Scaling laws for mixing and dissipation in unforced rotating stratified turbulence
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Scaling laws for mixing and dissipation inunforced rotating stratified turbulence
A. Pouquet , , D. Rosenberg , R. Marino and C. Herbert National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO, 80307, USA. Atmospheric & Space Physics Laboratory, University of Colorado, Boulder CO, 80309, USA. [email protected]. Laboratoire de M´ecanique des Fluides et d’Acoustique, CNRS, ´Ecole Centrale de Lyon,Universit´e de Lyon, ´Ecully, 69134, FRANCE. ´Ecole Normale Sup´erieure, 46 All´ee d’Italie, Lyon, F-69364, FRANCE.(Received xx; revised xx; accepted xx) We present a model for the scaling of mixing in weakly rotating stratified flows charac-terized by their Rossby, Froude and Reynolds numbers
Ro, F r , Re . It is based on quasi-equipartition between kinetic and potential modes, sub-dominant vertical velocity w , andlessening of the energy transfer to small scales as measured by a dissipation efficiency β = (cid:15) V /(cid:15) D , with (cid:15) V the kinetic energy dissipation and (cid:15) D = u rms /L int its dimensionalexpression, w, u rms the vertical and rms velocities, and L int the integral scale. Wedetermine the domains of validity of such laws for a large numerical study of the unforcedBoussinesq equations mostly on grids of 1024 points, with Ro/F r (cid:62) .
5, and with1600 (cid:54) Re ≈ . × ; the Prandtl number is one, initial conditions are either isotropicand at large scale for the velocity, and zero for the temperature θ , or in geostrophicbalance. Three regimes in Froude number, as for stratified flows, are observed: dominantwaves, eddy-wave interactions and strong turbulence. A wave-turbulence balance for thetransfer time τ tr = N τ NL , with τ NL = L int /u rms the turn-over time and N the Brunt-V¨ais¨al¨a frequency, leads to β growing linearly with F r in the intermediate regime, with asaturation at β ≈ . F r . The flux Richardson number R f = B f / [ B f + (cid:15) V ], with B f = N (cid:104) wθ (cid:105) the buoyancy flux, transitions for roughly thesame parameter values as for β . These regimes for the present study are delimited by R B = ReF r ≈ R B ≈ Γ f = R f / [1 − R f ] the mixing efficiency, puttingtogether the three relationships of the model allows for the prediction of the scaling Γ f ∼ F r − ∼ R − B in the low and intermediate regimes for high Re , whereas for higherFroude numbers, Γ f ∼ R − / B , a scaling already found in observations: as turbulencestrengthens, β ∼ w ≈ u rms , and smaller buoyancy fluxes altogether correspond to adecoupling of velocity and temperature fluctuations, the latter becoming passive.
1. Introduction
Mixing, which takes place for a large domain of flow parameters in fully devel-oped turbulence (FDT), in engineering and geophysical flows, has been analyzed ex-tensively (Peltier & Caulfield 2003; Dimotakis 2005; Ivey et al. et al. (2017)and references therein), for example as the ratio in the momentum equation of the buoy-ancy flux to the rate of kinetic energy dissipation (see § et al. a r X i v : . [ phy s i c s . f l u - dyn ] D ec A. Pouquet, D. Rosenberg, R. Marino and C. Herbert circulation of the ocean and atmosphere is modified by a combination of stratification anddissipation, but rotation can also play a role. In the oceanic context, data indicates anenhanced vertical mixing that can be compared with the Osborn (1980) model, statingthat the adimensionalized scalar diffusivity is proportional to the turbulence intensityparameter, R I = (cid:15) V / [ νN ], with an efficiency Γ f = B f /(cid:15) V ≈ . R I (cid:38) (cid:15) V isthe energy dissipation rate, ν the kinematic viscosity, B f the buoyancy flux, and N isthe Brunt-V¨ais¨al¨a frequency. Unsurprisingly, though, there is evidence that this efficiencycoefficient does depend on the intensity of the turbulence within a stratified flow (Smyth et al. et al. et al. e.g. , in the presence or not of shear,leading to a lack of monotonicity in its variation with the Richardson number (Peltier& Caulfield 2003). Thus, a one-parameter modeling of such flows based on stratificationalone may be insufficient.The amount of kinetic energy available for dissipation at small scales in rotatingstratified turbulence (RST) is a crucial quantity for sub-grid scale parameterizationsin oceanic and climate models, and depends on the amount of energy that is transferredto these small scales in the presence of inertia-gravity waves, through breaking at smallscales of the large-scale quasi-geostrophic (QG) balance (Lelong & Riley 1991; Staquet &Sommeria 2002; Riley & deBruynKops 2003), and a lowering of the Richardson numberbelow some critical value. Wave-turbulence interactions allow for coupling to the meanflow (Finnigan 1999); they have been measured in the stratosphere as well as in theupper ocean and lead to vertical mixing and enhanced dissipation (see e.g., Klymak et al. (2008); van Haren et al. (2016)). Few studies have considered mixing in decaying rotatingstratified flows. Lagrangian diffusion in RST is studied in Cambon et al. (2004); with
N/f taking the values of 0.1, 1 and 10, as well as 0 and ∞ , a reduced diffusivity is foundin the vertical, but not in the horizontal. The role of N/f , with f twice the rotation rate,as a governing parameter for the intensity of lateral mixing with geostrophic adjustmenthas also been stressed in Lelong & Sundermeyer (2005). Experimentally, it was shownin Praud et al. (2006) that structures develop an aspect ratio proportional to f /N (seealso Waite & Bartello (2006); Kurien & Smith (2014) for direct numerical simulationsfor the forced case): in RST, there is a progressive shift from a vertical scale due entirelyto stratification, the buoyancy scale L B = u rms /N , to one corresponding to QG whererotation and stratification are in balance with pressure gradients. Finally, Dritschel &McKiver (2015) studied the influence of N/f on large-scale quasi-geostrophic balance;they found it weak, the flows remaining balanced throughout the studied parameterregime, although vertical velocity increases with f /N for
F r << Ro << Re ≈ . × , an upper value comparable to that in theMesosphere and Lower Thermosphere (Liu et al. caling laws for mixing in unforced weakly rotating stratified turbulence
2. Problem setting
The Boussinesq equations, with constant rotation and stable stratification are: ∂ t u + ∇ p + u · ∇ u = f u × ˆ z − N θ ˆ z + νδ u , (2.1) ∂ t θ + u · ∇ θ = N w + κδθ , (2.2)with u = ( u, v, w ) the incompressible velocity field, ∇ · u = 0, and p the pressure; θ represents temperature (or density) fluctuations, in units of a velocity, since we want tostress the energetics of these flows. These fluctuations are super-imposed on on a stablystratified background with a linear vertical profile ¯ θ ( z ) = θ + z∂ z ¯ θ, ∂ z ¯ θ <
0. Introducingthe buoyancy b = N θ (note the choice of sign), with point-wise vertical flux b ( x ) w ( x ),as well as the integrated buoyancy flux B f = (cid:104) N θw (cid:105) , (2.3)one recovers the more standard form of the equations in terms of b . N = (cid:112) − ( g/θ ) ∂ z ¯ θ is the Brunt-V¨ais¨al¨a frequency, and ν = κ are the viscosity and thermal diffusivity.The Boussinesq equations are integrated using direct numerical simulations. A cubicbox of n p = 1024 points is used for 56 runs (see Table 1), with a linear dimension L box = 2 π , resulting in wave numbers in the range 1 (cid:54) k (cid:54) k M = n p / L box = 2 π , which of course can be rescaled to the physical problemwhen necessary. Another smaller set of runs, at lower resolutions, is also analyzed (seeTable 2). The pseudo-spectral code we use, GHOST (Geophysical High-Order Suite forTurbulence), is parallelized in a hybrid fashion with both MPI and Open-MP (Mininni et al. π/L = k = 2 . E V = 12 (cid:90) (cid:107) u ( x ) (cid:107) d x , E P = 12 (cid:90) | θ ( x ) | d x , E T = E V + E P ,E T being the total energy. They can also be written in terms of their respective isotropicFourier spectra, with (cid:82) E V,P ( k ) dk = E V,P . Similarly, the kinetic, potential and totalrates of energy dissipation are: (cid:15) V = ν (cid:90) (cid:107) ω ( x ) (cid:107) d x , (cid:15) P = κ (cid:90) (cid:107)∇ θ ( x ) (cid:107) d x , (cid:15) T = (cid:15) V + (cid:15) P . (2.4)Note that (cid:15) V can be measured relative to its dimensional evaluation of kinetic energydissipation for a fully turbulent flow as: β ≡ (cid:15) V /(cid:15) D , (cid:15) D ≡ u rms /L int , u rms = [ (cid:10) | u | (cid:11) ] / , L int = 2 π (cid:82) [ E V ( k ) /k ] dk (cid:82) E V ( k ) dk , (2.5)where L int is the integral scale (Monin & Yaglom 1979). β is a key parameter of thephenomenological and theoretical understanding of the interactions between waves andturbulence (Zakharov et al. § A. Pouquet, D. Rosenberg, R. Marino and C. Herbert strength of nonlinear interactions relative to dissipation, rotation and stratification; theyare the Reynolds (Re), Rossby (Ro) and Froude (Fr) numbers, defined as usual as: Re = u rms L int ν , Ro = u rms f L int , F r = u rms N L int , (2.6)with the Prandtl number P r = ν/κ taken equal to unity. The buoyancy Reynolds number,the Richardson number, and the turbulent intensity are defined as: R B ≡ ReF r , Ri ≡ [ N/ (cid:104) ∂ z u ⊥ (cid:105) ] , R I ≡ (cid:15) V / [ νN ] . (2.7) Ri is based on a shear time computed on vertical gradients of the horizontal wind, namely τ shear = [ (cid:104) ∂ z u ⊥ (cid:105) ] − . All these parameters are discussed further in the Appendix, § e.g. L B = 2 π (cid:112) E V /N , L Ell = 2 π (cid:112) E P /N , (cid:96) Oz = 2 π (cid:112) (cid:15) V /N , (2.8)or the buoyancy, Ellison and Ozmidov scales. In purely stratified flows, L B is the scalefor which the vertical Froude number becomes of order one (Billant & Chomaz 2001);it measures the thickness of the vertical layers. On the other hand, the Ellison scalecorresponds to the vertical distance traveled by a particle of fluid before being completelymixed, and it is thought to be significantly smaller than the integral scale in stronglystratified flows, as we shall show later (see Figs. 6b). L B and L Ell vary as 1 /N , but differby a (cid:112) E V /E P field-amplitude ratio. Finally, the Ozmidov scale is the scale beyond whichisotropy is thought to be recovered together with a classical Kolmogorov range.
