Potential Enstrophy in Stratified Turbulence

Transcription

1 Potential Enstrophy in Stratified Turbulence Michael Waite University of Waterloo 11 June 2013 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

2 Introduction Introduction Potential enstrophy integrated squared potential vorticity: V = 1 2 q2 neglecting forcing & dissipation: Dq/Dt = 0, V is conserved V -conservation important in QG turbulence (enstrophy cascade, inverse energy cascade) what happens at larger Ro atmospheric mesoscale & oceanic sub-mesoscale? Stratified turbulence homogeneous turbulence in stratified fluid with weak or no rotation model for geophysical turbulence at small-scale end of atmos meso and ocean sub-meso connects large-scale QG turbulence with small-scale isotropic turbulence waves, vortical modes, thin shear layers, K-H (reviews: Riley & Lelong 2000; Riley & Lindborg 2013) M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

3 Introduction Potential vorticity & enstrophy Ertel PV for Boussinesq fluid: q = (f ẑ + ω) (N 2 ẑ + b) = q 0 + q 1 + q 2, where q 0 = fn 2, q 1 = N 2 ω z + f zb, q 2 = ω b f = Coriolis, N = Brunt Väisälä freq, b =buoyancy q is quadratic in ω and b, so V = V 2 + V 3 + V 4 is a quartic invariant. no detailed conservation of V by wavenumber triads weird: viscosity & diffusion are not strictly dissipative (Herring, Kerr, Rotunno 1994) Dq Dt ( ) ( ) ( ) = N 2 ẑ + b ν 2 ω + κ (f ẑ + ω) 2 b M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

4 Introduction Potential vorticity & enstrophy But under certain conditions, is V approximately quadratic? (i.e. q linear?) yes, for QG turbulence what about for large Ro? Kurien, Smith & Wingate (2006), Aluie & Kurien (2011): V is quadratic for stratified turbulence how generic is this result? 4.5 x RS rs Rs time t Aluie & Kurien, EPL 96, 44006, 2011 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

5 So what? Introduction Cascade theories: quadratic V triad-by-triad conservation, like kinetic energy relationship between energy and p. enstrophy: e.g. for f = 0 have V (k) = N 2 k 2 h E R(k) joint conservation constrains cascade as in 2D, QG: inverse cascade? (Lilly 1983) Decomposition into waves and vortices: linear decomposition into vortical modes (with q 1 ) and gravity waves (no q 1 ) e.g. stratified turbulence (Lelong & Riley 1991), rotating-stratified turbulence (Bartello 1995) motivates decomp of KE spectra into horizontally rotational ( vortical) and divergent ( wave) popular/easy decomposition, but meaningless if higher-order V terms important M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

6 Introduction Scale analysis of potential vorticity Usual scaling of terms (Lilly 1983, Riley & Lelong 2000) gives: q 1 = N 2 ω z + f zb N 2 U ( ) ) max 1, Frv 2 /Ro (assuming b U 2 /L v, L h q 2 = ω b U3 L h L 2, v q 2 /q 1 min ( Frv 2, Ro ), where Fr v = U/NL v, Ro = U/fL h For strong rotation, q 2 /q 1 Ro 1, so V V 2 is quadratic For weak rotation q 2 /q 1 Frv 2 (W & Bartello 2006). How big is Frv? M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

7 Equations of motion Introduction Incompressible, Boussinesq, constant N Non-dimensionalize (e.g. Riley et al. 1981, Lilly 1983): Fr h U, Fr v U, NL h NL v Re UL h ν, α Lv L h Fr h Fr v. u t + u u + Frv 2 w u z + 1 Ro ẑ u = p + 1 Re Fr 2 h ( w t b t + u w + Fr 2 v w w z u + Fr 2 v ) w z = p z + b + Fr 2 h Re = 0, + u b + Fr 2 v w b z + w = 1 Re ( ) α 2 z 2 u, ( ) α 2 z 2 w, ( ) α 2 z 2 b. stratified turbulence means Fr h 1, Re 1. What about Fr v? Fr v 1 quasi-2d, Fr v O(1) anisotropic 3D M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

8 Introduction Vertical scales in geophysical turbulence Size of Fr v depends on L v : in QG turbulence, L v /L h f /N Fr v Ro 1 In stratified turbulence L v U/N Fr v 1 (e.g. Billant & Chomaz 2001) U/N = L b buoyancy scale, pancake thickness (W & Bartello 04) at which Ri O(1). but, need to be careful: assumes large Reynolds number Re = UL h /ν If Re not large enough, L v is set by viscosity: depends on buoyancy Reynolds number Re b = ReFr 2 h (e.g. Smyth & Moum 2000) turbulence requires large Re b (Riley & de Bruyn Kops 2003, Brethouwer et al 2007) for Re b 1, viscous effects small and Fr v 1 for Re b 1, viscous effects important and Fr v Re 1/2 b Suggests that quadratic potential enstrophy may only be realized for Re b 1 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

