Mean-Field Dynamo Model in Anisotropic Uniform Turbulent Flow with Short-Time Correlations

Round 1

Reviewer 1 Report

In the manuscript entitled "Anisotropy effects in mean-field dynamo-model" the authors Allahverdiyev et al. use the path integral approach to address new effects in magnetic field amplification by a mean-field dynamo. In comparison to other approaches, like the two-scale approach by Steenbeck, Krause and Rädler, the path integral approach has the advantage that it is not based on the assumption of isotropy.
The authors derive the a general mean-field dynamo model and test effects of anisotropy, as far as I know, for the first time in such an analytic way. The magnetic field evolution in anisotropic flows is discussed in a few specific examples. This study may have implications for understanding the generation of magnetic fields in galaxies.

I consider this work as interesting and worth publishing after a revision. In general, the analytical derivation of the anisotropic mean-field equation is well-explained and clearly presented. The manuscript, however, has a huge number of typos and grammatical mistakes that make reading difficult. Also, some of the figures could be improved slightly.

Below, I list all of my comments on the manuscript, with the order not representing the importance of the comments. I encourage the authors to include my suggestions for improving the paper.

 

Specific comments:

 

Title: "Anisotropy effects in mean-field dynamo-model" -> "Anisotropy effects in a mean-field dynamo-model"

L2: "First mean-field equations were obtained..." -> "The first mean-field equations were obtained..."

L4: "...short-correlated random velocity field in anisotropic streams." -> "...a short-correlated random velocity field in anisotropic streams."

L7: "isotropical" -> "isotropic"

L9 : "...on helicity tensor structure.." -> "...on the helicity tensor structure.."

L10: "...averaged magnetic field..." -> "...the averaged magnetic field..."

L17-18: "Averaged equation described the magnetic energy generation in helical rotated streams were
18 developed by Steenbeck, Krause and Radler in 60th,..." -> "Averaged equations describing the magnetic energy generation in helical rotated streams were developed by Steenbeck, Krause and Radler in the 60s,..."

L26: " of isotropic velocity field" -> " of the isotropic velocity field"

L28: "we go to the complicating of general approach" -> Rephrase.

L29: "of well-known model." -> "of the well-known model."

L30: "where B is averaged magnetic field ⟨H⟩ and averaged velocity parameters" -> "where B is the averaged magnetic field ⟨H⟩ and the averaged velocity parameters"

L34: "scales of magnetic field" -> "the scales of magnetic field"

Below L35: "There are exist various" -> "There are various"

L36: "used timeseries" -> "used the timeseries"

L37: Please define the magnetic Reynolds number Rm.

L37: "thought" -> "though"

L38: "base on two-scale assumption" -> "based on the two-scale assumption"

L39: "hypothesis" -> "hypotheses"

L40: "was far to reproduce" -> "failed to reproduce"

L41: " for short-correlated flow." -> "for a short-correlated flow."

L44: "in short-correlated field," -> "in a short-correlated field,"

L45: "Untying averaging process for magnetic " -> "Untying the averaging process for the magnetic "

L45: "defines anisotropic mean-field dynamo" -> "defines the anisotropic mean-field dynamo"

L50: "under curl in the right side. " -> "under the curl in the right side. "

L50: "one can present induction equation" -> "one can present the induction equation"

L51: "writing it in form" ->"writing it in the form"

L53: "get mean-field equation " -> "get the mean-field equation "

L53: "div(v) = 0": This implies that the fluid is incompressible. The authors claim that the main application of their work is astrophysical fluids, in particular galaxies. In galaxies, however, the gas can be incompressible. In the interstellar medium large Mach numbers have been reported, see e.g. Larson, Monthly Notices of the Royal Astronomical Society, 194, 4, 1981. If the dynamo model cannot be extended to the compressible case easily, a statement of caution would be useful here.

L54: "in form with" -> "in the form with a "

L55: "provides implementation of solenoidal condition for averaged magnetic field " -> "provides the implementation of a solenoidal condition for the averaged magnetic field "

L54: "in anisotropic case. Using vector potential " -> "in an anisotropic case. Using the vector potential "

L57: " in uniform field " -> " in an uniform field "

L58: "analyse dynamo growth rate " -> "analyse the dynamo growth rate "

L60: " if trace" -> " if the trace"

L60: "(similarly test" -> "(a similar test"

L61: "with very early stage" -> "with the very early stage"

L63: "for galactic" -> "for the galactic"

L64-65: " because the dynamo time scale in galaxies is smaller however comparable
65 with the age of galaxies" -> " because the mean-field dynamo time scale in galaxies is smaller however comparable
65 with the age of galaxies" [I would add "mean-field" here, because time scales form other types of dynamos operating in galaxies, such as the small-scale dynamo, can be much less than the galaxy age.]

L66: "Functional integral as solution " ->"Functional integral as a solution "

Below L66: "in random velocity field" ->"in a random velocity field"

First line of page 3:"Find a changing of vector potential, rewriting ideal equation (3) in the form of full time" -> "Find the changing of the vector potential by rewriting the ideal equation (3) in the form of the full time"

Below (4):
"Here full Euler derivative" -> "Here the full Euler derivative"
"Obtained expression " -> "The obtained expression "
"to write vector potential shift" -> "to write the vector potential shift"

Below (5):
"for ideal case can" -> "for the ideal case can"
"arbitrary time" -> "an arbitrary time"

Below (6):
"and vector potential in" -> "and the vector potential in"
"which is order to velocity correlation time" -> "which is of the order of the velocity correlation time"
"using theory of functional integrals," -> "using the theory of functional integrals,"

L67: "Obtained solution (7)" -> "The obtained solution (7)"

L68: "standard stochastic way" -> "the standard stochastic way"

L69: "re-change uniquely defined" -> "re-change the uniquely defined"

L70: " Feynman approach" -> "the Feynman approach"

L71: "on defined nonstochastic " -> "on a defined nonstochastic "

L72: " for stochastic velocity" -> " for a stochastic velocity"

Below 74:
"on stochastic one:" -> "to the stochastic one:"

Above (9):
"vector potential " -> "the vector potential "

Above (10):
"redefine them in the arbitrary point x by Taylor series" -> "redefined in the arbitrary point x by a Taylor series"

Below (12):
"Combining asymptotics (12) with functional integral (7) " -> "Combining the asymptotics (12) with the functional integral (7) "

Below (13):
"moving term Ai (x, t) from the right side to the left, we obtain equality:" -> "moving the term Ai (x, t) from the right side to the left, we obtain the equality:"

L74: "classical induction equation (3), written for each component of vector potential." -> "the classical induction equation (3), written for each component of the vector potential."

