The Lagrangian-Averaged Navier-Stokes alpha (LANS-α) Turbulence Model in Primitive Equation Ocean Modeling

Transcription

1 The Lagrangian-Averaged Navier-Stokes alpha (LANS-α) Turbulence Model in Primitive Equation Ocean Modeling Mark R. Petersen with Matthew W. Hecht, Darryl D. Holm, and Beth A. Wingate Los Alamos National Laboratory Outline POP ocean model & climate change assessment LANS-α implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic D2Hfest LA-UR July 24, 2007

2 CO2 ice core record currently: 380 ppm Homo sapiens neanderthalensis Homo sapiens archaic Homo sapiens sapiens Homo erectus domesticated plants & animals Siegenthaler et.al. Science 2005

3 Community Climate System Model Collaboration of: National Center for Atmospheric Research (NCAR) in Boulder, CO Los Alamos National Laboratory (LANL) Atmosphere (NCAR) Land Surface (NCAR) Flux Coupler Ocean POP (LANL) Sea Ice (LANL)

4 IPCC - Intergovernmental Panel on Climate Change Created in 1988 by World Meteorological Organization (WMO) and United Nations Environment Programme (UNEP) Role of IPCC: assess on a comprehensive, objective, open and transparent basis the scientific, technical and socio-economic information relevant to understanding: the scientific basis of risk of human-induced climate change its potential impacts and options for adaptation and mitigation. Main activity: Assessment reports Third Assessment Report: 2001 Fourth Assessment Report: 2007 Fifth: planned for 2013

5 IPCC scenarios of future emissions 1: global population levels off, declines after : continuously increasing population A: slower conversion to clean & efficient technologies A1FI: fossil intensive A1T: non-fossil intensive A1B: balance of F&T A2 B: faster conversion to clean & efficient technologies B1 B2 economic models IS92a: business as usual (extrapolation from current rates of increase) carbon cycle models

6 scenarios CO 2 emissions CO 2 concentration A: slower conversion to clean & efficient technologies B: faster conversion to clean & efficient technologies 1: global population levels off, declines after : continuously increasing population A1FI: fossil intensive A1T: non-fossil intensive A1B: balance of F&T A2 B1 B2 economic models carbon cycle models Final product average of ensemble Temperature change scenario A2 ensemble of climate models

7 Ensemble mean: temperature changes

8 IPCC: Estimates of confidence TAR p.82

9 Parallel Ocean Program (POP) Bryan-Cox type model, z-level vertical grid, finite difference model conservation of momentum " t u + u# $u % f & u = %' 0 %1 $p + A M $ h 2 u + " z µ" z u advection conservation of mass for incompressible fluid advection Coriolis " h # u + $ z w = 0 pressure gradient conservation of tracers (temperature, salinity) " t # + u$ %# = A H % h 2 # + " z &" z # + Q diffusion diffusion source/ sink grid u w ϕ t p ρ 0 T S hor. velocity vertical velocity tracer time pressure density temperature salinity hydrostatic in the vertical "p "z = #$g equation of state " = "( T,S, p)

10 POP: 0.1º resolution, speed

11 Parallel Ocean Program (POP) Resolution is costly, but critical to the physics Climate simulations low resolution: 1 deg (100 km) long duration: 100s of years fully coupled to atmosphere, etc. Eddy-resolving sim. high resolution: 0.1 deg (10 km) short duration: years ocean only Rossby Radius of deformation Potential temperature 0.8º x 0.8º grid Potential temperature 0.1º x 0.1º grid

12 low resolution: 0.8º What do you get with higher resolution? Small-scale turbulence and eddies transport energy and heat. cost of doubling horizontal grid is factor of º Reynolds decomposition: u = u + u " total 0.2º time average perturbation These become more realistic with higher resolution: eddy heat transport: u " 2 T " eddy kinetic energy: ( 1 2 u " u " ) 2 feedback of small-scale features on the largescale mean flow - important for oceanic jets vertical temperature profile high resolution: 0.1º

13 Outline POP ocean model & climate change assessment LANS-α implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

14 Lagrangian-Averaged Navier-Stokes Equation (LANS-α) Two ways to take averages: Lagrangian and Eulerian Lagrangian averaged velocity actual velocity v Lagrangian averaged trajectory particle trajectory Eulerian averaged velocity, u, is average at x o v Lagrangian averaged velocity u Eulerian averaged velocity smooth u = ( 1"# 2 $ 2 ) "1 v Helmholtz operator rough

15 Lagrangian-Averaged Navier-Stokes Equation (LANS-α) v rough u smooth larger α smooths more v Lagrangian averaged velocity u Eulerian averaged velocity smooth u = ( 1"# 2 $ 2 ) "1 v Helmholtz operator rough "v dt + u# $v + v j$u j % f & u = % 1 ' 0 $p + ($ 2 v + F advection extra nonlinear term Coriolis pressure gradient diffusion

