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  • 1.
    Augier, Pierre
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics.
    A New Formulation of the Spectral Energy Budget of the Atmosphere, with Application to Two High-Resolution General Circulation Models2013In: Journal of Atmospheric Sciences, ISSN 0022-4928, E-ISSN 1520-0469, Vol. 70, no 7, p. 2293-2308Article in journal (Refereed)
    Abstract [en]

    A new formulation of the spectral energy budget of kinetic and available potential energies of the atmosphere is derived, with spherical harmonics as base functions. Compared to previous formulations, there are three main improvements: (i) the topography is taken into account, (ii) the exact three-dimensional advection terms are considered, and (iii) the vertical flux is separated from the energy transfer between different spherical harmonics. Using this formulation, results from two different high-resolution GCMs are analyzed: the Atmospheric GCM for the Earth Simulator (AFES) T639L24 and the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecast System (IFS) T1279L91. The spectral fluxes show that the AFES, which reproduces quite realistic horizontal spectra with a k(-5/3) inertial range at the mesoscales, simulates a strong downscale energy cascade. In contrast, neither the k(-5/3) vertically integrated spectra nor the downscale energy cascade are produced by the ECMWF IFS.

  • 2.
    Augier, Pierre
    et al.
    LEGI, Université Grenoble Alpes.
    Mohanan, Ashwin Vishnu
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Shallow water wave turbulence2019In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 874, p. 1169-1196Article in journal (Refereed)
    Abstract [en]

    The dynamics of irrotational shallow water wave turbulence forced in large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the `four-fifths law' of Kolmogorov turbulence for a third order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks we develop a simple model predicting that the shock amplitude scales as (ϵd)1/3, where ϵ is the mean dissipation rate and d the mean distance between the shocks, and that the pth order displacement and velocity structure functions scale as (ϵd)p/3r/d, where r is the separation. Then we carry out a series of forced simulations with resolutions up to 76802, varying the Froude number, Ff=ϵ1/3/ckf1/3, where kf is the forcing wave number and c is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale as E∼√(ϵc/kf), and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson-Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as dFf1/2/kf. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, Ff→0, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given Ff, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

  • 3.
    Brethouwer, Geert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Billant, P.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Simulation of strongly stratified fluids2008Conference paper (Refereed)
    Abstract [en]

    Stably and strongly stratified turbulent flows have been studied by employing scaling analysis of the governing equations along the lines of [1], [2] and [3]. The scaling analysis suggests the existence of two different dynamical states. The parameter determining the state is R = ReF h 2, where Re and Fh are the Reynolds number and horizontal Froude number, respectively. If R≫1, viscous forces are negligible and the turbulence is strongly anisotropic but three-dimensional and causes a forward energy cascade. The vertical length scale lv scales as l v ∼ U/N (U is a horizontal velocity scale and N is the Brunt-Väisälä frequency). If R≪1, horizontal inertial forces are balanced by vertical viscous shearing and lv ∼ l hRe -1/2 (l h is a horizontal length scale). The scaling analysis has been confirmed by direct numerical simulations of homogeneous stratified turbulence. Spectra have been studied as well.

  • 4.
    Brethouwer, Geert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Billant, P.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Chomaz, J. M.
    Scaling analysis and simulation of strongly stratified turbulent flows2007In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 585, p. 343-368Article in journal (Refereed)
    Abstract [en]

    Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re >> 1 and horizontal Froude number F-h << 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter R = ReFh2. When R >> 1, viscous forces are unimportant and l(v) scales as l(v) similar to U/N (U is a characteristic horizontal velocity and N is the Brunt-Vaisala frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When R << 1, vertical viscous shearing is important so that l(v) similar to l(h)/Re-1/2 (l(h) is a characteristic horizontal length scale). The parameter R is further shown to be related to the buoyancy Reynolds number and proportional to (l(O)/eta)(4/3), where l(O) is the Ozmidov length scale and eta the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when R >> 1: the scales larger than l(O) are strongly influenced by the stratification while those between l(O) and eta are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and F-h, range and support the main parameter controlling strongly stratified turbulence being R. The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of R but they tend to be smooth for R < 1, while for R > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for R < 1 but tends to isotropy as R increases above unity. When R < 1, the horizontal and vertical energy spectra are very steep while, when R > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k(h)(-5/3)-power-law range and a clear forward energy cascade is observed.

  • 5.
    Brethouwer, Geert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Divergent and rotational modes in stratified flows2007In: ADVANCES IN TURBULENCE XI / [ed] Palma, JMLM; Lopes, AS, SPRINGER-VERLAG BERLIN: BERLIN , 2007, Vol. 117, p. 720-720Conference paper (Refereed)
  • 6.
    Brethouwer, Geert
    et al.
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Numerical study of vertical dispersion by stratified turbulence2009In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 631, p. 149-163Article in journal (Refereed)
    Abstract [en]

    Numerical simulations are carried Out to investigate vertical fluid particle dispersion in uniformly stratified stationary turbulent flows. The results are compared with the analysis of Lindborg & Brethouwer (J. Fluid Mech., vol. 614, 2008, pp. 303-314), who derived long- and short-time relations for the mean square vertical displacement of fluid particles. Several direct numerical simulations (DNSs) with different degrees of stratification and different buoyancy Reynolds numbers are carried out to test the long-time relation = 2 epsilon(P)t/N-2. Here, epsilon(P) is the mean dissipation of turbulent potential energy; N is the Brunt-Vaisala frequency; and t is time. The DNSs show good agreement with this relation, with a weak dependence on the buoyancy Reynolds number. Simulations with hyperviscosity are carried out to test the relation = (1 + pi C-PL)2 epsilon(P)t/N-2, which should be valid for shorter time scales in the range N-1 << t << T, where T is the turbulent eddy turnover time. The results of the hyperviscosity simulations come closer to this prediction with C-PL about 3 with increasing stratification. However, even in the simulation with the strongest stratification the growth of is somewhat slower than linear in this regime. Based on the simulation results it is argued that the time scale determining the evolution Of is the eddy turnover time, T, rather than the buoyancy time scale N-1, as suggested in previous studies. The simulation results are also consistent with the prediction of Lindborg & Brethouwer (2008) that the nearly flat plateau Of observed at t similar to T should scale as 4E(P)/N-2, where E-P is the mean turbulent potential energy.

