Conclusion: LANS-  produces turbulence statistics that resemble doubled-resolution simulations without LANS- . The Lagrangian-Averaged Navier-Stokes alpha (LANS-  ) 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 February 26, 2008 NCAR TOY Workshop LA-UR

Parallel Ocean Program (POP) Resolution is costly, but critical to the physics Climate simulations low resolution: 1 deg (100 km) long duration: centuries fully coupled to atmosphere, etc. Eddy-resolving sim. high resolution: 0.1 deg (10 km) short duration: decades ocean only Surface temperature 1.0º x 1.0º grid Surface temperature 0.1º x 0.1º grid Rossby Radius of deformation

The Rossby Radius is the kinetic energy forcing scale. At the scale of the Rossby Radius, energy is converted from potential energy to kinetic energy. Satellite measurement data from Scott and Wang 2005 Scales resolved by global ocean models

What do you get with higher resolution? low resolution: 0.8º 0.4º 0.2º high resolution: 0.1º Small-scale turbulence and eddies transport energy and heat. These become more realistic with higher resolution: eddy heat transport: eddy kinetic energy: feedback of small-scale features on the large- scale mean flow - important for oceanic jets vertical temperature profile Reynolds decomposition: totaltime average perturbation note: SST and thus heat flux is main influence on atmosphere for climate cost of doubling horizontal grid is factor of 10

Lagrangian-Averaged Navier-Stokes Equation (LANS-  ) rough smooth larger  smooths more Lagrangian averaged velocity u Eulerian averaged velocity rough smooth Helmholtz operator advection diffusion Coriolispressure gradient extra nonlinear term

The test problem: Idealization of Antarctic Circumpolar Current This test problem invokes the baroclinic instability. N up E solid boundary periodic bndry deep-sea ridge zonal wind surface thermal forcing 12ºC 2ºC

Test problem results, POP only Potential temperature - vertical cross section low resolution-0.8º standard POP low resolution-0.8º standard POP high resolution-0.1º standard POP high resolution-0.1º standard POP low resolution-0.8º standard POP high resolution-0.1º standard POP NSNS depth Surface temperature Depth of 6C isotherm Kinetic energy Eddy kinetic energy resolution: 0.8° 0.4° 0.2° 0.1° 0.1° (high res) 0.2° 0.4° 0.8° (low res) low res high res low res high res warm surf. forcing

POP-  with Helmholtz inversion: vary alpha… Kinetic energy POP-  with filters: vary the filter width… Eddy kinetic energy Kinetic energy Eddy kinetic energy filter width

Test Problem Results: Baroclinic Instability Vertical temperature profile 0.1° 0.2° 0.4° 0.8°(low res) 6C isotherm 0.1°(high res) 0.2° 0.4° 0.8°(low res)

Test Problem Results: Baroclinic Instability Vertical temperature profile 0.1° 0.2° 0.4° 0.8°(low res) 0.2° POP-  0.4° POP-  0.8° POP-  6C isotherm 0.1°(high res) 0.2° 0.4° 0.8°(low res) 0.2° POP-  0.4° POP-  0.8° POP- 

How does the Leray model compare to LANS-alpha? Vertical temperature profile Leray produces similar results as LANS-alpha, but to a lesser degree. extra nonlinear term not in Leray model

Gravity waves Dispersion Relation for LANS-  using linearized shallow water equations normally: with LANS-  : Rossby waves normally: LANS-  : normal with LANS-  normal with LANS-  LANS-  slows down gravity and Rossby waves at high wave number. frequency , wavenumber k, gravity g, height H Rossby radius Dispersion relation beta

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

Adding LANS-  increases computation time by <30% We can take larger timesteps with LANS- 

smooth u^2: 9.7 global avg POP-alpha in the North Atlantic Smith ea 200 p mKE: 13.2 total KE=~43 POP-  is still under development for realistic domains. Rough topography and high velocities in jets cause difficulties. POP-Leray 0.2º Eddy kinetic energy (3yr mean) The Leray model, a simplified version of LANS- , shows promising results: higher kinetic and eddy kinetic energy. POP, 0.2ºPOP-Leray, 0.2ºPOP, 0.1º KE EKE kinetic energy

cross section through North Atlantic current POP 0.2º EKEPOP-Leray 0.2º EKEPOP 0.1º EKE POP 0.2º EKEPOP-Leray 0.2º EKEPOP 0.1º EKE POP-alpha in the North Atlantic

Conclusions LANS-  produces turbulence statistics that resemble doubled-resolution simulations without LANS-  in:  Kinetic energy  Eddy kinetic energy  Temperature distributions associate with baroclinic instability LANS-  increases computation time by 30%, as opposed to a factor of 8-10 to double the resolution. Current work: simulations using LANS-alpha in realistic domains, such as the North Atlantic domain.

