Imbalance and Vertical Motion

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Chapter 11

• Imbalance and Vertical Motion

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(1) Wind-Parallel Accelerations

• In order for wind to flow in a cyclonically
  curved manner, (an acceleration) it has to be
  sub-geostrophic.
• In order for wind to flow in an anticyclonically
  curved manner (an acceleration) it must be
  super-geostrophic.
• We see in the real world that the wind also
  speeds up and slows down (also accelerations).

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• If the wind is blowing parallel to height
  contours or isobars in the absence of friction,
  how can the wind speed up or slow down since the
  Pressure

Gradient Force and the Coriolis Force are acting
at right angles to the wind direction (they can
only cause a change in direction)?
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• If the wind is accelerating (changing direction -
  even if parallel to contours or isobars, speeding
  up or slowing down), it cannot be in exact
  Geostrophic Balance.

Above the friction layer, air will speed up if it
has a component towards lower heights or lower
pressures. Air will slow down if it is drifting
toward higher heights or higher pressures.
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• Ageostrophic Wind Vector - a vector that
  represents the difference between what the wind
  is actually doing and what it would be doing if
  it were in perfect geostrophic balance.

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• Consider air flow about an anticyclone.
• The actual winds are stronger (super-geostrophic)
  than the geostrophic winds.

• The Coriolis force is stronger than the PGF.
• Thus, the Ageostrophic wind points the same way
  the wind is blowing.

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• Consider air flow about a cyclone.
• The actual winds are weaker (sub-geostrophic)
  than the geostrophic winds.

• The Coriolis force is weaker than the PGF.
• Thus, the Ageostrophic wind points the opposite
  direction from which the wind is blowing.

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• Consider air that is speeding up.
• The actual wind must be oriented toward lower
  pressures or lower contours, thus, the
  ageostrophic wind is pointed toward lower
  pressures at right angles to the geostrophic
  wind.

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• Consider air that is slowing down.
• The wind must be partly blowing toward higher
  pressures and would be oriented to the right of
  the geostrophic wind.
• The ageostrophic wind would be oriented at right
  angles, to the right of the geostrophic wind.

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• Putting it all together (for northern
  hemisphere). (opposite for S. H.)
• Anticyclonic curvature - air accelerating
  (turning) to the right. Wind faster than
  geostrophic and ageostrophic wind pointing in
  forward direction.
• Cyclonic curvature - air accelerating (turning)
  to the left. Wind slower than geostrophic and
  ageostrophic wind pointing in the backward
  direction.
• Air speeding up - air accelerating forward and
  ageostrophic wind pointed to the left.
• Air slowing down - air accelerating backward and
  the ageostrophic wind pointed to the right.
• In all four cases, the acceleration is 90o to the
  right of the ageostrophic wind.

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• Consider the vector wind equation from chapter 10
  and the geostrophic equation.
• Subtracting the equations gives
• Rearranging at substituting gives

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• If friction ( ) is negligible, the negative
  cross product of the unit vector k and the
  ageostrophic wind, shows that the acceleration
  will be 90o to the right of the ageostrophic wind
  in the northern hemisphere.

Correction
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(2) Wind Motion and System Motion

• Consider a jet streak - a localized region of
  very fast winds within a jet stream.
• Air entering the jet streak is speeding up
  (accelerating).
• Air leaving the jet streak is slowing down
  (accelerating).
• Where it is speeding up, there must be
  cross-contour flow toward lower heights (to the
  left of wind flow).
• Where it is slowing down, there must be
  cross-contour flow toward higher heights (to the
  right of the wind flow).

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• At entrance
• Ageostrophic wind toward left, acceleration to
  right of it (speeding up).
• At exit
• Ageostrophic wind toward right, acceleration to
  right of it (slowing down).

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What happens if the jet streak itself is moving
faster than the wind within it?

• At exit
• The streak is catching up to the air parcels and
  they would thus move faster. Should be
  cross-contour flow toward lower heights.
• At entrance
• Jet streak is moving away faster than air parcels
  are moving so parcels are essentially exiting the
  streak and slowing down.
• They should move toward higher contour heights.

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Which is happening?

• Compare successive maps to see if jet streak is
  moving faster than the geostrophic wind speed.
• Check wind direction with contour analysis to see
  how winds are crossing contours. (Need accurate
  analysis.)
• General rule Air moves faster than weather
  systems (e.g., jet streak) above 600 mb and
  slower than weather systems below 700 mb.

