The structure of thermal convection and geotherms beneath a continent and ocean

The formation of the thermal cross section of the lithosphere and mantle upon the interaction between the mantle convection and the immobile continent surrounded by the oceanic lithosphere is studied by numerical modeling. The convective temperature and velocity fields and then the averaged geotherms for subcontinental and suboceanic regions up to the boundary with the core are calculated from the solution of convection equations with a jump in viscosity in the continental zone. Using the experimental data on the solidus temperature in the rocks of the upper mantle, the average thickness of the continental and oceanic lithosphere is estimated at 190 and 30 km, respectively. The effect of a hot spot formed in the subcontinental upper mantle at a depth of 250–500 km, which has not been previously noted, is revealed. Although the temperature in this zone is typically assumed to be close to adiabatic, the calculations show that it is actually higher than adiabatic by up to 200°C. The physical mechanism responsible for this effect is associated with the accumulation of convective heat beneath the thermally insulating layer of the continental lithosphere. The revealed anomalies can be important in studying the phase and mineral transformations at the base of the lithosphere and in the regional geodynamical reconstructions.     

Three-dimensional numerical simulations of crustal deformation and subcontinental mantle convection

3-D simulations of mantle convection allowing for continental crust are explored to study the effects of crustal thickening on lithosphere stability and of continents on large-scale mantle flow. Simulations begin with a crustal layer within the upper thermal boundary layer of a mantle convection roll in a 1 × 1 × 1 Cartesian domain. Convective stresses cause crust to thicken above a sheet-like mantle downwelling. For mild convective vigor an initial crustal thickness variation is required to induce 3-D lithospheric instability below the zone of crustal convergence. The amplitude of the required variation decreases with increasing convective vigor. Morphologically, instability is manifest in formation of drip-like thermals that exist within the large-scale roll associated with initial crustal thickening. A strong surface signature of the drips is their ability to cause deviations from local Airy compensation of topography. After the initial thickening phase, the crustal accumulation that forms serves as a model analog to a continent. Its presence leads to mantle flow patterns distinctly different from the steady-state roll that results in its absence. Large lateral thermal gradients are generated at its edge allowing this region to be the initiation site for continued small-scale thermal instabilities. Eventually these instabilities induce a restructuring of large-scale mantle flow, with the roll pattern being replaced by a square cell. Although preliminary and idealized, the simulations do show the fluid dynamical plausibility behind the idea that significant mantle variations can be generated along the strike of a largely 2-D mountain chain by the formation of the chain itself. The ability of a model continent to cause a change in fundamental convective planform also suggests that the effects of continental crust on mantle convection may be low-order despite the seemingly trivial volume of crust relative to mantle.     

增厚大陆岩石层热边界层对流剥离的数值模拟

用数值模拟方法模拟了增厚大陆岩石层热边界层被对流地幔剥离并为软流层物质替代的动力学过程．结果表明,在初始温度分层分布、侧向均匀但存在微小热扰动的流场中,80km厚的增厚岩石层热边界层约需60Ma才能被完全剥离,剥离的速率微弱地依赖扰动的强度;在已建立好的流场中,同样厚度的增厚热边界层只需约10Ma就可被剥离．模拟结果暗示青藏高原地壳及岩石层在岩石层增厚和剥离以前就很热,其下伏地馒中可能已存在建立好的上地幔小尺度对流系统,而该尺度的对流系统很可能是由特提斯海洋岩石圈俯冲和消减诱发的     

Numerical models of subduction of the oceanic crust with basaltic plateaus

Light continents and islands characterized by a crustal thickness of more than 30 km float over a convective mantle, while the thin basaltic oceanic crust sinks completely in subduction zones. The normal oceanic crust is 7 km thick. However, anomalously thick basaltic plateaus forming as a result of emplacement of mantle plumes into moving oceanic lithospheric plates are also pulled into the mantle. One of the largest basaltic plateaus is the Ontong Java plateau on the Pacific plate, which arose during the intrusion of a giant superplume into the plate ～100 Myr ago. Notwithstanding its large thickness (averaging ～30 km), the Ontong Java plateau is still experiencing slow subduction. On the basis of numerical modeling, the paper analyzes the oceanic crust subduction process as a function of the mantle convection vigorousness and the density, thickness, viscosity, and shape of the crust. Even a simplified model of thermocompositional convection in the upper mantle is capable of explaining the observed facts indicating that the oceanic crust and sediments are pulled into the mantle and the continental crust is floating on the mantle.     

