the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Observed wavenumberfrequency spectrum of global, normal mode function decomposed, fields: a possible evidence for nonlinear effects on the wave dynamics
Abstract. The study of tropical tropospheric disturbances has led to important challenges from both observational and theoretical points of view. In particular, the observed wavenumberfrequency spectrum of tropical oscillations, also known as WheelerKiladis diagram, has helped bridging the gap between observations and the linear theory of equatorial waves. Here we have obtained a similar wavenumberfrequency spectrum for each equatorial wave type by performing a normal mode function (NMF) decomposition of global EraInterim reanalysis data, with the NMF basis being given by the eigensolutions of the primitive equations in spherical coordinates, linearized around a resting background state. In this methodology, the global multilevel horizontal velocity and geopotential height fields are projected onto the normal mode functions characterized by a vertical mode, a zonal wavenumber, a meridional quantum index and a mode type, namely Rossby, Kelvin, mixed Rossbygravity and westward and eastward propagating inertiogravity modes. The horizontal velocity and geopotential height fields associated with each mode type are then reconstructed on the physical space, and the corresponding wavenumberfrequency spectrum is calculated for the 200 hPa zonal wind. The results reveal some expected structures, such as the dominant globalscale Rossby and Kelvin waves constituting the intraseasonal frequency associated with the MaddenJulian Oscillation. On the other hand, some unexpected features such as westward propagating Kelvin waves and eastward propagating westward inertiogravity waves are also revealed by our observed 200 hPa zonal wind spectrum. These intriguing behaviours represent a large departure from the linear equatorial wave theory and can be a result of strong nonlinearities in the wave dynamics.
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RC1: 'Comment on wcd202121', Anonymous Referee #1, 23 May 2021
âReview of Observed wavenumberfrequency spectrum of global, normal mode function decomposed, fields: a possible evidence for nonlinear effects on the wave dynamics âby André Seiji Wakate Teruya et al.â, WCD MS No.: wcd202121Summary:âThis is an interesting paper that combines linear wave theory on the sphere using MODES software of Å½agar et al. 2015 with linear time filtering using the Fourier time series decomposition by Wheeler and Kiladis 1999 software.The paper is a combination of two parts. The first part presents outputs of the normalmode function decomposition of ERAInterim data by MODES, that has been explored in the previous studies by the same group. The second and main part of the paper applies the WheelerKiladis software to frequency decompose time series of Rosby and gravity modes including the equatorial Kelvin and mixed Rossbygravity waves.Not surprisingly, the authors find that the WheelerKiladis diagrams of 200 hPa zonal wind differ from expectations based on the linear propagation of wave signals. Unfortunately, the results provide little new understanding of dynamics. Conclusions that discrepancies between linear theory and WheelerKiladis diagrams must come from nonlinear processes can hardly be called a new result. It is therefore the opinion of this review that the paper is not publishable in WCD.Further comments are provided.Detailed comments:The authors combine the normalmode function decomposition of ERAInterim circulation âby MODES software to identify global circulation in terms of linear modes. This is a powerful method which was successfully applied in several earlier studies, including Å½agar et al. (2009, Mon Wea Rev), Castanheira and Marques (2015, QJRMS), Å½agar and Franzke (2015, GRL), Marques and Castanheira (2018, Math. Geosci.), Blaauw and Å½agar (2018, ACP), Kitsios et al. (2019, J Atmos Sci), Raphaldini et al (2020, GRL), etc. The classical method has recently experienced a revitalisation by the work of groups at Universities of Tsukuba, Aveiro and Hamburg. It has to be noted that this approach isolates the spatial structure of Rosby and gravity waves by assuming the basic state of rest. This is likely one of the reasons that associated frequencies from linear theory have not been much explored.There are a number of lacking definitions and inconsistencies in sections 23. For example, the expansion of ERAInterim data in section 3.1 is presented by infinite time series in all 3 directions. The parameter \alpha which distinguishes the wave types is not clearly defined in relation to meridional mode. The vertical mode m=0 is used in section 3 as the barotropic mode but omitted in the results. Even with all math corrected, it can be questioned if one