This fact alone indicates the importance of the equation of state in the problem of komatiite petrogenesis. The experimental measurement of the equation of state of a molten komatiite is presented in Chapter 1. The shock is first detected on the back surface of the driver, approximately at the same time as the plane shock wave enters the sample.
This separation into parts is a result of poor polishing of the driver next to the test well. The records show the arrival of the shock wave on the free surface of the driver plate. A similar impedance match can be made to determine the shock and particle velocity of.
In these cases, the high-pressure impedance characteristics of the shocked material (cHP and sHP). The shock wave EOS data specifies the internal energy, pressure and density of the high-pressure molten sample. The high-pressure isentropic bulk modulus was determined by differentiation of the Birch-Murnaghan EOS.
The first term in the numerator is the adiabatic gradient of the crystal-free liquid.
Note in particular that the melt fraction is nearly constant in the 200–400 km interval of the upper mantle (Fig. 15). In no case does an adiabat cross from liquidus to solidus in the pressure region of the upper mantle and transition zone, and as mentioned above, adiabatic diapirs with very large degrees of partial melting should originate in the transition zone or lower mantle. With the linear model, up to 50% partial melting is possible if a diapir crosses the solidus at the base of the upper mantle (400 km).
The circle represents the point of olivine-clinopyroxene cosaturation on a komatiite liquidus [Wei et al., 1990], and the dotted band is the projection of the trace of the komatiite liquidus. Note that melt fractions do not vary significantly in the lower 200 km of the upper mantle. This is comparable to the melting point of 34-35% at 5.2 GPa, 1780°C indicated by our calculations (Fig. 15), which, although dependent on the specific shape of the x(T) functions we have used, so they are also compatible. with this specific view of the genesis of komatiite.
If komatiitic fluids of this composition were generated by 10–30% partial melting of adiabatic KLB-1-like diapirs, then our calculations suggest that the source region of the diapirs (assuming they were initially unmelted) was in the 10–20 GPa range , and that they would have started to melt between 10 GPa, 1850°C (10%) and 20 GPa, 2150°C (30%). Estimates of the degree of partial melting of peridotite required to generate komatiitic fluid plus residual olivine (e.g., Arndt [1977] ) are typically on the order of 50–80%. Pyroxene geothermometry indicates that the potential temperature of the mantle beneath continents is about 1600°C [Mercier and Carter, 1975], so if this latter view of the conditions necessary for the production of komatiitic fluids is valid, it may be possible to generate komatiitic fluids in the modern mantle.
Many independent arguments suggest that it is likely that at least the upper mantle of the Earth was substantially melted in the Hadean. The recognition of this possibly molten state, together with the idea that ultrabasic fluids may be denser than olivine and other silicate phases at relatively shallow mantle depths, has given rise to speculation about the evolution of the upper mantle from a magma ocean [Nisbet and Walker , 1982; Ohtani, 1985; Agee and Walker, 1988b]. In this section, we consider applications of the equation of state for a komatiitic fluid to the evolution of the Hadean mantle from an initial fully molten state.
Since the mantle is assumed to be fluid and would have a low viscosity, we will also assume that the thermal state of the mantle was approximately adiabatic. These special circumstances may play an important role in the evolution of the Earth, as will be discussed below. Without relying on the details of the perovskite-magnesiowiistite phase diagram, however, two general conclusions can be drawn.
Viscous Perturbations
The essence of the perturbation treatment is to evaluate the viscous stresses given the inviscid solution. Since the third eigenvalue leads to exponential growth of the perturbations, the part of the solution (Eq. 64) weighted by this eigenvalue must be identically zero. The first term (Eq. 82a) is the Laplace transform of the inviscid solution, and the remaining terms are the viscous perturbations.
The initial slopes of the viscous disturbances can be determined from: where we use the fact that the initial value of the viscous perturbations is zero. That this is contrary to the behavior reported by Sakharov et al., [1965] may indicate the importance of the bulk viscosity in their experiments. This assumption is not limiting since we can resolve other, shorter wavelength, Fourier components of the solution.
The influence of the position of this boundary must be determined to assess the applicability of his approach. By treating viscosity as a perturbation of an inviscous solution, we restrict the shape of the solution to linear combinations of the inviscous eigenvectors. If the inviscous eigenvectors are demonstrably suitable, then higher-order viscous perturbations can be used successively to get increasingly accurate estimates of the fully viscous solution.
Applying these conditions to viscous materials requires that the front width of the shock be very small compared to the radius of curvature of the shock. The reason for this discrepancy will be shown to be a consequence of the initial conditions in §9 and § 10. As before, we require that the perturbations vanish as x-+oo, so the third component of the vector in the brackets must vanish.
The quantity a indicates how quickly the initial perturbations from the shock front decay, and the quantity S is the initial time decay of the shock front. 97 gives an estimate of the perturbation of the shock front in the inviscid boundary subject to the initial conditions of Eqn. Using these estimates, a new viscous perturbation can be derived that takes finite initial conditions into account.
