Download Qz.5.soln.S02

METR 402: Intro Atmos & Ocn Dynamics
Dr. Dave Dempsey
Spring 2002
Solutions to Quiz #5
(1) [6 pts] For each of the real and apparent forces/mass listed below, state what you can in
general terms about:
I. its magnitude; and
II. its direction.
(a) the pressure-gradient force/mass
Magnitude: Proportional to the magnitude of the pressure gradient, and inversely
proportional to the density.
Direction: Parallel to the pressure gradient, toward lower pressure.
(b) true gravity/mass (sometimes called “gravitation”)
Magnitude: Proportional to the mass of the earth; inversely proportional to the square of
the distance between the center of mass of the earth and the center of mass of the object
in question.
Direction: Toward the center of mass of the earth.
(c) friction (direction only)
Direction: Opposite the direction of motion of the object on which it acts. (There are
exceptions to this, though!)
(d) centrifugal force/mass when rotating with the earth.
Magnitude: Proportional to the distance from the axis of rotation and to the square of the
angular speed of the earth.
Direction: Outward from and perpendicular to the earth’s axis of rotation.
(e) effective force/mass of gravity
Magnitude: Equal to (at the poles) or slightly less than true gravity (minimum at the
equator).
Direction: Perpendicular to the earth’s surface, “downward”.
(f) the Coriolis force/mass
Magnitude: Proportional to the earth’s angular speed and to the component of an object’s
velocity that is perpendicular to the earth’s axis of rotation.
Direction: Perpendicular to the axis of rotation and to an object’s velocity relative to the
earth.
Answer your choice of only one of the following three questions:
(2) [4 pts] Explain why the rotating earth cannot be a perfect sphere but rather must be
slightly flattened at the poles and slightly bulging at the equator. (Adopt an inertial frame
of reference and invoke Newton’s Second Law appropriately. A diagram will help.)
Pieces of the earth exhibit perfect circular motion, which means they are accelerating directly
toward the earth’s axis of rotation at a rate proportional to their distance from the axis and to the
square of the earth’s angular speed. The only forces acting on pieces of the earth to produce this
acceleration are true gravity (gravitation), which pulls toward the center of mass of the earth (not
the axis of rotation); and pressure forces exerted by surrounding pieces of the earth. The net
effect of the pressure forces is to push in a direction perpendicular to the earth’s surface,
“upward”. If the earth were a sphere, there is no way that these two forces could combine to
produce a net force (and hence an acceleration) toward the axis of rotation. The only way that
these two forces can combine to produce the required acceleration is for the earth to be flattened
at the poles and bulge at the equator, to form an oblate spheroid. Then the normal force can
cancel the part of gravity normal to the earth’s surface and leave the remainder of true gravity to
pull pieces of the earth toward the axis of rotation, maintaining perfect circular motion.
(3) [4 pts] For a given pressure gradient, the pressure-gradient force/mass increases with
decreasing fluid density. Explain why, physically. Also, for a given fluid density, the
pressure-gradient force/mass increases with increasing pressure gradient. Explain why,
physically.
The pressure-gradient force per unit mass consists of the net force due to pressure on a fluid
parcel (or any object), divided by its mass. The net force due to pressure on the object of given
size depends on the pressure difference between opposite sides of the object and the surface area
of the object (greater pressure difference and greater surface area mean greater net force). For an
object with lower density but the same size, the net pressure force will be the same but it’s mass
is less, so the net force per unit mass will be greater.
The pressure gradient is a measure of how rapidly the pressure varies from place to place. For an
object of given size, the larger the pressure gradient the more rapidly pressure will vary from one
side of the object to the other. This implies a larger pressure difference from one side to the
other, and hence a larger net force on the object, and hence a larger pressure gradient force per
unit mass.
(4) [4 pts] Even in the absence of any torque, a fluid parcel that moves northward in the
Northern Hemisphere experiences an acceleration eastward relative to the earth’s surface.
Explain why, in terms of the conservation of (absolute) angular momentum.
Conservation of absolute angular momentum requires that the product of an object’s distance
from the axis of rotation, R, and its absolute tangential speed, Va, is conserved (that is, RVa
doesn’t change following the object). In our context, the object’s absolute tangential velocity
is the sum of the earth’s speed Ve = R and the tangential (east/west) component of the
object’s velocity relative to the earth, u (so Va = R
in the Northern Hemisphere is moving closer to the axis of rotation, which requires that it’s
absolute tangential speed, Va = R
speed of the earth actually decreases, so u must become more positive or less negative (that
is, the object must accelerate eastward) enough so that R
change in u occurs in the absence of any real forces acting in that direction (that is,
tangentially), consistent with the idea that the Coriolis force is an apparent force, not a real
one.
(5) [4 pts] In a frame of reference rotating with the earth, an object that moves eastward in
the Northern Hemisphere experiences an acceleration southward and upward. Explain
why.
In a frame of reference rotating with the earth, we observe an object at rest relative to the
earth not to be accelerating and hence must experience no net force acting on it. In particular,
true gravity/mass, the normal (vertical) force/mass (a pressure-gradient force/mass), and an
apparent centrifugal force/mass = R = (Ve/R)R, must sum to zero. However, an object
that moves eastward relative to the earth will experience a centrifugal force = ((Ve+u)/R)R
which is greater than that needed to balance gravity and the normal force. The extra
centrifugal force will make the object appear to accelerate outward, away from the axis of
rotation. In the Northern Hemisphere, this direction has components southward and upward.