the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Feb 2024
Dynamically-optimal models of atmospheric motion
Abstract. A derivation of a dynamical core for the dry atmosphere in the absence of dissipative processes based on the least action (i.e., Hamilton’s) principle is presented. This approach can be considered the finite-element method applied to the calculation and minimization of the action. The algorithm possesses the following characteristic features: (1) For a given set of grid points and a given forward operator the algorithm ensures through the minimization of action maximal closeness (in a broad sense) of the evolution of the discrete system to the motion of the continuous atmosphere (a dynamically-optimal algorithm); (2) The grid points can be irregularly spaced allowing for variable spatial resolution; (3) The spatial resolution can be adjusted locally while executing calculations; (4) By using a set of tetrahedra as finite elements the algorithm ensures a better representation of the topography (piecewise linear rather than staircase); (5) The algorithm automatically calculates the evolution of passive tracers by following the trajectories of the fluid particles, which ensures that all a priori required tracer properties are satisfied. For testing purposes, the algorithm is realized in 2D, and a numerical example representing a convection event is presented.
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RC1: 'Comment on egusphere-2024-303', Anonymous Referee #1, 27 Feb 2024
The author’s endeavor to apply Hamiltonian mechanics to atmospheric prediction, through minimizing the action over a set of irregularly spaced discrete points and introducing a new dynamic core, represents a noteworthy attempt to leverage Hamiltonian mechanics in the realm of atmospheric forecasting. While this approach offers an innovative perspective, the manuscript falls short in addressing critical concerns associated with applying such mechanics to atmospheric predictions. These concerns include the highly nonlinear and complex nature of atmospheric dynamics, the presence of dissipative processes, and the challenges related to computational efficiency. Without a thorough examination and resolution of these issues, it remains premature for researchers to invest in the development of new prediction systems based on this approach. The reviewer suggests a comparison with the conservative approach based on the Nambu bracket, as detailed in Salmon (2007, DOI: 10.1175/JAS3837.1), to highlight potential advantages of the proposed algorithm. Addressing these challenges and advantages comprehensively could significantly strengthen the manuscript, making it suitable for further review.
Citation: https://doi.org/10.5194/egusphere-2024-303-RC1 -
AC1: 'Reply on RC1', Alexander Voronovich, 29 Feb 2024
The author is grateful to the anonymous reviewer for drawing attention to (Salmon, 2007) paper which was overlooked. Two additional related papers by R. Salmon: “A general method for conserving energy and potential enstrophy in shallow-water models”, J. Atm. Sciences, 64, 515-531, 2007 and “A general method for conserving quantities related to potential vorticity in numerical models”, Nonlinearity, 18, R1-16, 2005 will be added to the references list, and the following sentence will be also added to the second paragraph of the Introduction:
“In (Salmon, 2005; 2007) a technique was developed which is based on discretization of a Nambu bracket (a generalization of the Poisson bracket) and allows derivation of spatially-discretized equations of motion which exactly conserve energy, potential enstrophy, mass, and circulation.”
The author totally agrees with the reviewer that suggesting development of a dycore for GCMs based on the approach presented in this paper would be at this point premature. Thorough comparisons of how this approach fares against more traditional ones in different practical situations is needed; however, this requires additional research and cannot be accomplished within a single publication.
The comparison of the model developed in (Salmon, 2007) and the model suggested in the paper is definitely of great interest, especially so since they are due to different approaches: approximation of the continuous Hamiltonian equations by a discrete analog on one hand, and approximation of the action of the continuous system by a discrete set of canonical variables on the other. Note also that they consider in fact different systems: shallow water equations and compressible atmosphere. It was mentioned in the concluding Section of (Salmon, 2007) that corresponding generalization for the nonhydrostatic primitive equations would be non-trivial; thus, re-formulation of the approach of this work for shallow-water equations might be easier. However, this requires an additional research and should be done in my view in a separate publication.
Citation: https://doi.org/10.5194/egusphere-2024-303-AC1
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AC1: 'Reply on RC1', Alexander Voronovich, 29 Feb 2024
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RC2: 'Comment on egusphere-2024-303', Anonymous Referee #2, 08 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-303/egusphere-2024-303-RC2-supplement.pdf
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AC2: 'Reply on RC2', Alexander Voronovich, 14 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-303/egusphere-2024-303-AC2-supplement.pdf
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AC2: 'Reply on RC2', Alexander Voronovich, 14 May 2024
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RC3: 'Comment on egusphere-2024-303', Anonymous Referee #3, 21 May 2024
The paper considers the derivation of a dynamical core from a variational principle. In contrast to most related work that used discrete variational principles, the proposed approach works with the continuous variational principle and uses a Lagrangian that depends on a finite number of spatial degrees of freedom (in fact, it works with the Hamiltonian, identified with the Lagrangian via the Legendre transform).
Major comments:
- l. 48 ff: The work by Gawlik and Gay-Balmaz (2021) is the closest in the literature that I am aware. Please discuss the relationship in more detail. It is stated that non-canonical variations can be avoided in the present manuscript because time is kept continuous. That's is not true--even in the fully continuous setting one has a non-canonical system. And also in the fully discrete case the system remains non-canonical. The work by Gawlik and Gay-Balmaz is considerably more complex because of the non-canonical nature of the system. It would be important to clarify why the present work does not need the setting.
- The paper seems to reinvent semi-Lagrangian advection; at least the method is described starting on p. 8 in much detail without mentioning the name or relating it to the large existing literature. However, it is a standard method for decades.
- The paper uses non-standard and confusing nomenclature in various places. For example, "forward operator" is used for interpolation; while I see the connection to the forward operations used in data assimilation for observations, to call it interpolation seems clearer. A second example is "a discrete set of parameters" on p. 2. This refers to the degrees of freedom induced by the spatial discretization, which is a standard mesh. It would be better, to be direct and explicit here so that the reader knows from the outset what is meant.
- p. 2: "the best approximation (in a broad sense) of the dynamic[s] of a continuous atmosphere" The proposed approach has merits but "best" can be measured in many ways; optimal approximation rates is, e.g., the most standard one in numerical analysis. Please qualify the statement, e.g. best because one obtains conservation properties and laws, or remove.
- The numerical results are insufficient, in my opinion, both in terms of the number of numerical experiments and the analysis that are presented. E.g., one could repeat one of the experiments from Gawlik and Gay-Balmaz (the code is online, if I remember correctly)
- l. 186 - 189: Here, conservation properties are discussed, which are typically one of the reasons for a geometric approach. But the formulation is vague and it's not clear what properties hold for the presented method.
l. 274: Adaptivity is a problem on its own. I think it should be removed from the present manuscript and then in detail be presented in a second paper.
l. 287: "For historical reasons" -> cite or clarify
l. 333-334 : "conservation of energy wasn't checked" : the standard reason for a geometric approach are (discrete) conservation properties. Please include an example without heating that shows this.
l. 358 : Very unclear to me why observation data (brightness temperatures...) would be needed here?
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AC3: 'Reply on RC3', Alexander Voronovich, 21 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-303/egusphere-2024-303-AC3-supplement.pdf
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AC3: 'Reply on RC3', Alexander Voronovich, 21 May 2024
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