March 23, 2009 Joint work with Rodrigo Cienfuegos.
Outline The Serre equations I. Derivation II. Properties III. Solutions IV. Solution stability V. Wave shoaling
Derivation of the Serre Equations Derivation of the Serre Equations
Governing Equations Consider the 1-D flow of an inviscid, irrotational, incompressible fluid. Let η(x, t) represent the location of the free surface u(x, z, t) represent the horizontal velocity of the fluid w(x, z, t) represent the vertical velocity of the fluid p(x, z, t) represent the pressure in the fluid ɛ = a 0 /h 0 (a measure of nonlinearity) δ = h 0 /l 0 (a measure of shallowness)
Governing Equations The 1-D flow of an inviscid, irrotational, incompressible fluid z Η, free surface a 0 z h 0 undisturbed level Η l 0 h 0 z x z 0 at the bottom
Governing Equations The dimensionless governing equations are u x + w z = 0, for 0 < z < 1 + ɛη u z δw x = 0, for 0 < z < 1 + ɛη ɛu t + ɛ 2 (u 2 ) x + ɛ 2 (uw) z + p x = 0, for 0 < z < 1 + ɛη δ 2 ɛw t + δ 2 ɛ 2 uw x + δ 2 ɛ 2 ww z + p z = 1, for 0 < z < 1 + ɛη w = η t + ɛuη x at z = 1 + ɛη p = 0 at z = 1 + ɛη w = 0 at z = 0
Depth Averaging The Serre equations are obtained from the governing equations by: 1. Depth averaging The depth-averaged value of a quantity f (z) is defined by f = 1 h h 0 f (z) dz where h = 1 + ɛη is the location of the free surface. 2. Assuming that δ << 1
Governing Equations After depth averaging, the dimensionless governing equations are η t + ɛ(ηu) x = 0 u t + η x + ɛu u x δ2 ( η 3( u xt + ɛu u xx ɛ(u x ) 2)) 3η = x O(δ4, ɛδ 4 )
The Serre Equations Truncating this system at O(δ 4, ɛδ 4 ) and transforming back to physical variables gives the Serre Equations η t + (ηu) x = 0 u t + gη x + u u x 1 ( η 3( u xt + u u xx (u x ) 2)) 3η = 0 x where η(x, t) is the dimensional free surface elevation u(x, t) is the dimensional depth-averaged horizontal velocity g is the acceleration due to gravity
Properties of the Serre Equations Properties of the Serre Equations
Properties of the Serre Equations The Serre equations admit the following conservation laws: I. Mass II. Momentum t (η) + x (ηu) = 0 t (ηu) + x ( 1 2 gη2 1 3 η3 u xt + ηu 2 + 1 3 η3 u 2 x 1 3 η3 u u xx ) = 0 III. Momentum 2 ( t u ηη x u x 1 ) 3 η2 u xx + x (ηη t u x +gη 1 3 η2 u u xx + 1 ) 2 η2 u 2 x = 0
Properties of the Serre Equations The Serre equations are invariant under the transformation η(x, t) = ˆη(x st, t) u(x, t) = û(x st, t) + s ˆx = x st where s is any real parameter. Physically, this corresponds to adding a constant horizontal flow to the entire system.
Solutions of the Serre Equations Solutions of the Serre Equations
Solutions of the Serre Equations η(x, t) = a 0 + a 1 dn 2( κ(x ct), k ) ( u(x, t) = c 1 h ) 0 η(x, t) κ = c = 3a1 2 a 0 (a 0 + a 1 )(a 0 + (1 k 2 )a 1 ) ga0 (a 0 + a 1 )(a 0 + (1 k 2 )a 1 ) h 0 h 0 = a 0 + a 1 E(k) K(k) where k [0, 1], a 0 > 0, and a 1 > 0 are real parameters.
