• Summary table

  The information here is summarized in a brief summary table

• Local rate of change

  • 10^-10 sec^-2 will be the unit of measure we use to compare terms.
  • 1 unit is: 0.1 mb per 3 hrs for a "square" wave with wavelength of about 2400 km (which is a rather short length).
  • for simple advection of a max, pattern is a dipole centered about the max of "Zeta": the vorticity
  • if there is amplification, then the area of positive tendency must be larger and vice-versa.

• Horizontal Advection

  • observed pattern is often a dipole centered about the max of the advected quantity (e.g. vorticity)
  • increases with height
    1. wind speeds generally increase with height
    2. pattern more favorable to contour intersections: low level Z pattern is "cellular" while at upper levels it is more "wavelike".
  • example shown: fig. 3.1: Note: contour interval of Z increases with height, so advection increases with height even faster than the figures look.
  • Horizontal advection has peak values at upper levels
  • peak values ~ 50 units (= 50 x10^-10 sec^-2 )
  • these peak values are the largest of all the vorticity eqn. terms
  • Subdivision of horizontal advection term:
    1. V_r * Del( Zeta): advection of relative vort. by rotational wind is largest contributor: 80-90%. Note: this term by definition must vanish at peak value of Zeta.
    2. v_r df/dy: advection of planetary vort. by rotational wind is next largest contributor: 10-15%
    3. V_d * Del( Zeta): advection of relative vort. by divergent wind has peak values similar to #2, but areal extent is much less, so the term has smaller impact
    4. v_d df/dy: advection of planetary vort. by divergent wind is negligible.
  • Since westerly winds generally increase with height, and developing cyclones tilt upstream (to the west) with height. Horizontal advection is trying to reduce the upstream tilt.

• Vertical Advection

  • Vertical motion
    • First need to know how vertical velocity is distributed in the vertical. See this link to the: "Bowstring" model:
    • max omega < 0 (upward) generally ahead of the surface low, but behind the upper ridge. (See "fig. 5" in Supplemental notes)
    • max omega > 0 (downward) generally ahead of the surface high, but behind the upper trough. (See "fig. 5")
    • horizontal distribution given by "fig. 6" (in supplemental notes) as:
      • "comma-shaped" area generally ahead of the cold front but extending over the warm front.
      • Largest upward motion north of surface warm front
      • largest downward motion behind upper trough
  • Vertical derivative of vorticity
    • IF: Zeta increases with height => d zeta/dP < 0
    • IF: Zeta has a tilt upstream with height for a developing low => d zeta/dP > 0 in the region ahead (east) of the trough but behind (west) of the ridge.
    • Both properties occur, need to know which property dominates.
    • In area ahead (and above) the surface low, d zeta/dP < 0.
  • Vertical advection of vorticity
    • pattern tends to be a dipole:
      • positive area east of trough, negative on west side
      • positive area tends to be larger => implies source of vorticity (i.e. growth)
    • larger values in middle troposphere -- omega is largest there.
    • peak values ~ 7 units (= 7 x10^-10 sec^-2 )

• Divergence term

  • can be a source or sink of vorticity, not just moving it around
  • distribution:
    • largest values near surface and near tropopause -- consistent with distribution of D in bowstring model.
    • dipole-like pattern in horizontal
    • dipole reverses sign with height -- consistent with Dines compensation
    • since convergence is a max just ahead (say, east) of the surface low so is div. term.
    • since convergence is a max just behind (say, west) of the upper level trough so is div. term.
    • dipole pattern tends to have larger positive area ahead of the surface low with larger negative area above.
  • estimating the size of the term:
    • could use AL MS border area in figs. 3.1a and 3.2b in text to estimate peak value: 16 units.
    • peak values in Grotjahn 1996 paper: 30 units for a composite of 15 developing cyclones.
  • implications:
    • term is 2^nd largest, consistent with -fD kept in the quasi-geostrophic system.
    • term opposes horizontal advection at upper levels, reinforces horizontal advection at low levels -- helps to maintain a fixed upstream tilt of developing cyclones.
    • asymmetric dipole pattern implies:
      • growth of the trough at low levels, less growth of upper trough
      • growth of the upper level ridge.
      • this is a form of "baroclinic growth"
      • many cyclones start off with weak surface low and stronger upper trough, so divergence term tries to equalize the two.
  • some diabatic processes have indirect effect on divergence term:
    • friction that leads to cross-isobaric flow which has nonzero D.
    • latent heat release may accelerate upward motion, which has corresponding change to distribution of D.

• Tilting term

  • vorticity eqn is just for the vertical component of vorticity, hence tilting term accounts for situations where a horizontal component of vorticity is "tilted" so as to create a vertical component.
  • fig. 3.3 shows case for wind increasing with height and omega increasing with latitude (say).
  • term is important near fronts and jet streaks
  • observed distribution:
    • tends to be largest in middle troposphere -- consistent with larger omega there (recall bowstring model).
    • horizontal pattern is complex,
      • might anticipate pattern from distribution of omega, since dV/dp is consistently negative
      • tendency to be large over smaller regions than advection and divergence terms.
      • some tendency for intensifying the low, but also shrinking its meridional extent.
  • estimating the size:
    • Carlson estimates peak value: 7 units.
    • peak values in Grotjahn 1996 paper: 15 units for a composite of 15 developing cyclones.

• Friction term

  • emphasis placed on friction due to surface stress
  • Carlson estimates peak values of 3-5 units
  • An aside: using the estimated friction, the e-folding time for decay of atmospheric winds is approximately 10 days.
  • can link omega and vorticity (in the boundary layer)
    • a low pressure center has positive relative vorticity
    • cross-isobar motion is from higher to lower pressure
    • that implies convergence at a low
    • low level convergence implies upward motion
    • hence, friction can cause upward motion for positive vorticity
    • vice-versa for sinking above a surface high
    • Mathematically, this can be derived from Ekman balance

• Summary table

  The information here is summarized in a brief summary table

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