  • 7.
    Brethouwer, Geert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Particle Diffusion in Stably Stratified Flows2010In: PROGRESS IN TURBULENCE III / [ed] Peinke, J.; Oberlack, M.; Talamelli, A., 2010, Vol. 131, p. 163-166Conference paper (Refereed)
    Abstract [en]

    Numerical simulations are used to study the vertical dispersion of fluid particles in homogeneous turbulent flows with a stable stratification. The results of direct numerical simulations are in good agreement with the relation for the long time fluid particle dispersion, = 2 epsilon(P)t / N-2, derived by [6], though with a small dependence on the buoyancy Reynolds number. Here, is the mean square vertical particle displacement, epsilon p is the dissipation of potential energy, t is time and N is the Brunt-Vaisala frequency. A simulation with hyperviscosicity is performed to verify the relation = (1 + pi C-PL)2 epsilon(P)t / N-2 for shorter times, also derived by [6]. The agreement is reasonable and we find that C-PL similar to 3. The onset of a plateau in is observed in the simulations at t similar to E-P / epsilon(P) which scales as 4E(P) / N-2, where E-P is the potential energy.

  • 8.
    Brethouwer, Geert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Passive scalars in stratified turbulence2008In: Geophysical Research Letters, ISSN 0094-8276, E-ISSN 1944-8007, Vol. 35, no 6Article in journal (Refereed)
    Abstract [en]

    Statistics of a passive scalar (or tracer) with a horizontal mean gradient in randomly forced and strongly stratified turbulence are investigated by numerical simulations. We observe that horizontal isotropy of the passive scalar spectrum is satisfied in the inertial range. The spectrum has the form E-theta(k(h)) = C-theta epsilon theta epsilon(-1/3)(K) k(h)(-5/3), where epsilon(theta), epsilon(K) are the dissipation of scalar variance and kinetic energy respectively, and C-theta similar or equal to 0.5 is a constant. This spectrum is consistent with atmospheric measurements in the mesoscale range with wavelengths 1 - 500 km. We also calculate the fourth-order passive scalar structure function and show that intermittency effects are present in stratified turbulence.

  • 9.
    Brethouwer, Gert
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Numerical simulations of particle dispersion in stratified flows2009In: ADVANCES IN TURBULENCE XII: PROCEEDINGS OF THE 12TH EUROMECH EUROPEAN TURBULENCE CONFERENCE / [ed] Eckhardt, B., 2009, Vol. 132, p. 51-55Conference paper (Refereed)
  • 10.
    Campagne, Antoine
    et al.
    LEGI, Université Grenoble Alpes.
    Alfredsson, Henrik
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Chassagne, Rémi
    LEGI, Université Grenoble Alpes.
    Micard, Diane
    LMFA, École Centrale de Lyon.
    Mordant, Nicolas
    LEGI, Université Grenoble Alpes.
    Segalini, Antonio
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Sommeria, Joel
    LEGI, Université Grenoble Alpes.
    Viboud, Samuel
    LEGI, Université Grenoble Alpes.
    Mohanan, Ashwin Vishnu
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mechanics.
    Augier, Pierre
    LEGI, Université Grenoble Alpes.
    First report of the MILESTONE experiment: strongly stratified turbulence and mixing efficiency in the Coriolis platform2016In: VIIIth International Symposium on Stratified Flows (ISSF), 2016, 2016Conference paper (Refereed)
    Abstract [en]

    Strongly stratified turbulence is a possible interpretation of oceanic and atmospheric mea-surements. However, this regime has never been produced in a laboratory experiment be-cause of the two conditions of very small horizontal Froude number Fh and large buoyancy Reynolds number R which require a verily large experimental facility. We present a new attempt to study strongly stratified turbulence experimentally in the Coriolis platform.The flow is forced by a slow periodic movement of an array of six vertical cylinders of 25 cm diameter with a mesh of 75 cm. Five cameras are used for 3D-2C scanned horizontalparticles image velocimetry (PIV) and stereo 2D vertical PIV. Five density-temperatureprobes are used to measure vertical and horizontal profiles and signals at fixed positions.The first preliminary results indicate that we manage to produce strongly stratified tur-bulence at very small Fh and large R in a laboratory experiment.

  • 11. Cho, J. Y. N.
    et al.
    Lindborg, Erik
    KTH, Superseded Departments, Mechanics.
    Horizontal velocity structure functions in the upper troposphere and lower stratosphere 1. Observations2001In: Journal of Geophysical Research-Atmospheres, ISSN 0747-7309, Vol. 106, no D10, p. 10223-10232Article in journal (Refereed)
    Abstract [en]

    We compute horizontal velocity structure functions using quasiglobal data accumulated by specially equipped commercial aircraft on 7630 flights from August 1994 to December 1997. Using the ozone concentration measurements, we classify the results as tropospheric or stratospheric. We further divide the results into four absolute latitude bands. For separation distance r between 10 and 100 km, the lower stratospheric diagonal third-order structure functions are proportional to negative r. This implies a downscale energy cascade, and we estimate the mean energy dissipation rate to be (E) approximate to 6 x 10(-5) m(2) s(-3). For r between 300 and 1500 km a positive r(3) dependence was visible for the polar stratospheric data. This may be the result of a two-dimensional (2D) turbulence downscale enstrophy cascade, and we estimate the average enstrophy flux to be IIomega approximate to 2 x 10(-15) s(-3) and the energy spectral constant to be K approximate to 2. The negative sign of these third-order functions at mesoscales in both the upper troposphere and lower stratosphere provide no support for an inverse energy cascade 2D turbulence. At scales above similar to 100 km, the second-order structure functions increase with latitude in the troposphere and decrease with latitude in the stratosphere. The off-diagonal third-order functions in the stratosphere show a remarkably clean negative r(2) dependency from 10 to 1000 km in scale.

  • 12.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Augier, Pierre
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Third order structure function in rotating and stratified turbulence: analytical and numerical results compared with data from the stratosphereManuscript (preprint) (Other academic)
  • 13.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Augier, Pierre
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Third-order structure functions in rotating and stratified turbulence: a comparison between numerical, analytical and observational results2014In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 755, p. 294-313Article in journal (Refereed)
    Abstract [en]

    First, we review analytical and observational studies on third-order structure functions including velocity and buoyancy increments in rotating and stratified turbulence and discuss how these functions can be used in order to estimate the flux of energy through different scales in a turbulent cascade. In particular, we suggest that the negative third-order velocity-temperature-temperature structure function that was measured by Lindborg & Cho (Phys. Rev. Lett., vol. 85, 2000, p. 5663) using stratospheric aircraft data may be used in order to estimate the downscale flux of available potential energy (APE) through the mesoscales. Then, we calculate third-order structure functions from idealized simulations of forced stratified and rotating turbulence and compare with mesoscale results from the lower stratosphere. In the range of scales with a downscale energy cascade of kinetic energy (KE) and APE we find that the third-order structure functions display a negative linear dependence on separation distance r, in agreement with observation and supporting the interpretation of the stratospheric data as evidence of a downscale energy cascade. The spectral flux of APE can be estimated from the relevant third-order structure function. However, while the sign of the spectral flux of KE is correctly predicted by using the longitudinal third-order structure functions, its magnitude is overestimated by a factor of two. We also evaluate the third-order velocity structure functions that are not parity invariant and therefore display a cyclonic-anticyclonic asymmetry. In agreement with the results from the stratosphere, we find that these functions have an approximate r(2)-dependence, with strong dominance of cyclonic motions.

