Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and
manipulating the equations of motion we can arrive at:
geostrophic balance
deviations from geostrophic balance (curvature and friction)
hydrostatic balance
hypsometric equation
thermal wind equation
Quasigeostrophic omega equation
DLA
Fig.10.2
The space and time scales of motion for a particular type of system are
the characteristic distances and times traveled by air parcels in the
system (or by molecules for molecular scales).
Example Scale Analysis
Horizontal Momentum Equation
Synoptic Scale:
U ≈ 10 m/s
W ≈ 10-2 m/s
L ≈ 106 m
H ≈ 104 m
T = L/U ≈ 105 s
R ≈ 107 m
fo ≈ 10-4 1/s
Po ≈ 1000 hPa
1 Pa = kg/(ms2)
ρ ≈ 1 kg/m3
Du
Dt
-2Wvsin f
U2
L
foU
10-4
10-3
-2Wvsin f
= -
+2Ww cos f
+
uw
a
-
uv tan f
a
foW
UW
a
U2
a
10-6
10-8
10-5
1 ¶p
r ¶x
=
-
1 ¶p
r ¶x
DP
rL
10-3
geostrophic balance
Forces Acting on the Atmosphere – Pressure Gradient Force
causes a net force
on air, directed
toward lower
pressure
DLA Fig. 7.5
Forces Acting on the Atmosphere
Coriolis Force
to the right of motion in the NH
strength determined by:
1.latitude
2.speed of motion
DLA Fig. 7.7A
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and
manipulating the equations of motion we can arrive at:
geostrophic balance
deviations from geostrophic balance (curvature and friction)
hydrostatic balance
hypsometric equation
thermal wind equation
Quasigeostrophic omega equation
Forces Acting on the Atmosphere
Centripetal Force & Gradient Wind Balance
force pointing
away the center
around which an
object is turning
centripetal acc = centrifugal force
(difference
beteeen PGF and
COR)
DLA Fig. 7.13
Geostrophic Approximation: Strengths and Weaknesses – curved flow
Winds and Heights at 500 mb
Geostrophic Approximation: Strengths and Weaknesses
Geostrophic Winds at 500 mb (determined using analyzed Z and geostrophic equations)
Geostrophic, Gradient, and Real Winds
Winds - Geostrophic Winds = Ageostrophic Winds (What’s Missing From Geostrophy)
Vg is too weak
Vg is too strong
Forces Acting on the Atmosphere
Friction
DLA Fig. 7.14
DLA Fig. 7.15
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and
manipulating the equations of motion we can arrive at:
geostrophic balance
deviations from geostrophic balance (curvature and friction)
hydrostatic balance
hypsometric equation
thermal wind equation
Quasigeostrophic omega equation
Example Scale Analysis
Vertical Momentum Equation
Synoptic Scale:
U ≈ 10 m/s
W ≈ 10-2 m/s
L ≈ 106 m
H ≈ 104 m
T = L/U ≈ 105 s
R ≈ 107 m
fo ≈ 10-4 1/s
Po ≈ 1000 hPa
1 Pa = kg/(ms2)
ρ ≈ 1 kg/m3
Dw
u2 + v 2
1 ¶p
- 2Wucosf = - g + Frz
Dt
R
r ¶z
UW
L
10 -7
f oU
U2
R
Po
rH
g
nWH -2
10 -3
10 -5
10
10
10 -15
1 ¶p
= -g
r ¶z
hydrostatic balance
Hydrostatic Balance
air parcel in
hydrostatic balance
experiences no net
force in the vertical
DLA Fig. 7.6
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and
manipulating the equations of motion we can arrive at:
geostrophic balance
deviations from geostrophic balance (curvature and friction)
hydrostatic balance
hypsometric equation
thermal wind equation
Quasigeostrophic omega equation
Geopotential, Geopotential Height, and the Hyposmetric Equation
