During its long residence at Earth's surface, continental lithosphere is shaped by tectonic events such as rifting (including supercontinent break-up) and plate collision, undergoing profound changes in its physical and chemical state. In some cases, previously stable (undeforming) portions of the continental lithosphere may be destabilized. Based on the close association of magma infiltration with these events there is growing speculation that, under certain circumstances, melt–rock interaction may somehow weaken and perturb the stability of continental lithosphere . Recent work in subduction settings suggests that heat advection by magma transport into the overriding lithosphere is a fundamental process that determines the thermal structure of arcs and possibly aids thinning of the overriding plate .

Fundamentally, this melt-enhanced weakening of continents is intertwined with the notion of thermal and chemical disequilibrium between infiltrating melts and the surrounding material, and therefore with transient processes that drive the lithosphere from one stable state to another. These processes, however, remain elusive. This paper explores one aspect of the problem, namely, thermal disequilibrium during infiltration of hotter-than-ambient melts into the base of the lithosphere, as a means of weakening and shaping the continental lithosphere from beneath. Specifically, I explore thermal disequilibrium between melt-rich channels and surrounding (melt-poor) material as a process to heat and modify the continental lithospheric mantle (CLM). The models explored here are a useful way to place constraints on the melt-transport scenarios for which a significant degree of disequilibrium heat exchange (driven by temperature differences between materials within and surrounding channels) may be important in the CLM at or near the lithosphere–asthenosphere boundary (LAB).

This study is inspired by evidence for the role of thermal disequilibrium from detailed field-based, petrologic, and geochemical studies in the Lherz and Ronda peridotite massifs . Important conclusions from studies in the Lherz and Ronda peridotite massifs include the following: (1) “lherzolite” (named after its type-section in the Lherz massif), commonly regarded as pristine, fertile sub-continental lithospheric mantle, may be derived from refertilization of a depleted, harzburigitic parent e.g.,; (2) careful microstructural, geochemical, and petrologic work has documented the dominant effect of a steep thermal gradient associated with the region of contact and interaction between partial-melt-rich regions and the surrounding lithosphere. These workers provide a quantitative estimate of a transient thermal gradient (≈230∘C km-1, or more than an order of magnitude larger than a typical equilibrium geothermal gradient expected at the LAB). Indeed, the authors recognize this as a transient LAB and coin the term “asthenospherization” for disequilibrium heat exchange processes modifying the LAB. The spatial scale over which this disequilibrium heating is observed in Ronda (∼1 km) guides the mesoscale modeling approach here.

Additionally, this work is motivated by observations from the western US, which has undergone extensive magma-infiltration in Cenozoic time. Pressures and temperatures of last equilibration of Cenozoic basalts consistently point to depths that are at or below the present LAB , suggesting that melt transport from those depths upward through the lower CLM occurs in thermal disequilibrium. In the Big Pine volcanic field, for example, the inferred depth of the LAB decreases by >10 km in a timespan of <1 Myr , suggesting that the processes associated with this migration may be transient (LAB vertical migration rates in excess of ≈10 km Myr-1). More recently, Cenozoic melt- or fluid-enhanced thinning of the CLM in the western US has also been inferred from geochemical and isotopic data from volcanic rocks . Such processes of thermally driven erosion and migration of the LAB are also inferred in arc settings e.g.,. Motivated by these observations, a primary goal of this work is to quantify the role of transient, disequilibrium heating by infiltrating channelized melt as a mechanism for modifying the LAB and the lowermost CLM.

