The Quasi-Geostrophic Omega Equation (without friction and diabatic terms)

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THE TRENBERTH (1978) INTERPRETATION
The Quasi-Geostrophic Omega Equation (without
friction and diabatic terms)
PROBLEM TERM 1 and 2 on the RHS are often large
and opposite leading to ambiguity about the sign
and magnitude of ? when analyzing weather maps
Trenberth (1978) argued that carrying out all of
the derivatives on the RHS on the ? Equation
could simplify the forcing function for ?.
We will now develop the Trenberth (1978)
modification to the QG Omega equation
Trenberth, K.E., 1978 On the Interpretation of
the Diagnostic Quasi-Geostrophic Omega Equation.
Mon. Wea. Rev., 106, 131137
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QG OMEGA EQUATION
EXPAND THE ADVECTION TERMS
USE THE EXPRESSIONS FOR THE GEOSTROPHIC WIND AND
GEOSTROPHIC VORTICITY
To Get
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EXPAND ALL THE DERIVATIVES THEN USE THE JACOBIAN
OPERATOR TO SIMPLIFY NOTATION
RESULT
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Same term opposite sign
Opposite term opposite sign same term
Deformation Terms in Sutcliff eqn
Trenberth Ignore deformation terms (removes
frontogenetic effects)
Approximate last term 2 ? last term
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Expand Jacobian terms
This result says that large scale vertical
motions can be diagnosed by Examining the
advection of absolute vorticity by the thermal
wind
RECALL SUTCLIFFS EQUATION
SAME INTERPRETATION!!
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The Geostrophic paradox
Confluent geostrophic flow will tighten
temperature gradient, leading to an increase in
shear via the thermal wind relationship..
but advection of geostrophic momentum by
geostrophic wind decreases the vertical shear in
the column
sogeostrophic flow destroys geostrophic balance!
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The geostrophic paradox a mathematical
interpretation
y momentum equation (QG)
Thermodynamic energy equation (QG)
For the moment, lets ignore the ageostrophy (no
uag and no ?)
Lets look at this equation
Take vertical derivative of first equation
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Expand the derivative
Substitute using the thermal wind relationship
to get
Remember equation in blue box
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The geostrophic paradox a mathematical
interpretation
y momentum equation (QG)
Thermodynamic energy equation (QG)
For the moment, lets ignore the ageostrophy (no
uag and no ?)
Now lets look at this equation
Take x derivative of second equation
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Expand the derivative and use vector notation
Now recall first saved equation
Lets take these two blue boxed equations and
compare them..
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thermal wind balance
Following the geostrophic wind the magnitude of
the temperature gradient and the vertical shear
have opposite Tendencies TIGHTENING THE
TEMPERATURE GRADIENT WILL REDUCE THE SHEAR!
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The Geostrophic paradox
RESOLUTION

• A separate ageostrophic
• circulation must exist that
• restores geostrophic balance that
• simultaneously
• Decreases the magnitude of the horizontal
  temperature gradient
• 2) Increases the vertical shear

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The Q-Vector interpretation of the Q-G Omega
Equation (Hoskins et al. 1978)
From consideration of the geostrophic wind, we
derived these equations
Lets denote the term on the RHS
If we start with our original equations, below,
y momentum equation (QG)
Thermodynamic energy equation (QG)
and perform the same operations as before, but
with the ageostrophic terms included.
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We arrive at
With ageostrophic terms
Only geostrophic terms
Note that the additional terms represent the
ageostrophic circulation that works to
reestablish geostrophic balance as air
accelerates in unbalanced flow.
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With ageostrophic terms
Lets multiply the bottom equation by -1 and add
it to the top equation, recalling that
Lets do the same operations with the x equation
of motion and the thermodynamic equation. If we
do, we find that
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Lets do the same operations with the x equation
of motion and the thermodynamic equation.
Where
and
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A
B
Take
Substitute continuity equation
And use vector notation to get
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COMPARE THIS EQUATION WITH THE TRADITIONAL QG ?
EQUATION!
We can write the Q-vector form of the QG ?
equation as
Where the components of the Q vector are
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Using the hydrostatic relationship, we can write
Q more simply as
or in scalar notation as
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First note that if the Q vector is convergent
Therefore air is rising when the Q vector is
convergent
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Lets go back to our jet entrance region
Q divergence
Q convergence
Note that there is no in this particular
jet
The Q vectors capture the sense of the
ageostrophic circulation and allow us to see
where the rising motion is occurring
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Resolution of the Geostrophic Paradox
Q divergence
The Q vectors capture the sense of the
ageostrophic circulation and allow us to see
where the rising motion is occurring
Q convergence
Q vectors diagnose a thermally direct circulation
Adiabatic cooling of rising warm air Adiabatic
warming of sinking cold air Counteracts the
tendency of the geostrophic temperature advection
in confluent flow
Under influence of Coriolis force,
horizontal branches tend to increase
shear Counteracts the tendency of the
geostrophic Momentum advection in the confluent
flow
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A natural coordinate version of the Q
vector (Sanders and Hoskins 1990)
Consider a zonally oriented confluent entrance
region of a jet where
Use non-divergence of geostrophic wind
or
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A natural coordinate version of the Q
vector (Sanders and Hoskins 1990)
Consider a meridionally oriented
confluent entrance region of a jet where
Use non-divergence of geostrophic wind
or
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A natural coordinate version of the Q
vector (Sanders and Hoskins 1990)
Using these two expressions, lets adopt A
natural coordinate expression for Q
Adopt a coordinate system where is
directed along the isotherms
is
directed normal to the isotherms
Q vector oriented perpendicular to the vector
change in the geostrophic wind along the
isotherms. Magnitude proportional to temperature
gradient and inversely proportional to pressure.
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rising motion
sinking motion
Simple Application 1 Train of cyclones and
anticyclones
At center of highs and lows Black
arrows Gray arrows Bold arrows
Note also that because of divergence/convergence,
train of cyclones and anticyclones propagates
east along direction of thermal wind
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sinking motion
rising motion
Simple Application 2 Pure deformation flow with
a temperature gradient
Along axis of dilitation Black arrows Gray
arrows Bold arrows
Increases toward east
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Simple Application 3 Homogeneous warm advection
No variation in Black arrows Gray arrows
Bold arrows
along an isotherm
No heterogeneity in the warm advection field No
rising motion!
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Note that the Q vector form of the QG ?-equation
contains the deformation terms (unlike the
Sutcliff and Trenberth forms) And combines the
vorticity and thermal advection terms into a
single diagnostic (unlike the traditional QG
?-equation)
Deformation term contribution to ?
Sutcliff/Trenberth approximation
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The along and across-isentrope components of the
Q vector
Begin with the hydrostatic equation in potential
temperature form
where
(which is constant on an isobaric surface)
And the definition of the Q vector
Substituting
This expression is equivalent to
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The Q-vector describes the rate of change of the
potential temperature along The direction of the
geostrophic flow
Lets consider separately the components of Q
along and across the isentropes
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Is parallel to and can only affect changes
in the magnitude of
Is perpendicular to and can only affect
changes in the direction of
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Returning to QG ? equation
Components of vertical motion can be distributed
in couplets across (transverse to) the thermal
wind (mean isotherms) and along (shearwise) the
thermal wind. We will see later that the
transverse component of Q is related to the
dynamics of frontal zones.