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. |

*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. |

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. |

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. |

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

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!

Thank you,it was very useful

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