needs a detailed repetition of the theory of normalmode function expansion from Å½agar et al. (2015) or Kasahara's earlier papers. A shorter summary would probably suffice in application studies like here.Equatorial wave filtering by Wheeler and Kiladis (1999, J Atmos Sci) and its variants or similar procedures, for example Roundy (2004, J Atmos Sci), Gehne and Kleeman (2012, J Atmos Sci), combine linear filtering in time and longitude to produce frequencywavenumber diagrams of equatorial waves.The application of the WheelerKiladis diagrams (WK) in this study is described very briefly, in a single paragraph section 3.2. It omits important details of the default setup of the software and their implications on the results. The authors refer almost exclusively to the previous WK analysis of OLR data, but relevant application studies are much wider and include reanalysis data and climate model simulations. If the work is continued or resubmitted, it would be useful to refer to previous studies using WK with the zonal wind and other dynamical variables.My main comment on the results is about the lack of discussion of the eastwardpropagating signals in WK diagrams using 200 hPa wind from the westwardpropagating linear modes, and vice versa. This is striking in all figures, and especially the fact that most of the signal has the barotropic tropospheric structure, such as in figure 7b.For example, how can one understand results of section 4.6 and figure 13 on mixed Rossbygravity waves? The authors say that they analyzed the zonal wind at 200 hPa, which is expected to be rather small for the mixed Rossbygravity mode, but the amplitude of its variance in figure 13 exceeds the variance in any other plot, including the total zonal wind. I assume that the same scaling is applied in all WK diagrams, as the amplitudes are not discussed and colour bars not explained, but I may be wrong. These things should be paid attention to.For a similar result with westward gravity modes, the authors write in the final discussion section that "a combination of moistconvective and nonlinear processes might explain the spectral peaks associated with equatorial Rossby waves, barotropic Rossby waves, MJO, and Kelvin waves found in the observed wavenumberfrequency spectrum of the westward inertiogravity wave field." Similarly, "gravity waves often have their propagation "slaved" to other modes" . This is appropriately a final discussion section, as there are no clear new results or explanation why WK diagrams look the way they do in this case.In the case of MJO, Kitsios et al. (2019, J Atmos Sci) used coherence of various modes to explain interactions of modes. Such or similar effort might be a way to proceed for a better process understanding.Equivalent depths are often missing in WK diagrams and are not mentioned in the discussion. For example, it is not mentioned what equivalent depth corresponds to the strongest variance in the eastwardpropagating sector that is associated with the Kelvin wave. A variance signal that looks like tha of the Kelvin wave appears in the analysis of other modes, even in the mixed Rossbygravity mode (figure 13b), but it is less obvious in the analysis of Kelvin wave zonal wind in figure 11. This should be explained.Visualisation of energy in Fig. 4,6,8,10,12 is unusual and it is unclear why these figures are relevant and what one should learn from them.âCitation: https://doi.org/
10.5194/wcd202121RC1 
AC1: 'Reply on RC1', Andre Teruya, 04 Jun 2021
In this reply we provide some preliminary responses to the main issues raised by Reviewer 1. Reviewer 1 raised someimportant questions, and we are confident to be able to address all of them. Also, we recognize that the incorporation ofthe revisions discussed here on the manuscript will significantly improve its quality. We provide here some elements on thetheory regarding the nonlinear correction of the linear eigenfrequencies, the socalled Stokes correction. We were planning toevoke this theory to explain the observational results presented in the present paper in a follow up article. However, in faceof the comments of Reviewer 1, we feel appropriate to introduce the Stokes correction theory here. We think that by furtherdiscussing a specific nonlinear mechanism to explain the departures of the observed time frequencies from their expectedvalues predicted by the linear theory might make more clear the original contribution of the present paper.The revisions discussed here are not to be considered as the final reply to all comments; we still plan to elaborate moreon them, particularly on the statistical tests on the coherence analysis. The intention here is to show that we are able toprovide a satisfactory answer to the main issues, while other minor issues will be addressed in the final answer, should we begiven the opportunity to do so.
