Asymptotic limits of the Shallow Water equations Carine Lucas MAPMO - PowerPoint PPT Presentation

Asymptotic limits of the Shallow Water equations Carine Lucas MAPMO - univ. Orl´ eans, France Work in collaboration with: Didier Bresch (LAMA, univ. Savoie Mont Blanc, France) Rupert Klein (Free University of Berlin, Germany).

Shallow Water equations Shallow Water equations: ∂ t h + div( hu ) = 0 , ∂ t ( hu ) + div( hu ⊗ u ) + gh ∇ h = − gh ∇ b. z u ( t, x ): flow velocity h ( t, x ): water height b ( x ): topography x ∈ Ω ⊂ R 2 Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 2 / 29

Shallow Water equations Shallow Water equations: ∂ t h + div( hu ) = 0 , ∂ t ( hu ) + div( hu ⊗ u ) + gh ∇ h = − gh ∇ b. z u ( t, x ): flow velocity h ref h ( t, x ): water height b ref b ( x ): topography x ∈ Ω ⊂ R 2 L Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 2 / 29

Shallow Water equations Dimensionless Shallow Water equations: L ∂ t h + div( hu ) = 0 , t ref u ref L ∂ t ( hu ) + div( hu ⊗ u ) + gh ref h ∇ h = − gh ref b ref h ∇ b. u 2 u 2 t ref u ref h ref ref ref z u ( t, x ): flow velocity h ref h ( t, x ): water height b ref b ( x ): topography x ∈ Ω ⊂ R 2 L Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 2 / 29

Shallow Water equations Dimensionless Shallow Water equations: Sr ∂ t h + div( hu ) = 0 , Sr ∂ t ( hu ) + div( hu ⊗ u ) + 1 Fr 2 h ∇ h = − 1 Fr 2 βh ∇ b. with L Sr (= St ) = the Strouhal number (vortex), t ref u ref u ref Fr = √ gh ref the Froude number (flow vs gravity waves velocities) and β = b ref . h ref Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 2 / 29

Shallow Water equations Dimensionless Shallow Water equations: Sr ∂ t h + div( hu ) = 0 , Sr ∂ t ( hu ) + div( hu ⊗ u ) + 1 Fr 2 h ∇ h = − 1 Fr 2 βh ∇ b. with L Sr (= St ) = the Strouhal number, t ref u ref u ref = ε α ( ε ≪ 1) the Froude number Fr = √ gh ref and β = b ref = 1 . h ref Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 2 / 29

Multiple scales in Shallow Water equations Low Froude number flows: velocities of the flow < speed of the gravity waves = ⇒ multiple length / time scales (depending on initial and boundary conditions). Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 3 / 29

Multiple scales in Shallow Water equations Low Froude number flows: velocities of the flow < speed of the gravity waves = ⇒ multiple length / time scales (depending on initial and boundary conditions). During t ref : √ gh ref = ( L/ Sr) /ε α t ref u ref = L/ Sr t ref distance of an advected particle < distance of gravity waves. Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 3 / 29

Multiple scales in Shallow Water equations Low Froude number flows: velocities of the flow < speed of the gravity waves = ⇒ multiple length / time scales (depending on initial and boundary conditions). During t ref : √ gh ref = ( L/ Sr) /ε α t ref u ref = L/ Sr t ref distance of an advected particle < distance of gravity waves. In a O ( L ) domain: L/ √ h ref = ε α Sr t ref L/u ref = Sr t ref time scales for advected particle > time scales for gravity waves. Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 3 / 29

Multiscale topography 4 2 0 20 50 60 90 100 0 10 30 40 70 80 ! 2 x x/ε : quick variations ( ε � 1) εx : slow variations Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 4 / 29

Multiscale topography 4 2 0 20 50 60 90 100 0 10 30 40 70 80 ! 2 x x/ε : quick variations ( ε � 1) εx : slow variations X = x χ = εx . ε Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 4 / 29

Outline Balanced flow, topography at the ‘normal’ scale: b = b ( x ) 1 Balanced flow, topography with quick variations: b = b ( X, x ) 2 Weakly nonlinear regime Fully nonlinear regime Topography with long scale variations: b = b ( x, χ ) 3 Formal derivations D. Bresch, R. Klein, C. L., 2011 Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 5 / 29

Balanced flow, topography at the ‘normal’ scale: b = b ( x ) Outline Balanced flow, topography at the ‘normal’ scale: b = b ( x ) 1 Balanced flow, topography with quick variations: b = b ( X, x ) 2 Weakly nonlinear regime Fully nonlinear regime Topography with long scale variations: b = b ( x, χ ) 3 Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 6 / 29

Balanced flow, topography at the ‘normal’ scale: b = b ( x ) b = b ( x ) , Fr = ε , Sr = 1 Flow on advective time scales. Shallow Water equations: ∂ t h + div( hu ) = 0 , ∂ t ( hu ) + div( hu ⊗ u ) + 1 ε 2 h ∇ h = 1 ε 2 h ∇ b. Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 7 / 29