3. Global behavior and scaling
Overview of the runs
Runs with an emphasis on realistic parameters for the mesosphere and lower thermo-sphere, and that overlap with the present data base, were investigated for the energypartition between waves and slow modes and the link with kinetic-potential energyexchanges in Marino et al. (2015 b ), as well as for parametric characteristic time-scalevariations in Rosenberg et al. (2016) (see also Rosenberg et al. (2017)). Here, the runs ongrids of 1024 points, cover the following parameter ranges (see Table 1): 0 . (cid:54) Ro (cid:54) (cid:54) Re (cid:54) . (cid:54) F r (cid:54) .
5, 0 . < R B < . × and 2 . (cid:54) N/f (cid:54) ≈ Re , with as high a value as can be realized on the chosen grid, thusbreaking large-scale balance toward isotropization, as studied already in Herring (1980)using a closure model of turbulence (see Pumir et al. (2016); Rubinstein et al. (2017)and Iyer et al. (2017) for recent references).Two other small series of runs at lower resolutions have been performed (see Table2). The first study ( Q runs) is focused on the role of initial conditions, taking nowgeostrophically balanced fields at t = 0, which should radiate waves much less initially.The second small set of ( Z ) runs deals with the variation of effective dissipation β definedin equation (2.5) with Reynolds number at fixed N/f = 5, with Re varying by a factorin excess of 10, between 1650 and 18590, when including runs of Table 1.All statistics are computed dynamically around the peak of dissipation, when the flowis most developed and starts its self-similar temporal decay. This is in contrast to whatis done in Stretch et al. (2010), where the data for mixing is taken when more than caling laws for mixing in unforced weakly rotating stratified turbulence ≈ . u rms and L int do not vary much across the first largeparametric study, from 0.66 to 0.89 for the former, and from 1.39 to 2.78 for the latter.Note that some of the runs tabulated in Rosenberg et al. (2016) have been removed fromthe data set in Table 1, which has been reduced from 65 to 56 runs. This is because ofvarious factors: archiving issues in view of the large data base that was created severalyears ago, or because the variation of enstrophy at peak was insufficient to satisfy theaveraging criterion, or because some of the data files were corrupted.The accuracy of the computations is quantified through the ratio of the maximumto the Kolmogorov dissipation wavenumber, k M /k η , with k η = [ (cid:15) V /ν ] / ; this is doneunder the assumption that the small scales have recovered a Kolmogorov spectrum, i.e. that the Ozmidov length scale is resolved, or for R B (cid:62)
1, which is the case for themajority of our runs. For all flows of Table 1, we have 0 . (cid:54) k η /k M (cid:54) .
3, with roughly17% slightly under-resolved runs which all have
N/f (cid:62)
10. We have also checked thatthe overall shape of the curves plotted in the figures did not depend on the resolution.Strong activity develops at small scales, with layer destabilization, as found as wellby a number of authors in the purely stratified case. An example of such structures isgiven in the visualizations found in Rosenberg et al. (2015) for a flow which correspondsrather closely to some of the runs computed in this data base (specifically, run Id=11,19, 33 and 43), but done on a grid of 4096 points, allowing for a substantially higherReynolds number. Prominent in this flow with N/f = 4 . F r ≈ . Re ≈ e.g. Figs. 10 and 11 in Rosenberg et al. (2015)).In all the figures in the present work, different symbols are used for different binning,mainly in Rossby number; stars/asterisks are used for runs with QG initial conditions(ICs), whereas the hollow shapes are always for the θ ( t = 0) = 0 ICs. Furthermore, thesizes of all symbols refer to the resolution and Reynolds number (see caption of Fig. 1).3.2. Energy ratios
We show in Fig. 1 the variation of central energetic quantities, as a function of Froudenumber, with binning in Rossby number. The widths of the bins are chosen so as tohave approximately the same number of data points in each bin, as for all other figures.The energy ratio r E ≡ E P /E V (left) varies roughly between 0.2 and 0.4, as long as thevelocity and temperature remain coupled through buoyancy, i.e. for F r <
1. At high
F r , E P /E V becomes negligible since the velocity is no longer constrained effectively bythe waves and we have a quasi passive scalar regime with θ ( t = 0) = 0 for most ofthe runs. The scaling in F r − at high F r , as advocated for oceanic turbulence in Wells et al. (2010) (see also Maffioli et al. (2016) for purely stratified flows), may be present aswell, although we have a scarcity of points in that domain. We thus conclude that in theintermediate regime of wave-vortex interactions: θ rms ∼ u rms . (3.1)The result r E ≈ et al. (2013)) is compatible with theexperimental, observational and numerical data compiled in Zilitinkevich et al. (2008). It A. Pouquet, D. Rosenberg, R. Marino and C. Herbert
Table 1.
Nomenclature of the runs performed on grids of 1024 points, with Id, F r, Ro and Re the identification of runs and their Froude, Rossby and Reynolds numbers, the runs beingordered by F r (see also Rosenberg et al. (2016)). Initial conditions are centered on the largescales and are isotropic for the velocity field, and zero for the temperature. Runs 62 to 65 arepurely stratified. Note that the 9 runs marked with a star are not included in the present study(see § Id F r Ro Re —— Id F r Ro Re —— Id F r Ro Re ∞ ∞ ∞ ∞ is also compatible with (cid:15) P /(cid:15) V ≈ /
3, as measured in the stratosphere (Lindborg 2006).In fact, such a quasi-equipartition of energy is found in a large range of wave-numbers,as shown in Marino et al. (2015 b ) in the forced case. It immediately implies that theEllison scale L Ell goes as L int F r , for which in fact the scaling is excellent (see Fig. 6bbelow), and that L Ell ∼ L B ∼ u rms /N . Note also that a pattern is discernible in E P /E V with, on average, higher relative vertical velocity and higher relative potential energy atlow Rossby number. As a function of Reynolds number, r E decreases on average for Re larger than ≈ , because of the initial conditions (not shown).The ratio of vertical to total kinetic energy (cid:10) w / (cid:11) /E V is given in Fig. 1(b). At low F r, R B , with weak nonlinear mode coupling, its high value for the runs of Table 1 isdue to initial conditions, taken with a rough equipartition between velocity modes inall directions, in order to let anisotropy develop dynamically. A similar reduction invertical velocity for rapidly rotating flows in the absence of stratification was observed inlaboratory experiments (van Bokhoven et al. F r ≈ .
1, there is a small plateau, the caling laws for mixing in unforced weakly rotating stratified turbulence Table 2.
