This is not an exhaustive guide to the quarter. You are responsible for all material discussed in lecture, not only the items listed here.

Waves in the extratropical atmosphere
• We studied basic wave properties, including phase and group velocity.
• I gave a "wave recipe" for solving wave problems.
• I argued that basic properties of extratropical weather systems can be understood by examining the linear waves in shallow water for constant Coriolis parameter. There are two sets of modes: Rossby waves and inertia--gravity waves.
• Rossby waves (RWs) have zero frequency, but we discussed how their frequency would be small if we allowed the Coriolis parameter to vary. The RWs have geostrophic balance, potential vorticity, and no divergence.
• Inertia-gravity waves move fast, are not in geostrophic balance, have no potential vorticity, and have significant divergence, which propagates the wave.
• We considered the evolution of an arbitrary localized initial condition, and showed that the solution evolves to a state of geostrophic balance as t->infinity. Potential vorticity is critical to understanding why this happens. In terms of the modes, the gravity waves propagate to infinity, leaving the stationary RWs.

Quasigeostrophic theory
• Based on observations, and our linear-wave analysis, we concluded that extratropical dynamics are dominated by RWs, not gravity waves. We set out to filter the gravity waves from the equations by linearizing the Ertel PV law, and assuming that the wind was well approximated by the geostrophic wind. This gave us the QG PV equation.
• The QG PV equation determines the time-evolution of extratropical weather systems. We wrote it in the form of a height tendency equation, which shows that heights fall where there is cyclonic PV advection.
• We derived a QG vorticity equation, which shows that vorticity changes following the flow due to vortex stretching of the planetary vorticity.
• Eliminating time derivatives gave a diagnostic equation for the w field. There were three forms of the w equation: traditional, Q-vector, and thermal-wind ("Sutcliffe").
• Comfort with Laplacians, quasi-Laplacians, and their inverse is critical to discussing these results.

Cyclone and front development
• Using the QG equations we showed the conditions necessary for the development of surface cyclones. From the PV perspective, pressure falls downwind of upper-level PV disturbances. From the w perspective, vortex stretching due to upward motion downwind of upper-level troughs increases the low-level vorticity.
• We talked about the importance of static stability, and how low static stability "stretches" the response of height tendencies and w in the vertical.
• Idealizing the extratropics in terms of a jet stream that increases linearly with height, we found exponentially growing disturbances that resemble to very good approximation the structure of developing cyclones in the atmosphere.
• We generalized the linear solution to include nonlinearity and the propagation of energy at the group velocity.
• The formation of fronts was considered by reviewing the linear variation of wind near a point. This revealed that only divergence and deformation can change the horizontal temperature gradient.
• An equation for the QG development of fronts shows two effects: Q-vector (deformation of existing temperature gradients by the geostrophic wind) and gradients in vertical motion. Thus the formation of temperature gradients is linked to vertical motion. Moreover, the ageostrophic circulation is needed to restore thermal wind balance, by changing the momentum and temperature fields. It also acts to oppose the action of the geostrophic (primary) circulation.

Tropical dynamics
• We generalized the shallow water linear wave analysis to include a variable Coriolis parameter. Applying this analysis around the equator, we have the tropical beta-plane.
• There are gravity and Rossby waves as in midlatitudes, but there structure is different. There are unique modes as well, including the Kelvin wave, and mixed Rossby-gravity waves. Regions of upward motion with these waves can help to organize convection. Observations show the existence of these waves, but the main signal in the Pacific Ocean comes from the Madden-Julian oscillation, which is a 30-60 mode that doesn't fit the linear analysis.
• Hurricanes are nearly axisymmetric disturbances that are conveniently described in terms of angular momentum and moist entropy (theta-e). Angular momentum surfaces flare outward in the eyewall due to thermal wind balance. Surface fluxes feed moisture into the storm and remove angular momentum.