Corresponding author: A. Schöpa, School of Earth Sciences, University of Bristol, Wills Memorial Building, Bristol BS8 1RJ, UK. (firstname.lastname@example.org)
 Incremental emplacement of large silicic intrusions limits the size and lifetime of any magma chamber if earlier injected magma pulses solidify before the next pulse is emplaced. Geochronology indicates that long-term average magma fluxes of plutons are too low for the accumulation of magma volumes larger than a single pulse. To better constrain the formation of large-volume magma chambers, we investigate the influence of a changing emplacement rate of successive sill injections that form a composite intrusion in the upper crust. A thermal model with an explicit finite difference scheme simulates periods of transient high magma fluxes keeping the long-term average flux small. Several scenarios regarding how fluxes vary were analyzed. A progressive increase in flux does not result in magma accumulation. Only a step-like flux increase of at least one order of magnitude above the background flux produces a large magma reservoir. To generate a magma chamber of a melt-crystal mix above solidus temperature with a volume of 500–2000 km3, transient high fluxes of at least 1.25 × 10 − 2km3/a, equivalent to vertical, one-dimensional accretion rates of a few cm/a, are required. The transient high flux range where magma accumulates is largest for model scenarios with one period of transient high flux, and therefore, this emplacement style is favored for the built-up of substantial magma chambers during pluton growth. Intrusion scenarios with three and four periods of transient high fluxes do not generate reservoirs of mobile magma, but magma mushes can be present for some hundred thousand years.
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 Incremental growth of batholiths and other intrusive bodies in the Earth's crust is now widely accepted to be one if not the fundamental mechanism of pluton construction [e.g., Pitcher, 1979; Hardee, 1982; Wiebe, 1988; Wiebe and Collins,1998; Saint Blanquat et al., 2001, 2006; de Silva and Gosnold, 2007; Lipman, 2007; Menand, 2008; Bartley et al., 2008; Vigneresse, 2008; Miller et al., 2011; Saint Blanquat et al., 2011]. Field relations, geochronology and geophysical data indicate that most plutons were constructed through repeated intrusions of sheet-like sills (Cruden and McCaffrey, 2001; Morgan et al. 2005; Pasquarè and Tibaldi, 2007; Horsman et al. 2009; Menand, 2011; Leuthold et al. 2012) with long-term average vertical growth rates of few millimeters per year (Coleman et al. 2004). However, numerical simulations show that these rates are too low to create large magma chambers because successive sills will solidify before the emplacement of the next sill (Annen, 2009). Consequently, the size of any magma chamber would not exceed the size of one sill intrusion. In simulations, emplacement rates that are at least one order of magnitude higher than the average emplacement rates of plutons are necessary to accumulate large volumes of magma able to feed the largest explosive eruptions (Brooks Hanson and Glazner, 1995; Yoshinobu et al. 1998; Annen, 2009), which questioned the relationship between plutonism and volcanism. Those numerical models assumed a constant flux into the magmatic system and did not take a fluctuating flux over time into account. It has been proposed that the eruption of large volumes of magma could be related to a progressive increase in magma production during a waxing-waning process (Lipman, 2007; de Silva and Gosnold, 2007). Thus, large magma reservoirs could possibly form in times of higher fluxes even when the long-term average flux is low (Druitt et al. 2012).
 To progress in the question about the link between plutonism and volcanism, this study tests model scenarios where the average magma flux is low and in agreement with geochronological data and where the effects of a non-constant magma flux are taken into account. For this purpose, a thermal model of magma emplacement was developed where the time interval between successive sill intrusions that build up a crustal pluton can be varied. Although the model does not intend to reproduce an entire plutonic system with all the complexities of nature, it was designed to fit the dimensions and age relationships known from the Tuolumne Intrusive Suite, California. This was done in order to investigate the physically possible conditions for a magma chamber to grow in the upper crust. For the Tuolumne Intrusive Suite, a zoned granitic intrusion in the Sierra Nevada Batholith, U-Pb geochronology gives insight into the age variations across the pluton. The age range of several millions of years from the rims to the core of the intrusive suite is used as a constraint in the numerical simulations. With the Tuolumne Intrusive Suite as a case study, the aim of the present paper is to test if large crustal magma chambers are related to the non-steady growth of plutons and buildup during periods of highest magma fluxes even though the long-term flux is small.
2 Geological Setting
 The Tuolumne Intrusive Suite is part of the Mesozoic Sierra Nevada Batholith in California. It covers an area of more than 1000 km2 and extends for about 70 km in N-S direction being 15–35 km wide. Like adjacent large-volume plutons, the Tuolumne Intrusive Suite is normally zoned both texturally and compositionally (Figure 1): the nested map units get progressively younger, more porphyritic, and more evolved toward the center of the intrusion (Kistler et al. 1986). This pattern led Bateman and Chappell (1979) to conclude that the Tuolumne Intrusive Suite formed through fractionation of a single parental magma at the emplacement level. However, this model was questioned by recent geochronological studies (Kistler and Fleck, 1994; Coleman and Glazner, 1997; Matzel et al. 2006; Miller et al. 2007; Memeti et al. 2010) suggesting an age variation of about 8Ma across the suite. In the Tuolumne Intrusive Suite, the 93.5 ± 0.7Ma granodiorite of Kuna Crest and the tonalite of Glen Aulin are interpreted to be the eastern and western part of one unit, separated by younger rocks. Toward the center, the Half Dome granodiorite, yielding ages between 92.8 ± 0.1Ma and 88.8 ± 0.8Ma, and the Cathedral Peak granodiorite are exposed. The core of the Tuolumne Intrusive Suite is made up by the 85.4 ± 0.1Ma Johnson Granite porphyry [Coleman et al., 2004, 2005]. These progressive changes in age of several million years toward the center of the Tuolumne Intrusive Suite resulted in the understanding that this pluton was emplaced incrementally (Coleman et al. 2004; Glazner et al. 2004). However, the shape and the size of the individual magma pulses remain controversial (Memeti et al. 2010).
 In addition to the geochronological data, the presence of K-feldspar megacrysts in the Cathedral Peak granodiorite supports the hypothesis of incremental emplacement of the Tuolumne Intrusive Suite. Repeated thermal cycles of heating and cooling produced by recurrent magma intrusion would have promoted the growth of megacrysts and thus could have caused the textural coarsening (Johnson and Glazner, 2010).
