Quasi-Normal Scale Elimination (QNSE) theory of anisotropic turbulence and waves in flows with stable stratification

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1 Quasi-Normal Scale Elimination QNSE theory of anisotropic turbulence and waves in flows with stable stratification Semion Suoriansy, Boris Galperin Ben-Gurion University of the Negev, Beer-Sheva, Israel University of South Florida, St. Petersburg, Florida Warwic worshop "Structures and Waves in Anisotropic Turbulence" Warwic University Mathematics Institute, - 7 November, 008

2 Introduction The aim is to develop a theory that systematically includes anisotropic turbulence and internal waves The difficulty: turbulence is a nonlinear, multi-scale, stochastic phenomenon. Analytical theories exist for simplest flows that are locally isotropic. Geophysical flows are anisotropic with waves. Reynolds averaging does not differentiate between scales and does not discern contributions from different processes. Reynolds stress closures employ the concept of invariant modeling and are not flexible enough Spectral approach is more suitable

3 The Quasi-Normal Scale Elimination QNSE theory of turbulence with stratification We consider fully D turbulent flow with imposed vertical temperature gradient dθ/d. Governing equations in Boussinesq approximation: momentum temperature u t t continuity u 0 P ρ u u αgt ˆ e ν 0 u f0 T dθ 0 u T u κ T d In linear approximation this system supports gravity waves with Brunt-Vaisala frequency dθ 1/ N αg d

4 Fourier-transformed velocity and temperature equations Velocity Green function becomes tensorial: ], [,, α β αβ αβ δ ω δ ω ω P A G G where 1 ] [, i G h ν h ν ω ω ν h, ν - horiontal and vertical eddy viscosities, P αβ - projection operator 1 ] [, i G h h T κ κ ω ω is temperature Green function, κ h and κ are horiontal and vertical eddy diffusivities φ κ κ ω ν ν ω ω sin, N i i N A h h φ is the angle between and the vertical, Eliminate pressure using continuity equation; obtain momentum equation in a self-contained form using formal solution to the temperature equation: complex poles > waves! Θ 4 4 dqˆ ˆ ˆ ˆ ˆ ˆ ˆ dqˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ π π ϕ α α ν µ βµν β αβ α q T q u i u d d G T q u q u P i G u T T ˆ f β ˆ f T

5 Quasi-Normal Scale Elimination Model QNSE As in other quasi-normal models, we see the solution in the form, ω G, ω f, ω u α αβ β LM f, ω dθ u, LT T, ω GT, ω T ω d In other words: mapping onto quasi-gaussian fields governed by the Langevin equations f α, ω is a stochastic force representing stirring of a given velocity mode by all other modes; postulated as quasi-gaussian, solenoidal, homogeneous in space and time: fα ω, fβ ω, ε Pαβ δ ω ω δ Effective viscosities and diffusivities in Green functions describe damping of a mode by nonlinear interactions with all other modes. Due to anisotropy, viscosities and diffusivities are different in the vertical and horiontal directions. Our goal is to calculate the mapping parameters, ν h,ν κ h,κ

6 Quasi-Normal Scale Elimination QNSE method Central problem is treatment of nonlinearity. Perturbative solution based on expansion parameter Re? It is strongly divergent! General idea: Re is small for smallest scales of motion > Derive perturbative solution for these small scales Using this solution and assumption of Quasi-Gaussianity perform averaging over infinitesimal band of small scales. Compute corrections to effective or eddy viscosity and heat diffusivity. Viscosity increases; Re for the next band remains small Repeat the above procedure for the next band of smallest scales. Partial scale elimination yields a subgrid-scale model for LES; complete scale elimination yields eddy viscosities and eddy diffusivities for RANS Reynolds-average Navier-Stoes models.

7 Corrections to viscosities and diffusivities Diagrammatic technique Momentum equation: Subdivision into slow large and fast small modes Additional iteration non-conventional, slow modes are iterated Replace the fast velocity modes by their Langevin equations and ensembleaverage over the fast modes After the averaging only diagrams 5 and 9 remain. Assumption of quasi-gaussianity means that triple correlators 6, 10 vanish

