Physical Processes in Earth and Environmental Sciences Phần 7 pps

190 Chapter 4

4.17.6 Earthquakes and strain

In the introduction to this chapter we noted that

earthquakes were marked by release of seismic energy

along shear fracture planes. This energy is released partly

as heat and partly as the elastic energy associated with rock

compression and extension. In the elastic rebound theory

of faults and earthquakes the strain associated with tec￾tonic plate motion gradually accumulates in specific zones.

The strain is measurable using various surveying

techniques, from classic theodolite field surveys to

satellite-based geodesy. In fact, the earliest discovery of

what we may call preseismic strain was made during inves￾tigations into the causes of the San Francisco earthquake

of 1906, when comparisons were made of surveys docu￾menting c.3 m of preearthquake deformation across the

San Andreas strike-slip fault. We have already featured the

results of modern satellite-based GPS studies in decipher￾ing ongoing regional plate deformation in the Aegean area

of the Mediterranean (Section 2.4). All such geodetical

studies depend upon the elastic model of steady accumu￾lating seismic strain and displacement. But then suddenly

the rupture point (Section 3.15) is exceeded and the

strained rock fractures in proportionate or equivalent mag￾nitude to the preseismic strain. This coseismic deformation

represents the major part of the energy flux and is dissi￾pated in one or more rupture events (order 102–101 m

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

energy is released. If the fault responsible breaks the

Earth’s surface then the coseismic deformation is that

measured along the exposed fault scarp whose length may

reach tens to several hundreds of kilometers.

Different types of faults give rise to characteristic first

motions of P-waves and it is this feature that nowadays

enables the type of faulting responsible for an earthquake to

be analyzed remotely from seismograms, a technique known

as fault-plane solution. Previously it was left to field surveys to

determine this, often a lengthy or sometimes impossible task.

The first arrivals in question are those up or down peaks

measured initially as the first P-wave curves on the seismo￾gram record (Fig. 4.142). It is the regional differences in the

nature of these records caused by the systematic variation of

compression and tension over the volume of rock affected by

the deformation that enables the type of faulting to be deter￾mined. This is best illustrated by a strike-slip fault where

compression and tension cause alternate zones of up (posi￾tive) or down (negative) wave motion respectively as a first

arrival wave at different places with respect to the orientation

of the fault plane responsible (Fig. 4.142). When plotted on

a conventional lower hemisphere stereonet (Cookie 19),

with shading illustrating compression, the patterns involved

are diagnostic of strike-slip faulting.

Down,

pull,

tension

Down,

pull,

tension

Up,

push,

compression Up,

push,

(b) compression

(c) (a)

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.

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Flow, deformation, and transport 191

We have so far discussed flow in terms of bulk movement

and mixing but there are also a broad class of systems in

which transport of some property is achieved by differen￾tial motion of the constituent molecules that make up a

stationary system rather than by bulk movement of the

whole mass. Such systems are not quite in equilibrium, in

the sense that properties like temperature, density, and

concentration vary in space. For example, a recently erupted

lava flow cools from its surfaces in contact with the very

much cooler atmosphere and ground. A second example

might be a layer of seawater having a slightly higher salin￾ity that lies below a more dilute layer. The arrangement is

dynamically stable in the sense that the lower layer has a

negative buoyancy with respect to the upper, yet over time

the two layers tend to homogenize across their interface in

an attempt to equalize the salinity gradient at the interface.

In both examples there is a long-term tendency to equal￾ize properties. In the first it is the oscillation of molecules

along a gradient of temperature and in the second the

motion of molecules down a concentration gradient. But

how fast and why do these processes occur?

4.18.1 Gases – dilute aggregates of

molecules in motion

The gaseous atmosphere is in constant motion due to its

reaction to forces brought about by changes in environ￾mental temperature and pressure. Volcanic gases also move

in response to changes brought about by the ascent of

molten magma through the mantle and crust. When we

study the dynamics of such systems we must not only pay

attention to such bulk motions but also to those of

constituent molecules that control the pressure and

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

gas as a relatively dilute substance in which its constituent

molecules move about with comparative freedom

(Section 2.1) is reinforced by the following logic:

1 A mole of a gas molecule is the amount of mass, in

grams, equal to its atomic weight. Nitrogen thus has a

mole of mass 28 g, oxygen of 32 g, and so on. Any quan￾tity of gas can thus be expressed by the number, n, of

moles it contains.

