Maxwell Times Research Articles

GPS-derived interseismic fault locking along the Jalisco–Colima segment of the Mexico subduction zone

SUMMARYNortheastward subduction of the oceanic Rivera and Cocos plates in western Mexico poses a poorly understood seismic hazard to the overlying areas of the North America plate. We estimate the magnitude and distribution of interseismic locking along the northern ∼500 km of the Mexico subduction zone, with a series of elastic half-space inversions that optimize the fits to the velocities of 57 GPS stations in western Mexico. All velocities were corrected for the co-seismic, afterslip and viscoelastic rebound effects of the 1995 Colima–Jalisco and 2003 Tecomán earthquakes. We explore the robustness of interseismic locking estimates to a variety of mantle Maxwell times that are required for the viscoelastic corrections, to the maximum permitted depth for locking of the subduction interface and to the location assigned to the Rivera–Cocos–North America plate triple junction offshore from western Mexico. The best-fitting locking solutions are associated with a maximum locking depth of 40 km, a triple junction location ∼50 km northwest of the Manzanillo Trough and a mantle Maxwell time of 15 yr (viscosity of 2 × 1019 Pa s). Checkerboard tests show that the locking distribution is best resolved at intermediate depths (10–40 km). All of our inversions define a gradual transition from strong locking (i.e. 70–100 per cent) of most (70 per cent) of the Rivera–North America subduction interface to strong but less uniform locking below the Manzanillo Trough, where oceanic lithosphere transitional between the Cocos and Rivera plate subducts, to weak to moderate locking (averaging 55 per cent) of the Michoacán segment of the Cocos–North America interface. Strong locking of the ∼125-km-long trench segment offshore from Puerto Vallarta and other developed coastal areas, where our modelling indicates an average annual elastic slip-rate deficit of ∼20 mm yr−1, implies that ∼1.8 m of unrelieved plate slip has accrued since the segment last ruptured in 1932, sufficient for an M ∼ 8.0 earthquake.

Nonequilibrium ensemble derivation of hydrodynamic heat transport and higher-order generalizations

Thermal transport in classical fluids is analyzed through higher-order generalized hydrodynamics (or mesoscopic hydrothermodynamics) depending on the evolution of the energy density and its fluxes of all orders. It is derived by a kinetic theory based on the nonequilibrium statistical ensemble formalism. A general system of coupled evolution equations is derived. Maxwell times, which are of significance to determine the character of the motion, are derived. They also have an important role in the choice of the contraction of description (limitation in the number of fluxes to be retained) in the studies on hydrodynamic motions. In a description of order 1, an analysis of the technological process of thermal prototyping is presented.

Comment on "Lagrangian formulation and symmetrical description of liquid dynamics".

Trachenko [Trachenko, Phys. Rev. E 96, 062134 (2017)2470-004510.1103/PhysRevE.96.062134] has argued for creating a Lagrangian formulation of liquid dynamics. This Comment shows that the proposed two-field Lagrangian does not describe dynamics of liquid in a correct way. Other claims on the symmetric description of liquid dynamics and on the similarity of Maxwell and Frenkel characteristic times are discussed.

Maxwell times in higher-order generalized hydrodynamics: Classical fluids, and carriers and phonons in semiconductors.

A family of what can be so-called Maxwell times which arises in the context of higher-order generalized hydrodynamics (HOGH; also called mesoscopic hydrothermodynamics) is evidenced. This is done in the framework of a HOGH built within a statistical formalism in terms of a nonequilibrium statistical ensemble formalism. It consists in a description in terms of the densities of particles and energy and their fluxes of all orders, with the motion described by a set of coupled nonlinear integro-differential equations involving them. These Maxwell times have a fundamental role in determining the type of hydrodynamic motion that the system would display in the given conditions and constraints. They determine a Maxwell viscous force not present in the usual hydrodynamic equations, for example, in Navier-Stokes equation.

Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)

We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group SH(2). We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the n-th conjugate time which is important in the study of sub-Riemannian wavefront. Based on our results of n-th conjugate time and n-th Maxwell time, we prove a generalization of Rolle's theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic.

Higher‐order generalized hydrodynamics of carriers and phonons in semiconductors in the presence of electric fields: Macro to nano

AbstractThe hydrodynamics of carriers (charge and heat motion) and phonons (heat motion) in semiconductors is analyzed in the presence of constant electric fields. This is done in terms of the so‐called higher‐order generalized hydrodynamics (HOGH), also referred to as mesoscopic hydro‐thermodynamics (MHT), that is, covering phenomena involving motions displaying variations short in space and fast in time and being arbitrarily removed from equilibrium, as it is the case in modern electronic devices. The particular case of an MHT of order 1 is described, covering wire samples from macro to nano sizes. Electric and thermal conductivities are obtained. As the size decreases toward the nanometric scale, the MHT of order 1 produces results that in some cases greatly differ from those of the usual hydro‐thermodynamics. The so‐called Maxwell times associated to the different fluxes present in MHT are evidenced and analyzed; they have a quite relevant role in determining the characteristics of the motion.

Higher-order generalized hydrodynamics: Foundations within a nonequilibrium statistical ensemble formalism.

Construction, in the framework of a nonequilibrium statistical ensemble formalism, of a higher-order generalized hydrodynamics, also referred to as mesoscopic hydrothermodynamics, that is, covering phenomena involving motion of fluids displaying variations short in space and fast in time-unrestricted values of Knudsen numbers, is presented. In that way, an approach is provided enabling the coupling and simultaneous treatment of the kinetics and hydrodynamic levels of descriptions. It is based on a complete thermostatistical approach in terms of the densities of matter and energy and their fluxes of all orders covering systems arbitrarily driven away from equilibrium. The set of coupled nonlinear integrodifferential hydrodynamic equations is derived. They are the evolution equations of the Gradlike moments of all orders, derived from a generalized kinetic equation built in the framework of the nonequilibrium statistical ensemble formalism. For illustration, the case of a system of particles embedded in a fluid acting as a thermal bath is fully described. The resulting enormous set of coupled evolution equations is of unmanageable proportions, thus requiring in practice to introduce an appropriate description using the smallest possible number of variables. We have obtained a hierarchy of Maxwell times, associated to the set of all the higher-order fluxes, which have a particular relevance in the process of providing criteria for establishing the contraction of description.

Thermal conductivity in higher-order generalized hydrodynamics: Characterization of nanowires of silicon and gallium nitride

An analysis of the influence of geometry and size on the thermal conductivity in semiconductors, particularized to the study in Si and GaN, is presented. This is done in the framework of a higher-order generalized hydrodynamics (HOGH) of phonons in semiconductors, driven away from equilibrium by external sources. This HOGH is derived by the method of moments from a generalized Peierls–Boltzmann kinetic equation built in the framework of a Non-Equilibrium Statistical Ensemble Formalism. We consider the case of wires (cylindrical geometry) exploring the effect of size (radius), particularly in the nanometric scale when comparison with experiment is done. Maxwell times, which are quite relevant to define the hydrodynamic movement, are evidenced and characterized.

Mesoscopic hydro-thermodynamics of phonons

A generalized Hydrodynamics, referred to as Mesoscopic Hydro-Thermodynamics, of phonons in semiconductors is presented. It involves the descriptions of the motion of the quasi-particle density and of the energy density. The hydrodynamic equations, which couple both types of movement via thermo-elastic processes, are derived starting with a generalized Peierls-Boltzmann kinetic equation obtained in the framework of a Non-Equilibrium Statistical Ensemble Formalism, providing such Mesoscopic Hydro-Thermodynamics. The case of a contraction in first order is worked out in detail. The associated Maxwell times are derived and discussed. The densities of quasi-particles and of energy are found to satisfy coupled Maxwell-Cattaneo-like (hyperbolic) equations. The analysis of thermo-elastic effects is done and applied to investigate thermal distortion in silicon mirrors under incidence of high intensity X-ray pulses in FEL facilities. The derivation of a generalized Guyer-Krumhansl equation governing the flux of heat and the associated thermal conductivity coefficient is also presented.

