Vorticity Advection and Vertical Motion

by

Chuck Doswell


Most recent update: 10 November 1999

Disclaimer: This document is provided without any endorsements by anyone other than me. It is for information purposes only and no one in my chain of command is responsible for its contents.


Introduction

A persistent myth that I see repeatedly in forecast discussions is that PVA (Positive Vorticity Advection) equates to upward vertical motion. The frequency at which this appears does not reflect a deep understanding of the process connecting vorticity and vertical motion, however! When I press forecasters to give me a physical explanation (no mathematics!) for how vorticity advection and vertical motion are connected, with only rare exceptions, they are unable to do so.

What seems to be happening is that forecasters have some vague memory of hearing some connection between vorticity advection and vertical motion, perhaps back in their Dynamics classes. Whatever physical process was at work was not understood clearly and the details were simply memorized for the exam and then quickly flushed from short-term memory. With their arrival at their new duty station, our novice forecasters heard their more established colleagues using "PVA=upward motion" [which I call the "Ignorant Person's Omega Equation"] in forecast discussions. Although not understanding how this concept might work, the novices adopted this without ever asking: "By the way, Mr./Ms. Long-Time Forecaster, how does this work?"

From my perspective, there is only one reason for using forecast "rules of thumb" without understanding them: no scientific explanation for the success of a forecasting rule exists. Otherwise, we are in the "Weather Lore School of Forecasting" described elsewhere. If you encounter a forecasting rule that seems to work, at least some of the time, it should be important to you to understand how it works! If you don't understand something, alarm bells should be going off in your head. How can you use a method you don't understand? Only on faith!! While faith might be useful in your spiritual life, it is anathema to your scientific life.

So how does this concept work? There are at least three different explanations, each from a different perspective, but I believe both offer insight into the process. First of all, let me offer a fairly technical discussion of the physical basis for understanding the connection between vorticity and vertical motion. Non-technical types can skip over this, but if you do, I believe that your ultimate understanding of the topic is correspondingly diminished. If you can't follow this technical talk, you should consider learning how to follow it.

 

Technical discussion of the Omega Equation.

As discussed elsewhere, the primary argument connecting vorticity and vertical motion is based on quasigeostrophic (QG) theory, in the form of the QG Omega Equation:

I am not going to derive this for you ... go back to your Dynamics course notes, or consult Holton's (1992, p 149 ff.) textbook presentation.

If this equation is completely unfamiliar to you, your meteorological education was a waste of time. The Omega Equation is an elliptic partial differential equation that must be solved for "omega" (vertical motion in p-coordinates). The first term on the right involves the vertical derivative of the geostrophic advection of geostrophic absolute vorticity. The second term is the Laplacian of the thermal advection. A quantitative evaluation of QG vertical motion requires the calculation of the right-hand side of this equation over some grid and then solving the differential equation for "omega" with appropriate boundary conditions.

If a quantitative solution is desired, current thinking suggests that the right-hand side (rhs) of this equation should be evaluated using the so-called "Q-vector" approach, in which the rhs becomes

where the Q-vector is defined as:

An excellent discussion of the Q-vector formulation, as well as an additional, intermediate formulation due to Trenberth, can be found in Durran and Snellman (1987). If the goal is to solve the Omega Equation quantitatively, the Q-vector formulation is preferred. Barnes (1985) has developed the Q-vector formulation for operational diagnostics.

I am not going to go into a full-blown discussion of Q-vectors! For physical insight, I prefer the "traditional" formulation. However, I agree with Durran and Snellman in saying that a strictly proper interpretation of vertical motion through QG processes should not describe vertical motion as being "caused" by terms involving vorticity and thermal advection. A strictly proper interpretation of QG-forced vertical motion is as follows.

The QG approximation assumes, among other things, geostrophic and hydrostatic balance ... with regard to geostrophic balance, the concept is that advection is dominated by the geostrophic contribution and only limited departures from geostrophic balance are permitted [hence, it is quasigeostrophic, and not simply geostrophic]. Advection of vorticity and the thermal field tend to disrupt geostrophic and hydrostatic balance, and the solution of the QG omega equation describes a hypothetical vertical motion field that exactly and instantaneously restores geostrophic and hydrostatic balance. Therefore, QG vertical motion is most properly thought of as a response to the disrupting influence of geostrophic advection on the system. QG omega has precisely the same character that the geostrophic wind has: it is a theoretical, imaginary flow. It has no physical reality and it cannot be measured as a real flow is measured!

