Journal of Petrology Advance Access originally published online on August 4, 2005
Journal of Petrology 2005 46(12):2569-2592; doi:10.1093/petrology/egi065
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Viscous Energy Dissipation and Strain Partitioning in Partially Molten Rocks
1 DEPARTMENT OF GEOLOGY AND GEOPHYSICS, UNIVERSITY OF MINNESOTA, 310 PILLSBURY DRIVE SE, MINNEAPOLIS, MN 55455, USA
2 DEPARTMENT OF EARTH AND ATMOSPHERIC SCIENCES, CORNELL UNIVERSITY, ITHACA, NY 14853-1504, USA
RECEIVED APRIL 19, 2004; ACCEPTED JUNE 20, 2005
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ABSTRACT |
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TOP
ABSTRACT
INTRODUCTIONWe develop a steady-state fluid-mechanical analysis describing the effect of strain partitioning on viscous energy dissipation. As observed in experimental studies of shear deformation of partially molten rocks, strain partitions when melt segregates because viscosity is reduced in regions of elevated melt fraction. The equations derived here are based on parameters measured in experiments, describing the evolution of melt distribution and rheological properties. We find that the dissipation depends strongly on the configuration of the melt-rich network of shear zones, including the average angle, volume fraction of melt and amplification of strain rate in the melt-rich bands. Minima in energy dissipation as a function of band angle develop, corresponding to configurations of melt networks that minimize the difference in mean stress between the band and the non-band regions. We propose that the organization of band networks occurs by the interplay between strain localization and viscosity variations associated with melt segregation. The band networks maintain a steady-state angle during shear by continuously pumping melt through the network. The development of strain partitioning in melt-rich networks will modify the energetics of melting and melt transport by efficiently extracting melt and reducing effective viscosity.
KEY WORDS: melt transport; rheology; self-organization; strain localization; strain partitioning
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INTRODUCTION |
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Most regions in the mantle where large volumes of melt are produced are also deforming intensely, such as mid-ocean ridges, subduction zones and sites of plume–lithosphere interactions. We have demonstrated in laboratory deformation experiments on partially molten rocks that the deviatoric stress causing deformation of the rock can also drive small fractions of melt to segregate and organize into melt-rich networks (Holtzman et al., 2003a
). We subsequently showed that melt transport and deformation are very closely coupled processes (Holtzman et al., 2003b). Similar coupled processes of weakening and strain localization occur in many systems, from metals to solid state rocks to granular materials, due to many causes of local weakening like grain size reduction or solid phase separation. One objective of this study is to quantify macroscopic effects of meso-scale strain partitioning on the effective viscosity, with the aim of developing a simple means of scaling the consequences of strain partitioning to rheological conditions relevant to the Earth. However, the primary goal is to derive an analysis of the steady-state process that helps us to understand the dynamics of the observed self-organization of melt and solid during strain partitioning.
For applications to processes in the Earth, melt segregation and strain partitioning will strongly influence seismic, fluid transport and rheological properties. In this paper, we will discuss the latter two. Numerous lines of evidence suggest that much of the transport of melt occurs in chemically isolated channels (Kelemen et al., 1997). Thus, large-scale melt transport must be preceded by segregation into channels. When melt segregates, viscosity becomes heterogeneous because it is highly melt fraction-dependent. Strain then partitions between weak and strong regions, modifying the stress field. The weak regions interact and self-organize into a connected network. Strain localization occurs at much smaller scales than are describable in current continuum geodynamic models of any of the regions in the Earth where they will be important. However, the effects of segregation on melt transport and on the effective viscosity of regions of the mantle and crust will likely be influential at the larger scales of these regional models (Buck and Su, 1989).
That melt segregation and channel formation can be driven by deviatoric stress was predicted near the end of the 20th century (Stevenson, 1989). An attempt at modeling the initiation of melt segregation and growth of melt-rich bands in simple shear has been performed by Spiegelman (2003), expanding the analysis of Stevenson (1989). Our study represents a very different approach. We do not attempt to solve for the growth rate of melt-rich perturbations; instead, we characterize the viscous energy dissipation in steady state as a function of the configuration of (melt-rich) weak zones. The analysis is written in terms of parameters that are measured directly from melt distributions in experimental samples, and thus should be amenable to comparison with experimental results.
Viscous energy dissipation, the energy produced by deformation of a fluid (or simply dissipation) is a quantity that connects fluid dynamics with thermodynamics (Mase, 1973). The concept appears in a range of contexts, including the energetics of plate tectonics and mantle convection (Froideveaux, 1973; Yuen and Schubert, 1977; Bercovici and Ricard, 2003), of mantle melting and melt/rock interactions (Asimow, 2002), and of self-organizing non-equilibrium systems (Nicolis and Prigogine, 1977; de Groot and Mazur, 1984). Dissipation is calculated to explore thermal weakening instabilities in deforming, temperature-sensitive materials with and without stress-dependent rheological properties (Yuen and Schubert, 1977; Kameyama et al., 1997). Here, we explore the effect of melt segregation and strain partitioning on dissipation in partially molten systems, with implications for a wide range of petrologic/geodynamic problems (Table 1).
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Summary of experimental studies
The observed melt distribution reflects the state of stress in the sample at the time of quenching, as shown in Fig. 1 and 2a. This statement is especially easy to justify for samples in which dissipation and effective viscosity reach a steady state, indicated by constant values of stress and strain rate for much of the duration of the deformation, as shown in Fig. 2b. Our aim is to relate the properties of this melt distribution with the measured rheological properties. A successful theory should explain the following observations:
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(1) The initial compaction length (of the system with unsegregated melt),
c, is an important characteristic length scale in the behavior of these systems, because it predicts the conditions under which melt segregation occurs and the spacing between largest bands, sp, is proportional to c in a range of melt-rock systems (Holtzman et al., 2003a). While the band spacing enters only implicitly, but not explicitly in this analysis (i.e. dissipation does not vary with band spacing), this observation is an essential part of the larger problem.
(2) When melt segregates, strain partitions between melt-rich networks and melt-poor lenses, strongly modifying the patterns of flow and orientations of the stress tensors in the sample, as revealed by measurements of lattice preferred orientations (LPO) (Holtzman et al., 2003b).
(3) Melt bands exist with a bimodal distribution of angles (0–30°) shown in Figs 1 and 2a. This pattern is common. In general, large bands are found at higher angles and narrow bands at lower angles. The thicker, higher angle population is rotated an average of 
20° from the shear plane regardless of finite shear strain (Holtzman et al., 2003a). We propose that this bimodal distribution is the signature of a connected network.
(4) The changes in effective viscosity in a sample, shown in Fig. 2b, begin with a gradual decrease and then reach a steady state at a shear strain of 
= 1. We hypothesize that the weakening is associated with the formation of a connected network of melt bands; steady state occurs when the kinetics of melt migration and reorganization are balanced by the kinetics of solid deformation during melt segregation.
The main purpose of this study is to understand what properties of the deforming system control the observed dissipation and enable the angular stability of a network to exist. To do so, we calculate the total viscous energy dissipation as the sum of the contributions from the band and non-band regions, in terms of parameters whose values are measured in our experimentally deformed samples, ab, 
b and 
. These equations offer an explanation for the observed preference of 20°. Even though the solutions below involve many simplifying assumptions, they provide useful insights and suggest directions for further study.
