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# Noble gas insights into early impact delivery and volcanic outgassing to Earth's atmosphere: A limited role for the continental crust

Xinmu J. Zhang, Guillaume Avice, Rita Parai

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Xinmu J. Zhang, Guillaume Avice, Rita Parai. Noble gas insights into early impact delivery and volcanic outgassing to Earth's atmosphere: A limited role for the continental crust. Earth and Planetary Science Letters, 2023, 609, 10.1016/j.epsl.2023.118083. insu-04155738

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Distributed under a Creative Commons Attribution 4.0 International License
1 Noble gas insights into early impact delivery and volcanic outgassing to

2 Earth's atmosphere: a limited role for the continental crust

3

4 Xinmu J. Zhang<sup>1,2*†</sup>, Guillaume Avice<sup>3</sup>, Rita Parai<sup>1,2</sup>

5

6 <sup>1</sup>Department of Earth and Planetary Sciences, Washington University in St. Louis, 1 Brookings

7 Drive, St. Louis, MO, 63130 USA

8 <sup>2</sup>McDonnell Center for the Space Sciences, Washington University in St. Louis, 1 Brookings

9 Drive, St. Louis, MO, 63130 USA

10 <sup>3</sup>Université Paris Cité, Institut de physique du globe de Paris, CNRS, F-75005 Paris, France

11 *corresponding author, email: xmjzhang@ucsd.edu

12 † now at Scripps Institution of Oceanography, UC San Diego

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14

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16 Keywords: noble gases; argon; neon; continental crust; mantle outgassing; volatile delivery

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18

19
20 Abstract

21 Earth's atmosphere, crust and mantle have evolved together through continuous geochemical
22 exchange throughout Earth's history. Constraints on the transport of volatile elements and
23 compounds between these reservoirs are crucial for understanding how Earth could have stayed
24 habitable for extended periods of time. Here we present a new forward model of He, Ne and Ar in
25 the mantle, crust and atmosphere. We explore concentrations of noble gases at the end of accretion,
26 bulk silicate Earth K/U ratios, crustal growth scenarios, and upper mantle processing rates
27 throughout Earth's history. We search for parameter combinations that simultaneously satisfy
28 observational constraints on present-day mantle ⁴He/³He and ⁴⁰Ar/³⁶Ar (sensitive to mantle
29 outgassing and continental crust growth, which depletes the mantle of U, Th and K), atmospheric
30 ²⁰Ne/²²Ne (which tracks the mix of outgassed vs. delivered Ne in the atmosphere), and atmospheric
31 ⁴⁰Ar/³⁶Ar in the past and today (sensitive to volatile delivery, mantle outgassing and crustal growth).
32 Leveraging this intertwined set of noble gas abundances and isotopic compositions yields new
33 constraints on initial noble gas abundances in the mantle and on the proportions of atmospheric
34 volatiles originating from delivery by impact degassing, mantle outgassing, and degassing of the
35 continental crust. We find that atmospheric Ar isotopic evolution is primarily sensitive to the
36 mantle processing rate history; the atmospheric Ar isotopic record should therefore not be used to
37 reconstruct continental crust growth, but instead provides valuable insights into mantle processing
38 rates. Our model predicts a measurably low ²⁰Ne/²²Ne ratio of ~9.7 in Archean atmospheric
39 samples. Most of atmospheric primordial ³⁶Ar was directly delivered by chondritic bodies and not
40 transferred to the atmosphere during an intense early episode of mantle outgassing. Nitrogen
delivered by impact degassing could account for the present-day atmospheric nitrogen inventory.
## 1. Introduction

Earth has experienced a vibrant planetary evolution since its accretion. The transport of volatiles (e.g., water, carbon, nitrogen, and noble gases) between deep Earth and surface reservoirs is an important aspect of the coupled evolution of the mantle, continental crust and atmosphere. Improved constraints on volatile origins and the history of volatile exchange are crucial to our understanding of Earth's transformation into a habitable environment for living organisms.

Noble gases are powerful tracers of the coupled geochemical evolution of terrestrial reservoirs over time. The noble gases are chemically inert, are incompatible during partial melting, exhibit low solubilities in magma, and are isotopically sensitive to radioactive decay. Each noble gas element has at least one isotope that is produced by nuclear reactions, and at least one isotope that is primordial (i.e., the nuclide is stable and not produced by any radioactive reactions). Each long-term geological process transporting volatile species among major geochemical reservoirs leaves an imprint on the isotopic composition of all of the noble gases and generates inherently linked geochemical signatures among the major terrestrial reservoirs. Therefore, isotopic systematics among multiple noble gas elements provide a rich record of the complex processes involved in the co-evolution of the mantle, the continental crust, and the atmosphere (e.g., Allègre et al., 1987).

Helium (He) and argon (Ar) isotopes provide an example of how volatile origins and transport are recorded by noble gas systems in Earth reservoirs. ³He is primordial, and ⁴He is a radiogenic nuclide produced by α-decay of ²³⁸U, ²³⁵U and ²³²Th. ³⁶Ar is a primordial nuclide and ⁴⁰Ar is produced from ⁴⁰K by electron capture. Noble gases and U, Th and K are incompatible during
partial melting, but the noble gases are largely degassed from melts to the atmosphere, generating
continuous fractionation of K/Ar and (U+Th)/He ratios in the solid Earth over time. As a mantle
reservoir is processed by partial melting, noble gases are transported from the mantle to the
atmosphere (called outgassing). Both primordial and radiogenic noble gas isotopes are lost from
the mantle, but only the radiogenic nuclides are subsequently produced by the decay of their
lithophile radioactive parent nuclides. Over time, outgassing leads to high radiogenic to primordial
noble gas isotopic ratios (e.g., ⁴He/³He and ⁴⁰Ar/³⁶Ar) in a mantle reservoir; the isotopic
composition of outgassed He and Ar supplied to the atmosphere evolves accordingly. Transport of
atmospheric gas into the mantle (called “regassing” hereafter) is not thought to be significant for
He, and its importance for mantle Ar is debated (Holland and Ballentine, 2006; Moreira and Raquin,
2007; Tucker et al., 2022; Péron and Mukhopadhyay, 2022). Thus, mantle and atmospheric
⁴He/³He and ⁴⁰Ar/³⁶Ar ratios are time-integrated results reflecting continuous, long-term volatile
transport between the deep Earth and surface reservoirs.

He and Ar isotopes in mid-ocean ridge basalts (MORB) have long been used to generate
important constraints on mantle outgassing history (Ozima, 1975; Fisher, 1975; Sarda et al., 1985;
Moreira et al., 1998; Gonnermann and Mukhopadhyay, 2009). While strongly affected by mantle
outgassing, mantle ⁴He/³He and ⁴⁰Ar/³⁶Ar are also sensitive to growth of the continental crust,
which depletes the mantle of U, Th and K (e.g., Allègre et al. 1983). A variety of studies have
modeled the growth of continental crust based on a combination of geophysical constraints (e.g.,
mantle thermal and convective evolution; Walzer and Hendel, 2017) and geochronology tools (e.g.,
U-Pb- and Hf-based zircon dating; Armstrong and Harmon, 1981; Belousova et al., 2010; Condie,
1998; Dhuime et al., 2012). The Ar isotopic composition of the atmosphere is closely tied to the evolution of the solid Earth (Ozima and Kudo, 1972; Sarda et al., 1985), and archives of atmospheric $^{40}$Ar/$^{36}$Ar potentially provide another way to track continental crust growth (Hamano and Ozima, 1978; Pujol et al., 2013; Stuart et al., 2016; Guo and Korenaga, 2020).

Argon is the third-most abundant (0.93% by volume) gas in Earth's atmosphere, with a $^{40}$Ar/$^{36}$Ar ratio of 298.56 ± 0.31 (Lee et al., 2006). The atmospheric $^{40}$Ar/$^{36}$Ar ratio is orders of magnitude higher than an initial solar or chondritic $^{40}$Ar/$^{36}$Ar ratio (Ott, 2002), reflecting outgassing of radiogenic $^{40}$Ar from the solid Earth to the atmosphere. However, atmospheric $^{40}$Ar/$^{36}$Ar is quite low compared to $^{40}$Ar/$^{36}$Ar in the solid Earth: mantle $^{40}$Ar/$^{36}$Ar is ~30,000 (Sarda et al., 1985; Moreira et al., 1998; Parai and Mukhopadhyay, 2021) and Ar in the continental crust should be largely $^{40}$Ar. The canonical explanation for the relatively low ratio of $^{40}$Ar/$^{36}$Ar in the present-day atmosphere compared to the mantle is a sudden, intense stage of gas loss ("catastrophic degassing") from the mantle (Fanale, 1971), early enough that radiogenic $^{40}$Ar ingrowth had not yet strongly affected the mantle composition. Subsequent mantle outgassing models adopted a two-stage setup, with a brief catastrophic outgassing stage followed by a continuous outgassing stage after the end of accretion, with several orders of magnitude reduction in the outgassing rate between the two stages (Alexander and Ozima, 1978; Hamano and Ozima, 1978; Allègre et al., 1987; Fisher, 1975; Ozima, 1975; Sarda et al., 1985). Hamano and Ozima (1978) presented one such two-stage model tracing mantle Ar outgassing, K extraction from the mantle in the process of building the continental crust, and direct outgassing of the continental crust. The authors compared model results to a compilation of crustal ages to argue for continuous growth of the
105 continental crust rather than rapid early crustal formation.

