Inertial Range Dynamics in Density-Stratified Turbulent Flows

Transcription

1 Inertial Range Dynamics in Density-Stratified Turbulent Flows James J. Riley University of Washington Collaborators: Steve debruynkops (UMass) Kraig Winters (Scripps IO) Erik Lindborg (KTH) Workshop on Inertial Range Dynamics and Mixing Isaac Newton Institute for Mathematical Sciences Cambridge, UK 29 September 3 October 2008

2 Examples of Instabilities Atmosphere Inertial Range Dynamics and Mixing P. Atsavapranee and M. Gharib, J. Fluid Mech., 342, 1997

3 Examples of Instabilities Atmosphere (cont d) P. G. Drazin and W. H. Reid, Hydrodynamic Stability, 1981

4 Examples of Instabilities Ocean Inertial Range Dynamics and Mixing J. D. Woods, J. Fluid Mech., 32, 1968

5 Examples of Instabilities Laboratory Inertial Range Dynamics and Mixing S. A. Thorpe, J. Fluid Mech., 46, 1971

6 Research Focus dynamics of larger-scale motions that lead to instabilities and turbulence horizontal scales of interest typically, in the ocean on horizontal scales from a few meters to several hundred meters (potentially up to 10 km) in strongly stable atmosphere boundary layers at horizontal scales from a few meters to several hundred meters (or more) in the upper troposphere, stratosphere at horizontal scales from tens of meters to kilometers (potentially up to 100 km)

7 Description of Motions Some general characteristics strongly affected by stable density stratification high Reynolds number (compared to laboratory experiments) usually there is a significant internal wave component classical, smaller-scale 3D turbulence is very intermittent, sporadic often horizontally meandering motions are observed e.g., plumes in stable atmospheric boundary layers larger-scale component velocities are highly non-isotropic larger-scale motions do not overturn Termed Stratified Turbulence (Lilly, 1983) as distinguished from classical 3D turbulence

8 Stratified Turbulence Controlling parameters Reynolds number: R l = u l H /ν u characteristic rms velocity l H horizontal scale of energy-containing motions Froude number: F l = u /Nl H T B /T F M N buoyancy frequency, N = g d ρ ρ o dz /[( ) 2 u Gradient Richardson Number: Ri = N 2 + z Stratified turbulence: F l O(1), R l 1 Ro = u /Ωl H 1 no effect of rotation (in this study) ( ) 2 ] v z

9 Scaling Arguments Turbulent flows are expected to exist, locally, when /[( ) 2 u Ri = N 2 + z ( ) 2 ] v z < O(1) This can be shown to imply that R b = F 2 l R l ɛ/νn 2 > O(1) buoyancy Reynolds number from oceanographic measurements, laboratory data, numerical simulations turbulence is found to exist when R b > O(10)

10 Stratified Turbulence (cont d) Inertial Range Dynamics and Mixing 1,000, ,000 Stratified Turbulence Kolmogorov turbulence R l 10,000 1,000 lab, numerical simulations F 2 l R l = ε/ν N2 = constant laboratory turbulence laminar? turbulent 100 laminar ,000 10,000 F l

11 Theoretical Arguments Stratified Turbulence Lilly (1983) used scaling arguments to suggest, for F l < O(1): flows in adjacent horizontal layers are somewhat decoupled leads to increasing vertical shearing of horizontal flow and to decreasing Richardson numbers Billant and Chomaz (1999) induced velocities lead to strong vertical inhomogeneities and layering Even though strong, stable stratification, i.e., F l < O(1), at high Reynolds numbers, both mechanisms lead to smaller vertical scales continually developing local instabilities and classical 3D turbulence intermittently occurring

12 Three-dimensional contour plots of the stream function for the case with F l = 0.6, R l = 3200 at t = 0 (left) and t = 15 (right).