Viscous energy dissipation
The viscous energy dissipation is the contribution to the total energy made by irreversible deformational work per unit time and volume, that is, the power per unit volume. We base our analysis on the dissipation because the energies of separate parts of a system are additive, whereas the viscosities are generally not (except in specific cases). We relate changes in melt distribution to changes in the energy of the system as a way of quantifying the effects on the rheological behavior of the system with time. An expression for the first law of thermodynamics (Batchelor, 1967, p. 153, modified from equations 3.4.4 and 3.4.5, respectively, pulling 
out of the definition of dissipation), describing the total change in internal energy, E, with respect to time, t, in a fluid is
 | (1) |
 | (2) |
is thermal conductivity, T is temperature, 
eff is effective shear viscosity, and 
is the ith, jth component of the strain rate tensor. [Note that Batchelor includes the density in the dissipation function while other sources do not (Mase, 1975; Ranalli, 1987)]. The first term on the RHS of equation (1) describes the contribution of the volumetric part of the deformation (i.e. due to changes in pressure). At the high T and P conditions of our experiments, ‘dilatancy’ is only local, accommodated by decompaction in one region and compaction in another during melt segregation, both of which are forms of work, so they contribute to the measured dissipation. (In Batchelor's definition of energy dissipation, this volumetric work is removed. However, in the measured data, we cannot remove the effects of this irreversible loss of energy. In the following analysis, we focus on shear-dominated deformation and assume that the volumetric part is negligible). The third term on the RHS is the rate of heat gain/loss, which is negligible during experiments at constant temperature, so 
T/xi = 0. A closely related equation describes the entropy production
 | (3) |
is the entropy. In our experiments, the main contribution to the entropy production comes from 
.
Measuring the dissipation and effective viscosity as a function of strain in experiments provides insight into the changing rheological behavior of the sample as the melt distribution evolves. Taking the simple expression for the shear stress, 
, where 
is the shear strain rate, the dissipation can be written as
 | (4) |
and eff from experimental data, we use the expressions
 | (5) |
and eff. |
STEADY STATE VISCOUS ENERGY DISSIPATION: GENERAL DERIVATION |
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Here, we present the most general equations in this analysis, describing the mass balance and rheological properties. The approach to solving for the strain rates is developed in the next section. To characterize the segregation of melt and the partitioning of strain rate between the melt-rich bands and the melt-depleted lenses, we make two main simplifications:
(1) The system is in a steady state (or a ‘stationary state’, de Groot and Mazur, 1984) such that the average heterogeneous spatial distributions of melt and strain rate are constant in time, though, the local distribution of melt is constantly changing. We do not assume that is minimized; rather, we demonstrate that geometric conditions exist for which it is minimized.
(2)The bands and the lenses are ‘boundary layers’ characterized by laminar flow (Batchelor, 1967, section 5.7), as illustrated in Fig. 3. In this approximation, we assume that each laminar flow field can be characterized by a vector representing the average velocity. These two vectors have compatibility conditions, but there is no spatially discretized compatibility between the margins of these velocity fields. From this simplification the strain rates are derived below.
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As implied by equation (1), contributions to the total energy are additive,
. Equation (2) can be simplified by assuming that the dilatancy term is much smaller than the shear term:
 | (6) |
 | (7) |
Melt configuration and mass conservation
In this analysis, as with the experiments, melt fraction is constant and the whole system is closed. The equations are written in terms of average properties of the melt distribution. Melt ‘configuration’ refers to the ensemble of measurable qualities describing the melt distribution in a rock, experimental or natural. Melt-rich bands and melt-depleted lenses are defined as regions with higher and lower values of than the average in the sample, t, respectively. The area fraction, ab, and average melt fraction in the bands, b, serve as rough measures of the effectiveness of melt segregation and allow us to calculate average rheological properties of the two types of regions. The non-dimensional area fraction of bands and non-bands are simply
 | (8) |
 | (9) |
n is the melt fraction in the melt-depleted, lens or ‘non-band’ regions, and the o indicates the dimensional area. Normalizing by 
, we obtain
 | (10) |
t,
 | (11) |
b as the segregation factor, a measure of the degree of melt segregation, as
 | (12) |
 | (13) |
b = (1/ab) [1 – (1 – ab)n], which says that as the melt fraction in the lenses approaches zero, n 
0, max 1/amax. From here on, we drop the primes from the notation.
Here, we define another dimensionless parameter that describes the degree to which melt is segregated. The total segregation factor, St = bab, ranges from 0 to 1. At St = 1, no melt remains in the lenses. We can measure this value in experimentally deformed samples. St does not uniquely describe the evolution of the melt configuration, but we can define paths that describe various evolutions in terms of St. From experimental observations, we know that ab and b increase with increasing strain. A simple equation that fits observed paths of b is
 | (14) |
b, so
 | (15) |
Melt fraction-dependent viscosity
The strain rate of a melt-free rock may be described by a flow law of the form
 | (16) |
is the shear stress, d is the grain size, E* is the activation energy for creep, P is the pressure, V* is the activation volume and R is the gas constant. The effective viscosity (
) can be expressed as (assuming grain size is independent of stress),
 | (17) |
1 to >3.)
In this paper, we are most interested in the non-linear weakening effect of melt on the viscosity, often parameterized with the expression
 | (18) |
). The value of the factor 
is 25 in the olivine + MORB system (Mei et al., 2002; Zimmerman and Kohlstedt, 2004, referred to therein as ). The value for the olivine + chromite + MORB samples is undetermined, but we assume that it is similar. The melt-dependent viscosity can be parameterized in terms of b as
 | (19) |
n, using equation (13). In this study, we only address the changes in dissipation due to the changing melt fraction, leaving to a future study the effects of non-linear dependence of viscosity on stress. This relationship implies that 
. Although we leave the degree of strain partitioning as a variable in the system, this relation does enter the derivation below. We emphasize that the part of the analysis dealing with the rheological properties and constitutive relations governing weak zones can be modified for any system, to account for other causes of viscosity variations, for example, by grain size reduction or thermal weakening.
Energy conservation
Ignoring the other contributions to the total energy from equation (1), the statement of partitioned dissipation becomes the energy conservation statement. The definitions of the parameters in this problem are such that most can be directly measured from the experiments: ab, b, t, , and 
. The unknowns are the strain rates in the band and the non-band regions, but strong constraints may be placed on these values in the following section. In sum, equation (7) becomes
 | (20) |
are the strain rate tensors for the bands and non-bands rotated into the reference frame of the sample. The ‘*’ indicates that these equations are not essential to the solution and are only used when solved in terms of St. |
SOLVING FOR STRAIN RATES |
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The essential remaining piece of this problem is to calculate the strain rate tensors for the band and non-band regions. The general approach is to define the average velocities (using the boundary layer approximation), derive velocity gradient tensors and then decompose those velocity gradient tensors into strain rate tensors. This procedure is outlined in detail in Appendix A, and the results surface at the end of this section. The velocity gradient tensors for the bands and non-bands are defined in their own reference frames. To add them, the tensors are rotated into the sample reference frame, as shown in Fig. 3. While only the total strain rate component is known, the other components can be either neglected or solved for, using simplifications stated below.