Having said this, I still believe that we can consider the terms on the rhs of the QG Omega Equation as "forcing" terms ... subject to the foregoing caveats. Physical insight can be obtained by considering the vorticity and thermal advection terms separately, as I will do in the next section. But before I do that, I want to address the issue of a qualitative interpretation of the Omega Equation. Holton has gone to some pains to make such a qualitative interpretation, and I do not want to repeat it here, but an issue that often comes up is the use of the Laplacian of the thermal advection. In Holton's qualitative interpretation of the Omega Equation, he does not use the Laplacian of the thermal advection. His qualitative interpretation requires a number of assumptions, including the representation of variables in a Fourier series expansion involving sinusoidal basis functions. In such an expansion, the Laplacian of a function can be approximated simply by putting a minus sign in front of the function itself. This is most decidedly not correct in a quantitative sense, but in a qualitative sense, it is not too bad. Since both the lhs and thermal advection term on the rhs of the Omega Equation involve Laplacians, in effect (for qualitative purposes only!!), the Laplacians "cancel." For a qualitative assessment of the influence of the thermal advection, you need not try to imagine what the Laplacian of the thermal advection looks like.

Another frequent complaint about not solving the Omega equation is that the boundary conditions used to integrate the Omega Equation can have a large impact on the solution, at least near the boundaries. What makes the boundary conditions problematic is that it is not always obvious what boundary conditions to use, especially (a) on the lateral boundaries of a bounded domain, (b) on the upper boundary (typically a pressure surface) and (c) on the lower boundary (where various boundary layer processes may need to be incorporated). This technical aspect of integrating the Omega Equation is yet another reason why I often prefer not to do the formal integration.

Finally, one commonly-voiced complaint about using the "forcing terms" to estimate the vertical motion is that the integral of the Omega Equation may be quantitatively different from the view of the vertical motion implied by the forcing terms ... that is, the integral of the function is not equivalent to the function! This truism is certainly a valid complaint, if the goal is to be quantitative. When the "forcing terms" on the rhs of the Omega Equation are spatially and temporally coherent, and the terms at any one level look more or less like the terms at other levels above and below, then the use of the "forcing terms" alone to obtain a qualitative sense of the QG vertical motion is not all that bad. If the field of forcing is so noisy that the integral is needed to ascertain the basic structure of the field of vertical motion, then by all means, do the integration! Of course, the resulting vertical motion may not be very strong (the integral of a noisy "forcing" field isn't going to be very large), so the QG signals are perhaps too weak to be of significance in such cases.

Frankly, I'm more than a bit tired of hearing these sorts of gripes about using the "forcing" terms in this qualitative sense. There's nothing wrong with nit-picking (It's obvious that I have to say this!), but at least be aware that from some perspectives, the nits just aren't all that important.

 

Non-technical discussion of the Omega Equation

This discussion is aimed at trying to get a physical picture of how vertical motions are implied by vorticity and thermal advection. All such interpretations are, to some extent, oversimplifications. The idea is to have a qualitative feeling for the physical connections between the mathematical terms and the physical processes.

The Vorticity Advection term:

Interpretation: Version 1

A way of looking at this is to consider a trough embedded in the westerlies (Fig. 1).

The trough, in a very real sense, is defined as the blob of enhanced vorticity. Now, consider a parcel entering the trough from behind (Fig. 2):

This parcel (the small circle) has some vorticity value as it enters the trough. Along its trajectory, it is encountering more and more cyclonic vorticity values. If it is to stay in equilibrium with its environment, it must increase its vorticity. How can it do so? For purposes of this non-technical discussion, I will use a "sort-of" equation:

where dz/dt denotes the change of vorticity following a parcel. I am going to assume that "other stuff" is negligible, so the only way an air parcel can increase its vorticity is to do what ice skaters do to increase theirs: the air must converge, so that conservation of angular momentum requires the spin to increase. By this reasoning, parcels entering the backside of the trough, in a region of anticyclonic vorticity advection (AVA) (that is, vorticity values are becoming more cyclonic along the flow) are having to converge. By the same argument, in the region of cyclonic vorticity advection (CVA) on the other side of the trough axis, parcels are diverging.

