The Coriolis Force
• The second part of the two apparent forces that arise
due to the use of a non-inertial, geocentric coordinate
system is the 2Ωu term.
• This term is called
the Coriolis force
and is directed
radially outward
(along RA) from the
axis of rotation if
u > 0 (westerly
flow) and radially
inward, toward the
axis of rotation, if
u < 0 (easterly).
• The figure below shows that the Coriolis force, 2Ωu,
associated with an eastward (u > 0) velocity (into the
page) can be resolved into a horizontal component in
the negative y-direction (towards the south), 2Ωu sinφ,
and a vertical component in the positive z-direction (up),
2Ωu cosφ.
• Since the parcel had no meridional or vertical velocity to
start with, the accelerations are:
dv dw φ
= −2Ωusin φ and = 2Ωucos φ
dt dt
• A parcel moving eastward is
deflected to€ the south and a
parcel moving west is
deflected to the north in the
Northern Hemisphere. Both
to the right of the initial motion.
• The vertical component of the Coriolis force is generally
much smaller than the gravitational force in the same
direction (and smaller than the other terms we will find
to be important in the vertical equation of motion), so
the small effect it has in changing the apparent weight of
an object is usually ignored.
g
• So far we have only considered the Coriolis force
associated with zonal motion, u. Can flow in the
meridional direction be influenced by the Coriolis force
as well?
• Taking a step back, consider the
image to the right. A pitcher
located at the North Pole is trying
to throw a ball to the catcher
located at home plate, X0.
• By the time the ball arrives at the
catcher, she will have rotated to
the pitcher’s left as the Earth
rotates.
• Thus, it appears to the pitcher as
if the ball was deflected to the
right (to the west).
• So, we have learned that northerly (v < 0) motion induced
a westward drift, du/dt < 0 (since there was no zonal flow
to begin with).
• Can we understand this from the equations we have
already derived for a geocentric coordinate system?
Yes, we can!
• Formally, we can understand the existence of the Coriolis
force in association with meridional flow from the
conservation of angular momentum.
• As a parcel with unit mass moves in
the meridional direction, its
distance from the axis of rotation,
RA, changes.
• Thus, as the parcel moves toward or away from the axis
of rotation, in the absence of any real forces acting
upon it, angular momentum is conserved. That is:
dω d
= [ RA (ΩRA + u)] = 0
dt dt
• Carrying out the differentiation, we find:
dRA dRA du
€ 2ΩRA +u + RA =0
dt dt dt
• If the zonal velocity, u, is set equal to 0 (no initial zonal
flow), the expression reduces to:
€ du dRA
= −2Ω
dt dt
• Therefore, if a parcel is moving radially outward from the
Earth’s axis of rotation, dRA/dt > 0 (as it would for motion
towards the equator,
€ v < 0), du/dt < 0 and the parcel will
experience an easterly acceleration, to the west, just like
our pitcher and ball example.
• Similarly, if a parcel is moving toward the axis of rotation,
dRA/dt < 0 (such as motion toward the pole, v > 0), then
du/dt > 0 and the parcel experiences a westerly
acceleration.
• Furthermore, to define the components of the Coriolis
force that arise from meridional flow, we can resolve the
y-direction wind (v) into a component, v cosφ, parallel to
the axis to rotation, and a
component, v sinφ, in the
plane perpendicular to the
axis of rotation (radially
outward along RA). RA v sinφ
• The first component is φ
directed into the ground v cosφ v
and generally small,
while the second induces a
force 2Ωv sinφ in the zonal direction.
• Thus, the two primary horizontal components of the
Coriolis force we are concerned with are
dv du
= −2Ωusin φ and = 2Ωv sin φ
dt dt
• Since the Coriolis force appears so frequently in the
equations of motion, the factor 2Ω sinφ is given the
€ €
symbol and called the Coriolis
f
parameter. By definition, f > 0 in
the Northern Hemisphere and
€f < 0 in the Southern Hemisphere.
• We should also note€ that vertical
motions do give rise to horizontal
€ Coriolis forces, but vertical
velocities on the large scale are
generally so small (< 1 m s-1) that
these effects are ignored.
• The above results describing the horizontal Coriolis
forces induced by horizontal motions in the geocentric
coordinate system can be summarized:
1) A parcel with horizontal velocity Vh experiences a
Coriolis force whose horizontal component has
magnitude |2ΩVh sinφ|.
2) The horizontal component of the Coriolis force is
directed perpendicular to the horizontal velocity
vector; toward the right in the Northern
Hemisphere where the planetary rotation is
counterclockwise (as viewed from above), and
toward the left in the Southern Hemisphere
where the planetary rotation is clockwise.
• Wait a minute…the sense rotation of the Earth is different
in the Northern and Southern Hemispheres? Yes, when
viewed from aloft looking down on the poles!
Dynamic tropopause pressure and vertical wind shear
hPa
Northern Hemisphere Southern Hemisphere
• The deflection by the Coriolis force is negligible for
atmospheric motions with time scales that are short
compared to the period of the Earth’s rotation, 1 day.
• So the Coriolis force is negligible for individual cumulus
clouds or a tornado, but fundamental to our
understanding of mid-latitude weather systems and
hurricanes.
