A Derivation and solution of the Reynolds-stress equation

# [

## Abstract

We discuss a mean-field theory of generation of large-scale vorticity in a rotating density stratified developed turbulence with inhomogeneous kinetic helicity. We show that the large-scale nonuniform flow is produced due to ether a combined action of a density stratified rotating turbulence and uniform kinetic helicity or a combined effect of a rotating incompressible turbulence and inhomogeneous kinetic helicity. These effects result in the formation of a large-scale shear, and in turn its interaction with the small-scale turbulence causes an excitation of the large-scale instability (known as a vorticity dynamo) due to a combined effect of the large-scale shear and Reynolds stress-induced generation of the mean vorticity. The latter is due to the effect of large-scale shear on the Reynolds stress. A fast rotation suppresses this large-scale instability.

Generation of large-scale vorticity in helical turbulence]Generation of large-scale vorticity in rotating stratified turbulence with inhomogeneous helicity: mean-field theory N. Kleeorin and I. Rogachevskii]N.\nsK\lsL\lsE\lsE\lsO\lsR\lsI\lsN and I.\nsR\lsO\lsG\lsA\lsC\lsH\lsE\lsV\lsS\lsK\lsI\lsI 1

## 1 Introduction

A large-scale nonuniform flow or differential rotation in a helical small-scale turbulence can result in generation of a large-scale magnetic field by or mean-field dynamo (see, e.g., Moffatt, 1978; Parker, 1979; Krause & Rädler, 1980; Zeldovich et al., 1983; Ruzmaikin et al., 1988; Rüdiger et al., 2013). The kinetic effect is related to a kinetic helicity produced, e.g., by a combined action of uniform rotation and density stratified or inhomogeneous turbulence. Formation of the nonuniform flows is caused, e.g., by a rotating anisotropic density stratified turbulence or turbulent convection. The latter effect is also related to a problem of generation of large-scale vorticity by a turbulent flow, and has various applications in geophysical and astrophysical flows (see, e.g., Lugt, 1983; Pedlosky, 1987; Chorin, 1994).

It has been suggested by Moiseev et al. (1983), that the generation of the large-scale vorticity in a helical turbulence occurs due to the kinetic alpha effect. This idea is based on an analogy between the induction equation for magnetic field and the vorticity equation (Batchelor, 1950). The latter implies that a large-scale instability is associated with the term in the equation for the mean vorticity, , similarly to the mean-field equation for the magnetic field, , where the key generation term is , see Moiseev et al. (1983). A mean-field equation for the vorticity has been derived by Khomenko et al. (1991) using the functional technique for a compressible helical turbulence. It has been shown there that the mean vorticity grows exponentially in time due to the kinetic alpha effect.

However, there are different arguments which are not in a favor of the above idea. One argument is that the analogy between the induction equation and the vorticity equation is not complete, because the vorticity , where the velocity is determined by the nonlinear Navier-Stokes equation. Another argument is related to the symmetry properties, implying that the term in the mean vorticity equation should originate from the Reynolds stress proportional to the mean velocity. The latter condition is in a contradiction to the Galilean invariance.

Frisch et al. (1987, 1988) have investigated the effect of a non-Galilean invariant forcing that causes a large-scale instability resulting in formation of a nonuniform flow at large scales (so called the anisotropic kinetic alpha effect or the AKA effect). A non-Galilean invariant forcing and generation of large-scale vorticity have been also investigated by Kitchatinov et al. (1994). There are various examples for turbulence driven by non-Galilean invariant forcing, e.g., supernova-driven turbulence in galaxies (Korpi et al., 1999) and the turbulent wakes driven by galaxies moving through the galaxy cluster (Ruzmaikin et al., 1989). Also presence of boundaries can break the Galilean invariance, see e.g., discussion in Brandenburg & Rekowski (2001), and references therein.