16 Lagrangian-Averaged Navier-Stokes Equation (LANS-α) "v dt + u# $v + v j$u j % f & u = % 1 ' 0 $p + ($ 2 v + F advection extra nonlinear term Coriolis pressure gradient diffusion $ v 1 # x u 1 + v 2 # x u 2 + v 3 # x u 3 ' & ) v j "u j = v 1 # y u 1 + v 2 # y u 2 + v 3 # y u 3 & ) % v 1 # z u 1 + v 2 # z u 2 + v 3 # z u 3 ( Extra nonlinear term is in alpha model, but not in Leray model. This term is required for conservation of PV.

17 tracer equation momentum equation tracer equation momentum equation advection Standard POP "# dt + u$ %# = D H (#) + D V (#) advection extra nonlinear term advection Coriolis advection Coriolis diffusion "u dt + u# $u % f & u + metric(u) = %' %1 0 $p + F H (u) + F V (u) Helmholtz inversion u = ( 1"# 2 $ 2 ) "1 v e.g. centrifugal POP-alpha "# dt + u$ %# = D H (#) + D V (#) diffusion e.g. centrifugal pressure gradient pressure gradient rough velocity, diffusion smooth velocity, "v dt + u# $v + u j$v j % f & u + metric(u) = %' %1 0 $p + F H (v) + F V (v) v u diffusion

18 The POP-alpha model Issues: 1. How do we implement the alpha model within the barotropic/baroclinic splitting of POP? full ocean (vertical section) level 1 level 2 level 3 level K fast surface gravity waves vertically integrated barotropic part single layer implicit time step slower internal gravity waves baroclinic part multiple levels explicit time step vertical mean = 0 level 1 level 2 level 3 level K

19 Outline of algorithm - Standard POP Start: variables known at step n+1 temperature, salinity, density, pressure at n n baroclinic velocities u k barotropic velocity U n Baroclinic computations levels k =1Kkm leap frog step: n u ˆ +1 k = u n"1 n k + 2#tRHS k where RHS contains: advection metric (centrifugal) Coriolis diffusion Barotropic computations implicit solve for the free surface height " n +1 new barotropic velocity: U n +1 = U n"1 + 2#tRHS where RHS contains: vertically integrated forcing "# n$1,"# n,"# n +1 subtract depth average: n u +1 n k = u ˆ +1 k " 1 H km # k=1 n u ˆ +1 k dz k add barotropic velocity back to baroclinic: n u +1 n k = u +1 k + U n +1 step n+1 complete

20 Outline of algorithm - POP-alpha Start: variables known at step n+1 temperature, salinity, density, pressure at n baroclinic smooth and rough velocities u n n k,v k barotropic smooth and rough velocities U n,v n Baroclinic computations levels k =1Kkm leap frog step: n v ˆ +1 k = v n"1 n k + 2#tRHS k where RHS contains: advection metric (centrifugal) Coriolis diffusion extra nonlinear term, subtract depth average: n v +1 n k = v ˆ +1 k " 1 H km # k=1 n u +1 n k = smooth(ˆ v +1 k ) n u +1 n k = u ˆ +1 k " 1 H km # k=1 "u T # v n v ˆ +1 k dz k n u ˆ +1 k dz k Barotropic computations implicit solve for the free surface height " n +1 new barotropic velocity: V n +1 = V n"1 + 2#tRHS where RHS contains: vertically integrated forcing "# n$1,"# n,"# n +1 U n +1 = smooth(v n +1 ) add barotropic velocity back to baroclinic: n u +1 n k = u +1 k + U n +1 n v +1 n k = v +1 k + V n +1 step n+1 complete

21 Barotropic Algorithm - Pop-alpha Simultaneously solve for: free surface height " n +1 and both velocities, U n +1,V n +1 Invert using iterative CG routine smoothing within each iteration is too costly! (" # H(1$% 2 " 2 ) $1 " $ 2 &g' )( n +1 = 2 1 $ % "#g HUn&1 & 2 ' n + 1 $ % #g" 2 #g H(1&(2 $ 2 ) &1 G n & BU n & #g$ ' n&1 + ' n V n +1 = V n"1 + 2#t[ G n " BU n " $g% (& n"1 + & n + & n +1 )] [ ( )] momentum forcing terms U n +1 = (1"# 2 $ 2 ) "1 V n +1 momentum forcing terms resolution:

22 (" # H(1$% 2 " 2 ) $1 " $ 2 &g' )( n +1 = 2 Barotropic Algorithm - Pop-alpha Simultaneously solve for: free surface height Invert using iterative CG routine 1 $ % "#g HUn&1 & 2 ' n + 1 $ % #g" 2 #g H(1&(2 $ 2 ) &1 G n & BU n & #g$ ' n&1 + ' n V n +1 = V n"1 + 2#t[ G n " BU n " $g% (& n"1 + & n + & n +1 )] " n +1 and both velocities, U n +1,V n +1 smoothing within each iteration is too costly! What if we eliminate just this one smoothing step? [ ( )] momentum forcing terms U n +1 = (1"# 2 $ 2 ) "1 V n +1 momentum forcing terms resolution:

23 Outline POP ocean model & climate change assessment LANS-α implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

24 up N E The test problem: Idealization of Antarctic Circumpolar Current surface 12ºC thermal forcing 2ºC periodic bndry solid boundary solid boundary zonal wind periodic bndry Surface temperature deep-sea ridge isotherms Baroclinic Instability: 1.Eastward zonal wind causes northward Ekman transport 2.Circulation tilts isotherms (lines of constant temperature) 3.Potential energy of baroclinic instability converted to kinetic energy. 4.Small scale turbulence (eddies) transport heat and kinetic energy Thermocline depth is determined by the eddy transport quantities.

25 The Baroclinic Instability Surface temperature Kinetic energy POP POP POP low resolution-0.8º standard POP high resolution-0.1º standard POP resolution: 0.2 (low) (high) Eddy kinetic energy POP Potential temperature - vertical cross section POP POP resolution: 0.2 (low) (high) Depth of 6C isotherm high res. depth low resolution-0.8º standard POP depth high resolution-0.1º standard POP med. res. S N S N low res.

26 POP-alpha Results Kinetic energy Eddy kinetic energy POP-α POP-α POP POP-α POP POP POP POP-α POP POP resolution: 0.8º 0.8º 0.4º 0.4º 0.2º Depth of 6C isotherm 0.2 POP resolution: 0.8º 0.8º 0.4º 0.4º 0.2º Computation time (log scale) POP 0.4 POP-α POP POP-α 0.4 POP 0.8 POP-α POP POP-α 0.8 POP resolution: 0.8º 0.8º 0.4º 0.4º 0.2º

27 Can you see more eddies with LANS-alpha? POP, 0.4º resolution Pot. Temperature POP-α, 0.4º resolution Pot. Temperature POP, 0.2º resolution Pot. Temperature Velocity Velocity Velocity All sections at a depth of 1600m rough velocity: red, smooth: black

28 Dispersion Relation for LANS-α using linearized shallow water equations normally: with LANS-α: Gravity waves " 2 = k 2 gh " 2 = k 2 gh 1+ # 2 k 2 frequency ω, wavenumber k, gravity g, height H Dispersion relation normal with LANS-α Rossby waves #k$ normally: " = k 2 +1/R 2 LANS-α: " = normal with LANS-α #k$ ( ) +1/R 2 k 2 1+ % 2 k 2 Rossby radius R = gh / f 0 beta " = # y f Dispersion relation LANS-α slows down gravity and Rossby waves at high wave number.

29 What does LANS-α do to the Rossby Radius? Solve for k R, the wavenumber of the Rossby Radius: Use that to find R*, the effective Rossby Radius, as a function of α: Dispersion relation k R k R * normal LANS-α, small α LANS-α, larger α LANS-α makes the Rossby Radius effectively larger.

30 The POP-alpha model Issues: 3. How do we smooth the velocity in an Ocean General Circulation Model? v rough u smooth Helmholtz inversion is costly! ( ) "1 v u = 1"# 2 $ 2 or: use a filter u(x) = # G(r)v(x " r)dr for example, a top-hat filter

31 1D Filter Instabilities filter, G(r) u i = bv i"1 + v i + bv i b If b = 0.5, smoothing filters out the Nyquist frequency. gridpoints v rough u smooth u i = bv i"1 + v i + bv i b = 1 ("1) ("1) = 0 The smooth velocity u is blind to this oscillation. Therefore, the free surface height " cannot counter it!

32 1D Filter Instabilities filter, G(r) u i = bv i"1 + v i + bv i b If b = 0.5, smoothing filters out the Nyquist frequency. v rough 370 m/s u smooth 30 m/s Condition for stability is b < 0.5

33 Filter: Conditions for stability rough velocity v filter: b < 1/2 stable b stable b stable b b < 1/2+c b+c < 1 c < 1/2 stable b,c b+d < 1/2+c b+c < 1+2d c < 1/2 d < 1/2+b b+d < 1/2+c+e b+c+e < 1+2d c < 1/2+e d+2e < 1+b

34 Filter analysis: Helmholtz inversion Green s function Take v to be a point source: Then compute u = (1" # $! Can use this to understand filter near boundaries: 2 2 "1 ) v

