the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 23 Apr 2024
Impact of solar cycle on the non-linearity of the relationship between the solar wind parameters and geomagnetic conditions
Abstract. Solar wind and its transients drive the dynamics of Earth’s magnetosphere. Interplanetary coronal mass ejections (ICMEs) induce the largest variations in the near-Earth space, but significant geomagnetic activity can also be driven by high-speed streams (HSSs) and stream interaction regions (SIRs). Solar wind – magnetosphere interaction may lead to fluctuations in the inner magnetosphere and, hence, impact the electrons in the outer radiation belt. In this study, we use mutual information from information theory to study the change in the statistical dependence between solar wind parameters and inner magnetospheric indices including ultra low frequency (ULF) waves in the Pc5 range and electrons in the outer radiation belt during solar cycle 23 (1998–2008). Unlike Pearson correlation coefficient, mutual information can be used to investigate non-linear statistical dependencies between different parameters. We calculate linear and non-linear correlation coefficients separately for each year during solar cycle 23 and define the non-linearity with the ratio between the linear and non-linear correlation coefficients. We find that the non-linearity between solar wind speed and electron flux index is higher during solar maximum when most of the geomagnetic activity is driven by ICMEs, while the non-linearity decreases during the declining phase, when a larger portion of the geomagnetic activity is driven by HSSs and SIRs. On the other hand, IMF Bz and solar wind electric field Ey = VswBz have smaller non-linearity with the geomagnetic indices during time periods of stronger geomagnetic activity.
To investigate further if the change of the ratio of ICMEs and SIRs/HSSs as the driver of geomagnetic activity is the possible cause of the changes in the non-linearity during the solar cycle, we calculate the correlation coefficients separately during ICMEs, HSSs/SIRs and quiet solar wind. We find that non-linearity for solar wind speed and inner magnetospheric electron flux and ULF wave indices is smallest and correlations (both linear and non-linear) highest and therefore, the non-linearity is the lowest during the quiet time, while other studied solar wind parameters correlate better either during HSSs or ICMEs. These results show that the selected time period (phase of the solar cycle, dominant driver of the geomagnetic activity during the selected time) for the correlation analysis can significantly impact the results. Results also indicate that during ICMEs the solar wind – magnetosphere coupling becomes more non-linear for the majority of the studied solar wind–magnetospheric index parameter pairs (velocity, density, dynamic pressure) but IMF Bz and solar wind electric field Ey = VswBz have smaller non-linearity during time periods of stronger geomagnetic activity.
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RC1: 'Comment on angeo-2024-3', Anonymous Referee #1, 29 Apr 2024
Review of “Impact of solar cycle on the non-linearity of the relationship between the solar wind parameters and geomagnetic conditions” by Hoilijoki, Kilpua, Osmane, Turc, Savola, Lipsanen, George and Kalliokoski
This paper covers geomagnetic activity at the Earth and their interplanetary causes over a solar cycle. Studies like this have been done since the 1980s and all of the physical findings presented here have been noted before. As far as I can tell there is nothing particularly new here. If the authors feel that there is a point or two that is new, please focus on those one or two points and shorten the paper considerably.
The referencing is inadequate. References lack coverage over particularly well-developed magnetospheric areas. If included, the authors/readers will note that the conclusions the paper draws are not informative or even worse, incorrect.
Data sets. The data sets for ICMEs, CIRs and ULF waves should be given in Appendices and should be verified as being correct by the authors. This should be a stand-alone paper and the authors are responsible for the accuracy of their data sets.
I cannot recommend this paper for publication in anything resembling its current form.
Comments on Abstract
Abstract line 3 and elsewhere. “SIRs” everywhere should be called “CIRs” as in the discovery paper and naming in GRL, 3, 3, 137-140, 1976. The name “Corotating Interaction Regions” had to do with the shape of the interaction region as it comes outward from the Sun. There was no implication that it had to reappear 27 days later. Please read the paper.
Lines 5-7. It has been well established that magnetospheric relativistic electron enhancements are primarily found during the declining phase of the solar cycle where high speed solar winds coming from coronal holes impact the magnetosphere (Quant.Model. Magne. Proc., Geophys. Monogr. Ser., 21, edited by W. Olsen, p. 21, AGU, Washington, D. C. 1979; JGR, 111, A07S01, doi:10.1029/2005JA011273, 2006). The mechanism is acceleration of substorm ~90-100 keV electrons by chorus waves (GRL 25, 3011, 1998). The substorms are caused by IMF Bsouth components of Alfvén waves embedded in the high speed streams (PSS, 35, 405, 1987; GRL, 41, 1876, 2014; ApJ, 799:39, 2015 doi:10.1088/0004-637X/799/1/39; Ext. Ev. Geosp., 373-400, https://doi.org/10.1016/B978-0-12-812700-1.00014-5, 2018). This scenario is believed to be the main mechanism for the cause of enhanced relativistic electrons in the magnetosphere. This is not mentioned in the abstract, only the effect of PC5 pulsations. Balance should be given to this paper and a statement about the relationship between chorus and relativistic electrons should be mentioned in the abstract. Since chorus and ULF waves are competing mechanisms (see the JGR 2006 paper) for relativistic electron acceleration, your abstract should make some conclusions on this issue.
The radial diffusion model is not described in detail. What does it assume, incoherent wave-particle interactions or coherent ones? Please state in the abstract.
Lines 10-12, “non-linearity” results. Please state what this mean in physical terms. Interpret the phrase: “the non-linearity decreases during the declining phase, when a larger portion of the geomagnetic activity is driven by HSSs and CIRs”. Is the implication: in the declining phase the geomagnetic activity is driven by HSSs and CIRs”? If so this has been discovered before and this is nothing new. See the second paragraph above. “Most of the geomagnetic activity is driven by ICMEs” is also well, well known. This goes back to JGR, 99 A4, 5771, 1994 and has been verified by others many, many times already.
Line 14 Ey = Vsw Bz correlation results with magnetic storms has been shown in many, many publications before. One of the first is the 1994 paper stated previously. There is nothing new here.
