Author Response to the Comments of Reviewer 2: “Dependence of the critical Richardson number on the temperature gradient in the mesosphere”

7 Maximum upper atmospheric turbulence results in the mesosphere from convective and/or 8 dynamic instabilities induced by gravity waves. For the first time, by comparing the vertical 9 accelerations induced by wind shear and the buoyancy force, it is shown that the critical 10 Richardson number Ric can be estimated. Dynamic instability is developed for Ri < Ric. This 11 new approach, for the first time, makes it is possible to establish and estimate the temperature 12 gradient impact on dynamic instability development. Regarding our results, Ric increases from 13 0.25 to 0.38 as the negative temperature vertical gradient increases from ∂T/∂z = 0 to ∂T/∂z ≤ -9 14 K/km. However, Ric for the temperature, independent of altitude, is 0.25, coinciding exactly with 15 the Ric commonly used and estimated in classical studies (Miles, 1961; Howard, 1961) and 16 subsequent papers without the temperature impact. The increase in the Ric value strongly 17 influences cooling, inducing the cooling rate increase. Also, our results show that criterion Ric < 18 0.25 can only be used for the turbulent diffusion, which is characterized by eddies with sizes much 19 smaller than the scale height of the atmosphere. The Ric value increases with the increasing size 20 of the eddies, but the term “eddy diffusion” cannot be applied to transport due to the large-scale 21 eddies (Vlasov and Kelley, 2015). 22 23 2


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and using the formulas This approach and these formulas cannot be used for the mesosphere.The thickness of the surface layer considered in the paper is less by a factor of 80 than the scale height of the atmosphere (about 8 km) and this condition is very different from the mesospheric conditions where the scale heights of 4 -6 km and the thickness of the turbulent layers may be larger than 1 km and the turbulence occupies a region of 40 km.Also, there are other important distinctions between the surface layer in the lower troposphere and the mesosphere.Apparently, the reviewer does not know the principal distinctions between the surface layers in the lower troposphere and the mesosphere.

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Reviewer comment: Btw. the study of Obukhov (1971) gives a rigorous summary of the Ri and Ric dependence on the temperature gradient and the authors need to explicitly cite this study and show where they give superior scientific information.
Author response: This reviewer's statement is wrong.There is only one sentence on estimating the Ricr value in the paper (page 15): "Corresponding processing of Sverdrup's data leads to Ricr = 1/11, which is used later in numerical calculations" and then the author states that, "The determination of the critical Ri number is an important problem for atmospheric physics and may be solved only experimentally on the basis of processing reliable data for simultaneous measurements of wind and temperature distributions in the lower layer of the atmosphere".
Thus, Obukhov uses the experimentally determined value (the only value) of Ric for a very rough estimate of the temperature gradient according to his statement (page 21): "Thus, the order of magnitude of the temperature gradient calculated according to K∞ agrees with the observations.In accordance with Sverdrup's observations, the value Ric = 1/11 was used during calculations of the gradient".It is necessary to emphasize that no dependence of the Ric value on the temperature gradient is presented because the author used the only value of Ric = 1/11 that was experimentally determined.This is exactly the opposite of what we have done in our paper.We theoretically define the Ric value and calculate the different Ric values for the different temperature gradients (see Figs. 3b and 4).
It is necessary to emphasize that Obukhov's result with a huge uncertainty in the temperature gradient calculated for the Ric fixed value strongly contradicts the direct and unique dependence of the Ric value on the temperature gradient presented in our paper.This contradiction and other problems with estimates using some formulas presented in the paper are explained in the paper by A.S. Monin and A.M. Obukhov, "Turbulent mixing in the atmospheric surface layer" (Trudy  Geophys.Inst., 1954, N o 24, 151 and "Turbulence and atmospheric dynamics", ed.J.L. Lumley, NASA, CTR Monograph, November 2001, p. 164).The authors of this paper state that "Obukhov used some insufficiently reliable data (the critical Richardson number was erroneously taken to be 1/11 on the basis of Sverdrup's results) and therefore we could not directly apply his formulas for the practical calculations".This statement is in good agreement with our attempt to use some of the formulas given in Obukhov's paper.
We are very confused by the reviewer's recommendation of this paper, which, according to the author's statement in his next paper, presents the wrong Ric value and the wrong formulas are used.
It should be noted that Obukhov's paper was published in 1946 by the journal Trudy Inst Teor.Geophys, vol. 1, 95-115.However, this publication was really inaccessible outside of the USSR.The reference given by reviewer 2 corresponds to a translation of this paper published by the journal Boundary-Layer Meteorol, 1971, 2, 7-29.In the introduction to this publication, J. A. Businger and A.M. Yaglom explain the reason for this publication: "Probably the major contribution of the paper is the introduction of the 'length scale of the dynamic turbulence sublayer', L. This length scale was later introduced independently by Lettau (1949), and at present, it is commonly known as the Monin-Obukhov length.Its fundamental role in the whole field of boundary-layer meteorology was most clearly explained in the well-known paper by Monin and Obukhov (1954)".The authors of the introduction do not mention the problem with the Richardson number in Obukhov's paper because of the comments in Monin and Obukhov (1954) discussed above.

