Francesca Quareni

Mean-field methods in mantle convection

The use of the single-mode, mean field method as a simple tool in understanding the physics of mantle convection is investigated. We present here a summary of steady-state results derived from both the incompressible (Boussinesq) and the compressible (anelastic) set of equations for cartesian geometry. We have employed various types of Newtonian rheology: constant, depth dependent, temperature dependent, and both temperature and pressure dependent viscosities. Mean-field methods can be quite useful in predicting heat transfer characteristics and the averaged interior temperature of the convecting medium for a wide variety of material properties and boundary conditions. Several new findings have resulted from the mean-field method. Of foremost geophysical interest are (1.) the strong non-linear coupling between the adiabatic heating parameters and the creep parameters, particularly the activation volume, (2.) the constraint on the amount of internal heating in the mantle from consideration of melting temperatures in variable viscosity convection and (3.) the important role played by dissipative heating in compressible convection with temperature and pressure dependent viscosity. In order to satisfy the constraints of mantle adiabaticity, the activation volume in the lower mantle can not be larger than about 4 cm3Jmole. Enhanced dissipation takes place when the compressibility is sufficiently high, corresponding to Grueneisen parameters less than about 1.5. Thus there seems to exist a self-regulating mechanism in the mantle for controlling the interior temperature, owing to the non-linear interactions among internal heating, variable viscosity and mantle compressibility.

Effects from equation of state and rheology in dissipative heating in compressible mantle convection

The thermal profile in the Earths interior is influenced by many factors. One of the least understood and studied processes is that resulting from adiabatic heating and viscous dissipation. With the exception of the work by Jarvis and McKenzie1 very little has been done on the effects of compressibility on mantle circulation. It is quite common for geophysicists to add the adiabatic temperature gradient a posteriori2 to temperature profiles derived from Boussinesq (incompressible) equations. Recently it has been shown3 with the mean-field method4 that there exists a strong coupling between the Theological parameters, such as the activation volume, and the thermodynamic constants governing adiabatic heating. Here we point out the important consequences brought about by incorporating the effects of equations of state and rheology in the dissipative heating term of the energy equation. We demonstrate explicitly the ways in which compression may raise the interior mantle temperature and illustrate how this effect can, in turn, be used for constraining some of the intrinsic parameters associated with the equation of state in the mantle.

Vertical ground displacement at Campi Flegrei (Italy) in the fifth century: Rapid subsidence driven by pore pressure drop

Campi Flegrei (Italy) caldera has experienced episodes of ground deformation throughout its geological history, alternating between uplift and subsidence phases. Although uplift periods are typically more alarming, here we focus on subsidence, looking for its driving mechanisms and its role in the caldera evolution. Historical and archaeological records constrain ground deformation over the last two millennia. Here we revise such records and combine them with published radiometric dating and with the simulation of sea level change. The resulting analysis highlights for the first time a rapid subsidence during the fifth century. We show that rate and magnitude of this subsidence are consistent with the compaction of porous material caused by a pressure drop of ~ 1 MPa within the hydrothermal system. We interpret this event as the decompression of the hydrothermal system following an unrecognized episode of unrest, during Roman times. These findings redefine the pattern of ground deformation and bear important implications for volcanic hazard assessment.

Time-dependent solutions of mean-field equations with applications for mantle convection

Abstract The set of time-dependent, single-mode, mean-field equations appropriate for mantle convection has been converted by means of the method of lines formulation to a set of non-linear ordinary differential equations. The resulting system has been integrated in time t to steady-state by means of differential/algebraic system F(t, y , d y d t ) = 0 , in which the solution vector y consists of the vertical velocity, the thermal perturbation and the mean temperature and the spatial operators are finite differenced by high order schemes (up to 4th order). This large differential system—up to 500 equations—can be solved conveniently and economically by employing variable step-size, variable order (up to 5th order) stiff method such as backward differentiation formulas. We have studied thermal convection of infinite Prandtl number fluids with constant viscosity for both base heated and constant internally heated configurations. Both stress-free and rigid boundaries have been examined for these two modes of heating. Solutions for Rayleigh numbers up to 104 times of the critical value have been obtained. Heat transfer coefficients in the relationship between the Nusselt and Rayleigh number of the form Nu = aRab are derived from the steady-state solutions. In general, the exponent b is ∼ 10% higher than the corresponding value obtained from asymptotic or numerical solutions of the full set of 2-dimensional equations. We have also examined the effects of time-dependent heating and different types of initial conditions on the subsequent thermal evolution. The role of time-dependent heating from radioactive sources is to lengthen substantially the period of time during which the system is still influenced by the initial conditions. This time of thermal equilibration is found to vary as cRa−d, where d = 0.24 and is related to the heat-transfer coefficient. The constant c depends on the decay time of the heat sources. For Ra between 106 and 107 this time of readjustment is one-tenth of the thermal diffusion time of the entire layer and is 0(109 y) for upper-mantle convection, thus suggesting that modeling thermal history in the Archean may depend on earlier primeval events of the Earth.

