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Physical Processes in Earth and Environmental Sciences Phần 2

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Chương 2 Magma có phân số nhỏ nhưng quan trọng của các chất khí hòa tan áp lực, bao gồm hơi nước. 5 sông nước có chứa chất rắn lơ lửng, trong khi bầu không khí mang theo các hạt bụi và bình xịt chất lỏng.

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  1. LEED-Ch-02.qxd 11/26/05 12:34 Page 20 20 Chapter 2 4 Magma has small but important fractions of pressurized ( 103 years) it flows, convecting most of Earth’s internal dissolved gases, including water vapor. heat production as it does so. Even the rigid lower crust is 5 River water contains suspended solids, while the atmos- thought to flow at depth, given the right temperature and phere carries dust particles and liquid aerosols. water content. 6 Seawater has c.3 percent by weight of dissolved salts and also suspensions of particulate organic matter. 2.1.5 Timescales of in situ reaction Solid Earth substances may break or flow: 1 Ice fragments when struck, yet deformation of boreholes drilled to the base of glaciers also shows that the ice there The lesson from the last of the above examples is that we flows, while cracking along crevasses at the surface. must appreciate characteristic timescales of reaction of 2 Earth’s mantle imaged by rapidly transmitted seismic Earth materials to imposed forces and be careful to relate waves behaves as a solid mass of crystalline silicate minerals. state behavior to the precise conditions of temperature and Yet there is ample evidence that in the longer term pressure where the materials are found in situ. 2.2 Thermal matters In each case temperature change signifies internal energy 2.2.1 Heat and temperature change. Changes of state between solid, liquid, and gas require major energy transfers, expressed as latent heats Heat is a more abstract and less commonsense notion than (Box 2.1). We shall further investigate the world of ther- temperature, the use of the two terms in everyday speech modynamics and its relation to mechanics later in this book being almost synonymous. We measure temperature with (Section 3.4). some form of heat sensor or thermometer. It is a measure Substances subject to changed temperature also change of the energy resulting from random molecular motions in volume, and therefore density; they exhibit the phenome- any substance. It is directly proportional to the mean non of thermal expansion or contraction (Box 2.1). This kinetic energy, that is, mean product of mass times velocity arises as constituent atoms and molecules vibrate or travel squared (Section 3.3), of molecules. Heat on the other around more or less rapidly, and any free electrons flow hand is a measure of the total thermal energy, depending around more or less easily. If changes in volume affect only again on the kinetic energy of molecules, and also on the discrete parts of a body, then thermal stresses are set up number of molecules present in any substance. that must be resisted by other stresses failing which a net It is through specific heat, c, that we can relate temperature force results. Temperature change can thus induce motion and heat of any substance. Specific heat is a finite capacity, or change in the rate of motion. Stationary air or water sometimes referred to as specific heat capacity, in that it is a when heated or cooled may move. Molten rock may move measure of how much heat is required to raise the temper- through solid rock. A substance already moving steadily ature of a unit mass (1 kg) of any substance by unit Kelvin may accelerate or decelerate if its temperature is forced to (K C 273). It is thus also a storage indicator – since change. But we need to consider the complicating fact that only a certain amount of heat is required to raise tempera- substances (particularly the flow of fluids) also change in ture between given limits, it follows that only this amount their resistance to motion, through the properties of vis- of heat can be stored. In Box 2.1, notice the extremely cosity and turbulence, as their temperatures change. We high storage capacity of water, compared to the gaseous investigate the forces set up by contrasting densities later atmosphere or rock. in this book (e.g. Sections 2.17, 4.6, 4.12, and 4.20). Temperature change induces internal changes to any substance and also external changes to surrounding environments, for example, 2.2.2 Where does heat energy come from? 1 Molten magma cools on eruption at Earth’s surface, turn- ing into lava; this in turn slowly crystallizes into rock. 2 Glacier ice in icebergs takes in heat from contact with There are two sources for the heat energy supplied to the ocean, expands, and melts. The liquid sinks or floats Earth (Fig. 2.4). Both are ultimately due to nuclear reac- depending upon the density of surrounding seawater. tions. The external source is thermonuclear reactions in 3 Water vapor in a descending air mass condenses and the Sun. These produce an almost steady radiance of heat is given out to the surrounding atmospheric flow. shortwave energy (sunlight is the visible portion), the
  2. LEED-Ch-02.qxd 11/26/05 12:34 Page 21 Matters of state and motion 21 Box 2.1 Some thermal definitions and properties of earth materials Thermal Diffusivity, k, units m2 s–1 x 10–6 at Specific heat capacities, cp , units standard T and P. of J kg–1 K–1, at standard T and P. Air 1,006 Air 21.5 Water vapor (100°C) 2,020 Water 0.143 Water 4,182 Mantle rock 1.1 Seawater 3,900 Thermal diffusivity indicates the rate of dissemination Olive oil 1,970 of heat with time. It is the ratio of rate of passage of Iron 106 heat energy (conductivity) to heat energy storage Copper 385 capacity (specific heat per unit volume) of any material Aluminum 913 Silica fiber 788 Carbon (graphite) 710 Mantle rock (olivine) 840 Heat flow required for vaporization, Lv, Limestone 880 units of kJ kg–1. Sometimes termed latent heat of vaporization, more correctly it is the Specific Heat Capacity , cp, cv, is the amount of heat specific enthalpy change on vaporization required to raise the temperature of 1 kg of substance (see Section 3.4). by 1 K . Subscripts refer to constant volume or pressure water to water vapor Coefficients (multiply by 10–6) of (and vice versa) 2,260 linear thermal expansion, al, units of K–1 at standard T and P. Heat flow required for fusion, Lf , units of kJ kg–1. Sometimes termed latent Iron 12 heat of fusion, more correctly it is the specific Copper 17 enthalpy change on fusion (see Section 3.4). Aluminum 23 Silica fiber 0.4 Ice 335 Carbon (graphite) 7.9 Mg Olivine 871 Crustal rock (to 373 K) 7–10 Na Feldspar 216 Basalt 308 Coefficients (multiply by 10–4) of cubical thermal expansion, av, units of K–1 at standard T and P. Heat flow produced by crystallization, (multiply by 104) units of J kg –1. Water 2.1 Olive Oil 7.0 Basalt magma Crustal rock 0.2–0.3 to basalt 40 Water to ice 32 Thermal conductivity, l, units of W m–1 K–1 at standard T and P. Air 0.0241 Water 0.591 Olive oil 0.170 Iron 80 Copper 385 Aluminum 201 Silica fiber 9.2 Carbon (graphite) 5 Mantle rock (olivine) 3–4.5 Limestone 2–3.4 Thermal Conductivity is the rate of flow of heat through unit area in unit time