It has been argued that a permeability contrast e.g., or a change in magma mobility e.g., across the LAB is likely to drive melts to pond and possibly drive the upward propagation of dikes that may freeze and heat the CLM e.g.,. Not all infiltrating melts would freeze, however, and some component of hotter-than-ambient melts may be transported in thermal and chemical disequilibrium into the CLM via established channels or pathways e.g.,. Thermal disequilibrium during melt transport is expected to become important within the CLM as the degree of channelization and the relative melt–solid velocity increases e.g.,. In this work, I am not concerned with the emergence and development of these channel networks within the lower CLM, nor the deeper processes that transport melt from a sub-lithospheric melt-generation zone e.g.,. Instead, the starting point of this study is the observation that high-porosity, melt-rich channels are an important part of melt–rock interaction in the CLM e.g.,. An important limitation of the models, therefore, is that channelization is imposed via parameters that control a heat transfer coefficient. The simplicity of the models, however, allows us to focus on the implications of significant thermal gradients between melt-rich channels and their surroundings. Although others have also argued for the important role of thermal disequilibrium in melt–rock interaction , this study provides a quantification of the role of thermal disequilibrium at the LAB based on observational constraints discussed above. This work builds on the 1D model in (which did not consider axial conduction) and includes both a thermal contrast that drives heat exchange and axial diffusion terms e.g., following in order to explore thermal equilibration over long timescales of melt infiltration (>103-4 years) into the lowermost 1–10 km of the CLM e.g., stage 3, large Péclet numbers in.

The 1D model abstracts the complex geometry of the melt–rock interface and therefore differs from other descriptions of disequilibrium heat exchange . Similar to chemical transport models that use a linear driving term for chemical disequilibrium e.g.,, the models below assume a linear thermal driving term e.g.,. In this study, however, only the role of thermal disequilibrium is included and I ignore important processes such as chemical disequilibrium e.g.,. The basic results of the 1D model are presented below, followed by further discussion of their limitations and implications. Although idealized, the models provide a first-order assessment of the temporal and spatial scales over which thermal disequilibrium can play a role in warming and therefore weakening the lowermost CLM. A key finding is that the lowermost portion of the CLM may evolve into a long-term thermal reworking zone (TRZ) driven by disequilibrium heat exchange in channelized melt transport. The factors that determine the rate at which the TRZ is formed and its spatial scale are estimated from the models and are compared to geologic observations within the western US, specifically geochemical and petrologic evidence for the upward migration of the LAB during Cenozoic melt–rock interaction .

The starting point for the models explored here is a simple, 1D theory of heat exchange in packed porous beds by , building on , where fluid moves within the pores of a matrix of solid grains, and the only variability in temperature (and temperature contrast between solid and fluid) is in the transport direction. In , the thermal evolution of the system is governed mainly by heat exchange across the solid–fluid interfacial surface. In this study, I present a re-interpretation of Schumann's model and of the effective heat transfer coefficient. Instead of considering fluid moving in pores between solid grains, the system of equations from may be used to describe thermal disequilibrium between material within high-porosity channels and outside channels. In other words, here “fluid” is interpreted to be in-channel material and “solid” is material outside channels (for simplicity, however, I retain the subscripts f (in-channel) and s (outside channels)). The goal here is to describe the relative importance of advective transport over length scales that are comparable to the channel spacings, so we define a transport velocity, vchannel, as an average relative velocity across channel walls. As described below, this reinterpretation also extends to the physical meaning of the effective heat transfer coefficient, where now the geometry across which the transfer occurs must take into account the channel geometry and spacing. Furthermore, following , heat exchange is assumed to be linearly proportional to the local temperature difference between solid and fluid (see also ). Unlike and , however, this study includes thermal diffusive effects to account for (axial) conduction within channels and within the material outside of the channels. These arguments lead to coupled equations for the temperature outside channels, Ts, and within channels, Tf: 1∂Tf∂t+vchannel∂Tf∂x=-Kϕcf(Tf-Ts)+λfcf∂2Tf∂x2≡-Kf(Tf-Ts)+λfcf∂2Tf∂x2,2∂Ts∂t=K(1-ϕ)cs(Tf-Ts)+λscs∂2Ts∂x2≡Ks(Tf-Ts)+λscs∂2Ts∂x2, where vchannel is the average transport velocity of material within melt-rich channels relative to the melt-poor surrounding material, ϕ is a fluid volume fraction, λf and λs are thermal conductivities, cf and cs are the heat capacities per unit volume (heat capacitances) at constant pressure (cf=cpfρf and cs=cpsρs), and x is the position coordinate in the transport direction. Note that the geometry of the solid–fluid interface is not treated in detail but is idealized in the channel volume fraction, ϕ, the channel spacing, d, both of which control an effective (across-channel-wall) heat transfer coefficient, K, discussed below.

