Download Coriolis Force

The Coriolis Force
A consideration of some of the basic forces that drive and constrain motion in the
atmosphere and oceans follows. The goal is to apply Newton’s second law,

 d 

dv
 ma ,
 F  mv   m
dt
dt
(6.1)
to a parcel of air or ocean, so we need to assemble the most important forces. Since it is
difficult to keep track of mass in observing the atmosphere and oceans, we write these
equations in terms of the forces per unit mass, or accelerations, acting on parcels of air or
ocean water according to:


dv  F

dt
m
(6.2)
Equations in the form of Eq. (6.2) are called “equations of motion”. In this chapter, we
will identify and derive expressions for the most important forces to include in Eq. 6.2 –
the forces that govern large-scale motion in the atmosphere and oceans.
Coriolis force
The Coriolis force is an "apparent force," arising because of the complicated
frame of reference from which we observe the atmosphere. The laws of Newtonian
physics are formulated in the absolute, or inertial, frame of reference, but we observe the
atmosphere and oceans within a noninertial frame of reference rotating with the earth.
The Coriolis force is defined and added to the equations of motion so Newton’s laws are
applicable in the rotating frame of reference.
Two different physical principles are involved in the definition of the Coriolis
farce, namely, conservation of absolute angular momentum and centrifugal accelerations
felt in the rotating frame.
Conservation of absolute angular momentum. Consider a parcel of air or water
with constant mass, m, moving with constant speed, v, in a circular orbit. According to
Newton’s second law, the parcel must conserve absolute angular momentum (i.e., angular
momentum in the absolute frame of reference). The magnitude of the absolute angular
momentum is mvr, where v is the tangential velocity, and its direction is perpendicular to
the plane of motion. The right-hand rule is applied for the sign convention, so for a
parcel rotating counterclockwise, the angular momentum is directed out of the paper.
Ground-based observations of motion in the atmosphere and oceans are taken
within the rotating frame of reference, i.e., when we conduct these measurements we are
not moving with respect to the earth's surface. However, the motion of a parcel is
constrained by conservation of angular momentum in the absolute frame of reference.
For the observer in the rotating frame of reference, a parcel will seem to have forces
acting on it when, in fact, there are no true forces acting on it and the parcel is simply
conserving angular momentum in the absolute frame of reference. Part of the "Coriolis
force" is an invention to make a parcel seem like it is obeying Newton's second law when
observed from the rotating frame of reference.
Consider a parcel of air in the earth’s atmosphere, diagramed in Fig. 6.1. The size of the
parcel as well as its distance about the earth’s surface are draw wildly out of scale. For a
parcel in the troposphere, for example, z ~ 10 km << a = 6371 km. So it is reasonable to
neglect the height of the parcel above the surface of the earth compared with the radius of
the earth, and to make the “thin atmosphere approximation”
az a .
(6.3)
In Figure 6.1, r is the moment arm, which is the distance from Earth’s axis of rotation to
the parcel, and r = a cos   is the angular velocity of the earth,
  2
24 hr
= 7.3 x 10-5 radians/sec.
(6.4)
The parcel does not have any meridional velocity relative to the earth’s surface. Define
UABS = total east/west velocity of the parcel in the absolute frame of reference; the
same as the tangential velocity in the angular momentum review above
UROT = east/west velocity of the earth under the parcel = velocity of a parcel that
is in solid body rotation with the earth. At the equator, UROT = 465 m/sec.
u = velocity of parcel relative to the rotating earth (this the velocity one would
observe from the earth’s surface)
Then,
UABS = UROT + u.
(6.4)
When visualized from the absolute frame of reference, the parcel is traveling in a circle of
radius r. UROT is the component of that velocity equal to the tangential velocity of the
earth’s surface, so
U ROT 
circumf . at latitude 
2a cos  2a cos 


 a cos  .
2
rotation period of Earth
24 hr

(6.5)


r = a cos 
z
a

equator
Figure 6.1. A parcel with zonal velocity in the earth’s atmosphere.

