# Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Reduced-Rotating Hydro-Dynamic Equations, R-RHD

#### 2.1. Geostrophic Inertial Waves and Eddies

^{^}represents Fourier coefficients from the Fourier series expansion of a function. In subsequent sections, we invoke the infinite box limit ($L\to \infty $) and switch to Fourier transform variables thusly ${\widehat{c}}_{\mathbf{k}}\to {\left(\frac{L}{2\pi}\right)}^{2}{\widehat{c}}_{\mathbf{k}}\equiv {c}_{\mathbf{k}}$ and $\delta (0)={(\frac{L}{2\pi})}^{2}$ (see ch. 5 and 6 in [7]). The power 2 arises from the fact that the analysis is performed in 2D, i.e., $\mathbf{x}=({\mathbf{x}}_{\perp},\widehat{\mathbf{z}})$, unless otherwise specified.

#### 2.2. Helical Basis for Circularly Polarized Inertial Waves

## 3. Dynamical Limits of Turbulence and Multiple Scales Order Parameters

#### 3.1. Time Period of Oscillations of Waves and Amplitudes

#### 3.2. Notations and New Slow Time Variables for the Multiple Scales Derivation Presented in This Paper

- $\tau \sim \mathcal{O}(1)$: slow time scale of weak amplitudes (slowest time scale),
- ${t}_{s}\sim \mathcal{O}(\frac{1}{\u03f5})$: slow R-RHD wave time scale, and
- ${t}_{f}\sim \mathcal{O}(\frac{1}{\u03f5\mathcal{R}o})$: fast inertial wave time scale (filtered out by R-RHD).

#### 3.3. Definitions of Turbulence Regimes

- WT:
- $\begin{array}{|c|}\hline {\omega}_{f}{T}_{\tau}\gg 1,\phantom{\rule{3.33333pt}{0ex}}{\omega}_{s}{T}_{f}\ll 1\\ \hline\end{array}$.
- AT:
- $\begin{array}{|c|}\hline {\omega}_{f}{T}_{\tau}\gg {\omega}_{s}{T}_{\tau}\gg 1,\phantom{\rule{3.33333pt}{0ex}}{\omega}_{s}{T}_{f}\ll 1\\ \hline\end{array}$. In essence, one may envision the AT dynamical regime for hydrodynamics as a zoomed in version of the WT limit in Galtier [8]. In this paper we investigate the subtle variations in the flow dynamic at this magnified scale concealed by the treatment in [8]. Further, in this paper, we perform the multiple scales analysis in the anisotropic regime defined by the order parameter $\u03f5=\frac{{k}_{Z}}{{k}_{\perp}}\ll 1$. So the region of validity of AT is also set by the limit where $\frac{{k}_{Z}}{{k}_{\perp}}\ll \mathcal{R}o\ll 1$.
- CB:
- $\begin{array}{|c|}\hline {\omega}_{f}{T}_{\tau}\sim 1\\ \hline\end{array}$.

## 4. Wave Amplitude Equations

#### 4.1. Dimensional Consistency of Wave Amplitude Equation (22)

#### 4.2. Spectral Tensor

## 5. Zero Helicity Dynamics

#### 5.1. The Hamiltonian

#### 5.2. Kinetic Equations

#### 5.2.1. Closure

#### 5.2.2. Dimensional Consistency of the kinetic Equation (43)

#### 5.2.3. Physical Realizability of Spectrum Prescribed by Equation (43) and Comparison with Quasi-Normal Type Closures

**cf.**p. 313, ch. IV), we envision a scenario where, starting with a positive initial condition for the energy spectrum, we assume

#### 5.2.4. Invariants of the Closed Kinetic Equations

#### 5.2.5. Kolmogorov Solution of the Kinetic Equations

- (i)
- ${x}_{1}=1$ and ${y}_{1}=-1$ (Rayleigh-Jeans (RJ) spectrum).
- (ii)
- ${x}_{2}=1$ and ${y}_{2}=0$.
- (iii)
- ${x}_{3}=(1+u)$ and ${y}_{3}=(2+v)$, where $2u$ and $2v$ are respectively the powers of ${k}_{Z}$ and ${k}_{\perp}$ in $|{\tilde{L}}_{kpq}{|}^{2}$. Clearly, in our case, $u=0,v=1$. Thus, ${x}_{3}=1$ and ${y}_{3}=3$.
- (iv)
- ${x}_{4}=1$ and ${y}_{4}=7/2$. This solution corresponds to the constant flux in the z-component of the momentum.

