The spreading rate dependence of the distribution of axial magma lenses along mid-ocean ridges

Significance

The best constraints on the size and depth of magma chambers on Earth come from seismic studies of quasihorizontal axial melt lenses (AMLs) seen all along fast-spreading mid-ocean ridges but only at the centers of slow-spreading segments. We present a model that explains the observed distribution of AMLs by assuming that lenses form as magma-filled tension fractures during the opening of magmatic dikes. This stress change can force open an AML below the axial lithosphere only if the average density of the lithosphere is less than or equal to the magma density. Seismic observations and thermal models suggest that this condition is met along most fast- and some intermediate-spreading ridges but only at the centers of slow-spreading segments.

Abstract

Seismically imaged axial melt lenses (AMLs) are seen almost everywhere along the axis of fast-spreading ridges but at only a few localized segment centers on slow-spreading ridges. Standard models assuming that AMLs form when melt percolating upward pools where freezing produces an impermeable cap do not explain this fundamental observation. To tackle this long-standing problem, we combine a crustal density model and a thermal model with a recent mechanical model for sill formation. The mechanical model predicts that AMLs form below the axial lithosphere but only if the average density of the axial brittle lithosphere is not greater than the magma density. For standard thermal models, crustal density structures inferred from seismic velocity data and normal crustal thicknesses, AMLs are found to be stable along all of a ridge segment for spreading rates greater than about 50 mm/y. To explain slow-spreading observations, we assume that a share of the melt produced by the mantle upwelling all along a segment is focused to the segment center. Some of this melt partially crystallizes, releasing latent heat, before the evolved magma flows along the axis to build the crust away from the segment center. This “extra” heat, beyond what is supplied by the magma that builds the crust near the segment center, results in the lithosphere thin enough for stable melt lenses at the segment center. Our results are consistent with observations and offer a quantitative explanation of the marked difference in the distribution of AMLs along fast- versus slow-spreading centers.

Two of the most systematic variations of Earth structure are seen at plate spreading centers. The most obvious variation is in across-axis bathymetric relief, with ~300- to 500-m axial highs seen for spreading rates greater than ~60 mm/y and ~1,000- to 2,000-m-deep axial valleys for rates less than ~40 mm/y (e.g., ref.1). The other variation is in magma chambers, or axial melt lenses (AMLs), that can only be seen with marine multichannel seismic (MCS) methods. AMLs about a kilometer wide are imaged at ~1 to 3 km depth almost everywhere at the axis of fast-spreading ridges and for some intermediate-spreading centers (2, 3) (Fig. 1B). In contrast, melt bodies are imaged at only a few localized segment centers on slow-spreading ridges (4)(Fig. 1A), although the overall seismic and bathymetric structure of slow-spreading ridges suggests that magma pools at the centers of most slow and ultraslow (<~20 mm/y) spreading segments (5).

Fig. 1.

Seismic reflection images of AMLs. (A) The AML reflector of the Lucky Strike ridge (4). The red circles represent the limits of the AML reflector, and the length of the AML along the axis (L_AML) is ~7 km, which is 1/10 of the entire length of the ridge segment. (B) An AML reflector along the 9°N segment of the Northern East Pacific Rise (3). To first approximation, the AML is distributed continuously along the ridge axis.

It is accepted that the spreading rate dependence of axial relief is related to the availability of magma to accommodate plate spreading (e.g., ref. 6). If there is always plenty of magma to fill dikes that open as plates separate, then an axial high forms (e.g., ref. 7). If there is not enough magma to fill that space, then fault slip results in an axial valley (e.g., refs. 8 and 9).

The formation of AMLs is more controversial. One idea is that AMLs form when melt percolates upward through partially molten mantle and crust until it pools at depth where freezing produces an impermeable cap (e.g., ref. 10). This process is likely to occur beneath ridges, but the model does not explain the irregular and sometimes overlapping structures of AMLs (e.g., ref. 2) since the depth of a thermally controlled “freezing front” should vary smoothly along and across a ridge axis, although we cannot rule out the possibility that along-axis variations in tectonics or hydrothermal processes could contribute to the complexity of the lens (3, 11). Also, freezing fronts should exist at both fast- and slow-spreading ridges, although at different depths, so this model does not explain the very different distribution of AMLs at fast- versus slow-spreading centers (Fig. 1). A recent model (12) that considers stress changes during dike intrusion suggests a way to form sills as tension fractures, consistent with the observed irregular structure of AMLs. For reasonable thermal and rheological structures of a spreading center, the model predicts that a melt lens could form only when the average density of the axial brittle lithosphere is close to or less than the density of the magma.