Lines 15-16. The geoeffectiveness of ICMEs, HSSs and CIRs for SC23 have been studied and reported before. Two good examples are: JGR 112, A10102, doi:10.1029/2007JA012321, 2007; JGR 113, A05221, doi:10.1029/2007JA012744, 2008. What new information has your study produced, if any?
Lines 16-17. “the nonlinearity for solar wind and magnetospheric electron flux and ULF wave indices is smallest… and correlations highest” is not new. Please see the second paragraph in this section.
Lines 19-20. “These results show that the selected time period (phase of the solar cycle, dominant driver of the geomagnetic activity during the selected time) for the correlation analysis can significantly impact the results”. This has been known for a long time. Please delete.
Lines 20-24. What does this mean physically and is there any new findings here?
Comments, Main Part of Text
Line 25. A better reference is JGR, 99, A4, 5771-5792, 1994. Add this here.
Lines 29-30 and elsewhere. SIRs should be changed to CIRs everywhere in the paper and the GRL 1976 reference should be added on line 30.
Line 31, “weak and moderate geomagnetic activity”. JGR 100, A11, 21727-21733, 1995 should be added to the references.
Line 32. The JGR 1994 reference should be added here. Also a more recent reference on intense magnetic storms: JGRSP 124, 3926-3948, 2019 https://doi.org/10.1029/2018JA026425.
Lines 35-37. These last two sentences should be deleted.
Line 37. For “magnetospheric current changes” a better reference is JGR 73, 17, 5549-5559, 1968. For “ionospheric current changes” a better reference is JSWSC, 2021, 11, 23 https://doi.org/10.1051/swsc/2021001.
Lines 39-40 needs references. For “high energy particle fluxes” see some of the references given previously. For “plasma wave activity” add the references JGR, 82, 32, 5112-5128, 1977 and JGR 106, A7, 13,165–13,178, 2001. And the waves are “chorus” and not generally unknown waves. Please be specific.
Line 41. A more recent reference should be added here: ApJ 946:17, 2023 https://doi.org/10.3847/1538-4357/acb143. It should be noted that solar wind energy is both stored and then released and also directly injected. So your sentence needs revision.
Lines 41-43 needs a reference. The discovery paper is JGR 76, 16, 3587-3611, 1971. Most people standardly state that the substorm electron energies range from ~10 to 100 keV, not up to “hundreds of keVs”. Please correct.
Line 44. Add above chorus references to “wave modes” and add the GRL 1998 reference to “accelerate them up to relativistic (MeV) energies”
Line 46. For ULF waves add the references Springer Nature Switz 2020, Ency. Sol. Ear. Geophys. https://doi.org/10.1007/978-3-030-10475-7_156-1 and JGR 111, A07S01, doi:10.1029/2005JA011273, 2006.
Line 48. Please state the assumptions of the radial diffusion model. Is it coherent or incoherent wave-particle interactions? For fast radial diffusion the assumption must be coherent interactions. There is actually no evidence for coherence in ULF ground observations.
Lines 66-74. This section is inaccurate and is dated. ULF wave activity is believed to be substorm related and the substorms are caused by IMF Bz fluctuation in high speed streams. See the JGR 2006 review paper mentioned before. For substorms during high speed streams see the JGR 1995 paper mentioned before. In parallel with this, the substorms inject ANISOTROPIC ~10 to 100 keV electrons (Wave Inst. Spa Plas. 55-62, 1979, D. Reidel), which in turn generate chorus, which is believed to accelerate the high energy portion of the electrons to relativistic energies. This should be mentioned here as well.
Lines 76-78. This sentence gives no physical meaning. Please delete.
Lines 89-81. It has been shown in the JGR 1994 and the JGR 2008 papers that it is IMF Bsouth that causes geomagnetic storms. What does your statement of “more non linear” mean in this context? Does it add any new information?
Lines 82-83. This “more linear during solar max of Kp and VBs” is just saying that large southward IMF is causing magnetic storms during solar max. This is again nothing new.
Citation: https://doi.org/10.5194/angeo-2024-3-RC1 -
AC1: 'Reply on RC1', Sanni Hoilijoki, 30 Aug 2024
Dear Reviewer,
We thank the reviewer for taking time to review our manuscript. Below we go through the reviewer comments in detail. The original reviewer comments are in italics.
Review of “Impact of solar cycle on the non-linearity of the relationship between the solar wind parameters and geomagnetic conditions” by Hoilijoki, Kilpua, Osmane, Turc, Savola, Lipsanen, George and Kalliokoski
This paper covers geomagnetic activity at the Earth and their interplanetary causes over a solar cycle. Studies like this have been done since the 1980s and all of the physical findings presented here have been noted before. As far as I can tell there is nothing particularly new here. If the authors feel that there is a point or two that is new, please focus on those one or two points and shorten the paper considerably.
Thank you for reading the manuscript. Based on the reviewer’s comments, we realized that the abstract, introduction and concluding remarks were not clear enough about the novelty of our results. We will modify the abstract, introduction and the conclusions sections to address this issue. Indeed, it has been known for a long time how the solar wind, ICMEs and HSSs impact the Earth’s magnetosphere. Our intention is not to claim that these would be the new findings, and we have multiple references in the text referring to the related previous studies. The key novelty of this study is that we make a comprehensive application of an information theory tool called mutual information to investigate the changes in the non-linearity of the statistical dependence between solar wind and inner magnetospheric parameters during different phases of a solar cycle. While the solar cycle dependence has already been studied by Johnson et al., 2005, we use here a different data set and extend the existing work by including more solar wind and magnetospheric parameters (e.g. ULF wave index) and by separating the data between different solar wind structures (ICME/HSS/QUIET). We acknowledge that previous studies have investigated the statistical dependencies between the solar wind and the magnetosphere and the changes during different drivers. To our knowledge the non-linear dependencies have not been investigated before using a similar set of solar wind parameters and magnetospheric indices as used in this study applying mutual information (but rather with linear Pearson correlation coefficient).
