Jet Streak Dynamics I: The four-quadrant model

As many frequent readers of my blog well know, I often use divergence and convergence patterns associated with jet streaks aloft to justify and explain weather patterns below.  However, recently a few people have asked me about the real reason why these divergence and convergence patterns are implied in certain parts of a jet streak.  So, I thought I'd do a quick two-part series of blogs to describe the mechanics behind these jet streak models (or, at least offer one interpretation behind them...there are many ways to look at it, actually). Right now I'll warn you that this does go into a bit of technical detail, but I hope people will be able to follow along.

First, what is a jet streak?  It's simply a localized maximum in wind speeds at a certain level.  This contrasts with a jet stream which is a large-scale, often global belt of enhanced winds that can often go all the way around the world.  Jet streaks are often found as local wind maxima within the larger jet stream.

Fig 1 -- 300mb geopotential heights and winds at 12Z, Jan. 6th, 2011.  From the HOOT website.

In figure 1 above, the jet streams are the long belts of enhanced winds--the big swaths of colors.  Jet streaks are embedded wind maxima within the jet stream, like the blobs off the southwest California coast or exiting the map to the right off the coast of New England.  We're focusing on the dynamics of jet streaks.

A lot of meteorology students (and hopefully most meteorologists) are familiar with the four-quadrant model that describes locations of convergence and divergence within an idealized straight jet streak.

Fig 2 -- Four-quadrant model of divergence and convergence within a straight jet streak.  The wind through the jet streak is blowing from left to right and is assumed to be in the mid-latitudes of the northern hemisphere.

But what causes this particular pattern to be inferred in the winds?  After all, they are just accellerating and decellerating--why don't we just see divergence as the winds accelerate into the jet and convergence as they decellerate out of the jet (here I use "jet" to mean jet streak)?

Let's look at the force balances that drive the winds in the first place.  Since this is a straight jet streak without curvature, we can neglect any centripetal accelerations.  What's left is a force balance that is a foundation of synoptic meteorology--the geostrophic balance.

Fig 3 -- Simple schematic of the geostrophic balance.

Here's how it works.  We start with a pressure gradient--in the figure above, high pressure is at the bottom and low pressure is at the top.  Naturally the air wants to flow from high pressure towards the low pressure, so it accelerates in that direction.  This force, that makes air move from high to low pressure, is called the Pressure Gradient Force, or the "PGF" as it is often abbreviated.  The red arrow above represents the direction of this force--from high to low pressure.  The PGF only depends on the density of the air and the strength of the pressure gradient--nothing else.

However, there's another "force" at work on this rotating planet of ours--the Coriolis "force".  This isn't really a true "force" but an apparent change in the direction of the wind due to the fact that the earth is rotating out from under it.  This effect depends on the Coriolis parameter "f" and the velocity of the wind (which is why I've abbreviated it as "fv" in the figures).  The faster the wind goes, the stronger this Coriolis turning effect becomes.  In the northern hemisphere, the Coriolis effect always acts to make the winds appear to turn to the right of their direction of movement--always. Simply a fact of the direction the earth rotates.

So as the air accelerates due to the pressure gradient force, the velocity increases and consequently the Coriolis effect (which depends on velocity) also increases.  This turns the winds more and more to the right the faster they go.  Finally, it gets to a point where the tendency of the Coriolis effect to turn the winds to the right exactly balances the pressure gradient force, like we see in figure 3 above where the red PGF arrow is exactly balanced by the blue Coriolis effect (fv) arrow. We see that the wind is no longer moving from high to low pressure, but perpendicular to the pressure gradient.  This balance between the Coriolis effect and the PGF is called the geostrophic balance.  Most of our upper-level winds are roughly in geostrophic balance--this is why we rarely see winds heading right into the middle of a trough or a ridge.  Instead they mostly parallel the contours of the troughs and ridges.  This is a result of the geostrphic balance.

So now we know what balance keeps the winds going in the directions that they are going.  So how does this affect jet streaks?  Jet streaks are inherently ageostrophic phenomena--they have accelerations in them which violate the geostrophic balance.  If everything were in geostrophic balance, there would be no wind maxima--we'd just have a homogenous nice jet of constant velocites and no wind maxima.  So let's look at what happens to the geostophic balance as the wind accelerates and enters a jet streak.

