2 Directionality and componentality of turbulent transport anisotropization under the action of rotation; horizontal and vertical eddy viscosities and eddy diffusivities

The anisotropy introduced by the system rotation causes the effective viscosity and diffusivity to transform differently in the directions parallel and orthogonal to the direction of rotation (i.e. along $k_{\Vert }$ and $k_{\bot }$ , respectively) whose unity vector is aligned vertically and given by $\boldsymbol{e}_{3}=\unicode[STIX]{x1D734}/|\unicode[STIX]{x1D734}|$ . In this coordinate system, vectors collinear with $k_{\bot }$ and $k_{\Vert }$ become $k_{h}$ (‘horizontal’) and $k_{z}\equiv k_{3}$ (‘vertical’), respectively, such that

In the process of small-scale elimination, there will emerge two types of anisotropization – by direction and by component (hence componentality). Utilizing the least restrictive and most general approach, we assume that the renormalized viscosities differ in different directions and for different components. For the horizontal velocity components, the effective viscosities in the horizontal and vertical directions are, respectively, $\unicode[STIX]{x1D708}_{h}$ and $\unicode[STIX]{x1D708}_{z}$ ,

and for the vertical velocity component $v_{3}$ they are

In (2.2) and (2.3), the terms with $\unicode[STIX]{x1D6FF}$ mark anisotropic corrections to viscosity. These corrections are small for weak rotation, the limit in the focus of this investigation. This smallness will be utilized later in small parameter expansions.

The spectral viscosity operator acting on the horizontal velocity components is

while that acting on $v_{3}$ is

The renormalized diffusivity is directionally dependent and given by

in the horizontal and vertical directions, respectively, while the spectral diffusivity operator is

3 Governing equations

Consider the dynamics of fully three-dimensional, incompressible turbulent flow in a coordinate system rotating with the angular velocity $\unicode[STIX]{x1D734}$ . The flow occupies an infinite domain and its dynamics is governed by the momentum (Navier–Stokes) and continuity equations,

where $v_{\unicode[STIX]{x1D6FC}}$ is the velocity vector, $P$ is the pressure divided by the constant density, and $\unicode[STIX]{x1D708}_{0}$ is the molecular viscosity. The external solenoidal force $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FC}}^{0}$ mimics the effect of large-scale stirring that maintains turbulence in a statistically steady state.

In the spectral domain bounded by the viscous dissipation (or Kolmogorov) wavenumber $k_{d}\propto (\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}_{0}^{3})^{1/4}$ , a space–time Fourier transform of the velocity is given by

where $d\,(=3)$ is the dimension of space.

In the original Navier–Stokes equation all scales are resolved, and the Fourier transform of the viscous term is simply $\unicode[STIX]{x1D708}_{0}k^{2}$ . In the rotating coordinate frame, however, this term undergoes anisotropization elaborated in § 2. Anticipating this anisotropization, we added new viscous terms in (3.4) with which the Fourier-transformed momentum equation becomes

where $\hat{k}=(\unicode[STIX]{x1D714},\boldsymbol{k})$ , $\hat{q}=(\unicode[STIX]{x1D71B},\boldsymbol{q})$ , $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}^{0}(\hat{k})$ is the Fourier transform of the external forcing and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}$ is the permutation tensor. Of course, in the limit $\unicode[STIX]{x1D6FA}\rightarrow 0$ , all anisotropic corrections disappear (see (7.26), (7.27), (7.45)–(7.47) below) and the process of small-scale elimination will only produce the isotropic eddy viscosity term, $\unicode[STIX]{x1D708}k^{2}$ , as in Sukoriansky et al. (Reference Sukoriansky, Galperin and Staroselsky2003).

The Fourier-transformed continuity equation is

The pressure term is evaluated by taking the divergence of (3.4) using the continuity equation (3.5),

where the standard nonlinear term on the left-hand side is complemented by contributions due to the Coriolis force and terms expected to emerge from eddy viscosity anisotropization.

Substituting (3.6) in (3.4) and symmetrizing the resulting equation using the tensor $P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}(\boldsymbol{k})$ (e.g. Lesieur Reference Lesieur1997), where

and

is the operator that projects any vector to a plane normal to $\boldsymbol{k}$ , one can transform (3.4) into

Here, the inverse Green function,

is a non-diagonal tensor conditioned by the rotation-imposed anisotropy,

is the inverse auxiliary scalar Green function and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ is the Kronecker delta.

The momentum equation (3.9) can be rearranged in the form of the canonical equation,

used in the procedure of coarse graining. This equation preserves its form in flows with different extra strains while the strains’ effect is absorbed in the tensorial Green function.

The canonical equation appears as the Langevin equation,

where $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\hat{k})$ represents nonlinear stirring of a mode $\hat{k}$ by all other modes (Kraichnan Reference Kraichnan1987; McComb Reference McComb1991; Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2003) and is associated with the ‘dressed’ force. Generally, replacing the nonlinear term in the Navier–Stokes equations by a random stochastic force has been one of the tools used in theories of turbulence (e.g. Lesieur Reference Lesieur1997).

We distinguish between the ‘bare’ and ‘dressed’ forces, $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}^{0}(\hat{k})$ and $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\hat{k})$ (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005). The ‘bare’ force mimics the effect of large-scale instabilities and provides forcing that maintains turbulence in statistically steady state. In 3-D isotropic flows, this force is isotropic; it generates direct energy cascade characterized by the Kolmogorov kinetic energy spectrum. Due to the anisotropy of the Coriolis force, rotating flows are inherently anisotropic. A bare forcing may introduce additional complications if it is also anisotropic and produces helicity on large scales. A flow of this kind, with Beltrami forcing, was studied in direct numerical simulations (DNS) by Mininni, Rosenberg & Pouquet (Reference Mininni, Rosenberg and Pouquet2012). They found, however, that on scales smaller than the Woods scale, $L_{\unicode[STIX]{x1D6FA}}=k_{\unicode[STIX]{x1D6FA}}^{-1}$ (the wavenumber $k_{\unicode[STIX]{x1D6FA}}$ is defined by (7.36); the naming of the scale $L_{\unicode[STIX]{x1D6FA}}$ will be elucidated in § 7.1.3), the helicity cascade breaks down and a flow returns to isotropic Kolmogorov turbulence. This result is consistent with a general study by Biferale & Procaccia (Reference Biferale and Procaccia2005) showing that on small scales, the isotropic solution is always of leading order. Being informed of these results, it is assumed here that the bare and the dressed forces are isotropic and that the forcing scale, $L_{\unicode[STIX]{x1D709}}$ , does not significantly exceed $L_{\unicode[STIX]{x1D6FA}}$ . The ensuing results are correct on scales smaller than $L_{\unicode[STIX]{x1D709}}$ . It is possible that they will remain correct for $L_{\unicode[STIX]{x1D709}}/L_{\unicode[STIX]{x1D6FA}}\gg 1$ but we do not know that a priori. In a study that follows up, we shall investigate the limits of validity of the theory and present comparisons with observations.

The ‘dressed’, or effective, force, $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\hat{k})$ , acts upon every mode $k$ and represents nonlinear stirring of that mode by all other modes. The Langevin equation (3.13) is used in the coarse-graining procedure for which $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\hat{k})$ needs to be determined. Unfortunately, a complete mathematical representation of this force cannot be derived from first principles at the present time (McComb Reference McComb1991; Canuto & Dubovikov Reference Canuto and Dubovikov1996). Since we consider an incompressible, homogeneous in space, stochastically steady-state turbulent flow in an infinite domain, the force $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\hat{k})$ must be solenoidal, zero mean and homogeneous in space and time. The rate of the energy injection into the mode $k$ must be proportional to the energy dissipation rate $\unicode[STIX]{x1D716}$ . These requirements yield the correlation function in the form

where the multiplicative factor $k^{-3}$ ensures correct dimensionality and numerical coefficients are introduced for convenience. The correlator’s amplitude $D$ is proportional to the mean rate of energy transfer through the mode $k$ . The relationship between $D$ and $\unicode[STIX]{x1D716}$ in neutrally stratified non-rotating flows can be calculated using the energy balance considerations (Yakhot & Orszag Reference Yakhot and Orszag1986) which yield (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005)

We shall use (3.15) as a zero-order approximation; the next-order correction will be considered in § 13.