To quantify how this model can explain the spreading rate dependence of the along-axis distribution of AMLs requires several efforts. First, we use a fairly standard thermomechanical model to estimate the long-term thermal and rheological structure of a spreading center as functions of spreading rate and magma supply. This structure, together with crustal density profile inferred from seismic velocity data (13), is then used as input for the mechanical model of Liu and Buck (2022) to determine whether AML could form for that ridge structure. Next, we consider how a concentration of melt at a segment center could lead to a stable melt lens only close to the segment center. Finally, we compare our predictions with available observations and discuss the limitations of our modeling approach.

Model Formulation

The formation of axial melt lens in the Liu and Buck (2022) model depends on the rheology (i.e., the elasticity, yield strength, and viscosity) and density structure of a spreading center. The viscosity of crust and mantle materials depends strongly on temperature (14). Thus, to map out the range of spreading rates and crustal thicknesses where an AML might form requires that we determine the thermal structure of a spreading center. We follow the procedure used in several recent studies of spreading center development (e.g., refs.  and 15–17), which combine the crustal accretion thermal model of Phipps-Morgan and Chen (1993) (18) with a 2D viscoelastic plastic numerical model (Methods A.1 and SI Appendix, Fig. S1). As in those studies, the thermal structure depends on the balance between the latent heat released by magma crystallization during crustal accretion and the heat removed by hydrothermal cooling. A key uncertainty in the thermal model involves the parameterization of the effects of hydrothermal circulation in terms of an enhanced conductivity or Nusselt number, Nu. Since cracks providing a pathway for hydrothermal flow are likely to close at depth or at high temperature (19), the effective conductivity is only increased for the shallowest 7 km of the model domain and where temperatures are less than 600 °C. For a given value of the Nusselt number, the main controls on the model thermal structure are the crustal thickness, H_C, and the full spreading rate, V_p.

The thermal structure of the model spreading center determines the strongly temperature-dependent effective viscosity of the crust and mantle (Methods A.1). Using the calculated rheological structures for a model spreading center, we show the effect of the intrusion of an ~1.5-m-wide dike on the stress state of axial ridge (12). We first let the model undergo 3 m of plate separation corresponding to an interval with no magma intrusion. During this stage, the brittle lithosphere behaves mainly as an elastic solid, accumulating a horizontal deviatoric stress of about 10 MPa, while the ductile sublithospheric region relaxes stress differences through viscous deformation (SI Appendix, Fig. S2A). During a magmatic intrusion stage, we use an adaptive boundary condition to simulate dike opening at the axis of spreading. Where the model horizontal stress is less than or equal to the magma pressure, the boundary normal stress is set equal to the magma pressure. Otherwise, the boundary condition is zero horizontal velocity. We assume that the magma pressure is hydrostatic pressure in a static dike and equals ρ_m g D, where ρ_m is the density of the magma, g is the acceleration of gravity, and D is the depth below the surface. An AML is assumed to open if there is a depth range at the spreading axis where the minimum (compressive) stress is close to vertical and that stress is lower than the pressure in a static column of magma reaching to the surface (SI Appendix, Fig. S2B). The density structure of the ridge axis has a strong influence on the stress changes that may or may not allow opening of such a quasihorizontal sill. We use a simplified density structure based on the seismic and rock mechanical analysis of Christeson, Goff, and Reece (13), as described in Methods A.2 and SI Appendix, Fig. S3.

