- Parcel (e.g., convective instability)
- Normal modes (e.g., wave instability: baroclinic instability)

For wave form ~ exp{ik(x-ct)}:- dispersion relation gives Im(c) nonzero
- growth rate = k Im(c)
- gives exponential growth for Im(c) > 0
- which is instability
- (perturbation does not disappear; system never returns to original state)

- Iinitial value problem (not covered in this class)

- Two layers
- Quasi-geostrophic
- Vorticity equation in upper & lower layers:
- allows vertical shear in wind
- thus horizontal temperature gradients (by thermal wind rel.)
- thus available potential energy

- Single thermodynamic equation, by upper-lower thickness

- Two-layer model
- Perturbation method
- Variables = Basic State + Perturbatin
- Basic State is a solution by itself
- Linearize equations about Basic State
- Assume wave form: exp{ik(x-ct)}
- Amplitudes nonzero implies dispersion relationship for c

- Im(c) > 0 implies exponential growth (instability)

- Thermal wind = 0
- stable
- two types of Rossby waves: barotropic & baroclinic

- beta = 0 (f-plane)
- instability requires nonzero thermal wind
- short-wave cutoff: only synoptic-scale and longer waves are unstable
- instability directly proportional to thermal wind
- instability inversely propotional to static stability

- general case (beta-plane)
- same short-wave cutoff as f-plane
- thermal wind must exceed a nonzero threshold for instability
- beta stabilizes longest waves
- critical wavelength (approx. wavelength of max. instability) from k*k = sqrt(2)*lambda*lambda, where k is wave number
- observed values for lambda give critical wavelength ~ 4000 km (e.g., synoptic scale)

- Available potential energy - present when there are horizontal temperature gradients
- wave (eddy) kinetic energy grows when
- unstable wave causes warm air rising & cold air sinking (net upward heat flux)
- which lowers center of mass of system
- lowering potential energy

- wave (eddy) available potential energy
- decreases from warm air rising & cold air sinking
- increases from warm air moving poleward & cold air moving equatorward (net poleward heat flux)

- Upward motion ~ poleward warm-air advection
- With east-west wave tilt, vortex stretching makes
- positive voriticity more positive (larger)
- negative vorticity more negative

- warm air moves upward and poleward, cool air moves downward and equatorward
- requirement of A.P.E. to K.E. for wave limits slope of motions to be between slope of zonal average isentropes and zero slope
- "wedge of instability"

- These features are consistent with developing synoptic systems seen on weather maps:
- upper-level trough lags lower-level low-pressure center
- upward motion ahead of surface low, downward behind
- warm-air advection ahead of surface low, cold-air advection behind
- vertical motion of modest intensity (weaker than horizontal motions)
- sizes of synoptic weather systems a few thousand kilometers across (separation of low centers)

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