Note that the assumption of isotropic $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D714},\boldsymbol{k})$ is well justified only in non-rotating neutral flows (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2003). Various extra strains may invalidate this assumption. Indeed, diverse numerical simulations point to a sensitivity of the results to the nature of the forcing. However, similarly to the case of stable stratification (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005), it is assumed here that flow anisotropization due to rotation will chiefly manifest itself through the anisotropization of the Green functions. This assumption is particularly well justified for the case of weak rotation which is the main subject of the present study.

Equation (3.12) is the starting point of the scale elimination procedure. It requires the inversion of the tensorial operator $[G_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\hat{k})]^{-1}$ given by (3.10)–(3.11). Since $\boldsymbol{f}$ is aligned with the vertical axis in the chosen coordinate system, the second term on the right-hand side of (3.10) is $P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E}}(\boldsymbol{k})\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}}f_{\unicode[STIX]{x1D6FE}}=fP_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FE}}e_{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}}\equiv fS_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})$ , where

and the matrix representation of the tensor $e_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ is

Retaining only the terms up to $O(f^{2})$ in the limit of weak rotation, one can invert (3.9) to obtain

where

In the derivation of (3.18) we took into account that the Green function $G_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\hat{k})$ acts upon the solenoidal vector field $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D714},\boldsymbol{k})$ and, as will be shown later, $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}_{3}=O(f^{2})$ .

4 The procedure of successive coarse graining

The scale elimination formalism consists of the following steps (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005):

1. (i) Introduction of the dynamic dissipation cutoff wavenumber, $\unicode[STIX]{x1D6EC}$ , a small shell $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D6EC}\ll 1$ , which is subject to elimination, and ‘slow’ and ‘fast’ modes.

2. (ii) Computing $O(\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC})$ correction to the inverse Green function resulting from the ensemble averaging of the fast modes over $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ . This correction generates $O(\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC})$ accruals to all renormalized viscosities defined in § 2 while otherwise preserving the analytical form of the governing equations.

3. (iii) All viscosities are updated and the process moves forward towards elimination of the next shell $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ .

These steps are similar to those in flows with stable stratification (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005) and so they will only be described briefly.

The wavenumber of the dynamic dissipation cutoff, $\unicode[STIX]{x1D6EC}$ , defines the current domain of definition of the canonical equation (3.12), $D=(0,\unicode[STIX]{x1D6EC}]$ . At the first iteration, when all scales are included, $\unicode[STIX]{x1D6EC}=k_{d}$ . A small shell $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ subdivides the domain onto $D^{{<}}=(0,\unicode[STIX]{x1D6EC}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}]$ and $D^{{>}}=(\unicode[STIX]{x1D6EC}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6EC}]$ , such that $D=D^{{<}}\cup D^{{>}}$ . At every step of the scale elimination procedure, modes within the shell $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ undergo ensemble averaging which results in shrinking of the domain of definition from $D$ to $D^{{<}}$ and corresponding modifications of the viscosity and diffusivity which renders them $\unicode[STIX]{x1D6EC}$ dependent or ‘dressed’. The ‘dressed’ Green functions employ these ‘dressed’ viscosities and diffusivities. After every step of this process, the governing equations reproduce their form precisely and ‘forget’ about the molecular values.

The ‘slow’ ( $\boldsymbol{v}^{{<}}$ ) and ‘fast’ ( $\boldsymbol{v}^{{>}}$ ) velocity modes are defined such that $k\in D^{{<}}$ for $\boldsymbol{v}^{{<}}(\hat{k})$ and $k\in D^{{>}}$ for $\boldsymbol{v}^{{>}}(\hat{k})$ . The velocity mode $v_{\unicode[STIX]{x1D6FC}}$ is represented as a sum, $v_{\unicode[STIX]{x1D6FC}}=v_{\unicode[STIX]{x1D6FC}}^{{<}}+v_{\unicode[STIX]{x1D6FC}}^{{>}}$ , substitution of which in the canonical equation (3.12) yields

This equation holds for either $k\in D^{{<}}$ or $k\in D^{{>}}$ and so (4.1) applies to both $v_{\unicode[STIX]{x1D6FC}}^{{<}}$ and  $v_{\unicode[STIX]{x1D6FC}}^{{>}}$ .

Schematically, the equation for $v^{{<}}$ contains $v^{{>}}$ in the second and third terms of the integrand. In order to derive a self-contained equation for $v^{{<}}$ , we iterate this equation once by substituting (4.1) itself, written for $v^{{<}}$ and $v^{{>}}$ , in the second and third terms, respectively. We then replace $v^{{>}}$ by the Langevin equation (3.13) and perform ensemble averaging over the force $\unicode[STIX]{x1D743}^{{>}}$ . Details of this procedure are given in Sukoriansky et al. (Reference Sukoriansky, Galperin and Staroselsky2005). The final expression for the correction to the inverse Green function produced by the elimination of the shell $D^{{>}}$ is

where

Equation (4.2) contains the velocity correlation tensor, $U_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D71B},\boldsymbol{q})$ , which can be evaluated using the Langevin equation (3.13) and (3.18),

where

Note that the tensor $Q_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})$ is antisymmetric.

The integral in (4.2) is calculated in the long-time limit, $\unicode[STIX]{x1D714}\rightarrow 0$ , and using the spectral gap approximation which is explained in the next section.

Concluding this section we remark that a requirement imposed upon the ‘dressed’ force $\unicode[STIX]{x1D743}$ in the Langevin equation (3.13) is that $\langle \unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{p})\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D6FD}}(\boldsymbol{q})\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D70E}}(\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\rangle =0$ for all vector triads $(\boldsymbol{p},\boldsymbol{q},\boldsymbol{k}-\boldsymbol{p}-\boldsymbol{q})\in \unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ being a shell subject to elimination (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2003, Reference Sukoriansky, Galperin and Staroselsky2005). This requirement is somewhat weaker than the global quasi-Gaussianity implied in classical theories such as the EDQNM. The quasi-Gaussianity in the shell $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ alone suffices to develop a rigorous self-contained mathematical procedure of successive coarse graining. There is no need for the force $\unicode[STIX]{x1D743}(\hat{k})$ to be quasi-Gaussian in the entire domain although such field would fulfil the above condition. In general terms, $\unicode[STIX]{x1D743}(\hat{k})$ could be characterized as quasi-normal. The combination of a quasi-normal forcing and eddy damping represented by the renormalized viscosity places the QNSE theory in the class of quasi-normal eddy-damped theories of turbulence elucidated by e.g. Orszag (Reference Orszag, Balian and Peabe1977), Chasnov (Reference Chasnov1991), McComb (Reference McComb1991), Sagaut & Cambon (Reference Sagaut and Cambon2008).

5 The spectral gap approximation

The concept of viscosity implies the existence of a spectral gap between the resolvable (explicit) and subgrid (implicit) scales which may be either molecular or computational. While in the former case the gap is real, in the latter it does not exist and as a result, the effect of the subgrid scales on the explicit ones in the vicinity of $\unicode[STIX]{x1D6EC}$ differs from that away from it. Recognizing this physics, Kraichnan (Reference Kraichnan1976) introduced a two-parametric viscosity denoted $\unicode[STIX]{x1D708}(k|\unicode[STIX]{x1D6EC})$ which depends on the local explicit wavenumber $k$ and the dynamic dissipation cutoff $\unicode[STIX]{x1D6EC}$ .

