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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 United States from 6:00:00Z to 18:00:00Z on November 8, 2005.  Although dynamical equations will be used to supplement physical interpretations, this module concentrates on visualizing the two forcing functions of the Q-G height tendency equation.  Conceptual analyses of each term will precede visual interpretation of the modeled height field’s evolution at fixed locations.


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 (CAA) and upper-level warm air advection (WAA) on the height of the 500-hPa surface:

 

 

Note that the height of the 500-hPa isosurface falls in response to CAA beneath and WAA above this level (neglecting Term B, the effects of geostrophic absolute vorticity advection).

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)

CONCLUSION


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 Maine).  The gray diamond located over western Virginia represents the fixed location at which changes in the height field will be considered.  Become oriented with this domain because subsequent steps will involve its features.  After you have familiarized yourself with the visual nature of IDV’s main window, proceed to the next section.


references


 

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