Jet Streak Dynamics II: Adding Curvature

I know...I know.  There are a lot of fascinating things going on in the weather right now and in the next few days.  Huge cold air outbreaks.  The possibility for a record-breaking snowfall in Seattle this week.  We've got time to discuss these...I promise.  But for now, I will finish up the topic I started discussing last time.

In my last post I talked about the four-quadrant model for straight jet streaks and how certain patterns of convergence and divergence could be determined from that model.  But what happens when flow becomes curved?

Any time there is movement in a circular or curved path, there is an ongoing acceleration oriented inward that's keeping the air (or whatever it is that's moving) turning around the curve.  If there weren't that acceleration, the air would just keep moving in a straight line.  This added acceleration is called the centripetal acceleration (or centripetal force if you include the mass).  Since our original four-quadrant model dealt with straight flow, we didn't have to worry about the centripetal acceleration.  But if the flow is curved (and it often is--just look at all the troughs and ridges on a weather map...), we do have to consider this acceleration and add it to our force balance.

Remember from last time that the direction and speed of our flow was governed by a balance two things:  the pressure gradient force pushing air from high to low pressure and the Coriolis effect which was always turning the wind to the right (and turned the wind more to the right the faster the wind was going.)  Now we have to add a third "force" to the balance--the centripetal acceleration, which also depends on the velocity and is always oriented in the inward direction as the air moves around a curve.  Its magnitude is found by taking the air velocity squared  and dividing it by the radius of curvature (V^2/R).  Let's look at how all these forces line up in troughs and ridges.

Fig 1 -- Force balances with curvature included.

Let's start with the trough on the left.  We see the beige PGF arrow pointing toward the low pressure center--this is oriented inward as the air goes around a trough.  We also see our centripetal acceleration, also oriented inward, as the red arrow.  However, the Coriolis effect always points to the right of the wind, so it points outward as the air goes around the trough.  This means that the centripetal acceleration and the Coriolis effect (both of which depend on the wind speed v) are pointed in opposite directions.  If we were to actually compute the force balance and put actual numbers in here, we would see that because those two forces are pointed in the opposite directions, the net effect is to slow the velocity down below the usual geostrophic speed as it rounds the base of the trough.  It involves some fancy calculations that typically involve the quadratic formula so I'm not going to show it here.  However, the key result you would find is that the wind speeds are slower than geostrophic around the base of the trough because the centripetal and Coriolis accelerations are in the opposite direction.  Since the winds would be slower than geostrophic, we say they are subgeostrophic.

What about around the ridge?  We see that here the PGF arrow is pointed once again from higher to lower pressure, or outward from the center of the ridge this time (away from the high pressure center toward the lower pressure outside the ridge).  The Coriolis effect, seen in the blue arrow, is still pointed to the right of the flow, but this time to the right of the flow means that it is pointed inward around the curve. The centripetal acceleration (seen in the red arrow) also always points inward.  So in the case of the ridge, the centripetal and Coriolis accelerations are both pointed in the same direction--both are pointed inward.  Since both of these depend on the wind speed and both point in the same direction, if we were to do those same force balance calculations, we would see that the winds would tend to be faster that geostrophic in this case.  So around a ridge, we say the winds should tend to be supergeostrophic--faster than geostrophic.

So we now know that we would expect to see slower, subgeostrophic winds in the base of a trough and faster, supergeostrophic winds around a ridge.  This implies that the winds are changing speed between troughs and ridges.  What does this mean for divergence and convergence?

Fig 3 -- Divergence and convergence pattern due to sub- and supergeostrophic winds around troughs and ridges.

Air must be accelerating to go from subgeostrophic in a trough to supergeostrophic in a ridge.  Therefore we would expect to find divergence in the region between a trough and a ridge (more air is going out at the top of the ridge than is entering at the base of the trough).  However between a ridge and a trough, the air must be decelerating to go from supergeostrophic in a ridge to subgeostrophic in a trough.  Therefore we would expect to find convergence in the region between a ridge and a trough (more air is entering from the top of the ridge than is leaving through the base of a trough).

So this does a lot to explain where our divergence and convergence come from in a curved air pattern.  But I've said nothing about jet streaks here--just curved air flow.  What if we add jet streaks in the troughs or ridges?

The common way to approach this problem is to just take the four-quadrant model and overlay it onto the pattern I've shown above.  For a refresher, here is the four-quadrant model for convergence and divergence around a jet streak.

Fig 4 -- Divergence and convergence patterns surrounding a straight jet streak according to the four-quadrant model.

Now lets overlay that onto our diagram from before.  To keep it clearer, I've just added the dividing lines that would separate the four-quadrants.  I'll overlay two jet streaks--one cyclonically curved around the base of the trough and one anticyclonically curved around the top of the ridge.

Fig 5 -- Curved air flow with jet streaks over the trough and ridge.  Areas of convergence and divergence due to both the curved flow and due to the four-quadrant model are indicated.

We see that the areas of convergence and divergence associated with the entrance and exit regions in the four-quadrant model occur in the same areas where there is large-scale convergence and divergence due to the curved flow pattern.  What does this all mean?  First, let's focus on the exit region of the cyclonically curved jet streak in the base of the trough.  We see there is divergence in the left exit region and convergence in the right exit region, all superimposed on large-scale divergence due to the location between a trough and a ridge. The net effects are additive--in this case, the divergence due to the curvature will tend to cancel out the convergence in the four-quadrant model and that same divergence will tend to reinforce and enhance the divergence in the four-quadrant model.  I know that's a lot of words trying to describe it--the next image will hopefully make it clearer.  We can apply this to all of these regions--the convergence and divergence due to the curved flow will either reinforce or cancel out the convergence and divergence due to the four-quadrant jet model.

Fig 6 -- Divergence due to curvature will tend to cancel out any jet streak convergence in the same areas.  Likewise, convergence due to curvature will tend to cancel out the jet streak divergence in the same areas. 

What does this mean for the final pattern?  Since anything that was not cancelled out is simply enhanced, we're left with this final pattern:

Fig 7 -- Divergence and convergence patterns in curved jet streaks expected due to a combination of curvature and the four-quadrant model.

And here finally we see the patterns that I've been using in all my blog posts.  In cyclonically curved jet streaks (like around the base of the trough) we tend to see convergence in the entrance region and divergence in the exit region.  For anticyclonically curved jet streaks (like around the top of the ridge) we expect to see divergence in the entrance region and convergence in the exit region.  There it is--the curved jet streak pattern.  Of course, these patterns can change even further if temperature advection is going on.  But I usually don't look that far into things...

I hope this has been helpful and finally explains these jet streak patterns I've been citing in all my blog posts.  I apologize if this seemed somewhat complex--the words "convergence" and "divergence" showed up a lot.  But I hope this was helpful.  As always, email me if you have any questions...

On to more real weather later this week!