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Lagrangian Averaged Turbulence Models for Ocean & Atmosphere Circulation Darryl D Holm Mathematics, Imperial College London CCS Division, Los Alamos National Lab CMG Summer School Aug 15-19, 2006 http//:www.lanl.gov/~LAScience/~Vol29 Collaborators at LANL Mark R. Petersen, Matthew W. Hecht, Darryl D. Holm, Beth A. Wingate LANL Ocean Modeling Group What can we predict about such circulations? Is it enough to predict only their average properties? Satellite observation of sea surface temperature Computational Resolution is an Issue (Part 1) • Resolution is a large part of the story (e.g., resolving eddies and topography ) • But resolution is not the whole story • Moreover, we have sparse data; so we will never know the initial & boundary conditions at the highest available numerical resolution LANL Ocean Modeling Group • LANL contributes the Ocean and Sea Ice components of the NCAR CCSM (Community Climate System Model) These run on 2048 processors at LANL • Two other Laboratories: GFDL and NASA GIS also contribute climate modeling programs to IPCC (Intergovernmental Panel on Climate Change) POP high resolution simulation: 0.1º grid Computational Resolution is an Issue (Part 2) • Fine resolution simulations can help by checking and guiding sub-coarse-scale parameterizations • Especially in recognizing coarse-scale phenomena that may emerge from the fine scales! (Backscatter) • Coarse-resolution simulations provide intuition, and the process of developing its parameterizations raises lots of new multi-scale physics questions • Some of these questions arise through averaging Eulerian averages: Time averages at a fixed location may extract global patterns of motion Lagrangian averages: Time averages along the trajectory of a fixed fluid parcel ride along with the fluctuations Lagrangian trajectories of floats in Labrador Bay Lagrangian Mean Trajectory Eulerian averages of DNS can be very illuminating Is it the same mechanism as for Jupiter’s Zonal Bands? Results and Goals of Averaging • As we saw, averaging numerical results may lead to new data processing methods & new observations. • We seek other opportunities, where averaging may lead to new theoretical views & numerical methods. • For this, we must derive closed evolution equations for the average quantities. • What sort of averaging should one apply? • How will we derive the equations? Eulerian vs Lagrangian means , EM 0 , LM 0 d d dt , EM 0 dt , LM 0 Closure is the difficulty • We need evolution equations for the averages of turbulent flow quantities, Eulerian, or Lagrangian. • However, the exact fluid equations are nonlinear and ( fg) f g • How will we derive the equations we need? • This is the problem of “closure” Obtaining Closure Eulerian-mean sub-grid scale parameterization: Eddy viscosity Reynolds velocity decomposition is u u u with the Eulerian mean u (Time average at a fixed point in space), The resulting EM Navier-Stokes equation is called RANS: u 1 p 2 u div(u u u u ) t Eulerian-mean closure schemes often assume the turbulent Reynolds stresses are linearly related to the Eulerian-mean velocity gradients. Smagorinsky-Lilly: div(u u divCS A 2 u ) where 2S u u T (2tr(S S ))1/ 2 and [C] L2 This models the divergence of Reynolds stress as artificial viscosity which enhances viscosity near the grid-scale and loses variability. turbulence does more than diffuse away small scales! But What about EM circulation laws? d c(u ) u dx dt u u u u u dx T c(u ) t 1 1 p u div(u u F dx 2 ) c(u ) 2 1 u A u F dx 2 c(u ) when div(u u A 2 u turbulence only damps ) Is this what we want? circulation What the alpha model does • The LANS- closure modifies the nonlinearity of the Navier-Stokes equation, not its dissipation. • The length alpha is the Lagrangian mean distance a fluid particle fluctuates away from its time-mean trajectory. • Alpha is the smallest eddy scale still participating actively in the cascade. • Fluctuating eddies at scales