#### 5.2.6. Locality of Interactions and Convergence of Collision Integral

## 6. Non-Zero Helicity Dynamics: Interplay of Energy and Helicity

#### 6.1. Coupled Equations for Energy and Helicity

**cf.**${e}_{\mathbf{k}}$ in Equation (43) is actually ${e}_{\mathbf{k}}^{+}\equiv {e}_{\mathbf{k}}^{-}$). Thus the closed form coupled energy-helicity equation becomes,

#### 6.2. Generalized Solution of Energy and Helicity Spectra

#### 6.3. Hierarchy of Slow Manifolds in Anisotropic Wave Turbulence Diverges from the Critical Balance Route towards Isotropy

#### 6.4. Comparison with Weak Turbulence Theory of Galtier

- The governing equations on which the wave turbulence theory [8,9] is developed are the Navier-Stokes equations (i.e., Equation (1)). In these equations, the Rossby number $\mathcal{R}o$ appears explicitly and is used as an order parameter in the perturbation analysis and the fast inertial wave dynamic is dominant. In contrast, the governing equations for the analysis presented here are the R-RHD (i.e., Equations (5) and (6)). The asymptotic limit of infinitesimally small $\mathcal{R}o$ is already accounted for in the multiple scales analysis to derive the R-RHD and hence do not explicitly appear in the R-RHD equations. This means that the effect of fast inertial waves is sub-dominant here and the R-RHD equations are hence suitably applicable for the slow manifold dynamics.
- The dynamical regime where the theory [8,9] is valid is ${\omega}_{f}{T}_{\tau}\gg 1,\phantom{\rule{3.33333pt}{0ex}}{\omega}_{f}{T}_{s}\gg 1$. This point has been elaborated in great detail in the work of Nazarenko and Scheckochihin [27] (see Section 2 in [27]). The dynamical regime of the R-RHD is ${\omega}_{f}{T}_{\tau}\gg {\omega}_{s}{T}_{\tau}\gg 1,\phantom{\rule{3.33333pt}{0ex}}{\omega}_{s}{T}_{f}\ll 1$ and ${k}_{z}\ll {k}_{\perp}$. It is the limit in which ${k}_{z}$ is so small that the turbulent dynamic is populated by slow inertial waves with dispersion relation ${\omega}_{k}=\frac{{k}_{Z}}{{k}_{\perp}}$ embodying slow oscillations when $\frac{{k}_{Z}}{{k}_{\perp}}\ll 1$ (also see Figure 5 above). In other words, within the slow manifold, ${k}_{z}$ is so small that $\frac{{k}_{Z}}{{k}_{\perp}}\ll \mathcal{R}o\ll 1$. The perturbative analysis presented here is applied to the R-RHD where the smallness (weakness) of the wave amplitude is measured by $\u03f5\ll 1$ and the slow dispersive three wave system undergoes weak non-linear exchanges at the asymptotic order ${\u03f5}^{2}$ as explained earlier in Section 4 and Section 5.
- In wave turbulence theory of Galtier [8] and Galtier [9], the kinetic equations for energy and helicity are derived by using multiple correlation functions to capture the energetics as well as the absence of symmetry due to helicity. This makes the calculations tediously lengthy. In contrast, the derivation for the helicity kinetic equation is presented here as a natural extension of the symmetrical non-helical system and bypasses the use of calculations using multiple correlation functions. This simpler approach follows a more general philosophy of Hamiltonian reduction exploiting symmetries in the system [25] and their natural extension to understanding asymmetrical phenomena.
- Finally, the spectral power laws obtained for the energy cascade are distinctly different as explained concisely in Figure 5.

