Matters of state and motion - Physical Processes in Earth and Environmental Sciences - page 29

to the observer and through which fluid or other mass
passes. Velocity measurements at different times are thus
gained from different fluid “particles” and must therefore
be averaged over time to give a time mean velocity.
The flow field seen by a moving viewer is known as the
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
flow. Velocity measurements at different times are thus
gained from the same fluid “particles” and the time aver-
age velocity is that gained over some downstream distance.
Most flow systems benefit by an Eulerian treatment.
Certainly for fluids, the mathematics is easier since we
consider dynamical results “at a point,” rather than the devi-
ous fate of a single fluid mass. Adopting a Eulerian stance,
any velocity is a function of spatial position coordinates x, y,
z , and time; we say in short (appendix), u

N S

2

Line of section

1

0

10 km

Volcano

20 km -20 km

Fig. 2.15 Vertical crustal velocity around Hualca Hualca volcano,
southern Peruvian Andes: surface deformation as seen by satellite
radar over about four years.

Note the high uplift rates and:
1 Concentric grayscale variations indicate uplift relative to
surrounding areas. Maximum uplift is seen due east of the volcanic
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/
water activity about 12 km below surface, but significantly offset
from volcano axis.
4 Volcano may be actively charging itself for a future eruption.

f ( x, y, z, t ).

2.4.8 Harmonic motion

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
motion deals with the periodic return of similar levels of
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-
trated in Fig. 2.17, of the passage of a fixed point on a
rotating wheel. The wave itself has various geometrical
terms associated with it, period, T , for example, and can be
considered mathematically most simply by reference to a
sinusoidal curve.

2.4.7 The observer and the observed: stationary
versus moving reference frames

You know the feeling; you are stationary in a bus or train
carriage and the adjacent vehicle starts to move away.
For a moment you think you are moving yourself. You are
confused as to exactly where the fixed reference frame is
located - in your space or your neighbors in the adjacent
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
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
another part is called relative motion . Examples are (1) the
relative motion of a crystal falling through a magma body
that is itself rising to the surface; (2) two lithosphere plates
sliding past each other (Fig. 2.16); (3) a mountain or vol-
cano rising (Fig. 2.15) due to tectonic forces but at the
same time having its surface lowered by erosion so that a
piece of rock fixed within the mountain is being both lifted
up and also exhumed (brought nearer to the surface) at
the same time.
The flow field seen by a stationary viewer is known as
the fixed spatial coordinate, or Eulerian , system. Analysis
is done with respect to a control volume fixed with respect

2.4.9 Angular speed and angular velocity

Consider curved (rotating) motion (Fig. 2.18a); in going
from a to b in unit time a particle sweeps out an arc of
length s , subtending an angle with the center of curva-

ture, radius r . We can talk about a constant quantity for
the traveling particle as

, usu-
ally measured in radians per second (a radian is defined as
360/2

/ t , the angular speed ,

degrees). The linear speed, u , of the rotating
particle is the product of angular speed of the particle and
its radial distance from the center of curvature, that is,
u

.
Angular velocity (Fig. 2.18b,c) has both magnitude and
direction and is thus a vector, denoted

r

. It has units of
radians per second. The angular velocity of rotation of