Scaling analysis of the forces at work in
our atmosphere
But first, let’s introduce a VERY IMPORTANT
PARAMETER, the CORIOLIS PARAMETER, f.
f=2!sin"
Scaling analysis allows us to eliminate terms (many terms,
as we’ll see) from the force balance that we’ve come up
with. The way it works is that each term in the balance is
examined individually and its magnitude is estimated by
using typical values for wind speed, pressure fluctuations,
length scales, etc.
In this manner the full equations may be tailored to any
given scenario. For the purposes of this course, we wish to
drastically simplify them in order to describe very large
scale, or synoptic, flows.
In scaling analysis, derivatives are approximated very
simply by the typical value of the variable in the numerator
divided by a typical value of the denominator.
For example, "U /"x would be approximated by a typical
value of U divided by a typical value of x. For synopticscale motion, x is on the order of 1000 km and U is on the
order of 10 m/s. So the zonal gradient of the zonal winds is
!
therefore
about 10/1000,000=10-5 s-1. Easy!
A typical time scale for synoptic flow is equal to a typical
length scale, L, divided by a typical flow speed, U. From
the above numbers, the time scale for synoptic flow is
therefore 105 s. Easy!
Let’s look at the values for all variable for synoptic-scale
flow at midlatitudes:
Horizontal wind speed: U~V~10 m/s
Vertical wind speed: W~1 cm/s
Horizontal length scale: L~106 m (1000 km)
Depth scale:
D~104 m (10 km; approximately
the depth of the troposphere)
Time scale:
L/U~105 s
Pressure:
P~105 Pa
Horizontal pressure gradient: #P/$~103 m2s-2 NOTE: P is
in Pascals!
Vertical pressure gradient: #P/$~105 m2s-2
Density:
$~1 kgm-3
Coriolis parameter:
f~10-4 s-1 (at midlatitudes, "~45o)
Now we’re ready to eliminate all but the most important
terms in our equations!
Let’s start with the zonal
force balance:
DU
1 $P
=%
+ fV + 2#W cos !
Dt
" $x
Scaled terms are:
U 2 "P
~
+ fV $ fW
L
#L
!
Plugging in typical values from above for synoptic-scale
motion, the magnitudes of these terms are, respectively
from left to right, 10-4 for the acceleration term, 10-3 for the
pressure gradient term, 10-3 for the horizontal Coriolis term,
and 10-6 for the vertical Coriolis term.
Now let’s look at the Meridional
For the scalings chosen, the pressure gradient term and the
horizontal Coriolis term are therefore an order of
magnitude GREATER than any other term in this equation.
Scaled terms are:
To a reasonable approximation, therefore, we can neglect
acceleration and neglect the vertical Coriolis term and say
that, for synoptic-scale motion, there is an approximate
balance between the pressure gradient and the (horizontal)
Coriolis force:
Force
Balance:
DV
1 "P
=!
! fU
Dt
# "y
V 2 "P
~
+ fU
L
#L
The magnitudes of these terms are, respectively from left to
right, 10-4 for the acceleration term, 10-3 for the pressure
!
gradient term, and 10-3 for the horizontal Coriolis term.
Once again, for the scalings chosen, the pressure gradient
term and the horizontal Coriolis term are an order of
magnitude GREATER than the acceleration term.
We can therefore say that, to a good approximation,
fV =
1 !P
" !x
This states simply that the meridional winds (V) are
proportional to the zonal pressure gradient. Note that the
Coriolis parameter, f, is an increasing function of latitude
so that, for a constant pressure gradient, the winds will
decrease with latitude.
QUESTION: for a constant zonal pressure gradient, would
the meridional winds in Edmonton be greater or less than
those in, e.g., Texas? Why?
fU = "
1 !P
# !y
The zonal winds (U) are therefore proportional to the
meridional pressure gradient.
Finally, let’s look at the balance of
forces in the vertical:
Subbing in scaled terms gives :
!
DW
1 %P
=$
$ g + 2#U cos !
Dt
" %z
WU "P
~
$ g + fU
L
#D
The acceleration term is therefore of order 10–7, the
pressure gradient term is of order 10, the gravity term is of
order 10, and the Coriolis term is of order 10-3. Therefore,
to a very good approximation we can neglect acceleration
in the vertical and the Coriolis term and write:
#P
= " !g
#z
which our familiar HYDROSTATIC BALANCE.
Therefore for the synoptic-scale motions under
consideration, hydrostatic balance is a very good
approximation because the other terms that contribute to
the vertical force balance are at least 4 orders of magnitude
smaller than either of the terms in the hydrostatic balance
relation. 10000+1~10000, right? (as a good
approximation!).