The reply including the figures and mathematical formulas is attached in the pdf file.

AC1: 'Reply on RC1', Andre Teruya, 04 Jun 2021

RC2: 'Comment on wcd202121', Anonymous Referee #2, 26 May 2021
This is a good contribution that uses two methods of analyzing proagating signals in the atmosphere, Hough normal mode decomposition and Wheeler Kiladis (WK) diagnostics. The analyses uncover some surprising results such as propagation of some linear normal modes in the opposite sense of their linar modal propagation and strong suggestions of slaving and nonlinear wave breaking. These results are independent of my request for a major revision which is related to the comparative utility of the WK analyses. I question why the authors have used different normalizations in each of the WK diagrams shown in manuscript. This choice of normalizing each energy with the background red noise spectrum for each individual modally filtered projection determines that the WK diagram for the Kelvin wave with all vertical modes included is NOT the superposition of the Kelvin wave WK diagrams of the barotropic and baroclinic parts. It also hides the true energy amplitude of every WK diagram. I believe this study would be much improved if either a uniform normalization was used for all the WK analyses or, at least, a uniform background normalization for each mode type (Rot, EIG, WG , MRG, K).
Very minor editorial suggestions :
Line 10 replace 'on the' with 'in'
Line 22 replace 'among others, each of these topics with crucial' with 'among others. With each of these topics there are associated crucial'
Line 143 The vertical resolution of the ERAI data should also be noted
Line 215 m=15 should planetaryscale not synopticscale
Line 227 states 'From Fig. 4 one observes that the spectrum is dominated by disturbances with large spatial scales (k = 1−5) and a barotropic structure in the troposphere m = 1−5, which agrees with the spatial structure displayed in Figure 2 that exhibits most part of total energy concentrated in the subtropical jets.' but Figure 2 shows that the subtropical jets are mostly baroclinic in structure. Which is it?
Citation: https://doi.org/10.5194/wcd202121RC2 
RC3: 'Comment on wcd202121', Anonymous Referee #3, 05 Jun 2021
General comments:
The present study investigates the zonal wavenumberfrequency spectrum of zonal wind field (at 200 hPa) as reconstructed for each type of wave (Rossby, westward/eastward inertia gravity, Kelvon, and mixed Rossby waves) based on the normal mode decomposition approach. The authors attribute the deviation of the observed spectrum from the theoretical expected behavior to nonlinear processes.
I would agree that this approach may be useful for classifying and understanding of the equatorial wave spectrum from a different perspective. However, I have a major concern about the interpretation of the results. Especially, (as mentioned in Introduction of this manuscript) the authors need to be aware that the normal modes used for this study are obtained under a quite ideal condition: motionless and frictionless atmosphere that has a rigid lid at the upper boundary. So, at least for me, it is no surprising that the results are different from the expected mode behavior. I would like to suggest that the authors make it clear what is the advantage of using this approach under such ideal assumption and that the interpretation of the results be carefully reexamined. Please see below for specific comments.
 One should note that the normal modes used here are obtained under a quite “ideal” condition: i.e., motionless and frictionless atmosphere without meridional gradient in background temperature and with a rigid lid at the upper boundary. In the real world, these assumptions are of course not valid: for example, even a presence of background zonal wind modifies the mode shape; there is no upper boundary (in this case the vertical structure equation is no more a StrumLiouville problem). Thus, it is no surprising that for example large Kelvin wave signals in real data contaminate other wave spectra such as WIG and EIG (Sections 4.3 and 4.4). I think not only nonlinear processes (see also my comment #3 for this point) but also other quite basic processes such as again the presence of background wind and dissipation processes, which could be considered within a “linear” theory, are all responsible for the deviation from the ideal mode behavior.