To this point, nothing has been said about the level where all of this is happening. Thus, the statement

AVA implies convergence and CVA implies divergence

must be true at any level, under the assumptions made so far.


Note (added 10 November 1999):

From the QG Height Tendency Equation, it can be shown that AVA implies height rises, whereas CVA implies height falls. When systems are not deepening or filling rapidly, their movement is associated with vorticity advection ... it is by vorticity advection that a steady-state QG system moves. See interpretation #2 for the relationship between the two primary QG equations. For extratropical cyclones that are not changing their intensity very rapidly, the Height Tendency Equation allows one to infer qualitatively the magnitude of vorticity advection at any level (not just 500 mb) by evaluating the height changes at that level.


What must be the case if CVA [I prefer this to PVA because it is not biased to the Northern Hemisphere] is to imply upward motion? The Law of Mass Continuity requires a particular physical picture (Fig. 3), where convergence at low levels changes to divergence at upper levels, connected by upward vertical motion. Between is the so-called Level of Non-Divergence (LND). This may not be a flat surface, of course! If CVA is to imply upward motion, then, the level where we measure vorticity advection (in their wisdom, the Great White Fathers inside the Washington Beltway have provided vorticity routinely only at 500 mb) must be above the LND.

Hence, "PVA equals upward motion" depends on a series of assumptions (not necessarily in an particular order!):

1. Parcels are moving faster than the troughs

2. Parcels seek to maintain vorticity values in equilibrium with their surroundings

3. The level where vorticity advection is being evaluated is above the LND

4. QG theory is a good approximation to the real flow

5. The only contribution to vertical motion is related to vorticity advection

To the extent that these assumptions are valid (and the last may be the one most frequently violated), the old rule of thumb might be said to work. I will have more to add on this later.


Note (added 10 November 1999), based on an e-mail from John Monteverdi:

Generally speaking, cyclonic vorticity advection (CVA) at 500 mb is typically associated with cyclonic vorticity advection increasing with height. Most of this is due to the increase of westerly wind speed with height ... hence, it is only applicable in mid-latitudes. Therefore, the common assumption is that CVA is associated with QG forcing for upward motion. However the sign of the vorticity advection at any particular level can be misleading, if the situation violates the "typical" arrangement. In his e-mail, John says:

I have saved the 12 UTC 9 November AVN VT 48 h (11 November [1999]) 500 mb/vort, surface pattern/1000-500 mb thickness and precip/vertical velocity prog. The 500 mb and vorticity pattern shows moderate to strong NEGATIVE VORTICITY ADVECTION over northern and north-central California. Yet the vertical velocity prog shows a pocket of relatively STRONG UPWARD MOTION over the same area extending offshore and lots of precip from the Bay Area north. Why?

The actual DYNAMIC forcing (meaning synoptic scale forcing) associated with the vorticity advection pattern is NOT associated with the advection at any level (say, 500 mb as is often used by forecasters) but with whether vorticity advection is becoming more positive (or less negative) with height. Here I am only considering the forcing for upward motion in the mid troposphere. NVA at 500 mb can be entirely consistent with strong forcing for upward motion in the mid troposphere IF stronger negative vorticity advection is occurring at lower levels and weaker at levels above 500 mb.

The MORE COMMON PROBLEM that forecasters have is often neglecting the often dominant TEMPERATURE ADVECTION term. Put simply, at levels where warm advection is significant, forcing for upward motion can be very strong. In fact, in patterns as in the upcoming one, the warm advection in the layer from the ground to 500 mb can be so strong that forcing can be much larger than the forcing from the vorticity advection term.

In the case of the Thursday forecast, warm advection is likely to occur from Monterey northward. In addition, I here are the values of vorticity advection that the ETA forecasts at various levels for Point Reyes on Thursday morning--

850 mb: -9.0 X 10-10 s-2, 500 mb: -5.0 X 10-10 s-2, 300 mb: -2.0 X 10-10 s-2.

