1. ATS/ESS 452: Synoptic Meteorology
Rossby Waves
QG Introduction
Cyclone Structure

2. Barotropic Vorticity and Rossby Waves
By making an assumption that the atmosphere is frictionless and barotropic, then we can greatly simplify the vorticity equation (above) and describe an extremely important atmospheric phenomenon, Rossby waves.
What are Rossby waves?
Quite simply, they are the wave-like perturbations you see in geopotential height contours throughout the atmosphere, but especially at 500-mb  troughs/ridges
Rossby waves are the major player in dictating weather on a daily (and longer time scales) basis.
They are critically important to the meridional transport of heat, moisture and momentum in the global energy balance
Afterall, the only reason we have weather is due to a global energy imbalance… weather redistributes this energy by transporting heat, moisture, and momentum from the equator to the poles and vice-versa.
Afterall, the only reason we have weather is due to a global energy imbalance. There is more heat and moisture at the equation then at the poles… ditto for atmospheric momentum. This causes a great instability in global energy… so weather redistributes this energy by transporting heat, moisture and momentum from the equation to the poles and vice-versa.

3. One of the assumptions we will make is of a barotropic atmosphere.
What is the barotropic assumption?
That the density depends only pressure, and following the ideal gas law, then temperature is constant on an isobaric surface.
In other words, temperature and density are constant on pressure surfaces and are only functions of height.
This implies that isotherms must parallels isoheights  no temperature advection!
No temperature advection  no thermal wind  no vertical wind shear.
We will also assume frictionless flow, which is actually a good assumption above the PBL.
But, for the flow to be barotropic, we must also assume nondivergent flow, which means no vertical motion.

4. Barotropic Vorticity and Rossby Waves
So based on those assumptions (barotropic atmosphere, no friction, no vertical motion), the above equation becomes:
OR,
Time-rate of change of absolute (relative + planetary) vorticity
Absolute vorticity advection
Middle equation is Eulerian form… if we combined the local term and the advection we get the Lagrangian form.
This equation is a statement of conservation of vorticity.
It says that absolute vorticity is conserved following the flow (i.e., absolute vorticity remains constant following the flow).

5. **Absolute vorticity is conserved following the flow.
What does this mean?
If an air parcel happens to move poleward to a region of larger planetary vorticity (f), then there must be a compensating change in the relative vorticity to keep the sum constant.
-board notes
In a vorticity-conserving flow, once an air parcel is displaced from its latitude of origin, an oscillation comes about with alternating anticyclonic and cyclonic curvature
It’s also important to note that planetary vorticity is only a function of the meridional direction, so we can write….. (board notes)

6. Through some substitutions and rearrangements of equations (see page 25), we can describe the behavior of Rossby waves.
We end up with:
Which is the Rossby wave phase speed equation, where c is the phase speed of Rossby waves (i.e., the speed of movement for trough, ridge axes).
This equation tells us the speed at which axes of troughs and ridges move in a barotropic atmosphere is given by:
- U  difference between the background zonal wind speed
- a term that involves the square of the wavelength (L) and the gradient of planetary vorticity (beta)
U is associated with the advection of vorticity by the background zonal flow
And beta results from the advection of planetary vorticity by the meridional wind

7. The two terms in the RHS of the phase speed equation “compete” with each other.
Competing effects:
- Advection by U wind moving system eastward
(if this term dominates then C is positive, so eastward movement)
- Advection of planetary vorticity moving system westward
(if this term dominates, then C is negative, so westward movement)
What do we need for the overall pattern to propagate eastward (prograde)?
- Strong westerly flow (large U)
- Small wavelength waves, such that the 2nd term is small  shortwaves!
This implies that longwaves tend to retrograde (propagate westward)
In reality, they remain pretty stagnate due to strong westerly flow.

8. Let’s use the advection of relative and planetary vorticity to physically understand how Rossby waves propagate.
-board notes
The tendency of Rossby waves to propagate westward relative to the flow is related to the “beta-effect”  motion due to the advection of planetary vorticity (the fact that f changes with latitude).

9. QG-Theory (An Overview)
So what have we done so far:
1.) Talked about the momentum equations & thermodynamic equations
- Specifically, geostrophic wind, advection, vorticity, etc….
2.) The momentum equations were combined to form the vorticity equation
in order to diagnose processes that lead to rotating systems
3.) Simplified the vorticity equation
to isolate the dynamics of Rossby waves, which dictate day-to-day weather, steering of tropical systems, etc.
Using a similar strategy, we will (eventually) simplify the entire set of governing equations to diagnose processes responsible for all synoptic systems
This simplified equation set is the “quasi-geostrophic” or QG set

10. QG-Theory (An Overview)
Paraphrased from Chapter 6 of Holton (Dynamic Meteorology):
A primary goal of dynamic meteorology is to interpret the observed structure of large-scale atmospheric motions in terms of physical laws governing the motions
These laws, which express the conservation of momentum, mass, and energy, completely determine the relationships among the pressure, temperature and velocity fields
Significant relationships between these laws
Pressure differences drive the wind, but temperature differences drive pressure