106 The approach of applying Ar isotopic evolution models to constrain the growth of
107 continental crust has expanded as records of past atmospheric $^{40}$Ar/$^{36}$Ar have been established.
108 Cadogan (1977) reported Ar isotopic compositions of ancient atmosphere trapped in Devonian-
109 aged Rhynie chert. Pujol et al. (2013) constrained the composition of the early atmosphere using
110 analyses of Archean hydrothermal quartz, and Avice et al. (2017) confirmed those results. Taking
111 into consideration how uncertainties in the age of the hydrothermal quartz (taken for example to
112 be 3.5 Ga, 3.0 Ga, or 2.7 Ga) would affect the inferred ancient atmospheric $^{40}$Ar/$^{36}$Ar ratio, Pujol
113 et al. (2013) applied the two-stage outgassing model described in Hamano and Ozima (1978) and
114 argued for catastrophic early outgassing and limited continental crust growth (<10%) in the first
115 170 million years. Stuart et al. (2016) integrated high precision measurements of ancient
116 atmospheric Ar in the Rhynie chert to the model of Pujol et al. (2013) to further limit the extent of
117 Ar outgassing early in the continuous outgassing stage. In light of the high estimated proportion
118 of the present-day flux of $^{40}$Ar from the solid Earth that comes from direct outgassing of the
119 continental crust (e.g., Allègre et al., 1987; Bender et al., 2008), Stuart et al. argued for minimal
120 continental crust growth prior to 4 Ga. However, the integrated atmospheric $^{40}$Ar contribution from
121 direct outgassing of continental crust may be small compared to mantle outgassing (e.g., Sarda et
122 al., 1985). Using constraints from ancient atmospheric Ar composition in a model of Earth's
123 thermal evolution and Ar outgassing history, Guo and Korenaga (2020) suggested that atmospheric
124 Ar isotope evolution was strongly sensitive to continental crust growth primarily through mantle
125 melting to generate continental crust, and to a lesser extent through crustal recycling and reworking.
Guo and Korenaga argued for intense early crustal growth that reached >80% of present-day continental crust volume during the early Archean.

All of the existing Ar isotopic models invoking a catastrophic early outgassing stage fail to account for Ne isotope systematics, which provide important constraints on atmospheric volatile origins. $^{20}$Ne and $^{22}$Ne are primordial isotopes of Ne. As the production of $^{20}$Ne or $^{22}$Ne is insignificant in the mantle, the mantle $^{20}$Ne/$^{22}$Ne ratio is not measurably affected by any radioactive processes. The present-day mantle $^{20}$Ne/$^{22}$Ne is high and solar-like (>12.5; Williams and Mukhopadhyay, 2019) compared to the atmosphere (9.8; Eberhardt et al., 1965). Catastrophic early outgassing would have released large amounts of solar-like Ne with high $^{20}$Ne/$^{22}$Ne to the atmosphere, and subsequent continuous mantle outgassing would only add Ne with high $^{20}$Ne/$^{22}$Ne to the atmosphere. Mass fractionation due to early hydrodynamic atmospheric loss (Lammer et al., 2020; Pepin, 2006), cannot explain the low $^{20}$Ne/$^{22}$Ne in the present-day atmosphere: while kinetic atmosphere loss can be driven to match the atmospheric $^{20}$Ne/$^{22}$Ne, it would yield unsatisfactorily low $^{36}$Ar/$^{40}$Ne compared to modern atmosphere (Marty, 2012; Marty, 2022). Rather, the atmospheric primordial Ne and Ar isotope signature is consistent with a mix between delivered primordial volatiles (similar to carbonaceous chondrites) and a small contribution of outgassed mantle volatiles with solar-like compositions (Marty, 2022). Therefore, a new model of noble gas isotopic evolution that takes volatile delivery to the early atmosphere by impact degassing (e.g., Zahnle et al., 1988) into account is needed.

The rich, multidimensional system of information available in mantle and atmospheric noble gas isotopes has yet to be harnessed in a self-consistent model. In order to investigate
atmospheric volatile origins, budgets of heat-producing elements in the solid Earth, and crust-
mantle-atmospheric differentiation, we present a numerical model of coupled He, Ne and Ar
isotopic evolution in the mantle, crust and atmosphere. We explore a wide variety of scenarios and
parameters regarding mantle degassing, continental crust growth and initial noble gases in Earth's
atmosphere. Our forward model of isotopic evolution identifies combinations of input parameters
that successfully reproduce constraints on present-day mantle He and Ar, atmospheric Ne and Ar,
and ancient (Archean and Devonian) atmospheric Ar isotopes. For reasonable combinations of
geochemical and geophysical parameter values, a simple model of primordial atmospheric delivery,
continuous continental crust growth, and mantle outgassing provides model outputs in agreement
with all observational constraints. Successful model realizations are found for a wide range of
input parameters given a bulk silicate Earth K/U estimate of 13,000; fewer successful parameter
combinations are found for higher and lower estimates of this ratio. The coupled He-Ne-Ar isotope
systematics provide constraints on initial primordial noble gas abundances in the mantle. We show
that atmospheric Ar isotope evolution is primarily controlled by the history of mantle outgassing
and is only weakly sensitive to net continental crust growth. Lastly, the results indicate that direct
delivery of volatiles by impact degassing is critically important to atmospheric origins, and that
contributions from delivery and outgassing can vary widely among volatile species and indeed
among isotopes of a given volatile element (e.g., ³⁶Ar and ⁴⁰Ar).

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166 **2. Methods**

To investigate the long-term co-evolution of the mantle, continental crust and the atmosphere
associated with mantle processing, we use a three box forward model (Figure 1) of noble gas isotopic evolution over the past 4.488 billion years, starting at 80 Myrs after Earth's formation (consistent with the model start time in Parai, 2022). Figure 2 compares our model setup to prior models of mantle outgassing and the evolution of the atmosphere over time. In each time step, we track the outgassing of mantle ³He, ⁴He, ²⁰Ne, ²²Ne, ⁴⁰Ar and ³⁶Ar to the atmosphere, the net growth of continental crust, the sequestration of incompatible radioactive parent isotopes ²³⁵U, ²³⁸U, ²³²Th and ⁴⁰K from the mantle in the continental crust, and the production of radiogenic nuclides in silicate reservoirs. Our goal is to explore a wide parameter space (Table S1) and find parameter combinations that satisfy constraints on the present-day MORB source He and Ar isotopic ratios and abundances, the present-day atmospheric Ne abundances and ²⁰Ne/²²Ne ratios, the present-day atmospheric Ar abundances, and atmospheric ⁴⁰Ar/³⁶Ar ratios of the present-day and the past (Cadogan, 1977; Pujol et al., 2013; Stuart et al., 2016). Exploring the bulk silicate Earth (BSE) K/U ratio, which affects estimates of the production of radiogenic heat and radiogenic noble gas nuclides ⁴He and ⁴⁰Ar in Earth's major geochemical reservoirs, is also key to refining the understanding of the details of mantle-crust-atmosphere evolution. The names, functions, and ranges of these parameters are shown in Table S1. Below we discuss the details of the model setup.

## 2.1 Box 1: the MORB source mantle

We consider the MORB source mantle as the primary source of outgassed atmospheric noble gases to reduce model complexity (Section 3). The first model parameter is the mass of the MORB source mantle reservoir ($M_{res}$); we explore the range from 50% to 90% of the total mantle mass.
189 The second parameter is the total number of mantle reservoir masses processed over the past 4.488
190 billion years (N<sub>res</sub>, ranging from 2 to 15; see SM and Parai and Mukhopadhyay (2018) for further
191 detail).

