Enstrophy Transfers in Helical Turbulence

Enstrophy transfers in helical turbulence

Shubhadeep Sadhukhan,1, ∗ Roshan Samuel,2, † Franck Plunian,3, ‡ Rodion Stepanov,4, § Ravi Samtaney,5, ¶ and Mahendra Kumar Verma6, ∗∗ 1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India 2Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India 3UniversitéGrenoble Alpes, UniversitéSavoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France 4Institute of Continuous Media Mechanics UB RAS, Perm, Russian Federation 5Mechanical Engineering, Division of Physical Science and Engineering, King Abdullah University of Science and Technology - Thuwal 23955-6900, Kingdom of Saudi Arabia 6Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016 (Dated: April 23, 2020) In this paper we study the enstrophy transers in helical turbulence using direct numerical simulation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra (∼ k−5/3) and fluxes (constants). We also calculate the separate contributions to enstrophy transfers via velocity to vorticity and vorticity to vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the fluxes due to vorticity stretching are larger than that due to advection. These transfers too are insensitive to helicity injection.

I. INTRODUCTION tensively studied [7–26]. Andréand Lesieur [9] studied the effect of helicity on the evolution of isotropic turbu- Turbulence is a classic and still open problem in fluid lence at high Reynolds numbers using a variant of Marko- dynamics. The energetics of homogeneous and isotropic vian eddy-damped quasi-normal (EDQNM) theory. Po- turbulence were explained through the celebrated works lifke and Shtilman [10] showed that the energy decay is of Kolmogorov [1, 2]. Kolmogorov argued that under sta- slowed down by large initial helicity. Waleffe [11] intro- tistically steady state, the rate of energy supplied by an duced helical decomposition of the velocity field and dis- external force is equal to the rate at which energy cas- cussed triadic interactions in helical flows. Yokoi and cades from large to smaller scales which is equal to the Yoshizawa [12] reported the effects of helicity in 3D in- rate of energy dissipation. Obukhov [3] and Corrsin [4] compressible inhomogeneous turbulence with the help of generalized Kolmogorov’s theory to isotropic turbulence a two-scale direct-interaction approximation (DIA). Zhou with statistically homogeneous fluctuations of tempera- and Vahala [13] showed that helicity does not alter renor- ture field embedded in a turbulent flow. The theories of malized viscosity. Ditlevsen and Giuliani [14], Chen et al. Kolmogorov, Obukhov, and Corrsin, collectively referred [15], and Lessinnes et al. [16] studied helicity fluxes for to as Kolmogorov-Obukhov theory, predict that the ki- different types of helical modes. Chen et al. [17] stud- 2 −5/3 ied intermittency in helical turbulence. Teitelbaum and netic energy spectrum E(k) = uk/k ∼ k , and the 2 Mininni [18] explored the effect of helicity in rotating spectrum of temperature fluctuations Eζ (k) = ζk /k ∼ k−5/3, where k = 2π/l is the wavenumber with l as the turbulent flows, while Pouquet et al. [19] reported ev- idences for three-dimensionalization recovered at small length scale, and uk and ζk are respectively the velocity and temperature fluctuations at wavenumber k. Re- scales. Stepanov et al. [20] explored the helical bottle- searchers have also studied the spectra of density and neck effect in 3D isotropic and homogeneous turbulence other passive scalars in the same framework. Refer to using a shell model. Maximum helicity states have been Lesieur [5] and Verma [6] for detailed references. It is studied by Biferale et al. [21], Kessar et al. [22], and Sa- important to note that the effect of helicity is absent in hoo and Biferale [23]. Mode-to-mode and shell-to-shell the above-described works. helicity transfers have been studied by Avinash et al. Energy and helicity are two inviscid conserved quanti- [24], and Teimurazov et al. [25, 26]. Helicity may also arXiv:2004.10621v1 [physics.flu-dyn] 22 Apr 2020 ties in three-dimensional (3D) turbulent flows. The effect play a crucial role in magnetohydrodynamics [27, 29, 30] of helicity in fully developed 3D turbulence has been ex- and dynamo action [31–34]. In this paper we derive formulas for mode-to-mode enstrophy transfers and their associated fluxes. These quantities arise due to the nonlinear interactions. In ∗ [email protected] Sec. II we derive the expressions of mode-to-mode en- † [email protected] ‡ [email protected] strophy transfers and the corresponding fluxes. In § [email protected] Sec. III, after introducing the helical forcing that we use, ¶ [email protected] the enstrophy fluxes are calculated from direct simula- ∗∗ [email protected] tions of 3D turbulence using the pseudo-spectral code 2