The referencing is inadequate. References lack coverage over particularly well-developed magnetospheric areas. If included, the authors/readers will note that the conclusions the paper draws are not informative or even worse, incorrect.
We will add more references to previous relevant studies where needed. As noted above, the investigation of non-linear dependencies between solar wind and magnetospheric parameters with mutual information during different solar cycle phases and drivers is novel, and information theory tools in general are still relatively little used in the field of space physics. We believe that our results, now that we will significantly improve the presentation in the paper, will be useful for the community and inspire further studies.
Data sets. The data sets for ICMEs, CIRs and ULF waves should be given in Appendices and should be verified as being correct by the authors. This should be a stand-alone paper and the authors are responsible for the accuracy of their data sets.
The existing lists of ICMEs (Richardson, I. and Cane, H.: Near-Earth Interplanetary Coronal Mass Ejections Since January 1996, doi: 10.7910/DVN/C2MHTH,
2024) and SIRs (Grandin, M., et al., Properties and Geoeffectiveness of Solar Wind High-Speed Streams and Stream Interaction Regions During Solar Cycles 23 and 24, doi: 10.1029/2018JA026396, 2019) used in this study have been created for the science community to utilize them. These lists have been applied in other studies before this and they have gone through peer-review process (e.g. SIR/HSS Grandin et al., 2019). Many papers rely on existing datasets, and we cannot take credit from these lists by compiling our own for this paper. We also emphasize that in any way done, identification of ICMEs etc. has some level of subjectivity, but with statistical studies like this, the effects should be relatively small to overall conclusions. This is indeed a valid point, and we will now mention this in the text.I cannot recommend this paper for publication in anything resembling its current form.
Now that we will significantly improve the presentation in the paper, we believe that our paper and results will be useful for the community and inspire further studies.
Comments on Abstract
Abstract line 3 and elsewhere. “SIRs” everywhere should be called “CIRs” as in the discovery paper and naming in GRL, 3, 3, 137-140, 1976. The name “Corotating Interaction Regions” had to do with the shape of the interaction region as it comes outward from the Sun. There was no implication that it had to reappear 27 days later. Please read the paper.
The corotating stream interaction regions (Smith & Wolfe, 1976) are generated as the corotating high speed streams lead to the compression of the slower solar wind possibly repeating at approximately 27-day period if the coronal hole persists for multiple solar rotations. The corotating stream interaction regions generally refers to the same repeating structure. We want to be more general in our description and call them stream interaction regions not taking account how many times they are encountered as the sun rotates. This notation follows many previous peer-reviewed literature (e.g. Grandin et al. 2019, doi: 10.1029/2018JA026396 and Richardson, 2018, doi: 10.1007/s41116-017-0011-z). This description will be modified also in the Introduction to explain better why we chose to call these structures SIR instead of CIR.
Lines 5-7. It has been well established that magnetospheric relativistic electron enhancements are primarily found during the declining phase of the solar cycle where high speed solar winds coming from coronal holes impact the magnetosphere (Quant.Model. Magne. Proc., Geophys. Monogr. Ser., 21, edited by W. Olsen, p. 21, AGU, Washington, D. C. 1979; JGR, 111, A07S01, doi:10.1029/2005JA011273, 2006). The mechanism is acceleration of substorm ~90-100 keV electrons by chorus waves (GRL 25, 3011, 1998). The substorms are caused by IMF Bsouth components of Alfvén waves embedded in the high speed streams (PSS, 35, 405, 1987; GRL, 41, 1876, 2014; ApJ, 799:39, 2015 doi:10.1088/0004-637X/799/1/39; Ext. Ev. Geosp., 373-400, https://doi.org/10.1016/B978-0-12-812700-1.00014-5, 2018). This scenario is believed to be the main mechanism for the cause of enhanced relativistic electrons in the magnetosphere. This is not mentioned in the abstract, only the effect of PC5 pulsations. Balance should be given to this paper and a statement about the relationship between chorus and relativistic electrons should be mentioned in the abstract. Since chorus and ULF waves are competing mechanisms (see the JGR 2006 paper) for relativistic electron acceleration, your abstract should make some conclusions on this issue.
We agree with the referee on that chorus waves are considered as one of the main drivers of relativistic electron enhancements. In this paper we, however, do not investigate chorus waves or their impact on the electrons due to the lack of similarly comprehensive data sets as, for example, is the case for ULF waves. Therefore, their role in electron acceleration is not mentioned in the abstract as we present there only the main concepts relevant to this particular study. We do not claim in the manuscript that the ULF waves would be the only mechanism for accelerating electrons. We will add a short description of the role of the chorus waves in electron acceleration and precipitation in the Introduction with relevant references. As an example, we will refer to a review article by Ripoll et al. (doi: 10.1029/2019JA026735) discussing the important mechanism for electron acceleration in the radiation belts including local acceleration (via chorus waves) and radial transport via ULF waves. We also note that ULF waves have been found to modulate the intensity and excitation of chorus waves (e.g., Li et al. 2011, doi: 10.1029/2010JA016313, Xia et al., 2016, doi: 10.1002/2016GL070280) and precipitation of the energetic electrons (e.g., Brito et al, 2012, 2015). In addition, ULF waves can directly contribute to the acceleration of electrons to relativistic energies, and some studies also suggest direct precipitation of electrons by ULF waves (Yin et al. 2023, doi: 10.1029/2022JA031163).
The radial diffusion model is not described in detail. What does it assume, incoherent wave-particle interactions or coherent ones? Please state in the abstract.
In this study, we do not study radial diffusion or use radial diffusion model. The electron flux index for 130keV electrons is calculated directly from spacecraft measurement at the geostationary orbit (see Borovsky et al., 2017, doi: 10.1002/2017JA024250), not from radial diffusion model. Therefore, no radial diffusion model is described in the abstract nor in the rest of the manuscript.
Lines 10-12, “non-linearity” results. Please state what this mean in physical terms. Interpret the phrase: “the non-linearity decreases during the declining phase, when a larger portion of the geomagnetic activity is driven by HSSs and CIRs”. Is the implication: in the declining phase the geomagnetic activity is driven by HSSs and CIRs”? If so this has been discovered before and this is nothing new. See the second paragraph above. “Most of the geomagnetic activity is driven by ICMEs” is also well, well known. This goes back to JGR, 99 A4, 5771, 1994 and has been verified by others many, many times already.
