Vorticity: Atmospheric Motion, Shortwaves, and the Vorticity Equation

I'm going to take a brief break from writing in my usual style of blog posts; i.e. forecasts and outlooks, and begin writing about certain meteorological topics; some technical, others not so much. But these topics will help those who want to understand forecasting more to understand certain atmospheric processes which are important to dynamic meteorology.

Vorticity Equation

In this series of blog posts, we will be looking at three different "versions" of the vorticity equation; examining the terms, and how they can be applied to our understanding of atmospheric motion and forecasting.

The first vorticity equation we will be looking at is the height-coordinate form of the vorticity equation. As you can see, it is really straightforward, and is in no need of converting from one coordinate system to the other. So, let's go through the terms and identify them first.

1.) First term on the left:

This term is the Lagrangian time derivative of absolute vorticity. Thus, it is time derivative equation of the change in absolute vorticity FOLLOWING the flow, instead of at a point location. In other words, it is showing the change in absolute vorticity of an parcel which is moving along with the "stream".

2.) First term on the right:

This is the convergence/divergence term. Convergence acting upon a moving column of air will cause its area to decrease, thus causing a correspondence of increasing relative vorticity (assuming that it remains at a constant latitude and therefore, the Coriolis parameter is constant). Increasing relative vorticity leads to stronger convergence, and thus, with no other forces acting upon the parcel, relative vorticity would increase ad infinitum. 

 

The term also shows that divergence causes a decrease in cyclonic relative vorticity, and an increase in anti-cyclonic relative vorticity. Divergence acting upon an air column will cause it to expand, and thus, due to the conservation of angular momentum, must have a corresponding decrease in relative vorticity.

 

3.) Second term on the right:

This portion of the vorticity equation outlines how vortex tube tilting affects vertical absolute vorticity.

Here is an example image:

From this, we can deduce that when the atmosphere has strong vertical shear, that horizontal vortex or vorticity tubes develop (theoretically). When differential vertical motion occurs in the vicinity, this will tilt the vorticity tubes into the vertical, introducing a number of effects, which cause the upper level pattern to modify (an increase in vorticity, by the geostrophic approximation, must be followed by a response to restore geostrophic balance. To do this, upper level height falls must occur, and the upper level wind patterns must acquire cyclonic curvature to compensate for the increase in vorticity).  

4.) Third term on the right:

This is the trickiest part of the vorticity equation, and perhaps the hardest one to understand. This term shows us that absolute vorticity can also change with time due to density differences in the x and y direction, and differences in the gradients. This term is positive when there is significant "density" advection, or if considering it on a constant pressure plane, thermal advection. This is the solenoidal term, and the circulation is related to the density of air. This last term is the main reason as to why the height-coordinate version of the absolute vorticity equation is not used in applied meteorology.

In coming blog posts, I will explain more of the isentropic form of the vorticity equation,  AND the pressure-coordinate form of the vorticity equation.

But what application can this have for meteorology? Well.. There are several things to note:

1.) The first term is significant, in that if we can identify regions of upward vertical motion, we can predict how the absolute vorticity tendencies will evolve with time. This is significant when forecasting cyclogenesis and the strengthening of a storm system. If upper level analysis indicates short-wave ridging ahead of a trough, its likely due to strong divergence AND warm-air advection at lower levels (because divergence causes surface convergence, which strengthens the low-level flow, and thus warm-air advection).

2.) The second term is significant, because this indicates to us how jet streaks, and changes in the thermal wind can cause shortwaves and troughs to develop out of originally straight, and "fast" upper level flow. When vorticity is tilted in the vertical by differential vertical motion, this leads to the development of shortwave troughs and "eddies" in the jet stream.

3.) The third term is not as significant for forecasting, because this equation is not widely used (and, if we apply scale analysis, we can see that this term is usually very small compared to the others), but when ever thermal advection occurs on the horizontal and even on the vertical scale, some sort of solenoidal circulation will develop; An excellent example of this is the sea-breeze circulation.

This is part 1 of the series on vorticity, shortwaves, and atmospheric motion. Tomorrow, we will be discussing the pressure-coordinate form of the equation, and how it can applied to forecasting. By the third article, we will then begin discussing the relationship between these equations, and the Quasi-geostrophic omega equation.

Heavy Rainfall Threat Around the Corner

Synopsis

 A certain situation is beginning to arise this weekend which will set up the possibility of heavy showers and thunderstorms from Thursday night through Sunday afternoon. There are several factors that will come into play:

1.) A closed low over the Central Plains and Midwest will begin to open-up in response to a digging shortwave in the Hudson Bay vortex. This will cause the ejection of a series of shortwaves around a positively-tilted trough, and allow for periods of thunderstorms.

2.)  In response to this deepening trough, the low-level "Gulf stream" or low-level jet will begin to strengthen, providing direct transport of a very moist and tropical air mass. Dew points will approach the upper 60's and lower 70's, with mid-level lapse rates approaching the moist adiabatic lapse rate (with at times a nearly saturated mid-level profile).

3.) The slow-moving nature of the trough, and frontal zone, as well as fairly weak upper level winds will allow for the possibility of slow-moving and training thunderstorm complexes. MCS development will also be likely Thursday night and Friday night as the low-level jet strengthens. 

Thus, these factors will come together to produce the possibility for a multi-day heavy rain event, culminating on Saturday night, and likely leading to the possibility of severe weather at times. 