Parameters for two other sets of DNS identified by Id and ordered, for each set,by their Froude number
F r , with Ro and Re the Rossby and Reynolds numbers computed atthe time of maximum of enstrophy for each run. R B = ReF r , and n p is the total number ofgrid points for each run. In runs Zx , x = [1 , Qx runs, x = [9 , N/f ≈
5, a value close to what is found in the ocean. Id n p F r Ro Re R B . Id n p F r Ro Re R B Z1
256 0.042 0.208 3458 6.1 . Z2
512 0.063 0.316 6202 24.6 Z3
256 0.064 0.321 3358 13.7 . Z4
512 0.064 0.321 6643 27.2 Z5
128 0.065 0.325 1694 7.1 . Z6
256 0.097 0.487 2951 27.8 Z7
256 0.651 3.255 1657 702 . Z8
256 3.296 16.481 1706 18578 Q9
256 0.007 0.036 5221 .25 .
Q10
256 0.015 0.073 4973 1.1
Q11
256 0.039 0.197 3706 5.6 .
Q12
128 0.067 0.335 1617 7.2
Q13
256 0.075 0.373 3130 17.6 .
Q14
512 0.076 0.382 6278 36.3
Q15
256 0.111 0.555 2537 31.2 .
Q16
256 0.577 2.817 2003 667
Q17
256 1.290 6.451 2008 3341 . stratification being strong enough to prevent most of the vertical motions; so, w rms (cid:47) u rms . (3.2)As turbulence strengthens with increasing F r , vertical motions develop slowly after thatplateau, with an approximate scaling (cid:10) w (cid:11) ∼ (cid:10) u ⊥ (cid:11) F r / . The origin of such a weakscaling is not clear, and no scaling is found in terms of N/f . We recall here that thevertical velocity w ˆ e z is also a direct measure of wave activity since, in a wave-vortexdecomposition as performed e.g. in Bartello (1995); Herbert et al. (2016), vortical modeshave vanishing w . The presence of rotation facilitates vertical motions in the form ofupward propagating inertial waves along Taylor columns that would form if there was nostratification, as clearly observed, including when the small scales develop strong vorticity(Davidson et al. et al. w rms /u rms ∼ F r , ruled out by the datawith isotropic ICs, but in agreement with the data with QG initial conditions at lowFroude numbers, with this energy ratio reaching the value obtained with the isotropicICs for
F r ≈ .
1. We know that strong vertical velocities develop for intermediate
F r when the buoyancy and nonlinear terms balance each other, leading to a “saturation”vertical energy spectrum E (cid:107) ∼ N k − (cid:107) . The model developed in Rorai et al. (2014),adding the buoyancy flux to a Vieillefosse description of intermittency for FDT, leads to E (cid:107) ≈ E P ≈ E V under the hypothesis that the characteristic vertical scale is the buoyancyscale. In these intermittent regions identified by low Richardson numbers, high verticalvelocities appear, due to strong turbulent stirring leading to overturning of layers. Notethat it is argued in Maffioli & Davidson (2016) that gradients are much larger in thevertical, or that w is really at small scale. Thus, these authors advocate (cid:10) w (cid:11) ∼ (cid:15) V /N .With (cid:15) V ∼ (cid:15) D F r (see § w/u ⊥ ∼ F r . On the other hand,in their paper, β is viewed as a constant denoted A k <
1, independent of dimensionlessparameters, so their estimate is rather w/u ⊥ ∼ F r / , unlike our data at high F r . A. Pouquet, D. Rosenberg, R. Marino and C. Herbert −3 −2 −1 E P / E V [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) −3 −2 −1 −5 −4 −3 −2 −1 Fr . w / E V (b) −3 −2 −1 −2 −1 I II IIII II IIII II IIII II IIII II III Fr β = ε V / ε D slope=0.91 (c) Figure 1. (Color online) Variation with Froude number of the ratio of potential to kinetic energy( a ), of vertical to kinetic energy ( b ), and of the rate of kinetic energy dissipation compared toits dimensional evaluation, β = (cid:15) V /(cid:15) D ( c ). Colors/symbols indicate binning in Rossby numberfor all runs: blue triangles for 0 < Ro (cid:54) .
3, black circles for 0 . < Ro (cid:54) .
9, green diamonds for2 . < Ro (cid:54) .
0, red squares for 6 . < Ro (cid:54) .
0, and magenta inverted triangles for
Ro > θ ( t = 0) = 0; their relativesize is proportional to viscosity, thus inversely proportional to Re and to numerical resolution.We define the three dynamical regimes as: I for strong waves ( F r (cid:46) . . (cid:46) F r (cid:46) . F r (cid:38) . β with F r in regime II, namely β ∼ F r . . Effective versus dimensional dissipation and the three regimes of RST
The dissipation efficiency of rotating stratified flows β is shown in Fig. 1(c) as a functionof F r ; it clearly displays three regimes. For small
F r up to
F r ≈ . (cid:15) V is low andconstant. Similar low dissipation efficiency, of the order of a few percents, is obtained forruns corresponding to the Upper Troposphere and Lower Stratosphere region, with lowFroude numbers, as analyzed for example in Paoli et al. (2014) using a sub-grid model.We find that, above F r ≈ . β grows quasi-linearly with F r , with a least-square fitgiving a slope of 0.91 after which β saturates, for F r (cid:38) .
2. Thus, β = (cid:15) V /(cid:15) D ∼ F r [ Intermediate regime , II ] , (3.3)thereby defining the three dynamical regimes of RST, namely I, II & III, ordered byincreasing F r . Such a scaling is also found when examining the small-scale energy flux inthe forced case in the presence of an inverse cascade (Marino et al. a ). β saturatesat a value close to unity for highly turbulent flows at higher Froude numbers, as found aswell in Maffioli & Davidson (2016), although their values at peak of enstrophy are a bithigher. It may be related to the fact that they compute in boxes of small aspect ratio,between 1 / /
6, a geometry that can favor vertical gradients and shear, and whichcan lead to a more active turbulence, as found in Mininni & Pouquet (2017). When theturbulence strengthens, so does the direct energy cascade through baroclinic instability,frontogenesis and nonlinear coupling of eddies (McWilliams 2016). It should finally benoted that we do not expect actual transitional values of β , between regimes I & II, andII & III, to be similar for similar control parameters in different flow geometries, but wedo expect the scaling β ∼ F r to hold, as shown here contrasting isotropic versus
QGinitial conditions (see also the dimensional argument in the Appendix showing how thetransfer time to small scales is moderated by the stratification). caling laws for mixing in unforced weakly rotating stratified turbulence −4 −2 −2 −1 Turbulent Intensity= β R B β = ε V / ε D [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) − − − − Turbulent Intensity= β R B ε D (b) −2 −2 −1 Rb N τ s h ea r slope=−0.56 (c) Figure 2. (Color online) Variation of β ( a ) and (cid:15) D = u rms /L int ( b ) with turbulent intensityparameter R I = (cid:15) V / [ νN ] . In (c) is shown Ri / ≡ Nτ shear as a function of the buoyancyReynolds number R B = β − R I , with τ shear = (cid:104) ∂ z u ⊥ (cid:105) . All plots have binning in Rossby number,and symbols are as described in the caption of Fig. 1. The scaling Nτ shear ≈ R − . B extendsthrough regimes I & II with lower R B and F r , and possibly regime III.
