ats/ess 452: synoptic meteorology -rossby waves -qg introduction -cyclone structure

ATS/ESS 452: Synoptic Meteorology

- Rossby Waves- QG Introduction- Cyclone Structure

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.

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.

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

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).

**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)

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

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.

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).

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 equationin order to diagnose processes that lead to rotating systems

3.) Simplified the vorticity equationto 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

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



- 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



- 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)


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


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


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


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

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

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


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


ats/ess 452: synoptic meteorology -rossby waves -qg introduction -cyclone structure

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