One advantage of “coarse-graining” in the model (from the pore scale to macroscopic channels) is its simplicity and that it has been investigated in numerous previous studies. There exist analytic solutions for Eqs. () and () in limiting cases, particularly for large Péclet numbers with axial diffusion terms ignored (without the last terms in Eqs.  and ) . This re-interpretation of the model must also be accompanied by an appropriate reinterpretation of the heat transfer coefficient, K, made possible because in the framework above the geometry of the interfacial surface is not explicitly specified. The reinterpreted model is applied to a semi-infinite domain where fluid transport occurs in high-porosity channels aligned in one direction (Fig. ). The high-porosity channels are assumed to occupy a constant volume fraction, ϕ, within which material moves with a constant (average) velocity vchannel relative to the surrounding stationary material outside the channel (volume fraction 1-ϕ). The model domain may be thought of as co-moving with the reference frame of material outside the channels. Because of the assumptions built into the 1D approach, the results below are applicable to physical situations where transport is dominantly in the along-channel direction and any motion of material outside channels is steady. The model assumes that the average channel geometry is unchanging within the domain and that the material outside the channels is initially at a uniform temperature. The models and their interpretations, therefore, are used here to assess the role of thermal disequilibrium as melts are transported (≈10–20 km or so) within the lowermost CLM above the LAB within intraplate settings.

In this model, the terms involving the thermal contrast (Tf-Ts) in Eqs. () and () represent heat transfer across the walls of the channels, between material within and outside the channels. This heat exchange depends on material parameters that govern thermal diffusion perpendicular to the transport direction, and on the geometry of the channels themselves (e.g., sinuosity, spacing). The detailed geometry of the channel walls (the relevant interfacial surface here) is not specified in this simple 1D treatment, but is parameterized by the heat transfer coefficient, K (Fig. ). Therefore, K is a proxy for the efficiency of conduction perpendicular to the transport direction, and the geometry of the channel wall interface, namely the wall area per unit volume, controlled by the spatial scale of channelization, d (see Sect. ).

As illustrated in Fig. , a large value of K may represent efficient heat exchange as in the case of many channels separated by a small distance. Conversely, a low value of K would represent inefficient exchange, as in the case of a larger characteristic separation between the channels. In the following sections, I consider the physical meaning of K and also present a non-dimensionalization of the Eqs. () and () based on characteristic length and timescales in the problem. The coefficients of the temperature-contrast terms on the right hand sides of Eqs. () and () specify the timescales of heat exchange within channels, tf=1/Kf=ϕcf/K, and outside channels, ts=1/Ks=(1-ϕ)cs/K. Additionally, a characteristic length scale emerges out of the relative motion across channel walls, vchannel/Kf. These characteristic length and timescales are used to non-dimensionalize Eqs. () and () (see Sect. ) and obtain the results presented below.

