Dynamics of magmatic intrusions in the upper crust: Theory and applications to laccoliths on Earth and the Moon

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, oi:1.129/21jb818, 211 Dynamics of magmatic intrusions in the upper crust: Theory an applications to laccoliths on Earth an the Moon Chloé Michaut 1 Receive 18 November 21; revise 11 February 211; accepte 23 February 211; publishe 17 May 211. [1] To unerstan the ynamics of shallow magmatic intrusions, I propose a theoretical moel of magma spreaing laterally below an elastic crust. Nonimensionalization of the flow equation leas to the ientification of characteristic scales for the intrusion: while the characteristic intrusion length is controlle by the elastic response of the crust, its characteristic thickness primarily epens on magma properties an injection rate. Three spreaing regimes are ientifie an characterize by ifferent morphologies as well as by scaling laws for thickness versus length an thickness versus time. The first spreaing regime is controlle by the elastic response of the crust an the shape of the flow is self similar. When the time, length, an thickness become larger than an elastic time, length, an thickness scale, the eges of the flow become steeper an the system transitions to a gravity current regime. When the intrusion is thick enough to accommoate the pressure hea, the flow enters a regime of lateral propagation an keeps a constant thickness. The intrusion shape in the elastic regime fits the observe shape of terrestrial laccoliths. The elastic scaling law for intrusion thickness versus length fits observations of laccoliths at Elba Islan, Italy, an provies for a physical explanation for the observe relationship between length an thickness on terrestrial laccoliths. Laccoliths are preicte to form over short time scales, epening on magma viscosity, that vary between approximately a month to several years for felsic magmas on Earth. On the Moon, several elongate low slope omes have recently been ientifie as possibly forme by laccolith intrusions at epth, although they are much larger than terrestrial laccoliths. Because the Moon has a smaller gravity than on the Earth, a eeper magma source, an a more mafic magma composition than for terrestrial laccoliths (implying smaller pressure graient an ike with), lunar intrusions have a larger characteristic length an a smaller characteristic thickness. After nonimensionalization, the morphologies (length versus thickness) of terrestrial an inferre lunar laccoliths follow the same curve an are well fitte by the elastic scaling law. This moel thus explains the size iscrepancy between terrestrial laccoliths an lunar low slope omes. Therefore, low slope omes ientifie on the Moon are goo caniates for laccolith type intrusions at epth. Citation: Michaut, C. (211), Dynamics of magmatic intrusions in the upper crust: Theory an applications to laccoliths on Earth an the Moon, J. Geophys. Res., 116,, oi:1.129/21jb Introuction 1 Equipe e Géophysique Spatiale et Planétaire, Institut e Physique u Globe e Paris, UMR 7154, CNRS, Université Paris Dierot, Sorbonne Paris Cité, Saint Maur es Fossés, France. Copyright 211 by the American Geophysical Union /11/21JB818 [2] Magma intrusions in the crust are the first steps of magma reservoir formation. Large magma reservoirs an plutons inee probably grow by small increments over long time scales [Petfor et al., 1993, 1994; Petfor an Gallagher, 21; Annen an Sparks, 22; Glazner et al., 24; Michaut an Jaupart, 26, 211]. Long term rates of magma transfer to the crust are quite well constraine [Crisp, 1984; Jellinek an e Paolo, 23]. But the formation of large magma reservoirs still poses some challenging questions as most of the thermal moels explaining the formation of large volumes of melt require large transient magma input rates, much larger than long term rates estimate from active an fossil systems [Annen, 29; Bachmann an Bergantz, 23; Michaut an Jaupart, 211]. The rate at which a single intrusion is transferre to the crust is thus crucial to etermine the valiity of these moels. [3] The ynamics of buoyancy riven ike propagation in an elastic meium has been well stuie [Lister an Kerr, 1991; Rubin, 1995]. In particular, Lister an Kerr [1991] have shown that, except at the ike tips where the elastic forces are important, the ynamics of magma in feeer ikes is controlle by a local balance between buoyancy forces an viscous pressure rop. At the level of neutral 1of19

2 buoyancy, ikes transition to sills or laccoliths. At large epth in the crust, the ynamics of sills an ikes are comparable [Lister an Kerr, 1991]. At shallow epth, however, geological an structural stuies on laccoliths have shown that roof lifting is the ominant process by which magma makes room for itself an that the elastic eformation of the overlying crust is crucial in controlling the shape of laccoliths [Pollar an Johnson, 1973; Jackson an Pollar, 1988]. Most of the research on shallow horizontal magmatic intrusions have thus focuse on the geometry of sill or laccolith formation, i.e., on the static elastic eformation inuce by the presence of a horizontal intrusion [Johnson an Pollar, 1973; Turcotte an Schubert, 1982]. Little attention has been given to the ynamics an evolution of such intrusions, i.e., to the fact that magma propagates an that flow length evolves with time. Crack propagation in a semi infinite elastic meium has been moele by Fialko [21] an Kavanagh et al. [26] have stuie sill formation an crack propagation experimentally as a function of the mechanical properties of the host meium. However, magma viscosity an injection rate necessarily influence the ynamics of sills an laccoliths. The own weight of the magma also inuces a hyrostatic pressure graient through thickening which as to the riving pressure an must influence the spreaing of the flow [Huppert, 1982]. [4] In the absence of a physical moel for shallow horizontal intrusion ynamics, the geometry of intermeiatescale intrusions can provie some insights into their emplacement an growth processes. Using the comprehensive ata sets of Corry [1988] on laccolith imensions, McCaffrey an Petfor [1997] propose an empirical power law istribution for laccolith thickness T as a function of length L of the form T =.12 L.88. The power law exponent value, smaller than 1, is interprete as reflecting the nee for a magma sheet to travel laterally some istance before unergoing vertical thickening [McCaffrey an Petfor, 1997]. This interpretation is in agreement with fiel stuies that escribe laccolith