We thank referee 2 for having looked once again carefully through the revisions. Our new changes are now shown in blue. 1. General remarks: >> In the resubmitted version of the paper, the concerns and proposals of >> my first report were fully addressed. Beyond that, the authors have added >> extended new sections (2.4 and 2.5) devoted to defining characteristic >> amplitude and energy density of gravitational waves and related spectral >> functions. Moreover, a new computational approach has been introduced in >> Sec. 2.7. As a result, the paper has been enlarged by 14 pages and the >> reader has over wide stretches no longer the impression that the topic of >> the paper is the time step constraint. I understand that having the >> definitions, given in Secs. 2.4 and 2.5, at hand may facilitate the >> presentation of the results. But I wonder whether this paper is really the >> first one to introduce the needed quantities in full clarity and detail, >> given earlier papers of the authors on GWs in the early universe and, of >> course, decades of other authors’ work on GW in general. As another aspect, >> having to go through six pages of definitions, not being accompanied by >> strong pragmatic motivations, is somewhat tedious, at least for somebody not >> working intensively on the topic. However, as the paper as a whole is >> carefully written and provides very useful information on the numerical >> simulation of GWs, sourced by turbulence, I don’t want to stand in the way >> and hence recommend publication after my detailed comments and proposals >> (see below) have been considered. Yet, I would appreciate if the authors were >> to find a way to shorten the paper, preferently by avoiding to repeat >> introduction of quantities whose definitions are easily accessible somewhere >> else. We have now moved much of that text to Appendix B. 2. Detailed comments and proposals: >> whole paper: “time of generation” is often used with the formal meaning >> of a time instant (with “at”), but as generation is happening during a >> time span, it is better to speak about either the “starting time of >> generation” or the “period of generation” according to appropriateness. This has been addressed properly and modified to make clear when we refer to the starting time of generation, and when we refer to the whole period when turbulence is present, and hence GWs are being generated. >> p.2, l.18: weaken “usually” to “often” changed >> l.37: better “convergence” -> “stability” changed >> l.48: add “with constant coefficients” behind “wave equations” added >> l.56/57: This sentence is somewhat contradictory to Eq. (2) which just >> indicates that the turbulent source and the sound wave are considered >> in the same region. Some text has been added in order to clarify this. Added reference to aeroacoustic book. Lighthill's wave equation describes the perturbations of density \delta \rho in a surrounding stationary medium (with constant sound of speed), due to the presence of a region of turbulence. Hence, the equation is evaluated in the turbulence region (where the sourcing term is non-zero due to the presence of a turbulent velocity field), but the density perturbations are only valid in the surrounding area (where the speed of sound is constant). >> p. 3, l.19/20: “expected value” -> “expectation value” changed >> l. 22.5: “The acoustic power is the radial intensity per unit time.” >> Strange as intensity has already the unit power/area, so why another >> “per time”? This sentence was wrong and it was not relevant so it has been removed. The acoustic power is obtained by integrating over a surrounding surface the intensity, as pointed out by the referee. >> “radial” unclear. Added sentence to clarify The radial intensity refers to the propagation away from the source, i.e., in the radial direction, where only the distance to the source is relevant. >> l.38.5: add “exhaustively” in front of “describes”. added >> l.40.5: add “(scaled)” behind “called” added >> l.43.5: “being δ ij -> “δ ij being changed >> l.56.5: “The total stress . . . expressed . . . as:”, but what is expressed >> is the energy-momentum tensor, hence the negative stress. modified >> l.61: “being ρ” -> “ρ being” changed >> p.4, l.17: better “denoting derivative w.r.t. physical time t phys, >> while the prime denotes derivative w.r.t. normalized conformal time” changed >> l.19.5: add “all taken” in front of “at the time” added >> p.5, l. 22.5: Zero should be bold. changed >> l.23.5: “because uniform fields . . . ” not very clear as already the >> projection operator is not well defined for k for 0. We have now removed "due to the traceless projection that rules out scalar factors" from that sentence, and have added: Since \Eq{GW_eqn} is the result of linearisation, we assume that the spatial average of $T_{ij}^{\rm TT}(\xx, t)$ vanishes. in the middle of the paragraph after Eq.