192 We parameterize the extent and rate of the long-term processing of the MORB mantle based
193 on M<sub>res</sub> and N<sub>res</sub>. Mantle outgassing at mid-ocean ridges, depletion of the MORB mantle due to
194 growth of continental crust, and radioactive decay are all processes that continuously alter the
195 MORB source mantle's chemical composition. Equations that track the concentrations of
196 radioactive <sup>235</sup>U, <sup>238</sup>U, <sup>232</sup>Th and <sup>40</sup>K, radiogenic <sup>4</sup>He and <sup>40</sup>Ar, and primordial <sup>3</sup>He, <sup>20</sup>Ne, <sup>22</sup>Ne and
197 <sup>36</sup>Ar in the MORB source mantle at a given time are given in the Supplementary Material (SM).

198 We determine the initial budgets of mantle primordial noble gas isotopes by allowing mantle
199 <sup>3</sup>He initial abundances to vary from 1×10<sup>9</sup> to 1×10<sup>12</sup> atoms/gram, a large range that encompasses
200 estimates from prior studies (Harper and Jacobsen, 1996; Parai and Mukhopadhyay, 2018; Parai,
201 2022). Corresponding initial mantle abundances of <sup>22</sup>Ne are determined using a depleted mantle
202 <sup>3</sup>He/<sup>22</sup>Ne ratio of 10 (Tucker and Mukhopadhyay, 2014). As the initial mantle <sup>20</sup>Ne/<sup>22</sup>Ne is debated
203 (Trieloff et al., 2000; Holland and Ballentine, 2006; Jaupart et al., 2017; Williams and
204 Mukhopadhyay, 2019), the MORB mantle <sup>20</sup>Ne/<sup>22</sup>Ne is treated as a stable isotope ratio fixed at the
205 modern depleted mantle value of 12.6 (Parai and Mukhopadhyay, 2021). The potential range of
206 initial mantle <sup>3</sup>He/<sup>36</sup>Ar ratios of 1.12 to 1.43 is computed using the depleted mantle <sup>36</sup>Ar/<sup>22</sup>Ne
207 values from 7 to 9 (Marty, 2012). Table S2 gives a summary of noble gas initial abundances and
208 isotope ratios. We use present-day BSE Th abundance of 85.3 ppb and BSE U abundance of 20.3
209 ppb from McCulloch and Bennett (1994). We test two BSE K/U ratios of 13,000 and 19,000, both
210 anchored to the BSE U abundance=20.3 ppb. The initial mantle $^{40}$K, $^{238}$U, $^{235}$U, and $^{232}$Th
211 concentrations are each computed using their respective present-day isotopic abundances and
212 decay constants (Table S3).

213 We assume that noble gases are perfectly incompatible during partial melting and are degassed
214 completely from the melt. For lithophile elements such as U, Th, and K, the net extraction from
215 the mantle is directly related to the net continental crust growth. The depletion of U, Th and K in
216 the mantle is therefore modeled to track the estimated volume of continental crust in 5 different
217 models (Section 2.2), independent of the mantle processing rate. We note that Guo and Korenaga
218 (2020) erroneously assumed that Parai and Mukhopadhyay (2018) used net continental crust
219 growth rates to calculate mantle degassing rates. That is an inaccurate description of the model of
220 Parai and Mukhopadhyay (2018); that work and the present study both explicitly separate mantle
221 degassing rate history (explored as a function of Nres and Mres) and prescribed models of net
222 continental crust growth that govern extraction of U, Th and K from the mantle.

223

224 2.2 Box 2: the continental crust

225 We model five plausible scenarios of net growth of continental crust over time (CC models,
226 Figure 3). The net growth in the volume of continental crust accounts for both the production of
227 new continental crust, and the destruction of existing continental crust through weathering, erosion,
228 and the subduction of sediments. The first continental crust growth scenario (CC1) is based on the
229 model of Fyfe (1978) with rapid crustal growth in late Hadean and Archean in excess of the
230 present-day continental crust volume, followed by net decline of crustal volume to the present.
The second scenario (CC2) models rapid growth within the first one billion years of Earth history (Armstrong and Harmon, 1981). The third scenario (CC3) is similar to CC2 except that net continental crust growth begins about 200 Myr after the start of the Solar System, accounting for the latest estimates for the timing of the Moon-forming giant impact (Borg et al., 2011). The fourth scenario (CC4) models a relatively even, protracted crustal growth from the start of Earth's history (Belousova et al., 2010; Dhuime et al., 2012; Pujol et al., 2013; Walzer and Hendel, 2017). The fifth scenario (CC5) models even, protracted growth that starts later in Earth's history: most of the continental crust is generated after the first 1.5 billion years in this model (Condie, 1998; Poupinet and Shapiro, 2008; Rino et al., 2004).

To determine the net extraction of each incompatible lithophile elements U, Th and K from the MORB mantle to the continental crust, we calculate extraction constants that satisfy mass balance in the bulk silicate Earth for each given combination of $M_{res}$ and CC model. Taking into consideration extraction of U, Th, and K from the MORB mantle, the in-situ decay of radiogenic parent isotopes accumulating in the continental crust, and crustal degassing, we determine the crustal concentrations of $^{235}$U, $^{238}$U, $^{232}$Th, $^{40}$K and $^{40}$Ar, with a parameter $f$ that describes the fraction of crustal radiogenic $^{40}$Ar degassed to the atmosphere in a given time step (SM).

## 2.3 Box 3: the atmosphere

The total amounts of $^{20}$Ne, $^{22}$Ne, $^{36}$Ar and $^{40}$Ar in the atmosphere at a given time are based on budgets in the previous time step, mantle degassing, and continental crust degassing (Table S4). Accumulation of $^{3}$He and $^{4}$He in the atmosphere is not tracked, as He escapes to space over time.
Catastrophic outgassing from the mantle during accretion and impact degassing of chondritic materials delivered as part of the late veneer would both contribute Ar with low $^{40}$Ar/$^{36}$Ar to the early atmosphere, but only chondritic delivery can explain both atmospheric Ne and Ar isotopes (Figure 2). Direct delivery of volatiles to the atmosphere by impact degassing (Zahnle et al., 2010, 1988) during accretion is an important process (Marty, 2012) that has not previously been accounted for in models of atmospheric noble gas isotopic evolution. In our model, the total amount of atmospheric Ar from delivery is calculated by assuming no delivered Ar to start, and then computing the amount of delivered Ar needed to give the present-day $^{40}$Ar/$^{36}$Ar ratio of 298.6 at the end of each model realization. The amount of atmospheric Ne from delivery is calculated using the chondritic $^{22}$Ne/$^{36}$Ar ratio (Marty, 2012). Calculations of the amounts of chondritic Ne and Ar delivered to the Earth's atmosphere are described in Table S4. The coupled atmospheric Ar-Ne systematics with delivery and mantle outgassing are constrained by the modern atmospheric $^{20}$Ne/$^{22}$Ne ratio, Archean $^{40}$Ar/$^{36}$Ar ratio from Pilbara cherts (Pujol et al., 2013), and Devonian atmospheric $^{40}$Ar/$^{36}$Ar from Rhynie cherts (Cadogan, 1977; Stuart et al., 2016) to assess model success.

Previous studies have demonstrated that atmospheric Ar is injected into the deep mantle via subducted materials (Holland and Ballentine, 2006; Jackson et al., 2013; Parai et al., 2019, 2012; Smye et al., 2017). However, the total amount of regassed Ar in the mantle is uncertain. In our model, if the final amount of atmospheric Ar (outgassed and delivered) exceeds the observed present-day Ar budget, then the excess atmospheric Ar is deducted from the atmosphere box and added to the MORB source mantle box as regassed atmospheric Ar (Table S4). The regassed Ar is
assumed to have the modern atmospheric $^{40}$Ar/$^{36}$Ar ratio and has an impact on the calculated present-day $^{40}$Ar/$^{36}$Ar ratio of the MORB source mantle, which is compared to the estimate of the present-day MORB source mantle $^{40}$Ar/$^{36}$Ar ratio (Moreira et al., 1998; Parai et al., 2012; Parai and Mukhopadhyay, 2021).

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## 2.4 Input parameters and criteria for model success

We explore three parameters (Table S1) using a simple Monte Carlo sampling method to seek the bounds on a successful parameter space: the initial mantle $^3$He abundance, the mantle $^3$He/$^{36}$Ar ratio, and the crustal degassing parameter *f*. Four other parameters are explored using discrete test values ($M_{res}$, $N_{res}$, CC model and the BSE K/U ratio) to examine the effect of varying these parameters (Table 1) on the successful parameter space. For each tested BSE K/U scenario (including a common estimate of 13,000 and higher estimate 19,000; Jochum et al., 1983; Arevalo et al., 2009; Gale et al., 2013), the model draws random values from a range with a uniform distribution for the three Monte Carlo parameters and performs parallel computations of isotopic evolution for discrete combinations of $M_{res}$, $N_{res}$, CC model. Table S1 gives the ranges tested for each parameter; model runs were repeated with progressively narrowed input ranges to get a well-defined solution space where possible.

