the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Energy conserving physics for nonhydrostatic dynamics in mass coordinate models
Mark A. Taylor
Peter A. Bosler
Christopher Eldred
Peter H. Lauritzen
Abstract. Motivated by reducing errors in the energy budget related to enthalpy fluxes with E3SM, we study several physicsdynamics coupling approaches. Using idealized physics, a moist rising bubble test case, and E3SM's nonhydrostatic dynamical core, we consider unapproximated and approximated thermodynamics applied at constant pressure or constant volume. Using timestep convergence studies, we show that the constant pressure update is more accurate at large timesteps despite being less consistent with the underlying equations. We reproduce the large inconsistencies between the energy flux internal to the model and the energy flux of precipitation when using approximate thermodynamics, which can only be removed by considering variable latent heats both when computing the latent heating from phase change as well as when applying this heating to update the temperature. Finally, we show that in the nonhydrostatic case, for physics applied at constant pressure, the general relation that enthalpy is locally conserved no longer holds. In this case, the conserved quantity is enthalpy plus an additional term proportional to the difference between the hydrostatic and full pressure.
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Oksana Guba et al.
Status: final response (author comments only)

RC1: 'Comment on gmd2023184', Thomas Bendall, 23 Oct 2023
General comments
This paper looks at different thermodynamics assumptions that can be used when coupling physical parametrizations and a dynamical core. Under the different assumptions, the updates to the dynamical prognostic variables change, thus changing the model's evolution. The authors then test the different approaches through a moist rising bubble simulation, in which the physics parametrization is the change of phase of moisture species, and the related latent heating effects.
This is an important and interesting question in the development of atmospheric models and has historically been neglected. The authors' work is a good contribution to investigating this question, and they have some nice maths to explain and derive the different approaches. The derivation of the constantpressure update that conserves energy is particularly satisfying, and the results demonstrating the lack of energy closure for their CPALHY and CPCLHY cases (without variable heat capacities) are convincing.
Besides some very minor points (listed below), I have a reservation about whether the authors' conclusions about the CVVLNH case (the constantvolume update) are fully supported by their numerical results. Otherwise, the paper is well written and communicated, so if this reservation can be addressed, I would be very happy to recommend this work for publication.Specific comments
1. My main concern is whether the authors' conclusions are fully supported by their experiments? The authors argue that the constant pressure update "is more accurate at large timesteps despite being less consistent with the underlying equations", and that it therefore "may be more computationally affordable".
This argument is based on the errors in the final solution of the test case, when compared with a reference state (calculated with very small Δt). The errors recorded are larger for the constantvolume update than the constant pressure update, at the larger values of Δt that were investigated.My worry is that rising bubble test cases, like the one used here, can be inherently unstable (even in the absence of moist physics). These typically look like a mushroom or pimple at the top of the main bubble (let's call this a "positive" instability), but can also be the opposite situation with the bubble's top having collapsed (a "negative" instability). I have previously seen that small changes in the model configuration can trigger these instabilities in bubble test cases or dramatically alter their appearance  for instance just changing the number of horizontal cells by 1 can flip whether there is a positivity or negative instability.
Figure 4(e) shows the final state of the rising bubble that corresponds to the larger errors that underpin the authors' conclusion, but to me this seems like a situation where this "negative" instability has occurred, and I believe that this is the dominant contribution to the error measurement. Because I think that the error is dominated by the effect of the instability, and because I suspect that the formation of such instabilities in this test case may be very sensitive to modeling choices, I am not (yet!) convinced that we can conclude that the constantvolume update may be less accurate at large time steps. This is particularly true because an instability in a smallscale rising bubble may not be representative of the much largerscale motions in a climate model (which is the authors' primary motivation).
I'd like to propose several different actions that the authors could take to address this concern:
(a) In my opinion, the work would still be worthy of publication if the conclusions relating to the constantvolume update were softened, and if more explanation is given as to why the constant pressure update might appear to be more accurate. I note that the authors give some explanation in lines 366368, but I think this could be expanded.
(b) However if the authors want to further support their conclusions, I think it would be helpful to know how robust their numerical results are. For instance, would they find the same result (that for large timesteps the constant pressure update gives a lower error) for the same bubble test but at different spatial resolutions? Or do they find the same result for similar bubble tests with slightly different initial conditions? Or do they find the same result for different numerical schemes used in the dynamical core?
