QG Analysis: System Evolution

Advanced Synoptic M. D. Eastin

QG Analysis: System Evolution


QG AnalysisQG Theory• Basic Idea• Approximations and Validity• QG Equations / Reference

QG Analysis• Basic Idea• Estimating Vertical Motion

• QG Omega Equation: Basic Form• QG Omega Equation: Relation to Jet Streaks• QG Omega Equation: Q-vector Form

• Estimating System Evolution• QG Height Tendency Equation

• Diabatic and Orographic Processes• Evolution of Low-level Cyclones• Evolution of Upper-level Troughs


Forecast Needs:• The public desires information regarding temperature, humidity, precipitation,

and wind speed and direction up to 7 days in advance across the entire country• Such information is largely a function of the evolving synoptic weather patterns

(i.e., surface pressure systems, fronts, and jet streams)

Forecast Method: Kinematic Approach: Analyze current observations of wind, temperature, and moisture fields

Assume clouds and precipitation occur when there is upward motion

and an adequate supply of moisture QG theory

QG Analysis:• Vertical Motion: Diagnose synoptic-scale vertical motion from the observed

distributions of differential geostrophic vorticity advection and temperature advection

• System Evolution: Predict changes in the local geopotential height patterns fromthe observed distributions of geostrophic vorticity

advectionand differential temperature advection

QG Analysis: Basic Idea


Recall: Two Prognostic Equations –Two Unknowns:

• We defined geopotential height tendency (X) and then expressed geostrophic vorticity (ζg)

and temperature (T) in terms of the height tendency.

2

0

1fg

2

0

2

0

11

ffttg

pRpT

pRp

pRp

ttT

pffV

t ggg

0)(p

fff

Vf o

go

022 11

RpTV

tT

g

Rp

pRpV

pRp

g

QG Analysis: A Closed System of Equations

t


The QG Height Tendency Equation:

We can also derive a single prognostic equation for X by combining our modified vorticity and thermodynamic equations (the height-tendency versions):

To do this, we need to eliminate the vertical motion (ω) from both equations

Step 1: Apply the operator to the thermodynamic equation

Step 2: Multiply the vorticity equation by

Step 3: Add the results of Steps 1 and 2

After a lot of math, we get the resulting prognostic equation……


pff

fV

f og

o

022 11

Rp

pRpV

pRp

g

pR

pf

20

0f

VorticityEquation

AdiabaticThermodynamic

Equation

TVpRf

pfVf

pf

go

ggo

2

2

2202


The QG Height Tendency Equation:

• This is (2.32) in the Lackmann text• This form of the equation is not very intuitive since we transformed geostrophic vorticity and temperature into terms of geopotential height.• To make this equation more intuitive, let’s transform them back…


p

Vfp

ff

Vfp

fg

o

ogo

22

2

2202 1

2

0

1fg pR

pT


The BASIC QG Height Tendency Equation:

Term A Term B Term C

• To obtain an actual value for X (the ideal goal), we would need to compute the forcing terms (Terms B and C) from the three-dimensional wind and temperature fields, and then invert the operator in Term A using a numerical procedure, called “successive over-relaxation”, with appropriate boundary conditions

• This is NOT a simple task (forecasters never do this)…..

Rather, we can infer the sign and relative magnitude of X simple inspection of the three-dimensional absolute geostrophic vorticity and temperature fields (forecasters do this all the time…)

Thus, let’s examine the physical interpretation of each term….

TVpRf

pfVf

pf

go

ggo

2

2

2202





Term A: Local Geopotential Height Tendency

This term is our goal – a qualitative estimate of the synoptic-scale geopotential height change at a particular location

• For synoptic-scale atmospheric waves, this term is proportional to –X• Thus, if we incorporate the negative sign into our physical interpretation, we can just think of this term as local geopotential height change

TVpRf

pfVf

pf

go

ggo

2

2

2202





Term B: Geostrophic Advection of Absolute Vorticity (Vorticity Advection)

Recall for a Single Pressure Level:

• Positive vorticity advection (PVA) PVA → causes local vorticity increases

• From our relationship between ζg and χ, we know that PVA is equivalent to:

therefore: PVA → or, since: PVA →

Thus, we know that PVA at a single level leads to height falls Using similar logic, NVA at a single level leads to height rises

TVpRf

pfVf

pf

go

ggo

2

2

2202


0tg

2

0

1p

g

ft

02 p 0 2




Trough AxisInitial Time

NVA

Expect Height Rises

PVA

Expect Height Falls

Expect the trough to move east

Initial Time


Full-PhysicsModel

Analysis


Trough AxisTrough AxisInitial Time

12 Hours Later




Generallyconsistent

with expectations!




Generally Consistent…BUT…Remember!• Only evaluated one level (500mb) → should evaluate multiple levels• Used full wind and vorticity fields → should use geostrophic wind and vorticity• Mesoscale-convective processes → QG focuses on only synoptic-scale (small Ro)• Condensation / Evaporation → neglected diabatic processes• Did not consider differential temperature (thermal) advection (Term C)!!!