11. QG-Theory (An Overview)
Paraphrased from Chapter 6 of Holton (Dynamic Meteorology):
Atmospheric behavior can be extremely complicated. Fortunately, we can simplify these behaviors quite a bit in the mid-latitudes when dealing with synoptic-scale motions
Since synoptic-scale horizontal motions in the mid-latitudes (i.e. away from the ground) are approximately geostrophic, we can apply quasi-geostrophic relationships to these motions
 winds blow like they are in geostrophic balance but not quite
For synoptic scale motions, the wind is approximately in rotational balance…. In other words, the wind blows nearly parallel to height contours (see 500-mb)

12. QG-Theory (An Overview)
Paraphrased from Chapter 6 of Holton (Dynamic Meteorology):
Quasi-geostrophic behavior on the synoptic scale allows many simplifications to be made to the laws of motion
The above statement is important because synoptic systems are of primary importance in short-range weather forecasting
 Can’t use QG approximations on mesoscale features
 Need large scale and long time period features (i.e., those that occur on the synoptic scale)
On the mesoscale, winds are NOT rotationally balanced, so QG does not work on the mesoscale or smaller scales. The winds need to at least start out being geostrophic, and that’s not always the case on the mesoscale.

13. QG-Theory (An Overview)
So why do we need QG-theory… besides making the equations simple to use on the synoptic scale?
Because geostrophic flow has limitations!
- No friction allowed
- Must have sufficient Coriolis
- No divergence allowed (no weather!)
- No curved flow allowed (curved flow is very important – look at the flow around troughs/ridges at 500-mb)
QG allows for some of the above

14. QG-Theory (An Overview)
QG framework is the cornerstone of synoptic analysis & forecasting and a good deal of climate dynamics work as well
For synoptic-scale motions, we utilize the fact that midlatitude atmosphere is approximately geostrophic to simplify equations
We will eventually develop two equations that will explain A LOT
QG omega equation:
Relates vertical air motion to thermal and vorticity advections
QG height-tendency equation:
Illustrates processes leading to development and movement of weather systems
Remember – WAA = lift and CAA = sinking air… well this is the equation that will say why that is the case.

15. QG-Theory (An Overview)
Two main requirements for applying Q-G theory:
- Hydrostatic Balance
- Geostrophic Balance
Result:
- The structure and subsequent evolution of a synoptic- scale weather system can be determined by the distribution of geopotential height on an isobaric surface

16. But before we really talk in depth about QG-theory, it is important to understand the basic structure of extratropical (or midlatitude) cyclones.
First… what is an “extratropical” cyclone?
Cyclones defined as synoptic scale low pressure weather systems that occur in the middle latitudes of the Earth and are connected with fronts and horizontal temperature gradients (baroclinic zones).
They are the everyday phenomena which drive the weather over much of the Earth

17. Observed Structure of Extratropical Systems
Fig. 6.1 (Meridional Cross Sections of Temperature and Zonal Winds)
Meridional temperature gradient characteristics
Compare winter vs. summer in NH
Temperature gradient
Larger in NH winter, less in NH summer
Not as much seasonal change in SH
Can see migration of ITCZ (Thermal Equator)
Sfc temps are warmer in NH summer than SH summer
SH has a lot of water, which warms up slower and cools off slower (high specific heat) when compared to land. NH has a lot more land… it heats up very quickly.

18. Observed Structure of Extratropical Systems
Fig. 6.1 (Meridional Cross Sections of Temperature and Zonal Winds)
Zonal wind characteristics
Jet stream much stronger in NH winter than in NH summer
Smaller seasonal difference in SH winter vs. summer
Jet core is located just below the tropopause
Jet core is found at the latitude where the thermal gradient (as averaged through the troposphere) is greatest:
30-35 degrees N during winter
40-45 degrees N during summer
**Jet stream found right along the largest temperature gradients in mid-latitudes
** We can deduce that the jet stream has a lot to do with frontal boundaries!

19. Observed Structure of Extratropical Systems
Fig. 6.2 (Longitudinal Distribution of Average Zonal 200 mb Winds for NH Winter)
Characteristics
Large differences in zonal 200-mb wind speed with longtiude
Two NH jet maxima in mid latitudes
East Asian Jet and Eastern North America Jet
Synoptic disturbances frequently develop near these maxima
Semi-permanent low surface pressure is observed near the left exit region of each jet maximum (Aleutian and Icelandic Lows)
Semi-permanent high pressure is observed near the left entrance region of each jet maximum (Siberian and NW Canada Highs)
Develop 4-quadrant jet theory for straight flow… correlate with semi-permanent surface pressure features.
Two NH jet minima in mid-latitudes
**Stongest winds found near 30 deg N
Variations in windspeed along latitude
**Jet streams are exhibitive of strong thermal gradients… notice where both of the average jet maxes are located. Strong warm ocean currents riding along continential boundaries create strong thermal gradients in those locations!