In the models below, I consider a 1D domain (x≥0) that is initially in equilibrium Ts′=Tf′=0; material outside the channels is stationary, and only the in-channel material is moving, at speed vchannel. At x=0 and t=0+, the in-channel material is subjected to a thermal perturbation in Tf. The non-dimensional Eqs. (), (), and () are used to determine the transient thermal equilibration between material inside and outside channels, but we limit consideration to models where advective transport dominates and Pe>10. I consider transient thermal evolution with three scenarios of influx of hotter-than-ambient melt or fluid, in order of increasing complexity (Fig. ): (I) a step-function increase in temperature of the in-channel material; (II) a sinusodal temperature perturbation; and (III) a finite-duration, constant-amplitude thermal pulse. The perturbation in Tf disturbs the initial steady-state condition in the domain starting at t=0+ as material with a perturbed temperature enters channels at x=0 with a positive perturbation amplitude ΔT (Fig. ). The perturbation “front”, the farthest extent of channel material that entered at x=0 with perturbed temperature, is at xpert=vchannelt (or xpert′=t′/ζ; where the dimensionless velocity inside channels is 1/ζ; see Sect. ). (Note that the location of maximum disequilibrium heat exchange (Tf′-Ts′)max) moves at a different speed, Vdiseqm, discussed below.) In each case, the physical, user-specified quantities that specify the model are the channel spacing d, channel volume fraction ϕ, the in- and out-channel thermal conductivities λf and λs, heat capacitances cf, and cs, and the speed of the in-channel material vchannel (see values in Table ). Since the material properties cs and cf are held constant, there is a unique mapping between ζ and ϕ, the channel volume fraction (for ϕ=0.01 to 0.20, ζ=0.0096 to 0.2376; Table ).

The model scenarios considered above are idealized and therefore limited in their representation of the complexities of deformation and fluid–rock interactions within the CLM. In particular, the effective thermal properties and the geometry of the fluid- and melt-rich channels are abstracted into a single number, the heat transfer coefficient, K, strongly controlled by the channel spacing, d. Sinuosity and other aspects of the geometry of channelization are abstracted, and the details of processes at and below the scale of an average channel spacing, d, are ignored. Instead, the focus here is on the effective behavior at mesoscopic spatial scales ≫d. Even at these scales, this formulation ignores spatial variations in transport, including variability in the channel volume fraction ϕ, in-channel velocity vchannel, and effective heat transfer coefficient K. Time-dependent variability in transport is also ignored, e.g., feedbacks due to possible phase changes during disequilibrium heating or cooling which would affect the geometry of the channels . Finally, this 1D model ignores the 3D nature of relative motion between material inside and outside channel walls even on the mesoscale (≫d).

Given these limitations, the models above are a way to frame first-order questions and develop arguments related to the consequences of disequilibrium heating, particularly when the behavior is dominated by downstream effects in the direction of transport. Taking the model domain to be analogous to the lowermost lithosphere, where melt or fluid transport may be channelized (Fig. ), x<0 corresponds to a melt-rich sub-lithospheric region e.g., a decompaction layer,, whereas the domain x>0 represents an initially sub-solidus lowermost CLM, and x=0 is the initial LAB (Fig. ). Melt infiltration into the lithosphere may be episodic, controlled by timescales associated with transport from the melt-generation zone to the LAB e.g.,, processes of fracturing and crystallization in a dike boundary layer e.g., and melt supply from a deeper region of melt production e.g.,.

Three key results emerge from the models above: (1) disequilibrium heating, estimated using the heat transfer coefficient, may be a significant portion of the heat budget at the LAB and the lowermost CLM; (2) there is a material-dependent velocity associated with transient disequilibrium heating; and (3) there is a region of spatiotemporally varying disequilibrium heat exchange, a thermal reworking zone (TRZ). Below I discuss each of these within the context of episodic melt infiltration into the CLM in an intra-plate setting, specifically the Basin and Range province of the western US where deformation and 3D melt transport may be simplified by neglecting plate-boundary effects. In this case, dominantly vertical heat transport within a slowly deforming lithosphere is a reasonable first-order assumption.

i. Disequilibrium heating and the heat budget at the LAB. The relative importance of disequilibrium heat exchange at the LAB may be established by considering the effective heat transfer coefficient, K, and the parameter which most strongly controls it, namely the average spacing of channels, d. For the material parameters in Table , and channel volume fraction ϕ≈0.1, channel spacing of d=500 to 1000 m, K is of order 10-5 W m-3 K-1 (Sect. ), corresponding to Péclet numbers on the order of 101. Physically, K corresponds to across-channel heat transfer per unit time, per unit volume, and per unit difference in temperature . Therefore, if we assume that the average thermal contrast in the TRZ is roughly 2 % to 5 % of ΔT (e.g., (Tf′-Ts′)max in Fig. ), for ΔT=100 K excess temperature of the infiltrating melt, disequilibrium heat exchange might contribute around 10-4 to 10-5 W m-3 to the heat budget at the LAB. This is a conservative estimate, given that the temperature difference between magma and the surrounding material may be larger (e.g., in Lherz the inferred contrast is >200 K, , and up to 1000 K in crust, ). Similarly, plume excess temperatures are estimated to be as large as 250 K .