Thermal evolution of Venus

The theory of three-dimensional and finite-amplitude convection in a viscous spherical shell with temperature and pressure dependent physical parameters is developed on the basis of a modified Boussinesq fluid assumption. The lateral dependences of the variables are resolved through their spherical harmonic representations, whereas their radial and time dependences are determined by numerical procedures. The theory is then applied to produce thermal evolution models for Venus. The emphasis is on illustrating the effects of certain physical parameters on the thermal evolution rather than proposing a specific thermal history for the planet. The main conclusions achieved in this paper are (1) a significant portion of the present temperature in the mantle and heat flux at the surface of Venus is probably owing to the decay of a high temperature established in the planet at the completion of its core formation, (2) the effective Rayleigh number of the mantle is so high that even the lower order modes of convection cool the planet sufficiently and maintain an almost adiabatic temperature gradient in the convecting region and high temperature gradients in the thermal boundary layers, (3) the convection is oscillatory with avalanche type properties which induces oscillatory features to the surface heat flux and the thickness of the crustal layer, and (4) a planetary model with a recycling crust cools much faster than those with a permanently buoyant crust.The models presented in this paper suggest that Venus has been highly convective during its history until ～ 0.5 Ga ago. The vigorous convection was bringing hot and fresh material from the deep interior to the surface and dragging down the crustal slags, floating on the surface, in to the mantle. The rate of cooling of the planet was so high that its core has solidified. In the last 0.5 Ga the vigour of convection diminished considerably and the crustal slags developed into a global and permanently buoyant crustal layer. The tectonic style on Venus has, consequently, changed from the recycling of crustal plates to hot spot volcanics. At the present time the planet is completely solid, except in the upper part of its mantle where partial melting may occur.     

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν₀ + ν₁ exp(?γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10⁷ and Rayleigh numbers of up to 10⁸ are studied.For intermediate viscosity contrasts (around 10³), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts.For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z_r (the rigid lid thickness) such that this number is maximum.The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10¹⁵?5 × 10¹⁶ m² s^?1 for viscosity in the thermal boundary layer below the plate.     

Stability of the boundary layer between the lithosphere and convecting mantle and the steady-state lithospheric geotherm

In the steady state, the convective boundary layer (CBL) (the transition from the lithosphere to the convecting mantle, the lithosphere-asthenosphere boundary) is on the verge of stability. This determines its depth, thickness, and the steady-state temperature distribution in the lithosphere. Had the mantle been homogeneous, the base of the lithosphere at the current potential temperature would lie globally at the same depth H _rh of 50 to 70 km. Actually, the regime of interaction of the mantle convection with the lithosphere is determined by the relationship between this depth and the thickness H _depl of the chemical boundary layer including the crust and the layer of the depleted rock. If the thickness of the chemical boundary layer is small H _depl < H _rh, as it is the case in the present-day oceanic mantle, the suboceanic regime is established with the mantle convection that does not reach the base of the chemical boundary layer. In this case, the top of CBL is located at depth H _rh, while the oceanic heat flow and the depth of the seafloor only depend on the potential temperature T _p and, within the areas where the crust is older than 60 to 70 Ma, are the same everywhere far from the disturbed territories (the hot points and the subduction zones). The absence of noticeable distinctions between the heat flow in the different oceanic basins suggests a global constancy of the potential temperature. If H _depl > H _rh, the subcontinental regime of the interaction of the mantle convection with the lithosphere is established. In this case, the CBL is immediately adjacent to the depleted lithosphere, its top is located at depth H _depl, and the surface heat flow q(T _p, H _depl) not only depends on the potential temperature T _p but also on the the thickness of the depleted lithosphere H _depl; it decreases with increasing H _depl and, therefore, with the age of the lithosphere. Given the potential temperature, the dependence q(T _p, H _depl) agrees well with the envelope of the results of kimberlite xenolith thermobarometry presented in the diagram of the deepest xenolith depth as a function of the heat flow. It is likely that in the lowest part of the continental lithosphere there is a zone of horizontal shear deformation, from where kimberlites entrain the strongly deformed and, at the same time, the deepest xenoliths. Besides, the azimuthal anisotropy of seismic velocities can be associated with this zone. The change in its direction with depth can be observed as the Lehmann discontinuity.     