In a homogeneous non-helical and incompressible turbulence with an imposed mean velocity shear, the large-scale vorticity can be generated due to an excitation of a large-scale instability, referred as a vorticity dynamo and caused by a combined effect of the large-scale shear motions and Reynolds stress-induced generation of perturbations of mean vorticity. This effect has been studied theoretically by Elperin et al. (2003, 2007) and detected in direct numerical simulations by Yousef et al. (2008); Käpylä et al. (2009). To derive the mean-field equation for the vorticity, the spectral approach which is valid for large Reynolds numbers has been applied by Elperin et al. (2003). The linear stage of the large-scale instability which is saturated by nonlinear effects has been investigated by Elperin et al. (2003), but not a finite time growth of large-scale vorticity as described by Chkhetiany et al. (1994). In particular, a first order smoothing (a quasi-linear approach) has been used in the latter study to derive equation for the mean vorticity in a compressible random flow with an imposed large-scale shear. The latter approach is valid only for small Reynolds numbers, and this is the reason why the large-scale instability has not been found by Chkhetiany et al. (1994). Importance of the vorticity dynamo has been demonstrated by Guervilly et al. (2015), where they suggested a mechanism for the generation of large-scale magnetic fields based on the formation of large-scale vortices in rotating turbulent convection.

Formation of large-scale non-uniform flow by inhomogeneous helicity in a rotating incompressible turbulence has been studied theoretically (Yokoi & Yoshizawa, 1993) and in direct numerical simulations (Yokoi & Brandenburg, 2016). The theoretical study and numerical simulations show that a non-uniform large-scale flow is produced in the direction of angular velocity. Recent direct numerical simulations have demonstrated formation of large-scale vortices in rapidly rotating turbulent convection for both, compressible (Chan, 2007; Käpylä et al., 2011; Mantere et al., 2011) and Boussinesq fluids (Guervilly et al., 2014; Rubio et al., 2014; Favier et al., 2014). These large-scale flows consist of depth-invariant, concentrated cyclonic vortices, which form by the merger of convective thermal plumes and eventually grow to the size of the computational domain. Weaker anticyclonic circulations form in their surroundings.

In the present study we develop a mean-field theory of the generation of large-scale vorticity in a rotating density stratified turbulence with inhomogeneous helicity and large Reynolds numbers. To derive the mean-field equation for the vorticity, the spectral approach was applied here for a large Reynolds number turbulence. We have shown that a non-uniform large-scale flow is produced in a rotating fully developed turbulence due to either inhomogeneous kinetic helicity or a combined effect of a density stratified flow and uniform kinetic helicity. An interaction of the turbulence with the formed large-scale shear causes an excitation of the large-scale instability resulting in the generating of the mean vorticity (vorticity dynamo). On the other hand, a fast rotation suppresses this large-scale instability. The present study of the dynamics of large-scale vorticity in a rotating density stratified helical turbulence demonstrates that the mean-field equation for the large-scale vorticity does not contain the term as was previously suggested by Moiseev et al. (1983).

## 2 Effect of rotation on the Reynolds stress

To study an effect of rotation on the Reynolds stress in a rotating, density stratified and inhomogeneous turbulence, we apply a mean-field approach and use the Reynolds averaging. In the framework of this approach, the velocity and pressure are separated into the mean and fluctuating parts.

### 2.1 Equation for velocity fluctuations

To determine the Reynolds stress, we use equation for fluctuations of velocity , which is obtained by subtracting equation for the mean field from the corresponding equation for the instantaneous field:

 ∂u∂t = −(¯¯¯¯U⋅\boldmath∇)u−(u⋅\boldmath∇)¯¯¯¯U−\boldmath∇pρ0+2u×Ω+UN. (1)

Here are fluctuations of fluid pressure, is the mean fluid velocity, and Eq. (1) is written in the reference frame rotating with the angular velocity . The fluid velocity for a low-Mach-number fluid flow satisfies the continuity equation written in the anelastic approximation: and . The mean fluid density and pressure with the subscript correspond to the hydrostatic basic reference state, given by the equation: . The nonlinear term which includes the molecular viscous force, , is given by

 UN = ⟨(u⋅\boldmath∇)u⟩−(u⋅\boldmath∇)u+Fν(u).