If we now examine the variations of β with R I = (cid:15) V / [ νN ], we see in Fig. 2(a) that weagain have good scaling throughout, with β ∼ R / I ∼ β / R / B . It is easy to show thatthis is compatible with equation (3.3), since β ∼ R / B ∼ F r , omitting a dependency in Re / , although it appears clearly in Fig. 2 that, at fixed R I and lower Re , β is measurablylarger since F r is larger. This indicates that care must be taken when interpreting data asa function of dimensionless parameters. In R B , the transitions between the three regimesoccur respectively for ≈ ≈ R B ≈ −
100 (Fleury & Lueck 1994), as well as in lakes in whichan average for R B is ≈
200 (Bouffard & Boegman 2013).The dimensional kinetic energy dissipation (cid:15) D (Fig. 2b) is constant across parameters,and across initial conditions, except for very small or large F r values. A slight trendtowards smaller values at higher R I , which can be attributed to smaller rms velocities, isdiscernible. Finally, we show in Fig. 2(c) that Ri / ∼ R − / B , a scaling compatible with Ri ∼ F r − at constant Reynolds number. At low F r , this relation gives the strengthof vertical gradients (slanted because of rotation), and at higher
F r , it indicates aprogressive return to isotropy and to only a single time-scale determining the dynamics,transfer and dissipation of such turbulent flows. Note that the three results in equations(3.1–3.3) may not be entirely new but, taken together, they define the key ingredientsfor establishing the scaling of the mixing efficiency which is discussed in §
4. Mixing and dissipation
Definitions of mixing efficiency and flux Richardson number
Irreversible mixing is found in the laboratory to be triggered by merging Kelvin-Helmoltz billows (Patterson et al. Re increases. In the absenceof rotation, parameter space has been separated into three regions in terms of F r and Re (Luketina & Imberger 1989): for small Re and Fr, waves are dominant and thereare no turbulent motions, whereas for high Fr and Re, isotropic turbulence prevails. Theintermediate region with roughly F r (cid:54) , R B (cid:62)
10 is where turbulence is anisotropic andstrongly interacting with waves. The data on which these conclusions are based comes0
A. Pouquet, D. Rosenberg, R. Marino and C. Herbert −3 −2 −1 −2 −1 Fr R f = | B f / | ( | B f | + ε V ) slope=−0.57[ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) − − − − − Fr Γ f = R f / ( − R f ); R f = | B f | / ( | B f | + ε V ) (b) −3 −2 −1 −2 −1 Fr | < w θ > | / κ N slope=0.97 (c) Figure 3. (Color online) Variation with Froude number of the flux Richardson number R f and of the mixing efficiency Γ f ( a,b ), both defined in equation 4.1, as well as of the effectivediffusivity κ ρ /κ ( c ), with κ ρ = N − (cid:104) wθ (cid:105) . Binning is performed in Ro (see Fig. 1 for symbols).A transition at F r ≈ .
02 is seen in all three plots. Scalings are given as indications. from the analysis of turbulent plumes active in tidal estuary flows (Stillinger et al. et al. § B f = N (cid:104) wθ (cid:105) with the dissipation rates, the Coriolisforce not affecting the energy balance. Performing space-averaging, one can write: D t E V = − B f + (cid:15) V , D t E P = B f + (cid:15) P . In order to quantify the relative magnitudes of these terms, several expressions havebeen introduced in the literature. Concerning the momentum equation, one traditionallydefines the flux Richardson number R f and its associated mixing efficiency Γ f as: R f = B f B f + (cid:15) V , Γ f = R f − R f = B f (cid:15) V . (4.1)The functional variation of R f with gradient Richardson number is central to numericalstudies of geophysical flows. The mixing efficiency Γ f is singular for R f = 1, i.e. forfully mixed potential and kinetic modes (see Mashayek & Peltier (2013); Salehipour& Peltier (2015); Mashayek et al. (2017) for a discussion on the definitions of mixingefficiency). This corresponds to negligible kinetic energy dissipation, i.e. a limit of zeroFroude number. As we shall see in Fig. 3, Γ f does reach high values at low F r , in excessof 10 . Many recent works indicate variations with parameters as Re grows.4.2. Mixing and effective diffusivity as a function of parameters
We evaluate Γ f at peak of dissipation, whereas in Stretch et al. (2010), it is computedas the ratio of two time integrals after more than 90% of the energy has dissipated. Sinceenergy decay in turbulence is self-similar, these two methods should lead to comparablescalings. We show in Figs. 3(a,b) R f and Γ f as a function of Froude number. At low F r , (cid:15) V is negligible compared to B f , and R f ≈
1. A sharp transition occurs for
F r ≈ . R f decreasing continuously thereafter, and no visible saturation. The decline in R f begins in the intermediate range in which waves and vortices interact strongly. Similarly, Γ f shows a transition for F r ≈ .
02, with a change in slope from Γ f ≈ F r − to ≈ F r − ,as F r grows (approximate scalings), and with a variation of several orders of magnitude.At higher
F r (regimes II & III), a power-law scaling seems likely, as for R f . The variation caling laws for mixing in unforced weakly rotating stratified turbulence −2 −1 Re Γ f = R f / ( − R f ); R f = | B f | / ( | B f | + ε V ) [0.001,0.007][0.007,0.038][0.038,0.196][0.196,1.021][1.021,5.316] (a) −3 −2 −1 −3 −2 −1 Fr | < w u ⊥ > / < w θ > | [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (b) Figure 4. (Color online) (a):
Variation of mixing efficiency Γ f with Reynolds number, withbinning in Froude number (see insert). (b): Variation with
F r of the ratio of the vertical flux ofhorizontal velocity to the vertical flux of buoyancy, with binning in Rossby number. The size ofsymbols for both plots is described in the caption of Fig. 1. of R f with Richardson number mirrors its variation in terms of F r (not shown). Anotherexample of strong variation of Γ f is found in Bluteau et al. (2013) where Γ f = 0 . < R B < (cid:46) R B (cid:46) Γ f with F r and R B is similar to that found in the absence ofrotation but with shear (Mater & Venayagamoorthy 2014). These authors further notethat the centroid of such a curve depends on what flows are studied, as for example in thedata of Lozovatsky & Fernando (2013) where the centroid is shifted to higher R B . Thispresents a challenge, since parameterization schemes are mostly based on DNS, whichmay still be at too low a value of R B , and since using R B implies studying variations interms of both stratification (through F r ) and of turbulence (through Re ). The variationswith Reynolds and buoyancy Reynolds numbers are discussed further in § κ ρ , relative to the molecular diffusivity κ .It is proportional to the buoyancy flux B f . Taking the notation in Ivey et al. (2008): κ ρ = B f /N , κ ρ /κ = (cid:104) wθ (cid:105) / [ N κ ] ; (4.2) κ ρ is comparable to κ at low F r , and its increase with
F r is close to a linear variation,a least-square fit giving κ ρ /κ ≈ F r . . Finally, a saturation begins to occur at high F r .This behavior can be interpreted as being due to an increase in buoyancy flux because ofmore vigorous stirring when R B increases, at relatively constant Re . When comparing thehigh- F r values of κ ρ /κ to the model proposed in Barry et al. (2001), a rough agreement isobtained. Similarly, when examining a compilation of observational data for both salinityand temperature in the ocean together with numerical data for purely stratified flows,it is found in Bouffard & Boegman (2013) that a transition occurs in κ ρ , at R B ≈ R / B and a R / B scaling (see also Shih et al. (2005) for the latter). Thisenhancement of dissipation and of transport coefficients, such as anomalous diffusivity,is expected in turbulent flows as shown in numerous studies of FDT (Ishihara et al. et al. et al. et al. Γ f as a function of Reynolds number.2 A. Pouquet, D. Rosenberg, R. Marino and C. Herbert −3 −2 −1 N/f | N < w θ > | [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) −2 −1 N/f β = ε V / ε D (b) Figure 5. (Color online) Variation with
N/f = Ro/F r of the buoyancy flux B f = (cid:104) Nwθ (cid:105) ( a ),and of the dissipation efficiency β = (cid:15) V /(cid:15) D ( b ), with binning in Ro (see insert); symbols are asin Fig. 1. For a given N/f , both can take a large range of values.
Around Re ≈ , which covers many of the runs of Table 1, Γ f takes on a variety ofvalues corresponding to how strongly the flow is stratified, with variations in excess of1000. However, at lower Re (for most of the runs of Table 2), Γ f remains at lower values,the peak in Γ f being linked to the development of small-scale instabilities as Re and R B grow. Finally, another measure of the small-scale mixing efficiency of a flow is the ratio ofthe vertical fluxes of horizontal velocity to that of temperature fluctuations, as analyzedin Zilitinkevich et al. (2013). It is shown in Fig. 4(b) as a function of F r ; no clear scalingemerges although one could advocate a
F r − decrease for regimes II and III. Also, thisratio appears to be higher in the intermediate regime, on average.
5. The combined roles of rotation and stratification
The addition of rotation leads to the propagation of inertia-gravity waves whosedispersion relation depends on
N/f . Thus, the atmosphere, with
N/f ∼ N/f (cid:46)
10 may differ in their statistical properties. For all runs of this paper,
N/f (cid:62) .
5, so that stratification dominates. It is thus not surprising that the classicalpicture of mixing in stratified flows has not been changed in a significant way when weakrotation is included but in the absence of scale separation and of forcing. We do seean effect of rotation on the magnitude of the potential energy (see e.g.
Fig. 1a), strongrotation altering the large scales where the energy is contained. It was shown in Marino et al. (2015 a ) that rotation and stratification play complementary roles in the relativestrength of the direct and inverse constant-flux energy cascades in the forced case: thesmall-scale cascade is weaker the smaller the Froude number, and conversely the large-scale cascade is stronger the smaller Ro is, in both cases affecting the effective dissipationof energy in the small scales and thus, presumably, the mixing properties of such flows.For the flows of Table 1, all micro Rossby numbers, R ω = ω rms /f , are larger than 3 . Ro (cid:62) .