 The exact emplacement mechanisms of the Tuolumne Intrusive Suite are still not entirely understood. Glazner (1991) suggested that the space for the growing pluton was created in the releasing bands of dextral strike-slip faults. This faulting resulted from the right-oblique convergence of the North American and the Farallon plate in the Mesozoic. Coleman et al. (2005) mentioned the possibility that the emplacement of the Tuolumne Intrusive Suite was facilitated by an intrabatholithic shear zone. This is supported by field relations, gravity surveys, and an anisotropy of magnetic susceptibility study of the Johnson Granite porphyry (Titus et al. 2005) suggesting intrusion in a tension gash of a regional shear zone.
 The initial geometry of the Tuolumne Intrusive Suite before exhumation and erosion took place and sculptured the pluton to how it is exposed today continues to be speculative. Field exposures at the recent erosion level reveal a minimum thickness of the Johnson Granite porphyry of 1.5 km (Titus et al. 2005), whereas the entire intrusive suite is vertically exposed for about 2.5 km. Specific gravity measurements in the Sierra Nevada showed that lateral gravity residual variations can be explained by the exposed granitoids if they extend down to a depth of approximately 10 km (Oliver et al. 1993). This depth coincides well with seismic studies: below 12 km depth, a zone of lower seismic velocities than those of the overlying and underlying materials is explained to represent the roots of the upper crustal plutons in the Sierra Nevada (Fliedner et al. 2000). An estimated thickness of 10 km for the Tuolumne Intrusive Suite is in line with the thickness of other, large-volume intrusions of the Sierra Nevada Batholith [e.g., the Mount Givenspluton, McNulty et al., 2000].
 Geochemical data suggest that isotopic, major, and trace element variations in the granitoids of the Tuolumne Intrusive Suite originate from crystallization-differentiation in the lower crust or upper mantle (Kistler and Fleck, 1994; Ratajeski et al. 2005). This agrees with models of silicic magma generation taking place at a deeper level than the emplacement level [e.g., Grove and Kinzler, 1986; Hildreth and Moorbath, 1988; Musselwhite et al., 1989; Rogers and Hawkesworth, 1989; Müntener et al., 2001; Grove et al., 2003; Annen et al., 2006]. Generation of silicic melts requires a substantial volume of basalt either as a parent or as a heat source if crustal melting is involved and a significant amount of residual mafic cumulates of high-density and high-seismic velocities are left behind (Annen et al. 2006). Although the production of silicic magma at the crust-mantle boundary may be continuous (Vigneresse, 2008), segregation, ascent, and emplacement of this magma are likely to be episodic processes (Solano et al. 2012). This is concurrent with the concept that the Tuolumne Intrusive Suite grew incrementally through the amalgamation of magma pulses. Additional arguments pointing toward lower crustal or upper mantle formation of the silicic magmas of the Tuolumne Intrusive Suite are field and seismic observations. Mafic rocks make up only 1% of outcropping intrusives in the Sierra Nevada Batholith (Coleman and Glazner, 1997), and crustal velocities of up to 6.6 km/s are measured down to a Moho depth of 30–35 km (Fliedner et al. 1996), which is interpreted to be caused by a crust predominately composed of granitoids (Ducea, 2001). An alternative concept of high-silica magma generation includes differentiation of mantle-derived basalt in a shallow magma reservoir [e.g., Sisson and Grove, 1993; Grove et al., 1997; Pichavant et al., 2002]. However, this concept is problematic regarding the formation of the Tuolumne Intrusive Suite because the high-density cumulates would have to be present in the upper crust and would influence seismic wave velocities. Nevertheless, we cannot exclude that this process might have played a minor role during the formation of the Tuolumne Intrusive Suite.
 To address the problem of incremental pluton growth, a numerical model was used to simulate the repeated emplacement of sills in the upper crust. As shown by previous studies, a constant magma flux would not lead to the accumulation of magma if the average flux is low (Brooks Hanson and Glazner, 1995; Yoshinobu et al. 1998; Annen, 2009). The aim of this work is hence to test if for the same low average flux, transient high magma recharge rates would produce a magma reservoir and if that is the case to quantify the magma reservoir's lifetime and size.
 Although magmas with a low melt fraction sometimes erupt (Murphy et al. 2000), more generally magma is considered to be mobile when crystals are suspended in melt, which usually occurs at a melt fraction higher than 60% (Marsh, 1981; Lejeune and Richet, 1995). When crystals start to touch each other, in the range of 60–40% melt, the properties of a crystal-melt mixture, such as viscosity, change rapidly, as does the mixture's behavior from a liquid-like manner to a solid-like manner (Caricchi et al. 2007; Huber et al. 2010a). The mixture is then regarded as a crystal mush where crystals form a rigid, connected network (Vigneresse et al. 1996; Petford, 2003; Bachmann and Bergantz, 2004; Huber et al. 2009). The effect crystal shape has on magma rheology is not included in these considerations (Mueller et al. 2011). Segregation of melt from a mush can produce a layer of mobile melt (Jackson et al. 2003; Bachmann and Bergantz, 2004). Many models that explain the occurrence of large eruptions rely on the reactivation of a mush [Bachmann and Bergantz, 2006; Huber et al., 2011, 2012; Karlstrom et al., 2012]. Even though we speak of mobile magma (melt fraction > 60%), our model does not involve melt removal or segregation and we do not model eruptions.
 This section first describes the model geometry and then gives the physical parameters adopted in the model before introducing the reader to the different intrusion modes used to vary the time between successive sill intrusions and thus change the magma emplacement rate over time. Finally, some model limitations are listed.
 The numerical model used in this study calculates temperatures and melt fractions in the Earth's crust. The model is axisymmetric and computations are performed on a two-dimensional slice of a three-dimensional space [cf. Annen et al., 2008; Annen, 2009]. This geometry and the use of cylindrical coordinates means that heat is transferred in the three dimensions of space. The numerical domain of the model is 20 km wide and 18 km deep. It is discretized over a squared grid with a spatial resolution of 50m × 50m. Temperatures and melt fractions are calculated at the nodes of the grid. A convergence test was performed to ensure that the grid size and the type and position of the boundaries do not affect the modeling results.
 A growing pluton in the Earth's crust is simulated by the repeated emplacement of hot sills. Several geometries can be envisaged for the successive pulses such as blobs, sills, and dikes. We use sills in the model because it is the most common geometry observed in the field in general [e.g., McCaffrey, 1992; Michel et al., 2008; Searle, 1999; Searle et al., 2003] although clear field evidences have not been found in the Tuolumne Intrusive Suite (Žák and Paterson, 2005). We note that emplacing dikes results in the same thermal evolution than emplacing sills (Annen et al. 2008). However, blobs would give different results because of different volume : surface ratios. The sills forming the pluton are disc shaped and vertically piled up to create a cylindrical intrusion (Figure 2). The model neglects magma transport through the crust and detailed sill emplacement mechanisms by simulating an instantaneous emplacement at the intrusion level. This assumption is reasonable in the framework of geological timescales of several millions of years as used in the model. Individual sills and hence the intruded pluton have a radius of 18 km, corresponding to the aerial extent of the Tuolumne Intrusive Suite of 1000 km2.