  3. LEED-Ch-02.qxd 11/26/05 12:34 Page 22 22 Chapter 2 by only 1 m2 area of the outer atmosphere and equivalent GEOTHERMALHEAT 65 mW m–2 to the output of a small domestic electric bar heater. The heat energy available to drive plates is thus minuscule (though quite adequate for the purpose) by comparison with that provided to drive external Earth processes like life’s metabolism, hydrological cycling, oceanographic circulations, and weather. 2.2.3 How does heat travel? Radiative heat energy is felt from a hot object at a distance, for example, when we sunbathe or bask in the glow of a fire, in the latter case feeling less as we move further HEAT ENERGY is required away. The heat energy is being transported through space for life, plate motion, water and atmosphere at the speed of light as electromagnetic cycling, weather, and SOLAR HEAT waves. convectional circulations 1,367 W m–2 Conductive heat energy is also felt as a transfer process Fig. 2.4 Heat energy available to drive plates is minuscule by directly touching a hot mass, like rock or water, because when compared with that provided by solar sources for life, the the energy transmits or travels through the substance to hydrological cycle, weather, etc. be detected by our nervous system. In liquids we feel the effects of movement of free molecules possessing kinetic average magnitude of which on an imaginary unit surface energy, in metals the transfer of free electrons, and in the placed at the uppermost surface of Earth’s atmosphere solid or liquid state as the atoms transmit heat energy by facing the sun is now approximately 1,367 W m 2. This solar vibrations. constant is the result of a luminosity which varies by Convection is when heat energy is transferred in bulk >0.3 percent during sunspot cycles, possibly more during motion or flow of a fluid mass (gas or liquid) that has been mysterious periods of negligible sunspots like the Maunder externally or internally heated in the first place by radiation Minimum (300–370 years BP) coincident with the Little Ice or conduction. Age. At any point on Earth’s surface, seasonal variations in received radiation occur due to planetary tilt and elliptical 2.2.4 Temperature through Earth’s atmosphere orbit, with longer term variations up to 1 percent due to the Croll–Milankovitch effect (Section 6.1). Internal heat energy comes from two sources. A minority, The mean air temperature close to the land surface at sea about 20 percent, comes from the “fossil” heat of the level is about 15 C. Commonsense might suggest that the molten outer core. The remainder comes from the radioac- mean temperature increases the further we ascend in the tive decay of elemental isotopes like 238U and 40K locked atmosphere: like Icarus, “flying too close to the sun,” up in rock minerals, especially low density granite-type more radiant energy would be received. In the lower rocks of the Earth’s crust where such elements have been atmosphere, this commonsense notion, like many, is soon concentrated over geological time. However, the total proved wrong (Fig. 2.5) either by direct experience of mass of such isotopes has continued to decrease since the temperatures at altitude or from airborne temperature origin of the Earth’s mantle and crust, so that the mean measurements. The “greenhouse” effect of the lower internal outward heat flux has also decreased with time. atmosphere (Sections 3.4, 4.19, and 6.1) keeps the surface Today, the mean flux of heat issuing from interior Earth is warmer than the mean – 20 C or so, which would result in around 65 mW m 2 (Fig. 2.4), though there are areas of the absence of atmosphere. Although a little difficult to active volcanoes and geothermally active areas where the compare exactly, since the Moon always faces the same way flux is very much greater. The mean flux outward is thus toward the Sun, mean Moon surface temperature is only some 4.8 10 5 of the solar constant. To make this of about this order (varying from 130 C on the sunlit contrast readily apparent, the total output of internal heat side to 158 C on the dark side). Due to the declining from the area enclosed by a 400 m circumference racetrack greenhouse effect, as Earth’s atmosphere thins, tempera- would be about 1 kW, of the same order as that received ture declines upward to a minimum of about 55 C above
  4. LEED-Ch-02.qxd 11/26/05 12:34 Page 23 Matters of state and motion 23 Sub-Antarctic Antarctic front front 110 0.0 Free electrons and 7.0 5.0 ionized ice particles 2.5 100 MESOSPHERE 4.0 3.0 2.0 0.5 be here 2.5 90 mesopause Ionization (–160ºC at poles) 1.0 energy 80 2.0 1.5 Ocean water depth (km) Ozone decreasing 70 upward Height (km) 1.5 2.0 THERMOSPHERE 60 1.0 2.5 50 0.5 stratopause 3.0 40 STRATOSPHERE 0.25 Ozone heating 3.5 0.1 30 by solar radiation 4.0 20 Subtle T changes define tropopause distinct water bodies 4.5 0 Greenhouse 10 separated by frontal TROPOSPHERE effect regions of high gradient 5.0 0 57ºS 58ºS 59ºS 60ºS 61ºS 62ºS –80 –60 –40 –20 0 20 Temperature (°C) Fig. 2.6 Section across Drake Passage between South America and Fig. 2.5 Mean temperature gradients for atmosphere. Antarctic to show oceanic temperature ( C): depth field. widely by ocean currents. Thermal energy is lost as water is the equator at 12–18 km altitude. The mean lapse rate is thus some 4 C km 1. The temperature minimum is the evaporated (see latent heat of evaporation explained in Section 3.4) by the overlying tropospheric winds but this tropopause. Above this, temperature steadily rises is eventually returned as latent heat of condensation through the stratosphere at about half the tropospheric (Section 3.4) to heat the atmospheres of more frigid lapse rate, to a maximum of about 5 C at 50 km above climes. But it is a mistake to assume that the oceans are of the equator. This is because stratospheric temperatures homogenous temperature. Distinct ocean water masses are depend on the radiative heating of ozone molecules by present that have small but significant variations in direct solar shortwave radiation. Another rapid dip in ambient temperature (Fig. 2.6), which control the density, temperature through the mesosphere to the mesopause at and hence buoyancy of one ocean water mass over about 85 km altitude reflects the decrease in ozone concentration. Above this the positive 1.6 C km 1 lapse another. Those illustrated for the Southern Ocean show the subtle changes that define fronts of high temperature rate in the thermosphere (ionosphere) to 400 km altitude is gradient. due to the ionization of outer atmosphere gases by incoming ultra-shortwave radiation in the form of -rays and x-rays. Beyond that, in space at 32,000 km, the 2.2.6 Temperature in the solid Earth temperature is around 750 C. The gradient of temperature against depth in the Earth is 2.2.5 Temperature in the oceans called a geotherm. The simplest estimate would be a linear one and it is a matter of experience that the downward gradient is positive. We could either take the geotherm Earth’s oceans have an important role in governing to be the observed gradient in rock temperature or that climate, since the specific heat capacity of water is very measured in deep boreholes (below c.100 m) and extrapo- much greater than that of an equivalent mass of air. late downward, or take the indirect evidence for molten So, ocean water has a very high thermal inertia, or low dif- iron core as the basis for an extrapolation upward. The fusivity, enabling heat energy produced by high radiation mean near-surface temperature gradient on the continents levels in low-latitude surface waters to be transferred
