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2D Turbulence
Paul Cueva and Sean Seyler
Cornell University
pdc23@cornell.edu, sls374@cornell.edu
December 14, 2011
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 1 / 28
Overview
1 Why 2D turbulence?
2 2D turbulence: the inverse cascade of energy to long length-scales
3 A brief review of fluid dynamics
4 Band-limited forcing and power laws
5 Energy transfer in 2D versus 3D turbulence
6 Applying RG methods to 2D turbulence?
7 Stochastic forcing of 2D N-S
8 A nice way to conclude our discussion
9 Acknowledgments
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 2 / 28
Approximate 2D systems: atmospheric dynamics
Turbulence on the sphere
Assume surface effects dominate
Weather prediction
Figure: Earth’s atmosphere
Source: The New Republic (website)
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 3 / 28
2D turbulence: the inverse cascade of energy to longlength-scales
Bubbles
Immiscible liquids
Figure: Turbulence on a soap film
Source: Fotolog website
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 4 / 28
Tractable problem: conformal solution by Polyakov
Shown to exist for every case
Doesn’t mean it’s easy to understand...
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 5 / 28
Tractable problem: allegorical/numerical solution byOnsager, Kraichnan
Problem related to solvable ones: lattice/vortex gas
Can get scaling and statistics
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 6 / 28
Tractable problem: statistical solution by Miller and Cross
Equilibrium solution maximizes entropy
Quite different from standard stat physics (more on this soon)
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 7 / 28
2D turbulence: the inverse cascade of energy to longlength-scales
〈 vor nf early.mov 〉 〈 psi nf early.mov 〉
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 8 / 28
2D turbulence: the inverse cascade of energy to longlength-scales
〈 psi nf late.mov 〉
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 9 / 28
A brief review of fluid dynamics
2D Navier-Stokes:
∂u
∂t+ u · ∇u = −∇p
ρ+ ν∇2u
Incompressibility:∇ · u = 0
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 10 / 28
A brief review of fluid dynamics
2D Navier-Stokes and incompressibility
∂u
∂t+ u · ∇u = −∇p + ν∇2u ∇ · u = 0
In 2D, the vorticity is a scalar (in the z-direction):
ω = (∇× u) · z
For 2D only, we can define the stream function ψ, where:
u = −z×∇ψ
ω = −∇2ψ
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 11 / 28
A brief review of fluid dynamics
Fourier transformed 2D vorticity equation
∂ωk
∂t+∑k′
Mk,k′ωk′ωk−k′ = νk2ωk k · u(k, t) = 0
Mk,k′ takes care of the nonlinear term:
Mk,k′ =z · (k× k′)
2
(1
k ′− 1
k
)
Nonlinear advection term responsible for energy transfer to differentlength-scales
Energy transfer T (k , q, p) occurs between a triad of wave numbers:
k → k k ′ → q k − k ′ → p
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 12 / 28
Constants of motion for 2D turbulence
Infinite number of constants of motion of the form:∮u · dl =
∫ω · dS = const
The only other invariants are quadratic in ω:
Conservation of energy
∂E
∂t=
∂
∂t〈|uk |2〉 = 〈ω
2k
k2〉 = 0
Conservation of enstrophy
∂
∂tΩ =
∂
∂t〈k2 |uk |2〉 =
∂
∂t〈ω2
k〉 = 0
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 13 / 28
Power laws in the band-limited forcing (BLF) energyspectrum
Figure: Long-time energy spectrum of 2D N-Swith BLF
Figure: Energy pileup at small wavenumbersdue to finite grid size
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 14 / 28
The inverse cascade as seen from the BLF energy spectrum
Inverse cascade?