smaller than alpha are slaved to the LM motions of larger ones (Taylor’s Hypothesis). • Thus, Taylor’s Hypothesis finally closes the GLM theory of Andrews and McIntyre (1978). • In the limit alpha goes to zero, the LANS- solution converges to the Navier-Stokes equation Derivation of the alpha Model Taylor’s Hypothesis (1938) (x,t) q U t) q (x • Turbulent flows often have high power at large scales • Sufficiently small fluctuations are too weak to influence their own nonlinear evolution • Instead, they are swept along by the larger scales, • Hence, sufficiently small scales of turbulence are “frozen” into the (Eulerian) mean flow • Question #1 How small is “sufficient” ?? • Question #2 Why sweeping by the Eulerian mean flow? Why not by the Lagrangian mean flow? Lagrangian Mean Trajectory Taylor’s Hypothesis for x x d is 0 dt along dx u ˜ dt LANS- Circulation Law L d 1 c(u) v dx c(u) F dx ˜ ˜ ˜ D dt ˜ 1 L v ˜ u p u u u O( ) ˜ ˜ ˜ 2 ˜ 4 D u By Taylor’s hypothesis ˜ d d c(u) u u dx c(u) u dx ˜ ˜ 2 ˜ ˜ dt dt u ˜ u u u u dx ˜ ˜ T c(u) t modified nonlinearity Lagrangian Mean Kelvin Theorem ˜ c( u ) LANS- vs RANS Circulation LANS- L d 1 c(u) u u dx c(u) F dx ˜ ˜ 2 ˜ ˜ D dt modified nonlinearity where ˜ 2u u O( 4 ) u ˜ (divu 0) ˜ And RANS (divu 0) d 2 1 E ) u dx c(u ) u u F dx c(u A 2 dt modified dissipation Supplementary discussion slides The Lagrange to Euler Map Forward map Inverse map The Lagrangian Mean Map Eulerian & Lagrangian Paths of Averaging to Obtain New Models Cube of Lagrange-to-Euler Maps Lagrangian Action picture principles Eulerian picture Vary Motion equations Lagrangian views of Newton’s Law & Kelvin’s Theorem dx d 3 x F dx d 3 x d m dt Kinematics Physics D d 3 x 0 d Mass Conservation dt d m 1 dx F dx Kelvin Thm dt D D Kelvin + Stokes yields Vortex and PV dynamics Vortex dynamics is written as d m 1 curl dS curl F dS dt D D db PV follows when 0 dt d 1 m 1 1 b curl b curl F dt D D D D Taking Lagrangian averages yields L L d m 3 x 1 F dx d 3 x dx d dt D D d d L dt , LM 0 d 3 x 0 D dt Hence, LM preserves the forms of Kelvin / Newton, PV & Continuity. But what about its closure? The alpha-model equations The LANS- equations for incompressible flow in a rotating frame are u 1 2 2 2 u u u u u f p u ˜ ˜ T ˜ ˜ u 2 u F ˜ t 2 2 u u . modified nonlinearity 1 ˜ 2 2 u 0 ˜ This is a prognostic equation for the Eulerian mean velocity u ˜ The Lagrangian mean velocity, u , computed diagnostically by the inversion of the Helmholtz operator, is smoother than u The smoothed Lagrangian mean velocity is used in the advection term, the Coriolis term, and the additional nonlinear term. Thank You! The End Lect # 1 Mark R. Petersen, Matthew W. Hecht, Darryl D. Holm, Beth A. Wingate Kelvin’s circulation theorem for LANS- The LANS- model is derived from Hamilton’s principle, which preserves the basic properties of the flow and enables processes at scales greater than alpha to be modeled accurately. For example, the LANS- model Kelvin’s circulation theorem is: d x c(u) u fz dx c(u) u F dx ˜ ˜ 2 dt 2 ˜ where c(u) is a material loop moving with the smoothed velocity ˜ u d x c( u˜ ) u fz dx dt 2 u c( u ) u u u u u f dx. ˜ ˜ T ˜ ˜ t modified nonlinearity Experimental &Numerical tests of LANS-alpha model • Steady solutions match Channel and Pipe turbulence data at the highest experimental Reynolds number (Chen, Foias, Holm, Olson, Titi, Wynne, 1998, 1999a,b) • Homogeneous forced turbulence numerics show the predicted computational acceleration with LANS-alpha (Chen, Holm, Margolin, Zhang, 1999) • Transient simulations capture correctly the spread of turbulent mixing in a shear layer (Holm & Kerr, 2001, Geurts & Holm, 2001, 2003) • Double-gyre simulations enable time-step increase with increased variability (Holm & Nadiga, 2003) • MHD simulations show correct intermittency and anomalous scaling (Graham, Holm, Mininni & Pouquet, 2006 PoF) Comparison to Pipe turbulence data Steady alpha model solution matches high Re pipe flow data Steady LANS- solutions