Why do we want to know the surface
pressure tendency?
Pressure Tendency Equation
Reading Carlson: pp 93-105
Lecture Outline
QG Review
Geopotential Height Tendency Equation
Surface Pressure Tendency Equation
Examples and Applications
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QG Refresher: The Vorticity Equation
1
3
Refresher: Omega
5
∂F ∂F
∂ (ζ + f )
∂ζ
∂u ∂v
∂ω ∂v ∂ω ∂u
= −V ⋅ ∇(ζ + f ) − ω
− (ζ + f )( + ) − (
−
)+( y − x)
∂t
∂P
∂x ∂y
∂x ∂P ∂y ∂P
∂x
∂y
[
]
How did we make sense of this equation?
4
2
Rd ~
s 2
f 2 ∂ 2ω
f ∂
∂Z
∇ ω+ 0
=− 0
− Vg ⋅ ∇( f + ζ g ) + ∇ 2 (−Vg ⋅ ∇ )
gp
g ∂p 2
g ∂p
∂p
6
Brief Descriptor of Terms in the Vorticity Equation
1. Local rate of change of spin of a fluid
2. Advection of vorticity by the horizontal flow
3. Advection of vorticity by the vertical flow
4. Source/Sink associated with Divergence
5. Source/Sink associated with Twisting/Tilting
6. Frictional Force
Impt: Note that we have a prognostic equation based on diagnostic fields
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Refresher: Omega
Map Analysis: Vorticity
From the vorticity advection term only, where is there rising/sinking
motion?
⎛ ∂2
⎞
f ∂
R
⎜⎜ α1 2 + α 2∇ 2 ⎟⎟ω = − 0
− Vg ⋅ ∇( f + ζ g ) − ∇ 2 (−Vg ⋅ ∇T )
g ∂p
P
⎝ ∂p
⎠
[
L(ω ) = Fv + FT
]
∂Z
R 2
2
Note: ∇ (−Vg ⋅ ∇ ∂p ) ≈ − P ∇ (−Vg ⋅ ∇T )
Simplification:
(1) Laplacian of omega is proportional to negative omega…
So if forcing terms of RHS are positive, then vertical motion is…
(2) Forcing terms on RHS occur simultaneously, and are referred to as
vorticity advection and temperature advection terms
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More detailed Map Analysis
Other QG Review Items
From the thermal forcing term only, where is there rising/sinking motion?
Let’s recall the following items we’ve covered thus far
• Vertical distribution of divergence determines vertical motion
(bow string model)
• Vertical motion is related to vorticity tendency
(vorticity equation)
• Surface vorticity tendency is innately related to surface divergence
(scale analysis of vorticity equation at surface, 3.6b)
• And therefore we would expect surface pressure tendency to be
related to vertical velocity near the level of non-divergence
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Derivation of Pressure Tendency
•
•
⎛ 2
∂2 ⎞
f
∂
∂Q&
⎜⎜ ∇ + k1 2 ⎟⎟ χ = 0 − Vg ⋅ ∇( f + ζ g ) − k 2
− Vg ⋅ ∇(T ) − k3
∂p ⎠
g
∂p
∂p
⎝
[
Similar to how we derived the omega equation, we employ the QG temperature
and vorticity equations
What falls out is the geopotential height tendency equation
]
Laplacian of Height = Vorticity Advection +
⎛ 2
⎛ f 02 p ∂ ⎞ ⎡
f 02 p ∂ 2 χ ⎞ f 0
f 02 ∂Q&
∂Z ⎤
⎜⎜ ∇ χ + R s ∂p 2 ⎟⎟ = g − Vg ⋅ ∇ ( f + ζ g ) + ⎜⎜ R s ∂p ⎟⎟ ⎢− Vg ⋅ ∇ ∂p ⎥ − gc s ∂p
⎦
d
p
⎝
⎠
⎝ d
⎠⎣
[
•
•
]
Where χ is the geopotential height tendency ( dZ/dt)
Let’s break this down term-by-term
– LHS: Laplacian term
– RHS: Term 1: vorticity advection
– RHS: Term 2: thermal advection differentiated w/respect to pressure
– RHS: Term 3: diabatic heating differentiated w/respect to pressure
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[
]
d/dp (-temp adv) + d/dp (diabatic)
Tendency
So from the above equation, how should height change in time if we have
Right hand side is positive
PVA
Increasing warm air advection with height (or decreasing WAA w/ pressure)
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The trouble with geopotential height
Examples on Board
• Height needs to be fixed at boundaries, so if we are solving for
500hPa height tendency we assume there is no change at bottom
(1000hPa) and top (10hPa)
• Let’s make this assumption for the time being
• Let’s set vort adv to 0 by evaluating at the vort max
• Differential temperature advection then drives height tendency
(d/dp(WAA)>0)
P=10hPa
P=500hPa
z
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P=1000hPa
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General Rules
General Rules Cont.
Vortcity Advection
• NVA at a location acts to produce height rises
• PVA at a location acts to produce height falls
Temperature Advection
• Negative thickness (CAA) advection that decreases with height
produces height falls, amplification of 500hPa trough
• Positive thickness (WAA) advection that decreases with height
produces height rises, amplification of 500hPa ridge
Properties of Temperature Advection
1. Temperature decreases with height
2. Due to the Thermal Wind, temperature advection at upper levels is
much smaller as geostrophic wind blow parallel to isotherms
3. Temperature gradient decreases with elevation (below 300hPa)
4. Therefore temperature advection largest at lower levels
5. Temperature Gradient largest on cold air side of surface fronts
Diabatic Heating
• Below heat source dQ/dP<0, so diabatic term is positive, χ?
• Above heat source dQ/dP>0, so diabatic term is negative, χ?
• Is this true for surface heating???
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Geopotential Height to Surface Pressure
Steps
• Surface of 1000hPa vorticity equation
• Relate vorticity to surface (1000hPa) height
• Relate 1000hPa height to surface pressure
• Utilize omega equation
−
[
] [
∂p
= a ' − Vg 5 ⋅ ∇( f + ζ g 5) + b' − Vg 0 ⋅ ∇h
∂t
]
Surface Pressure Falls in response to
PVA and WAA
Surface Pressure Rises in response to
NVA and CAA
[
] [
∂p
−
= a' − Vg 5 ⋅ ∇( f + ζ g 5) + b' − Vg 0 ⋅ ∇h
∂t
]
- Surface Pressure Tendency = Vort Adv + Thickness Advection
This looks quite a bit like our omega equation, but it is prognostic
Meaning: surface pressure tendency is at the whim of QG flow aloft
Application: pressure tendency informs you of surface cyclone propagation
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−
∂p
= a ' − Vg 5 ⋅ ∇( f + ζ g 5) + b' − Vg 0 ⋅ ∇h
∂t
[
] [
500hPa vort and height
]
Phase Lag
MSLP and 1000-500 thickness
1. Phase lag needed between 500hPa trough and surface low for
intensification
2. Upper level PVA occurs ahead of upper-trough to drive low level
development
3. Thermal term stretches even further upstream to extend tilt of
system
4. Thickness advection terms are generally not symmetric, H tends to
move south, and L to the north
Where do WAA + PVA act in the same direction?
Where do WAA and PVA opposed one another?
Where do we have surface pressure rises and falls?
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Movement of surface systems
•
•
•
•
Surface systems move in response to upper level forcing
Lows move towards areas of pressure falls
Highs move towards areas of pressure rises
They do not move as a unit, although it appears that way
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