Abstract
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[1] The lunar steepsided Gruithuisen and Mairan domes are morphologically and spectrally distinctive structures and appear similar to terrestrial extrusive volcanic features characterized by viscous magma. We use the basic morphologic and morphometric characteristics of the domes to estimate the yield strengths (∼10^{5} Pa), plastic viscosities (∼10^{9} Pa s), and effusion rates (∼50 m^{3}/s) of the magmas which formed them. These values are similar to those of terrestrial rhyolites, dacites, and basaltic andesites and support the hypothesis that these domes are an unusual variation of typical highlands and mare compositions. The dikes which formed them are predicted to have had widths of ∼50 m and lengths of about 15 km. The magma rise speed implied by this geometry is very low, ∼7 × 10^{−5} m/s, and the Reynolds number of the motion is ∼2 × 10^{−8}, implying a completely laminar flow regime. Estimates of emplacement duration range from one to several decades. These new calculations confirm the unusual nature of these features and support previous qualitative suggestions that they were formed from magmas with significantly higher viscosity than those typical of mare basalts.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[2] The Gruithuisen and Mairan domes, located in northern Oceanus Procellarum (Figure 1), represent examples of topographically, morphologically and spectrally distinctive structures on the Moon [Malin, 1974; Head and McCord, 1978; Chevrel et al., 1999; Head and Wilson, 1992, and references therein] that appear to be candidates for the sites of extrusion of very viscous magma about 3.7–3.85 Gyr ago [Wagner et al., 2002]. In this analysis, we use the basic morphologic and morphometric characteristics of the domes as a basis for estimation of their yield strength, plastic viscosity, eruption rates, dike feeder geometry (e.g., dike width and length), and eruption duration. We then compare these values with those characterizing terrestrial environments and compositions.
2. Initial Estimates of Rheological Parameters
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[3] The Gruithuisen and Mairan domes are sufficiently symmetrical that it is more appropriate to regard them as being the result of the extrusion of magma onto a flat plane, across which it spreads in all directions from the vent, than onto an inclined surface down which it flows in one main direction. Available treatments of such domeforming eruptions are those of Huppert [1982], Huppert et al. [1982], Blake [1990], Fink and Griffiths [1990], and Sakimoto and Zuber [1995]. Huppert [1982] and Huppert et al. [1982] treated aspects of the motion of a Newtonian magma of constant viscosity and did not consider how cooling of the flow surface would limit the motion of such a flow. Fink and Griffiths [1990] considered cooling mainly in so far as it affected the development of surface textures on a flow rather than the rheology of the flow interior. Sakimoto and Zuber [1995] treated a Newtonian fluid with timevariable viscosity and showed that a sufficiently large increase in viscosity could effectively cause motion to cease; however their numerical treatment does not lend itself easily to modeling the dimensions of a dome produced by a given effusion rate. The most suitable model in this respect is that of Blake [1990], which treats the cooling magma as a Bingham plastic characterized by a yield strength, τ, and plastic viscosity, η. The yield strength is given by
where ρ is the lava density (we initially adopt 2000 kg/m^{3} and discuss the reason for, and significance of, this later), g the acceleration due to gravity (1.63 m/s^{2}), r_{m} the radius of the dome and d_{tot} its maximum height. Table 1 gives the measured values of r_{m} and d_{tot}: we treat the Gruithuisen domes γ and NW and all three of the Mairan domes as being circular for this purpose and give the radial extent (i.e. the halfwidth) at rightangles to the long axis for the elongate dome Gruithuisen δ. Table 1 also gives the inferred values of τ. They are of order 3 × 10^{5} Pa for the Gruithuisen domes and 1.0 × 10^{5} Pa for the Mairan features, all larger than the values found for the distinctive festoon structure on Venus [Head et al., 1992; Pavri et al., 1992; Moore et al., 1992; Head and Hess, 1996], mainly due to the greater thickness of the domes on the Moon.