match the measured mean velocity for high Reynolds pipe-flow experiment, over many orders of magnitude in wall units (Chen et al. 1999a). This result gave us our first hope for the model. Computing homogeneous forced turbulence Lagrangian-Averaged Navier-Stokes Equation (LANS-) Two ways to take averages: Lagrangian vs Eulerian Lagrangian averaged velocity u Lagrangian averaged trajectory at fixed x0 actual velocity particle trajectory Eulerian averaged velocity, ˆ u is the average at fixed x ˆ u Eulerian averaged velocity u 1 2 u 2 1 ˆ u Lagrangian averaged velocity smooth Helmholtz operator rough u ˆ u u uT u f u p 2u F ˆ ˆ ˆ dt advection extra Coriolis pressure diffusion nonlinear term gradient DNS vs LANS-alpha Compare NS-Alpha Simulation 2563 0 1/8 1/32 1/ 8 2563 643 Shear turbulence in mixing layers LANS- Simulation captures the onset of shear mixing The LANS- solution was more accurate than a dynamic large-eddy simulation (LES), when both were compared to a high-resolution direct numerical simulation (DNS) (Geurts and Holm 2002). The figure compares the momentum thickness as a function of time for shear turbulence in a mixing layer driven by the Kelvin-Helmholtz instability. Eddy-driven mean effects in the Double Gyre configuration In the Quasi-Geostrophic Double-Gyre Problem, the alpha model preserves mean effects of subgrid-scale eddy activity The QG- model, developed in Holm and Nadiga (2003), showed that the alpha model can capture the correct eddy-driven time-mean circulation in the double-gyre problem at coarse resolution, better than a traditional QG model. Rotating shallow water simulations of the double gyre show the alpha-model represents correct variability at coarser resolution The kinetic energy (a) and potential energy (b) as a function of frequency, from shallow water simulations with different resolutions and values of alpha (Holm et al. 2005). The variability of the fine-grid case is still captured when alpha is the size of the Rossby deformation radius (50 km). Neutral curves for Baroclinic Instability in the alpha model show onset at larger scales with the same forcing The onset of baroclinic instability (1) Is resolvable with fewer grid points in the LANS- model. (2) Occurs at the same value of the forcing for any value of alpha. Intermittency and anomalous scaling in 2D MHD turbulence Anomalous scaling in 2D MHD Summary of the Alpha Model tests LANS- has shown it can represent eddy- driven sub-grid scale turbulent mean effects • QG and shallow water models -- realistic variability and mean structures • Homogeneous turbulence -- realistic spectra and Kármán-Howarth statistics • Correct dynamics of spreading mixing layers • Correct intermittency for MHD in 2D Numerical Approaches Compatible with the Alpha Model • Vortex blob method (Roberts, Leonard) • Particle methods: SPH (Monaghan), HPM (Reich, Cotter, Frank, Gottwald), VPM (Cotter) • SemiLagrange: UK Met Office Unified Model • Variational SemiLagrange schemes, using the Inverse Map (M Dixon) • DEC (Discrete Exterior Calculus) (Leok et al.) POP-alpha model LANL is implementing a LANS- version of the POP ocean model. (Mark Petersen, Matthew Hecht, Beth Wingate, DDH, CCS-2) POP uses a split baroclinic/barotropic formulation to solve the primitive equations. The barotropic velocity is the vertically averaged velocity, and its timestep is severely limited by fast gravity waves. The 2D barotropic velocity is computed with an iterative implicit solver, while the 3D baroclinic velocity uses an explicit leap-frog time-step (Dukowicz & Smith 1994). At each time-step four velocities are computed in POP-alpha: u(x,y,z) the BaroClinic Lagrangian mean (smoothed) velocity ˜ u(x,y,z) the BaroClinic Eulerian mean velocity U(x,y) the BaroTropic Lagrangian mean (smoothed) velocity ˜ U(x, y) the BaroTropic Eulerian mean velocity Implementing POP-alpha Potential advantages for full ocean POP-alpha model: • Coarse-grid solutions generate realistic eddies, without overly