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Reviewer comment: A) Most importantly, I have serious concern about the validity of the methodology and flawlessness of the analytical derivations in this paper: The crucial point of this study is that the authors assume adiabatic expansion.While this can be a good assumption for the GW induced perturbations, it is completely irrelevant for the background, where e.g., the solar tides govern a significant part of the mesospheric variability.Also, the authors use this assumption to connect the vertical gradient of full (background + disturbed) density distribution to the full temperature and its gradient and wind shear (Eqs. 6,7,8,9,10).Also in the light of tides, this assumption crucial for the paper needs to be properly justified, ideally by referencing observational studies.
Author response: "Turbulence is generated by waves breaking in the MLT through mechanisms such as convective and dynamic instabilities (e.g., Hodges, 1969;Lindzen, 1981;Zhao et al., 2003;Liu et al., 2004;Williams et al., 2006;Hecht et al., 2014) . Geophys. Res., 72, 3455-3458, 1967) pointed out that it is unlikely to have conditions for dynamic instability without gravity waves.Tides alone are not sufficient to induce dynamic or convective instabilities, but the tides can influence the conditions for dissipation of the gravity waves and the development of dynamic instability due to change in the temperature gradient.In any case, adiabatic expansion is a fundamental process for dynamic instability and the adiabatic lapse rate is a very important parameter.This assumption is used to derive the buoyancy frequency formula (see, for example, Peixoto, J. P., and Oort, A. H.: Physics of Climate.New York: Springer-Verlag, 1992), which is included in the chain of equations ( 6)-( 10).The Richardson number depends directly on the adiabatic lapse.Unfortunately, the reviewer does not explain why adiabatic expansion cannot exist for the tides.We do not consider the mesospheric background parameters' variability induced by the different processes.We only consider the dependence of dynamic instability on the temperature gradients in the mesosphere.Unfortunately, the reviewer does not explain what kind of observational studies he means.In our paper, the results of the experimental data (Bishop et al., 2004;Kelley et al., 2003;Larsen, 2002;Lubken, 1997) are used.

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Reviewer comment: But more than just general doubts about the validity of this assumption, the authors make errors also in analytical description, where in eq. 8, which shows partial derivative of T with altitude they refer to it as (P4L81) "temperature gradient in the parcel (sic) with upward motion and adiabatic expansion" -but for this, total derivative would have to be shown.

Author response:
We are very surprised by this comment.Eq. 8 is the result of the simple combination of generally accepted Eqs. 2, 6, and 7 with partial derivatives and it is impossible to obtain this formula with total derivatives in only one equation in this combination.Eq. 6 is the key formula and presents the temperature gradient corresponding to adiabatic expansion due to upward parcel displacement.This result does not depend on the kinetics of parcel motion.This is the generally accepted approach for estimating the effect of parcel displacement on the temperature for adiabatic expansion/compression.Unfortunately, the reviewer's statement is too general without an explanation or a reference.

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Reviewer comment: Most importantly, on their way from eq. 6 to 10 they use in P4L80 an equation for Ri based on different assumption (they don't tell anything about this formula, which is crucial) and then they consider this Ri (general?) to be equal to the Ri in eq.7 (adiabatic expansion) for deriving eq. ( 10).
Author response: The derivation of Eq. 6 was given in Appendix 1. Taking this comment into account, an additional explanation is included in the text (page 3) and Appendix 1.The main point is that Eq. 4 corresponds to incompressible fluid and ωB 2 =(-g/ρ0)∂ρ0/∂z, but Eq. 6 corresponds to compressible fluid (adiabatic expansion) and ωB 2 = (g/T)(∂T/∂z + g/Cp) should be used, so in this case, Eq. 7 and Eq. 8 must correspond to compressible fluid.
Revision in the paper: page 3, lines -57, 59, 63 page 4, lines 72, 73, 80, 81, 83, 90 page 16, lines -262, 263, 264, 266 Reviewer comment: A similar situation takes place in section 3, where they give equation 13b (P6L110) without properly discussing how they derived this equation and the underlying assumptions (polytropic atmosphere?).This formula (13b) and the formula for wind shear (eq.10) are the crucial parts of the paper, because every other result then presented is only a trivial evaluation of Ri based on those formulas.
The derivation of eq.13b is now given in Appendix 4.