L77: "on random velocity field, " -> "on a random velocity field, "

L80: "Average" -> "The average"

L81: "Do it in" -> "We proceed in"

L83: "then average velocity field" -> "then average the velocity field"

L85: "account nonzero plasma diffusivity" -> "account a nonzero plasma diffusivity"

Page 5, line 1:
"saving old denotation" -> "saving the old denotation"

L93: "decreasing slower" -> "decreasing slower than"

Above (16):
"Using asymptotic representation" -> "Using an asymptotic representation"

Below (18):
" thus first four terms " -> " thus the first four terms "

Above (20):
"For averaged terms introduce helicity" -> "For averaged terms we introduce helicity"

Above (21):
"and substituting them into (19), rewrite mean-field system:" -> "and substitute them into (19) to rewrite the mean-field system:"

Below (21):
" use gradient uncertainty" -> " use the gradient uncertainty"

Above (22):
"obtain mean-field system " -> "obtain the mean-field system "

Below (22):
"Emphasize that obtained " -> "We emphasize that the obtained "

L95: "equation (22) transforms in standard" -> "equation (22) transforms in the standard"

Below (24):
"Note that vector" -> "Note that the vector"
"in nonuniform" -> "in a nonuniform"

Above (26):
"transforms partial differential system" -> "transform the partial differential system"

Below (26):
"are diagonal components" -> "are the diagonal components"

Above (27):
"proportional exp" -> "proportional to exp"

Below (29):
"In isotropic stream" -> "In an isotropic stream"

Middle of page 7:
"(1) For large t hyperbolic functions in (29) and (30) grow like exponents," -> "(1) For large t the hyperbolic functions in (29) and (30) grow like exponentials,"
"anisotropical flows maximal growth" -> "anisotropic flows the maximal growth"
"for instance, harmonic along z-axis k = ?0, 0, √αxxαyy/2βzz? can grow with rate αxxαyy/4βzz, which depends only on helicity " ->
"for instance, the harmonic along z-axis k = ?0, 0, √αxxαyy/2βzz? can grow with rate αxxαyy/4βzz, which depends only on the helicity "
"maximum of function (−δ + γ) gives two possible extremums" -> "the maximum of the function (−δ + γ) gives two possible extrema"

L104: "when generation will" -> "when the generation will"

Below (32):
"for anisotropic field, and for isotropic field" -> "for an anisotropic field, and for an isotropic field"

L107: "changing of magnetic field along the current density for isotropic" -> "changing of the magnetic field along the current density for an isotropic"

L109: "isotropic random field " -> "an isotropic random field "

L110: "because magnetic field" -> "because the magnetic field"

Above (34):
"x-harmonic and magnetic field localized in xy-plain with current density along z-axis" ->
"x-harmonic and a magnetic field localized in the xy-plain with a current density along z-axis"

Below (36):
"Growth rate is " -> "The growth rate is "
"Anisotropic case (36)" -> "The anisotropic case (36)"
" on x-helicity component " -> " on the x-helicity component "

Above (38):
"For such function the original field can be calculated only for isotropic case" -> "For such a function the original field can be calculated only for the isotropic case"

Equation (38): There are two "=" signs.

Bottom of page 8: "For anisotropic flow " -> "For an anisotropic flow "

Top of page 9: "periodical structures of averaged magnetic field" -> "periodic structures of the averaged magnetic field"

Above (39):
"redistribution exponent power" -> "redistribution the exponent power"

L118: " neither analysis at large times," -> " nor analysis at large times,"

L119: "do not give" -> "do give" [Please check if that is what is meant.]

L123: " to B02(r)-field is not force-free, Lorentz " -> " to the B02(r)-field being not force-free, the Lorentz"

L124: " in isotropic case (" -> " in the isotropic case ("

L129: "Therefore dynamo successfully" -> "Therefore the dynamo successfully"

Section 6:
"to get well-known dynamo model " -> "to get the well-known dynamo model "

L135: "Obtained model" -> "The obtained model"

L136: "main advantage of obtained equation" -> "the main advantage of the obtained equation"

L138: "such form of equation" -> "such a form of equation"

L142-143: "the governing equation for A has a more complicated and less comfortable for path integral
142 method form, but have advantage in gradient uncertainty." -> "the governing equation for A has a more complicated and less comfortable form for the path integral method, but has the advantage of gradient uncertainty."

Page 10, top: " of anisotropic dynamo " -> " of the anisotropic dynamo "

L143: "and dependence " -> "and the dependence "

L148: "Using obtained solution " -> "Using the obtained solution "

L149: "consider magnetic field" -> "consider a magnetic field"

L150: "is solution" -> "is a solution"

L151: "for isotropic flow " -> "for an isotropic flow "

L152: "because magnetic field generates" -> "because the magnetic field generates"

L153: "than Lorentz force, of course, exponentially increase." -> "then the Lorentz force, of course, increases exponentially."

L155: "in anisotropic flow magnetic field" -> "in an anisotropic flow the magnetic field"

L158: " initially r0-localized field. Obtained " -> " the initially r0-localized field. The obtained "

L159: "that averaged magnetic" -> "that the averaged magnetic"

L160: "proportional exp" -> "proportional to exp"

Figure 1: A legend indicating the density of lines in the plot would be useful. Also please add labels "(a)", "(b)", and "(c)" in the plot panels.

Figure 2: "It is clearly seen the oscillated structure on the left panel, and superexponential increasing at the initial moment transforming in exponential growth on the right panel." -> "The oscillating structure on the left panel, and superexponential increasing at the initial moment transforming in exponential growth on the right panel are seen clearly."

Figure 3: "Initial distribution is B02(r)." Please reference the equation where B02(r) is defined.

 

General comments:

1) Full stops are missing after almost all equations.