9 Introduction Vertical scales in geophysical turbulence Rotating Stratified: L v N/U vs Ro Stratified only: L v N/U vs Re b N = (b) 100 1/Ro u L v f /NL h L v U/N l v N/u R Brethouwer et al., J. Fluid Mech. 585, , Ro W & Bartello, J. Fluid Mech. 568, , 2006 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

10 Introduction Geophysical vs. lab/dns regimes Typical values for atmospheric mesoscale: Fr h = 10 3 Re = 10 10, Re b = 10 4 Lab experiments and (most) DNS: Re b 1. A & O simulations may have smaller effective Re b from eddy or numerical viscosity Viscosity-affected stratified flow atmosphere 1/F h P,F&S P,F&S ocean DNS Strongly stratified turbulence R&dBK 10 S&G Weakly stratified turbulence L&VA Classical Kolmogorov turbulence Re Brethouwer et al., J. Fluid Mech. 585, , 2007 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

11 What we do Introduction Direct numerical simulations of stratified turbulence with Re b 4 Questions: how important are higher-order contributions to potential enstrophy? test hypothesis that potential enstrophy quadratic only for Re b 1 implications for using idealized experiments/simulations as proxy for a & o? M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

12 Approach Approach Numerical model periodic BCs, constant N spectra, FFT, de-aliased DNS: x = z Kolmogorov scale Experimental set-up: lab-scale units domain: L = 2π force large-scale vortical modes gives U 0.02, L h 4, T 200 run for 2000 time units; average over set κ = ν Vary N and ν to get: Fr h Re Re b 4 not geophysical, but at least O(1) resolution: 512 3, (SciNet) M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

13 Results Time series of V and V 2 for Fr h = 0.01 V and V 2 for Fr h = 0.01 with two different Re b relative size of V 2 depends on Re b, even at fixed Fr h. higher-order terms important for Re b 1 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

14 Results Relative contributions of V 2 and V 4 V 2 /V and V 4 /V vs Fr h and Re b no collapse with Fr h see collapse with Re b for small enough Re b M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

15 Results Relative contributions of V 2 and V 4 V 2 /V and V 4 /V vs Fr h and Re b no collapse with Fr h see collapse with Re b for small enough Re b M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

16 Potential enstrophy spectra Results Horizontal wavenumber spectra of V, V 2, and V 4 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

17 Results Snapshots: Fr h = 0.01, Re b = 0.5 ω y (x, z) ( zu) q 1 (x, y) q(x, y) Intermittent KH instabilities (as in Laval, McWilliams & Dubrulle 2003, etc.) show up in ω z field, which contributes to q 1 but not (much) in q field larger Re b : more KH, transitions to small-scale 3D turb M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

18 Larger Re b = 2 Results ω y (x, z) ( zu) q 1 (x, y) q(x, y) Intermittent KH instabilities (as in Laval, McWilliams & Dubrulle 2003, etc.) show up in ω z field, which contributes to q 1 but not (much) in q field larger Re b : more KH, transitions to small-scale 3D turb M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

19 Smaller Re b = 0.1 Results ω y (x, z) ( zu) q 1 (x, y) q(x, y) Intermittent KH instabilities (as in Laval, McWilliams & Dubrulle 2003, etc.) show up in ω z field, which contributes to q 1 but not (much) in q field larger Re b : more KH, transitions to small-scale 3D turb M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

20 Energy spectra E(k h ) Results Bumps due to KH inst (Laval et al 2003) Position of bump at k h N/U (Waite 2011) Laval, McWilliam & Dubrulle, Phys. Rev. E 68, 03608, 2003 Waite, Phys. Fluids 23, 06602, 2011 M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

21 Results Relative contributions of V 2 and V 4 from large scales only Compute potential enstrophy from large horizontal scales (filter out KH billows) nice collapse when plotted against Re b for small Re b, V V 2 higher-order contributions to V grow with increasing Re b consistent with KH interpretation, since L b /L h Re b /Re M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

