Research on Rotating Stratified Turbulent (RST) Flows
Creation of Helicity in Rotating-Stratified Turbulence
Kinetic helicity \(H_V = v \cdot \omega\) (velocity-vorticity correlation) is an invariant of the non-dissipative Navier-Stokes equations, including in the presence of rotation, but not with stratification. The connection between \(H_V\) and geostrophic balance (i.e. the equilibrium that develops between pressure gradients, Coriolis and gravity forces at scales large enough for non-linearities to be neglected) was shown by Hide in 1976 (1) who derived what can be considered the helical version of the geostrophic balance equations predicting the creation of helicity in rotating stratified flows. In order to verify the validity of Hide’s model, I performed a study of decaying turbulence by means of DNS (up to \(512^3\) ) to investigate the possibility of observing the spontaneous emergency of helicity in RST. My results show that, in the rotating Boussinesq framework, kinetic helicity is indeed created in decaying DNS runs and follows the linear relation predicted by Hide (see Fig. 1) in a specific range of the parameters (2). Based on this study, in order to fulfill Hide’s prediction, generation of helicity requires a balance between stratification and rotation, that is to say order of unity values of the ratio between the Brunt-Väisälä frequency (\(N\)) and the angular velocity (\(\Omega\)); more precisely I found that \(N/f < 3\), where \(f = 2\Omega\) is the Coriolis parameter (see Fig. 1). It should be pointed out that a linear relation for the net helicity created obtains as well when the analysis is conditioned to moderate values of the Buoyancy Reynolds number (combination of the Froude and Reynolds numbers as follows: \(R_B = ReFr^2\) ) and its equivalent for rotation (which involves the Rossby number: \(ReRo^2\) ); in particular \(ReFr^2 < 20\) and \(ReRo^2 < 20\). The fact that rotating stratified flows can spontaneously generate helicity opens new lines of research and questions; e.g., in nature creation of helicity might indeed be relevant to the persistence of large-scale convective storms and to the onset phase of hurricanes (3; 4). Furthermore, it is also known that the generation of large-scale magnetic fields occurs due to small-scale kinetic helicity in a wide variety of astrophysical settings (5; 6).
References
[1] R. Hide, A note on helicity, Geophys. Fl. Dyn. 7, 157 (1976)[2] R. Marino, P.D. Mininni, D.Rosenberg & A. Pouquet, Phys. Rev. E 87, 033016 (2013).
[3] E. Rassmussen & D. Blanchard, Weath. For. 13, 1148 (1998)
[4] G. Levina & M. Montgomery, Dokl. Earth Sc. 434, 1285 (2010)
[5] A. Brandenburg & K. Subramanian, Phys. Rep. 417, 1 (2005)
[6] M. Miesch & J. Toomre, Ann. Rev. Fluid Mech. 41, 317 (2009)
Inverse Cascade in Rotating-Stratified Turbulence
We examined features of the inverse cascade in RST as it occurs in the climate system and in laboratory experiments, while varying the relative frequency of gravity to inertial waves \(N/f\) . Using direct numerical simulations (DNS) forced at small scale (on grids up to \(1024^3\)) we have found that the inverse cascade of kinetic energy to the large scales is more efficient than in the equivalent purely rotating case when a moderate level of stratification is added (1). More specifically, for a given value of \(f\), for a rotating-stratified flow with \(1/2 < N/f < 2\), the growth of the kinetic energy was found to be faster than in the correspondent purely rotating case (thus with same \(f\) and \(N = 0\)). The above range is peculiar as it contains values of \(N/f\) for which the triadic wave resonances disappear in rotating-stratified flows. For values of \(N/f > 2\) the efficiency of the inverse cascade decreases monotonically and it goes to zero in the purely stratified case (thus for \(N/f \sim \infty\), see Fig. 1). Indeed also in purely stratified flows is observed some growth of the kinetic energy at large scales, as the latter piles up in the so called Vertically Sheared Horizontal Winds (i.e. modes for the horizontal velocity with \(k_{\perp} = 0\) and with strong vertical gradients). However, this growth is not accompanied by a negative constant flux in a wide range of scales, as expected in presence of an inverse cascade. Instead, the isotropic kinetic energy flux tends to be small at scales larger the forcing scale in simulations, while the perpendicular energy flux becomes negative and the parallel energy flux becomes positive. In other words, the growth of energy at large scales in stratified flows is not the result of a self-similar cascade but rather of a very strong anisotropization of the flow.