Fig 4 -- Disruptions in the geostrophic balance at the entrance region of a jet streak.

The colored ovals in the image above indicate different wind speeds, kind of like you would see on a weather map.  Just like above, we have high pressure below and low pressure above and as air enters the jet streak we assume that it is in geostrophic balance.  The arrows on the left edge of the jet streak show this--the two "forces" are exactly balanced.  But as the wind enters the jet streak, what does it do?  It accelerates.  But as the wind accelerates, its velocity is increasing.  Now, our pressure gradient force doesn't depend on the velocity, but the Coriolis effect does.  So as our air accelerates into the jet streak, the Coriolis effect is going to increase.  This means our forces will no longer be in balance--the same PGF will not equal the larger Coriolis effect.

This should mean that our winds would start curving to the right--if the Coriolis force is stronger than the PGF, the balance is no longer there and the winds should move in the direction that the Coriolis force is pushing them. But we've already prescribed this as a straight jet streak--we know that winds are not curving away like that.  Thus, if our winds are straight, there must be some component of them that is violating this geostrophic adjustment--there must be an accelerating component of the wind that is opposing the increase in the Coriolis effect to keep the jet moving straight.  This acceleration is directed against the increased Coriolis effect and contributes to what is called the "ageostrophic" (or NOT geostrophic) component of the wind.  Basically this implies that there has to be an acceleration upward on the drawing to keep the jet straight.

Similarly we can look at the exit region of the jet:

Fig 5 -- Disruptions in the geostrophic balance in the exit region of a jet streak.

Here we assume that in the core of the jet, the winds have once again returned to their happy geostrophic balance--the PGF equals the Coriolis effect.  But as the air leaves the jet, it decellerates.  As its velocity slows, the Coriolis effect will now weaken.  Logically, we would then assume that the winds would start curving upward on the drawing since the PGF is now stronger than the Coriolis effect.  But this is a straight jet streak and we know that's not happening.  Therefore once again there must be an acceleration keeping the wind moving straight.  This time, however, the acceleration must be in the opposite direction of the now-larger PGF.  Therefore we have an acceleration, and consequently an ageostrophic component of the wind, pointing downward, as seen above.

One more detail--these ageostrophic components of the wind will be strongest where the wind is accelerating or decelerating the most.  Therefore, the strongest ageostrophic wind will be along the axis of the jet, and the further above or below the jet axis you get (on the drawing), the weaker the ageostrophic wind will be.  We then can look at this pattern of ageostrophic wind components that we're left with:

Fig 6 -- Patterns of divergence and convergence implied by changes in the ageostrophic wind (black arrows) around a jet streak.

As I just said, the strongest ageostrophic winds will be found along the axis of the jet--where the acceleration or deceleration in the direction the air in the jet is moving is the largest. The ageostrophic wind weakens the further you move from the jet axis.  We can see this in the black arrows of the figure above. 

This is finally where the convergence and divergence is revealed!  Notice with this pattern of ageostrophic winds, there is implied convergence and divergence surrounding the jet.  In the right entrance region (on the drawing, the lower-left quadrant), the ageostrophic wind is accelerating toward the jet axis--this is an area where the wind is diverging (more air is heading toward the jet axis than is entering from outside the jet).  In the left exit region (in the drawing, the upper-left quadrant), however, the ageostrophic wind is decellerating away from the jet axis.  Therefore the wind is converging in this region (more air is leaving the jet axis than is actually leaving the jet itself on the outside).  The opposite patterns are found in the exit region because the ageostrophic wind is directed in the opposite direction.  This matches the patterns we saw in figure 2 above--and explains this four-quadrant model.

This is only one way of explaining why the four-quadrant model of a straight jet streak works.  It has to do with motions perpendicular to the jet (the ageostrophic component of the wind) that are induced by the air accelerating into and decelerating out of the jet.  Pretty amazing stuff.

For some other ways of explaining the four quadrant model (including a fun and even more complicated one using the quasi-geostrophic equations), North Carolina State University has a nice webpage here.

In my next post, I'll finish up by discussing what happens when a jet streak becomes curved.  It's actually not too bad once you have the basic geostrophic framework that we saw here today.

As always, if you have any questions, please email me or post a comment (on Facebook, here or otherwise).  Thanks for reading, and I hope this helped some of you!