growth as a two stage process, implying, first, lateral spreaing of a sill with very large aspect ratio an, secon, vertical thickening of the intrusion by roof lifting [Johnson an Pollar, 1973; Corry, 1988]. [5] Recent ata on laccolith imensions also show the same kin of power law relationships, but with an exponent larger than 1 an up to 1.5 which has been interprete as reflecting the vertical inflation stage uring laccolith growth [Rocchi et al., 22; Mazzarini et al., 24; Westerman et al., 24; Corazzato an Gropelli, 24]. [6] Cruen an McCaffrey [22] gathere ata on intrusion imensions from small scale elastic cracks to large scale batholithic intrusions an propose that the thickness versus length of felsic to intermeiate intrusions follows an S shape istribution (in a logarithmic scale) with a maximum slope of 1.5 for intrusion lengths typical of laccoliths. However, there is no physical framework to explain the origin of this istribution. [7] At the surface of terrestrial planets, shallow magmatic intrusions have been propose to explain the relatively high topography an fracture floor of some craters on the Moon, Mars an recently on Mercury [Schultz, 1976; Hea et al., 29] as well as low slope omes on the Moon [Wöhler et al., 29]. The topographic eformation that coul be cause by shallow intrusions can be constraine by observations of planetary surfaces; that is, volume, shape an other imensions of intrusions can be quantifie. But such observations must be linke to moels of magma intrusion ynamics in orer to provie insights into magma physical properties an injection rate. [8] In this paper, I evelop a theoretical moel for the spreaing of shallow epth intermeiate scale intrusions, where magma, that is continuously injecte at the intrusion center, is accommoate by roof lifting. The moel iffers from the classical static laccolith moel of Turcotte an Schubert [1982] in that the length of the flow is self consistently etermine. The own weight of the magma was neglecte in previous moels on laccolith growth [Kerr an Pollar, 1998]; it is taken into account here as it also inuces pressure graients that rive the flow horizontally. [9] Depening on the injection rate, the overlying crust thickness an elasticity as well as magma viscosity an ensity, three ifferent spreaing regimes are ientifie an characterize by specific morphologies an scaling laws for intrusion thickness versus length an time. Available ata on terrestrial laccolith morphologies as well as on caniate intrusive omes on the Moon are compare with these scaling laws an interprete within the physical framework of this moel. Implications of this moel are iscusse in terms of intrusion thickness an time scale of emplacement. The effects of cooling on flow ynamics are also evaluate. 2. Shallow Magmatic Intrusions [1] At shallow epth in the upper crust, roof lifting is the ominant process by which magma makes room for itself [Pollar an Johnson, 1973; Jackson an Pollar, 1988], which leas to the eformation an bening of the overlying strata. [11] Numerous fiel an geochronological stuies inicate that large scale intrusions such as plutons, batholiths an magma chambers form an grow by increments of small magma pulses over very long time scales of 1 5 to >1 6 years [Petfor et al., 1993, 1994; Metz an Mahoo, 1991; Sisson et al., 1996; Coleman et al., 24]. In this paper, I focus on intermeiate scale intrusions such as sills, laccoliths an bysmaliths. Some geological stuies also suggest that intermeiate scale intrusions form by amalgamation of small magma sheets [Habert an e Saint Blanquat, 24; Horsman et al., 25]. However, such stuies also show that the emplacement of these intrusions must occur over a short enough time scale for the intrusion to keep a melte core. In the Black Mesa intrusion for instance, soli state textures aroun internal contacts between sheets are absent an contact metamorphism at the periphery of the intrusion is not significant; Habert an e Saint Blanquat [24] thus estimate that this boy forme in less than 1 years. Hence, if intermeiate scale intrusions form by amalgamation of small sheets, with a short repose time interval between intrusions, (i.e., such that a melte zone still exists), then the whole intrusion behaves an eforms as a single flow. A characteristic injection rate for the intrusion is 2of19

3 Figure 1. Sketch showing the 2 D geometry an physical properties of the system. Parameters are liste in Table 1. then simply an average over the whole uration of the emplacement process. 3. A Two Dimensional Moel for Magmatic Intrusions at Shallow Depth [12] The moel evelope herein treats the spreaing of an isoviscous magma along a weak being plane, below a thin elastic crust of constant thickness an above a rigi layer, situate at z =. Magma is injecte continuously at the base an center of the intrusion an the initial rate of injection Q is etermine by the initial overpressure riving the flow in the feeer conuit. As the intrusion weight increases, the overpressure riving magma flow an hence the rate of injection ecrease. In orer to take this effect into account, the feeer conuit is schematically represente by a ike of with a an length Z c, infinitely long in the y irection (Figure 1). Intrusion thickness h varies with time an horizontal coorinates. In Appenix B, I consier the case of an axisymmetric intrusion, with injection through a circular conuit Equations of Motion Momentum Equations [13] Horizontal motion is similar to the flow in a thin channel, with a characteristic length L much larger than the characteristic thickness H of the flow. Consequently, the classical lubrication theory applies, as in the gravity current problem [Huppert, 1982]. Horizontal velocities are much larger in magnitue than vertical velocities an horizontal variations in stress an velocity are negligible compare to vertical variations. [14] Consequently, many simplifications can be mae from the full Navier Stokes equations (see Huppert [1982] an Michaut an Bercovici [29] for more etails), an conservation of momentum in the horizontal (x) an vertical (z) irection mg ¼ where P is pressure, r m is magma ensity, g gravity, u horizontal velocity an m viscosity. The viscosity is consiere constant. In reality, magma viscosity is temperature epenent, an, as the intrusion spreas an cooling procees, the flow viscosity increases. Cooling, an hence the viscosity increase, o not procee homogeneously within the flow an this might also influence the ynamics of the flow. In this moel, the viscosity m is thus an effective viscosity for the whole intrusion, for the uration of the flow [Griffiths, 2]. Intrusion cooling an its effects on flow ynamics are iscusse in section Pressure [15] As in the gravity current theory, the pressure graient across the layer is hyrostatic (2) an is easily integrate: P ¼ m gz þ P z¼ At the base of the flow, the pressure P z= is the sum of the hyrostatic pressure P h ue to the weight of both the magma an crust, an the elastic pressure P e, ue to the eformation of the elastic crust. The hyrostatic pressure at z = is given by, with r r the crust ensity: P h ¼ m gh þ r g þ P a where P a is the atmospheric pressure. In the absence of horizontal forces within the elastic plate, the elastic pressure ð1þ ð2þ ð3þ ð4þ 3of19

