Physical Processes in Earth and Environmental Sciences Phần 7

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LEED-Ch-04.qxd 11/26/05 14:08 Page 190 190 Chapter 4 4.17.6 Earthquakes and strain slip). The remainder dissipates over weeks or months by aftershocks as smaller and smaller roughness elements on the fault plane shear past each other until all the strain In the introduction to this chapter we noted that energy is released. If the fault responsible breaks the earthquakes were marked by release of seismic energy Earth’s surface then the coseismic deformation is that along shear fracture planes. This energy is released partly measured along the exposed fault scarp whose length may as heat and partly as the elastic energy associated with rock reach tens to several hundreds of kilometers. compression and extension. In the elastic rebound theory Different types of faults give rise to characteristic ﬁrst of faults and earthquakes the strain associated with tec- motions of P-waves and it is this feature that nowadays tonic plate motion gradually accumulates in speciﬁc zones. enables the type of faulting responsible for an earthquake to The strain is measurable using various surveying be analyzed remotely from seismograms, a technique known techniques, from classic theodolite ﬁeld surveys to as fault-plane solution. Previously it was left to ﬁeld surveys to satellite-based geodesy. In fact, the earliest discovery of determine this, often a lengthy or sometimes impossible task. what we may call preseismic strain was made during inves- The ﬁrst arrivals in question are those up or down peaks tigations into the causes of the San Francisco earthquake measured initially as the ﬁrst P-wave curves on the seismo- of 1906, when comparisons were made of surveys docu- gram record (Fig. 4.142). It is the regional differences in the menting c.3 m of preearthquake deformation across the nature of these records caused by the systematic variation of San Andreas strike-slip fault. We have already featured the compression and tension over the volume of rock affected by results of modern satellite-based GPS studies in decipher- the deformation that enables the type of faulting to be deter- ing ongoing regional plate deformation in the Aegean area mined. This is best illustrated by a strike-slip fault where of the Mediterranean (Section 2.4). All such geodetical compression and tension cause alternate zones of up (posi- studies depend upon the elastic model of steady accumu- tive) or down (negative) wave motion respectively as a ﬁrst lating seismic strain and displacement. But then suddenly arrival wave at different places with respect to the orientation the rupture point (Section 3.15) is exceeded and the of the fault plane responsible (Fig. 4.142). When plotted on strained rock fractures in proportionate or equivalent mag- a conventional lower hemisphere stereonet (Cookie 19), nitude to the preseismic strain. This coseismic deformation with shading illustrating compression, the patterns involved represents the major part of the energy ﬂux and is dissi- are diagnostic of strike-slip faulting. pated in one or more rupture events (order 10 2–101 m (c) (a) Down, pull, Up, tension push, Up, compression push, compression (b) Down, pull, tension Fig. 4.142 To illustrate the use of ﬁrst motion polarity in determining the type of fault slip, in this case the right-lateral San Andreas strike-slip fault; (b) 1906 San Francisco quake ground displacement; (c) San Andreas dextral strike-slip fault and schematic ﬁrst P-wave arrival traces.
LEED-Ch-04.qxd 11/26/05 14:08 Page 191 Flow, deformation, and transport 191 4.18 Molecules in motion: kinetic theory, heat conduction, and diffusion We have so far discussed ﬂow in terms of bulk movement mole of any gas. This astonishing property has come to be and mixing but there are also a broad class of systems in known as Avagadro’s constant, Na, in honor of its discov- which transport of some property is achieved by differen- erer. It implied to early workers in molecular dynamics that tial motion of the constituent molecules that make up a molecules of different gases must have masses that vary stationary system rather than by bulk movement of the directly according to atomic weight, for example, oxygen whole mass. Such systems are not quite in equilibrium, in molecules have greater mass than nitrogen molecules. the sense that properties like temperature, density, and 3 Following on from Avagadro’s development, it became concentration vary in space. For example, a recently erupted obvious that Boyle’s law (Section 3.4) relating the pres- lava ﬂow cools from its surfaces in contact with the very sure, temperature, volume, and mass of gases implies that much cooler atmosphere and ground. A second example for any given temperature and pressure, one mole of any might be a layer of seawater having a slightly higher salin- gas must occupy a constant volume. This is 22.4 L (22.4 10 3 m3) at 0°C and 1 bar. ity that lies below a more dilute layer. The arrangement is dynamically stable in the sense that the lower layer has a 4 It follows that each molecule of gas within a mole volume can occupy a volume of space of some 4 10 26 m3. negative buoyancy with respect to the upper, yet over time 5 Typical molecules have a radius of some 10 10 m and the two layers tend to homogenize across their interface in an attempt to equalize the salinity gradient at the interface. may be imagined as occurring within a solid volume of some 4 10 30 m3. In both examples there is a long-term tendency to equal- ize properties. In the ﬁrst it is the oscillation of molecules From these simple considerations it seems that a gas molecule only takes up some 10 4 of the volume available along a gradient of temperature and in the second the motion of molecules down a concentration gradient. But to it, reinforcing our previous intuition that gases are how fast and why do these processes occur? dilute. The phenomenon of molecular diffusion in gases, say of smell or temperature change, occurs extremely rapidly in comparison to liquids because of the extreme 4.18.1 Gases – dilute aggregates of velocity of the molecules involved. Also, since gaseous molecules in motion temperature can clearly vary with time, it must be the collisions between faster (hotter) and slower (cooler) The gaseous atmosphere is in constant motion due to its molecules that bring about thermal equilibrium. And since reaction to forces brought about by changes in environ- heat is a form of energy it follows that the motion of mental temperature and pressure. Volcanic gases also move molecules must represent the measure of a substance’s in response to changes brought about by the ascent of intrinsic or internal energy, E (Section 3.4). Let us examine molten magma through the mantle and crust. When we these ideas a little more closely. study the dynamics of such systems we must not only pay attention to such bulk motions but also to those of 4.18.2 Kinetic theory – internal energy, temperature, constituent molecules that control the pressure and and pressure due to moving molecules temperature variations in the gas. Compared to any speed with which bulk processes occur, the internal motions of stationary gases involve much higher speeds. The view of a It is essential here to remember the distinction between gas as a relatively dilute substance in which its constituent velocity, u, and speed, u. If we isolate a mass of gas in a molecules move about with comparative freedom container then it is clear that by deﬁnition there can be no (Section 2.1) is reinforced by the following logic: net molecular motion, as the motions are random and will 1 A mole of a gas molecule is the amount of mass, in cancel out when averaged over time (Fig. 4.143). Neither grams, equal to its atomic weight. Nitrogen thus has a can there be net mean momentum. In other words gas mole of mass 28 g, oxygen of 32 g, and so on. Any quan- molecules have zero mean velocity, u 0. However, the tity of gas can thus be expressed by the number, n, of randomly moving individual molecules have a mean moles it contains. speed, u, and must possess intrinsic momentum and there- 2 A major discovery at the time when molecular theory fore also mean kinetic energy, E. In a closed volume of any was still regarded as controversial, was that there are always gas the idea is that molecules must be constantly bom- exactly the same number of molecules, 6 1023, in one barding the walls of the container – the resulting transfer