enhancing the eddy viscosity or diffusivities. • Realistic variability at coarse scales • Time-step is controlled by alpha, not by gridpoint resolution • Baroclinic instability is represented well at coarse scales Test problem results at four resolutions Parallel Ocean Program (POP) Resolution is costly, but critical to ocean physics Climate simulations Eddy-resolving sims • low resolution: 1 deg (100 km) • high resolution: 0.1 deg (10 km) • long duration: 100s of years • short duration: 50-100 years • fully coupled to atmosphere, etc. • ocean only Rossby Radius of deformation Potential temperature Potential temperature 0.8º x 0.8º grid 0.1º x 0.1º grid 0.8º Cost vs Benefit for higher resolution? Benefit: Small-scale turbulence & eddies transport significant energy and heat. Reynolds decomposition: 0.4º u u u perturbation total time average Cost of doubling horizontal grid is factor of 10 0.2º Higher resolution produces more realistic: • Eddy heat transport v T 0.1º • Eddy kinetic energy 1 2 u v 2 2 • Feedback of small-scale features on the large-scale mean flow • Vertical temperature profile note: SST and thus heat flux is main influence on atmosphere for climate Traditional sub-grid scale parameterization: Eddy viscosity Reynolds velocity decomposition u u u u 1 u u u u v w u u p u 2 t x y z 2 A u Assume that the stresses are related to the mean velocity gradients by a turbulent, or eddy viscosity A, i.e. u u u u A , u A , u A u v w x y z These types of closure schemes enhance the viscosity near the grid- scale, so small coherent structures dissipate away and the energy spectrum falls off at this scale. Lagrangian-Averaged Navier-Stokes Equation (LANS-) ˆ u rough u smooth larger smooths more ˆ u Lagrangian averaged velocity u 1 2 u 2 1 ˆ u Eulerian averaged velocity smooth Helmholtz operator rough u ˆ u u uT u f u p 2u F ˆ ˆ ˆ dt advection extra Coriolis pressure diffusion nonlinear term gradient Lagrangian-Averaged Navier-Stokes Equation (LANS-) u ˆ u u uT u f u p 2u F ˆ ˆ ˆ dt Chen et al., 1998. The Camassa-Holm Equations as a Closure Model for turbulent channel and pipe flows. Phys. Rev. Lett. 81: 5338 Holm. 1999. Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid Holm, Marsden, Ratiu. 1998. The Euler-Poincare Equations in Geophysical Fluid motion. Physica D. 133: 215 Dynamics. (book chapter) Foias, Holm, Titi, 2001. The Navier-Stokes-alpha model of fluid turbulence. Physica D. 152: 505 The LANS- model satisfies Kelvin’s circulation theorem: For any closed loop embedded in and moving within a fluid, the circulation about that loop only changes if work is done on it: D x Dt c(u )u fz 2 dx ˆ 2u F c(u ) ˆ dx where c(u) is a material loop moving with the smoothed velocity u Issues: The POP-alpha model 1. The smooth flow, u, must be non-divergent: u 0 2. How do we implement the alpha model within the barotropic/baroclinic splitting of POP? full ocean level 1 level 2 level 3 (vertical section) level K fast external gravity waves level 1 slower internal gravity waves level 2 vertically integrated level 3 level K barotropic part baroclinic part • single layer • multiple levels • implicit time step • explicit time step • vertical mean = 0 Issues: The POP-alpha model 3. How do we smooth the velocity in a General Circulation Model? ˆ u rough u smooth Helmholtz inversion or: use a filter u 1 2 u 2 1 ˆ u(x) G(r)u(x r)dr ˆ is costly! for example, a top-hat filter POP alpha model In the original formulation of the LANS- model, the Lagrangian ˜ mean velocity u is calculated from the Eulerian mean velocity u by the inversion of a Helmholtz operator, u 1 u 2 2 ˜ This is not feasible in a full ocean model like POP, so a simple smoothing filter is used instead. This filter has the form u(x) ˜ G(r)u (x r)dr G * u where the filter function G satisfies the normalization condition G(r)dr 1 (Geurts and Holm 2002). The current implementation uses a simple nine-point averaging stencil for this smoothing filter. Standard POP tracer equation advection