Calculation of complex singular solutions to the 3D ...-100 1 1 Singularities detected through asymptotics of Fourier components (Sulem, Sulem, Frisch 1983) rolled using high-precision
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Calculation of complex singular solutionsto the 3D incompressible Euler equations
Michael SiegelDepartment of Mathematical SciencesNew Jersey Institute of Technology
Russel CaflischDepartment of MathematicsUCLA
Supported by NSF
Numerical Studies
•Axisymmetric flow with swirl and 2D Boussinesq convection-Grauer & Sideris (1991, 1995),Pumir & Siggia (1992)Meiron & Shelley (1992), E & Shu (1994) Grauer et al (1998), Yin & Tang (2006)
• High symmetry flows-Kida-Pelz flow: Kida (1985), Pelz & coworkers (1994,1997)-Taylor-Green flow: Brachet & coworkers (1983,2005)
• Antiparallel vortex tubes-Kerr (1993, 2005)-Hou & Li (2006)
•Pauls et al(2006).: Study of complex space singularities for 2D Euler in short time asymptotic regime
Axisymmetric flow with swirl
•Annular geometry
1 2 , 0 2r r r z π< < < <•Steady background flow
(0, , )( )zu u rθ=uchosen to satisfy Rayleigh’s criterion for instabilityand an unstable eigenmode
•Caflisch (1993), Caflisch & Siegel (2004)
1ˆ ( ) σ+iz tr eu
Background flow
0Background flow is smoothed vortex sheet at (motivated by Caflisch, Li, Shelley 1991)
ri
1 2
Pure swirling flow is unstable if (Rayleigh criterion)Γ > Γ
i
Traveling wave solution
iConstruct complex, upper-analytic traveling wave solution
σiTraveling wave with speed in Im(z) direction
σ
+
∞−
+=
= +
= ∑ ( )
1
( ) ( , , ) in which
ˆ ( ) ik z i tk
k
r r z t
r e
u u u
u u
Baker, Caflisch & Siegel (1993)Caflisch(1993), Caflisch & Siegel (2004)
1ˆ is linearly unstable eigenmode with eigenvalue Traveling wave speed is thus determined from linear
eigenvalue problem and is independent of the amplitude
σσ
uii
Motivation for traveling wave form
Construction of solution is greatly simplified One way coupling among wavenumbers so mode depends only o
-Degrees of freedom reduced
-Computational errors minimized since no truncationn
k k k′ ′<
i
i
ˆEquation f
or aliasing errors in r
or has form
is second order ODE operato
estriction to finite number of Fourier components
ˆ ˆ ˆ( , , , )rk
k
LL −=
k
k k 1 k 1u F u u uu
i…
i
Motivation (cont’d)
Singularities at travel with speed in Im z direction, reach real z line in finite time (for 0)Singularities detected through asymptotics of
ˆ Fourier coefficients (Sulem, Sulem & Fris
u
r i
i
z z i zz
σ= +
≤
i
i
Provide information on generic form of singuch 19
lari83)
tiesi
Perturbation construction of real singular solution
* *Consider where ( ) , , are exact solutions of Euler equations satisfies system of equations in which forcing
terms are quadratic, i.e.,
z+ − − +
+ −
+ − −
= + + + =
+ +
⋅∇ +
u u u u u u uu u u u uu
u u u
iii
2
2sin
We want , ( ) ( )
Full construction requires analysis showingthat singularity
( ), ( ) ( )
of is same or weaker than that of ,
reg gu O T u O T O
O O
ε
ε
ε
ε+
+ −
+ −
⋅∇
= ⇒ =
+
uu u u
uu u
i∼
i∼
Real remainder
•Similar approach used in studies of singularityformation on vortex sheets-Caflisch & Orellana (1989), Duchon & Robert (1988)-Siegel, Caflisch, Howison (2004)-Cordoba(2006)
•For vortex sheets, singularity formation is associatedwith ill-posedness
•For Euler equations, traveling wave solution comes from balance between instability and nonlinearity
2
Numerical construction in Caflisch (1993) was for 0Singularity position depends on , i.e., Im z= (r) Result: ( ) where 1/ 3 Amplitude of (i.e., ) is (1)
r
r
ru c z i t i r
u c O
α
ρσ α+
+
=
−
− − = −
zui
ii ∼i
γ
γ
= = ⇒
i
i
i
- vortex sheet strength A -
Vortex sheets in Boussinesq approximation Siegel (1992), (1995)
Pure Boussinesq ( 1, 0) traveling waves of O(1) amplitud
density difference
e Pure vortex sh
A
γ
γ εε
= = ⇒
<< =→ →
i
eet ( 0, 1) no traveling waves due to conservation of vorticity on sheet For 1, 1 small amplitude traveling waves
0 as A 0.