We do not say that the geomagnetic activity being driven by ICMEs during solar maxima and by SIR/HSS during the declining phase are new findings, but they are likely the cause for the changes in the non-linear statistical dependence between solar wind parameters and inner magnetospheric parameters studied in this paper. For the solar wind speed and the 130 keV electron flux our results show that the Pearson correlation coefficient decreases during solar maxima while the adjusted correlation remains relatively at the same level. This indicates that while these two parameters share the same level of mutual information their linear dependence decreases and, therefore, their dependence becomes more non-linear. However, depending on the parameters investigated, the physical causes of changes in the non-linear dependence may vary. We will modify the introduction, abstract and discussion to describe better what the changes in the non-linearity can indicate.
Line 14 Ey = Vsw Bz correlation results with magnetic storms has been shown in many, many publications before. One of the first is the 1994 paper stated previously. There is nothing new here.
We fully agree with the reviewer that Ey is well known to correlate with magnetospheric activity proxies, and that is precisely why we selected this parameter in our study. References to previous studies using this parameter are listed at lines 165-168: “The solar wind parameters used in this study that are commonly accounted to be important in the solar wind-magnetosphere coupling are … represents the solar wind electric field in the y direction (e.g. Vasyliunas, 1975, Dungey, 1961; Newell et al., 2007; Wing et al., 2021).” We will clarify the abstract to highlight better the novel findings of this study.
Lines 15-16. The geoeffectiveness of ICMEs, HSSs and CIRs for SC23 have been studied and reported before. Two good examples are: JGR 112, A10102, doi:10.1029/2007JA012321, 2007; JGR 113, A05221, doi:10.1029/2007JA012744, 2008. What new information has your study produced, if any?
We would like to clarify here that we do not study the geoeffectiveness of ICMEs, HSSs and CIRs in this paper, but rather the changes in the non-linear statistical dependence between the solar wind drivers and inner magnetosphere over the solar cycle and during different solar wind structures. We will rewrite parts of the abstract to highlight and summarize better and more clearly our aims and the new information found in this study.
Lines 16-17. “the nonlinearity for solar wind and magnetospheric electron flux and ULF wave indices is smallest… and correlations highest” is not new. Please see the second paragraph in this section.
We will reformulate to better clarify the novelty of our findings.
Lines 19-20. “These results show that the selected time period (phase of the solar cycle, dominant driver of the geomagnetic activity during the selected time) for the correlation analysis can significantly impact the results”. This has been known for a long time. Please delete.
We will clarify and rewrite the conclusions to highlight the novelty of the results.
Lines 20-24. What does this mean physically and is there any new findings here?
It is difficult to point to exact physical meanings behind statistical results as they do not reveal the causality. Some possible physical meanings will be discussed in the Discussion.
Comments, Main Part of Text
Line 25. A better reference is JGR, 99, A4, 5771-5792, 1994. Add this here.
Thank you for suggesting this reference, we will add it after the sentence.
Lines 29-30 and elsewhere. SIRs should be changed to CIRs everywhere in the paper and the GRL 1976 reference should be added on line 30.
We will add the reference on the suggested line. We will explain in this paragraph why we chose to call these structures SIR instead of CIR similarly as done by Grandin et al. 2019 (doi: 10.1029/2018JA026396) and Richardson 2018 (doi: 10.1007/s41116-017-0011-z).
Line 31, “weak and moderate geomagnetic activity”. JGR 100, A11, 21727-21733, 1995 should be added to the references.
We will add the suggested reference.
Line 32. The JGR 1994 reference should be added here. Also a more recent reference on intense magnetic storms: JGRSP 124, 3926-3948, 2019 https://doi.org/10.1029/2018JA026425.
We will add the suggested reference.
Lines 35-37. These last two sentences should be deleted.
We will rephrase these two sentences to explain better why we chose to call these structures SIR instead of CIR similarly as done by Grandin et al. 2019 (doi: 10.1029/2018JA026396) and Richardson 2018 (doi: 10.1007/s41116-017-0011-z).
Line 37. For “magnetospheric current changes” a better reference is JGR 73, 17, 5549-5559, 1968. For “ionospheric current changes” a better reference is JSWSC, 2021, 11, 23 https://doi.org/10.1051/swsc/2021001.
The first suggested reference discusses DP 2 fluctuations, and the second suggested reference discusses ground induced currents. Both are in our opinion too specific compared to the general statements made in this part of the introduction (we discuss ionospheric currents in a general manner and not focusing necessarily on geomagnetically-induced currents. Therefore, we believe that the references currently included in the text are better suited.
Lines 39-40 needs references. For “high energy particle fluxes” see some of the references given previously. For “plasma wave activity” add the references JGR, 82, 32, 5112-5128, 1977 and JGR 106, A7, 13,165–13,178, 2001. And the waves are “chorus” and not generally unknown waves. Please be specific.
We will add more references here. We are not stating that the plasma waves are unknown, but we do not specify chorus waves as the only wave activity in the magnetosphere as during geomagnetic activity also other magnetospheric plasma wave modes intensify including ULF waves, EMIC waves and hiss (e.g., Elkington et al, 2003,doi:10.1029/2001JA009202, Meredith et al., 2003, doi:10.1029/2002JA009700).
Line 41. A more recent reference should be added here: ApJ 946:17, 2023 https://doi.org/10.3847/1538-4357/acb143. It should be noted that solar wind energy is both stored and then released and also directly injected. So your sentence needs revision.
In this part we are specifically describing the substorm process and not the division of solar wind energy input to the magnetosphere. Even this suggested citation describes that “some solar wind energy is stored in the magnetotail system”. The description of substorms in our manuscript is more general and the McPherron reference covers the basic information that we are writing about. The suggested reference about the supersubstorms is more specific and, therefore, outside of the scope of our study.