In climate dynamics, the concepts of Newtonian physics are applied to a parcel of
air or water of unit mass, i.e., 1 kg. The absolute angular momentum for a parcel with
unit mass is
M  U ABSr  a cos   u a cos  = constant.
(6.6)
According to the laws of physics, M is be conserved, i.e., M cannot change as the parcel
moves around in the atmosphere or ocean. If a parcel moves to a higher latitude, , then
cos  will decrease and u must increase. Thus, parcels moving meridionally within the
atmosphere and ocean attain zonal velocities relative to the rotating earth.
Is this an important factor to consider? Is it important for determining features of
the general circulation? To evaluate its importance, consider a parcel of air that moves
from the equator to 30N latitude due to some impulsive meridional force. At the
equator, the parcel’s (initial) absolute angular momentum is


M i  a cos 0 o  0 a cos 0 o  a  465 m / s .
(6.7)
Note how large this meridional velocity is compared with zonal velocities, u, observed in
the earth’s troposphere (see Ch. 2). When the parcel reaches its final position at 30N, it
must have the same absolute angular momentum because no force acted on the parcel in
the zonal direction. Therefore,


M f  a cos 30 o  u f a cos 30 o  M i  a .
(6.8)
Solving Eq. 6.8 for the final zonal velocity of the parcel relative to the rotating earth at
30N,
uf 
a 2
 a cos 30 o  134 m / sec .
o
a cos 30
(6.9)
Again, this is a very large zonal velocity – larger than those observed. This
indicates that the Coriolis force is an important factor determining atmospheric wind
speeds. For an observer standing on the Earth’s surface at 30N, it looks as if the parcel
was accelerated to the east as it approached from the south. However, in the absolute
frame of reference, the parcel is simply moving in the meridional direction, and the
observer at 30°N is simply moving slower than the observer on the equator for whom the
parcel had no zonal velocity. The zonal component of the Coriolis force is formulated to
capture this apparent east/west acceleration of the parcel as viewed from the rotating
frame of reference.
With this conceptual understanding, a mathematic expression for this component
of the Coriolis force can be derived. The equation
dM
0
dt
(6.10)
expresses the principle of conservation of absolute angular momentum. The material, or
total, derivative is used to indicate that the derivative is taken “following the parcel”.
This is the framework in which the laws of physics are formulated. For example, in
applying F = ma to a block on an incline plane in a traditional physics application, we
conceptually follow the block down the plane, considering the forces acting on the block
at any instance.
Substituting Eq. 6.6 into Eq. 6.10, carrying through the differentiation, and
solving for du/dt results in a mathematical expression for the apparent acceleration (i.e.,
force per unit mass) that causes a parcel moving meridionally in the atmosphere to attain
a zonal velocity as seen by the observer in the rotating frame of reference. To carry out
the differentiation, note the following:
  = constant
   constant. The total differential indicates motion following the parcel,
and the latitude of the parcel changes as it moves northward or southward.
 u  constant.
 a  constant. Remember that a is the distance from the center of the earth to
the parcel and it equals the radius of the earth because we have neglected the
height of the parcel above the surface. But just because we have made the
thin atmosphere approximation a  z  a doesn’t mean that the parcel can’t
move vertically in the atmosphere. So we will write
da d a  z dz


w
dt
dt
dt
(6.11)
where w is vertical velocity.


