Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 40

the upstream velocity. You can check the dimensions to see
that it really is acceleration.

place so that d u /d t 0 everywhere and the velocity is

steady (Section 2.4). From the definition given above, you
would therefore expect no acceleration. But there is accel-
eration along these passageways. Why? The velocity poten-
tial lines and streamline constructs in Fig. 2.20 show that
velocity must increase or decrease as cross-sectional area
changes. Therefore, something is wrong with our simple
definition of acceleration. We must allow for accelerations
due to spatial changes in velocity that affect a fluid cell as
it goes from a to b , where the velocity is different. In this
case, we are letting a fluid cell see a change in the velocity
field as it travels, in addition to any local velocity change.
This raises some complications in fluid analysis, since we
need to know something about the upstream flow history
of any fluid in order to understand its state as it arrives in
front of the local observer.
Spatial acceleration , sometimes called advective acceler-
ation , the change in velocity, d u /d s , along the flow (i.e. as
discussed in Section 2.4 for a Lagrangian observer travel-
ing with the flow), is given generally by u d u /d s where u is

3.2.3 Total acceleration in moving fluids

It is common to find that both time and space acceleration
occur at the same time. To allow for this we make use of
term 1 in the equation of Fig. 3.6, designated as total accel-
eration , written D u /D t , the substantive or total derivative
as we follow the fluid (substantive is used in the same sense
as in the “substantive motion” in political debate). It com-
prises the sum of both time (term 2) and spatial (term 3)
accelerations: flows may show either acceleration, or both,
or none. It is sometimes also termed the Lagrangian deriva-
tive. Term 4 is shorthand for terms 2 and 3 and is explained
in the appendix. An important analysis of turbulent flows,
done originally by Reynolds (Section 4.5), makes much use
of this expression and, although it looks long and cumber-
some, it contains a wealth of information about a fluid flow.

3.3 Force, work, energy, and power

We have previously hinted (Section 3.2) that any accelera-
tion or change of momentum implies that an equivalent
force must be acting to cause the change. Physical Earth
and environmental processes cannot be understood with-
out an appreciation of what forces are, how they arise, and
how they operate upon Earth materials.

that a substance does not need to be moving for gravity to
exert a force: gravity acts upon everything: moving or
stationary. A very accurate spring balance in a constant
temperature room at sea level at the equator will thus
record a different “weight” for a standard kilogram at the
North Pole, or on the top of Everest: in each case because
the distance from the center of the Earth to the balance,
and hence gravity, is different. The big moonboots of
1960s astronauts had the same mass on the Moon as they
had when they were manufactured on Earth or when they
were tried out in the desert landscape of New Mexico.
It was the vastly reduced gravity on Moon that gave them
less weight. Similarly, an average sand grain (mean
diameter 1 mm) made of silicate mineral dropping onto
the surface of the Martian desert at its terminal velocity has
a weight of ratio g mars / g earth

3.3.1 Weight as a gravity force

We may generalize our definition of force, F , as causing an
acceleration, a , to act upon a mass, m . In symbols, F

m a .
It is clear from this definition that despite a mass being in
motion, if there is no acceleration there can be no net force
acting, though every moving substance, whether accelerat-
ing or not, has momentum. We made a fuss about the
appropriate use of the term mass in an earlier section. Spring
balances are calibrated by standard masses: their action of
measurement is not relative, as in a beam balance, but due
to the balance of forces between the effect of gravity on the
mass suspended by the torsion in the elastic spring.
Weight , the action of gravity on mass (Fig. 3.7), is
perhaps the easiest concept of force to begin with, pro-
vided we carefully avoid discussion of the true origin of
gravity! It is given by the product m g . You can check the
dimensions of force from this expression: MLT 2 ,
designated unit, N, for Newtons. This definition means

3.69/9.78, about 0.4, to an
identical grain in a Sahara desert sandstorm.

3.3.2 Gravitational forces

Forces are due to gravity acting from a distance on the
partial or total mass of any moving or stationary substance,
the total force being the sum of those acting on all the partial
masses. Thus, we could speak of the force exerted by grav-
ity on an individual apple, volcano, or ice sheet. The con-
cept is of great use when surveying the precise values of