In source regions of magmatic systems the temperature is above solidus, and melt ascent is assumed to occur predominantly by two-phase flow, which includes a fluid phase (melt) and a porous deformable matrix. Since McKenzie (1984) introduced equations for two-phase flow, numerous solutions have been studied, one of which predicts the emergence of solitary porosity waves. By now most analytical and numerical solutions for these waves used strongly simplified models for the shear- and bulk viscosity of the matrix, significantly overestimating the viscosity or completely neglecting the porosity dependence of the bulk viscosity. Schmeling et al. (2012) suggested viscosity laws in which the viscosity decreases very rapidly for small melt fractions. They are incorporated into a 2-D finite difference mantle convection code with two-phase flow (FDCON) to study the ascent of solitary porosity waves. The models show that, starting with a Gaussian-shaped wave, they rapidly evolve into a solitary wave with similar shape and a certain amplitude. Despite the strongly weaker rheologies compared to previous viscosity laws, the effects on dispersion curves and wave shape are only moderate as long as the background porosity is fairly small. The models are still in good agreement with semi-analytic solutions which neglect the shear stress term in the melt segregation equation. However, for higher background porosities and wave amplitudes associated with a viscosity decrease of 50 % or more, the phase velocity and the width of the waves are significantly decreased. Our models show that melt ascent by solitary waves is still a viable mechanism even for more realistic matrix viscosities.
Magmatic phenomena such as volcanic eruptions on the earth's surface show,
among others, that melt is able to ascend from partially molten regions in
the earth's mantle. The melt initially segregates through the partially
molten source region and then ascends through the unmolten lithosphere until
it eventually reaches the surface. Within supersolidus source regions at low
melt fractions, melt is assumed to slowly percolate by two-phase porous flow
within a deforming matrix (McKenzie, 1984; Schmeling, 2000; Bercovici et
al., 2001), followed by melt accumulation within rising high-porosity waves
(Scott and Stevenson, 1984; Spiegelman, 1993; Wiggins and Spiegelman, 1995;
Richard et al., 2012) or focusing into channels which can possibly penetrate
into subsolidus regions. Stevenson (1989) carried out a linear stability
analysis and found conditions at which flow instabilities may arise, which
may result in different 3-D shapes like elongated pockets, channels or
porosity waves (Richardson, 1998; Wiggins and Spiegelman, 1995). Formation
of 3-D channels within a deforming matrix has been demonstrated in Omlin et
al. (2018) or Räss et al. (2014). Here we focus on the supersolidus
source region and in particular on the dynamics of porosity waves. An
essential parameter controlling the width and phase velocity of porosity
waves is the effective shear and bulk matrix viscosity (Simpson and
Spiegelman, 2011; Richard et al., 2012). Most of the porosity wave model
approaches used either equal bulk and shear viscosities or simple laws in
the form of
The mathematical formulation of differential movement between solid matrix
and melt basically builds on that described in Schmeling (2000) and
Schmeling et al. (2019) and is applied here to a porosity wave. We solve the
equations for mass and momentum conservation for melt and solid. The
formulation of the governing equations for the melt-in-solid two-phase flow
dynamics is based on McKenzie (1984), Spiegelman and McKenzie (1987) and
Schmeling (2000), and it is valid for infinite Prandtl number (i.e., neglecting
inertia terms in the momentum equations), and small fluid-to-matrix-viscosity ratios. In the following all variables associated with the fluid
(melt) have the subscript
The fluid pressure in Eqs. (5) and (6) is the same and can be eliminated
by merging the two equations. Inserting the density of the mixture, and
using Eq. (7), Eq. (5) is recast into
Following Šrámek et al. (2007) the matrix velocity,
Taking the curl of the matrix momentum (Eq. 6) eliminates the pressure.