 

 Rainfall threat/probability

As we can see from the above "factors and ingredients".. The possibility remains for flooding and heavy rainfall this weekend. Let's take a look at what the models are saying.


1.) A very rich and deep GOM tropical air mass will be entrenched across the area Friday through Saturday. PWAT (Precipitable Water) values will exceed 1.5-2 inches, indicating a rather incredibly moist atmosphere, and the potential for heavy rainfall, if deep-layer moisture convergence and efficency is also involved. 

2.) A positively-tilted, slow-moving upper level trough will be the main "controller" of this weather situation. The slow-moving trough will only allow for weaker low-level convergence along the stalling surface front, while upper level divergence from the upper level jet core could allow for multiple thunderstorm complexes to develop parallel to the front in a moderately-unstable and moist air mass. Weaker winds aloft will allow for training thunderstorms, especially in tandem with strong storm-relative inflow from the low-level jet.

3.) These periods of thunderstorms will not likely be continuous, given the lack of strong upper level forcing and only modest instability at times. They will develop at peak heating times, and organize into complexes in regions of stronger wind-shear. And with this, comes the possibility of highly localized flash-flooding.. Some areas will see lower rainfall amounts near a half inch, while other areas could see as much as four inches of rain (depending on how thunderstorms train).


Models currently vary on how much rainfall will fall across the region, and given the isolated nature of thunderstorms, its going to be hard to pinpoint a general range of rainfall amounts, but at this time, I will have to agree with the WPC (former HPC), who are forecasting a general 2-3 inches of rainfall areawide. While I do not see such rainfall amounts being "widespread", I do see a wide variety of rainfall amounts, ranging from a half an inch to over 4 inches possible. Heavy rainfall on this order could cause regions of minor to moderate flooding.

 

Severe Weather

Although I do not expect the severe weather threat to be nearly as substantial as the threat over the Central Plains today or tomorrow, there is still the possibility for hail-wielding or thunderstorms with damaging winds from Thursday night through Saturday. This will all hinge upon the moisture-content of the lower atmosphere, solar insolation, and the depth of the PBL on both Thursday, Friday and Saturday.

1.) The #1 factor in this event will be instability. Dew points are indicated to approach the upper 60's and lower 70's, with a very moist profile throughout the entire atmosphere by Friday. Although mid-level lapse rates will be lower due to the tropical influence, surface heating and mixing will allow for MLCAPE to approach 2000 J/kg both tomorrow and possibly Friday. This will be heavily dependent on surface heating, as mentioned before. Models continue to develop scattered numerous showers and thunderstorms Thursday afternoon through Saturday, but I disagree with their assessment. The models, the NAM especially, seem to be overdeveloping convection, and developing convection too far away from the front itself. Thus, it seems to me that models are too cool with surface temperatures. I believe that if cloud cover is not widespread tomorrow or Friday, temperatures could reach the upper 80's on both days. If this is the case, expect substantial instability.


2.) 0-6 km bulk shear will be lacking through most of the event (unless, like I described before, a jet streak develops along the eastern flank of the trough), and thus storm organization will be limited to localized shear along boundaries, and the development of shear due to storm-scale processes. Thus, any kind of organized storm development will also be limited to multi-cellular storms and bow-echoes.


3.) With the above factors, I think the severe weather threat in itself will be limited, especially if we see a longer duration of cloud cover tomorrow, Friday, and Saturday. But even so, we need to be on the look out for strong to severe thunderstorms.

Warm Week Ahead After Cooler Weekend...

After a very cool, and somewhat rainy, Memorial Day weekend. expect a nice reprieve from this dreary weather into a very warm, sunny, and possibly humid weather pattern. Temperatures will increase into the mid and upper 80's under the influence of a strengthening upper level ridge, and with increasing flow from the Gulf of Mexico, the warmer temperatures combined with higher moisture could lead to widespread thunderstorm development by the weekend.

Since we all know that good things can't last forever, I'm here to also inform you that this period of "hot" weather will not likely last all that long. A weak trough looks to become "stuck" in an increasingly "split-flow" pattern, which will allow for a cooler "pocket" of air to drift around the Great Lakes region by Sunday and Monday. This will cause temperatures to drop back into the mid-70's for the end of the weekend.

BUT, do not despair. Long-range models have consistently indicated that an even warmer blast of air is on the way, starting next Tuesday. As the weak trough exits the region, ridging will build in from the West, sending the hot EML eastward. This will override a cooler, but moist surface air mass, but as soon as full insolation commences, mixing in the PBL will draw some of this warmer air downward, and allowing hot temperatures to prevail once again. It appears very likely that 850 millibar temperatures may approach or exceed 20 degrees Celsius during this next surge of heat; making it all the more likely that we exceed 90 degrees.

But what does this mean for the rest of the summer? Based on the current trends, its becoming apparently obvious that the rest of the summer may not be as "incredibly hot" as it has been in the previous three years. Although this is subject to change, its likely that instead of seeing average temperatures above normal by 5-7 degrees, its likely that our summertime average will only be 1-3 degrees above average. Now, it is too early to judge how the rest of the summer will turn out, especially in light of the fact of what happened last year (very, very cold beginning to June; but the rest of the summer turned out to be outrageously hot), but I think its safe to assume that this type of pattern will continue for the foreseeable future. This means that June will likely see a fair number of 90 degree days (3-5 days?), while July will probably see slightly more than normal 90 degree days (12-15 days?). Even so... I expect that this summer will feature a "backloaded" summer; i.e. it will remain quite hot even into September, unlike the previous three years.