  5. LEED-Ch-02.qxd 11/26/05 12:34 Page 24 24 Chapter 2 is about 25 C km 1 and although linear for the very upper Temperature (K) 1000 2000 3000 4000 5000 part of the crust directly penetrated by humans, such a gradient cannot be extrapolated further downward since Lithosphere plate widespread lower crustal and mantle melting would result 410 km Discontinuity (or even vaporization in the mantle!) for which there is no 500 evidence. We therefore deduce that (Fig. 2.7) Upper mantle 660 km Discontinuity 1 The geothermal gradient decreases with depth in the crust; that is, it becomes nonlinear. Lower mantle 2 The high near-surface heat flow must be due to a 1000 concentration of heat-producing radioactive elements there. Curve (b) Depth (km) Concerning the temperature at the 3,000 km radius assumes Curve (a) separate core–mantle boundary (CMB), metallurgy tells us that assumes upper and iron melts at the surface of the Earth at about 1,550 C. whole-mantle lower mantle convection Allowing for the increase of this melting temperature with convection pressure, the appropriate temperature at the CMB may be layers 2000 approximately 3,000 C, yielding a conveniently easy to remember (though quite possibly wrong) mantle gradient of c.1 C km 1. Core–mantle interface 3000 Outer core, Fe–liquid Fig. 2.7 Mean temperature gradient (geotherm) for solid Earth. 2.3 Quantity of matter century English translation of the original Latin: 2.3.1 Mass “Quantity of matter is the measure of it arising from its density and bulk conjointly,” that is, gravity does not come We measure all manner of things in everyday life and into it. express the measured portions in kilograms; we usually say that the portions are of a certain “weight.” On old-fashioned beam balances, for example, kilogram or pound “weights” 2.3.2 Density are used. These are of standard quantity for a given material so that comparisons may be universally valid. In science, however, we speak of all such estimates of bulk The amount of mass in a given volume of substance is a measured in kilograms as mass (symbol m). The bigger the fundamental physical property of that substance. We portion of a given material or substance, the larger the define density as that mass present in a unit volume, the mass. We can even “measure” the mass of the Earth and unit being one cubic meter. The units of density are thus kg m 3 (there is no special name for this unit) and the the planets (see Section 1.4). We must never speak of dimensions ML 3. The unit cubic meter can comprise air, “weight” in such contexts because, as we shall see later in this book, weight is strictly the effect of acceleration due to freshwater, seawater, lead, rock, magma, or in fact any- gravity upon mass. Mass is independent of the gravita- thing (Fig. 2.8). In this text will usually symbolize fluid tional system any substance happens to find itself in. So density and , solid density (though beware, for we also when we stand on the weighing scales we should strictly use as a symbol for stress, but the context will be obvi- speak of being “undermass” or “overmass.” ous and well explained). Sometimes the density of a Newton defined mass, what he termed “quantity of substance is compared, as a ratio, to that of water, matter” succinctly enough (Fig. 2.8). Here is a nineteenth- the quantity being known as the specific gravity, a rather
  6. LEED-Ch-02.qxd 11/26/05 12:35 Page 25 Matters of state and motion 25 Pressure (bars) 0 250 500 750 1000 Quantity of matter is the 1080 measure of it arising from its FRESHWATER density and bulk conjointly Constant T 1040 20°C REPRESENTATIVE DENSITIES Density (kg m–3) (all in kg m–3) 1000 Air at top Everest 0.467 Air at sea level 15°C 1.225 Water at 20°C Constant P 998 Seawater at 0°C 1 bar 1,028 960 Ice 917 Average crustal rock at surface 2,750 920 Average mantle rock 0 25 50 75 100 at surface 3,300 Temperature (°C) Mean solid Earth 5,515 Pressure (bars) 0 250 500 750 1000 Typical basalt magma 1.6 at 90 km depth 3,100 AIR Ditto near surface 2,620 Fig. 2.8 Density may vary with state, salinity, temperature, pressure, 1.2 and content of suspended solids. Density (kg m–3) Constant P 1 bar 0.8 confusing term. Density is regarded as a material property of any pure substance. The magnitude of such a property 0.4 Constant T under given conditions of temperature and pressure is 20°C invariant and will not change whether the pure substance 0 is on Moon, Mercury, or Pluto, as long as the conditions 0 125 250 375 500 are identical. Neither does the value change due to any Temperature (°C) flow or deformation taking place. Fig. 2.9 Variation of density of freshwater and air with temperature and pressure. 2.3.3 Controls on density also has significant effects on both liquid (Fig. 2.9) and Note the emphasis on “given conditions” in Section 2.3.2, solid density whereas pressure has smaller to negligible for if these change then density will also change. effects upon liquid and solid density in most near-surface Temperature (T ) and pressure (p) can both have major environments, becoming more important at greater effects on the density of Earth materials. We have already depths. There are also important effects to consider in sketched the magnitudes of temperature change with cold lakes due to the anomalous expansion of pure water height and depth in the atmosphere, ocean, and within below approximately 4 C. This means that water is less solid Earth (Section 2.2). These variations come about dense at colder temperatures. As salinity increases to that due to variable solar heating by radiation, radioactive heat of seawater the temperature of maximum density falls to generation, thermal contact with other bodies, changes of about 2 C. In the deep oceans and deep lakes, for example, physical state, and so on. Pressure varies according to Lake Baikal, an additional effect must be considered, the height or depth in the atmosphere, ocean, or solid Earth thermobaric effect. This is the effect of pressure in decreasing (Section 3.5). All of these factors exert their influence on the temperature of maximum density. the density of Earth materials. Why is this? Referring to The case of seawater density is of widespread interest in Section 2.1, you can revisit the role of molecular packing oceanography since natural density variations create buoy- upon the behavior of the states of matter. The loose ancy and drive ocean currents. Its value depends upon molecular packing of gases means that they are compressi- temperature, salinity (Fig. 2.10), and pressure. The covari- ble and that small changes in temperature and pressure ation with respect to the former two variables is shown in have major effects upon density (Fig. 2.9). Temperature Fig. 2.11. It is convenient to express ocean water density,
  7. LEED-Ch-02.qxd 11/26/05 12:35 Page 26 26 Chapter 2 Salinity g kg–1 1040 at 0°C 20 30 40 AVERAGE Brine density (kg m–3) and 1 atm 30 30 1030 SEAWATER 12 st 14 16 18 20 22 24 26 1020 –3 1,028 kg m 20 20 at salinity Temperature (ºC) 28 35 g kg–1 1010 1000 30 10 10 90% of ocean 0 10 20 30 40 50 Salinity (g kg–1) Fig. 2.10 Variation of seawater density with salinity. freezing point 0 0 20 30 40 Salinity (g kg–1) , as the excess over that of pure water at standard condi- Fig. 2.11 Covariation of seawater density (as t) with salinity and tions of temperature and pressure. This is referred to as t temperature. 1,000) kg m 3. This variation is usu- and is given by ( ally quite small, since over 90 percent of ocean water lies at Freshwater temperatures between 2 and 10 C and salinities of 1.5 suspension 20–40 parts per thousand (g kg 1) when the density t of solids, density Density of freshwater suspension (×103 kg m–3) ranges from 26 to 28 (Fig. 2.11). It is difficult to measure 2,750 kg m–3 density in situ in the ocean, so it is estimated from tables 1.4 or formulae using standard measurement data on temper- ature, salinity, and pressure. Detailed measurements reveal that the rate of increase in seawater density with decreasing 1.3 temperature slows down as temperature approaches freez- ing: this is important for ocean water stratification at high latitudes when it is more difficult to stratify the very cold, almost surface waters without changes in salinity. 1.2 Finally, our definition of density deliberately refers to the “pure” substance. As noted in Section 2.1, many Earth materials are rather “dirty” or impure, due to nat- 1.1 Seawater density reached by ural suspended materials or human pollutants. The tur- fractional mass of bid suspended waters of a river in flood, a turbidity 0.01 mineral solids current, or the eruptive plume of an explosive volcanic 1.0 eruption are cases in point. The changed density of such 0 0.1 0.2 0.3 suspensions (see Fig. 2.12) is a feature of interest Fractional mass of mineral solids and importance in considering the flow dynamics of such Fig. 2.12 Variation of freshwater density with concentration of systems. suspended mineral solids. 2.4 Motion matters: kinematics 2.4.1 Universality of motion hydrosphere. Glaciers and ice sheets move, as do the per- mafrost slopes of the cryosphere during summer thaw. The slow motion of lithospheric plates may be tracked by GPS All parts of the Earth system are in motion, albeit at and by signs of motion over plumes of hot material rising radically different rates (Box 2.2); the study of motion in from the deeper mantle. Magma moves through plates to general is termed kinematics. We may directly observe reach the surface, inflating volcanoes as it does so. The motion of the atmosphere, oceans, and most of the
