Revisiting Bolgiano-Obukhov scaling for stably stratified turbulence

According to the celebrated Bolgiano-Obukhov phenomenology for moderately stably stratified turbulence, the energy spectrum in the inertial range shows a dual scaling; the kinetic energy follows (i) $\sim k^{-11/5}$ for $kk_B$, where $k_B$ is Bolgiano wavenumber. The $k^{-5/3}$ scaling akin to passive scalar turbulence is a direct consequence of the assumption that buoyancy is insignificant for $k>k_B$. We revisit this assumption, and using constancy of kinetic and potential energy fluxes and simple theoretical analysis, we demonstrate an absence of the $ k^{-5/3}$ spectrum. This is because the velocity field at small scales is too weak to establish a constant kinetic energy flux as in passive scalar turbulence. Our findings may have important implications in the modelling of stably stratified turbulence.


Introduction
Stable density stratification is ubiquitous in oceans and planetary atmospheres (Davidson 2013; Sagaut & Cambon 2008;Turner 2009). Both atmospheric and oceanic flow can often be turbulent; such turbulence, commonly known as "stably stratified turbulence" (SST), is different from the classical "Kolmogorov turbulence", which is applicable for homogeneous and isotropic flows. Two important non-dimensional parameters that quantify SST are-Reynolds number Re, which is the ratio of inertia and viscous forces, and Richardson number Ri, which is the ratio of buoyancy and flow shear term (Davidson 2013;Lindborg 2006Brethouwer et al. 2007;Verma 2018;Rosenberg et al. 2015): (1 a,b) Here U and L respectively denote the large scale velocity and length scale, ν is the kinematic viscosity, u is the velocity, ρ m andρ are respectively the overall mean density in the domain and local mean density and g is the acceleration due to gravity. Based on the competition between Re and Ri, SST can be classified into three broad regimes: (i) Re O(1) and Ri O(1) (weakly SST or w SST)-here the nonlinearity u · ∇u is far stronger than buoyancy, hence density acts like a passive scalar. These flows, as expected, show Kolmogorov's spectrum. (ii) Re O(1) and Ri ≈ O(1) (moderately SST or mSST)-here buoyancy and non-linearity have comparable strength. Although inhomogeneous, these flows are nearly isotropic. (iii) Re O(1) and Ri O(1) (strongly SST or sSST)-here buoyancy is far stronger than the non-linearity, making the flow quasi two-dimensional in nature (Davidson 2013;Lindborg 2006Brethouwer et al. 2007;Verma 2018). In the present paper we consider mSST only.
According to Bolgiano (1959) and Obukhov (1959), buoyancy force in mSST converts kinetic energy into potential energy. Bolgiano-Obukhov (BO) phenomenology reveals that the kinetic energy flux Π u (k) in the inertial range decreases with the wavenumber k as ∼ k −4/5 , however the potential energy flux Π b (k) in the inertial range is assumed to be a constant. Consequently, the kinetic energy spectrum, E u (k), varies as ∼ k −11/5 , which is steeper than ∼ k −5/3 as derived for hydrodynamic turbulence in Kolmogorov's theory. On the other hand, the Kolmogorov spectrum is steeper than the potential energy spectrum, E b (k), which varies as k −7/5 . It was further argued in the BO phenomenology that for latter half of the inertial range, the energy supply rate by buoyancy becomes negligible, implying that density acts as a passive scalar, as it does in w SST. Hence, both kinetic and potential energy fluxes follow the Kolmogorov spectrum: Direct numerical simulation results of Kimura & Herring (1996) and Kumar, Chatterjee & Verma (2014), shell-model results of Kumar & Verma (2015), and global energy balance analysis of Bhattacharjee (2015) have unequivocally shown that the kinetic energy flux indeed scales as ∼ k −11/5 for mSST. However, we are not aware of any numerical or experimental work that convincingly demonstrate the 'dual scaling' of BO phenomenology.
In this paper we start with constancy of total energy flux (kinetic + potential) and demonstrate that for large wavenumbers, the velocity field is always weaker than the density field. Hence, the assumption that buoyancy becomes weaker at large wavenumbers leading to k −5/3 spectra is incorrect. Thus we show that E u (k) ∼ k −11/5 for k > 1/L, where L is the system size, with no crossover to k −5/3 spectra. As an aside, we recover E u (k) = k −5/3 for k < 1/L, which may be possible in systems with large aspect ratio. Thus, we provide a revision of the celebrated Bolgiano and Obukhov phenomenology.
The outline of the paper is as follows: In §2, the equations governing SST are introduced and in §3, the BO phenomenology is described. In §4.1 and §4.2, respectively, numerical solution and asymptotic analysis of the equation for the total energy flux (a fifth order equation) are presented. Finally the work is concluded in §5.