13 Volume-Averaged Gradient Richardson Number versus t 10 8 F4R8 F4R16 F4R32 F4R64 6 Ri V t Ri V = ( u z N 2 ) 2 + ( ) 2 v z

14

15 Horizontal Kinetic Energy Spectra Inertial Range Dynamics and Mixing

16 Ozmidov Scale Inertial Range Dynamics and Mixing

17 Scaled Horizontal Energy Spectra Inertial Range Dynamics and Mixing Scaled horizontal energy spectra, Lindborg (2005).

18 Horizontal Kinetic Energy Spectra Lindborg Scaling Horizontal kinetic energy spectra at t = 18.5 for R l = 9600 case.

19 Scaling Arguments Assume F l 1 (strong stratification) implies l O /l H F 3/2 l 1 Fl 2R l > R B crit 40 implies η/l O < R b crit 3/4 < 0.25 ɛ u 3 /l H, independent of N, ν ( or F l, R l ) Implications for horizontal spectra E = E(k H, ɛ), so, analogous to Kolmogorov, E = Cɛ 2/3 k 5/3 H

20 Evidence for Stratified Turbulence Inertial Range Numerical simulations (all with constant N) Riley & debruynkops, DNS, no shear decaying flow Lindborg, LES, no shear, forced flow Diamessis, DNS, wake of sphere Field measurements Ocean Klymak & Moum, off Hawaiian ridge, shear, temperature, ɛ Ewart, various locations, temperature Hollbrook & Fer, displacement spectra Bouruet-Aubertot, van Haren & Lelong, temperature Atmosphere Frehlich, very stable atmospheric boundary layer velocity, temperature

21 Shear Spectra Ocean (Klymak, 2005) Inertial Range Dynamics and Mixing

22 Shear Spectra Ocean (Klymak, 2005) Inertial Range Dynamics and Mixing 1/3

23 Temperature spectra Ocean (Ewart, 1976) Power spectra of temperature off the coast of San Diego (30 o N, 124 o W).

24 Displacement spectra Ocean (Hollbrook & Fer, 2005 ) Vertical displacement spectra from open ocean (squares) and near slope (dots)

25 Temperature spectra Ocean Inertial Range Dynamics and Mixing Bouruet-Aubertot, van Haren & Lelong, 2008

26 Temperature Spectrum Atmosphere (Frehlich et al., 2007 ) Temperature frequency (horizontal wave number) spectrum in very stable atmospheric boundary layer

27 Velocity Spectrum Atmosphere (Frehlich et al., 2007 ) Velocity frequency (horizontal wave number) spectrum in very stable atmospheric boundary layer

28 Conclusions Stratified turbulence F l O(1), R l 1, R b O(10) e.g., at oceanic horizontal scales larger than a few meters strong tendency for vertical shearing of horizontal motion leads to intermittent, 3D turbulence intermittent occurence in space leads to a strong downscale transfer of energy potential for stratified turbulence inertial cascade highly nonisotropic inertial subrange possible explanation of field data replaces notion of a buoyancy subrange

29 Some Open Issues Dynamics in inertial range? E.g., except is 3-D turbulent regions, vortex lines are mainly horizontal Role of internal waves, especially in the ocean Are the density perturbations slaved to velocity? How does energy cascade down from the scales dominated by rotation? Is there both upscale and downscale propagation of energy? What are the vertical spectra? Does horizontal particle separation goes as: D 2 = Cɛt 3? Does shear dispersion dominate?

30

31

32

33

34 Mean Square Patch Size Ocean Okubo, Deep-Sea Research, 1971 Inertial Range Dynamics and Mixing

35 Spectra of Available Potential Energy Ocean Spectra of available potential energy in horizontal (Dugan et al., 1986)

36 Temperature Structure Function Ocean Voorhis and Perkins, Deep-Sea Research, 1966

37 Temperature Spectrum Ocean Inertial Range Dynamics and Mixing Lafond and Lafond, 1967, Marine Technical Society