2) The authors explore the dynamics based on the vector potential rather than the magnetic field. It would be useful to comment explicitly on the choice of the gauge.

3) Is there any comparison with direct numerical simulations possible? The authors should discuss the current state of the art in dynamo simulations and, if available, compare their analytical predictions with simulation results.

4) In all figures, no units are given. In what units are B, k, and t potted?

 

 

Author Response

Review 1.

 

In the manuscript entitled "Anisotropy effects in mean-field dynamo-model" the authors Allahverdiyev et al. use the path integral approach to address new effects in magnetic field amplification by a mean-field dynamo. In comparison to other approaches, like the two-scale approach by Steenbeck, Krause and Rädler, the path integral approach has the advantage that it is not based on the assumption of isotropy.

 
The authors derive the general mean-field dynamo model and test effects of anisotropy, as far as I know, for the first time in such an analytic way. The magnetic field evolution in anisotropic flows is discussed in a few specific examples. This study may have implications for understanding the generation of magnetic fields in galaxies. I consider this work as interesting and worth publishing after a revision. In general, the analytical derivation of the anisotropic mean-field equation is well-explained and clearly presented. The manuscript, however, has a huge number of typos and grammatical mistakes that make reading difficult. Also, some of the figures could be improved slightly.

Below, I list all of my comments on the manuscript, with the order not representing the importance of the comments. I encourage the authors to include my suggestions for improving the paper.

 

Dear Reviewer, thank you very much for all your comments and corrections! We try to take them into account, improving this work as best as we can. All new corrections in the text and figures are marked by blue color. In addition to grammatical and specific corrections, we provide a detailed answer to the general comments.

 

General comments:

1) Full stops are missing after almost all equations.

Thank you! We add stops after all equations and remove extra mathematical signs “+, =, -”.

2) The authors explore the dynamics based on the vector potential rather than the magnetic field. It would be useful to comment explicitly on the choice of the gauge.

Thank you for this comment, of course, the vector potential application needs a more detailed explanation. We use the induction equation in the form of the vector potential (3) to distance from the question about solenoidal condition of averaged magnetic field. After averaging we take curl-operation from both sides of equation (21), obtaining equality (22). Such way exclude the question, because from (22) follows that div(B) does not change in time – div from right side is zero. Therefore, we use equation (3), which, of course, has uncertainty in potential gauge. As we show in (4) this gauge allows us to write ideal induction equation in the form of full derivative, the gauge is chosen in the way then the last term in (4) is zero. Note that after curl-operation this gauge does not affect on the final form of mean-field equation for B. We add the explanations in the text.

3) Is there any comparison with direct numerical simulations possible? The authors should discuss the current state of the art in dynamo simulations and, if available, compare their analytical predictions with simulation results.

We agree with reviewer that it would be very interesting to compare the current state of the art in dynamo simulations with obtained anisotropic results. Therefore, we add in the text some references on the numerical dynamo studies, e.g., on detailed review-paper Brandenburg&Subramanian(2005), where anisotropy effects are discussed in sections 5-6. However, main result of our paper is the obtaining of mean-field anisotropic system and study of some general effects. Comparison of theoretical and observational results will be fulfilled in the following works.

4) In all figures, no units are given. In what units are B, k, and t potted?

Yes, indeed, here we obtain the mean-field equation in SI, however, it’s linear, so magnetic field can be renormalized for convenient units. Similarly, spatial and time scales can be also renormalized for particular comparison with astrophysical objects. But we analyze the system without strict detailing and comparisons, and define B, k, and t in conventional dimensionless units. In the same way we define initial distributions in both examples: not for particular astrophysical systems, but to demonstrate general anisotropic effects. We agree that the detailed comparisons of anisotropic solutions with experimental and numerical data should be also fulfilled, however we plan to do that in the next investigation.

 

Specific comments:

Title: "Anisotropy effects in mean-field dynamo-model" -> "Anisotropy effects in a mean-field dynamo-model"

Thank you for this title correction! We think about it and at last change the title on “Mean-field dynamo model in anisotropic uniform turbulent flow with short time correlations.”, because we think that such title is less broad and more concrete for obtained results.

L2: "First mean-field equations were obtained..." -> "The first mean-field equations were obtained..."

L4: "...short-correlated random velocity field in anisotropic streams." -> "...a short-correlated random velocity field in anisotropic streams."

L7: "isotropical" -> "isotropic"

L9 : "...on helicity tensor structure.." -> "...on the helicity tensor structure.."

L10: "...averaged magnetic field..." -> "...the averaged magnetic field..."

L17-18: "Averaged equation described the magnetic energy generation in helical rotated streams were
developed by Steenbeck, Krause and Radler in 60th,..." -> "Averaged equations describing the magnetic energy generation in helical rotated streams were developed by Steenbeck, Krause and Radler in the 60s,..."

L26: " of isotropic velocity field" -> " of the isotropic velocity field"

L28: "we go to the complicating of general approach" -> Rephrase.

L29: "of well-known model." -> "of the well-known model."

L30: "where B is averaged magnetic field ⟨H⟩ and averaged velocity parameters" -> "where B is the averaged magnetic field ⟨H⟩ and the averaged velocity parameters"

L34: "scales of magnetic field" -> "the scales of magnetic field"

L35: "There are exist various" -> "There are various"

L36: "used timeseries" -> "used the timeseries"

L37: Please define the magnetic Reynolds number Rm.

L37: "thought" -> "though"

L38: "base on two-scale assumption" -> "based on the two-scale assumption"

L39: "hypothesis" -> "hypotheses"

L40: "was far to reproduce" -> "failed to reproduce"

L41: " for short-correlated flow." -> "for a short-correlated flow."

L44: "in short-correlated field," -> "in a short-correlated field," 

L45: "Untying averaging process for magnetic " -> "Untying the averaging process for the magnetic "

L45: "defines anisotropic mean-field dynamo" -> "defines the anisotropic mean-field dynamo"

L50: "under curl in the right side. " -> "under the curl in the right side. "

L50: "one can present induction equation" -> "one can present the induction equation"

L51: "writing it in form" ->"writing it in the form"

L53: "get mean-field equation " -> "get the mean-field equation "

L53: "div(v) = 0": This implies that the fluid is incompressible. The authors claim that the main application of their work is astrophysical fluids, in particular galaxies. In galaxies, however, the gas can be incompressible. In the interstellar medium large Mach numbers have been reported, see e.g. Larson, Monthly Notices of the Royal Astronomical Society, 194, 4, 1981. If the dynamo model cannot be extended to the compressible case easily, a statement of caution would be useful here.