11. Thus, the small-scale vorticity created by the nonlinear dynamics of theflow, including in the presence of strong waves, is dominant at small scale, compared tothe imposed rotation; note that such values for R ω are plausible for geophysical flows.Figs. 5(a,b) give the variations with N/f of the buoyancy flux B f and of the dissipationefficiency β . At a given N/f , there is less buoyancy flux the higher the Rossby number,and for a given bin in Ro , B f is larger the higher N/f : as the Froude number increases, caling laws for mixing in unforced weakly rotating stratified turbulence − − N/f L E lli s on / L O z [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) −3 −2 −1 −4 −3 −2 −1 Fr L E lli s on / L i n t slope=1.04 (b) −3 −2 −1 −5 −4 −3 −2 −1 Fr L O z / L i n t slope=1.93 slope=1.48 (c) Figure 6. (Color online) Ellison scale L Ell = θ rms /N relative to the Ozmidov scale (cid:96) Oz ( a )or the integral scale L int ( b ), as a function of N/f ( a ), and of F r ( b) . In (c) is given (cid:96) Oz /L int as a function of F r . All plots have binning in Ro (and see caption of Fig. 1). Note the smalldispersion in the scaling of length scales versus Froude number (approximate scalings are given). the buoyancy flux decreases. The efficiency of dissipation is higher the higher the Rossbynumber (Fig. 5b), again at fixed N/f . Perhaps the scaling of β with F r is somewhatmodified by rotation, leading to the slightly sub-linear law observed in Fig. 1(c): theremay be less direct energy transfer for strong rotation, as is observed in the forced case(Marino et al. a ), and thus it takes a higher Froude number to reach a given level ofdissipation. A clear effect of the Rossby number on dissipation in RST was also shownin the forced case in Pouquet et al. (2017).Thus, the absence in the overall statistical properties of clear scaling in Ro simply showsthat the energy transfer to the small scales is dominated by a combination of stratificationand turbulence. However, the primary purpose of this study is not to examine the roleof rotation directly. Such a study would be helped by analyzing flows with N/f <
N/f >
1, the most important role of rotation in such flows is the triggering of an inversecascade of energy to the large scales, attenuating the energy transfer to the small scales(Pouquet & Marino 2013; Marino et al. a ; Pouquet et al.
6. The Ellison scale and a generalized mixing efficiency
Similarly to characteristic time scales, one can also examine characteristic length scales.A comparison of the Thorpe and Ellison scales in stratified turbulence was performedin Mater et al. (2013) (see also Dillon (1982)). The Thorpe scale L T corresponds to thevertical distance a parcel of fluid must be moved to produce a stable density profile,suppressing inversions, and as such gives an idea of the size of local mixing structures inthe fluid; it was computed as a function of R B for the purely stratified case in M´etais& Herring (1989). In Mater et al. (2013), L T is found to be strongly linearly correlatedwith the Ellison scale L Ell . Furthermore, the Thorpe length normalized by the Ozmidov4
A. Pouquet, D. Rosenberg, R. Marino and C. Herbert scale is found to vary as
F r − / for F r >
F r − / for F r <
1, the latter withan excellent scaling for 0 . (cid:54) F r (cid:54) .
3. Thus, L T ∼ L B for F r <
1, whereas L T ∼ L int for F r >
1. In the latter case, stratification is weak and structures in density follow theintegral length scale, whereas in the former case of strong stratification, density changesoccur over the vertical layer width, i.e. the buoyancy scale. Note that this also impliesthat, at small Froude number, L T /L int ∼ F r and that E P ∼ E V (Mater et al. L Ell normalized by the Ozmidov scale (a) andintegral scale (b), as a function of
N/f (a), and of
F r (b), with binning in Rossbynumber. We conclude that the Ellison scale is larger, the larger
F r is, as expected. As afunction of Froude number, L Ell /(cid:96) Oz decreases, in a linear fashion for the intermediateregime, and with little dispersion among the runs (not shown). One could argue thatwith (cid:96) Oz = [ (cid:15) V /N ] / , for small Froude number, (cid:96) Oz /L int ∼ F r in the intermediateregime in which β ∼ F r , whereas for high
F r , (cid:96) Oz /L int ∼ F r / , in rough agreementwith scaling laws, as indicated in Fig. 6(c) giving (cid:96) Oz /L int = f ( F r ) with least-squarefits of respectively 1 .
93 and 1 .
48, the transitions taking place for
F r ≈ .
01 and
F r ≈ . β (see Fig. 1c).On the other hand, the linear variation L Ell /L int ∼ F r in Fig. 6(b), with a least-square fit giving ∼ F r . , is a direct consequence of the scaling law θ rms ∼ u rms . Theonly transition in this power-law behavior takes place for F r = O (1), in which case L Ell (cid:46) L int , with L Ell remaining smaller than L int because of the 1 /k factor in thedefinition of L int . There is also an indication of a slight saturation at low F r .Another measure of the relative importance of terms in the Boussinesq equations isdefined through the ratio of the two dissipative terms for momentum and temperature,which can differ even at unit Prandtl number. We use the following definitions (see alsoOsborn (1980); Venayagamoorthy & Koseff (2016)): R ∗ f = (cid:15) P (cid:15) T , Γ ∗ f = R ∗ f − R ∗ f = (cid:15) P (cid:15) V , (6.1)with (cid:15) T = (cid:15) V + (cid:15) P already defined in equation (2.4). Γ ∗ f , called the irreversible mixingefficiency in Mater & Venayagamoorthy (2014), relates to the partition of energy betweenthe kinetic and potential modes, i.e. to the importance of the waves versus nonlineareddies at small scales. R ∗ f is shown in Fig. 7(a) as a function of Ri . In the first regimeof strong waves ( Ri > Ri regime in which the potential-kinetic exchanges are inefficient,leading to an abrupt decrease in R ∗ f . For intermediate values of Ri , almost from 10 − toroughly 10, R ∗ f stays rather constant in a range between 0 . .
4. Note also that thisdata is consistent with the variations of (cid:15) V and (cid:15) P with F r studied in Sozza et al. (2015)in a thin layer box.We finally examine in Fig. 7(b) the variations of Γ ∗ f with Rif /N = [ τ shear f ] , thuscombining the effects of stratification and rotation. A rather small variation of Γ ∗ ≈ . f . There is a slow decline forsmall Ro and large Ri which may be related to sensitivity to initial conditions at low F r :in regime I, (cid:15) V ≈ (cid:15) P is compatible with E P ≈ E V since, when the waves are strong tomoderate, there is little nonlinear transfer and the dissipation is mostly contained in thelarge scales. Also, the highest value of Γ ∗ f ≈ . F r ≈ .
01, corresponds obviously toflows with comparable kinetic and potential energy dissipation. This is likely associatedin that regime to strong waves and intermittent bursts which are due to wave breakingwhich temporarily relax the flow to a quasi-equipartition of kinetic and potential energiesacross a wide range of scales, the more so the smaller the scale, as observed for example caling laws for mixing in unforced weakly rotating stratified turbulence − − − − Ri R f * = ε P / ( ε P + ε V ) [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) −6 −4 −2 −2 −1 Ri f /N Γ f * = ε P / ε V [0.001,0.007][0.007,0.038][0.038,0.196][0.196,1.021][1.021,5.316] (b) Figure 7. (Color online) Potential energy dissipation normalized by total dissipation, R ∗ f = (cid:15) P /(cid:15) T , vs. Richardson number Ri ( a ), and Γ ∗ f = (cid:15) P /(cid:15) V vs. Rif /N = [ τ shear f ] (b) .Binning is performed in Ro in (a) and in F r in (b) (see inserts, and caption of Fig. 1). in Rosenberg et al. (2015). Our results corroborate those of Venayagamoorthy & Koseff(2016): different measures of mixing, such as Γ f or Γ ∗ f , give rather equivalent information,although it is not clear if this result will persist in the presence of forcing, at much higherbuoyancy Reynolds numbers.