A theoretical study of hydrothermal convection and the origin of the ophiolitic sulphide ore deposits of Cyprus

Hydrothermal circulation of seawater has been suggested as a mass transport mechanism for the formation of sulphide ore deposits in the ophiolitic rocks of Cyprus. Since ophiolitic sequences are generally regarded as fragments of oceanic crust and upper mantle, hydrothermal circulation of a form inferred from geological observations on Cyprus may be analogous to that thought to occur in oceanic crust at spreading ridges. The hypothesis that ore deposits were formed in ascending plumes of hot, buoyant fluid is examined by considering thermal convection in a permeable medium. To match the inferred pattern of circulation, finite amplitude convection in a cylindrical geometry is studied using finite difference approximations. These results combined with available geological and geochemical data are applied to understand better the physical controls on mineralisation.A simple model for the formation of the hydrothermal ore deposits of Cyprus is discussed. The model is semi-quantitatively reasonable in terms of vertical fluid flow rate, thermal structure, permeability and basal heat flow, and predicts volumes of maximum mineralisation similar to those observed. Three factors are identified which were important in confining mineralisation to a small volume immediately beneath the sea water/rock boundary: (1) hot fluid was confined to a narrow core zone of a rising plume, (2) the upward fluid flux was greatest in this same core zone, and (3) significant temperature decrease occurred within a thin surface boundary layer.     

Numerical model of the supercontinental cycle stages: Integral transfer of the oceanic crust material and mantle viscous shear stresses

We compute the transfer of oceanic lithosphere material from the surface of the model to the inner convective mantle at successive stages of the supercontinental cycle, in the time interval from the beginning of convergence of the continents to their complete dispersal. The sequence of stages of a supercontinental cycle (Wilson cycle) is calculated with a two-dimensional numerical model of assembling and dispersing continents driven by mantle flows; in turn, the flows themselves are forming under thermal and mechanical influence of continents. We obtain that during the time of the order of 300 Myr the complete stirring of oceanic lithosphere through whole mantle does not occur. This agrees with current ideas on the circulation of oceanic crust material. Former oceanic crust material appears again at the Earth’s surface in the areas of mantle upstreams. The numerical simulation demonstrates that the supercontinental cycle is a factor which intensifies stirring of the material, especially in the region beneath the supercontinent. The reasons are a recurring formation of plumes in that region as well as a global restructuring of mantle flow pattern due to the process of joining and separation of continents. The computations of viscous shear stresses are also carried out in the mantle as a function of spatial coordinates and time. With a simplified model of uniform mantle viscosity, the numerical experiment shows that the typical maximal shear stresses in the major portion of the mantle measure about 5 MPa (50 bar). The typical maximal shear stresses located in the uppermost part of mantle downgoing streams (in a zone that measures roughly 200 × 200 km) are approximately 8 times greater and equal to 40 MPa (400 bar).     

Approximate solutions for the formation of the lithosphere

The lithosphere is interpreted as a thermal boundary layer. Approximate solutions of the boundary layer cooling problem are developed which include mantle radioactivity, partial melt in the asthenosphere, a temperature gradient in the asthenosphere, and a non-zero lithospheric thickness at the ridge crests. The cooling history of oceanic lithosphere is found to be remarkably insensitive to assumptions about the amount of radioactivity in the upper mantle and the extent of melting in the asthenosphere. Determinations of the thickness of oceanic lithosphere and the depths of oceans as a function of age are in excellent agreement with boundary layer predictions which include a heat flux from the asthenosphere. However, the determinations do not resolve how much of the total asthenospheric heat flux might be caused by a temperature gradient in the asthenosphere. Simple thermal arguments indicate that the initial lithospheric thickness, L₀, at ridge crests should depend on the local half-spreading rate, V, as L₀ = 3 km/V(cm/year).     

