The
Quasigeostrophic Assumption and the Inertial-Advective Wind

The
quasigeostrophic omega and height tendency equations
can be derived directly from the equations of horizontal motion. Going backwards from these two
equations without removing the synoptic-scaling provides some insight to what ÒquasigeostrophicÓ means.

The
two equations of horizontal motion in Cartesian coordinates are

(1)

x component

(2)

y component

The
u-component of the geostrophic wind is obtained from
(2)

(3)

divide both sides of equation (2) by f and substitute (3)
into the result.

(4)

This
equation states that there will be northward or southward acceleration of the
air parcel if the real wind differs from the geostrophic
wind.

LetÕs
assume that we have initially only a west wind (a jet stream in the upper
troposphere that lies along a line of latitude). For the sake of argument,letÕs also assume that there are no west-east
accelerations, so that (1) is zero, and the equation of horizontal motion
reduces to the geostrophic wind (for equation (1) and
and an acceleration given by (4).

The
real wind can always be broken into a geostrophic and
an ageostrophic component

u = u_{g}
+ u_{a}_{ }(5a)

v = v_{g} + v_{a}_{ }(5b)

Substitute
(5a) into (4)

(6)

Equation
(6) gives the ageostrophic wind that occurs if the
wind is not in geostrophic balance. Note that the far right hand side says
that subgeostrophic flow will be associated with
northward accelerations and vice versa.

The
quasigeostrophic assumption implied by (6) is that we
try to Òadd backÓ some of the acceleration to the left hand side of the
equation that is assumed to be zero in geostrophic flow.
To do that, we replace V in the equation of motion on the left hand side
with the geostrophic wind (essesntially,
assuming that the ageostrophic wind along the y axis
is small).

This
gives, when you expand out the Lagrangian derivative

(7)

By
replacing the real wind with the geostrophic wind on
the left hand side, weÕre saying that the geostrophic
wind ÒalmostÓ is the same as the real wind. This allows us to add back a bit of the acceleration (which
we have found out is related to divergence) that the geostrophic
assumption strictly does not allow (since, except for the effect of the
northward variation of the Coriolis parameter, the geostrophic wind is non-divergent).

The
first term to the right of the equals sign in (7) is the local change of the geostrophic wind.
This is due to changes in pressure gradients (in turn due to isallobaric effects), and the motion of troughs and ridges,
for example. If we start
with the assumptions that the wind flow is zonal, this term is small or
zero.

The
far right hand term, called the convective-advective
term, is zero since w is zero in the upper troposphere (weÕll look at the
concept of this term another time).

The
second term to the right of the equals sign is essentially the advection of the
geostrophic wind by itself. In other words, as an air parcel in geostrophic balance moves into a region with a different
pressure gradient, it will initially have its initial speed
which will be out of balance with the pressure gradient. (This is sort of like a ball rolling
down an inclined plane to a flat surface, proceeding along the flat surface,
where there is no gradient, by its own momentum). This is called the Òinertial advective
windÓ

With
these assumptions, equation (7) becomes

(8)

Putting
(8) back into (6) (since there is no du/dt) we get

(9)

WeÕll
now apply this to the real atmosphere by taking your first look at jet streak
dynamics.