the glacial–interglacial temperature transition in the noble gas temperatures can be used as a stratigraphic marker to check the derived ages." A thermometer that checks the clock that dates the thermometer: circular only if you forget which assumption each step rests on.
13 · Excess air as a rain gauge

The nuisance that became a proxy

For twenty years excess air was "a disturbance for which the measured data had to be corrected." Heaton and Vogel had suggested in 1981 that it might record recharge conditions — sporadic, heavy rainfall in semi-arid climates should trap and dissolve more air — and Heaton et al. (1983) found peaks of excess air in the Kalahari at ages of about 10 and 30 kyr, "coinciding with independently reconstructed periods of more humid climate and flooding." Then the temperate aquifers showed no systematic variation at all, and the idea went quiet. The CE model revived it, because it gave the signal a physical meaning: ΔNe measures pressure, pressure means water-table rise, and water-table rise means recharge — its intensity and variability more than its mean.

ΔNe against noble-gas temperature in three semi-arid aquifers — after Kipfer et al. Fig. 24 (trends only)

Niger (solid), Brazil (dotted), Australia (dashed): "all three aquifers show strong and very similar decreasing trends of ΔNe with temperature (about −10 % per °C)", r = 0.71, 0.83 and 0.63. In the Brazilian aquifer, glacial water carried roughly two to three times the ΔNe of Holocene water (Stute et al. 1995b).

Niger (r = 0.71)Brazil (r = 0.83)Australia (r = 0.63)

What can be said, and what cannot

  • Five semi-arid aquifer systems — Stampriet Auob (Namibia), Serra Grande and Cabeças (Brazil), the Continental Intercalaire (Niger), the Continental Terminal (Niger), and the Great Artesian Basin (Australia) — show past periods of strongly increased excess air, "in several of these studies … correlated to known periods of more humid climate." Hence the hypothesis "that excess air is a proxy for infiltration conditions, in particular the intensity and variability of recharge in semi-arid regions."
  • Temperate, humid aquifers show "little evidence for systematic variations of excess air."
  • Lithology matters less than it once seemed: Wilson and McNeill (1997) found the Ne excess increasing from granites through sandstones to limestones, but the CE model "argues against a major influence of lithology, unless very fine pores cause the surface tension to become a major source of excess pressure on the trapped bubbles, or very low porosity severely limits air entrapment."
  • The missing link: "A verification of a direct link between water table fluctuations and ΔNe or q under field conditions is still missing." Sand columns (Holocher et al. 2002) support the q–amplitude correlation, and show that fluctuations of less than 1 m can generate excess air. That is a laboratory, not an aquifer.
Carry this to Week 6's ledger: the excess N₂ that measures denitrification is what is left after subtracting equilibrium N₂ and excess-air N₂, and "depends on the model of excess air formation." The bubble that confuses the thermometer also confuses the denitrification budget — and the same four noble gases fix both.
14 · When the thermometer lies

A diagnosis board

Each row is a symptom you might meet in a noble-gas data set, the physical cause the readings offer, and the test or fix. Click a row for the detail.

15 · Reading the three sources

Three readings, three blocks, one bubble

Kipfer et al. supply the whole apparatus and the case studies; Aeschbach-Hertig et al. supply the statistical method and its first field tests; White supplies the geochemical backdrop. The guides below are maps, not substitutes: what each asks, where its load-bearing figures live, which equations to recognise, which numbers to be able to quote, and where a careful reader might press. The notes and the critique are yours to write.

Block 1 · Core reading

Kipfer, R., Aeschbach-Hertig, W., Peeters, F., & Stute, M. (2002). Noble gases in lakes and ground waters. Reviews in Mineralogy and Geochemistry, 47, 615–700.

Eighty-six pages: the components, the solubilities and the analytics; the four excess-air models and how to separate components by hand or by inverse fit; lakes (mixing, ³H–³He, helium fluxes, mantle gases); groundwater (young-water dating, old-water dating, the noble-gas thermometer, excess air as a climate proxy); and a coda on noble gases in ice. doi:10.2138/rmg.2002.47.14

Block 2 · Supplementary reading

Aeschbach-Hertig, W., Peeters, F., Beyerle, U., & Kipfer, R. (1999). Interpretation of dissolved atmospheric noble gases in natural waters. Water Resources Research, 35(9), 2779–2792.