Lines 41-43 needs a reference. The discovery paper is JGR 76, 16, 3587-3611, 1971. Most people standardly state that the substorm electron energies range from ~10 to 100 keV, not up to “hundreds of keVs”. Please correct.
Thank you for pointing out the lack of references at this point. However, the suggested reference describes hot plasma being injected during substorms not specifically electrons. We will add citations to studies by Baker et al. 1986 (doi:10.1029/JA091iA04p04265), Cayton et al. 1989 (doi:10.1029/GL016i002p00147), Ingraham et al. 2001 (doi:10.1029/2000JA000458), and Turner et al. 2015 (doi: 10.1002/2015GL063225) that are saying that the substorm injection electron energies range from 10 to hundreds of keVs and is not limited to 100keV.
Line 44. Add above chorus references to “wave modes” and add the GRL 1998 reference to “accelerate them up to relativistic (MeV) energies”
Thank you for the suggestion, we will add the GRL 1998 reference in addition to the paper by Shprits, et al. 2008 (doi: 10.1016/j.jastp.2008.06.008) showing that also ULF waves accelerate electrons to relativistic energies.
Line 46. For ULF waves add the references Springer Nature Switz 2020, Ency. Sol. Ear. Geophys. https://doi.org/10.1007/978-3-030-10475-7_156-1 and JGR 111, A07S01, doi:10.1029/2005JA011273, 2006.
We think that the list of existing references is more comprehensive than the first suggested citation that is not focused on ULF waves. The latter suggested reference does not even mention ULF waves. Therefore, they will not be added at this point.
Line 48. Please state the assumptions of the radial diffusion model. Is it coherent or incoherent wave-particle interactions? For fast radial diffusion the assumption must be coherent interactions. There is actually no evidence for coherence in ULF ground observations.
In this study, we do not study radial diffusion or use any radial diffusion model. The electron flux index for 130keV electrons is calculated directly from spacecraft measurement at the geostationary orbit (see Borovsky et al., 2017), not from radial diffusion model. Therefore, the radial diffusion model is not described in the abstract or in the rest of the manuscript.
Lines 66-74. This section is inaccurate and is dated. ULF wave activity is believed to be substorm related and the substorms are caused by IMF Bz fluctuation in high speed streams. See the JGR 2006 review paper mentioned before. For substorms during high speed streams see the JGR 1995 paper mentioned before. In parallel with this, the substorms inject ANISOTROPIC ~10 to 100 keV electrons (Wave Inst. Spa Plas. 55-62, 1979, D. Reidel), which in turn generate chorus, which is believed to accelerate the high energy portion of the electrons to relativistic energies. This should be mentioned here as well.
We will add a short description in the Introduction on the role of the chorus waves in the inner magnetosphere. However, we are not going to describe them in more detail as they are out of scope of this study. Regarding the relationship between the ULF waves and solar wind parameters we have extended the Introduction and added some new references. ULF waves can indeed be generated in the magnetosphere by several mechanisms, one of which is substorm activity. The different generation mechanisms of toroidal and poloidal ULF wave modes in the inner magnetosphere are already described in the Introduction of the paper (lines 50-63) with appropriate references: “The toroidal components are usually observed with large extent in the azimuthal direction (small wave number m) and they have been found to be driven by external sources i.e. direct driving by the solar wind (e.g. dynamic pressure pulses, Kelvin-Helmholtz instability driven by the velocity shear, reconnection and formation and propagation of flux transfer events at the dayside magnetopause) (e.g. Hudson et al., 2004; Claudepierre et al., 2008). Poloidal modes have shorter azimuthal extent, and they have been found to be driven by internal sources for example, generation via magnetospheric processes such as bounce-drift resonance and other wave-particle interactions (e.g. Southwood et al., 1969).”
Lines 76-78. This sentence gives no physical meaning. Please delete.
We will rephrase the sentence to be more understandable and have physical meaning:
“For example, the results by Hoilijoki et al. (2022) indicated that the statistical dependence between the AE index and electron precipitation at different MLT sectors is more non-linear during the solar maximum year 2004 than during the solar minimum year 2007.”
Lines 89-81. It has been shown in the JGR 1994 and the JGR 2008 papers that it is IMF Bsouth that causes geomagnetic storms. What does your statement of “more non linear” mean in this context? Does it add any new information?
In this particular part of the Introduction, we are briefly describing the results of the study. We are not saying that the results reveal the driver of the geomagnetic activity. We do cite many previous studies showing that the HSS and ICMEs drive geomagnetic activity and the ICMEs drive the most intense geomagnetic storms. In this paper, we investigated how the statistical dependence between the solar wind parameters (known from previous studies to be important for geomagnetic activity) and inner magnetospheric parameters and activity indices varies over the solar cycle and during ICMEs, HSS and quiet solar wind periods. In the statistical analysis we compare the linear Pearson correlation coefficient to the non-linear adjusted correlation coefficient obtained from mutual information as is described in Section 2 of the paper. To the knowledge of the authors, the parameters chosen and separating the data to different solar wind drivers have not been done before using mutual information tools, and hence, our study provides new information. Our results both validate the previous statistical results and add to the previous work by expanding the data sets used. We will modify the Introduction and Discussion sections describing better the possible physical meaning of the non-linear dependence between the solar wind and magnetosphere.
Lines 82-83. This “more linear during solar max of Kp and VBs” is just saying that large southward IMF is causing magnetic storms during solar max. This is again nothing new.
This sentence is referring to the paper by Johnson and Wing (2005) that was mentioned in the previous sentence. “They concluded that the non-linearities in the Kp index and solar wind driving with VBs…”. This is not presented as a new finding of our study. We will reformulate this sentence to clarify this.
