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.