华北平原邢台地震活动地区的上地幔结构和地幔低速层

邢台地震活动地区,地壳底部莫霍界面之下尚有纵波速度为8.1、7.6、7.8和8.1公里/秒的四层介质,它们的下面有速度为7.2-7.4公里/秒的地幔低速层,低速层顶部的深度为83公里左右。 本文所得低速层与世界其他地区所得低速层参数的比较似乎说明:在构造活动地区、火山活动地区和地震活动地区,低速层中的速度值一般均较正常状态下（海洋和大陆地区）的地幔低速层为低,且埋深相对较浅。     

Topography on the D′′ region from analysis of a thin dense layer beneath a convecting cell

In this study, we examine the development of topography on a thin dense layer at the base of the lower mantle. The effect of the convecting mantle above is represented as a traction acting on the upper surface of the layer. Topography on the layer boundaries is predicted by a balance of dynamic flow stress and external traction. The nature of boundary topography depends on the magnitude of the driving tractions and the density variation within the layer. If we assume that the layer density is greatest beneath areas of mantle downwelling and decreases to a minimum beneath areas of mantle upwelling (the layer is thermally coupled to the convection in the overlying mantle) then its upper boundary develops a cusp-like peak beneath the upwelling mantle. The height of this peak is potentially much greater than the layer thickness. If, however, the layers are effectively coupled by viscous shear then internal density gradients of the opposite sign may be established. In this case, we observe solutions where the layer is completely swept away beneath areas of mantle downwelling leaving steep-sided ‘islands’ of dense material. This mechanism therefore provides a possible explanation for steep-sided anomalously slow regions at the base of the mantle observed by seismic methods (e.g. beneath south Africa) or for discrete ultralow velocity zones detected at the core-mantle boundary beneath locations of surface hotspots. The magnitude of the upper boundary driving tractions compared to the density gradient within the layer is the key parameter that determines the nature of flow in, and consequently boundary topography of, the layer. The deflection of the core-mantle boundary is small compared with that of the top of the dense layer, but a change in sign of the ratio of these deflections is observed as the magnitude of the driving tractions changes relative to the magnitude of the internal density gradient. We compare seismic measurements of core-mantle boundary topography and D′′ topography with the predictions of this model in an attempt to constrain model parameters, but no clear correlation seems to exist between D′′ thickness and CMB topography.     

On the geodynamic setting of kimberlite genesis

The emplacement of kimberlites in the North American and African continents since the early Palaeozoic appears to have occurred during periods of relatively slow motion of these continents. The distribution of kimberlites in time may reflect the global pattern of convection, which forces individual plates to move faster or slower at different times. Two-dimensional numerical experiments on a convecting layer with a moving upper boundary show two different regimes: in the first, when the upper boundary velocity is high, heat is transferred by the large-scale circulation and in the second, when the upper boundary velocity is lower, heat is predominantly transferred by thermal plumes rising from the lower boundary layer. For a reasonable mantle solidus, this second regime can give rise to partial melting beneath the moving plate, far from the plate boundaries. The transition between these modes takes place over a small range of plate velocities; for a Rayleigh number of 10⁶ it occurs around 20 mm yr^?1. We suggest that the generation of kimberlite magmas may result from thermal plumes incident on the base of a slowly moving plate.     

