On the quasi-steady vorticity balance in the mature stage of hurricane <em>Irma (2017)</em>

de Jong, Jasper; Baatsen, Michiel L. J.; van Delden, Aarnout J.

doi:https://doi.org/10.5194/egusphere-2023-1259

Preprints

https://doi.org/10.5194/egusphere-2023-1259

Preprints

17 Jul 2023

| 17 Jul 2023

On the quasi-steady vorticity balance in the mature stage of hurricane Irma (2017)

Jasper de Jong, Michiel L. J. Baatsen, and Aarnout J. van Delden

Abstract. Tropical cyclone (TC) intensification is a process depending on many factors related to the thermodynamical state and environmental influences. It remains a challenge to accurately model TC intensity due to the role of unsteady features like deep convective bursts, boundary layer dynamics and eddy processes. The impermeability theorem for potential vorticity substance, PVS, on isentropic surfaces provides a way to analyze the absolute vorticity structure and tendency in TCs. We will examine this theorem in a numerical simulation of hurricane Irma (2017) near lifetime-peak intensity. Hurricane Irma was a very intense hurricane that persisted as a category five hurricane on the Saffir-Simpson intensity scale for three consecutive days, the longest for any Atlantic hurricane since satellite observations. During this period the intensity of Irma was remarkably constant. According to the impermeability theorem, the radially outward vorticity flux due to divergence above the atmospheric boundary layer must be compensated by an equally strong radially inward vorticity flux due to the effect of diabatic heating in the presence of vertical wind shear. The model results agree with this theorem and we find a strong anticorrelation between the advective and diabatic components of the radial vorticity flux. The impact of parametrized turbulence on the vorticity balance is found to be weak and does not explain the residual flux that would otherwise close the vorticity balance.

Received: 16 Jun 2023 – Discussion started: 17 Jul 2023

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Jasper de Jong, Michiel L. J. Baatsen, and Aarnout J. van Delden

Status: closed (peer review stopped)

RC1:
'Comment on egusphere-2023-1259', Anonymous Referee #1, 21 Aug 2023

Please see the attached file.

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC1
- AC1: 'Reply on RC1', Jasper de Jong, 04 Dec 2023
  
  Thank you for your helpful comments. We acknowledge that the vorticity flux due to turbulence does not close the budget, especially when integrated over time. This might be in part due to disagreement between our offline turbulence calculation and that in the model (they should be close to identical but are not completely) and the lack of data to employ this calculation on (we only have hourly time steps and just one turbulence-related variable, turbulent kinetic energy, which we used to validate our offline calculation). We will investigate the former more closely for the next article version, while the latter is an inherent limitation to our research. Despite this limitation we believe that a vorticity flux due to turbulence derived from hourly data should give a fair representation of the actual flux because of the steady dynamics. If we do find our current vorticity flux due to turbulence to be accurate after revision, its negligible magnitude contributes greatly to remaining a strong vortex for so long, since the smoothening of the tangential wind profile you pointed out is strongly inhibited.
  We will work through all minor comments you provided and work on reframing our story. You ask whether we performed a quantitative analysis of the modelled cooling in the eye. Despite qualitatively looking at vertical moisture and temperature profiles, we did not perform such an analysis yet. We will incorporate this in the revised version.
  Specific comments:
  Items 1, 3, 4, 8, 9 and typos: we will rectify these issues.
  2: This has not been done, except a qualitative inspection of vertical profiles of temperature and humidity.
  5: This sentence refers to a panel that was there in an earlier draft, we will update the text.
  6: ‘increase’ should be ‘decrease’.
  7: Figure 5c does not show the total vorticity flux (not shown in the article) but its turbulent component. At 310 K, the maximum advective flux (fig 5a) is higher than the absolute minimum diabatic flux (fig 5b), leading to a (positive) maximum total flux near r=30km, neglecting the turbulent flux. Therefore, the radial flux divergence is positive (negative) inward (outward) from this radius (fig 6b). We will indicate figure numbers in this sentence for clarity.
  With your points in mind, we hope to improve the article and make it a much more interesting and sound piece of literature.
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC1
RC2:
'Comment on egusphere-2023-1259', Anonymous Referee #1, 21 Aug 2023

I appologize no reference in my specific comment#2. I revised the comment as follows:
L49-50: Did the authors quantitatively confirm the radiative cooling for the warm core extension? Previous studies indicated that adiabatic processes associated with subsidence in the eye can be a major contribution to the development of the warming in the eye (e.g., Stern and Zhang, 2013; Ohno and Satoh, 2015).

