1. Transition in Energy Spectrum for Forced Stratified Turbulence
Workshop on Petascale Computing :
Its Impact on Geophysical Modeling and simulation
5-7 May 2008, NCAR Mesa Laboratory
Transition in Energy Spectrum for Forced Stratified Turbulence
Yoshifumi Kimura
Graduate School of Mathematics, Nagoya University
Today, I’m gonna talk
in collaboration with
Jack Herring (NCAR)

2. Transition in Energy Spectrum for Rotating and Stratified turbulence-3 -3
-5/3
-5/3
We simulate the Navier-Stokes equation with the Boussinesq approximation. The vertical velocity and temperature fluctuation are coupled together by the buoyancy frequency and horizontal velocity components are coupled by the angular velocity of rotation.
Nastrom-Gage’s atmospheric
observation (1985)
(JAS )
Mark Taylor’s climate model simulation (2008)
Both stratification
and rotation are essential
(CCSM project at NCAR)
-3 : enstrophy cascade for Quasi-Geostrophic turbulence (~2D)
-5/3 : Kolmogorov turbulence (3D)

3. The objective of this talk is
to see if there is also a transition in the energy spectrum
for flows with only stratification.
We simulate the Navier-Stokes equation with the Boussinesq approximation. The vertical velocity and temperature fluctuation are coupled together by the buoyancy frequency and horizontal velocity components are coupled by the angular velocity of rotation.

4. Navier-Stokes equation with the Boussinesq approximation
where
We simulate the Navier-Stokes equation with the Boussinesq approximation. The vertical velocity and temperature fluctuation are coupled together by the buoyancy frequency and horizontal velocity components are coupled by the angular velocity of rotation.

5. Numerical Methods forced simulations
2-periodic box with up to 5123 grid points
3rd order time-marching scheme
Initial energy spectrum : E(k) = 0
Force horizontal velocity components
Add red noise to modes within a wave number band

6. Forced turbulence simulation
Starting from energy 0.
Putting stochastic noise (O-U process)
at low wave number modes.
Using spectral eddy viscosity,
cutoff wave number
for 2563 simulation
Lesieur, M. & Rogallo, R.
Phys. Fluids (1989) A1,

7. “Craya-Herring” decomposition
orthnormal coordinates
new variables
linear Navier-Stokes eqs. in Fourier space

8. Enstrophy contours for 5123 simulation side view (forced turbulence)
Thanks to
John Clyne
(NCAR)

9. Enstrophy contours for forced turbulence
About 50% of the maximum

10. History of energy spectra (1 & 2 , N2=100)
transition point
(horizontal,vortical)
(turbulent)
forcing wave band -3 -2
-5/3
-5/3

11. Transition in stratified turbulence spectra
Buoyancy- to inertial range transition in forced stratified turbulence
Carnevale, Briscolini & Orlandi
J. Fluid Mech. (2001) 427 pp
buoyancey spectrum
Inertial range spectrrum
equi-density surface

12. Energy spectra for various N-3
N2=100
-5/3
depends on N
so much?
Don’t much
depend on N. 1
N2k-3 is questionable.
2/3k-3 ?
: enstrophy dissipation

13. Energy and Enstrophy dissipations
0.625 X 10-2
0.143 X 102
N2 = 10
0.626 X 10-2
0.124 X 102
N2 = 50
0.629 X 10-2
0.815 X 101
N2 = 100
0.575 X 10-2
0.615 X 101
Not big change!
Both  and  are small scale quantities !

14. Compensated 1 spectra (high wave number)
1.6
1.0
0.35
0.15
 alone cannot explain
the shift of spectra.
1-2/3 k 5/3
N2=1
N2=10
N2=50
N2=100

15. Compensated 1 spectra (low wave number)
1-2/3 k 3
0.025
Maybe too small.
Perhaps because  is overestimated.
N2=1
N2=10
N2=50
N2=100

16. 2 spectra -2 2
Isotropic  cannot
explain the tendancy

17. Difficulties in the problem
The spectra are extended in the wide range of wave number
space which have different features due to different physical
(or instability) mechanisms.
Parametrization of the large scale spectra needs small scale information (on dissipation.)
Slow processes (such as spectrum building, or large scale structure formation) exist in the calculation.
Those are some reasons we need “petascale computing”