Note that negative vorticity advection is forecast at each of the levels, but that the negative vorticity advection is DECREASING with height (or, becoming more positive). So despite the negative vorticity advection at 500 mb, the dynamic forcing is for upward motion.

That is, AVA decreasing with height is equivalent to CVA increasing with height! John's point about the often-ignored thermal advection is considered below. I would add to John's comments that it is not all that uncommon for the contribution at a point from differential vorticity advection to be opposed to the contribution from thermal advection. Since the differential vorticity advection often dominates in the mid- to upper-troposphere, whereas the thermal advection usually dominates in the lower troposphere, you can still get clouds and even precipitation in regions where differential vorticity advection suggests descent (even ignoring non-QG effects).


Interpretation: Version 2

There is an alternative physical explanation for how vorticity advection relates to upward motion. Consider a vertically-aligned column of vorticity embedded in a sheared flow (Fig. 4). What might you expect that column to look like after some time has elapsed? Is it not likely to expect it to be tilted over? How might such a column remain upright in this environment? To see how this might happen, first remove the mean wind from the profile (since it only contributes to moving the column from left to right and does not alter the tilt), as in Fig. 5. The effect of this flow is to give a revised picture (Fig. 6), But we have seen above that this implies yet another picture (Fig. 7). This implied vertical motion, therefore, is just that which will keep this column of vorticity upright! Divergence downstream and convergence upstream (relative to the column) counteracts the tendency for advection to tilt the tube ... and the associated vertical motion is a direct consequence of that convergence and divergence pattern. It is this process that keeps systems more or less aligned vertically in spite of the presence of shear. Basically, in terms of QG theory, a system that is aligned vertically is neither amplifying nor decaying. If a system's intensity is changing, that has other implications (associated with the 2nd term on the rhs of the QG Height Tendency Equation, below). This discussion also can be developed more rigorously (as in Spencer et al. 1996).

Interpretation: Version 3

I have just heard of yet another interpretation of the relationship between vorticity advection and vertical motion ... it is attributed by the person from whom I heard it to Prof. Ken Crawford at the OU School of Meteorology. It goes like this:

Vorticity advection, per se, is directly associated with height changes, provided we can assume the system is not deepening or filling from differential thermal advection, as can be seen from the QG Height Tendency Equation:

In this equation, F is the geopotential height. According to this equation, cyclonic vorticity advection produces height falls, proportional to the magnitude of the vorticity advection. If cyclonic vorticity advection increases with height, this means that heights are falling more with height, so that the depth of the atmospheric column between the surface and some upper level is decreasing. This is shown schematically here, where increasing CVA with height results in a shrinking column. According to the hypsometric equation, the thickness of a layer between two pressure levels is related to the mean temperature in between the pressure levels. If the thickness decreases, the mean temperature is also decreasing. In an adiabatic atmosphere, in the absence of thermal advection, the way a column must cool is by ascent. Very neat and very succinct ... I like this interpretation.

The Thermal Advection term

In many situations (a small portion of which are described in Maddox and Doswell 1982), the thermal advection term, especially associated with low-level flow, is actually the dominant physical effect associated with large-scale ascent. To use the Ignorant Person's Omega Equation is to ignore what often is the dominant signal for large-scale ascent. To the extent that it is a significant contributor to vertical motion, ignoring the contribution to vertical motion from thermal advection seems quite obviously to be a perilous assumption.

The contribution associated with thermal advection typically (not always!) decreases with height, as the normal situation is for the atmosphere to become more equivalent barotropic with height. This may be counteracted, to some extent, by the typical increase of westerly wind speeds with height, so a quantitative evaluation of thermal advection may be required, at times when the situation is not entirely obvious.

How, physically, does thermal advection contribute to vertical motion? In QG theory, the flow is assumed to be adiabatic. This means that a parcel's potential temperature does not change, so if it originates on a particular isentropic surface, it must remain on that surface. An illustration of this can be found here. Basically, for adiabatic flow, warm advection on an isobaric surface implies ascent, while cold advection implies descent. In fact, the adiabatic assumption often is a good one, even on scales smaller than implied by the full set of QG assumptions. Hence, QG implied vertical motion may be reasonable on scales where the QG assumptions are not met, largely because of the importance of the thermal advection term and its connection to adiabatic flow.


