Modified shallow water equations for significant bathymetry variations D IDIER CLAMOND University of Nice – Sophia Antipolis, France D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 1 / 25
Motivation Understanding waves and surface flows ⇒ Simplified models. Saint-Venant (SV) simple model is often good enough, but not always. Can SV be improved at minimum cost (keeping the hyperbolicity)? We propose a modified Saint-Venant (mSV) model when bathymetry gradient is significant. D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 2 / 25
Hypothesis Physical assumptions: • Fluid is ideal, homogeneous & incompressible; • Flow is irrotational, i.e., � V = grad φ ; • Free surface is a graph; • Above free surface there is void; • Atmospheric pressure is 0.5 z η (x,y,t) constant. 0 a y O −0.5 • Surface and bottom are both h(x,y,t) x h 0 −1 impermeable. −1.5 • Bottom can vary in space and −2 4 3 2 1 0 time. 4 −1 3 2 −2 1 −1 0 −3 −2 −3 −4 −4 D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 3 / 25
Definition Sketch y η ( x , t ) O x h ( x , t ) d ( x , t ) D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 4 / 25
Notations • x = ( x 1 , x 2 ) : Horizontal Cartesian coordinates. • y : Upward vertical coordinate. • t : Time. • u = ( u 1 , u 2 ) : Horizontal velocity. • v : Vertical velocity. • φ : Velocity potential. • y = η ( x , t ) : Equation of the free surface. • y = − d ( x , t ) : Equation of the seabed. • ∇ : Horizontal gradient. • Over tildes : Quantities at the surface, e.g., ˜ u = u ( y = η ) . • Over check : Quantities at the surface, e.g., ˇ u = u ( y = − d ) . • Over bar : Quantities averaged over the depth, e.g., � η u = 1 ¯ h = η + d . u d y , h − d D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 5 / 25
Basic shallow water models (2D + flat bottom) Columnar flow: uniform horizontal velocity u ( x , y , t ) ≈ ¯ u ( x , t ) Vertical velocity (2 classical possibilities): (1) Incompressibility: u x + v y = 0 ⇒ v ( x , y , t ) ≈ − ( y + d ) ¯ u x v x − u y = 0 ⇒ v ( x , y , t ) ≈ 0 (2) Irrotationality: Kinetic and Potential energy densities: u 2 + v 2 � η � η 2 + h 3 ¯ u 2 u 2 g ( y + d ) d y = gh 2 d y ≈ h ¯ x K = V = , 2 2 6 − d − d D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 6 / 25
Equations of motion Lagrangian density: L = K − V + { h t + [ h ¯ u ] x } φ φ : Lagrange multiplier. Euler–Lagrange equations: h t + ∂ x [ h ¯ u ] = 0 � u 2 + 1 2 g h 2 + 1 � 3 h 2 γ ∂ t [ h ¯ u ] + ∂ x h ¯ = 0 u 2 2 h ¯ x − h ∂ x [ ¯ u t + ¯ u ¯ u x ] = γ With red terms: Serre–Green–Naghdi equations. Without red terms: Saint-Venant equations. D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 7 / 25
Serre equations Pros: - Dispersive. - Admit permanent solutions. - Regular. Cons: - High-order derivatives. - Hard to solve numerically. - Not hyperbolic. D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 8 / 25
Saint-Venant equations Pros: - Hyperbolic. - Characteristics. - Fast numerical solvers. Cons: - Non-dispersive. - No smooth permanent solutions. - Limited to very slowly varying bottoms. D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 9 / 25
Modified Saint-Venant (mSV) equations Choice of the ansatz (columnar flow): u ≈ ¯ u ( x , t ) , v ≈ ˇ v ( x , t ) = − d t − ¯ u · ∇ d Equations of motion: 0 = ∂ t h + ∇ · [ h ¯ u ] u 2 + 1 2 g h 2 ] = ( g + γ ) h ∇ d + h ¯ ∂ t [ h ¯ u ] + ∇ [ h ¯ u ∧ ( ∇ ˇ v ∧ ∇ d ) D UTYKH & C LAMOND 2011. J. Phys. A: Math. & Theor. 44, 332001. D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 10 / 25
Model properties Hyperbolic equations. Method of characteristics is usable. Waves propagation speed in SV and mSV: √ gh � c SV = c mSV = gh � 1 + | ∇ d | 2 D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 11 / 25
Wave propagation over oscillating bottom Initial surface: η ( x , t = 0 ) = b sech 2 ( κ x ) , u ( x , t = 0 ) = 0 . Bottom profile: d ( x ) = d 0 + a sin ( kx ) . D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 12 / 25