Anisotropic Energy Transfer in Stably Stratified Flows
In the case of purely stratified flows evidence for the existence of an inverse cascade is less clear (1), and conclusions from simulations are sometimes contradictory. In fact, while purely rotating flows, and rotating and stratified flows at moderate values of \(N/f\) develop isotropic inverse cascades, as mentioned in the previous section, purely stratified flows have almost zero isotropic energy flux at large scales (\(\Pi_T(k) = 0\), with \(k < k_{forcing}\)). Extending the previous numerical study on RST, I was able to prove that in the purely stratified case there is indeed constant negative perpendicular flux towards small \(k_{\perp}\) in Fourier space (2). Unlike in previous works, my results show that when sufficient scale separation is allowed between the forcing scale and the domain size, the reduced kinetic energy spectrum (\(E_V(k_{\perp})\)) displays at large scales a power-law behavior in the perpendicular direction compatible with \(\sim k_{\perp} \sim 5/3\). I was able to show how this spectrum is the result of the combined inverse transfer in the perpendicular direction (\(k_{\perp}\)) and a direct cascade of the large-scale horizontal winds manifested by a positive flux of energy in the parallel direction at the largest scales (\(k\)), see Fig. 1. This highly anisotropic transfer in stratified flows can result in the buildup of a power-law spectrum at large scales (albeit up to some wave number), with positive flux in some directions in spectral space.
Dual Energy Cascade in Rotating-Stratified Turbulence
While it has been known for some time that when rotation and stratification are of comparable strength inverse energy cascades take place, it was perhaps not as well recognized that the 3D Boussinesq equations, including rotation, can produce concurrently large and small-scale energy excitation, both with constant flux. Numerous numerical studies on RST suffer from a lack of resolving both the large and the small eddies: because of the inherent cost of such computations, a divide-and-conquer approach has been followed in the literature, analyzing either the direct or the inverse cascade, but not, convincingly, both. Taking advantage of HPC and exploiting the high scalability of the GHOST code (Geophysical High Order Suite for Turbulence, developed within my former team at NCAR), we were able to provide unambiguous numerical evidence of the simultaneous generation of large-scale and small-scale flows both with constant flux, using high-resolution DNS of the Boussinesq equations (1; 2). In particular, we showed how rotating stratified flows forced at intermediate scale (\(k_{forcing} = [10, 11]\)) support a bi-directional cascade of the total energy with a robust Kolmogorov at scales larger than the forcing (characteristic of a 2D cascade). This numerical evidence points to the possibility terms of a co-existence in geophysical fluids of idealized large-scale dynamics dominated by quasi-geostrophic motions, together with the production of small scales, essential to mixing (3). Indeed, a puzzle of oceanic dynamics is the contrast between the observed large scale geostrophic balance and the necessary turbulent transport at the smaller scales. Recent studies of the Southern Ocean analyzing Sea-Surface Height (SSH) – a proxy for the horizontal velocity field of near-surface currents – and numerical modeling introducing a positive eddy viscosity to represent the effect of the direct energy cascade, show that the flux ratio \(R_{\Pi} = \epsilon_L/\epsilon_S\) (of the inverse to the direct energy cascade rate) remains typically between the values 3 and 7. To check this evidence in our numerical and theoretical framework we have conducted an unprecedented investigation by means of high resolution DNS of the Boussinesq equations in the stably-stratified rotating case (on grids from \(1024^3\) to \(2048^3\) points). The values of \(N/f\) achieved in this study are compatible with the abyssal Southern Ocean at mid-latitude, where stratification is stronger than rotation and the observed \(N/f\) spans from 5 to 10. Our research proves that within such realistic range of values of \(N/f\) , the flux ratio \(R_{\Pi}\) computed from the DNS agrees with observations (see Fig. 1 left). This evidence allows one to estimate that globally the small-scale dissipation is between 20% and 25% of the available energy, thus alleviating the long-standing issue in ocean and climate dynamics concerning the amount of energy dissipation. The results also confirm the validity of a phenomenological model we developed (Fig. 1 right), based on wave turbulence phenomenology, that predicts the scaling of \(R_{\Pi}\) with the inverse of the Froude and Rossby numbers: \(R_{\Pi} \sim (FrRo) \sim 1\) . Considering the estimates of \(R_{\Pi}\) and the parameters typical of the abyssal Southern Ocean – except for the Reynolds number, adapted to the value used in the DNS, and assuming topographic gravity waves as a small-scale energy source (as described in 1), our model shows a remarkable agreement with the observation (green star Fig. 1 right). Another important outcome of this study is that, with an assortment of parameters, a dual energy transfer is also observed separately for the kinetic and the potential modes, with negative fluxes at large scales and positive at small scales.