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Seven noble gas observational constraints were used to assess model success (Table S5). To identify the combinations of input parameters generating the most satisfactory outputs, a weighted scoring system was developed based on residuals between the model outputs and the observational constraints (Table S4). The total score (Score$_{total}$) is a sum of squared residuals normalized to the
uncertainties: the smaller the total score, the better a model realization output meets the observational constraints. Model realizations with Score$_{total}$ smaller than 7 translate to being within one standard deviation (1σ) on average for each of seven tests, and realizations with Score$_{total}$ smaller than 28 are within 2σ on average.

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299 **3. Model Results**

The model generates time series for He, Ne, Ar, U, Th, and K abundances and He, Ne, and Ar isotopic compositions in the mantle, continental crust and atmosphere. Tables 1 and S1 identify the input parameter ranges that yield successful model realizations as defined above. Below we discuss the modeled isotopic evolution and outcomes compared to model constraints.

The net extraction of incompatible elements U, Th, and K from the MORB source mantle to the continental crust over time is calculated using mass balance (Section 2.2). All but one combination of continental crust growth models and $M_{res}$ can satisfy mass balance for the present-day U and Th concentrations of the continental crust: for the CC1 model, where the total volume of continental crust once exceeded the present-day volume by about 30%, the smallest $M_{res}$ (50% the total mass of the mantle) would not contain sufficient U and Th. The CC1 growth model would also require more K than is available even in the largest $M_{res}$ of 90% of the mantle given the largest tested BSE K/U of 19,000 (the combination that maximizes K available to go into the continental crust). This indicates that if large volumes of crust accumulated early in Earth's history (as in CC1), that crust cannot have resembled the modern continental crust in its trace element content (e.g., Tucker et al., 2022). Because CC1 fails to satisfy mass balance for U and K, we do not discuss it
further; the rest of the model results and discussion only include models CC2 to CC5.

Time evolution of the abundances of <sup>238</sup>U, <sup>232</sup>Th, and <sup>40</sup>K in the MORB source mantle (Figure 4) reflects radioactive decay and the progressive depletion of incompatible elements directly reflecting extraction to the growing continental crust. Outgassing of the depleted MORB source mantle is reflected in the increase in <sup>4</sup>He/<sup>3</sup>He and <sup>40</sup>Ar/<sup>36</sup>Ar ratios over time (Figure 5): all noble gas isotopes are released from the MORB source mantle to the atmosphere due to outgassing during partial melting, but loss of <sup>4</sup>He and <sup>40</sup>Ar to the atmosphere is counterbalanced by ingrowth from the in-situ decay of radioactive U, Th and K isotopes in the MORB source mantle. Higher mantle processing rates (corresponding to higher numbers of reservoir masses processed over Earth history, N<sub>res</sub>) will lead to significantly more radiogenic noble gas isotopic signatures in the MORB source mantle reservoir at a given timestep, as well as larger total amounts of mantle <sup>40</sup>Ar outgassed to the atmosphere. For model realizations where the resulting atmospheric <sup>40</sup>Ar budget exceeds what is observed, atmospheric Ar must have been regassed into the mantle to compensate (Figure 5).

MORB source mantle <sup>4</sup>He/<sup>3</sup>He ratios and <sup>40</sup>Ar/<sup>36</sup>Ar ratios at the present-day for successful model realizations (Score<total<28) are compared with present-day observational constraints (Figure 6). Among successful model scenarios, we found both some that require Ar regassing and others with no regassing of Ar (Figure S1). Most of the successful simulations fall in the range of 80,000 to 100,000 for the MORB source mantle <sup>4</sup>He/<sup>3</sup>He ratio (Graham, 2002), and 24,000 to 40,000 for the <sup>40</sup>Ar/<sup>36</sup>Ar ratio (Figure 6; Holland and Ballentine, 2006; Moreira, 1998; Parai and Mukhopadhyay, 2021). In successful model realizations, 98-99% of primordial noble gas budgets
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are lost from the mantle by continuous outgassing. The outgassing rates for ³⁶Ar over time are shown in Figure S2.

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Atmospheric $^{20}$Ne/$^{22}$Ne and $^{40}$Ar/$^{36}$Ar ratios are tracked from an initial composition similar to chondrites (Marty, 2022, 2012) to the present-day compositions. For atmospheric Ne isotopes, only the initial and present-day compositions are constrained. For $^{40}$Ar/$^{36}$Ar, additional constraints are available in the Archean and Devonian. $^{40}$Ar/$^{36}$Ar time series in successful model realizations show that although both BSE K/U of 13,000 and 19,000 can generate past atmospheric Ar isotopic compositions that are consistent with measurements from Archean and Devonian samples (Figure 7), a relatively narrow set of input parameters yields success for high BSE K/U.

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We note that while the model explores the contribution of Ar from the upper mantle with 50-90% of total mantle mass ($M_{res}$), it does not track the isotopic evolution of the plume source mantle or its contributions to the atmosphere by degassing. For CC models with early crustal growth (CC2 and CC3), successful model realizations are found with $M_{res}$ of 90%, in which case the plume mantle is only 10% of total mantle mass and makes a relatively small outgassing contribution to the atmosphere. For CC models with late crustal growth (CC4 and CC5), successful model realizations are only found with relatively low $M_{res}$ (62.5%; Figure S3); given the model setup, the rest of the mantle is not outgassed. Parai (2022) found that while the plume mantle is less degassed than the MORB source mantle, it is an open system that has experienced significant outgassing. The late crustal growth scenarios likely would no longer satisfy atmospheric Ne and Ar observational constraints if degassing of the rest of the mantle were taken into account.

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349

350

351

352

353

354

355

356
357 **4. Discussion**

358 The three-reservoir box model identifies combinations of parameters that enable good fits to
359 all observational criteria. By accounting for delivery of chondritic noble gases, mantle outgassing
360 and atmospheric regassing of volatiles, our model setup provides successful isotopic evolution
361 paths and budgets simultaneously for He, Ne, and Ar in the mantle and atmosphere. Below we
362 discuss the impact of considering atmospheric Ne isotopes, and the implications of our model on
363 Earth's initial volatile budgets, continental crust growth, the bulk silicate Earth K/U ratio, and the
364 coupled evolution of the terrestrial atmosphere, crust, and mantle over time.
365

366
4.1 *The initial terrestrial noble gas budgets*

367 The initial budgets of primordial He and Ar in the mantle were established during accretion.
368 Primordial isotope ratios in the mantle are consistent with a mixture of He- and Ne-rich solar
369 nebular gas (likely ingassed into a magma ocean) with a planetary component relatively rich in Ar,
370 Kr and Xe, similar to chondrites (Broadley et al., 2022; Parai, 2022; Parai and Mukhopadhyay,
371 2021; Péron et al., 2021; Williams and Mukhopadhyay, 2019). Catastrophic mantle outgassing due
372 to giant impacts affected primordial noble gas budgets through the end of accretion. Harper and
373 Jacobsen (1996) computed an initial mantle ³He budget assuming closed system evolution from
374 solar He composition to the present-day composition of the less processed mantle component
375 sampled at ocean islands. However, this estimate is a lower limit; any mantle outgassing would
376 require higher initial ³He concentrations to compensate. An upper limit can be determined by
377 combining He and Xe isotope systematics to find an upper bound on Nres for the MORB source
378 mantle ((Parai and Mukhopadhyay, 2018; Parai, 2022).

