**LEAD-TO-LEARN**

__APPLYING
THE QUASI-GEOSTROPHIC HEIGHT TENDENCY EQUATION TO A DEEPENING SHORTWAVE THROUGH
THE USE OF AN IDV BUNDLE__

*WORKSHEET*

__Introduction__

This module will guide you
through a pre-structured IDV bundle that permits an in-depth analysis of the
Q-G height tendency equation. The model
output used in the IDV bundle illustrates the deepening of a 500-hPa trough
over the eastern

__Objectives__

In this module, you will
observe the factors that contribute to height changes of the 500-hPa
isosurface. Upon completion, you should
have a full understanding of:

·
The Q-G height
tendency equation’s importance in meteorological forecasting

·
The role of
vorticity advection in governing height falls/rises

·
Three useful
methods to determine vorticity advection from model output

·
The role of
differential thickness advection in forcing height falls/rises

·
Three viable
techniques to determine differential thickness advection from model output

·
The underlying
assumptions of the Q-G height tendency equation and precautions of using its
two forcing functions in certain atmospheric scenarios

__Background__

Quasi-Geostrophic (Q-G)
theory assumes an entirely hydrostatic (balance between vertical pressure
gradient and gravity) and geostrophic (balance between horizontal pressure
gradient and Coriolis) atmosphere. Perturbations
to these balances can occur via 1) thermal advection and 2) geostrophic
absolute vorticity advection. When
perturbed from either hydrostatic or geostrophic balance, the atmosphere
responds via ageostrophic motions (i.e., divergence and vertical
velocity).

After eliminating ω (ageostrophic
vertical motion) between the Q-G thermodynamic equation and the Q-G vorticity
equation, the __prognostic__ Q-G height tendency equation (involving a __time
derivative__ of the height field,_{}) is found:

Term A:
three-dimensional Laplacian of height tendency

Term B:
advection of absolute geostrophic vorticity by the geostrophic wind

Term C:
vertical variation of geostrophic thickness advection

In brief, the Q-G height
tendency equation is used to deduce change in a height field (Term A) based on the sign
(+ or -) and the magnitude of two forcing functions:

·
Geostrophic
advection of absolute vorticity (Term B)

·
Differential
thickness (temperature) advection (Term C)

The simultaneous analysis of
these two forcing functions allows meteorologists to predict change in a height
field and thus the evolution of extratropical cyclones.

__Term A__

Recall that the
Laplacian of a variable is proportional to the negative of its magnitude
(because of the sinusoidal properties of waves). In the case of Term A, it is important to notice, therefore, that the sign of _{} (and thus the sign of
the sum of Term B and Term C) is
proportional to the negative of _{}, or height tendency.
In summary, if the sum of Term B and Term C is* positive*, the
following is true:

·
_{} is +

·
_{} is –

·
There
is a height *fall*.

Conversely, if
the sum of Term B and Term C is *negative*, the following is true:

·
_{} is –

·
_{} is +

·
There
is a height *rise*.

__Term
B__

From the Q-G
vorticity equation,

_{},

it is apparent
that local changes in relative vorticity can only be induced by two parameters:

·
Horizontal
advection of absolute vorticity

·
Convergence/divergence
(vertical stretching)

When written as
the total derivative in absolute vorticity, the Q-G vorticity equation clearly
shows that the only way a wave can amplify (change intensity) is via
convergence/divergence:

_{}

Conversely,
horizontal advection of absolute vorticity acts only to propagate a wave. Therefore, Term B in the Q-G height tendency equation (geostrophic advection of
absolute vorticity) does not act to amplify a shortwave trough/ridge system;
rather, it serves the sole purpose of propagating waves. In conclusion, the deepening of a shortwave trough
axis can only be determined by analyzing Term C, the
vertical variation of thickness advection.
This concept will be discussed in greater detail in this module.

*General Rules:*

·
PVA
at a location acts to produce height *falls*.

·
NVA
at a location acts to produce height *rises*.

A STEP-BY-STEP
INVESTIGATION:

1) The atmosphere
is originally in geostrophic balance, thus inhibiting any divergence.

2) If positive
geostrophic absolute vorticity advection (PVA) occurs, then the geostrophic
relative vorticity must *increase* at
the focus of the advection.

3) Via

_{},

the geostrophic heights _{} are *decreasing* at the focus of the geostrophic
absolute vorticity advection.

4) The converse is true for
NVA.

Some ASSUMPTIONS to keep in
mind:

- Geostrophic relative vorticity is changing only via the
advection of geostrophic absolute vorticity.

- NO thermal advection

__Term
C__

In Term C, the differential
operator _{}signifies a vertical variation (in terms of pressure) of
thickness (_{}). Whereas the
Laplacian of height tendency (Term A) depends on the geostrophic advection of vorticity
restricted to the isobaric level under consideration (500 hPa in this module),
thickness advection must be considered as a vertically differentiated
parameter. Recall that thickness of a
layer is proportional to the mean layer temperature via the hypsometric
equation:

_{}

The diagram below shows the
effect of low-level cold air advection (

Note that the height of the
500-hPa isosurface *falls* in response
to

*General Rules:*

·
Negative
thickness (cold) advection decreasing with height acts to produce height *falls*, thus amplifying a 500-hPa trough.

·
Positive thickness
(warm) advection decreasing with height acts to produce height *rises*, thus amplifying a 500-hPa ridge.

Some ASSUMPTIONS to keep in
mind:

- Warming
and cooling are solely due to thermal advection (i.e., NO adiabatic temperature
changes because the atmosphere is originally in geostrophic balance, thus inhibiting
vertical velocity).

- NO
geostrophic absolute vorticity advection

__USING INDIRECT MODEL
OUTPUT TO DETERMINE VORTICITY ADVECTION (TERM ____B)__

__USING A MATHEMATICAL
APPROACH TO DETERMINE VORTICITY ADVECTION (TERM ____B____)__

__USING DIRECT MODEL OUTPUT
TO DETERMINE VORTICITY ADVECTION (TERM ____B____)__

** USING INDIRECT MODEL
OUTPUT TO DETERMINE DIFFERENTIAL THICKNESS ADVECTION (TERM **C

__USING A MATHEMATICAL
APPROACH TO DETERMINE DIFFERENTIAL THICKNESS ADVECTION (TERM ____C)__

** USING DIRECT MODEL OUTPUT
TO DETERMINE DIFFERENTIAL THICKNESS ADVECTION (TERM **C

__VISUALLY DEPICTING HEIGHT
TENDENCY (TERM ____A____)____
__

__DETERMINING HEIGHT
TENDENCY AT THE REGION OF STRONGEST POSITIVE VORTICITY ADVECTION (PVA)
__

__DETERMINING HEIGHT
TENDENCY AT THE TROUGH AXIS (ABSOLUTE VORTICITY MAXIMUM)
__

__Loading the IDV Bundle__

Click on the following link to load the IDV bundle associated with this module: QG_HEIGHT_BUNDLE. After the bundle finishes loading, the screen
should look very similar to the one pictured below:

WARNING: Do not alter the
original appearance of the bundle; this will lead to confusion later on. Simply observe the screen without clicking or
manipulating any of its content.

NOTE: The main window is
titled “Unidata Workshop IDV”. On the
right of this window is the “Displays” menu.
The “VCR” control is located at the top right corner of the display
(i.e. directly north of