diffusion momentum equation advection Coriolis e.g. centrifugal pressure diffusion gradient POP-alpha rough velocity, smooth velocity, tracer equation advection diffusion momentum equation advection extra Coriolis e.g. centrifugal pressure diffusion nonlinear term gradient Helmholtz inversion u 1 2 u 2 1 ˆ The test problem: Topography & Baroclinic Instability up solid boundary surface temperature, °C N surface 12ºC thermal zonal wind E forcing 2ºC solid boundary deep-sea ridge Specs • Channel model, cyclic, with a N/S ridge • At 60ºS, +/- 8° • 32º zonal width (re-entrant) • Meridional resolutions of 0.1º, 0.2º, 0.4º, 0.8º • 1:1 grid aspect ratio at 60ºS • Vertical res: 10m@surface, 250m@depth as in CCSM ocean • 4000m max depth, N/S ridge rises to 2500m • Buoyancy forcing through restoring of SST 2ºC at 68ºS, 12º at 52ºS • Zonal wind stress The test problem Baroclinic Instability causes mixing: 1.Eastward zonal wind causes northward Ekman transport 2.Circulation tilts isopycnals (lines of constant density) 3.Baroclinic instability converts potential energy into kinetic energy. 4.Small scale turbulence (eddies) transports heat and kinetic energy Eddy transport of heat due to baroclinic instability determines thermocline depth. Climate is sensitive to thermocline depth. But how to get it right at coarse resolution? Neutral curves for Baroclinic Instability in the alpha model show onset at larger scales with the same forcing Value of thermocline depth is determined by onset of baroclinic instability. LANS- shifts onset of baroclinic instability to larger scales at the same value of the forcing. Hence, onset of baroclinic instability is resolvable with fewer grid points in the LANS- model. Global Statistics for the Thermocline 0.2º standard POP 0.4º standard POP 0.8º standard POP 0.4º, =1 0.4º, =2 0.8º, =0.4 0.8º, =1 0.8º, =2 Benefit in speed of 2X coarser horizontal grid in POP-alpha is a factor of 10X Global Statistics Kinetic Energy: KE 1 u2 v 2 2 KE 1 uu vv 2 ˆ ˆ Eddy Kinetic Energy: EKE 2 u v 1 2 2 EKE 1 u v 2 ˆ u v ˆ 0.2º 0.4º 0.8º 0.4º 0.4º 0.8º 0.8º 0.8º standard POP =1 =2 =0.4 =1 =2 POP-alpha, Helmholtz inversion Potential Temperature - Vertical Sections 0.8º 0.1º standard standard POP POP Vertical cross-sections show that as resolution increases from 0.8º to 0.1º, the isotherms become less tilted. This is because meso-scale eddies convert the potential energy of the baroclinic instability into kinetic energy. When this potential energy is released, the isotherms flatten out. 0.8º The isotherms in the POP- 0.8º run are POP- less tilted than the standard POP 0.8º run, indicating that the meso-scale eddies are more active in POP-. Comparison of simulations: Potential Temperature at mid-depth (1600 m) Standard POP, 0.4 deg res. Standard POP, 0.2 deg res. POP-alpha, 0.4 deg res. Note: color scales are different. Stay Tuned for more! • More POP-alpha results for the Test Problem • Variational SemiLagrange (VSL) and Variational Particle Mesh (VPM) numerical schemes preserving mass and satisfying the Kelvin Circulation Law • Full ocean circulation results at 0.2º with POP-alpha versus at 0.1º with POP • Further references at: http//:www.lanl.gov/~LAScience /~Vol29 SIAM News, September 2005 Simulations should answer the question How will the alpha model interact with the Gent- McWilliams (GM) diffusion scheme? GM was intended for tracer equations, to transport and mix temperature, salinity and passive tracers. GM has a diffusive component, as well as an advective component. The alpha model modifies the momentum equation and uses the resulting filtered Lagrangian velocity to advect the tracer. We believe this separation in physics between the alpha model and the GM scheme will allow them to be used together and thus will improve both the turbulent dynamics and the eddy transport of tracers and buoyancy. Thank You! The End Mark R. Petersen, Matthew W. Hecht, Darryl D. Holm, Beth A. Wingate Satellite observation of sea surface temperature