379 Our model explores the abundance of mantle ³He at the end of accretion (taken to be 80 Myrs after the start of the Solar System) as an input parameter, which then ties to the initial budgets of 380 ²²Ne and ³⁶Ar respectively using the mantle ³He/²²Ne ratio (Tucker and Mukhopadhyay, 2014) and 381 a relatively narrow range of potential mantle ³He/³⁶Ar ratios to account for the possible regassing 382 of atmosphere ³⁶Ar. The initial ³He budget needs to satisfy coupled He-Ne-Ar constraints in 383 multiple reservoirs. When the initial mantle ³He concentration is high, more vigorous mantle 384 outgassing is needed to satisfy the present-day observed upper mantle ⁴He/³He. However, a high 385 initial mantle ³He concentration directly translates to a high initial mantle ²²Ne concentration. With 386 enhanced mantle outgassing and high mantle noble gas abundances, the MORB source mantle 387 would outgas more Ne with high ²⁰Ne/²²Ne to the atmosphere, which would yield present-day 388 atmospheric ²⁰Ne/²²Ne ratios higher than observed. Consideration of atmospheric Ne and Ar 389 isotopes thus enables a stricter constraint on initial mantle ³He. 390

391 Successful model realizations (Score<28) with BSE K/U=13,000 are found with initial 392 mantle ³He concentrations from 1.0 to 2.5×10¹¹ atoms/gram. Realizations with Score<7 occur 393 in a narrower initial ³He range of 1.2 to 2.1×10¹¹ atoms/gram (Table 1). For realizations with BSE 394 K/U=19,000, successful initial mantle ³He has a tighter range of 2.2 to 2.7×10¹¹ atoms/gram (Table 395 S1). These ranges are consistent with MORB source mantle initial ³He concentrations in successful 396 scenarios from the coupled He-Ne-Xe model of Parai (2022). Compared with the prior lower limit 397 of 7.8×10¹⁰ atoms/gram (Harper and Jacobsen, 1996) and upper bound of 8.0×10¹² atoms/gram 398 inferred from Xe isotope systematics (Parai and Mukhopadhyay, 2018), the model significantly
narro ws down the estimate of initial mantle $^{3}$He that would be retained in the mantle at the end of
the accretion (and corresponding $^{22}$Ne and $^{36}$Ar) by leveraging combined He-Ne-Ar isotope
systematics. This has implications for the pressure of the nebular atmosphere in equilibrium with
the terrestrial magma ocean, and for the amount of gas retained through the giant impact stage of
the accretion.

404

405 4.2 Atmospheric Ar isotopes are controlled by mantle outgassing and are only weakly sensitive to
406 growth of the continental crust

The timing and rate of net continental crust growth influence the abundance and isotopic
composition of Ar outgassed from the MORB mantle to the atmosphere and crustal $^{40}$Ar outgassed
407
408
to the atmosphere. Prior studies have argued that ancient and present-day records of atmospheric
409
$^{40}$Ar/$^{36}$Ar are proxies for continental crust growth over time (e.g., Pujol et al., 2013; Stuart et al.,
410
2016; Guo and Korenaga, 2020).
411

Figure 8a shows atmospheric $^{40}$Ar/$^{36}$Ar ratio time series for the model realization with the lowest Score$_{total}$, along with a set of curves generated by varying the CC model while holding all other model parameters constant. Variation of the CC model generates only minor differences in atmospheric $^{40}$Ar/$^{36}$Ar ratio time evolution, with variations manifesting most strongly before ~2.5 Ga. Compared to models with more gradual net continental crust growth that started later into Earth history (CC4 and CC5), models with early, rapid net continental crust growth during Hadean and Archean (CC2 and CC3) show slightly higher atmospheric $^{40}$Ar/$^{36}$Ar ratios early in Earth's history due to crustal outgassing (~2% more at ~1 Gyr after the start). However, differences
between atmospheric $^{40}$Ar/$^{36}$Ar time series for different CC models (with all other parameters kept the same) are small compared to the uncertainties reported for the paleo-atmospheric $^{40}$Ar/$^{36}$Ar ratio (on the order of ~10%; Pujol et al., 2013). The existing atmospheric Ar isotopic record thus has little power to distinguish among crustal growth models.

We find instead that the mantle processing rate history exerts primary control on the evolution of atmospheric $^{40}$Ar/$^{36}$Ar, as this history dictates the rate of outgassing of highly radiogenic mantle Ar to the atmosphere over time (Figure 8b). Our model setup assumes an exponentially decreasing mantle processing rate. A linearly decreasing processing rate and a constant rate are shown for comparison, which yield significantly larger (~7% and ~24% respectively) variations at 1 Gyr after the start than the ~2% differences generated by continental crust growth models (Figure 9). The geochemical-geodynamical model of Tucker et al. (2022) gives atmospheric $^{40}$Ar/$^{36}$Ar over time with either constant convective vigor or enhanced early convective vigor, corresponding to processing rates that are near-constant or linearly decreasing. Our results demonstrate that Tucker et al. (2022) yielded significantly lower Archean atmospheric $^{40}$Ar/$^{36}$Ar than the observed because their mantle processing rates are too low in the past. Instead, the mantle processing rate must have decreased near-exponentially over time.

Guo and Korenaga (2020) presented a model in which the total mantle processing rate is split into three components: mantle processing to generate oceanic crust, to generate hotspots, and to generate continental crust. The authors presented results in which atmospheric $^{40}$Ar contributions from mantle melting to generate the continental crust far exceed any other contributing flux, including $^{40}$Ar from mantle processing at mid-ocean ridges, both in a cumulative sense and at any
441 instantaneous point in Earth's history after initiation of crustal growth. It is difficult to envision
442 how mantle processing to generate continental crust could contribute twice the instantaneous flux
443 of $ ^{40}Ar $ to the atmosphere as that from melting at mid-ocean ridges even in the present-day (Figure
444 4 in Guo and Korenaga, 2020). Nonetheless, the dominant $ ^{40}Ar $ contribution suggested by Guo and
445 Korenaga (2020) is still derived from mantle outgassing, which is consistent with our result.

446 We note that Guo and Korenaga (2020) also argued that crustal recycling and reworking
447 contributed considerable proportions (~25%) of atmospheric $ ^{40}Ar $. In our model, all crustal $ ^{40}Ar $
448 contributions are captured via the crustal degassing coefficient, *f*. The proportion of atmospheric
449 $ ^{40}Ar $ with a crustal outgassing origin was computed (Figure S4) for all model simulations with
450 $ \text{Score}_{\text{total}} < 7 $. Crustal outgassing accounts for ~3 to 6% of atmospheric $ ^{40}Ar $ with a late onset of net
451 crustal volume increase (CC4 and CC5), while CC models with net crustal growth starting early
452 (CC2 and CC3) can provide up to 13% of the present-day atmospheric $ ^{40}Ar $, similar to the estimate
453 from Sarda et al. (1985). The discrepancy potentially stems from the lack of consideration in Ne
454 isotope systematics in Guo and Korenaga (2020). To explain the atmosphere Ne abundance and
455 $ ^{20}Ne / ^{22}Ne $ ratio, outgassing of mantle-derived Ne to the atmosphere is required, which
456 simultaneously transports mantle $ ^{40}Ar $ to the atmosphere. Without taking atmospheric Ne isotopes
457 and abundances into account, crustal contributions to atmospheric $ ^{40}Ar $ can be overestimated.
458 Invoking hypothetical catastrophic early mantle outgassing also drastically decreases the average
459 $ ^{40}Ar / ^{36}Ar $ of the integrated Ar contribution from mantle outgassing, leading to the overestimation
460 of $ ^{40}Ar $ contribution from crustal processes.

461 Our model uses He-Ne-Ar systematics to provide new constraints on mantle outgassing rates,
which in turn limits the potential range of crustal degassing rates. For Score<7 model realizations with BSE K/U of 13,000, the crustal degassing coefficient must be smaller than $2.05×10^{-9}$ yr$^{-1}$ and larger than $1.98×10^{-11}$ yr$^{-1}$. For Score<28 model realizations with BSE K/U=19,000, the crustal degassing rate is constrained to a much narrower range of $1.85-1.89×10^{-10}$ yr$^{-1}$. In both tested BSE K/U scenarios, the extent of crustal degassing of successful model realizations is broadly consistent with the crustal degassing coefficient estimate of $3.7×10^{-10}$ yr$^{-1}$ (Hamano and Ozima, 1975). The apparent narrow range of crustal degassing coefficients for successful BSE K/U=19,000 simulations is likely caused by the limited combinations of CC models, MORB source mantle reservoir mass, and mantle outgassing rate (Table S1, Figure S3) that simultaneously accommodate for high K abundances while satisfying tight constrains from He and Ne isotope systematics.

Overall, instead of crustal growth and degassing, mantle outgassing is the dominant process that transports $^{40}$Ar to the atmosphere. Therefore, we caution against using atmospheric $^{40}$Ar/$^{36}$Ar to reconstruct continental crust growth over time (Figure 8a). Instead, archives of ancient atmospheric $^{40}$Ar/$^{36}$Ar have a greater potential to shed light on past mantle processing rates.