Several lines of evidence suggest that at slowly spreading or melt poor ridges much of the magma to build all the crust of an entire segment is first concentrated at the segment center and then flows along the axis, either as dikes or as ductile lower crust (e.g., refs. 20, 21–23). We use this idea to consider how a melt lens might form along only a portion of the central part of a segment. If some of the magma that adds to the distal crust of a segment crystallizes at the segment center, releasing “extra” latent heat, then the 1,000 °C isotherm at the segment center may be shallow enough for a lens to form. We quantify the effect of this extra heat by computing the length L_AML of the central part of a segment where an AML could form at the maximum depth of mechanical stability. The entire segment is taken to have a length L₀ and a fraction f of the crust forming the segment away from the central AML crystallizes in the segment center. We define an “effective crustal thickness” to mean the total crustal thickness that would crystallize in a segment-centered magma chamber in this scenario. As described in Methods A.3, we are then able to define the ratio of L_AML to L₀ as functions of the spreading rate and average crustal thickness along a segment.

Results

Fig. 2 shows the model axial thermal structure of spreading centers. Fig. 2A shows three typical model cases to visualize the effects of spreading rate and crustal thickness. We subject all models to 30 km of extension (the width of the model box) with constant axial magma injection to ensure that the crust presented is newly accreted and to avoid disturbances from the initial model setup. We assume that the depth of on-axis solidus temperature (D_Solidus) marks the shallowest depth that a magma lens is thermally viable. D_Solidus increases nonlinearly as the spreading rate decreases. For instance, for cases with Nu = 8, D_Solidus gradually increases as the spreading rate decreases for rates greater than 50 mm/y, while for spreading rates less than 50 mm/y, D_Solidus rapidly increases as the spreading rate decreases (Fig. 2B.2). Fig. 2B.1 shows that increasing the crustal thickness also decreases D_Solidus nonlinearly and that a small change in crustal thickness may cause a sharp change in D_Solidus. The nonlinear relationship between D_Solidus and both spreading rate and crustal thickness is related to the assumed depth dependence of hydrothermal cooling and is similar to previous modeling results (18, 24). The model results with a Nusselt number of ~8 are consistent with observed axial brittle lithosphere thicknesses inferred from the depth of AML reflectors (25) and the maximum depth of microearthquakes (26) (Fig. 2B.2). The results of our thermal models assuming Nu = 8 are comparable to the temperature structure obtained from the previous hydrothermal flow simulations (e.g., ref. 24) assuming that the on-axis permeability is 4 × 10⁻⁵ m² at the seafloor and decays exponentially toward 0 at Moho.

Fig. 2.

Thermal modeling results. (A) Three typical examples demonstrating model temperature structures and velocity vectors. Panels A.2, A.1, and A.3 show the effect of increasing crustal thickness and spreading rate, respectively. The cases shown in a.1 and A.3 do not form an axial valley, while that shown in A.2 develops faults and forms an axial valley. (B) Plots of the effects of crustal thickness and spreading rates on the depth of on-axis 1,000 °C isotherm. The depth of axial magma lens in B.2 is from (25), and the maximum depth of microearthquakes is from (26).

The thermal structure from the long thermal model runs is then used as a starting point for the short-term runs that look at the opening of a dike. Dike opening leads to an increase in the horizontal stress of the ambient rock, which gradually approaches the magma pressure. In addition, the vertical stress of the sublithospheric region decreases during this process as the surface uplifts (SI Appendix, Fig. S2C). For opening of an ~1.5-m dike, the reduction in the vertical stress is about ~5 Mpa (SI Appendix, Fig. S2B). The magnitude of vertical stress reduction is linearly proportional to dike width (12). During dike opening, the direction of maximum principal stresses in the sublithosphere rotates, and stress conditions conducive to sill formation can develop (SI Appendix, Fig. S2C). The formation of a sill requires two things: that the maximum (compressive) principal stress be horizontal and that the vertical stress be less than the magma pressure.

Given that the reduction in vertical stress during dike opening is limited, the vertical stress before dike opens should not substantially exceed the hydrostatic pressure of the magma column (i.e., the average density of the axial brittle lithosphere should be close to or less than the density of the magma). This is why our calculations show that AMLs only form where the axial lithosphere is relatively thin (e.g., less than ~4 km; Fig. 3A). For a set value of the Nusselt number, the thickness of the brittle layer is determined by the spreading velocity and magma supply (Fig. 2). This allows us to map out the necessary conditions for the formation of AMLs as functions of spreading rate and magma supply (effective crustal thickness) (Fig. 3B). The results (Fig. 3) show that AMLs are stable at spreading centers with spreading rate greater than ~50 mm/y for reasonable hydrothermal cooling intensities (i.e. Nu = 8) and segment-average crustal thicknesses. For spreading centers with spreading rates less than ~50 mm/y, extra heat is needed to produce stable AMLs (Fig. 4). For example, for cases with full spreading rate of 20 mm/y, stable AMLs can form only when latent heat is enough to build an effective crustal thickness greater than ~40 km.