 The thermal evolution of the system is controlled by the sill emplacement rate, i.e. the sill thickness divided by the time interval between two sill injections (Annen and Sparks, 2002). The exact thickness and time interval are not relevant as long as their ratio is maintained. To optimize computation efficiency and accuracy, we opted for sills that are 200m thick (Vsill = 204 km3) after having verified that the same results were obtained with 100m and 400m thick sills (Figure A1). Fifty sills are injected during each model run accreting to a total intrusion thickness of 10 km. We used this value in our simulations because gravity and seismic data reveal an approximate thickness of about 10 km for the Tuolumne Intrusive Suite (Oliver et al. 1993; Fliedner et al. 2000).
 The first sill injection is emplaced with its top at 5 km depth. This is consistent with the emplacement depth of the Tuolumne Intrusive Suite. It was inferred based on an Al-in-hornblende geobarometer that gave 1–3kbar as intrusion pressures (Ague and Brimhall, 1988) corresponding to 3.5–7 km of overburden (Bateman, 1992) removed through erosion in the Sierra Nevada since the Cretaceous (Saleeby et al. 2003).
 If the previous sill contains any liquid with a melt fraction of more than 40%, the successive sill is injected at the lower mush-liquid boundary (melt fraction 40%) of the previous sill (van der Molen and Paterson, 1979; Marsh, 1981; Huppert and Sparks, 1988; Lejeune and Richet, 1995). This emplacement style is supported by field observations (Blundy and Sparks, 1992; Wiebe and Collins, 1998; Morgan et al. 2005; Miller et al. 2011) and field studies that shallow crustal magma chambers can evolve from sills (Gudmundsson, 2011). If the previous sill does not contain any liquid with a melt fraction of more than 40%, the successive sill is emplaced above the older sill because the rigidity contrast at the interface between sills and country rock favors intrusion emplacement [Kavanagh et al., 2006; Menand, 2008, 2011]. We note, however, that for our purposes, the relative position of successive sills is not important (Annen et al. 2008).
 The volume of the sills is accommodated by floor subsidence where underlying material is displaced downward. This type of pluton emplacement is inferred from mechanical models and field studies (Paterson et al. 1996; Cruden, 1998; Wiebe and Collins, 1998; Cruden and McCaffrey, 2001; Grocott et al. 2009) although space for the Tuolumne Intrusive Suite is assumed to have been created by strike-slip faulting (Glazner, 1991) or within a shear zone (Coleman et al. 2005). However, the space-creation process has no influence on the model outputs because we focus here on the thermal evolution and not on the mechanical state of the system.
 In the numerical model, an explicit finite difference scheme computes temperatures according to the equation of conductive heat transfer (Appendix B). The model takes heat production of radioactive decay and of phase changes into account.
 The boundary condition of the model is constant at the surface with a temperature t of 0°C. The right boundary of the model is an insulating boundary where no heat is allowed to flux through. The left boundary is the axis of symmetry. At depth, infinity-like conditions are modeled with a far-field temperature constrained by the geothermal gradient. This configuration assumes that the magma reservoir feeding the growing intrusion from depth is too far away to thermally influence the temperature at the emplacement level. Over the timescale of the simulation, 7.3Ma, this assumption is valid if the source is more than a few of kilometers below the emplacement level.
 The initial thermal condition of the crust in the numerical simulations is calculated with a steady state geothermal gradient, which corresponds to a surface heat flow of 90mW/m2 (Turcotte and Schubert, 2002). This high value was chosen to give an upper bound of the initial temperatures of the host rock. Radiogenic heat production A of the country rock and the intruding magma is 4.8 × 10 − 6W m − 3, a value representative for acid intrusives like granite (Wollenberg and Smith, 1987; van Schmus, 1995).
 Thermal conductivity k of the country rock and the magma was set to an initial value of k0 = 3.0 Wm − 1K − 1 which is in the range for crystalline rocks of the upper crust [Roy et al., 1968a, 1968b; Rao et al., 1976; Chapman, 1986; Jaupart and Provost, 1985; Robertson, 1988] and has been used in crustal intrusion models (Fountain et al. 1989; Petford and Gallagher, 2001; Annen et al. 2006). During the model run, k depends on pressure and temperature (Chapman and Furlong, 1992) and varies by less than 3%. The modeling results are dependent on k0 in that respect that a higher k0 leads to faster cooling of the emplaced magma whereas more magma can accumulate with a lower k0 due to more insulating conditions. In the range of realistic values for the upper crust between k0 = 2.5 − 3.5 Wm − 1K − 1 (Roy et al. 1968a; Roy et al. 1968b; Rao et al. 1976; Chapman, 1986; Jaupart and Provost, 1985; Robertson, 1988), assembled magma volumes change by ± 15%. Similarly, the use of a strong dependence between conductivity and temperature (Whittington et al. 2009) might increase the magma volume by 20% (Gelman et al. 2012). We note here that regarding fluxes, we are aiming at estimating their order of magnitude; thus, the uncertainty in conductivity and geothermal gradient is of limited consequence in our conclusions.
 A tonalitic composition is assumed for both the magma emplaced and the host rock. The melt fraction-temperature relation for this composition is provided by experimental petrology (Piwinskii and Wyllie, 1968; Martel et al. 1999) and was used in this study as published in Annen et al. (2008). The injection temperature of the magma is 1000°C. Although this temperature is high and might in fact have been lower, even below the liquidus if we assume that the intruded magma already contained crystals, this value represents an upper limit of possible intrusive conditions.
 The liquidus temperature TL and the solidus temperature TS of both the magma and the host rock are 990°C and 760°C, respectively. The model allows melting of the country rock if its temperature rises above the solidus. Further physical parameters of the magma and the country rock are given in Table 1.