Reference
Ohno, T., and M. Satoh, 2015: On the warm core of a tropical cyclone formed near the tropopause. J. Atmos. Sci., 72, 551–571, doi:10.1175/JAS-D-14-0078.1.
Stern, D. P., and F. Zhang, 2013: Howdoes the eyewarm? Part I:Apotential temperature budget analysis of an idealized tropical cyclone. J. Atmos. Sci., 70, 73–90, doi:10.1175/JAS-D-11-0329.1.

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC2
- AC2: 'Reply on RC2', Jasper de Jong, 04 Dec 2023
  
  See 'Reply on RC1'
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC2
RC3:
'Comment on egusphere-2023-1259', Anonymous Referee #2, 28 Sep 2023

Please see attachment

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC3
- AC4: 'Reply on RC3', Jasper de Jong, 04 Dec 2023
  
  Thank you for your helpful comments. We acknowledge that our current storyline does not emphasize the message we want to convey to the reader enough. While the vorticity budget analysis on its own is not special, we use it to investigate why Irma was able to remain an intense hurricane for so long. Specifically, we try to expose the important vorticity flux components that are crucial in sustaining Irma’s hollow PV tower structure in isentropic coordinates. Even though the impermeability theorem relies on the use of isentropic surfaces, few authors have examined these vorticity fluxes in tropical cyclones using this vertical coordinate. We shall incorporate these ideas more closely in our revised version of the article.
  We will include more relevant information on the model run in the article. Details about the turbulent length scale calculation according to the HARATU scheme are provided by (Lenderink and Holtslag, 2004; Bengtsson et al. 2017), but will be summarized in the appendix as they are vital to the results.
  We agree there are examples throughout the text that are not well explained. We will further elucidate on points that lack clarification and iterate through the text until all topics are explained more clearly to the reader.
  The conclusions shall be adapted such that they match the reframed research questions in the introduction. Some of the error sources will be addressed in more detail such as the turbulence calculations and radiative effects.
  Specific/minor comments:
  You are right about the paragraph on vortical hot towers, which is not necessary to mention as we are interested in the average (balanced) dynamics.
  Middle and Overworld is a term from Hoskins, 1991 (with far earlier origins). We will add references.
  The turbulence calculations are performed using the model output after simulation as the turbulent length scale has not been saved in the model output. Hence, we checked the outcome using the turbulent kinetic energy (which is a saved variable). The factor 100 is quite large so we will revise these computations. Possibly related to the note on eq. A6.
  In the introduction a few lines will be devoted to the validation of the model.
  More detail will be provided to the vertical interpolation procedure, specifically in cases where the atmosphere is hydrostatically unstable, which is mainly in the boundary layer but not exclusively. We shall provide further arguments for the validity of this method in these regions.
  The centre definition section will be improved for readability.
  It has been shown (Willoughby, 1990) that gradient wind balance is a very reasonable assumption on the azimuthally averaged wind field. We view the system as a whole to be in a state of global balance and believe convection to play a minor role.
  Scaling of the transverse velocity in fig 4 is indeed for visual purposes, this will be mentioned in the article. The arbitrary threshold of 0.006 is chosen for the same reason.
  The radius of maximum wind is a little far out as you mention. The reasons for this have not been investigated. We will do a check in existing literature to see if others have had similar issues and/or if it is worthwhile investigating further.
  Positive divergence leads to a decrease in absolute vorticity, so lines 217-218 indeed contain a typo (increase  decrease).
  References to region I and II/panel a and b in fig. 6 will be checked for mistakes.
  The comment on line 237 seems valid and we will revise the given explanation.
  The integrated vorticity flux components and tendency of absolute vorticity are an order of magnitude different. One important reason is that the fluxes are based on instantaneous wind fields at an hourly resolution, which are integrated over 36 time steps, while the model uses a far smaller time step. Any time derivative used in the calculations is therefore an approximation to the modelled value. This error accumulates over time.
  Line 247 might be confusing as the error due to turbulent diffusion is small. The error explained above likely plays a more important role.
  Comment on eq A6 is related to third specific comment. We will revise this section as needed. After the derivation an elaboration on the calculation of the turbulent length scale will be provided.
  With all changes above in consideration, as well as addressing minor points, we hope the upcoming revised version will live up to your expectation and make a more interesting contribution to our knowledge on Irma.
  Hoskins, B. J. (1991). Towards a PV-θ view of the general circulation. Tellus A: Dynamic Meteorology and Oceanography, 43(4), 27-36.
  Willoughby, H. E. (1990). Gradient balance in tropical cyclones. Journal of the Atmospheric Sciences, 47(2), 265-274.
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC4
RC4:
'Comment on egusphere-2023-1259', Anonymous Referee #3, 28 Sep 2023