A
A
Vortex sheet analogue
Numerical method
0 0 sin , cos2 2
0 1
-Instability co
Pseudospectral in , 4th order discretization (in r) for Background velocities
Numerical method is accurate but unstablent
kz L
γ γθ θ
γ
θ π θ π
θ
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠< <
z zu u u u
ii
i
-100
1
1
Singularities detected through asymptotics of Fourier components (Sulem, Sulem, Frisch 1983)
rolled using high-precision arithemetic (10 )
( ( ))ˆ exp( (
))k
u c z i
u c k k i
α
α
µ δ
δ µ
−+
−
≈ − −
≈ − +
i
Caflisch & Siegel (2004)
0
0
0
Shift in time by mult. of
Fourier coeff. by shift in imag.component of sing. position by
kt
t kth
et
σ
σ
≡
≡
1ˆAdjustable parameters: u , γθ
•Square root singularity does not satisfyBeale, Kato, Majda theorem
sup
Singularity formation at time T
( , ) T
t dtω
⇔
= ∞∫ xx
3D traveling wave
Control numerical instability Look for traveling wave solution, periodic in ( , , )
ˆ exp ( )
( , , ), ( , , )
Simplify construction -Base flow -Instability driven by f
k
x y z
x y zi i t
k l m σ σ σ>
= ⋅ −
= =
=
∑k 0
u u k x σ
k σ
u 0
ii
i
orcing termˆ ( )= exp ( )
Euler equations ˆ ˆ ˆ ˆ ( , , , )
, 1, ,k
j
i i t
L
j n
⋅ −
=
< =
∑
1 2 n
kk<N
k k k k k
F x F k x σ
u G u u uk k
i…
…
{ }( )
( ){ } ( )
2 2 ( ) ( ) ( )1
( ) 2 2 ( ) ( )
1 ( ) ( ) 1
ˆ ˆ ˆˆ( )( )
ˆ ˆ ˆ ˆ
ˆ ˆˆ ˆ w ( )
where 0
Small amplitude singularity by choice of
, 0ˆ ˆ ˆ
x y z
x y z
z x
l m M lkM kmMuv lkM m k M lmM
k kM mM mk u
k
−
− −
⎛ ⎞+ − −⎛ ⎞ ⎜ ⎟= ⋅ ⋅⎜ ⎟ ⎜ ⎟− + + −⎝ ⎠ ⎝ ⎠
= ⋅ − +
⋅ ≠ ≠
= +
k k kk
k k k k
k k k k
k k k
σ k k k
σ k
σ kM F Ni
forcing Introduce into forcing; when =0, solution
is entire. For small , singularity amplitude is O( )
ε ε
ε ε
ui
i
( )℘ − ⋅∇k u u
Numerical method
kˆ Nonlinear terms N evaluated by pseudospectral method
No truncation error in restriction to finite Since N is quadratic, padding with zeroes eliminates
aliasing error from pseudospectral part
kiii
of calculation Extreme numerical instability eliminated; very mild
instability controlled by spectral filtering i
We compute traveling wave , is real+++ +++ −−−+u u ui
k
δ
α
1
ˆ1D fit: ( , )
log( ( , ))
ikxk
ku u y z e
u c x i t y zσ ρ
∞
=
+++
=
− +
∑∼
Fit of singularity parameters (1,0,0), 1σ ε= =
c
•BKM satisfied
k
δ
α
Fit of singularity parameters 0.1ε =
c
δ
α
Fit of singularity parameters 0.01ε =
k
c
Singularity amplitude
max u+++
ε
c
yz
(1,0,0)=σIm ( , )x y zρ− =
Im x−
Singular surface
•Geometry of singular surface is useful for analysis
Conclusion
•Introduced new method to compute singular solutions to3D Euler equations with complex velocity
•Eliminated numerical instability observed in earlier calculations;introduced techniques to achieve small amplitude singularity
•Results suggest a traveling wave singularity to 3D complex Euler equations in which the velocity blows up;satisifies Beale, Kato, Majda theorem, smooth singular surface
• Easily generalized to other problems,e.g., 2D and 3D MHD, quasi-geostrophic equation, etc.
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