  • 14.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. DAMTP, England.
    Boffetta, Guido
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Musacchio, Stefano
    Dimensional transition in rotating turbulence2014In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 90, no 2Article in journal (Refereed)
    Abstract [en]

    In this work we investigate, by means of direct numerical hyperviscous simulations, how rotation affects the bidimensionalization of a turbulent flow. We study a thin layer of fluid, forced by a two-dimensional forcing, within the framework of the "split cascade" in which the injected energy flows both to small scales (generating the direct cascade) and to large scale (to form the inverse cascade). It is shown that rotation reinforces the inverse cascade at the expense of the direct one, thus promoting bidimensionalization of the flow. This is achieved by a suppression of the enstrophy production at large scales. Nonetheless, we find that, in the range of rotation rates investigated, increasing the vertical size of the computational domain causes a reduction of the flux of the inverse cascade. Our results suggest that, even in rotating flows, the inverse cascade may eventually disappear when the vertical scale is sufficiently large with respect to the forcing scale. We also study how the split cascade and confinement influence the breaking of symmetry induced by rotation.

  • 15.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. Centre for Mathematical Sciences, Cambridge, England.
    Brethouwer, Geert
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Schlatter, Philipp
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    A numerical study of the unstratified and stratified Ekman layer2014In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 755, p. 672-704Article in journal (Refereed)
    Abstract [en]

    We study the turbulent Ekman layer at moderately high Reynolds number, 1600 < Re = delta(E)G/v < 3000, using direct numerical simulations (DNS). Here, delta(E) = root 2v/f is the laminar Ekman layer thickness, G the geostrophic wind, v the kinematic viscosity and f is the Coriolis parameter. We present results for both neutrally, moderately and strongly stably stratified conditions. For unstratified cases, large-scale roll-like structures extending from the outer region down to the wall are observed. These structures have a clear dominant frequency and could be related to periodic oscillations or instabilities developing near the low-level jet. We discuss the effect of stratification and Re on one-point and two-point statistics. In the strongly stratified Ekman layer we observe stable co-existing large-scale laminar and turbulent patches appearing in the form of inclined bands, similar to other wall-bounded flows. For weaker stratification, continuously sustained turbulence strongly affected by buoyancy is produced. We discuss the scaling of turbulent length scales, height of the Ekman layer, friction velocity, veering angle at the wall and heat flux. The boundary-layer thickness, the friction velocity and the veering angle depend on Lf/u(tau), where u(tau) is the friction velocity and L the Obukhov length scale, whereas the heat fluxes appear to scale with L+ = Lu-tau/v.

  • 16.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. Centre for Mathematical Sciences, Cambridge, England.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Helicity in the Ekman boundary layer2014In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 755, p. 654-671Article in journal (Refereed)
    Abstract [en]

    Helicity, which is defined as the scalar product of velocity and vorticity, H = u . omega, is an inviscidly conserved quantity in a barotropic fluid. Mean helicity is zero in flows that are parity invariant. System rotation breaks parity invariance and has therefore the potential of giving rise to non-zero mean helicity. In this paper we study the helicity dynamics in the incompressible Ekman boundary layer. Evolution equations for the mean field helicity and the mean turbulent helicity are derived and it is shown that pressure flux injects helicity at a rate 2 Omega G(2) over the total depth of the Ekman layer, where G is the geostrophic wind far from the wall and Omega = Omega e(y) is the rotation vector and e(y) is the wall-normal unit vector. Thus right-handed/left-handed helicity will be injected if Omega is positive/negative. We also show that in the uppermost part of the boundary layer there is a net helicity injection with opposite sign as compared with the totally integrated injection. Isotropic relations for the helicity dissipation and the helicity spectrum are derived and it is shown that it is sufficient to measure two transverse velocity components and use Taylor's hypothesis in the mean flow direction in order to measure the isotropic helicity spectrum. We compare the theoretical predictions with a direct numerical simulation of an Ekman boundary layer and confirm that there is a preference for right-handed helicity in the lower part of the Ekman layer and left-handed helicity in the uppermost part when Omega > 0. In the logarithmic range, the helicity dissipation conforms to isotropic relations. On the other hand, spectra show significant departures from isotropic conditions, suggesting that the Reynolds number considered in the study is not sufficiently large for isotropy to be valid in a wide range of scales. Our analytical and numerical results strongly suggest that there is a turbulent helicity cascade of right-handed helicity in the logarithmic range of the atmospheric boundary layer when Omega > 0, consistent with recent measurements by Koprov, Koprov, Ponomarev & Chkhetiani (Dokl. Phys., vol. 50, 2005, pp. 419-422). The isotropic relations which are derived may facilitate future measurements of the helicity spectrum in the atmospheric boundary layer as well as in controlled wind tunnel experiments.

  • 17.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Schlatter, Philipp
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Brethouwer, Gert
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Direct numerical simulations of stratified open channel flows2011In: 13th European Turbulence Conference (ETC13): Wall-Bounded Flows And Control Of Turbulence, 2011, p. 022009-Conference paper (Refereed)
    Abstract [en]

    We carry out numerical simulations of wall-bounded stably stratified flows. We mainly focus on how stratification affects the near-wall turbulence at moderate Reynolds numbers, i.e. Re-tau = 360. A set of fully-resolved open channel flow simulations is performed, where a stable stratification has been introduced through a negative heat flux at the lower wall. In agreement with previous studies, it is found that turbulence cannot be sustained for h/L values higher than 1.2, where L is the so-called Monin-Obukhov length and h is the height of the open channel. For smaller values, buoyancy does not re-laminarize the flow, but nevertheless affects the wall turbulence. Near-wall streaks are weakly affected by stratification, whereas the outer modes are increasingly damped as we move away from the wall. A decomposition of the wall-normal velocity is proposed in order to separate the gravity wave and turbulent flow fields. This method has been tested both for open channel and full channel flows. Gravity waves are likely to develop and to dominate close to the upper boundary (centerline for full channel). However, their intensity is weaker in the open channel, possibly due to the upper boundary condition. Moreover, the presence of internal gravity waves can also be deduced from a correlation analysis, which reveals (together with spanwise spectra) a narrowing of the outer structures as the stratification is increased.

