Holm by pengxiang

VIEWS: 3 PAGES: 93

									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  divCS  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       ˜   2u  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

								
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