the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Fluid Models Capturing FarleyBuneman Instabilities
Enrique Rojas
Keaton Burns
David Hysell
Abstract. It is generally accepted that modeling FarleyBuneman instabilities require resolving ion Landau damping to reproduce experimentally observed features. Particleincell (PIC) simulations have been able to reproduce most of these, but at a computational cost that severely affects their scalability. This limitation hinders the study of nonlocal phenomena that require three dimensions or coupling with largerscale processes. We argue that a form of the fivemoment fluid system can recreate several qualitative aspects of FarleyBuneman dynamics such as density and phase speed saturation, wave turning, and heating. Unexpectedly, these features are still reproduced even without using artificial viscosity to capture Landau damping. Comparing the proposed fluid models and a PIC implementation shows good qualitative agreement.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
 Preprint
(2082 KB)  BibTeX
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 Final revised paper
Journal article(s) based on this preprint
Enrique Rojas et al.
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221264', Anonymous Referee #1, 05 Jan 2023
Lines 6667: Why did the authors artificially increase the electron mass? This should only apply to models that need to resolve the plasma frequency, which is what places the constraint on the simulation time step. If the authors simply wanted to mimic the parameter values used in previous works, they may say so. Otherwise, they should justify this choice.
Lines 7879: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastestgrowing wave modes dominant other than the fact that they're growing fastest?
Line 79: What is the geometry of E and B? It would help the reader to not have to refer to previous work in order to determine the ExB direction.
Lines 7983: Have the authors made any attempt to quantify the angle of deflection from ExB? Comparing that angle between regularized and unregularized models, as well as to previous results from PIC simulations, would support the authors' main argument. The presentation would benefit from a figure that shows the 2D wavenumber spectrum corresponding to each stage.
Line 82: The note about "a slight turning of the waves in the direction consistent with linear theory" shouldn't apply to the turbulent regime. Did the authors mean to associate this with the linear (or possibly mixing) regime?
Have the authors considered changing the resolution while maintaining the simulation domain size, to see if the dominant wavelength changes?
Citation: https://doi.org/10.5194/egusphere20221264RC1 
AC1: 'Reply on RC1', Enrique Rojas Villalba, 20 Jan 2023
Thank you for your questions and comments. Here you will find our replies.
Lines 6667: Why did the authors artificially increase the electron mass? This should only apply to models that need to resolve the plasma frequency, which is what places the constraint on the simulation time step. If the authors simply wanted to mimic the parameter values used in previous works, they may say so. Otherwise, they should justify this choice.
Reply: We chose the same simulation parameters as Oppenheim(2008) to compare our results to theirs.Lines 7879: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastestgrowing wave modes dominant other than the fact that they're growing fastest?
Reply: We just wanted to point out that there is a clear dominant mode in the linear regime, but we can see now that this was redundant because the dominant mode, by definition, grows faster.Line 79: What is the geometry of E and B? It would help the reader to not have to refer to previous work in order to determine the ExB direction.
Reply: The electric and magnetic fields are parallel to y and z, respectively. We have added this clarification to the text.Lines 7983: Have the authors made any attempt to quantify the angle of deflection from ExB? Comparing that angle between regularized and unregularized models, as well as to previous results from PIC simulations, would support the authors' main argument. The presentation would benefit from a figure that shows the 2D wavenumber spectrum corresponding to each stage.
Reply: The main goal of this paper is to show that the fivemoment fluid system can remain stable and capture several qualitative features of the FarleyBuneman instabilities. We think showing the small but visible tilt in the density structures from Figure 1 is enough for a qualitative argument. A systematic quantitative analysis of this and other metrics is a topic we are currently working on, but it will be published in the future.
Line 82: The note about "a slight turning of the waves in the direction consistent with linear theory" shouldn't apply to the turbulent regime. Did the authors mean to associate this with the linear (or possibly mixing) regime?
Reply: Even though the analytical estimations of the waveturning effect have been derived for the linear regime, we are assuming that the preferential direction criteria should still hold approximately as the system goes through saturation. Nevertheless, we agree that the text is ambiguous, so we have modified it to restrict the comment for the linear regime.Have the authors considered changing the resolution while maintaining the simulation domain size, to see if the dominant wavelength changes?