  • 18.
    Deusebio, Enrico
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Vallgren, Andreas
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    The route to dissipation in strongly stratified and rotating flows2013In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 720, p. 66-103Article in journal (Refereed)
    Abstract [en]

    We investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, R0 and Fr, we show that the PEs can be obtained from the BQs by taking the limit Fr-2/R0(2)-> 0. When Fr-2/R0(2) is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from k(h)(-3) to k(h)(-5/3) h at the small-scale end. The vertical spectra show a similar transition as the horizontal spectra and we find that Charney isotropy is approximately valid also at larger wavenumbers than the transition wavenumber. The high resolutions employed allow us to capture both ranges within the same simulation. At the transition scale, kinetic energy in the rotational and in the horizontally divergent modes attain comparable values. The divergent energy is several orders of magnitude larger than the quasi-geostrophic divergent energy given by the Omega-equation. The amount of energy cascading downscale is mainly controlled by the rotation rate, with a weaker dependence on the stratification. A larger degree of stratification favours a downscale energy cascade. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. 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 involving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. Dividing the ageostrophic motions into two classes depending on the sign of the linear wave frequency, we show that the energy transfer is for the largest part supported by interactions within the same class, ruling out the wave-wave-vortex resonant triad interaction as a mean of the downscale energy transfer. The role of inertia-gravity waves is studied through analyses of time-frequency spectra of single Fourier modes. At large scales, distinct peaks at frequencies predicted for linear waves are observed, whereas at small scales no clear wave activity is observed. Triad interactions show a behaviour which is consistent with turbulent dynamics, with a large exchange of energy in triads with one small and two large comparable wavenumbers. The exchange of energy is mainly between the modes with two comparable wavenumbers.

  • 19.
    Li, Qiang
    et al.
    Stockholm Univ, Dept Phys Geog, Stockholm, Sweden..
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Weakly or Strongly Nonlinear Mesoscale Dynamics Close to the Tropopause?2018In: Journal of Atmospheric Sciences, ISSN 0022-4928, E-ISSN 1520-0469, Vol. 75, no 4, p. 1215-1229Article in journal (Refereed)
    Abstract [en]

    Recently, it has been discussed whether the mesoscale energy spectra in the upper troposphere and lower stratosphere are generated by weakly or strongly nonlinear dynamics. A necessary condition for weak non- linearity is that the Rossby number Ro vertical bar zeta(z)vertical bar/f << 1, where zeta(z) is the vertical vorticity and f is the Coriolis parameter. First, it is shown that Ro can be estimated by integration of the rotational wavenumber energy spectrum E-r. Then divergence and rotational energy spectra and their ratio, R E-d/E-r, are calculated from the Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) dataset, and it is shown that at least 1000 flight segments are needed to obtain converged results. It is found that R < 1 in the upper troposphere, ruling out the hypothesis that the spectra are produced by inertia-gravity waves with frequencies larger than f. In the lower stratosphere R is slightly larger than unity. An analysis separating between land and ocean data shows that E-d and temperature spectra have somewhat larger magnitude over land compared to ocean in the upper troposphere-a signature of orographically or convectively forced gravity waves. No such effect is seen in the lower stratosphere. At midlatitudes the Rossby number is on the order of unity and at low latitudes it is larger than unity, indicating that strong nonlinearities are prevalent. Also the temperature spectra, when converted into potential energy spectra, have larger magnitude than predicted by the weakly nonlinear wave hypothesis.

  • 20.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    A Helmholtz decomposition of structure functions and spectra calculated from aircraft data2015In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 762, p. R4-Article in journal (Refereed)
    Abstract [en]

    Longitudinal and transverse structure functions, D-ll = <delta u(l)delta u(l)> and D-tt = delta u(t)delta u(t), can be calculated from aircraft data. Here, d denotes the increment between two points separated by a distance r, u(l) and u(t) the velocity components parallel and perpendicular to the aircraft track respectively and < > an average. Assuming statistical axisymmetry and making a Helmholtz decomposition of the horizontal velocity, u = u(r) + u(d), where u(r) is the rotational and u(d) the divergent component of the velocity, we derive expressions relating the structure functions D-rr = delta u(r). delta u(r) and D-dd = delta u(d). delta u(d) to D-ll and D-tt. Corresponding expressions are also derived in spectral space. The decomposition is applied to structure functions calculated from aircraft data. In the lower stratosphere, D-rr and D-dd both show a nice r(2/3)-dependence for r epsilon [2, 20] km. In this range, the ratio between rotational and divergent energy is a little larger than unity, excluding gravity waves as the principal agent behind the observations. In the upper troposphere, D-rr and D-dd show no clean r(2/3)-dependence, although the overall slope of D-dd is close to 2/3 for r epsilon [2, 400] km. The ratio between rotational and divergent energy is approximately three for r < 100 km, excluding gravity waves also in this case. We argue that the possible errors in the decomposition at scales of the order of 10 km are marginal.

  • 21.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics.
    A note on acoustic turbulence2019In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 874, article id R2Article in journal (Refereed)
    Abstract [en]

    We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as (epsilon d)(1/3) and the pth-order structure function scales as (epsilon d)(p/3)3r/d, epsilon being the mean energy dissipation per unit mass, d the mean distance between the shocks and r the separation distance. However, for the acoustic field, epsilon should be replaced by epsilon + chi, where chi is associated with entropy production due to heat conduction. In particular, the third-order longitudinal structure function scales as <delta u(r)(3)> = -C(epsilon + chi)r, where C takes the value 12/5 (gamma + 1) in the weak shock limit, gamma = c(p)/c(v) being the ratio between the specific heats at constant pressure and constant volume.

  • 22.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Comment on "turbulence-condensate interaction in two dimensions"2009In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 102, no 14, p. 149401-Article in journal (Refereed)
    Abstract [en]

    A Comment on the Letter by H. Xia et al., Phys. Rev. Lett. 101, 194504 (2008). The authors of the Letter offer a Reply.