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

Kinetic Energy Eddy Kinetic Energy How should we measure kinetic energy with LANS-alpha?

B.C.Equation I’ve tried many! What are my boundary conditions? waterlandwaterland Option 1: shrink filter at boundary Option 3: make filter weights=0 on land waterland Option 2: shrink filter near boundary

A Possibility: Use variable alpha We are thinking about this…

Parallel Ocean Program (POP) Resolution: 0.1° global. Color shows speed.

Outline Resolution of eddies in ocean simulations LANS-  implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

Bryan-Cox type model, z-level vertical grid, finite difference model Parallel Ocean Program (POP) conservation of mass for incompressible fluid conservation of momentum advection diffusion advection diffusion Coriolispressure gradient conservation of tracers (temperature, salinity) u hor. velocity w vertical velocity  tracer t time p pressure  0 density T temperature S salinity hydrostatic in the verticalequation of state grid source/ sink

Outline Resolution of eddies in ocean simulations LANS-  implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

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

Barotropic Algorithm - Pop-alpha Simultaneously solve for: free surface height and both velocities, Invert using iterative CG routine smoothing within each iteration is too costly! momentum forcing terms resolution:

Barotropic Algorithm - Pop-alpha Simultaneously solve for: free surface height and both velocities, Invert using iterative CG routine smoothing within each iteration is too costly! momentum forcing terms What if we eliminate just this one smoothing step? resolution:

Wider filters result in: Stronger smoothing Effects are like larger  More computation More ghostcells filter width 3 filter width 5 filter width 7 filter width 9 Helmholtz inversion is costly! Common to use a filter instead: Issue #2: How should we compute the smooth velocity u ? The POP-alpha model

Outline Resolution of eddies in ocean simulations LANS-  implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

Outline Resolution of eddies in ocean simulations LANS-  implementation in POP Idealized test case: the channel domain The real thing: the North Atlantic

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

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

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

IPCC scenarios of future emissions A: slower conversion to clean & efficient technologies B: faster conversion to clean & efficient technologies 1: global population levels off, declines after 2050 A1FI: fossil intensive A1T: non-fossil intensive A1B: balance of F&T B1 2: continuously increasing population A2B2 IS92a: business as usual (extrapolation from current rates of increase) economic models carbon cycle models

A: slower conversion to clean & efficient technologies B: faster conversion to clean & efficient technologies 1: global population levels off, declines after 2050 A1FI: fossil intensive A1T: non-fossil intensive A1B: balance of F&T B1 2: continuously increasing population A2B2 economic models carbon cycle models scenarios CO 2 emissions CO 2 concentration ensemble of climate models Temperature change scenario A2 average of ensemble Final product

Ensemble mean: temperature changes

TAR p.82 IPCC: Estimates of confidence

Standard POP POP-alpha tracer equation momentum equation advection diffusion advection diffusion Coriolis pressure gradient e.g. centrifugal advection diffusion advection diffusion Coriolispressure gradient e.g. centrifugal tracer equation momentum equation extra nonlinear term rough velocity, smooth velocity, Helmholtz inversion

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

Issues: 3. How do we smooth the velocity in an Ocean General Circulation Model? The POP-alpha model Helmholtz inversion is costly! or: use a filter for example, a top-hat filter rough smooth

Filter Instabilities 1D filter, If, smoothing filters out the Nyquist frequency. rough smooth gridpoints 11 1 The smooth velocity is blind to this oscillation. Therefore, the free surface height cannot counter it!

Condition for stability is rough smooth 370 m/s 30 m/s 1D filter, If, smoothing filters out the Nyquist frequency. Filter Instabilities

Filter: Conditions for stability filter: rough velocity b < 1/2 stable  b b < 1/2+cb+c < 1c < 1/2 stable  b,c b+d < 1/2+cb+c < 1+2d c < 1/2d < 1/2+b b+d < 1/2+c+eb+c+e < 1+2d c < 1/2+e d+2e < 1+b

Filter analysis: Helmholtz inversion Green’s function Take v to be a point source: Then compute Can use this to understand filter near boundaries:

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

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

B.C.Equation I’ve tried many! What are my boundary conditions? waterlandwaterland Option 1: shrink filter at boundary Option 3: make filter weights=0 on land waterland Option 2: shrink filter near boundary

A Possibility: Use variable alpha We are thinking about this…

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-  in the North Atlantic domain, and shows promising signs, like higher eddy activity  Both POP-  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.

When eigenvalues are negative, filter matrix is not positive definite, and kinetic energy may be negative: 1D filter, Eigenvalue analysis of Filter Spread of eigenvalues indicates degree of smoothing by filter. Follow the same process in 2D, up to a 9x9 filter.