#### 6.5. Comparison with Weak Turbulence Theory of Newell: Cumulant Hierarchy vs. Wave Amplitude Hierarchy

**cf.**Newell et. al. [39], rather than for the Fourier amplitudes.

## 7. Coupling between Wave and 2D Modes Through Non-Resonant Interactions

**cf.**presence of $\tau $ in the higher order terms) are inherently embedded in the full system that is not restricted to resonating triadic wave interactions only. These terms account for the coupling with the 2D modes. This is evident by re-writing Equation (22) with the higher order terms as follows,

## 8. Conclusions

**cf.**equivalently a ${k}_{\perp}^{-2}$ spectrum for the cylindrically symmetric spectrum, ${E}_{{k}_{\perp}}$) that is in agreement with results from experimental and numerical simulations [17,21,22,23] as has been stated earlier. An asymptotically reduced system spans a hierarchy of slow manifold regimes and thereby captures the gradual transfer of energy towards the 2D modes. Interestingly, a similar power law solution can also be obtained by applying a critical balance phenomenology (where fast inertial wave time scale balances the nonlinear advection time scale) to the system of rotating turbulence as has been shown in Nazarenko and Scheckochihin [27]. This is the realm of strong turbulence where the nonlinear interactions are strong, meaning ${\omega}_{f}{T}_{{\tau}_{NL}}\sim \mathcal{O}(1)$. However, the anisotropic limit of rapidly rotating turbulence is farther away from modes where critical balance holds, this has been shown through numerical simulations in Di Leoni et al. [59]. In addition to the discussion in Section 6.3 above, the reader is referred to [27,38] for a detailed discussion on a wave turbulence and critical balance schematic of the energy cascade process. It is important to note that in the analysis presented in this paper, any physical artifact induced by boundary condition is not considered. Interested readers are referred to the work of [60] that describes discrete boundary effects on wave turbulence formalism.

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Notations and Definitions of Wave Vectors and Gradient Operators

#### Appendix A.2. Derivation of the Amplitude Equation: Supplementary Calculations

**cf.**since the R-RHD limit is $\omega \tau =\frac{\tau}{t}\ll 1$, we chose $\tau =\u03f5t,\u03f5\ll 1$, also $t\ll $ fast inertial time scale that has been filtered out by the R-RHD),

**1st term**: $\frac{1}{2}({\mathbf{h}}_{\mathbf{k}}^{-{s}_{k}}\xb7{\mathcal{L}}_{\mathcal{H}}{\mathbf{U}}_{2\mathbf{k}}^{{s}_{k}}):=\frac{1}{T}{\int}_{T}\langle \frac{1}{2}{\mathbf{h}}_{\mathbf{k}}^{-{s}_{k}},{\mathcal{L}}_{\mathcal{H}}{\mathbf{U}}_{2\mathbf{k}}^{{s}_{k}}\rangle dt=\frac{1}{T}{\int}_{T}0dt=0$. The 0 integrand follows from application of the solvability criterion which is basically a Fredholm alternative in the context of the adjoint problem explained in Section 4. This makes the left hand side of Equation (21) null upon application of the inner product.

**2nd term**: $\frac{1}{T}{\int}_{T}{\partial}_{\tau}{c}_{\mathbf{k}}^{{s}_{k}}(\tau )\langle \frac{1}{2}{\mathbf{h}}_{\mathbf{k}}^{-{s}_{k}},{\mathbf{h}}_{\mathbf{k}}^{{s}_{k}}\rangle dt={\partial}_{\tau}{c}_{\mathbf{k}}^{{s}_{k}}\frac{1}{T}{\int}_{T}1dt={\partial}_{\tau}{c}_{\mathbf{k}}^{{s}_{k}}(\tau )$. Note that the term ${\partial}_{\tau}{c}_{\mathbf{k}}^{{s}_{k}}(\tau )$ can be factored out of the integration over t because t and $\tau $ are independent variables as has been explained above. Also note that the integrand is equal to one because we have used the fact that $\frac{1}{2}\langle {\mathbf{h}}_{\mathbf{k}}^{-{s}_{k}},{\mathbf{h}}_{\mathbf{k}}^{{s}_{k}}\rangle =1$ as has been explained earlier in Section 4.