 The 2D spectrum results are all shown as the ratio to the background spectrum. This could be misleading if the background spectrum is very small: i.e., Even if the signals appear to be significant for the ratio, the actual wave energy could be very small. I suggest that the authors should also show and examine the raw (or background) spectrum as well. For instance, I would suspect that the background spectrum for the eastward components in Fig. 7 (ideal WIG modes) may be small.
 (Following my comment #1) The authors’ arguments about the nonlinear interaction (e.g., L264267, L326330) sound just a speculation and not so convincing and I think some more quantitative analyses and/or discussion are necessary. For L265267, for example, most readers including me may not be familiar with plasma physics and I cannot understand why such specific process (of other many processes) is considered the most important. I would guess, again, that there are many possibilities for the deviation from the ideal normal modes (i.e., background wind etc.).
Citation: https://doi.org/10.5194/wcd202121RC3
Status: closed

RC1: 'Comment on wcd202121', Anonymous Referee #1, 23 May 2021
âReview of Observed wavenumberfrequency spectrum of global, normal mode function decomposed, fields: a possible evidence for nonlinear effects on the wave dynamics âby André Seiji Wakate Teruya et al.â, WCD MS No.: wcd202121Summary:âThis is an interesting paper that combines linear wave theory on the sphere using MODES software of Å½agar et al. 2015 with linear time filtering using the Fourier time series decomposition by Wheeler and Kiladis 1999 software.The paper is a combination of two parts. The first part presents outputs of the normalmode function decomposition of ERAInterim data by MODES, that has been explored in the previous studies by the same group. The second and main part of the paper applies the WheelerKiladis software to frequency decompose time series of Rosby and gravity modes including the equatorial Kelvin and mixed Rossbygravity waves.Not surprisingly, the authors find that the WheelerKiladis diagrams of 200 hPa zonal wind differ from expectations based on the linear propagation of wave signals. Unfortunately, the results provide little new understanding of dynamics. Conclusions that discrepancies between linear theory and WheelerKiladis diagrams must come from nonlinear processes can hardly be called a new result. It is therefore the opinion of this review that the paper is not publishable in WCD.Further comments are provided.Detailed comments:The authors combine the normalmode function decomposition of ERAInterim circulation âby MODES software to identify global circulation in terms of linear modes. This is a powerful method which was successfully applied in several earlier studies, including Å½agar et al. (2009, Mon Wea Rev), Castanheira and Marques (2015, QJRMS), Å½agar and Franzke (2015, GRL), Marques and Castanheira (2018, Math. Geosci.), Blaauw and Å½agar (2018, ACP), Kitsios et al. (2019, J Atmos Sci), Raphaldini et al (2020, GRL), etc. The classical method has recently experienced a revitalisation by the work of groups at Universities of Tsukuba, Aveiro and Hamburg. It has to be noted that this approach isolates the spatial structure of Rosby and gravity waves by assuming the basic state of rest. This is likely one of the reasons that associated frequencies from linear theory have not been much explored.There are a number of lacking definitions and inconsistencies in sections 23. For example, the expansion of ERAInterim data in section 3.1 is presented by infinite time series in all 3 directions. The parameter \alpha which distinguishes the wave types is not clearly defined in relation to meridional mode. The vertical mode m=0 is used in section 3 as the barotropic mode but omitted in the results. Even with all math corrected, it can be questioned if one needs a detailed repetition of the theory of normalmode function expansion from Å½agar et al. (2015) or Kasahara's earlier papers. A shorter summary would probably suffice in application studies like here.Equatorial wave filtering by Wheeler and Kiladis (1999, J Atmos Sci) and its variants or similar procedures, for example Roundy (2004, J Atmos Sci), Gehne and Kleeman (2012, J Atmos Sci), combine linear filtering in time and longitude to produce frequencywavenumber diagrams of equatorial waves.The application of the