Exchanges Kinetic-Potential Energy: Evidence for the Bolgiano-Obukhov Scaling
Together with the investigation of RST flows in the bi-directional and inverse cascade regimes (forced at intermediate or small scale: \(k_f \sim 10, 20, 40\)), we designed and ran the still to date largest DNS of a rotating-stratified flow, on grid of \(4096^3\) (Fig. 2). This simulation was performed on the Titan supercomputer (U.S. Department of Energy) using the pseudo-spectral code GHOST. Because of its huge computational cost the spin-down (or decaying) configuration was the only possible option. We decided as well to initialize the flow with random 3D-isotropic (in the Fourier space) initial conditions for the velocity field, centered this time in the large scales (\(k_{forcing} = 2, 3\)) and null initial conditions for the temperature field. Other parameters of this run, rendered in Fig. 2 are \(Re = 5.4 \times 10^4\) , \(Fr = 0.0242\), \(N/f = 4.95\) and then \(R_B = ReFr^2 = 32\), \(Ro = 0.12\), appropriate to model the dynamics of the abyssal Southern Ocean at mid latitude (although with a Reynolds number still low compare to reality). These choices, including the large scale initial conditions, have set up the optimal scale separation allowing for the first observation of the Bolgiano-Obukhov (BO) scaling in a DNS of RST (1). The BO scaling obtains under the assumption that buoyancy acts as a reservoir of energy at the large scale, with constant nonlinear transfer rate of the potential energy (\(\epsilon_P = |dE_P|/dt\)) and negligible advection terms in the momentum equation. This leads to the following prediction for kinetic and potential energy spectra: \(E_V(k) \sim \epsilon^{2/5}_P k^{−11/5}\), \(E_P(k) \sim \epsilon^{4/5}_P k^{−7/5}\) . In BO phenomenology the scalar actively modifies the velocity field. It was originally derived for the purely stratified case but it must be noted that the Coriolis force does not contribute to the energy balance in rotating stratified flows. Thanks to the remarkable scale separation of the \(4096^3\) run, we were able to provide evidence for the BO scaling in both \(E_V(k)\) and \(E_P(k)\) (Fig. 1) over about a decade of scales, up to the wave vector \(k_c = 12\). The latter roughly corresponds to our estimate of the characteristic Bolgiano scale (\(K_{BO} \sim \epsilon^{3/4}_P \epsilon^{-5/4}_V\)) for which we found \(k_{BO} = 11\). A Kolmogorov law is instead observed for \(k_c \le k \le 100\) in the kinetic energy spectrum.