Reply: As mentioned before, a more systematic evaluation of diagnostics and numerics is part of a current project. And although we have done this analysis and haven't seen a significant change, we decided not to elaborate on this because we were focusing on the qualitative aspects of the results. One of the main contributions of this work is to challenge the standard assumption that simulating FarleyBuneman instabilities in the fluid domain would quickly produce smallscale structures growing faster than the larger ones. In our simulations, we see a dominant mode, so the dependence between growth rate and wavenumber is not monotonically increasing as the standard linear theory predicts. Even though the exact growth rate could change with the resolution of the system, we are assuming that it will maintain the same general behavior.Citation: https://doi.org/10.5194/egusphere20221264AC1 
RC2: 'Reply on AC1', Anonymous Referee #1, 24 Jan 2023
I am satisfied that the authors have sufficiently addressed my concerns.
Citation: https://doi.org/10.5194/egusphere20221264RC2 
AC3: 'Reply on RC2', Enrique Rojas Villalba, 28 Feb 2023
Thank you.
Citation: https://doi.org/10.5194/egusphere20221264AC3

AC3: 'Reply on RC2', Enrique Rojas Villalba, 28 Feb 2023

RC2: 'Reply on AC1', Anonymous Referee #1, 24 Jan 2023

AC1: 'Reply on RC1', Enrique Rojas Villalba, 20 Jan 2023

RC3: 'Comment on egusphere20221264', Anonymous Referee #2, 13 Feb 2023
Lines 2819: The authors used Hassan's artificial viscosity, including thermal effects. Could the authors elaborate more on the following:
 What thermal effects do the authors refer to in the "including thermal effects" statement?
 Why do the authors treat the stress tensor divergence as "artificial viscosity"?
 Did the authors learn about the role that the stress tensor divergence plays as a good sink for the energy sources to stratify the conservation of energy in Hassan’s 2015 fluid model as shown in Figure 8 in Hassan E, Hatch DR, Morrison PJ, Horton W. Multiscale equatorial electrojet turbulence: Energy conservation, coupling, and cascades in a baseline 2‐D fluid model. Journal of Geophysical Research: Space Physics. 2016 Sep;121(9):912745?
 Could the inclusion of the energy equation in the fluid model remove the need for the divergence of stress tensor closure?
 Including the thermal effects in studying the FarleyBunemann instability in the aurora region is important, but how important is the inclusion of the thermal effects in simulations at the equatorial electrojet?
Lines 6972: The authors stated that using a smaller simulation box size will not influence the nonlinear features, and the scalability assumption is still preserved. Could the authors justify the following concerns:
 Using a small simulation box size couldn’t justify the authors’ claim about the scalability of the fluid model as running simulations in larger boxes shouldn’t use large computational resources.
 Using a small simulation box limits the destabilization of modes of longer wavelength by energy cascading which might the comparison of the presented results in this manuscript and both the PIC model in Oppenheim et al. 2008 and Hassan et al. 2015.
Figure 2: Would the authors explain the difference in the ion and electron heating in the unregularized and regularized cases while having the rootmeansquare of the density and electric field almost the same?
Citation: https://doi.org/10.5194/egusphere20221264RC3 
AC2: 'Reply on RC3', Enrique Rojas Villalba, 28 Feb 2023
Thank you for your questions and comments. Here you will find our replies.
Lines 2829: The authors used Hassan's artificial viscosity, including thermal effects. Could the authors elaborate more on the following:
What thermal effects do the authors refer to in the "including thermal effects" statement?
Reply: We meant to say that the system solved in this work includes an energy equation, therefore, relaxing the isothermal assumption. We now see that "including thermal effects" might mislead the reader into thinking that Hassan did not include a pressure term in the momentum equation. We have modified the text to be more explicit about this.Why do the authors treat the stress tensor divergence as "artificial viscosity"?
Reply: Given that the precise form of the term was not justified and its similarity to the neutral viscosity term, it is our impression that this term was added as a numerical proxy for Landau damping. Nevertheless, we now think that it would be more appropriate just to mentioned that this was the operator proposed by the author. The text has now been changed accordingly.