Reviewer comment:
The authors need to carefully rewrite all of their analytical derivations, distinguish properly between local and total derivatives, list the assumptions made an ensure consistency between the assumptions and also distinguish in their formulas between constants and functions of altitude (f(z)).Without this it doesn't make sense to discuss any results given later in the text (poor evaluation of the derived formulas), because my personal opinion (the authors are welcomed to prove otherwise) is that the results are dominated by flaws in their analytical construct.
Author comment: The reviewer's negative comments are too general without any evidence, examples, or references.For instance, the reviewer says that "the results are dominated by flaws" but does not prove his/her mere allegations.Moreover, the reviewer has stated (in two separate instances) that the assumptions have not been explicitly listed in the paper, whereas in fact, they were provided on pages P2L27,28; P3L58-64; P4L72,73; P5L96,97; P6L 111-113; P7L114,115 and L126,127; and P13L204-206 of the submitted manuscript.Also, it is totally unclear why the reviewer insists on using "total derivatives" while all the well-known formulas are customarily defined in terms of partial derivatives.

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Reviewer comment: Language: Non-scientific language is used frequently, with weird phrases like: we could find just one paper… or acceleration in wind shear the authors write that some study is wrong, but do not prove it.Just to list: What is the?P5L92 Does wind shear really induce vertical accelerations?(no, you have to replace the word induce by e.g., support) Page 3, L 67 not wind shear nor stability are forces.Those were the most striking ones.I am not listing all the typos made in the manuscript because I expect major changes before it can be assessed for publication.
Author response: Note that reviewer 1 did not have a problem with the language used in our paper.We made a few language corrections to the text.The reviewer's statement, "the authors write that some study is wrong, but do not prove it," is incorrect.The explanation was presented in detail in Appendix 3. Note that this reviewer's statement does not demonstrate a language problem.Our paper stated, "The goal of this paper is to estimate the critical Richardson number,   , corresponding to the equilibrium between the buoyancy force and the force induced by wind shear in the mesosphere.Dynamic instability is developed for  <   .Our approach considers the acceleration corresponding to both forces, taking into account the mesospheric temperature height distributions".It is not clear why the reviewer objects to the word "force".Again, note that reviewer 1 did not have a problem with the language used in our paper.
In general, reviewer 2's apparent lack of understanding concerning the distinction between the surface layer in the troposphere and the mesosphere, the unproven statements about the important role of tides for dynamic instability development, the use of total derivatives in commonly used formulas, and his/her request to present the derivation of the well-known and commonly used formula of density distribution in the upper atmosphere clearly demonstrate that the reviewer is not adequately familiar with the physics of the upper atmosphere and dynamic instability.One obvious evidence of this is the reviewer's persistent recommendation of a paper that, according to the author's statement in his next paper, presents the wrong Ric value and uses the wrong formulas.
Revision in the paper: Changes were made, including on page 3.