8 Corrections to viscosities and diffusivities Correction to viscose term: Correction to diffusive term G G 1 αβ 1 T whereu ω,, Analytical expressions for the corrections are given by ω,, µσ qˆ P αβϑ α qˆ G qˆ G q U ˆ dqˆ qˆ π qˆ P The integrals are calculated using Distant Interaction Approximation <<Λ in which only the terms up to O/ L are retained spectral gap β Dq > U G > αβ αµ P νσβ T βσ αβ µσ qˆ G q 4 ϑν ˆ dqˆ qˆ π Final result: a coupled system of 4 nonlinear ordinary differential equations for all corrections. The system is solved analytically for wea and numerically for arbitrary stratification yielding expressions for scale-dependent horiontal and vertical eddy viscosities and eddy diffusivities 4 Λ

9 Theoretical results Wea stable stratification Expansion in powers of spectral Froude number, O α ν, n N Pr κ 1 t n ε ν n Omidov wavenumber κ 0.7 inverse turbulent Prandtl number n eddy viscosity and diffusivity in neutral flows ν n 0.46 ε 1 4

10 Scale- dependent horiontal and vertical eddy viscosities and diffusivities Figure. 1: Normalied horiontal and vertical eddy viscosities and diffusivities as functions of / 0. Dashed vertical line indicates the maximum wave number threshold of internal wave generation in the presence of turbulence; O N /ε 1/

11 Some interpretations For << O strong stratification : ν ε κ ε κ 1 1 a Only scales up to -1 O contribute to mixing of scalar, while the mixing of momentum continues on larger scales dominated by waves b 4 4 O ε N 1 c The vertical turbulent Prandtl number is proportional to RiN /S ν κ Ri, Ri > 1

12 E Turbulence spectra Due to anisotropy traditional D energy spectrum provides only limited information. Various 1D spectra are computed analytically for wea stratification: 8 / 5/ 1 4 U 11 ω,dωd1d 0.66ε π N Spectrum ~ - is generated! Transition from -5/ to - spectrum at large scales; coefficients are in a good agreement with experimental data and LES Carnevale et al, JFM, 001. The Gargett et al normalied spectrum of the vertical shear: This scaling presents the normalied vertical shear spectrum as a universal function of / O. The QNSE theory provides rigorous theoretical basis for this universal scaling

13 Temperature spectra Temperature spectra , 0.7, 0.068, Corrsin constant, 0.6 where / 5/ 1/ 1 4 / / 1 1/ 1 Θ Θ C C C C C d d C C C E d d C C C E O T O T θ θ θ θ θ ε ε ε ε Assuming that for relatively strong stratification ε θ ΓεdΘ/d /N where Γ~0. is the mixing efficiency, one gets, 1, 5/ 1, 1/, 1 0. Θ d d C E T ε θ ε θ

14 Comparison with experimental data The spectrum of the vertical shear of the horiontal velocity in the ocean; data from Gregg, Winel, Sanford, JPO 199. The theoretical prediction is shown by a gray line. This is the first time that these spectra are derived within an analytical theory.

15 Another example - Atmospheric spectra Our theory predicts the vertical spectrum which is in a good agreement with the spectra observed in the stratosphere, troposphere, mesosphere, and thermosphere our prediction is well approximated by the dashed line. S. Smith et al., JAS, 1987

16 E E Other 1D spectra 8 / 5 / 1 4 U ω, dω d d 0.66ε π N 8 / 5/ 4 U ω, dωd1d 0.47ε π N Recall the vertical spectrum of horiontal velocity, The anisotropiation manifests itself as energy increase in the horiontal velocity components at the expense of their vertical counterpart

17 RANS modeling Invoing energy balance equation, the eddy coefficients are recast in terms of Richardson number Ri N / S or Froud number Fr ε / NK Figure : Normalied eddy viscosities and diffusivities as functions of Ri and Fr. For Ri>0.1, both vertical viscosity and diffusivity decrease, with the diffusivity decreasing faster than the viscosity residual mixing due to effect of IGW? Horiontal mixing increases with Ri. The model accounts for flow anisotropy. The crossover from neutral to stratified flow regime is replicated.