2 A major discovery at the time when molecular theory

was still regarded as controversial, was that there are always

exactly the same number of molecules, 6  1023, in one

mole of any gas. This astonishing property has come to be

known as Avagadro’s constant, Na

, in honor of its discov￾erer. It implied to early workers in molecular dynamics that

molecules of different gases must have masses that vary

directly according to atomic weight, for example, oxygen

molecules have greater mass than nitrogen molecules.

3 Following on from Avagadro’s development, it became

obvious that Boyle’s law (Section 3.4) relating the pres￾sure, temperature, volume, and mass of gases implies that

for any given temperature and pressure, one mole of any

gas must occupy a constant volume. This is 22.4 L

(22.4  103 m3) at 0°C and 1 bar.

4 It follows that each molecule of gas within a mole

volume can occupy a volume of space of some 4  1026 m3.

5 Typical molecules have a radius of some 1010 m and

may be imagined as occurring within a solid volume of

some 41030 m3.

From these simple considerations it seems that a gas

molecule only takes up some 104 of the volume available

to it, reinforcing our previous intuition that gases are

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

velocity of the molecules involved. Also, since gaseous

temperature can clearly vary with time, it must be the

collisions between faster (hotter) and slower (cooler)

molecules that bring about thermal equilibrium. And since

heat is a form of energy it follows that the motion of

molecules must represent the measure of a substance’s

intrinsic or internal energy, E (Section 3.4). Let us examine

these ideas a little more closely.

4.18.2 Kinetic theory – internal energy, temperature,

and pressure due to moving molecules

It is essential here to remember the distinction between

velocity, u, and speed, u. If we isolate a mass of gas in a

container then it is clear that by definition there can be no

net molecular motion, as the motions are random and will

cancel out when averaged over time (Fig. 4.143). Neither

can there be net mean momentum. In other words gas

molecules have zero mean velocity, u 0. However, the

randomly moving individual molecules have a mean

speed, u, and must possess intrinsic momentum and there￾fore also mean kinetic energy, E. In a closed volume of any

gas the idea is that molecules must be constantly bom￾barding the walls of the container – the resulting transfer

4.18 Molecules in motion: kinetic theory, heat conduction, and diffusion

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192 Chapter 4

or flux of individual molecular momentum is the origin of

gaseous pressure, temperature, and mean kinetic energy

(Fig. 4.144). These properties arise from the mean speed

of the constituent molecules: every gas possesses its own

internal energy, E, given by the product of the number of

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

to a kinetic version of the ideal gas laws, together with a

statistical view of the distribution of gas molecular speed.

The resulting kinetic theory of gases depends upon the

simple idea that randomly moving molecules have a proba￾bility of collision, not only with the walls of any container,

but also with other moving molecules. Each molecule thus

has a statistical path length along which it moves with its

characteristic speed free from collision with other mole￾cules: this is the concept of mean free path. Since gases are

dilute the time spent in collisions between gas molecules is

infrequent compared to the time spent traveling between

collisions. Thus the typical mean free path for air is of

order 300 atomic diameters and a typical molecule may

experience billions of collisions per second. Similar ideas

have informed understanding of the behavior, flow, and

deformation of loose granular solids, from Reynolds’ con￾cept of dilatancy to the motion of avalanches (Section 4.11).

4.18.3 Heat flow by conduction in solids

In solid heat conduction, it is the molecular vibration

frequency in space and time that varies (Fig. 4.145). Heat

energy diffuses as it is transmitted from molecule to mole￾cule, as if the molecules were vibrating on interconnected

springs; we thus “feel” heat energy transfer by touch as it

transmits through a substance. In fact, all atoms in any

state whatsoever vibrate at a characteristic frequency about

their mean positions, this defines their mean thermal

energy. Vibration frequency increases with increasing

temperature until, as the melting point is approached,

the atoms vibrate a large proportion of their interatomic

separation distances. Conductive heat energy is always

transferred from areas of higher temperature to areas of

lower temperature, that is, down a temperature gradient,

dT/dx, so as to equalize the overall net mean temperature.

u

deformable elastic wall

urms = (u 2) 0.5 _

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

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

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

root, the result being termed the root-mean-square velocity, or

urms in the present notation.

This is NOT the same as the mean speed, a feature you can

easily test by calculating the mean and rms values of , say, 1, 2,

and 3.

The internal energy, E, of any gas is the sum of all the

molecular kinetic energies. In symbols, for a gas with N

molecules:

E = N(0.5 mu2

rms)

Or we may alternatively view the molecular velocity as a

direct function of the thermal energy:

u2

rms = 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.

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