  8. LEED-Ch-02.qxd 11/26/05 12:35 Page 27 Matters of state and motion 27 Earth’s surface has tiny, but important, vertical motions 2.4.3 Velocity arising from deeper mantle flow. Spectacular discoveries relating to motions of the interior of the Earth have come A practical analysis of motion needs extra information to from magnetic evidence for convective motion of the that provided by speed; for example, (1) it is of little use to outer core and, more recently, for differential rotation of determine the speed of a lava flow without specifying its the inner core. Some Earth motions may be regarded as direction of travel; (2) a tidal current may travel at 5 ms 1 steady, that is to say they are unchanging over specified time but the description is incomplete without mentioning that periods, for example, the movement of the deforming plates it is toward compass bearing 340 . Velocity (symbol u, and, presumably, the mantle. Other motion, as we know units LT 1) is the physical quantity of motion we use to from experience of weather, is decidedly unsteady, either express both direction and magnitude of any displace- through gustiness over minutes and seconds or from day to ment. A quantity such as velocity is known generally as a day as weather fronts pass through. How we define vector. A velocity vector specifies both distance traveled unsteadiness at such different timescales is clearly important. over unit of time and the direction of the movement. Vectors will usually be written in bold type, like u, in this text, but you may also see them on the lecture board or → 2.4.2 Speed other texts and papers underlined, u, with an arrow, u or a circumflex, û. Any vector may be resolved into three Faced with the complexity of Earth motions we clearly orthogonal (i.e. at 90 ) components. On maps we repre- need a framework and rigorous notation for describing sent velocity with vectorial arrows, the length of which are motion. The simplest starting point is rate of motion proportional to speed, with the arrow pointing in the measured as speed; generally we define speed as increment direction of movement (Fig. 2.13). With vectorial arrows of distance traveled, s, over increment of time, t. Speed it is easy to show both time and space variations of veloc- is thus s/ t, length traveled per standard time unit (usu- ity, and to calculate the relative velocity of moving objects. ally per second; units LT 1). In physical terms, speed is a Further comments on vectors are given in the appendix. scalar quantity, expressing only the magnitude of the motion; it does not tell us anything about where a moving 2.4.4 Space frameworks for motion object is going. Thus a speeding ticket does not mention the direction of travel at the time of the offense. Further comments on scalars are given in the appendix. Both scalars and vectors need space within which they can be placed (Fig. 2.14). Nature provides space but in the lab a simple square graph bounded by orthogonal x and y Box 2.2 Typical order of mean speeds for some Earth coordinates is the simplest possibility. The points of the flows (m s 1) compass are also adequate for certain problems, though many require use of three-dimensional (3D) space, with Jet stream 30–70 three orthogonal coordinates, x, y, z. This 3D space (also High latitude front 7–10 any two-dimensional (2D) parts of this space) is termed Gale force wind 19 Storm force wind >26 Cartesian, after Descartes who proposed it; legend has it Hurricane 33 that he came up with the idea while lazily following the Hurricane grade 4 46–63 path of a fly on his bedroom ceiling. Using the example of Gulf stream 1–2 the velocity vector, u, we will refer to its x, y, z components Thermohaline flow 0.5–1 Tidal Kelvin wave at coast 15 as u, v, w. The motion on a sphere taken by lithospheric Equatorial ocean surface plates and ocean or atmospheric currents is an angular Kelvin wave 200 one succinctly summarized using polar coordinates Tsunami 200 (Fig. 2.13c) or in the framework provided by a latitude Spring tidal flow 2 Mississippi river flood 2 and longitude grid. 3.2 . 10–6 (10 m a–1) Alpine valley glacier 3.2 . 10–4 (1,000 m a–1) Antarctic ice stream 1.6 . 10–9 (0.05 m a–1) Lithospheric plate 2.4.5 Steadiness and uniformity of motion Pyroclastic flow >100 8.3 . 10–3 (30 m h–1) Magma in volcanic vent 10–3 ( 3.6 m h–1) Magma in 3 m wide dyke Consider a stationary observer who is continuously 10–8 (0.3 m a–1) Magma in pluton measuring the velocity, u, of a flow at a point. If the
  9. LEED-Ch-02.qxd 11/26/05 12:36 Page 28 28 Chapter 2 (a) y (a) Object 1 u constant P 5 t1 Speed–time graph 6y 5 t2 t3 5 3x x Speed, u –x 5 t4 5 Steady motion Any position, P, can 5 t5 be described by 2 measures of length t1 t2 t3 t4 t5 Time, t –y (b) (b) Object 1 u constant y 5 x1 Speed–distance graph P 5 x2 5y u x3 r 5 Speed, u 5 x4 5 Uniform motion –3x –x O x 5 x5 If we regard P as x1 x2 x3 x4 x5 directed from the Distance, x origin, O, then the line OP may also be Fig. 2.14 (a) Vectors for steady west to east motion at velocity specified by its length u 5 ms 1 for times t1 t5. (b) Vectors for uniform west to east r and angle u. OP is a motion at velocity u 5 ms 1 for positions x1 x5. –y position vector The description of steadiness depends upon the frame of reference being fixed at a local point. We may take instan- z (c) taneous velocity measurements down a specific length, s, P of the flow. In such a case the flow is said to be uniform when there is no velocity change over the length, that is, y 6z u u/ s 0 (Fig. 2.14b). r –x This division into steady and uniform flow might seem O pedantic but in Section 3.2 it will enable us to fully explore 3x –3y the nature of acceleration, a topic of infinite subtlety. f x –y 2.4.6 Fields Vector OP is either: (3x, –3y, 6z) or (r, u, f) A field is defined as any region of space where a physical scalar or vector quantity has a value at every point. Thus –z we may have scalar speed or temperature fields, or, a Fig. 2.13 Coordinate systems: (a) Two dimensions; (b) two dimen- vectorial velocity field. Crustal scale rock velocity sions with polar notation, and (c) three dimensions. (Figs 2.15 and 2.16), atmospheric air velocity, and labora- tory turbulent water flow are all defined by fields at various velocity is unchanged with time, t, then the flow is said to scales. Knowledge of the distribution of velocities within a be steady (Fig. 2.14a). Mathematically we can write that flow field is essential in order to understand the dynamics the change of u over a time increment is zero, that is, of the material comprising the field (e.g. Fig. 2.16). u/ t 0.