Wave propagation over oscillating bottom: low freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 2 . 00 mSV 0.14 SV 0.12 0.1 η ( x, t ) [m] 0.08 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 2 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 13 / 25
Wave propagation over oscillating bottom: low freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 5 . 00 mSV 0.14 SV 0.12 0.1 η ( x, t ) [m] 0.08 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 5 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 13 / 25
Wave propagation over oscillating bottom: low freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 9 . 00 mSV 0.14 SV 0.12 0.1 η ( x, t ) [m] 0.08 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 9 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 13 / 25
Wave propagation over oscillating bottom: low freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 20.00 SV++ 0.14 SV 0.12 0.1 0.08 η (x,t) 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x Figure: t = 20 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 13 / 25
Wave propagation over oscillating bottom: low freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 24.00 SV++ 0.14 SV 0.12 0.1 0.08 η (x,t) 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x Figure: t = 24 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 13 / 25
Wave propagation over oscillating bottom: high freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 2 . 00 mSV 0.14 SV 0.12 0.1 η ( x, t ) [m] 0.08 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 2 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 14 / 25
Wave propagation over oscillating bottom: high freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 5 . 00 mSV 0.14 SV 0.12 0.1 η ( x, t ) [m] 0.08 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 5 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 14 / 25
Wave propagation over oscillating bottom: high freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 9 . 00 mSV 0.14 SV 0.12 0.1 η ( x, t ) [m] 0.08 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 9 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 14 / 25
Wave propagation over oscillating bottom: high freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 20.00 SV++ 0.14 SV 0.12 0.1 0.08 η (x,t) 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x Figure: t = 20 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 14 / 25
Wave propagation over oscillating bottom: high freq. Comparison with classical Saint-Venant equations Free surface elevation at t = 24.00 SV++ 0.14 SV 0.12 0.1 0.08 η (x,t) 0.06 0.04 0.02 0 −0.02 −10 −8 −6 −4 −2 0 2 4 6 8 10 x Figure: t = 24 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 14 / 25
Moving bottom test-case: slow uplift Comparison with classical Saint-Venant equations Free surface elevation at t = 0 . 50 0.5 mSV SV Bathymetry 0 η ( x, t ) [m] −0.5 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 0 . 5 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 15 / 25
Moving bottom test-case: slow uplift Comparison with classical Saint-Venant equations Free surface elevation at t = 1 . 00 0.5 mSV SV Bathymetry 0 η ( x, t ) [m] −0.5 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 1 . 0 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 15 / 25
Moving bottom test-case: slow uplift Comparison with classical Saint-Venant equations Free surface elevation at t = 2 . 00 0.5 mSV SV Bathymetry 0 η ( x, t ) [m] −0.5 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 2 . 0 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 15 / 25
Moving bottom test-case: slow uplift Comparison with classical Saint-Venant equations Free surface elevation at t = 5 . 00 0.5 mSV SV Bathymetry 0 η ( x, t ) [m] −0.5 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 x [m] Figure: t = 5 . 0 s D. C LAMOND (LJAD) Modified St-Venant equations AMoSS, July 8th, 2015. 15 / 25
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