18 Comparison with data: Pr t as a function of Ri Inverse Prandtl number, κ /ν, as a function of Ri. Solid line - theoretical prediction by QNSE theory Suoriansy et al Blac and white squares and blac tringles - data from Halley Base, Antarctica collected in 1986 Yague et al., 001. White circles show data by Monti et al. 00 and grey rectangles are data by Strang and Fernando 001. Small dots show data from Halley Base British Antarctic Survey

19 Dispersion relation for internal waves with turbulence Complex poles yield the secular equation det [ 1 ω, ] 0 G αβ Waves exist if the solution of the secular equation has a real part. Identifying this real part with the wave frequency ω we obtain the dispersion relation The limit of strong stratification > classical dispersion relation for linear waves, ωn sinθ. Turbulence dominates at small scales. Criterion for wave generation is ω 0 giving

20 Curve fitting functions For implementation in PBL schemes, the theoretically derived stability functions, α M K M /K 0 and α H K H /K 0, were approximated by a fraction-polynomial fit K 0 is the eddy viscosity at Ri0: α α M H 1 8Ri 1.Ri 5Ri Ri 1.9 Ri 1.44Ri 19.8Ri The fitting functions are valid for Ri<1.5. For larger values, flux Richardson number R f approaches limiting value <0.5

21 QNSE-based K-l model [ ] 1/ * 0 0 1/ /,, *********************************** / 0.55,, ; ;, 1 ; 0 c le K K K K K f Bu c c c l l l c N K c l l l c K K K t E H H M M N B N N N B E q H g V U M E Θ α α λ λ ε ε ε ε θ

22 QNSE-based surface layer parameteriation Using theoretically derived stability functions α M, α H and approximations of constant flux layer, we derived the drag coefficients for momentum and heat, C D, C H, that replace the Louis formulation. The corresponding expressions are: C C ψ ψ Pr D H M H 0 ln ln 0 0 ζ ψ ζ ζ.5ζ 0.ζ ζ Pr ζ 0.1 ζ ψ ψ 0 M M κ M 0 ζ ψ ζ Pr ln ψ ζ ψ ζ M 0T turbulentprandtl number for neutral flow, κ ζ / L, 0, 0T ζ H 0 0 / L H 0T

23 Testing of the model - neutral ABL Comparison with Leipig wind profile

24 Testing in WRF BASE temperature profiles Comparison with LES results Stroll and Porte-Agel, BLM, 008

25 Testing in WRF BASE velocity profiles Comparison with LES results Stroll and Porte-Agel, BLM, 008

26 Effect of surface layer parameteriation QNSE surface scheme eliminates warm temperature bias. With other schemes, increase in resolution may decrease the bias but does not eliminate it.

27 Unstable stratification Convection

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30 Conclusions Derivation of the QNSE model of turbulence is maximally proximate to first principles Theory explicitly resolves horiontal-vertical anisotropy Accounts for the combined effect of turbulence and waves Predicts correct behavior of Pr t as a function of Ri Anticipates the absence of the critical Ri Yields modification of the classical dispersion relation for internal waves that accounts for turbulence Yields analytical expressions for various 1D and D spectra; captures transition from the -5/ to the N - vertical spectrum of the horiontal velocity and recovers Gargett et al. scaling Provides subgridscale closures for both LES and RANS The QNSE theory has been implemented in K-ε and K-l models of stratified ABL Good agreement with CASES-99 and other data sets has been found for cases selected for the GABLS experiment The new stability functions improve predictive sills of HIRLAM in 4h and 48h weather forecasts Is being incorporated in WRF Weather Research and Forecasting a new model developed at NCAR. Will become a standard module of WRF

31 Acnowledgments This research has been supported by the US Army Research Office, Office of Naval Research and Israel Science Foundation References S. Suoriansy, B. Galperin and I. Staroselsy, Cross-term and ε-expansion in RNG theory of turbulence, Fluid Dynamics Research, Vol., No.4, 00, pp Suoriansy, S., B. Galperin and I. Staroselsy, A quasi-normal scale elimination model of turbulent flows with stable stratification. Physics of Fluids, 17, , 005. Suoriansy, S., B. Galperin, and V. Perov, Application of a new spectral theory of stably stratified turbulence to atmospheric boundary layers over sea ice. Boundary-Layer Meteorology, 117, 1 57, 005. Suoriansy, S., B. Galperin, and V. Perov, A quasi-normal scale elimination model of turbulence and its application to stably stratified flows. Nonlinear Processes in Geophysics, 1, 9, 006. B. Galperin, S. Suoriansy, and P. S. Anderson, On the critical Richardson number in stably stratified turbulence, Atmospheric Science Letters, 8: , DOI: /asl.15.

32 Validation of the QNSE theory in modeling of atmospheric boundary layers and numerical weather prediction Validation was conducted for models in both K-ε and K-l format Data from numerous observation campaigns was employed CASES99 Data-asteriss, model - lines Potential temperature Wind speed