  10. LEED-Ch-02.qxd 11/26/05 12:36 Page 29 Matters of state and motion 29 to the observer and through which fluid or other mass S N Vertical ground velocity (cm yr–1) passes. Velocity measurements at different times are thus 2 gained from different fluid “particles” and must therefore Line of section be averaged over time to give a time mean velocity. The flow field seen by a moving viewer is known as the 1 moving spatial coordinate, or Lagrangian, system. Analysis is done with respect to Cartesian axes and flow control volumes moving with the same velocity as the 0 flow. Velocity measurements at different times are thus 10 km Volcano gained from the same fluid “particles” and the time aver- age velocity is that gained over some downstream distance. 20 km –20 km Most flow systems benefit by an Eulerian treatment. Fig. 2.15 Vertical crustal velocity around Hualca Hualca volcano, Certainly for fluids, the mathematics is easier since we southern Peruvian Andes: surface deformation as seen by satellite radar over about four years. consider dynamical results “at a point,” rather than the devi- ous fate of a single fluid mass. Adopting a Eulerian stance, Note the high uplift rates and: any velocity is a function of spatial position coordinates x, y, 1 Concentric grayscale variations indicate uplift relative to surrounding areas. Maximum uplift is seen due east of the volcanic z, and time; we say in short (appendix), u f (x, y, z, t). edifice. Note symmetrical uplift rate and constant uplift gradients. 2 Uplift appears steady over the four years. 3 Surface swelling is due to melting, magma recharge, or hot gas/ 2.4.8 Harmonic motion water activity about 12 km below surface, but significantly offset from volcano axis. 4 Volcano may be actively charging itself for a future eruption. We speak of harmony in everyday life as the experience of mutually compatible levels of being. In music the term applies to the contrasting levels or frequencies of sound that bring about a harmonious combination. Harmonic 2.4.7 The observer and the observed: stationary motion deals with the periodic return of similar levels of versus moving reference frames some material surface relative to a fixed point; it is best appreciated by reference to the displacement of surface water level during passage of a surface wave, or as illus- You know the feeling; you are stationary in a bus or train trated in Fig. 2.17, of the passage of a fixed point on a carriage and the adjacent vehicle starts to move away. rotating wheel. The wave itself has various geometrical For a moment you think you are moving yourself. You are terms associated with it, period, T, for example, and can be confused as to exactly where the fixed reference frame is considered mathematically most simply by reference to a located – in your space or your neighbors in the adjacent sinusoidal curve. vehicle. Well, both spaces are equally valid, since all space coordinate systems are entirely arbitrary. The important thing is that we think about the differences in the velocity 2.4.9 Angular speed and angular velocity fields witnessed by both stationary and moving observers and understand that one can be exactly transformed into another. Motion of one part of a system with reference to Consider curved (rotating) motion (Fig. 2.18a); in going another part is called relative motion. Examples are (1) the from a to b in unit time a particle sweeps out an arc of relative motion of a crystal falling through a magma body length s, subtending an angle with the center of curva- that is itself rising to the surface; (2) two lithosphere plates ture, radius r. We can talk about a constant quantity for sliding past each other (Fig. 2.16); (3) a mountain or vol- the traveling particle as / t, the angular speed, , usu- cano rising (Fig. 2.15) due to tectonic forces but at the ally measured in radians per second (a radian is defined as same time having its surface lowered by erosion so that a 360/2 degrees). The linear speed, u, of the rotating piece of rock fixed within the mountain is being both lifted particle is the product of angular speed of the particle and up and also exhumed (brought nearer to the surface) at its radial distance from the center of curvature, that is, the same time. u r. The flow field seen by a stationary viewer is known as Angular velocity (Fig. 2.18b,c) has both magnitude and the fixed spatial coordinate, or Eulerian, system. Analysis direction and is thus a vector, denoted . It has units of is done with respect to a control volume fixed with respect radians per second. The angular velocity of rotation of
  11. LEED-Ch-02.qxd 11/26/05 12:36 Page 30 30 Chapter 2 PLATE 1 (EURASIAN) the stationary reference frame BLACK SEA PLATE 1 (EURASIAN) the stationary reference frame PLATE 2 (ANATOLIA–AEGEA) AEGEAN SEA PLATE 4 (ARABIAN) MEDITERRANEAN SEA 0 200 400 600 km SURFACE CRUSTAL VELOCITY VECTORS 20 mm a-1 PLATE 3 PLATE BOUNDARIES (AFRICAN) Fig. 2.16 Horizontal surface velocities of the lithospheric plates making up the eastern Mediterranean and Asia Minor. Data derived from satellite geodesy platforms (GPS) averaged over a few years and stated with reference to a stationary Eurasian plate reference frame. Notes: 1 Contrasts in velocity vectors between different plates and sharp discontinuities present across plate boundaries. 2 Evidence for systematic east to west acceleration (implying crustal strain) and anticlockwise spin (vorticity) of the Anatolia–Aegea plate. particle by means of the position vector, r. This leads to Displacement the important result that the angular velocity vector, , and the linear velocity vector, u, of the water at position +a vector, r, are at right angles to each other (Fig. 2.18c). Time 0.5p p 1.5p 2p 0 Vector geometry relates the linear velocity vector, u, to –a Period T 2p Radians the vector product of the angular velocity vector and the position vector (i.e. u r). Fig. 2.17 Harmonic motion. A wave has periodic, often sinusoidal, motion. The example is a curve traced out in time, best imagined as the track to a point on a moving wheel. 2.4.10 Vorticity Earth is 7.29 10 5 rad s 1. In order to give angular Vorticity is related to angular motion and is best envisaged as “spin,” or rotation; it is the tendency for a parcel of fluid velocity its vectorial status, the direction is conventionally or a solid object to rotate. It is sometimes given the sym- taken as a normal axis to the plane of the rotating bol, , but in oceanographic contexts more usually, , a substance, pointing toward the direction in which a convention we follow subsequently. Vortical motions right-handed screw would travel if screwed in by rotating occur all around us: the whole solid planet possesses vor- in the same direction as the rotating substance ticity (appropriately termed planetary vorticity), on (Fig. 2.18b). For example, in the case of clockwise flow in account of spin about its own axis; lithospheric plates and the xy plane, the axis is in the vertical sense, pointing crustal blocks may also slowly spin (Fig. 2.16); the whole downward and thus of negative sign. Vice versa for anti- atmosphere and atmospheric cyclones and anticyclones clockwise flow. We can denote the position of any rotating