WheelerKiladis diagrams (WK) in this study is described very briefly, in a single paragraph section 3.2. It omits important details of the default setup of the software and their implications on the results. The authors refer almost exclusively to the previous WK analysis of OLR data, but relevant application studies are much wider and include reanalysis data and climate model simulations. If the work is continued or resubmitted, it would be useful to refer to previous studies using WK with the zonal wind and other dynamical variables.My main comment on the results is about the lack of discussion of the eastwardpropagating signals in WK diagrams using 200 hPa wind from the westwardpropagating linear modes, and vice versa. This is striking in all figures, and especially the fact that most of the signal has the barotropic tropospheric structure, such as in figure 7b.For example, how can one understand results of section 4.6 and figure 13 on mixed Rossbygravity waves? The authors say that they analyzed the zonal wind at 200 hPa, which is expected to be rather small for the mixed Rossbygravity mode, but the amplitude of its variance in figure 13 exceeds the variance in any other plot, including the total zonal wind. I assume that the same scaling is applied in all WK diagrams, as the amplitudes are not discussed and colour bars not explained, but I may be wrong. These things should be paid attention to.For a similar result with westward gravity modes, the authors write in the final discussion section that "a combination of moistconvective and nonlinear processes might explain the spectral peaks associated with equatorial Rossby waves, barotropic Rossby waves, MJO, and Kelvin waves found in the observed wavenumberfrequency spectrum of the westward inertiogravity wave field." Similarly, "gravity waves often have their propagation "slaved" to other modes" . This is appropriately a final discussion section, as there are no clear new results or explanation why WK diagrams look the way they do in this case.In the case of MJO, Kitsios et al. (2019, J Atmos Sci) used coherence of various modes to explain interactions of modes. Such or similar effort might be a way to proceed for a better process understanding.Equivalent depths are often missing in WK diagrams and are not mentioned in the discussion. For example, it is not mentioned what equivalent depth corresponds to the strongest variance in the eastwardpropagating sector that is associated with the Kelvin wave. A variance signal that looks like tha of the Kelvin wave appears in the analysis of other modes, even in the mixed Rossbygravity mode (figure 13b), but it is less obvious in the analysis of Kelvin wave zonal wind in figure 11. This should be explained.Visualisation of energy in Fig. 4,6,8,10,12 is unusual and it is unclear why these figures are relevant and what one should learn from them.âCitation: https://doi.org/
10.5194/wcd202121RC1 
AC1: 'Reply on RC1', Andre Teruya, 04 Jun 2021
In this reply we provide some preliminary responses to the main issues raised by Reviewer 1. Reviewer 1 raised someimportant questions, and we are confident to be able to address all of them. Also, we recognize that the incorporation ofthe revisions discussed here on the manuscript will significantly improve its quality. We provide here some elements on thetheory regarding the nonlinear correction of the linear eigenfrequencies, the socalled Stokes correction. We were planning toevoke this theory to explain the observational results presented in the present paper in a follow up article. However, in faceof the comments of Reviewer 1, we feel appropriate to introduce the Stokes correction theory here. We think that by furtherdiscussing a specific nonlinear mechanism to explain the departures of the observed time frequencies from their expectedvalues predicted by the linear theory might make more clear the original contribution of the present paper.The revisions discussed here are not to be considered as the final reply to all comments; we still plan to elaborate moreon them, particularly on the statistical tests on the coherence analysis. The intention here is to show that we are able toprovide a satisfactory answer to the main issues, while other minor issues will be addressed in the final answer, should we begiven the opportunity to do so.
The reply including the figures and mathematical formulas is attached in the pdf file.