Table 1. Measured Values of the Maximum Radial Extent r_{m} and Total Thickness d_{tot} for All of the Domes Studied, and of the Implied Yield Strength τ and Plastic Viscosity, η, of the Material Forming the DomesDome Name  r_{m}/m  d_{tot}/m  τ/Pa  η/(Pa s) 

Gruithuisen γ  10000  1200  1.5 × 10^{5}  1.6 × 10^{9} 
Gruithuisen δ  6500  1550  3.9 × 10^{5}  15.6 × 10^{9} 
Gruithuisen NW  4000  1100  3.2 × 10^{5}  9.7 × 10^{9} 
Mairan T  6500  900  13.1 × 10^{4}  11.5 × 10^{8} 
Mairan “middle”  5500  600  6.9 × 10^{4}  2.5 × 10^{8} 
Mairan “south”  5000  500  5.3 × 10^{4}  1.3 × 10^{8} 
[4] Next we use an empirical formula given by Moore and Ackerman [1989] to relate the plastic viscosity, η, to the yield strength, τ:
where Q = 6 × 10^{−4} when η is expressed in Pa s and τ is in Pa. On this basis (see Table 1), the most likely value of η for the Gruithuisen domes is ∼ 1 × 10^{10} Pa s and for the Mairan domes is within a factor of about two of 5 × 10^{8} Pa s. The above analysis is oversimplified, however, in that all of the Gruithuisen domes show evidence for having been formed from more than a single flow unit. The δ and NW domes have very small secondary domes on top of the main dome and the γ dome has three flow lobes descending from near its summit [Head et al., 1978] (Figure 2). We therefore analyze each of the major Gruithuisen domes in more detail in the following sections.
4. Analysis of the Gruithuisen δ and NW Domes
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[6] Each of these features (Figure 1) consists of a large dome on which is superimposed one (in the case of NW) or two (in the case of δ) smaller domes (Figure 2). Let the maximum radial extents of the larger, lower dome and the smaller, upper dome be r_{ml} and r_{mu}, respectively, and let their maximum thicknesses be correspondingly d_{ml} and d_{mu}. In the case of NW the radial distances are true radii since both upper and lower domes are essentially circular. In the case of δ, the underlying dome is strongly elongate and the two superimposed domes are less strongly elongate but with the same orientations: in these cases the “radial distances” are the halfwidths measured at right angles to the long axis. The measured values are given in Table 2. For the lower dome in each case we can write the equivalent of equation (1):
and for the upper dome
where we assume, as before, that both domes are made of material with the same rheological properties. The sum of d_{ml} and d_{mu} is the total measured height of the composite dome, d_{tot}, and so we can use the above expression for d_{ml} and d_{mu} to give
where β is defined by
and it follows that
Table 2. Measured Values of the Maximum Radial Extents, r_{ml} and r_{mu}, of the Lower and Upper Components, Respectively, of Two of the Gruithuisen Domes^{a}Dome Name  r_{ml}/m  r_{mu}/m  β/m  τ/Pa  d_{ml}/m  d_{mu}/m  η/(Pa s) 


Gruithuisen δ (NW minor dome)  6500  4000  116.1  12.2 × 10^{4}  869  681  9.7 × 10^{8} 
Gruithuisen δ (SE minor dome)  6500  2750  135.7  14.2 × 10^{4}  939  611  13.9 × 10^{8} 
Gruithuisen NW  4000  1500  116.4  12.3 × 10^{4}  682  418  9.9 × 10^{8} 
[7] Table 2 shows the values of β found for each pair of domes by inserting the total thickness d_{tot} (from Table 1) and the measured values of r_{ml} and r_{mu} into equation (7). Also given are the implied values of τ from equation (8) and the corresponding values of d_{ml} and d_{mu} from equation (9) and the plastic viscosity η implied by equation (2). As was the case for the Gruithuisen γ dome, the values found for the rheological parameters are smaller than the first approximations given in Table 1.
6. Eruption Rate Estimates
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[9] We now turn to the estimation of the eruption rates of the magmas forming the various dome units. To do this we assume that the advance of the flow front of each flow unit or domeforming episode was limited by cooling. Pinkerton and Wilson [1994] show that a variety of types of lava flows cease to move when the Grätz number, a dimensionless measure of the depth of penetration into the flow of the cooled boundary layer, has decreased from its initially very high value to a critical value of about 300. This is shown to correspond to assuming that the flow has been moving for a time T such that
where κ is the thermal diffusivity of the lava (∼10^{−6} m^{2}/s) and d_{f} is the thickness of the flow near its front.