The idea of a spectral gap is used to simplify the computation of the correction to the inverse Green function given by (4.2). In this computation, only the terms up to $O[(k/\unicode[STIX]{x1D6EC})^{2}]$ are retained. This is equivalent to introducing a virtual spectral gap and taking all renormalized viscosities constant and equal to their values at $\unicode[STIX]{x1D6EC}$ . This method will be named a spectral gap approximation, or SGA. Note that in Yakhot & Orszag (Reference Yakhot and Orszag1986), as well as in our earlier publications, this method was referred to as the distant interaction approximation, or DIA. This abbreviation conflicts with DIA introduced by Kraichnan (Reference Kraichnan1959) for the direct interaction approximation. The new name mitigates this conflict.

To illustrate the effect of the SGA, figure 1 shows the two-parametric viscosity, $\unicode[STIX]{x1D708}(k|\unicode[STIX]{x1D6EC})$ , computed following Kraichnan (Reference Kraichnan1976) for 3-D non-rotating flows and normalized with $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6EC})$ evaluated using the QNSE theory and SGA. The two-parametric viscosity displays a cusp-like behaviour at $k\rightarrow \unicode[STIX]{x1D6EC}$ the nature of which was elaborated by Kraichnan (Reference Kraichnan1976). Even though the cusp is lost in QNSE, it gives reliable subgrid scale parameterization in 3-D flows featuring direct energy cascade.

Figure 1. Plot of $\unicode[STIX]{x1D708}(k|\unicode[STIX]{x1D6EC})/\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6EC})$ (solid line) in non-rotating 3-D flows. Here $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6EC})$ is evaluated using QNSE with SGA.

In flows with inverse energy cascade, the SGA becomes problematic because $\unicode[STIX]{x1D708}(k|\unicode[STIX]{x1D6EC})$ changes sign and grows negative at $k\rightarrow 0$ , as shown in figure 2 (also see Chekhlov et al. Reference Chekhlov, Orszag, Sukoriansky, Galperin and Staroselsky1994). A problem of this kind arises in 3-D flows with a solid body rotation where the energy is gradually channelled into a manifold occupying a conical region orthogonal to the axis of rotation giving rise to the inverse energy cascade (Sagaut & Cambon Reference Sagaut and Cambon2008). In such flows, the QNSE formalism can only be applied on relatively small scales in the assumption that the forcing acts on scales where the eddy viscosity remains positive. In other words, in flows with rotation, the QNSE theory can only be used in the limit of weak rotation or on scales where the effect of rotation is weak. Although this limitation constrains the applicability of the QNSE theory to rotating flows, it does not preclude the possibility of developing an analytical theory of transition from Kolmogorov to rotation-dominated turbulence which is a subject of the forthcoming sections. Note in this respect that the QNSE theory was also used to develop an analytical theory of the transition from Kolmogorov to stable stratification-dominated turbulence where it provided a deep physical insight of flow anisotropization and turbulence–internal wave interaction (Sukoriansky & Galperin Reference Sukoriansky and Galperin2013).

Figure 2. Normalized two-parametric eddy viscosity in isotropic 2-D flows from DNS (dots), the test field model (dashed line) and the renormalization group theory (solid line). (After Sukoriansky & Zemach Reference Sukoriansky and Zemach2016, © The Royal Swedish Academy of Sciences. Reproduced by permission of IOP Publishing. All rights reserved.)

6 Passive-scalar diffusion equation and its successive coarse graining

The procedure of successive quasi-normal coarse graining can be developed for the scalar diffusion equation. The resulting equation and (4.2) will be used later to compute scale-dependent eddy viscosities and eddy diffusivities.

The diffusion of a passive scalar $c$ is described by a standard diffusion equation

where $\unicode[STIX]{x1D705}_{0}$ is the molecular diffusivity. Space–time Fourier-transformed representation of (6.1) is

where $G_{D}(\hat{k})$ is the diffusion Green function,

where following § 2 an anisotropic correction is included.

Similar to (3.13), (6.2) can be written in the form of the Langevin equation for a scalar $c$ ,

where $\unicode[STIX]{x1D709}_{c}(\hat{k})$ represents nonlinear stirring of a mode $\hat{k}$ by all other modes.

Successive coarse graining of the diffusion equation (6.2) is performed using a two-part procedure similar to that outlined in § 4 for the momentum equation. Namely, ‘slow’, $c^{{<}}$ , and ‘fast’, $c^{{>}}$ , scalar modes are introduced and the expansion $c=c^{{<}}+c^{{>}}$ is substituted in (6.2) yielding

Similarly to (4.1), this equation is valid for either $k\in D^{{<}}$ or $k\in D^{{>}}$ .

Then, an equation for $c^{{<}}(\hat{k})$ is derived by manipulating (6.5) written for $c^{{<}}(\hat{k})$ . The second and third terms are iterated by substituting the ‘slow’ mode equations for $c^{{<}}$ and $v^{{<}}$ , respectively, while the term $v^{{>}}c^{{>}}$ is iterated by substituting the equation for $c^{{>}}$ . Then, replacing the ‘fast’ variables $v^{{>}}$ and $c^{{>}}$ by their respective Langevin equations (3.13) and (6.4) and ensemble averaging the result over the fast modes $\unicode[STIX]{x1D743}^{{>}}$ and $\unicode[STIX]{x1D743}_{c}^{{>}}$ , we compute corrections to the inverse Green function produced by the elimination of the shell $D^{{>}}$ ,

Similarly to (4.2), the integral in (6.6) is calculated in the long-time limit, $\unicode[STIX]{x1D714}\rightarrow 0$ , and using the spectral gap approximation.

Note that (6.6) depends on the velocity correlator $U_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D71B},\boldsymbol{q})$ and so the statistics of the scalar forcing $\unicode[STIX]{x1D709}_{c}(\hat{k})$ is immaterial. The closure is achieved by neglecting the odd velocity-scalar moments of the fast modes in the iterated (6.5) (see Sukoriansky et al. (Reference Sukoriansky, Galperin and Staroselsky2005) for details).

7 Computation of the effective viscosities and diffusivities

Evaluation of the integrals in (4.2) and (6.6) is accomplished in the following steps:

1. (i) Contour integration over the frequency $\unicode[STIX]{x1D71B}$ .

2. (ii) Invoking the spectral gap approximation.

3. (iii) Integration over the spherical shell $D^{{>}}$ which is a two-step operation combining multiplication by $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ and the angular integration over the surface of a sphere with the radius $\unicode[STIX]{x1D6EC}$ . The inherent anisotropy due to the system rotation complicates the integration. However, by utilizing the isotropy of the integrand in the plane normal to $\unicode[STIX]{x1D734}$ one can carry out all calculations analytically.

In the next sections it will be shown that the corrections to the inverse velocity and diffusion Green functions caused by the elimination of the fast shell $D^{{>}}$ result in $O(\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC})$ corrections to eddy viscosities and eddy diffusivities such that both $G_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}^{-1}(0,k,k_{z})+\unicode[STIX]{x0394}G_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}^{-1}(0,k,k_{z})$ and $G_{D}^{-1}(0,k,k_{z})+\unicode[STIX]{x0394}G_{D}^{-1}(0,k,k_{z})$ preserve their forms. The procedure also generates a correction proportional to $fk^{2}S_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(k,k_{z})$ . This correction corresponds to renormalization of $\unicode[STIX]{x1D6FA}$ . It should be discarded as being $O(k^{2})$ small whereas the original term is $O(1)$ . Finally, by dividing $\unicode[STIX]{x0394}G_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}^{-1}(0,k,k_{z})$ and $\unicode[STIX]{x0394}G_{D}^{-1}(0,k,k_{z})$ by $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}$ and taking the limit $\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC}\rightarrow 0$ , we obtain a system of coupled ordinary differential equations for all eddy viscosities and eddy diffusivities.