Table 1. Model Input Parameters for the Intruding Magma and the Surrounding Crust
W m − 1 K − 1
kg m − 3
Specific heat capacity
J kg − 1 K − 1
4.8 x 10 − 6
W m − 3
3.0 x 10 5
J kg − 1
 Within the errors of the geochronological data (Coleman et al. 2004), the minimum age difference between the oldest and youngest unit of the Tuolumne Intrusive Suite is 7.3Ma. We are aware of the fact that these data cover about 1.5 km of relief and that we do not know which ages those rocks below the current erosion level would yield. With an emplacement time ttot of 7.3Ma, a total thickness of 10 km and a surface area of 1000 km2, thus a total volume Vtot of about 10,000 km3, the average magma input flux
of the Tuolumne Intrusive Suite is 1.4 × 10 − 3km3/a corresponding to a vertical, one-dimensional accretion rate of 1.4 × 10 − 3m/a. For intrusions with a high width : thickness ratio like our model, the controlling parameter of the thermal evolution of the magmatic system is this one-dimensional emplacement rate and not the volumetric magma flux (Annen et al. 2008). To change the emplacement rate, it is either possible to vary the time between magma pulses or the thickness of these pulses. The former possibility is used in our model.
3.3 Intrusion Modes
 Incremental pluton growth can be envisaged by a variety of intrusion scenarios if the size of the single intrusion pulses and the time between these pulses is varied (Saint Blanquat et al. 2011). This results in a changing magma flux during the construction of a crustal pluton even though the average flux might be the same for different intrusion scenarios. To simulate diverse intrusion scenarios with fluctuating magma fluxes in the numerical model, the time between successive sill injections that construct the pluton incrementally is varied in different ways (Figures 3 and 4), keeping the volume of a single sill injection, their total number, and the total duration of the simulation ttot constant in all scenarios. Hence, the average emplacement flux QAV is the same for each intrusion scenario simulated.
3.3.1 Gradual Flux Changes
 For each intrusion scenario tested, the first injection is emplaced at the start of the model run at t = 0. The injection time of subsequent sills is calculated according to different mathematical functions. Combining one of those with the constraint of the age range of the Tuolumne Intrusive Suite, different intrusion modes used in the numerical model include the following:
 A constant magma flux of 1.4 × 10 − 3km3/a (broken line in Figures 3 and 4) corresponding to QAV.
 A needle-shaped flux-time curve displaying lower fluxes at the beginning and the termination of the model run and a maximum flux of 2.8 × 10 − 2km3/a at half of the total emplacement time at 3.65Ma (Figure 3a) where the injection times follow half a period, one π, of a sine function.
 A trough-like flux-time curve with highest fluxes of 2.8 × 10 − 2km3/a at the start and the end of the model run, with a lower flux of 8.9 × 10 − 4km3/a in the central part of the total emplacement time (Figure 3b), calculated with half a period of a cosine function.
 A scenario where the flux decelerates by more than one order of magnitude over the whole model run (Figure 3c), achieved by a reversed logarithmic function.
 An intrusion mode where the flux accelerates by more than one order of magnitude (Figure 3d) and the injection times are calculated by a logarithmic function.
 Apart from the vertex of needle-shaped function, these scenarios adopt a gradual increase or decrease in magma flux over time. The slope of the accelerating and decelerating flux scenarios can be varied to simulate a faster or slower increase or decrease of the magma flux with time (see section C1 for further details).
3.3.2 Peak Scenarios
 Furthermore, we simulated step-like changes of the magma flux over time (Figure 4) where the injection times of subsequent sills are obtained through the use of linear functions with different slopes. The step scenarios are characterized by two fluxes, a lower flux QLF which is below QAV and lasts for tLF and a higher flux QHF which is above QAV and lasts for tHF. The changes between QLF and QHF are nearly instantaneous.
 Step-like flux scenarios comprise the following intrusion scenarios:
 One peak of QHF sustained for tHF (Figure 4a).
 Two peaks of QHF, each lasting for tHF (Figure 4b).
 Three peaks of QHF, each sustained for tHF (Figure 4c).
 Four peaks of QHF, each maintained for tHF (Figure 4d).
 The two, three, and four peaks scenarios are symmetric with respect to ttot / 2 resulting in identical tLF before the first peak and after the last peak of QHF (see section C2 for further details).
 With the given constraints of the step scenarios, tHF and the magnitude of QHF were systematically varied to investigate the conditions when magma can accumulate. In order to minimize QHF which is required to assemble large magma volumes in the model, the periods of tHF needed to be either (a) positioned at the end of the model run at the end of the intrusion history when the system is already heated up (one peak scenarios) or (b) placed as closely together as possible (scenarios two, three, and four peaks). This approach was used because it is more likely that magma accumulates in a hot and hence viscous environment [cf. Jellinek and DePaolo, 2003].
 We explore the conditions required for the accumulation of magma above 0% (magma above TS, presence of melt), 40%, and 60% melt fraction. As reported in the beginning of this section, 40% and 60% melt volume correspond to transitions in magma rheology: the break down of the crystal framework of a magma mush and complete suspension of crystals in a mobile magma, respectively. For each level of magma accumulation, tHF is varied incrementally to be 0.5%, 1%, 2%, 3%, etc. of ttot, corresponding to time spans of 36.5 ka, 73 ka, 146 ka, 219 ka, etc. The magnitude of QHF is limited by the number of high flux peaks and by tHF. At each time increment of tHF, the highest possible transient magma flux Qmax is calculated (section C2). As one sill needs to be emplaced at the start of the model run to simulate the oldest ages of the pluton, the spectrum of possible QHF is restricted. Starting with Qmax, QHF is systematically decreased by increments of 1 × 10 − 2km3/a, 2.5 × 10 − 3km3/a, 1 × 10 − 3km3/a, and 2.5 × 10 − 4km3/a until we reach the condition where no magma over more than one sill is accumulated. This is tested for the three different melt fraction thresholds: 0%, 40%, and 60%. For simulations where magma accumulation is obtained, the size and lifetime of the magma reservoir are recorded.
 To give first-order approximations of the thermal evolution of a growing pluton, heat transfer in the numerical model is only by conduction. No convective process, neither magma convection nor hydrothermal circulation in the wall rock, is considered in the simulations. The model also does not include advected heat that the crystallizing magma would lose through exsolved fluids. All these mechanisms would accelerate cooling of the emplaced magma and thus would require higher QHF for magma accumulation. Therefore, our results give a lower limit of QHF necessary to form a magma chamber.
 It is beyond the scope of this study to model the mechanism that would produce the large volumes of tonalitic magma emplaced in our model. From the several processes to generate hot that are currently discussed, we favor differentiation of a primitive magma in the lower crust or upper mantle to produce the magmas of the Tuolumne Intrusive Suite. Geochemical, field, and seismic data argue for this process and against the generation of large volumes of silicic magmas at shallow crustal levels (section 2). The mechanisms at the source of the magma that would produce transient high fluxes [e.g., Solano et al., 2012] are also beyond the scope of this paper.
