29 Jun 2021
29 Jun 2021
A vorticityandstability diagram as a means to study potential vorticity nonconservation
 Institute for Atmospheric and Climate Science, ETH Zürich, Zürich, Switzerland
 Institute for Atmospheric and Climate Science, ETH Zürich, Zürich, Switzerland
Abstract. The study of atmospheric circulation from a potential vorticity (PV) perspective has advanced our mechanistic understanding of the development and propagation of weather systems. The formation of PV anomalies by nonconservative processes can provide additional insight into the diabatictoadiabatic coupling in the atmosphere. PV nonconservation can be driven by changes in static stability, vorticity or a combination of both. For example, in the presence of localized latent heating, the static stability increases below the level of maximum heating and decreases above this level. However, the vorticity changes in response to the changes in static stability (and vice versa), making it difficult to disentangle stability from vorticitydriven PV changes. Further diabatic processes, such as friction or turbulent momentum mixing, result in momentumdriven, and hence vorticitydriven, PV changes in the absence of moist diabatic processes. In this study, a vorticityandstability diagram is introduced as a means to study and identify periods of stability and vorticitydriven changes in PV. Potential insights and limitations from such a hyperbolic diagram are investigated based on three case studies. The first case is an idealized warm conveyor belt (WCB) in a baroclinic channel simulation. The simulation allows only condensation and evaporation. In this idealized case, PV along the WCB is first conserved, while stability decreases and vorticity increases as the air parcels move poleward near the surface in the cyclone warm sector. The subsequent PV modification and increase during the strong WCB ascent is, at low levels, dominated by an increase in static stability. However, the following PV decrease at upper levels is due to a decrease in absolute vorticity with only small changes in static stability. The vorticity decrease occurs first at a rate of 0.5 f per hour and later decreases to approximately 0.25 f per hour, while static stability is fairly well conserved throughout the period of PV reduction. One possible explanation for this observation is the combined influence of diabatic and adiabatic processes on vorticity and static stability. At upper levels, largescale divergence ahead of the trough leads to a negative vorticity tendency and a positive static stability tendency. In a dry atmosphere, the two changes would occur in tandem to conserve PV. In the case of additional diabatic heating in the mid troposphere, the positive static stability tendency caused by the dry dynamics appears to be offset by the diabatic tendency to reduce the static stability above the level of maximum heating. This combination of diabatically and adiabatically driven static stability changes leads to its conservation, while the adiabatically forced negative vorticity tendency continues. Hence, PV is not conserved and reduces along the upper branch of the WCB. Second, in a fullfledged real case study with the Integrated Forecasting System (IFS), the PV changes along the WCB appear to be dominated by vorticity changes throughout the flow of the air. However, accumulated PV tendencies are dominated by latent heat release from the largescale cloud and convection schemes, which mainly produce temperature tendencies. The absolute vorticity decrease during the period of PV reduction lasts for several hours, and is first in the order of 0.5 f per hour and later decreases to 0.1f per hour when latent heat release becomes small, while static stability reduces moderately. PV and absolute vorticity turn negative after several hours. In a third case study of an air parcel impinging on the warm front of an extratropical cyclone, changes in the horizontal PV components dominate the total PV change along the flow and thereby violate a key approximation of the twodimensional vorticityandstability diagram. In such a situation where the PV change cannot be approximated by its vertical component, a higherdimensional vorticityandstability diagram is required. Nevertheless, the vorticityandstability diagram can provide supplementary insights into the nature of diabatic PV changes.
Gabriel Vollenweider et al.
Status: final response (author comments only)

RC1: 'Comment on wcd202131', Anonymous Referee #1, 30 Jul 2021
This paper presents a new construct, the 'vorticityandstability' diagram as a tool to provide better understanding of Lagrangian changes in potential vorticity (PV), due to nonconservative physical processes such as diabatic heating or turbulent transport of heat and momentum.