In the process of computation, the integral in (6.6) will be evaluated first. This integral is simpler of the two and its evaluation provides a good illustration for calculation of the other integral. In all the forthcoming derivations, the continuity equation (3.5) will be routinely enforced. This means that if a term, such as the integral (4.2), is factored with $v_{\unicode[STIX]{x1D6FD}}(\hat{k})$ then any part of this term proportional to $k_{\unicode[STIX]{x1D6FD}}$ will be set to zero automatically. Other simplifying relationships are related to the symmetry and solenoidality properties of the projection operator, i.e.  $P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})=P_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}}(\boldsymbol{k})$ and $k_{\unicode[STIX]{x1D6FC}}P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})=0$ , and zeroing out the contractions of symmetric and antisymmetric tensors (e.g.  $P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})$ and $Q_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})$ or $P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})$ and $e_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ ). All related simplifications will usually be implemented without specific mentioning.

8 The forcing amplitude

It is relatively straightforward to determine the forcing amplitude $D$ in (3.14) in the non-rotating case in which the velocity correlation tensor is $U_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(k,\unicode[STIX]{x1D714})=|G_{0}(k,\unicode[STIX]{x1D714})|^{2}D(k)P_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{k})$ and the 3-D energy spectrum is

( $E_{0}(k)$ denotes the energy spectrum in a non-rotating flow). Let $k_{c}$ be the dynamic dissipation cutoff bounding the range of eliminated (or implicit) modes $p>k_{c}$ . The effective viscosity $\unicode[STIX]{x1D708}_{n}(k_{c})$ is a result of this scale elimination. It acts on the resolved (explicit) modes in such a way as to ensure that the energy loss in the range $0<p<k_{c}$ is equal precisely to the dissipation rate $\unicode[STIX]{x1D716}$ ,

Substituting (7.28) and (8.1) into (8.2), we find

which makes it possible to relate the eddy viscosity and the energy spectrum of non-rotating turbulence to $\unicode[STIX]{x1D716}$ ,

where the Kolmogorov constant $C_{K}\simeq 1.46$ . This value agrees well with the experimental high Reynolds number estimate, $C_{K}=1.5\pm 0.15$ , reported by Sreenivasan (Reference Sreenivasan1995). Variation of $\unicode[STIX]{x1D716}$ due to rotation will be considered in § 13.

9 Eddy viscosities, eddy diffusivities and flow regimes in rotating and stably stratified turbulence

The solution $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}_{3z}=0$ simplifies (7.26), (7.27) and (7.45)–(7.47) which now can be solved analytically in the lowest order of $Ro(k)^{-1}$ . Designating the independent variable as $k$ (instead of $\unicode[STIX]{x1D6EC}$ ) one obtains the scale-dependent eddy viscosities,

With no rotation, i.e. in the limit $Ro^{-1}\rightarrow 0$ , the flow becomes isotropic, $\unicode[STIX]{x1D708}_{h}=\unicode[STIX]{x1D708}_{z}=\unicode[STIX]{x1D708}_{3}=\unicode[STIX]{x1D708}_{3z}$ . For $k/k_{\unicode[STIX]{x1D6FA}}\gtrsim 10$ , the deviations of $\unicode[STIX]{x1D708}_{h}$ , $\unicode[STIX]{x1D708}_{z}$ , $\unicode[STIX]{x1D708}_{3}$ and $\unicode[STIX]{x1D708}_{3z}$ from their values in non-rotating flows are within 5 %. On larger scales, rotation causes gradual decrease of the first three ratios, $\unicode[STIX]{x1D708}_{h}/\unicode[STIX]{x1D708}_{n}$ , $\unicode[STIX]{x1D708}_{z}/\unicode[STIX]{x1D708}_{n}$ and $\unicode[STIX]{x1D708}_{3}/\unicode[STIX]{x1D708}_{n}$ , and the increase of the fourth one. The decreasing viscosities, especially $\unicode[STIX]{x1D708}_{h}\rightarrow 0$ , point to a decreasing energy flux to small scales. The concurrent energy redirection to large scales can be interpreted as energy backscatter or the inverse cascade. From (9.1)–(9.3) one infers that the horizontal viscosity decreases at the fastest rate and may become zero at $k/k_{\unicode[STIX]{x1D6FA}}=O(1)$ . Numerical solution of the system (7.45)–(7.48), which can be carried out for larger values of $\unicode[STIX]{x1D6FA}$ than it is feasible in the weak rotation approximation, indicates that $\unicode[STIX]{x1D708}_{h}\rightarrow 0$ at considerably smaller values of $k/k_{\unicode[STIX]{x1D6FA}}$ . This result is consistent with numerical simulations by Smith & Waleffe (Reference Smith and Waleffe1999) and Lindborg (Reference Lindborg2005) who showed that at $k_{\unicode[STIX]{x1D709}}/k_{\unicode[STIX]{x1D6FA}}\simeq 0.01$ , $k_{\unicode[STIX]{x1D709}}$ being the wavenumber of the forcing, a 3-D flow develops strong inverse energy cascade and acquires features of a ‘negative viscosity phenomenon’ (Starr Reference Starr1968).

Have we reached a dead end by arriving at the negative eddy viscosity? The answer is ‘no’ because, as explained in Galperin et al. (Reference Galperin, Sukoriansky, Orszag and Staroselsky1993), Sukoriansky et al. (Reference Sukoriansky, Chekhlov, Orszag, Galperin and Staroselsky1996) and Sukoriansky, Galperin & Chekhlov (Reference Sukoriansky, Galperin and Chekhlov1999), the negative Laplacian viscosity in quasi-2-D flows represents only part of the physics. The other part can be gleaned from figure 2 showing that, on smaller scales, the effect of the negative viscosity is compensated by a biharmonic and, possibly, higher-order dissipative hyperviscosities. Thus emerging SGS parameterization was coined a stabilized negative viscosity, or SNV by Sukoriansky et al. (Reference Sukoriansky, Chekhlov, Orszag, Galperin and Staroselsky1996) and Sukoriansky et al. (Reference Sukoriansky, Galperin and Chekhlov1999). The negative Laplacian viscosity in SNV assumes the role of the unresolved subgrid-scale forcing which otherwise would be unaccounted for in estimating the large-scale eddy kinetic energy and the global energy balance. It is well possible that an SNV representation can be developed within the QNSE formalism for 3-D flows to alleviate the problems posed by the use of the spectral gap approximation as elaborated in § 5.

As was already noted, $\unicode[STIX]{x1D708}_{h}$ decreases faster than $\unicode[STIX]{x1D708}_{z}$ and $\unicode[STIX]{x1D708}_{3}$ with the increasing scale. This result is consistent with the experiments by Jacquin et al. (Reference Jacquin, Leuchter, Cambon and Mathieu1990). Numerical simulations by Yeung & Zhou (Reference Yeung and Zhou1998) demonstrated that the spectral energy transfer in the direction of rotation is similar to that in non-rotating flows, but in the normal direction, the transfer weakens significantly. The ratios, $\unicode[STIX]{x1D708}_{z}/\unicode[STIX]{x1D708}_{h}$ and $\unicode[STIX]{x1D708}_{3}/\unicode[STIX]{x1D708}_{h}$ ,

can be viewed as another evidence of increasing flow two-dimensionalization as they are related to the increase of the vertical correlation length scale compared to its horizontal counterpart. The anisotropization of the integral length scale tensor in the physical space was detected in simulations by Yeung & Zhou (Reference Yeung and Zhou1998). As they put it, ‘…the large-scale motions are lengthened along the axis of rotation but are more compact within the plane orthogonal to this axis’.

