1. 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

2. 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).

3. geostrophic balance Horizontal Momentum Equation
Example Scale Analysis
Horizontal Momentum Equation
Synoptic Scale:
U ≈ 10 m/s
W ≈ m/s
L ≈ 106 m
H ≈ 104 m
T = L/U ≈ 105 s
R ≈ 107 m
fo ≈ /s
Po ≈ 1000 hPa
1 Pa = kg/(ms2)
ρ ≈ 1 kg/m3
geostrophic balance

4. causes a net force on air, directed toward lower pressure
Forces Acting on the Atmosphere – Pressure Gradient Force
causes a net force on air, directed toward lower pressure
DLA Fig. 7.5

5. Forces Acting on the Atmosphere Coriolis Force
to the right of motion in the NH
strength determined by:
latitude
speed of motion
DLA Fig. 7.7A

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

7. Forces Acting on the Atmosphere
Centripetal Force & Gradient Wind Balance
force pointing away from the center around which an object is turning
centripetal acc = - centrifugal force (difference beteeen PGF and COR)
DLA Fig. 7.13

8. Geostrophic Approximation: Strengths and Weaknesses – curved flow
Winds and Heights at 500 mb

9. Geostrophic Approximation: Strengths and Weaknesses
Geostrophic Winds at 500 mb (determined using analyzed Z and geostrophic equations)

10. Geostrophic, Gradient, and Real Winds
Winds - Geostrophic Winds = Ageostrophic Winds (What’s Missing From Geostrophy)
Vg is too weak
Vg is too strong

11. Forces Acting on the Atmosphere
Friction
DLA Fig. 7.14
DLA Fig. 7.15

13. 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

14. hydrostatic balance Vertical Momentum Equation Example Scale Analysis
Synoptic Scale:
U ≈ 10 m/s
W ≈ m/s
L ≈ 106 m
H ≈ 104 m
T = L/U ≈ 105 s
R ≈ 107 m
fo ≈ /s
Po ≈ 1000 hPa
1 Pa = kg/(ms2)
ρ ≈ 1 kg/m3
hydrostatic balance

15. Hydrostatic Balance
air parcel in hydrostatic balance experiences no net force in the vertical
DLA Fig. 7.6

16. 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

17. 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
Hypsometric Equation
The hypsometric equation:
provides a quantitative measure of the geometric distance between 2 pressure surfaces – it is directly proportional to the temperature of the layer
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

18. 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

19. A non-zero horizontal T gradient leads to vertical wind shear
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

20. VT also provides information about T advection (backing vs. veering)
Thermal Wind - Concepts
VT is therefore a useful tool for analyzing the relationship between T, ρ, p and winds
VT also provides information about T advection (backing vs. veering)

21. Note similarity to geostrophic wind, except T replaces Φ
The Thermal Wind Equation
VT is derived by combining the hypsometric equation and the geostrophic equation
Note similarity to geostrophic wind, except T replaces Φ
VT ‘blows’ parallel to isotherms, with low T on the left

22. cold warm vT is positive vg increases w/ height Thermal Wind
Spatial relationships between horizontal T and thickness gradients, horizontal p gradient, and vertical geostrophic wind gradient.
cold
warm
H, Fig. 3.8
vT is positive
vg increases w/ height

23. Thermal Wind – Climatological Averages
North y
South
WH Figure 1.11

24. Thermal Wind – Extratropical Cyclone
we can apply the same logic to the instantaneous picture in an extratropical cylcone NW SE
Vertical cross section from Omaha, NE to Charleston, SC. WH Figure 3.19

25. 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

26. following air parcel motion:
Term B – Relationship of Upper Level Vorticity to Divergence / Convergence
DLA Fig. 8.31
following air parcel motion:
divergence occurs where ζa is decreasing
convergence occurs where ζa is increasing

27. quasigeostrophic vorticity equation (1)
Omega Equation – Derivation
quasigeostrophic vorticity equation
(1)
quasigeostrophic thermodynamic equation
(2)
quasigeostrophic relative vorticity can be expressed as the Laplacian of geopotential
(3)
plug (3) into (1)  (4)
re-arrange (2)  (5)

28. no wind observations necessary no info regarding vorticity tendency
Omega Equation – Derivation
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

29. C B A Omega Equation – Derivation Differential Vorticity Advection
+ B = + vorticity adv.
 rising
B = - vorticiy adv.
 sinking
Rising/Sinking
A ≅ - sign
LHS ≅ - ω
+ RHS = rising motion
- RHS = sinking motion
Thickness Advection
+ C = warm adv.
 rising
- C = cold adv.
 sinking

30. Above Surface L PVA the column is cooling
Term B – Differential Vorticity Advection
Above Surface L
PVA
the column 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)
500 mb Height
H Fig. 6.11
1000 mb Height
PVA

31. Above Surface H NVA the column is warming
Term B – Differential Vorticity Advection
Above Surface H
NVA
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)
500 mb Height
H Fig. 6.11
1000 mb Height
NVA

32. 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 mb layer would not remain hydrostatic

33. Term C – Thickness Advection
At the 500 mb Ridge
WAA
anticyclonic vorticity must increase  at the 500 mb ridge, vorticity advection cannot produce additional anticyclonic vorticity  divergence is required (rising)
500 mb Height
H Fig. 6.11
1000 mb Height
WAA

34. Term C – Thickness Advection
At the 500 mb Trough
CAA
cyclonic vorticity must increase  at the 500 mb trough, vorticity advection cannot produce additional cyclonic vorticity  convergence is required (sinking)
500 mb Height
H Fig. 6.11
1000 mb Height
CAA

35. 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