the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 01 Mar 2024
Tropical Cyclone Asymmetric Eyewall Evolution and Intensification in a Two-Layer Model
Abstract. Radar and satellite imagery of numerous intensifying tropical cyclones (TCs) depict an appearance of polygonal eyewall structure where deep convection is often located near the polygonal vertices. A recent observational study of Hurricane Michael’s (2018) polygonal eyewall evolution suggests that the vorticity asymmetries are coupled with the reflectivity asymmetries during rapid intensification. Conceptual theory of a polygonal eyewall structure has been linked to vortex Rossby waves (VRWs) and the breakdown of an enhanced potential vorticity (PV) ring, but how the asymmetries affect TC intensification remains unclear. Non-divergent barotropic models have been previously employed to study polygonal eyewall dynamics, but this approach has limitations due to the importance of diabatic heating to PV generation and the intensification process. Results from prior studies motivate us to explore the nature of the relationship between asymmetric vorticity and vertical velocity in the free atmosphere and the boundary layer and their compound impacts on the TC intensification process. Here we use a simple two-layer model framework with a shallow water model on top of a slab boundary layer model (SBL) to simulate a frictional boundary layer underneath the free atmosphere. Results from simulating a wavenumber-two elliptical asymmetry suggest the VRW in the free atmosphere can organize the updrafts in the SBL, which is consistent with radar observations of enhanced reflectivity at the polygonal vertices. Free atmospheric divergence in the shallow water layer does not explain the coupling between vorticity and reflectivity. The coupling can be explained to first order by the one-way boundary layer response to the pressure gradient associated with the free atmospheric vorticity asymmetries, consistent with prior studies. Further simulations that allow two-way interaction between the layers show that the organization of the updrafts out of the SBL plays a critical role in the growth of a PV ring and intensification of the mean vortex. In this framework, diabatic heating in the shallow-water layer parameterized by a mass sink driven by the free-atmosphere/SBL interaction leads to rapid intensification of the vortex, thinning of the PV ring, and eventual barotropic instability and PV mixing. The simplified modeling framework with two-way interactions captures many of the essential dynamics of rapid intensification in the presence of evolving asymmetries similar to those seen in the observations from Hurricane Michael (2018), which provides new insight into the complex interactions between dynamics and convection during hurricane intensification.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(16755 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2024-505', Anonymous Referee #1, 07 Apr 2024
The two-layer model (modeling boundary layer and free atmosphere), developed by the authors, can express the effect and feedback from the boundary layer to the free atmosphere. The effect and feedback cannot be expressed by the previous one-way configuration. The model has the potential to understand essential processes of the TC vortex development. The model is also suitable for the scaling law study. A scaling law is a power law that asserts a proportional relationship between relevant quantities. In general, the finding of the scaling law for a phenomenon in fluid systems can be largely helpful for understanding the phenomenon. It will be very fascinating to apply the two-way setting model to the scaling law studies in many different typhoon cases. Overall, the paper is well-written and can be published.
Â
 [Major] My major concern with the paper is a potential gap between the authors' model and the real observations or full-physics model simulations. Specifically, the momentum transport in the two-layer model may need to be considered on some occasions. As the authors also pointed out, even in the two-way configuration, the updraft from the SBL does not provide momentum to the SWM (L332-340). That is, the vortex or tangential wind in the free atmosphere in the two-way configuration may be enhanced through gradient wind adjustment by the change in the pressure distribution of the free atmosphere associated with the updraft from the SBL. If so, there may be some differences in the intensifying process of the vortex between the authors' model and the full physics model. In the full physics model, the momentum transport associated with the updraft from the boundary layer to the free atmosphere greatly affects the development of the vortex in the free atmosphere (e.g., Fig. 6 of Wang et al. 2016). This suggests that the TC vortex can develop by a process that cannot be described by the authors' model. To what extent the evolution of the vortex represented by the authors' model is valid for the evolution of real TC vortices does not seem to be discussed in the current manuscript. If the authors can quantify or estimate the validity of their model, it may be useful to add a discussion in the text.