Did the authors learn about the role that the stress tensor divergence plays as a good sink for the energy sources to stratify the conservation of energy in Hassan's 2015 fluid model as shown in Figure 8 in Hassan E, Hatch DR, Morrison PJ, Horton W. Multiscale equatorial electrojet turbulence: Energy conservation, coupling, and cascades in a baseline 2‐D fluid model. Journal of Geophysical Research: Space Physics. 2016 Sep;121(9):912745?
Reply: Yes. Even though the viscosity term used in that work is important for weaklyionized plasmas, it is not crucial for the paper's central argument. We want to show that the fluid equations without any viscosity (either physical or numerical) can avoid a growth rate of k^2.Could the inclusion of the energy equation in the fluid model remove the need for the divergence of stress tensor closure?
Reply: As we mentioned in line 93, we observed a similar behavior using an isothermal system, namely, no k^2 growth and a dominant band of wave numbers similar to the PIC simulation. Nevertheless, you are right to assume that the energy equation provides an additional source of stabilization to the system. A more comprehensive numerical analysis of these terms is a topic of current work.
Including the thermal effects in studying the FarleyBunemann instability in the aurora region is important, but how important is the inclusion of the thermal effects in simulations at the equatorial electrojet?
Reply: Even though we don't expect thermal effects to dominate in the equatorial electrojet, they could be relevant if the experiments are precise enough to measure the details of its dynamics. Nevertheless, this paper aimed not to argue that the presented model is more realistic but to compare its results with PIC simulations.Lines 6972: The authors stated that using a smaller simulation box size will not influence the nonlinear features, and the scalability assumption is still preserved. Could the authors justify the following concerns:
Using a small simulation box size couldn't justify the authors' claim about the scalability of the fluid model as running simulations in larger boxes shouldn't use large computational resources.
Reply: As we mentioned in the conclusions, our claim of computational efficiency relies on the assumption that fluid simulations are computationally cheaper than kinetic simulations if the parameters are similar and kinetic effects are not required.Using a small simulation box limits the destabilization of modes of longer wavelength by energy cascading which might the comparison of the presented results in this manuscript and both the PIC model in Oppenheim et al. 2008 and Hassan et al. 2015.
Reply: Even though the energy cascading to longer wavelengths is an issue, the paper's primary goal is to show that a simple fluid system avoids the k^2 growth rate. Our argument for scalability is mainly related to the computational complexity of the kinetic solvers. Nevertheless, we have added one sentence to comment on the long wavelength issue.Figure 2: Would the authors explain the difference in the ion and electron heating in the unregularized and regularized cases while having the rootmeansquare of the density and electric field almost the same?
Reply: This might be related to the regularized system restricting the velocity magnitude, making the collisional heating term in the energy equation smaller.Citation: https://doi.org/10.5194/egusphere20221264AC2
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221264', Anonymous Referee #1, 05 Jan 2023
Lines 6667: Why did the authors artificially increase the electron mass? This should only apply to models that need to resolve the plasma frequency, which is what places the constraint on the simulation time step. If the authors simply wanted to mimic the parameter values used in previous works, they may say so. Otherwise, they should justify this choice.
Lines 7879: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastestgrowing wave modes dominant other than the fact that they're growing fastest?
Line 79: What is the geometry of E and B? It would help the reader to not have to refer to previous work in order to determine the ExB direction.
Lines 7983: Have the authors made any attempt to quantify the angle of deflection from ExB? Comparing that angle between regularized and unregularized models, as well as to previous results from PIC simulations, would support the authors' main argument. The presentation would benefit from a figure that shows the 2D wavenumber spectrum corresponding to each stage.
Line 82: The note about "a slight turning of the waves in the direction consistent with linear theory" shouldn't apply to the turbulent regime. Did the authors mean to associate this with the linear (or possibly mixing) regime?
Have the authors considered changing the resolution while maintaining the simulation domain size, to see if the dominant wavelength changes?
Citation: https://doi.org/10.5194/egusphere20221264RC1 
AC1: 'Reply on RC1', Enrique Rojas Villalba, 20 Jan 2023
Thank you for your questions and comments. Here you will find our replies.
Lines 6667: Why did the authors artificially increase the electron mass? This should only apply to models that need to resolve the plasma frequency, which is what places the constraint on the simulation time step. If the authors simply wanted to mimic the parameter values used in previous works, they may say so. Otherwise, they should justify this choice.