The method paper: a general, error-weighted least-squares inversion of Ne, Ar, Kr and Xe (and He where it can be trusted) for temperature, salinity, pressure, excess air and re-equilibration — with Monte Carlo error analysis, then tested on air-equilibrated water, lakes, rivers, springs and aquifers. It works with the UA and PR models; the CE model came a year later. doi:10.1029/1999WR900130

Block 3 · Supplementary reading

White, W. M. (2015). Noble gas isotope geochemistry. Ch. 12 in Isotope Geochemistry (pp. 418–452). Wiley.

The textbook chapter: what noble gases are and where they came from (solar and planetary patterns), then helium, neon, argon, krypton and xenon in turn — production, isotope ratios, the crust–mantle–atmosphere story — and the origin and evolution of Earth's noble-gas inventory. Hydrology gets a few paragraphs; the rest is the context that makes those paragraphs intelligible.

Cross-paper synthesis, in one line: White explains why the gases do nothing and what their isotopes remember; Aeschbach-Hertig et al. turn four concentrations into a temperature with an error bar and a test; Kipfer et al. show what the temperatures and the excess air have said about climate and recharge — and how much of it rests on the choice of a bubble model. The question that carries forward: for the water your proposal cares about, what would you need to assume about the bubble, and what measurement would test the assumption instead of making it?
16 · Practice

Four situations to reason through

Each scenario gives the kind of evidence a Week 7 reader should now be able to interrogate. Choose the reading you would defend first, then compare with the debrief.

Habit to keep: before quoting a noble-gas temperature, say aloud which excess-air model produced it, what its χ² and p were, what altitude and salinity were assumed, and whether helium was fitted or left out. A temperature that cannot survive those four questions is a solubility table wearing a lab coat.
17 · Self-check

Ten questions before the meeting

Immediate feedback, no grade, no record. If you miss one, the linked section is the fix.

18 · Vocabulary

Glossary

The Week 7 working vocabulary, with a few terms you will meet again in Weeks 9 and 10. Search or browse.

19 · Where to go next

The Week 7 readings — and what follows

Papers are not posted for this course: retrieving them from the citation and DOI — via the UGA Libraries, GALILEO, or the publisher — is part of the training. The review and the article are behind publisher walls; the chapter is in the library's e-book collection.

CORE

Kipfer, R., Aeschbach-Hertig, W., Peeters, F., & Stute, M. (2002). Noble gases in lakes and ground waters. Reviews in Mineralogy and Geochemistry, 47, 615–700.

Components and solubilities (Table 1, Figs. 1–4); the excess-air models (Fig. 5, Table 2, eqs. 13–28); component separation and the inverse method (eqs. 29–32, Figs. 7–8, Table 3); lakes (Figs. 9–16, Tables 4–5); dating young and old groundwater (Figs. 17–20); the noble-gas thermometer and its records (Figs. 21–24); ice. doi:10.2138/rmg.2002.47.14

ARTICLE

Aeschbach-Hertig, W., Peeters, F., Beyerle, U., & Kipfer, R. (1999). Interpretation of dissolved atmospheric noble gases in natural waters. Water Resources Research, 35(9), 2779–2792.

The model (eqs. 1–11); the inverse method and its errors (eqs. 12–13, Tables 1–2, Fig. 1); air-equilibrated water, lakes, the Caspian Sea, glacier ponds, rivers, groundwater of the Töss valley and Botswana, altitude in the Alps (Figs. 2–6). doi:10.1029/1999WR900130

CHAPTER

White, W. M. (2015). Noble gas isotope geochemistry. In Isotope Geochemistry (Ch. 12, pp. 418–452). Wiley-Blackwell.

Chemistry and abundances (Tables 12.1–12.3, Fig. 12.1); helium in atmosphere, crust, oceans and mantle (Figs. 12.2–12.5); neon (Figs. 12.6–12.7); argon (Fig. 12.8); krypton and xenon (Figs. 12.9–12.12); mantle–atmosphere evolution and the xenon paradox (Figs. 12.13–12.16); problems.

Next week — the Picarro laboratory

Week 8 leaves the desk for the bench: laser spectroscopy of water isotopes in practice — standards and the VSMOW–SLAP scale, memory and carry-over, organic contamination, and the QA/QC that makes a data set defensible. The lesson of this week travels with you: a measurement's precision (1 % on a noble gas; 0.1 ‰ on δ¹⁸O) is only the beginning of its uncertainty, and the assumptions in the reduction — the excess-air model here, the calibration and drift correction there — usually dominate.

Carry-forward question, for your proposal: if your question turns on when or under what conditions water was recharged, which of this week's quantities would answer it — the noble-gas temperature, the excess air, the tritiogenic helium, the radiogenic helium — and which assumption in its derivation would a hostile reviewer attack first?