[10] Blake's [1990] model of Bingham plastic domes shows that the radius of the dome, r, grows as a function of time, t, given by
where E is the volume eruption rate. For a symmetrical dome, it is not trivial to define the thickness d_{f} which corresponds to the distal thickness of a lava flow lobe. We assume that a reasonable approximation is to take the thickness of the dome halfway between its center and its edge. The parabolic dome profiles implied by Blake's [1990, equation (23)] model are such that d_{f} defined in this way is equal to ∼0.7d_{m}. Substituting this relationship and equation (10) into equation (11) we obtain an expression relating the radius of the dome when it ceases spreading (i.e. r_{m}) to its maximum thickness at this time, d_{m}, and the effusion rate feeding it:
Eliminating τ from this equation using equation (1) and rearranging:
Table 4 summarizes the values of r_{m} and d_{m} for all of the dome components and the values of effusion rate that they imply. It must be pointed out that the assumption that flow lobes are coolinglimited always leads to effusion rate estimates that are lower bounds. This is because any flow unit assumed to be coolinglimited may in fact have been volumelimited (i.e. ceased to flow simply because the magma supply was exhausted), implying that it had the potential to travel further than the observed distance and therefore had a higher effusion rate than that deduced. For each of the Gruithuisen domes (Figures 1 and 2) we have either a second smaller dome or a flow lobe superimposed on a larger, earlier dome. It is very tempting to assume that the second unit in each case is a breakout at the vent caused by the magma supply continuing after the first unit has reached its coolinglimited length. This automatically implies that the effusion rates deduced from the large dome geometries (∼48, 24 and 119 m^{3}/s for Gruithuisen δ, NW and γ, respectively) are the realistic estimates and that the rates found from the smaller units are underestimates. For the Mairan domes we are not able to resolve multiple lobe structures and must regard the (as it happens remarkably similar) effusion rates of ∼50 m^{3}/s as lower limits on the true rates.
Table 4. Summary of the Radii, r_{m}, and Thicknesses, d_{m}, of the Various Dome Components and the Implied Volume Eruption Rates, E, of Magma if the Flow Units Are Cooling LimitedDome Name  r_{m}/m  d_{m}/m  E/(m^{3}/s) 

Gruithuisen γ  10000  856  119.3 
Gruithuisen δ (NW lower dome)  6500  869  49.7 
Gruithuisen δ (NW upper dome)  4000  681  24.0 
Gruithuisen δ (SE lower dome)  6500  939  46.0 
Gruithuisen δ (SE upper dome)  2750  611  12.6 
Gruithuisen NW (lower dome)  4000  682  24.0 
Gruithuisen NW (upper dome)  1500  418  5.5 
Mairan T  6500  900  48.0 
Mairan “middle”  5500  600  51.5 
Mairan “south”  5000  500  51.1 
[11] It is possible to obtain useful information by considering the motion of the flow lobes on the flanks of Gruithuisen γ. The relevant theory is outlined by Hulme [1974]: a Bingham plastic flow with total width w_{t} emplaced on a plane inclined at an angle ϑ to the horizontal has stationary margin levees of width w_{b} such that
The width of the channel within which lava moves is then w_{c} = w_{t} − 2w_{b}. This channel width can be related to the other variables, including the effusion rate E, by
which is a slight simplification of Hulme's original formula suggested by Wilson and Head [1983]. Table 5 gives examples of the solution of equations (14) and (15) using our estimate of 5 km for w_{t} and a range of plausible values of sin ϑ. The range is chosen to encompass the mean slope of the flanks of the underlying dome which, for the shallow angles involved, is of order sin ϑ ≅ tan ϑ ≅ (856 m/10 km) ≅ 0.085. The analysis clearly suggests that E is ∼400 m^{3}/s, which is not greatly at variance with the 119 m^{3}/s deduced by applying the coolinglimited criterion to the underlying dome. However, because of the strong dependence of E on the values used for sin ϑ and w_{c}, this estimate must be regarded as very inaccurate: if 6 km were adopted for w_{t} instead of the 5 km used above, the estimate of E would increase by a factor of more than three.