## 4.3 The BSE K/U ratio

The BSE K/U ratio has crucial implications for accretionary processes that led to the depletion of terrestrial moderately volatile elements (MVE) compared to chondritic precursors (Wang et al., 2021), terrestrial radiogenic heat production, and compositional evolution of the mantle and atmosphere through outgassing. Although K and U behave similarly during silicate partial melting,
it is challenging to estimate the terrestrial BSE K/U ratio. One complexity comes from dehydration
reactions at subduction zones that fractionate the more fluid-mobile element K from U (Becker et
al., 2000, Hanyu et al., 2011; Lassiter et al., 2003). Recycled oceanic crust and lithosphere that are
not yet well-mixed into the ambient mantle could lower estimates of BSE K/U computed from
weighted averages inferred from globally sampled MORBs, OIBs, and continental crust. Neither
the mass of isolated patches of recycled materials nor the extent of K/U fractionation in subduction
zone settings is well-understood, which generates uncertainties in mass balance among major
silicate reservoirs. Based on the existing global K/U datasets, recent studies have accounted for
effects of K/U fractionation to infer magma source K/U compositions (Farcy et al., 2020) or have
decreased uncertainty from terrestrial K/U fractionation by filtering global oceanic and continental
crust datasets, selecting only those with similar relative abundances of strongly incompatible
lithophile refractory elements compared to CI chondrites (Dauphas et al., 2022).

Most recent BSE K/U estimates fall between ~10,000 and 14,000 (Arevalo et al., 2009;
Dauphas et al., 2022; Farcy et al., 2020; Gale et al., 2013; Jochum et al., 1983; O'Neill et al., 2020;
Palme and O'Neill, 2013). However, we explored the extreme case where K/U is not fractionated
by partial melting and the BSE K/U is the same as the MORB source mantle estimate of 19,000
(Arevalo et al., 2009). Model success is affected by the choice of BSE K/U ratio, and strongly
depends on the corresponding combined choices of other parameters including Nres, Mres, and
continental crust growth model.

For both high (=19,000) and low (=13,000) BSE K/U ratios, we identify model realizations
that meet all observational constraints (Table S1, Figure S3). Given the same total number of
simulations, a high BSE K/U=19,000 generates a very limited number of successful (Score<28)
model realizations with a very restricted range of input parameters (Table S1). For BSE
K/U=13,000, fine tuning of the other model parameters is not necessary for generating model
success, and the minimum total scores are lower, indicating better overall fits. We also explored
an even lower estimate of BSE K/U=9,000, which produces some successful model realizations
(Table S1; Figure S5). Decreased <sup>40</sup>Ar production in silicate reservoirs leads to smaller amounts
of chondritic Ar and Ne delivery allowed to explain atmospheric Ar composition, which means
limited mantle Ne with high <sup>20</sup>Ne/<sup>22</sup>Ne can be outgassed to the atmosphere. As a result, only low
mantle outgassing rates can be tolerated, which yield lower atmospheric <sup>40</sup>Ar abundances than
observed. Overall, our model finds the greatest number of successful realizations using a moderate
BSE K/U=13,000.

515

4.4 *The origin and evolution of atmospheric Ar and Ne*

Three sources with distinct Ne and Ar isotopic compositions contribute to the present-day
atmosphere: (1) impact degassing of chondritic materials with low $^{20}$Ne/$^{22}$Ne, low $^{40}$Ar/$^{36}$Ar, and
high $^{36}$Ar/$^{22}$Ne ratios delivered; (2) outgassing to the atmosphere from the MORB source mantle
with a solar-like high $^{20}$Ne/$^{22}$Ne ratio and progressively increasing $^{40}$Ar/$^{36}$Ar ratio over time; and
(3) degassing of $^{40}$Ar produced in-situ by the decay of $^{40}$K in the continental crust. We note that
our model approach of combining He, Ne and Ar systematics enables distinction between the
different processes contributing to the atmospheric composition over time (Figure 2).

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518

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523

Model realizations with Score<7 indicate that the amount of <sup>36</sup>Ar contributed by the delivery

524
of chondritic materials accounts for 87% to 91% of present-day atmospheric $^{36}$Ar after accounting
for regassing to the mantle. Thus, our model supports primordial isotope arguments that chondritic
Ar delivered to Earth's atmosphere dominates the present-day atmospheric $^{36}$Ar budget (Marty,
2012; Broadley et al., 2022), instead of an early stage of catastrophic mantle outgassing and
provides a range on the relative proportions of outgassed and delivered $^{36}$Ar. In contrast, more than
87% of atmospheric $^{40}$Ar is outgassed from the mantle. It is thus important to note that for different
atmospheric elemental and isotopic species (in this case, $^{36}$Ar and $^{40}$Ar), there are distinct
breakdowns in the contribution from degassing or chondritic delivery. Given the model estimate
of ~4.8 to $5.0 \times 10^{15}$ moles of $^{36}$Ar contributed to the atmosphere through chondritic delivery, ~1.5
to $1.6 \times 10^{22}$ moles of $^{14}$N would be delivered to the atmosphere at the same time assuming a
chondritic $^{36}$Ar/$^{14}$N of about $3.2 \times 10^{-7}$ (Bogard et al., 1971; Marty, 2012; Mazor et al., 1970).
Chondritic delivery could thus contribute two orders of magnitude more nitrogen than is presently
in the atmosphere. The excess delivered nitrogen may have been dissolved into a more reducing
early mantle (Kadik et al., 2011), sequestered in the core at high pressures (Kadik et al., 2011;
Roskosz et al., 2013), or progressively sent back to the mantle via subduction (e.g., Bekaert et al.,
2021; Stüeken et al., 2016). No nitrogen contribution from mantle outgassing to the atmosphere is
required to account for the most abundant species in the present-day atmosphere. Therefore, the
common notion that the terrestrial atmosphere was outgassed from the mantle may be inaccurate.

The time evolution of atmospheric $^{20}$Ne/$^{22}$Ne and $^{40}$Ar/$^{36}$Ar ratios for successful model
realizations (Score$_{total}<7$) is shown in Figure 10. The model atmospheric $^{20}$Ne/$^{22}$Ne ratio starts at
8.9 (similar to Ne in CI chondrites, Williams and Mukhopadhyay, 2019; Table S3) and increases
546 to the present-day value of 9.8±0.16, with ~90% of the increase occurring in the first Gyr.

547 Atmospheric $^{40}$Ar/$^{36}$Ar ratio starts at 0.1 (estimated for CI chondrites based on Ott, 2002) and rises

548 to the present-day value of 298.6, but only 33% of the increase took place in the first Gyr. This is

549 because the increase in atmospheric $^{20}$Ne/$^{22}$Ne ratio is only sensitive to the rate of mantle

550 outgassing, which decays exponentially over time (Table S4), whereas a combination of processes

551 including mantle outgassing and the decay of $^{40}$K in the MORB source mantle and continental

552 crust all contribute to increases in the atmospheric $^{40}$Ar/$^{36}$Ar ratio. Our model indicates that, given

553 the current precision attainable with noble gas mass spectrometers, it would be more difficult to

554 resolve past atmospheric $^{20}$Ne/$^{22}$Ne ratios from the present compared to $^{40}$Ar/$^{36}$Ar in post-Archean

555 rocks. Our model predicts an Archean atmospheric $^{20}$Ne/$^{22}$Ne ratio of around 9.7. This value is >1%

556 lower than the modern ratio (9.8) and could be detected given current analytical precision in

557 $^{20}$Ne/$^{22}$Ne (e.g., Parai and Mukhopadhyay, 2021). Future high-precision measurements of ancient

558 Ne and Ar will thus provide further model constraints to refine our understanding of the

559 geophysical parameters that have exerted strong control on volatile transport throughout Earth's

560 history.

561

562 **5. Conclusion**

563 We developed a forward model of noble gas abundances and isotopic evolution in the MORB

564 source mantle, continental crust, and the atmosphere. We investigated initial terrestrial budgets of

565 noble gases and long-lived radioactive lithophile elements, as well as histories of mantle

566 processing and continental crust growth. We explored a wide parameter space and evaluated model
realizations by implementing a scoring system based on present-day and ancient (Archean and Devonian) observational constraints on noble gas isotopic compositions in the mantle and atmosphere. The three-reservoir model can simultaneously account for mantle He, Ne, Ar as well as atmospheric Ne and Ar abundances and isotopic compositions and provide insight into initial budgets, atmospheric chondritic noble gas delivery, and long-term mantle outgassing. We are able to narrowly estimate Earth's mantle initial budget of ³He to 1.0-2.5×10¹¹ atoms/gram, providing a basis for future evaluation of the retention of nebular gas throughout the accretion process. Our model outcomes demonstrate that the atmospheric ⁴⁰Ar/³⁶Ar ratio is largely insensitive to histories of continental growth and instead records primarily the mantle processing rate history. The pairing of radiogenic ⁴He and ⁴⁰Ar provides a novel way to investigate the BSE K/U ratio. An estimate of BSE K/U of 13,000 satisfies model observational constraints for a wide range of other input parameters. Model results also suggest that the ²⁰Ne/²²Ne ratio of atmospheric neon evolved with time with a predicted Archean atmospheric ²⁰Ne/²²Ne ratio of around 9.7. We find that chondritic delivery is the major contributor of atmospheric primordial ³⁶Ar instead of mantle outgassing throughout Earth history. Ample nitrogen would be delivered along with ³⁶Ar by impact degassing to account for the present-day atmospheric nitrogen inventory. Future efforts to combine the full suite of noble gases, including Kr and Xe, would further constrain mantle outgassing and the atmospheric evolution over time.