Reply: We chose the same simulation parameters as Oppenheim(2008) to compare our results to theirs.Lines 7879: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastestgrowing wave modes dominant other than the fact that they're growing fastest?
Reply: We just wanted to point out that there is a clear dominant mode in the linear regime, but we can see now that this was redundant because the dominant mode, by definition, grows faster.Line 79: What is the geometry of E and B? It would help the reader to not have to refer to previous work in order to determine the ExB direction.
Reply: The electric and magnetic fields are parallel to y and z, respectively. We have added this clarification to the text.Lines 7983: Have the authors made any attempt to quantify the angle of deflection from ExB? Comparing that angle between regularized and unregularized models, as well as to previous results from PIC simulations, would support the authors' main argument. The presentation would benefit from a figure that shows the 2D wavenumber spectrum corresponding to each stage.
Reply: The main goal of this paper is to show that the fivemoment fluid system can remain stable and capture several qualitative features of the FarleyBuneman instabilities. We think showing the small but visible tilt in the density structures from Figure 1 is enough for a qualitative argument. A systematic quantitative analysis of this and other metrics is a topic we are currently working on, but it will be published in the future.
Line 82: The note about "a slight turning of the waves in the direction consistent with linear theory" shouldn't apply to the turbulent regime. Did the authors mean to associate this with the linear (or possibly mixing) regime?
Reply: Even though the analytical estimations of the waveturning effect have been derived for the linear regime, we are assuming that the preferential direction criteria should still hold approximately as the system goes through saturation. Nevertheless, we agree that the text is ambiguous, so we have modified it to restrict the comment for the linear regime.Have the authors considered changing the resolution while maintaining the simulation domain size, to see if the dominant wavelength changes?
Reply: As mentioned before, a more systematic evaluation of diagnostics and numerics is part of a current project. And although we have done this analysis and haven't seen a significant change, we decided not to elaborate on this because we were focusing on the qualitative aspects of the results. One of the main contributions of this work is to challenge the standard assumption that simulating FarleyBuneman instabilities in the fluid domain would quickly produce smallscale structures growing faster than the larger ones. In our simulations, we see a dominant mode, so the dependence between growth rate and wavenumber is not monotonically increasing as the standard linear theory predicts. Even though the exact growth rate could change with the resolution of the system, we are assuming that it will maintain the same general behavior.Citation: https://doi.org/10.5194/egusphere20221264AC1 
RC2: 'Reply on AC1', Anonymous Referee #1, 24 Jan 2023
I am satisfied that the authors have sufficiently addressed my concerns.
Citation: https://doi.org/10.5194/egusphere20221264RC2 
AC3: 'Reply on RC2', Enrique Rojas Villalba, 28 Feb 2023
Thank you.
Citation: https://doi.org/10.5194/egusphere20221264AC3

AC3: 'Reply on RC2', Enrique Rojas Villalba, 28 Feb 2023

RC2: 'Reply on AC1', Anonymous Referee #1, 24 Jan 2023

AC1: 'Reply on RC1', Enrique Rojas Villalba, 20 Jan 2023

RC3: 'Comment on egusphere20221264', Anonymous Referee #2, 13 Feb 2023
Lines 2819: The authors used Hassan's artificial viscosity, including thermal effects. Could the authors elaborate more on the following:
 What thermal effects do the authors refer to in the "including thermal effects" statement?
 Why do the authors treat the stress tensor divergence as "artificial viscosity"?
 Did the authors learn about the role that the stress tensor divergence plays as a good sink for the energy sources to stratify the conservation of energy in Hassan’s 2015 fluid model as shown in Figure 8 in Hassan E, Hatch DR, Morrison PJ, Horton W. Multiscale equatorial electrojet turbulence: Energy conservation, coupling, and cascades in a baseline 2‐D fluid model. Journal of Geophysical Research: Space Physics. 2016 Sep;121(9):912745?
 Could the inclusion of the energy equation in the fluid model remove the need for the divergence of stress tensor closure?
 Including the thermal effects in studying the FarleyBunemann instability in the aurora region is important, but how important is the inclusion of the thermal effects in simulations at the equatorial electrojet?
Lines 6972: The authors stated that using a smaller simulation box size will not influence the nonlinear features, and the scalability assumption is still preserved. Could the authors justify the following concerns:
 Using a small simulation box size couldn’t justify the authors’ claim about the scalability of the fluid model as running simulations in larger boxes shouldn’t use large computational resources.
