# Presentation: 054650

## The -model of sub-gridscale turbulence in the Parallel Ocean Program (POP)

- Matthew Hecht1, Beth Wingate1
- and Mark Petersen1 with
- Darryl Holm1,2 and Bernard Geurts3
- 1Los Alamos
- 2Imperial College, Great Britain
- 3Twente University, Netherlands
- LA-UR-05-0887

**Notes:**
Lagrangian avg’ing of NS
Still a work-in-progress
Could yet fail. Important that we pursue work of this sort, tapping strengths of the Lab.
Alpha has been taken from theory to model to a number of perhaps simpler applications.  Time for one more step.
“Holy Grail” -- realistic ocn jet dynamics at moderate resolution (NW Corner, Kuroshio, Agulhas retroflection...).
In the future we may talk about bringing another Lab-supported devel effort into HYPOP...

## Ocean Modeling

- Ocean models for climate are based on the Primitive Equations
- Shallow approximation
- Hydrostatic

**Notes:**
In terms of eqn sets...

## -model of sub-gridscale turbulence

- -model developed within (un-approximated) Navier-Stokes Eqns
- What if the velocity in the discretized NS eqns were really a smoother, time-averaged representation of what could exist if finer scales were resolved?
- Leray had proposed something like this -- in 1934
- Use of a filtered, smoother advecting velocity led to a regularization of the NS eqs:

**Notes:**
But breaks Kelvin’s Circulation Theorem (on next page)

## Kelvin’s circulation theorem

- For any closed loop embedded in and moving within a fluid, the fluid circulating around that loop only spins up or down if work is done on it:

- Where (**v**) is some closed fluid loop moving with v(**x**,t).

## Now, consider a smoother, filtered velocity, as Leray did:
u = g * v

and a closed fluid loop which follows this smooth velocity u:

**u** = g * **v**
- and a closed fluid loop which follows this smooth velocity **u**:

- After manipulation, get the Kelvin-filtered Navier-Stokes Eqn

- Just like Leray, but with one additional term! The difference between this and the NS eqns is what we call the -model of turbulence.

**Notes:**
One way to derive the Lagrangian-Averaged Navier Stokes alpha Model
(Darryl first derived it from dynamical, variational principles.)
Consider 2 velocities, the original, containing relatively fine scales, and a smoother, filtered velocity.
Notice that the filter contains a length scale alpha.

## Eulerian Averaging

- Tracer concentration is averaged over some neighborhood around fixed-space cells

**Notes:**
This is a movie which will play in the PowerPoint version of slide show, not in html version!
From quasi-2D Lab experiments, Ecke’s group (Overflow/Mixing PI, with Beth).
They derived a picture of the flow field -- that’s what was produced in laboratory.
Then, after the fact, these numerical simulations of tracer spreading were performed.

## Lagrangian Averaging

- Tracer concentration is averaged over some neighborhood which follows the flow

**Notes:**
This is a movie which will play in the PowerPoint version of slide show, not in html version!

## Some Applications

**Notes:**
An incomplete but illustrative list.
Three applications will be touched on  here.

## Turbulent decay, direct and modeled 

-  Kang, Chester & Meneveau (KCM) at JHU newly    performed a classic wind-tunnel experiment in    turbulence decay, at 10X higher Reynolds number than   was previously possible
-  TWG at Los Alamos provided computational support    by simulating their experimental results at 2048-cubed
-  This was the largest-ever computational simulation of a    turbulence experiment ever performed  (It produced 11 *Tbytes of data *for 3 1/2 eddy turnover times)

**Notes:**
First app (dramatic).
Laboratory study of turbulent decay.
TWG is Darryl and Beth’s Turbulence Working Group -- stimulating group of people.

## TWG Simulation of the KCM Experiment 

- Pseudo spectral and spectral methods
- Resolution: 20483
- 8B grid points
- 11 ***TB*** of data (192***GB*** per snapshot)
- 2048 CPUs
- 1 CPU century* on ASCI-Q
*R*** = 220  (*  *  = 100,000)

- _______________

## 20483 – DNS vs 2563 LANS-

**Notes:**
On right, amount to which these features bend limited by length alpha.

## Holm and Nadiga, JPO 2003

**Notes:**
Not so BIG, but more directly applicable.
Investigated effectiveness of LANS alpha model in a problem where mesoscale variability strongly determines the solution.

## Holm & Nadiga: high res soln

- secondary gyres, generated by mesoscale eddies

**Notes:**
Simple zonal 2-gyre wind stress, get 4 gyre solution. Would call these Fofonoff Gyres?
Might not always see these in snapshots, but they’re seen in the time-mean.

## Holm & Nadiga: 1/4 res

- Secondary gyres are lost

**Notes:**
At lower resolution, with not so much mesoscale variability...

## Holm & Nadiga:1/4 res with -model

- Secondary gyres recovered (but too strong)

## Holm & Nadiga:1/8 res with -model

- Secondary gyres are reasonable, even at 1/8 of fully-resolved res.

**Notes:**
Recover this feature which is generated by mesoscale turbulence, even at 1/8 of “full” resolution.

## What to expect in 3-D ocean model?

- Baroclinic instability occurs within the curve
- Onset occurs at lower wavenumber with , even without increased forcing

- k2

- forcing

- -model

- Eddy viscosity model

**Notes:**
Eddy viscosity model requires increased forcing to get bc instability at lower wavenumber.

## Larger time steps may be possible

- Wingate showed an easing of time step limitation in a shallow water model with increasing 
- “The maximum allowable time step for the shallow water -model and its relation to time implicit differencing”, Mon. Weather Review, to appear 2004.

## How does this fit in with Gent-McWilliams?

- GM was intended for “tracer eqns”
- transport and mixing of temperature, salinity and also passive tracers
- GM has a diffusive component, as well as an advective component
- though it’s non-dissipative in terms of density, adiabatic

**Notes:**
My take on it -- have talked with Beth and Darryl about it -- I’d like to hear if someone else has more insight on this.

##  and GM, continued

-  comes into momentum and tracer eqns
- completely non-dissipative for constant alpha
- GM has been a major advance in ocean modeling for climate, particularly in terms of poleward heat transports
- We believe the -model can be used with GM to improve the turbulent dynamics

## Test problem for -model in POP

- 4-gyre problem of Holm and Nadiga is excellent, but more “inertial” than one would see in the real ocean
- Antarctic circumpolar-like problem motivated by Karsten, Jones and Marshall, JPO, 2002:
- “We argue that the eddies themselves are fundamental in setting the stratification -- both in the horizontal and vertical.”
- Also influenced by work of Henning and Vallis (private communication).

**Notes:**
Fofonoff problem should be done, too.

## Eddy transport across the ACC

- Karsten, Jones and Marshall, JPO, 2002

**Notes:**
Eddy transports act to flatten isopycs, balance surface buoyancy forcing.

## Meridional fluxes: Ekman and Eddy vs surface buoyancy flux

**Notes:**
What we’ll look at to distinguish between eddy-resolving and eddy ignoring...

## and vertical transports

## the test problem

- 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

## just a glance at test problem...

## Conclusions

- Not ready for conclusions

- Discussion?

- -model is on a very solid footing in terms of theory and application
- We aim to find out what it will give us in terms of the effects of unresolved turbulence on the larger scale circulation