LEED-Ch-04.qxd 11/26/05 14:08 Page 192 192 Chapter 4 In this thought experiment the container has its right hand wall as an elastic membrane. Individual gas molecules are shown approximately to scale so that the average separation distance between neighbors is about 20 times molecular u radius. The individual molecules all have their own instanta- neous velocity, u, but since the directions are random the sum of all the velocities, Σu, and therefore the average velocity must deformable elastic wall _ urms = (u 2) 0.5 be zero. This is true whether we compute the average velocity of an individual molecule over a long time period or the instantaneous average velocity of a large number of individual molecules. The arrows denote instantaneous velocities. Nevertheless the gas molecules have a mean speed, u, that is not zero. This is because although the directions cancel out the magnitudes of the molecular velocities, that is, their speeds, do not. In such cases we compute the mean velocity by finding the value of the mean square of all the velocities and taking the square The internal energy, E, of any gas is the sum of all the root, the result being termed the root-mean-square velocity, or molecular kinetic energies. In symbols, for a gas with N urms in the present notation. molecules: This is NOT the same as the mean speed, a feature you can E = N(0.5 mu2rms) easily test by calculating the mean and rms values of , say, 1, 2, Or we may alternatively view the molecular velocity as a and 3. direct function of the thermal energy: u2rms = 2E/mN Fig. 4.143 Molecular collisions and the internal thermal energy of a gas. One molecule is shown striking the elastic wall, which responds by displacing outward, signifying the existence of a gaseous pressure force and hence molecular kinetic energy transfer. or ﬂux of individual molecular momentum is the origin of have informed understanding of the behavior, ﬂow, and gaseous pressure, temperature, and mean kinetic energy deformation of loose granular solids, from Reynolds’ con- (Fig. 4.144). These properties arise from the mean speed cept of dilatancy to the motion of avalanches (Section 4.11). of the constituent molecules: every gas possesses its own internal energy, E, given by the product of the number of 4.18.3 Heat ﬂow by conduction in solids molecules present times their mean kinetic energy. In a major development in molecular theory, Maxwell calcu- lated the mean velocity of gaseous molecules by relating it In solid heat conduction, it is the molecular vibration to a kinetic version of the ideal gas laws, together with a frequency in space and time that varies (Fig. 4.145). Heat statistical view of the distribution of gas molecular speed. energy diffuses as it is transmitted from molecule to mole- The resulting kinetic theory of gases depends upon the cule, as if the molecules were vibrating on interconnected simple idea that randomly moving molecules have a proba- springs; we thus “feel” heat energy transfer by touch as it bility of collision, not only with the walls of any container, transmits through a substance. In fact, all atoms in any but also with other moving molecules. Each molecule thus state whatsoever vibrate at a characteristic frequency about has a statistical path length along which it moves with its their mean positions, this deﬁnes their mean thermal characteristic speed free from collision with other mole- energy. Vibration frequency increases with increasing cules: this is the concept of mean free path. Since gases are temperature until, as the melting point is approached, dilute the time spent in collisions between gas molecules is the atoms vibrate a large proportion of their interatomic infrequent compared to the time spent traveling between separation distances. Conductive heat energy is always collisions. Thus the typical mean free path for air is of transferred from areas of higher temperature to areas of order 300 atomic diameters and a typical molecule may lower temperature, that is, down a temperature gradient, experience billions of collisions per second. Similar ideas dT/dx, so as to equalize the overall net mean temperature.
LEED-Ch-04.qxd 11/26/05 14:09 Page 193 Flow, deformation, and transport 193 Before collision 2D elastic collision between a molecule and wall u1 = ux + uy Signs and coordinates – ux After collision uy u2 = –ux + uy +y u2 Momentum change is thus u1 ∆P = mu2 – mu1 = m(–ux + uy) – m(ux + uy) –x +x uy or ∆P = –mux –mux = –2mux –y ux And Momentum transfer is ∆P = –(–2mux) = 2mux The overall pressure, force per unit area, acting on any surface is given by the contribution of all molecules colliding with the wall in unit time. This number will be half of the total molecules, N, in any volume, V (the other half traveling away from the wall over the same time interval). The pressure is 0.5(N/V)(2mux). An N is given by ux dt and p = mux2 N/V. Finally, since ux2 = 1/3urms2 and urms2 = 2E/mN, we have the important result that: pV = 2/3(E). Fig. 4.144 Origin of molecular pressure and its relation to internal thermal energy: link between mechanics and thermodynamics. Atomic vibration Hotter Cooler Heat flow Fig. 4.145 Conductive heat ﬂow in solids is movement of heat energy in the form of atomic vibrations from hot areas to cool areas so as to reduce temperature. A steady-state condition of heat ﬂow occurs when the quantity of heat arriving and leaving is equal. Many natu- ral systems are not in steady state, for example, the cooling of molten magma that has risen up into or onto the crust (Fig. 4.146; Section 5.1) and in such cases the physics is a little more complicated (Cookie 20). The rate of movement of heat by conduction across unit area, Q, is controlled by a bulk thermal property of the substance in question, the thermal conductivity, k, so that overall, for steady-state conditions when all temperatures are constant with time, Q kdT/dx (Fig. 4.147). Conductivity relates to spatial rate of transfer, the ef ﬁ- ciency of a substance to transfer its internal heat energy from one point to another. Heat transfer may also be Fig. 4.146 Bodies of molten magma intruded into the crust like the dyke shown here (see Section 5.1) or extruded as lava ﬂows cool by expressed via a quantity known as the thermal diffusivity, conduction of heat energy outward into adjacent cooler rocks (or (kappa; dimensions L2T 1), deﬁned as k/ c, where is the the atmosphere in the case of lava). The rate of cooling and the density and c is the speciﬁc heat (Section 2.2). It indicates gradual decay of temperature with time may be calculated from the time rate of heat energy dissemination, being the ratio variants of Fourier’s law of heat conduction (see Cookie 20).
LEED-Ch-04.qxd 11/26/05 14:09 Page 194 194 Chapter 4 between conductivity (rate of spatial passage of heat energy must be transferred. This might be the thickness of energy) and thermal energy storage (product of speciﬁc a lava ﬂow or dyke, the whole Earth’s crust, an ocean cur- heat capacity per unit mass and density, that is, speciﬁc rent, or air mass. The conductive time constant, , is then given by l 2/ . heat per unit volume). Thermal diffusivity gives an idea of how long a material takes to respond to imposed tempera- ture changes, for example, air has a rapid response and 4.18.4 Molecular diffusion of heat and mantle rock a slow one. This leads to a useful concept con- concentration in ﬂuids cerning the characteristic time it takes for a system that has been heated up to return to thermal equilibrium. Any system has a characteristic length, l, across which the heat In ﬂuids it is the net transport of individual molecules down the gradient of temperature or concentration that is respon- sible for the transfer; the process is known as molecular Heat axis diffusion. As before, the process acts from areas of high to HIGH LOW For 1D variation of heat at any instant low temperature or concentration so as to reduce gradients T + δT T the flux, Q , goes from high to low and equalize the overall value (Fig. 4.148). For temperature temperature. the rate of transfer depends upon the thermal conductivity, as for solids, but the process now occurs by collisions Q = heat flux between molecules in net motion, the exact rate depending upon the molecular speed of a particular liquid or gas at par- k = thermal conductivity ticular temperatures. For the case of concentration the over- Q Q = –kδT/δx all rate depends on both the concentration gradient and upon molecular collision frequency and is expressed as a dif- This is the heat conduction equation fusion coefﬁcient. The rate of molecular diffusion in gases is rapid, reﬂecting the high mean molecular speeds in these Applies when conditions do not change with time. substances, of the order several hundred meters per second. The rapidity of the process is best illustrated by the passage of smell in the atmosphere. By way of contrast the rate of x + δx x x-axis molecular diffusion in liquids is extremely slow. ID heat conduction. Fig. 4.147 (a) (b) n + δn concentration axis n δn For 1D variation of molecular HIGH LOW HIGH LOW concentration at any instant n + δn n δn/δt = 0 the flux J, goes from high to low Jin Jout concentration Jin = Jout area unit n = no mols./unit vol. = conc. J = –Dδn/δx J = no particles crossing unit area per sec. in direction >x (c) J D = a diffusion coefficient HIGH LOW measuring the rate of diffusion Jin = Jout / Jx Jx + dx J = –Dδn/δx δn/δt = 0 / This is Fick´s law of diffusion. Dδ2n/δx2 = δn/δt x + δx Applies when conditions do not x change with time. x + δx x Particles can accumulate or be lost; x-axis there may be a gradient of J across x Fig. 4.148 Molecular diffusion occurs in liquids and gases as translation of molecules from high concentration/temperature areas to low concentration/temperature areas so as to eliminate gradients. The rate of diffusion is rapid for gases and slow for liquids (a) Fick’s law of 1D diffusion, (b) Derivation: Steady state diffusion (time independent), and (c) time variant diffusion (time/space dependent).
