Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 61

For curved flows we can make use of the coordinate
system shown in Fig. 3.37, with s along the flow direction
and n in the plane of rotation normal to s and positive
inward toward the center of rotation. V is the local fluid
speed and r is the radius of the curved flow. The vertical
vorticity component,

twice the angular velocity, therefore for a given latitude,
f . With respect to local normal directions from

the surface, we realize that only at the Pole does the verti-
cal vorticity axis align exactly normal to the plane of the
rotation. In fact, vertical vorticity, necessarily defined as
parallel to the Earth's axis of rotation, must decrease to
zero at the equator when the local normal to the surface is
in the plane of the rotation. We commonly call Earth's
vorticity, f , the Coriolis parameter . For southern latitudes

2 sin

z , is now the sum of the shear

( n ) and the curvature ( V / r ) components, both of

which are positive.

V /

is taken as negative and thus cyclonic vorticity is nega-
tive, vice versa for anticyclonic vorticity. The magnitude of
the Coriolis parameter is quite small, of order 10 4 ms 2
between latitudes 45

3.8.4 Toward a physical explanation; Second,
solid vortical motions

and 90 .

Solid vorticity pertains to solid Earth rotation or to plate
and crustal block rotation. It also applies to the rapidly
rotating cores of tropical cyclones like hurricanes. It is best
investigated initially as curved solid flow, as in the last
example (Fig. 3.37), with a rotating disc or turntable
setup. In the disc case, both velocity components are gen-
erally nonzero. Consider first the shear term, (

3.8.5 Finally: Absolute fluid vorticity on
a rotating Earth

Any unbounded fluid, be it water or air, moving slowly
over the Earth, must possess not only its own relative or
shear vorticity,

V / n ).

, but also the Earth's vorticity, f . This is
the absolute vorticity,

In solid rotation, V r , and the shear term is

A , given by the sum, A f . In

( n ). Since V is increasing outward with n chosen

positive inward,

r /

the slow-moving and slow-shearing oceans, f . Just as

we have to conserve angular momentum so we also have
to conserve absolute vorticity. The poleward increase in
absolute vorticity explains why the slow flows of ocean and
atmosphere are turned by the Coriolis effect, the fluid
motion is turned in the direction of angular velocity
increase as extra angular momentum is obtained from the
spinning Earth, that is, to the right in the Northern
Hemisphere and to the left in the Southern Hemisphere.
This, finally, is why earthquakes are better than winds for
punishing transgressive minor goddesses.

r / n 1, the term becomes simply

. The contribution, ( V / r ), due to curvature flow is also

r by definition. Thus for solid body rota-
tions we have the simple result that the shear and curva-
ture components contribute equally to the total vorticity,
and this is equal to 2

, since V

.
Now consider the vorticity, f , of a solid sphere like
Earth. Viewed from the North Polar rotation axis
(Figs 3.38 and 3.39) Earth spins anticlockwise, with each
successive latitude band,

, increasing in angular velocity

poleward by sin . Since the vorticity of a solid sphere is

3.9 Viscosity

Viscosity, like density, is a material property of a substance,
best illustrated by comparing the spreading rate of liquid
poured from a tilted container over some flat solid surface
or the ease with which a solid sphere sinks through the liq-
uid. Viscosity thus controls the rate of deformation by an
applied force, commonly a shearing stress. Alternatively,
we can imagine that the property acts as a frictional brake
on the rate of deformation itself, since to set up and main-
tain relative motion between adjacent fluid layers or
between moving fluid and a solid boundary requires work
to be done against viscosity. An analog model combining
these aspects (the idea was first sketched as a thought
experiment by Leonardo) is illustrated in Fig. 3.40.

3.9.1 Newtonian behavior

Newton himself called viscosity (the term is a more mod-
ern one, due to Stokes) defectus lubricitatis or, in collo-
quial translation, “lack of slipperiness.” While pondering
on the nature of viscosity, Newton originally proposed that
the simplest form of physical relationship that could
explain the principles involved was if the work done by a
shearing stress acting on unit area of substance (fluid in
this case) caused a gradient in displacement that was
linearly proportional to the viscosity (Fig. 3.41). He
defined a coefficient of viscosity that we variously know as
Newtonian, molecular, or dynamic viscosity, symbol (mu)