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-1259/egusphere-2023-1259-RC4-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC4
- AC3: 'Reply on RC4', Jasper de Jong, 04 Dec 2023
  
  Thank you for your review, indicating weaknesses of the current article version. We acknowledge that the article unfortunately does not emphasize the reason for the investigation and the novelty of our results very well. We intend to explain why hurricane Irma was able to sustain its intense core above the frictional boundary layer. The intense core of Irma is reflected in an intense "vorticity cylinder” coinciding with the eye wall. This “vorticity cylinder” is maintained in a stationary state, despite divergent (outward) advective isentropic vorticity fluxes in the eye wall, which singly would very quickly reduce the intensity of Irma.
  An evaluation of the vorticity budget above the frictional boundary layer in isentropic coordinates explains the stationary intense phase of Irma. Due to the impermeability of isentropic surfaces to vorticity, this evaluation can be restricted to fluxes along isentropic surfaces. Therefore, as long as the isentropic surfaces lie in the “free atmosphere”, there is no need to take the frictional boundary layer into consideration. To our knowledge, a vorticity budget analysis in isentropic coordinates for a tropical cyclone has not yet been presented. From this analysis, we discover the approximate balance between the advective and diabatic radial vorticity fluxes along isentropic surfaces above the frictional boundary layer. We also find that the isentropic vorticity flux due to turbulence above the frictional boundary layer is very weak. Our study represents a new view on the stationary state of a mature and intense tropical cyclone. It is important to note that the balance between advective and diabatic vorticity fluxes works only with latent heating in a warm core balanced cyclone, not in a cold core balance cyclone. By balance we mean both in gradient wind balance and in hydrostatic balance, i.e. in thermal wind balance. In a warm core cyclone in thermal wind balance the tangential wind decreases with height, as is observed in hurricane Irma in its stationary state phase on September 6, 2017. A decreasing tangential wind with height is required to get an inward (relative) radial diabatic vorticity flux, which counters the advective vorticity flux.
  In our view, a mature tropical cyclone is basically in hydrostatic balance. Even though hydrostatic imbalance and attendant convection may occur on relatively small scale's, this is, we think, not of much importance to understand the quasi-balanced dynamics (growth, decay and stationary state) of a well developed tropical cyclone, which is governed by the existence of a secondary radial circulation needed to maintain gradient wind balance, in the presence of latent heating in the upward branch of the secondary circulation.
  You mention that the existence of a global steady state is controversial. While this is true and imposes issues in the boundary layer, the impermeability theorem dictates that any vorticity anomaly is due to vorticity fluxes along these surfaces. The effect of surface drag does not affect the vorticity flux balance in the higher atmosphere.
  With your points in mind, we hope to improve the article and make it a much more interesting and a sound piece of literature.
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC3

Status: closed (peer review stopped)

RC1:
'Comment on egusphere-2023-1259', Anonymous Referee #1, 21 Aug 2023

Please see the attached file.

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC1
- AC1: 'Reply on RC1', Jasper de Jong, 04 Dec 2023
  
  Thank you for your helpful comments. We acknowledge that the vorticity flux due to turbulence does not close the budget, especially when integrated over time. This might be in part due to disagreement between our offline turbulence calculation and that in the model (they should be close to identical but are not completely) and the lack of data to employ this calculation on (we only have hourly time steps and just one turbulence-related variable, turbulent kinetic energy, which we used to validate our offline calculation). We will investigate the former more closely for the next article version, while the latter is an inherent limitation to our research. Despite this limitation we believe that a vorticity flux due to turbulence derived from hourly data should give a fair representation of the actual flux because of the steady dynamics. If we do find our current vorticity flux due to turbulence to be accurate after revision, its negligible magnitude contributes greatly to remaining a strong vortex for so long, since the smoothening of the tangential wind profile you pointed out is strongly inhibited.
  We will work through all minor comments you provided and work on reframing our story. You ask whether we performed a quantitative analysis of the modelled cooling in the eye. Despite qualitatively looking at vertical moisture and temperature profiles, we did not perform such an analysis yet. We will incorporate this in the revised version.
  Specific comments:
  Items 1, 3, 4, 8, 9 and typos: we will rectify these issues.
  2: This has not been done, except a qualitative inspection of vertical profiles of temperature and humidity.
  5: This sentence refers to a panel that was there in an earlier draft, we will update the text.
  6: ‘increase’ should be ‘decrease’.
  7: Figure 5c does not show the total vorticity flux (not shown in the article) but its turbulent component. At 310 K, the maximum advective flux (fig 5a) is higher than the absolute minimum diabatic flux (fig 5b), leading to a (positive) maximum total flux near r=30km, neglecting the turbulent flux. Therefore, the radial flux divergence is positive (negative) inward (outward) from this radius (fig 6b). We will indicate figure numbers in this sentence for clarity.
  With your points in mind, we hope to improve the article and make it a much more interesting and sound piece of literature.
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC1
RC2:
'Comment on egusphere-2023-1259', Anonymous Referee #1, 21 Aug 2023