LEED-Ch-04.qxd 11/26/05 14:09 Page 195 Flow, deformation, and transport 195 4.18.5 Fourier’s famous law of heat conduction where conduction or molecular diffusion depends upon time. In the latter case, some mathematical development leads to a relationship in which the temperature of a Illustrated (Fig. 4.148) are the two cases of heat cooling body varies as the square root of time elapsed conduction and molecular diffusion for (1) steady state, (see Cookie 20). with no variation in time and (2) the more complex case 4.19 Heat transport by radiation 4.19.1 Solar radiation: Ultimate fuel for the distance traveled. The fraction of monochromatic energy climate machine transmitted is given by the Lambert–Bouguer absorption law stated opposite (Box 4.4). Further latitude dependence of incoming solar energy received by Earth’s surface arises Solar energy is transmitted throughout the Solar System as from the simple fact that oblique incident light must warm electromagnetic waves of a range of wavelengths, from a larger surface area that can be warmed by normally inci- x-rays to radio waves, all traveling at the speed of light. dent light. In addition to mean absorption of energy by The Sun’s maximum energy comes in at a short wave- atmospheric gases, radiative energy is also reﬂected, scat- length of about 0.5 m in the visible range. Much shorter tered, and absorbed by wind-blown and volcanic dust and wavelengths in the ultraviolet range are absorbed by ozone natural and pollutant aerosol particles in the atmosphere. and oxygen in the atmosphere. The magnitude of incom- The amount of dust varies over time (by up to 20 percent ing radiation is represented by the solar constant, deﬁned or more), exerting a strong control on the magnitude of as the average quantity of solar energy received from normal- incoming solar radiation. Because of scattering, absorp- incidence rays just outside the atmosphere. It currently tion, and reﬂection, it is usual to distinguish the direct has a value of about 1,366 W m 2, a value which has ﬂuc- radiation received by any surface perpendicular to the Sun tuated by about 0.2 percent over the past 25 years. As from the diffuse radiation received from the remainder of discussed below it is possible that over longer periods the atmospheric hemisphere surrounding it. Continuous the irradiance might vary by up to three times historical cloud cover reduces direct radiation to zero, but some variation. radiation is still received as a diffuse component. Although the outer reaches of the atmosphere receive equal amounts of solar radiative energy, speciﬁc portions of the atmosphere and Earth’s surface receive variable 4.19.2 Sunspot cycles: Variations in solar energy levels (Fig. 4.149). One reason is that solar radia- irradiance and global temperature ﬂuctuations tive energy is progressively dissipated by scattering and absorption en route from the top of the atmosphere The extraordinary dark patches on the face of the downward. Since light has to travel further to reach all otherwise bright sun are visible when a telescopic image is surface latitudes north and south of a line of normal projected onto a screen and viewed. The dark blemishes incidence, it is naturally weaker in proportion to the 1,366 W m–2 on perpendicular surface x Solar constant = incoming solar irradiance Local Thickness outside earth´s atmosphere path of atmosphere length 1,366 W m–2 on perpendicular surface Fig. 4.149 Higher latitude radiation travels further through the atmosphere and is thus attenuated and scattered more. The more attenuated higher latitude radiance must also act upon a larger earth surface area.
LEED-Ch-04.qxd 11/26/05 14:09 Page 196 196 Chapter 4 Box 4.4 Box 4.4(a) Lambert–Bouguer absorption law. Box 4.4(b) Other relevant aspects regarding Solar radiation d = exp(-bx) 1 The solar radiation “constant” has probably decreased The fraction of energy, d, transmitted through the over geological time since Earth nucleated as a planet. This atmosphere depends on the path length, x, and an has severe implications for estimates of geological palaeo climates. absorption coefficient, b, whose value at sea level is 2 Sunspots cause variations in the incoming solar energy. about 0.1 km–1. 3 The number of sunspots seem to vary over about an In 10 km of travel, only 1/e (37%) of energy remains. 11-year cycle. There is increasing evidence that a longer term variability has severe effects on the global climate system for example, the 80-year long Maunder Minimum in sunspots coincides with the “Little Ice Age” of northern Europe. are not ﬁxed and though cooler than surrounding areas incident radiation. Thus solids like ice, rocks, and sand are the sun’s irradiance is increased due to unusually high opaque and the short wavelength solar radiation is either bursts of electromagnetic activity from them, with solar reﬂected or absorbed. Water, on the other hand, is translu- ﬂaring generating intense geomagnetic storms. The dark cent to solar radiation in its surface waters, although when patches were well known to ancient Chinese, Korean, and the angle of incidence is large in the late afternoon or early Japanese astronomers and to European telescopic morning, or over a season, the amount of reﬂected radia- observers from the late-Medieval epoch onward: nowadays tion increases. It is the radiation that penetrates into the they are termed sunspots. We owe this long historical shallow depths of the oceans that is responsible for the record to the dread with which the ancient civilizations energy made available to primary producers like algae. It is regarded sunspots, as omens of doom. Systematic visual useful to have a measure of the reﬂectivity of natural observations over a c.2 ky time period reveals distinct surfaces to incoming shortwave solar radiation. This is the waxing and waning of the area covered by sunspots. albedo, the ratio of the reﬂected to incident shortwave An approximately 11-year waxing and waning sunspot radiation. Snow and iceﬁelds have very high albedos, cycle is well established, with a longer multidecadal reﬂecting up to 80 percent of incident rays, while the Gleissberg cycle of about 90 years also evident. Because the equatorial forests have low albedos due to a multiplicity of electromagnetic effects of sunspot activity reach all the way internal reﬂections and absorptions from leaf surfaces, into Earth’s ionosphere, where they interfere with (reduce) water vapor, and the low albedo of water. The high albedo the “normal” incoming ﬂux of cosmic rays, longer-term of snow is thought to play a very important feedback role proxies gained from measuring the abundance of cosmo- in the expansion of snowﬁelds during periods of global cli- genically produced nucleides (like 14C preserved in tree- mate deterioration. rings) accurately push back the radiation record to 11 ka. What emerges is a fascinating record of solar misbehavior, 4.19.4 Earth’s reradiation and the “greenhouse” culminating in the record-breaking solar activity of the last concept 50 or so years, which is the strongest on record, ever. This increased irradiance is thought to contribute about one- third to the recent global warming trend. But this estimate Incoming shortwave solar radiation in the visible is model driven: what if the models are wrong? A chilling wavelength range has little direct effect upon Earth’s thought is the fact that the global “Little Ice Age” of atmosphere, but heats up the surface in proportion to the 1645–1715 correlates exactly with the sunspot minimum magnitude of the incoming energy ﬂux, the surface albedo, named the Maunder minimum. and the thermal properties of the surface materials. It is the reradiated infrared radiation (Fig. 4.150) that is responsible for the elevation of atmospheric temperatures 4.19.3 Reﬂection and absorption of radiated energy above those appropriate to a gray body of zero absolute temperature. It was the savant, Fourier, who ﬁrst postu- The Sun’s radiation falls upon a bewildering array of natu- lated this loss of what he called at the time, chaleur obscure, ral surfaces; each has a different behavior with respect to in 1827. We now know that the reradiated infrared energy