Equations (9.1)–(9.3) make it possible to solve (7.26) and (7.27) analytically and derive expressions for the horizontal and vertical eddy diffusivities in the limit of weak rotation,

where $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D705}_{n}/\unicode[STIX]{x1D708}_{n}=Pr_{t0}^{-1}\simeq 1.39$ is the inverse turbulent Prandtl number in non-rotating flows (Yakhot & Orszag Reference Yakhot and Orszag1986; Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005).

On the first sight, the increase of the horizontal diffusivity versus the decrease of its vertical counterpart seems to contradict the data. For instance, using analytical and numerical tools, Cambon et al. (Reference Cambon, Godeferd, Nicolleau and Vassilicos2004) and Rodriguez Imazio & Mininni (Reference Rodriguez Imazio and Mininni2013) show that in rotating flows, the vertical diffusivity increases while the horizontal diffusivity decreases; eventually, their ratio attains the value of approximately 2. Their analysis, however, pertains not to scale-dependent eddy diffusivities, such as $\unicode[STIX]{x1D705}_{h}(k)$ and $\unicode[STIX]{x1D705}_{z}(k)$ , but the total eddy diffusivities obtained by averaging over all fluctuating scales. Those diffusivities grow with the integral scales in the respective directions. Since the integral length scale is larger in the vertical direction (Yeung & Zhou Reference Yeung and Zhou1998; Müller & Thiele Reference Müller and Thiele2007; Godeferd & Moisy Reference Godeferd and Moisy2015), the total vertical diffusivity may become larger than the horizontal one.

The increase of the horizontal spectral diffusivity in rotating flows undergoing two-dimensionalization pointed to by (9.7) can be better understood by comparing the values of $\unicode[STIX]{x1D705}_{h}$ in isotropic turbulence with 2-D versus 3-D geometry. Solving (7.30) for $d=2$ and $d=3$ and taking a note that for $d=2$ , $A_{d}=(16\unicode[STIX]{x03C0})^{-1}$ and $K_{d}=(4\unicode[STIX]{x03C0})^{-1}$ , one derives

such that $\unicode[STIX]{x1D705}_{n}^{2D}/\unicode[STIX]{x1D708}_{n}$ is approximately 1.4 times larger than $\unicode[STIX]{x1D705}_{n}^{3D}/\unicode[STIX]{x1D708}_{n}=\unicode[STIX]{x1D6FC}\sim 1.4$ . In other words, the ratio $\unicode[STIX]{x1D705}_{h}/\unicode[STIX]{x1D708}_{n}$ increases from $\unicode[STIX]{x1D6FC}=1.39$ to approximately 2 with increasing scale and expanding two-dimensionalization.

Figure 3. Vertical and horizontal eddy viscosities and eddy diffusivities normalized with $\unicode[STIX]{x1D708}_{n}$ for turbulence with rotation (a) and stable stratification (b) (after Sukoriansky & Galperin Reference Sukoriansky and Galperin2013). Horizontal and vertical eddy viscosities and vertical eddy diffusivities exhibit opposite tendencies.

Figure 3 compares the behaviours of the eddy viscosities and eddy diffusivities in rotating versus stably stratified flows as analysed by Sukoriansky & Galperin (Reference Sukoriansky and Galperin2013). Immediately obvious is the similarity between the scalings with $k/k_{\unicode[STIX]{x1D6FA}}$ in the former and $k/k_{O}$ in the latter which highlights the affinity between the Woods and Ozmidov scales in flows with different extra strains.

On relatively small scales, the effect of either extra strain is miniscule and practically not felt by the flow. Rotational effects become sizeable at respective scales of approximately 0.1 $L_{\unicode[STIX]{x1D6FA}}$ and 0.1 $L_{O}$ where one observes increasing with scale anisotropization of both flows. The tendencies to anisotropization expressed by the horizontal and vertical eddy viscosities and eddy diffusivities under the action of rotation and stable stratification are switched around as $\unicode[STIX]{x1D708}_{h}/\unicode[STIX]{x1D708}_{n}$ decreases with decreasing $k/k_{\unicode[STIX]{x1D6FA}}$ in the former case and increases with decreasing $k/k_{O}$ in the latter. On the other hand, while $\unicode[STIX]{x1D705}_{z}/\unicode[STIX]{x1D708}_{n}$ stays nearly constant with $k/k_{\unicode[STIX]{x1D6FA}}$ , it decreases and tends to zero with $k/k_{O}$ . These differences are stipulated by the action of different types of extra strains whose directions of zero frequency are orthogonal.

On larger scales, while the horizontal viscosity in rotating flows decreases due to inverse energy cascade, the eddy diffusivities always remain positive definite regardless of the direction of energy transfer (e.g. Kraichnan Reference Kraichnan1976). The increase in $\unicode[STIX]{x1D705}_{h}/\unicode[STIX]{x1D708}_{n}$ from approximately 1.4 to approximately 2 confirms the tendency towards flow two-dimensionalization under the action of rotation, and is quantitatively consistent with (9.9). On the other hand, the ratio $\unicode[STIX]{x1D705}_{z}/\unicode[STIX]{x1D708}_{n}$ is almost $\unicode[STIX]{x1D6FA}$ -independent.

Although QNSE implies the arbitrarily small molecular viscosity, consideration of the viscous effects is necessary for practical purposes. On very small scales, of the order of the Kolmogorov microscale, $L_{K}=(\unicode[STIX]{x1D708}_{0}^{3}/\unicode[STIX]{x1D716})^{1/4}$ , both rotating and stably stratified flows are strongly impacted by the effect of the molecular viscosity. For the latter class of flows, the range of scales between $L_{O}$ and $L_{K}$ is quantified by the buoyancy Reynolds number $Re_{b}$ (also known as the isotropy index, Thorpe Reference Thorpe2005),

The value $Re_{b}\simeq 10^{2}$ is an approximate threshold below which the viscous dissipation becomes increasingly anisotropic (e.g. Galperin & Sukoriansky Reference Galperin and Sukoriansky2010, and references therein).

A similar parameter can be introduced for rotating flows,

where $Re_{\unicode[STIX]{x1D6FA}}$ can be referred to as the rotational Reynolds number. It is plausible to assume that the threshold of the viscous dissipation anisotropization in rotating flows is about the same as in stratified flows, i.e.  $Re_{\unicode[STIX]{x1D6FA}}\simeq 10^{2}$ . Since $Re_{\unicode[STIX]{x1D6FA}}\gg Re_{b}$ in geophysical flows, this criterion is of little consequence. It may be important in computer simulations of rotating flows, however, where the Reynolds numbers are still relatively low.

10 Dispersion relation for inertial waves in the presence of turbulence

Consider now the effect of turbulence on inertial waves. Sukoriansky et al. (Reference Sukoriansky, Galperin and Staroselsky2005) showed that turbulence modifies the classical dispersion relation for linear internal waves and it is important to know whether or not turbulence has a similar effect on the inertial waves. This can be done using the Langevin equation (3.13) that describes a linear, forced, stochastic oscillator and whose eigenfrequencies are given by the real parts of the roots of the secular equation

These roots are the poles of the Green function $G_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D71B},\boldsymbol{k})$ given by (7.40). Recalling (7.7), one writes the eigenfrequencies as

where

$\unicode[STIX]{x1D703}$ being the angle between the wave vector $\boldsymbol{k}$ and the vertical axis, is the classical dispersion relationship for linear inertial waves (Landau & Lifshitz Reference Landau and Lifshitz1993). Thus, turbulence does not change the linear wave frequency. The imaginary part of $\unicode[STIX]{x1D71B}$ is related to turbulent dissipation and, as elaborated by Galperin, Sukoriansky & Dikovskaya (Reference Galperin, Sukoriansky and Dikovskaya2010) in the appendix A, it broadens the spectral peak associated with the wave.