To solve for the components of the strain rate tensors analytically, we derive relationships between the components of these tensors that allow the total number of unknowns to be reduced to three:
- We use a boundary layer approximation to define the average velocity vectors and the relationships between average velocity fields.
- We impose simple boundary conditions on the above relations and derive a relationship between the shear strain rate in the bands and that in the non-bands.
- We define a parameter that describes the degree of strain rate partitioning occurring in the system. This dimensionless parameter, pb, allows us to further reduce the number of unknowns in the system.
- Finally, we derive a relationship between the band angle and the non-band angle ß.
Boundary layer approximation
The boundary layer approximation, illustrated in Fig. 3, holds that if the velocity gradients normal to the boundary are much greater than the gradients parallel to the wall (and the shear direction), the flow can be described as laminar and a simple average velocity vector can be assigned (Batchelor, 1967, section 5.7). These average velocity vectors for each region sum to the total average velocity vector, vt, which is the basis for the vector diagram shown in Fig. 3. This rule specifies the interactions between the two velocity fields; there is no other coupling between these fields. In this boundary layer approximation, the stress and velocity compatibility conditions at the interface are not satisfied point-wise. Instead, we derive compatibility conditions for the average velocity fields, which is constrained by the divergence and curl of the average velocity fields. Mass conservation based on the boundary layer approximation, using the average velocity vectors for each boundary layer, yields
 | (21) |
b = abt; likewise, the density of the non-bands is n = (1 – ab)t (assuming that the variations in densities due to differing melt fractions are negligible). Dividing by t, yields
 | (22) |
Performing the following operations on this equation yields several constraints necessary for deriving relations between various components of the velocity gradient tensors.
 | (23) |
· vt = 0. This constraint is not used in the present analysis.
(2) The curl in the 1, 3 plane (the ‘flow plane’),
 | (24) |
Boundary conditions
The actual boundary conditions of the experiments are complicated; there is no change in total volume in the high-pressure experiments, but there is a small component of flattening and sidewards extrusion during shear (Holtzman et al., 2003b). Thus, in the following approximations, the first is straightforward, but the second is a simplification of reality:
(1) The divergence of the flow must be zero, since no mass is gained or lost in the system:
 | (25) |
 | (26) |
(2) The curl of the velocity field, equation (24), provides another relation between components of the velocity gradient tensors:
 | (27) |
 | (28) |
Strain partitioning
Up to this point in this analysis the equations are symmetric in the sense that the bands and lenses are treated equally. Here, we introduce an asymmetry, an equation that prescribes the amount of strain partitioned into the bands, and a ‘response’ propagates into the strain rates in the lenses. In these experiments, the shear strain rate, , is known, but the strain rates in the bands, 
, and lenses, 
, are not. Because these two unknown strain rates are coupled, we solve for in terms of . Here, we define a partitioning factor, pb, that relates the total shear strain rate to the shear strain rate in the bands. The definition of pb includes the constraint that the shear strain rate in a band must be maximum when the band is parallel to the shear plane ( = 0) and zero when it is parallel to the principle compressive stress, = 45° in simple shear:
 | (29) |
 | (30) |
, p. 354). The factor pb embodies the principal unknown in this formulation. We substitute equation (29) into equation (28) for b13 and rearrange
 | (31) |
The evolution of pb during melt segregation is one of the broader issues in this study. Because pb is defined in terms of components of the velocity gradient tensor, via the strain rates (see Appendix B), we can write pb in terms of the constitutive relation for melt-bearing rocks, equation (19):
 | (32) |
b = 1 + (max – 1)(St)c. In the solutions, we will compare the evolution of dissipation for paths of pb as a function of St with three different values of c.
There is a clear physical limitation to the value of pb. The vector diagram in Fig. 3c states that the velocity of any particle in the band (unweighted by ab) cannot be larger than the velocity applied to the walls of the sample, that is, |vb| 
|vt|. When this equality is solved, |vb| = hbb13 and |vt| = htt13. When plugged in to the inequality statement and the substitutions made for
 |
 | (33) |
|vt|. This limit is ensured in the calculation by only solving dissipation for values of for which the respective values of pb and ab satisfy these conditions.
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The relationship between
and ßAn interesting piece that falls out of these equations is the relationship between the angle of the band and non-band shear planes, or an expression for ß in terms of only
, pb and t13 (defined in Fig. 3). This relationship between and ß is derived in Appendix A and plotted in Fig. 5. There are several constraints on this relationship, which are also illustrated in Fig. 5:
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(1) Based on crystallographic fabrics, when bands form at
20°, the b-planes of olivine are rotated in the opposite sense away from the sample shear plane (i.e. the sample walls) by 20°. The assumption that the b-planes are roughly parallel to the local ‘non-band’ shear plane is discussed in Holtzman et al. (2003b). Thus, we infer that when 20°, then ß –20°.
(2) If the bands are parallel to the sample shear plane ( 0°), the shear plane in the melt-depleted lenses should also be parallel to the sample shear plane, so ß 0°.
(3) If the bands reached an angle of 45°, where they would be parallel to the maximum compressive stress, no shear stress is resolved on them such that they could only open in purely tensile mode. The shear in the non-band regions should then be parallel to the sample shear plane, such that when 45°, ß 0°.
The theoretical predictions plotted in Fig. 5 match these three constraints very well. As discussed in the Appendix, this relationship is not needed to solve the strain rates because, in this version, we use only the (second) strain rate invariant, which, by definition is a scalar quantity derived from a tensor that is independent of orientation. However, we include the derivation here because it lends physical insight to the equations and may be useful in future modifications to the analysis.
Synthesis
The following array of equations comprise the analytical solution for the components of the strain rate tensor. The first two, derived in Appendix A, give the scalar second invariant of the strain rate tensor in the band and non-band regions. The second three were derived above:
 | (34) |
Again, the ‘*’ indicates that this equation is only used when the equations are solved in terms of St and when pb is not fixed. In this set of equations the constants are , t, t13 and c. The dependent variables are b13(pb, t13, ), n13(pb, t13, ab, ), ß(), and from equations (20), b,n(St), ab(St) and b(St). The independent variables are then and St. We plot t as a function of , initially treating ab, b and pb as independent variables, and then all three as a function of St.
From the above equations we also calculate an approximate shear stress in the bands and lenses by using the following definition,
 | (35) |
b and n,
 | (36) |
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RESULTS |
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We solve equations (20) and (34) for total dissipation,
t. In general, t develops a minimum at the band angle at which the contributions to the total from the bands and lenses are equal, that is, where dissipated energy is equipartitioned between the two regions, as shown in Fig. 6. The location of this minimum depends on ab, b and pb. To develop these ideas we first illustrate the behavior of the equations by varying these parameters independently. Then, we present a more global picture of the behavior of the system as a function of the total segregation factor, St, which couples these parameters. We also demonstrate how the minimum in dissipation corresponds to the band angle at which shear stress is the same in the two regions.