Governing Equations
The governing Navier-Stokes equations for stably stratified flows (density stratification in the vertical (z) direction) under the Boussinesq approximation are (Davidson 2013;Lindborg 2006;Verma 2018) where u = (u x , u y , u z ) and σ are respectively the velocity and the pressure fields, ν and κ are respectively the kinematic viscosity and diffusivity of the density, F u is the external force (in addition to the buoyancy), and b is the density fluctuation variable in the unit of velocity, which is achieved by the following transformation (Davidson 2013;Lindborg 2006;Rosenberg et al. 2015 where g is the acceleration due to gravity, ρ m is the mean, and ρ is the density fluctuation. The parameter is the Brunt-Väisälä frequency, which is a measure of the density stratification. We also note here that −N b is buoyancy. Our interest is in mSST; in this regard it is convenient to describe the flow behaviour in Fourier space since it captures the scale-by-scale energy transfer and interactions. The one-dimensional kinetic spectrum, E u (k), and the potential energy spectrum, E b (k), are defined as Henceforth, the explicit time dependence in E u and E b are suppressed for brevity. The nonlinear energy transfers across modes are quantified using energy fluxes or energy cascade rates.The kinetic (potential) energy flux, Π u(b) (k 0 ), for a wavenumber sphere of radius k 0 is the total kinetic (potential) energy leaving the said sphere due to nonlinear interactions. The fluxes are computed using (Dar, Verma & Eswaran 2001;Verma 2004Verma , 2018) where k + p + q = 0, the giver Fourier modes (with wavenumbers p) are within the sphere, while the receiver Fourier modes (with wavenumbers k ) are outside the sphere. The dynamical equations for modal kinetic energy (E u (k) = 1 2 |u(k, t)| 2 ) and potential energy (E b (k) = 1 2 |b(k, t)| 2 ) respectively can be derived from (2.1a) and (2.1b), and are as follows: Here T u(b) (k) and D u(b) (k) are respectively the nonlinear kinetic (potential) energy transfer rate and dissipation rate, while F B , F ext denote respectively the energy feed rate by the buoyancy and external force. These quantities are defined as follows (Verma, Kumar & Pandey 2017;Verma 2018): where k = p + q. The kinetic and potential energy fluxes are related to nonlinear energy transfer terms as We write (2.7a) and (2.7b) for sphere of radii k and k + dk and take the difference that yields (2.11b) Now taking the limit dk → 0 yields The above energetics is illustrated in Figure 1. Let us consider a statistically steady state (∂/∂t → 0). In the inertial range, F ext = 0, and the dissipative effects are negligible, i.e. D u → 0 and D b → 0. Hence the equations for the kinetic and potential energies simplify to Figure 1. (a) The kinetic energy contents of a wavenumber shell changes due to the kinetic energy flux difference Πu(k+dk)−Πu(k), energy supply rate by external force Fext(k)dk, energy removal rate by buoyancy FB(k)dk, and viscous dissipation rate Du(k)dk. (b) The potential energy changes due to potential energy flux difference Π b (k + dk) − Π b (k), energy supply rate by buoyancy FB(k)dk, and dissipation rate D b (k)dk.
The sum of these two equations yield (2.15) Hence the total energy flux is constant in the inertial range. Using direct numerical simulations, Kumar et al. (2014) found that both in mSST and w SST, F B (k) < 0 in the inertial range. This implies that in the inertial range of w SST and mSST, buoyancy converts kinetic energy into potential energy. If this was not the case, the kinetic energy of the system would grow, making it unstable. Thus, according to (2.14a)-(2.14b), Π u (k) decreases with k, while Π b (k) increases with k. As a result, the kinetic energy spectrum E u (k) is steeper than the hydrodynamic counterpart (k −5/3 ), while the potential energy spectrum E b (k) is shallower than k −5/3 spectrum.