Introduction
In general, the Richardson number  can be defined as the ratio of the destruction of turbulent kinetic energy by buoyancy forces and the production of turbulent energy by the wind shear flow.
This determination leads to the relation where   is the buoyancy frequency, and T is the temperature, g is the acceleration of gravity,   is the heat capacity at constant pressure, and is the vertical shear of the horizontal wind with the velocity V(z) height profile.It is generally accepted that a dynamic instability develops when the Richardson number is less than ¼, i.e., the parcel's vertical motion induced by wind shear dominates the motion induced by the buoyancy force.The former creates and the latter destroys these perturbations.Most authors use the critical Richardson number   < ¼ without references.Some authors refer to Miles (1961) andHoward (1961).They consider the stable-stratified, horizontal shear flows of an ideal fluid.A set of studies takes into account the time-dependent shear flow and the results of laboratory experiments (Peixoto and Oort, 1992;Galperin et al., 2007).However, we could not find papers on the critical Richardson number that take the mesospheric conditions into account.Miles and other authors (Abarbanel et al., 1984;Ligniéres et al., 1999;Galperin et al., 2007) did not consider the temperature's influence on the   value.However, the eddy turbulence peak is observed in the mesosphere or the lower thermosphere where the large negative and positive gradients of the temperature occur.We could find just one paper [Hysell et al., 2012] on the estimate of the   value in the lower thermosphere.Using the data on observations of the sporadic E layer, Hysell et al. (2012) inferred the parameters of wind shear corresponding to the irregularities observed in the layer and estimated the   value of 0.75.However, the authors used the wrong formula for the background density, resulting in densities much larger than the observed atmospheric density corresponding to the hydrostatic equilibrium.It is shown in Appendix 3 how 0.7 <   < 0.8 can be found due to the background density used by Hysell et al. (2012).
The principal measure of stability regarding the buoyancy effects of the density gradient for overriding its inertial effects in the incompressible fluid is the Richardson number given by formula (1) in Miles (1961), which can be written as where ρ0 is the density and V is the horizontal wind velocity.This formula can be rewritten as This will be used here to estimate the accelerations due to wind shear and the buoyancy force in compressible fluid under mesospheric conditions.
The goal of this paper is to estimate the critical Richardson number,   , corresponding to the equilibrium between the buoyant force and the force supported by wind shear in the mesosphere.
Dynamic instability is developed for  <   .Our approach considers the acceleration corresponding to both forces, taking into account the mesospheric temperature height distributions.

Acceleration Induced by Wind Shear
We start from formula ( 5 where is the acceleration in wind shear.As can be seen from Fig. 1, this acceleration increases with the increase of the vertical size of the wind shear layer.Note that this size cannot exceed 1-2 km according to the experimental data (Larsen, 2002).The   dependence on the altitude is linear because ( −  0 ) ≪  0   (1 + /2) for − 0 < 2 km.

Acceleration Induced by the Buoyancy Force
The buoyancy force is   = (  −   ) where   and   are the background atmospheric density and the disturbed density, respectively.The acceleration is given by The atmospheric density distribution can be given by for   / = 0 in the mesopause and the formula for   / =  < 0 below the mesopause, and   =  0 /mg is the scale height of the atmospheric gas.By integrating equation ( 6) with the temperature and temperature gradient given by formulas ( 8) and ( 9), it is possible to get the disturbed density distribution ( 0 =  0 ), and the acceleration corresponding to the buoyancy force can be written as for   / = 0.As seen from Fig. 1, there is very good agreement between the   and   absolute values for   = 0.25, and  0 = 140 K and  0 = 200 K for the vertical size of a stable wind shear layer that is less than 400 m.The   value becomes larger than the   value for  −  0 > 400 m, which means that the   value should be increased.The turbulence develops if   is larger than the   that corresponds to  <   .We emphasize that the perturbation scale sizes induced by wind shear do not exceed 1-2 km, according to the observations (see Lübken (1997)).
Note that formula (13b) should be used instead of formula (13a) in the nominator of formula ( 15) for atmospheric temperature distribution with    < 0. As can be seen from Fig. 2, the   values significantly decrease in this case, since the atmospheric density given by formula (13b) is larger and the density gradient is less than the density and gradient corresponding to formula (13a).The small buoyancy force corresponds to the small density gradient.This dependence explains the   reduction with the   decrease.

Estimating the Richardson Number
Using formulas ( 11) and ( 15) in the equation   +   = 0, the formula for   can be inferred: The   values calculated by formula ( 16) and this formula with {[ 0 − ( −  0 )]/ 0 } (/−1) (see formula (13b) instead of the exponential term) are shown in Figs.3a and 3b.The   values increase with increasing altitude, corresponding to the vertical expansion of the region of the stable wind shear.However, according to the experimental data (Larsen, 2002;Kelley et al., 2003;Bishop et al., 2004), the wind shears are very unstable.As mentioned above, the size scales of the density perturbations do not exceed 1 -2 km, according to the observations.A more accurate consideration of eddy turbulence (Vlasov and Kelley, 2015) concludes that the scale size of density perturbations l should be much less than the scale height of atmospheric gas, l << HA and l << 4 km for   =  0 = 140 K and l << 5.7 km for   =  0 = 200 K.However, this restriction can only apply to turbulence corresponding to the eddy diffusion approximation (Vlasov and Kelley, 2015).
As seen from Fig. 3a   Thus, turbulence can develop with   > 0.25 for wind shears with a vertical size of 1-2 km, but this turbulence may not correspond to eddy diffusion.The scales of the density fluctuations are very small (for example, see Lübken (1997)) that correspond to z → z0.However, the   value estimation for z → z0 is problematic because, in this case, the numerator and denominator in formula ( 16) try to attain zero.This uncertainty can be solved using L'Hospital's rule, leading to the formula (see Appendix 2) for the   limit value for  →   .This formula corresponds to the limit value formula ( 16