Exchanges Wave-Turbulent/ Vortical Modes
The Boussinesq equations (with or without rotation) can support waves with their eigenmodes being determined introducing an orthonormal basis of the the linearized dynamics in the Fourier space (2): \(X_0(k)\), \(X_+(k)\), \(X_−(k)\), with \(X^{\dagger}_r(k)X_s(k) = \delta_{rs}\) . These modes have a natural physical interpretation: \(X_{\pm}(k)\) are the modes associated to (pure) inertia-gravity waves that develop with a ”fast” dynamics, while the mode \(X_0(k)\) corresponds to ”slow” turbulent eddies advected by the nonlinear term. The latter contribute to potential vorticity and can also be referred to as the balanced (quasi-geostrophic) or the vortical modes. We designed two studies to quantify the partition of the energy between waves and turbulent modes in RST in the forward and inverse cascade regime, separately. The transition from vortex to wave dominated dynamics is a quite general feature of atmospheric and oceanic flows. By performing an unprecedented parametric study (51 DNS runs of decaying RST flows up to \(3072^3\) grid points) we were able to show that the wavenumber \(k_R\) at which this transition occurs does not depend on the Reynolds number (1), suggesting that the partition of energy between wave and vortical modes does not depend on how vigorous the turbulence is but rather represents multi-scale interactions between slow and fast modes (see Fig. 1). We also showed that \(k_R\) is compatible with the wavenumber at which kinetic and potential energy, respectively \(E_V\) and \(E_P\) , are close to the equipartition (insert Fig. 1) and how in some cases \(1/k_R\) is comparable to the characteristic BO scale, emphasizing the role of potential energy, as conjectured in the atmosphere and oceans. Finally, we provided evidences that \(k_R\) varies as the inverse of the Froude number, consistently with recent observations and modeling. Later on we examined as well the case of RST flows forced a small scale to investigate the inverse cascade regime. As an important result we found that the inverse cascade is dominated by the slow/quasi-geostrophic modes. In other words, when an inverse cascade of kinetic energy develops this is mostly associated to the vortical modes (2), while little or no energy piles up in the wave modes (Fig. 2, left), in agreement with the theoretical predictions based on statistical mechanics. To quantify the relevance of wave modes over turbulent modes in the forward cascade regime, in the range \(1/2 < N/f < 2\), we also performed a spatio-temporal spectal analysis (Fig. 2, right), which provided a way to decompose the velocity field in waves, mean winds and eddies (3). As for the previous study based on the normal modes decomposition of frozen in time fields, spatio-temporal spectra confirmed that quasi-geostrophic motions dominate the dynamics in the range \(1/2 < N/f < 2\), as estimated from the broadening of the energy near the dispersion relation, expected in a regime where no resonant triadic interaction are available.
Dissipation and Mixing in Rotating-Stratified Turbulent Flows
Only a few studies have considered mixing in stratified flows with rotation. Strong rotation alters the large scales where the energy is contained. Furthermore, the dispersion relation for inertia-gravity waves involves both the parallel and the perpendicular directions (referring to the vertical). Thus, the atmosphere (with \(N/f \sim 100\) or more) and the ocean where this ratio is more than ten times smaller, may differ in how strong the effect of rotation is. The mixing efficiency can be characterized in several ways. A dimensionless measure of mixing can be defined as \(\Gamma_f = R_f/[1 − R_f \)] with \(\Gamma_f\) being the flux Richardson. Another possibility is to evaluate the so called irreversible mixing efficiency \(\Gamma = \epsilon_P/\epsilon_V\) (where \(\epsilon_{P,V}\) are respectively the potential and kinetic dissipation rate), whose definition is based on the assumption that the mixing efficiency should only account for the irreversible conversion of available potential energy into background potential energy (quantified by \(\epsilon_P\)). It is customary in literature to take \(\Gamma\) as constant a priori for all flows and in particular \(\Gamma \simeq 0.2\) (or \(R_f = 0.17\)). By extending the parametric study introduced in the previous section up to 65 high-res DNS runs (all spin-down, this time selecting only those on grids of \(1024^3\) points) covering a vast range of values in \(R_B = ReFr^2\), the so called Buoyancy Reynolds number (with an emphasis on parameters realistically compatible with