Fig. 3.

Stress distribution after a 1.5-m dike opening. (A) Three typical examples demonstrating stress state. In each panel, density and dike width versus depth are shown on the left. The density profile is inferred from the seismic velocity structure of Christeson, Goff, and Reece (13); on the right, the color shows the magma pressure minus vertical stress, with the small black line representing the direction of maximum principal stress. There are two criteria for determining whether an AMC would form: 1) the direction of maximum principal stress should be horizontal and 2) magma pressure > vertical stress. Based on these criteria, cases a.1 and a.3 can form AMLs, while case a.2 does not have the stress conditions to form an AML. (B) Plots show the effects of crustal thickness and spreading rates on the predicted depth of an AML for two values of Nusselt number.

Fig. 4.

Schematic shows different AML distribution patterns along slow- and fast-spreading ridges. Seafloor bathymetry data are from the GMRT (27).

Fig. 5 shows the ratio of the along-axis length of an AML to the length of a segment, L_AML/L₀, for a range of spreading rates and segment-average crustal thicknesses. Fig. 5B was constructed assuming that 60% of the latent heat of that material making up the crust of the segment away from the centralized AML is released at the AML (i.e f = 0.6) and with Nu = 8. Segment-average crustal thickness, ${H_C}^A$, is assumed to be 7 km at places where the spreading rate is greater than 20 mm/y and 4 km for ultraslow-spreading ridges (28). The results show that at fast-spreading ridges, the ratio of L_AML/L₀ is 1, indicating the AMLs would be continuously distributed along the segment. At slow-spreading ridges, the ratio of L_AML/L₀ ranges from 0.025 to 1, which implies that the AMLs are only locally distributed along the central parts of the segment. For the case of ultraslow mid-ocean ridges, the L_AML/L₀ ratio is less than 0.025, which means that it is difficult to develop an AML greater than 1 km in length on a 40-km-long ridge segment. Taking different values for f affects the predicted value of L_AML/L₀ but will not affect the overall trend with spreading rate, as shown in SI Appendix, Fig. S4 and Fig. 5A shows how variations in ${H_C}^A$ and Nu affect L_AML/L₀ for a given value of full spreading rate (50 mm/y) and f (0.6).

Fig. 5.

Distribution of AMLs along a spreading axis. (A) The effect of Nu and segment-average crustal thickness, H_C^A. The solid green dots indicate modeling cases, and the pentagrams show observation data. All models run with full spreading rate equal 50 mm/y. E. GSC-Galapagos Spreading Center east of 93°W; W. GSC-Galapagos Spreading Center from 95.5°W to 93°W (29). (B) L_AML/L₀ versus spreading rate with H_C^A = 7 km and Nu = 8. We assume that the heat during the formation of the lower crust of a ridge segment with length L₀ is released in an axial magma chamber with length L_AML (details in Methods). For L_AML/L₀ < 0.025, the calculations suggest that a kilometer-long AML cannot be formed in a 40-km-long ridge segment; for 0.025 < L_AML/L₀ <1, a localized AML can be formed in the center of the ridge segment, and for L_AML/L₀ close to 1, a continuous AML along ridge segment will be formed. Green circles show the modeling results, and red hexagons indicate data. SWIR50°E-Speculation AML at 50°E segment of the Southwest Indian Ridge (5); Lucky Strike-AML reflector after (4); JdFR-Juan de Fuca Ridge (30); SEIR-Southeast Indian Ridge (31); EPR15-17°N segments of the East Pacific Rise (32); EPR9°N-9°N segments of the East Pacific Rise (2).