To put this in perspective, we now compare this estimated heat budget to the heat budget due to deposition of latent heat during crystallization of melt transported in channels in the lithosphere e.g.,. The contribution from freezing of melt may be estimated using scaling arguments made in . Assuming that melt and rock are in equilibrium, estimate that the heat released by a crystallization front would contribute around ρHSdike, where ρ is the melt density, H is the latent heat of crystallization, and Sdike is a volumetric flow rate out of a decompacting melt-rich LAB boundary layer due to diking. For a representative dike porosity of ϕ=0.1 within the dike, estimate Sdike≈0.2 mm3 s-1. Taking ρ=3000 kg m3, and H=3×105 J kg-1, the heat source due to the moving crystallization front would be around 102 W per dike. If we assume that this heating takes place within a volume that is roughly the dike height × dike spacing × dike length, we can determine the volumetric power generated due to crystallization. For example, assuming dike heights of about 1 km and dike spacing large enough for non-interacting dikes (as estimated by , a porosity of 0.1 would require a dike spacing of ∼1 km), the heat source due to a crystallizing dike boundary layer would be <10-4 W m-3 (per unit length along strike). These arguments corroborate the idea that disequilibrium heat exchange during melt–rock interaction could be an important portion of the heat budget at the LAB as compared to other expected processes, such as heating due to crystallization of melt in channels see also.

ii. Progression of a disequilibrium heating zone or front at a rate Vdiseqm. The disequilibrium heating front is associated with a migration rate Vdiseqm that is less than the in-channel material velocity, vchannel (Eq. ). Therefore, Vdiseqm limits the rate at which the lowermost CLM may be modified by thermal disequilibrium during migration of a disequilibrium front (e.g., Figs. b, ). Although in this work I considered a fixed vchannel=1 m yr-1, note that Vdiseqm (Eq. ) is independent of temperature contrast between the CLM and infiltrating melt and, for fixed material properties and channel volume fraction ϕ, depends linearly on vchannel. However, Eq. () also illustrates a tradeoff between vchannel and ϕ. Assuming a channel volume fraction of ϕ=0.1 at the LAB, and material properties in Table , Vdiseqm would be around 10 % of the in-channel velocity (Fig. c). For in-channel velocity in the range of 0.01 to 1 m yr-1, the disequilibrium heating front at the LAB would migrate upward at a rate of ≈1 to 102 km Myr-1 (Fig. ) This is comparable to rates of CLM thinning predicted by heating due to the upward motion of a dike boundary layer (1 to 6 km Myr-1 in ). Interestingly, an upward-moving disequilibrium heating zone with Vdiseqm≈1 to 102 km Myr-1 also brackets the 10–20 km Myr-1 rate of upward migration of the LAB inferred from the pressure and temperature of last equilibration of Cenozoic basalts in the Big Pine volcanic field in the western US . An implication of the models here, therefore, is that disequilibrium heating may produce lithosphere modification at geologically relevant spatial and temporal scales provided that the material velocity in channels at the LAB is on the order of 10-1 to 1 m yr-1 e.g., Fig. ;; higher transport rates would require lower ϕ to drive a similar rate of CLM modification.