I appologize no reference in my specific comment#2. I revised the comment as follows:
L49-50: Did the authors quantitatively confirm the radiative cooling for the warm core extension? Previous studies indicated that adiabatic processes associated with subsidence in the eye can be a major contribution to the development of the warming in the eye (e.g., Stern and Zhang, 2013; Ohno and Satoh, 2015).

Reference
Ohno, T., and M. Satoh, 2015: On the warm core of a tropical cyclone formed near the tropopause. J. Atmos. Sci., 72, 551–571, doi:10.1175/JAS-D-14-0078.1.
Stern, D. P., and F. Zhang, 2013: Howdoes the eyewarm? Part I:Apotential temperature budget analysis of an idealized tropical cyclone. J. Atmos. Sci., 70, 73–90, doi:10.1175/JAS-D-11-0329.1.

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC2
- AC2: 'Reply on RC2', Jasper de Jong, 04 Dec 2023
  
  See 'Reply on RC1'
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC2
RC3:
'Comment on egusphere-2023-1259', Anonymous Referee #2, 28 Sep 2023

Please see attachment

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC3
- AC4: 'Reply on RC3', Jasper de Jong, 04 Dec 2023
  
  Thank you for your helpful comments. We acknowledge that our current storyline does not emphasize the message we want to convey to the reader enough. While the vorticity budget analysis on its own is not special, we use it to investigate why Irma was able to remain an intense hurricane for so long. Specifically, we try to expose the important vorticity flux components that are crucial in sustaining Irma’s hollow PV tower structure in isentropic coordinates. Even though the impermeability theorem relies on the use of isentropic surfaces, few authors have examined these vorticity fluxes in tropical cyclones using this vertical coordinate. We shall incorporate these ideas more closely in our revised version of the article.
  We will include more relevant information on the model run in the article. Details about the turbulent length scale calculation according to the HARATU scheme are provided by (Lenderink and Holtslag, 2004; Bengtsson et al. 2017), but will be summarized in the appendix as they are vital to the results.
  We agree there are examples throughout the text that are not well explained. We will further elucidate on points that lack clarification and iterate through the text until all topics are explained more clearly to the reader.
  The conclusions shall be adapted such that they match the reframed research questions in the introduction. Some of the error sources will be addressed in more detail such as the turbulence calculations and radiative effects.
  Specific/minor comments:
  You are right about the paragraph on vortical hot towers, which is not necessary to mention as we are interested in the average (balanced) dynamics.
  Middle and Overworld is a term from Hoskins, 1991 (with far earlier origins). We will add references.
  The turbulence calculations are performed using the model output after simulation as the turbulent length scale has not been saved in the model output. Hence, we checked the outcome using the turbulent kinetic energy (which is a saved variable). The factor 100 is quite large so we will revise these computations. Possibly related to the note on eq. A6.
  In the introduction a few lines will be devoted to the validation of the model.
  More detail will be provided to the vertical interpolation procedure, specifically in cases where the atmosphere is hydrostatically unstable, which is mainly in the boundary layer but not exclusively. We shall provide further arguments for the validity of this method in these regions.
  The centre definition section will be improved for readability.
  It has been shown (Willoughby, 1990) that gradient wind balance is a very reasonable assumption on the azimuthally averaged wind field. We view the system as a whole to be in a state of global balance and believe convection to play a minor role.
  Scaling of the transverse velocity in fig 4 is indeed for visual purposes, this will be mentioned in the article. The arbitrary threshold of 0.006 is chosen for the same reason.
  The radius of maximum wind is a little far out as you mention. The reasons for this have not been investigated. We will do a check in existing literature to see if others have had similar issues and/or if it is worthwhile investigating further.
  Positive divergence leads to a decrease in absolute vorticity, so lines 217-218 indeed contain a typo (increase  decrease).
  References to region I and II/panel a and b in fig. 6 will be checked for mistakes.
  The comment on line 237 seems valid and we will revise the given explanation.
  The integrated vorticity flux components and tendency of absolute vorticity are an order of magnitude different. One important reason is that the fluxes are based on instantaneous wind fields at an hourly resolution, which are integrated over 36 time steps, while the model uses a far smaller time step. Any time derivative used in the calculations is therefore an approximation to the modelled value. This error accumulates over time.
  Line 247 might be confusing as the error due to turbulent diffusion is small. The error explained above likely plays a more important role.
  Comment on eq A6 is related to third specific comment. We will revise this section as needed. After the derivation an elaboration on the calculation of the turbulent length scale will be provided.
  With all changes above in consideration, as well as addressing minor points, we hope the upcoming revised version will live up to your expectation and make a more interesting contribution to our knowledge on Irma.
  Hoskins, B. J. (1991). Towards a PV-θ view of the general circulation. Tellus A: Dynamic Meteorology and Oceanography, 43(4), 27-36.
  Willoughby, H. E. (1990). Gradient balance in tropical cyclones. Journal of the Atmospheric Sciences, 47(2), 265-274.
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC4
RC4:
'Comment on egusphere-2023-1259', Anonymous Referee #3, 28 Sep 2023