Earthquake cycle deformation in the Tibetan plateau with a weak mid‐crustal layer

AbstractGeodetic observations of interseismic deformation across the Tibetan plateau contain information about both tectonic and earthquake cycle processes. Time‐variations in surface velocities between large earthquakes are sensitive to the rheological structure of the subseismogenic crust, and, in particular, the viscosity of the middle and lower crust. Here we develop a semianalytic solution for time‐dependent interseismic velocities resulting from viscoelastic stress relaxation in a localized midcrustal layer in response to forcing by a sequence of periodic earthquakes. Earthquake cycle models with a weak midcrustal layer exhibit substantially more near‐fault preseismic strain localization than do classic two‐layer models at short (<100 yr) Maxwell times. We apply both this three‐layer model and the classic two‐layer model to geodetic observations before and after the 1997 MW = 7.6 Manyi and 2001 MW = 7.8 Kokoxili strike‐slip earthquakes in Tibet to estimate the viscosity of the crust below a 20 km thick seismogenic layer. For these events, interseismic stress relaxation in a weak (viscosity ≤1018.5 Pa⋅s) and thin (height ≤20 km) midcrustal layer explains observations of both preseismic near‐fault strain localization and rapid (>50 mm/yr) postseismic velocities in the years following the coseismic ruptures. We suggest that earthquake cycle models with a localized midcrustal layer can simultaneously explain both preseismic and postseismic geodetic observations with a single Maxwell viscosity, while the classic two‐layer model requires a rheology with multiple relaxation time scales.

Brownian motion in a Maxwell fluid

The equilibrium dynamics of a spherical particle immersed in a complex Maxwell fluid is analyzed in terms of velocity autocorrelation function (VACF), mean-square displacement (MSD), and power spectral density (PSD). We elucidate the role of hydrodynamic memory and its interplay with medium viscoelasticity for a free and a harmonically confined particle. The elastic response at high frequencies introduces oscillations in the VACF, which are found to be strongly damped by the coupling to the fluid. We show that in all Maxwell fluids hydrodynamic memory eventually leads to a power-law decay in the VACF as is already known for Newtonian fluids. The MSD displays asymptotically an intermediate plateau reflecting the elastic restoring forces of the medium. In the frequency domain, the PSD exhibits at high frequencies a step due to the trapping, whereas the low-frequency decay reflects the viscoelastic relaxation. Our results suggest that high-frequency microrheology is well-suited to infer the elastic modulus, which is sensitive over a wide range of Maxwell times.

Diffusionally assisted grain-boundary sliding and viscoelasticity of polycrystals

Motivated by recent attenuation experiments on finely grained samples, we reanalyse the Raj–Ashby model of grain-boundary sliding. Two linearly elastic layers having finite thickness and identical elastic constants are separated by an interface (grain boundary) whose location is a given periodic function of position. Dissipation is confined to that interfacial region. It is caused by two mechanisms: a slip (boundary sliding) viscosity, and grain-boundary diffusion, with corresponding Maxwell relaxation times t v and t d . Owing to the assumption of a given, time-independent interface, the resulting boundary-value problem (b.v.p.) is linear and time-separable. The response to time-periodic forcing depends on angular frequency ω , on the ratio M = t v / t d of Maxwell times, and on the characteristic interface slope. The b.v.p. is solved using a perturbation method valid for small slopes. To relate features of the mechanical loss spectrum previously studied in isolation, we first discuss the solution as a function of M . Motivated by experiments, we then emphasize the case M ≪ 1 in which the relaxation times are widely separated. The loss spectrum then always has two major features: a frequency band 1 ≪ ω t d ≪ M - 1 within which the loss varies relatively weakly with ω ; and a loss maximum at ω t d ∼ M - 1 due to the slip viscosity. If corners on the interface are sufficiently rounded, those two universal features are separated by a third feature: between them, there is a strong minimum whose location is (entirely) independent of slip viscosity. The existence of that minimum has not previously been reported. These features are likely to occur even in solutions for finite interface slopes, because they are a consequence of the separation of timescales. The precise form of the spectrum in the weakly varying band must, however, be slope-dependent because it is controlled by stress singularities occurring at corners, and the strength of those singularities depends on the angle subtended by the corner.