(3) and restrict Eq.(4) only to |k|>0. >> l.44-49: Contradictory as it is first said that e_{ij}^{+,x} must be >> even, but later that they could also be defined to be odd. I think >> its the base vectors e^{1,2,3} which can be even or odd. This sentence has been corrected. As pointed by the referee, it is e^{1,2} that can be defined even or odd with respect to parity transformation k -> -k, but they have to both (e^{1,2}) have same parity, such that e_{ij}^{+,x} are both even operators, as required to reproduce the required modes. >> p.6, l.21.5: add “physical” in front of “strains” added >> l.23.5: set h^0_{\rm rms} between commas added >> Eq. (20): provide an explanation for the factor a^{\rm end} added definition >> l.28.5: better: “scaled such that the scale factor at the time of >> generation, a_*, is unity” changed >> Eq. (22): provide a definition for E_{\rm rad}^* (which should be analogous >> to that of E^0_{\rm crit}) added reference to the equation where this is defined >> l.51: better “event” -> “epoch” changed >> p.7, l.32: Using the same symbol for two different quantities is not >> reader–friendly. changed to avoid using the same symbol without explicit time dependence as a new variable for this quantity. Same has been done for following expressions (old eqs. 26, 27, 59, 60; new eqs. B.6, 7, 36, 37). >> below Eq. (28): “over all directions of \frac{1}{2} ...” -> “of \frac{1}{2} >> ... over all directions of \pmb{k}” changed >> p.8, l.13/14: “characteristic strain h_{\rm rms}(k,t) -> “characteristic >> amplitude h_{\rm rms}(t)” changed 'characteristic strain' to 'characteristic amplitude' >> Eq. (31), 1st l.: A factor 1/L^3 seems to be missing in front of the >> last integral. In practice, we work with discrete Fourier modes, so the Parseval theorem is = \sum htilde^2. We have changed this now. >> l.35.5: “The wavenumber k ... ” already said on p. 7. “normalized”: >> In (7) where the wavenumber vector is used first, there is no indication >> of its normalization. this line has been removed and the corresponding text slightly modified >> Eq. (36): Normally a scale should be 2\pi/wavenumber. We really only work with k_GW and 1/k_GW, which is a length scale. We do this similarly for the correlation length, which is now in Eq.(B.39). It would then be awkward to include a 2pi in the definition and then keep working with xi/2pi all the time. To clarify this, we have now added the following just after Eq.(B.16). Note that we have defined here xi_GW without 2pi factor, which is analogous to our definition of the magnetic and kinetic correlation lengths; see Kahniashvili et al. (2010). >> p.9, l.1: Using the same symbol for two different quantities is not >> reader–friendly. In this case, for hrms, \Omega_{GW} and \Xi_{GW}, it is common in cosmology to use the same symbol for both the spectral function per logarithmic wavenumber interval, and the total integrated (over all wavenumbers) quantity, that is why we have decided to use this nomenclature in the paper and to use it (in analogy) for \Omega_{K, M}. >> l.35: remove doubled “tensor” removed >> Eq. (40): What’s the meaning of the angle brackets? This equation has now been moved to the appendix as Eq.(B.18). We have now changed it into a shell integration, which is really what is done here. >> l.49: “Due to ...” This sentences could be confusing as it is implicitly >> clear that the basis vectors used in (11) and (41) should be the same. this sentence has been removed, since it was repeating a clear statement >> l.57.5: “relation” -> “analogy” changed >> p.10, l.31.5: “is expressed ...” is just repeating the formula before, >> thus dispensable. deleted repetitive sentence, and modified accordingly >> Eq. (50): second prime too small changed >> Eq. (51): Shouldn’t there be an asterisk at \tilde{h'_x}? yes, this has been corrected >> p.11, l.23: “are” -> “can be” changed >> l.56.5: “characteristic length ... corresponding to the source” Here, >> the term “integral scale” could be mentioned. added sentence >> p.12, l.11: set “normalized” between quotation marks added >> l.36.5: If \nu and \eta are constant in the simulations it could be >> mentioned already here. added sentence stating this >> p.13, l.9: “onto” -> “into” ? changed >> l.14.5: “obtained” does formally refer to “gauge” while supposedly >> referring to h_{ij}. modified >> Eq. (71): Better to use another symbol for the index of the substep. >> Indicate that it is running from 1 to 3. the symbol i to indicate the iterative process has been changed to l, and the final solution is changed to be q_3 for the third-order scheme, with q_0 being the previous timestep solution. >> l.49.5: delete “this” removed >> l.60: What is the meaning of “normalized” here? when we talk about normalised quantities along the paper we refer to those quantities normalised according to what it is described in appendix A. Hence, omega (normalised) = omega (physical)/H_* added reference to appendix to make this clear >> p.14, l.41: “the time ... to be the” -> “this time to be at the” changed >> l.44/45: delete “where ... the Beltrami field” as already said deleted and modified >> l.47: “In this case ...” The case of dynamo-generated fields would >> require an initial seed field. added some text to clarify this. Magnetic field could be generated if \eta \neq 0, even though initial magnetic field is zero >> p.16, Eq. (86): indicate time average at symbols h_{\rm rms} and >> Ω_{\rm GW} (e.g., by overbar); factor 1/2 missing in Ω_{\rm GW} overbar added to identify time average >> l.42: “negative k” strange as k is the modulus of the wave vector changed to bold k, since this refers to our definition of positive and negative k vectors for the projection. >> Eq. (90): Formally, P is undefined for k other than \pm 2k_0. True. The equation has been removed since it is repetitive, and now is just stated inline. We've added that P = \pm 1, for \pm 2k_0, and undefined for other values of k. >> p.17, l.20: k -> k_0 modified >> p.18, Fig. 1: Instead of giving the abscissa and ordinate value ranges >> in words, I would highly recommend to put logarithmic ticks onto the axes. >> Eases reading the plot a lot. We have done this now. >> l.57: “we consider ...” already said in Sec. 3.1 removed >> p.19, Fig. 3: Perhaps Ω_{\rm M}, Ω_{\rm GW} at the ordinate axes is helpful. We have done this now for Figures 3 and 4. >> p.20, l.46: “dashed” -> “solid” (I guess) modified >> p.21, l.25: If k_* means the same as in Fig. 3, no explanation for it is >> needed. removed >> l.27: “magnetic initial” -> “initial magnetic” modified >> l.34: “wavenumber” -> “wavenumbers” modified >> l.35: Due to the new text on Fig. 4, “degradation” is no longer clear, >> refer again to Fig. 3 or so. added reference to figure 3 and to approach I >> p.22, l.51: “linear wavenumber interval” unclear this has been removed to avoid confusion and slightly modified >> l.58: “BHLS showed that ...” hard to understand We have now skipped all excessive detail, because it was not critical to this paper. >> p.23, Fig. 7: Shouldn’t it read k^{-5/3} \epsilon_{\rm K}^{-2/3} at the >> ordinate? the usual linear spectrum is equivalent to the logarithmic spectrum over k, i.e., E_K (k)/rho_rad = \Omega_K (k)/k, which leads to k^{-5/3} k = k^{-2/3}, which is what is shown in the ordinate >> l.27: “compensated kinetic spectrum” -> “kinetic spectrum compensated” modified >> l.29: ‘theoretically expected” needs reference. We have now removed "theoretically" and added 2 references. >> l.37: “above” unclear. If it means Sec. 3 this should be indicated. changed >> l.45: “relative to the usual case ...” better to understand when >> “relative to Run D” changed to indicate that this refers to Run D >> p.24, Fig. 8: Ordinate annotation should indicate compensation. changed >> l.27: move “compensated” to front of “with”; “as in figure 7” dispensable modified >> l.31.5: “similarly rapid variation” unclear added text to clarify that this refers to oscillations compared to GW >> p.28, Eq. (A5): seems to me equivalent to (A4), as T_* is still a variable, >> not specified to the value at the electroweak phase transition. The point of eq. (A5) is to give a specific value of the Hubble parameter at the electroweak phase transition. It is very common to add the factors of variable T* and g*, to be able to directly obtain values of H* for different temperatures, since the exact value of the temperature at the electroweak phase transition is not completely certain. However, T* = 100 GeV is a commonly used value for the phase transition, as well as g* = 100. >> p.29, l.26.5: role of “and” not clear. It can be said more clearly that >> time of generation and initial time of simulation are equal. second line has been removed since this has been clearly stated in the paper, and also stated some lines below.