We arrive at the hypsometric equation by using scale analysis (hydrostatic balance)
and by combining the hydrostratic equation and the equation of state
Dp
Df = -RT
p
DZ = -
Hypsometric Equation
RT Dp
g p
The hypsometric equation:
1. provides a quantitative measure of the geometric distance between 2 pressure
surfaces – it is directly proportional to the temperature of the layer
2. Shows that the gravitational potential energy gained when raising a parcel is
also proportional to the temperature of the layer
We can quantitatively see what we intuitively know:
a warm layer will be thicker than a cool layer
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and
manipulating the equations of motion we can arrive at:
geostrophic balance
deviations from geostrophic balance (curvature and friction)
hydrostatic balance
hypsometric equation
thermal wind equation
Quasigeostrophic omega equation
Thermal Wind - Concepts
• Horizontal T gradients  horizontal p gradients  vertical
variations in winds (e.g. geostrophic winds)
 A non-zero horizontal T gradient leads to vertical wind shear
• Thermal wind (VT) describes this vertical wind shear:
→ not an actual wind
→ it represents the difference between the geostrophic wind at 2
vertical levels
→ specifically, VT relates the horizontal T gradient to the vertical
wind shear
Thermal Wind - Concepts
• VT is therefore a useful tool for analyzing the relationship
between T, ρ, p and winds
¶Vg
¶z
Û ÑH T
• VT also provides information about T advection
(backing vs. veering)
The Thermal Wind Equation
• VT is derived by combining the hypsometric equation and the
geostrophic equation
const DT
uT » f Dy
const DT
; vT »
f Dx
• Note similarity to geostrophic wind, except T replaces Φ
• VT ‘blows’ parallel to isotherms, with low T on the left
Thermal Wind
Spatial relationships between horizontal T and thickness gradients,
horizontal p gradient, and vertical geostrophic wind gradient.
cold
warm
H, Fig. 3.8
( ) - (v )
vT = vg
top
g bot
const DT
»
f Dx
DT
>0
Dx
Þ
vT is positive
vg increases w/ height
Thermal Wind – Climatological Averages
¶ug
DT
>0 Û
<0
Dy
¶z
¶ug
DT
»0 Û
»0
Dy
¶z
¶ug
DT
<0 Û
>0
Dy
¶z
North
South
y
( ) - (u )
uT = ug
top
g bot
const DT
»f Dy
WH Figure 1.11
Thermal Wind – Extratropical Cyclone
we can apply the
same logic to the
instantaneous
picture in an
extratropical
cylcone
N
W
SE
Vertical cross section from Omaha, NE to Charleston, SC. WH Figure 3.19
Synoptic Scale Balance Equations
Using scale analysis (to identify the dominant ‘forces at work’) and
manipulating the equations of motion we can arrive at:
geostrophic balance
deviations from geostrophic balance (curvature and friction)
hydrostatic balance
hypsometric equation
thermal wind equation
Quasigeostrophic omega equation
Term B – Relationship of Upper Level Vorticity to Divergence / Convergence
DLA Fig. 8.31
æ ¶u ¶v ö
Dz a
= - foç + ÷
Dt
è ¶x ¶y ø
following air parcel motion:
- divergence occurs where ζa is decreasing
- convergence occurs where ζa is
increasing
Omega Equation – Derivation
(1)
(2)
(3)
¶z g
¶w
= -Vg iÑ z g + f + fo
¶t
¶p
(
)
æ ¶f ö
¶ æ ¶f ö
- ÷ = -Vg iÑ ç - ÷ + sw
ç
è ¶p ø
¶t è ¶p ø
1 2
zg = Ñ f