The Chi-Chi 1999 earthquake: Correlation between the spatio-temporal distribution of aftershocks and viscoelastic stress changes

Abstract The Chi-Chi 1999 (ML = 7.3) earthquake generated a large number of aftershocks in the vicinity of the rupture plane. The spatial-temporal distribution of these aftershocks was recorded with high precision and thus provided a unique possibility to study whether the correlation between aftershocks and stress changes are primary due to coseismically induced stress changes (static), or whether stress relaxation processes (viscoelastic) in the lower crust contribute significantly to this correlation. From our analysis of a 3D finite element model simulating the viscoelastic stress changes due to the coseismic displacement and tectonic loading we found that the aftershocks are highly correlated with the stress variations (static and viscoelastic) caused by the main shock. Although we found that the correlation between seismicity rate changes and viscoelastic stress fluctuation is slightly better than that of the static stress changes, these differences can only be identified well in the lower crust. As a result, it is reasonable to conclude that static stress changes are the key mechanism for triggering early and shallow aftershocks in the upper crust. It is reasonable to infer that the viscoelastic relaxation in the lower crust does affect the occurrence of early aftershocks in the deep crust, but it does not significantly affect the shallow aftershocks. However, the stress changes induced from the lower crust gradually transfer to the upper crust and may influence the occurrence of aftershocks after a longer time period (>four Maxwell times).

Complete description of the Maxwell strata in the generalized Dido problem

The generalized Dido problem is considered — a model of the nilpotent sub-Riemannian problem with the growth vector (2,3,5). The Maxwell set is studied, that is, the locus of the intersection points of geodesics of equal length. A complete description is obtained for the Maxwell strata corresponding to the symmetry group of the exponential map generated by rotations and reflections. All the corresponding Maxwell times are found and located. The conjugate points that are limit points of the Maxwell set are also found. An upper estimate is obtained for the cut time (time of loss of optimality) on geodesics.

Large time step numerical modelling of the flow of Maxwell materials

SUMMARY Maxwell viscoelastic materials are commonly simulated numerically in order to model the stresses and deformations associated with large-scale earth processes, such as mantle convection or crustal deformation. Both implicit and explicit time-marching methods require that the time steps used be small compared with the Maxwell relaxation time if accurate solutions are to be obtained. For crustal tectonic modelling, where Maxwell times in a ductile lower crust may be of order of a decade or less, the large number of time steps required to model processes lasting many millions of years imposes a huge computational burden. This burden is avoidable. In this paper I show that, with the appropriate formulation of the problem, time steps may be taken which are much larger than the Maxwell time without loss of accuracy, as long as they are not large compared with the times over which strain rates vary significantly (‘tectonic’ timescales) in the model. The method relies on explicit analytic integration of the Maxwell constitutive relation for the stress over time intervals, which may be longer than the relaxation time as long as they are short compared with the timescale over which crustal stresses and geometries change. The validity of the formulation is also demonstrated numerically by comparison with the analytic solutions for three simple plane-strain models: extension of a uniform block, shear of a composite layer and development of a Rayleigh‐Taylor instability.