4 require for bening the plate is given by the force per unit area that is necessary for a vertical isplacement h of the thin elastic plate [Turcotte an Schubert, 1982]: P e ¼ 4 where D = E 3 /12 (1 n 2 ) is the flexural rigiity of the plate which represents the resistance to bening of the plate; D epens on the crust elastic thickness, Young s moulus E an Poisson s ratio n. The total pressure at height z within the magma is thus given by P ¼ P a þ r g þ m gh ð Horizontal Velocity [16] Bounary conitions are such that the horizontal velocity u is equal to zero at both z = h an z =. Using (6) in (1) an integrating twice, I obtain the following expression for 5 h D ux; ð z; tþ ¼ 1 2 z2 5 [17] Because magma spreas along a weak being plane, no fracture criterion is here consiere. However, Lister an Kerr [1991] have shown that the pressure necessary for fracture extension is negligible in comparison with the elastic pressure in the case of sill propagation. In Appenix C, I also show a posteriori that the conitions for failure are met in the cases consiere Mass Conservation [18] The flow is assume to be incompressible although, at shallow epth, the magma may contain a non negligible fraction of gas. With this assumption, conservation of mass integrate over the intrusion @x Z h ð5þ ð6þ ð7þ ux; ð z; tþz þ wx; ð tþ ð8þ where w is the influx of magma into the intrusion, i.e., magma vertical velocity within the feeer ike (Figure 1). [19] The effective overpressure DP* riving the flow in the feeer conuit ecreases as the intrusion thickens an is given by DP* ðþ¼dp t c m gh ðþ t where h (t) is the maximum intrusion thickness at x =, r m gh is the intrusion weight at the origin an DP c is the initial riving pressure or the overpressure at the base of the ike (z = Z c ). [2] In (9), the elastic pressure term ) 4 x= has been neglecte. This term scales as Dh /l 4 ; it is thus maximum when the intrusion initiates the flow length, l is then minimum, an ecreases with time to become negligible as l increases. This term represents the elastic pressure to overcome in orer to initiate the flow, it vanishes as long as the flow spreas. This moel assumes a large aspect ratio for the ð9þ intrusion an oes not consier the initiation of a lateral intrusion, where the vertical flow transitions to horizontal. On the contrary, the intrusion weight at the center r m gh is equal to zero at the beginning an increases with time. [21] The total injection rate is thus given by V t ¼ Q 1 mgh ðþ t ð1þ DP c where Q is the injection rate in case of a constant riving pressure DP c. Within the feeer conuit, Poiseuille flow applies, Q an the vertical velocity w are relate to ike an magma properties through Q ¼ DP ca 3 12Z c ( wx; ð tþ ¼ DP* a 2 2Z c 4 x2 if x a=2 if x > a=2 ð11þ ð12þ [22] The equation for the intrusion thickness evolution h in cartesian coorinates is then obtaine using (7) ¼ m gh 12 þ E 3 12ð1 2 Þ 5 þ wx; ð tþ ð13þ This evolution equation for the flow thickness is compose of three ifferent terms on the right han sie. The first term represents gravitational spreaing of the flow: except for a constant arising from a no slip bounary conition at the intrusion roof, this term is the same as for a gravity current [Huppert, 1982]. The secon term on the right represents the squeezing of the flow ue to the elastic response of the crust. Both terms are negative an inuce spreaing. The last term represents magma injection an is positive Nonimensionalization [23] Equations (13) an (1) are nonimensionalize using a horizontal scale L, a vertical scale H, an a time scale t given by E 3 1=4 L ¼ 12ð1 2 ð14þ Þ m g H ¼ 12Q 1=4 1=4 L 1=4 ¼ a3 DP c L 1=4 ð15þ m g Z c m g 12 1=4 ¼ m gq 3 L 5=4 ¼ 12 Z 3=4 c ð m gþ 1=4 a 3 L 5=4 ð16þ DP c where scales are chosen such that Q = HL. These characteristic scales are efine by equating the nonimensional groups in front of the gravity current an elastic terms to 1, i.e., such that H3 m g = H3 D = 1. The length scale L (14) is 12L 2 12L 6 the flexural wavelength or flexural parameter as efine by Turcotte an Schubert [1982], i.e., the characteristic length 4of19

5 Table 1. Moel Parameters Parameters an Physical Properties Symbols Range of Values Depth of intrusion 1 m 5 km Young s Moulus E 1 1 GPa Poisson s ratio n.25 Gravity g 1.62 or 9.81 m.s 2 Magma ensity r m kg/m 3 Magma viscosity m Pa s feeer ike with a 1 2 m feeer ike height Z c 5 5 km Initial overpressure DP c 5 5 MPa Crust flexural rigiity D Crust ensity r r Injection rate Q Intrusion length scale L Intrusion thickness scale H Intrusion time scale t Maximum imensionless h thickness Dimensionless length L Dimensional thickness h Dimensional length scale over which the crust can support elastic stresses. The flexural wavelength epens primarily on the parameters characterizing the overlying elastic crust (thickness an elastic properties); it varies between 1.5 an 15 km for an elastic thickness or epth of intrusion between 5 an 5 m an E between 1 an 1 GPa (see Table 1 for moel parameters an their ranges of values). The thickness scale (15) represents the thickness for which the gravity current an the elastic terms are of similar importance; it epens primarily on the injection rate an magma physical properties (weight an viscosity). The time scale (16) is the characteristic time to fill up a laccolith an is mainly a function of the injection rate or of magma viscosity an ike with a. [24] Two imensionless parameters, g an s, are efine: ¼ DP c m gh ¼ ¼ a L DP 3=4 c Z c m ga L ð17þ 1=4 ð18þ The number g is the normalize source with an represents the effect of the ike with a, whereas s is the normalize pressure hea, i.e., the rate between the overpressure in the reservoir below an the weight of a characteristic thickness H of magma. [25] The imensionless equations for the thickness an volume are then, for @x þ 5 l þ 6 1 x2 4 2 V t ¼ 1 h 1 h ð19þ ð2þ For x > g/2, the last term on the right han sie of (19), i.e., the source term, is replace by zero. The sixth spatial erivative of h in (19) causes classical numerical metho use for gravity currents to be unstable. This equation is thus solve with a spectral metho along a gri of finite length. More etails are given in Appenix A Range of Values for the Dimensionless Numbers [26] For a Young s moulus value between 1 an 1 GPa, a epth of injection between 1 an 5 m an a gravity g = 9.81 or 1.62 m.s 2, the characteristic length L, given by (14), varies between several hunre meters an 2 km. For a ike with between 1 an 2 m, values for g = a/l range between 1 3 an.2. [27] In terrestrial settings, the overpressure in magma reservoirs riving ike an conuit flows is typically between a few to several tens of MPa [Barmin et al., 22; Stasiuk et al., 1993; Tait et al., 1989]. Conuit or ike lengths Z c are typically several kilometers an are not likely to excee several tens of kilometers. Hence, for terrestrial settings, values for s are in the range 1 to 1. [28] In lunar settings, the magma probably comes from much eeper, up to a epth of several hunres of kilometers [Shearer et al., 26]. Lunar magmas were probably rier than terrestrial ones an gas pressure builup is probably negligible at several kilometer epth. Thus, magma buoyancy is the most likely cause of overpressure [Wieczorek et al., 21]. The parameter s can thus be reuce to: s = DZ c 3/4 Zc 1/4 m a L. For Dr = 5 kg.m 3 an Z c up to 5 km, s may be as high as Scaling Analysis [29] I evelop a simple scaling analysis of (19) for the maximum flow thickness h s in which erivatives in x are approximate by 1/L s, where subscript s stans for scale. This type of analysis is suitable for behavior close to the flow center x = [Michaut an Bercovici, 29] an is use to compare the strength of the ifferent terms of (19). Equation (19) becomes h s t h4 s L 2 h4 s s L 6 þ 3 s 2 1 h s V s t 1 h s ð21þ ð22þ [3] The gravity current term (first term on the right of (19) an (21)) becomes ominant over the elastic term (secon term on the right of (19) an (21)) when h 4 s L 2 s > h4 s L 6 ) L s > 1 ð23þ s This scaling analysis preicts thus that the uplift an spreaing is first controlle by the elastic response of the crust an then by the own weight of the current. Hence, a gravity current regime follows an elastic regime when the imensionless length of the flow is of orer 1. 5of19

6 Figure 2. Shape of the flow, i.e., thickness as a function of horizontal coorinate, at four ifferent times inicate on the graph. Variables are imensionless, an one nees to multiply by the characteristic scales (thickness, length, or time given by (15)), (14), an (16)) to obtain the imensional values (g =.25, s = 1). [31] When h s s, the source term (thir term on the right of (19) an (21)) goes to zero as the intrusion weight at the center compensates for the overpressure riving the flow; hence h s cannot be larger than s. As h s s, a thir spreaing regime is reache where the maximum thickness remains constant. 4. Results [32] Numerical results show three spreaing regimes: an elastic regime, a gravity current regime an a regime of lateral propagation that are escribe in this section an in section 5. Calculations show that the flow shape, thickness at the center an length epen only on the normalize pressure hea s, the normalize source size g has no effect on the ynamics an shape of the flow (see Appenix A) Flow Shape as a Function of Time [33] Initially, as the intrusion grows with injection of more magma, the flow is self similar an both the flow thickness an length increase with time (Figure 2). As t becomes larger than 1, h 2 an L 4, the thickness of the flow oes not increase as much as in the first phase an the shape of the flow changes: the intrusion inflates on the sies an the intrusion fronts become steeper. This transition reflects the change in the spreaing an growth regime, when the gravity current term becomes ominant over the elastic one Effect of the Normalize Pressure Hea s [34] During the first phase where the elastic term is ominant, the effect of s on the thickness evolution is negligible (Figure 3): for ifferent values of s, all calculations form a single line of slope 5/9 (in logarithmic scale), preicting that h / t 5/9 uring this regime. This first phase ens when the thickness is close to 2 an the time reaches 1 (Figure 3). This change in the spreaing regime correspons to the change in flow morphology escribe in section 4.1. [35] When h reaches the value s, the intrusion weight at the center compensates for the overpressure riving the flow an the thickness at the center remains constant (Figure 3) while the flow propagates laterally. [36] For t > 1, the thickness increases much more slowly with time (i.e., with a smaller slope in logarithmic scale, Figure 3), even if the flow is not in the laterally propagating solution (as in the case for s = 1). [37] The evolution of the imensionless thickness h as a function of length L also shows the same behavior with ifferent spreaing regimes (Figure 4). The first elastic phase oes not epen on the value of s an is characterize by a slope larger than 1. During the secon phase, the increase in thickness is much slower, an the flow grows mostly laterally. When h s, the regime of lateral propagation is reache as L. [38] In section 5, scaling laws are propose for the evolution of the thickness at the intrusion center as a function of time an length. 5. Three Regimes: Characteristics an Scaling Laws 5.1. Elastic Regime Scaling Laws in the Elastic Regime [39] When both h an L are less than 1, the elastic term (secon term on the right sie of (19)) is ominant over the gravity current term (first term on the right sie of (19)). [4] As long as h s, the overpressure DP c in the reservoir below is much larger than the weight of the intrusion r m gh, an the injection rate can be consiere constant. If the injection occurs through a point source at the intrusion center, (19) an (2) reuce to (in h h 5 ð24þ 6of19