AC1: 'Reply on RC1', Andre Teruya, 04 Jun 2021

RC2: 'Comment on wcd202121', Anonymous Referee #2, 26 May 2021
This is a good contribution that uses two methods of analyzing proagating signals in the atmosphere, Hough normal mode decomposition and Wheeler Kiladis (WK) diagnostics. The analyses uncover some surprising results such as propagation of some linear normal modes in the opposite sense of their linar modal propagation and strong suggestions of slaving and nonlinear wave breaking. These results are independent of my request for a major revision which is related to the comparative utility of the WK analyses. I question why the authors have used different normalizations in each of the WK diagrams shown in manuscript. This choice of normalizing each energy with the background red noise spectrum for each individual modally filtered projection determines that the WK diagram for the Kelvin wave with all vertical modes included is NOT the superposition of the Kelvin wave WK diagrams of the barotropic and baroclinic parts. It also hides the true energy amplitude of every WK diagram. I believe this study would be much improved if either a uniform normalization was used for all the WK analyses or, at least, a uniform background normalization for each mode type (Rot, EIG, WG , MRG, K).
Very minor editorial suggestions :
Line 10 replace 'on the' with 'in'
Line 22 replace 'among others, each of these topics with crucial' with 'among others. With each of these topics there are associated crucial'
Line 143 The vertical resolution of the ERAI data should also be noted
Line 215 m=15 should planetaryscale not synopticscale
Line 227 states 'From Fig. 4 one observes that the spectrum is dominated by disturbances with large spatial scales (k = 1−5) and a barotropic structure in the troposphere m = 1−5, which agrees with the spatial structure displayed in Figure 2 that exhibits most part of total energy concentrated in the subtropical jets.' but Figure 2 shows that the subtropical jets are mostly baroclinic in structure. Which is it?
Citation: https://doi.org/10.5194/wcd202121RC2 
RC3: 'Comment on wcd202121', Anonymous Referee #3, 05 Jun 2021
General comments:
The present study investigates the zonal wavenumberfrequency spectrum of zonal wind field (at 200 hPa) as reconstructed for each type of wave (Rossby, westward/eastward inertia gravity, Kelvon, and mixed Rossby waves) based on the normal mode decomposition approach. The authors attribute the deviation of the observed spectrum from the theoretical expected behavior to nonlinear processes.
I would agree that this approach may be useful for classifying and understanding of the equatorial wave spectrum from a different perspective. However, I have a major concern about the interpretation of the results. Especially, (as mentioned in Introduction of this manuscript) the authors need to be aware that the normal modes used for this study are obtained under a quite ideal condition: motionless and frictionless atmosphere that has a rigid lid at the upper boundary. So, at least for me, it is no surprising that the results are different from the expected mode behavior. I would like to suggest that the authors make it clear what is the advantage of using this approach under such ideal assumption and that the interpretation of the results be carefully reexamined. Please see below for specific comments.
 One should note that the normal modes used here are obtained under a quite “ideal” condition: i.e., motionless and frictionless atmosphere without meridional gradient in background temperature and with a rigid lid at the upper boundary. In the real world, these assumptions are of course not valid: for example, even a presence of background zonal wind modifies the mode shape; there is no upper boundary (in this case the vertical structure equation is no more a StrumLiouville problem). Thus, it is no surprising that for example large Kelvin wave signals in real data contaminate other wave spectra such as WIG and EIG (Sections 4.3 and 4.4). I think not only nonlinear processes (see also my comment #3 for this point) but also other quite basic processes such as again the presence of background wind and dissipation processes, which could be considered within a “linear” theory, are all responsible for the deviation from the ideal mode behavior.
 The 2D spectrum results are all shown as the ratio to the background spectrum. This could be misleading if the background spectrum is very small: i.e., Even if the signals appear to be significant for the ratio, the actual wave energy could be very small. I suggest that the authors should also show and examine the raw (or background) spectrum as well. For instance, I would suspect that the background spectrum for the eastward components in Fig. 7 (ideal WIG modes) may be small.
 (Following my comment #1) The authors’ arguments about the nonlinear interaction (e.g., L264267, L326330) sound just a speculation and not so convincing and I think some more quantitative analyses and/or discussion are necessary. For L265267, for example, most readers including me may not be familiar with plasma physics and I cannot understand why such specific process (of other many processes) is considered the most important. I would guess, again, that there are many possibilities for the deviation from the ideal normal modes (i.e., background wind etc.).
Citation: https://doi.org/10.5194/wcd202121RC3
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