the Mesosphere and Lower Thermosphere – MLT, see Fig. 1), we were able to show that \(\Gamma_f\) is actually far from being constant in RST and it exhibits a strong variation with \(Fr\) (1). and a rather sharp transition for \(Fr \simeq 0.02\) (Fig. 2, left), with a change in slope from \(\Gamma_f \sim Fr^{−2} to \Gamma_f \sim Fr^{−1}\) . This study allowed as well a quantitative assessment of the dissipative properties of RST flows showing that the dissipation efficiency \(\beta = \epsilon_V/\epsilon_D\) (with \(\epsilon_D = u^3_{rms}/L_{integral}\) being the dimensional expression of the kinetic energy dissipation) depends mostly on \(Fr\), with possibly three identifiable regimes: transitions at \(Fr \simeq 0.01\) (\(R_B \simeq 2\)) and around \(Fr \simeq 0.2\) (\(R_B \simeq 200\)), Fig. 2, right. Small-scale properties of the flows are associated to the exchange of kinetic and potential energy. In further understanding how energy is distributed between these modes, it is customary to define \(T_V = E_V/\epsilon_V\) and \(T_P = E_P/\epsilon_P\) (\(E_{V,P}\) being kinetic and potential energies) which are dimensional evaluations of the time it takes in a given flow to dissipate its kinetic and potential energies respectively. As such, they give a measure of the effective dissipation that can occur in flows submitted to both linear waves and nonlinear turbulent transfer to small scales. Based on the same pool of runs used to study the variation of the mixing efficiency \(\Gamma_f\), we conducted an extensive exploration of the variation of characteristic time scales \(T_{V,P}\) in RST flows with dimensionless parameters, focusing on the role played by the partition of energy between the kinetic and potential modes, as a key ingredient for modeling the dynamics of such flows (2).
References
[1] A. Pouquet, D. Rosenberg, R. Marino, C. Herbert, J. Fluid Mech., 844 519 (2018)[2] D. Rosenberg, R. Marino, C. Herbert, A. Pouquet, European Phys. J. E (2016)
Intermittency in Purely Stratified and Rotating-Stratified Flows
Intermittency is a hallmark of fully developed turbulence in fluids. Contrary to the predictions of Kolmogorov original theory, both experiments and numerical simulations show that dissipation exhibits intense fluctuations, localized in space and time. This phenomenon, known as small-scale intermittency, is widely observed in the atmosphere and in the ocean in the form of highly concentrated and sporadic dissipation. Intermittency, however, is not only present at the smallest scales. In the problem of mixing of a passive scalar by a turbulent flow, for stratified flows as in the Earth’s atmosphere and in the oceans, non-stationary energetic bursts at scales comparable to that of the mean flow are also observed. We investigated the large-scale intermittency of vertical velocity and temperature, and the mixing properties of stably stratified turbulent flows using both Lagrangian and Eulerian fields from DNS of the Boussinesq equations with periodic boundary conditions, in a parameter space relevant for the atmosphere and the oceans. That was done in the frame of the PhD thesis (in co-tutelle with the University of Calabria, IT, with a scolarship provided by the école doctorale MEGA at ECL) of one of my students, Fabio Feraco (A.Y. 2018/2019 – 2020/2021), who conducted the analysis that follows. Over a range of Froude numbers of geophysical interest (\(\simeq 0.05 − 0.3\)) we observed very large fluctuations of the vertical components of the velocity and the potential temperature (diagnosed through their kurtosis \(K_x = (x^4/x^2\) , with \(x = v, \theta\)), with a sharp transition leading to non-Gaussian wings of the probability distribution functions (1, see Fig. 1). This behavior is captured by a simple model we developed, representing the competition between gravity waves on a fast time-scale and nonlinear steepening on a slower time-scale. The existence of a resonant regime characterized by enhanced large-scale intermittency, as understood within the framework of the proposed model, was then linked to the emergence of structures in the velocity and potential temperature fields, localized overturning and mixing. In the same regime we observed a linear scaling of the mixing efficiency with the Froude number and an increase of its value of roughly one order of magnitude. In this context, we also evaluate quantitatively the link between mixing and dissipation, anisotropy and intermittency in the presence of both rotation and stratification, and as a function of the intensity of the turbulence and the Richardson number defined as \(Ri = (N/\partial_z u_{\perp})^2\) . A lack of isotropy can be in fact associated with intermittency, as well as with the long-range interactions between large-scale coherent structures and small-scale dissipative eddies. Anisotropy has been studied extensively for a variety of flows and many diagnostics have been devised. Signatures of anisotropy can be obtained through the properties of the following tensors of velocity (u) and potential temperature (\(\theta\)):