The energy ”cascades” or ”is pumped” from shorter to longer length scales
〈 bl spec 256.mov 〉
This phenomenon is due solely to the nonlinear term in N-S
If dissipative (viscous) forces dominate (the nonlinear term):
There will be no cascade
The power law will be flat for k < kforcing
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 15 / 28
Features of 2D turbulence
Locality of energy transfer
Small eddies swept along by large eddies
Similar-sized eddies coalesce
(Board algebra to introduce 2D energy transfer)
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 16 / 28
Comparison with features of 3D turbulence
3D vortex stretching
Kolmogorov energy cascade
Equilibrium (direct cascade)
Energy goes to high wavenumbers
Viscosity damps out high wavenumbers
For numerical solutions, grid cell size can act as viscous cutoff
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 17 / 28
Power law prediction through dimensional analysis
(More board work to reproduce energy and enstrophy cascade powerlaws)
2D gives ”inverse energy cascade” and ”direct enstrophy cascade”
→ energy cascades to small wavenumbers
→ enstrophy cascades to large wavenumbers
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 18 / 28
Power law prediction through dimensional analysis
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 19 / 28
Onsager/Kraichnan’s point vortex gas
Considers point vortices interacting with near neighbors
End up with vortex segregation with disorder/”temperature” comingfrom forcing
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 20 / 28
Point vortex simulation using particle-in-cell method
Source: Joyce and Montgomery
Red and green regionscorrespond to oppositely-signedpoint vortices
Left snapshot at t = 0 withchecker board initial condition
Right shows like-signed pointvortices clustering at t = 50
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 21 / 28
Stochastic forcing of 2D N-S
FT 2D stream function equation with stochastic forcing
ψk +∑k′
Mk,k′ψk′ψk−k′ + νk2ψk = Wk(t)
The stochastic (white noise) term Wk has:
an amplitude selected at random from a time-scaled Gaussiandistribution at each time step
a randomly-selected phase at each time step
→ Scaled random walk in time applied at each k (Brownianmotion)
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 22 / 28
Stochastic forcing of 2D N-S
FT 2D stream function equation with stochastic forcing
ψk +∑k′
Mk,k′ψk′ψk−k′ + νk2ψk = Wk(t)
To get the stochastic term Wk:
Using theory of hydrodynamic fluctuations, consider fluctuations inpressure tensor δP
δP turns out to be anti-symmetric
Taking curl of N-S (to get vorticity equation) gives white noisefluctuation term: ∇2W
ω = −∇2ψ → only have W in equation of motion for stream function
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 23 / 28
White-noise Forcing of 2D N-S
〈 vor wn.mov 〉 〈 psi wn.mov 〉
Ising-model-like scale invariance
Like-signed vortices cluster → correlation with neighbors
Striking similarities to MD simulations of Onsager’s discrete vortexmodel
Thermal fluctuations give rise to discrete vortices on dissipation scale?
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 24 / 28
Energy spectrum for white noise forced fluid
Figure: Energy spectrum shows evidence of k−5/3 power law
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 25 / 28
Miller and Cross and a thermodynamic explanation of theinverse cascade
Final state of 2D turbulent system should maximize entropy
Clustering of local regions of vorticity maximizes number ofmicrostates
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 26 / 28
Acknowledgments
Charles Seyler - Mentor, architect and designer of code used forsimulation
Jane Wang - Informal guide in the world of turbulence
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 27 / 28
References
C. Seyler (1975)
Guiding-Center Plasmas and Inviscid Navier-Stokes Fluids in Two Dimensions
Ph.D. thesis, University of Iowa
G. Joyce and D. Montgomery (1973)
Negative Temperature States for the Two-Dimensional Guiding-Centre Plasma
J. Plasma Physics 10, 1, pp. 107-121
J. Miller, P. Weichmann, and M. Cross (1992)
Statistical Mechanics, Euler’s Equation, and Jupiter’s Red Spot
Phys. Rev. A 45, 2328-2359
M. Bajic (2010)
Onsager Theory of Hydrodynamic Turbulence
Seminar, University of Ljubljana
M. Rivera (2000)
The Inverse Energy Cascade of Two-Dimensional Turbulence
Ph.D. Thesis, University of Pittsburgh
R. Kraichnan (1967)
Inertial Ranges in Two-Dimensional Turbulence
Phys. Fluids 10, 1417
R. Kraichnan and D. Montgomery (1980)
Two-dimensional Turbulence
Rep. Prog. Phys. 43, 43547
Paul Cueva and Sean Seyler (Cornell) 2D Turbulence December 14, 2011 28 / 28
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