Trivial Solution of the Serre Equations If k = 0, η(x, t) = a 0 + a 1 u(x, t) = 0
Periodic Solutions of the Serre Equations The water surface if 0 < k < 1. 1 a 0 a 1 1 k 2
Soliton Solution of the Serre Equations If k = 1, η(x, t) = a 0 + a 1 sech 2 (κ(x ct)) ( u(x, t) = c 1 a ) 0 η(x, t) 3a1 κ = 2a 0 a0 + a 1 c = g(a 0 + a 1 ) h 0 = a 0
Soliton Solution of the Serre Equations The corresponding water surface a 1 a 0
Stability of Solutions of the Serre Equations Stability of Solutions of the Serre Equations
Stability of Solutions of the Serre Equations Transform to a moving coordinate frame The Serre equations become χ = x ct τ = t η τ cη χ + ( ηu ) χ = 0 u τ cu χ + u u χ + η χ 1 ( η 3( u χτ cu χχ + u u χχ (u χ ) 2)) 3η = 0 χ and the solutions become η = η 0 (χ) = a 0 + a 1 dn 2( κχ, k ) ( u = u 0 (χ) = c 1 h ) 0 η 0 (χ)
Stability of Solutions of the Serre Equations Consider perturbed solutions of the form where η pert (χ, τ) = η 0 (χ) + ɛη 1 (χ, τ) + O(ɛ 2 ) u pert (χ, τ) = u 0 (χ) + ɛu 1 (χ, τ) + O(ɛ 2 ) ɛ is a small real parameter η 1 (χ, τ) and u 1 (χ, τ) are real-valued functions η 0 (χ) = a 0 + a 1 dn 2 (κχ, k) ( ) u 0 (χ) = c 1 h 0 η 0 (χ)
Stability of Solutions of the Serre Equations Without loss of generality, assume η 1 (χ, τ) = H(χ)e Ωτ + c.c. u 1 (χ, τ) = U(χ)e Ωτ + c.c. where H(χ) and U(χ) are complex-valued functions Ω is a complex constant c.c. denotes complex conjugate
Stability of Solutions of the Serre Equations This leads the following linear system ( ) ( ) H H L = Ω M U U where ( u L = 0 + (c u 0 ) χ η 0 η ) 0 χ L 21 L 22 ( ) 1 0 M = 0 1 η 0 η 0 χ 1 3 η2 0 χχ and prime represents derivative with respect to χ.
Stability of Solutions of the Serre Equations where L 21 = η 0(u 0) 2 cη 0u 0 2 3 cη 0u 0 + η 0u 0 u 0 2 3 η 0u 0u 0 + 2 3 η 0u 0 u 0 + ( η 0 u 0 u 0 g η 0 (u 0) 2 cη 0 u 0) χ L 22 = u 0 + η 0 η 0u 0 + 1 3 η2 0u 0 + ( c u 0 2η 0 η 0u 0 1 3 η2 0u 0) χ + ( η 0 η 0u 0 cη 0 η 0 1 3 η2 0u 0 ) χχ + ( 1 3 η2 0u 0 1 3 cη2 0 ) χχχ
Stability of Solutions of the Serre Equations L ( ) ( ) H H = Ω M U U Solved numerically using the Fourier-Floquet-Hill Method. 8 6 4 2 0 0.000 For a 0 = 0.3, a 1 = 0.1, k = 0.99
Stability of Solutions of the Serre Equations Qualitative observations: Not all solutions are stable If k and/or a 1 is large enough, then there is instability Most/all instabilities have complex growth rates As k increases, so does the maximum growth rate As a 1 increases, so does the maximum growth rate As a 0 decreases, the maximum growth rate increases As a 1 and/or k increase, the number of bands increases
Wave Shoaling in the Serre Equations Wave Shoaling in the Serre Equations
Wave Shoaling So far we ve assumed shallow water and a horizontal bottom. What happens if the bottom varies (slowly)?
Wave Shoaling The Serre equations for a non-horizontal bottom η t + (hu) x = 0 ( hu t + huu x + ghη x + h 2( 1 3 P + 1 ( 1 ) 2 Q)) + ξ xh x 2 P + Q = 0 P = h ( u xt + uu xx (u x ) 2) Q = ξ x (u t + uu x ) + ξ xx u 2 where z = ξ(x) is the bottom location (ξ 0 for all x) z = η(x, t) is the location of the free surface h(x, t) = η(x, t) ξ(x) is the local water depth u = u(x, t) is the depth-averaged horizontal velocity
Wave Shoaling We consider a slowly-varying, constant-slope bottom of the form ξ(x) = 0 + ɛx + O(ɛ 2 )
Wave Shoaling In order to deal with the slowly-varying bottom, ξ(x) = 0 + ɛx + O(ɛ 2 ) we assume h(x, t) = h 0 (x, t) + ɛh 1 (x, t) + O(ɛ 2 ) u(x, t) = u 0 (x, t) + ɛu 1 (x, t) + O(ɛ 2 )
Wave Shoaling The solution to the leading-order problem is h 0 (x, t) = a 0 + a 1 sech 2( κ(x ct) ) ( u 0 (x, t) = c 1 a ) 0 h 0 (x, t) 3a1 κ = 2a 0 a0 + a 1 c = g(a 0 + a 1 ) Note: we only consider solitary waves here.
Wave Shoaling At the next order in ɛ, the equations are a big mess. However, the system can be solved analytically!
Wave Shoaling
Wave Shoaling