Yes, of course, we use here the incompressible assumption, which is usually used in mean-filed dynamo investigations, where variations of averaged velocity field are not taken into account. We consider here a velocity field with constant second moments, so we think that the incompressibility is reasonable for this work. However we agree that sonic and supersonic processes play the very important role in interstellar medium and galaxy formation, and, therefore, dynamo effects in compressible turbulence are very very interesting. We add some references and some explanations in the text.

L54: "in form with" -> "in the form with a "

L55: "provides implementation of solenoidal condition for averaged magnetic field " -> "provides the implementation of a solenoidal condition for the averaged magnetic field "

L54: "in anisotropic case. Using vector potential " -> "in an anisotropic case. Using the vector potential "

L57: " in uniform field " -> " in an uniform field "

L58: "analyse dynamo growth rate " -> "analyse the dynamo growth rate "

L60: " if trace" -> " if the trace"

L60: "(similarly test" -> "(a similar test"

L61: "with very early stage" -> "with the very early stage"

L63: "for galactic" -> "for the galactic"

L64-65: " because the dynamo time scale in galaxies is smaller however comparable
with the age of galaxies" -> " because the mean-field dynamo time scale in galaxies is smaller however comparable
with the age of galaxies" [I would add "mean-field" here, because time scales form other types of dynamos operating in galaxies, such as the small-scale dynamo, can be much less than the galaxy age.]

Thanks a lot!

L66: "Functional integral as solution " ->"Functional integral as a solution "

Below L66: "in random velocity field" ->"in a random velocity field"

First line of page 3:"Find a changing of vector potential, rewriting ideal equation (3) in the form of full time" -> "Find the changing of the vector potential by rewriting the ideal equation (3) in the form of the full time"

Below (4):

"Here full Euler derivative" -> "Here the full Euler derivative"

"Obtained expression " -> "The obtained expression "

"to write vector potential shift" -> "to write the vector potential shift"

Below (5): 

"for ideal case can" -> "for the ideal case can" 

"arbitrary time" -> "an arbitrary time"

Below (6):

"and vector potential in" -> "and the vector potential in" 

"which is order to velocity correlation time" -> "which is of the order of the velocity correlation time"

"using theory of functional integrals," -> "using the theory of functional integrals,"

L67: "Obtained solution (7)" -> "The obtained solution (7)"

L68: "standard stochastic way" -> "the standard stochastic way"

L69: "re-change uniquely defined" -> "re-change the uniquely defined"

L70: " Feynman approach" -> "the Feynman approach"

L71: "on defined nonstochastic " -> "on a defined nonstochastic "

L72: " for stochastic velocity" -> " for a stochastic velocity"

Below 74: 

"on stochastic one:" -> "to the stochastic one:"

Above (9):

"vector potential " -> "the vector potential "

Above (10):

"redefine them in the arbitrary point x by Taylor series" -> "redefined in the arbitrary point x by a Taylor series"

Below (12):

"Combining asymptotics (12) with functional integral (7) " -> "Combining the asymptotics (12) with the functional integral (7) "

Below (13):

"moving term Ai (x, t) from the right side to the left, we obtain equality:" -> "moving the term Ai (x, t) from the right side to the left, we obtain the equality:"

L74: "classical induction equation (3), written for each component of vector potential." -> "the classical induction equation (3), written for each component of the vector potential."

L77: "on random velocity field, " -> "on a random velocity field, "

L80: "Average" -> "The average"

L81: "Do it in" -> "We proceed in"

L83: "then average velocity field" -> "then average the velocity field"

L85: "account nonzero plasma diffusivity" -> "account a nonzero plasma diffusivity"

Page 5, line 1: "saving old denotation" -> "saving the old denotation"

L93: "decreasing slower" -> "decreasing slower than"

Above (16): "Using asymptotic representation" -> "Using an asymptotic representation"

Below (18): " thus first four terms " -> " thus the first four terms "

Above (20): "For averaged terms introduce helicity" -> "For averaged terms we introduce helicity"

Above (21): "and substituting them into (19), rewrite mean-field system:" -> "and substitute them into (19) to rewrite the mean-field system:"

Below (21): " use gradient uncertainty" -> " use the gradient uncertainty"

Above (22): "obtain mean-field system " -> "obtain the mean-field system "

Below (22): "Emphasize that obtained " -> "We emphasize that the obtained "

L95: "equation (22) transforms in standard" -> "equation (22) transforms in the standard"

Below (24): "Note that vector" -> "Note that the vector" "in nonuniform" -> "in a nonuniform"

Above (26): "transforms partial differential system" -> "transform the partial differential system"

Below (26): "are diagonal components" -> "are the diagonal components"

Above (27): "proportional exp" -> "proportional to exp"

Below (29): "In isotropic stream" -> "In an isotropic stream"

Middle of page 7:

"(1) For large t hyperbolic functions in (29) and (30) grow like exponents," -> "(1) For large t the hyperbolic functions in

(29) and (30) grow like exponentials,"

"anisotropical flows maximal growth" -> "anisotropic flows the maximal growth"

"for instance, harmonic along z-axis k = ?0, 0, √αxxαyy/2βzz? can grow with rate αxxαyy/4βzz, which depends only on helicity " ->"for instance, the harmonic along z-axis k = ?0, 0, √αxxαyy/2βzz? can grow with rate αxxαyy/4βzz, which depends only on the helicity "

"maximum of function (−δ + γ) gives two possible extremums" -> "the maximum of the function (−δ + γ) gives two

possible extrema"

L104: "when generation will" -> "when the generation will"

Below (32): "for anisotropic field, and for isotropic field" -> "for an anisotropic field, and for an isotropic field"

L107: "changing of magnetic field along the current density for isotropic" -> "changing of the magnetic field along the current density for an isotropic"

L109: "isotropic random field " -> "an isotropic random field "

L110: "because magnetic field" -> "because the magnetic field"

Above (34): "x-harmonic and magnetic field localized in xy-plain with current density along z-axis" ->

"x-harmonic and a magnetic field localized in the xy-plain with a current density along z-axis"

Below (36): "Growth rate is " -> "The growth rate is "

"Anisotropic case (36)" -> "The anisotropic case (36)" 

" on x-helicity component " -> " on the x-helicity component "

Above (38): "For such function the original field can be calculated only for isotropic case" -> "For such a function the original field can be calculated only for the isotropic case"

Equation (38): There are two "=" signs.

