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Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral Charalampos Markakis Mathematical Sciences, University of Southampton work in progress in collaboration with: John Friedman, Masaru Shibata, Niclas Moldenhauer, David Hilditch, Sebastiano Bernuzzi, Koutarou Kyutoku, Bernd Brüegmann

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Page 1: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Hamiltonian Fluid Dynamics &

Irrotational Binary Inspiral

Charalampos MarkakisMathematical Sciences, University of Southampton

work in progress in collaboration with: John Friedman, Masaru Shibata, Niclas Moldenhauer, David Hilditch, Sebastiano Bernuzzi, Koutarou Kyutoku, Bernd Brüegmann

Page 2: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Introduction

• Gravitational waves from neutron-star and black-hole binaries carry valuable information on their physical properties and probe physics inaccessible to the laboratory.

• Although development of black-hole gravitational wave templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged.

• Recent progress by groups in Kyoto (SACRA), Caltech-Cornell-CITA-AEI (SpEC), Frankfurt (Whisky), Jena (BAM), Illinois, etc.

• The Valencia scheme has been a workhorse for hydro in numerical relativity…

1( ) ( ) 0

1( ) 0

u g ug

T gT Tg

a aa a

b b g bb a b a ab g

r r = ¶ - =-

= ¶ - -G =-

Page 3: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Introduction

• Gravitational waves from neutron-star and black-hole binaries carry valuable information on their physical properties and probe physics inaccessible to the laboratory.

• Although development of black-hole gravitational wave templates in the past decade has been revolutionary, the corresponding work for double neutron-star systems has lagged.

• Recent progress by groups in Kyoto (SACRA), Caltech-Cornell-CITA-AEI (SpEC), Frankfurt (Whisky), Jena (BAM), Illinois, etc.

• The Valencia scheme has been a workhorse for hydro in numerical relativity, but considering alternative hydrodynamic schemes can lead to further progress…

• Hamiltonian methods have been used in all areas of physics but have seen little use in hydrodynamics

Page 4: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Introduction

• Constructing a Hamiltonian requires a variational principle• Carter and Lichnerowicz have described barotropic fluid motion

via classical variational principles as conformally geodesic

00 1

b

a

dx dx dph g d h

d d

a b r

abd tt t r

- = = +ò ò

Page 5: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Introduction

• Constructing a Hamiltonian requires a variational principle• Carter and Lichnerowicz have described barotropic fluid motion

via classical variational principles as conformally geodesic

• Moreover, Kelvin’s circulation theorem

implies that initially irrotational flows remain irrotational.

00 1

b

a

dx dx dph g d h

d d

a b r

abd tt t r

- = = +ò ò

0c

dhu dx

d t

aat

=ò 0S

tS

Page 6: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Introduction

• Constructing a Hamiltonian requires a variational principle• Carter and Lichnerowicz have described barotropic fluid motion

via classical variational principles as conformally geodesic

• Moreover, Kelvin’s circulation theorem

implies that initially irrotational flows remain irrotational.

• Applied to numerical relativity, these concepts lead to novel Hamiltonian or Hamilton-Jacobi schemes for evolving relativistic fluid flows, applicable to binary neutron star inspiral.

00 1

b

a

dx dx dph g d h

d d

a b r

abd tt t r

- = = +ò ò

0c

dhu dx

d t

aat

=ò 0S

tS

Page 7: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Carter-Lichnerowicz variational principlesfor barotropic flows

• Carter’s Lagrangian:

• Canonical momentum:

• Carter’s superHamiltonian:

(on shell2

)2

hh h

g u ua bab = -= -

;p huua aa

¶= =

¶ dx

ud

aa

t=

10

2 2h

p u g p ph

a aba a b= == - +

Page 8: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Carter-Lichnerowicz variational principlesfor barotropic flows

• Carter’s Lagrangian:

• Canonical momentum:

• Carter’s superHamiltonian:

• Euler equation in Carter-Lichnerowicz form:

2 2u h

h hg ua bab= - = -

£ 0 (Euler-Lagrange)u

dpp

d xa

a aat¶

- = - =¶

;p huua aa

¶= =

¶ dx

ud

aa

t=

( ) 0 (Hamilton)dp

u p pd x

bab a a b aat

¶+ = - + =

10

2 2h

p u g p ph

a aba a b= == - +

Page 9: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Constrained Hamiltonian approach

