Fluid Models Capturing Farley-Buneman Instabilities

Rojas, Enrique; Burns, Keaton; Hysell, David

doi:https://doi.org/10.5194/egusphere-2022-1264

Preprints

https://doi.org/10.5194/egusphere-2022-1264

Preprints

01 Dec 2022

| 01 Dec 2022

Fluid Models Capturing Farley-Buneman Instabilities

Enrique Rojas, Keaton Burns, and David Hysell

Abstract. It is generally accepted that modeling Farley-Buneman instabilities require resolving ion Landau damping to reproduce experimentally observed features. Particle-in-cell (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 non-local phenomena that require three dimensions or coupling with larger--scale processes. We argue that a form of the five-moment fluid system can recreate several qualitative aspects of Farley-Buneman 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.

Received: 15 Nov 2022 – Discussion started: 01 Dec 2022

Enrique Rojas et al.

Status: closed

RC1:
'Comment on egusphere-2022-1264', Anonymous Referee #1, 05 Jan 2023

Lines 66-67: 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 78-79: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastest-growing 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 79-83: 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/egusphere-2022-1264-RC1
- AC1:
  'Reply on RC1', Enrique Rojas Villalba, 20 Jan 2023
  
  Thank you for your questions and comments. Here you will find our replies.
  Lines 66-67: 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 78-79: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastest-growing 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 79-83: 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 five-moment fluid system can remain stable and capture several qualitative features of the Farley-Buneman 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 wave-turning 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 Farley-Buneman instabilities in the fluid domain would quickly produce small-scale 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/egusphere-2022-1264-AC1
  - 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/egusphere-2022-1264-RC2
    
    AC3: 'Reply on RC2', Enrique Rojas Villalba, 28 Feb 2023
    
    Thank you.
    
    Citation: https://doi.org/10.5194/egusphere-2022-1264-AC3
RC3:
'Comment on egusphere-2022-1264', Anonymous Referee #2, 13 Feb 2023
Lines 28-19: 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):9127-45?

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 Farley-Bunemann instability in the aurora region is important, but how important is the inclusion of the thermal effects in simulations at the equatorial electrojet?

Lines 69-72: 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 root-mean-square of the density and electric field almost the same?
Citation: https://doi.org/10.5194/egusphere-2022-1264-RC3
- AC2: 'Reply on RC3', Enrique Rojas Villalba, 28 Feb 2023
  
  Thank you for your questions and comments. Here you will find our replies.
  Lines 28-29: 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):9127-45?
  
  Reply: Yes. Even though the viscosity term used in that work is important for weakly-ionized 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 Farley-Bunemann 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 69-72: 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 root-mean-square 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/egusphere-2022-1264-AC2

Status: closed

RC1:
'Comment on egusphere-2022-1264', Anonymous Referee #1, 05 Jan 2023

Lines 66-67: 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 78-79: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastest-growing 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 79-83: 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/egusphere-2022-1264-RC1
- AC1:
  'Reply on RC1', Enrique Rojas Villalba, 20 Jan 2023
  
  Thank you for your questions and comments. Here you will find our replies.
  Lines 66-67: 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 78-79: Can the authors elaborate on "we see dominant wave modes growing distinctly faster"? What makes the fastest-growing 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 79-83: 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 five-moment fluid system can remain stable and capture several qualitative features of the Farley-Buneman 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 wave-turning 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 Farley-Buneman instabilities in the fluid domain would quickly produce small-scale 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/egusphere-2022-1264-AC1
  - 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/egusphere-2022-1264-RC2
    
    AC3: 'Reply on RC2', Enrique Rojas Villalba, 28 Feb 2023
    
    Thank you.
    
    Citation: https://doi.org/10.5194/egusphere-2022-1264-AC3
RC3:
'Comment on egusphere-2022-1264', Anonymous Referee #2, 13 Feb 2023
Lines 28-19: 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):9127-45?

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 Farley-Bunemann instability in the aurora region is important, but how important is the inclusion of the thermal effects in simulations at the equatorial electrojet?

Lines 69-72: 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 root-mean-square of the density and electric field almost the same?
Citation: https://doi.org/10.5194/egusphere-2022-1264-RC3
- AC2: 'Reply on RC3', Enrique Rojas Villalba, 28 Feb 2023
  
  Thank you for your questions and comments. Here you will find our replies.
  Lines 28-29: 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):9127-45?
  
  Reply: Yes. Even though the viscosity term used in that work is important for weakly-ionized 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 Farley-Bunemann 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 69-72: 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 root-mean-square 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/egusphere-2022-1264-AC2

Enrique Rojas et al.

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Short summary

The standard linear fluid theory of Farley-Buneman predicts that kinetic physics is required to avoid the artificial growth of smaller structures. We decided to explore the possibility of simulating Farley-Buneman using, for the first time, a fully fluid five-moment model. This is the first time a fully fluid model has been used to simulate the Farley-Buneman instability. The results obtained with both models are qualitatively consistent with the ones from kinetic simulations.


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