**3rd term**: For the third term we interchange the order of integration, i.e., we swap the operations $\frac{1}{T}{\int}_{T}(\xb7)dt$ and $\langle \frac{1}{2}{\mathbf{h}}_{\mathbf{k}}^{{s}_{k}},\xb7\rangle $. Note that all terms except the exponential term are functions of $\tau $ (and not t) and hence can be factored out of the integration over t. So we now concentrate only on the averaging of the exponential term, comprising of the $\omega $ terms, as follows. Recall, for sake of brevity, we use the following notation $\varphi (\omega ):=({s}_{k}{\omega}_{k}-{s}_{p}{\omega}_{p}-{s}_{q}{\omega}_{q})$. Now, in the limit of large $t\sim \mathcal{O}(\frac{1}{\u03f5})$ as $\u03f5\to 0$, $\varphi (\omega )\sim \mathcal{O}(\u03f5)$ becomes proportionally small thereby $\varphi (\omega )t=\u03f5{\varphi}_{1}(\omega )\frac{\tau}{\u03f5}={\varphi}_{1}(\omega )\tau \sim \mathcal{O}(1)$ in the leading order approximation. Here ${\varphi}_{1}(\omega )\sim \mathcal{O}(1)$. This means we have

#### Appendix A.3. Energy Conservation in Wave Triads

#### Appendix A.4. Summary of Kinetic Wave Turbulence Regimes with Their Dynamical and Physical Properties and Solutions

Reduced HD | Reduced MHD | Full HD | Critical Balance HD | |
---|---|---|---|---|

(Current Paper) | (Ref. [7,34]) | (Ref. [8]) | (Ref. [7,27]) | |

name | anisotropic wave | anisotropic wave | weak wave | strong wave |

of model | turbulence (AT) | turbulence (AT) | turbulence (WT) | turbulence (CB) |

dynamical | strong ${\mathsf{\Theta}}_{0}$ & | strong ${B}_{0}$ & | strong ${\mathsf{\Theta}}_{0}$ | waves balance |

features | strong anisotropy | strong anisotropy | nonlinearity | |

waves | slow inertial | slow Alfven | fast inertial | fast inertial |

waves | waves | waves | waves | |

anisotropy | very | very | moderate | moderate |

(smallness of $\frac{{k}_{Z}}{{k}_{\perp}}$) | strong | strong | to isotropic | to isotropic |

governing | kin. eq. deduced | kin. eq. deduced | kin. eq. deduced | unknown |

equation | from reduced HD | from reduced MHD | from full HD | |

(Navier Stokes) | ||||

wave amplitude | weak | weak | weak | large |

& nonlinearity | ||||

order | $\u03f5=\frac{{k}_{Z}}{{k}_{\perp}}\ll 1$ | $\u03f5=\frac{{c}_{k}^{\prime}}{{B}_{0}}\ll 1$ | $\u03f5=\mathcal{R}o\ll 1$ | — |

parameter | (also, $\frac{{k}_{Z}}{{k}_{\perp}}\ll \mathcal{R}o)$ | (also, $\frac{{k}_{Z}}{{k}_{\perp}}\ll 1)$ | ||