Application Tips: Often the primary forcing in the upper troposphere (500 mb and above)

• Term is equal to zero at local vorticity maxima / minima• If the vorticity maxima / minima are collocated with trough / ridge axes, (which is often the case) this term cannot change system strength by increasing or decreasing the amplitude of the trough / ridge system Thus, this term is often responsible for system motion [more on this later…]





Term C: Vertical Derivative of Geostrophic Temperature Advection (Differential Thermal Advection)

• Consider a three layer atmosphere where the warm air advection (WAA) is strongest in the upper layer

• The greater temperature increase aloft will produce the greatest thickness increase in the upper layer and lower the pressure surfaces (or heights) in the lower levels

Therefore an increase in WAA advection with height leads to height falls

TVpRf

pfVf

pf

go

ggo

2

2

2202


WAA

WAA

WAA

Z-top

Z-400mb

Z-700mbZ-bottom

ΔZ increasesΔZHeight Falls

occurbelow the

level ofmaximum

WAA





• Possible height fall scenarios: Strong WAA in upper levelsWeak WAA in lower levels

WAA in upper levelCAA in lower levels

No temperature advection in upper levelsCAA in lower levels

Weak CAA in upper levelsStrong CAA in lower levels

TVpRf

pfVf

pf

go

ggo

2

2

2202






• Consider a three layer atmosphere where the warm air advection (CAA) is strongest in the upper layer

• The greater temperature increase aloft will produce the greatest thickness increase in the upper layer and lower the pressure surfaces (or heights) in the lower levels

Therefore an increase in CAA advection with height leads to height rises

TVpRf

pfVf

pf

go

ggo

2

2

2202


Height risesoccur

below thelevel of

maximumCAA

CAA

CAA

CAA

Z-topZ-400mb

Z-700mb

Z-bottom

ΔZ decreasesΔZ





• Possible height rise scenarios: Strong CAA in upper levelsWeak CAA in lower levels

CAA in upper levelWAA in lower levels

No temperature advection in upper levelsWAA in lower levels

Weak WAA in upper levelsStrong WAA in lower levels

TVpRf

pfVf

pf

go

ggo

2

2

2202






InitialTrough Axis

Strong WWA

Weaker WAA aloft(not shown)

Expect Height Rises

Initial Time850 mb

Strong CAA

Weaker CAA aloft(not shown)

Expect Height Falls

Full-PhysicsModel

Analysis


InitialTrough Axis

12 Hours Later850 mb

Trough “deepened”Ridge ”rose”

slightly




Generallyconsistent

with expectations!




Generally Consistent…BUT…Remember!• Used full wind field → should use geostrophic wind • Only evaluated one level (850mb) → should evaluate multiple levels/layers **• Mesoscale-convective processes → QG focuses on only synoptic-scale (small Ro)• Condensation / Evaporation → neglected diabatic processes• Did not consider vorticity advection (Term B)!!!

Application Tips: Often the primary forcing in the lower troposphere (below 500 mb)

• Term is equal to zero at local temperature maxima / minima Since the temperature maxima / minima are often located between the trough / ridge axes, significant temperature advection (or height changes) can occur at the axes and thus amplify the system intensity





Important: You should evaluate the vertical structure of temperature advection!!!





Important: You should evaluate the vertical structure of temperature advection!!!


N-S Cross-section of Temperature Advection

WAA = Warm ColorsCAA = Cool Colors


The BASIC QG Omega Equation:

Application Tips:

Remember the underlying assumptions!!!

You must consider the effects of both Term B and Term C at multiple levels!!!

If the vorticity maxima/minima are not collocated with trough/ridge axes, then Term B will contribute to system intensity change and motion If the vorticity advection patterns change with height, expect the system

“tilt” to change with time (become more “tilted” or more “stacked”) If differential temperature advection is large (small), then expect

Term C to produce large (small) changes in system intensity

Opposing expectations from the two terms at a given location will weaken the total vertical motion (and complicate the interpretation)!!!

The QG height-tendency equation is a prognostic equation:

• Can be used to predict the future pattern of geopotential heights• Diagnose the synoptic–scale contribution to the height field evolution• Predict the formation, movement, and evolution of synoptic waves



ReferencesBluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume I: Principles of Kinematics and Dynamics.

Oxford University Press, New York, 431 pp.

Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume II: Observations and Theory of WeatherSystems. Oxford University Press, New York, 594 pp.

Charney, J. G., B. Gilchrist, and F. G. Shuman, 1956: The prediction of general quasi-geostrophic motions. J. Meteor.,13, 489-499.

Durran, D. R., and L. W. Snellman, 1987: The diagnosis of synoptic-scale vertical motionin an operational environment. Weather and Forecasting, 2, 17-31.

Hoskins, B. J., I. Draghici, and H. C. Davis, 1978: A new look at the ω–equation. Quart. J. Roy. Meteor. Soc., 104, 31-38.

Hoskins, B. J., and M. A. Pedder, 1980: The diagnosis of middle latitude synoptic development. Quart. J. Roy. Meteor.Soc., 104, 31-38.

Lackmann, G., 2011: Mid-latitude Synoptic Meteorology – Dynamics, Analysis and Forecasting , AMS, 343 pp.

Trenberth, K. E., 1978: On the interpretation of the diagnostic quasi-geostrophic omega equation. Mon. Wea. Rev., 106,131-137.

QG Analysis: System Evolution

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