(c) I recognise how challenging it is to choose a test case to investigate this question  I don't know of any idealised moist test cases with analytic solutions that aren't steadystate problems! However there are also other moist tests that may further support the authors' conclusions (for instance moist versions of the SkamarockKlemp gravity wave in a vertical slice, moist versions of baroclinic wave tests or moist HeldSuarez configurations). I do realise that this would be much more work, and do not think that this is a requirement for this work to be published.2. A key part of the authors' argument is that using the timesplit physics equations (12)(17), it is not possible to simultaneously satisfy both δp=0 and δφ=0 (for instance this is discussed in lines 128129 of page 5). Sorry if I am missing something obvious here, but this actually wasn't clear to me just from looking at equations (12)(17), so I think this could benefit from further explanation. I can see that if δφ=0 and δu=0, then to satisfy δe=0, we require a constant volume update. Is there anything more than that?
3. Are the latent heats L_{l} and L_{v} dependent on temperature or constant? The notation used in Section 4 includes "VL"" to describe variable latent heats, but it is not clear to me if the analysis takes this into account. I think it would be helpful to clarify this in the text. If L_{l} and L_{v} are in fact constants, how would the analysis change if they were temperature dependent? Would the derivation of (24) and (28) still follow?
4. Similarly, the configurations are named either by variable latent heats or constant latent heats, but are these actually refering to the heat capacities rather than the L_{l} and L_{v} terms? Or do variable heat capacities and variable latent heats come handinhand?
5. Page 6, lines 151152. The authors discuss that traditionally the equation for φ does not include physics terms. Is this a simplification, and so something that could be reconsidered in order to derive a constant pressure approach that is still consistent with the underlying equations?
Technical comments
1. Page 3, line 69, first sentence. Is the tense correct in this sentence? Maybe "is using" should be "uses"?
2. Page 3, equation 4. The authors use D to represent the dynamical terms in the momentum equation, which is also used for the material derivative. I think choosing a different letter here might be help avoid confusion.
3. Page 6, line 153. Should δφ=0 actually be δφ≠0? Or are the authors making the point that δφ=0 will hold irrespective of the choice of physicsdynamics coupling?
4. Page 6, line 160. It might be worth reminding the readers here why p=π (since π is the hydrostatic pressure).
5. Page 8, line 199200. I didn't quite follow the point that the authors are trying to make here: "instead, we reinterpret the first law as the general statement that the energy of the system must be conserved up to fluxes." Could there be more explanation here? Or could this point be reworded?
6. Page 9, section 4.2. What form does the initial perturbation take (e.g. a cosine bell or a Gaussian)?
7. Page 11, line 285. Is \tilde{E} missing a subscript t?
8. Page 13, lines 321322. The authors describe the plots as being "visually identical". The pedant inside of me doesn't agree! I think there are small differences can be spotted, so that they are almost visually identical.
9. Page 14, line 329. Should VP be VL here?
10. Page 16, line 399, "properly modeling". Should this be "proper modeling"?Citation: https://doi.org/10.5194/gmd2023184RC1 
RC2: 'Comment on gmd2023184', Anonymous Referee #2, 06 Nov 2023
This is really impressive work. The authors have described an important (and surprisingly longstanding) problem and, more importantly, a constructive way of fixing it with minimal modifications to the model. I only have a few comments.
1. You have the line "A key difference between the two approaches is that with constant pressure, latent heat release results in only vertical transport (by changing the position of the layers, φ), while with constant volume, latent heat release increases the pressure leading to gradients that can result, through the dynamical terms, in both vertical and horizontal mass transport." I would recommend that this is more prominently highlighted, perhaps in the abstract. Physics inaccuracies can lead to a spurious pressure gradient force, which affects dynamics and transcends the splitting.
2. I would like to see a mention in the conclusions of how widespread this problem is (or might be). I know this is a paper focusing on solutions for EAM, but are there other models that are affected? Or is this a unique EAM feature?
Citation: https://doi.org/10.5194/gmd2023184RC2 
RC3: 'Comment on gmd2023184', Anonymous Referee #3, 08 Nov 2023
This manuscript derives and evaluates several different methods of updating physics tendencies in an atmospheric model. One method assumes that volume is constant, another that hydrostatic pressure is constant, and a third that nonhydrostatic pressure is constant. The manuscript is a useful contribution but parts of it could be clarified.
I have no major suggestions, but listed below are some minor ones.
Line 7: “variable latent heats”. What quantity does latent heat vary with? Temperature?
Line 70: Could you define phi and pi more clearly? What are their dimensions?
