Chapter 8

Key Concepts

1. Instability methods

• Parcel (e.g., convective instability)
• Normal modes (e.g., wave instability: baroclinic instability)
  For wave form ~ exp{ik(x-ct)}:
  1. dispersion relation gives Im(c) nonzero
  2. growth rate = k Im(c)
  3. gives exponential growth for Im(c) > 0
  4. which is instability
  5. (perturbation does not disappear; system never returns to original state)
• Iinitial value problem (not covered in this class)

2. Minimal Model for Baroclinic Instability

• Two layers
• Quasi-geostrophic
• Vorticity equation in upper & lower layers:
  1. allows vertical shear in wind
  2. thus horizontal temperature gradients (by thermal wind rel.)
  3. thus available potential energy
• Single thermodynamic equation, by upper-lower thickness

3. Instability computation

• Two-layer model
• Perturbation method
  1. Variables = Basic State + Perturbatin
  2. Basic State is a solution by itself
  3. Linearize equations about Basic State
  4. Assume wave form: exp{ik(x-ct)}
  5. Amplitudes nonzero implies dispersion relationship for c
• Im(c) > 0 implies exponential growth (instability)

4. Properties of Two-Layer Model Dispersion Relation

• Thermal wind = 0
  1. stable
  2. two types of Rossby waves: barotropic & baroclinic
• beta = 0 (f-plane)
  1. instability requires nonzero thermal wind
  2. short-wave cutoff: only synoptic-scale and longer waves are unstable
  3. instability directly proportional to thermal wind
  4. instability inversely propotional to static stability
• general case (beta-plane)
  1. same short-wave cutoff as f-plane
  2. thermal wind must exceed a nonzero threshold for instability
  3. beta stabilizes longest waves
  4. critical wavelength (approx. wavelength of max. instability) from k*k = sqrt(2)*lambda*lambda, where k is wave number
  5. observed values for lambda give critical wavelength ~ 4000 km (e.g., synoptic scale)

5. Energetics

• Available potential energy - present when there are horizontal temperature gradients
• wave (eddy) kinetic energy grows when
  1. unstable wave causes warm air rising & cold air sinking (net upward heat flux)
  2. which lowers center of mass of system
  3. lowering potential energy
• wave (eddy) available potential energy
  1. decreases from warm air rising & cold air sinking
  2. increases from warm air moving poleward & cold air moving equatorward (net poleward heat flux)

6. Coordination of motions in unstable wave

• Upward motion ~ poleward warm-air advection
• With east-west wave tilt, vortex stretching makes
  1. positive voriticity more positive (larger)
  2. negative vorticity more negative
• warm air moves upward and poleward, cool air moves downward and equatorward
  1. 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
  2. "wedge of instability"
• These features are consistent with developing synoptic systems seen on weather maps:
  1. upper-level trough lags lower-level low-pressure center
  2. upward motion ahead of surface low, downward behind
  3. warm-air advection ahead of surface low, cold-air advection behind
  4. vertical motion of modest intensity (weaker than horizontal motions)
  5. sizes of synoptic weather systems a few thousand kilometers across (separation of low centers)

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