7 Figure 3. Maximum imensionless thickness h as a function of imensionless time t. Bol lines, results from the numerical calculations; ashe line, h / t 5/9 an h / t 1/5 ; otte lines, thickness at t. Results are for four ifferent values of s: 1/2,1,4/3,an1,ang =.25 (although calculations o not epen on g). Z Lt ðþ hx; ð tþx ¼ t ð25þ [43] Thus, I obtain h / t 5=9 ð31þ where L(t) is the flow length. [41] The variable h = x t is introuce an I look for a similarity solution by ecomposing h into hx; ð tþ ¼ f ðþt ð26þ where f an b are to be etermine. [42] Introucing (26) in (25), one fins, with h L a constant equal to the value of h at x = L: Z L f ðþt t ¼ t ð27þ which gives b =1 a. Using this result an (26) in (24) one obtains t ð1 Þf f ¼ t 4 1 f 3 5 f 5 ð28þ A similarity solution exists if t 4 9a = 1, i.e., a = 4/9, which gives hð; tþ ¼ f ðþt 5=9 ) Lt ðþ¼ L t 4=9 ð29þ ð3þ where f (h) characterizes the shape of the flow an is given by the solution of (28) with a = 4/9 although no analytical solution exists for this equation. h / L 5=4 ð32þ These analytical solutions for the maximum thickness as a function of time (31) an length (32) an for the length as a function of time (3) closely fit the numerical results uring the first spreaing regime (Figures 3, 4, an 5). Numerical calculations give the value of the constants of proportionality, see Table 2. [44] This analytical result confirms that the elastic response of the crust controls the flow uring the first phase of the intrusion Shape of the Flow [45] During the elastic regime, a simple function fits the numerical results for the shape of the flow (Figure 6): hx; ð tþ h ðþ t! 2 x2 ¼ 1 Lt ðþ 2 ð33þ where L an h are the imensionless flow length an thickness at the center, that vary with time t. [46] Equation (33) is the solution for the vertical isplacement of a thin elastic plate of length 2L(t) pinne at its ens an submitte to a homogeneous pressure [Turcotte an Schubert, 1982]; this inicates that the viscous flow of magma ue to the elastic response of the crust homogeneously istributes the available overpressure from the source over the length scale of the flow below the elastic crust. Because of the elasticity of the crust, its flexure is 7of19

8 Figure 4. Maximum imensionless thickness h as a function of imensionless length L. Bol lines, results from the numerical calculations; ashe line, h / L 5/4 an h / L 1/4 ; otte lines, thickness at L. Results are for four ifferent values of s: 1/2, 1, 4/3, an 1, an g =.25 (although calculations o not epen on g). Figure 5. Flow length as a function of time from the numerical calculations (bol lines) for four ifferent values of s = 1/2, 1, 4/3, an 1, inicate on the graphs. Calculations are for g =.25. Scaling laws obtaine in the elastic (ashe otte line), gravity current (ashe line), an lateral propagation (otte lines) regimes are also inicate an fit the calculations. For small values of s = 1/2, 1, an 4/3, the transition to the laterally propagating solution is reache early an the gravity current phase is not visible. 8of19

9 Table 2. Scaling Laws in 2 D an Axisymmetric Geometries Two Dimensional Axisymmetric Elastic Regime h =.7 t 5/9 h / t 1/3 L= 1.5 t 4/9 R / t 1/3 h =.4 L 5/4 h / R Gravity Current Regime h = 1.3 t 1/5 h = C te L =.6 t 4/5 R / t 1/2 h = 1.5 L 1/4 h = C te Lateral Propagation Regime L = 1.1 s t 1/3 R / s 3/4 t 1/4 istribute over the whole length of the flow, i.e., over a few times the flexural wavelength L; an hence the shape of the flow oes not epen on the normalize size of the source (parameter g) for g 1, on the contrary to gravity currents [see Michaut an Bercovici, 29]. [47] The thickness of the flow is very well fitte by a fourth orer polynomial (33), for which the elastic term is zero in (19). If (33) is only a fit an not the self similar solution (i.e., it is not solution of (28) with a = 4/9), it shows however that the shape of the flow is such that the elastic term is small, i.e., that in the absence of injection an gravity current terms, the flow woul be very slow Gravity Current Regime Scaling Law [48] In this regime, the gravity current term becomes ominant over the elastic term. As long as h s, the intrusion rate can still be consiere constant. Therefore, the equation for a two imensional gravity current with constant flux applies @x Z Lt ðþ ð34þ hðx; tþx ¼ t ð35þ The same exercise as above, an as evelope by Huppert [1982], shows that a similarity solution to these equations exists an gives h / t 1=5 L / t 4=5 h / L 1=4 ð36þ ð37þ ð38þ [49] The exponents relating the thickness to the length an time are both much smaller than 1, showing that the uplift is much slower uring the gravity current regime than uring the elastic regime. Scaling laws in the gravity current regime for the thickness at the intrusion center as a function of time (36) an length (38) as well as for the length as a function of time (37) are in goo agreement with the numerical results (Figures 3, 4, an 5 for s = 1). Numerical calculations give the value of the constants of proportionality (see Table 2) Shape of the Flow [5] During the gravity current regime, the normalize shape of the flow evelops steeper fronts (Figure 6). Steep Figure 6. Characteristic shape of the intrusion in the elastic an in the gravity current regime: flow thickness normalize by the maximum thickness as a function of horizontal axis normalize by flow length. Soli lines, results from the numerical calculations in the elastic an gravity current regime; ashe line, shape given by (33). During the elastic regime the flow shape remains unistinguishable from (33); uring the gravity current regime the shape progressively evolves from (33) towar a shape with steeper eges. 9of19

10 fronts are characteristics of gravity currents [Huppert, 1982]. At the intrusion roof, the curvature becomes important towar the tip where the resistance to bening (5), an hence the elastic response of the crust, become nonnegligible. Gravity currents show a vertical slope at their fronts but in this case, the elastic eformation of the crust always imposes a zero erivative of the flow thickness h at the tip. Hence the gravity current shape is never completely reache. In this case also the shape of the flow oes not epen on g Lateral Propagation [51] Once h s, the flow is thick enough to accommoate the overpressure below. The thickness at the center remains constant an the flow enters the regime of lateral propagation, where only its length increases. In this regime, the elastic term is negligible. However, this term ensures that the magma overpressure is homogeneously istribute over a length scale of the orer of the characteristic length of the flow L (see section 5.1.2), which is equivalent to g 1. Using these assumptions with g = 1, scaling analysis of (19) gives h s t ¼ h4 s L 2 þ 3 s 2 1 h s ¼ ð39þ where, as in section 3.4, erivatives by x are approximate by 1/L s an subscript s enotes the scales. The shape of the flow is approximate to a rectangle, hence V s h s L s, an, using h s s an (2), one obtains h 4 s L 2 s ¼ h s ) L 2 s L s t ¼ 3 2 h L s s t ¼ 2=3 3 ð4þ ð41þ ) L s / ð2tþ 1=3 ð42þ Numerical calculations for s =.5, 1 an 4/3 show inee that the imensionless length L goes as s t 1/3 once the lateral propagation regime is reache (Figure 5) Summary of the Results [52] The characteristic length of a viscous flow below an elastic crust is controlle by the resistance to bening of the crust (the flexural wavelength). However, the characteristic thickness of the flow mostly epens on the injection rate an magma physical properties, i.e., magma weight an viscosity, an aspect which has been neglecte in previous moels relate to laccolith growth in particular. [53] In a first stage, intrusion growth is ominate by the flow of magma ue to the elastic eformation of the overlying crust an the intrusion keeps a self similar shape as it grows. In two imensional geometry, the imensionless shape of the flow is well approximate by (33) (Figure 6); together with the imensionless relation h =.4 L 5/4 (Figure 4) an (15) this expression gives the imensional shape of the flow as a function of x: h ðþ¼:4 x l 5=4 L 1 a 3 1=4 2 DP c 1 x2 ð43þ m gz c where h an l are the imensional thickness an length of the flow, x is also imensional an L is given by (14). During the elastic regime, the relatively large value of the power law exponent relating the flow thickness to length (equal to 1 in the axisymmetric case, see Appenix B, an 1.25 in the two imensional case) suggests that uplift an spreaing occurs simultaneously. [54] In the two imensional case, the transition to the gravity current regime occurs when the intrusion length an thickness become larger than 2H an 4L, respectively (Figure 7), an when the time becomes larger than 1t (Figure 3). [55] During the gravitational spreaing regime, the intrusion mainly experiences lateral spreaing as the uplift is negligible or very slow, with an exponent relating flow thickness to flow length equal to in the axisymmetric case an.25 in the two imensional case (Figure 7) [Huppert, 1982]. The intrusion shape evelops steeper fronts which inuces thickening on the intrusion sies (Figure 6), although the elastic eformation of the crust always ensures a progressive ecrease of the thickness at the front. [56] When the thickness reaches h = DP c /r m g, it remains constant an the flow enters a thir regime of lateral propagation where the flow length increases as st 1/3 in the two imensional case an as s 3/4 t 1/4 in the axisymmetric case (see Appenix B). 6. Discussion 6.1. Intrusion Shape an Thickness [57] Laccolith shapes are usually moele by the upwar eflection of an overlying elastic strata ue to a given magma pressure P using the a priori assumption that the with of the laccolith is fixe [Pollar an Johnson, 1973; Turcotte an Schubert, 1982]. In this theory, the magma pressure P available for lifting the plate is a free parameter an it cannot be relate to any pressure relate physical parameters such as the riving pressure for the flow or magma weight. Using plate bening theory, the thickness of the laccolith in 2 D uner constant magma pressure P is given by h ðþ¼ x P 24D l4 l x2 l 2 ð44þ [58] However, laccolith with evolves as the intrusion grows. Kerr an Pollar [1998] propose a treatment of laccolith evolution but prescribe a horizontal pressure istribution in the magma. In the moel presente here, the shape, length an thickness of the flow evolve with time, an flow length an horizontal pressure istribution within the flow are self consistently etermine. The imensional thickness of the flow as a function of imensional length is, however, well escribe by (43). Equation (43) correspons to (44) but where the magma pressure P epens on the 1 of 19

11 Figure 7. The ynamics of a magmatic intrusion spreaing below an elastic crust shows three spreaing regimes: elastic, gravity current, an lateral propagation, that are characterize by ifferent scaling laws an imensions (length an thickness). length scales of the flow as well as on the overpressure in the feeer ike: P ¼ 24 :4 L 11=4 DP c L 1=4 ð m gaþ 3=4 ð45þ l Z c P ¼ 24 :4 L 11=4 ð12q LÞ 1=4 ð m gþ 3=4 ð46þ l where L =(D/r m g) 1/4 is the flexural wavelength or parameter given by (14). Hence, this moel provies insights into the magma pressure available for eforming the plate in terms of magma an crust physical properties an riving pressure or injection rate. In particular, expressions (45) an (46) show that the homogeneous pressure available for eforming the elastic plate increases with the injection rate an in particular with the ike with an pressure graient riving the flow; it ecreases as the flow spreas an l increases, as the available overpressure is then istribute over a larger with. [59] From (43) the imensional thickness h at the center of the intrusion in the elastic regime is h ¼ :4 l 5=4 L 1 a 3 1=4 DP c ¼ :4 l 5=4 H ð47þ m gz c L The intrusion thickness epens thus on the crust elastic properties but also on magma physical properties an injection rate, aspects that were not consiere in the previous moels on laccolith shape [Pollar an Johnson, 1973; Turcotte an Schubert, 1982]. [6] In the gravity current regime, the flow thickness oes not increase much more. Hence this expression can be compare to the maximum thickness of intrusions. The flexural wavelength L oes not epen on magma physical properties an, in the elastic regime, l 4 L. However, both the ike with a an overpressure graient DP c /Z c ten to increase as magma composition becomes more evolve an viscosity increases. Inee, observations an physical analysis show that mafic magmas with low viscosities ( Pa s) form ikes about 1 m wie whereas felsic magmas, with viscosities Pa s, form ikes of the orer of several tens of meters to 1 m wie [Bruce an Huppert, 1989; Waa, 1994; Kerr an Lister, 1995]. [61] Using an overpressure graient 1 orer of magnitue larger an a ike with 2 orers of magnitue larger for a felsic intrusion (relative to a mafic one) leas to a maximum thickness 56 times larger for the felsic intrusion. This moel thus implies that, in the upper crust, felsic intrusions ten to form laccoliths with a maximum thickness of a few hunre meters, while mafic magmas ten to form thin elongate intrusions less than 1 m thick, in agreement with observations. [62] Intrusion shapes evelop steeper fronts when they reach the gravity current regime. Hence, one shoul be able to ifferentiate between an intrusion that has reache the gravity current regime from one which has stoppe in the elastic one from their shapes. However, this consieration might be complicate by the fact that magma viscosity is not constant, as assume here, but temperature epenent; an this will also play in the same way, i.e., inuces steeper fronts (see section 6.4) in both regimes. Another way of istinguishing a gravity current type intrusion from an elastic one is to estimate its characteristic length L an to compare with the intrusion length l.ifl is larger than 4L, then the intrusion is likely to have reache the gravity current regime. 11 of 19