Table 5. Values of the Total Levee Width, (2 w_{b}), the Width of the Channel in Which Lava Flowed, w_{c}, and the Implied Volume Eruption Rate of Magma, E, for the Southwest Flow Lobe on the Flank of Gruithuisen γ, Calculated for a Series of Assumed Values of sin ϑ for the Flank Slopesin ϑ  (2 w_{b})/m  w_{c}/m  E/(m^{3}/s) 

0.125  1512  3488  5407 
0.100  2362  2638  1795 
0.090  2916  2084  801 
0.085  3269  1731  441 
0.080  3691  1309  187 
0.075  4199  801  44 
7. Geometries of Feeder Dikes
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[12] To explore the conditions under which the magmas erupted we assume that in each case the melt reached the surface from a source region at least as deep as the base of the mainly anorthositic crust. Assuming a mean crustal density ρ_{c} = 2800 kg/m^{3} and recalling the assumed magma density of ρ = 2000 kg/m^{3}, the pressure gradient driving the magma rise is ∼[g (ρ_{c} − ρ)] = ∼130 Pa m^{−1}. The magma rise speed is found by balancing the driving pressure gradient against the wall friction which, allowing for the need to overcome the yield strength, leads to [Wilson and Head, 1981; Johnson and Pollard, 1973]
The rise speed, dike geometry and volume eruption rate are related by
but horizontal length L and width W of the dike are not independent; their ratio is a function of the elastic and plastic properties of the crustal rocks and also of the viscosity of the magma. Rubin [1993] showed that for magmas with the relatively high viscosity found here, and assuming a mean viscosity of ∼10^{18} Pa s for the hot lower crustal rocks through which the dike passes, the ratio L/W probably lies within a factor of 2 of 200. Using this value and combining the above expressions, we have
from which W can be obtained recursively from any initial estimate.
[13] Using the mean rheological properties for all of the domes, τ = 10 × 10^{4} Pa and η = 6 × 10^{8} Pa s, together with an eruption rate of 50 m^{3}/s (typical of the majority of the values in Table 4), we find W = 154 m and so L = 31 km. The magma rise speed implied by this geometry is very small, 11 × 10^{−6} m/s, and the Reynolds number of the motion is ∼10^{−8}, implying a completely laminar flow regime. At first sight the implied fissure length of ∼30 km seems large judged against the facts that (i) the δ dome is elongate with its main vents ∼12 km apart and (ii) the γ and NW domes together could be regarded as defining an underlying fissure about 18 km long (Figures 1 and 2). However, it is common on Earth for eruptions to localize so that only part of an initial fissure remain active throughout an eruption [Richter et al., 1970; Wolfe et al., 1988; Wilson and Head, 1988].
[14] We have repeated the solution of equation (18) with a range of magma densities. Increasing the magma density (i.e. reducing the buoyancy) makes U smaller and W and L larger. Table 6 shows the results of using ρ = 2400, 2000 and 1600 kg m^{−3}. The solutions W = 103 m and L = ∼21 km, found for the very small magma density of 1600 kg m^{−3}, imply a dike length close to that evidenced by the observations. Thus, one possible explanation for the unusual properties of the magma involved in building these domes is that it is an extremely vesicular foam.
Table 6. Values of Mean Dike Width, W, Horizontal Dike Length, L, Magma Rise Speed, U, and Reynolds Number of Magma Motion, Re, as a Function of Magma Density, ρ, for Magma Feeding Dome Eruptions at a Fixed Volume Flux of 50 m^{3} s^{−1}ρ/(kg m^{−3})  W/m  L/km  U/(m s^{−1})  Re 

2400  307  61  3 × 10^{−6}  0.5 × 10^{−8} 
2000  154  33  11 × 10^{−6}  1.0 × 10^{−9} 
1600  103  21  24 × 10^{−6}  1.6 × 10^{−9} 
[15] In order to see if a better fit to the inferred dike geometries can be found, we give in Table 7 the individual best estimates of τ, η and E for each of the domes, the corresponding calculated values of W and L using the original magma density estimate of 2000 kg/m^{3}, and for comparison the estimates of active fissure length L_{e} based on the morphology and arrangement of the domes. Examination of this table confirms that it is plausible to assume that Gruithuisen NW and Gruithuisen γ are fed by the same dike. It also seems likely that Mairan “middle” and Mairan “south” share a common feeder. However, Mairan T appears to be anomalous in respect of the geometry of the dike needed to supply it. Indeed, Mairan T appears to be significantly different from Mairan “middle” and Mairan “south” in terms of its rheological properties, largely as a result of its large thickness to diameter ratio, and it seems likely that it, like the Gruithuisen domes, is a composite feature. If Mairan T consists of two phases of extrusion which produced domes with radii 6500 (the mapped radius) and 3000 m (an arbitrary but plausible value), the superposed dome treatment used earlier would yield β = 44.2 m, τ = 4.7 × 10^{4} Pa and η = 1.0 × 10^{8} Pa s; the implied effusion rate for the main phase would be 81 m^{3}/s, and the width and length of the feeding dike would be 72 m and 14 km, respectively. This value of the dike length is much more consistent with the observed geometry of this dome.