The dispersion relationship (10.2) underscores the difference between the inertial and internal waves. Recall that in the case of stable stratification, turbulence does modify the linear wave frequency. Moreover, internal waves are completely eradicated on scales where turbulence contribution renders $\unicode[STIX]{x1D714}$ imaginary (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005).

We emphasize that even though turbulence does not shift the frequency of the inertial waves in the QNSE theory, turbulence and waves are considered as one entity in exactly the same fashion as it was done in the case of flows with stable stratification (Sukoriansky et al. Reference Sukoriansky, Galperin and Staroselsky2005).

We also notice that though turbulence affects the dispersion relation for internal waves, their range of existence, $\unicode[STIX]{x1D714}\leqslant N$ , remains the same as in the case of purely linear waves. As shown above, turbulence does not affect the dispersion relation for inertial waves at all. Therefore, the range of existence of the mixed internal–inertial waves, $f\leqslant \unicode[STIX]{x1D714}\leqslant N$ (Phillips Reference Phillips1977), should remain the same as in the case of linear waves even if the waves coexist with turbulence.

11 Spectra of kinetic energy and scalar variance

Laboratory experiments, numerical simulations, ocean currents and various layers of the atmosphere exhibit a remarkable and ubiquitous vertical spectrum of the horizontal velocity obeying a spectral law $E_{1}(k_{z})\propto N^{2}k_{z}^{-3}$ (Smith, Fritts & VanZandt Reference Smith, Fritts and Vanzandt1987; Galperin & Sukoriansky Reference Galperin and Sukoriansky2010; Sukoriansky & Galperin Reference Sukoriansky and Galperin2013). The horizontal layering in such flows can be associated with modes with zero frequency, i.e. slow modes (e.g. Sagaut & Cambon Reference Sagaut and Cambon2008). This spectrum is peculiar because, while being aligned in the direction of zero frequency, it nevertheless depends on $N$ . In fact, this spectrum depends on $N$ and $k_{z}$ only and is, thus, independent of the forcing, seasons and altitude, if the atmospheric realization is considered (Smith et al. Reference Smith, Fritts and Vanzandt1987). In QNSE, this spectrum was derived in the approximation of weakly stratified turbulence which reduces to Kolmogorov turbulence on small scales. Since the large-scale spectrum also depends on $N$ and $k_{z}$ only, the QNSE results appear applicable well beyond their domain of validity. The importance of this result is underscored by the fact that the non-dimensional coefficient in the spectrum expression turned out to be invariant across the scales. In fact, this is the only existing theoretical derivation of this coefficient. Note also that the QNSE theory attributes this spectrum to the interaction between turbulence and internal waves rather than accepting the traditional view that the spectrum is a product of internal wave saturation (see e.g. Dewan Reference Dewan1979; Dewan & Good Reference Dewan and Good1986).

In analogy to turbulence with stable stratification, various 1-D and 3-D spectra of turbulence with rotation will be derived in the approximation of weak rotation. Being a product of interaction between turbulence and inertial waves, they are of considerable interest, particularly if they also turn out to be applicable beyond the domain of their derivation.

In computing the spectra, we follow the nomenclature in Monin & Yaglom (Reference Monin and Yaglom1975) and define by $E_{1}(k_{1})$ a 1-D longitudinal spectrum of the velocity component $v_{1}$ , while $E_{1}(k_{2})$ and $E_{1}(k_{3})$ are the transverse spectra of $v_{1}$ in the horizontal and vertical directions. In the case of weak rotation, all 1-D spectra can be calculated analytically. For instance, for the 1-D longitudinal horizontal spectrum $E_{1}(k_{1})$ one finds

Generally, any other 1-D spectrum $E_{i}(k_{j})$ can be computed in a similar way by integrating the spectral tensor $U_{ii}(\unicode[STIX]{x1D714},\boldsymbol{k})$ , no summation over $i$ is implied, over the frequency $\unicode[STIX]{x1D714}$ and the two wavenumbers different from $k_{j}$ .

Those other 1-D spectra are

Note that the effect of rotation on the vertical velocity component is weaker than that on the two horizontal ones.

The 3-D energy spectrum is computed as

where $E_{0}(k)$ is given by (8.5) and $\unicode[STIX]{x1D6FF}E(k)$ is the lowest-order correction to $E_{0}(k)$ due to rotation. Here, $\int _{\unicode[STIX]{x2202}S}\text{d}\unicode[STIX]{x1D6F4}$ denotes integration over the surface of a 3-D unit sphere. These equations demonstrate that within QNSE, rotation provides positive contributions to all Kolmogorov 1-D and 3-D spectra; they are all proportional to the respective factors $f^{2}k_{i}^{-3}$ , $i=1,2,3$ .

Comparing these results with numerical simulations recall that turbulence isotropization on scales smaller than the Woods scale was observed by Mininni et al. (Reference Mininni, Rosenberg and Pouquet2012) and Sen et al. (Reference Sen, Mininni, Rosenberg and Pouquet2012). The relevance of the Coriolis term-related additions $\propto f^{2}k_{\bot }^{-3}$ to the isotropic Kolmogorov horizontal spectra can be assessed via comparison with computer simulations. Two factors ought to be kept in mind in making such comparisons. (i) It is well known that physical and spectral space characteristics of rotating turbulence profoundly depend on the way it is being forced (e.g. Smith, Chasnov & Waleffe Reference Smith, Chasnov and Waleffe1996; Bourouiba et al. Reference Bourouiba, Straub and Waite2012; Sen et al. Reference Sen, Mininni, Rosenberg and Pouquet2012). The QNSE formalism developed in this paper implies isotropic ‘bare’ forcing which is then projected onto isotropic ‘dressed’ modal forcing specified by (3.14). Thus, numerical results based upon isotropic forcing are most suitable for comparison with QNSE predictions. In principle, QNSE could be extended to anisotropic and other types of forcing by modification of (3.14) but this is a separate effort which is outside the scope of the present study. (ii) The resolution of simulations performed to date was sufficient to attain only relatively small values of the rotational Reynolds number, $Re_{\unicode[STIX]{x1D6FA}}\lesssim 70$ , and oftentimes, they were much smaller. QNSE, on the other hand, implies arbitrarily large $Re_{\unicode[STIX]{x1D6FA}}$ such that some differences in flow regimes are unavoidable.

Early simulations by Smith & Waleffe (Reference Smith and Waleffe1999) provided evidence of the $k_{\bot }^{-3}$ spectral scaling but the transition to the small-scale Kolmogorov range was unresolved.

Mininni et al. (Reference Mininni, Rosenberg and Pouquet2012) performed simulations with the resolution of 3072 $^{3}$ points in which the Woods scale was more than an order of magnitude larger than the dissipation scale thus allowing for the ranges of the large-scale inverse energy cascade, direct cascades of energy and helicity, and the Kolmogorov cascade to small-scale dissipation to be resolved reasonably well. Even though these simulations employed the Beltrami forcing that evokes a helical flow regime, they presented clear evidence of flow three-dimensionalization and establishing the Kolmogorov $k_{\bot }^{-5/3}$ spectral laws for both energy and helicity on scales smaller than $L_{\unicode[STIX]{x1D6FA}}$ .