Note (added 24 June 1999): John Locatelli has written:

... your discussion entitled "The Thermal Advection Term" brings up one of my pet peeves. That is, that air parcels are somehow constrained to move along isentropic surfaces, like a train must follow the already nailed down train tracks. I feel that this phrase gives the wrong impression as to what's happening in the atmosphere. I don't think it is the shape of the isentropic surfaces that determines the air motion of the parcels, but it is the motion of the parcels that determines the isentropic surfaces. Upward sloping isentropic surfaces are associated with upward velocity within warm-frontal regions because the parcels move slantwise upward which then (since they mostly conserve their theta , or theta-e values) produces the typical upward sloping isentropic surfaces.

to which I responded:

Interesting thought ... it seems to me that you're right in pointing out that the phrasing might well be subject to some misinterpretations. It certainly is true that isentropic surfaces are not some solid "given" structure, like rails, along which air parcels ride. Further, I'm quite willing to accept that the evolution of isentropic surfaces is attributable, in part, to the airflow. However, I stop short of agreeing that the slope of isentropic surfaces in frontal zones is entirely an artifact of the airflow ... it seems to me that we have a bit of a "chicken-egg" problem here, and I think that your interpretation might be too rigid toward one aspect of the issue, and my interpretation too rigid toward the other.

Let me expand on this some. When viewing an instantaneous map of the flow on an isentropic surface, the airflow on that surface might well be directed toward lower (or higher) pressure, and therefore the implication is that the air on that surface is ascending (or descending). However, this presumes that the isentropic surface is not changing. As John has pointed out, the airflow might well be deforming that surface and, of course, the flow may not be entirely adiabatic. Hence, this interpretation is not a complete description of the processes.

Nevertheless, considering that the goal is simply to have a qualitative sense of vertical motion along the lines that I have described, I believe that the picture implied by this interpretation is not often at great variance with the large-scale motion. That is, I believe that the foregoing physical interpretation of thermal advection is typically consistent with what is going on , at least on synoptic scales. To some extent, my interpretation ignores the process of how the isentropic surfaces have come to have their instantaneous configuration (and how they might be changing in the future) ... the idea in this essay is to have a qualitative, physically-based sense of the relationship between thermal advection and vertical motion and, subject to the caveat that any such explanation is to some extent an oversimplification, I maintain my belief that this is an acceptable interpretation.

Interestingly, when considering the contribution from differential vorticity advection, the flow must still be adiabatic (in QG theory, at least!). Therefore, the same concepts must apply in that case, as well. That is, if the flow is assumed to be adiabatic, it must "remain on the isentropic surfaces" in situations involving differential vorticity advection, as well. It's possible that when differential vorticity advection is operating, the contribution to the process from deformation of the isentropic surfaces (attributable to the airflow) is more important than in cases of "pure" thermal advection. This is not something I've considered and might be interesting to look at. Are isentropic surfaces deforming more rapidly in circumstances where strong differential vorticity advection is occurring than in cases of "pure" thermal advection? Note that it is typical for thermal advection to dominate at low levels and differential vorticity advection to dominate in the middle and upper troposphere. There are reasons for this, well beyond the scope of this essay. It also is typical for the contributions from thermal advection and differential vorticity advection to not be co-located in time and space, although they may be co-located occasionally. When they are co-located, they often (not always!) contribute to vertical motion in opposite senses.


References:

 

Barnes, S.L., 1985: Omega diagnostics as a supplement to LFM/MOS guidance in weakly forced convective situations. Mon. Wea. Rev., 114, 2121-2141.

Durran, D.R., and L.W. Snellman, 1987: The diagnosis of synoptic-scale vertical motion in an operational environment. Wea. Forecasting, 2, 17-31.

Holton, J.R., 1992: An Introduction to Dynamic Meteorology (3rd Ed.), Academic Press, 511 pp.

Maddox, R.A., and C.A. Doswell III, 1982: An examination of jet stream configurations, 500 mb vorticity advection and low-level thermal advection patterns during extended periods of intense convection. Mon. Wea. Rev., 110, 184-197.

Spencer, P.L., F.H. Carr and C.A. Doswell III, 1996: Diagnosis of an amplifying and decaying baroclinic wave using wind profiler data. Mon. Wea. Rev., 124, 209-223.