Section 1: movement at 1 level

• consider barotropic vorticity equation (6.1a):
  (lcl chg of zeta) = (horiz. advection of zeta) - beta * v
• illustrates the "beta effect" where beta = df/dy >0 in Northern Hemisphere.
• We used a trough to illustrate this
  1. ahead (east) of the trough: PVA, but beta term negative since v>0
  2. behind the trough: NVA, but beta term positive (v<0)
• beta term a dipole pattern consistent with motion towards the west.
• can recover the Convergence (C) and Divergence (D) pattern we have seen before for developing systems
  • consider QG vorticity equation:
    d zeta_a/dt = - f D
  • wind speed increases with height
  • system moves with velocity of a middle atmospheric pressure level. Call that the steering level. (In dynamics this is a "critical level")
  • to deduce the total derivative on the LHS, need to consider motions relative to the trough.
  • for levels above the critical level:
    1. parcels move faster than the trough and so blow through it
    2. parcels ahead of the trough move away from it, so vorticity following the motion is decreasing
       from QG vort. eqn, it must be that D>0
    3. parcels behind the trough are overtaking the trough, so vorticity following the motion is increasing.
       from QG vort. eqn, D<0 for parcels behind the trough
       1. parcels below the critical level move slower than the trough, so the trough overtakes parcels ahead (east) of it. The trough moves away from parcels west of the trough.
       2. these relative motions imply vorticity changes, that imply D>0 behind the low level trough and D<0 ahead of it.

Section 2: Equivalent Barotropic Vorticity Eqn. (EBVE)

• General comments
  • simple vertical profiles of V and thus zeta, Given by A(p)
  • no turning of wind
  • integrate QG vort. eqn w.r.t. pressure to find mean values indicated by subscript "m".
  • designate a "star" level where vertical mean of A² = a value of A(p). P = P^* there.
  • integral of divergence term obtains Omega at the surface.
  • result is (6.6) p. 141.
• Advantages
  • looks like BVE, so it is easy to understand and interpret
  • Omega term can be expressed in terms of several processes:
    • local change of geopotential height (this term slows down long waves in particular. The BVE has problem that planetary (long) waves move rapidly to the west. The EBVE does not have this fault.
    • slope flow
    • Ekman pumping or suction from friction-driven surface con/divergence.
    • see supplemental notes for analysis of first 2 items.
• Disadvantages
  • many approximations were used -- so accuracy not good.
  • * level assumed fixed. such a level would vary
  • vertical profile too simple in various ways. The most important is that no turning is allowed. No turning -> no T advection -> no development.

Sections 3 and 4: vertical motion in EBVE & 500 mb flow steering

• Nomenclature:
  • CAA = backing = CCW turning of wind barb as go up
  • WAA = veering = CW turning of wind barb as go up
• eqn (6.8b) is like an "omega" eqn. we see vorticity advection, slope flow, but no T advection term on RHS.
• deduce trough motion by looking at its symmetry. It will tend to move in direction perpendicular to axis of symmetry.
  Ex: asymmetric trough with Z contours more closely spaced on the east side than on the west side likely will move to NE.
• these rules apply to troughs that are not developing!

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