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• Consider an upper-level trough.
• The trough axis typically moves with a component
  toward the east. (E, NE, SE)
• Behind the trough air moves from the northwest
  toward the southeast.
• Thus, by the time the northwest air reaches the
  point it should recurve toward the northeast, the
  trough axis has moved so it continues moving from
  the northwest.
• Even more so if the trough is deepening.

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• Therefore, in this situation for the air parcels
  moving from the northwest
• Their acceleration is weaker (not changing
  direction toward the northeast).
• The ageostrophic wind is less (not pointed as
  strongly into the geostrophic wind - which makes
  the acceleration which is always to the right of
  the ageostrophic wind not as strong toward lower
  height contours - thus, it is not diverging from
  its path).

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(3) Convergence and Ageostrophic Wind

• Consider the definition of the ageostrophic wind.
• Taking the divergence of both sides gives
• However, the divergence of the geostrophic wind
  is almost exactly nondivergent, so this becomes

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• Therefore, any divergence or convergence of the
  horizontal wind is almost entirely accounted for
  by the divergence or convergence of the
  ageostrophic wind.
• If you can infer the ageostrophic wind and, using
  your understanding of the accelerations that are
  occurring, you can make a good guess about the
  patterns of divergence and convergence in the
  upper air and from this, the patterns of vertical
  motion that are occurring.

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• Consider the trough again.
• Air moves from northwest, through the trough
  axis, and then toward the northeast.
• The greatest curvature is in the trough axis.
• Winds should depart most from geostrophic balance
  because that is where the greatest acceleration
  (changing direction) is occurring.
• Thus, the ageostrophic wind should be greatest in
  the trough axis and weaker on either side.
• The wind in the trough axis is subgeostrophic (as
  about a low).
• The ageostrophic wind is pointed into the
  horizontal wind (downstream toward upstream).
• Acceleration is to the right (toward lower
  contour heights).

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• Therefore
• Upstream of the trough axis (where ageostrophic
  wind pointed into the horizontal wind is
  greatest) there should be convergence.
• Downstream of the trough axis (where ageostrophic
  wind is weaker) there should be divergence.
• Also, upstream of a ridge axis there should be
  divergence.
• Downstream of a ridge axis there should be
  convergence

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• Since vertical motion is restricted by the ground
  and stratosphere,
• Regions of convergence are associated with
  downward motion in the interior of the
  troposphere.
• Regions of divergence are associated with upward
  motion in the interior of the troposphere.

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• Consider again the jet streak.
• The entrance region would have air flowing across
  contours toward lower heights (ageostrophic wind
  directed from high heights toward low heights) -
  (and air is speeding up).
• The exit region would have air flowing across
  contours higher heights (ageostrophic wind
  directed toward higher heights - (acceleration
  opposite to wind flow - slowing down).

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• Entrance Side
• Thus, there should be convergence on the low
  height side of the entrance to the jet streak.
• There should be divergence on the high height
  side of the entrance to the jet streak.
• Exit Side
• There should be convergence on the high height
  side of the exit to the jet streak.
• There should be divergence on the low height side
  of the exit to the set streak.

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(4) Isallobaric Wind

• An isallobar is a line of equal change in
  atmospheric pressure.
• When pressure changes, the initial response of
  air molecules is to move toward the region of
  lower pressure.
• Then, the Coriolis Force for this movement begins
  to act to bring the forces into balance,
• But, there is continual changes in pressure, so
  there is a continual attempt to arrive at
  balance.
• The motion of the air at balance is considered
  the steady-state response to a pressure change.
• The instantaneous response is the to the pressure
  change.
• The Isallobaric wind is the instantaneous
  response.
• Mathematically, it is a component of the
  ageostrophic wind.

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(4) Isallobaric Wind

• Consider the horizontal equations of motion given
  in chapter 10, where v is the actual horizontal
  wind.
• If we ignore friction, we have
• We can write
• The minus sign is used since when dzgt0, dplt0.

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• Then since from the hydrostatic equation,
• we have and
• Then we have
• Which becomes

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• All we did was change the equation from one
  typically expressed for upper-air maps (change in
  height) to one typically for surface maps (change
  in pressure).
• We can see that
• The term on the right (using any of these
  horizontal wind equations) can be expressed using
  the geostrophic wind.

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• The geostrophic wind components are given by

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• This can be written using the ageostrophic wind
  definition as