Inserting the viscous stress tensor (Eq. 8), the density (Eq. 9) and the
matrix velocity (Eq. 11) into the resulting equation gives the momentum
equation in terms of the stream function
The effective viscosity laws proposed by Schmeling et al. (2012) assume
ellipsoidal melt inclusions, or melt films if the inclusions are flat, or
melt tubules embedded within an effective viscous medium. This
self-consistent assumption is able to predict viscous weakening of a solid
matrix with a disaggregation melt porosity on the order of 50 % or less
depending on the assumed melt geometry. From their numerical models,
Schmeling et al. (2012) derive approximate formulas for the porosity
dependence of the dimensional effective matrix shear and bulk viscosities
for a melt network geometry consisting of 100 % films,
For a melt network consisting of 50 % tubes and 50 % films, the following
approximate equations have been derived from the model of Schmeling et al. (2012):
Parameters to calculate the viscosities for a melt network consisting of 50 % tubes and 50 % films using Eqs. (24) and (25).
Figure 1 shows the effective shear and bulk viscosities for different aspect ratios together with the simplified previous laws (1) and (2).
Shear (solid) and bulk (dashed) viscosity for several aspect
ratios as a function of the melt fraction.
Takei and Holtzman (2009) and Rudge (2018) suggest that in the presence of
an infinitesimal amount of connected melt the effective viscosity undergoes
a finite drop on the order of a few tens of percents of the intrinsic matrix
viscosity. In our approach we always have a finite melt porosity, and thus we
may identify the zero porosity viscosity
For the model we use a square box (
At the sides of the symmetric box boundaries and at the top and the bottom, free slip boundaries are used. The in- and outflow velocities of matrix and melt at the top and bottom are prescribed in terms of the analytical solution for the background porosity.
The influence of the boundaries on the ascending wave was investigated and found to be fairly small. In Fig. 3 one can see the effect of the upper boundary on the phase velocity. At the end, as the waves approach the upper boundary, the dispersion curves slightly deviate from the supposed line. This error is smaller than 0.5 % as long as the distance from the center of the wave to the upper boundary is greater than 1.5 times its 10 % radius. This radius is defined as the radius at which the porosity has decreased to 10 % of the amplitude of the wave. For the side boundaries this distance has to be larger. For distances greater than 3 times the 10 % radius, this error is smaller than 1 %. In our models the waves have distances of 7–10 times the 10 % radius, which correspond to errors between 0.2 % and 0.05 %.
The equations are solved on a
The amplitude and phase velocity of the evolving porosity wave are obtained at every time step by quadratic interpolation of the porosity values on the FD grid and determining the value and velocity of the position of the maximum of the quadratic function. The resulting phase velocity shows small oscillations in time, which are probably due to the interaction of the 1st-order error in time when solving Eqs. (3) and (4) and the 2nd-order error of the interpolation. These oscillations are smoothed by applying a moving average including 50 neighboring points. The resulting time series of porosity amplitude and phase velocity can be plotted as a curve with time as curve parameter in an amplitude–phase velocity plot. This curve can be understood as a dispersion curve because the phase velocity depends on amplitude and thus implicitly on the width or wavelength of the porosity wave.
For the model series presented below the width and the amplitude of the
initial wave, the background porosity, and the rheology law have been varied.
All models were carried out using
As the shape of a two-dimensional porosity wave for a certain wave amplitude
is not known, the initial width is varied. In Fig. 2a we show a porosity
wave of amplitude 8 initially positioned at
To analyze the evolution of the ascending solitary wave the phase velocity
and the amplitude are tracked over the full rising time and plotted into a
dispersion diagram. In Fig. 3 the dispersion curves of a model with a
starting wave width which is initially larger than the resulting solitary
wave, a model with a similar width and a model with a smaller initial width
are shown. The curves start with high velocities for the Gaussian bell-shaped wave and then rapidly slow down until they approach a specific point
visible as a sharp kink from which they slowly follow a line. For the bigger
and optimal width models, after this kink, the wave is expected to have
reached the solitary wave stage. For the bigger initial width this stage is
reached at a higher amplitude than initially assumed. It is important to
note that, independent of the initial wave width, after reaching a solitary
wave stage the velocities and shapes of waves of a certain amplitude are
always equal, i.e., the three curves merge on one dispersion curve. For
comparison with semi-analytic 2-D solitary porosity wave solutions, the
dashed curves in Fig. 3 and later figures show dispersion curves with
different power law
Dispersion curves for three models with an initial width
bigger, smaller and approximately equal to the resulting solitary wave. Each
model was carried out for a melt network geometry consisting of 100 %
films and an aspect ratio of 0.1. The background porosity is 0.005 and
Based on this result one can carry out many models with different initial wave widths and different initial amplitudes and get one empirical steady-state solitary wave dispersion curve for one viscosity law for a wide range of amplitudes.