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In Fig. 6, the separate contributions (weighted by ab) to
t from the bands and the lenses are plotted as functions of . The results are normalized by the reference dissipation for a material of the same total, but unsegregated, melt fraction, 0. The bands and lenses make inverse contributions to the total dissipation. The band contribution is highest at low angles when the bands are well oriented to take up the largest amount of slip and lenses need to deform relatively little. As increases, the contribution from the lenses increases because they have to deform more as the amount of deformation accommodated in the bands decreases. At the value of where these contributions are equal in magnitude (*), that is, where energy is equipartitioned, a minimum exists in the total dissipation, defined where (d/d)|pb = 0 and (d2/d2)|pb > 0. In the following section, we explore how variations in each of the variables describing melt distribution, ab and b, influence the position of the minima in dissipation.
Dissipation, varying ab and b
In each panel of Fig. 7, t is plotted as a function of with the partitioning factor pb as the varied parameter. As discussed above, a minimum value of t occurs at * for a given value of pb. For a given value of pb, with increasing , dissipation decreases until a minimum is reached. As pb values increase, these minima migrate towards higher values of *. Interestingly, for any fixed values of ab, b and , these minima all occur at the same value of t for all values of pb. This situation occurs because, in this set of solutions, pb is defined independently of ab and b. (In the 3D plots discussed below pb does depend on ab and b.)
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The six sets of
t – curves plotted in Fig. 7 explore the effects of varying ab and b. The left and right columns illustrate the effects of increasing ab when b = 2 and b = 3, respectively. In both columns, increasing the value of ab results in greater reduction in dissipation (and greater spread between maximum and minimum values of ) and a higher value of * at the minimum in dissipation for a given value of pb. In other words, pb can be larger because the bands are much weaker owing to a higher melt fraction. In the right column, the magnitude of the reduction in dissipation is much greater, i.e. 
50%, than in the left column. In Appendix D, we describe in detail the dependence of t on variations in ab, b and pb.
The evolution of dissipation with melt segregation
We can simplify the above results by coupling ab, b and pb to the melt segregation parameter St, which is described by equations marked with a ‘*’ in equations (20) and (34) [also (14), (15) and (32)]. In Fig. 8, various paths for the evolution of the ab, b and pb are plotted as functions of St, which are parameterized only by the exponent c. From here on, t is calculated as a function of St and , these producing a surface of t at different states. Figure 9a illustrates an example of one such surface for one set of paths with c = 0.5, as in Fig. 8. As St increases t gradually begins to decrease, and the tightness of the valley increases, i.e., the dependence of t on strengthens. Thus, with increasing melt segregation and strain partitioning, the system tends to exert a stronger preference for a particular value of *.
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In Fig. 10, top row, the 3D surface is reduced to 2D by contouring values of normalized dissipation. In these plots, as St increases along the x-axis, the contours tighten and the values of
t drop towards the minimum at *. The path of the minima with increasing St, shown by the gray dots, provides an easy means of visually comparing the effects of varying c. As c increases from 0.1 to 1, as shown in Fig. 8, the melt fraction in the bands is higher at any given value of St, consequently pb is correspondingly higher and ab is correspondingly lower. Therefore, t tend to be lower, and * tends to be higher, in agreement with the results shown in Fig. 7.
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) and the bottom row is the shear stress (Shear stress and the mean pressure difference,
b–nAs discussed above, we can calculate the shear stress,
, in the band and non-band regions, and b–n, to provide further insight into the meaning of the equations derived here. In Fig. 9b, in the bands and the non-bands is plotted as a function of for the same path of ab, b and pb as used in Fig. 9a. First, the shear stress in the bands is independent of St because pb depends on b such that shear strain rate increases proportionally to the decrease of viscosity in the bands as melt segregates. As increases, the shear stress in the bands decreases, because the bands can accommodate less and less strain as they approach 45°. The stress in the lenses is more complicated, and depends on ab and b in addition to pb.
For any value of St, the point where the two curves cross, that is, b–n = 0, corresponds to the band angle * at which t is minimized. In other words, the configuration at which dissipation is equipartitioned is also the set of conditions at which shear stress is uniform across the material, or [equation].
 | (37) |
with increasing St, the effective viscosity is decreasing (because the total strain rate is constant).
The fact that * corresponds to both the minima in t and the point where b–n = 0 has significant implications for the dynamics of the organization of the melt bands and the stability of the observed value of . In the Discussion and Appendix C, we will explore this condition further in a speculative way, while not knowing the relationship between b–n, the pressure difference in the solid 
, and the pressure difference in the melt, 
.
Summary
(1) As b and ab increase, min decreases; as pb increases, the angle at which min occurs, *, increases.
(2) As melt segregates (i.e. as St increases), the preference for a specific * becomes stronger (i.e. the potential wells become steeper). If b, ab and pb increase roughly as described in Fig. 8, then * should increase as melt segregates, thus approaching an angle similar to those observed in experiments.
(3) At *, both min occurs and b–n = 0. An implication is that the bands may organize around an angle at which shear stress is constant in the sample and dissipated energy is minimized.
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DISCUSSION |
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The above analysis is a partial description of the thermodynamics of melt segregation and strain partitioning, and is incomplete in several basic ways: (1) We look only at the steady state dissipation as a function of configurational parameters, but do not solve for growth or decay in these properties with time. (2) We do not incorporate length scales or statistics of melt distribution, only average properties of melt distribution. (3) Velocity fields are subject to very simplified constraints, in order to be solved analytically. (4) Viscosities are Newtonian. Nonetheless, the analysis captures some general trends in the changes of states of melt distribution and clues to the dynamics of the deforming system. It is intended to bridge experiments and fully dynamic numerical models and to provide a framework for analyzing and interpreting the experimental observations, written in terms of parameters that describe average properties (i.e.
b, ab, and ), which are directly measurable from experiments.
When dissipation is calculated as a function of the (coupled) degrees of melt segregation and strain partitioning, as well as the band angle, clear local and global minima develop. The implication is that, if a deforming system has the possibility of minimizing its energy dissipation or entropy production rate, it will tend in that direction. In the context of this analysis, variations in t can be viewed as small incremental movements between short-term steady states. The minimization at one value of St and transition to the next value of St occur by changing the melt distribution (band angle and melt content in bands) and strain partitioning such that the dissipated energy is equipartitioned between the bands and the lenses. The analysis predicts that the minima in dissipation can easily occur at the observed band angles. The main parameter that controls the value of at which min occurs (*) is the degree of strain partitioning, pb. However, in real (experimental and natural) systems, some other property of the system may influence the observed , such as the ratio of shear to normal stress. The values of pb may be closely coupled to spatial variations in dilation and compaction rates. In this analysis we do not explore such interplay.
We also demonstrate that the local minimum in dissipation corresponds to the equalization of stress and thus the elimination of stress gradients between the bands and the lenses. This finding is a more physically intuitive way of thinking about the stability of an average band angle in a constantly reorganizing melt-network; it leads to an answer to ‘how?’, whereas the energetics provides an answer to ‘why?’. Below, we will develop an idea for how the system constantly readjusts melt distribution around the optimal band angle, by a pumping mechanism driven by the departures from constant stress. Also, this tendency towards constant stress suggests that the ‘Reuss bound’ approximation of constant stress during strain partitioning is a good one, and will allow for the derivation of a much simpler set of equations that describe effective viscosity changes during melt segregation and strain partitioning.