On behalf of the authors,
Sanni Hoilijoki
Citation: https://doi.org/10.5194/angeo-2024-3-AC1
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AC1: 'Reply on RC1', Sanni Hoilijoki, 30 Aug 2024
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RC2: 'Comment on angeo-2024-3', Anonymous Referee #2, 28 May 2024
This paper discusses non-linearities between several solar wind and magnetospheric parameters during solar cycle 23, making use of mutual information based on information theory. The results suggest that the non-linearity depends on the phase of the solar cycle and the type of solar wind driver (interplanetary coronal mass ejections, ICMEs) or stream interaction regions/high speed streams, SIRs/HSSs). The paper is generally well-written, but I have a few comments/suggestions that the authors should consider. In particular, there are many minor errors in the text, some of which are indicated below. Also, aside from an exercise in statistics, it’s not clear how this work might be applied to our understanding of solar wind-magnetospheric coupling. Perhaps the authors could discuss this further?
Specific comments:
The order of authors’ first/last names does not follow the usual publishing convention.
Line 18: There are several long sentences in the text and some would benefit from more accurate punctuation or splitting into short sentences. For example, here, adding a comma would make the meaning clearer: … lowest, during quiet times…
Line 20: Suggest: can significantly impact the results of the correlation analysis. Also: ICMEs, (comma added).
Line 23: Suggest: whereas [rather than but].
Line 26: The Richardson and Cane reference is the only one to include the authors’ first names.
Line 50: divided into
Line 51: radial magnetic [spelling]
Line 63: Brackets around reference.
Lines 70-71: Suggest: activity that more frequently injects the few tens of keV electrons that provide a source for further acceleration.
Line 72: effects on
Line 82: cycles
Line 86: the majority
Line 88: Suggest: discussed in Section 4 and the conclusions are summarized in Section 5.
Line 92: If the authors consider "ICMEs" to include post-shock sheaths, there will be other events in which just the sheath is encountered that are not in the R+C list. Would these then contribute to the “quiet time” sample?
Line 94: compiled by
Line 101: obtained from
Line 110: are of particular
Equations 1-4: Only equation 3 has log_2 indicated but from equation 5, it appears that all the logs should be base 2. Is this correct? If so, presumably all the equations should indicate log_2.
Line 147: An explanation of the “surrogate data” would be useful – it’s unclear to the reader what this refers to.
Figure 1 caption: This refers to events that are “driving [unspecified] geomagnetic activity” but Section 2 suggests that all the events in the ICME or SIR/HSS catalogs are included with no requirement related to geomagnetic activity. So, is there a selection here?
Line2 157-9: “Most of the geomagnetic activity …” - This isn't shown in the figure if it’s just a count of events with no selection with respect to geomagnetic activity. Also, if there is a selection then "most of the geomagnetic activity" could be defined more carefully, e.g., does this refer to storms that exceed a threshold in a certain geomagnetic index?
Line 163: Is there any reason for the 169 hr limit, or is it arbitrary?
Line 167: Is this Ey or -Ey? Is V_sw a scalar, in which case this is an approximate relationship (there’s a cross product involved), or does V_sw mean the x component of V? Also, how is the sign of Bz treated in the correlations, e.g., the result will be different if just southward (geoeffective) intervals are considered vs. including both north and south/positive and negative values in the correlation. Also, there are places where Bs is mentioned, which only includes the southward field, while Bz is used in other places. Are the authors making a distinction here? Furthermore, is it a problem in this type of analysis that the magnetospheric parameter may not be expected to be correlated with the instantaneous value of the solar wind parameter, even allowing for a time shift to get the maximum correlation. E.g., Dst, though not considered here, depends predominantly on the time history of Ey, not the instantaneous value, and can still be large (negative) when B has returned northward. So correlating time series of Ey and Dst doesn’t really make sense, though some studies do so. Are there similar issues with the parameters selected?
Line 168: Brackets for references.
Line 198: This sentence has no verb. Also, it mentions solar wind density, whereas the preceding sentence and figure indicate V_sw.
Lines 210-11: 2003 was dominated by strong HSSs, though this isn’t reflected in the number of SIRs/HSS in Figure 1 which doesn’t tell us about the strength of the streams. Is this responsible for the peak? There are also the CME-driven Halloween storms of course, but these might not be dominant in annual averages.
Line 270 and following: Can the statistical results really be accounted for in terms of the detailed effects (e.g., due to the sheath or ejecta) discussed in this paragraph, or are these "smeared out" by considering many events, which seems more likely? If so, what is the impact of the results on understanding the relationship between the solar and magnetospheric parameters?
Line 286 and following: Similarly, the discussion largely interprets the results in terms of what is already known, but doesn't really demonstrate that this interpretation is correct. I worry that annualized results including many events may not "behave" as expected based on ideas from individual events.
Citation: https://doi.org/10.5194/angeo-2024-3-RC2 -
AC2: 'Reply on RC2', Sanni Hoilijoki, 30 Aug 2024
Dear Reviewer,
We thank the reviewer for taking time to review our manuscript. Below we go through the reviewer comments in detail. The original reviewer comments are in italics.
This paper discusses non-linearities between several solar wind and magnetospheric parameters during solar cycle 23, making use of mutual information based on information theory. The results suggest that the non-linearity depends on the phase of the solar cycle and the type of solar wind driver (interplanetary coronal mass ejections, ICMEs) or stream interaction regions/high speed streams, SIRs/HSSs). The paper is generally well-written, but I have a few comments/suggestions that the authors should consider. In particular, there are many minor errors in the text, some of which are indicated below. Also, aside from an exercise in statistics, it’s not clear how this work might be applied to our understanding of solar wind-magnetospheric coupling. Perhaps the authors could discuss this further?
