Hamiltonian structure for an incompressible Euler two-layer fluid - - PowerPoint PPT Presentation

Hamiltonian structure for an incompressible Euler two-layer fluid

joint work with R. Camassa, S. Chen, G. Falqui, M. Pedroni Giovanni Ortenzi Dipartimento di Matematica e Applicazioni Universit` a di Milano Bicocca Bi-Hamiltonian Systems and All That Conference in honour of Franco Magri’s 65th birthday Milano, IX-2011

The physical two-layer system

ζ(x,t)

x z

2 2

ρ

1 1

ρ

Ω Ω η (x,t)

1

η (x,t)

2

P(x,t) ζ(x,t)

2

ρ

1

ρ

η (x,t)

1

η (x,t)

2

(a) (b) n n

A model for the study of the internal waves in a stratified fluid.

The model

The equations of motion for an incompressible two-layer Euler fluid with velocities of upper and lower layer uj = (uj, wj), j = 1, 2 are uj x + wj z = 0, uj t + ujuj x + wjuj z = −pj x/ρj, wj t + ujwj x + wjwj z = −pj z/ρj − g, with suitable boundary conditions

The model

The equations of motion for an incompressible two-layer Euler fluid with velocities of upper and lower layer uj = (uj, wj), j = 1, 2 are uj x + wj z = 0, uj t + ujuj x + wjuj z = −pj x/ρj, wj t + ujwj x + wjwj z = −pj z/ρj − g, with suitable boundary conditions

• w1(x, z = h1, t) = 0,

w2(x, z = −h2, t) = 0 top and bottom

• ζ = 0, ui = 0, pi z = −gρi

x → ±∞

• ζt + uiζx = wi, i = 1, 2, p1 = p2 at the interface z = ζ(x, t)

The layer mean equations

η1(x, t), η2(x, t) height of the upper and lower layer. The means of a quantity f are f i(x, t) := 1 ηi

top

bottom f (x, z, t) dz,

i = 1, 2 . The equations for the horizontal component of the system become (ηiui)t + (ηiuiui)x = −ηi pi x ρi ηi t + (ηiui)x = 0, i = 1, 2 η1 + η2 = h.

The layer mean equations

η1(x, t), η2(x, t) height of the upper and lower layer. The means of a quantity f are f i(x, t) := 1 ηi

top

bottom f (x, z, t) dz,

i = 1, 2 . The equations for the horizontal component of the system become ui t + uiui x = −Px ρi − (−1)igηi x ηi t + (ηiui)x = 0 i = 1, 2 η1 + η2 = h. At the leading order of the the long wave asymptotics the hydrostatic approximation holds everywhere: pi(x, z, t) = P(x, t) − gρi(z − ζ(x, t))

Main question addressed in this talk

Is the two-layer system Hamiltonian?

Main question addressed in this talk

Is the two-layer system Hamiltonian? Answer: Yes

Decoupling the system

For a one-layer system with free surface the equations for the mean quantities (omit overline) are ut + uux + gηx = −Px ρ ηt + (ηu)x = 0 where P is an external assigned pressure. This system is Hamiltonian in the variables m = ρηu, η w.r.t. the Poisson structure Π = − m∂ + ∂m η∂ ∂η

• and the Hamiltonian

H = m2 2ρη − gρ 2 η2 + ηP

Coupling two one-layer systems

The system of a two-layer fluid in a channel can be obtained as a (suitable) coupling of two copies of one-layer system

• the total height of the fluid is fixed: η1 + η2 = h
• the interface pressure of the two fluid has to be the same:

P1 = P2

Coupling two one-layer systems

The system of a two-layer fluid in a channel can be obtained as a (suitable) coupling of two copies of one-layer system

• the total height of the fluid is fixed: η1 + η2 = h
• the interface pressure of the two fluid has to be the same:

P1 = P2 Decoupled u1t + u1u1x − gη1x = −P1x ρ1 η1t + (η1u1)x = 0 u2t + u2u2x + gη2x = −P2x ρ2 η2t + (η2u2)x = 0

Coupling two one-layer systems

The system of a two-layer fluid in a channel can be obtained as a (suitable) coupling of two copies of one-layer system

• the total height of the fluid is fixed: η1 + η2 = h
• the interface pressure of the two fluid has to be the same:

P1 = P2 Coupled u1t + u1u1x − gη1x = −Px ρ1 η1t + (η1u1)x = 0 u2t + u2u2x + gη2x = −Px ρ2 η2t + (η2u2)x = 0 η1 + η2= h

Coupling two one-layer systems

The system of a two-layer fluid in a channel can be obtained as a (suitable) coupling of two copies of one-layer system

• the total height of the fluid is fixed: η1 + η2 = h
• the interface pressure of the two fluid has to be the same:

P1 = P2 Two field system ρ1 ρ2 + η1 h − η1

• u1
• t

+ ρ1 ρ2 − η2

1

(h − η1)2 u2

1

2 + (1 − ρ1 ρ2 )gη1

• x

= 0 η1t + (η1u1)x = 0.