Bottom of page 8: "For anisotropic flow " -> "For an anisotropic flow "

Top of page 9: "periodical structures of averaged magnetic field" -> "periodic structures of the averaged magnetic field"

Above (39): "redistribution exponent power" -> "redistribution the exponent power"

L118: " neither analysis at large times," -> " nor analysis at large times,"

L119: "do not give" -> "do give" [Please check if that is what is meant.]

L123: " to B02(r)-field is not force-free, Lorentz " -> " to the B02(r)-field being not force-free, the Lorentz"

L124: " in isotropic case (" -> " in the isotropic case ("

L129: "Therefore dynamo successfully" -> "Therefore the dynamo successfully"

"to get well-known dynamo model " -> "to get the well-known dynamo model "

L135: "Obtained model" -> "The obtained model"

L136: "main advantage of obtained equation" -> "the main advantage of the obtained equation"

L138: "such form of equation" -> "such a form of equation"

L142-143: "the governing equation for A has a more complicated and less comfortable for path integral
142 method form, but have advantage in gradient uncertainty." -> "the governing equation for A has a more complicated and less comfortable form for the path integral method, but has the advantage of gradient uncertainty."

Page 10, top: " of anisotropic dynamo " -> " of the anisotropic dynamo "

L143: "and dependence " -> "and the dependence "

L148: "Using obtained solution " -> "Using the obtained solution "

L149: "consider magnetic field" -> "consider a magnetic field"

L150: "is solution" -> "is a solution"

L151: "for isotropic flow " -> "for an isotropic flow "

L152: "because magnetic field generates" -> "because the magnetic field generates"

L153: "than Lorentz force, of course, exponentially increase." -> "then the Lorentz force, of course, increases exponentially."

L155: "in anisotropic flow magnetic field" -> "in an anisotropic flow the magnetic field"

L158: " initially r0-localized field. Obtained " -> " the initially r0-localized field. The obtained "

L159: "that averaged magnetic" -> "that the averaged magnetic"

L160: "proportional exp" -> "proportional to exp"

Figure 1: A legend indicating the density of lines in the plot would be useful. Also please add labels "(a)", "(b)", and "(c)" in the plot panels.

Thank you for this comment! We add color map and A,B,C-labels to the panels.

Figure 2: "It is clearly seen the oscillated structure on the left panel, and superexponential increasing at the initial moment transforming in exponential growth on the right panel." -> "The oscillating structure on the left panel, and superexponential increasing at the initial moment transforming in exponential growth on the right panel are seen clearly."

Figure 3: "Initial distribution is B02(r)." Please reference the equation where B02(r) is defined.

Author Response File: Author Response.pdf

Reviewer 2 Report

Please see the attached pdf file.

Comments for author File: Comments.pdf

Author Response

Review 2.

 

This article presented an interesting calculation of anisotropic turbulent transport coefficients of meanfield dynamo models using a path-integral approach,. The authors have also considered two specific examples of mean-field dynamo models with anisotropic coefficients, and found a few features that are distinct from the common isotropic models. I think the manuscript can and should be improved in a revised version; please see my comments below.

Dear Reviewer, thank you very much for all your comments and corrections! We try to take them into account, improving this work as best as we can. All new corrections in the text and figures are marked by blue color. In addition to grammatical and specific corrections, we provide a detailed answer to the general comments.

 

Overall comments:

1) The title seems too broad and can be improved. Also “dynamo-model” should be “dynamo model(s)”.

Thank you for this title correction! We think about it and at last change the title on “Mean-field dynamo model in anisotropic uniform turbulent flow with short time correlations”, because we think that such title is less broad and more concrete for obtained results.

2) The reader might want to see more references. Many claims are stated without a reference, especially in the first section.

We agree and try to add additional important references in the text on particular works and on generals reviews.

3. One of the main themes of the work is about the path-integral approach, but except for equation (6), I don’t see too many calculations involving path integrals or functional analysis; in fact, the small parameter expansion turns the calculations directly into algebraic ones. As the path-integral method hasn’t been widely applied in the community, can the authors comment on this, and probably make it more clear what are its advantages? Can any technique from quantum mechanics or field theory be borrowed and applied here?

Yes, of course, path integral approach is quite frequently used in dynamo investigations, see, e.g., 1985JETP_Zeldovich, 1987SovPhysUsp_Zeldovich, 1987AcadSciUSA_Zeldovich and literature therein. After these pioneering studies the  approach was used in small- and large-scale problems for obtaining averaged systems, like SKR system, Kazantsev system, VainshteinKichatinov system and so on. Here we show that it allows one to obtain mean-field equation in anisotropic case and, indeed, we start from (6) as from start point – it is clue moment that solution can be presented in multiplication form -- and then there are just traditional path integral expansion and averaging over wiener noise. It is not the new approach, but new results, so we add some explanation and additional references in the text. Note also that we do not claim that the approach here is the direct following from quantum mechanics technique, strictly saying from the mathematical point of view there are different integrals, however the idea about integrating over set of trajectories is from quantum mechanics. Is it possible to apply other approaches of field theory in mhd averaging? We do not know, but, we agree, that it is very interesting question.

4. As the article is submitted to Galaxies, it would be appropriate to have more discussion or examples directly relevant to galaxies or galactic dynamos. For example, what would the discussed examples possibly imply for astrophysical dynamos?