iii. Thermal reworking zone (TRZ). A key result that may be relevant to the evolution of the LAB is that episodic infiltration of melts that are hotter than the surrounding CLM would lead to a finite region of disequilibrium heating within a thermal reworking zone or TRZ. The TRZ would undergo a phase of transient widening (at a rate given by Vdiseqm), reaching a maximum width δ that is proportional to ϕvchan(τ/d)n (n≈1 for individual perturbations, but <2 for multiple; Fig. ; also Fig. d). Here d is a characteristic scale of channelization and τ is a timescale associated with the episodicity of melt infiltration. This scaling gives us a way to conceptualize the modification of the lowermost CLM as a zone that may encompass a variable thickness TRZ, depending on variability in transport velocity and in the timescale of melt infiltration (Fig. b). Regions where the timescale of episodic melt infiltration is longer are predicted to have a thicker zone of modification at the LAB (Fig. ). For example, for ϕ≈0.12, a channel spacing of d=500 m, disequilibrium heating by melt pulses that last around 10 kyr implies a maximum thickness of roughly 10 km for the zone of modification (Fig. ; also Fig. d). In this scenario, the TRZ grows to this maximum width over a timescale governed by δ/Vdiseqm; for Vdiseqm=10 km Myr-1, which corresponds to melt velocity of roughly 0.1 m yr-1 (see (ii) above), the 10 km wide TRZ would be established within about 1 Myr (Fig. ), comparable to rates of CLM modification inferred from observations in .

These scaling arguments lead to the idea that perhaps the TRZ represents a zone of thermal modification at the base of the CLM that may also correspond to (or encompass) a zone of rheologic weakening and/or in situ melting if the infiltrating fluids are hotter than the ambient material. The dynamic evolution of the LAB during episodic pulses of melt infiltration is beyond the scope of the simple models above (which assume a stationary, undeforming matrix). However, assuming mantle material obeys a temperature and pressure-dependent viscosity scaling relation such as in at an LAB depth of about 75 km, where the ambient mantle is cooler than the dry solidus (e.g., 1100 ∘C + 3.5 ∘C km-1;, , and assuming a wet dislocation creep mechanism), we would expect a viscosity reduction by a factor ≈1/1.4 to 1/12 during heating (e.g., ≈1/2.3 for a temperature increase of 20 K (0.2ΔT for a 100 K perturbation amplitude)). This effect is weaker, but still important for a deeper LAB; e.g, at 125 km depth, the viscosity reduction would be a factor ≈1/1.2 for a temperature increase of 20 K.

Geochemical evidence from Cenozoic basalts from the western US, particularly space–time variations in volcanic rock Ta/Th and Nd isotopic compositions, suggests that the timescale of modification and removal of the lowermost CLM is on the order of 101 Myr . These authors argue that the observed transition from low to intermediate to high Ta/Th ratios indicates a change from arc- or subduction-related magmatism, to magmatism associated with in situ melting of a metasomatized CLM (the “ignimbrite flare-up”), to magmatism due to decompression and upwelling after removal of the lowermost CLM. At a minimum, the observed timescale of the transition in Ta/Th ratios in volcanic rocks (101 Myr) in the western US should be comparable to the timescales of degradation of the CLM. If correct, these interpretations and observations are promising and provide an important avenue for exploring the role of thermal and chemical disequilibrium during melt–rock interaction and destabilization of the CLM in an intra-plate setting.

In summary, I have presented arguments supporting the role of disequilibrium heating in the modification of the base of the continental lithospheric mantle (CLM) during melt infiltration into and across the lithosphere–asthenosphere boundary (LAB). Infiltration of pulses of hotter-than-ambient material into the LAB should establish a thermal reworking zone (TRZ) associated with disequilibrium heat exchange. The spatial and temporal scales associated with the establishment of the TRZ are comparable to those for CLM modification inferred from geochemical and petrologic observation intra-plate settings, e.g., the western US. Disequilibrium heating may contribute around 10-5 W m-3 to the heat budget at the LAB and, for transport velocity of 0.1 to 1 m yr-1 in channels that are roughly 102 m apart, a 10 km wide TRZ may be established within 1 Myr. Disequilibrium heating during melt infiltration may therefore be an important process for modifying the CLM. Further work is needed to explore its role in the rheologic weakening that must precede mobilization (and possibly removal) of the lowermost CLM.