22 Discussion Discussion Quadratic potential enstrophy is not a good approximation when Re b 1, even for Fr h 1 regime of weakly (or marginally) viscous stratified turbulence layerwise structure with KH instabilities and small-scale turbulence breakdown of quadratic approximation occurs at small horizontal scales: KH instabilities? Quadratic potential enstrophy is a good approximation when Re b < 0.4 regime of viscously coupled layerwise pancakes no KH instabilities or transition to small-scale turbulence likely that Aluie & Kurien (2011) is in this regime But Re b does not tell the whole story V 2 /V does not collapse w.r.t. Re b unless small scales are filtered More info: Waite (2013), Potential enstrophy in stratified turbulence, JFM 722, R4. M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

23 Discussion Discussion Implications for atmospheric and ocean: back-of-the-envelope: atmospheric meso Re b 10 4, oceanic sub-meso Re b quadratic approx seems doubtful here Atmospheric models may have small effective Re b Brune & Becker (2013) computed mesoscale U/N 80 m, not resolved artificially small mesoscale Fr v quadratic V? mesoscale cascade in these models probably not stratified turbulence lack of consensus on decomposition of mesoscale spectrum into waves and vortical modes M. L. Waite (UWaterloo) Fields Sub-Meso June / 25

24 Discussion HAMILTON ET AL.: MESOSCALE SPECTRUM D W avelength (km) Lindborg (1999) eqn 71 k ARW mb avg rotational component divergent component 106 E(k) (m3/s2) 8110 Discussion deformational component k-5/ h forecast valid 0 UTC 5 June 2005 JANUARY 2013 BRUNE AND BECKER Figure 3. Solid lines show the total wave number kinetic energy spectrum at the 200-hPa level Hamilton, & Ohfuchi, GRL 113, 2008 computed from control runs with Takahashi T639L24 and T319L24 versions of the AFES model. The dotted lines show the same spectrum computed for the divergent component of the wind. The lines in the upper right corner are plotted for reference and have slopes of "5/3 and "3. vided into so many components. However, the pattern is in the eddy field is the strong meridional shear of the zonal ite clear. For total wave number greater than about 100 wind in the large-scale jets. Eddies with m! n would have orresponding to horizontal wavelengths less than about phase lines oriented almost nearly north-south and so might 0 km) the spectrum appears to be a strong function of n, possibly be particularly subject to tilting by the strong t almost independent of m. This is what would be meridional shear (a process which in 2D spectral space pected if the statistical properties of the flow at this scale would appear as an energy transfer from m! n to other ere horizontally isotropic and geographically homoge- components). ous [Boer, 1983]. For n < 10 there is very strong pendence on both n and m, indicating large deviations 3.3. Total Wave Number Spectrum, Convergence of om isotropy and homogeneity. This is expected, of course, Results, and Subgrid-Scale Dissipation [17] The analysis in the remainder of the paper will focus nce the large-scale features in the circulation (jets, anetary waves, etc.) clearly are shaped by processes that on the KE spectrum as a function of total wave number, En, e strongly anisotropic. It is interesting that the region which is computed by summing the En,m over all m (less tween about n = 10 and n = 100 is characterized by near than or equal to n) for each n. The solid curves in Figure 3 dependence of En,m on m except for a narrow range where show the 200-hPa KE spectrum computed in this way from becomes close to n, which has lower KE content. Boer 100 snapshots for AFES integrations conducted at T319 and d Shepherd [1983] computed a similar KE spectrum from T639 spectral resolution and with 24 model levels. The obal observational analyses (though only out to n = 32). horizontal hyperdiffusion coefficient, Kh, in each run has r n > 10 in their results they noted without explanation been adjusted by trial-and-error to produce power law e same near independence of En,m on m, except for the spectra that agree with observations and converge reasonop when m! n. KH computed a KE spectrum for the ably well with resolution, as described in THO. THO found their70, trial-and-error Brune Becker,by JAS 2013"3.22procedure results in Kh values that per troposphere/lower stratosphere from their& SKYHI ntrol simulation (in this case out to n = 450). KH s results scale roughly as nt, where nt is the truncation wave ee their Figure 3) are quite similar to those seen in the number. Table 1 shows the diffusion coefficient Kh determined by THO to be appropriate for T79, T159, T319 and esent Figure 2, and also the low(uwaterloo) values for m! n M.display L. Waite Fields Sub-Meso T639 resolutions. Also given are the timescales for dissipa W avenumber (radians m-1) Skamarock & Klemp, JCP 227, 2008 Waite & Snyder, JAS 70, June / 25

25 Acknowledgments Discussion Funding: NSERC Computing: SciNet, Compute Canada M. L. Waite (UWaterloo) Fields Sub-Meso June / 25