The solutions provide several insights into the behavior of a deforming partially molten rock in which melt segregates and thus strain partitions. To illustrate these insights, first we describe this behavior in terms of the effect of strain partitioning on the total dissipation and begin to relate the conclusions to experimental observations. Second, we demonstrate simple inferences into the mechanics of melt segregation and self-organization of the networks of melt-rich bands. Finally, we discuss the implications of melt segregation and strain partitioning for the rheological properties of partially molten mantle and the irreversible thermodynamics of melt–rock interaction.
Relating theory and experimental observation: viscosity and dissipation
Magnitudes of dissipation change
In the data from the experiment shown in Fig. 2, viscosity is reduced by 30% as melt segregates and strain partitions, corresponding to a reduction in dissipation of 20%. For this sample, the value of St is 0.5. Thus, we have one point in the space described in Fig. 9, and it falls very close to the surface [at St = 0.5, = 20° and t/o = –0.1 = log(0.8)]. However, this result does not validate the analysis, but it is simply an illustration of how we will proceed in comparing the analysis to experimental data. Essentially, by measuring the parameters describing the melt distribution (ab and b, and thus St), , and the final dissipation value, we can fit the data by varying the dependence of pb on St.
The evolution of dissipation
The implication of the above paragraph is that the evolution of strain partitioning controls completely the evolution of dissipation. As shown in Fig. 2, within increasing strain, the dissipation (and the effective viscosity) decreases until about = 1, after which it flattens out. This flattening indicates that a steady state has been reached in the system, presumably in the statistical variations in the melt distribution and the strain partitioning. We suggest that the decrease in dissipation and effective viscosity is caused by an increase in strain partitioning, associated with increasing melt segregation. The onset of steady state may reflect the achievement of a balance between the processes of melt reorganization relative to the shearing solid and strain partitioning in the melt-rich network. As discussed below, the transition to steady state may coincide with an increased degree of connectedness of bands within the melt networks. When this connectivity increases, melt can redistribute and strain can partition more effectively than when bands are not connected; thus, the degree of connectedness and the potential for strain partitioning may be coupled.
Another aspect of the experiments not described in the analysis is the non-linear dependence of viscosity on stress. The degree of dislocation creep depends on stress and melt fraction (to the extent that melt fraction influences the effectiveness of grain boundary diffusion creep and also influences the grain size). As melt segregates, the stress magnitudes in the lenses will increase. Thus, in the right regime, the activity of dislocations may increase with melt segregation, causing weakening in the lenses. This weakening will couple to the deformation conditions in the band networks. These aspects can be quantified and accounted for in future versions of this analysis.
The melt distribution
In Fig. 1, the widest bands traverse the sample at higher angle relative to the shear plane than the thinner, more numerous bands. In the bimodal distribution of band angles, as shown in Figs 2b and 11, these wider bands constitute the smaller population of bands at higher angles. The narrow bands at lower angles connect the wider bands. We suggest that this pattern of band angle and thickness distribution is caused by the same set of processes occurring at a smaller scale; because of the strain partitioning, the stress field and thus the shear plane are back-rotated relative to the sense of shear by an angle of ß, as illustrated in Fig. 11 and discussed in Holtzman et al. (2003b). In this view, the ‘lower angles’ actually have the same angle relative to their local shear plane as the thicker, higher angle bands do to the sample walls. Thus, the mechanical preference for a 15–20° angle propagates downward in scale. This hypothesis explains the bimodal distribution of band angles. The lower angle bands tend to be narrower than high angle bands because they are forming in a region with lower background melt fraction. Another part of this story of the distribution of band angles involves rotation and growth of bands during shear, which is discussed below.
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Mechanics of network formation and stability
Calculations of the dissipation do not reveal the mechanical process by which the system achieves any given state. However, when we look at the variations in stress between these two regions, as in Figs 9, 10 and 12, this simple picture of the variations in stress distribution with
explains qualitatively many aspects of the behavior of the networks during deformation. In the following, we discuss why bands nucleate, how networks form and maintain a dynamic steady state, and how they evolve. We want to understand how the melt moves through the sample, how these patterns of movement couple to the effective viscosity of the system, and ultimately how to extrapolate these dynamics to conditions in the Earth.
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Why and how do bands nucleate and grow?
The notion of stress-driven melt segregation was first proposed by Stevenson (1989)
. In a 1D model of a deforming partially molten material, small perturbations in melt fraction cause small variations in viscosity because of the dependence of viscosity on melt fraction. In the limit where strain rate is spatially invariant, these low-viscosity regions are also low-pressure regions, so more melt tends to flow into them, leading to an instability, spontaneous melt segregation and band formation on length scales less than the compaction length, c. Richardson (1998) extended the problem to 2D pure shear. None of these studies have been able to predict a characteristic spacing, other than that it will be less than a compaction length. Hall and Parmentier (2000) attempted to define a constraint on the band spacing by incorporating the effect of water removal during melting on increasing the matrix viscosity.
Rabinowicz and Vigneresse (2004) and Speigelman (2003) extended Stevenson's analysis to a simple shear boundary condition, in 1D and 2D, respectively. In a linear stability analysis, Spiegelman found that the melt segregation instabilities (bands) form at a range of angles and rotate with the shear flow and those at = 45° grow fastest. In the linear analysis with Newtonian melt fraction-dependent viscosity, shear perturbations on the bands do not reduce rotation rate because they are not coupled to the background simple shear flow. Both Holtzman et al. (2003a) and Spiegelman (2003) suggested that the observed band angle is a compromise between the fastest growing orientation (45°) and the orientation of maximum shear (0°). In this analysis, consideration of strain partitioning and not growth rate leads us to a different, but not necessarily exclusive, conclusion.
Here, we directly couple the flows in the band and non-band regions, but are limited to assuming a steady state; however, we can speculate on the conditions that cause the initial segregation of melt, using the illustration in Fig. 12. In the plot of as a function of , to the right of *, at which b–n = 0 (i.e. the crossover of the () curves for the band and the non-band), melt flows down the pressure gradient, from non-band to band regions. This curve implies that, as bands rotate with increasing shear strain to higher angles, melt flows from bands at low angle to ones at high angle, consistent with the (instantaneous) observed correlation between band angle and thickness, and with the results of Speigelman (2003). The value of * depends on St and pb, so bands should be observed at angles equal to or less than *, where the condition for the growth of bands is met. However, they tend to rotate to higher angles, up to an observed limit of 30° not 45°, as discussed below. An attempt to explain this upper limit is made below. The existence of bands at lower angles than * may be explained by the strain partitioning and multiple scales of segregation, as discussed above and in Fig. 11.
How is a steady state maintained?
As discussed in Holtzman et al. (2003a), in order to maintain a steady state and a constant average band angle, melt must constantly move relative to the solid. The following hypothesis is based on the analysis, but is an inference based more on the observations than the calculations. We propose that the steady state average band angle is maintained by the following ‘pumping’ cycle, illustrated in Fig. 12:
(1) Bands nucleate by the mechanism discussed above at some low value of .
(2) As a band rotates with the overall shear flow, it grows (i.e. melt fraction and thickness increase) up to a limit, which is observed at 30°. The orange arrow in Fig. 12a shows melt flowing from a band at low angle to one at slightly higher angle. In Fig. 12b, this flow is represented by the orange arrow that tracks the decreasing shear stress in the band with increasing band angle.