  • 23.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Horizontal wavenumber spectra of vertical vorticity and horizontal divergence in the upper troposphere and lower stratosphere2007In: Journal of Atmospheric Sciences, ISSN 0022-4928, E-ISSN 1520-0469, Vol. 64, no 3, p. 1017-1025Article in journal (Refereed)
    Abstract [en]

    The author shows that the horizontal two-point correlations of vertical vorticity and the associated vorticity wavenumber spectrum can be constructed from previously measured velocity structure functions in the upper troposphere and lower stratosphere. The spectrum has a minimum around k = 10(-2) cycles per kilometer (cpkm) corresponding to wavelengths of 100 km. For smaller wavenumbers it displays a k(-1) range and for higher wavenumbers, corresponding to mesoscale motions, it grows as k(1/3). The two-point correlation of horizontal divergence of horizontal velocity and the associated horizontal spectrum is also constructed. The horizontal divergence spectrum is of the same order of magnitude as the vorticity spectrum in the mesoscale range and show similar inertial range scaling. It is argued that these results show that the mesoscale motions are not dominated by internal gravity waves. Instead, the author suggests that the dynamic origin of the k(1/3) range is stratified turbulence. However, in contrast to Lilly, the author finds that stratified turbulence is not a phenomenon associated with an upscale energy cascade, but with a downscale energy cascade.

  • 24.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    The effect of rotation on the mesoscale energy cascade in the free atmosphere2005In: Geophysical Research Letters, ISSN 0094-8276, E-ISSN 1944-8007, Vol. 32, no 1Article in journal (Refereed)
    Abstract [en]

    A set of numerical simulations of the full Boussinesq equations including system rotation is carried out, in order to fully solve the problem of the origin of the k(-5/3) mesoscale wave number energy spectra measured in the free atmosphere. In a companion paper ( E. Lindborg, The energy cascade in a strongly stratified fluid, submitted to Journal of Fluid Mechanics, 2005, hereinafter referred to as Lindborg, submitted manuscript, 2005) it is shown that such a spectrum can arise as a result of a special type of forward energy cascade seen in the limit of strong stratification. In this letter we study the effect of rotation on such an energy cascade and show that it can prevail in the presence of system rotation as long as the Rossby number is above a critical value, which is determined to Ro(crit) approximate to 0.1.

  • 25.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    The energy cascade in a strongly stratified fluid2006In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 550, p. 207-242Article in journal (Refereed)
    Abstract [en]

    A cascade hypothesis for a strongly stratified fluid is developed on the basis of the Boussinesq equations. According to this hypothesis, kinetic and potential energy are transferred from large to small scales in a highly anisotropic turbulent cascade. A relation for the ratio, l(v)/l(h), between the vertical and horizontal length scale is derived, showing how this ratio decreases with increased stratification. Similarity expressions are formulated for the horizontal and vertical spectra of kinetic and potential energy. A series of box simulations of the Boussinesq equations are carried out and a good agreement between the proposed hypothesis and the simulations is seen. The simulations with strongest stratification give horizontal kinetic and potential energy spectra of the form EKh = C1 is an element of E-K(2/3) k(h)(-5/3) and E-Ph = C-2 is an element of(P)k(h)(-5/3)/is an element of(1/3)(k), where k(h) is the horizontal wavenumber, EK and ep are the dissipation of kinetic and potential energy, respectively, and C-1 and C-2 are two constants. Within the given numerical accuracy, it is found that these two constants have the same value: C-1 approximate to C-2 = 0.51 +/- 0.02.

  • 26.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Third-order structure function relations for quasi-geostrophic turbulence2007In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 572, p. 255-260Article in journal (Refereed)
    Abstract [en]

    We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

  • 27.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Two Comments on the Surface Quasigeostrophic Model for the Atmospheric Energy Spectrum2009In: Journal of Atmospheric Sciences, ISSN 0022-4928, E-ISSN 1520-0469, Vol. 66, no 4, p. 1069-1072Article in journal (Refereed)
    Abstract [en]

    The horizontal wavenumber spectra of wind and temperature in the upper troposphere and lower stratosphere display a narrow k(-3) range at scales on the order of 1000 km and a broad k(-5/3) range at mesoscales on the order of 1 to 500 km. Recently, Tulloch and Smith suggested that a surface quasigeostrophic (SQG) turbulence model can explain the observed spectra. Here, it is first argued that the mesoscale spectra are not likely to be explained by any quasigeostrophic model because the Rossby number corresponding to the mesoscale dynamics is on the order of unity or larger. Then it is argued that the SQG model in particular cannot explain the observations because its mesoscale spectrum displays a k(-5/3) dependence only in a very thin layer just below the tropopause. The thickness of this layer can be estimated to be of the order of 10 m, whereas aircraft measurements are typically performed several hundred meters away from the tropopause.

  • 28.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Vorticity and divergence spectra in the upper troposphere and lower stratosphere2007In: ADVANCES IN TURBULENCE XI / [ed] Palma, JMLM; Lopes, AS, BERLIN: SPRINGER-VERLAG BERLIN , 2007, Vol. 117, p. 585-587Conference paper (Refereed)
  • 29.
    Lindborg, Erik
    et al.
    KTH, Superseded Departments, Mechanics.
    Alvelius, K.
    The kinetic energy spectrum of the two-dimensional enstrophy turbulence cascade2000In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 12, no 5, p. 945-947Article in journal (Refereed)
    Abstract [en]

    A direct numerical simulation of forced two-dimensional turbulence with hyperviscosity is performed at resolution 4096(2). A stage is reached at which the flux of enstrophy from large to small scales is approximately constant in time. The cubic and quintic relations for the third-order velocity structure function derived by Lindborg [J. Fluid Mech. 388, 259 (1999)] are verified. The calculated kinetic energy spectrum in the constant enstrophy flux range has the form E(k)=K epsilon(omega)(2/3)k(-3), where epsilon(omega) is the enstrophy dissipation. This is in accordance with the prediction of Kraichnan [Phys. Fluids 10, 1417 (1970)] and Batchelor [Phys. Fluids 12, II233 (1969)]. The logarithmic correction, suggested by Kraichnan [J. Fluid Mech. 47, 525 (1970)], is not present in the calculated spectrum.

