Chapter 11

Imbalance and Vertical Motion

(1) Wind-Parallel AccelerationsIn 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).

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 PressureGradient Force and the Coriolis Force are acting at right angles to the wind direction (they can only cause a change in direction)?

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.

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.

v a

v

v g

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.

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.

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.

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.

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.

Consider the vector wind equation from chapter 10 and the geostrophic equation.

Subtracting the equations gives:

Rearranging at substituting gives:

Dv h

Dt f k

v h goh Z

F h

f k v g goh Z

Dv h

Dt f k

v h

v g

F h

Dv h

Dt f k

v a

F h

v a

v h

v g

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.

Dv h

Dt f k

v a

F h

F h

Correction

(2) Wind Motion and System MotionConsider 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).

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).

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.

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.

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.

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).

(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:

v a

v h

v g

v a

v h

v g

v a

v h

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.

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).

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

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.

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).

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.

(4) Isallobaric WindAn 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.

(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 δz>0, δp<0.

Du

Dt fv go

Z

x Fx

Dv

Dt fu go

Z

y Fy

Du

Dt fv go

Z

x

Dv

Dt fu go

Z

y

p

x

z

p

z

x

z

x

p

Then since from the hydrostatic equation,we have: and:Then we have:

Which becomes:

Du

Dt fv go

Z

x

Dv

Dt fu go

Z

y

p

x

z

p

z

x

z

x

p

p

z go

p

xgo

z

x

1

go

p

x

z

x

Du

Dt fv go

1

go

p

x

Dv

Dt fu go

1

go

p

y

Du

Dt fv

1

p

x

Dv

Dt fu

1

p

y

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.

Du

Dt fv

1

p

x

Dv

Dt fu

1

p

y

1

p

x

g

z

x

The geostrophic wind components are given by:

Du

Dt fv fvg

Dv

Dt fu fug

Du

Dt f v - vg Dv

Dt f u ug

fvg 1

p

x fug

1

p

y

Du

Dt fv

1

p

x

Dv

Dt fu

1

p

y

This can be written using the ageostrophic wind definition as:

Du

Dt f v - vg Dv

Dt f u ug

Du

Dt f va Dv

Dt f ua