 The model does not take deformation of the material into account although modeled emplacement rates higher than strain rates reported on accommodating material transfer in pluton aureoles [as high as 10 − 7s − 1, Johnson et al., 2001] might be unrealistic. In fact, the model does not intend to reproduce intricate natural processes but is designed to give first-order approximations of pluton formation. Although it is just one possible model, it is the most likely based on our current knowledge of plutonism.
4.1 Thermal Evolution
 Our thermal model investigates the temperature evolution in a growing pluton and in the crust that surrounds it. The pluton grows through the emplacement of hot sills that bring heat into the system. This heat is conducted away from the sills, predominately toward the Earth's surface. At the beginning of the simulations, the sills are intruded into cold crust and solidify completely before the injection of the next sill thereby transferring all their heat to the country rock. With maturation of the system and/or shorter time intervals between subsequent sill injections, more heat is advected than can be conducted away and magma (mixture of melt and crystals in varying portions) of previous sills persists until the emplacement of the next sill. The controlling parameters of this competition between conduction and advection are the rock thermal conductivity and the magma emplacement rate. This study investigates the effect of a changing emplacement rate with time simulated by varying the time interval between magma injections. Regarding the whole duration of the model run ttot, QAV is the same for the different scenarios of emplacement rate fluctuations that are simulated for a given pluton total thickness. For each scenario, the volume, lifetime and recurrence of an active magma chamber are quantified.
4.2 Conditions for Magma Accumulation
 Given the geometrical and age constraints of the Tuolumne Intrusive Suite, the long-term average magma flux QAV is 1.4 × 10 − 3km3/a. If QAV is held constant for the whole emplacement time (broken line in Figures 3 and 4), no magma is left in the injected sills when the next sill is emplaced at any time of the simulation and thus no magma accumulates. Magma is also not assembled in the four scenarios of a gradually changing magma flux with time (Figure 3). Each sill cools down below solidus temperature in about 1 ka before the next sill is injected. This happens for the intrusion modes where the flux-time curve is needle or trough shaped as well as for the scenarios where the magma flux increases or decreases over the total emplacement time, regardless of the slope of the magma flux-time curve. No magma persists in these scenarios although QHF was as high as 2.8 × 10 − 2km3/a in the needle scenario, more than one order of magnitude higher than QAV, and sustained for tHF 20 ka.
 In contrast, magma subsists between two sill injections and accumulates in several scenarios with steps in the flux-time plot (Figure 4). Mobile magma with a melt fraction of more than 60% can persist in some intrusion scenarios. This is dependent on two conditions, the magnitude of QHF and the time tHF at which it is maintained. Higher QHF or longer tHF are needed to accumulate more magma volume or magma with a higher melt : crystal ratio. However, the magnitude of QHF and its duration tHF are constrained by QAV and ttot so that a maximum high transient magma flux Qmax (section C2) exists for each tHF and only a limited number of intrusion scenarios are possible to simulate.
 Figures 5, 6, 7, and 8 show the conditions for magma accumulation in the one peak and two, three, and four peaks scenarios. In general, QHF needs to be at least one order of magnitude higher than QAV so that mobile magma can accumulate. QLF for those scenarios can be one order of magnitude lower than QAV. For longer tHF, smaller QHF are required to accumulate magma and hence QLF drops accordingly, following the same trend as QHF (Figures 5, 6, 7, and 8). If more peaks of QHF are simulated, higher QHF are required for mobile magma to persist in the growing pluton. This is due to the fact that more peaks of QHF result in lower QLF for the same QAV.
 There is only a small flux window of about half an order of magnitude over which magma can accumulate between the highest possible flux Qmax and the minimum, higher transient flux QHF below which no magma persists. For an increasing number of QHF peaks, the window of QHF where magma subsists and the range of tHF decrease. We note that for some scenarios, more than 50% of Vtot is injected at QHF but such high volumes of intrusive rocks emplaced at high flux have not been found in the geological record. Furthermore, it would be mechanically difficult to create several kilometers of vertical space in the crust for the growing intrusion in the short periods of tHF. In the following, we concentrate only on one peak and two peaks scenarios where less than 50% of Vtot is injected at QHF to be conform with geochronological data.
4.3 Peak Scenarios
 When having a closer look at the occurrence of magma during the whole emplacement time ttot, it is evident that if QHF is high enough, magma starts to accumulate at the end of the QHF period when the system is heated up and magma supply is high. Magma volumes reach maximum values at the end of the QHF period. Then all magma will start to cool down below solidus when QLF resumes.
 For the intrusion scenarios with more than one peak of QHF, the last peak is the most favorable for magma accumulation because the system is hotter. Magma can also accumulate in the first and second peaks if QHF is marginally higher than QHF necessary for persistent magma in the last peak, but more magma accumulates in the later peak than in the earlier peaks. However, no magma persists from one peak of QHF to the next peak. During QLF, the system cools down so that eventually each sill injection is below solidus temperature before the next sill is emplaced.
 We tested for the one peak scenarios whether the relative timing of QHF, when it starts and ends with respect to ttot, affects the volume of accumulated magma. This was done by starting QHF at different times when the magnitudes of QHF and QLF as well as tHF were kept constant. Although the effect is small, more magma accumulates when QHF is closer to the end of the model run (Figure 9) and the system is hotter. This implies that QHF needs to be higher to generate the same volume of magma if it is placed further toward the start of the model run, i.e., starts earlier within ttot. When not mentioned explicitly, the modeling results of this study for the one peak scenarios where obtained with QHF placed as close to the end of the model run as possible to give minimum QHF for magma accumulation.
 More magma volume is assembled in the system for higher QHF if tHF is constant. More magma also persists if tHF is longer for a fixed QHF. No magma volumes larger than one single sill injection are present for QHF smaller than 1 × 10 − 2km3/a, which is about one order of magnitude higher than QAV. Maximum magma volumes above TS can reach 2000 km3, almost 20% of Vtot (see Figure 10a for the one peak scenarios). Mobile magma volumes of up to 15% of Vtot can be present if QHF is higher than 5 × 10 − 2km3/a. With more peaks of QHF, less magma volume can accumulate for the same QHF and the same tHF because QLF is lower (compare Figure 5 with Figures 6, 7, and 8). Hence, a single peak of QHF is the most favorable scenario for magma accumulation reaching the highest volumes of eruptible magma.