Sen et al. (Reference Sen, Mininni, Rosenberg and Pouquet2012) investigated rotating flows with inverse cascade at a fixed small Rossby number and a linear resolution of 384 points. They considered a flow in a periodic box with the aspect ratio of unity and employed both helical and non-helical forcing functions. To attain reasonably high Reynolds numbers, they employed LES in which the EDQNM-based SGS parameterization included the effect of helicity. Following Mininni et al. (Reference Mininni, Rosenberg and Pouquet2012), this model assumed that turbulence returns to 3-D isotropy on scales smaller than $L_{\unicode[STIX]{x1D6FA}}$ . In addition, following Baerenzung et al. (Reference Baerenzung, Politano, Ponty and Pouquet2008), it included energy backscatter. The simulated large-scale energy spectrum attained either the $k_{\bot }^{-5/3}$ power law consistent with the inverse energy cascade of quasi-2-D turbulence or a steeper $k_{\bot }^{-3}$ slope in which 3-D modes release significant amount of energy to 2-D modes. The spectral shape ultimately depended on forcing isotropy: the $k_{\bot }^{-5/3}$ law was associated with the forcing that selectively injected energy into 2-D modes while the $k_{\bot }^{-3}$ law was more endemic to isotropic forcing.

Thiele & Müller (Reference Thiele and Müller2009) found that turbulence, driven by a large-scale forcing, exhibits a 1-D stationary scaling $\propto k_{\bot }^{-2}$ in the rotation-dominated inertial range. The spectrum was computed in analogy to (11.1) or (11.2) with the integration over $k_{3}$ . The individual spectra, without the integration over $k_{3}$ , exhibited the spectral slope $\propto k^{-3}$ .

Generally, the horizontal spectra (11.1) and (11.2) are equivalents of the spectrum $E_{1}(k_{3})$ in the case of stable stratification. Similar to the latter, they demonstrate the transition from the Kolmogorov spectrum to a steeper spectrum $\propto f^{2}k_{\bot }^{-3}$ that depends on $f$ and $k\bot$ only. Unfortunately, almost no spectrum obtained in simulations and exhibiting the $k_{\bot }^{-3}$ slope was checked for the $f^{2}$ dependence as well and so the nature of the $k_{\bot }^{-3}$ scaling is difficult to ascertain even though it may be important for explaining the horizontal spectra of atmospheric winds and ocean currents that often exhibit such slopes.

One may also inquire about a possible connection between the present results and studies of turbulence in the limit of fast rotation. For instance, Galtier (Reference Galtier2003) considered rotating turbulence in the limit of small Rossby number using weak turbulence theory which yielded the energy spectrum in the form $E(k)\propto k_{\bot }^{-5/2}k_{\Vert }^{-1/2}$ . This distribution is quite different from those obtained here. Note, however, that Galtier (Reference Galtier2003) considered a limit of weak turbulence and strong wave interaction while the regime considered here implies strong turbulence and weak wave interaction. The former regime does not converge to Kolmogorov turbulence for $\unicode[STIX]{x1D734}\rightarrow 0$ and for this reason, there is no analytical transition between the two limits.

Spectra of a passive-scalar variance (labelled with $\unicode[STIX]{x1D703}$ ) can be computed using the passive-scalar Langevin equation (6.4) and scalar variance dissipation equation that, in analogy to (8.2), equates the loss of the scalar variance due to the action of the effective diffusivity $\unicode[STIX]{x1D705}(k_{c})$ in the resolved range $0<p<k_{c}$ to the scalar dissipation rate $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$ ,

Unlike the energy spectra, all rotation-related terms in (11.7)–(11.9) are negative such that all three spectra fall below the Kolmogorov law with increasing scale. Rodriguez Imazio & Mininni (Reference Rodriguez Imazio and Mininni2011, Reference Rodriguez Imazio and Mininni2015) advanced a simple phenomenological scaling argument for rotating non-helical flows according to which, if the energy spectrum follows a power law $E(k)\propto k^{-n}$ , then the spectrum of a passive-scalar variance is $E_{\unicode[STIX]{x1D703}}(k)\propto k^{-m}$ , where $m=(5-n)/2$ . Accordingly, since all energy spectra, (11.1)–(11.6), undergo the transition from the Kolmogorov $k^{-5/3}$ to $k^{-3}$ power law with the increasing scale, then the slope of the corresponding scalar variance spectra can be expected to flatten from the $k^{-5/3}$ to $k^{-1}$ which indeed (11.7)–(11.9) may be indicative of since the coefficients $C_{1}$ , $C_{13}$ and $C_{11}$ are all negative. The results do not extend far enough, however, to determine whether or not the $k^{-1}$ slope is attained.

12 The anisotropy of spectral energy transfers

Equations (11.1)–(11.5) make it possible to analyse the energy transfers between different directions and different velocity components in rotating turbulence. One observes that contributions to the horizontal spectra exceed those to their vertical counterparts, which is true for both the transverse, i.e.  $E_{1}(k_{2})>E_{1}(k_{3})$ , and the longitudinal, i.e.  $E_{1}(k_{1})>E_{3}(k_{3})$ , spectra. These tendencies are produced by turbulence energy redistribution between different directions and concentration in the equatorial plane in the spectral space. This concentration is consistent with the Taylor–Proudman constraint that causes the appearance of columnar vortices or ‘cigars’. The coherence inside these ‘cigars’ increases while vertical gradients diminish leading to decimation of the vortex stretching and development of the inverse cascade on large scales in the horizontal planes. Even though the flow remains three-dimensional, in the sense that it possesses all three velocity components, it virtually depends on two horizontal coordinates and develops properties of 2-D turbulence. This behaviour is opposite to the case of stable stratification in which the energy concentrates inside vertically aligned cone and the slow modes are comprised of horizontal sheets, or ‘pancakes’ with strong gradients between them. These gradients enhance vortex stretching and facilitate vorticity generation on ever smaller scales thus precluding the development of the inverse energy cascade. Extensive studies of the energy redistribution in rotating and stably stratified flows was undertaken over the years by the scientists of the Lyon group (see extensive reviews by Sagaut & Cambon Reference Sagaut and Cambon2008; Godeferd & Moisy Reference Godeferd and Moisy2015); their results are schematically presented in figure 4 adopted from Sagaut & Cambon (Reference Sagaut and Cambon2008).

Figure 4. Angular energy drain for stably stratified (a) and rotating (c) turbulence; spectral transfers (a,c) versus structures in physical space (b,d). (After Sagaut & Cambon (Reference Sagaut and Cambon2008); reprinted with the permission of Cambridge University Press.)

13 Direct and inverse energy transfers

The tendency of the rotating flows to develop the inverse energy cascade on large scales is so prolific that doubts have been raised as to whether or not the large-scale energy of the atmospheric and oceanic circulation can ever be delivered to the scales at which it can become involved in the direct cascade and dissipate (Pouquet & Marino Reference Pouquet and Marino2013). On the other hand, the co-existence of the direct and inverse cascades in rotating flows has been known at least since the early studies by Smith et al. (Reference Smith, Chasnov and Waleffe1996). It was also evident in simulations of oceanic flows by Klein et al. (Reference Klein, Hua, Lapeyre, Capet, Gentil and Sasaki2008). Pouquet & Marino (Reference Pouquet and Marino2013) showed that a dual, direct and inverse, energy cascade at constant rates $\unicode[STIX]{x1D716}_{D}$ and $\unicode[STIX]{x1D716}_{I}$ , respectively, can develop in rotating stratified turbulence. Furthering these findings, Marino, Pouquet & Rosenberg (Reference Marino, Pouquet and Rosenberg2015) performed high-resolution DNS of rotating stratified flows and identified a double cascade of the kinetic and potential energies to large and to small scales. All these efforts alleviate the paradox of the small-scale mixing in the presence of the inverse energy cascade. QNSE results provide an additional framework that can be used to analyse this paradox from a different viewpoint.