  • 30.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Brethouwer, Geert
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Stratified turbulence forced in rotational and divergent modes2007In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 586, p. 83-108Article in journal (Refereed)
    Abstract [en]

    We perform numerical box simulations of strongly stratified turbulence. The equations solved are the Boussinesq equations with constant Brunt-Vaisala frequency and forcing either in rotational or divergent modes, or, with another terminology, in vortical or wave modes. In both cases, we observe a forward energy cascade and inertial-range scaling of the horizontal kinetic and potential energy spectra. With forcing in rotational modes, there is approximate equipartition of kinetic energy between rotational and divergent modes in the inertial range. With forcing in divergent modes the results are sensitive to the vertical forcing wavenumber K-v(f) If k(v)(f) is sufficiently large the dynamics is very similar to the dynamics of the V V simulations which are forced in rotational modes, with approximate equipartition of kinetic energy in rotational and divergent modes in the inertial range. Frequency spectra of rotational, divergent and potential energy are calculated for individual Fourier modes. Waves are present at low horizontal wavenumbers corresponding to the largest scales in the boxes. In the inertial range, the frequency spectra exhibit no distinctive peaks in the internal wave frequency. In modes for which the vertical wavenumber is considerably larger than the horizontal wavenumber, the frequency spectra of rotational and divergent modes fall on top of each other. The simulation results indicate that the dynamics of rotational and divergent modes develop on the same time scale in stratified turbulence. We discuss the relevance of our results to atmospheric and oceanic dynamics. In particular, we review a number of observational reports indicating that stratified turbulence may be a prevalent dynamic process in the ocean at horizontal scales of the order of 10 or 100m up to several kilometres.

  • 31.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Brethouwer, Geert
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Vertical dispersion by stratified turbulence2008In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 614, p. 303-314Article in journal (Refereed)
    Abstract [en]

    We derive a relation for the growth of the mean square of vertical displacements, delta z, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times (delta z(2)) goes to a constant value 2(E-P(0) + aE(0))/N-2, where E-P(0) and E(0) are the initial mean potential and total turbulent energy per unit mass, respectively, a < 1 and N is the Brunt-Vaisala frequency. In the case of stationary turbulence, we find that (delta z(2)) = /N-2 + 2 epsilon(P)t/N-2, where epsilon(P) is the mean dissipation of turbulent potential energy per unit mass and is the Lagrangian structure function of normalized buoyancy fluctuations. The first term is the same as that obtained in the case of adiabatic fluid particle dispersion. This term goes to the finite limit 4E(P)/N-2 as t -> infinity. Assuming that the second term represents irreversible mixing, we show that the Osborn & Cox model for vertical diffusion is retained. In the case where the motion is dominated by a turbulent cascade with an eddy turnover time T >> N-1, rather than linear gravity waves, we suggest that there is a range of time scales, t, between N-1 and T, where = 2 pi C-PL epsilon(P)t, where C-PL is a constant of the order of unity. This means that for such motion the ratio between the adiabatic and the diabatic mean-square displacement is universal and equal to pi C-PL in this range. Comparing this result with observations, we make the estimate C-PL approximate to 3.

  • 32.
    Lindborg, Erik
    et al.
    KTH, Superseded Departments, Mechanics.
    Cho, J. Y. N.
    Determining the cascade of passive scalar variance in the lower stratosphere2000In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 85, no 26, p. 5663-5666Article in journal (Refereed)
    Abstract [en]

    Using aircraft data from 7630 commercial flights, we determine the flux of temperature and ozone variance from large to small scales in the lower stratosphere. The relation that we use for this purpose is a form of the classical Yaglom relation [A. M. Yaglom, Dokl. Akad. Nauk SSSR 69, 743 (1949)] for the third-order scalar-velocity structure function. We find that this function is negative and that it depends linearly on separation distance in the mesoscale range for temperature as well as ozone.

  • 33.
    Lindborg, Erik
    et al.
    KTH, Superseded Departments, Mechanics.
    Cho, J. Y. N.
    Horizontal velocity structure functions in the upper troposphere and lower stratosphere 2. Theoretical considerations2001In: Journal of Geophysical Research-Atmospheres, ISSN 0747-7309, Vol. 106, no D10, p. 10233-10241Article in journal (Refereed)
    Abstract [en]

    The Kolmogorov equation for the third-order velocity structure function is derived for atmospheric mesoscale motions on an f plane. A possible solution is a negative third-order structure function, varying linearly with separation distance and mean dissipation, just as in three-dimensional turbulence, but with another scaling constant. On the basis of the analysis and the observed stratospheric third-order structure function, it is argued that there is a forward energy cascade in the mesoscale range of atmospheric motions. The off-diagonal part of the general tenser equation is also studied. In this equation there is an explicit Coriolis term that may be crucial for the understanding of the kinetic energy spectrum at scales larger than 100km.

  • 34.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Fedina, Ekaterina
    Vertical turbulent diffusion in stably stratified flows2009In: Geophysical Research Letters, ISSN 0094-8276, E-ISSN 1944-8007, Vol. 36Article in journal (Refereed)
    Abstract [en]

    Based on a Lagrangian description of fluid particle dispersion we suggest that there is a single expression for the vertical eddy diffusivity for all scalars following fluid particles in stably stratified flows. This expression is the same as the Osborn-Cox diffusivity for buoyancy. To test this hypothesis we carry out turbulence simulations with stable background stratification by solving the Boussinesq equations with random forcing together with the equation for a passive scalar with an initial vertical Gauss profile. The development of the mean scalar concentration is studied for three different values of the width of the profile, sigma. It is found that the passive scalar diffuses in very good agreement with the classical diffusion equation if the ratio between s and a turbulent length scale is large enough. The associated eddy diffusivity agrees exactly with the Osborn-Cox diffusivity for buoyancy. Citation: Lindborg, E., and E. Fedina (2009), Vertical turbulent diffusion in stably stratified flows, Geophys. Res. Lett., 36, L01605, doi:10.1029/2008GL036437.

  • 35.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Mohanan, Ashwin Vishnu
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    A two-dimensional toy model for geophysical turbulence2017In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 29, no 11, article id 111114Article in journal (Refereed)
    Abstract [en]

    A toy model for large scale geophysical turbulence is constructed by making two modifications of the shallow water model. Unlike the shallow water model, the toy model has a quadratic expression for total energy, which is the sum of Available Potential Energy (APE) and Kinetic Energy (KE). More importantly, in contrast to the shallow water model, the toy model does not produce any shocks. Three numerical simulations with different forcing are presented and compared with the simulation of a full General Circulation Model (GCM). The energy which is injected cascades in a similar way as in the GCM. First, some of the energy is converted from APE to KE at large scales. The wave field then undergoes a forward energy cascade displaying shallow spectra, close to k−5/3, for both APE and KE, while the vortical field either displays a k−3-spectrum or a more shallow spectrum, close to k−5/3, depending on the forcing. In a simulation with medium forcing wave number, some of the energy which is converted from APE to KE undergoes an inverse energy cascade which is produced by nonlinear interactions only involving the rotational component of the velocity field. The inverse energy cascade builds up a vortical field at larger scales than the forcing scale. At these scales, coherent vortices emerge with a strong dominance of anticyclonic vortices. The relevance of the simulation results to the dynamics of the atmosphere is discussed as in possible continuations of the investigation.

