Physical Processes in Earth and Environmental Sciences Phần 3 pot

differing density does not “feel” the same gravitational

attraction as it would if the ambient medium were not

there. For example, a surface ocean current of density 1

may be said to “feel” reduced gravity because of the posi￾tive buoyancy exerted on it by underlying ambient water

of slightly higher density, 2. The expression for this

reduced gravity, g , is g g(2 1)/2. We noted earlier

that for the case of mineral matter, density m, in atmos￾phere of density a

, the effect is negligible, corresponding

to the case m  a

.

3.6.3 Natural reasons for buoyancy

We have to ask how buoyant forces arise naturally.

The commonest cause in both atmosphere and ocean is

density changes arising from temperature variations acting

upon geographically separated air or water masses that

then interact. For example, over the c.30 C variation in

near-surface air or water temperature from Pole to equa￾tor, the density of air varies by c.11 percent and that of

seawater by c.0.6 percent. The former is appreciable, and

although the latter may seem trivial, it is sufficient to drive

the entire oceanic circulation. It is helped of course by

variations in salinity from near zero for polar ice meltwater

to very saline low-latitude waters concentrated by evapora￾tion, a maximum possible variation of some 4 percent.

Density changes also arise when a bottom current picks up

sufficient sediment so that its bulk density is greater than

that of the ambient lake or marine waters (Fig. 2.12); these

are termed turbidity currents (Section 4.12).

Motion due to buoyancy forces in thermal fluids is

called convection (Section 4.20). This acts to redistribute

heat energy. There is a serious complication here because

buoyant convective motion is accompanied by volume

changes along pressure gradients that cause variations of

density. The rising material expands, becomes less dense,

and has to do work against its surroundings (Section 3.4):

this requires thermal energy to be used up and so cooling

occurs. This has little effect on the temperature of the

ambient material if the adiabatic condition applies: the net

rate of outward heat transfer is considered negligible.

3.6.4 Buoyancy in the solid Earth:

Isostatic equilibrium

In the solid Earth, buoyancy forces are often due to

density changes owing to compositional and structural

changes in rock or molten silicate liquids. For example, the

density of molten basalt liquid is some 10 percent less than

that of the asthenospheric mantle and so upward

movement of the melt occurs under mid-ocean ridges

(Fig. 3.27). However we note that the density of magma is

also sensitive to pressure changes in the upper 60 km or so

of the Earth’s mantle (Section 5.1).

In general, on a broad scale, the crust and mantle are

found to be in hydrostatic equilibrium with the less dense

54 Chapter 3

crust

mantle

rc

rm

hmr

hcr

o

ocean, rwrw

mountain range

crustal

“root”

Moho

har

ho

thickness of iceberg root

hir = ri /(rw - ri

)

ri

thickness

of

crustal

antiroot,

har = (rc – rw) /(rm – rc)

crustal equilibrium

thickness of crustal root,

hcr = rc / ( rw – rc )

level of buoyancy compensation: all pressures are equal

antiroot

rw

hir

Fig. 3.28 Sketches to illustrate the Airy hypothesis for isostasy, analogos to the “floating iceberg” principle.

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crust either “floating” on the denser mantle or supported

by a mantle of lower density. This equilibrium state is

termed isostasy; it implies that below a certain depth the

mean lithostatic pressure at any given depth is equal.

As already noted (Section 3.5.3), above this depth a

lateral gradient may exist in this pressure. In the Airy

hypothesis, any substantial crustal topography is balanced

by the presence of a corresponding crustal root of the

same density; this is the floating iceberg scenario

(Fig. 3.28). In the Pratt hypothesis, the crustal topography

is due to lateral density contrasts in the upper mantle (at

the ocean ridges) or in separate floating crustal blocks

(Fig. 3.29). Sometimes the isostatic compensation due to

an imposed load like an ice sheet takes the form of a down￾ward flexure of the lithosphere, accompanied by radial

outflow of viscous asthenosphere (Fig. 3.30). The reverse

process occurs when the load is removed, as in the isostatic

rebound that accompanies ice sheet melting.

An important exception to isostatic equilibrium occurs

when we consider the whole denser lithosphere resting on

the slightly less dense asthenosphere, a situation forced by

the nature of the thermal boundary layer and the creation

of lithospheric plate at the mid-ocean ridges (Sections 5.1

and 5.2). Lithospheric plates are denser than the astheno￾sphere and hence at the site of a subduction zone, a low￾angle shear fracture is formed and the plate sinks due to

negative buoyancy (Fig. 3.27).