The Grüneisen parameter and adiabatic gradient in the Earth's interior

Abstract The most straightforward approach to derive thermodynamic properties for the Earths interior is to express them in terms of mechanical parameters. These are directly available from seismology, and represent so far the only detailed information relative to the interior of the Earth. This goal can be achieved through a number of approximate theories of unproven practical validity. Focusing our attention on the Gruneisen parameter and using the available data, such a test is possible only at room pressure and for 19 solids (five metals, seven alkali halides, five synthetic minerals and two rocks). It shows unequivocally the superiority of the Debye-Brillouin formulation, which provides good agreement with experiment in all the cases examined. While its validity at high pressures cannot be demonstrated, under all the possible test conditions it appears much more accurate than alternatives which enjoyed great popularity in modeling the Earths interior, such as the free-volume formulation. Applying the Debye-Brillouin theory to the Preliminary Reference Earth Model, yields the result that the Gruneisen parameter in the mantle is essentially constant around 1.2, although the approximate nature of this result does not rule out other possibilities such as γρ = constant . Thermal expansion appears remarkably constant through the lower mantle, decreasing approximately by only a factor of two. This yields a virtually flat mantle adiabat with a temperature increase of only 350 ± 150 K through the lower mantle. The Gruneisen parameter is also constant with depth in the inner core and has values > 1.5. By analogy, both the adiabat and thermal expansion in the inner core are constant.

Modeling the closure of volcanic conduits with an application to Mount Vesuvius

The eruptive activity of a volcano is controlled by the opening and closure of conduits through which magma ascends to the surface. Assuming that the conduit is filled with magma only immediately before and during an eruption, and disregarding any cooling of magma in this state, we develop a model to study the deformation of a cylindrical conduit surrounded by a viscoelastic cylindrical region in an infinite, elastic, homogeneous space. The viscoelastic behavior of the zone around the conduit is due to heat conduction from the hot magma, which raises the temperature beyond the brittle-ductile transition point. The effect of a tectonic regional stress which favors (compressive) or acts against (tensile) conduit closure is taken into account. The sensitivity of the model is checked with respect to the governing parameters, namely size of the viscoelastic region, ratio between its rigidity and that of the elastic medium, Poissons ratio, and tectonic stress. Conduit closure is found to be ruled essentially by the extension of the viscoelastic region and by the ratio between its rigidity and the rigidity of the surrounding elastic medium, while tectonic stress is much less important. The model is applied to the last eruptive cycle of Mount Vesuvius. Using realistic values for the elastic and viscoelastic moduli, we find that an open conduit condition has been possible from 1631 to 1944, while the quiescence from 1944 on implies a closed conduit state.

Some rheological properties of the Earth's mantle

Abstract We estimate the viscosity in the Earths mantle through a material science approach. Since refined theoretical formulations are still difficult to apply without drastic approximations due to the very scarce knowledge of the material properties of the mantle, we rely on a semi-empirical approach. The most effective semi-empirical formulation accounts for the effect of temperature and pressure on creep utilizing the melting temperature as a scaling parameter. There is no well established method of estimating the effect of polymorphic phase transitions (which in any case are not definitely known in the mantle) and only very rough estimates are possible. Inaccuracies of a few orders of magnitude must be allowed. We apply it to the mantle and analyze the stability of the results against the feasible range of rheology types, geothermal profile/melting curves, presence of boundary layers and related temperature jump, and material constants. In spite of a large uncertainty in the results, a comparison with the available studies on global Earth rheology (with the proviso that the two sets are not always compatible) based on post-glacial rebound, geoid anomalies and post-seismic deformation allows to derive the combination rules for the different variables and to depict the possible scenarios for the thermodynamic state and the rheology of the mantle. For example, Newtonian rheology is compatible only with some particular geothermal profiles and with some given sets of material parameters.

Quasi-harmonic and leading order anharmonic parameters for some minerals estimated from available thermodynamic data

Abstract The thermodynamics of the Earths interior is usually approached through the quasi-harmonic theory of solids, relying on the tacit assumption that explicit anharmonic contributions, which are difficult to calculate even for the most simple solids, are not important. The basic qualitative idea assumes that, as anharmonicity increases with temperature but decreases with pressure, in the Earths interior the two effects are self-cancelling. A safer approach is to consider not the absolute but the homologous temperature, i.e. to take the melting temperature as a scaling factor. Since the geotherm lies rather close to melting, the relative importance of anharmonic terms in the thermodynamics of the Earths interior is presumably high. At room pressure, the leading order anharmonic coefficients and their volume derivatives, as well as the quasi-harmonic terms from available thermodynamic data, can be estimated by fitting to experiment the theoretical expressions to the fourth order in the Hamiltonian of three functions of compressibility, heat capacity and thermal expansion, tailored to minimize the effect of experimental errors. The capability of this procedure to produce stable and accurate results has been previously tested through extensive applications to NaCl, and to several metals, chosen because of the large number of available independent data, which allowed the study in detail of the effect of external as well as internal errors. Comparatively accurate estimates were found to be possible. The procedure is here applied to forsterite, periclase, rutile and pyrope, allowing the estimation of the relative magnitude of the quasi-harmonic and leading order anharmonic terms. At temperatures of half the melting temperature the difference between the two is of the order of 20% on the Gruneisen parameter and just a few per cent on the heat capacity, which is nevertheless typically 10% from the classical 3R limit.