Discussion and Conclusion

Our simple model using standard thermal and rheological parameters is remarkably consistent with available observations (Fig. 5B). The model predicts that for fast-spreading ridges, there should be AMLs everywhere (i.e., L_AML/L₀ is 1), and seismic reflection data for all surveyed fast-spreading segments show this is true except close to some ridge offsets (e.g., refs. 2 and 3). For intermediate-spreading rates, the model predicts that decreasing the spreading rate and/or crustal thickness would lead to a diminishing proportion of segments being underlain by AMLs and this is what is seen seismically for the Juan de Fuca (e.g., ref. 30) and the Galapagos spreading centers (e.g., ref. 29). The data from the Galapagos Spreading Center are particularly important constraints since the surveyed segments have nearly the same full spreading rate (52 ± 3 mm/y) but show gradual changes in crustal thickness with distance from the Galapagos hotspot (33). An overlapping spreading center divides the ridge where the MCS survey was conducted in two segments. For the segments east of 93 °W, the segment-average crustal thickness is about 7 km, AMLs are imaged all along the segments, while for the segments from 95.5 °W to 93 °W, the segment-average crustal thickness is about 6 km, AMLs are seen discontinuous only along an eastern part of the segments (29). There is some trade-off between Nu and f, the data from the Western Galapagos Spreading Center require that the model Nu be larger than 8. For Nu = 10, the model fits the observed melt lens distribution and crustal thickness with f = 0.6.

At slow-spreading ridges, the only data that allow us to make a comparison is from the Lucky Strike of Mid-Atlantic ridge, where multichannel seismic data (4) indicate that a ~70-km segment of the ridge has an AML along approximately 7 km along the axis, which is in good agreement with our predictions. For ultraslow ridges, there are no seismic reflection profiles that provide a good indication of the presence of AML reflectors, but wide-angle seismic velocity data suggest that AMLs may exist in center of the one magmatically robust segment at 52 °E on the Southwest Indian Ridge (5). More geophysical surveys in the future, especially for slow and ultraslow ridges, have the potential to test the predictions of our model.

A uniform density–depth structure is used in our model, which is consistent with the compilation of global seismic velocity profiles (13). However, for some extreme spreading centers, the density–depth structure may be different. For example, the huge magma supply caused by the hot mantle plume under Iceland may explain that thicker-than-normal extrusive layer there (34). Since extrusives are less dense than intrusives (13), the density of the shallow lithosphere of the Iceland spreading center may be lower than we used here. This may explain why the seismically estimated depth of some AMLs in Iceland is ~5 km (35).

Similar thermal and density conditions needed for formation of an AML in our model are consistent with the long-term stability of magma chambers. If the long-term pressure in the AML is lithostatic, then it depends on the average density of the overlying rock. If overburden density is greater than the magma density and the magma has a pathway to the surface, then it should be pushed out of the magma sill, potentially emptying the sill. This simple argument leads to the same predictions as our sill creation model, if one assumes long-term sills form where temperatures are at the basalt solidus temperature. Many studies of continental volcanism, including a recent paper considering large igneous provinces (36), consider that crustal densities affect intrusion of magma sills.

There has been much discussion about whether a variety of ophiolite and other observations are better explained by a single shallow sill (18, 37) or by multiple sills (38) below a spreading axis. Although we assume that all of the lower crust crystallizes in a shallow magma lens and releases the corresponding heat to approximate the ridge thermal structure, it is only the depth of the on-axis D_solidus that has a major controlling effect on the final results (i.e., whether or not there can be a stress condition for the formation of the AMLs after a dike opening). Previous numerical simulations (e.g., refs. 39 and 40) have shown that assuming multiple AMLs versus a single AML has a negligible effect on the depth of the on-axis D_solidus. Interestingly, our model predicts that the stress condition for sill opening occurs over a depth range of several kilometers below the axial lithosphere (Fig. 3B.3). Thus, it may explain the multiple sills observed below some part of spreading centers. Multiple sills have also been suggested as a way to explain layering of gabbro in oceanic lower crust (e.g., ref. 41) and new seismic evidence for multiple sills (42).

Our simulation of the concentration of heat at segment centers from magma that then flows along the spreading axis is highly idealized and may neglect important three-dimensional processes. Eventually, it should be possible to add the processes treated in our 2D models, such as short-term opening of dikes and magma flow in dikes, and in three-dimensional models of spreading centers (43). However, the present model explains the radically different distributions of melt lenses along various spreading centers (Fig. 4). The fact that our model predictions fit existing data leads us to believe that our model approximates some of the key geological processes at mid-ocean ridges.