TARANG [35, 36]. We conclude in Sec. IV. with k0 + p + q = 0. Using the incompressibility condition, we can show that Sωω(k0|p, q) + Sωω(p|q, k0) + Sωω(q|p, k0) = 0, (9) II. ENSTROPHY TRANSFERS AND FLUXES where Sωω represents the advection of vortices, thus in- The motion of an incompressible fluid is described by volving enstrophy exchange among the vorticity modes 0 the Navier-Stokes equations ω(k ), ω(p), and ω(q). On the other hand we have ωu 0 ωu 0 ωu 0 ∂u S (k |p, q) + S (p|q, k ) + S (q|k , p) 6= 0, (10) = −(u · ∇)u − ∇p + ν∇2u + F, (1) ∂t where Sωu represents the stretching of vortices, thus ei- ∇ · u = 0, (2) ther increasing or decreasing the enstrophy Eω. Inci- dentally (10) implies that enstrophy is not a conserved where u and p are the velocity and pressure fields, ν the quantity in 3D hydrodynamics. kinematic viscosity, and F an external force. Now we can split the above transfers into individual The vorticity ω = ∇ × u obeys the following equation: contributions from modes p and q: ∂ω Sωω(k0|p, q) = Sωω(k0|p|q) + Sωω(k0|q|p) (11) = (ω · ∇)u − (u · ∇)ω + ν∇2ω + ∇ × F, (3) ∂t Sωu(k0|p, q) = Sωu(k0|p|q) + Sωu(k0|q|p), (12) where the first two terms in the right hand side corre- with spond to vorticity stretching and vorticity advection re- Sωω(k0|p|q) = −=[(k0 · u(q))(ω(p) · ω(k0))], (13) spectively. In Fourier space the vorticity, which is defined ωu 0 0 0 as ω(k) = ik × u(k), satisfies S (k |p|q) = +=[(k · ω(q))(u(p) · ω(k ))]. (14) Here Sωω(k0|p|q) and Sωu(k0|p|q) denote two kinds of d X ω(k) = i (k · ω(q))u(p) − (k · u(q)) ω(p) mode-to-mode enstrophy transfer, both being from p to dt 0 p k with q acting as a mediator. They arise due to advec- − νk2ω(k) + ik × F(k), (4) tion and stretching respectively. Here we note that Sωω(k0|p|q) = −Sωω(p|k0|q) (15) where q = k − p and k = |k|. due to the incompressibility condition, k · u(k) = 0; this result is consistent with (9). Substitution of (13-14) into A. Enstrophy transfers (5) yields d X X Enstrophy, which is defined as W (k) = |ω(k)|2/2, sat- W (k0) = Sωω(k0|p|q) + Sωu(k0|p|q) dt isfies p p − 2νk02W (k0) + k02<{u∗(k0) · F(k0)}, (16) d X W (k) = ={(k · u(q))(ω(p) · ω∗(k))} dt 0 p and q = −(k + p). − ={(k · ω(q))(u(p) · ω∗(k))} B. Enstrophy fluxes − 2νk2W (k) + k2<{u∗(k) · F(k)}, (5) Summing the previously discussed mode-to-mode en- where q = k − p. Setting k0 = −k, the above equation strophy transfers over p and k0 and depending on 0 can be rewritten in the form whether p and k belong to the sphere of radius k0 or not, we can define the five following enstrophy fluxes d 0 X ωω 0 ωu 0 W (k ) = S (k |p, q) + S (k |p, q) ω< X X ωω 0 dt Πω>(k0) = S (k |p|q) (17) p 0 |p|≤k0 |k |>k0 02 0 02 ∗ 0 0 − 2νk W (k ) + k <{u (k ) · F(k )}, (6) u< X X ωu 0 Πω>(k0) = S (k |p|q) (18) ωω 0 ωu 0 0 where S (k |p, q) and S (k |p, q) are the combined |p|≤k0 |k |>k0 0 transfers of enstrophy from modes p and q to k , defined u< X X ωu 0 Πω<(k0) = S (k |p|q) (19) as 0 |p|≤k0 |k |≤k0 Sωω(k0|p, q) = −={(k0 · u(q))(ω(p) · ω(k0))} u> X X ωu 0 Πω>(k0) = S (k |p|q) (20) 0 0 0 −={(k · u(p))(ω(q) · ω(k ))}, (7) |p|>k0 |k |>k0 Sωu(k0|p, q) = +={(k0 · ω(q))(u(p) · ω(k0))} u> X X ωu 0 Πω<(k0) = S (k |p|q), (21) 0 0 0 +={(k · ω(p))(u(q) · ω(k ))} (8) |p|>k0 |k |≤k0 3