7. Role of Reynolds number
The variation of the intensity of the turbulence in rotating stratified flows can bemeasured by the Reynolds number, as well as by the buoyancy Reynolds number. Havinghigh-enough Re and R B is known to be important for turbulent flows, to allow forcoherent structures to develop including in the presence of strong stratification (see e.g. Laval et al. (2003)). However, from a numerical point of view, having high
Re, R B forlow F r is quite challenging and remains a goal for the near future. When taking the datafor the runs of Tables 1 and 2 for a possible scaling of the dissipation efficiency β with R B , we observe some scatter (see Fig. 8a). Specifically, we see that in the intermediateregime ( R B between 10 and a few 100), at fixed R B , there is a measurable variationin β , by contrast to regimes I, and to a lesser extent regime III. This scatter in regimeII is larger than when examining variations with the Froude number itself, irrespectiveof the rotation (see Fig. 1(c)). In Fig. 8b, we see that overall, there is markedly lessscatter when plotting β as a function of the parameter [ N T L ] − as discussed in Mater &Venayagamoorthy (2014) (see the Appendix, § T L ≡ E V /(cid:15) V is the effective kineticenergy transfer time. Expressing (cid:15) V = β(cid:15) D , we see that N T L = [ βF r ] − ; thus, the choiceof the abscissa in Fig. 8(b) is to be able to make a direct comparison with R B whichscales as F r at fixed Re . We also note that the regime transitions in terms of Richardsonnumber occur for Ri ≈ . Ri ≈
10 (not shown).The difference in data points scatter between Fig. 8(a,b) can be attributed to thevariations with Reynolds number, as displayed in Figure 8(c). It shows clearly that theReynolds number alone does not allow for predicting the effectiveness of dissipation,and consequently that of mixing efficiency, with a wide scattering of data points for β ,irrespective of the initial conditions tested in this paper. However, we note that at agiven Re , QG initial conditions lead to a substantially lower dissipation efficiency. Wefurther note that Reynolds numbers are still quite low for these runs, when comparing6 A. Pouquet, D. Rosenberg, R. Marino and C. Herbert −2 −2 −1 Rb β = ε V / ε D [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) −10 −5 −2 −1 (NT L ) −2 β = ε V / ε D slope=0.25 (b) −2 −1 Re β = ε V / ε D (c) Figure 8. (Color online) Dissipation efficiency β as a function of (a) R B , (b) [ NT L ] − , and(c) Re (see § NT L = NE V /(cid:15) V ). Note the scaling β ∼ [ NT L ] − / in (b). with geophysical flows. Similar conclusions can be drawn for the variation with Re of theratio of the Ellison scale normalized by the integral scale (not shown).
8. Discussion, conclusion and perspectives
A parametric study of mildly rotating stratified turbulence without forcing leads to arather systematic quantitative assessment of its mixing and dissipative properties whichdepend on the Froude number provided the Reynolds number is high enough. Threedifferent regimes are observed, in agreement with previous studies of purely stratifiedflows. These regimes are also identifiable in terms of the interaction parameter R I .The three basic laws illustrated in Fig. 1 are compatible with an intermediate regimecharacterized by the dynamics of waves and eddies interacting nonlinearly weakly, eventhough the full weak turbulence formalism leading to a set of closed integro-differentialequations in terms of energy spectra cannot work for stratified flows because of a Froudenumber in the vertical of order unity (Billant & Chomaz 2001). It is thus somewhatremarkable that the simple phenomenology embodied in the parameter β ∼ F r , i.e. theefficiency of the turbulent dissipation, based on a ratio of characteristic time scales (seeequation (9.4)), may still apply on average for such flows.Together with θ rms ∼ u rms and a scaling for w/u ⊥ going as a quasi-constant atintermediate F r , these laws imply that the mixing efficiency Γ f ∼ F r − as soon as F r > .
01, and Γ f ∼ F r − ∼ R − / B for F r (cid:46) O (1). We emphasize that the actual values of thecontrol parameter for the change of regimes may depend on the geometry and topologyof such flows. In the intermediate regime, β ∼ F r , showing the connection betweenbuoyancy flux and nonlinear transfer leading to dissipation, with Γ f β ∼
1. Note that,with R I ∼ F r R B , this scaling law in the intermediate regime differs when expressed using R I . Finally the mixing efficiency measured in terms of the ratio of potential to kineticenergy dissipation, is shown to be rather constant. The scaling Γ f ∼ F r − ∼ R − / B ,in regime III at high F r and R B , simply stems from the decoupling of the velocity andtemperature, together with β ≈
1, leading to B f ∼ N .Note also that Γ f and R f seem to be more sensitive to parameters with a clear indi-cation of the three physical regimes in terms of F r , R B or R I , than either (cid:104) wu ⊥ (cid:105) / (cid:104) wθ (cid:105) or E P /E V . Furthermore, if w /u ⊥ ∼ F r as advocated in Maffioli & Davidson (2016),then the phenomenological arguments developed in our paper lead straightforwardly to caling laws for mixing in unforced weakly rotating stratified turbulence Γ f ∼ R − / B , which cannot be entirely ruled out given the scatter in data points for Γ f , although it is not compatible with the data of Fig. 1(b) with w rms /u rms ∼
1. Weverified that taking as initial conditions geostrophically balanced flows did not alter ourconclusions; similarly, having non-zero potential energy, but still unbalanced and with w (cid:54) = 0 ICs, we obtained the same fundamental scalings.Local variations in Richardson number may trigger local density micro-structures, asobserved in the ocean (Phillips 1972; Peltier & Caulfield 2003). If the agreement of ourresults, without shear but with rotation, with previous results mainly for sheared purelystratified flows is striking, it remains to be seen whether it will persist in the presence offorcing, i.e. in the presence of a strong inverse cascade. We note that Waite & Bartello(2006) already observed three regimes in the presence of forcing, with a switch for theenergy cascade from predominantly inverse to direct.Within the confines of the present parametric study with a wide range of buoyancyReynolds numbers, Γ f is in fact rather variable, as in the purely stratified case. Rotationis essential for the existence of a dual constant-flux cascade of energy, implying two-dimensional (horizontal) lateral mixing as well as vertical. If such mixing occurs inproportion to the ratio of the inverse to direct cascade, it will scale as [ RoF r ] − (Marino et al. a ). In the absence of forcing, with large-scale initial conditions and withrotation weaker than stratification, all effects associated with the presence of rotation areseverely quenched. As discussed in Mashayek & Peltier (2013), shear can induce vortexpairing at moderate Reynolds number, reinforcing the potential for an inverse cascadein the presence of rotation but, on the other hand, as Re increases, 3D instabilities takeover and shearing leads to enhanced dissipation.Another issue, when incorporating rotation or stratification, will be to consider therole of anisotropy on statistics, spectra and structures, the role of nonlocal interactionsamong scales, and the role of potential vorticity P V and the magnitude of its nonlinearpart; these will be the topic of future work. Several other extensions of the presentstudy are desirable. On the one hand, a larger scanning in terms of Reynolds numbers isneeded, but only feasible today at values comparable to or lower than what is presentlyachieved in this paper, without using modeling such as eddy viscosity or hyper-viscosity,or some other partial truncation of modes such as computing in boxes with small aspectratio. From the numerical standpoint, the condition (cid:96) Oz >> η, R I >> X ≈ et al. points with a TaylorReynolds number of 1300). In the context of the turbulent planetary boundary layer (see e.g. Sukoriansky et al. (2005)), one can write simplified expressions for vertical mixing,governed by vertical velocity, and horizontal mixing on the basis of a return to isotropymodel; this leads to agreement between these models, and laboratory and atmosphericdata. One can also model the time-evolution of a mixing event by following characteristiclength scales (Smyth et al. R I,B embodied in the Γ f ∼R − / B scaling, might imply the lessening of water mass motions in the ocean, by at leasta factor of 2 as found for the Antarctic Bottom Water (de Lavergne et al. et al. A. Pouquet, D. Rosenberg, R. Marino and C. Herbert propagating upward, in particular at mid latitudes where rotation plays a role, to scalesof the order of 1 km . This may lead to exchanges of light and dense waters and abyssalsinking (Ferrari et al. Acknowledgments: AP is thankful to LASP and Bob Ergun for support, and to ColmCaulfied for a useful discussion while at IPAM. RM acknowledges support from thePRESTIGE program coordinated by Campus France (co-financed under Marie CurieFP7 PCOFUND-GA-2013-609102) and the PALSE program at the University of Lyon.Computations were performed at the National Center for Atmospheric Research, throughan ASD allocation, as well as a new (2017) allocation of background time. NCAR issupported by the National Science Foundation. Finally, we also acknowledge requests bythe reviewers to perform more runs (see Table 2), as well as to simplify the text.
9. Appendices
Many parameters and characteristic time-scales and length scales have been definedin the literature for rotating stratified turbulence, and we regroup some of them here forcompleteness. They allow for the definition of slightly different dimensionless parametersfor which we also give an overview.9.1.