Table 7. Values of the Best Estimates of Yield Strength, τ, Plastic Viscosity, η, and Volume Eruption Rate, E, for Each of the Domes and of the Corresponding Values Calculated for the Width, W, and Length, L, of the Underlying Feeder Dike^{a}Feature  τ/Pa  η/(Pa s)  E/(m^{3}/s)  W/m  L/km  L_{e}/km 


Gruithuisen γ  7.7 × 10^{4}  3.2 × 10^{8}  119  119  24  5–10 
Gruithuisen δ  13.2 × 10^{4}  11.8 × 10^{8}  48  203  41  12–15 
Gruithuisen NW  12.3 × 10^{4}  9.9 × 10^{8}  24  189  38  2–3 
Gruithuisen NW and γ combined  10.0 × 10^{4}  6.0 × 10^{8}  143  154  31  18–20 
Mairan T  13.1 × 10^{4}  11.5 × 10^{8}  24  201  40  1–2 
Mairan “middle”  6.9 × 10^{4}  2.5 × 10^{8}  52  106  21  2–3 
Mairan “south”  5.3 × 10^{4}  1.3 × 10^{8}  51  82  16  ∼1 
Mairan “middle” and “south” combined  6.1 × 10^{4}  1.8 × 10^{8}  52  94  19  8–10 
8. Discussion and Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Initial Estimates of Rheological Parameters
 3. Analysis of the Gruithuisen γ Dome
 4. Analysis of the Gruithuisen δ and NW Domes
 5. Summary of Rheological Parameters
 6. Eruption Rate Estimates
 7. Geometries of Feeder Dikes
 8. Discussion and Conclusions
 Acknowledgments
 References
 Supporting Information
[16] In summary, we find typical values of yield strength, τ, of order 10^{5} Pa, plastic viscosity, η, of order 10^{9} Pa s, and effusion rate, E, of order 50 m^{3}/s. What do these values mean? Values of τ ∼ 3 × 10^{5} occur with terrestrial rhyolites, dacites and basaltic andesites [see Blake, 1990, Table 3]. Effusion rate values of up to 10 m^{3}/s for the Mt. St. Helens dacite dome lobes [Anderson and Fink, 1992], and about 5 m^{3}/s for the Soufriere basaltic andesite [Blake, 1990; Huppert et al., 1982] have been documented.
[17] Typical dike geometries are predicted to be of width ∼100–200 m and length about 20–40 km. The magma rise speed implied by this geometry is very low, ∼10^{−5} m/s, and the Reynolds number of the motion is ∼10^{−8}, implying a completely laminar flow regime. If the viscosity/yield strength relationship is unreliable, the W and L values will not be very reliable (though W depends on η raised to a power close to one quarter so this will not change the values dramatically). Total duration of formation of the domes can be estimated by using the data on effusion rates and volumes (Table 8). The range of formation times is estimated to be ∼10 to 50 years.
Table 8. Summary of the Key Values for the Gruithuisen and Mairan Domes^{a}Dome Name  r_{m}/m  d_{m}/m  V/km^{3}  E/(m^{3}/s)  D/years 


Gruithuisen γ  10000  856  135.9  119.3  38.0 
Gruithuisen δ (NW lower dome)  6500  869  57.7  49.7  38.7 
Gruithuisen δ (NW upper dome)  4000  681  17.1  24.0  23.8 
Gruithuisen δ (SE lower dome)  6500  939  62.3  46.0  45.1 
Gruithuisen δ (SE upper dome)  2750  611  7.3  12.6  19.3 
Gruithuisen NW (lower dome)  4000  682  17.1  24.0  23.8 
Gruithuisen NW (upper dome)  1500  418  1.5  5.5  9.1 
Mairan T  6500  900  59.7  48.0  41.5 
Mairan “middle”  5500  600  28.5  51.5  18.4 
Mairan “south”  5000  500  19.6  51.1  12.8 
[18] In summary, these new calculations confirm the unusual nature of these features and support previous qualitative suggestions that they were formed from magmas with significantly higher viscosity than typical of mare basalts.