Download Horizontal Equation of Motion Notes

1
Notes to accompany Chapter 8 Material:
Well the earth rotates and so our coordinate system is embedded in a rotation frame
of reference which then gives rise to apparent of ctitious forces.
The angular period of the earth is
2radiansperday = 7:3x10;5s;1
If we consider a parcel of air that is moving due East and West (i.e. purely zonal
velocity, u), then to an observer external to the earth the velocity is really
v = Re + u
The acceleration (remember, the external observer is seeing the parcel rotating around
the center of the earth) is then
a = vr = (ReR+ u)
2
2
e
expand this out to get
2 2
2
a = RRe + Ru + 2uRRe
e
e
e
or
2
a 2 Re + 2u + Ru
e
The rst time, is what causes the oblateness of the earth. Think of it this way. g is
really g + 2Re where g is what g would be if the earth did not rotate. How big is the
2Re correction?
not very ...
6300x103m (7:3x10;5 )2 0:033msec;2
which is about a .3% correction. So the earth is non-spherical by about this amount
(equatorial radius .3% larger than the polar radius).
2
The second term 2u is the Coriolis force term and it is an acceleration which depends
on the velocity u. So, bigger zonal velocity means a larger Coriolis force. The Coriolis force
is therefore an induced force caused by horizontal motion. It is directed perpendicular to
the velocity. As a function of latitude, the horizontal component is
2usin = fu
where f = 2sin
Real Forces:
The real force, of course, is the pressure gradient force. When dealing with pressures
in a uid one must formally user partial derivatives which specify the change of one variable
at particular place in the uid, and not throughout the uid.
Partial derivatives are well beyond the background needed for this class so we will
only deal with them briey.
I think the derivation in the book (pages 372-373) is simple enough that I need not
repeat it. The general result is that the pressure gradient force can be expressed as
@p
Pn = ; 1 @n
The meaning of this (refer to gure 8.9) is that there is a change in pressure, as a
function of the horizontal coordinate, n, at a particular value of n (somewhere else, the
pressure change will be dierent - larger or smaller).
Problem 8.3 is a straightforward application of this idea. Go through it in some detail.
It is not hard. And it shows that the vertical migration is so small that the atmosphere
really is highly planar.
Other Forces:
In principle there is a fricitonal force operating all the time. This friction force is
basically related to constant momentum exchange between a rising parcel of air and its
surroundings. This is very complicated. The amplitude of the force will depend on the
velocity itself as it represents a kind of drag force. We will ignore this force and when it
appears it only formally appears.
The Total Horizonatl Equation of Motion is then:
3
dV = P + C + F
n
dt
A Scaling Argument:
For large-scale horizontal motion, typical horizontal velocities are 10-30 m/s. Over
this scale there are typically signicant changes ov velocity of order one day ( 105 seconds).
So that the eective horizontal acceleartion is, say
20m=s = 2x10;4 ms;2
105 s
This can be compared to the coriolis acceleration at mid latitude of
20m=s 2sin 2x10;4 ms;3
to see that the Coriolis force dominates the acceleration of a typical parcel of air. Since
the real force is provided by the pressure gradient force, this means that the only possible
way the forces can be balanced is that the pressure gradient force must be balanced by
the Coriolis force. This produces the geostrophic wind which causes the wind to blow
perpendicular to the isobars.
So the geostrophic condition is one where the horizontal pressure gradient force is
balanced by the coriolis force
@p
fv = 1 @x
The eects of friction, will generally complicated, would obviously produce subgeostrophic ow when they become an important part of the force balancing act.
Finally, vorticity introduces gradient ow as geostrophic ow around a curved trajectory will introduce a centrigual accleration (refer to Figure 8.12).
This is nicely illustrated in Problem 8.5