Destabilization of a 650 km chemical boundary layer and its bearing on the evolution of the continental crust

A mechanism for the production of a chemical change in the mantle, from primordial silicate compositions above the 650 km discontinuity to differentiated compositions below, is reviewed. Some consequences of this are the stabilization of two layer convection with a temperature contrast between the anhydrous mantle solidus and the geotherm which, at 650 km depth, is lower than any other location in the mantle. With thermal contributions from the concentration of the heat producing elements U, Th and K below the 650 km mantle boundary layer and the higher geotherms in the past, widespread or catastrophic melting may have taken place at this location. An episodic breach of this boundary layer by extensive heat and mass transport may have periodically destroyed any simple two-layer convection geometry in the mantle. Such episodic injections of mass and energy into the upper mantle from below may have been the mechanism responsible for episodes of enhanced surface tectonism and thermal activity which appear to be recorded in apparent polar wandering paths and radiometric ages of continental rocks.     

Two-dimensional mantle flow beneath a rigid accreting lithosphere

This study considers two-dimensional mantle flow beneath a rigid lithosphere. The lithosphere which forms the upper boundary of a convecting region moves with a prescribed uniform horizontal velocity, and thickens with distance from the accreting plate boundary as it cools. Beneath the lithosphere, the mantle deforms viscously by diffusion creep and is heated radiogenically from within. Solutions for thermal convection beneath the lithosphere are obtained by finite-difference methods. Two important conclusions have resulted from this study: (1) convective patterns of large aspect ratio are stable beneath a rigid moving lithosphere; (2) even for a lithosphere velocity as small as 3 cm/yr. and a Rayleigh number as large as 10⁶, mantle circulation with large aspect ratio is driven dominantly by the motion of the lithosphere rather than by temperature gradients within the flow. Gravity, topography and heat flow are determined and implications for convection in the upper mantle are discussed.     

Dynamics of the mid-ocean ridge plate boundary: Recent observations and theory

The global mid-ocean ridge system is one of the most active plate boundaries on the earth and understanding the dynamic processes at this plate boundary is one of the most important problems in geodynamics. In this paper I present recent results of several aspects of mid-ocean ridge studies concerning the dynamics of oceanic lithosphere at these diverging plate boundaries. I show that the observed rift valley to no-rift valley transition (globally due to the increase of spreading rate or locally due to the crustal thickness variations and/or thermal anomalies) can be explained by the strong temperature dependence of the power law rheology of the oceanic lithosphere, and most importantly, by the difference in the rheological behavior of the oceanic crust from the underlying mantle. The effect of this weaker lower crust on ridge dynamics is mainly influenced by spreading rate and crustal thickness variations. The accumulated strain pattern from a recently developed lens model, based on recent seismic observations, was proposed as an appealing mechanism for the observed gabbro layering sequence in the Oman Ophiolite. It is now known that the mid-ocean ridges at all spreading rates are offset into individual spreading segments by both transform and nontransform discontinuities. The tectonics of ridge segmentation are also spreading-rate dependent: the slow-spreading Mid-Atlantic Ridge is characterized by distinct bulls-eye shaped gravity lows, suggesting large along-axis variations in melt production and crustal thickness, whereas the fast-spreading East-Pacific Rise is associated with much smaller along-axis variations. These spreading-rate dependent changes have been attributed to a fundamental differences in ridge segmentation mechanisms and mantle upwelling at mid-ocean ridges: the mantle upwelling may be intrinsically plume-like (3-D) beneath a slow-spreading ridge but more sheet-like (2-D) beneath a fast-spreading ridge.     

流变边界层及其对华北克拉通热/地震岩石圈底界差异的意义

二维热传导/对流数值模型显示,纯传导的固体岩石圈与纯对流的流体软流圈之间存在一过渡层,即流变边界层,其间传导与对流共同作用来传递热量.流变边界层厚度主要由软流圈黏性系数（η）控制,而受固体岩石圈厚度及热状态影响很小.随着η从1×10²¹Pa·s降低至1×10¹⁹ Pa·s,流变边界层也随之减薄,流变边界层的厚度与lg（η）成正比. 流变边界层的存在是造成热岩石圈与地震岩石圈厚度差异的重要因素. 全球典型克拉通岩石圈的对比结果表明,地震岩石圈厚度普遍大于热岩石圈厚度,二者的差异多数在70~90 km,很好地验证了流变边界层的存在. 研究发现二者的差异在华北克拉通自西向东逐渐减小：由西部鄂尔多斯的约80 km减少至渤海湾盆地的约20 km. 反映出华北克拉通岩石圈下部流变边界层厚度自西向东减薄,意味着软流圈黏性系数自西向东逐渐降低.这可能与中生代太平洋俯冲脱水形成的低黏大地幔楔有关,从一侧面印证了太平洋俯冲对华北克拉通破坏的影响.     