LEED-Ch-04.qxd 11/26/05 14:09 Page 197 Flow, deformation, and transport 197 uv Visible Infrared Stefan–Boltzmann law: 5.0 Energy of radiation from a body is proportional Suns blackbody radiation at 6,000 K to the 4th power of absolute temperature. 2.0 Wein´s displacement law: Wavelength of maximum energy from a body is 1.0 Extraterrestrial solar radiation Energy of radiation: LY min–1 mm–1 inversely proportional to absolute temperature. 0.5 Diffuse solar radiation at Earth surface 0.2 Direct beam normal incidence solar radiation at Earth’s surface 0.1 Earth’s blackbody radiation at 300 K 0.05 0.02 Infrared radiation lost to space 0.01 The serrated nature of the grayscale radiation curves 0.005 is due to selective absorption of certain wavelengths by particular atmospheric and stratospheric gases. 0.002 0.001 0.1 0.2 0.5 1.0 2.0 5.0 10 20 50 Radiation wavelength: microns, µm Chief absorption bands by greenhouse gases O2 O3 H2O CO2 H2O O3 CO2 H2O uv radiation filter Fig. 4.150 The great energy transfer from solar short wave to reradiated long wave radiation. flux is of the same order as that received from the Sun at demonstrated by the geologist de Saussure who exposed the Earth’s surface. Some of this energy is lost into space a black insulated box with a glass lid to sunlight, then for ever but a significant proportion is absorbed and comparing the elevated internal temperature of the trapped by the gases of the atmosphere and emitted back closed box with that of the box when open. Thus it is the to Earth as counter radiation where together with absorption spectra of our atmospheric gases that ulti- absorbed shortwave radiation it does work on the atmos- mately drives the atmospheric circulation (Fig. 4.150). phere by heating and cooling it. During this process Water vapor is the most important of these gases, water vapor may condense to water, or vice versa, and the strongly absorbing at 5.5–8 and greater than 20 m effects of differential heating give rise to density differ- wavelengths. Carbon dioxide is another strong absorber, ences, which drive the general atmospheric circulation. but this time in the narrow 14–16 m range. The 10 per- The insulating nature of Earth’s atmosphere, like that of cent or so of infrared radiation from the ground surface the glass in a greenhouse, is nowadays referred to as the that escapes directly to space is mainly in the 3–5 and “greenhouse” effect. The general concept was originally 8–13 m wavelength ranges. 4.20 Heat transport by convection Convection is the chief heat transfer process above, on and how do these motions relate to convection? We shall within Earth. We see its effects most obviously in the return to the question below and in later chapters atmosphere, for example, in the majestic cumulonimbus (Sections 5.1 and 5.2). clouds of a developing thundercloud or more indirectly in the phenomena of land and sea breezes. It is fairly obvious 4.20.1 Convection as energy transfer by bulk motion in these cases that convection is occurring, but what about within Earth? It is now widely thought that Earth’s silicate mantle also convects, witnesses the slow upwelling of man- We have seen previously that the heat transfer processes of tle plumes and motion of lithospheric plates. But exactly radiation and conduction cause the temperature and internal
LEED-Ch-04.qxd 11/26/05 14:09 Page 198 198 Chapter 4 energy of materials to change. Convection depends upon is conventionally considered as negligible by a dodge these transfer processes causing an energy change that is known as the Boussinesq approximation. This assumes that suf ﬁcient to set material in motion, whereby the moving all accelerations in a thermal ﬂow are small compared to substance transfers its excess energy to its new surround- the magnitude of g. ings, again by radiation and conduction. We stress that the 2 The gradient in viscosity on the other hand will cause convection process is an indirect means of heat transfer; a change in the viscous shear resistance once convective convection is not a fundamental mechanism of heat ﬂow, motion starts. The extreme complexity of free convec- but is the result of activity of conduction or radiation. tion studies arises from considering both gradients of When convection results from an energy transfer suf ﬁcient density and viscosity at the same time; the Boussinesq to cause motion, as for example in a stationary ﬂuid approximation assumes that only density changes are heated/cooled from below or heated/cooled at the side, considered. we call this free (or natural) convection. Alternatively, it The magnitude of density change is given by o T, may be that a turbulent ﬂuid is already in motion due to where is the coef ﬁcient of thermal expansion and o is external forcing independent of the local thermal condi- the original or a reference density. The term g o T then tions. Here ﬂuid eddies will transport any excess heat signiﬁes the buoyancy force (Section 3.6) available during energy supplied along with their own turbulent momen- convection and is an additional force to those already tum. Convective heat transfer, such as that accompanying familiar to us from the dynamical equations of motion eddies forming in the turbulent boundary layer of an developed previously (Section 3.12). When the ﬂuid is already moving ﬂuid over a hotter surface is termed forced warmer than its surroundings the buoyancy force is overall convection (or sometimes as advection). positive: this causes the ﬂuid to try to move upward. When the net buoyancy force is negative the ﬂuid tries to sink downward. 4.20.2 Free, or natural, convection: Basics In detail it is extremely dif ﬁcult to determine the velocity or the velocity distribution of a freely convecting The fundamental point about convection is that it is a ﬂow. This is because of a feedback loop: the velocity is buoyant phenomenon due to changed density as a direct determined by the gradient of temperature but this gradi- consequence of temperature variations. We have seen previ- ent depends on the heat moved (advected) across the ously (Section 2.1) that values for ﬂuid density are highly velocity gradient! So we must turn to experiment and sensitive to temperature. Thus if we consider an interface the use of scaling laws and dimensionless numbers such as between ﬂuids or between solid and ﬂuid across which there the Prandtl and Peclet numbers discussed below. is a temperature difference, T, caused by conduction or radiation, then it is obvious that the heat transfer will cause 4.20.3 The nature of free convection gradients in both density and viscosity across the interface. These gradients have rather different consequences. 1 The gradient in density gives a mean density contrast, A simple example is convection in a ﬂuid that results from , and a gravitational body force, g per unit motion adjacent to a heated or cooled vertical wall. In the volume, that plays a major role in free convection. former case, illustrated for heating in Fig. 4.151, the ther- The density contrast should also apply to the acceleration- mal contrast is maintained as constant and the heat is related term in the equation of motion (Box 4.5) but since transferred across by conduction. As the ﬂuid warms up this complicates matters considerably, any effect on inertia immediately adjacent to the wall it expands, decreases in Box 4.5 Equation of motion for a convecting Boussinesq fluid. ACCELERATION = PRESSURE FORCE + VISCOUS FORCE + BUOYANCY FORCE Time : Temperature balance equation for a convecting Boussinesq fluid ∆T = CONDUCTION IN + INTERNAL HEAT GENERATION – HEAT ADVECTION OUT.