  • 36.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Riley, James J.
    A condition on the average Richardson number for weak non-linearity of internal gravity waves2007In: Tellus. Series A, Dynamic meteorology and oceanography, ISSN 0280-6495, E-ISSN 1600-0870, Vol. 59, no 5, p. 781-784Article in journal (Refereed)
    Abstract [en]

    A condition on the average Richardson number, Ri, for weak non-linearity of an internal gravity wavefield is derived using a quasi-normal assumption. For weak non-linearity to be satisfied it is required that Ri(-1) << 0.5. This condition is very rarely satisfied in the ocean at vertical scales up to the order of 100 m, for which it is often found that Ri(-1) similar to 1. The analysis suggests that non-linear effects are of no less importance than linear effects in the dynamics of the interior of the ocean at these scales.

  • 37.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tung, K. K.
    Nastrom, G. D.
    N. Cho, J. Y.
    Gage, K. S.
    Comment on "reinterpreting aircraft measurement in anisotropic scaling turbulence" by Lovejoy et al. (2009)2010In: Atmospheric Chemistry And Physics, ISSN 1680-7316, E-ISSN 1680-7324, Vol. 10, no 3, p. 1401-1402Article in journal (Refereed)
    Abstract [en]

    Recently, Lovejoy et al. (2009) argued that the steep ∼k-3 atmospheric kinetic energy spectrum at synoptic scales (&amp;ge;1000 km) observed by aircraft is a spurious artefact of aircraft following isobars instead of isoheights. Without taking into account the earth's rotation they hypothesise that the horizontal atmospheric energy spectrum should scale as k-5/3 at all scales. We point out that the approximate k -3-spectrum at synoptic scales has been observed by a number of non-aircraft means since the 1960s and that general circulation models and other current models have successfully produced this spectrum. We also argue that the vertical movements of the aircraft are far too small to cause any strong effect on the measured spectrum at synoptic scales.

  • 38.
    Lindborg, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Vallgren, Andreas
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Testing Batchelor's similarity hypotheses for decaying two-dimensional turbulence2010In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 22, no 9, p. 091704-Article in journal (Refereed)
    Abstract [en]

    We carry out three high resolution direct numerical simulations of the two-dimensional Navier-Stokes equation to test Batchelor's similarity hypotheses of an equilibrium spectral range and an inertial subrange where the enstrophy wave number spectrum has the form Phi(k)=C chi(2/3)k(-1), where chi is the mean enstrophy dissipation rate and C is a constant. We use very different initial conditions in the three simulations and find that Batchelor's hypotheses are well satisfied in each simulation. However, there is a small but significant difference between the equilibrium range spectrum of one of the simulations as compared to the spectra of the other two. We suggest that the difference is linked to the stronger degree of large scale variation of the enstrophy dissipation which is observed in this simulation as compared to the other two.

  • 39.
    Maffioli, Andrea
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Brethouwer, Geert
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Mixing efficiency in stratified turbulence2016In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 794, article id R3Article in journal (Refereed)
    Abstract [en]

    We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number (Formula presented.), where (Formula presented.) is the kinetic energy dissipation, (Formula presented.) is a turbulent horizontal velocity scale and (Formula presented.) is the Brunt–Väisälä frequency. For (Formula presented.), in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as (Formula presented.), where (Formula presented.) and (Formula presented.) is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with (Formula presented.), we argue that (Formula presented.) should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of (Formula presented.) and confirm that at high (Formula presented.), (Formula presented.), while at low (Formula presented.) it approaches a constant value close to (Formula presented.). The parametrization of (Formula presented.) based on (Formula presented.) due to Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of (Formula presented.) in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value (Formula presented.) found in our simulations.

  • 40.
    Read, Peter L.
    et al.
    Univ Oxford, Dept Phys, Clarendon Lab, Atmospher Ocean & Planetary Phys, Oxford, England..
    Tabataba-Vakili, Fachreddin
    Univ Oxford, Dept Phys, Clarendon Lab, Atmospher Ocean & Planetary Phys, Oxford, England.;Jet Prop Lab, Pasadena, CA USA..
    Wang, Yixiong
    Univ Oxford, Dept Phys, Clarendon Lab, Atmospher Ocean & Planetary Phys, Oxford, England..
    Augier, Pierre
    LEGI, BP 53, F-38041 Grenoble 9, France..
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Valeanu, Alexandru
    Univ Oxford, Dept Phys, Clarendon Lab, Atmospher Ocean & Planetary Phys, Oxford, England..
    Young, Roland M. B.
    Univ Oxford, Dept Phys, Clarendon Lab, Atmospher Ocean & Planetary Phys, Oxford, England.;Sorbonne Univ, LMD, IPSL, Paris, France..
    Comparative terrestrial atmospheric circulation regimes in simplified global circulation models. Part II: Energy budgets and spectral transfers2018In: Quarterly Journal of the Royal Meteorological Society, ISSN 0035-9009, E-ISSN 1477-870X, Vol. 144, no 717, p. 2558-2576Article in journal (Refereed)
    Abstract [en]

