• Some cautions:
  • This section is best viewed as collecting together the various processes that occur, essentially simultaneously, as a front intensifies.
  • There are multiple comments needed to understand the principal figures used in this section: figs. 14.5 and 14.6
    • in 14.5: think of this as the flow at 700 mb (to match caption in fig 14.6)
    • in 14.6: need to add a few more dashed contours (isentropes) at upper levels of the "warm" (upper left) side of the diagram.
    • in 14.6: the vertical axis is elevation (z; increasing upwards on the page), not P! Hence the solid and dot-dashed lines can be viewed as constant P surfaces.

• Carlson identifies 8 "Steps".

    Except for "Step 0", view these as "parts" of the
              frontogenesis; the "Steps" organize the presentation, and
              step "n" does not need to finish before step "n+1" can commence.

      0.   Initial state isentropes might be oriented with
              a steeper slope (i.e. closer to vertical) than in later
              diagrams.

  1. Confluent flow frontogenesis. Confluence causes the isentropes in the warm and cold sides of the frontal zone to approach each other. The flow can be geostrophic. Similar to confluence term in eqn (13.2)

  2. Cross-isobar view of ageostrophic flow.
     • Having a horizontal gradient: / y' <0 implies that there must be a thickness gradient h/ y' <0 in the y' direction as well.
     • frontogenesis means both gradients are increasing
     • Consider first the upper level.
       • Falling h in the cold air and rising h in the warm air
       • the height gradient between the warm and cold sides shall be increased (the change is in the same fashion as that between the upper solid line becoming the upper dot-dashed line in fig. 14.6)
       • Where there is a horizontal pressure gradient, we might expect ageostrophic wind:
       • ageostrophic wind normally blows from higher to lower pressure.
       • Hence an ageostrophic wind is formed at the top which drives some mass from the warm side into the cold side.
     • Now consider the lower level.
       • The addition of mass at upper levels raises the surface pressure on the right side. (note change of lower solid line becoming lower dot-dashed line: the curve rises).
       • The loss of mass at upper levels lowers the surface pressure on the left side.
       • Note the h changes are still as we expect: smaller h on the cold (right) side, larger h on the warm (left) side.
       • A pressure gradient at low levels implies an ageostrophic wind at low levels as well
       • V_ag directed to warm air side at low levels; opposite to upper levels
     • Understanding where these V_ag winds comes from is key to the rest of what follows.

  3. Unbalanced change creates V_ag
     • This step and the next are a bit weird because Carlson is trying to split apart his discussion of a simultaneous process. So, in these two steps a different thing is held constant.
     • hold u constant
     • y' momentum equation states that: dv/dt = -_y - fu.
       Since the gradient in y direction must be increasing (step 2) then the v motion must be accelerating, since we are holding u constant.
     • if V_ag is initially 0, then the result that dv/dt >0 implies V_ag >0 must develop.
     • this is Carlson's way to get the upper V_ag. It is just another way to reach the same conclusion we stated in step 2.

  4. Change in u creates V_ag.
     • now hold constant.
     • The flow is confluent, and is crossing isotachs (as depicted in fig. 14.5). Therefore, from the standpoint of an air parcel, the air parcels must be accelerating. i.e. du/dt >0
     • x' momentum equation states that: du/dt = -_x + fv.
       Since du/dt >0, using the x' momentum eqn Carlson then concludes that v must be positive since gradient has been held constant. i.e. V_ag >0 as before.
     • My suggestion: don't place too much emphasis on this chain of events to bring formulas back into balance.
     • Instead: View the discussion here as showing how the ageostrophic circulation set up maintains the momentum balances.

  5. Frontogenesis from convergence due to V_ag
     • Carlson finally looks at those ageostrophic motions: V_ag
     • towards larger y' at upper levels; opposite at lower levels. That means:
       towards cold air at upper levels; towards warmer air at lower levels.
     • Since height gradient largest in middle, expect largest V_ag there.
     • V_ag/ y' implies areas of divergence and convergence
     • V_ag decellerates in upper right part of diagram, so V_ag/ y' <0 there which means convergence there. Convergence also takes place in the lower left part of the chart.
     • Divergence in the upper left and lower right.
     • Intuitively, where there is convergence you might expect the isentropes to be moving closer together. The isentropes would spread apart where there is divergence.
     • Hence, convergence fosters frontogenesis, divergence inhibits it.
     • The V_ag field creates an asymmetric, tilted frontal zone (see fig. 14.6) with isentropes concentrated in a diagonal band from lower left to upper right on the figure.
     • Note 1: divergence & convergence imply vertical motions.
       • Here that means sinking on the cold side, rising on the warm side.
       • Carlson does not comment on this, but the tilting term indicates that this works against frontogenesis
       • vertical motions are trying to tilt the isentropes in a more horizontal orientation, thus weakening the horizontal gradient of
       • to put it another way: sinking means adiabatic warming on the cold side; rising means adiabatic cooling on the warm side. This reduces the horizontal gradient of
     • Note 2: the pattern of V_ag, divergence, and U here are all consistent with what we saw in the jet streak (entrance region) discussion. Again, that is consistent with fig. 14.5 as well.

  6. Changes in u maintain thermal wind balance
     • here we are working the analysis in step 4 in the opposite order. Changes in V_ag cause accelerations in u.
     • Note how V_ag reverses sign between upper and lower elevations.
     • From the x' momentum equation, this means that du/dt >0 in upper levels and du/dt <0 in lower levels. Hence the vertical shear must be increasing.
     • Increasing vertical shear is consistent with thermal wind balance, since the horizontal gradient is increasing.

  7. Three important conclusions
     1. Convergence patterns augment the confluence and are asymmetric. Hence, the axis of dilatation rotates: toward warmer air at low levels; toward colder air at upper levels. The frontal zone becomes more tilted.
     2. The jet tends to migrate towards the warm air side of the frontal zone. Consider what happens at the upper levels:
        • Assme u is initially constant in y' direction (no jet)
        • Consult vorticity eqn. d/dt = - fD.
        • Assume vorticity changes are from shear vorticity, not curvature vorticity:
        • Thus: convergence increases vorticity which increases -u/ y'.
        • Opposite for areas of divergence.
        • Next remember that a straight jet (having no curvature vorticity) would have its maximum speed where the vorticity is zero. There would be positive vorticity on the left side (looking downwind) and negative vorticity on the right side.
        • Thus, the convergence/divergence fields cause the jet to migrate to a location in between the upper level divergence and convergence. Since the front develops a tilt back over the cold air, then the jet has migrated to the warm air side of the frontal zone. (see for example, fig. 14.6; look for the circled J)
     3. Since the confluence term depends upon the y' gradient of , then the process accelerates. The rate of intensification is proportional to the intensity of the front. As the front gets stronger its rate of intensification keeps getting faster.

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