1. 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
February 26, 2008

2. 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

3. 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

4. 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

5. 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

6. 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.

7. Test problem results, POP only
Depth of 6C isotherm
Kinetic energy
Eddy kinetic energy
resolution: 0.8° 0.4° ° °
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

8. 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

9. 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)

10. 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-

11. 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.

12. 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.

13. 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.

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

15. 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

16. 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

17. 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.

18. 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

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

20. 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

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

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

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

24. 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

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

26. 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

27. 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:

28. 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:

29. 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

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

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

32. 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

33. 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)

34. 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

35. 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

36. 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

37. Ensemble mean: temperature changes

38. IPCC: Estimates of confidence
TAR
p.82

39. 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

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

41. 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!

42. 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!

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

44. 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

45. 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:

46. 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

47. 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-

49. 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

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

51. 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.

52. Eigenvalue analysis of Filter1D
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.