The Bolgiano-Obukhov phenomenology
According to Bolgiano (1959) and Obukhov (1959), a force balance between the nonlinear term and buoyancy in (2.1a) yields where u k and b k are respectively the velocity and density fluctuations at wavenumber k. We can non-dimensionalize the above equation using division with U 2 /L, where U and L are respectively the large scale velocity and length scales. Thus, N in (3.1) is nondimensionalized Brunt-Väisälä frequency, which is equal to √ Ri (Ri has been defined in §1). Henceforth, all equations will be viewed in the non-dimensional sense. Furthermore, we recall here that for mSST, N ∼ O(1) and Ri ∼ O(1). Also note that k = 1 corresponds to 1/L. Hence, k 1 is possible only when the transverse length scale is much larger than the vertical scale (L).
The BO phenomenology assumes that in the inertial range, Π b (k) ≈ constant, and it equals the dissipation rate of the potential energy ( b ): Equations (3.1) and (3.2) yield the following relations: (3.3d) Bolgiano and Obukhov argued that the above-mentioned behaviour of the inertial range is true only for lower wavenumbers (k < k B , where k B will be defined below). For higher wavenumbers (k > k B ) in the inertial range, the buoyancy effects are weak and hence cannot balance the inertial term (which is balanced by the pressure gradient). Hence in this region, the scaling of passive scalar (i.e. Kolmogorov) turbulence should be valid. The energy and flux relations obtained here are: where u is the viscous dissipation rate, and K Ko , K OC are Kolmogorov's and Obukhov-Corrsin's constants. The behavioural transition from one regime to another occurs near the Bolgiano wavenumber k B , which is obtained by matching Π u (k) in the two regimes: The nature of kinetic and potential energy fluxes, as well as dual scaling of mSST as predicted by Bolgiano and Obukhov are illustrated in Figure 2.

Revision of Bolgiano-Obukhov phenomenology
A crucial assumption made in the BO phenomenology is that Π b (k) ≈ constant in the inertial range (refer to (3.2)), and this needs a closer examination. A more rigorous approach would be to start with the constancy of total energy flux i.e., (2.15), that follows from the conservation of total energy (kinetic + potential) in the inviscid limit.
We start with (2.15), and equate it to the total dissipation rate . That is, (4.1) In the above equation we eliminate b k using (3.1) that yields the following fifth-order polynomial in u k : There is no analytical solution for a fifth order algebraic polynomial. Hence, we resort to numerical solution, and later in the section to an asymptotic analysis. These two results are consistent with each other.