LEED-Ch-04.qxd 11/26/05 14:09 Page 199 Flow, deformation, and transport 199 T2 >T1 z Vertical slot δ d (d) Horizontal slot Tw T1 T ∆T d T2 ∆T T1 To h T2 l w Side view Heated wall Fluid Counter-rotating cells at reservoir w=0 Ra > c.1,700 at To (e) Horizontal reservoir T1 Free surface ∆T T2 Thermal boundary layer thickness, d, temperature, Side T, velocity, w. (a) (b) (c) view single multiple turbulent cell cells cell y Plan Fig. 4.151 Development of a free convective thermal boundary layer > Rayleigh No. view in a wide ﬂuid reservoir adjacent to a vertical heated wall. Fig. 4.152 Convection in vertical slots and in horizontal slots and density, and when the buoyancy force exceeds the resisting reservoirs. force due to viscosity it moves upward along the wall at constant velocity, with the overall negative buoyancy force by conduction the moving ﬂuid takes on extraordinary in balance with pressure and viscous forces. At this time, forms. We illustrate convective ﬂows within vertical or hor- the background heat being continuously transferred across izontal wall-bounded slots and in open containers the wall by conduction, a portion is now transporting (Fig. 4.152). Here the convection takes the form of single upward by convection within a thin thermal boundary (Fig. 4.152a) or multiple (Fig. 4.152b) vertical cells, tur- layer. The general form of the boundary layer and of the bulent vertical cells (Fig. 4.152c), nested counter rotating temperature and velocity gradients across it are illustrated cells seen as polygons in plan view (Figs 4.152d, e and in Fig. 4.151. This situation encourages us to think about 4.153) or multiple parallel convective cells or rolls that the possible controls upon convection and upon the adjust to both the shape of the containing walls and the nature of the associated boundary layers, for it must be the presence of a free surface (Figs 4.154 and 4.155). The balance between a ﬂuid’s viscosity and thermal diffusivity polygonal convective cells may form under the inﬂuence of that controls the degree and rate of conduction versus variations in surface tension caused by warming and cooling convection of heat energy and therefore the rate of trans- and are termed Bérnard convection cells. Perhaps the com- fer of temperature and velocity. We might imagine that monest form of convection in nature involves the heating of when the viscosity: diffusivity ratio is high then the veloc- a ﬂuid by a point, line, or wall source to produce laminar or ity boundary layer is thick compared to the temperature turbulent thermal plumes (Figs 4.156 and 4.159). Such one, vice versa for a low ratio. In detail the prediction of plumes play an important role in the vertical transport of boundary layer properties depends critically upon whether heat in the Earth’s mantle, oceans, and atmosphere. the ﬂows are laminar or turbulent, hence the consideration of a thermal equivalent to Re. The foregoing analysis has been rather dry and a little 4.20.4 Forced convection through a boundary layer abstract and does scant justice to the interesting patterns and scales of free convection. That the process is hardly In forced convection, the motive force for ﬂuid movement predictable and achievable by molecular scale motions is comes from some external source; the ﬂuid is forced to illustrated by the great variety of natural thunderclouds or transfer heat as it ﬂows over a surface kept at a higher by laboratory ﬂow visualizations. Once heated or cooled
LEED-Ch-04.qxd 11/26/05 14:14 Page 200 200 Chapter 4 Fig. 4.153 View from above of Bénard convection cells in a thin layer of oil heated uniformly below: the convection is driven by inhomogeneities in surface tension rather than buoyancy. The Fig. 4.154 Circular buoyancy-driven convection cells in silicone oil hexagonal cells with ﬂow out from the centers are visualized by light heated uniformly from below in the absence of surface tension. reﬂected from Al-ﬂakes. Fig. 4.155 Rayleigh–Bénard convection cells in a rectangular box ﬁlled with silicone oil being heated uniformly from below. The convection is due to buoyancy in this case. Fig. 4.156 Isotherms in a plume sourced from a heated wire and shown by an interferogram. Plume grows outward as the 2⁄5 power of Fig. 4.157 Isotherms of a laminar plume formed by convection around a heated cylinder in air. height.
LEED-Ch-04.qxd 11/26/05 14:14 Page 201 Flow, deformation, and transport 201 δ y z T u2 T To Turbulent burst Fluid eddy to u1 Tw c Fluid heated wall x reservoir w Heated wall at Tw at To c = specific heat w=0 rate of change of momentum per unit mass is of order to/(u2 – u1) rate of change of internal energy per unit mass is of order Thermal boundary layer c(T – Tw) thickness, 2δ, temperature, for Prandtl number of about 1, heat flow rate is of order T, velocity, w. c(T – Tw)to/(u2 – u1) Fig. 4.158 Development of a thermal plume generated from a heated Fig. 4.160 Visualization of Reynolds’ analogy between thermal and point source, Tp. momentum ﬂux. passage will be resistance to convective motion established by the viscous shear layer. Laminar ﬂows at low Re, where there is no motion normal to the boundary surface, must transfer the excess heat entirely by conduction. They con- sequently have very much lower heat transfer coef ﬁcients than high Re turbulent ﬂows, which have very thin viscous sublayers. In such turbulent ﬂows, once through the thin sublayer barrier, heat is rapidly disseminated as convective turbulence by upward-directed ﬂuid bursts (Section 4.5) shed off from the wall layer of turbulence (Fig. 4.160). 4.20.5 Generalities for thermal ﬂows Reynolds himself established the relationship between heat ﬂow and ﬂuid shear stress. Known now as “Reynolds’ anal- ogy” this involves a comparison of the roles of kinematic Fig. 4.159 The starting head vortex and the feeding axial column of viscosity and thermal diffusivity when these two properties a laminar plume. of ﬂuids have approximately similar values (Box 4.6). Reynolds could proceed with his analogy because, as we mentioned in Section 3.9, Maxwell had previously viewed molecular viscosity as a diffusional momentum transport temperature than the ﬂuid itself (Fig. 4.160). The process coef ﬁcient, analogous to the transport of conductive heat is highly important in many engineering situations when by diffusion. What is more natural than to express the ratio relatively cool ﬂuids are forced through or over hotter of kinematic viscosity, , to thermal diffusivity, Dtd, as a pipes, ducts, and plates. In natural situations we might characteristic property of any ﬂuid: /Dtd, is termed the envision heat transfer into a cool wind forced by regional Prandtl number, Pr (Fig. 4.160), whose value is usually pressure gradients to ﬂow over a hot desert surface. In quoted for thermal ﬂows of particular ﬂuids. To compare such convection the buoyancy force is small compared to the behavior of different ﬂuid ﬂows, not just the ﬂuids that due to ﬂuid inertia and thus the ﬂow of heat has neg- themselves, we make a more direct analogy with Re ligible effect on the ﬂow ﬁeld or the turbulence. Heat sup- (remember this expression is uL/ ). The required thermal plied by conduction to the boundary of ﬂowing ﬂuid must equivalent to Re, uL/Dtd, is termed the Peclet number, Pe, pass through the boundary layer. The major barrier to
LEED-Ch-04.qxd 11/28/05 10:14 Page 202 202 Chapter 4 Box 4.6 Some Prandtl numbers Box 4.7 Rayleigh number: Ratio of for common ﬂuids. buyoancy to viscous and thermal diffusivity Fluid Prandtl no Ra = ga(∆T)d 3 / nk Air 0.71 Steam 0.93 a = expansion coefficient, Water 7.0 ∆T = temperature difference across fluid, Crude oil 1000 d = distance across fluid, n = kinematic viscosity, k = thermal diffusivity. giving the ratio of advection to conduction of heat. possible at all (Box 4.7). This is useful for remotely deter- At small values of Pe the ﬂow has a negligible effect on the mining whether convection can occur in Earth’s mantle, temperature distribution, which can be analyzed as if the for example (Section 5.2). For convection in a horizontal ﬂuid were stationary. Finally, there is a criterion, the slot Ra must exceed about 2,000, a value thought to be far Rayleigh number, that establishes whether convection is exceeded in the mantle. Further reading Fishbane et al. (cited for Part 3) is again useful for basic dealt with in J. Simpson’s elegant and clearly written physics. Basic concepts in fluid mechanics have never (with many superb photographs) Gravity Currents been better explained than by A. H. Shapiro in Shape and (Cambridge, 1997). Folds and faults are related to stress Flow (Doubleday, New York, 1961). Introductory fluid and strain as in G. H. Davies’ and S. J. Reynolds’s dynamics presented in a careful, rigorous way, but with- Structural Geology of Rocks and Regions (Wiley, 1996), R. out undue mathematical demands, features in B. S. J. Twiss and E. M. Moores’ Structural Geology (Freeman, Massey’s Mechanics of Fluids (Van Nostrand Reinhold, 1992), and J. G. Ramsay’s and M. I. Huber’s The 1979) and M. W. Denny’s Air and Water (Princeton, Techniques of Modern Structural Geology, vol. 2 1993). Beautiful and inspirational photos of fluid flow (Academic Press, 1993). Seismology is clearly introduced visualization may be found in M. Van Dyke’s An Album and explained in B. A. Bolt’s Inside the Earth (Freeman, of Fluid Motion (Parabolic Press, 1982) and M. Samimy 1982) and the concepts beautifully illustrated in his more et al.’s A Gallery of Fluid Motion (Cambridge, 2003). popular Earthquakes and Geological Discovery (Scientiﬁc The topic of gravity currents in all their various forms is American Library, 1993).