The idea is to consider an air parcel (or the mean of an ensemble of air parcels) over some period and to display the evolution in a two dimensional space, with the coordinates being two factors, one absolute vorticity and the other static stability, that multiplied together give the PV. The argument of the paper is that the new construct is 'a means to study and identify periods of stability and identify periods of stability and vorticitydriven changes in PV'.
One certainly cannot dispute the usefulness of studies that look in detail at the processes that change PV, e.g. in forecast models, as are exploited in the later part of this paper. But my reservation about this paper is over whether or not it provides a genuine advantage for such studies. The particular shortcoming of the approach proposed here is that the diagram does not by itself provide any information on the physics operating.
Using a highly abbreviated notation, let the PV be Q, the absolute vorticity be Z and the static stability be S. Using dQ to denote Lagrangian change in Q, for example, we have
dQ = S dZ + Z dS
Now suppose that dZ = dZ_P + dZ_D where the first term on the righthand side is due to nonconservative physics and the second is due to conservative dynamics, with corresponding notating for dS. Then S dZ_D + Z dS_D = 0 because there is no change in Q is conserved under the effects of conservative dynamics alone.
The diagram gives us information about dZ and dS (i.e. two pieces of information). Even though we have the constraint S dZ_D + Z dS_D = 0 we cannot determine dZ_P and dQ_P separately  so whether or not there are nonconservative diabatic processes (acting on temperatures) or nonconservative mechanical processes (acting on velocities) cannot be determined.
To me then, it seems that the behaviour seen in the diagram can, of course, be explained in terms of the physical processes acting if those are known, but the information given in the diagram is NOT sufficient to determine what those processes are (i.e. how they are partitioned between diabatic or mechanical in sense used above). In your case studies described in Sections 3.2 and 3.3 you have (and present) the detailed information about how different physical processes contribute to PV changes  how does the use of the diagram add anything?
Therefore I am skeptical that these diagrams add anything genuinely useful to the analysis of PV and its evolution in weather systems.
I have made some further detailed comments on the paper below.
Detailed comments:
Abstract: very long  seems to miss the point of an abstract which is to provide a brief summary of the aim, methodology and findings of the paper. In fact the text in general is overlong  there is a lot of background material.
l10: 'hyperbolic'  term will be meaningless to reader without explanation
l61: '"latent vorticity" generation'  first time I had come across this term  which essentially seems to mean forcing of PV by diabatic processes (with the 'direct' effect, so to speak, on the temperature/stability) together with the very familiar principle that the partitioning of PV between relative/absolute vorticity and stability can change through purely reversible conservative processes. The term seems to be very rarely used and, with all respect to Chagnon and Gray (2009), I'm not convinced that it helps general understanding to perpetuate it.
l62: 'adjusts to a new balanced state in the process of hydrostaticgeostrophic adjustment ... during which inertia, gravity and sound waves radiate away from the heating perturbation'  there is a question  perhaps it is a matter of taste  about whether it is appropriate to describe evolution of a balanced flow as a continuous process of geostrophic adjustment. One subtlety is that the amount of emitted wave activity is determined not just by the difference between the two states A and B, say, at different times, but by the time that elapses between A and B (see Vanneste 2013). There are some advantages to restricting the term geostrophic adjustment to an initial value problem or a problem with 'impulsive' forcing.
(But as I have noted  this is partly a matter of taste  I'm not insisting on a change.)l86: 'moist diabatic processes'  not all processes that affect PV, even in the troposphere, are moist.
Figure 1: My understanding is that the colours of the dots here are not providing 'extra' information  they are simply displaying information that could be deduced from the diagram  since (using the notation I have introduced above) what you are indicating is  S dZ / Z dS   which can be deduced from the position in the diagram and the slope of the curve. (This is not a criticism of the use of the colours  but I think it is important to be clear on what is 'new' information and what is not.
l99: 'vertical component'  I realise that taking account only of the vertical component of absolute vorticity is a useful simplification, and I don't have any particular problem with that, but I do think that the term 'vertical component of PV', which you use subsequently at various points in the paper, is a unfortunate. PV is a scalar, so it doesn't have a vertical component in the sense that absolute vorticity, as a vector, has a vertical component. Your terminology muddles use of 'component' with respect to a vector, with the more general use of 'component' as meaning 'part of'. It is not a serious problem, but it is not very elegant or precise.
