with Matthew W. Hecht, Darryl D. Holm, and Beth A. Wingate 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 Conclusion: LANS- produces turbulence statistics that resemble doubled-resolution simulations without LANS-. NCAR TOY Workshop LA-UR-05-0887 February 26, 2008

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 Rossby Radius of deformation Surface temperature 1.0º x 1.0º grid Surface temperature 0.1º x 0.1º grid

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. Scales resolved by global ocean models Satellite measurement data from Scott and Wang 2005

What do you get with higher resolution? low resolution: 0.8º What do you get with higher resolution? Small-scale turbulence and eddies transport energy and heat. Reynolds decomposition: 0.4º perturbation cost of doubling horizontal grid is factor of 10 total time average 0.2º 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 high resolution: 0.1º note: SST and thus heat flux is main influence on atmosphere for climate

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

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

Test problem results, POP only 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 Surface temperature low resolution-0.8º standard POP high resolution-0.1º standard POP Potential temperature - vertical cross section warm surf. forcing warm surf. forcing depth depth low resolution-0.8º standard POP high resolution-0.1º standard POP S N S N

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

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

Test Problem Results: Baroclinic Instability 6C isotherm Vertical temperature profile 0.1°(high res) 0.2° POP- 0.2° 0.4° POP- 0.4° 0.8° POP- 0.8°(low res) 0.1° 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? extra nonlinear term not in Leray model Vertical temperature profile Leray produces similar results as LANS-alpha, but to a lesser degree.

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

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

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

POP-alpha in the North Atlantic POP- is still under development for realistic domains. Rough topography and high velocities in jets cause difficulties. The Leray model, a simplified version of LANS-, shows promising results: higher kinetic and eddy kinetic energy. POP-Leray 0.2º Eddy kinetic energy (3yr mean) kinetic energy POP, 0.2º POP-Leray, 0.2º POP, 0.1º KE 17.3 22.1 42.6 EKE 11.1 13.1 29.4 Smith ea 200 p 1550 0.1 mKE: 13.2 total KE=~43 smooth u^2: 9.7 global avg

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

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 resolution POP-, 0.4º resolution POP, 0.2º resolution Pot. Temperature Pot. Temperature Pot. Temperature Velocity Velocity Velocity All sections at a depth of 1600m rough velocity: red, smooth: black

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

Option 1: shrink filter at boundary What are my boundary conditions? B.C. Equation I’ve tried many! 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

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

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

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

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

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 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! What if we eliminate just this one smoothing step? momentum forcing terms momentum forcing terms resolution:

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

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

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

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)

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 A2 B2 economic models IS92a: business as usual (extrapolation from current rates of increase) carbon cycle models

scenarios CO2 emissions CO2 concentration Temperature change 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 A2 B2 economic models carbon cycle models ensemble of climate models Temperature change scenario A2 average of ensemble Final product

Ensemble mean: temperature changes

IPCC: Estimates of confidence TAR p.82

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

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

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

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

Filter: Conditions for stability rough velocity filter: b < 1/2 stable  b b < 1/2+c b+c < 1 c < 1/2 stable  b,c b+d < 1/2+c b+c < 1+2d d < 1/2+b b+d < 1/2+c+e b+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 1 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 Temperature - hor. mean med. res POP high res POP filter width 7 filter width 9

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

What are my boundary conditions? B.C. Equation I’ve tried many! 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

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.

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