Performing a sum over all modes inside the sphere of radius k0 yields X d ( + 2νk02)W (k0) − k02<{u∗(k0) · F(k0)} dt 0 < > |k |≤k0 u u ω> u = (Πω< + Πω< + Πω<)(k0). (23) Adding (22) and (23) and noticing that (15) implies ω< ω> Πω> + Πω< = 0, we obtain X d ( + 2νk02)W (k0) − k02<{u∗(k0) · F(k0)} dt k0 u u = (Πω> + Πω> + Πω< + Πω<)(k0). (24) ω< ω> As the left hand side of (24) does not depend on k0, it implies

d u u Πω> + Πω> + Πω< + Πω< = 0 (25) dk0 which is valid at all times. That is, the following sum FIG. 1. (color online) Illustration of the five enstrophy fluxes u> u u in addition to the kinetic energy flux Πu< ≡ ΠE which is Πω>(k0) + Πω>(k0) + Πω<(k0) + Πω<(k0) = const (26) defined in Eq. (39). is independent of k0. Under a steady state, we obtain where the superscript and subscript of Π represent re- X 02 0 X 02 ∗ 0 0 spectively the giver and the receiver modes, and the sym- 2νk W (k ) − k <{u (k ) · F(k )} k0 F 0 F bols < and > denote respectively the inside and outside k0 <|k |≤k1 of the sphere of radius k . Accordingly, Πx<(k ) denotes u u 0 y> 0 = (Πω> + Πω> + Πω< + Πω<)(k0), (27) the flux of enstrophy from all x modes inside the sphere of radius k0 to all y modes outside the sphere. An illus- where the forcing is employed to the wavenumber band F F tration of these fluxes is given in Figure 1. (k0 .k1 ]. A physical interpretation of the above equa- Customarily a turbulent flux is associated with some tion is that the enstrophy injected by the external force conserved quantity [5]. For example, in two-dimensional is transferred to the enstrophy fluxes and the enstrophy hydrodynamic turbulence, the energy and enstrophy dissipation. fluxes are connected to the corresponding conservation laws. The concept of turbulent flux could be generalized to a more complex scenario when a certain quan- III. SIMULATION METHOD AND RESULTS tity is transferred from one field to another. Here, these fluxes are cross transfers among the two fields. Though The Navier-Stokes equations (1) are solved numeri- enstrophy is not conserved, we define enstrophy fluxes cally using a fully de-aliased, parallel pseudo-spectral of Eqs. (18- 21) as enstrophy transfers from large/small code Tarang [35, 36] with fourth-order Runge-Kutta time scale velocity field to large/small vorticity field. The en- stepping. For de-aliasing purpose, the 3/2-rule has been ω< −3 strophy flux Πω>(k0) is related to the conservation law chosen [37, 38]. The viscosity is set to ν = 10 and the related to Eq. (9). Similar issues arise in magnetohydro- simulations are performed with a resolution of 5123. dynamic turbulence where fluxes for kinetic and magnetic energies are defined even though kinetic and magnetic energies are not conserved individually. We define A. Helical forcing kinetic-to-magnetic energy fluxes as the energy transfers from large/small scale velocity field to from large/small In Fourier space the velocity field satisfies scale magnetic field; such fluxes are crucial for dynamo d X action [27–29]. u(k) = −i (k · u(q))u(p) − νk2u(k) + F(k), (28) dt Summing (16) over all modes outside the sphere of p radius k0, we obtain where q = k − p. Then the energy and helicity, which X d ( + 2νk02)W (k0) − k02<{u∗(k0) · F(k0)} are defined as dt 0 |k |>k0 |u(k)|2 u(k) · ω(k)∗ ω< u E(k) = ,H(k) = , (29) = (Πω> + Πω> + Πω>)(k0). (22) 2 2 4 satisfy the following equations 0.2 d X E(k0) = − ={(k0 · u(q))(u(p) · u(k0))} (a) dt p