12 Table 3. Dimensional Parameters of the Intrusive Layers at Elba Islan, Italy, From Rocchi et al. [22] Unit Layer Thickness (m) Length (km) Depth (km) Central Elba Flysch (Complex V), San Martino layer Flysch (Complex V), San Martino layer Flysch (Complex V), San Martino layer Flysch (Complex V), Portoferraio layer Capo Bianco layer Western Elba Hornfels (Complex IV), Portoferraio layer Capo Bianco layer Ophiolite (Complex IV), Portoferraio layer Ophiolite (Complex IV), Portoferraio layer Scaling Laws for Laccolith Morphology [63] Rocchi et al. [22] stuie a series of 9 laccolith layers at Elba Islan, Italy, an foun a characteristic power law relationship linking intrusion thickness T an length L that iffers from the general power law relationship of McCaffrey an Petfor [1997], with an exponent larger than 1, which is interprete as reflecting the recor of vertical inflation stage in laccolith growth: T ¼ :26L 1:36 ð48þ [64] Rocchi et al. [22] provie ata for the intrusion epths (see Table 3), which are ifferent for each laccolith layer. Using these epths an characteristic values for various properties of the system (see Figure 8), the length an thickness scales can be estimate for each laccolith such that the ata can be nonimensionalize. For this example, each laccolith is part of a larger intrusive system, an hence variability of the moel parameters shoul be limite, except for the elastic crust thickness, taken to be the intrusion epth, whose variation between laccoliths is accounte for in the nonimensionalization. In this case, the best fit for the imensionless thickness h is proportional to L 1.25, in very goo agreement with the elastic scaling law in two imensions (32) (Figure 8). The imensionless intrusion lengths are smaller than 4, also supporting a flow in the elastic regime for all intrusions. The geometry of these laccoliths is not well known an probably not perfectly two imensional or circular but Rocchi et al. [22] observe that laccolith layers are connecte by feeer ikes, the major one being 1.5 km long, inicating a 2 D contribution. [65] Therefore, application of the moel to terrestrial ata shows that laccolith morphologies are well explaine by the elastic spreaing regime. [66] The power law relationship T =.12 L.88 propose by McCaffrey an Petfor [1997], shows an exponent close to 1 but smaller. The ifference in exponents between this relation an the elastic scaling law in axisymmetric or even 2 D geometry might result from fitting imensional ata with a scaling law, since ata shoul first be nonimensionalize to account for intrinsic scales of ifferent settings. An exponent smaller than 1 woul also result from fitting all the ata with a single scaling law; however at least two scaling laws, characterizing two ifferent spreaing regimes (elastic an gravity current), may be require. [67] The assumptions of the moel are such that they restrain its applicability to intrusions that have alreay initiate an are sufficiently long that the conitions of thin elastic plate an small aspect ratio are met. The thin elastic plate conition is probably the most restrictive, hence, flow length must be at least of the orer of the plate thickness for the moel to be applicable, i.e., larger than several hunres of meters to about a kilometer long. This moel provies however interesting insights into the mechanism of laccolith formation. In particular, the power law relation between thickness an length is well explaine by viscous magma flow below an elastic crust. Furthermore, this moel sug- Figure 8. Dimensionless maximum thickness h as a function of imensionless half length for laccolith layers from Elba Islan, Italy. Thickness, length an epth of intrusions, or elastic crust thickness are taken from Rocchi et al. [22]. In the moel the flow length is the length from the intrusion center to the front (Figure 1); the lengths provie by Rocchi et al. [22] are the total intrusion lengths, which are thus ivie by 2 for comparison with the moel. Other parameters use for calculating H an L from (15) an (14) are a = 15 m, E = 1 GPa, n =.25, r m = 26 kg.m 3, DP c = 5 MPa, Z c = 5 km, an g = 9.81 m.s 2. For epth of intrusions between 1.9 an 3.7 km, H is between 26 an 3 m an L is between 4 an 6.5 km. Dashe line, h / L 5/4 as for the elastic scaling law. 12 of 19

13 Figure 9. Apollo 15 orbital image AS , oblique view of the Valentine ome. The iameter of this ome is 3 km in average. gests that the main phase of laccolith growth an spreaing oes not require a two stage process, with horizontal spreaing of a sill followe by vertical inflation, as often propose [Johnson an Pollar, 1973; Corry, 1988], or by sill stacking [Menan, 28], but that spreaing an inflation occur simultaneously Low Slope Domes on the Moon [68] On the Moon, thirteen elongate low slope omes have recently been ientifie as possibly forme by laccolith type intrusions [Wöhler et al., 29, 21], although their iameters are larger an their thicknesses smaller than most terrestrial laccoliths: for a given intrusion thickness, lunar low slope omes are at least twice as wie as terrestrial laccoliths (Figure 1a). Because these omes are subtle structures with low elevation an low slopes, the etermination of their morphometric properties is complex; Wöhler et al. [29] use an image base 3 D reconstruction approach which relies on a combination of photoclinometry an shape from shaing techniques. The error in the etermination of the ome height is the largest one an amounts to 1% [Wöhler et al., 26]. The Valentine ome is one of the largest of these omes (Figure 9); it has an elliptical shape, i.e., a structure in between 2 D an cylinrical. These caniate intrusive omes are locate in regions of extensive effusive activity, in general near the borers of Mare basalts. [69] Using (14) an (15), both the characteristic length an thickness can be calculate in lunar versus terrestrial settings. On the Moon, the smaller gravity (g = 1.62 m.s 2 versus 9.81 m.s 2 for the Earth) leas to a characteristic length for intrusions 1.6 times larger than on Earth, i.e., close to twice as long as terrestrial ones, as observe by Wöhler et al. [29]. For instance, using a Young s