Appendix A: Characteristic time scales
The four global control parameters of the Boussinesq equations written in § Re = τ diss τ NL , Ro = τ wr τ NL , F r = τ wg τ NL , (9.1)with τ diss = L int /ν, τ NL = L int /u rms , τ wg = 1 /N and τ wr = 1 /f respectively thedissipation and eddy turn-over times, and the gravity and inertial wave periods; finally, P r = τ κ /τ diss with τ κ = L int /κ . The integral scale L int was defined in §
2. Whenlinearizing the Boussinesq equations, one obtains inertia-gravity modes of frequency ω k = ± (cid:113) N k ⊥ + f k (cid:107) /k , with k (cid:107) , ⊥ referring to the vertical and horizontal directions(see e.g. Bartello (1995)). However, for the sake of simplicity, we define the aboveparameters using isotropy, i.e. omitting [ k ⊥ , k (cid:107) ] factors which would appear throughthe dispersion relation. One can define an effective transfer time for the kinetic energy,measured directly from observational or numerical data, as: T L ≡ E V /(cid:15) V , (9.2)whereas τ NL = E V /(cid:15) D = βT L is based on a-priori large-scale characteristics of the flow,with β = (cid:15) V /(cid:15) D as defined in equation (2.5).In the absence of imposed shear, the Richardson number is based on a shear time τ shear built from the vertical gradient of the mean horizontal wind u ⊥ : τ shear = 1 / (cid:104) ∂ z u ⊥ (cid:105) , Ri = [ N τ shear ] . (9.3)As such, Ri can be viewed as measuring the strength, in terms of time-scales, of theformation of internal turbulent shear layers due to nonlinear interactions to that of thevertical layers due to the gravity waves, omitting the effect of rotation. caling laws for mixing in unforced weakly rotating stratified turbulence (cid:15) V , and its dimensional evaluation, (cid:15) D , through their ratio β = (cid:15) V /(cid:15) D .It is traditional in wave turbulence (Zakharov et al. a priori on dimensional grounds as: τ tr ≡ τ NL τ NL τ wg = τ NL F r , (9.4)using the small parameter adequate for the problem at small scale, here
F r <<
1. Thus, τ tr > τ NL , as expected. In the purely rotating case, one would use τ wr = τ NL Ro (Cambon& Jacquin 1989).It is then deduced that consistency between the two definitions of a transfer time,namely taking T L and τ tr to be proportional, immediately implies that one must have: β ∼ F r (9.5)in the intermediate range. This scaling, confirmed by numerical data (Fig. 1c), can beextended to
F r = 1; then, τ tr = τ NL : the energy is transferred to small scales in an eddyturn-over time, the hallmark of FDT.Note that a second characteristic time was also introduced in Mater & Venayagamoor-thy (2014), based again on (cid:15) V , and now on viscosity, namely: T λ = [ ν/(cid:15) V ] / = τ NL / [ βRe ] / . (9.6)The dependence of T λ on √ ν indicates that this time is linked to the Taylor scale λ V = u rms /ω rms = (cid:112) u rms ν/(cid:15) V , by writing λ V = u rms T λ . Other relevant scales are discussedin Barry et al. (2001); Davis & Monismith (2011); Mater & Venayagamoorthy (2014).9.2. Appendix B: A note on dimensional analysis
Control parameters can be defined using the characteristic times mentioned above: R g ≡ [ τ tr /T λ ] = u rms / [ ν(cid:15) V ] = β − Re , (9.7) F g ≡ τ w /τ tr = [ N T L ] − = βF r , (9.8) R I ≡ R g F g = (cid:15) V / [ νN ] = β R B , (9.9)with R B ≡ ReF r already defined in §
2. The difference between the two formulationsin terms of [
Re, F r ] and [ R g , F g ] is the appearance of the measured efficiency of energydissipation in turbulent flows interacting with waves in the latter case as opposed to apurely dimensional expression in the former case. F g and R g correspond to the choice ofdefinition of Froude and Reynolds numbers in Maffioli et al. (2016) in terms of (cid:15) V (with (cid:15) V / [ N u rms ] = F g ). Note that extending this second set of parameters to the rotatingcase, one will define Ro g as [ f τ tr ] − and thus N/f = Ro/F r = Ro g /F g will remain thesame in both formulations. It is not clear whether considering these different parametersand characteristic scales allows for a better assessment of these flows. For example, whentaking this second set of parameters, τ tr varies by a factor 20 within the confine of thedata base in Table 1, whereas, as noted in § u rms and L int , and thus τ NL , vary by ofthe order of a factor of 2.To illustrate this point, we examine in Fig. 9 the variation with [ N T L ] − = F +2 g of0 A. Pouquet, D. Rosenberg, R. Marino and C. Herbert − − − − (NT L ) − L E lli s on / L O z [ 0.0, 0.3][ 0.3, 3.0][ 3.0, 6.0][ 6.0,10.0][10.0, Inf] (a) −10 −5 −2 −1 (NT L ) −2 Γ f = R f / ( − R f ); R f = | B f | / ( | B f | + ε V ) (b) −10 −5 −2 −1 (NT L ) −2 | < w θ > | / κ N slope=0.45 (c) Figure 9. (Color online) Variation with [ NT L ] − = F g of L Ell /L Oz ( a ) and of Γ f ( b ), as wellas of the normalized eddy diffusivity κ ρ /κ ( c ), all with binning in Ro (symbols as in Fig. 1). L Ell /(cid:96) Oz (a), of Γ f (b), and of the normalized diffusivity κ ρ /κ (c), with κ ρ = (cid:104) wθ (cid:105) N . Thechoice of power of F g on the abscissa is to be able to compare with variations in R B , R I .The least-square fit for L Ell /(cid:96) Oz ≈ F − / g for regimes I and II is in agreement with thedata in Fig. 6, with in the intermediate regime (II), L Ell /(cid:96) Oz ≈ F r − , and F g ∼ F r .However, it extends now through two regimes, showing that the F g parameter allows tocross smoothly through these regimes, with only saturation when F g (cid:38) , F r (cid:38)
1, nowmoving into the third regime of stratified but strong turbulence.The two scalings that can be identified for the mixing efficiency Γ f are in agreementwith previous figures, in particular with Γ f ∼ F − g at small N T L , and Γ f ∼ F − g at large N T L , with a cross-over at F g ≈ − .Finally, the normalized eddy diffusivity again shows scaling behavior for F g sufficientlylarge. At lower F g , it saturates to values close to unity except for three data points whichare both at low Froude and low Rossby numbers, with Richardson numbers between5 and 10 and buoyancy Reynolds numbers close to 5: these flows are in a transitionalregime sensitive to fluctuations close to the threshold of instabilities.9.3. Appendix C: Derived dimensionless parameters
One can also define micro-Froude and micro-Rossby numbers, F ω and R ω , based onthe effective kinetic energy dissipation rate (cid:15) V , with ω rms the rms vorticity, ω = ∇ × u : F ω ≡ (cid:20) (cid:15) V νN (cid:21) / = ω rms N , R ω ≡ (cid:20) (cid:15) V νf (cid:21) / = ω rms f . (9.10)The runs of Table 1 have 11 . (cid:54) R ω (cid:54) R I ≡ F ω = (cid:15) V / [ νN ]is called any of: the buoyancy Reynolds number (Ivey et al. et al. et al. R B is what R I would be under the assumptionthat the turbulence has reached its full potential, and that the dissipation rate is equalto its dimensional expression, (cid:15) D . Thus, R I is an expression that is compatible with asmall-scale Kolmogorov isotropic energy spectrum, with R B and R I differing by a factor β , namely R B ≡ ReF r = (cid:15) D / [ νN ] = β − R I . In terms of ratio of characteristic length caling laws for mixing in unforced weakly rotating stratified turbulence E V ( k ) ∼ (cid:15) / V k − / , one can also write: R I = [ (cid:96) Oz /η ] / , R ω = [ (cid:96) Ze /η ] / = β / Re / Ro , (9.11)with as usual the dissipation and Ozmidov scales defined as η = 2 π [ (cid:15) V /ν ] − / , (cid:96) Oz =2 π (cid:112) (cid:15) V /N (see § (cid:96) Ze = 2 π (cid:112) (cid:15) V /f . It isthus clear that R I represents a dimensional estimate of the ratio of inertial to dissipativeforces for stratified turbulence. REFERENCESBarry, M., Ivey, G., Winters, K. & Imberger, J.
J. Fluid Mech. , 267–291.
Bartello, P.
J. Atmos. Sci. , 4410–4428. Billant, P. & Chomaz, J. M.
Phys.Fluids , 1645–1651. Bluteau, C. E., Jones, N. L. & Ivey, G. N.