2 0 ∗ 0 0 s s

− 2νk E(k ) + <{u (k ) · F(k )}, (30) i 0.1 d

d X E H(k0) = <{u(q) · (ω(p) × ω(k0))} dt p 2 0 ∗ 0 0 − 2νk H(k ) + <{ω (k ) · F(k )}, (31) 0.0 0.3 where q = −(k0 + p). (b) Following Carati et al. [39, 40] and Teimurazov et al. 0.2 s

[25, 26], the forcing is taken to be of the form s i

d 0.1

F(k) = αu(k) + βω(k) (32) H 0.0 such that 220 0.1 X ∗ (c) <{u (k) · F(k)} = E, (33) kF <|k|≤kF 0 1 160.9 s

≈ s i and 110 d ²H = 0.0 Ω X ∗ <{ω (k) · F(k)} = H . (34) ²H = 0.1 F F k0 <|k|≤k1 ²H = 0.2 Here, the forcing is applied in the wavenumber band 0 F F 0 5 10 15 20 25 (k0 , k1 ], and E and H are the prescribed energy and helicity injection rates. From the above three equations t (32-34) we deduce that 1 W − H 1 E − H α = F E F H , β = F H F E , (35) FIG. 2. (color online) Time evolution of (a) energy dissipa- 2 2 tion, (b) helicity dissipation, and (c) enstrophy dissipation for 2 EF WF − HF 2 EF WF − HF the same energy injection rate E = 0.1 and three different where EF , HF and WF are respectively the energy, ki- helicity injections H = 0, 0.1, 0.2 . For t ≥ 10 a steady state netic helicity and enstrophy in the forcing band, is reached with a mean enstrophy dissipation about 160.9. X (EF ,HF ,WF ) = (E(k),H(k),W (k)). (36) F F k0 <|k|≤k1 B. Energy and helicity spectra and fluxes F In our simulations, the forcing band corresponds to k0 = F 2 and k1 = 3. The main advantage of using the forcing From (30) and (31) and following steps analogous to given by (32-35) is that the injection rates of energy and (5-14), the following expression for mode-to-mode energy helicity can be set independently. The realizability condi- and helicity transfers can be derived: tion |H(k)| ≤ kE(k) directly comes from the definitions E 0 0 0 of energy and helicity and is therefore always satisfied, S (k |p|q) = −={(k · u(q))(u(p) · u(k ))}, (37) H 0 0 whatever the values of E and H . For example even in S (k |p|q) = +<{u(q) · (ω(p) × ω(k ))}, (38) the case , which was studied by Kessar et al. [22], E H leading to the following flux definitions the realizability condition is satisfied, reaching a state close to the maximal helical state |H(k)| = kE(k). X X E,H 0 In Fig. 2 the time evolution of (a) the total energy ΠE,H (k0) = S (k |p|q). (39) 0 dissipation 2ν P k2E(k), (b) the total helicity dissipa- |k |>k0 |p|