Figure 3 caption: 'vocticity'.
l22931: goes back to my earlier comment re 'geostrophic adjustment'  evolution of the balanced state does not require substantial emission of gravity waves  it may or may not. A better statement in my view would be something like 'the vorticity decrease occurs as part of the evolution of the balanced state (under conservative dynamics)'. I'm not convinced that the 'geostrophic adjustment' sentence is needed. ', which seems unlikely' could simply be added to the previous sentence. Certainly 'desired' is not the correct word to use.
l339: 'In the absence of diabatic processes, this vorticity reduction and stability increase (i.e, column shrinking) would occur in tandem to conserve PV. However, the lesson learned from the vorticityandstability diagram is that it seems as if the largescale divergence drives a vorticity reduction, but the diabatic and adiabatic influences on static
stability are engaged in a tugofwar, such that PV is not conserved.' The first sentence, of course, is simply describing PV conserving dynamics, in the absence of, say, diabatic processes. If diabatic processes act then there will, unless the diabatic forcing term in the PV equation is zero, be a change in the PV. Some of this change will appear in vorticity, some will appear in static stability  that is all well known and it depends on nonlocal effects  it can't be determined simply from what happens in a single air parcel. I don't really see how the diagram is helping  apart from showing that the two quantities change  where does one go from that?VANNESTE, J. 2013 Balance and spontaneous wave generation in geophysical flows. Annu. Rev. Fluid Mech. 45, 147–172.

RC2: 'Comment on wcd202131', Anonymous Referee #2, 17 Aug 2021
Synopsis:
This paper takes a potential vorticity (PV) perspective on atmospheric dynamics. Given that PV is broadly speaking a product of static stability and absolute vorticity, a material rate of change of PV can occur through nonconservative terms in either the heat equation or the momentum equation or both. However, it is well possible that both static stability and vorticity suffer a nonzero material rate of change, but PV is, at the same time, conserved: this is exactly what happens in the event of purely conservative flow with vortex stretching.
Major concern:
The authors analyse the situation with the help of a novel diagram that represents the motion of an air parcel in a twodimensional phase space spanned by absolute vorticity and static stability. This is an interesting approach. At the same time, I think that the paper does not live up to the expectations, and it seems to me that essential aspects of “adjustment to balance” need to be discussed more lucidly in order to make this a useful contribution to the literature. In particular, it seems to me that there is a fundamental flaw in the argument. Consider the following thought experiment which was originally suggested by (I believe) M. McIntyre and/or B. Hoskins quite some time ago (sorry, I cannot find the respective reference). Assume that initially a parcel is instantaneously being subject to differential heating such that its static stability increases; the point in phase space would move straight upward. The ensuing adjustment process is thought to be adiabatic such that the parcel moves along one of the red hyperbolas. During this adjustment process, part of the original material increase in static stability is reduced and converted into a material increase of absolute vorticity (such that PV is conserved during the adjustment process). Where exactly the point ends on the diagram during the adjustment process essentially depends on the ratio of static to inertial stability and the aspect ratio of the heating (see, e.g., the work of Eliassen 1952). Of course, in reality the (initial) diabatic change and the (ensuing) adjustment process cannot be separated from each other, rather they occur more or less simultaneously. In addition, the occurrence or absence of inertiogravity waves depends on the time scale during which the initial nonconservative process is applied.
Now consider a second thought experiment where initially there is only an (impulsive) nonconservative material tendency on absolute vorticity, followed by the adiabatic adjustment process. Both thought experiments may lead to the same end point in the phase space diagram. Thus, considering only the change of the point in phase space from the initial to the end state does not really tell us anything about the nature of the nonconservative processes – they may be diabatic (nonconservative heat equation), friction (nonconservative momentum equation), or a mixture of both.