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-1259/egusphere-2023-1259-RC4-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2023-1259-RC4
- AC3: 'Reply on RC4', Jasper de Jong, 04 Dec 2023
  
  Thank you for your review, indicating weaknesses of the current article version. We acknowledge that the article unfortunately does not emphasize the reason for the investigation and the novelty of our results very well. We intend to explain why hurricane Irma was able to sustain its intense core above the frictional boundary layer. The intense core of Irma is reflected in an intense "vorticity cylinder” coinciding with the eye wall. This “vorticity cylinder” is maintained in a stationary state, despite divergent (outward) advective isentropic vorticity fluxes in the eye wall, which singly would very quickly reduce the intensity of Irma.
  An evaluation of the vorticity budget above the frictional boundary layer in isentropic coordinates explains the stationary intense phase of Irma. Due to the impermeability of isentropic surfaces to vorticity, this evaluation can be restricted to fluxes along isentropic surfaces. Therefore, as long as the isentropic surfaces lie in the “free atmosphere”, there is no need to take the frictional boundary layer into consideration. To our knowledge, a vorticity budget analysis in isentropic coordinates for a tropical cyclone has not yet been presented. From this analysis, we discover the approximate balance between the advective and diabatic radial vorticity fluxes along isentropic surfaces above the frictional boundary layer. We also find that the isentropic vorticity flux due to turbulence above the frictional boundary layer is very weak. Our study represents a new view on the stationary state of a mature and intense tropical cyclone. It is important to note that the balance between advective and diabatic vorticity fluxes works only with latent heating in a warm core balanced cyclone, not in a cold core balance cyclone. By balance we mean both in gradient wind balance and in hydrostatic balance, i.e. in thermal wind balance. In a warm core cyclone in thermal wind balance the tangential wind decreases with height, as is observed in hurricane Irma in its stationary state phase on September 6, 2017. A decreasing tangential wind with height is required to get an inward (relative) radial diabatic vorticity flux, which counters the advective vorticity flux.
  In our view, a mature tropical cyclone is basically in hydrostatic balance. Even though hydrostatic imbalance and attendant convection may occur on relatively small scale's, this is, we think, not of much importance to understand the quasi-balanced dynamics (growth, decay and stationary state) of a well developed tropical cyclone, which is governed by the existence of a secondary radial circulation needed to maintain gradient wind balance, in the presence of latent heating in the upward branch of the secondary circulation.
  You mention that the existence of a global steady state is controversial. While this is true and imposes issues in the boundary layer, the impermeability theorem dictates that any vorticity anomaly is due to vorticity fluxes along these surfaces. The effect of surface drag does not affect the vorticity flux balance in the higher atmosphere.
  With your points in mind, we hope to improve the article and make it a much more interesting and a sound piece of literature.
  
  Citation: https://doi.org/10.5194/egusphere-2023-1259-AC3

Jasper de Jong, Michiel L. J. Baatsen, and Aarnout J. van Delden

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Short summary

Tropical cyclones often embed a ring-shaped vorticity tower, instead of a centre maximum. Inspired to identify mechanisms in the conservation of such a vorticity structure, we examined the vorticity budget in a simulation of hurricane Irma (2017) near lifetime-peak intensity. Hurricane Irma persisted as a category five hurricane for three consecutive days. We find that vertical exchange of momentum by diabatic heating compensates the advective vorticity loss and eddy activity plays a minor role.


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