  12. LEED-Ch-02.qxd 11/26/05 12:36 Page 31 Matters of state and motion 31 photograph of a continuously introduced dye will yield a A (a) streakline, the locus of all fluid elements that pass through. A photograph of an instantaneously introduced dye or of reflective particles will yield a pathline. For a steady flow it is r s possible to construct an overall flow map by drawing streamlines. These are lines drawn such that the velocity of Centre of every particle on the line is in the direction of the line at that curvature f B point. Numerous examples of flow visualization are given in r the text that follows (see in particular Figs 3.53–3.55). Angular speed, v = df/dt u u Linear speed, u = rv 2.4.12 Flow without dynamics: “Ideal” flow along streamlines (b) a –c From the definition of a streamline quoted above it is c obvious that streamlines cannot cross and that it is possible b b to define a volume of fluid bounded by streamlines along its length. Such an imaginary volume is termed a stream- tube (Fig. 2.20). If the discharge into and out of a stream- a tube of any shape is constant, areas of streamline axb=c b x a = –c convergence indicate flow acceleration and areas of diver- gence indicate deceleration. Thus areas of close spacing (c) have higher velocity than areas with wide spacing. Some v progress may be made concerning the prediction of v streamline positions rather than the experimental visualiza- IvI p tion considered previously by using concepts of ideal r o (potential) flow as applied to fluids in which the molecular viscosity (see Section 3.9) is considered zero. Although such frictionless fluids are far from physical reality, ideal Fig. 2.18 To illustrate curved motion angular speed and velocity. flow theory may be of great help in analyzing motions (a) Angular speed, (b) angular velocity conventions, and (c) angular distant from solid boundaries (i.e. away from boundary lay- velocity. ers; see Section 4.3) and in flows where viscous effects are rotate; spinning eddies of fluid turbulence are readily negligible (at very high Reynolds’ numbers; see Section 4.5). observed in rivers and from satellite images in ocean cur- As subsequent discussions will show, in the absence of rents. Fluid vorticity is termed relative or shear vorticity shearing stresses in an ideal fluid there can be no rotational and is due to velocity differences, termed velocity gradi- motion (vorticity), that is, all ideal flows are considered ents, across a fluid element (Section 1.19). It can be shown irrotational. (Section 3.8) that rigid body vorticity is twice the angular Considering any ideal flow past a bounding (solid) velocity, that is, 2 . Finally, vorticity must be surface, it is apparent that discharge between the boundary conserved according to the principle of the Conservation and a given streamline must be constant. Thus it is possi- of Absolute Vorticity (see Section 3.8). ble to label streamlines according to the magnitude of the discharge that is carried past themselves and a distant 2.4.11 Visualization of flow boundary. This discharge is known as the stream function, , of a streamline (Fig. 2.20). The magnitude of is obviously unique to any particular streamline and must be No dynamical analysis may be confidently begun without constant along the streamline. Velocity is higher when some idea of actual flow pattern. In everyday life the gusting streamline spacing is closer and vice versa (Cookie 2.1). eddies of a wind are picked out by the motion of autumn Another useful method of analyzing ideal flow arises leaves or by the swirling pattern of snow or sleet across a from the concept of velocity potential lines, symbol . These road or field. In the same way in the lab, flow visualization imaginary lines are drawn normal to streamlines (Fig. 2.20). introduces some marker into a flow which is then pho- They define a flow field, as defined in Section 2.4 and are best tographed (Fig. 2.19). Considering the Eulerian case, a
  13. LEED-Ch-02.qxd 11/26/05 12:45 Page 32 32 Chapter 2 (a) (b) Cylinder axis normal to page Fig. 2.19 Flow visualization photos. (a) Dye introduced continuously into flow through jets at left define streamlines of laminar flow around a stationary solid cylinder. (b) Streak photograph of aluminum flakes on the surface defines a pattern of convection in a counterclockwise rotating cylinder pan that is being heated at the outside rim and cooled in the center. Flow pattern is analogous to the circulation of the upper atmosphere. A streamtube is an imaginary, rigid, impermeable Equipotential lines, f, are drawn normal to streamlines, Ψ, with tube that transmits the same discharge out as their spacing proportional to velocity. The closer the lines the received in. It allows velocity to have 3D faster the flow. The combination of streamlines and equipo- components tential lines defines a flow net IN OUT f5 f4 f3 f2 f1 Ψ2 Streamlines, Ψ1–2 define a 2D section through the streamtube. They allow velocity, u, to have two components: u and v in this case. Ψ1 Ψ2 u w, z w IN OUT Components u of the velocity, Ψ1 u u, x The discharge in and the discharge out are identical. As the streamlines diverge the flow velocity must lessen down tream, vice versa for convergence. So velocity is proportional to streamline spacing Fig. 2.20 Streamtubes, streamlines, and potentials. compared to contour lines on a map where the direction of If the distance between equipotential lines and greatest rate of change of height with distance is along any streamlines is made close and equal, then the resultant local normal to the contours (gradient of the scalar height). pattern of small squares is known as a flow net (Fig. 2.20). The velocity is the gradient of (Cookie 1). Construction of flow nets for flow through various 2D
  14. LEED-Ch-02.qxd 11/26/05 12:45 Page 33 Matters of state and motion 33 shapes may considerably aid physical analysis. The grid is useful because, for example, from the streamline construc- built up by trial and error from an initial sketch of stream- tion one may deduce velocity and, with a knowledge of lines between the given boundaries. Then the equipoten- Bernoulli’s equation (Section 3.12), pressure variations. tial lines are drawn so that their spacing is the same as the However, it will be obvious to the reader that flow nets are streamline spacing. Continuous adjustments are made only a rather simple imitation of natural flow patterns. until the grid is composed (as nearly as possible) of Experimental studies will reveal patterns of flow that cannot squares, and the actual streamlines are then obtained. This is be guessed at by potential approaches (e.g. Fig. 2.19b). 2.5 Continuity: mass conservation of fluids A fundamental principle in fluid flow is that of conservation, to calculate the effects of decelerating or accelerating flow the interaction between the physical parameters that deter- (Section 3.2). mine mass between adjacent fluid streamlines. The trans- To be applicable, continuity of volume has important port of mass, m, along a streamline involves the parameters conditions attached: velocity, u, density, , and volume, V. These determine the 1 The fluid is incompressible, so no changes in density due to conservation of mass discharge, termed continuity. this cause are allowed. 2 Fluid temperature is constant, so there is no thermally induced change in density. 2.5.1 Continuity of volume with constant density 3 Fluid density due to salinity or suspended sediment con- tent also remains unchanged. River, sea, and ocean environments essentially comprise 4 No fluid is added, that is, there is no source, like a incompressible fluid. They contain layers, conduits, chan- submarine spring or oceanic upwelling. nels, or straits that vary in cross-sectional area, a, while a 5 No fluid is subtracted, that is, there is no sink, like a discharge, Q (units L3 T 1) of the constant density fluid permeable bounding layer or thirsty fish. through them remains steady, being supplied from else- One natural environment where most of these condi- where due to a balance of applied forces at a constant rate tions are satisfied is a length of river channel, where cross- (Fig. 2.21). Generally, if there is cross-sectional area a1 sectional area changes downstream (e.g. Section 3.2). and mean velocity u1 upstream, and area a2 and mean velocity u2 downstream, the product Q ua must remain 2.5.2 Continuity of mass with variable density constant (you can check that the product Q has dimen- sions of discharge, or flux, L3 T 1). We then have the Consider now a steady discharge of fluid with a variable equality u1a1 u2a2 so that any change in cross-sectional density that flows into, through, and out of any fixed vol- area is accompanied by an increase or decrease of mean ume containing mass, m (Fig. 2.22). If that mass changes velocity and there is no change in Q that is, Q 0. Any then the difference, m, may be due to a change of fluid changes in u naturally result in acceleration or decelera- density, , of the fluid within the volume over time tion. This simplest possible statement of the continuity and/or space. The fact that density is now free to vary, as equation may be used in very many natural environments a = area Q1= Q2 = a1u1 = a2u2 r = variable a = area a2 > a1 u1 > u 2 m1= m2 = a1r1u1 = a2r2u2 a2 > a1 r = constant a1 a1 Q2 r2 r1 m2 m1 u1 u2 u2 u1 a2 a2 Q1 Fig. 2.21 Continuity of volume: constant density case in 1D. Fig. 2.22 Continuity of mass: variable density case in 1D.
