Matters of state and motion - Physical Processes in Earth and Environmental Sciences - page 27

Earth's surface has tiny, but important, vertical motions
arising from deeper mantle flow. Spectacular discoveries
relating to motions of the interior of the Earth have come
from magnetic evidence for convective motion of the
outer core and, more recently, for differential rotation of
the inner core. Some Earth motions may be regarded as
steady, that is to say they are unchanging over specified time
periods, for example, the movement of the deforming plates
and, presumably, the mantle. Other motion, as we know
from experience of weather, is decidedly unsteady, either
through gustiness over minutes and seconds or from day to
day as weather fronts pass through. How we define
unsteadiness at such different timescales is clearly important.

2.4.3 Velocity

A practical analysis of motion needs extra information to
that provided by speed; for example, (1) it is of little use to
determine the speed of a lava flow without specifying its
direction of travel; (2) a tidal current may travel at 5 ms 1
but the description is incomplete without mentioning that
it is toward compass bearing 340 . V elocity (symbol u ,

units LT 1 ) is the physical quantity of motion we use to
express both direction and magnitude of any displace-
ment. A quantity such as velocity is known generally as a
vector . A velocity vector specifies both distance traveled
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
other texts and papers underlined, u , with an arrow, → or
a circumflex, û . Any vector may be resolved into three
orthogonal (i.e. at 90

2.4.2 Speed

Faced with the complexity of Earth motions we clearly
need a framework and rigorous notation for describing
motion. The simplest starting point is rate of motion
measured as speed ; generally we define speed as increment
of distance traveled,

) components. On maps we repre-
sent velocity with vectorial arrows , the length of which are
proportional to speed, with the arrow pointing in the
direction of movement (Fig. 2.13). With vectorial arrows
it is easy to show both time and space variations of veloc-
ity, and to calculate the relative velocity of moving objects.
Further comments on vectors are given in the appendix.

s , over increment of time, t . Speed

is thus t , length traveled per standard time unit (usu-

ally per second; units LT 1 ). In physical terms, speed is a
scalar quantity , expressing only the magnitude of the
motion; it does not tell us anything about where a moving
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.

s /

2.4.4 Space frameworks for motion

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
coordinates is the simplest possibility. The points of the
compass are also adequate for certain problems, though
many require use of three-dimensional (3D) space, with
three orthogonal coordinates, x, y, z . This 3D space (also
any two-dimensional (2D) parts of this space) is termed
Cartesian, after Descartes who proposed it; legend has it
that he came up with the idea while lazily following the
path of a fly on his bedroom ceiling. Using the example of
the velocity vector, u , we will refer to its x, y, z components
as u, v, w . The motion on a sphere taken by lithospheric
plates and ocean or atmospheric currents is an angular
one succinctly summarized using polar coordinates
(Fig. 2.13c) or in the framework provided by a latitude
and longitude grid.

Box 2.2 Typical order of mean speeds for some Earth
flows (m s 1 )

Jet stream 30-70

High latitude front 7-10

Gale force wind 19

Storm force wind >26

Hurricane 33

Hurricane grade 4 46-63

Gulf stream 1-2

Thermohaline flow 0.5-1

Tidal Kelvin wave at coast 15

Equatorial ocean surface
Kelvin wave

200

Tsunami 200

Spring tidal flow 2

Mississippi river flood 2

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
measuring the velocity, u , of a flow at a point. If the

Magma in pluton 10 -8 (0.3 m a -1 )