The Influence of 𝑹𝒊 𝒄 Dependence on G on Cooling in the Mesosphere
The eddy turbulence heating/cooling rate can be given by the equation ( where eh K is the coefficient of the eddy heat transport, ρ is the undisturbed gas density, and b is a dimensionless constant given by the relation obtained using the results of Gordiets et al. (1982), where P is the turbulent Prandtl number.According to equation ( 18), the Qed value is given in units erg×cm -3 ×s -1 .The  ℎ value is given by where ε is the energy dissipation rate, and b can be given by formula (19).The vertical distribution of the ε value in the turbulent layer can be approximated by the Gaussian function where h is half of the layer thickness and   is the ε value at the altitude of the layer peak   .
Using this approximation, dividing equation ( 18 ( Using the   dependence on the temperature gradient given by formula (17), the impact of the Richardson number on the cooling rates can be estimated.According to the results in Fig. 5, the cooling rates increase by a factor of 2.2 for 0.25 <   < 0.38 corresponding to 0 ≤ G ≤ -9 K/km, but the G value influence on the cooling for   = const = 0.25 is very small (curves near the thick solid curve).Note that the turbulence induced by the large wind shear may not correspond to the eddy diffusion heat transport.The values of   ,   , and h correspond to the experimental data (Lübken, 1997).

Conclusions
For the first time, by comparing the accelerations in wind shear and the buoyancy force, it is shown that the critical Richardson number, corresponding to the equilibrium of these forces, can ) corresponding to compressible fluid, and adiabatic expansion should be taken into account in the mesosphere.Differentiating the adiabatic relation  −/(−1) =  corresponding to Poisson's equation where  = / and p is the pressure; m is the mean molecular mass;  =   /  ;   and   are the heat capacities at constant pressure and volume; /( − 1) = 1 + /2;  = 5 is the number of degrees of freedom for diatomic gas; and κ is the Boltzmann's constant, it is possible to get the adiabatic expansion equation derivation of this formula in Appendix 1).It is necessary to note that formula (6) corresponds to compressible fluid and, according to (5): to compression fluid.Taking into account (/) 2 =   2 = (/)(/ + /  ) and using formula (6), the temperature gradient in the parcel with upward motion and adiabatic expansion can be given by the equation By substituting formulas (8) and (9) in formula (7) multiplied by ( −  0 ), it is possible to obtain the formula   =  2 (− 0 ) 2[ 0   (1+/2)−(− 0 )]

Figure 2 .
Figure 2. The height profiles of the acceleration of the buoyancy force calculated by formula (15) with the nominator  0 {[ 0 − ( −  0 )]/ 0 } (/−1) for  0 =  0 = 140 K and 200 K (thick , the   value of 0.25 corresponds to perturbations with scales less than 10 m, and the   values reach 0.256 and 0.263 for l = 200 m and 400 m and for  0 = 140 K and 0.254 and 0.257 for  0 = 200 K, respectively.The   value of 0.25 corresponds to the mean value l = 27.3 m obtained by Lübkin (1997), using the measured spectrum of the density fluctuation.Vlasov and Kelley (2015) reconsidered the results ofKelley et al. (2003) and found that the spectrum scale fluctuations inferred from the meteor train turbulence observations can be approximated by Heisenberg's formula with l = 119 m, and eddies with very large scales may occur in the narrow layer of localized turbulence.As can be seen from Fig.3b, the   values increase with the increase in the negative gradient of the temperature and can reach almost 0.36.

Figure 3a .
Figure 3a.The height profiles of the critical Richardson number calculated by formula (16) with

Figure 3b .
Figure 3b.The height profiles of the critical Richardson number calculated by formula (16) with {[  − ( −   )]/  } (/−) instead of the exponential term for the  0 = 140 K with / = ) with the term {[  − ( −   )]/  } (/−) instead of the term [−( −  0 )/  ].The   dependence on the negative temperature gradient, given by formula (17), is shown in Fig. 4. The G increase improves the conditions for the dynamic instability development.Note that the   value for G = 0 coincides with the results of Miles (1961) and the commonly used value of   .