Special Relativistic Inviscid Fluid Turbulence in 2+1 · 2018. 7. 11. · John Ryan...
Transcript of Special Relativistic Inviscid Fluid Turbulence in 2+1 · 2018. 7. 11. · John Ryan...
Special Relativistic Inviscid Fluid Turbulence in2+1
John Ryan Westernacher-Schneider, Luis Lehner
Guelph-Waterloo Physics Institute
October 25, 2013
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Motivation
Broad: probe gravitational physics through fluid/gravityduality conjecture.
Specific: test analytic results by Fouxon, Oz (2009)
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Background: Five-Thirds Law
Etot =
∫E (k)dk
[E ] =energy
mass= (
length
time)2
[dk] =1
length
⇒ [E (k)] =length3
time2
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Background: Five-Thirds Law
Ansatz: energy flow ε scale-free (ε 6= ε(k)), so E (k) ∝ εpkq, where
[ε] =energy
mass
1
time=
length2
time3
[k] =1
length
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Background: Five-Thirds Law
Now match units on both sides of E (k) ∝ εpkq:
length3
time2=
length2p−q
time3p
⇒ p =2
3
⇒ q = −5
3
Thus, E (k) ∝ k−5/3... this is true in the inertial range.Scale-free enstrophy flow leads instead to E (k) ∝ k−3.
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Five-Thirds Law
Champagne (1978)JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Relativistic Concerns
length contraction ... whose k are we talking about?
geometrized units ... are dimensional arguments still valid?
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Our Setup
spacetime dimension = 2+1 ... computationally inexpensive,useful for 4D gravity
perfect fluid ... T ab = (P + ρ)uaub + Pηab
equations of motion given by conservation of T ab, ∂aTab = 0
edges of computational grid are periodically identified ... i.e. a2-torus
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Lack of scale implies an equation of state...
To be useful for fluid/gravity duality, we impose that T ab has nointrinsic scale. This implies tracelessness, T a
a = 0. Thus,
0 = T aa = (P + ρ)uau
a + Pδaa
= −(P + ρ) + 3P
⇒ P =1
2ρ.
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Turbulence onset through destabilization...
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Five-Thirds law is recovered up to v ≈ 0.6c ...
Carrasco, Lehner et al. (2012)
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Fouxon, Oz results...
Impose random forcing with stationary statistics: ∂aTab = f b.
Pick two points: label ~0 and ~r .
Suppose |~r | is in the inertial range.⟨T0j(~0, t)Tij(~r , t)
⟩∝ ri , no sum on j.
〈.〉 means ensemble average.
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Preparation for testing Fouxon, Oz results...
Argument relies on approximations. It’s worth checkingnumerically.
Must satisfy mathematical assumptions as best as possible.
Simplify ensemble average by imposing spatiallyhomogeneous, isotropic statistics for f.
Must restrict forcing to narrow range of scales to allow |~r | tobe far from forcing scale.
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
How we construct f
Build it in Fourier space, then Fourier transform to real space...Real condition: f ∗(~k) = f (−~k)
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Non-triviality of the task...
Previous non-relativistic simulations of forced turbulence arevery different.
Usually incompressible... ⇒ ∇ · ~v = 0... This simplifiesequations analytically, and vorticity is forced directly.
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Unforeseen difficulty
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
Summary
Kolmogorov scalings seem to hold in relativistic regime.
Satisfying assumptions of Fouxon and Oz results requiresforcing the fluid.
Appropriate forcing method is not given straightforwardly byprevious work.
Work in progress...
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1
End
Thanks for listening.
JRWS Special Relativistic Inviscid Fluid Turbulence in 2+1