35 Filters filter width 3 filter width 5 Wider filters result in: Stronger smoothing Effects are like larger α More computation More ghostcells Temperature - hor. mean filter width 7 high res POP filter width 9 med. res POP

36 Filters filter width 3 filter width 5 Wider filters result in: Stronger smoothing Effects are like larger α More computation More ghostcells Temperature - hor. mean filter width 7 high res POP filter width 9 med. res POP med. res POP-α width 9 width 7 width 5 width 3

37 Helmholtz inversion: vary alpha Kinetic energy Filters: vary the filter width Kinetic energy Eddy kinetic energy Eddy kinetic energy

38 Adding LANS-α increases computation time by <30% We can take smaller timesteps with LANS-α

39 How does LANS-α compare with other turbulence models? POP Hypervisc. only POP-α Hypervisc, LANS- α GM Hypervisc, Gent-McW. GM-α Hypervisc, Gent-McW, LANS-α 2ν Hypervisc, 2x viscosity coef. 4ν Hypervisc, 4x viscosity coef. Kinetic Energy POP POP-α POP GM-α 2ν 4ν GM Depth of 6C isotherm POP, higher res. GM GM-α POP-α POP POP-α 0.4 resolution Eddy Kinetic Energy 0.2 res. POP 2ν 4ν POP GM-α 2ν 4ν GM 0.4 resolution 0.2 res.

40 Leray Model is about half as strong as LANS-α Kinetic Energy Depth of 6C isotherm Eddy Kinetic Energy

41 Outline POP ocean model & climate change assessment LANS-α implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

42 POP simulations of the North Atlantic POP, 0.28º resolution Sea surface height POP, 0.1º resolution Sea surface height More realistic Gulf Stream and Northwest corner at high resolution Smith et. al jpo

43 POP simulations of the North Atlantic POP, 0.28º resolution Eddy kinetic energy POP, 0.1º resolution Eddy kinetic energy Higher eddy kinetic energy at high res. Smith et. al jpo

44 LANS-α in POP: North Atlantic simulations The punch lines: 1. Due to rough boundaries and fast jets in realistic domains, both POP-α and Leray have numerical instabilities in the North Atlantic, particularly at boundaries. 2. The Leray Model runs longer in the North Atlantic, and has higher EKE and visibly more vortices (but still boundary issues). 3. The key is to use the right boundary conditions for the smoothing step (Helmholtz or filter). spurious oscilations in rough vel. filter weights: filter weights:

45 LANS-α in POP: North Atlantic simulations Even with lower filter weights, numerical instabilities at boundary stop code after 6000 time steps (70 days):

46 Leray Model runs stably in North Atlantic "v dt + u# $v + v j$u j % f & u = % 1 extra nonlinear term not in Leray model ' 0 $p + ($ 2 v + F POP 0.2º 3yr mean temp. & velocity POP-Leray 0.2º 3yr mean temp. & velocity Leray Model has visibly more vortices, but still boundary issues.

47 POP-Leray has higher KE and EKE than POP POP 0.2º Eddy kinetic energy (3yr mean) POP-Leray 0.2º Eddy kinetic energy (3yr mean) cross section through North Atlantic current Globally averaged EKE (3 yr mean): POP 0.2º: 11.1 POP-Leray 0.2º: 13.1 POP 0.1º: 29.4 (Smith et. al. 2000) POP-Leray

48 POP-Leray has higher KE and EKE that POP POP 0.2º Eddy kinetic energy POP 0.1º EKE, varying viscosity high visc. POP-Leray 0.2º Eddy kinetic energy more realistic low visc. (Bryan et. al. 2007)

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50 What are my boundary conditions? B.C. v = 0 u = 0 I ve tried many! Equation " t v + u# $v + v j $u j % f & u = % 1 ' 0 $p + ($ 2 v + F u = ( 1"# 2 $ 2 ) "1 v u = filter(v) Option 1: shrink filter at boundary Option 2: shrink filter near boundary Option 3: make filter weights=0 on land water land water land water land

51 A Possibility: Use variable alpha " t v + u# $v + $u 2 $% 2 (x) + v j $u j & f ' u = & 1 ( 0 $p + )$ 2 v + F u = ( 1" 2#(x)$#(x) % $ "# 2 $ 2 ) "1 v We are thinking about this

52 Summary Higher resolution can solve all of your problems. You can t possibly have high enough resolution to solve your problems. The LANS-alpha model captures higher-resolution effects in our test problem, where eddies near the grid-scale are important. POP-Leray runs longer than POP- α in the North Atlantic domain, and shows promising signs, like higher eddy activity Both POP-α and POP-Leray have problems with the rough boundaries and topography of the North Atlantic. Further work on boundary conditions for LANS-alpha, with Helmholtz or filter smoothing, is required.