(3) An upper limit of the band angle (30°) is reached at which melt is driven back to lower angle bands, as shown by the red arrows in Fig. 12a and b. The upper limit of band angles exists because the system will only tolerate a certain shear or mean stress difference between the bands and non-band regions. As the bands rotate to higher angle, the lenses are forced to deform at higher rates, increasing the stress in them, as plotted in Fig 12b. As they deform more, they squeeze the bands, increasing the stress in the solid framework in a band, thus increasing the melt pressure in the bands. This increased melt pressure drives melt from high to low angle bands. The red arrow in Fig. 12b moves from the shear stress in the band to that in the lens, representing the local increase in stress as a band at high angle starts to be squeezed between two lenses. This change in stress does not fall out of the analysis because we only consider the simple shear components of deformation. Thus, this idea is testable in theory by calculating the flattening components of the solid deformation in the lenses, and coupling these modes to the stress in the band. The key to the pumping mechanism is that the slope of the shear stress gradient with band angle switches from negative (orange line) to positive (red).
This pumping cycle drives a net melt flow relative to the solid through the network of melt-rich bands, such that the average band angle remains constant as the sample deforms in shear. This process relies on the existence of a highly connected network, as observed in experiments (Figs 1 and 2) and illustrated in Fig. 11.
Another mechanism and pathway by which melt migrates relative to the solid may exist, as proposed by Holtzman et al. (2003a). In this mechanism referred to as ‘wave migration’, the melt in a band migrates against the solid flow by doing work against the solid to open porosity on the ‘upstream’ side and close porosity on the ‘downstream’ side, moving as a wave of porosity through the solid matrix. This process does not require the presence of a network; every band struggles on its own.
The variations in the evolution of with increasing strain may represent variations in the dominant mechanism by which bands are moving relative to the solid. We propose that the energetics of the two band migration mechanisms actually differ, although this point is not considered in the analysis. The wave migration mechanism may both consume more energy (i.e. make eff higher) in the opening and closing of melt pockets and partition strain less effectively than a network can. The network migration requires longer distances of melt travel than wave migration, but these pathways have much higher permeability than non-band regions. Both mechanisms may occur at different locations in the network at any given moment and/or time. If the energetics of the two mechanisms are significantly different, the initial increase in effective viscosity in some experiments (not shown) may indicate a dominance of the wave migration mechanism over the network migration mechanism at the start of an experiment, before the connected network develops. If so, the decrease in eff with increasing strain results from a subsequent increase in the effectiveness of the mechanism of network migration.
Implications for the Earth
Increasingly, we are able to quantitatively study partially molten regions in the Earth as complete dynamic and thermodynamic system. In this view, the formation and persistence of melt-rich networks will characterize the close coupling between dynamics of mantle flow and thermodynamics of melting. However, for now, we separate the discussion into implications for rheological properties and for melting thermodynamics.
Rheological properties
When strain partitioning in networks occurs, the macroscopic effective rheological properties reflect mesoscopic variations in viscosity as well as the microscopic deformation mechanisms that determine those viscosities. Thus, to determine these effective rheological properties (i.e. eff) constitutive equations must be written in terms of the geometric and rheologic properties of the various regions combined in a physically meaningful way. Most simply, 
, where is calculated with equations (20) and (44). Although we made some progress towards this aim, this approach is probably limited. Since the assumption of constant stress appears to be a good one, deriving an effective viscosity from a Reuss-bound calculation will be useful.
As discussed above, in the tightly confined conditions of our experiments the reduction of dissipation (or effective viscosity) is less than an order of magnitude. The effective viscosity for reasonable earth-like values of many of the parameters for a segregated system (1.8 x 1018 Pa.s) gives a 50% reduction of the melt-present viscosity (3.7 x 1018 Pa.s for t = 0.04), that is, an almost order-of-magnitude reduction of the melt-free viscosity (1 x 1019 Pa.s). However, in natural systems with less rigid boundaries (and boundary conditions) than in the experiments, systems may have more degrees of freedom to organize into configurations that allow greater reductions in viscosity.
Thermodynamics of melting and melt–rock reaction
The formation of melt-rich networks and the onset of strain partitioning have interesting implications for the thermodynamics of melt formation and extraction processes. Both deformation due to mantle flow and melt transport in networks can influence the entropy budget, as stated by the relation between energy and entropy in equation (3). The very fast and effective segregation and transport of melt provided by the network will lead to both positive and negative contributions to the energetics of melt–rock reaction. Asimow (2002) expanded the energy equation of McKenzie (1984, equation A37) to include the irreversible terms and the energy contributions of melt–solid disequilibrium in addition to the enthalpy of isentropic decompression. In his scaling analysis, Asimow compares the energies of all irreversible terms to the energy available to drive melting due to adiabatic (isentropic) decompression, or the enthalpy H. This value is WCp(T/z) 3 x 10–6 W/m3 where is the density (3.3 x 103 kg/m3), W is the upwelling velocity (10–1 m/year), Cp is the isobaric heat capacity [1000 J/(K.kg)] and (T/z) is the geotherm (0.3 K/km). He isolates these irreversible contributions as the (1) thermal conduction and radiogenic heat production; (2) viscous dissipation in the fluid due to buoyancy driven compaction (including gravitational energy release and frictional heating); (3) viscous dissipation in the matrix as a result of compaction; (4) advection of heat by migrating melt; and (5) compositional disequilibrium between solid and melt. In the following discussion, we address the influence of the formation of melt-rich connected networks on several of these contributions. For the sake of discussion, we assume that stress-driven networks form deep in the melting column, and other melt transport/segregation mechanisms occur subsequently. To explore that assumption, a greater understanding of the relative kinetics of stress-driven and reaction-driven segregation is required, as discussed further below.
Deformation and dissipation
Asimow's analysis examines viscous dissipation in the migrating melt and solid due to compaction. He concludes that both of these terms are insignificant. However, the contribution to the total dissipation may be much greater from the shear deformation than from compaction. Because his analysis is 1D, Asimow does not include deformation due to corner flow. As shown in Fig. 13, the dissipation has a wide range of values for reasonable mantle strain rates and effective viscosities beneath a mid-ocean ridge; these values broadly straddle isentropic decompression value for power available to drive melting, implying that, in some locations, the heat produced by dissipation may contribute to melting.
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As illustrated in Fig. 14, for perfectly passive upwelling, the stress, strain rate and thus the dissipation will be zero in the center and will increase towards the flanks of the melting region, with maximum values in the region of tightest corner flow for the case of a Newtonian viscous flow. Dissipation will be highest where stresses are highest in a slab-pull-driven spreading center, where a constant global strain rate is imposed. If equipartitioning and minimization of dissipation are achieved, then the entropy production would be reduced relative to that of a homogeneously deforming system. However, this reduction may still leave the dissipation large enough to contribute considerably to the total energy of the system available to drive melting. For example, for
= 1 x 1019 Pa.s and 
, = 2.5 x 10–6 Pa/s, or 83% of H due to upwelling. A 50% reduction in due to melt segregation would cause a 50% reduction in or 42% of H.