 Using a small simulation box limits the destabilization of modes of longer wavelength by energy cascading which might the comparison of the presented results in this manuscript and both the PIC model in Oppenheim et al. 2008 and Hassan et al. 2015.
Figure 2: Would the authors explain the difference in the ion and electron heating in the unregularized and regularized cases while having the rootmeansquare of the density and electric field almost the same?
Citation: https://doi.org/10.5194/egusphere20221264RC3 
AC2: 'Reply on RC3', Enrique Rojas Villalba, 28 Feb 2023
Thank you for your questions and comments. Here you will find our replies.
Lines 2829: The authors used Hassan's artificial viscosity, including thermal effects. Could the authors elaborate more on the following:
What thermal effects do the authors refer to in the "including thermal effects" statement?
Reply: We meant to say that the system solved in this work includes an energy equation, therefore, relaxing the isothermal assumption. We now see that "including thermal effects" might mislead the reader into thinking that Hassan did not include a pressure term in the momentum equation. We have modified the text to be more explicit about this.Why do the authors treat the stress tensor divergence as "artificial viscosity"?
Reply: Given that the precise form of the term was not justified and its similarity to the neutral viscosity term, it is our impression that this term was added as a numerical proxy for Landau damping. Nevertheless, we now think that it would be more appropriate just to mentioned that this was the operator proposed by the author. The text has now been changed accordingly.
Did the authors learn about the role that the stress tensor divergence plays as a good sink for the energy sources to stratify the conservation of energy in Hassan's 2015 fluid model as shown in Figure 8 in Hassan E, Hatch DR, Morrison PJ, Horton W. Multiscale equatorial electrojet turbulence: Energy conservation, coupling, and cascades in a baseline 2‐D fluid model. Journal of Geophysical Research: Space Physics. 2016 Sep;121(9):912745?
Reply: Yes. Even though the viscosity term used in that work is important for weaklyionized plasmas, it is not crucial for the paper's central argument. We want to show that the fluid equations without any viscosity (either physical or numerical) can avoid a growth rate of k^2.Could the inclusion of the energy equation in the fluid model remove the need for the divergence of stress tensor closure?
Reply: As we mentioned in line 93, we observed a similar behavior using an isothermal system, namely, no k^2 growth and a dominant band of wave numbers similar to the PIC simulation. Nevertheless, you are right to assume that the energy equation provides an additional source of stabilization to the system. A more comprehensive numerical analysis of these terms is a topic of current work.
Including the thermal effects in studying the FarleyBunemann instability in the aurora region is important, but how important is the inclusion of the thermal effects in simulations at the equatorial electrojet?
Reply: Even though we don't expect thermal effects to dominate in the equatorial electrojet, they could be relevant if the experiments are precise enough to measure the details of its dynamics. Nevertheless, this paper aimed not to argue that the presented model is more realistic but to compare its results with PIC simulations.Lines 6972: The authors stated that using a smaller simulation box size will not influence the nonlinear features, and the scalability assumption is still preserved. Could the authors justify the following concerns:
Using a small simulation box size couldn't justify the authors' claim about the scalability of the fluid model as running simulations in larger boxes shouldn't use large computational resources.
Reply: As we mentioned in the conclusions, our claim of computational efficiency relies on the assumption that fluid simulations are computationally cheaper than kinetic simulations if the parameters are similar and kinetic effects are not required.Using a small simulation box limits the destabilization of modes of longer wavelength by energy cascading which might the comparison of the presented results in this manuscript and both the PIC model in Oppenheim et al. 2008 and Hassan et al. 2015.
Reply: Even though the energy cascading to longer wavelengths is an issue, the paper's primary goal is to show that a simple fluid system avoids the k^2 growth rate. Our argument for scalability is mainly related to the computational complexity of the kinetic solvers. Nevertheless, we have added one sentence to comment on the long wavelength issue.Figure 2: Would the authors explain the difference in the ion and electron heating in the unregularized and regularized cases while having the rootmeansquare of the density and electric field almost the same?
Reply: This might be related to the regularized system restricting the velocity magnitude, making the collisional heating term in the energy equation smaller.Citation: https://doi.org/10.5194/egusphere20221264AC2
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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