fo
plug (3) into (1)
re-arrange (2)
quasigeostrophic
vorticity equation
quasigeostrophic
thermodynamic equation
quasigeostrophic relative vorticity can be
expressed as the Laplacian of geopotential
(4)
(5)
1 2 æ ¶f ö
¶w
Ñ ç ÷ = -Vg iÑ z g + f + fo
fo è ¶t ø
¶p
(
)
æ ¶f ö
¶ æ ¶f ö
- ç ÷ = -Vg iÑ ç - ÷ + sw
è ¶p ø
¶p è ¶t ø
Omega Equation – Derivation
¶
delete t derivates by : fo ( 4 ) + Ñ2 ( 5 ) Þ Omega Equation
¶p
æ ¶f ö ù
¶
æ 2 fo ö
2 é
s ç Ñ + ÷ w = fo éëVg iÑ z g + f ùû + Ñ êVg iÑ ç - ÷ ú
è
sø
¶p
è ¶p ø û
ë
(
)
the QG Omega Equation is a diagnostic equation used to
determine rising and sinking motion based solely on the 3D
structure of the geopotential
• no wind observations necessary
• no info regarding vorticity tendency
• no T structure
• downside: higher order derivates
Omega Equation – Derivation
æ ¶f ö ù
¶
æ 2 fo ö
2 é
s ç Ñ + ÷ w = fo éëVg iÑ z g + f ùû + Ñ êVg iÑ ç - ÷ ú
è
sø
¶p
è ¶p ø û
ë
(
A
Rising/Sinking
A ≅ - sign
LHS ≅ - ω
+ RHS = rising motion
- RHS = sinking motion
)
B
Differential
Vorticity Advection
+ B = + vorticity adv.
rising
-B = - vorticiy adv.
sinking
C
Thickness Advection
+ C = warm adv.
rising
- C = cold adv.
sinking
Term B – Differential Vorticity Advection
¶
w µ éë -Vg iÑ z g + f ùû
¶z
(
PVA
¶z g
¶f
>0Û
<0
¶t
¶t
500 mb Height
1000 mb Height
PVA
)
z g µ Ñ f µ -f
2
¶z g
¶f
\
µ¶t
¶t
Above Surface
the columnL is cooling
there is very little temperature
advection above the L center the
only way for the layer to cool is
via adiabatic cooling (rising)
H Fig. 6.11
Term B – Differential Vorticity Advection
¶
w µ éë -Vg iÑ z g + f ùû
¶z
(
NVA
¶z g
¶f
<0Û
>0
¶t
¶t
500 mb Height
1000 mb Height
NVA
)
z g µ Ñ f µ -f
2
¶z g
¶f
\
µ¶t
¶t
Above Surface H
the column is warming
there is very little temperature
advection above the H center the
only way for the layer to warm is
via adiabatic warming (sinking)
H Fig. 6.11
Term B – Differential Vorticity Advection
the ageostrophic circulation (rising/sinking) predicted in
the previous slides maintains a hydrostatic T field (T and
thickness are proportional) in the presence of differential
vorticity advection
without the vertical motion, either the vorticity changes at
500 mb could not remain geostrophic or the T changes in
the 1000-500 mb layer would not remain hydrostatic
Term C – Thickness Advection
æ ¶f ö
w µ -Vg iÑ ç - ÷
è ¶p ø
WAA
¶z g
¶f
>0Û
<0
¶t
¶t
500 mb Height
1000 mb Height
WAA
z g µ Ñ f µ -f
2
¶z g
¶f
\
µ¶t
¶t
At the 500 mb Ridge
anticyclonic vorticity must
increase at the 500 mb ridge,
vorticity advection cannot produce
additional anticyclonic vorticity
divergence is required (rising)
H Fig. 6.11
Term C – Thickness Advection
æ ¶f ö
w µ -Vg iÑ ç - ÷
è ¶p ø
CAA
¶z g
¶f
<0Û
>0
¶t
¶t
500 mb Height
1000 mb Height
CAA
z g µ Ñ f µ -f
2
¶z g
¶f
\
µ¶t
¶t
At the 500 mb Trough
cyclonic vorticity must increase
at the 500 mb trough, vorticity
advection cannot produce
additional cyclonic vorticity
convergence is required (sinking)
H Fig. 6.11
Term C – Thickness Advection
the predicted vertical motion pattern is exactly that
required to keep the upper-level vorticity field
geostrophic in the presence of height changes caused by
the thermal advection