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.