1

2

1 0

( ) fluid velocity measured by normal observers

/ fluid velocity measured in local coordinates

canonic o1

al m

b ba b

aba a

a a a

a a

aa aa

dx dxh g dt h dt

dt dt

v

v dx dt

v

Lp h hu

a b

abd d a g

a b-

- = - n n =

n = +

=n

n

¶= = =

¶ -

ò ò

mentum of a fluid element

Page 10: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Constrained Hamiltonian approach

1

2

1 0

( ) fluid velocity measured by normal observers

/ fluid velocity measured in local coordinates

canonic o1

al m

b ba b

aba a

a a a

a a

aa aa

dx dxh g dt h dt

dt dt

v

v dx dt

v

Lp h hu

a b

abd d a g

a b-

- = - n n =

n = +

=n

n

¶= = =

¶ -

ò ò

2

mentum of a fluid element

Constrained Hamiltonian: a aba a a b t

ap p h p p hv L uH b a g= - + = -= +-

Page 11: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Constrained Hamiltonian approach

1

2

1 0

( ) fluid velocity measured by normal observers

/ fluid velocity measured in local coordinates

canonic o1

al m

b ba b

aba a

a a a

a a

aa aa

dx dxh g dt h dt

dt dt

v

v dx dt

v

Lp h hu

a b

abd d a g

a b-

- = - n n =

n = +

=n

n

¶= = =

¶ -

ò ò

2

“ ” Dire Strai

mentum of a fluid element

Constrained Hamiltonian:

t

gh s

a abaa a a b t

You've got the action, you've got the motio

p pH v L h p hu

n

pb a g+ + = -

-

= - = -

Page 12: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Constrained Hamiltonian approach

1

2

1 0

( ) fluid velocity measured by normal observers

/ fluid velocity measured in local coordinates

canonic o1

al m

b ba b

aba a

a a a

a a

aa aa

dx dxh g dt h dt

dt dt

v

v dx dt

v

Lp h hu

a b

abd d a g

a b-

- = - n n =

n = +

=n

n

¶= = =

¶ -

ò ò

2

mentum of a fluid element

Constrained Hamiltonian:

Euler-Lagrange equation:

Hamilton equation:

= ( £ ) 0

=

a a

at

ba a a b

a

a aa

bat aa

t

dp Lp L

dt x

dp Hp v

dt x

p p h p p hH v L u

u

b a g

¶- ¶ + - =

¶+ ¶

+ + = -= -

- =

( ) 0

(independent of gravity theory)

b a a b ap p H - + =

Page 13: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Conservation of circulation

0

( £ ) ( £ ) 0

orticity 2-form:

Eul

( £

er-Lagrange equation:

V

Kelvin's theor )em: 0t t

t a a t ab

ab b a

a a b a ba ab t ab

a b

p L

p p

d dp dx dx x dx x

dt dtd d

u u

u

w

w

w w

¶ + = ¶ + =

=

=

-

= ¶ + =ò ò ò

• The most interesting feature of Kelvin's theorem is that, since its derivation did not depend on the metric, it is exact in time-dependent spacetimes, with gravitational waves carrying energy and angular momentum away from a system. In particular, oscillating stars and radiating binaries, if modeled as barotropic fluids with no viscosity or dissipation other than gravitational radiation exactly conserve circulation

• Corollary: flows initially irrotational remain irrotational.

0S

tS

Page 14: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Irrotational hydrodynamicsIrrotational flow: 0

Hamilton equation: 0

Hamilton-Jacobi equation: 0

Example: In the dust limit on a Minkowsky background, on

( )

e obt

a a

t a a

t

bb a

b a a

a b

bp p Sp

p

H

p

S

p Hv+

¶ +

- = =

=-

+ =

2 2 2

ains a relativistic Burgers equation:

0

Obtained noncovariantly by LeFloch, Makhlofand and

( / 1 ) ( 1 ) 0 1 (

Okutmustur, SINUM 50, 2136 (2012) by

algebraic manipu

)t a a t

S Su u u¶ - +¶ + = ¶ + + =

lation of the Euler equation in Minkowski and Schwarzschild charts. The fact that

these are Hamilton equations and can be obtained covariantly for arbitrary spacetimes is unnoticed.