Yes, it is very interesting moment. We think that one of the important problems of astrophysical dynamos in the frame of mean-field approach is a sufficient unhomogeneity of galaxies plasma, its nonuniformity on boundaries and rotational anisotropy. We try to find the method here, which allows one to study dynamos, taking into account such effects. As we see integration over set of trajectories in path-integral approach looks promising, especially if we can define trajectories with different features in and outside the galaxy disc. This work, of course, will be continue and we make comparison of anisotropy effects with observed and numerical astrophysical applications, however now we show that this method allows us to obtain mean-field system, including anisotropy and nonuniformity conditions. Moreover, this path-integral method is interesting for galaxy dynamo studying, because the main assumption about short correlated velocity field is quite reasonable for systems, in which the typical rotational time is much less than memory time, defined by supernovas appearance. We try to improve the test and present this idea more clearly.

More detailed comments:

1) First line of the abstract: Probably “one of the basic models” is more appropriate.

Yes, of course, it is more appropriate, so we correct the abstract.

2) Page 2, Line 31: Here α and β are indeed isotropic and homogeneous, because when the mean-field induction equation is written in the form of equation (1), their isotropy and homogeneity has already been assumed. In the next sentence, “anisotropy scale” is ambiguous: do the authors mean “the scale of the inhomogeneity of the anisotropy”?

We try to say here that the using of equation (1) in the astrophysical problems, where inhomogeneity or anisotropy is considered, looks a little bit incorrect, because the equation (1) is obtained for uniform and isotropic turbulence. However, it is common way in dynamo investigations, when equation (1) is solving for helicity or diffusivity, depending on r or anisotropically distributed in space, in particular when systems are considered in some bounded regions, then inhomogeneity of parameters on the boundary is quite clear. Generally speaking, such way can be reasonable, if the scale of the averaged inhomogeneity is larger than the scale of the inhomogeneity of turbulence, but more correctly is to reobtain equation (1) for nonuniform and anisotropic case, see (25), and then use (25) for helicity or diffusivity, depending on r or anisotropically distributed in space. We add some correction in the text and hope that or idea is clear now.

3. Page 2, Line 38: The sentence “Other works are ... non-spherical objects”. Clarifying the statement a little bit more could be helpful; for example why “incorrect” and why “non-spherical objects”? Also adding some references to the next sentence (“More recent ... medium”) could be helpful.

We want to tell here that the turbulence in galactic discs can have at least three typical scales: diameter and thickness of disc and correlation scale for plasma velocity field or typical scale of turbulence vortices, for example. So assumption about two-scale turbulent structure can be questionable. We add some remark in the text.

4. Page 2, Line 48: “Gradient uncertainty” is not clear. The authors possibly mean that there is freedom to do integration by parts in equation (3) because a curl will ultimately be taken to get the induction equation. A more important issue here is the gauge dependence: In writing equation (3) the authors has dropped a gauge term ∇φ. It is not immediately obvious to me how this missing term will contribute to their path integral approach. Is dropping it at this point really appropriate?

Thank you for this comment, of course, the vector potential application needs a more detailed explanation. We use the induction equation in the form of the vector potential (3) to distance from the question about solenoidal condition of averaged magnetic field. After averaging we take curl-operation from both sides of equation (21), obtaining equality (22). Such way exclude the question, because from (22) follows that div(B) does not change in time – div from right side is zero. Therefore, we use equation (3), which, of course, has uncertainty in potential gauge. As we show in (4) this gauge allows us to write ideal induction equation in the form of full derivative, the gauge is chosen in the way then the last term in (4) is zero. We do not check here how such gauge can effect on path-integral method; however note that after curl-operation this gauge seems like not affecting on the final form of mean-field equation for B, in isotropic case we obtain well-known mean-field equation. We add some explanations in the text.

 

5. Section 2, first paragraph, second line: The authors might want to be more precise about the “very short correlation time”; for example, how short is it compared to other time scales in the system? Also, does this imply a scale separation between mean and turbulent fields? This would contradict the authors’ claim that the calculation does not assume a two-scale field.

Yes, it is good comment! We forget to explain this in the frame of astrophysical dynamo. We mean here that time of vortices rotation is much smaller then turbulence memory time. In other words, that intervals between supernova’s birthdays are much smaller that vortices rotation time. Of course, it is scale separation, but the separation of time scales, not spatial scale, which is commonly used. Thanks! We add explanation in the text.

6. Below equation (4): The last term in equation (4) is dropped, but it is not clear that it wouldn’t contribute non-trivially in the path integrals to give a term with non-vanishing curl.

See the answer on the comment 4 and the additional explanation after the equation (4).

7. Equation (6) and (7): Is time ordering taken care of when transforming (6) to (7)? I would have expected some thing like Pn i=1 Pi j=1 in (7). Also, in equation (7), it seems like the sums can now be trivially carried out because n∆t = ∆, and the equation looks like expanding equation (4) with a small correlation time ∆ [assuming the last term in (4) can be dropped]. Am I correct?

Unfortunately the time ordering is important here, because the velocity components are defined in various times, see (6). That’s why we can not use just n∆t = ∆ here and should use much more complex expandings (16) and (17). In fact, we write here not sums, but integrals, and sums are just convenient for this presentation and usually used in path integrals approaches, see references.

8. Equation (23): Both scalars α and β differ from the common results by a factor of 2 [for example Sokoloff and Yokoi [2018] who used the same path-integral method; see also Brandenburg and Subramanian [2005]]. Is there anything missing in the calculations or this is a new discovery?

That is not a new discovery, that is the general problem which connects with the questions, what we call the memory time for velocity turbulent field. We use here the memory time as the time from one time point to other time point, when turbulent field correlation disappears. But usually, such memory time is assumed as a sum of intervals from the particular moment to the next and previous moments when correlation lost. We add some explanation to the text.

9. Page 8, Line 115: “velocity field transformation” is probably not the most suitable phrase; “Lorentz force back reaction” or something like that could be better.

Yes, thank you. We correct the text.

10. Equation (39) and the discussions on the superexponential growth: (i) First, equation (39) is only valid at r = 0; for any r 6= 0, the exponent will include zeroth and linear terms in t. (ii) Even at r = 0, I got O(t) terms in the Taylor expansion of equation (38). This would be inconsistent with the author’s claim that the field grows superexponentially (and also contradicts Line 118). (iii) Furthermore, even if the leading-order terms of equation (38) was O(t2), it is still insufficient to conclude that the growth is superexponential (a counter-example is 1 − cos x = x 2/2 + · · ·). More detailed analysis and a clearer indicator is needed. In addition, I think the right panel in figure 2 does not provide conclusive evidence that the field grows superexponentially: One cannot distinguish a superexponential growth from a fast yet exponential one by inspecting this plot; probably a more suitable way is to plot ln B v.s. t 2 , or something like that. (iv) After all, the Fourier solution (30) implies exponential growths at all scales with fixed growth rates. Why would their sum, being the real-space magnetic field, have superexponential growth? It would be helpful to have a more detailed discussion on this point if the authors find the superexponential growth to be real.