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Melting and melt–rock reaction
The effects of stress-driven segregation on the transport properties of a partially molten region may tend to increase the disequilibrium and the irreversible contributions to the total energy. Continuing with the mid-ocean ridge example, we offer the hypothesis that melting may be more ‘batch’-like in the center of the melting column where stresses are lower and stress-driven segregation is less effective, and more ‘fractional’ at the flanks of the melting region, where stress drives melt segregation more efficiently. When melt segregates easily at small melt fractions, the melting is more fractional than batch-like in nature. Fractional melting generally leads to lower melt productivity than a batch melting model in which the melt has time to equilibrate with its source (Asimow et al., 1997
). The degree of fractional melting will depend on the relative kinetics of the stress-driven and reaction-driven mechanisms. Estimating these kinetics is beyond the scope of this paper. As illustrated in Fig. 14, the compositional lithosphere (dark green) forms where enough water, Fe and melt have been extracted that the remaining rock is significantly stronger than adjacent asthenosphere material (Hirth and Kohlstedt, 1996; Phipps Morgan, 1997). In this idea, much of the upwelling peridotite will pass through a region of more batch-like melting followed by more fractional melting in regions where segregation is driven by stress. Both Kelemen et al. (1997) and Asimow (1999) concluded that most parcels of melting peridotite have to undergo part of their melting history in a batch melting mode and part in a fractional melting mode, but placed no constraints on the spatial or temporal sequence. We would predict that the fractional melting follows an initial period of batch melting.
In more detail, we also suggest further spatial variations in the melting thermodynamics. Fast and efficient melt segregation by stress-driven network formation in the deep parts of the melting region will lead to greater melt–rock disequilibrium at shallower levels in the melting region. A highly connected network allows fast melt transport to advect heat upward and to cause greater differences in chemical potentials between melt and solid, thus contributing more energy to the melt–rock reactions at shallower levels in the melting region. These reactions lead to an instability called the ‘reaction infiltration instability’ (RII) (Aharonov et al., 1995; Daines and Kohlstedt, 1993). This mechanism can cause efficient melt segregation, channel formation and transport, but requires initial melt–rock disequilibrium and subsequent dissolution in order to occur. As melt is quickly segregated and chemically isolated from its source rock, the degree of melt–rock disequilibrium increases as the melt ascends, thus increasing the efficiency of the RII as a mechanism of channel formation. The greater the degree of disequilibrium between melt and peridotite, the more rapid will be the dissolution reaction kinetics that produce dunite from peridotite. Thus, stress-driven and dissolution driven melt segregation mechanisms will probably interact closely in the mantle. In this scenario, the RII may be more efficient at shallower levels in the melting column and may act to stabilize channels formed by stress-driven melt segregation.
In conclusion, there may be large spatial variability in the controls on the thermodynamics of melting beneath a ridge and other melting, deforming regions in the Earth. Where stresses are low, melting may be more batch-like, and thus productivity will be higher. As stresses increase, stress-driven segregation causes more fractional melting, implying lower productivity. However, in these regions, both the dissipation due to deformation and the increasing thermal and chemical disequilibrium may lead to increased productivity. Thus, it is possible that these two opposite trends will tend to balance each other such that an isentropic approximation would be reasonable. This conjecture can be explored quantitatively in further studies.
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CONCLUSIONS AND FURTHER QUESTIONS |
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Although simple, this analysis provides insight into a possible mechanism by which networks of melt-rich bands form and maintain an average configuration in a dynamic steady state, by achieving an equipartition of energy dissipation between melt-rich and melt-poor regions. The values of energy dissipation in the partitioned system are very sensitive to several parameters that characterize the melt distribution, and these predictions can be compared quantitatively with experimental observations. Numerous variations could be added to this analysis, but we present it here in its skeletal form. The success it has when compared with experimental data will determine which modifications are made. However, several possible directions are worth pointing out for their value in describing unanswered questions.
(1) Compaction and decompaction. In the analysis above, we assume that the simple shear velocity gradient contributes far more to the total viscous energy dissipation than the work done in accommodating local changes in melt fraction (i.e. compaction and decompaction). Though the paths along which melt flows are not solved explicitly (i.e. the compaction equations are not solved), the volumetric change may be accounted for in the energy lost in changing the melt fraction, as the bands migrate relative to the solid. To maintain stability, the bands must constantly be adjusting. To adjust, they must be able to do work against the solid matrix. If we can estimate the driving force for local melt migration, then we can constrain the ability of the solid to resist decompaction, that is, the bulk viscosity. Including these terms may modify the preference for an angle associated with a certain degree of strain partitioning.
(2) Length scales and statistical distribution of melt. At present, the analysis is written in very simple average boundary layer approximations with no coupling on interfaces between the boundary layers. However, deforming systems with identical average properties may have different local distributions of melt. Will the energy dissipation of a system with one band of area fraction a equal that of a system with 10 bands of 0.1a? If not (as seems likely), we need to incorporate the spacings between bands and mechanical interactions between band and non-band regions at their interfaces.
(3) The need for non-linear dynamic models. A complete continuum mechanical description of a melt-segregation process requires coupled mass, momentum, and energy conservation equations for two fluids with very different rheological properties (McKenzie, 1984; Scott and Stevenson, 1984; Fowler, 1990; Bercovici et al., 2001). The viscoelastic properties of the matrix are generally, though not always (Connolly and Podladchikov, 1998) ignored, but may be important. Non-linear solutions arise because of the dependence of both permeability and viscosity on melt fraction, as well as the effects of stress-dependent viscosity. We must seek the simplest theory that can describe the full richness of behavior exhibited in experiments. Fully dynamic models will be an essential part of extrapolating the experimental results to natural settings.
Ultimately, we aim to be able to map St onto partially molten regions of the Earth, in spatial and temporal variations. Knowing the degree of melt segregation and all the related parameters, we can constrain the effects of meso-scale self-organization of melt on the rheological, seismic and transport properties of these regions. By calculating seismic properties (anisotropy and attenuation) directly to St, direct comparisons of model and observation may be possible.
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APPENDIX A: VELOCITY GRADIENT TENSORS |
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Here, we solve for the velocity gradient and strain rate tensors in the bands and non-bands by using the reference frames shown in Fig. 3. In the experimentally deformed samples, the deformation is 3D (Holtzman et al., 2003b
), and thus there are no null components of the strain rate tensor. However, for simplicity as discussed above, we assume that the dominant components of the tensor are the 13 components (i.e. b31 = n31 = 0), simplifying the problem to 2D. We further reduce these tensors to only their 13 components, prior to rotation, based on the assumption that the local shear (13) components are much greater than any dilatational (ii) components. Therefore,
 | (38) |
v1/x3)b, n13 = (v1/x3)n and t13 = (v1/x3)t. The band and non-band velocity gradient tensors are then rotated into the sample (t) reference frame, using the rotation tensor
 | (39) |
= or ß. [In 3D, this rotation occurs about the two-axis, normal to the shear direction (1), in the shear plane (1–2)]. The rotated matrix is calculated as, for example,
 | (40) |
is >0 and the non-band angle ß is <0, as counterclockwise angles are positive. The velocity gradient tensor must first be rotated into the total (sample) reference frame and then decomposed into the strain rate form. (These operations are not dependent on the order in which they are performed.) The following derivation is for the strain rate in the bands; the exact parallel method applies for the non-bands.