${E}_{{k}_{\perp}}$ | ${k}_{\perp}^{-2}$ | ${k}_{\perp}^{-2}$ | ${k}_{\perp}^{-5/2}$ | ${k}_{\perp}^{-2}$ |

spectra |

#### Appendix A.5. Computer Program Used for the Analysis in Section 5.2.6

(* Definition of terms in the integrand*) ------------------------------------------- perp[k_] := {First@k, k[[2]], 0} par[k_] := {-k[[2]], First@k, 0} norm[k_] := Sqrt[k[[1]]^2 + k[[2]]^2 + k[[3]]^2] (*Lkpq[k_,p_,q_]:=(par[p].perp[q](perp[q]-perp[p]).(perp[k] + \ perp[q]+perp[p]))/(2*norm@perp@k norm@perp@p norm@perp@q)*) Lkpq[k_, p_, q_] := (par[p].perp[q] ( norm@perp@q - norm@perp@p) ( norm@perp@k + norm@perp@p + norm@perp@q))/(2* norm@perp@k norm@perp@p norm@perp@q) FullSimplify[Lkpq[{kx, ky, kz}, {px, py, pz}, {qx, qy, qz}]] alpha0 = 3; beta0 = 1; ee[k_, alpha_: alpha0, beta_: beta0] := (norm@perp@k)^-alpha k[[3]]^-beta I1b[k_, p_, q_, alpha_: 3, beta_: 1] := Lkpq[k, p, q]^2 (ee@p ee@q - ee@k ee@p - ee@k ee@q) Simplify[I1b[{kx, ky, kz}, {px, py, pz}, {qx, qy, qz}]] I2b[{kx_, ky_, kz_}, {px_, py_, pz_}, {qx_, qy_, qz_}] := 2 I1b[{px, py, pz}, {kx, ky, kz}, {qx, qy, qz}] I0b[{kx_, ky_, kz_}, {px_, py_, pz_}, {qx_, qy_, qz_}] := I1b[{kx, ky, kz}, {px, py, pz}, {qx, qy, qz}] - I2b[{kx, ky, kz}, {px, py, pz}, {qx, qy, qz}] (*Define surface of integration*) ------------------------------------- surfpz[k_, p_] := (norm@perp[p] k[[3]]/ norm@perp[k]) (Sqrt[(k[[1]] - p[[1]])^2 + (k[[2]] - p[[2]])^2] - norm@perp[k])/(Sqrt[(k[[1]] - p[[1]])^2 + (k[[2]] - p[[2]])^2] - norm@perp[p]) surf[{kx_, ky_, kz_}, {px_, py_, pz_}] := pz == surfpz[{kx, ky, kz}, {px, py, 0}] (*Definition of the Jacobian, computing Jacobian*) ----------------------------------------------------- jaco[{kx_, ky_, kz_}, {px_, py_, pz_}] := \[Sqrt](1 + D[surfpz[{kx, ky, kz}, {px, py, 0}], px]^2 + D[surfpz[{kx, ky, kz}, {px, py, 0}], py]^2) Simplify[jaco[{kx, ky, kz}, {px, py, 0}]] (*Display surface of integration and value ofsub- integrand*) ----------------------------------------------------- windowx = -10; windowy = -10; With[{kx = 20, ky = 20, kz = 1}, SliceContourPlot3D[ Log@Abs@I0b[{kx, ky, kz}, {px, py, pz}, {kx - px, ky - py, kz - pz}], surf[{kx, ky, kz}, {px, py, pz}] == 0, {px, -kx + windowx, kx - windowx}, {py, -ky + windowy, ky - windowy}, {pz, -4, 4}, ColorFunction -> "Rainbow", PlotLegends -> All, AxesLabel -> {px, py, "pz"}]] With[{kx = 20, ky = 20, kz = 1}, SliceContourPlot3D[ I0b[{kx, ky, kz}, {px, py, pz}, {kx - px, ky - py, kz - pz}], surf[{kx, ky, kz}, {px, py, pz}] == 0, {px, -kx + windowx, kx - windowx}, {py, -ky + windowy, ky - windowy}, {pz, -4, 4}, ColorFunction -> "Rainbow", PlotLegends -> All, AxesLabel -> {px, py, "pz"}]] (*Define the integrand, set it to zero at singular points which are outside the region of validity of AT (R-RHD)*) -------------------------------------------------------------------------- tolerance1 = 0.5; tolerance2 = 0.5; kz = 1; I0bjacos[{kx_, ky_, kz_}, {px_, py_, pz_}] := If[ Boole[Abs[Sqrt[(kx - px)^2 + (ky - py)^2] - Sqrt[(px^2 + py^2)]] > tolerance1] Boole[ Abs[Sqrt[(kx - px)^2 + (ky - py)^2] - Sqrt[(kx^2 + ky^2)]] > tolerance2] Boole[Abs[Sqrt[(px^2 + py^2)]] > tolerance1] Boole[Abs[Sqrt[(kx - px)^2 + (ky - py)^2]] > tolerance1] Boole[Abs[kz - surfpz[{kx, ky, kz}, {px, py, 0}]] > tolerance1] == 0, 0, I0b[{kx, ky, kz}, {px, py, surfpz[{kx, ky, kz}, {px, py, 0}]}, {kx - px, ky - py, kz - surfpz[{kx, ky, kz}, {px, py, 0}]}] jacos[{kx, ky, kz}, {px, py, 0}]] I0bjacosFULL[{kx_, ky_, kz_}, {px_, py_, pz_}] := I0b[{kx, ky, kz}, {px, py, surfpz[{kx, ky, kz}, {px, py, 0}]}, {kx - px, ky - py, kz - surfpz[{kx, ky, kz}, {px, py, 0}]}] jacos[{kx, ky, kz}, {px, py, 0}] (*Perform the numerical integration and then display*) ----------------------------------------------------------------------- txy = Table[NIntegrate[ I0bjacos[{kx, ky, kz}, {px, py, 0}], {px, 10, 150}, {py, 10, 150}, Method -> {"AdaptiveMonteCarlo", "MaxPoints" -> 100000}, WorkingPrecision -> MachinePrecision, MaxRecursion -> 100], {kx, 10, 150, 20}, {ky, 10, 150, 20}] (*txy = Table[ NIntegrate[ I0bjacos[{kx, ky, kz}, {px, py, 0}], {px, 10, 150}, {py, 10, 150}, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 3000}, WorkingPrecision -> MachinePrecision, MaxRecursion -> 30], {kx, 10, 150, 20}, {ky, 10, 150, 20}]*) ListPlot3D[txy, PlotLegends -> Automatic, AxesLabel -> {kx, ky, "I(kx,ky)"}, ClippingStyle -> None, DataRange -> {{10, 150}, {10, 150}}]