The derivation of the equation for the Reynolds stress includes the following steps: (i) use new variables for fluctuations of velocity ; (ii) derivation of the equation for the second moment of the velocity fluctuations in the space; (iii) application of the spectral approach (see Sect. 2.3) and solution of the derived equation for in the space; (iv) returning to the physical space to obtain formula for the Reynolds stress as the function of the rotation rate . Here the angular brackets denote ensemble averaging.

### 2.2 Equation for the Reynolds stress

Applying a multi-scale approach (Roberts & Soward, 1975) and using Eq. (20) derived in Appendix A, we obtain equation for the correlation function: , where , , and correspond to the large scales, while to the small ones. The equation for is given by

 ∂fij(k,K,t)∂t = (IUijmn+LΩijmn)fmn+^Nfij, (2)

where

 IUijmn = [2kiqδmpδjn+2kjqδimδpn−δimδjqδnp−δiqδjnδmp+δimδjnkq∂∂kp]∇p¯Uq −δimδjn[div¯¯¯¯U+¯¯¯¯U⋅\boldmath∇], LΩijmn = DΩim(k1)δjn+DΩjn(k2)δim,DΩij(k)=2εijmΩnkmn.

Here is the Kronecker tensor, and is the Levi-Civita tensor. The correlation function is proportional to the fluid density and are the third-order moments appearing due to the nonlinear terms:

 ^Nfij=⟨Pim(k1)vNm(k1)vj(k2)⟩+⟨vi(k1)Pjm(k2)vNm(k2)⟩.

### 2.3 τ approach

Equation (2) for the second-order moment contains high-order moments and a closure problem arises (see, e.g., Monin & Yaglom, 2013; McComb, 1990). To simplify the notation, we do not show the dependencies on and in the correlation function . We apply the spectral approximation, or the third-order closure procedure (see, e.g., Orszag, 1970; Pouquet et al., 1976; Kleeorin et al., 1990; Rogachevskii & Kleeorin, 2004). The spectral approximation postulates that the deviations of the third-order-moment terms, , from the contributions to these terms afforded by the background turbulence, , are expressed through the similar deviations of the second moments, :

 ^Nfij(k)−^Nf(0)ij(k)=−fij(k)−f(0)ij(k)τr(k), (3)

where is the characteristic relaxation time of the statistical moments, which can be identified with the correlation time of the turbulent velocity field for large Reynolds numbers. In Eq. (3) the quantities with the superscript correspond to the background turbulence (i.e., a turbulence with . We apply the -approximation (3) only to study the deviations from the background turbulence, which is assumed to be known (see below). Validation of the approximation for different situations has been performed in various numerical simulations and analytical studies (Brandenburg & Subramanian, 2005; Rogachevskii & Kleeorin, 2007; Rogachevskii et al., 2011, 2012; Brandenburg et al., 2012a; Käpylä et al., 2012).

## 3 Effects of rotation and kinetic helicity on the Reynolds stress

In this section we consider a combined effect of rotation and kinetic helicity on the Reynolds stress. To this end we consider a model for the background rotating helical turbulence.

### 3.1 Model for the background turbulence

We use the following model of the background rotating, density stratified, and inhomogeneous turbulence with inhomogeneous kinetic helicity:

 f(0)ij = E(k)[1+2kεuδ(^k⋅^Ω)]8πk2(1+εu){[δij−kij+ik2(~λikj−~λjki)]ρ0⟨u2⟩(0) (4) Missing or unrecognized delimiter for \Big

where , is the Dirac delta function, is the kinetic helicity, , , , the turbulent correlation time is given below.

We assume that the background turbulence is the Kolmogorov type turbulence with constant fluxes of energy and kinetic helicity over the spectrum, i.e., the kinetic energy spectrum , the function with being the exponent of the kinetic energy spectrum for Kolmogorov spectrum), and is the integral scale of turbulent motions.