  15. LEED-Ch-02.qxd 11/26/05 12:45 Page 34 34 Chapter 2 VERTICAL MIXING River water OUT River water IN Qrwout Qrwin Sea water IN Qswout Qswin Sea water OUT Fig. 2.23 Estuarine circulation: example of mass conservation in action. 2.5.3 Examples of volume and mass continuity Plan Section 1 Delta or estuary channels are informative environments within which to consider the workings of continuity (Fig. 2.23). For any control volume the upstream dis- charge of seawater decreases while the downstream input of fresh river water decreases. A mass balance is brought Surface DIVERGENCE from a point is a source, causes upwelling about by vertical mixing of seawater upward and freshwa- ter downward. 2 It is instructive to apply the 3D volume continuity Plan Section expression for an incompressible fluid such as that found in an idealized portion of fast-moving ocean, river, or tidal shelf. It is usually fairly straightforward to measure the two mean surface components of the local velocity but more difficult to measure the time mean vertical velocity. We compute this useful parameter from the basic conservation expression in Cookie 3. Surface CONVERGENCE to a point is a sink, 3 We finally touch upon divergence and convergence with causes downwelling respect to sources and sinks. We stated that the continuity Fig. 2.24 Sources and sinks. expression depends upon the lack of sources or sinks linked to the system in question. Two important cases arise in hydrological, oceanographical, and meteorological flows (Fig. 2.24; see also Cookie 3). Surface divergence of streamlines, most obviously seen when flow is diverging in the case of compressible gas flow or a thermally varying from a point implies that a source is present below the sur- flow, means there is one more degree of freedom than in face, leading to a mass influx. Surface convergence of the case considered previously; we have: u1A1 1 streamlines to a point implies a sink is present and that u2A2 2, so that any change in net mass outflow per unit downwelling is occurring. An added complication for time (check the expression gives units MT 1) is now meteorological flows is that vertical motions of fluid in caused by a change in density and/or velocity. downwelling or upwelling situations also cause changes of The full algebraic expression for 3D continuity is given temperature and density, which cause feedback relevant to in Cookie 2 (the algebra looks hideous but is quite the stability of a moving air mass. logical).
  16. LEED-Ch-02.qxd 11/26/05 12:45 Page 35 Matters of state and motion 35 Further reading Everyone has their favorite college physics text that explains The math and physics appendices in S. Pond and things to their satisfaction. Our “bible” is P.M. Fishbane G. L. Pickard’s Introduction to Dynamical Oceanography et al.’s Physics for Scientists and Engineers: Extended Version (Pergamon, 1983) and R. McIlveen’s Fundamentals of (Prentice-Hall, 1993). Flowers and Mendoza’s Properties of Weather and Climate (Stanley Thornes, 1998) are excep- Matter (Wiley, 1970) is erudite. Massey’s Mechanics of tionally clear. More advanced physical derivations are set out Fluids (Van Nostrand Reinhold, 1979) is exceptionally clear. in D. J. Furbish’s Fluid Physics in Geology (Oxford, 1997).
  17. LEED-Ch-03.qxd 11/27/05 3:56 Page 36 3 Forces and dynamics b 3.1 Quantity of motion: momentum In Spanish, however, cantidad de movimiento or “quantity 3.1.1 Linear momentum of motion” is a commonly expressed synonym for momen- tum. We see immediately the significance of the word con- Momentum, symbol p, is the product of the mass, m, of any jointly in Newton’s definition, for similar values of p mu substance (gas, liquid, or solid) and its velocity, u. Hence may be achieved as the consequence of either large mass dimensions are MLT 1; there are no special units for and small velocity or vice versa. It is thus instructive to momentum. We get the importance of the concept most calculate the momentum of various components of the directly from Newton’s Definition 2, translated from the Earth system; the dual roles of mass and velocity playing original Latin into the elegant English of the mid-nineteenth off each other can produce some unexpected results century: (Figs 3.2 and 3.3). For this reason it is also often instruc- tive to express momentum per unit volume, given by The quantity of motion of a body is the measure of it arising from its velocity and the quantity of matter conjointly. u. Momentum can also be easily related to kinetic p energy, Ek (Section 3.3). You may agree with us that the phrase “quantity of Linear momentum is a vector and is orientated through motion” (Fig. 3.1) is a good deal more expressive and a mass in the same direction as its velocity vector, u. Each unequivocal than the term in modern English language of the three Cartesian components of the velocity vector usage, “momentum”; the obvious semantic confusion for will have its component part of momentum attached to it, the beginner is with moment, as in moments of forces. that is, u, v, and w. 0.5 kg “Bramley” cooking apple falling at velocity 10 m s–1 … is p = 5 kg m s–1 I called momentum quantity of motion – a much more suitable 1 mm diameter spinning sand grain name, don´t you think? impacts onto rocky desert floor… Velocity 2 m s–1 Mass 1.15 . 10–5 kg Rebound … is p = 2.3 . 10–5 kg m s–1 (Elastic collision) Fig. 3.1 Newton and his definition of momentum. Fig. 3.2 On the momentum of apples and sand grains.