101 103 157.3 ≈ 5/3 k − 10-1 1 k 1

H 10 ²

/ 2 ) k k ( 1 H -3 10 ω -1 , 10 10 Π E u u ² Πω > + Πω < + Πω < + Πω > /

) 0

k ω u >

( Π Π Π 10-1 ω > ω < ω > E 10-5 -1 u Πω <

10-3 101 102 10-7 -101 100 101 102 101 102 k k0

FIG. 3. (color online) Normalized energy spectra (solid FIG. 4. (color online) The five enstrophy fluxes given in (17- curves) and helicity spectra (dashed curves) with the same 21), and the sum Πu< + Πu> + Πu< + Πu> are plotted for color code as in Fig. 2. In the inset, normalized energy fluxes ω> ω< ω< ω> E = 0.1 and H = 0. (solid curves) and helicity fluxes (dashed curves). The black dashed curves correspond to the analytical formula (40-43) . ω< The flux Πω> is positive suggesting a direct cascade of enstrophy. In the inertial range it obeys a k2 scaling They take the following form u< law. However we observe that in the inertial range, Πω> ω< E(k) K 3 is larger than Πω>, showing that the enstrophy flux due = E k−5/3exp − K (k/k )4/3 , (40) 1/3 E d to stretching is larger than the one due to advection. We E 2 u> E also observe that Πω< is always negative, indicating that H(k) K 3 the small-scale velocity fluctuations squeeze (not stretch) = H k−5/3exp − K (k/k )4/3 , (41) 1/3 H d 2 H 2 the large scale vorticity. It also obeys a k scaling law in E the inertial range. ΠE(k) 3 4/3 u< = exp − KE(k/kd) , (42) The flux Πω<(k0) is an accumulated sum of all the E 2 enstrophy transferred from u< to ω< up to the radius ΠH (k) 3 4/3 k0. All the wavenumber shells have positive sign for = exp − KH (k/kd) , (43) u< 2 these transfer, hence Πω<(k0) is an increasing function H u< of k0. Note that Πω<(∞) is the net enstrophy transfer. u> where KE is Kolmogorov’s constant, and KH is another The complementary flux, Πω>(k0), has an opposite be- 3 1/4 non-dimensional constants, and kd = (E/ν ) is the haviour; that is, it decreases with k0. All the fluxes other u< Kolmogorov’s wavenumber. Note however that Pao’s than Πω<(k0) vanish for k0 > kd because the fluctuations spectrum slightly overestimates the dissipation range vanish in this range. u u spectrum [42], and we expect similar discrepancies for the It is found that (Πω> + Πω< + Πω< + Πω>)(k0) ≈ −10 kinetic helicity that may show up in high-resolution simu- 157.3, independently of k0 to a precision of about 10 . lations. These issues may be related to intermittency and This value is comparable to the enstrophy dissipation enhanced dissipation due to bottleneck effect [43, 44]. which is about 160.9 (Fig. 2); the difference between the two quantities is P |k0|2<{u∗(k0) · F(k0)} ≈ 3.6.; F 0 F k0 <|k |≤k1 C. Results of enstrophy fluxes these computations are consistent with (27).

The enstrophy fluxes given in (17-21) have been calcu- IV. CONCLUSIONS lated for H = 0, 0.1, 0.2. They are found to be insensitive to H . Therefore only the results corresponding to H = 0 are presented in Fig. 4. The manifestation of this Using direct numerical simulations and varying the in- independence is consistent with many numerical results jection rate of helicity we find that injecting helicity does which showed that the forward flux of kinetic energy is not change the results in terms of energy and helicity also insensitive to kinetic helicity injection at large scale spectra and fluxes, and also in terms of enstrophy fluxes. [45]. Crucial changes can be expected when significant The energy and helicity spectra and fluxes follow rather imbalance between helical modes of different signs is im- well the formula of Pao [41] that we extended to helic- posed by helicity forced over a wide range of scales [22]. ity. Of course, some correction due to intermittency and 6 bottleneck effect should be added, but these issues are to all vorticity modes and therefore should not depend beyond the scope of the present paper. These results are on the wavenumber, that we confirm numerically with a consistent with the predictions that the inertial range precision better than 10−10. properties of hydrodynamic turbulence is independent of kinetic helicity [13, 24]. The main objective of this paper is to introduce the derivation of mode-to-mode enstrophy transfers using which we compute the enstrophy fluxes. There are five ACKNOWLEDGMENTS u< u< different enstrophy fluxes. Four of them, Πω<,Πω>, u> u> Πω<,Πω>, correspond to vorticity stretching , the fifth Our numerical simulations have been performed on ω< one Πω> is due to the advection of vorticity by the Shaheen II at Kaust supercomputing laboratory, Saudi u< flow. It is remarkable that in the inertial range Πω> Arabia, under the project k1052. This work was sup- ω< is larger than Πω> implying that the enstrophy flux ported by the research grants PLANEX/PHY/2015239 due to vorticity stretching is larger than the one due to from Indian Space Research Organisation, India, and advection of vorticity. The sum of the first four fluxes by the Department of Science and Technology, India corresponds to the contribution from all velocity modes (INT/RUS/RSF/P-03).