Thank you for your careful review of our paper and pointing out the incomplete discussion on the importance of the results on the solar wind-magnetosphere coupling. We will add paragraphs to the Introduction and Discussion section discussing the possible implications and prospects of the results of our study. We will add more text to the introduction and the discussion along the lines of the following:
In the introduction:
Correlation of solar wind parameters and geomagnetic activity have been studied intensively for decades. Different coupling functions have been created to find the highest correlation between solar wind parameters and geomagnetic activity. Coupling functions are usually combinations of solar wind velocity and interplanetary magnetic field with varying powers (e.g., Newell et al., 2007, doi:10.1029/2006JA012015, Borovsky et al., 2008, doi:10.1029/2007JA012646). Guo et al. (2011, 10.1029/2011JA016490) showed that Borovsky and Newell coupling functions perform differently during the sheath and ICME-driven geomagnetic storms that feature different dynamic pressure and Mach number values. These changes in the performance of the coupling function could indicate changes in the non-linear dependence between the solar wind and inner magnetosphere during varying solar wind conditions. Non-linear statistical dependence and the changes in the non-linearity can also be seen in the saturation of certain values like cross polar cap potential as a function of solar wind Ey. Wilder et al., (2011, doi: 10.1029/2011JA016924) showed that the saturation of the non-linearity of the cross polar cap potential is dependent on the Alfvénic Mach number. A study by Myllys et al. (2016, doi: 10.1002/2016JA022407) shows that the efficiency of the coupling functions to predict the geomagnetic indices and cross polar cap potential can be dependent on the Mach number. Changes in the coupling efficiency under different solar wind drivers indicate changes in the non-linear dependence between the solar wind parameters and geomagnetic activity. The causes for the changes in the non-linearity may lie at the dayside reconnection that may become less efficient during low Mach number as the magnetosheath flow is able to divert around the magnetopause (Lavraud and Borovsky, 2008, doi: 10.1029/2008JA013192). On the other hand, during high solar wind pressure the high compression of the magnetopause may cause the X-line to shorten decreasing the coupling efficiency, even though the driving electric field increases (e.g., Raeder and Lu, 2005, doi: 10.1016/j.asr.2004.05.010). The cause of the non-linearity may also lie in the coupling of multiple physical processes in the inner magnetosphere.
In the Discussion:
Correlation analysis can be used to help deducing the main physical mechanism at play in solar wind-magnetosphere coupling. For example, the relatively steady non-linear dependence (and adjusted correlation of ~0.7) over the solar cycle and during different solar wind structures between dynamic pressure and Tgeo suggest that the compressions of the magnetosphere are the physical process connecting these two parameters. However, when the system becomes more complicated and multiple physical processes are working simultaneously, the coupling becomes more non-linear. The high correlation between Bz and AE can be interpreted in a straightforward manner as the manifestation of the Dungey cycle, as enhanced dayside reconnection leads to nightside reconnection, which is then measured as an increase in the auroral electrojet index. The correlation between Bz and AE is high and independent of the solar cycle and solar wind conditions. The AE index is also always highly (and quite linearly) correlated with the ULF wave indices. However, the correlation and non-linear dependence between solar wind Bz and ULF indices shows a dependence on the solar cycle and different solar wind drivers. This suggest that different physical processes are at play connecting Bz to AE and AE to Tgeo/gr at different times.
In this paper, we expanded the parameter set to study the changes in the non-linear dependence over one solar cycle. Our results validated the behavior of previously widely investigated parameters (Bz, Ey) but our key aim was to obtain new results regarding non-linearities in dependencies between different parameters. To the knowledge of the authors, the investigation of the non-linear statistical dependencies under different solar wind structures has not been investigated before. In order to improve the space weather forecasts it is important to know the (non-linear) statistical dependencies and how they change during different solar wind conditions. The non-linear dependence of solar wind speed and interplanetary magnetic field with proxies of magnetospheric activity (e.g. AE) also seem to have an opposite dependence on the solar cycle. Our results indicate that phase of the solar cycle might also need to be accounted for in the coupling functions and models predicting geomagnetic activity.
Specific comments:
The order of authors’ first/last names does not follow the usual publishing convention.
We will fix the author's names to follow the convention.
Line 18: There are several long sentences in the text and some would benefit from more accurate punctuation or splitting into short sentences. For example, here, adding a comma would make the meaning clearer: … lowest, during quiet times…
Thank you for pointing this out. We will add the suggested comma here and also try to shorten/add punctuation to long sentences throughout the paper.
Line 20: Suggest: can significantly impact the results of the correlation analysis. Also: ICMEs, (comma added).
We will include the suggested changes.
Line 23: Suggest: whereas [rather than but].
We will make the suggested correction.
Line 26: The Richardson and Cane reference is the only one to include the authors’ first names.
We will fix this to match the rest of the references.
Line 50: divided into
We will make the suggested correction.
Line 51: radial magnetic [spelling]
We will correct this.
Line 63: Brackets arond reference.
Ww will correct the brackets.
Lines 70-71: Suggest: activity that more frequently injects the few tens of keV electrons that provide a source for further acceleration.
We will make the suggested change.
Line 72: effects on
Line 82: cycles
Line 86: the majority
We will make the corrections suggested above.
Line 88: Suggest: discussed in Section 4 and the conclusions are summarized in Section 5.
We will make the suggested change.
Line 92: If the authors consider "ICMEs" to include post-shock sheaths, there will be other events in which just the sheath is encountered that are not in the R+C list. Would these then contribute to the “quiet time” sample?
If the events are not in these two lists or belong to the one-day recovery period after the solar wind events, they are considered quiet time. We, however, note that R+C list for example contains also several weaker ICMEs and edge encountered cases, giving therefore a relatively comprehensive coverage of ICME-related plasma and field in the solar wind. Some “driverless” shocks / sheaths cases could have indeed been missed in the analysis, but their impact is likely negligible, considering that our data set covers an entire solar cycle. We will now mention this in the text.
Line 94: compiled by
Line 101: obtained from
Line 110: are of particular
We will make the corrections suggested above.
Equations 1-4: Only equation 3 has log_2 indicated but from equation 5, it appears that all the logs should be base 2. Is this correct? If so, presumably all the equations should indicate log_2.
Yes, all equations are in base 2. We will correct the equations in the paper.
Line 147: An explanation of the “surrogate data” would be useful – it’s unclear to the reader what this refers to.
We will clarify in the text that by the surrogate data we mean the randomly shuffled data of the original data sets used in the study. Discretization of the data sets will introduce a small error to the mutual information. We shuffle the original data 100 times and calculate MI for each (surrogate) data set. We calculate the standard deviation and the average from 100 shuffle test and the average MI is defined as the zero baseline. If the mutual information from the original data set is larger than the zero baseline+3*standard deviation, the result is considered significant.