Figure 4 shows the time-dependent dispersion curves of models with four different initial amplitudes (4 to 10) and 11 different initial widths each. Depending on the initial widths, they either gain amplitudes as they approach the solitary wave stage or they monotonously loose amplitude. Depending on the initial amplitude and width, each case is characterized by a certain total melt volume, corresponding to a specific steady-state solitary wave with a specific amplitude. Therefore the 44 models finally reach one steady-state solitary wave dispersion curve at different amplitudes. As discussed in Sect. 2, the amplitudes of the waves slowly continue to decrease due to some small amount of numerical diffusion. Yet, they continue following the steady-state solitary wave dispersion curve.
Dispersion curves for 44 models with four different initial amplitudes (4 to 10) and 11 different initial widths each. All models were carried out for a melt network geometry consisting of 100 % films with an aspect ratio of 0.1.
Although we use a different rheology law and do not apply the
simplifications mentioned above, the steady-state dispersion curve of our
model is in general agreement with the
To investigate the effect of different viscosity laws, two melt network
geometries are chosen. The first one consists of 50 % films, or ellipsoidal
melt pockets, and 50 % tubes; the second of 100 % films or ellipsoidal melt
pockets. Furthermore, the aspect ratio
Waves with these different viscosity laws give only minor differences in the dispersion curves (Fig. 5a, b). Especially with the films and tubes case, the curves for different aspect ratios (Fig. 5a) are not distinguishable, both during the transient and final stages. In contrast, the analytic viscosity case (Eqs. 1 and 2) propagates along a different path and converges to a 4 %–6 % higher final phase velocity curve. With 100 % films the differences among curves with the different viscosity laws in the final velocity are higher and lie on the order of 6 %. These differences are surprisingly small if compared to the actual differences in effective shear viscosities of about 13 % and bulk viscosity of about a factor of 4 (at 4 % melt corresponding to a porosity amplitude of 8). It is also to be noted that the steady-state part of our dispersion curve calculated with the analytical viscosity (Eqs. 1 and 2) excellently agrees with the semi-analytical solution (dashed) by Simpson and Spiegelman (2011) for the same viscosity law, if we account for the 10 % numerical overestimation of our model phase velocity (see Sect. 2.2). Thus, their neglect of shear stresses and other simplifications have only a very minor effect compared to the effect of different viscosity laws. The overall effect of weakening of matrix viscosity due to decreasing aspect ratio is to slow down the phase velocity slightly.
Dispersion curves of solitary waves with
Changing
While the ascending phase velocity of the wave is only slightly affected by
the different viscosity laws, the width of the wave changes more strongly.
In Fig. 6 the half widths of the solitary waves of amplitude of 8 are
plotted against the corresponding wave velocities for the different
viscosity laws. For
Non-dimensional half width plotted against non-dimensional
phase velocity for a porosity wave of amplitude 8 for different viscosity
laws. The numbers give the aspect ratios of the films or melt pockets. The
background porosity is 0.5 %.
Another interesting phenomenon to observe is the matrix velocity in the
center of the wave, which increases for all geometries with aspect ratio
(Fig. 7). While for 100 % films this increase is stronger, for both
geometries the velocities are approximately equal at aspect ratios between
0.2 and 0.3. For
Matrix velocity in the center of a wave with an amplitude of 8 as a function of the aspect ratio of the films for
In the previous models the scaling background porosity of 0.005 and maximum
wave amplitudes of 10 to 12 imply maximum melt fractions of 5 % to 6 %.