  18. LEED-Ch-03.qxd 11/27/05 3:56 Page 37 Forces and dynamics 37 … a lithosphere plate … a moving, floating iceberg mean density 3,000 kg m–3 mean density 1,000 kg m–3 mean thickness 75 km mean thickness 1,000 m area 4.0 . 106 km2 area 100 km2 velocity 50 mm year–1 velocity 0.1 ms–1 … is p = c.1.4 . 1012 kg m s–1 … is p = 1.0 . 1012 kg m s–1 Ocean Mid-ocean ridge Plate Plate Fig. 3.3 The momentum of the iceberg (not drawn to scale) is of the same order of magnitude as that of the plate. … Earth´s angular momentum, L = Iv, is 3.1.2 Angular momentum v In considering the momentum of rotating solid objects, such as planets, sand grains, or figure skaters, it is necessary to determine the angular momentum (Fig. 3.4) arising v = 7.3 10–5 rad s–1 r from the rotational motion, rather than the linear momen- m = 6 1024 kg tum of the mass. The momentum is thus considered as m r = 6.4 106 m I = 10 1037 kg m2 that arising about a rotational axis. The angular momen- L = 7 1033 kg m2 s–1 tum, L, is given by the product of rotational inertia (often called moment of inertia) about its rotation axis, I, and its vectorial angular velocity, . Thus L I . Notice that the mass term relevant to the determination of linear momen- … spinning sand grain´s angular momentum, L = Iv, is tum is here replaced by a rather unfamiliar quantity, rotational inertia. This is a subtle concept arising from v = 50 rad s–1 the notion of rotational kinetic energy (Section 3.3) and m = 1.15 10–5 kg the fact that in rigid body rotation each small element of r = 1 10–3 m I = 4.6 10–12 kg m2 mass, m, of a solid can be considered to have its own angu- L = 2.3 10–10 kg m2 s–1 lar velocity of rotation, , and therefore kinetic energy, about any rotation axis. All motion is considered about Kinetic energy of rotation this rotation axis: every small element has its own defined measurable perpendicular distance, R, from this axis and a Ek = 0.5 mR2 v2 = 0.5 Iv2 characteristic speed of R . The rotational kinetic energy of Fig. 3.4 Angular momentum. The rotational inertia, I, of a solid the small element is also related to its angular momentum. sphere is 2/5mr2. The rotational inertia and angular momentum of the whole body must be taken as the sum or integral (appendix) of each small element. Particular regular 3.1.3 Dynamic significance of momentum and inertia shapes have specific integral solutions, for example, the rotational inertia of a uniform density solid sphere of radius, r, is given by 2/5mr2. Making use of this expres- The big clue concerning the significance of momentum can sion we can easily calculate the approximate rotational be approached simply from first principles. We have seen that inertia and angular momentum of Earth (ignoring its mass gives us a measure of the quantity of matter present in internal density layering) or of a spinning sand grain a solid or fluid; this helps determine an object’s inertia, its (Fig. 3.4). tendency to carry on in the same line of motion or to resist
  19. LEED-Ch-03.qxd 11/27/05 3:56 Page 38 38 Chapter 3 changes in motion. This can be the inertia of a stationary magma that is losing pressure and exsolving gas in object or that of a steadily moving object to any accelera- bubbles, and so on. The example of colliding solid bodies tion. In many relevant physical situations the mass of a such as sand grains violently impacting on a desert floor or given volume element is constant with time and therefore colliding in a granular fluid brings us to one definition of it is the velocity term that determines the conservation of the conservation of linear momentum for such solid–solid momentum. When velocity changes in magnitude or direc- interactions: “. . . the sum of momenta of an isolated tion, momentum changes. As we shall see a little later, any system of two bodies that exert forces on one another is a such change in momentum over time is due to an equiva- constant, no matter what form the forces take . . . .” lent force, F dp/dt. We expect momentum changes to In other words, the collision of bodies or their interaction arise in Nature very frequently: in fluids when an air or leads to no change in overall energy (the production of water mass changes direction and/or speed due to changes collisional heat energy is included in the balance). This in external conditions; in fluid flow over solid boundaries principle of the conservation of momentum forms the where a velocity gradient is set up; in the ascent of molten basis of Newton’s Third Law (see Section 3.3). 3.2 Acceleration 3.2.1 A simple introduction Object 1 Object 2 1.5 u proportional t u proportional t Acceleration is a very obvious physical phenomenon; we 5 5 feel it driving or being driven. In both cases the effect is due to a change of velocity; the more sudden the change, 6.0 6.0 the more reaction we feel. The very fact that a body can 7.0 7.8 “feel” acceleration and is forced to move in response to it means that the phenomenon is somehow connected to 8.0 10.2 that of force. Another kind of universal acceleration con- cerns a falling body through a frictionless medium, such as 13 9.0 a solid through a vacuum, or through air whose resistance to motion is low (and may sometimes be neglected). Here Speed–time graph the falling solid is attracted by Earth’s gravity field. Some 2. Nonlinear examples of the use of uniform acceleration appropriate to 1. Linear acceleration such falling bodies are given in Cookie 4. In other more acceleration “resistant” liquids, a steady rate of fall is achieved after an interval such that the downward acceleration due to grav- Speed, u ity is quickly balanced by the resistance of the liquid medium. Acceleration (from now on we use the term without regard for the sign, positive or negative) of the kinds men- t1 t2 t3 t4 t5 tioned is most simply imagined as change of velocity over Time, t time (Fig. 3.5). Thus in differential form (appendix), a du/dt, with dimensions of LT 2. The standard accel- Fig. 3.5 Acceleration. Vectors for W to E motion at velocity, u, for eration due to gravity, g, at sea level is 9.81 ms 2. We stress times t1–t5. that natural accelerations may be extreme compared with this; for example, turbulent eddies are subject to accelera- 3.2.2 Complications in moving fluids 103g) and the tions of many times gravity (order Earth’s surface is subject to several g acceleration during Now we consider constant flow or discharge of fluid earthquake motions. By way of contrast, a slow-moving through conduits, channels, cols, or gates when the pas- lithospheric plate may change velocity so slowly over such a long time period (of the order of 106 years) that the sageway has varying cross-sectional area along its length (Fig. 3.6). There is no change of velocity over time at any acceleration can be practically neglected.
  20. LEED-Ch-03.qxd 11/27/05 3:56 Page 39 Forces and dynamics 39 C D Qout A B Qin A downstream-narrowing channel with constant discharge (Q) The field situation depicted involves absolutely steady discharge, so that In symbols at each section AB and CD there is no variation of discharge or velocity du/dt = 0 with time … but there is a change in space C D s Water A surface B Bed of channel Discharge in, Qin , equals discharge out, Qout. Area of cross-section AB >> area of cross-section CD. By continuity: mean flow velocity uAB
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