Solutions to HJ equation are NOT unique.

Nevertheless, 'viscosity' solutions to HJ equation are unique.

Page 15: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral
Page 16: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Irrotational hydrodynamics

2

2

0

Analytic 1+1 solution for homogeneously

1 ( )

t

ranslating flow:

1) ( , ) ( )

Numerical soluti

(

o

,1

n:

t

x

S

x S t x t x

S

tu u uu

=

= =

-- +

+ +

Page 17: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Irrotational hydrodynamics

2 2

2

Analytic 1+1 solution for homogeneously tra

1 ( )

nslating flow:

1) ( ( ,

1, ) ( )

Numerical 'viscosity' solution:

t x x

x

S

x S

S S

t t x t xu u uu

e=

= = - +

¶ + + ¶ ¶

-

Page 18: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Irrotational hydrodynamicsIrrotational flow: 0

Hamilton equation: 0

Hamilton-Jacobi equation: 0

For barotropic fluids, the above equation is coupled to the c

( )

a a b a a

t a a

t

a

b

bb a b

Sp p p

p p

S H

p Hv+

¶ +

- = =

+

=-

=

1,2

0

where : det( )

Characteristics : 0

ontinuity equation, resulting in a system

,

k

kt ki i

t tij

k

p H

g u u

r r u

d

r r a g r g g

l

æ öæ ö ÷÷ çç ÷÷ çç¶ + ¶ =÷÷ çç ÷÷ çç ÷ ÷ç çè ø è ø

= - = =

=

2 2 1 2 2 1/2 2 2 2 2 1/23,4 s s s s s

s

(1 ) { (1 ) (1 ) [(1 ) (1 )( )

Complete eigenbasis The system is strongly hyperbolic (for fini e

}

t

]

)

k k kk k kc c c c c

c

l a n n n n g n b-

= - - - - - - -

Page 19: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

ConclusionsNotable features:

• Unlike Valencia, recovery of primitives from conservatives requires no atmosphere: ui is recovered via dividing pi=hui by specific enthalpy h which is 1 at the surface (no division by zero)

• Like Valencia, strong hyperbolicity is lost when cs = 0: eigenbasis not complete, system becomes weakly hyperbolic instability on surface

• Instead of artificial atmosphere, can use crust EOS with small but nonzero cs near surface: sound speed in a realistic NS crust (outer 1 km) cs ~ 0.05

0k

kt ki i

p H

r r u

d

æ öæ ö ÷÷ çç ÷÷ çç¶ +¶ =÷÷ çç ÷÷ çç ÷ ÷ç çè ø è ø

Page 20: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

Sound speed profile of a TOV star

0 2 4 6 8 10 120.0

0.1

0.2

0.3

0.4

0.5

2S

c

(km)R

S~ 0.05c

Credit: Christian Krüger

Page 21: Hamiltonian Fluid Dynamics Irrotational Binary Inspiralcompact-merger.astro.su.se/MICRA2015/slides/CharalamposMarkak… · Hamiltonian Fluid Dynamics & Irrotational Binary Inspiral

ConclusionsNotable features:

• Unlike Valencia, recovery of primitives from conservatives does requires no atmosphere:ui is recovered via dividing pi=hui by specific enthalpy h which is 1 at the surface (no division

by zero)• Like Valencia, strong hyperbolicity is lost when cs = 0: eigenbasis not complete, system

becomes weakly hyperbolic instability on surface• Instead of artificial atmosphere, can use crust EOS with small but nonzero cs near

surface: sound speed in a realistic NS crust (outer 1 km) cs ~ 0.05. Then, extrapolating the EOS to the exterior (h<1) allows one to evolve smooth fields and obtain pointwiseconvergence on the surface, which is unattainable with an artificial atmosphere.

• Scheme may be combined with symplectic integration or constraint damping methods that preserve symplecic structure and circulation

• SPH schemes based on the Lagrangian or Hamiltonian formulation possible• Extension beyond irrotational flows also possible

ReferenceC. Markakis, arXiv:1410.7777

0k

kt ki i

p H

r r u

d

æ öæ ö ÷÷ çç ÷÷ çç¶ +¶ =÷÷ çç ÷÷ çç ÷ ÷ç çè ø è ø