To derive Eq. (4) we use the following conditions:
(i) the anelastic approximation: , which implies that and , where and ;
(ii) ;
(iii) ;
(iv) .

To introduce anisotropy of the background turbulence due to rotation, we consider an anisotropic turbulence as a combination of a three-dimensional isotropic turbulence and two-dimensional turbulence in the plane perpendicular to the rotational axis. The degree of anisotropy is defined as the ratio of turbulent kinetic energies of two-dimensional to three-dimensional motions. In this model we neglect effects which are quadratic in , and . Different contributions in Eq. (4) have been discussed by Batchelor (1953); Elperin et al. (1995); Rädler et al. (2003).

The effect of rotation on the turbulent correlation time is described just by an heuristic argument. In particular, we assume that , that yields:

 τΩ=τ0[1+(C−1ΩΩτ0)2]1/2, (5)

For a fast rotation, , the parameter tends to a finite value, , where and is the characteristic turbulent velocity at the integral scale .

### 3.2 The Reynolds stress in a rotating and helical turbulence

In this section we determine the contribution to the Reynolds stress caused by either rotation and stratification in helical turbulence or rotation and inhomogeneous kinetic helicity. For a slow rotation, , the function is given by

 f(Ω,χ)ij=∫τ[~Lijmnf(0,~λ)mn+(L∇ijmn+Lλijmn)f(0,χ)mn]dk, (6)

where we use Eq. (24) derived in Appendix A, the tensors and determine corresponding terms in the model (4) of the background turbulence, , and all other definitions are given in Appendix A. After integration in space in Eq. (6), we obtain contribution to the Reynolds stress caused by either rotation and stratification in helical turbulence or rotation and inhomogeneous kinetic helicity for a slow rotation, :

 f(Ω,χ)ij=(q−1)2qρ0τ0ℓ20[Ωiλj+Ωjλi+415(Ωi∇j+Ωj∇i)]χ(0). (7)

For a fast rotation, , the contribution to the Reynolds stress caused by either rotation and stratification in helical turbulence or rotation and inhomogeneous kinetic helicity is given by

 f(Ω,χ)ij=∫τ[L∇ijmn+Lλijmn]f(0,χ)mndk. (8)

After integration in space in Eq. (8), we obtain

 f(Ω,χ)ij=CΩ(q−1)4qρ0ℓ20{^Ωiλj+^Ωjλi+^Ωi^Ωj[^Ω⋅(λ+∇)]}χ(0), (9)

where . In the derivation of Eq. (9) we take into account that the turbulent time for a fast rotation, is determined by Eq. (5). To integrate over the angles in -space for a fast rotation we use the integrals given at the end of Appendix A.

### 3.3 Formation of the mean velocity shear

Let us consider the case when the angular velocity, , is perpendicular to the density stratification axes, . For simplicity, also consider the case when the gradient of the kinetic helicity is parallel to , i.e., . In this case, is only one nonzero contribution to the Reynolds stress caused by either rotation and stratification in helical turbulence or rotation and inhomogeneous kinetic helicity for :

 f(Ω,χ)xy(x)=f(Ω,χ)yx(x)=(q−1)2qρ0(x)(Ωτ0)ℓ20(λχ(0)+415∇χ(0)). (10)

The last term in Eq. (10) is in agreement with Eq. (30) of Yokoi & Brandenburg (2016).

The steady-state solution of the momentum equation for the -component of the mean velocity reads:

 ∇x[ρ0(x)νT(∇x¯¯¯¯U(S)y)−f(Ω,χ)yx(x)]=0, (11)

where is the turbulent viscosity (Elperin et al., 2002) and we take into account that the gradient of the mean pressure along vanishes. Integrating Eq. (11) over , we determine the formed large-scale shear:

 ¯¯¯¯S≡∇x¯¯¯¯U(S)y=f(Ω,χ)yxρ0νT=15(q−1)2q(q+3)Ωτ20(λχ(0)+415∇χ(0)). (12)

It follows from Eq. (12) that the large-scale shear is produced in rotating turbulence due to either inhomogeneous kinetic helicity or a combined action of a density stratified flows and uniform kinetic helicity.

In the present study we assume that large-scale shear does not affect the background turbulence. For large values of the shear rate of large-scale motions, the background turbulence and turbulent correlation time can be affected by the large-scale shear. In this case the quenching of the correlation time can be increased by the large-scale shear. The inclusion of these effects in the background turbulence is a subject of a separate study. On the other hand, the solution of Eq. (2) determines the deviations from the background turbulence, and the obtained solution of this equation yields Eq. (24), that describes the effect of shear on turbulence.

## 4 Generation of the large-scale vorticity

The formed large-scale shear in a turbulent flow causes an excitation of the large-scale instability resulting in the generation of the mean vorticity due to the vorticity dynamo. The linearized equation for the small perturbations of the mean vorticity is given by

 ∂¯¯¯¯¯¯W∂t=\boldmath∇×[¯¯¯¯U(S)×¯¯¯¯¯¯W+¯¯¯¯U×¯¯¯¯¯¯W(S)+2¯¯¯¯U×Ω+ρ−10(\boldmathF(S)+%\boldmath$F$(νT))], (13)

where and are perturbations of the mean velocity and mean vorticity, while and are the equilibrium mean velocity and mean vorticity related to the formed large-scale shear , given by Eq. (12). Here is the effective force caused the shear effect on the Reynolds stress, determines the turbulent viscosity, and we neglect small kinematic viscosity. Let us consider for simplicity small perturbations of the mean vorticity, , so that Eq. (13) reads:

 ∂¯¯¯¯¯¯Wx∂t = ¯¯¯¯S¯¯¯¯¯¯Wy+νT¯¯¯¯¯¯W′′x, (14) ∂¯¯¯¯¯¯Wy∂t = −β¯¯¯¯Sℓ20¯¯¯¯¯¯W′′x−2Ωλ¯¯¯¯Ux+νT¯¯¯¯¯¯W′′y. (15)

(see Appendix B), where and the coefficient has been determined in (Elperin et al., 2003): . For Kolmogorov energy spectrum , the coefficient . In Eqs. (14) and (15) we take into account the Coriolis force and the density stratification. In the presence of the density stratification due to the gravity field that is directed perpendicular to the angular velocity, we can neglect a weak centrifugal force. In Eq. (14) we take into account that the characteristic scale of the mean vorticity variations is much larger than the maximum scale of turbulent motions . Since , Eq. (15) can be rewritten as

 ∂¯¯¯¯¯¯W′y∂t = −β¯¯¯¯Sℓ20¯¯¯¯¯¯W′′′x−2Ωλ¯¯¯¯¯¯Wy+νT¯¯¯¯¯¯W′′′y. (16)

We seek for a solution of Eqs. (14) and (16) in the form , where the growth rate of the large-scale instability and the frequency of the generated waves are given by

 γ = [β(¯¯¯¯Sℓ0Kz)2−(ΩλKz)2]1/2−νTK2z, (17) ω = ΩλKz. (18)

Equation (17) implies that rotation in a density stratified turbulence decreases the growth rate of the large-scale instability. Since we consider the case when the angular velocity is perpendicular to the wave vector of the mean vorticity perturbations, large-scale inertial waves are absent in the system. In the absence of rotation and density stratification, the expression (17) for the growth rate of the large-scale instability coincides with that obtained by Elperin et al. (2003). Equation (18) describes three-dimensional slow Rossby waves in rotating density stratified flows which are similar to those studied by Elperin et al. (2017), see Eq. (28). We remind that the system considered in this study is a three-dimensional one, where the angular velocity, , stratification, , and the wave number, , are perpendicular each other.

