2D Fluid-PIC Simulations of Hall Thrusters with Self-Consistent Resolution of the Space-Charge Regions

Round 1

Reviewer 1 Report (Previous Reviewer 1)

This revision addressed  all of my previsous  questions.   Congratulations  for your  excellent work., 

Reviewer 2 Report (Previous Reviewer 2)

The authors have addressed the comments 

Reviewer 3 Report (Previous Reviewer 3)

The authors have answered satisfactorily to all the points I raised in my previous review and I thus recommend the publication of the manuscript.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

please  see my report

Comments for author File: Comments.pdf

Reviewer 2 Report

Authors report nonneutral effects (sheath) of the Hall thruster simulations. 

- The authors are strongly suggested to cite other numerical models so that the community can improve as a whole. For instance, there are fluid models that capture nonneutral regimes in Hall thrusters (e.g., Physics of Plasmas 27 113505 (2020), Physics of Plasmas 29, 083903 (2022)). Additionally, there are other hybrid models other than HPHall and the Italian's group (e.g., Boeuf and Garrigues, Ecole Polytechnique, Boyd's group). 

- "Without the appropriate mesh refinement, Poisson’s equation cannot resolve the potential gradient inside the sheath." While mesh refinement is definitely a plus, I recommend this statement to be modified. Many other nonneutral models using drift-diffusion (for ICPs) do not fully resolve the sheath (plasma gradient) but it works. The authors must elaborate on why such mesh refinement is required (e.g., based on experiences or it is just based on a belief?) In addition, the authors shall cite some nonneutral plasma modeling papers (even from other communities) if necessary. 

- "We assume a Maxwellian distribution yielding the well-known Boltzmann relation" and equation 2 is only applicable when the sheath is ion attracting. Please elaborate. If needed the authors are encouraged to read and cite the section about sheath in Plasma Sources Sci. Technol. 28 044001 (2019). 

- "It is well-known that magnetic field lines in Hall thrusters are isothermal in the vicinity of the acceleration region". If it is well known what references do mention these?  Additionally which direction are the authors referring to? Along magnetic field? The sheath with the oblique magnetic field is not well understood in my opinion; therefore, the authors are encouraged to state these are "assumptions". 

- "Ohm’s law": Assumptions for equation (3) must be written out.

- "The resistive term can be neglected since the resistivity along magnetic field lines is very low". My understanding is that Hall2De does not neglect the resistive term. Why do the authors need to / decide to neglect this term for the sheath? 

- In addition, why are the authors only talking about parallel direction? Looking at Fig 1, the sheath region is 2D, thus, the electron transport in cross-field direction must also be accounted for? Why is the "nonmagnetized sheath" assumption valid? If not, the authors shall address their thoughts and mention that these are "assumptions". 

- equation 2 is an accepted description of the electron density within the sheath. Equation 2 comes from a simplification of the electron momentum equation. However, it is unclear what equation 4 is and how it is derived from the fluid (or kinetic) equations. Why is the left hand side the quasineutral density and the right hand side has ne0, which is electron density? What is phi? 

- equation 6 seems like an assumption based on a physical argument. However, for the truncated Maxwellian can be directly obtained using kinetic theory. see Plasma Sources Sci. Technol. 27 065004 (2018). 

- it is unclear what equation 7 is doing. The correct way is that ni evolves in space and time and ne evolves in space and time. Why are there different potential values (phi_sc and phi_F) located on the same cell center?? How is phi_w determined?? 

- Is equation 8 correct? Form a dimensional argument, V [m^3], A [m^2] and (r / |r|) is a unit vector (thus no unit). In the denominator there is only a length scale of [m]. For \nabla^2 f, the denominator should be [m^2]. 

- "such as Newton-Raphson A Taylor series" sentence seems to need a punctuation. 

- "We note that Eq. (10) is solved in the whole computational domain and not only in the refined regions" This should be elaborated much earlier. If one applies this to the entire domain, do authors observe that the large potential drop region exhibits non-neutrality? 

 - boundary condition. Where are the cells located? cell center of nodes (e.g., finite volume or finite difference)? 

- the validity of equation 11 must be discussed carefully. The electron flux shall be determined by the sheath potential (potential difference between the sheath edge and the wall). Equation 11 uses \phi_ek and \phi_Fk, which seem to suggest that these are different potential values  at the same location. In addition, this equation assumes a Maxwellian distribution function locally. I thought that equation 6 was introduced to account for truncated VDF. Please elaborate. 

- "ne=3x10^16" unit is missing. 

- "The target number of PIC particles" What does the "target number" means? Why not just report the spatially averaged or maximum/minimum number of macroparticles per cell? 

- fig 3: how long did it take to achieve a steady state result? please mention about the time-dependent nature or transient aspect.

- "The electron fluid equations were not solved" I thought that the model relies on the fluid estimate (phi) from which a correction (phi_sc) is obtained. Please elaborate. 

- What is "\Delta \phi_sh"? 

- How is the boundary condition of V= 300 V imposed in this toy problem? The Vlasov Poisson solver requires an elliptic PDE treatment, while the authors only used an explicit RK4 scheme. Do the authors just pick an arbitrary large domain so that the potential can go down to 0V from 300V and just cut the domain there? please elaborate. 

- figure 4 and figure 5: as there is not much change in the bulk plasma, the authors may consider plotting a more "zoomed in" figure where the differences can be more clearly seen for the readers to understand the work. 

- From the results it seems like the sheath due to conducting surface can be clearly seen. However, the sheath of the dielectric material seems unmodified. Why? For a floating sheath, there should be a 5Te potential drop from the sheath edge in the absence of SEE. Hence, if Te ~ 10 eV, there should be a 50 eV sheath near the dielectric material. 

- Why is the pole piece biased to 0V? They are not floating? In the space operation, what will be the potential of pole piece? 

- figure 7 and 8. Is there any experimental evidence that the finite sheath results are closer to the experiments? 

Reviewer 3 Report

This manuscript propose a method to automatically relax the thin-sheath approximation near solid boundaries where this is not appropriate in 2D Fluid-PIC simulations of Hall Thrusters. The resulting scheme can resolve the conditions of ionised gas in space-charge regions of any thickness, recovering the thin-sheath approximation where mesh resolution is coarse and resolving the sheath region where the Debye length is of the order or higher than the local grid resolution

In my opinion the manuscript is interesting, well written and organised, and generally very clear. I only have some minor comments that the authors may consider in a revised version of the manuscript, listed below.

1) I would ask the authors an effort, if possible, in expanding the literature review on the subject in order to reduce the ratio of self citations with respect to the total number of references; at present 8 references over 14 are  authored by one (or both) among the authors of the present manuscript.

2) I suggest the authors to comment about the assumption of a Maxwellian energy distribution function of electrons in the sheath, referring to related works in the literature. Also, I wonder if there are references in the literature related to the truncation of the distribution function to grant potential monotonicity  as the one operated by the authors.

3) If I understood properly the symbols of the equation (14), I think that a plus instead of a minus should be used within brackets.