Mathematical consequences of the coupling

P1 = P2 and η1 + η2 = h implies η1u1 + η2u2 = Q(t). If we choose the velocities zero at infinity, then η1u1 + η2u2 = 0.

Mathematical consequences of the coupling

P1 = P2 and η1 + η2 = h implies η1u1 + η2u2 = Q(t). If we choose the velocities zero at infinity, then η1u1 + η2u2 = 0. RELEVANT PROPERTY The constraints φ1 := η1 + η2 − h = 0, φ2 := η1u1 + η2u2 = 0 satisfy

• φ1t = −(φ2)x and φ1t = 0 =

⇒ φ2 = 0

• all the higher dynamical consequences of φ1 are compatible

with the equations of motion

The Dirac reduction

What happens to the Hamiltonian structure?

The Dirac reduction

What happens to the Hamiltonian structure? We can perform a Dirac reduction. ΠD

m1m1 =

• Πm1m1 − ∑

i,j=1,2

Πm1φiΠ−1

φi φj Πφjm1

• φ1=φ2=0

ΠD

m1η1 =

• Πm1η1 − ∑

i,j=1,2

Πm1φi Π−1

φi φj Πφjη1

• φ1=φ2=0

ΠD

η1η1 =

• Πη1η1 − ∑

i,j=1,2

Πη1φi Π−1

φi φj Πφjη1

• φ1=φ2=0

where {f (x), g(y)} = Πf ,gδ(x − y),

The reduced Hamiltonian structure

Finally, after some algebraic manipulation (integration by parts) the Dirac-Poisson structure will be ΠD = − M∂ + ∂M N∂ ∂N

• .

where M =ρ2

1(h − η1)2 − ρ1ρ2η2 1

(ρ2η1 + ρ1(h − η1))2 m1 N = ρ1 ρ1 η1 − ρ2 h − η1 −1 and the Hamiltonian hD := m2

1

2ρ1η1 + ( ρ2

ρ1 m1)2

2ρ2(h − η1) + gρ2 (h − η1)2 2 − gρ1 η2

1

2 .

The translational invariance along the channel

In view of the study of symmetries we change the variables m =

• 1 +

η1/ρ1 (h − η1)/ρ2

• m1,

η = η1. In these new variables the Poisson structure becomes ΠD(m, η) = − m∂ + ∂m η∂ ∂η

• .

and the Hamiltonian hD(m, η) = m2 2ρ1η

• 1 +

η/ρ1 (h − η)/ρ2 −1 + gρ2 (h − η)2 2 − gρ1 η2 2 .

The reduced system is mt+ 2ρ1(h − η)2 + ρ2η(h − 2η) (ρ1(h − η) + ρ2η)2 m2 2η + g(ρ1 − ρ2) 2 η2

• x

= 0, ηt+

• 1 +

η/ρ1 (h − η)/ρ2 −1 m ρ1

• x

= 0. m is the conserved density related to the x-invariance. It is not the physical momentum which is not, in general, a conserved quantity.

The Boussinesq reduction

When 1 − ρ1 ρ2 << 1 we can consider ρ1 = ρ2 only in the terms where g appears mt + 2h − 3η 2ρ1hη m2 + g(ρ2 − ρ1) 2 η2

• x

= 0, ηt + m(h − η) hρ1

• x

= 0. This sytem can be diagonalized [Boonkasame et al.(2011)] and it is equivalent to the dispersionless AKNS sytem which is bi-Hamiltonian. One of the two AKNS Hamiltonian structures is a restriction of our Poisson bracket when ρ1 = ρ2.

Conclusions and work in progress

Hamiltonian structure of a two-layer incompressible Euler fluid in a channel Two decoupled one-layer Dirac reduction

• Two-layer system

Boussinesq approximation

• dispersionless AKNS

Conclusions and work in progress

Hamiltonian structure of a two-layer incompressible Euler fluid in a channel Two decoupled one-layer Dirac reduction

• Two-layer system

Boussinesq approximation

• dispersionless AKNS
• Are the Hamiltonian structures preserved in the

non-hydrostatic case?

• Can the bi-Hamiltonian structure be lifted to the

non-Boussinesq system?

Bibliography

• R. Camassa, S. Chen, G. Falqui, G.O., M. Pedroni, An inertia

‘paradox’ for incompressible statified Euler fluids, submitted for publication

• R. Camassa, S. Chen, G. Falqui, G.O., M. Pedroni, Hamiltonian

structure for a two-layer incompressible Euler fluid, in progress

• R. Camassa, D. Holm, Levermore, Long-time effects of bottom

topography in shallow water, Physica D 98 258-286 (1996)

• T. B. Benjamin, On the Boussinesq model for two-dimensional

wave motions in heterogeneous fluids, J . Fluid Mech., 165, 445-474 (1986)

• T. Wu, Long Waves in Ocean and Coastal Waters, Journal of

Engineering Mechanics, 107, 501-522 (1981)

• A. Boonkasame, P. Milewski, The stability of large-amplitude

shallow interfacial non-Boussinesq flows, Stud. Appl. Math., DOI: 10.1111/j.1467-9590.2011.00528.x.(2011)