For this reason, I cannot follow the basic argument that underlies the reasoning of this paper. The argument first occurs on line 110: “…. PV changes in regions where grey hyperbolas are oriented more vertically …. tend to be driven by changes in static stability.” Not accounting for the problem that I am not sure whether “changes in static stability” here are meant to be conservative or nonconservative, I think that any such statement cannot be made based on the trajectory of the parcel on the diagram alone.
In the end it does not become clear to me what we have really learned from the analysis using this novel phase diagram. As far as I understand the text, the authors themselves are not very clear about that, and this materializes in the fact that the abstract is very long and very detailed. If one has so many results to report, this raises the suspicion in me that there is not really any true result.
Further issues:
I had a problem with this manuscript in that I could sometimes not really evaluate the validity of individual statements because I did not fully understand them. For instance, one should very carefully distinguish between (1) observed (material) tendencies in vorticity and static stability (which may be due to either conservative or nonconservative processes) and (2) nonconservative (material) tendencies in vorticity and static stability (which could be obtained by analysing the corresponding nonconservative terms in the momentum and heat equation (although the authors consider this to be beyond the scope of the paper).
Another important concern of mine are the many occurrences of formulations (A “drives” B) that suggest a direction of causality where (as far as I can tell) the authors do not provide any prove of such causality. I suggest to simply replace the word “drive” or “driven” by a more appropriate word, and often this more appropriate formulation would simply be “A is associated with B”. To give an example: I could not follow your interpretation of the diagram in Fig 3d: what do you mean when you say that a PVver change is “driven” by …, and what does this mean? You should be more explicit here. Especially it is not clear to me whether a “change” in stability or vorticity is meant to be conservative or nonconservative.
PVver is a strange variable. What do we know about it? It is not necessarily materially conserved for conservative flow. This is dangerous, since the impact of nonconservative processes is at the heart of your analysis. A few lines later (and in the remainder of the text) PVver and PV are essentially treated as synonymous….
I provide an annotated manuscript in which I point to several issues which are partly summarized above, plus some further issues (e.g., with terminology such as the use of the word “diabatic”, “component of a scalar”, etc.).
Reference:
Eliassen, A. 1952. Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norv. 5, No 2, 19–60.
 AC1: 'Reply on wcd202131', Sebastian Schemm, 27 Sep 2021

EC1: 'Comment on wcd202131', Michael Riemer, 09 Oct 2021
Dear authors, thank you for considering ACP for your publication. Your manuscript presented a thoughtprovoking decomposition of changes of potential vorticity in vorticity and stability changes and conservative and nonconservative changes. The manuscript has raised insightful contributions to the online discussion by two highly qualified referees.
Both referees raise fundamental concerns, and they agree on a very basic issue: The problem that you attempt to solve with your proposed diagram is underdetermined. This fact is lucidly demonstrated at the beginning of the review by referee #1. In other words, referee #1 provides a simple mathematical proof that the proposed diagram cannot contain the information that you claim to be depicted.
In your response, you provide a revised version of the diagram that seemingly overcomes this issue. The seeming solution is achieved by the tacit assumption that the conservative and nonconservative PV changes form an orthogonal basis. This assumption, however, is incorrect because any nonconservative change in stability or vorticity could have been partly achieved by conservative dynamics. For further illustration: Assume that both the conservative vorticity and stability change is zero and that only a frictional process operates, which leads to a nonconservative change of the vorticity component (for full clarity, in the notation of referee #1: dZ_D=0, dS_D=0, dS_P=0, dZ_P ≠ 0). The orthogonality assumption implied in your revision would obscure this simple process. Instead, the erroneous interpretation of the diagram under the orthogonality assumption would be that the evolution had included all four changes: dZ_D ≠ 0, dS_D ≠ 0, dS_P ≠ 0, dZ_P ≠ 0.
Thank you again for having brought an interesting discussion to WCD. Due to the fundamental flaw raised by the referees, however, I need to reject this manuscript.
Gabriel Vollenweider et al.
Gabriel Vollenweider et al.
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