J. Geophys. Res. , 1–11. van Bokhoven, L.J.A., Clercx, H.J.H., van Heijst, G.J.F. & Trieling, R.R.
Phys. Fluids , 096601. Bouffard, D. & Boegman, L.
Dyn. Atmosph. Oc. , 14–34. de Bruyn Kops, S.M.
J. Fluid Mech. , 436–463.
Cambon, C., Godeferd, F. S., Nicolleau, F. & Vassilicos, J. C.
J. Fluid Mech. , 231–255.
Cambon, C. & Jacquin, L.
J. Fluid Mech. , 295–317.
D’Asaro, E., Lee, C., Rainville, L., Harcourt, R. & Thomas, L.
Science , 318–322.
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B.
J. Fluid Mech. , 135–144.
Davis, K. A. & Monismith, S. G.
J. Phys. Oceano. , 2223–2241. de Lavergne, C., Madec, G., Le Sommier, J., Nurser, A. J. G. & Garabato, A.C. Naveira J. Phys. Ocean. , 663–681. Dillon, T.M.
J.Geophys. Res. , 9601–9613. Dimotakis, P.E.
Ann. Rev. Fluid Mech. , 329–356. Dritschel, D.G. & McKiver, W.J.
J . Fluid Mech. , 569–590.
Ferrari, R., Mashayek, A., McDougall, T.J., Nikurashin, M. & Campin, J.M.
J. Phys. Oceano. , 2239–2261. Finnigan, J.
Bound. Lay. Met. , 529–539. Fleury, M. & Lueck, R.G.
J. Phys.Ocean. , 801–818. van Haren, H., Cimatoribus, A. A., Cyr, F. & Gostiaux, L. Geophys. Res. Lett. (DOI:10.1002/2016GL068032), 1–7. Herbert, C., Marino, R., Pouquet, A. & Rosenberg, D.
J. FluidMech. , 165–204. A. Pouquet, D. Rosenberg, R. Marino and C. Herbert
Herring, J. R.
J. Atmos. Sci. ,969–977. Ishihara, T., Gotoh, T. & Kaneda, Y.
Ann. Rev. Fluid Mech. , 165–180. Ivey, G., Winters, K. & Koseff, J.
Ann. Rev. Fluid Mech. , 169–184. Iyer, K. P., Sreenivasan, K. R. & Yeung, P. K.
Phys. Rev. E , 021101(R). Karimpour, F. & Venayagamoorthy, S.K.
Phys. Fluids , 046603. Kimura, Y. & Herring, J.R.
J. Fluid Mech. , 253–269.
Klymak, J. M., Pinkel, R. & Rainville, L.
J. Phys. Oceano. , 380–399. Kurien, S. & Smith, L. M.
J. of Turb. , 241–271. Laval, J.-P., McWilliams, J. C. & Dubrulle, B.
Phys. Rev. E , 036308. Lelong, M-P. & Riley, J.J.
J. Fluid Mech. , 1–19.
Lelong, M-P. & Sundermeyer, M.
J. Phys. Oceano. ,2352–2367. Lindborg, E.
J. Fluid Mech. , 207–242.
Lindborg, E. & Brethouwer, G.
J. FluidMech. , 303–314.
Liu, H.L., Yudin, V. & Roble, R.
Geophys. Res. Lett. , 665–670. Lozovatsky, I.D. & Fernando, H.J.S.
Phil. Trans.A , 20120213.
Luketina, D. & Imberger, J.
J. Geophys. Res. , 12619–12636. Maffioli, A., Brethouwer, G. & Lindborg, E.
J . Fluid Mech. , R3.
Maffioli, A. & Davidson, P.A.
J. Fluid Mech. , 210–233.
Marino, R., Pouquet, A. & Rosenberg, D. a Resolving the paradox of oceanic large-scale balance and small-scale mixing.
Phys. Rev. Lett. , 114504.
Marino, R., Rosenberg, D., Herbert, C. & Pouquet, A. b Interplay of waves andeddies in rotating stratified turbulence and the link with kinetic-potential energy partition.
EuroPhys. Lett. , 49001.
Mashayek, A. & Peltier, W. R.
J. Fluid Mech. , 216 – 261.
Mashayek, A., Salehipour, H., Bouffard, D., Caulfield, C.P., Ferrari, R.,Nikurashin, M., Peltier, W.R. & Smyth, W.D.
Geophys. Res. Lett. , 6296–6306. Mater, B.D., Schaad, S.M. & Venayagamoorthy, S.K.
Phys. Fluids , 076604. Mater, B.D. & Venayagamoorthy, S.K.
Geophys. Res.Lett. , 4646–4653. McWilliams, J.
Proc. Roy. Soc. A , 2016.0117.
M´etais, O. & Herring, J.
J. Fluid Mech. , 117–148.
Mininni, P.D. & Pouquet, A.
Submitted to Phys. Rev. F, ArXiv 1706.10287 . caling laws for mixing in unforced weakly rotating stratified turbulence Mininni, P.D., Rosenberg, D. & Pouquet, A.
J. Fluid Mech. , 263–279.
Mininni, P.D., Rosenberg, D., Reddy, R. & Pouquet, A.
ParallelComputing , 316–326. Monin, A. S. & Yaglom, A. M.
Statistical Fluid Mechanics . MIT Press, Cambridge.
Osborn, T.R.
J. Phys. Oceano. , 83–89. Paoli, R., Thouron, O., Escobar, J., Picot, J. & Cariolle, D.
Atm. Chem. Phys. , 5037–5055. Patterson, M.D., Caulfield, C.P., McElwaine, J.N. & Dalziel, S.B.
Geophys. Res. Lett. , L15608. Peltier, W. & Caulfield, C.
Ann. Rev. FluidMech. , 135. Phillips, O.M.
Deep Sea Res. ,79–81. Pouquet, A. & Marino, R.
Phys. Rev. Lett. , 234501.
Pouquet, A., Marino, R., Mininni, P. D. & Rosenberg, D.
Phys. Fluids (111108). Praud, O., Sommeria, J. & Fincham, A.
J. Fluid Mech. , 389–412.
Pumir, A., Xu, H. & Siggia, E. D.
J. Fluid Mech. , 5–23.
Riley, J. J. & deBruynKops, S. M.
Phys. Fluids , 2047–2059. Rorai, C., Mininni, P.D. & Pouquet, A.
Phys. Rev. E , 043002. Rosenberg, D., Marino, R., Herbert, C. & Pouquet, A.
Eur.Phys. J. E , 8. Rosenberg, D., Marino, R., Herbert, C. & Pouquet, A.
Eur. Phys. J. E , 87. Rosenberg, D., Pouquet, A., Marino, R. & Mininni, P.D.
Phys. Fluids , 055105. Rubinstein, R., Clark, T. T. & Kurien, S.
Comp. Fluids , 108–114.
Salehipour, H. & Peltier, W.R.
J. Fluid Mech. , 464–500.
Shih, L., Koseff, J., Ivey, G. & Ferziger, J.
J. Fluid Mech. , 193–214.
Smyth, W.D., Moum, J.N. & Caldwell, D.R.
J. Phys.Oceano. , 1969–1992. Sozza, A., Boffetta, G., Muratore-Ginanneschi, P. & Musacchio, S.
Phys. Fluids ,035112. Stacey, M., Monismith, S. & Burau, J.
J. Phys. Oceano. , 1950–1970. Staquet, C. & Sommeria, J.
Ann. Rev. Fluid Mech. , 559. A. Pouquet, D. Rosenberg, R. Marino and C. Herbert
Stillinger, D., Helland, K. & van Atta, C.
J. Fluid Mech. , 91–122.
Stretch, D.D., Rottman, J., Venayagamoorthy, S.K., Nomura, K. & Rehmann, C. R.
Dyn. Atm. Oc. , 25–36. Sukoriansky, S., Galperin, B. & Staroselsky, I.
Phys. Fluids , 085107. Thorpe, S.A.
J. Geophys. Res. , 5231–5248. Venayagamoorthy, S.K. & Koseff, J.R.
J. Fluid Mech. , R1–R10.
Waite, M. & Bartello, P.
J.Fluid Mech. , 89–108.
Wells, M., Cenedese, C. & Caulfield, C.P.
J. Phys. Oceano. , 2713–2727. Zakharov, V. E., L’vov, V. S. & Falkovich, G.
Springer, Non-linear dynamics . Zilitinkevich, S.S., Elperin, T., Kleeorin, N., Rogachevskii, I. & Esau, I.
Bound.-Layer Meteorol. , 341–373.
Zilitinkevich, S. S., Elperin, T., Kleeorin, N., Rogachevskii, I., Esau, I., Mauritsen,T. & Miles, M. W.
Quart. J. Met. Roy. Soc.134