Denote the energy dissipation in the non-rotating case, (8.2), as

In the lowest order in $Ro(k)^{-1}$ , the total energy flux through a wavenumber $k$ can be computed as a sum of two terms,

where $\unicode[STIX]{x1D6FF}E(k)=E(k)-E_{0}(k)$ is the correction to the total spectrum as defined in (11.6) and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}(p)=\unicode[STIX]{x1D708}(p)-\unicode[STIX]{x1D708}_{n}(p)$ . While $\unicode[STIX]{x1D716}_{0}$ is computed by integration over the interval $(0,k)$ as it quantifies the energy transfer from the explicit to the implicit, or SGS domain $(k,\infty )$ , the correction $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D716}$ is computed by integration over the interval $(k,\infty )$ as it quantifies the energy transfer from the implicit to explicit domain. Since $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D716}<0$ , one infers that it effectively quantifies energy backscatter due to rotation which, in fact, is the $k$ -dependent rate of the inverse energy cascade.

Based upon the other two corrections to the vertical viscosity, we can compute the angular and component-wise redistributions of the energy flux,

and

Note that both $\unicode[STIX]{x1D716}_{z}(k)$ and $\unicode[STIX]{x1D716}_{3}(k)$ are negative meaning that the energy is withdrawn both from the vertical component and the $z$ (or $k_{\Vert }$ ) direction of all components. The energy loss of the third component (which is rather minor, (13.4)) is the gain of the two other horizontal components. More significant is the energy transfer from the $k_{\Vert }$ direction to the horizontal plane (for which $k_{\Vert }=0$ ) given by (13.3). This picture agrees well with the process of energy redistribution as presented schematically on figure 4. This analysis also indicates that while the direct cascade diminishes with the increasing scale, the inverse cascade gradually increases, consistently with the evidence of the dual cascade discussed in Pouquet & Marino (Reference Pouquet and Marino2013) and Marino et al. (Reference Marino, Pouquet and Rosenberg2015). Clearly, this reasoning is qualitative only and valid for $\unicode[STIX]{x1D708}_{h}>0$ .

14 Discussion and conclusions

The QNSE theory has been extended to include the case of forced rotating flows in a steady state. The process of small-scale elimination produces componentality which manifests via eddy viscosities acting in different ways on different velocity components and in different directions. In fact, the eddy viscosity $\unicode[STIX]{x1D708}_{3}$ acting on the vertical velocity component alone has been obtained here for the first time. All but $\unicode[STIX]{x1D708}_{3z}$ eddy viscosities decrease with increasing scales compared to their values in non-rotating flows. In the vicinity of the Woods scale, $L_{\unicode[STIX]{x1D6FA}}$ , the horizontal eddy viscosity, $\unicode[STIX]{x1D708}_{h}$ , tends to zero and the process of successive scale elimination cannot proceed beyond that point. The decrease in eddy viscosities is associated with gradual reorientation of the energy flux from small to large scales, i.e. increasing energy backscatter or inverse energy transfer inherent to turbulence with rotation. This behaviour of the eddy viscosities sets apart QNSE results for the cases of stably stratified and rotating turbulence. While in the former, the QNSE procedure produced positive viscosities for any stratification and so it could be carried out to any scale, in the latter, the procedure can only be used in the limit of weak rotation for as long as $\unicode[STIX]{x1D708}_{h}>0$ . The results are nevertheless important because they provide an analytical closed-form description of the transition from the isotropic 3-D Kolmogorov turbulence to anisotropic, rotation-dominated, quasi-2-D, three-component flow field emerging under the action of rotation. With regard to the frequencies of the inertial waves, turbulence appears not to affect them and so they remain unchanged compared to the linear case.

The QNSE-generated eddy viscosities and eddy diffusivities can be used as SGS parameterization for rotating flows on scales with positive eddy viscosities. Several LES studies of rotating turbulence used the EDQNM formalism to develop the SGS parameterization. Since the spectra in this approach are unknown, it was assumed that in the SGS range the deviations from the Kolmogorov spectrum are small (e.g Baerenzung et al. Reference Baerenzung, Politano, Ponty and Pouquet2008; Sen et al. Reference Sen, Mininni, Rosenberg and Pouquet2012; Pouquet et al. Reference Pouquet, Sen, Rosenberg, Mininni and Baerenzung2013). QNSE could possibly improve such SGS schemes because it provides simple analytical expressions for the response functions needed by EDQNM. Furthermore, an adequate resolution of the Kolmogorov inertial subrange in simulations of turbulence with rotation requires that the values of the rotational Reynolds number, $Re_{f}=\unicode[STIX]{x1D716}/(\unicode[STIX]{x1D708}_{0}f^{2})$ , be relatively large, well in excess of $10^{2}$ . QNSE perfectly addresses this requirement as it implies an arbitrarily large $Re_{f}$ . Clearly, QNSE offers many advantages which are yet to be tested within EDQNM.

Planetary rotation and density stratification significantly influence the dynamics of the Earth’s atmosphere and oceans as well as many other planetary circulations (Read Reference Read2011). Their combined effect is scale dependent and can be characterized in terms of the ratio of the Ozmidov and Woods scales, $L_{O}/L_{\unicode[STIX]{x1D6FA}}=(f/N)^{3/2}$ , where $f/N$ is sometimes referred to as the Prandtl’s ratio (e.g. Dritschel & McKiver Reference Dritschel and McKiver2015). The paramount role of this ratio in the dynamics of the Earth’s atmosphere was elaborated by Charney (Reference Charney1971). The complicated and often opposing tendencies of the rotation- and stratification-affected eddy viscosities and eddy diffusivities shown in figure 3 can be expected to have different and even opposing effects on flows on different scales. Recent numerical investigation by Marino et al. (Reference Marino, Mininni, Rosenberg and Pouquet2013) showed that the energy transfer from 3-D to 2-D modes is most efficient for $1/2\leqslant f/N\leqslant 2$ , or $0.35\lesssim L_{O}/L_{\unicode[STIX]{x1D6FA}}\lesssim 2.8$ . Within this interval, the stratification is most effective at creating large-scale structures and facilitating inverse energy cascade. On the other hand, the inverse cascade diminishes when stratification becomes predominant.

One of the most important outcomes of the competition between the effects of rotation and stratification is the behaviour of the energy spectra. As was elaborated in Galperin & Sukoriansky (Reference Galperin and Sukoriansky2010), in the case of stratification with no rotation, a flow tends to develop a universal spectrum $\propto N^{2}k_{z}^{-3}$ for the horizontal velocity in the vertical direction, i.e. the direction along which the phase speed of the internal waves is zero. It is quite possible that a similar universal spectrum emerges for the horizontal spectrum of the horizontal velocity in rotating flows. Note in this respect that the energy spectral laws and changes in the power laws exponents are often explained in terms of cross-overs between 3-D and 2-D turbulence. For the Earth’s atmosphere and ocean, $f/N\sim 10^{-2}$ , and so, according to Marino et al. (Reference Marino, Mininni, Rosenberg and Pouquet2013), such a transition would be quite inefficient. In addition, the dimensional cross-overs, if such exist, are difficult, if at all possible, to capture within the paradigms of isotropic or geostrophic turbulence. The QNSE formalism is free from this difficulty because it operates with the full set of 3-D Navier–Stokes and continuity equations with extra strains; no approximations regarding the character of a flow are made. In this 3-D framework, the emergence of flow two-dimensionalization and inverse energy cascade originate from anisotropic energy transfers reflected in the anisotropization of the Green function and velocity correlation tensor, the basic processes within the anisotropic turbulence paradigm. This paradigm has also been promoted by Schertzer & Lovejoy (Reference Schertzer and Lovejoy1985) and Lovejoy et al. (Reference Lovejoy, Tuck, Schertzer and Hovde2009) for atmospheric flows.