The mechanism of the large-scale instability studied here is as follows. The first term, , in Eq. (14) describes a stretching of the mean vorticity component by non-uniform motions, which produces the component . On the other hand, the first term, , in Eq. (15) determines a Reynolds stress-induced generation of perturbations of the mean vorticity by turbulent Reynolds stresses. In particular, this term is determined by , where describes the effective force caused the shear effect on the Reynolds stress. The growth rate of the instability is caused by a combined effect of the sheared motions and the Reynolds stress-induced generation of perturbations of the mean vorticity (Elperin et al., 2003, 2007). On the other hand, the equilibrium large-scale shear is produced either rotating turbulence and inhomogeneous kinetic helicity or a combined effect of a density stratified rotating turbulence and uniform kinetic helicity (see Sect. 3.2).

The physical explanation for why the rotation quenches the vorticity growth is the following. In the presence of the density stratified rotating turbulence, there are three effects: (i) the three-dimensional slow Rossby waves; (ii) the Reynolds stress-induced generation of perturbations of the mean vorticity ; (iii) turbulent viscosity which decreases both, the energy of the Rossby waves and the Reynolds stress-induced generation of perturbations of the mean vorticity . When rotation is fast, the Reynolds stress-induced generation of perturbations of the mean vorticity is suppressed. A slow rotation just decreases the latter effect, so there is a competition between the generation of perturbations of the mean vorticity and the Rossby waves.

Note that additional terms in Eqs. (14) and (15) caused by a combined effect of kinetic helicity and large-scale shear, are much smaller than the terms which are taken into account in these equations. The combined effects of the uniform kinetic helicity, rotation and stratification or non-uniform kinetic helicity and rotation are only important for the production of the background large-scale velocity shear.

## 5 Conclusions

In the present study, the following effects are investigated: (i) the effect of density stratification on the production of the large-scale vorticity by the helical rotating turbulence; (ii) the large-scale instability (vorticity dynamo) suggested by Elperin et al. (2003) for incompressible non-helical turbulence with a large-scale shear, has been generalised for the case of density stratified rotating and helical turbulence. In particular, we show that the large-scale flow is produced in rotating turbulence due to inhomogeneous kinetic helicity or a combined action of a density stratified flows and uniform kinetic helicity. This results in the formation of a large-scale shear determined by the balance between turbulent viscous force and the effective force caused by the modification of Reynolds stress by either rotation and inhomogeneous kinetic helicity or a combined action of rotation and a density stratified turbulence with a uniform kinetic helicity. This large-scale shear interacting with a turbulent flow results in an excitation of the large-scale instability generating the mean vorticity due to the vorticity dynamo, while fast rotation suppresses this instability.

###### Acknowledgements.
This work was supported in part by the Research Council of Norway under the FRINATEK (grant No. 231444). The authors acknowledge the hospitality of NORDITA.

## Appendix A Derivation and solution of the Reynolds-stress equation

In this Appendix we derive and solve the equation for the Reynolds-stress. To this end, Eq. (1) is rewritten in the new variables for fluctuations of velocity :

 1√ρ0∂v(x,t)∂t = −\boldmath∇(pρ0)+1√ρ0[2v×Ω−(v⋅\boldmath∇)¯¯¯¯U−GUv]+vN, (19)

where and are the nonlinear terms which include the molecular viscous terms. The fluid velocity fluctuations satisfy the equation , where . To derive equation for the Reynolds-stress, we rewrite the momentum equation in a Fourier space:

 dvi(k)dt = [DΩim(k)+JUim(k)]vm(k)+vNi(k), (20)

where

 JUij(k) = 2kin∇j¯¯¯¯Un−∇j¯¯¯¯Ui−[12div¯¯¯¯U+i(¯¯¯¯U⋅k)]δij.

To derive Eq. (20), we multiply the momentum equation written in -space by to exclude the pressure term from the equation of motion. Here we also use the following identities:

 Missing or unrecognized delimiter for \big √ρ0[\boldmath∇×[\boldmath∇×u]]k=−[Λ2δij−Λiλj]vj(k),

where and .