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**Figure 1.**Anisotropic energy spectra of rotating helical turbulence simulations published in [21,22]. (

**a**) ${e}_{\mathbf{k}}\sim {k}_{\perp}^{-3}$ and (

**b**) ${E}_{{k}_{\perp}}\sim {k}_{\perp}{e}_{\mathbf{k}}\sim {k}_{\perp}^{-2}$ comport with the spectra obtained analytically in the current article as is explained in later sections. The inertial range in the above plots span roughly over an order of magnitude in the log scale, i.e., roughly 70 to 90 wave numbers. The line showing the $-5/3$ slope is just for reference as is clearly mentioned in [21].

**Figure 2.**Helical wave basis: $({\widehat{\mathbf{k}}}^{\perp},\widehat{\mathit{J}},{\widehat{\mathbf{k}}}^{\prime})$ forms a right-handed coordinate system with $\langle {\widehat{\mathbf{k}}}^{\prime}{\widehat{\mathbf{k}}}^{\perp}\rangle =\langle {\widehat{\mathbf{k}}}^{\prime}\widehat{\mathit{J}}\rangle =0$ where $\widehat{\mathit{J}}=\frac{{{\mathbf{k}}^{\prime}\times \mathbf{k}}^{\perp}}{{k}_{\perp}^{2}}$. The wave propagation direction is given by the wave vector, ${\widehat{\mathbf{k}}}^{\prime}$ (which we simply call $\mathbf{k}$ in the body of the manuscript). Within the slow manifold where ${k}_{z}=\mathcal{R}o{k}_{Z}$, $({\widehat{\mathbf{k}}}^{\perp},\widehat{\mathit{J}},{\widehat{\mathbf{k}}}^{\prime})\to ({\widehat{\mathbf{k}}}^{\perp},\widehat{\mathbf{z}},{\widehat{\mathbf{k}}}_{\perp}^{\prime})$.