4.4 Magma Chamber Lifetime
 The total lifetime of a magma chamber
includes the time tacc where magma volumes Vmagma are accumulated until maximum volumes of magma Vmax are reached, and the time tcool, it takes Vmax to cool down below solidus temperature TS. In the scenarios with step-like changes of the magma flux over time, Vmax is attained at the end of the QHF periods.
4.4.1 One Peak Scenarios
 In the one peak scenarios, tHF can be up to 365 ka and QHF needs to be at least 1.25 × 10 − 2km3/a so that magma above TS subsists (Figure 5). A flux of at least 6 × 10 − 2km3/a during 73 ka is required for mobile magma accumulation. As mentioned earlier, higher QHF and longer tHF produce more magma. Maximum tacc are about 135 ka for magma volumes above TS (Figure 10b) and not more than 40 ka for mobile magma. However, larger Vmagma are not necessarily assembled over longer time spans, and Vmax of almost 2000 km3 with about 1500 km3 of mobile magma can be accumulated in about 50 ka.
 Overall, tcool is longer for higher Vmagma and can reach 230 ka for Vmagma above TS (Figure 10c). Mobile magma cools down in less than 80 ka. In general, tcool is twice as long as tacc, and thus, tmagma is mainly controlled by tcool in the one peak scenarios. The total lifetime tmagma of a magma reservoir with more than 0% melt can be as long as 290 ka and tmagma for mobile magma with more than 60% melt can reach 115 ka. In other words, the time over which mobile magma is present is much smaller than the time spent by the magma above TS, which could be thought as the “active” pluton time scale during which petrological and textural re-equilibrium occurs. We note that the scenario with the longest tacc is not necessarily the scenario with the longest tcool. This might be a modeling effect reflecting when the magma volumes are recorded during the model run.
4.4.2 Two Peaks Scenarios
 If more peaks of QHF are modeled, magma only persists for shorter tHF and higher QHF when compared to one peak scenarios (Figures 5 and 6). For magma above TS to accumulate in the two peaks scenarios, tHF needs to be shorter than 73 ka with QHF of up to 7 × 10 − 2km3/a high. Usually more than 25% of Vtot is emplaced at QHF. Fewer scenarios with two peaks in the flux-time curve than with one peak satisfy the condition that less than 50% of Vtot is injected at QHF. Magma above TS reaches smaller Vmax of 1000 km3 in 35 ka , and mobile magma of Vmax of about 480 km3 is assembled in 21 ka in the two peaks scenarios.
 Here again, Vmax does not necessarily coincide with the longest tacc or tcool. tacc, tcool and tmagma are shorter in the two peak scenarios, in comparison to the one peak scenarios, because less Vmax accumulates. tacc can be up to 50 ka long for magma above TS and as long as 27 ka for mobile magma. The time tcool it takes Vmagma to cool down completely is shorter than 95 ka for magma above TS and shorter than 24 ka for mobile magma. Whereas tcool for magma above TS is about twice as long as tacc, tcool and tacc are about the same for mobile magma. The whole lifetime tmagma of a magma mush above TS can be as long as 120 ka and tmagma of mobile magma is 51 ka.
4.4.3 Three and Four Peaks Scenarios
 The three and four peaks scenarios can be in agreement with geochronological data if more than 50% of Vtot is emplaced during QHF. The peaks of QHF could be imagined to represent the main geological units found in the Tuolumne Intrusive Suite separated by a time of lower intrusive flux.
 For the scenarios of three QHF peaks, tHF has to be shorter than 146 ka with a QHF of 1.8 × 10 − 2km3/a that magma above TS can be assembled (Figure 7). Mobile magma accumulates if QHF is at least 7 × 10 − 2km3/a and maintained for tHF 36.5 ka. Vmax of a magma mush above TS can reach up to 1300 km3 and Vmax of mobile magma is about 800 km3. These Vmax accumulate in tacc of up to 108 ka and 51 ka, respectively. It takes tcool of about 284 ka for magma mush and 185 ka for mobile magma to cool down below TS. The entire time tmagma is above TS can be 312 ka and tmagma for mobile magma is up to 196 ka.
 In the four peaks scenarios, magma above TS is assembled for QHF higher than 2.7 × 10 − 2km3/a if sustained for 73 ka and mobile magma accumulates for QHF higher than 3.7 × 10 − 2km3/a if tHF is 36.5 ka (Figure 8). Vmax of magma above TS can reach 570 km3 at the end of the fourth peak. The accumulation time tacc for these Vmax can be as long as 119 ka, comparable to the timescales of tcool of up to 103 ka. Magma above TS is present for a total time tmagma of up to 222 ka. Mobile magma with a melt fraction higher than 60% does not accumulate in the four peaks scenarios.
4.5 Example of Pluton Evolution
 Figure 11 shows the evolution of Vmagma in a growing pluton for a one peak scenario (tHF is 36.5 ka with a QHF of 7 × 10 − 2km3/a). During the first 7Ma of the model run, a QLF of 1 × 10 − 3km3/a is modeled and each sill intrusion cools down before the emplacement of the next sill. When the flux rises to QHF and is maintained, there sill injections follow quickly one after another and Vmagma increases rapidly. Vmax of about 1000 km3 is present at the end of the QHF period, assembled in a tacc of 44 ka. After the flux returns to QLF, all magma cools down below solidus in a tcool of about 73 ka. Adding tacc and tcool, magma above TS persists for a tmagma of approximately 117 ka whereas tmagma for mobile magma is 48 ka in this scenario.
5.1 Previous Modeling of the Tuolumne Intrusive Suite Emplacement and Cooling
 Various authors have modeled the emplacement and the cooling history of the Tuolumne Intrusive Suite. Glazner et al. (2004) used a two-dimensional model representing a vertical cross section of the pluton. They found that a 5 km thick and 20 km wide magmatic body with an initial temperature of 900°C that is instantaneously emplaced with its top at 15 km depth will cool down by conduction below 750°C in approximately 0.5Ma. They argued that this time span is in conflict with the 9Ma of age variation across the entire pluton. It is even too short to match the 3Ma given for the emplacement time of the Half Dome unit of the Tuolumne Intrusive Suite. Therefore, they concluded that a single-pulse intrusion cannot account for the U-Pb ages of the Tuolumne Intrusive Suite, and an incremental emplacement style is suggested for this pluton.
 In contrast to this modeling approach, Memeti et al. (2010) used a two-dimensional model of a horizontal section of the Tuolumne Intrusive Suite at the recent exposure level. Their solutions show that the 10–60 km2 lobes of the pluton have a lifetime of up to 2Ma. From these results, they draw the conclusion that the main magma chamber was even longer lived. However, their model does not take heat loss to the Earth's surface into account because it is infinite in the vertical dimension.