The solution for the velocity gradient tensor in the band, rotated into the sample reference frame, is
 | (41) |
vi/xj) + (vj/xi)] and a rotational part, the vorticity, [(vi/xj) – (vj/xi)] (Mase, 1970, p. 112), such that
 | (42) |
 | (43) |
, which means that all the elements of the tensor are squared and summed (Ranalli, 1995, equation 4.20). Thus, after much algebra, the invariants reduce to (as they should be independent of orientation),
 | (44) |
 | (45) |
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APPENDIX B: THE RELATIONSHIP BETWEEN AND ß
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The following section describes the derivation of the relationship between
and ß that is essential for solving the strain rates and thus the dissipation calculation. All constraints on the relationships between ab, pb, and the velocities and strain rates are derived from the vector diagram drawn in Fig. 3. We seek a relationship between and ß in terms of the parameters ab and pb. Using the trigonometric relations drawn in Fig. 3b,
 | (46) |
 | (47) |
 | (48) |
 | (49) |
 | (50) |
 |
 | (51) |
), we get the workable version
 | (52) |
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APPENDIX C: THE RELATIONSHIP BETWEEN SHEAR STRESS AND MELT PRESSURE |
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The shear stress in the bands and lenses can be calculated from the relative strain rates and viscosities in the band and lenses. The values of these stresses provide perhaps a more physically intuitive view of the behavior of these equations than the dissipation does, and provide an avenue into further questions and problems of the dynamics of melt segregation. The solid pressure influences the fluid pressure and melt flows directly in response to gradients in the fluid pressure. Here, we outline a few simple relationships between the shear stress and the local melt and solid pressures. The aim is to motivate the questions of what pressure gradients may exist in the deforming systems that allow the melt to organize and continually readjust to maintain a constant average angle and a steady state.
As illustrated with a Mohr circle in Fig. A1, we define the various stress and pressure terms that we discuss below. Within the band and non-band regions, we define solid and fluid pressures, 
, 
, 
and 
, respectively, where the overline indicates that the pressure is the mean stress, or
 | (53) |
33 is equal to the confining pressure, Pc, and 22 33, then
 | (54) |
is the shear stress. Since Pc is constant in the sample, 
, where the shear stress difference is what we calculate in this analysis. We use the invariant of the deviatoric strain rate tensor as an approximation of the shear strain rate, such that 
and 
.
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So the remaining question is how this shear stress or mean stress relates to the local fluid pressure? The fluid pressure is not defined as a mean stress because the shear stress in the fluid will be negligible (Fowler, 1990
). Ultimately, when we calculate , we must relate it to 
in order to understand how deformation will affect melt flow. However, the relationship between these two differentials is not clear. Within one element of a two-phase continuum, there exists a solid and a fluid pressure, for example, and . These two pressures are related by
 | (55) |
· vs is the compaction rate and 
is the bulk viscosity, which is probably strongly dependent on the melt fraction (McKenzie, 1984; Scott and Stevenson, 1986; Fowler, 1990). The form of this dependence is often assumed to be 
–1. If 
, the melt pockets dilate, and if 
, they compact. Thus, if we knew the compaction rate and the bulk viscosity, we could determine a local melt pressure from this equation. However, the pressure gradient depends on the gradient of the shear strain rate and on the compaction/decompaction rate (Spiegelman, 2003, equation 4). Because we are only looking at the differences in the contribution of shear deformation, in the present analysis, we cannot calculate this relationship, but only speculate in the Discussion. For the purpose of the analysis presented here, we make the simplest assumption that spatial gradients in fluid pressure will have the same sign as those in solid pressure in the same location.
Another factor affecting the fluid pressure is the surface tension, which relates the Pf to the curvature of melt-pocket walls (Stevenson, 1986; Cooper, 1990). When fluids tend to wet the grain boundaries (i.e. 2sf > ss), surface tension tends to flatten perturbations in melt distribution and resist melt segregation. These effects are also not considered in this analysis.
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APPENDIX D: SENSITIVITY TESTS |
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We systematically vary ab,
b and pb to explore the influence of a wide range of parameters on the values of the minima in dissipation and their dependence on band angle. The results, illustrated in Fig. A2, are plotted according to the following scheme. In the top row (Fig. A2a and b), the effects of varying b are shown in two ways. In the bottom row (Fig. A2c and d), the effects of varying ab are shown. In the left column (Fig. A2a and c), we plot the values of associated with each minimum in dissipation, as a function of pb for different values of b (top) and for a (bottom). In the right column (Fig. A2b and d), we plot the normalized values of the dissipation where the minima occur, as functions of b (top) and ab (bottom), exploiting the observations above that the value of dissipation at the minima does not depend on or pb, but only on ab and b. The figures in the right column show the most basic and intuitive results: (1) the more that melt segregates into the bands (b), the lower the energy dissipation; and (2) the larger the volume fraction of bands (ab), the lower the energy dissipation. In other words, melt segregation and strain partitioning are energetically favorable; the more the melt segregates, the lower the energy becomes and the weaker the rock becomes.
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Varying
b. In Fig. A2a, we plot the influence of the partitioning factor pb on the band angle at which is minimized, *. The general trend of all slopes indicate that, as pb increases, * increases, as seen in Fig. 7, for a fixed value of ab and b. As b increases, a higher value of pb is required to reach a minimum in dissipation at a given angle. In other words, at a given value of , the weaker the bands are relative to the lenses, the more strain must concentrate in the bands to minimize the dissipation. As shown in Fig. A2b, as strain rate in the bands increases relative to the total, the absolute value of the dissipation drops quickly and almost linearly. The deviation from linearity comes from the fact that b appears in the exponential term in the viscosity– relation, equation (19). For ab = 0.2, a twofold increase in b leads to a twofold decrease in dissipation.
Varying ab. When ab is varied (Fig. A2c and d), the curves of * vs pb have a similar curvature as those when b is varied. However, increasing ab has the inverse effect on * than increasing b. At a constant value of pb, increasing ab causes an increase in *; or at a constant *, a decrease in ab corresponds to an increase in pb. To clarify the latter, the more the strain rate is concentrated in the bands, a smaller volume of (or fewer) bands is (are) needed to attain a minimum dissipation level at a certain angle. In Fig. A2d, min decreases linearly with increasing ab, because ab is simply a weighting factor on the dissipation values for the bands and lenses in equation (20) and for pb in equation (34d and e). The total amount of dissipation reduction is much more sensitive to b than to ab. Finally, the effect of varying (not shown) on the relation of to pb is very similar to that caused by the variation of b, because an increase in both essentially weakens the bands relative to the solid.
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ACKNOWLEDGEMENTS |
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We would like to thank numerous people for discussions that contributed to this paper, including Reid Cooper, Saswata Hier Majumder, Marc Hirschmann and members of the Kohlstedt group. Yasuko Takei's very thoughtful critique greatly improved this analysis. After much scrituny and agony, she finally agreed to let me (B.K.H.) continue with this paper. The authors would also like to thank Julian Mecklenburgh, Jean Louis Vigneresse and Dave Stevenson for their thorough reviews. The research was supported by NSF grants OCE-0327143 and INT-0123224 to D.L.K. and a Fulbright Fellowship to France for B.K.H.
* Corresponding author. Present address: Lamont Doherty Earth Observatory, Columbia University, Palisades, NY, USA. Fax: 845 365 8150. E-mail: benh{at}ldeo.columbia.edu
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