LEED-Ch-05.qxd 11/27/05 2:18 Page 203 5 Inner Earth processes and systems 5.1 Melting, magmas, and volcanoes The ancient Greeks supposed that a river of melt, shifting ﬁrst hand evidence for localized accumulations of abun- according to Poseidon’s whims, ran under the Earth’s dant magma not far below the surface. Magma is a high surface, periodically rising to cause volcanic eruptions and temperature, multiphase mixture of crystals, liquid, and violent earthquakes. We have seen evidence (Section 4.17) vapor (gas or supercritical ﬂuid). It is impossible to meas- that most of the mantle and crust of the outer Earth is ure its temperature or other physical properties directly, solid, exhibiting elastic or plastic behavior and transmit- for once it has ﬂowed out of a volcanic vent as lava it will ting P and S waves. Yet the Low Velocity Zone marking have cooled somewhat, begun to crystallize, and would the top of the asthenosphere has a tiny amount of melt, have lost dissolved gas phases. We have to make recourse sufﬁcient to slow seismic waves somewhat and to enable to experiments that show at atmospheric pressure, typical plate motion over it (see Section 5.2). On the other hand, basalt magma is at about 1,280 C with a viscosity of more than 1,500 Holocene-active volcanoes (Fig. 5.1) give around 15 Pa s. Iceland Kamchatka Aleutians Vesuvius Mt St Yell´stone Santorini Azores Helens Jemez Fuji M´serrat Canaries S´boli Etna Unzen St Pierre Hawaii Phillipines Andes New Hebrides Kili´jaro Tonga Taupo seismic zone Holocene-active volcano or volcanic arc midocean ridge Fig. 5.1 Map showing summary world seismic belts (14 year record of M 4.5) and the location of selected Holocene-active volcanoes and the major volcanic arcs.
LEED-Ch-05.qxd 11/27/05 2:18 Page 204 204 Chapter 5 5.1.1 Difﬁcult initial questions and early clues We need to ask a number of exploratory questions about magma genesis. Why, where, and how does melting of Earth’s crust and mantle occur? Does magma exist as con- tinuous or discontinuous pockets? Why and how does magma rise to the surface? We know heat escapes from the Earth at a mean ﬂux of some 65 mW m 2 (Chapter 8). But this global mean value allows for local areas of much higher ﬂux. The geographi- cal distribution of active volcanoes and geothermal areas shows that the local production of enhanced heat energy and subsurface melting is far from accidental or random: it usually occurs associated with areas of plate creation along the midocean ridges (Iceland) or destruction along the subduction zone trenches (Section 5.2; Fig. 5.1). Therefore we conclude that melting is also associated with these large-scale processes. Exceptions, as always, disprove 1 km this rule and so we also need to look with particular inter- est at those prominent volcanic ediﬁces that occur far from Fig. 5.2 Thermal imaging view of three cinder cones and associated plate boundaries, like the Canary Islands and Hawaii. Why breaching lava ﬂow A. Note the lava levees bordering the upper does melting occur there? channel conduit and ﬂow wrinkles on the lobate lava fan margin. A younger ﬂow (black) has breached the end of the levee system at We can gather clues as to the nature of magma from B. C–E are older ﬂows. Kamchatka, Russia. observing different styles of volcanic activity. Quiescent volcanoes often gently discharge gases like steam, CO2, and SO2 from craters or subsidiary vents called fumaroles. So, we infer that magma must also contain such gas phases, presumably in dissolved form under pressure, and that the gases can discharge passively. Volcanic eruptions of lava (Fig. 5.2) are themselves often passive; thus a Hawaiian volcano emits molten lava easily as rapidly moving ﬂows. On the other hand, eruption may be far from passive; Vesuvian or Surtseyan explosions (Fig. 5.3) blast material vertically into the stratosphere as massive plumes or later- ally as horizontal jets hugging the ground. Strombolian eruptions (Fig. 5.4) shower molten material periodically skywards for a few hundred meters in a ﬁre fountain. Why this diversity of volcanic behavior into ﬂow, blast, and fountain? A ﬁrst clue came from observations made by geologists of the types of rock produced by these various Fig. 5.3 Explosive eruption column (2 km high) and accompanying styles of eruption. There is a wide range of possible chem- base surge blast, Capelinhos volcano, Azores, October 1957. The central part of the Surtseyan eruption column is an internal core-jet ical composition of magma, with more than a dozen main rich in dark-colored volcanic debris. The base surge is steam- chemical elements and a score or more of minor (trace) dominated. elements involved, for our purposes we need simply to divide magmas and igneous rocks into three types (Fig. 5.5), according to their silica content – acid, interme- lattices, tend to occur as the products of violent blasts. diate, and basic. Acidic volcanic rocks rich in silica ( 63 Rocks solidiﬁed from melts that passively ﬂow as lavas tend percent SiO2), called rhyolites, are comparatively rare as vol- to have the lowest amount of silica ( 52 percent SiO2); canic ﬂows. Rocks with intermediate amounts of silica these are the ubiquitous basalts. Basalt ﬂows are also the (52–63 percent SiO2), called dacites or andesites, often with products of submarine volcanoes at midocean ridges. minerals containing tiny amounts of water in their atomic
LEED-Ch-05.qxd 11/27/05 2:19 Page 205 Inner Earth processes and systems 205 Although hidden from our direct view by thousands of generated in the mantle and crust remains below surface meters of ocean, these contribute by far the most voluminous forming slow-cooled plutonic igneous rock in the form of proportion of volcanic products to the surface each masses called plutons. Some is squirted from consolidating year. The overall proportion of acid : intermediate : basic plutons into vertical or subvertical cracks as dykes, or volcanics erupted each year is about 12 : 26 : 62 percent. nearer the surface as horizontal sills, both of which may Despite the obvious surface manifestations of volcanic feed surface volcanoes. Plutons, dykes, and sills are very activity, the majority of melt (around 90 percent) common in the upper crust, as seen in deeply eroded mountainous terranes like the Andes or Rockies. We would like to know why such large volumes of former melt remain below the surface. 5.1.2 Melting processes We have seen in our consideration of the states of matter (Section 3.4) that thermal systems transfer energy by changing the temperature or phase of an adjacent system or by doing mechanical work on their local environment. For melting to occur, a solid phase may be converted to a liquid by (1) application of temperature or pressure, (2) temperature retention with only minor heat loss due to work done by internal energy on expansion during adia- batic ascent, and (3) reduction in local melting point by addition of aqueous or volatile ﬂuxes. We further amplify these reasons below. Concerning heat energy, a certain amount, the latent heat of fusion, Lf (Section 3.4), is needed to melt crys- talline rock. This amount can be measured in a calorimeter Fig. 5.4 Typical nightime view of Stromboli ﬁre fountain erupting apparatus by comparing the heat released on melting from vent three, May 1979. Note parabolic ballistic trajectories of silicate crystals or rock with amorphous silicate glass of volcanic ejecta. Two Figures silhouetted for scale. (a) (b) (c) ol fp qz fp px ol px px qz fp Granite with coarse equant Andesite lava showing well- Two half-views of olivine basalts, crystals of clear quartz (qz) and developed phenocrysts of feldspar with well-developed phenocrysts shaded alkali feldspars (the (fp) and pyroxenes (px) set in a of olivine (ol) and lath-like feldspars laminae in the latter are twin very finely crystalline to glassy set in finely crystalline to glassy planes or compositional layers) groundmass groundmass Fig. 5.5 Sketches of microscopic fabric (ﬁelds of view about 5 mm diameter) and mineral phases of common igneous rocks that have crystallized from cooling melts.