Â
[Minor] There are some typographical errors in the model equations. Please correct them. For example, it may be necessary to revise the sign of the pressure gradient force in Eq. (5), u (u_b correctly) in the last term of the right side of Eq. (5), and the specific expression of the Laplacian in Eqs. (4) - (5).
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Reference: Wang, H., C. Wu, and Y. Wang, 2016: Secondary Eyewall Formation in an Idealized Tropical Cyclone Simulation: Balanced and Unbalanced Dynamics. JAS, 73, 3911-3930, https://doi.org/10.1175/JAS-D-15-0146.1
Citation: https://doi.org/10.5194/egusphere-2024-505-RC1 -
AC1: 'Reply on RC1', Ting-Yu Cha, 17 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-505/egusphere-2024-505-AC1-supplement.pdf
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AC1: 'Reply on RC1', Ting-Yu Cha, 17 May 2024
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RC2: 'Comment on egusphere-2024-505', Rupert Klein, 19 Apr 2024
Report on Tropical Cyclone Asymmetric Eyewall Evolution and Intensification in a Two-Layer Model
by Ting-Yu Cha and Michael M. Bell
Referee: Rupert KleinGeneral comments:
The authors investigate the effect of asymmetric perturbations on the intensification of tropical storms based on a reduced two-layer model. The model consists of a slab boundary layer (SBL) and a superimposed layer with shallow water-type (SWM) dynamics. The SBL model is to represent the near-surface layer of the atmosphere influenced strongly by vertical turbulent transport, while the SWM represents about the lower 2-3km of the troposphere. The rest of the troposhere up to the tropopause is modelled implicitly by assuming that it has no dominant effect on vortex intensification and that it takes up and redistributes any vertical mass fluxes that may emerge out of the shallow water layer due to convection. These model components, including parameterizations of unresolved scale processes are adopted from the established literature, where they have already been used and argued for in similar contexts. In this sense, I consider the ingredients of this two-layer model to have stood the test of time as qualitative representations of some important aspects of large-scale atmospheric vortices. I do have some questions regarding the layer coupling, which I will post below.
The numerical scheme implemented to solve the model equations judiciously borrows from spectral and finite difference discretizations and is solidly state of the art.
The paper provides a detailed numerical study that juxtaposes model results with one- and two-way coupling of the two layers. In the one-way version, the SWM influences the SBL but not vice versa. The study clearly reveals that two-way coupling is crucial for reproducing key observed features of accelerating storms, such as a shrinking of the radius of maximum wind RMW during the intensification period, and -- more importantly for the present paper -- the interplay of Fourier mode one and two asymmetries during the process. Plausible physical interpretations of the processes observed in a series of model runs are provided, yielding an interesting set of hypotheses regarding the mechanisms behind what is called "rapid intensification" of tropical storms.
The paper is very well written, with a concise and clear literature review, well-structured technical descriptions of both the mathematical model used and of its numerical discretization, and with clear discussions of the simulation results.
Specific comments
1. I do have one concern regarding the structure of the two-layer model. In lines 174, 175, the authors state that "Lack of strict mass conservation is not a problem for the length of time integration and the aims of the study considered here, but the model is not expected to reach a steady state with this numerical approach." I urge the authors to provide an extended argument leading to this conclusion for the following reason: The spin-up of a vortex is largely driven by the conservation of angular momentum and the fact that in the boundary layer mass is moving inwards, thereby inducing acceleration of the primary circulation. The inward-moving mass must, for conservation reasons, go somewhere. If I understand it correctly, it is assumed here that the mass more or less slips through the SW layer and then disappears in the implicitly modelled bulk of the troposphere. What justifies assuming that the SW layer does not pick up at least part of that mass - an effect that would counteract that of the assumed "mass sink" attributed to convection and entrainment? And why would the implicitly modelled upper part of the atmosphere, which does absorb the upward mass flux and should, therefore, reveal a slow-down, not influence the shallow water layer at all?