[1] A. N. Kolmogorov, “The local structure of turbulence “Dissipation scales of kinetic helicities in turbulence,” in incompressible viscous fluid for very large Reynolds Phys. Fluids 23, 035108 (2011). numbers,” Dokl Acad Nauk SSSR 30, 299 (1941). [17] Q. Chen, S. Chen, G.L. Eyink, and D.D. Holm, “In- [2] A. N. Kolmogorov, “Dissipation of energy in the locally termittency in the joint cascade of energy and helicity,” isotropic turbulence,” Dokl Acad Nauk SSSR 32, 16–18 Phys. Rev. Lett 90, 214503 (2003). (1941). [18] T. Teitelbaum and P.D. Mininni, “The decay of turbu- [3] A. M. Obukhov, “Structure of the temperature field in lence in rotating flows,” Phys. Fluids 23, 065105 (2011). turbulent flows,” Isv. Akad. Nauk. SSSR, Ser. Geogr. [19] A. Pouquet, J. Baerenzung, P.D. Mininni, D. Rosenberg, Geophys. Ser. 13, 58–69 (1949). and S. Thalabard, “Rotating helical turbulence: three- [4] S. Corrsin, “On the spectrum of isotropic temperature dimensionalization or self-similarity in the small scales?” fluctuations in an isotropic turbulence,” J. Appl. Phys. J. Phys.: Conference Series 318, 042015 (2011). 22, 469–473 (1951). [20] R. Stepanov, E. Golbraikh, P. Frick, and A. Shes- [5] M. Lesieur, Turbulence in Fluids (Springer-Verlag, Dor- takov, “Helical bottleneck effect in 3d homogeneous drecht, 2008). isotropic turbulence,” Fluid Dynamics Research 50, [6] M.K. Verma, Physics of Buoyant Flows: From Instabili- 011412 (2018). ties to Turbulence (World Scientific, Singapore, 2018). [21] L. Biferale, S. Musacchio, and F. Toschi, “Inverse Energy [7] R.H. Wasserman, “Helical fluid flows,” Quarterly of Ap- Cascade in Three-Dimensional Isotropic Turbulence,” plied Mathematics 17, 443–445 (1960). Phys. Rev. Lett. 108, 164501 (2012), arXiv:1111.1412 [8] A. Brissaud, U. Frisch, J. Leorat, M. Lesieur, and [physics.flu-dyn]. A. Mazure, “Helicity cascades in fully developed isotropic [22] M. Kessar, F. Plunian, R. Stepanov, and G. Balarac, turbulence,” Phys. Fluids 16, 1366–1367 (1973). “Non-Kolmogorov cascade of helicity-driven turbulence,” [9] J.C. Andréand M. Lesieur, “Influence of helicity on the Phys. Rev. E 92, 031004(R) (2015). evolution of isotropic turbulence at high reynolds num- [23] G. Sahoo and L. Biferale, “Energy cascade and intermit- ber,” J. Fluid Mech. 81, 187–207 (1977). tency in helically decomposed Navier–Stokes equations,” [10] W. Polifke and L. Shtilman, “The dynamics of helical de- Fluid Dyn. Res. 50, 011420 (2018). caying turbulence,” Phys. Fluids A: Fluid Dyn. 1, 2025– [24] V. Avinash, M.K. Verma, and A.V. Chandra, “Field- 2033 (1989). theoretic calculation of kinetic helicity flux,” Pramana-J. [11] F. Waleffe, “The nature of triad interactions in homoge- Phys. 66, 447–453 (2006). neous turbulence,” Physics of Fluids A: Fluid Dynamics [25] A. Teimurazov, R. Stepanov, M.K. Verma, S. Barman, 4, 350–363 (1992). A. Kumar, and S. Sadhukhan, “Direct numerical simu- [12] N. Yokoi and A. Yoshizawa, “Statistical analysis of the lation of homogeneous isotropic helical turbulence with effects of helicity in inhomogeneous turbulence,” Phys. the Tarang code,” Computational Continuum Mechanics Fluids 5, 464–477 (1993). 10, 474 (2017). [13] Y. Zhou and G. Vahala, “Reformulation of recursive- [26] A. Teimurazov, R. Stepanov, M.K. Verma, S. Barman, renormalization-group-based subgrid modeling of turbu- A. Kumar, and S. Sadhukhan, “Direct numerical simu- lence,” Phys. Rev. E 47, 2503–2519 (1993). lation of homogeneous isotropic helical turbulence with [14] P. D. Ditlevsen and P. Giuliani, “Dissipation in helical the tarang code,” J. Appl. Mech. Tech. Phys. 59, 1279 turbulence,” Phys. Fluids 13, 3508–3509 (2001). (2018). [15] Q. Chen, S. Chen, and G.L. Eyink, “The joint cascade [27] M.K. Verma, “Statistical theory of magnetohydrody- of energy and helicity in three-dimensional turbulence,” namic turbulence: recent results,” Phys. Rep. 401, 229– Physics of Fluids 15, 361–374 (2003). 380 (2004). [16] T. Lessinnes, F. Plunian, R. Stepanov, and D. Carati, [28] G. Dar, M. K. Verma, and V. Eswaran, “Energy transfer 7