## Acknowledgments

We thank Rajdeep Dasgupta for efficient editorial handling and Philippe Sarda for constructive
588 comments that improved the manuscript. We gratefully acknowledge the McDonnell Center for

589 the Space Sciences for support to XJZ. This work constituted XJZ's senior honors thesis and was

590 supported by Department of Energy National Nuclear Security Administration grant DENA003911

591 to RP.

592

593 Main text: 7356 words

594
595 **Main Text Figure captions:**

596

597 **Figure 1. Schematic diagram of the three-box model.** The three geochemical reservoirs of concern are the MORB source mantle, continental crust and the atmosphere. Incompatible lithophile elements U, Th and K are transported to the growing continental crust (controlled by model parameters *CC* and *M*<sub>res</sub>). Through partial melting of the MORB source mantle, noble gas isotopes are lost to the atmosphere (controlled by *M*<sub>res</sub> and *N*<sub>res</sub>). Long-term decay of U, Th and K takes place in the MORB source mantle and the continental crust, leading to in-situ ingrowth of radiogenic daughter isotopes <sup>4</sup>He and <sup>40</sup>Ar. Part of the <sup>40</sup>Ar produced in the continental crust reservoir is lost to the atmosphere, controlled by the crustal degassing parameter *f*. Oceanic crust is omitted in this model due to its negligible mass and impact on Ar production. Reservoir sizes shown are not to scale.

607

608 **Figure 2. Conceptual model setup for mantle outgassing and atmosphere evolution over time.**

Catastrophic outgassing of the mantle occurred during the giant impact phase of accretion. Prior models intended to explain the present-day atmospheric <sup>40</sup>Ar/<sup>36</sup>Ar through the retention of Ar outgassed from the mantle during the catastrophic outgassing stage. However, retention of such gases, which have solar-like high <sup>20</sup>Ne/<sup>22</sup>Ne, would fail to yield the low present-day atmospheric <sup>20</sup>Ne/<sup>22</sup>Ne without invoking mass fractionation that would generate an overly low <sup>36</sup>Ar/<sup>22</sup>Ne ratio (Marty, 2012, 2022). Therefore, the model assumes that such gases released during accretion were lost to space during the giant impact stage. Instead, the model accounts for the delivery of
chondritic volatiles to the atmosphere by impact degassing of the late veneer (Zahnle et al., 1988, 2010) and mantle outgassing over time after the last giant impact. The mantle outgassing rate at ridges decreases exponentially with time. Mantle $^{20}$Ne/$^{22}$Ne remains the same while $^{40}$Ar/$^{36}$Ar increases over time due to mantle outgassing and the decay of $^{40}$K. The model accounts for the atmosphere Ne and Ar compositions, and subsequently provides constraints on chondritic delivery and continuous mantle outgassing through the combined He-Ne-Ar systematics.

**Figure 3.** Tested models of net continental crust growth over time. CC1 (green), CC2 (red), and CC3 (yellow) feature various types of early rapid crustal growth. CC1 reached ~130% of today's continental crust volume, which is followed by a net decrease of continental crust volume over the past 2 billion years (Fyfe, 1978); CC2 and CC3 test different timings of net crustal growth initiation based on Armstrong and Harmon (1981); CC4 (blue) and CC5 (purple) simulate two different sigmoidal curves where net continental crust growth both continue into later stages in Earth history (CC4: Belusova et al., 2010; Dhuime et al., 2012; Pujol et al., 2013; Walzer and Hendel, 2017; CC5: Condie, 1998; Poupinet and Shapiro, 1998; Rino et al., 2004).

**Figure 4.** Time evolution of MORB source mantle $^{238}$U, $^{232}$Th, and $^{40}$K abundances. The MORB source mantle is assumed to have a BSE starting composition. Abundances of (a) $^{238}$U, (b) $^{232}$Th, and (c) $^{40}$K in the MORB source mantle decrease with time due to extraction and radioactive decay. Green dashed curves show the isolated effect of radioactive decay. U, Th, and K are also extracted from the model MORB source mantle reservoir to the growing continental crust,
depleting the MORB mantle of incompatible elements over time (see Supplementary Material for model mass balance setup). Orange and purple time series are produced with continental crust net growth models CC3 and CC5, respectively, with $M_{res}$, (mass of the MORB source mantle reservoir) equal to 90% of the total mantle mass.

**Figure 5.** Time evolution of MORB source mantle (a) $^{4}$He/$^{3}$He and (b) $^{40}$Ar/$^{36}$Ar ratios with variation in mantle processing rates. He isotopes start at the solar composition ($^{3}$He/$^{4}$He=120 R$_{A}$, where R$_{A}$ is today's atmospheric $^{3}$He/$^{4}$He ratio; Mahaffy, 1998). The mantle $^{40}$Ar/$^{36}$Ar ratio initiates at an estimated chondritic value of 0.1 (Ott, 2002). MORB source mantle $^{4}$He/$^{3}$He and $^{40}$Ar/$^{36}$Ar increase with time because all He and Ar isotopes are lost to the atmosphere upon mantle partial melting but only radiogenic $^{4}$He and $^{40}$Ar accumulate over time owing to the decay of radioactive $^{235}$U, $^{238}$U, $^{232}$Th, and $^{40}$K. For each simulation, regassing of Ar is invoked only if the model present-day atmospheric $^{40}$Ar abundance is higher than the model constraint (Table S5). The present-day mantle $^{40}$Ar/$^{36}$Ar ratios after the regassing calculation are shown as squares. Time evolution curves are color-coded by $N_{res}$ (the number of reservoir masses processed in the past 4.488 Gyr), which characterizes mantle processing rates. As the color changes from purple to orange, $N_{res}$ increases from 3 to 6. For $N_{res}$ = 3 (purple), mantle $^{4}$He/$^{3}$He and $^{40}$Ar/$^{36}$Ar are too low (demonstrating a failed model realization), and the atmospheric $^{40}$Ar abundance is less than observed such that no regassing is invoked. When $N_{res}$ is larger, the present-day MORB source mantle is more processed and outgassed, and the MORB source mantle has a significantly higher $^{4}$He/$^{3}$He ratio (a). For the larger $N_{res}$ values shown here (4.05, 5 and 6), outgassed Ar is
overabundant and some amount of Ar regassing is needed to match the present-day atmospheric
$^{40}$Ar abundance constraint. For $N_{res}$ = 4.05 (light blue; success), mantle $^{4}$He/$^{3}$He and mantle
$^{40}$Ar/$^{36}$Ar after regassing both match observational constraints at the present-day. For $N_{res}$ = 5 and
6 (green and orange), a greater amount of atmospheric regassing leads to mantle $^{40}$Ar/$^{36}$Ar after
regassing that are too low compared to the present-day mantle $^{40}$Ar/$^{36}$Ar constraint. Variations in
mantle processing rate create a large range of modeled $^{4}$He/$^{3}$He and $^{40}$Ar/$^{36}$Ar evolution curves
over time, and exert determining constraints on model successes.

