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Direct Numerical Simulations of geostrophic turbulence Enrico Deusebio Erik Lindborg Andreas Vallgren [email protected] [email protected] [email protected] Linn´ e FLOW Centre, KTH Mechanics, Stockholm, Sweden Swedish e-Science Research Centre (SeRC) Introduction Flows in the atmosphere and in the oceans develop over an extremely wide range of scales, both in time and space. At synoptic scales (∼ 1000 km), at which most of the energy is injected by baroclinic instabilities, the Earth’s rotation and the stable density stratification strongly affect the flow. On the other hand, at mesoscales (∼ 100 km) rotation becomes of secondary importance whereas stratification still influ- ences the dynamics. Since the Reynolds number of geophysical flows is gener- ally very large, they are turbulent. The study of turbulence in strongly rotating and stratified systems is therefore cru- cial in order to understand geophysical processes. How energy can cascade from the largest scale of baroclinic in- stability to the smallest scales where energy is dissipated is, at the present point, not understood and has been the subject of a number of recent studies. Numerical code We carry out numerical simulations of turbulence strongly affected by rotation and stratification. Very high numer- ical resolutions are needed in order to simulate transi- tions from a geosphysical turbulent dynamics to a strongly stratified dynamics within the same simulation. The Boussinesq system Du Dt = − ∇p ρ 0 − f e z × u + Nbδ i3 , (1a) Db Dt = −Nw, (1b) ∇· u =0 , (1c) is rescaled using geostrophic scaling (Charney 1971) and it is rewritten in terms of the Charney potential vorticity q and the two ageostrophic components a 1 and a 2 (Vallis 2006), q = − ∂u ∂y + ∂v ∂x + ∂b ∂z , (2a) a 1 = − ∂v ∂z + ∂b ∂x , (2b) a 2 = ∂u ∂z + ∂b ∂y . (2c) Here, δ i3 is the Kronecker’s delta, u is the velocity vector, p is the pressure, N is the Brunt-V ¨ ais¨ al¨ a frequency and f the Coriolis parameter. b = gρ/(Nρ 0 ) is the rescaled buoyancy, where ρ and ρ 0 are the fluctuating and background den- sities, respectively. The x,y are the horizontal coordinates whereas z represents the vertical coordinate. We consider a triply-periodic domain, such that Fourier representation of the variables is possible in all the three spatial directions. The box is cubic in a space where the vertical coordinate is stretched by a factor of N/f , accorid- ing to the geostrophic scaling. 10 2 10 3 10 -1 10 0 Number of processors speed-up Figure 1 Scaling of the code. Inverse of the time step versus the number of processors. Tests have been run on two SNIC parallel machines: Lindgren and Ekman. Resolutions: 512 3 and 512 3 . A pseudo-spectral method is used. Nonlinear terms, cal- culated in physical space, are advanced in time using a low-storage fourth-order Runge-Kutta scheme. In order to prevent aliasing errors, the 2/3-dealiasing rule is ap- plied. Linear terms are solved separately using an exact implicit procedure. The code has been heavily parallelized by means of Message Passage Interface (MPI) protocols in order to be run on distributed memory parallel machines. The code has been benchmarked on up to 4096 processors (figure 1). We carry out a set of high-resolution numerical simula- tions on 1024 3 grids. A random forcing is introduced at the largest scales. Rotation and stratification are imposed and they are quantified by the Rossby number Ro = U/fL and the Froude number Fr = U/NL. Energy spectra and fluxes In figure 2a, the three-dimensional total energy spectrum, which is the sum of potential and kinetic energy, is shown for several Ro. At Ro =0, QG dynamics with a spectrum of the form k −3 is obtained. As Ro increases, departures are observed at small scales at which a k −5/3 spectrum is found. Such a transition is consistent with measurements in the atmosphere (Nastrom & Gage 1985). Such agree- ment is further supported by analises of third order struc- ture functions (Vallgren et al. 2011). In figure 2b, the total spectral energy flux and the potential enstrophy flux are shown. Unlike QG dynamics, we find that for finite rotation rates only a fraction of the injected energy goes into an upscale cascade. The remaining part cascades instead towards small scales and increases with increasing Ro. As Ro increases, a larger amount of the in- jected energy cascades towards small scales. At small Ro, a constant energy and enstrophy flux range coexist within the same range of scales. Further stud- ies show that in this range, the enstrophy flux is a result of triad interactions involving three geostrophic modes, while the energy flux is a result of triad interactions in- volving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. k h E(k h ) 10 0 10 1 10 2 10 -8 10 -6 10 -4 10 -2 10 0 k Π η /η, Π E /P 10 1 10 2 10 -2 10 0 Figure 2 a) Total energy spectra for Ro =0.2, Ro =0.1 and Ro =0.2. b) Energy ( ) and enstrophy ( ) fluxes made dimensionless with the energy and enstrophy injection rates. Ro =0.1 and Ro =0.05. a) b) Flow fields In figure 3, a horizontal cut of the potential vorticity flow field q is shown for a run with Ro =0.1 and Fr =0.01. The overall dynamics at large scales resemble QG dynam- ics with a large scale vortex surrounded by small-scale fil- aments. Patches of small scale turbulence can also be ob- served all over the flow. A particularly intense small-scale chaotic region is found in the top left corner. In such a re- gion, the intensity of ageostrophic motions increases about one order of magnitude. The inset of figure 3 shows a close-up of the local vertical Froude number Fr L = ω 2 x + ω 2 y /2N , where ω i is the i−th component of the vorticity, in the turbulent patch of fig- ure 3. Fr L shows small-scale structures which follow very similar patterns and attains values on the order of unity. This suggests the possible presence of Kelvin-Helmotz in- stability and of a stratified turbulent dynamics. Figure 3 Snapshot of the instanteneous potential vorticity q flow field. In the close-up, the local Froude number Fr L is depicted. The color map for Fr L ranges from 0 to 2. Conclusion We have carried out a set of high-resolution numerical simulations on 1024 3 grids, aiming at investigating the en- ergy transfer in stratified and rotating turbulence. The un- derstanding of turbulent processes in such conditions is crucial in a geophysical perspective both from a funda- mental and a practical point of view. At large scales, ageostrophic dynamics superimpose onto the dominant geostrophic dynamics and result in a leak- age of energy towards small scales. At small-scales, ageostrophic motions become more important and lead to a shallowing of the energy spectrum from k −3 to k −5/3 , consistent with observations (Nastrom & Gage 1985). The amount of energy cascading downscale increases with de- creasing rate of rotation. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. Computer time provided by SNIC (Swedish National Infrastruc- ture for Computing) is gratefully acknowledged. References [1] Nastrom G. D. & Gage K. S. A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42, 950–960, 1985. [2] Charney J. G. Geostrophic turbulence. J. Atmos. Sci. 28 (6), 1087–1094, 1971. [3] Vallis G. K. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Leiden: Cambridge Univ. Press, 2006. [4] Vallgren A., Deusebio E. & Lindborg E. A possible explanation of the atmospheric kinetic and potential energy spectra. Phys. Rev. Lett. 99, 99–101, 2011.