in two-dimensional magnetohydrodynamic turbulence: Fourier Transform and a pseudo-spectral fluid solver up formalism and numerical results,” Physica D 157, 202– to 196608 cores,” J. Parallel Distrib. Comput. 113, 77–91 225 (2001). (2018). [29] A. Alexakis, P.D. Mininni, and A. Pouquet, “On the [37] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. inverse cascade of magnetic helicity,” ApJ 640, 335–343 Zang, Spectral Methods in Fluid Dynamics (Springer- (2006). Verlag, Berlin Heidelberg, 1988). [30] T. Lessinnes, F. Plunian, and D. Carati, “Helical shell [38] J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd models for MHD,” Theor. Comput. Fluid Dyn. 23, 439– ed. (Dover Publications, New York, 2003). 450 (2009). [39] D. Carati, O. Debliquy, B. Knaepen, B. Teaca, and M.K. [31] F Krause and K.-H. Rädler, Mean-Field Magnetohydro- Verma, “Energy transfers in forced MHD turbulence,” J. dynamics and Dynamo Theory (Pergamon Press, Oxforc, Turbul. 7, 1–12 (2006). 1980). [40] D. Carati, S. Ghosal, and P. Moin, “On the represen- [32] H.K. Moffatt, Magnetic Field Generation in Electrically tation of backscatter in dynamic localization models,” Conducting Fluids (Cambridge University Press, Cam- Phys. Fluids 7, 606–616 (1995). bridge, 1978). [41] Y.-H. Pao, “Structure of Turbulent Velocity and Scalar [33] D. Sokoloff, P. Akhmetyev, and E. Illarionov, “Magnetic Fields at Large Wavenumbers,” Phys. Fluids 8, 1063– helicity and higher helicity invariants as constraints for 1075 (1965). dynamo action,” Fluid Dynamics Research 50, 011407 [42] M.K. Verma, A. Kumar, P. Kumar, S. Barman, (2017). A.G. Chatterjee, R. Samtaney, and R.A. Stepanov, “En- [34] Y.B. Zeldovich, A.A. Ruzmaikin, and D.D. Sokoloff, ergy Spectra and Fluxes in Dissipation Range of Turbu- Magnetic fields in astrophysics (Gordon and Breach, lent and Laminar Flows,” Fluid Dyn. 53, 862–873 (2018). 1983). [43] G. Falkovich, “Bottleneck phenomenon in developed tur- [35] M.K. Verma, A.G. Chatterjee, R.K. Yadav, S. Paul, bulence,” Phys. Fluids 6, 1411 (1994). M. Chandra, and R. Samtaney, “Benchmarking and scal- [44] M. K. Verma and D. A. Donzis, “Energy transfer and ing studies of pseudospectral code Tarang for turbulence bottleneck effect in turbulence,” J. Phys. A: Math. Theor. simulations,” Pramana-J. Phys. 81, 617–629 (2013). 40, 4401-4412 (2007). [36] A.G. Chatterjee, M.K. Verma, A. Kumar, R. Sam- [45] A. Alexakis and L. Biferale, Phys. Reports 767, 1 (2018), taney, B. Hadri, and R. Khurram, “Scaling of a Fast arXiv:1808.06186 [physics.flu-dyn].