 The model presented in this paper is different to those mentioned above because it uses axisymmetric coordinates, thus is quasi three-dimensional. Consequently, it simulates heat loss in all dimensions although the intrusion geometry is a simple right circular cylinder and does not follow the outlines of the Tuolumne Intrusive Suite like the model of Memeti et al. (2010). Despite that, the basal area of the cylinder matches the areal extend of the Tuolumne Intrusive Suite of about 1000 km2. A model with the undulating contours of the Tuolumne Intrusive Suite would have a larger perimeter than our model with a simpler geometry and consequently would cool down faster. Therefore, our model gives only a lower limit of QHF necessary to create a magma chamber. QHF needs to be higher or tHF longer for more complex intrusion geometries in order to form a magma reservoir. We used a simple intrusion geometry because the continuation of the map units of the Tuolumne Intrusive Suite at depth is uncertain.
 If we use an intrusion scenario where all the volume of the Tuolumne Intrusive Suite, about 10,000 km3, is emplaced at once and not incrementally, mobile magma is present for 800 ka, and all magma cools down below TS in about 3Ma. These results contradict the age range of the Tuolumne Intrusive Suite of minimum 7.3Ma. In addition, scenarios where all the pluton's volume is instantaneously emplaced (see also Glazner et al. (2004)) are unrealistic because the crust has to accommodate several kilometers of vertical pluton thickness at once. Moreover, it would be problematic keeping this amount of magma in the crust without erupting even if the wall rocks of a magma chamber are warm and of a low effective viscosity to withstand high magma pressures (Jellinek and DePaolo, 2003; de Silva and Gosnold, 2007). In contrast, we show that large magma chambers can be generated with transient fluxes of a few 10 − 2km3/a. For an intrusion radius of 18 km, this corresponds to a thickening of few centimeters per year, a value that is comparable to the ground deformations currently observed in some volcanic areas (Pritchard and Simons) and that can be accommodated by the crust.
5.2 Fluctuating Magma Fluxes
 Apart from the Tuolumne Intrusive Suite, episodic construction of upper crustal intrusions in the Sierra Nevada Batholith has been reported from the Mount Givens pluton (McNulty et al. 2000), the Mount Whitney Intrusive Suite (Hirt, 2007), the intrusive suite of Yosemite Valley (Ratajeski et al. 2001), and the John Muir Intrusive Suite (Davis et al. 2012). An incremental intrusion style was also described for other plutons and laccoliths from different locations and tectonic settings, like the Torres del Paine intrusion in Patagonia [Michel et al.2008, Leuthold et al. 2012] or the Spirit Mountain batholith, Colorado River region, Nevada (Miller et al. 2011).
 If intrusion pulses do not follow after each other at the same rate, an incremental intrusion style results in fluctuating intrusive fluxes during pluton construction. High-precision geochronology revealed changing and transient higher magma fluxes that cause pulsed pluton growth (Matzel et al. 2006; Burgess and Miller, 2008). For instance, the 5.5Ma intrusive history of the Mount Stuart batholith, North Cascades, Washington, saw four periods of high magma fluxes of 10 − 3km3/a lasting for a few hundred thousand years, whereas the average emplacement flux is calculated to be 2.2 × 10 − 4km3/a, one order of magnitude lower (Matzel et al. 2006). This average flux is in agreement with long-term emplacement rates of other plutons and batholiths several thousand cubic kilometers in volume (Crisp, 1984; Miller et al. 2011).
 For a variety of subvolcanic systems, time-averaged intrusion fluxes into the upper crust span a wide range over several orders of magnitude from less than 1 × 10 − 4km3/a up to 1 × 10 − 0km3/a (Saint Blanquat et al. 2011). With an average emplacement flux QAV of 1.4 × 10 − 3km3/a, assuming a pluton thickness of 10 km, the Tuolumne Intrusive Suite falls into the lower part of this range. Previous numerical studies of incremental pluton emplacement showed that this long-term QAV is one order of magnitude too low for magma to accumulate in the growing intrusion if the magma intrusion rate is constant throughout the whole emplacement time (Brooks Hanson and Glazner, 1995; Yoshinobu et al. 1998; Annen, 2009). However, if a fluctuating magma flux is simulated, our results show that with the low QAV of the Tuolumne Intrusive Suite magma can accumulate in the crust but this requires periods of elevated transient magma fluxes.
 The accumulation of melt necessitates a transient flux of at least 1.25 × 10 − 2km3/a during 365Ma, and the presence of mobile magma requires a flux of at least 5 × 10 − 2km3/a during 73 ka. These fluxes are similar to those obtained by Annen (2009) when she tried to determine the minimum long-term fluxes needed to build up large magma chambers implying that a preheating episode of lower fluxes QLF before a period of QHF is of minor importance. Our transient high fluxes are in the range reported from injection fluxes related to active upper crustal intrusions of magma, which vary from about 10 − 2km3/a at Uturuncu volcano, Bolivia, (Sparks et al. 2008) to 6–8 × 10 − 3km3/a for the Socorro magma body, New Mexico, (Fialko and Simons, 2001).
 Our numerical modeling results show that the accumulation of large volumes of magma in the crust is quite difficult during pluton growth if the geochronological data are taken into account. However, magma mixing and crystal fractionation can only occur in the portion of the pluton molten at the same time and thus cannot explain the compositional zoning over the whole pluton. Therefore, the overall gradient in composition might reflect processes happening at deeper crustal or mantle levels (Glazner et al. 2004). That these processes can affect a large area is evidenced by the Tuolumne Intrusive Suite and plutons of similar ages in the Sierra Nevada Batholith which display a compositional zoning from an outer, older, and more mafic facies to a younger and more felsic unit in the center (Tikoff and Teyssier, 1992). That the chemical zoning of the Tuolumne Intrusive Suite is not unique but in fact a case example makes our modeling results and conclusions drawn from them applicable to a wide range of intrusive bodies.
 Volcanic systems experience changes in their magma supply fluxes with higher fluxes before eruptions. With a long-term magma production flux of about 1 × 10 − 3km3/a, a transient recharge flux of more than 5 × 10 − 2km3/a is calculated prior to the Minoan eruption of Santorini (Druitt et al. 2012) consistent with observations of extensive magma recharge just before other large eruptions (Morgan et al. 2006; Wark et al. 2007; de Silva et al. 2008; Saunders et al. 2010) and in agreement with our finding that the formation of a magma chamber requires a sudden and sharp increase in magma fluxes. In contrast, a progressive and steady increase in magma flux (acceleration scenario) does not result in the accumulation of magma although this scenario was hypothesized for super-eruptions (Lipman, 2007; de Silva and Gosnold, 2007).