Balanced flow, topography at the ‘normal’ scale: b = b ( x ) b = b ( x ) , Fr = ε , Sr = 1 Flow on advective time scales. Shallow Water equations: ∂ t h + div( hu ) = 0 , ∂ t ( hu ) + div( hu ⊗ u ) + 1 ε 2 h ∇ h = 1 ε 2 h ∇ b. Asymptotic development: � ε i h i ( t, x ) , h ( t, x, ε ) = i � ε i u i ( t, x ) . u ( t, x, ε ) = i Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 7 / 29

Balanced flow, topography at the ‘normal’ scale: b = b ( x ) b = b ( x ) , Fr = ε , Sr = 1 O ( ε − 2 ) h 0 ∇ ( h 0 + b ) = 0 , O ( ε − 1 ) h 1 ∇ ( h 0 + b ) + h 0 ∇ h 1 = 0 , O ( ε 0 ) ∂ t h 0 + div( h 0 u 0 ) = 0 , ∂ t ( h 0 u 0 ) + div( h 0 u 0 ⊗ u 0 )+ h 2 ∇ ( h 0 + b ) + h 1 ∇ h 1 + h 0 ∇ h 2 = 0 . Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 8 / 29

Balanced flow, topography at the ‘normal’ scale: b = b ( x ) b = b ( x ) , Fr = ε , Sr = 1 O ( ε − 2 ) h 0 ∇ ( h 0 + b ) = 0 , h 0 + b ≡ c 0 ( t ) O ( ε − 1 ) h 1 ∇ ( h 0 + b ) + h 0 ∇ h 1 = 0 , h 1 ≡ c 1 ( t ) O ( ε 0 ) ∂ t h 0 + div( h 0 u 0 ) = 0 , ∂ t ( h 0 u 0 ) + div( h 0 u 0 ⊗ u 0 )+ h 2 ∇ ( h 0 + b ) + h 1 ∇ h 1 + h 0 ∇ h 2 = 0 . � dc 0 ∂b div( h 0 u 0 ) = − d dt = − 1 h 0 u 0 · n dσ dtc 0 ( t ) , ∂t = 0 : | Ω | Ω ∂ t ( h 0 u 0 ) + div( h 0 u 0 ⊗ u 0 ) + h 0 ∇ h 2 = 0 . Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 8 / 29

Balanced flow, topography at the ‘normal’ scale: b = b ( x ) b = b ( x ) , Fr = ε , Sr = 1 Shallow Water limit when b = b ( x ) , Fr = ε , Sr = 1: Lake equations ∂ t ( h 0 u 0 ) + div( h 0 u 0 ⊗ u 0 ) + h 0 ∇ h 2 = 0 , � dc 0 dt = dh 0 − 1 h 0 u 0 · n dσ = dt | Ω | Ω + initial / boundary conditions on h 0 , c 0 . see D. Bresch, G. M´ etivier, AMRX, 2010 for a rigorous justification of the limit. Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 9 / 29

Balanced flow, topography with quick variations: b = b ( X, x ) Outline Balanced flow, topography at the ‘normal’ scale: b = b ( x ) 1 Balanced flow, topography with quick variations: b = b ( X, x ) 2 Weakly nonlinear regime Fully nonlinear regime Topography with long scale variations: b = b ( x, χ ) 3 Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 10 / 29

Balanced flow, topography with quick variations: b = b ( X, x ) b = b ( X, x ) ! $ % % !# #% "# $%% & ! $ ! ! X ∈ T 2 Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 11 / 29

Balanced flow, topography with quick variations: b = b ( X, x ) Weakly nonlinear regime b = b ( X, x ) , Fr = ε 3 / 2 , Sr = ε − 1 Characteristic lengths too short to support gravity waves. Weakly nonlinear regime. Shallow Water equations: ∂ t h + ε div( hu ) = 0 , ∂ t ( hu ) + ε div( hu ⊗ u ) + 1 ε 2 h ∇ h = 1 ε 2 h ∇ b. Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 12 / 29

Balanced flow, topography with quick variations: b = b ( X, x ) Weakly nonlinear regime b = b ( X, x ) , Fr = ε 3 / 2 , Sr = ε − 1 Characteristic lengths too short to support gravity waves. Weakly nonlinear regime. Shallow Water equations: ∂ t h + ε div( hu ) = 0 , ∂ t ( hu ) + ε div( hu ⊗ u ) + 1 ε 2 h ∇ h = 1 ε 2 h ∇ b. Asymptotic development: � ε i h i ( t, X, x ) , h ( t, x, ε ) = i � ε i u i ( t, X, x ) . u ( t, x, ε ) = i Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 12 / 29

Balanced flow, topography with quick variations: b = b ( X, x ) Weakly nonlinear regime b = b ( X, x ) , Fr = ε 3 / 2 , Sr = ε − 1 O ( ε − 3 ) X ( h 0 + b ) = 0 , h 0 ∇ O ( ε − 2 ) x ( h 0 + b ) + h 1 ∇ X ( h 0 + b ) + h 0 ∇ X h 1 = 0 , h 0 ∇ Carine Lucas (MAPMO - Orl´ eans) Asymptotic limits of the Shallow Water equations 6 nov. 2015 - 13 / 29