LEED-Ch-05.qxd 11/27/05 2:19 Page 206 206 Chapter 5 identical composition. A selection of values for Lf is shown similar size and charge. However, since the Mg2 ion in in Box 5.1. Because, melting of a given volume of solid forsterite is somewhat smaller than the Fe2 ion in fayalite, cannot be achieved instantaneously, even if a homogenous it is held more tightly by atomic bond energy into the mineral or elemental solid is involved, we need concepts silicate crystal lattice and therefore melts at a higher to express the onset of melting and its completion: these temperature; olivines composed of pure Mg2 and Fe2 are solidus and liquidus respectively. We generally draw the thus melt at about 700 C apart. Now, take a 50 : 50 solidus and liquidus as lines on temperature : pressure combination of Fe2 and Mg2 silicate in an olivine solid graphs or on phase diagrams. The solidus line thus indi- volume and heat it up at atmospheric pressure to 1400 C cates the temperature at which a rock begins to melt (Figs 5.7 and 5.8). The composition of the initial melt, or (or conversely becomes completely solid on cooling) and partial melt, produced from such an olivine will tend to be the liquidus line is the temperature at which melting is complete (or conversely at which solidiﬁcation begins on Temperature (°C) cooling). As an example, we can follow the solidus of Melt basalt on the P–T diagram of Fig. 5.6. collection Depth (cm) Since most rocks are chemically different and may be Onset comprised of various mineral species or minerals free to melting vary in composition, the onset of melting or the process of crystallization on cooling is complex. Major progress in Upwelling understanding the processes of melting and crystallization of natural silicates were made by N.L. Bowen in experi- Liquidus Solidus ments conducted in the early twentieth century (Figs 5.7 and 5.8). To illustrate this, consider one of Bowen’s earli- Fig. 5.6 To show solidus, liquidus, and an adiabatic melting curve as est triumphs, an explanation of the variation in behavior of mantle rock is elevated by convection, partially melts and rises to surface. the simplest possible rock made up of only olivine, an iron–magnesium silicate, whose composition is free to vary between 100 percent iron silicate (representing a mineral phase called fayalite) and 100 percent magnesium silicate (the mineral forsterite). The olivine system is obviously of major importance because it makes up a major mineral phase of the Earth’s ocean crust. Minerals like olivine that are able to vary in their solid composition between two end-members like this are quite common in nature (the common feldspar minerals are another) and are said to exhibit solid solution. A solid solution is like any alloy, bronze, solder, or pewter for example, where the metal N.L. Bowen ions can mix freely in most proportions since they are of Mineral phase A Mineral phase B Initial Box 5.1 Latent heat of melting melt Liquidus (cal g 1) for some important silicate minerals. Solidus Mg-olivine 208 Fe-olivine 108 Clinopyroxene 146 Orthopyroxene 85 Garnet 82 Ca-Feldspar 67 100% A 100% B 50 : 50 Na-Feldspar 52 Mixture K-Feldspar 53 Fig. 5.7 Melting relations in a binary silicate solid solution series.
LEED-Ch-05.qxd 11/27/05 2:19 Page 207 Inner Earth processes and systems 207 richer in Fe2 than Mg2 . As melting proceeds, the whole Thus far, we have considered melting temperatures as if melt progressively enriches in Mg2 until it matches they were unaffected by pressure. In fact, for mantle rock there is a strong change of dry solidus temperature with the initial 50 : 50 mixture and melting of the initial solid pressure. dT/dP is positive for the dry solidus of most key volume has become total at the liquidus. Experiments over silicate minerals of the Earth’s mantle (e.g. Fig. 5.6) and a range of initial compositions enable us to deﬁne a phase for the garnet peridotite composition (this is equivalent to diagram showing the range of solidus and liquidus appro- an ultramaﬁc rock with c.90 percent of Fe- and Mg-bearing priate to a whole solid solution series. Similar principles minerals) that best seems to satisfy constraints for mean govern the behavior of binary or ternary mixtures of mantle composition. mineral phases. 5.1.3 Water, melting, and the terrestrial water cycle 2000 At 1 atm P Water exerts a profound inﬂuence on both the melting Liquid silicate melt 1800 point (Fig. 5.9c) and strength of crustal and mantle rocks. Temperature (°C) The presence of H2O in silicate melts is thought to cause 1600 depolymerization by breaking the Si–O–Si bonds, leading Ol + liquid to the marked decreases in viscosity and melting tempera- Solid ture observed experimentally. For example, in order to 1400 olivine give a 20 percent melt fraction, the temperature of anhydrous granite at 10 kbar pressure has to be about 900 C; the addition of 4 percent by weight of water 1200 decreases the required temperature to about 600 C. 50 Mg2SiO4 Fe2SiO4 For basalt, the effect is even more startling for the positive Weight % Forsterite Fayalite gradient of the dry solidus noted above is reversed and at Moho depths of 35 km the saturated wet solidus Fig. 5.8 Phase relations in the olivine solid solution series at temperature is reduced from c.1150 C to 650 C. 1 atm pressure. Temperature (°C) Temperature (°C) (a) (b) 1200 1400 1000 1200 1400 1000 0 0 Adiabatic upwelling in convection limb or Solidus Depth (km) Depth (km) stretched mantle Geotherm 50 50 Melt Solidus Mantle is heated, geotherms increase gradient, melting 100 100 occurs Path Path 2 1 Temperature (°C) 1200 1400 (c) 1000 0 (a) The situation in the rising limb of a major convection cell under a midocean ridge or in stretched lithosphere. Depth (km) (b) Mantle heating above a plume head causes geotherms to intersect solidus. 100 Solidus (c) The asthenosphere above a subduction zone may melt if Water acts as there is sufficient flux of water from mineral dehydration a flux to lower reactions, especially the breakdown of serpentinite minerals. the melting temperature 1. 200 2. of mantle rock Fig. 5.9 Various scenarios for the production of melt from mantle rocks.
