Trenberth's Approximation

Reading Bluestein, Vol. 1, , pp. 346-350

Term A Term B Term C

Equation 1

Quasigeostrophic Omega Equation

which can be expanded and rewritten (using the Thermal Wind Relation and the hydrostatic equation)

Equation 2

and, placing into Equation (1) yields:

Equation 3

This expression may be simplified on the basis of scaling (Bluestein, p. 349). Note also that the expression has a number of terms in common, illustrating the pitfall of assuming that the two forcing functions in (1) are distinct and that the term with ∂f/∂p is dropped on an order of magnitude basis.

The result is the so-called Trenberth Approximation to the QG omega equation is:

Equation 4

Note the following: (a) the left hand side is proportional to -w; (b) for a trough with a given amplitude and for short time frames, the static statibility parameter can be considered constant. Also, ∂V_g/∂p is the vertical shear of the geostrophic wind in x, y, p. Recall that -∂V_g/∂p and ∂V_g/∂z are the expressions for the Thermal Wind in x, y, p and x, y, z components, respectively. Thus,

Equation 5

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 1000 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 non-divergence, 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 deformation terms are very important near jet streaks, and fronts. Hence Equation 5 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.