5.3 Magma Reservoir Dynamics
 Crystal residence times in magmas prior to eruptions have been inferred from several analytical techniques like dating of radioactive isotopes and can be up to 10 5 years [Turner and Costa, 2007, and references therein]. For crystals from evolved igneous rocks, residence times are in the order of 10 3–10 5years (Hawkesworth et al. 2000). For instance, zircon stalled for a few tens of thousands of years in a pre-eruptive magma reservoir prior to the caldera-forming eruption of the Bishop Tuff (Reid and Coath, 2000). However, caution is needed when interpreting zircon ages because the crystals might have started to grow in a deeper reservoir (Miller et al. 2007). This range reported from residence times of crystals in magmas covers the time span mobile magma is present in our numerical model simulations.
 If three or four peaks of QHF are simulated, only minor volumes of mobile magma accumulate but substantial magma mush reservoirs can subsist for a few hundreds of thousands of years. The timescales for the reactivation of a crystal mush through gas percolation (Bachmann and Bergantz, 2006) and through the injection of new magma (Huber et al. 2010a) and the timescales for segregation of melt from a magma mush reservoir (Bachmann and Bergantz, 2004) are in the range of 104 − 105 years and in agreement with our numerical results for magma chamber lifetimes. The reactivation of long-lived magma mushes was proposed as one mechanism to generate mobile magma available for volcanic eruptions (Bachmann and Bergantz, 2003; Girard and Stix, 2009; Wilson and Charlier, 2009; Huber et al. 2010b; Wright et al. 2011; Burgisser and Bergantz, 2011). Viewed from this perspective, it seems possible that parts of the magma reservoir, which might have been present during the formation of the Tuolumne Intrusive Suite, had the potential to erupt. However, cogenetic volcanic deposits in the Sierra Nevada have not been found (Coleman et al. 2012).
 A set of intrusion scenarios was created in a numerical model to investigate if a fluctuating magma flux during pluton growth would lead to the formation of large magma chambers. If the long-term average flux is kept at a low value (10 − 3km3/a), no magma accumulates if the flux increases gradually. Large magma chambers form only if the flux increases suddenly and significantly to values that are one order of magnitude above the average flux. Transient magma fluxes of more than 1.25 × 10 − 2km3/a that last for up to 365 ka are required to generate a magma reservoir with a melt-crystal mix above solidus temperature in the range of 500–2000 km3. Magma above solidus temperature is present for 290 ka, and mobile magma persists for up to 115 ka. Magma volumes larger than 2000 km3 are only assembled if almost the total vertical pluton thickness of 10 km is emplaced very fast, which is unrealistic with regards to geochronology and mechanical considerations. Magma chambers can only form during pluton growth for very specific conditions when the transient flux is high enough for magma to accumulate but low enough that magma is still injected throughout the total emplacement time of the pluton to satisfy geochronological data. This window of realistic transient high fluxes where magma accumulates is largest for model scenarios with only one peak of high flux. Consequently, this emplacement style is preferred for the built-up of large-volume magma chambers during pluton growth when the long-term magma flux is small. Intrusion scenarios with three and four peaks of QHF do not create substantial reservoirs of mobile magma but magma mushes can be present for up to 300 ka.
Appendix A: Sill Thickness
 Tests with different sill thicknesses showed that the modeling results are not dependent on this parameter (Figure A1).
Appendix B: Conduction
 The model of conductive heat transfer takes radioactive and latent heat production into account
where ρ is density, cp is heat capacity, t is temperature, t is time, q is heat flux, x is a spatial coordinate, and A and L are radioactive and latent heat production, respectively.
Appendix C: Intrusion Modes
 For each intrusion mode, the first sill injection is emplaced at t = 0. If the magma flux is constant over the total simulation time ttot (broken line in Figures 3 and 4), the time ti of each sill injection i is calculated with
where n is the total number of sill injections. The magma flux is varied with time in all other intrusion scenarios according to the following formulas.
C1 Gradual Flux Changes
 For the peak-shaped and trough-shaped flux-time curve (Figures 3a and 3b), the injections follow half a period, one π, of a sine or cosine function, respectively. Each of the subsequent injection times ti is than obtained by
 A deceleration of the flux (Figure 3c) is achieved by a reversed logarithmic function
whereas an acceleration of the flux (Figure 3d) is obtained when the sill injection times are calculated by a logarithmic function
where the factor r varies the slope of the flux-time curve.
 Starting with a gentle slope corresponding to a factor r = 1, r is incrementally increased until the point where only one sill is injected during ttot / 2. All steeper slopes would not be in line with the geochronological data suggesting repeated magma recharge.
C2 Peak Scenarios
 The step-like changes of the flux over time result from the use of linear functions where m1 is the slope of the first linear function and m2 is the slope of the second linear function
with the number of high flux peaks nHF, the duration of one high flux peak tHF and the total emplacement time ttot.
 The highest possible transient magma flux Qmax at each time increment tHF is calculated with
where nmax is the maximum number of sill injections at QHF,
rounded down to the nearest integer.
 A flux Qp where px100% of the intrusion are emplaced at QHF is calculated with
where nHF is the number of sill injections at QHF,
rounded down to the nearest integer.
 The one peak scenarios are calculated according to
where tx1 is the time when the first linear function ends and the second one starts, and tx2 is the time when the second linear function ends and the first one is applied again.
 The scenarios with two peaks in the flux-time curve are calculated according to
 The scenarios with three peaks in the flux-time curve are calculated according to
 The scenarios with four peaks in the flux-time curve are calculated according to
where tmin is the minimum time QLF needs to last between tHF
 We are grateful to A.Burgisser and an anonymous referee for their constructive reviews, which significantly helped to improve this manuscript. We thank A.Revil and J.-F.Lénat for their careful editorial work. Comments by J.Kavanagh, T.Menand and R.S.J.Sparks strengthened earlier versions of this manuscript. D.Coleman, A.Glazner and J.Bartley are acknowledged for the stimulating discussions in the field. A.S. and C.A. were financially supported by the ERC Advanced Grant “VOLDIES.” This work was carried out using the computational facilities of the Advanced Computing Research Center, University of Bristol—http://www.bris.ac.uk/acrc/. Some figures were created with the help of GMT (Wessel and Smith, 1991).