Figure 1 caption: This refers to events that are “driving [unspecified] geomagnetic activity” but Section 2 suggests that all the events in the ICME or SIR/HSS catalogs are included with no requirement related to geomagnetic activity. So, is there a selection here?
The reviewer is correct, this is indeed confusing. We include all the events in the ICME and SIR/HSS lists regardless of the impact on the geomagnetic activity levels. We will correct this in the paper.
Line2 157-9: “Most of the geomagnetic activity …” - This isn't shown in the figure if it’s just a count of events with no selection with respect to geomagnetic activity. Also, if there is a selection then "most of the geomagnetic activity" could be defined more carefully, e.g., does this refer to storms that exceed a threshold in a certain geomagnetic index?
We will correct this in the paper. We mean the count of events, there is no threshold for geomagnetic activity.
Line 163: Is there any reason for the 169 hr limit, or is it arbitrary?
The correct hour limit is 168h which is one week, this will be corrected in the paper. We chose to use up to a week of delay because the information transfer from solar wind parameters to the inner magnetosphere between some parameters can peak with delays up to few days (e.g., Wing et al., 2016, 2022, doi: 10.1002/2016JA022711, doi: 10.1029/2021JA030246). We will add this same explanation to the paper.
Line 167: Is this Ey or -Ey? Is V_sw a scalar, in which case this is an approximate relationship (there’s a cross product involved), or does V_sw mean the x component of V? Also, how is the sign of Bz treated in the correlations, e.g., the result will be different if just southward (geoeffective) intervals are considered vs. including both north and south/positive and negative values in the correlation. Also, there are places where Bs is mentioned, which only includes the southward field, while Bz is used in other places. Are the authors making a distinction here? Furthermore, is it a problem in this type of analysis that the magnetospheric parameter may not be expected to be correlated with the instantaneous value of the solar wind parameter, even allowing for a time shift to get the maximum correlation. E.g., Dst, though not considered here, depends predominantly on the time history of Ey, not the instantaneous value, and can still be large (negative) when B has returned northward. So correlating time series of Ey and Dst doesn’t really make sense, though some studies do so. Are there similar issues with the parameters selected?
V_sw is the scalar flow speed from OMNIdata, whereas for the calculation of Ey we used the Vx component. We will correct this in the paper. For Bz both negative and positive (sign) values are included in the correlation calculations. Bs is mentioned only in reference to previous work, but in our study we only used Bz.
We are using hourly averaged data, and do not consider longer time history of the solar wind parameters. One hour averaging evens out the instantaneous fluctuations of the solar wind parameters. However, the geomagnetic activity proxies we use (SYM-H and AE) are intrinsically 1-minute indices that should respond relatively quickly to variations in solar wind driving. For example, Myllys et al. (2017, doi: 10.1002/2017GL072676) indicated that correlations between AE and solar wind Ey peaks at 46-60 minutes lag, while for SYM-H the lag is a bit larger, about 120 minutes (2 hours). Myllys et al. (2016, doi: 10.1002/2016JA022407) showed that increasing the time window for correlation between solar wind parameters and geomagnetic indices increases the correlation, but the high correlation is spread over wide area of different lengths of time windows for a specific time shift. In their study Myllys et al. (2016) used time windows only up to 120 min, therefore, we think that one hour average is long enough to filter out the smallest fluctuations that would otherwise decrease the correlations. We will add a short discussion of this and its possible effect on our results in the text. In a future study we could explore further how varying the length of time window impacts the correlation results.
Line 168: Brackets for references.
We will correct the brackets.
Line 198: This sentence has no verb. Also, it mentions solar wind density, whereas the preceding sentence and figure indicate V_sw.
We added a verb and corrected density to speed.
Lines 210-11: 2003 was dominated by strong HSSs, though this isn’t reflected in the number of SIRs/HSS in Figure 1 which doesn’t tell us about the strength of the streams. Is this responsible for the peak? There are also the CME-driven Halloween storms of course, but these might not be dominant in annual averages.
In 2003 the average solar wind speed remained high throughout the whole year (e.g. Hynönen et al., 2020, doi: 10.1051/swsc/2020046). Also the levels of Pc5 ULF wave activity of the interplanetary magnetic field and the ground-based ULF waves maintained a high monthly average during 2003 compared to other years during the solar cycle. It is likely that these unusually high solar wind speeds and ULF wave levels are the cause of the small peak in the non-linearity. We will add this discussion to the paper.
Line 270 and following: Can the statistical results really be accounted for in terms of the detailed effects (e.g., due to the sheath or ejecta) discussed in this paragraph, or are these "smeared out" by considering many events, which seems more likely? If so, what is the impact of the results on understanding the relationship between the solar and magnetospheric parameters?
The referee is right that the results cannot be used to identify the impact of ICME substructures as we did not study the correlations separately for sheath and ejecta. We will rewrite this part of the Discussion and remove the description of the different impacts seen caused by sheath and ejecta as we cannot separate those. This is an interesting idea for a future study. We will rewrite parts of the abstract, Introduction and Conclusions to discuss more about the impact and novelty of the results.
Line 286 and following: Similarly, the discussion largely interprets the results in terms of what is already known, but doesn't really demonstrate that this interpretation is correct. I worry that annualized results including many events may not "behave" as expected based on ideas from individual events.
Thank you for bringing up this concern. We will add more discussion about the interpretations, impacts and novelty of the results. However, demonstrating the exact physical causes behind the statistical results is not a straightforward task to do. Even investigating correlation during individual events can vary from one another. Therefore, making accurate conclusions out of statistical results is complicated. For example, statistical results do not tell about the causality. These results are needed as preliminary step to see when and between what parameters high dependencies either linear or non-linear come up. For the next steps of future study, one can include causal effects, for example, applying causal interference.
On behalf of the authors,
Sanni Hoilijoki
Citation: https://doi.org/10.5194/angeo-2024-3-AC2
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AC2: 'Reply on RC2', Sanni Hoilijoki, 30 Aug 2024
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