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 η (x,t) ρ Ω 1 1 1 η (x,t) z ρ 1 1 n n ζ( x,t ) P ( x,t ) x ζ( x,t ) η (x,t) ρ 2 2 η (x,t) ρ 2 2 Ω 2 (a) (b) 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 u j = ( u j , w j ) , j = 1, 2 are u j x + w j z = 0, u j t + u j u j x + w j u j z = − p j x / ρ j , w j t + u j w j x + w j w j z = − p j z / ρ j − 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 u j = ( u j , w j ) , j = 1, 2 are u j x + w j z = 0, u j t + u j u j x + w j u j z = − p j x / ρ j , w j t + u j w j x + w j w j z = − p j z / ρ j − g , with suitable boundary conditions • w 1 ( x , z = h 1 , t ) = 0, w 2 ( x , z = − h 2 , t ) = 0 top and bottom • ζ = 0, u i = 0, p i z = − g ρ i x → ± ∞ • ζ t + u i ζ x = w i , i = 1, 2, p 1 = p 2 at the interface z = ζ ( x , t )

The layer mean equations η 1 ( x , t ) , η 2 ( x , t ) height of the upper and lower layer. The means of a quantity f are � top f i ( x , t ) : = 1 bottom f ( x , z , t ) dz , i = 1, 2 . η i The equations for the horizontal component of the system become p i x ( η i u i ) t + ( η i u i u i ) x = − η i ρ i η i t + ( η i u i ) x = 0, i = 1, 2 η 1 + η 2 = h .

The layer mean equations η 1 ( x , t ) , η 2 ( x , t ) height of the upper and lower layer. The means of a quantity f are � top f i ( x , t ) : = 1 bottom f ( x , z , t ) dz , i = 1, 2 . η i The equations for the horizontal component of the system become u i t + u i u i x = − P x − ( − 1 ) i g η i x ρ i η i t + ( η i u i ) x = 0 i = 1, 2 η 1 + η 2 = h . At the leading order of the the long wave asymptotics the hydrostatic approximation holds everywhere: p i ( x , z , t ) = P ( x , t ) − g ρ i ( 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 u t + uu x + g η x = − P x ρ η t + ( η u ) x = 0 where P is an external assigned pressure. This system is Hamiltonian in the variables m = ρη u , η w.r.t. the Poisson structure � m ∂ + ∂ m � η∂ Π = − ∂η 0 and the Hamiltonian H = m 2 2 ρη − g ρ 2 η 2 + η P

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: η 1 + η 2 = h • the interface pressure of the two fluid has to be the same: P 1 = P 2

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: η 1 + η 2 = h • the interface pressure of the two fluid has to be the same: P 1 = P 2 Decoupled u 1 t + u 1 u 1 x − g η 1 x = − P 1 x ρ 1 η 1 t + ( η 1 u 1 ) x = 0 u 2 t + u 2 u 2 x + g η 2 x = − P 2 x ρ 2 η 2 t + ( η 2 u 2 ) 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: η 1 + η 2 = h • the interface pressure of the two fluid has to be the same: P 1 = P 2 Coupled u 1 t + u 1 u 1 x − g η 1 x = − P x ρ 1 η 1 t + ( η 1 u 1 ) x = 0 u 2 t + u 2 u 2 x + g η 2 x = − P x ρ 2 η 2 t + ( η 2 u 2 ) x = 0 η 1 + η 2 = 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: η 1 + η 2 = h • the interface pressure of the two fluid has to be the same: P 1 = P 2 Two field system �� ρ 1 η 1 � � + u 1 ρ 2 h − η 1 t � u 2 η 2 �� ρ 1 2 + ( 1 − ρ 1 � 1 1 + − ) g η 1 = 0 ρ 2 ( h − η 1 ) 2 ρ 2 x η 1 t + ( η 1 u 1 ) x = 0.

Mathematical consequences of the coupling P 1 = P 2 η 1 + η 2 = h and implies η 1 u 1 + η 2 u 2 = Q ( t ) . If we choose the velocities zero at infinity, then η 1 u 1 + η 2 u 2 = 0.

Mathematical consequences of the coupling P 1 = P 2 η 1 + η 2 = h and implies η 1 u 1 + η 2 u 2 = Q ( t ) . If we choose the velocities zero at infinity, then η 1 u 1 + η 2 u 2 = 0. RELEVANT PROPERTY The constraints φ 1 : = η 1 + η 2 − h = 0, φ 2 : = η 1 u 1 + η 2 u 2 = 0 satisfy • φ 1 t = − ( φ 2 ) x and φ 1 t = 0 = ⇒ φ 2 = 0 • all the higher dynamical consequences of φ 1 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. � � Π m 1 m 1 − ∑ Π m 1 φ i Π − 1 Π D m 1 m 1 = φ i φ j Π φ j m 1 � � φ 1 = φ 2 = 0 i , j = 1,2 � � Π m 1 η 1 − ∑ Π m 1 φ i Π − 1 Π D m 1 η 1 = φ i φ j Π φ j η 1 � � φ 1 = φ 2 = 0 i , j = 1,2 � � Π η 1 η 1 − ∑ Π η 1 φ i Π − 1 Π D η 1 η 1 = φ i φ j Π φ j η 1 � � φ 1 = φ 2 = 0 i , j = 1,2 where { f ( x ) , g ( y ) } = Π f , g δ ( x − y ) ,

The reduced Hamiltonian structure Finally, after some algebraic manipulation (integration by parts) the Dirac-Poisson structure will be � M ∂ + ∂ M � N ∂ Π D = − . ∂ N 0 where 1 ( h − η 1 ) 2 − ρ 1 ρ 2 η 2 M = ρ 2 1 ( ρ 2 η 1 + ρ 1 ( h − η 1 )) 2 m 1 � ρ 1 � − 1 ρ 2 N = ρ 1 − η 1 h − η 1 and the Hamiltonian ( ρ 2 ρ 1 m 1 ) 2 ( h − η 1 ) 2 η 2 m 2 h D : = 1 1 + 2 ρ 2 ( h − η 1 ) + g ρ 2 − g ρ 1 2 . 2 ρ 1 η 1 2

The translational invariance along the channel In view of the study of symmetries we change the variables � η 1 / ρ 1 � m = 1 + η = η 1 . m 1 , ( h − η 1 ) / ρ 2 In these new variables the Poisson structure becomes � m ∂ + ∂ m � η∂ Π D ( m , η ) = − . ∂η 0 and the Hamiltonian � − 1 h D ( m , η ) = m 2 ( h − η ) 2 η 2 � η / ρ 1 1 + + g ρ 2 − g ρ 1 2 . 2 ρ 1 η ( h − η ) / ρ 2 2

The reduced system is � m 2 �� 2 ρ 1 ( h − η ) 2 + ρ 2 η ( h − 2 η ) 2 η + g ( ρ 1 − ρ 2 ) � η 2 m t + = 0, ( ρ 1 ( h − η ) + ρ 2 η ) 2 2 x � − 1 m �� � η / ρ 1 η t + 1 + = 0. ( h − η ) / ρ 2 ρ 1 x 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 − ρ 1 << 1 ρ 2 we can consider ρ 1 � = ρ 2 only in the terms where g appears � 2 h − 3 η 2 ρ 1 h η m 2 + g ( ρ 2 − ρ 1 ) � η 2 m t + = 0, 2 x � m ( h − η ) � η t + = 0. h ρ 1 x 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 ρ 1 = ρ 2 .

� � 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?

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