**Figure 3.**Contour plot showing the surface of integration $S:{p}_{Z}({p}_{x},{p}_{y})$ of the collision integral (without the constant $\pi $) in Equation (43) projected on the $({p}_{x},{p}_{y})$ plane. Colors represent the value of the integrand. Figures on the left are in logarithmic scale. (

**a**) and (

**b**) correspond to $\alpha =1,k=20$ while (

**c**) and (

**d**) correspond to $\alpha =0.1,k=100$ with ${k}_{Z}=1$ fixed such that ${k}_{Z}/{k}_{\perp}\ll 1$. (

**e**) is top view of (a). The singular regions are either outside the region of validity of AT (R-RHD) or correspond to identically null value of the integrand as explained in this section.

**Figure 4.**Numerical integration of the collision integral in Equation (43) normalized by the constant $\pi $. The plots show $I({k}_{\perp},{k}_{Z})$ vs. ${k}_{\perp}=({k}_{x},{k}_{y})$ for fixed ${k}_{Z}=1$. The integration is computed at the nodes ${k}_{x},{k}_{y}\in [10,150]$ in intervals of 20. (

**a**) Method: Gauss Kronrod quadrature; (

**b**) method: adaptive Monte Carlo.

**Figure 5.**A sketch of cascade paths for rotating turbulence shows the different flow regimes and the corresponding energy spectra. Here, ${k}_{i}$ is the isotropic wavenumber, ${k}_{\perp c}$ is the classical critical balance wavenumber, ${k}_{0}$ is the injection wavenumber corresponding to an initial wave field. Three distinct regimes are shown: (i) WT (Galtier) corresponding to the wave-turbulence regime with ${\omega}_{f}{T}_{\tau}\gg \phantom{\rule{3.33333pt}{0ex}}1,\phantom{\rule{3.33333pt}{0ex}}{\omega}_{f}{T}_{s}\gg 1$, (ii) CB w/ pol. (i.e., critical balance with polarization alignment) as explained in [27] leading towards isotropy, and (iii) AT (R-RHD) corresponding to the anisotropic wave turbulence dynamics of the R-RHD equations with ${\omega}_{f}{T}_{\tau}\gg {\omega}_{s}{T}_{\tau}\gg 1,\phantom{\rule{3.33333pt}{0ex}}{\omega}_{s}{T}_{f}\ll 1$ and ${k}_{Z}\ll {k}_{\perp}$. As we move along the horizontal axis from left to right, the flow traverses a hierarchy of slow manifolds with successively rescaled (decreasing) ${k}_{Z}/{k}_{\perp}$ wave number ratio. Also shown are possible explanation of 2D-3D coupling by non-resonant interactions. The AT (R-RHD) does not explain inverse cascade phenomena as the energy flux obtained is direct. Here ${k}_{0}$ is the wavenumber corresponding to the initial condition.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sen, A.
Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves. *Fluids* **2017**, *2*, 28.
https://doi.org/10.3390/fluids2020028

**AMA Style**

Sen A.
Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves. *Fluids*. 2017; 2(2):28.
https://doi.org/10.3390/fluids2020028

**Chicago/Turabian Style**

Sen, Amrik.
2017. "Anisotropic Wave Turbulence for Reduced Hydrodynamics with Rotationally Constrained Slow Inertial Waves" *Fluids* 2, no. 2: 28.
https://doi.org/10.3390/fluids2020028