moulus E = 1 GPa an an intrusion epth of 15 m, the characteristic length for lunar intrusions is 5 km but it is 3.2 km for terrestrial ones. [7] On Earth, laccoliths are generally forme by relatively evolve lavas that may have ifferentiate from primitive magma in eep crustal magma chambers, locate some 5 to 15 km below the surface. The overpressures riving magmatic ascent are typically 2 to 5 MPa, [Tait et al., 1989; Barmin et al., 22; Stasiuk et al., 1993] which gives overpressure graients of 1 4 Pa.m 1. [71] On the Moon, Wilson an Hea [28] show that because of the low ensity of the lunar crust, the overpressure graient riving magma flow within the feeer ikes of mare basalts lies within a factor of 5 of 2 Pa.m 1. Because of the smaller gravity an coler interior, the likely magma source for these intrusions is much eeper than for terrestrial laccoliths: most of the Mare basalts are thought to be a prouct of melting initiate eep in the lunar mantle, eeper than 4 km [Shearer et al., 26; Elkins Tanton et al., 23]. Using the same value for the riving pressure on the Moon than on the Earth, DP c = 5MPa, although lunar magmas are likely to be more mafic an contain less volatiles implying smaller riving pressure, an a epth of 5 km for the magma source region, the overpressure graient is inee only 1 Pa.m 1. [72] Dike with also increases with viscosity [Waa, 1994; Kerr an Lister, 1995]; Bruce an Huppert [1989] have shown that the critical ike with for magma propagation increases with viscosity to the power 1/4. Using a with a equal to 25 m for the feeer ike of lunar intrusions an 1 m for terrestrial felsic ikes (as well as other parameters values liste in Figure 1), a characteristic thickness for lunar laccoliths is 35 m; but is 19 m for terrestrial intrusions. [73] The same ratio between the characteristic thicknesses of lunar an terrestrial intrusions is obtaine if one consiers the same rate of injection Q in (15) but a viscosity value 5 times lower for lunar magmas than for terrestrial ones. [74] After nonimensionalization, terrestrial an lunar ata nearly collapse to the same curve an the elastic scaling law of the moel fits both lunar an terrestrial ata in a 2 D geometry (Figure 1b) as well as in an axisymmetric geometry which requires a ifferent nonimensionalization (see Figure 1c). Given that the same intrusion epth has been arbitrarily chosen for all intrusions, the fit is surprisingly accurate. The gravity current regime oes not provie a goo fit to the ata (Figure 1b). [75] This moel therefore explains the ifference in morphology between terrestrial laccoliths an low slope omes on the Moon; it also suggests that lunar low slope omes are inee goo caniates for laccolith intrusions at shallow epths. [76] After nonimensionalization, the largest lunar omes plot at the limit between the elastic an the gravity current regimes (i.e., their imensionless lengths are close to 4). Hence it is possible that the largest lunar omes have reache the gravity current regime, although this woul epen on their epths of intrusion. If they are inee gravity current type intrusions, they shoul show steeper fronts than the smaller omes. 13 of 19

14 6.4. Intrusion Cooling [77] This moel assumes an isothermal, isoviscous flow, whereas, in reality, intrusions cool as they sprea an magma viscosity is temperature epenent. Cooling leas to an important increase in viscosity which might influence flow ynamics as well as its shape. A more complex theory is necessary to stuy the effects of cooling; however, some conclusions can be reaily erive. Cooling of the intrusion is much faster at the fronts, where the flow is thin, than at the center. Hence the magma will be more viscous at the flow fronts which can act as a plug an slow own the intrusion. Bercovici [1994] has shown that cooling viscous gravity currents with temperature epenent viscosity grow more by thickening an evelop steeper fronts than isoviscous ones that grow more by spreaing. Thus, taking into account the temperature epenence of magma viscosity might lea to increase the thickening rate of the intrusion in the elastic as well as the gravity current regime, i.e., to larger slopes for the relationships between thickness an time an thickness an length in logarithmic scale in both regimes. An intrusion characterize by a variable viscosity might also evelop steeper fronts, in all three ynamic regimes; this might partly explain why seimentary strata are nearly horizontal an little eforme on top of laccoliths but are steeply bent over their peripheries [Koch et al., 1981]. [78] The ecrease in temperature inuces crystallization an soliification of the intrusion, which might stop it from spreaing. In a first approximation, the time scale for conuctive cooling of an intrusion provies for an estimate for the characteristic time for flow soliification. As avective cooling is not negligible, this time scale is an upper limit. On Earth, the circulation of grounwater through hyrothermal systems within the crust largely participates in the cooling of shallow magmatic intrusions, which leas to even shorter time scales for the cooling of terrestrial intrusions. On the Moon, this is obviously not the case an cooling of the intrusion is limite to conuction of heat in the wall rock an conuction an avection of heat in the magma itself. Hence, this ifference might also contribute to the fact that inferre lunar laccoliths sprea further than terrestrial ones: after nonimensionalization, lunar intrusive omes plot on the upper part of the elastic scaling law (Figures 1b an 1c), whereas terrestrial laccoliths generally show smaller imensionless length. Figure 1. (a) Thickness as a function of half length or raius for terrestrial laccoliths from Elba Islan, Italy (circles) [Rocchi et al., 22], an for caniate lunar intrusive omes (iamons) [Wöhler et al., 29]. Uncertainties for lunar ome heights amounts to 1%. (b) Dimensionless thickness as a function of imensionless half length or raius, characteristic thickness, an length are calculate from (15) an (14). Dashe line, 2 D elastic scaling law; otte line, 2 D gravity current scaling law. (c) Dimensionless thickness as a function of imensionless half length or raius, characteristic thickness, an raius are calculate from (B8) an (B7). Dashe line, axisymmetric elastic scaling law. In both geometries, I use, for terrestrial laccoliths, g = 9.81 m.s 2, DP c /Z c = 1 Pa/m, r m = 26 kg.m 3 ; a = 1 m, an for lunar intrusive omes, g = 1.62 m.s 2, DP c /Z c = 1 Pa.m 1, r m = 29 kg.m 3, a = 25 m. In all cases, E = 1 GPa, = 15 m, n = of 19