665

Figure 6. Present-day MORB source mantle ⁴He/³He and ⁴⁰Ar/³⁶Ar (with regassed Ar) ratios from Score<28 simulation outputs. Each dot represents the present-day MORB source mantle ⁴He/³He ratio and the present-day MORB source mantle ⁴⁰Ar/³⁶Ar ratio (including the regassed atmospheric Ar) for one model realization. Dots are color-coded by Score<28, from the lowest (darkest blue, best performance) up to 28 (lightest blue; see Section 2.4). The rectangle shows observational constraints for both ratios: the present-day MORB source mantle ⁴He/³He ratio ranging from 80,000 to 100,000 (Graham, 2002) and the present-day MORB source mantle ⁴⁰Ar/³⁶Ar ratio ranging from ~24,000 (Parai and Mukhopadhyay, 2021) to ~40,000 (Holland and Ballentine, 2006). The dense cluster of dots in the upper left of panel (a) are model realizations that do not invoke any regassing of atmospheric Ar (Figure S1). Mantle outgassing and decay of ²³⁵U, ²³⁸U, and ²³²Th influence the present-day MORB source mantle ⁴He/³He ratio on the x-axis, while outgassing, decay of ⁴⁰K and regassing of Ar influence the present-day MORB source mantle ⁴⁰Ar/³⁶Ar ratio on the y-axis. Simulations in panel (a) and (b) are generated using BSE K/U ratio

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679 of 13,000 and 19,000 respectively. With a BSE K/U of 19,000, a very restricted set of realizations
680 have Score<total>28, whereas a BSE K/U of 13,000 yields more successful realizations and better
681 overall fits (Score<total> as low as 1.16).

682
683 **Figure 7. Time evolution of the atmospheric <sup>40</sup>Ar/<sup>36</sup>Ar ratio for successful model simulations.**

684 Blue and red curves are model realizations with Score<total><28 and Score<total><7 respectively. The
685 atmosphere is assumed to have a chondrite-like starting <sup>40</sup>Ar/<sup>36</sup>Ar ratio, reflecting delivery of
686 atmospheric Ar from accreted material. Outgassing through mantle processing contributes Ar to
687 the atmosphere with the instantaneous mantle <sup>40</sup>Ar/<sup>36</sup>Ar ratio, which increases due to degassing
688 and decay of <sup>40</sup>K over time. Chondritic delivery also contributes a significant amount of Ar to the
689 atmosphere with a <sup>40</sup>Ar/<sup>36</sup>Ar ratio of 0.1. The present-day <sup>40</sup>Ar/<sup>36</sup>Ar ratio is anchored at 298.6 (Lee
690 et al., 2006). The orange lines illustrate the bounds on observational constraints for Archean (Pujol
691 et al., 2013) and Devonian (Stuart et al., 2016) atmospheric <sup>40</sup>Ar/<sup>36</sup>Ar. Simulations in panel (a) and
692 (b) are generated using BSE K/U ratios of 13,000 and 19,000 respectively. Both BSE K/U ratios
693 are able to generate successful simulated atmospheric <sup>40</sup>Ar/<sup>36</sup>Ar time evolution series with minimal
694 residuals compared to observations from the Archean and Devonian, but many more successful
695 model realizations are found using the BSE K/U of 13,000.

696
697 **Figure 8. Evolution of the atmospheric <sup>40</sup>Ar/<sup>36</sup>Ar ratio with variations in CC model, or with**
698 **different mantle processing history.** (a) Atmospheric <sup>40</sup>Ar/<sup>36</sup>Ar ratio time evolution curves are
699 generated by using the set of model parameters that generate the smallest Score<total> and only
varying the choice of CC models. Changing the CC model generates very similar atmospheric $^{40}$Ar/$^{36}$Ar ratio time series. The atmospheric $^{40}$Ar/$^{36}$Ar ratio over time is not significantly sensitive to the style of continental crust growth. Other model constraints (i.e., He and Ne) are responsible for the CC model selectivity shown in Figure S3. Green lines show the Archean and Devonian atmospheric Ar compositions (Cadogan, 1977; Pujol et al., 2013; Stuart et al., 2016). (b) Atmospheric $^{40}$Ar/$^{36}$Ar ratio time evolution curves are generated by using the same set of initial model parameters that generate the lowest Score$_{total}$ and only varying the mantle processing history.

The red curve is generated using the exponentially decreasing mantle processing rate pinned to $Q_p$, the present-day processing rate at mid-ocean ridges (this is the model default; see SM). The dashed curves are similar to mantle processing rates explored in Tucker et al. (2022). The light grey dashed curve reflects a linearly decreasing mantle processing rate pinned at $Q_p$, with the same integrated mantle processing as the exponentially decreasing scenario (same $N_{res}$). A linearly decreasing model degrades the fit to Archean constraints, as a less-vigorous early mantle outgassing does not raise the atmospheric $^{40}$Ar/$^{36}$Ar ratio fast enough. The dark grey dashed curve, produced by using a constant mantle processing rate $Q_p$ throughout Earth history, does not satisfy any past atmospheric $^{40}$Ar/$^{36}$Ar constraints. The rate of continuous mantle outgassing needs to have been significantly higher in the past to explain the change in atmospheric Ar composition over time (Figure S2).

<caption>**Figure 9.** Evolution of atmospheric $^{40}$Ar/$^{36}$Ar and $^{20}$Ne/$^{22}$Ne ratios in model realizations with Score$_{total}$<7. The atmosphere is assumed to have a chondritic starting composition, reflecting</caption>
delivery of Ne and Ar from accreted material. Mantle outgassing contributes Ne with higher

$^{20}$Ne/$^{22}$Ne reflecting solar nebular contributions to the Earth's interior noble gas budget (Williams

and Mukhopadhyay, 2019) and contributes Ar with the instantaneous mantle $^{40}$Ar/$^{36}$Ar, which

evolves due to mantle outgassing and decay of $^{40}$K over time. Time series are color-coded from

light yellow at the start time to dark blue after 4.488 Gyr of evolution starting at 80 Myr after

Earth's formation, and magenta squares denote one-billion-year intervals after the start time for

each model realization. The model results predict a small but resolvable difference between the

Archean atmospheric $^{20}$Ne/$^{22}$Ne and the present-day composition. Further measurements of the Ar

and Ne isotopic composition of the atmosphere over time will provide a test of the model

predictions.
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<figure><img src="image_3.png" /><figcaption>Figure 1</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4</figcaption></figure>
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Figure 5</figcaption></figure>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Figure 6</figcaption></figure>
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Figure 7</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 8</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 9</figcaption></figure>
<table><caption>**Table 1.** Model parameters and results for successful model realization with BSE K/U=13,000.</caption><thead><tr><th>Names</th><th>Tested Input Range</th><th>Score<sub>total</sub> &lt;7 Model Output Range</th><th>Score<sub>total</sub> &lt;28 Model Output Range</th></tr></thead><tbody><tr><td><sup>3</sup>He<sub>m,init</sub></td><td>1.0 × 10<sup>9</sup><br>to 1.0 × 10<sup>13</sup><br>(atoms/gram)</td><td>1.2 × 10<sup>11</sup><br>to 2.1 × 10<sup>11</sup><br>(atoms/gram)</td><td>1.0 × 10<sup>11</sup><br>to 2.5 × 10<sup>11</sup><br>(atoms/gram)</td></tr><tr><td><sup>3</sup>He<sub>m,init</sub><br>/<sup>36</sup>Ar<sub>m,init</sub></td><td>1.12 to 1.43*</td><td>1.12 to 1.43</td><td>1.12 to 1.43</td></tr><tr><td><i>f</i></td><td>1.0 × 10<sup>-10</sup><br>to 1.0</td><td>1.98×10<sup>-11</sup>yr<sup>-1</sup> to<br>2.05×10<sup>-9</sup> yr<sup>-1</sup></td><td>full range</td></tr><tr><td><i>N</i><sub>res</sub></td><td>2.00 to 15.00</td><td>3.90 to 4.80</td><td>3.60 to 5.10</td></tr><tr><td><i>M</i><sub>res</sub></td><td>2.0 × 10<sup>27</sup><br>to 3.6 × 10<sup>27</sup> grams<br>(50% to 90% of the<br>total mantle mass)</td><td>CC2 &amp; <i>M</i><sub>res</sub>= 90%;<br>CC3 &amp; <i>M</i><sub>res</sub>=82.5% or<br>90%;<br>CC4 &amp; <i>M</i><sub>res</sub>=62.5%;<br>CC5 &amp; <i>M</i><sub>res</sub>=62.5%;</td><td>CC2 &amp; <i>M</i><sub>res</sub>= 82.5% or<br>90%;<br>CC3 &amp; <i>M</i><sub>res</sub>=82.5% or<br>90%;<br>CC4 &amp; <i>M</i><sub>res</sub>=62.5%;<br>CC5 &amp; <i>M</i><sub>res</sub>=62.5%;</td></tr><tr><td>CC models</td><td>CC1 to CC5</td><td></td><td></td></tr></tbody></table>

<footnote>*Calculated using <sup>3</sup>He/<sup>22</sup>Ne=10 and <sup>36</sup>Ar/<sup>22</sup>Ne from 7 to 9 (see Section 2.1)</footnote>