Thus, the matrix shear viscosity decrease was only small, on the order of 10 %
for the aspect ratio 0.1 models and on the order of 5 % for the stiffer
analytical viscosity laws (1) and (2). This explains the rather mild rheology
effect when comparing the effect of the different viscosity laws. With the
aim to reach higher maximum melt fractions associated with stronger
rheological effects, we carried out a model series with increased background
porosities, both applying the analytical viscosity law (
It is interesting to note that although the semi-analytic solutions of
Simpson and Spiegelman (2011) neglect the shear term in the matrix momentum
equation and in the separation flow equation, they are in good agreement with
the low
Recently, Rudge (2018) developed a diffusion creep model based on
microscopic diffusion calculations in the presence of melt in textural
equilibrium with truncated octahedrons. Assuming infinite diffusivity in the
melt phase, Rudge (2018) obtains a somewhat stronger weakening of the shear viscosity
at smaller melt fractions than in our model but comparable disaggregation
porosities as in Fig 1. However, due to the infinite diffusivity assumption,
the bulk viscosity remains finite (equal to
It should be noted that in our study the viscosity law has been varied by
assuming various melt geometries of melt films and films or melt pockets
superimposed with tubes, while the permeability–porosity relation has been varied
independently between
In Richard et al. (2012) it was observed that with increasing background
porosities the waves will widen and the phase velocities will slow down. In
our models we observe faster velocities with increasing background porosity
if the analytical viscosity is used. This can be explained by the different
scaling which was used by Richard et al. (2012). They used just the shear
viscosity to calculate the compaction length and not the sum of shear and
bulk viscosity. If the same scaling is used, we get the same behavior for
the phase velocity (Fig. S1b, the Supplement). In contrast to Richard et al. (2012), we observe a narrowing effect of the waves for larger background
porosities, which cannot be explained by scaling (Fig. S1a). As
Richard et al. (2012) used a 1-D model, we suspect that 2-D effects such as
including the incompressible flow velocity,
As the shape of a solitary wave in our models cannot be described analytically, we start with a Gaussian wave which develops quite rapidly into a solitary wave with a similar shape and a certain amplitude, depending on the initial width of the wave.
Even though the rheologies used are much weaker than the simplified analytical ones, the effect on dispersion curves and wave shape are only moderate as long as the shear viscosity does not drop below about 80 % of the intrinsic shear viscosity. This value corresponds to a melt fraction of 5 %, equivalent to 20 % of the disaggregation value. At this porosity the bulk viscosity is approximately 5–7 times the intrinsic shear viscosity. In this case the phase velocity changes just slightly for all cases, while the waves broaden in the absence of tubes with increasing aspect ratio.
In contrast, for higher melt fractions of about 12 %, equivalent to 50 % of the disaggregation values, the shear viscosity decreases to 50 % of the intrinsic viscosity, and the bulk viscosities are on the order of the intrinsic shear viscosity. Then, our models predict significant narrowing of the porosity waves and slowing down of the phase velocities. For such conditions a strong discrepancy in solitary wave behavior between our viscosity law and the analytical ones is found.
For low melt fractions our models are in good agreement with semi-analytic solutions which neglect the shear stress term, because the matrix shear contribution of the downward segregation flow is taken over by the increase in the compaction contribution.
The used code and all model results, namely the melt porosity, segregation velocity and matrix velocity fields, are available on request.
The supplement related to this article is available online at:
The idea for this project came from HS. The models were carried out by JD. JPK carried out the calculations for the viscosity laws. JD and HS prepared the article. All authors have contributed to the discussion and article review.
The authors declare that they have no conflict of interest.
We would like to thank the reviewers Guillaume Richard and Viktoriya Yarushina for their detailed and thoughtful reviews, which helped to improve the article.
This paper was edited by Susanne Buiter and reviewed by Guillaume Richard and Viktoriya Yarushina.