Lower‐Crust Ductile Flow and its Mechanical Relation with Syn‐Collision Crustal Extension in Orogenic Belt

AbstractIn attempt to reveal the mechanical relation between lower‐crust ductile flow and upper‐crust extension in collisional orogenic belt, the present paper makes numerical simulation using the Finite Element Method (FEM), which is based on stress‐strain constitution of incompressional Non‐Newtonian rheology. Results show that, under the tectonic compression and the gravitational loading conditions, the crust in mountain‐building process undergoes complex ductile flow. Kinematics of ductile flow is time‐dependent, and influenced by the shape of the crust transition zone (CTZ) between orogenic belt and frontal stable plate. Under the mechanic balance of tectonic compression and gravitational loading, the ductile flow is facilitated first in mountain root. However after several Maxwell times, it will be localized narrowly to the CTZ, where the flow pattern is featured by a downward flow near the mountain side and an upward flow near stable‐plate side. This flow pattern causes stress partition and displacement differentiation spatially in orogenic belt. In details, the minimum principle stress in upper crust near the mountain center will change from horizontal compression to horizontal tension. The magnitude of tensile stress is mainly related with the height of mountain belt above reference surface. It indicates that crustal extension in syn‐collision orogenic belt has a mechanic relation with gravitational collapse, but the extension is probably intensified further by lower‐crust ductile flow. For this consideration, it is concluded that mechanics of compression‐extension coupling in orogenic belt is derived from the lower‐crust ductile flow that is initiated by the gravitational loading and horizontal tectonic compression. Using model result, the north‐south crustal extension in the southern Qinghai‐Xizang Plateau has been discussed.

Geodetic constraints on postseismic deformation following the 1990 Ms 7.8 Luzon earthquake and implications for Luzon tectonics and Philippine Sea plate motion

We have obtained three epochs of high‐quality dual‐frequency GPS data at 15 stations in the Luzon, Philippines, region since the July 16, 1990, Ms 7.8 earthquake, from which we generate horizontal velocity fields at the Earth's surface for the 1993–1996 and 1996–1998 periods. We use these velocities, plus 1996–1998 GPS velocity results reported elsewhere from 13 additional stations, to show that present‐day deformation in Luzon is dominated by strike‐slip motion along the Philippine fault system. The measured strike‐slip rate across central Luzon is faster than the expected long‐term slip rate on the fault, which we attribute to postseismic deformation following the 1990 earthquake. The 1993–1996 and 1996–1998 deformation patterns cannot be statistically distinguished within the ∼10% uncertainties of the data. If the observed deformation is interpreted using an elastic half‐space model with uniform slip below a subsurface locking depth, we find a fault slip rate of 40 mm/yr and a locking depth of ∼15 km. This is too fast to be a steady interseismic rate, as it disagrees with what is known from paleoseismic and other evidence about earthquake recurrence and long‐term slip rate on the Philippine fault system in central and northern Luzon. Also, a 15 km locking depth for steady interseismic slip is probably too shallow compared to the depth of rupture in the 1990 earthquake. If the deformation is instead interpreted with two types of viscoelastic model, we find that the observed velocities can be fit well for a range of values of lower lithosphere viscosity and long‐term slip rate. Using a two dimensional (2‐D) viscoelastic coupling model, the minimum allowable lower lithosphere viscosity is 0.5 × 1019 Pa s, with an associated long‐term slip rate of 15–22 mm/yr. Faster long‐term slip rates also fit well, with correspondingly higher values of viscosity. With a possibly more realistic model that includes 3‐D viscoelastic effects we find preferred long‐term velocities in the range 20–35 mm/yr and viscosities in the range 2–6 × 1019 Pa s. Our preferred viscosities correspond to Maxwell times on the order of 7–20% of the typical earthquake recurrence interval and are several times higher than values found for a number of Californian strike‐slip earthquakes. Other implications of our 1993–1998 surface velocities are that the Philippine Sea‐Eurasia (PH‐EU) Euler vector of Seno et al. [1993] is a better description of the PH‐EU convergence rate than a Euler vector recently estimated from GPS velocities in the northern PH and that the great majority of the normal component of PH‐EU plate convergence is taking place west of Luzon, presumably largely at the Manila Trench. We also find that Luzon is rotating counterclockwise relative to PH at 1°–2°/Myr, only ∼25% of the rate recently suggested by other workers on the basis of limited GPS data.