Numerical Solution
We numerically solve (4.2) using fsolve function of Python. We choose N = 1.0 and = 1.0, and vary k from 10 −6 to 10 10 in logarithmic scale. Using the numerically evaluated u k and b k we evaluate E u (k) = u 2 k /k, E b (k) = b 2 k /k, Π u (k) = ku 3 k , and Π b (k) = ku k b 2 k . The quantities are plotted in Figure 3. Figure 3 exhibits both the fluxes and spectra. For 1 < k < 10 10 , Π b ≈ 1, Π u (k) ∼ k −4/5 , E u ∼ k −11/5 , and E b ∼ k −7/5 , which are the predictions of Bolgiano-Obukhov phenomenology for k < k B . Surprisingly, there is no crossover to k −5/3 scaling of passive scalar turbulence. This is because u k b k , hence u k cannot induce a constant kinetic energy flux. We will show a more rigorous derivation in the next subsection.
Interestingly, for k 1, we obtain Π u ≈ 1, Π b ∼ k 4/3 , E u ∼ k −5/3 and E b ∼ k −1/3 . That is, u k dominates b k at small k's that leads to Kolmogorov's scaling for the velocity field. Note however that k = 1 corresponds to 1/L. Hence, k 1 is possible in SST when the transverse length scale is much larger than the vertical scale (L), however this regime may lead to inverse cascade of kinetic energy as in two-dimensional and quasitwo-dimensional turbulence. This prediction needs to be tested in numerical simulations.
In the next two subsections we will perform asymptotic analysis of (4.1).

Asymptotic analysis
We examine the dominant balance for the two extreme limits of (4.2).
4.2.1. Case 1: mSST for k 1 In this situation, Π u Π b , hence the balance is between Π b and : 10 -6 10 -2 10 2 10 6 10 10 k 10 -8 10 -5 10 -2 10 -6 10 -2 10 2 10 6 10 10 k 10 -12 10 -8 10 -4 10 0 10 4 10 8 10 12  Using (3.1), b k is found to be b k ≈ 2/5 N −1/5 k −1/5 . (4.4) Therefore the kinetic and potential energy spectra and fluxes, as well as the energy feed by buoyancy are given below: Π u (k) ≈ 3/5 N 6/5 k −4/5 , (4.5a) 3/5 N 6/5 k −9/5 , (4.5c) E u (k) ≈ 2/5 N 4/5 k −11/5 , (4.5d) Note that u k ∼ k −3/5 decreases faster that b k ∼ k −1/5 . Therefore, buoyancy is strong enough so as to yield E u (k) ∼ k −11/5 for the whole of inertial range. Note that dissipation range starts after the inertial range. 4.2.2. Case 2: mSST for lower wavenumbers (k 1) Equations (4.3)-(4.4) indicate that u k ≈ b k near k = 1. For k 1, Π u Π b implying that the dominant balance has to be between Π u and : With the above u k and b k , the evaluated energy spectra and fluxes, as well as the energy feed by buoyancy in this situation are given below: However, we are not certain that the above scaling can be observed in realistic systems. The range k 1 is possible in a large aspect ratio box, but such systems could exhibit two-dimensional or quasi-two-dimensional turbulence for which (4.1) is not valid. Hence this prediction needs to be tested thoroughly in future. Schematic diagram of the behaviour of kinetic and potential energy fluxes found out using our revised Bolgiano-Obukhov phenomenology is shown in Figure 4.

Conclusions
In this short paper, we revisit the celebrated Bolgiano-Obukhov (BO) phenomenology for stably stratified turbulence under moderate stratification. BO phenomenology proposes that there exists a dual scaling for the energy spectra -kinetic energy varies as ∼ k −11/5 for k < k B , and as ∼ k −5/3 for k > k B , where k B is the Bolgiano wavenumber. The potential energy varies as ∼ k −7/5 and ∼ k −5/3 respectively in the respective regimes. The transition to k −5/3 scaling is based on the argument that the energy supply rate from buoyancy becomes negligible when k is large, thus making density a passive scalar (such passive scalar behavior of density is observed in weakly stratified turbulence). In the present paper we show that the second scaling is not observable in moderately stratified turbulence (mSST). This is because u k is too weak at large wavenumbers to be able to start a constant energy cascade. Our arguments are based on numerical computation and asymptotic analysis, which are in turn based on the constancy of the total energy flux in the inertial range. Our revised scaling for stable stratification may have important consequences in the modelling of buoyancy-driven flows.