Yes, it is very interesting moment! Of course, all magnetic field harmonics increase or decrease exponentially, however it does not tell a lot about magnetic field behavior in the particular point or in the particular interval. We understand (and write about it in the text) that at large time growth will be always exponential; from the other side, at small times magnetic field can be always expanded in the Taylor series, see as examples (32-33), then the growth will be defined by terms linear to t. But such features do not define the magnetic field growth at various interval, for example, at interval when the field redistributes in space from some initial distribution. We show for particular example, see case 2, that the magnetic field components increase like (39). There are no doubts that (39) is linear to t at small times and grows exponentially and large times, however, there is also well seen, that if 4*\beta*t<<r0^2<<(\alpha*t)^2 than the growth is proportional to exp(\alpha^2*t^2/\beta^2), so at this interval we observe superexponential growth. And it does not depend on considered point, if we consider non zero r, then the growth will be also proportional to exp(\alpha^2*t^2/\beta^2) multiplied on some near constant value. See figure 2, where we chose \beta=\alpha=1 and r0=200, we change it little bit from the pervious variant and chose the nonzero point r. In this case for times 200<<t<<10000 we should observe superexponential behavior. Indeed, we hope that the new figure clearly shows that between 250<t<1500 growth is faster than exponential (parabolic in log scale), while at large times t>1500 it transforms in exponential one (linear in log scale). We add some remarks in the text.

11. Section 6, first paragraph, first sentence: I think the authors meant a scale separation between the turbulence and the mean fields, rather than “two-scale turbulence”. Also, I disagree with the claim that the authors have not assumed scale separation and high Reynolds numbers. In fact, the calculations have assumed (i) short correlation time and a large scale of variation for the mean fields [equation (9), then used this in (15) for the mean field], which implies a scale separation, and (ii) stochastic velocity fields, which implies a high Reynolds number.

Yes, we agree, that in this work we assume high Reynolds number, because we assume that mhd-dynamo works. However we don’t use any direct assumptions about Reynolds number and, moreover, don’t suggest that that it is small, like in Krauze-Radler works. About two-scale structure, we also agree, but it is delicate moment, because we don’t suggest spatial scale separation, only time separation, which looks quite correct, see answer 5 and comment about short-time correlated velocity field. We correct the text and add some remarks.

12) Equation (24) does not agree with the first equation in section 6. Also it might be worth labeling all the equations.

Yes, thanks! The formula is corrected and labeled.

13. Page 10, Line 156 and 157: This sentence does not seem precise enough. I understand the authors may want to say the magnetic fields back-reacts on the flow in a different way, but that cannot be captured by the kinematic treatment here. The wording can be improved.

Thanks! We indeed want to say here that anisotropy can influence on Lorentz back-reaction. We show that isotopic flow magnetic field in initial times generates along the magnetic field. So force-free construction with currents along filed lines remains force free. In anisotropic flow, magnetic field generates inclined to current density that leads to exponentially growing Lorentz forces. We correct the text and try to explain that more clear.

14. The authors might want to refer to Zhou and Blackman [2019] (for example in the introduction section), which is on a similar theme. Dynamo growth rate with anisotropic α coefficient (as in section 4) is also in Moffatt [1978]. 3 Minor points/Typos 1. The notations could be improved. For example the cross product is denoted by [v, H] in most places, but there are a few exceptions, for example v×H (Page 2 Line 51). It would be helpful if the notations are consistent. Also the authors might wish to use a sentence to clarify their notations; for example (v, H) means dot product and [v, H] means cross product, etc. 2. Wiener process, noise, set ... should start with a capital “W”. 3. Page 5, Line 90: “is is” → “it is”. 4. Section 5, first paragraph: “xy-plain” → “xy-plane”. 5. Equation (32) and (33) can be directly obtained from the induction equation. 6. Page 8, Line 113: “mhd” is not defined. 7. Page 9, Line 117 to 119: It should be “neither analysis ..., nor analysis at large times, ..., gives ...”. 8. Page 10, Line 153: “than” → “then”.

We add references and definitions, correct notations and other mistakes. We check all points, except the point 5: we don’t see how (32) and (33) can be obtained directly from the induction equation: in (32) and (33) evolution of averaged magnetic field is defined by averaged velocity parameters, while in induction equation evolution of non-averaged magnetic field is defined by non-averaged velocity, however we check this equations once more time – it is right. Thank you very much for all corrections! We hope that the work becomes better!

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have carefully reviewed their manuscript. I think most of their response to my earlier concerns are reasonable, but there remain a few points that I wish to address.

Line 43. I think having multiple large scales is ok for two-scale approaches. For example if one has r as the radial scale and d as the thickness of a galactic disk, wouldn't the assumption r > d >> l suitable for a two-scale approach? (l is the turbulent length scale)

The authors' response to my earlier point 5 may have a typo. It was stated first that "the time of vortices rotation is much smaller than the turbulence memory time" but later it said  "the intervals between supernova's birthdays are much small than the vortices rotation time".

Regarding the blue comments below equation (4), can the authors confirm that the last term in equation (4) (or a gauge term) will only contribute a pure gradient term after both sides of the equation are path-integrated? That would be my main worry.

Regarding the factor of 2 in equation (23), the reasoning given by the authors in blue is not quite convincing. For example, the authors claimed that their \Delta has chosen to be twice smaller (shouldn't it be twice larger?) than that in the usual definition, but I found that equation (11) in this manuscript looks the same as equation (4.2) in Sokoloff (2018), yet the two approaches arrive at different results. I'm curious that at which point does the present calculation start to deviate from that in Sokoloff (2018)? What is the reason or advantage here to use a different time scale \Delta for anisotropic calculations?

In the conclusion, caution would better to be taken when the authors comment on the super-exponential growth. To my understanding this behavior crucially depends on the initial field geometry, whereas for galaxies the geometry of the initial seed fields is rather uncertain.