LEED-Ch-05.qxd 11/27/05 2:19 Page 208 208 Chapter 5 Magma The amount of ambient water present in the mantle as a MOR chamber whole is thought to be c.0.03 weight percent, so the aver- age basaltic melt produced at the midocean ridges is Crust largely anhydrous. Most interstitial water taken with ocean Lithosphere Magma ascent crust and sediments into subduction “factories” is rather efﬁciently processed back into the atmosphere and terres- Partial melting trial environment via arc volcanoes and subsurface magma Asthenosphere bodies. It has been calculated that of about 1012 kg of water taken into the subduction zones of the world every year, 92 percent is recycled in arc volcanism. This is just as well, because without recycling, the water-rich oceanic crust would effectively drain the oceans in only 109 years. How exactly the majority of this water is recycled by Upwelling convection Cybertectonica (Section 1.6.7) shall be brieﬂy explored loop below. Fig. 5.10 Decompression melting under a midocean ridge magma chamber. Volcanism at the midocean ridges is by far the most voluminous on Earth. 5.1.4 Why and where does melting occur in the Earth’s crust and mantle? The mobility of the Earth’s convecting mantle (Sections capacity. Computed values of dT/dz for mantle peridotite are about 0.4 C km 1. In the rising decompressing 4.20 and 5.2) means that there are ample opportunities for large-scale circulation to cause hotter material to rise up mantle, numerical calculations indicate that substantial from below. The process may be part of the large-scale partial melt fractions (25 percent) can be produced over ﬂow to the midocean ridges (Section 5.2; Fig. 5.10), with 20 km or so near the surface. The partial melt fraction pro- melt volumes produced at rates of c.25 km3 a 1. Or it may duced at the solidus is of anhydrous basalt composition be on a more regional scale, around thermal plumes and its intrusion and extrusion at the Earth’s midocean (Fig. 5.9b) whose head area may be up to of order ridges leads to the formation of new lithospheric plate 104 km2. In either case, the melting associated with the (Section 5.2). slow upward motion by plastic ﬂow, of order 10 2 mm a 1, The second cause of melting (Figs 5.9c and 5.11) is coincidental. It occurs because of what has been termed explains the large-scale distribution of volcanoes and melt decompression melting under conditions approximating the zones associated with volcanic arcs, such as those around adiabatic thermal transformation discussed in Section 3.4. the Paciﬁc “ring-of-ﬁre” (Fig. 5.1) and returns to the sub- Remember that a volume undergoing adiabatic transfor- theme of water in melts outlined above. Melting in arcs is mation is treated as being thermally isolated from its associated with the sinking of lithosphere plates back into surrounding environment. In the adiabatic rise and thus the mantle via subduction zones (Section 5.2); but it is not decompression of deep mantle rock, despite some energy the sliding process and frictional heat generation (see loss due to work done in expansion, the rising and expand- below) that causes the melting, for the descending plate is ing hot rock loses so little heat that it eventually intersects actually quite cool, and remains so for considerable the mantle solidus (Fig. 5.9a) thereby causing melting. depths. Rather, it is the transformation of the oceanic In this case, the adiabatic transformation is possible mantle of the descending slab that causes melting in the because of the very low thermal diffusivity (Section 4.18) overriding plate. The transformation involves a mineral of mantle material. The work done in expansion, as the group called serpentinite, which forms in the suboceanic pressure decreases upward, requires a certain amount mantle as olivine is altered by deep penetration of water of internal heat energy to be expended but this has very lit- along fracture zones and by subsea convection. tle effect on the temperature of a rock volume. The tem- Serpentinite contains up to 12 percent by weight of water perature path illustrated in Fig. 5.9a slopes gently negative in its mineral lattice. As the descending slab heats up, but to illustrate the point, with the actual solid adiabatic still well below the limits of the mantle solidus, it loses its gradient, dT/dz, given by the expression g T/cp, where structural water at 400–800 C under pressures of T is the initial solid temperature, volume coefﬁcient 3–6 GPa. The water percolates upward, perhaps aided by of thermal expansion and cp is the isobaric speciﬁc heat pressure changes in fractures opened during deep
LEED-Ch-05.qxd 11/27/05 2:19 Page 209 Inner Earth processes and systems 209 dissipation of heat upward along a fault zone may decrease VOLCANIC ARC TRENCH the efﬁciency and occurrence of the mechanism. Sea level A ﬁnal mechanism is thought to be responsible for OVERIDING PLATE brit widespread deep melting and continental crustal fusion in Magma tle she mountain belts caused by massive overthrusting of one ascent ar Bas alt/ MANTLE gab crustal terrane upon another on deep thrust faults. This bro Amphibo process acts quickly, at horizontal velocities appropriate to lite WEDGE Ec colliding plates (order of 10 1 m a 1), and places crustal lo gi rocks rich in radioactive elements under other crustal rocks te Partial melting whose ambient temperature is that of their truncated sub- from serpentinite DIPPING dehydration surface geotherms. As always, any crustal melting that might result will be aided by the presence of water in the system and also by the rapidity of the faulting movements in relation to the thermal diffusivities of the rocks SLAB involved. The process is thought to have caused the fusion of continental crust under mountain belts like the Himalayas and the production of viscous acidic magmas that slowly crystallize to granitic rocks rich in potassium- bearing radioactive minerals like the mica and muscovite. The approximate annual amounts of melt produced and Fig. 5.11 Volcanic arc magmatism results from the ﬂuxing effects of attributed to the ﬁrst two mechanisms above are indicated water released into the overiding plate as serpentinite dehydrates in a in Box 5.2. Fluxes from third and fourth are unknown descending lithospheric slab. since the melt remains subsurface. 5.1.5 Melt material properties earthquakes (called “seismic pumping”) and mixes with the plastic mantle olivine of the continental lithosphere of Adjacent quadrivalent silicon cations, Si4 , in silicate melts the overriding plate. This causes the melting point of the enter into shared coordination with four surrounding mantle to fall and its mechanical strength to drop drasti- oxygen ions to form silica–oxygen tetrahedra. Adjacent cally. The resulting partial melting and melt migration tetrahedra share O ions and also join to aluminum ions in eventually leads to generation of water-rich intermediate linked rings. The linked groups are said to be in a state of magmas characteristic of volcanic arcs. polymerization and are a feature of silicate melts. It is the The third melting mechanism notes the local coinci- continuous, polymer-like, linkage of oxygen ions (up to 15 dence of certain magmatic bodies, chieﬂy ancient mag- or so tetrahedral lengths may be involved) that seems to matic plutons exposed by deep erosion, with strike-slip control important physical properties; the greater the silica faults and appeals to the transformation of mechanical content and degree of group polymerization, the greater work to heat energy during deep faulting to cause melting. the viscosity and higher the solidus temperature. Alkali The magnitude of thermal energy produced is given by the and alkali earth cations like Ca, Na, and K, together with mechanically equivalent acceleration times the velocity of nonbridging O anions and OH reduce the degree of the fault surface motion. This shear heating during earth- quakes is of order u, in Watts, where is the frictional shear stress on the fault surface and u is the mean velocity of its motion. As long as the heat energy is retained locally due to low thermal diffusivities of the rocks involved, then Global melt ﬂuxes. Box 5.2 the temperature can build up with the possible occurrence Total volume of oceanic plate added as melt at MORs: of local melting. Temperature build up is aided by a ther- c.25 km3 a 1 mal feedback process such that any increase in local strain Total volume of oceanic plume-related intraplate volcanic melt: rate caused by lowering of viscosity at the heightened tem- c.1–2 km3 a 1 perature releases even more heat and this continues Total of volcanic arc melt: c.2.9–8.6 km3 a 1 until melting occurs after a few million years. However, Total of continental intra-plate melt: c.1.0–1.6 km3 a 1 the presence of circulating ﬂuids and their role in the

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