    The energetics of possible global atmospheric circulation patterns in an Earth-like atmosphere are explored using a simplified global General Circulation Model (GCM) based on the University of Hamburg's Portable University Model for the Atmosphere (designated here as PUMA-S), forced by linear relaxation towards a prescribed temperature field and subject to Rayleigh surface drag and hyperdiffusive dissipation. Results from a series of simulations, obtained by varying planetary rotation rate with an imposed equator-to-pole temperature difference, were analysed to determine the structure and magnitude of the heat transport and other contributions to the energy budget for the time-averaged, equilibrated flow. These show clear trends with rotation rate, with the most intense Lorenz energy cycle for an Earth-sized planet occurring with a rotation rate around half that of the present-day Earth (i.e., =/(E)=1/2, where (E) is the rotation rate of the Earth). Kinetic energy (KE) and available potential energy (APE) spectra, E-K(n) and E-A(n) (where n is total spherical wavenumber), also show clear trends with rotation rate, with n(-3) enstrophy-dominated spectra around =1 and steeper (approximate to n(-5)) slopes in the zonal mean flow with little evidence for the n(-5/3) spectrum anticipated for an inverse KE cascade. Instead, both KE and APE spectra become almost flat at scales larger than the internal Rossby radius, L-d, and exhibit near-equipartition at high wavenumbers. At <<1, the spectrum becomes dominated by KE with E-K(n)approximate to(2-3)E-A(n) at most wavenumbers and a slope that tends towards n(-5/3) across most of the spectrum. Spectral flux calculations show that enstrophy and APE are almost always cascaded downscale, regardless of rotation rate. KE cascades are more complicated, however, with downscale transfers across almost all wavenumbers, dominated by horizontally divergent modes, for less than or similar to 1 / 4 . At higher rotation rates, transfers of KE become increasingly dominated by rotational (horizontally nondivergent) components with strong upscale transfers (dominated by eddy-zonal flow interactions) for scales larger than L-d and weaker downscale transfers for scales smaller than L-d.

  • 41. Riley, J. J.
    et al.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Recent progress in stratified turbulence2010In: Ten Chapters in Turbulence, Cambridge University Press, 2010, p. 269-317Chapter in book (Other academic)
    Abstract [en]

    Introduction. Stable density stratification can have a strong effect on fluid flows. For example, a stably-stratified fluid can support the propagation of internal waves. Also, at large enough horizontal scales, flow in a stably-stratified fluid will not have enough kinetic energy to overcome the potential energy needed to overturn; therefore flows at this horizontal scale and larger cannot overturn, greatly constraining the types of motions possible. Both of these effects were observed in laboratory experiments of wakes in stably-stratified fluids (see, e.g., (Lin and Pao, 1979)). In the wake experiments, generally the flow in the near wake of the source, e.g., a towed sphere or a towed grid, consisted of three-dimensional turbulence, little affected by the stable stratification. As the flow decayed, however, the effects of stratification became continually more important. After a few buoyancy periods, when the effects of stable stratification started to dominate, the flow had changed dramatically, and consisted of both internal waves and quasi-horizontal motions. Following Lilly (1983), we will call such motions, consisting of both internal waves and quasi-horizontal motions due to the domination of stable stratification, as “stratified turbulence”. It has become clear that such flows, while being strongly constrained by the stable stratification, have many of the features of turbulence, including being stochastic, strongly nonlinear, strongly dispersive, and strongly dissipative. A primary interest in stratified turbulence is how energy in such flows, strongly affected by stable stratification, is still effectively cascaded down to smaller scales and into three-dimensional turbulence, where it is ultimately dissipated.

  • 42. Riley, James J.
    et al.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics, Turbulence.
    Stratified turbulence: A possible interpretation of some geophysical turbulence measurements2008In: Journal of Atmospheric Sciences, ISSN 0022-4928, E-ISSN 1520-0469, Vol. 65, no 7, p. 2416-2424Article in journal (Refereed)
    Abstract [en]

    Several existing sets of smaller-scale ocean and atmospheric data appear to display Kolmogorov-Obukov-Corrsin inertial ranges in horizontal spectra for length scales up to at least a few hundred meters. It is argued here that these data are inconsistent with the assumptions for these inertial range theories. Instead, it is hypothesized that the dynamics of stratified turbulence explain these data. If valid, these dynamics may also explain the behavior of strongly stratified flows in similar dynamic ranges of other geophysical flows.

  • 43.
    Vallgren, Andreas
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Deusebio, Enrico
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Possible Explanation of the Atmospheric Kinetic and Potential Energy Spectra2011In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 107, no 26, p. 268501-Article in journal (Refereed)
    Abstract [en]

    We hypothesize that the observed wave number spectra of kinetic and potential energy in the atmosphere can be explained by assuming that there are two related cascade processes emanating from the same large-scale energy source, a downscale cascade of potential enstrophy, giving rise to the k(-3) spectrum at synoptic scales and a downscale energy cascade giving rise to the k(-5/3) spectrum at mesoscales. The amount of energy which is going into the downscale energy cascade is determined by the rate of system rotation, with negligible energy going downscale in the limit of very fast rotation. We present a set of simulations of a system with strong rotation and stratification, supporting these hypotheses and showing good agreement with observations.

  • 44.
    Vallgren, Andreas
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Charney isotropy and equipartition in quasi-geostrophic turbulence2010In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 656, p. 448-457Article in journal (Refereed)
    Abstract [en]

    High-resolution simulations of forced quasi-geostrophic (QG) turbulence reveal that Charney isotropy develops under a wide range of conditions, and constitutes a preferred state also in beta-plane and freely decaying turbulence. There is a clear analogy between two-dimensional and QG turbulence, with a direct enstrophy cascade that is governed by the prediction of Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525) and an inverse energy cascade following the classic k(-5/3) scaling. Furthermore, we find that Charney's prediction of equipartition between the potential and kinetic energy in each of the two horizontal velocity components is approximately fulfilled in the inertial ranges.

  • 45.
    Vallgren, Andreas
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindborg, Erik
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    The enstrophy cascade in forced two-dimensional turbulence2011In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 671, p. 168-183Article in journal (Refereed)
    Abstract [en]

    We carry out direct numerical simulations of two-dimensional turbulence with forcing at different wave numbers and resolutions up to 32768^2 gridpoints. In the absence of large scale drag, a state is reached where enstrophy is quasistationary while energy is growing. In the enstrophy cascade range the energy spectrum has the form $ E(k) = K \epsilon_{\omega} ^{2/3} k^{-3} $, without any logarithmic correction, where$ \epsilon_{\omega} $ is the enstrophy dissipation and K is of the order of unity. However, K is varying between different simulations and is thus not a perfect constant. This variation can be understood as a consequence of large-scale dissipation intermittency, following the argument by Landau (Landau \& Lifshitz 1959).  In the presence of a large scale drag, we obtain a slightly steeper spectrum. When forcing is applied at a scale which is somewhat smaller than the computational domain no vortices are formed and the statistics remain close to Gaussian in the enstrophy cascade range. When forcing is applied at a smaller scale, long lived coherent vortices form at larger scales  than the forcing scale and intermittency measures become very large at all scales, including the scales of the enstrophy cascade. We conclude that the enstrophy cascade with a $ k^{-3} $-spectrum, is a robust feature of the two-dimensional Navier-Stokes equations. However, there is a complete lack of universality of higher order statistics of vorticity increments in the enstrophy cascade range.

1 - 45 of 45
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