\(\begin{equation} b_{ij}=\frac{\langle u_iu_j\rangle}{\langle u_ku_k\rangle}-\frac{\delta_{ij}}{3},\ d_{ij}=\frac{\langle \partial_k u_i \partial_k u_j\rangle}{\langle \partial_k u_m \partial_k u_m\rangle}-\frac{\delta_{ij}}{3},\ g_{ij}=\frac{\langle \partial_i \theta \partial_j \theta\rangle}{\langle \partial_k \theta \partial_k \theta\rangle}-\frac{\delta_{ij}}{3},\ v_{ij}=\frac{\langle \omega_i \omega_j\rangle}{\langle \omega_k \omega_k\rangle}-\frac{\delta_{ij}}{3} \end{equation}\)
These tensors are equal to zero in the isotropic case. We define as usual the second and third-order invariants of a tensor \(T_{ij}\) as \(T_{II} = T_{ij}T_{ji}\) and \(T_{III} = T_{ij} T_{jk} T_{ki}\). For the tensors above, they are denoted respectively \(b_{II,III}\) , \(d_{II,III}\) , \(g_{II,III}\) , and \(v_{II,III}\). They refer in particular to the geometry of the fields (1D vs. 2D, 3D, and axisymmetric, oblate or prolate). Combining these mathematical tools with dimensional analysis and estimates of the adimensional numbers, we have shown that in rotating stratified turbulence, a sharp increase in dissipation and mixing efficiency is associated, in an intermediate regime of parameters, with large-scale anisotropy, as seen in the tensors defined above, and large-scale intermittency, as observed in the vertical velocity through its kurtosis (2), see Fig. 2.
Observing Mesospheric Turbulence
The atmosphere represents indeed the largest domain where it is possible to study the complex dynamics of stratifies flows, including the interplay between gravity waves and turbulent motions, and in particular the mesosphere and lower thermosphere MLT (from 70 km to 120 km). While in situ observations (such as the ones performed by TILDÆ) allow to collect measurements with a very high temporal cadence but usually only in a limited spatial domain, RADAR observations provide instead spatially extended snapshots of the atmosphere but usually with poor temporal and spatial resolutions. However, the study of phenomena developing on time and length scales characteristic of turbulence does require high cadence acquisitions, possibly over many points in the space. Observing the the velocity field in the MLT is also crucial to understanding the energetics and coupling processes of the upper atmosphere. Optical remote sensing instruments are sparsely-located giving only a local view of the atmosphere, and are mostly limited to nighttime observations. Recent advances in meteor RADARs, which observe winds from about 80 to 100 km altitude through the Doppler shift of drifting meteor trails (schematic in Fig. 1), hold the promise of temporally and spatially continuous coverage, with increasing resolutions. Together with colleagues from the Leibniz Institute of Atmospheric Physics – IAP (DE) we developed a novel method to perform high order spatio-temporal statistics within the inertial range of turbulence in the MLT. That was done in the frame of the PhD thesis of Harikrishnan Charuvil (in co-tutelle with the IAP), whose PhD project was awarded in France with the prestigious Eiffel fellowship (from Campus France). The novel methodology relies on using pairs of meteor events – in a way to maximize the spatial coverage without sacrificing the temporal resolution – to estimate the correlation function of the wind with different spatial and temporal lags. The preliminary analysis we conducted allowed us to compute high-resolution correlation functions with temporal, horizontal, and vertical lags (1), see Fig. 2. The temporal correlation function was then used to retrieve the kinetic energy spectrum, while from the horizontal and vertical correlation functions of the wind we were able to derive second-order structure functions of the velocity field. The latter were found to be compatible with the Kolmogorov prediction for spectral distribution of kinetic energy in the turbulent inertial range. The presented method can be used to extend the capabilities of specular meteor radars and our goal is that of applying it to achieve a fully resolved three-dimensional velocity field from meteor radar observations that can be used in a synergistic approach with high resolution numerical investigations by means of our DNS.