To derive equation for the Reynolds stress, we apply a standard multi-scale approach (Roberts & Soward, 1975), i.e., the non-instantaneous two-point second-order correlation function is written as follows:

 ⟨vi(x,t)vj(y,t)⟩ = ∫dk1dk2⟨vi(k1,t)vj(k2,t)⟩exp[i(k1⋅x+k2⋅y)] (21) = ∫fij(k,R,t)exp(ik⋅r)dk,

where we use large-scale variables: and ; and small-scale variables: , and . Mean-fields depend on the large-scale variables, while fluctuations depend on the small-scale variables, where

 fij(k,R,t) = ∫⟨vi(k1,t)uj(k2,t)⟩exp(iK⋅R)dK, (22)

and . Applying a multi-scale approach and using Eq. (20), we derive equation for the correlation function: , see Eq. (2), where .

To solve Eq. (2) we extract in tensor the parts which depend on large-scale spatial derivatives or the density stratification effects, i.e.,

 LΩijmn=~Lijmn+L∇ijmn+Lλijmn+O(λ2,∇2), (23)

where

 ~Lijmn = 2Ωq(εimpδjn+εjnpδim)kpq,L∇ijmn=−2Ωq(εimpδjn−εjnpδim)k∇pq, Lλijmn = −2Ωq[(εimpδjn−εjnpδim)kλpq+ik2(εilqδjnλm−εjlqδimλn)kl], k∇ij = i2k2[ki∇j+kj∇i−2kij(k⋅\boldmath∇)],kλij=i2k2[kiλj+kjλi−2kij(k⋅% \boldmathλ)].

Equation (2) in a steady state and after applying the spectral approximation (3), reads

 fij(k) = L−1ijmn[f(0)mn+τ(IUmnpq+L∇mnpq+Lλmnpq)fpq], (24)

where we neglected terms . Here the operator is the inverse of , and it is given by

 L−1ijmn(Ω) = 12[B1δimδjn+B2kijmn+B3(εimpδjn+εjnpδim)^kp+B4(δimkjn+δjnkim) (25) +B5εipmεjqnkpq+B6(εimpkjpn+εjnpkipm)],

where , , , , , , , and . Note that for a slow rotation, .

To integrate in Eq. (8) over the angles in -space for a fast rotation, we use the following integrals:

 ∫k⊥ijdφ=πδ(2)ij,∫k⊥ijmndφ=π4Δ(2)ijmn,

where and .

## Appendix B Effect of shear on Reynolds stress

There are two effects of shear on Reynolds stress. First effect is related to the contribution due to the turbulent viscosity: , and the second contribution determines the Reynolds stress-induced generation of perturbations of mean vorticity by the effect of large-scale shear on turbulence (Elperin et al., 2003):

 ⟨uiuj⟩(S)=−l20[4C1Mij+C2(Nij+Hij)+C3Gij], (26)

where is the turbulent viscosity, ,

 Mij = (∂¯¯¯¯U(S))im(∂¯¯¯¯U)mj+(∂¯¯¯¯U(S))jm(∂¯¯¯¯U)mi,Gij=¯¯¯¯¯¯W(S)i¯¯¯¯¯¯Wj+¯¯¯¯¯¯W(S)j¯¯¯¯¯¯Wi, Hij = ¯¯¯¯¯¯W(S)n[εnim(∂¯¯¯¯U)mj+εnjm(∂¯¯¯¯U)mi],Nij=¯¯¯¯¯¯Wn[εnim(∂¯¯¯¯U(S))mj+εnjm(∂¯¯¯¯U(S))mi],

and are perturbations of the mean velocity and mean vorticity, while and are the equilibrium mean velocity and mean vorticity related to shear , the coefficients, , , , depend on the exponent of the energy spectrum. When small perturbations of the mean velocity, and the mean vorticity, , depend only on , the effective force is given by

 ρ−10F(S)i=−¯¯¯¯Sℓ20(β¯¯¯¯¯¯W′x,β0¯¯¯¯¯¯W′y,0), (27)

where and . Here we used the following identities:

 ∇jMij = −(¯¯¯¯S/4)(¯¯¯¯¯¯W′x,−