The Myasnikov global theory of the evolution of planets and the modern thermochemical model of the Earth’ls evolution

The thermochemical model of the authors is shown to be naturally related to the general theory of V.P. Myasnikov. A heterogeneous modification of this homogeneous theory is described in light of the present ideas on the differentiation of the mantle substance at the boundary with the core and its eclogitization during submersion from the outer boundary and at the endothermic phase transition at a depth of 670 km. The Earth’ls evolution from an initial hot state is numerically modeled. The evolution is shown to start with an abrupt mantle overturn followed by a long period of steady evolution. Global mantle overturns recur a few times, gradually weaken, and are transformed into regional avalanches. The spatial configuration of overturns is represented by a predominant funnel-shaped sink and a few (three to five) ascending superplumes, which convincingly explains the causes of the formation of supercontinents, the opening of oceans, and the observed asymmetry of the planet. The times of overturns remarkably correlate with geological data on the existence of supercontinents. The processes of core growth, mantle cooling, and crust formation exhibit a clearly expressed stepwise behavior. The supplementation of the endothermic phase transition by chemical transformations favors the overcoming of the phase barrier between the upper and lower mantle, enhances the nonlinearity of mantle convection, and imparts a heterocyclic pattern to the process of evolution. It is shown that the lower mantle plume of chemical origin is fragmented by the phase transition into parts that, interacting with the thermal convection, generate a system of upper mantle plumes. This modeling provides an explanation of the coeval systems of oceanic plateaus and continental traps observed on the surface.     

Long wavelength thermal convection between non-conducting boundaries

Non-linear Rayleigh-Bénard convection in a fluid layer is considered as a model of convection in the Earth's upper mantle. Previous studies have shown that when the temperature is held fixed at one of the boundaries of the layer, convection takes place in cells of width of the order of the layer depth or less. We investigate the effects of a different thermal boundary condition, in which the flux of heat is held fixed on both layer boundaries; then if this flux is just greater than that required for the onset of convection, motion takes place on horizontal scales much greater than the layer depth. An analytical treatment of the equations, based on an expansion in the depth-to-width ratio of the cells, shows that cells of a definite horizontal scale are the fastest growing according to linearised theory, but that these cells are unstable to ones of larger wavelength than themselves. Thus the dominant wavelength lengthens with time. The results hold whether the heat flux is generated internally of comes from beneath the layer. These results produce flow patterns similar to those found when the heat flux is much greater than the critical value. The results have important consequences for the understanding of mantle convection.     

Convection and gravity waves in two layer models. I. Overstable modes driven in conducting boundary layers

Abstract

Stability analysis is formulated for a two-layer fluid model in which the upper and lower layers are convectively stable and unstable, respectively. With discontinuities in viscosity and conductivity at the interface, the exchange of stability does not generally hold and overstability is possible. A detailed analytical treatment is presented for the case of small viscosity and conductivity in which viscous and conducting boundary layers are formed at the interface.

The usual damping effect due to the energy dissipation by viscosity and thermal conductivity exists irrespective of whether the mode is the convection or the gravity wave, but, for larger horizontal wave lengths, the effect of the boundary layer can become more important. The jump in the thermal conductivity in the boundary layer can give rise to overstability of the gravity wave in agreement with Souffrin and Spiegel (1967). The jump in the viscosity provides a self-catalytic action for the unstable flow if the viscosity is assumed to be the nonlinear turbulent viscosity due to the motion itself. The effect, however, is not strong enough to overcome the usual viscous damping.