d
 v  a
 

(6.12)
 dt 
Then, one can derive
F 1cor 
du
dt
 2v sin  
coriolis
uv
uw
.
tan   2wcos  
a
a
(6.13)
This is part of the zonal component of the Coriolis force.
Centrifugal accelerations and the Coriolis force
The Coriolis force also accounts for centrifugal forces that act on parcels traveling
in curved paths in the absolute frame of reference. For a parcel of air or ocean water in
solid body rotation with the earth (u = v = 0), the centrifugal force per unit mass, C, felt
by the parcel within the rotating frame of reference – another “apparent force” – is
directed perpendicular outward from the axis of rotation of the earth and given by
 U abs 2
C
r̂ ,
r
(6.14)
where r̂ is the unit vector pointing away from the axis of rotation. (Remember: In the
absolute frame of reference, the parcel in solid body rotation is traveling in a circle of
radius r = a cos and completing one circuit in 24 hours.) Bold arrows in the diagram
below show the forces acting on this parcel. Some balance of forces exists to hold the
parcel in the solid body orbit; the state of solid body rotation is not a motionless state in
the absolute frame of reference. FM is some arbitrary “mystery” force (per unit mass)
acting on the parcel to maintain its position over the same spot on the earth’s surface.


FM
r̂
g

C


Now consider a second parcel at the same latitude, but with some zonal velocity, u
> 0, relative to the earth’s surface. In the absolute frame of reference, this parcel is
traveling around the same circle as the parcel in solid body rotation, but at a faster rate, so
the centrifugal force that it “feels” within the rotating frame of reference is greater than
that felt by the parcel in solid body rotation. This difference in the centrifugal forces
acting on the two parcels – i.e., the centrifugal force associated with velocity relative to
the rotating earth, is another part of the Coriolis force.
Mathematical form of the centrifugal contribution to the Coriolis force
In formulating the equations of motion for the atmosphere and oceans, we absorb
the centrifugal acceleration (or force per unit mass) associated with solid body rotation,
Csb , into gravity, g , and define effective gravity, geff , as
geff  g  Csb .
(6.15)

If the parcel has u  0 , then the centrifugal force, C , acting on the parcel does not equal

C sb . Rather,
C  Csb  C ,
(6.16)
where
C  0 for u  0 , or westerly flow
(the case of super-rotation)
C  0 for u  0 , or easterly flow
(the case of sub-rotation).
and
C  can be decomposed into a meridional component, Cy ĵ , and a vertical
component C k̂ , as diagrammed in Figure 2. (Remember that ˆj is the unit vector in the
z
north/south direction, and k̂ is the unit vector in the vertical direction.)
Cz

C

k
 
 
C  Csb

C y
Unit vector r̂ indicates the direction perpendicular to the axis of rotation, i.e., in the same
direction as Csb and C :
r̂   sin ĵ  cos k̂ .
(6.17)
Substituting Eqs. 6.4 and 6.5 into Eq. 6.14,
 UA2
U ROT  u 2 r  u 2  2
u2 
r̂
C
r̂ 
r̂ 
r̂    r  2u 
r
r
r
r 

(6.18)
If a parcel is in solid body rotation, then u = 0 and C  Csb . Setting u = 0 in the above
expression, then, yields
C sb   2 rrˆ .
(6.19)
Therefore, from Eq. (6.16), it must be that
 
u 2 

C   2u 
rˆ .

r 

(6.20)
Using the definitions r̂ of and r,
 

u 2 tan   
u2 
 ĵ   2u cos  
k̂
F 2 cor  C    2u sin  
 
a
a 

 
(6.12)
A parcel with meridional velocity, v, relative to the earth’s surface will also experience a
centrifugal acceleration. This acceleration will be oriented upward. Since there is no
component of the earth rotation in the meridional direction, this centrifugal acceleration is
simply,
3
v2 ˆ
Fcor

k.
a
Adding all three components, the full Coriolis force is
(6.22)

uv
uw  
u 2 tan  

 ĵ 
Fcor   2v sin  
tan   2w cos  
î    2u sin  

a
a  
a


(6.23)

u 2  v2 
 2u cos  
k̂


a


In middle latitudes, for large-scale (1000’s of km) motion, under typical atmospheric
conditions away from the earth’s surface, the full Coriolis force can be approximated by
retaining only two terms in Eq. 6.23:

Fcor  2v sin  î  2u sin  ĵ
(6.24)