2. The authors report to impose homogeneous Neumann inner boundary conditions for vertical velocity, w, and boundary layer height, h. While I can see how that can be justified for the height from radial momentum balance, I don't see (i) why this condition should hold for w and (ii) why there should be a boundary condition for w in the first place. According to (6), the vertical velocity is the product of the boundary layer height and the horizontal divergence. Even if the height satisfies a homogeneous Neumann condition, I don't think the horizontal divergence would do so. If I am right, the radial gradient of w is height times the radial gradient of the divergence. Moreover, nowhere in the governing equations does the radial derivative of w occur, so why should a boundary condition for w be needed at all. What am I missing?
Minor comments:
l. 113:Â shallow watter -> shallow waterÂ
l. 146: later -> layer
l. 374:Â evoluti99on ->Â evolutionCitation: https://doi.org/10.5194/egusphere-2024-505-RC2 -
AC2: 'Reply on RC2', Ting-Yu Cha, 17 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-505/egusphere-2024-505-AC2-supplement.pdf
-
AC2: 'Reply on RC2', Ting-Yu Cha, 17 May 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-505', Anonymous Referee #1, 07 Apr 2024
The two-layer model (modeling boundary layer and free atmosphere), developed by the authors, can express the effect and feedback from the boundary layer to the free atmosphere. The effect and feedback cannot be expressed by the previous one-way configuration. The model has the potential to understand essential processes of the TC vortex development. The model is also suitable for the scaling law study. A scaling law is a power law that asserts a proportional relationship between relevant quantities. In general, the finding of the scaling law for a phenomenon in fluid systems can be largely helpful for understanding the phenomenon. It will be very fascinating to apply the two-way setting model to the scaling law studies in many different typhoon cases. Overall, the paper is well-written and can be published.
Â
 [Major] My major concern with the paper is a potential gap between the authors' model and the real observations or full-physics model simulations. Specifically, the momentum transport in the two-layer model may need to be considered on some occasions. As the authors also pointed out, even in the two-way configuration, the updraft from the SBL does not provide momentum to the SWM (L332-340). That is, the vortex or tangential wind in the free atmosphere in the two-way configuration may be enhanced through gradient wind adjustment by the change in the pressure distribution of the free atmosphere associated with the updraft from the SBL. If so, there may be some differences in the intensifying process of the vortex between the authors' model and the full physics model. In the full physics model, the momentum transport associated with the updraft from the boundary layer to the free atmosphere greatly affects the development of the vortex in the free atmosphere (e.g., Fig. 6 of Wang et al. 2016). This suggests that the TC vortex can develop by a process that cannot be described by the authors' model. To what extent the evolution of the vortex represented by the authors' model is valid for the evolution of real TC vortices does not seem to be discussed in the current manuscript. If the authors can quantify or estimate the validity of their model, it may be useful to add a discussion in the text.
Â
[Minor] There are some typographical errors in the model equations. Please correct them. For example, it may be necessary to revise the sign of the pressure gradient force in Eq. (5), u (u_b correctly) in the last term of the right side of Eq. (5), and the specific expression of the Laplacian in Eqs. (4) - (5).
Â
Reference: Wang, H., C. Wu, and Y. Wang, 2016: Secondary Eyewall Formation in an Idealized Tropical Cyclone Simulation: Balanced and Unbalanced Dynamics. JAS, 73, 3911-3930, https://doi.org/10.1175/JAS-D-15-0146.1
Citation: https://doi.org/10.5194/egusphere-2024-505-RC1 -
AC1: 'Reply on RC1', Ting-Yu Cha, 17 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-505/egusphere-2024-505-AC1-supplement.pdf
-
AC1: 'Reply on RC1', Ting-Yu Cha, 17 May 2024
-
RC2: 'Comment on egusphere-2024-505', Rupert Klein, 19 Apr 2024
Report on Tropical Cyclone Asymmetric Eyewall Evolution and Intensification in a Two-Layer Model
by Ting-Yu Cha and Michael M. Bell
Referee: Rupert KleinGeneral comments:
The authors investigate the effect of asymmetric perturbations on the intensification of tropical storms based on a reduced two-layer model. The model consists of a slab boundary layer (SBL) and a superimposed layer with shallow water-type (SWM) dynamics. The SBL model is to represent the near-surface layer of the atmosphere influenced strongly by vertical turbulent transport, while the SWM represents about the lower 2-3km of the troposphere. The rest of the troposhere up to the tropopause is modelled implicitly by assuming that it has no dominant effect on vortex intensification and that it takes up and redistributes any vertical mass fluxes that may emerge out of the shallow water layer due to convection. These model components, including parameterizations of unresolved scale processes are adopted from the established literature, where they have already been used and argued for in similar contexts. In this sense, I consider the ingredients of this two-layer model to have stood the test of time as qualitative representations of some important aspects of large-scale atmospheric vortices. I do have some questions regarding the layer coupling, which I will post below.