In the right panel of figure 2, would it be possible to add some reference parabolic curves to show more clearly the super-exponential growth? From the current version I still cannot tell it from e.g. an exponential growth with growing growth rate.

Overall, I would strongly suggest again to improve the English language and style in the manuscript.

Author Response

Review 2 (second round)

 

Comments and Suggestions for Authors

The authors have carefully reviewed their manuscript. I think most of their response to my earlier concerns are reasonable, but there remain a few points that I wish to address.

 

Dear Reviewer, thank you again for your comments! We try to correct the text and give point-to-point answers below; in the text we mark corrections and explanations by blue. We again slightly change the figure 2 to make the superexponential growth more clear and hope that the paper becomes a little bit better!

 

• Line 43. I think having multiple large scales is ok for two-scale approaches. For example if one has r as the radial scale and d as the thickness of a galactic disk, wouldn't the assumption r > d >> l suitable for a two-scale approach? (l is the turbulent length scale)

Yes, apparently you are right and for the assumption r > d >> l the traditional method of mean-field averaging would be applicable, as we suppose. Meanwhile we don’t know investigations, where possible delicate details of application of two-scale approach for multiscale systems were tested. However, here we use the path-integral approach, mostly because it allows us to obtain the mean-field system for nonuniformly and anisotropic turbulence and the required suggestion of short-correlated field looks reasonable for astrophysical systems. Is it possible to obtain similar anisotropic system by two-scale approach? – it’s very interesting question; but we can’t give the clear answer now, so it will be interesting to study it in the following works! We add some comment in the text.

• The authors' response to my earlier point 5 may have a typo. It was stated first that "the time of vortices rotation is much smaller than the turbulence memory time" but later it said"the intervals between supernova's birthdays are much small than the vortices rotation time".

Yes, of course! It was a misprint, we meant the time of vortices rotation is much larger than the short turbulence memory time. Thank you, we check this moment through the text!

• Regarding the blue comments below equation (4), can the authors confirm that the last term in equation (4) (or a gauge term) will only contribute a pure gradient term after both sides of the equation are path-integrated? That would be my main worry.

Yes, we understand the question: this term indeed gives the pure gradient term after averaging, because the equation is linear and the solution can be presented as a combination of two parts: the first one is solution without this gradient gauge, and the second is solution of the equation dA/dt=grad(v,a). Averaging the second one we can swap gradient operation (as spatial derivatives) and averaging operation. As a result at the end we obtain pure gradient averaged term, and after transition to the magnetic field (after curl-operation) this term disappears. Note that usually for isotropic case path-integral approach is applied for magnetic field, not for vector potential, so there are no questions about gauge and it is very important, that in usual way the approach gives completely the same equation – we tell about that in the text – so it also can be the confirmation that there are no mistakes in our calculations for the vector potential.

• Regarding the factor of 2 in equation (23), the reasoning given by the authors in blue is not quite convincing. For example, the authors claimed that their \Delta has chosen to be twice smaller (shouldn't it be twice larger?) than that in the usual definition, but I found that equation (11) in this manuscript looks the same as equation (4.2) in Sokoloff (2018), yet the two approaches arrive at different results. I'm curious that at which point does the present calculation start to deviate from that in Sokoloff (2018)? What is the reason or advantage here to use a different time scale \Delta for anisotropic calculations?

It is very important remark and the point here is the following: in the obtained formulas (23) for turbulent diffusivity and helicity \Delta is the double memory time. It is the standard assumption discussed particularly in the first path-integral dynamo investigations see, e.g., 1983MHD_MolchanovRuzmaikinSokolof or 1985UPN_MolchanovRuzmaikinSokolof. It follows from the idea that memory time \tau, after which there are no velocity correlation, should be measured back and forth, that guarantees the independence between velocities on the \Delta interval, therefore \Delta=2\tau. In the paper 2018JPP_SokoloffYoukoi the authors attention was focused on the dynamo nonlinear effects, thus the path-integral method was presented quite briefly via links to early works, the question had been omitted and very inconvenient notation was used – the same for \Delta interval and for the memory time. Such identical designations, for which authors apologize, lead to misunderstanding. However, path-integral approach is the same in 1983Molchanov, 2018Sokoloff and in this paper. We add some comments and explanation in the text. Thanks a lot!

• In the conclusion, caution would better to be taken when the authors comment on the super-exponential growth. To my understanding this behavior crucially depends on the initial field geometry, whereas for galaxies the geometry of the initial seed fields is rather uncertain. In the right panel of figure 2, would it be possible to add some reference parabolic curves to show more clearly the super-exponential growth? From the current version I still cannot tell it from e.g. an exponential growth with growing growth rate.

We agree and also think that the superexponential growth depends on particular initial condition, because such behavior, as it appears, connects with the redistribution of initial averaged field (we write about that in the text). We show it on the particular example in the paper and also consider more additional examples, not presented in the text, – we have no doubts that such superexponential evolution can be observed. However, we don’t know is it possible to obtain the full class of initial distributions and parameters that leads to such behavior. Here we just show that such growth can be demonstrated. The main obstacle here (we tell about this in the text) that large-time asymptotics can give us only exponential evolution, small times asymptotics – only power evolution, while superexponential growth is observed at intermediate interval. For considered example we present the figure 2, that shows faster than exponential (faster that linear in log-scale) increasing, we add blue point that correspond to the superexponential growth exp(\alpha^2 t^2/r0^2), analytically obtained in (39). We add comments into the text, where we explain that such growth should be observed for \alpha t>r0 (for considered case it correspond to the time t>100). Moreover, when r0^2 becomes comparable with 4\beta t such superexponential growth should be transformed in exponential (for considered case this border is near t=2500) and we observe such transformation from t>500 (parabolic growth in log-scale transforms to the linear one). We add some comments in the text and hope that explanation is clear now.

• Overall, I would strongly suggest again to improve the English language and style in the manuscript.

Dear Reviewer, we’ve read the text again and tried to improve the language. We also write to the Journal Editor and ask if it is possible to apply the journal editing system for our work by our review vouchers. We hope that the text will be better! Thank you very much!

Author Response File: Author Response.pdf