Methods

A1. 2D Thermomechanical Spreading Center Model.

To approximate the thermal structure of spreading centers, we follow the procedure used in several recent studies of spreading center development (e.g., refs. 15–17), which combine the crustal accretion thermal model of Phipps-Morgan and Chen (1993) (18) with a 2D viscoelastic plastic numerical model. Here, we assume that the magma supply per length of the ridge and per unit time is H_C · V_p, where H_C is the crustal thickness and V_p is the full spreading rate. This simple approach ignores details of the complex magma extraction process in the mantle (44), and the two input parameters, H_C and V_p, can be constrained with observations. We assume that accretion of the crust in the axis of the brittle layer occurs within a dike region from the surface to the brittle–ductile transition depth (SI Appendix, Fig. S1). The accretion rate of the dike is calculated as follows:

$$U_{\mathit{dike}}=\min \left(\frac{H_C}{H_L},,,1\right)·V_p,$$ [1]

where H_L is the thickness of the axial brittle layer, described here as the depth of the 600°C isotherm. We track the value of H_L at each numerical time step and update the rate of dike opening in Eq. 1. If H_C is less than H_L, then the remaining crust accretes in an AML 1 km wide and 250 m thick, the depth of the AML is set to be the depth where the axial temperature equals the solidus temperature, and the rate of dilation of the AML, V_AML is calculated as follows:

$$V_{\mathit{AML}}=\max \left(\frac{H_C-H_L}{W_{\mathit{AML}}},,,0\right)·V_p,$$ [2]

where W_AML is the width of the AML.

The dilation of the dike and AML is achieved by varying the stress in the intrusion zones (8, 15), assuming 2D plane strain the stress can be expressed as follows:

$${\sigma }_{\mathit{ii}}^{\mathit{dila}}=U_{\mathit{dila}}\left(\frac{E\left(\nu ,-,1\right)}{\left(\nu ,+,1\right)(1-2v)}\right),$$ [3.1].

$${\sigma }_{\mathit{jj}}^{\mathit{dila}}=\frac{{v\sigma }_{\mathit{ii}}^{\mathit{dila}}}{1-v},$$ [3.2].

$${\sigma }_{\mathit{kk}}^{\mathit{dila}}=v\left({{\sigma }_{\mathit{ii}}^{\mathit{dila}}+\sigma }_{\mathit{jj}}^{\mathit{dila}}\right),$$ [3.3].

$${\sigma }_{\mathit{ij}}^{\mathit{dila}}=0,$$ [3.4].

where E is Young’s modulus, and v is Poisson ratio. The rate of dilation U_dila is U_dike in the dike intrusion zone and V_AML in AML.

Crustal accretion affects the thermal structure due to heating from cooling of the high-temperature magma and by the release of latent heat during magma crystallization. In the numerical calculations, both components are treated as a source term in the temperature equation as follows:

$$\frac{\mathit{DT}}{\mathit{Dt}}=\mathrm{\nabla }\left(k,\mathrm{\nabla },\mathrm{T}\right)+\left[\left(T_{\mathit{liq}},-,T\right),+,\frac{L}{C_p}\right]\frac{U_{\mathit{dila}}^T}{W},$$ [4]

where T is the temperature at a given point, t is numerical time, k is thermal effective diffusivity, T_liq is liquidus temperature, L is latent heat of crystallization, and C_p is specific heat of melt. For the dike zone, W is the width of the 2 elements over which the dike heat input is distributed. For the magma lens, W is the 250 m assumed thickness of the lens. $U_{\mathit{dila}}^T$ is $U_{\mathit{dike}}^T$ in the dike intrusion zone and $V_{\mathit{AML}}^T$ in magma lens. Here, we assume that all melts would crystallize and accrete into the crust rather than only a fraction of the magma being able to crystallize (15). Following numerous other workers (e.g., refs. 18 and 45), the cooling effect of seawater circulation within the crust on thermal structure is approximated by increasing thermal diffusivity k by a factor, Nu, in the region where depths are less than 7 km and temperatures are less than 600 °C.