The numerical scheme implemented to solve the model equations judiciously borrows from spectral and finite difference discretizations and is solidly state of the art.
The paper provides a detailed numerical study that juxtaposes model results with one- and two-way coupling of the two layers. In the one-way version, the SWM influences the SBL but not vice versa. The study clearly reveals that two-way coupling is crucial for reproducing key observed features of accelerating storms, such as a shrinking of the radius of maximum wind RMW during the intensification period, and -- more importantly for the present paper -- the interplay of Fourier mode one and two asymmetries during the process. Plausible physical interpretations of the processes observed in a series of model runs are provided, yielding an interesting set of hypotheses regarding the mechanisms behind what is called "rapid intensification" of tropical storms.
The paper is very well written, with a concise and clear literature review, well-structured technical descriptions of both the mathematical model used and of its numerical discretization, and with clear discussions of the simulation results.
Specific comments
1. I do have one concern regarding the structure of the two-layer model. In lines 174, 175, the authors state that "Lack of strict mass conservation is not a problem for the length of time integration and the aims of the study considered here, but the model is not expected to reach a steady state with this numerical approach." I urge the authors to provide an extended argument leading to this conclusion for the following reason: The spin-up of a vortex is largely driven by the conservation of angular momentum and the fact that in the boundary layer mass is moving inwards, thereby inducing acceleration of the primary circulation. The inward-moving mass must, for conservation reasons, go somewhere. If I understand it correctly, it is assumed here that the mass more or less slips through the SW layer and then disappears in the implicitly modelled bulk of the troposphere. What justifies assuming that the SW layer does not pick up at least part of that mass - an effect that would counteract that of the assumed "mass sink" attributed to convection and entrainment? And why would the implicitly modelled upper part of the atmosphere, which does absorb the upward mass flux and should, therefore, reveal a slow-down, not influence the shallow water layer at all?
2. The authors report to impose homogeneous Neumann inner boundary conditions for vertical velocity, w, and boundary layer height, h. While I can see how that can be justified for the height from radial momentum balance, I don't see (i) why this condition should hold for w and (ii) why there should be a boundary condition for w in the first place. According to (6), the vertical velocity is the product of the boundary layer height and the horizontal divergence. Even if the height satisfies a homogeneous Neumann condition, I don't think the horizontal divergence would do so. If I am right, the radial gradient of w is height times the radial gradient of the divergence. Moreover, nowhere in the governing equations does the radial derivative of w occur, so why should a boundary condition for w be needed at all. What am I missing?
Minor comments:
l. 113:Â shallow watter -> shallow waterÂ
l. 146: later -> layer
l. 374:Â evoluti99on ->Â evolutionCitation: https://doi.org/10.5194/egusphere-2024-505-RC2 -
AC2: 'Reply on RC2', Ting-Yu Cha, 17 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-505/egusphere-2024-505-AC2-supplement.pdf
-
AC2: 'Reply on RC2', Ting-Yu Cha, 17 May 2024
Peer review completion
Journal article(s) based on this preprint
Data sets
Scythe model Version 1.0.0 Michael M. Bell and Ting-Yu Cha https://doi.org/10.5281/zenodo.10668054
Model code and software
Scythe model Version 1.0.0 Michael M. Bell and Ting-Yu Cha https://doi.org/10.5281/zenodo.10668054
Interactive computing environment
Scythe model Version 1.0.0 Michael M. Bell and Ting-Yu Cha https://doi.org/10.5281/zenodo.10668054
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Michael M. Bell
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(16755 KB) - Metadata XML