Trenberth's Approximation
The results of your homework assignment should have included the
following:
 2 f o2  2 
g   f 0 v g
g 
f 0 v g








v







v



 p





g
g
g
g
 p 2 
 p
p   p
p 


(1)
Note that the expression has a number of terms in common, illustrating the
point made in Bluestein and in class that although the two forcing functions
in the QG omega equation appear distinct (differential vorticity advection
and temperature advection), they really are related to one another to some
extent.
The result is the so-called Trenberth Approximation to the QG omega
equation is:
 2 f o2  2 

f 0 v g
  2   g 
 p 
2 
 p 
 p



Note the following: (a) the left hand side is proportional to -; (b) for a
trough with a given amplitude and for short time frames, the static
statibility parameter can be considered constant. Also, -∂Vg/∂p is the
vertical shear of the geostrophic wind in x, y, p. Recall that -∂Vg/∂p and
∂Vg/∂z are the expressions for the Thermal Wind in x, y, p and x, y, z
components, respectively. Thus, equation (2) simplifies to:
  kvT g   kvT g 

(2)
(3)
This expression says that quasigeostrophically diagnosed omega at given
level (say,700 mb) is proportional to the advection of the absolute
geostrophic vorticity at 700 mb by the thermal wind centered at that level
(say the 850 to 500 mb layer). This is often referred to as "isothermal
vorticity advection". Where the thermal wind "flows" from high values of
absolute vorticity at a given level to low values of absolute vorticity, it is
said that positive isothermal vorticity advection (PIVA) is occurring, which
implies qg "forcing" for upwards omega.
A common way to infer areas of PIVA at 700 mb is to overlay the 1000-500
mb thickness field on the 700 mb vorticity pattern. In reality, the absolute
vorticity at 700 mb is not geostrophic (unless you plot that specific
variable). However, it is close enough to infer the NIVA and PIVA areas.
An alternative os to overlay the 1000-500 mb thickness field on the 500 mb
absolute vorticity field. Since the 500 mb level is near the level of nondivergence, the vorticity is nearly geostrophic on that level, and probably
reflects the geostrophic absolute vorticity field at 700 mb as well, as a first
guess.
*Please also note that the “small terms” dropped out the full expansion leading to
Equation (1) are very important near jet streaks, and fronts. Hence Equation 3 will only
be approximately accurate in those cases.
Dave Dempsey has devised a script that allows one to plot the 850 to 500
mb thickness contours overlain with the 700 mb absolute vorticity: isothva
jmont_derecho[401]% isothva -help
#---------------------------------------------------------------------# This script overlays absolute vorticity at a specified level (default
# 700 mb) on thickness of a layer centered at that level (default is
# 850 to 500 mb) from eta (or other) model output.
#---------------------------------------------------------------------#
# The syntax for script "isothva" is:
#
# isothva [help] [yymmddhh or -cu [hours]] [-lev plev]
#
[-mod model] [-time fcsttime]
#
[-con_in_th thcint] [-con_in_av avcint]
#
[-file_param fileprm] [-gif] [-gif_dir gifdir]
#
# where:
#
# help
simply prints this info to the screen;
#
# yymmddhh
specifies the model analysis time (truncated to the
#
nearest 12Z or 00Z model run);
#
# (-cu) hours specifies is the number of hours before the current
#
time;
#
# (-lev) plev the pressure level at which the vorticity will be
#
plotted;
#
# (-mod) model the model to use (default eta; others are mrf, ngm,
#
ruc);
#
# (-time) fcsttime the model forecast time (default=init, the initial
#
analysis; other times have form h12, h24, etc.,
#
d3 for 3-day, etc.)
#
# (-con_in_th) thcint is the contour interval for the thickness field
#
(default 60 meters);
#
# (-con_in_av) avcint is the contour interval for the absolute
#
vorticity field (default is 4x10^(-4)/sec);
#
# (-file_param) fileprm is an instruction about whether to reuse or
#
calculate from scratch a grid file containing the
#
desired thickness field; possible values are:
#
use (use existing thickness field, if
#
possible);
#
calc (calculate thickness field, even if a
#
file already exists; this is the
#
default);
#
# -gif
saves the image as a GIF file.
#
# (-gif_dir) gifdir specifies the directory where a GIF image will be
#
saved.