Eqn. 10: Could you please indicate, with a subscript, e.g., which variables are being held constant in each of these partial derivatives?
Line 84: What is pi, the mass coordinate/hydrostatic pressure? How is pi related to p? Is there an equation that relates the two?
Line 85: What are the units of pi and s?
Eqn. (12): It might be worth mentioning that by assuming no change in wind, dissipative heating is neglected.
Line 119: In this equation, what do the subscripts mean? Are they derivatives? Or do they indicate that a variable is held constant?
Line 136: “we simplify the algebra by neglecting momentum tendencies by taking fu = 0” This omits frictional heating.
Line 140: How can a physics update keep both pressure and volume (and hence density) constant? A parameterization will increase temperature, thus changing density by the ideal gas law.
Lines 150–153: “An update which holds pressure constant while allowing the volume to change is impossible to derive via timesplitting for the nonhydrostatic equations, since the prognostic equation for layer positions does not have any traditional physics tendency terms, and thus any dynamics/physics timesplit approach will lead to δϕ = 0 for the update.” Why can’t an equation for dphi be derived based on, e.g., the expansion of the layer thickness due to heating at constant pressure? In this way, the physics parameterizations can indirectly affect phi even if they don’t directly modify phi.
Lines 153–154: “a diagnostic equation for ϕ”: What is the diagnostic equation for phi? Could you write out an example of such an equation?
Lines 170171: “In the nonhydrostatic 170 case, one cannot derive a δp = 0 update consistent with the timesplit equations since the combination δp = 0 and δϕ = 0 prohibits changes to any state variables.” What does dphi=0 have to do with time splitting? Where is the inconsistency with timesplit equations?
Lines 186–187: “As noted above, the nonhydrostatic constant pressure update is not consistent with our original timesplit equations, since it induces a change in volume which in our original equations is only allowed through dynamical terms.” Why can’t a change in volume occur through physics terms?
Lines 190–192: “A key difference between the two approaches is that with constant pressure, latent heat release results in only vertical transport (by changing the position of the layers, ϕ), while with constant volume, latent heat release increases the pressure leading to gradients that can result, through the dynamical terms, in both vertical and horizontal mass transport.” Please elaborate this sentence. It is important enough and subtle enough to deserve its own paragraph.
Eqn. (24): Shouldn’t the RHS have another term that equals f_T?
Eqns. (31)(35); These equations for the physics updates include various zero updates, e.g., delta_p = 0 in (35). Could you please explain how you derive which quantities have a zero update?
Line 329: “two constant pressure VP updates” What is a “VP” update?
I don’t understand why, in Fig. 2, the solutions at different time steps differ from each other early in the simulation (200 to 450 s) and then magically collapse onto identical solutions before the peak precipitation is reached at 550 s, and remain identical thereafter. Why do diverging solutions come back into agreement with each other? Does something constrain the precipitation, or is there a bug in the simulations?
Fig. 3: The line colors are not distinct enough. CVVLNH and CPCLHY both look green. CPVLHY and CPALHY look similar too.
Fig. 3(b) shows several purple lines. Please describe their meaning in the legend or the caption.
Citation: https://doi.org/10.5194/gmd2023184RC3 
RC4: 'Comment on gmd2023184', Anonymous Referee #4, 09 Nov 2023
General comments:
The paper written by Oksana Guba presented energy conserving physics in a nonhydrostatic dynamical core. The result is clearly presented and this development is also useful for other applications. I have some questions and several minor comments for improving the presentation quality.
Specific comments:
Line 95: It will be better to separately express these three eqs., or please add a comma for clear reading.
Line 157: Eq. (25) seems to be the same as Eq. (17). Please confirm.
Line 159: It will be better to separately express eH in Eq. (26), or use a comma for clear reading.
Line 168: Why no eq. nos. here?
Line 179: Again, Eq. (29) seems to be the same as Eq. (17). Please confirm.
Line 180: It will be better to separately express these three eqs., or please add a comma for clear reading.
Line 271: Please move the caption at the top of this table. This caption seems to be long, so it could partly remain as a footnote for this table.
Line 315: Why fails to converge for CPALHY and CPCLHY? These would be the best performance (minimized errors) at 10^{1} timestep.
Line 327: Is the difference unitless? Please confirm.
Technical comments:
Abstract: Need to define “E3SM” within the abstract.
Line 196: The accessed date is shown in the reference list, so there maybe no need to show it in the main text.
Citation: https://doi.org/10.5194/gmd2023184RC4
Oksana Guba et al.
Oksana Guba et al.
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