The rheology of a material is described by the power law relationship between strain rate and stress as follows:

$$\dot{\epsilon }=A{\sigma }^n\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{Q}{RT}\right),$$ [5]

where $\dot{\epsilon }$ is strain rate, A is preexponential factor, σ is stress, n is dislocation exponent, Q is activation energy, R is gas constant, and T is the absolute temperature. Here, we use the rheological parameters for dry diabase provided by (46). All parameters are shown in SI Appendix, Table S1.

A2. Crustal Density Structure.

Oceanic crustal density is an important factor that affects whether sills can open due to dike-induced stress changes (12). Laboratory studies indicate that the density of oceanic crustal rocks is strongly correlated with their seismic velocity (13, 47, 48), and seismic refraction studies constrain the p-wave seismic velocity structure of the crust at spreading centers. Christeson, Goff, and Reece (13) used seismic velocity profiles from a variety of spreading centers to estimate the density structure of nascent oceanic crust. Their study found a remarkably small dependence of velocity and so inferred density structure on spreading rate (SI Appendix, Fig. S3A). At both fast- and slow-spreading ridges, the velocity structure of the oceanic crust shows an approximate exponential relationship with depth. We express the relationship between velocity and depth as a two-part exponential function as follows:

$$S= \left\{\begin{array}{c}a1·x^n+\mathrm{b }x<2\\ a2·x^n+\mathrm{c }x\ge 2\end{array}\right. ,$$ [6]

where S is seismic velocity in km/s, and x is depth in km. We found that when n = 0.4, a1 = 2.65, a2 = 1.165, b = 2.8, and c = 4.8 (red dashed lines in SI Appendix, Fig. S3 A and B), we are able to capture the relationship between seismic velocity and depth to first order for both slow- and fast-spreading ridges (SI Appendix, Fig. S3B).

We then translate the seismic velocity into density by a simple equation (48) as follows:

$${\rho }_c=1000·\left(3.5-\frac{3.8}{S}\right),$$ [7]

where ρ_c is crustal density, and S is seismic velocity calculated in Eq. 6. Thecrustal density versus depth given by a combination of Eqs. 6 and 7 was used to set the crustal density for all models. The upper mantle density used in all models is 3,250 kg/m³.

A3. 2.5-Dimensional Model of AMLs.

If we find that an AML is not stable everywhere for a given spreading rate and crustal thickness, then we assume some concentration of heat at the segment center (Fig. 4). For a segment of length L₀, we compute the maximum length L_AML of the central part of the segment where an AML could be mechanically stable. A fraction f of crust forming the part of a segment without a melt lens (of length L₀−L_AML) is assumed to release its latent heat along the segment center. This increases the effective crustal thickness, $H_C^T$, controlling thermal evolution of the segment center (i.e., Eq. 4). To simplify the problem, we ignore along-axis variations in crustal thickness allowing us to define the effective crustal thickness along the segment center as follows:

$$H_C^T={H_C}^A+f{H_C}^A\left(\frac{L_0-L_{\mathit{AML}}}{L_{\mathit{AML}}}\right),$$ [8]

$H_C^T$ is then substituted for H_C in Eqs. 1 and 2 to calculate $U_{\mathit{dike}}^T$ and $V_{\mathit{AMC}}^T$, respectively. For a given spreading rate, we compute the minimum effective crustal thickness, ${H_C}^{\mathit{TAML}}$, needed to support a stable AML. Setting $H_C^T={H_C}^{\mathit{TAML}}$ in Eq. 8 allows us to define the fraction of a segment where an AML could form at the maximum depth of mechanical stability as follows:

$$\frac{L_{\mathit{AML}}}{L_0}=\frac{{H_C}^A·f}{{H_C}^{\mathit{TAML}}-{H_C}^A+{H_C}^A·f},$$ [9]

Eq. 9 shows that when ${H_C}^{\mathit{TAML}}$ is close to or less than ${H_C}^A$, L_AML/L₀ approaches 1, while when ${H_C}^{\mathit{TAML}}$ is much larger than ${H_C}^A$, L_AML/L₀ gradually approaches 0.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix. The numerical modelling code used here can be downloaded for free at https://github.com/tan2/geoflac.