On Maxwell singularities in postglacial rebound

We investigate the problem of finding the numerous relaxation times associated with the postglacial rebound of a layered Maxwell earth model. In general, these relaxation times are the roots of a secular polynomial. When a numerical approach is followed, this polynomial can be very ill behaved, with a number of singularities that coincide with the Maxwell times associated with the model rheology. This problem becomes dramatically evident when the rheological profile of the model is continous or includes a large number of uniform layers (these two cases are basically the same when the solution is computed numerically). In order to understand the physical meaning of such Maxwell singularities, we perform a comparison between the numerical approach and the existing analytical solution to the problem of the postglacial relaxation of an incompressible, self-gravitating, N-layer, spherical Maxwell earth. We show that the analytical method does not suffer from the Maxwell singularity problem, and give a theoretical explanation of the ill behaviour of the secular ploynomial computed in numerical studies.

Crater relaxation on Ganymede: Implications for ice rheology

Controversy has existed over whether or not viscous relaxation is an important process on the icy satellites. Previous models involved large extrapolations of Newtonian flow laws for ice, whereas ice is known to exhibit non‐Newtonian behavior. Recently, the flow law parameters for ice at the appropriate temperatures and stresses have been measured. Numerical modelling of the viscous relaxation of basins on Ganymede using these parameters has given implausibly short relaxation times. However, this model treated ice as a purely viscous substance, so that no elastic lithosphere could develop near the surface. Here we treat ice as a Maxwell visco‐elastic material and numerically model the relaxation of basins on Ganymede. We find that realistic Young's moduli lead to little relaxation occurring in basins even 4.0 Ga after their formation. Further, we demonstrate that within broad limits the near surface temperature gradient has little effect on this result. Finally, through examination of the distribution of Maxwell times in the vicinity of the crater, we show that most viscous relaxation occurs early in the basin's history when stresses are high, and thus Maxwell time is short. As stresses are relieved, the Maxwell time becomes long, and relaxation essentially ceases.

Postseismic deformation due to subcrustal viscoelastic relaxation following dip‐slip earthquakes

The deformation of the earth following a dip‐slip earthquake is calculated by using a three‐layer rheological model and finite element techniques. The three layers are an elastic upper lithosphere, a standard linear solid lower lithosphere, and a Maxwell viscoelastic asthenosphere. Attention is focused on the magnitude of the postseismic subsidence and the width of the subsidence zone that develops due to the viscoelastic response to coseismic reverse slip. Detailed analysis for a fault extending from the surface to 15 km with a 45° dip reveals that postseismic subsidence is sensitive to the depth to the asthenosphere but is only weakly dependent on lower lithosphere depth. The greatest subsidence occurs when the elastic lithosphere is about 30 km thick, and the asthenosphere lies just below this layer (asthenosphere depth = 2 times the fault depth). The extremum in the subsidence pattern occurs at about 5 km from the surface break of the fault and lies over the slip plane. At this location the subsidence reaches 60% of the coseismic uplift after about 30 Maxwell times. Unlike the horizontal deformation following a strike‐slip earthquake, significant vertical deformation due to asthenosphere flow persists for many times the Maxwell time of the asthenosphere. As a consequence, the relaxation of the lower lithosphere has only a secondary effect on uplift and subsidence; it does, however, substantially increase horizontal straining. The width of the postseismic subsidence zone also depends strongly on the depth to the asthenosphere and shows little sensitivity to the presence of a slowly relaxing lower lithosphere. Parametric adjustment of the lithosphere thickness and fault dip angle over a considerable range of values in a simple model of an elastic lithosphere and viscoelastic asthenosphere reveals that the postseismic subsidence is greatest when the fault penetrates through about one half the lithosphere. Previous estimates of the asthenosphere viscosity (∼1019 Pa s) and lithosphere thickness (∼30–40 km) for the region of Japan affected by the 1896 Riku‐u earthquake are confirmed by model calculations.