Forces and dynamics 55

Crust

Mantle

rc r0 rc r2 r1

rm

hmr Nivel del Mar

D

2,900 kg m–3

3,350 kg m–3

3,000 kg m–3

Partial melt

hc

Moho

Ocean

Midocean ridge or rift uplift

rm > r0> rc> r1> r2

Fig. 3.29 Sketches to illustrate the Pratt hypothesis for isostasy. Here topography is supported by lateral density contrasts in the upper mantle

(left) and crust (right).

Fig. 3.30 Sketch to illustrate the Vening–Meinesz hypothesis for

isostatic compensation by lithospheric bending and outward flow

due to surface loading.

Lithosphere

Asthenosphere

r1

rm

h0

w0

l

Ice sheet

or

structural load

3.7 Inward acceleration

In our previous treatment of acceleration (Section 3.2),

we examined it as if it resulted solely from a change in

the magnitude of velocity. In our discussion of speed and

velocity (Section 2.4), we have seen that fluid travels at a

certain speed or velocity in straight lines or in curved

paths. We have introduced these approaches as relevant

to linear or angular speed, velocity, or acceleration.

Many physical environments on land, in the ocean, and

atmosphere allow motion in curved space, with substance

moving from point to point along circular arcs, like the

river bend illustrated in Fig. 3.31. In many cases, where

the radius of the arc of curvature is very large relative to

the path traveled, it is possible to ignore the effects of cur￾vature and to still assume linear velocity. But in many

flows the angular velocity of slow-moving flows gives rise

to major effects which cannot be ignored.

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3.7.1 Radial acceleration in flow bends

Consider the flow bend shown in Fig. 3.31. Assume it to

have a constant discharge and an unchanging morphology

and identical cross-sectional area throughout, the latter a

rather unlikely scenario in Nature, but a necessary restric￾tion for our present purposes. From continuity for

unchanging (steady) discharge, the magnitude of the

velocity at any given depth is constant. Let us focus on

surface velocity. Although there is no change in the length

of the velocity vector as water flows around the bend, that

is, the magnitude is unchanged, the velocity is in fact

changing – in direction. This kind of spatial acceleration

is termed a radial acceleration and it occurs in every

curved flow.

3.7.2 Radial force

The curved flow of water is the result of a net force being

set up. A similar phenomena that we are acquainted with is

during motorized travel when we negotiate a sharp bend

in the road slightly too fast, the car heaves outward on its

suspension as the tires (hopefully) grip the road surface

and set up a frictional force that opposes the acceleration.

The existence of this radial force follows directly from

Newton’s Second Law, since, although the speed of

motion, u, is steady, the direction of the motion is con￾stantly changing, inward all the time, around the bend and

hence an inward angular acceleration is set up. This

inward-acting acceleration acts centripetally toward the

virtual center of radius of the bend. To demonstrate this,

refer to the definition diagrams (Fig. 3.32). Water moves

uniformly and steadily at speed u around the centerline at

90 to lines OA and OB drawn from position points A and

B. In going from A to B over time t the water changes

direction and thus velocity by an amount u uB uA

with an inward acceleration, a u/t. A little algebra

gives the instantaneous acceleration inward along r as

equal to u2/r. This result is one that every motorist

knows instinctively: the centripetal acceleration increases

more than linearly with velocity, but decreases with

increasing radius of bend curvature. For the case of the

River Wabash channel illustrated in Fig. 3.31, the

upstream bend has a very large radius of curvature,

c.2,350 m, compared with the downstream bend, c.575 m.

For a typical surface flood velocity at channel centerline of

u c.1.5 m s1, the inward accelerations are 9.6 · 104

and 4.5 · 103 m s2 respectively.

3.7.3 The radial force: Hydrostatic force imbalance

gives spiral 3D flow

Although the computed inward accelerations illustrated

from the River Wabash bends are small, they create a flow

pattern of great interest. The mean centripetal acceleration

must be caused by a centripetal force. From Newton’s

56 Chapter 3

A

B f

Angular speed, v = df/dt

Linear speed, u, at any point along AB = rv

Center of curvature

r = Radius of curvature

Meander bends, R. Wabash,

Grayville, Illinois, USA

v

u

100 m

r

Fig. 3.31 The speed of flow in channel bends.

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