The shallow water equations Lecture 8. (photo due to Clark Little /SWNS)

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1 The shallow water equations Lecture 8 (photo due to Clark Little /SWNS)

2 The shallow water equations This lecture: 1) Derive the shallow water equations 2) Their mathematical structure 3) Some consequences 4) Some open questions

3 Derive shallow water equations 1. Commonly used in large-scale ocean models 2. Start with Euler s equations, no surface tension p = 0, D" Dt = # t" +! u $ %" = w, on z =!(x,y,t) D! u Dt + 1 " #p + gˆ z = 0, " # u! = 0, -h(x,y) < z <!(x,y,t)! u " #(z + h(x, y)) = 0 = w +! u " #h(x, y), on z = -h(x,y) mass conservation

4 Derive shallow water equations 1. Commonly used in large-scale ocean models 2. Start with Euler s equations, no surface tension D" Dt = # t" +! p = 0, u $ %" = w, on z =!(x,y,t) D! u Dt + 1 " #p + gˆ z = 0, " # u! = 0, -h(x,y) < z <!(x,y,t)! u " #(z + h(x, y)) = 0 = w +! u " #h(x, y), on z = -h(x,y) 3. Step one: global mass conservation mass conservation

5 Derive shallow water equations " #! u = 0, -h(x,y) < z <!(x,y,t) mass conservation

6 Derive shallow water equations " #! u = 0, -h(x,y) < z <!(x,y,t) % & dz = [" $h x u + " y v + " z w]dz #h [" # u! ] mass conservation $ % = 0

7 Derive shallow water equations " #! u = 0, -h(x,y) < z <!(x,y,t)! % & dz = [" $h x u + " y v + " z w]dz #h [" # u! ] $ $ mass +" conservation y [v]dz #h $ % = 0 " % x [u]dz "u z=# $ x # +u "h # x ("h) #h % "v # $ y # +v "h # y ("h) +w " "w "h = 0.

8 Derive shallow water equations " #! u = 0, -h(x,y) < z <!(x,y,t) % & dz = [" $h x u + " y v + " z w]dz #h [" # u! ] $ $ mass +" conservation y [v]dz #h $ % = 0 " % x [u]dz "u z=# $ x # +u "h # x ("h) #h % "v # $ y # +v "h # y ("h) +w " "w "h = 0.

9 Derive shallow water equations " #! u = 0, -h(x,y) < z <!(x,y,t) % & dz = [" $h x u + " y v + " z w]dz #h [" # u! ] $ " % x [u]dz "u z=# $ x # #h $ mass +" conservation y [v]dz #h $ % "v # $ y # % = 0 +w " = 0.

10 Derive shallow water equations " #! u = 0, -h(x,y) < z <!(x,y,t) % & dz = [" $h x u + " y v + " z w]dz #h [" # u! ] $ " % x [u]dz "u z=# $ x # #h $ +" % y [v]dz "v # $ y # #h # $ % = 0 +w " = 0. # + " % x [u]dz + " % y [v]dz = 0.! (exact) $h # $h

11 Derive shallow water equations Summary so far (no approximations): Conservation of mass: 3 momentum eq ns: # # + " % x [u]dz + " % y [v]dz = 0. $h # $h D! u Dt + 1 " #p + gˆ z = 0, -h(x,y) < z <!(x,y,t)

12 Derive shallow water equations Step 2: Assume long waves, but not small amplitudes. D! u Dt + 1 " #p + gˆ z = 0, Neglect vertical accelerations Dw Dt + 1 " # z p + g = 0,! Hydrostatic pressure p(x, y,z,t) = g" #{$(x,y,t) % z} -h(x,y) < z <!(x,y,t)

13 Derive shallow water equations Hydrostatic pressure in fluid p(x, y,z,t) = g"{#(x, y,t) $ z} u + u u + v" y u + w" z u + g" x # = 0, v + u v + v" y v + w" z v + g" y # = 0. Assume no vertical variation in (u, v) u + u u + v" y u + w" z u + g" x # = 0, v + u v + v" y v + w" z v + g" y # = 0. # # + " % x [u]dz + " % y [v]dz = 0. $h # $h

14 Shallow water equations # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. h(x,y): known bottom topography Allows for variable depth in natural way Want: {!(x,y,t), u(x,y,t), v(x,y,t)} Similar to equations for gas dynamics in 2-D Common variation: include (Earth s) rotation Q: What is mathematical structure of eq ns?

15 Math structure #1: Hyperbolic PDEs # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. $ #' & ) u & ) + % v( * u # + h 0- $, / # ', g u 0 / " & ) x u +, 0 0 u. / & ) + % v( * v # + h 0- $, / # ' $ u" x h + v" y h', 0 v 0 / " & ) u y +, g 0 v. / & ) = 0 & ) 0 & ) % v( % 0 (

16 Math structure #1: Hyperbolic PDEs # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. $ #' * u # + h 0- $ & ) u & ) +, / # ' * v # + h 0- $, g u 0 / " & ) u x % v( +, 0 0 u. / & ) +, / # ' $ u" x h + v" y h', 0 v 0 / " & ) u y % v( +, g 0 v. / & ) = 0 & ) 0 & ) % v( % 0 ( No dispersion! Solutions are non-dispersive. Eq ns admit discontinuous (weak) solutions, that approximate breaking waves [How do waves break?] CLAWPACK ( developed by Randy Leveque (U of W) and others

17 Math structure #2 (if v " 0, " y # 0): Method of characteristics (Stoker, 1948) (# + h) + u" x (# + h) + (# + h)" x u = 0, u + u" x u + g" x (# + h) = g" x h.

18 v " 0, " y # 0 Math structure #2 (if ): Method of characteristics (Stoker) " g% $ '( # 1& (# + h) + u" x (# + h) + (# + h)" x u = 0, u + u" x u + g" x (# + h) = g" x h. Define c 2 (x, y,t) = g " (# + h)

19 v " 0, " y # 0 Math structure #2 (if ): Method of characteristics (Stoker) " g% $ '( # 1& (# + h) + u" x (# + h) + (# + h)" x u = 0, u + u" x u + g" x (# + h) = g" x h. Define c 2 (x, y,t) = g " (# + h) c "[# t (2c) + u# x (2c) + (c)# x u] = 0, # t u + u# x u + (c)# x (2c) = g# x h,! (u + 2c) + u" x (u + 2c) + (c)" x (u + 2c) = g" x h, (u # 2c) + u" x (u # 2c) # (c)" x (u # 2c) = g" x h.

20 Math structure #2 (if ): Method of characteristics (Stoker) Along curves in the x-t plane defined by v " 0, " y # 0 (u + 2c) + u" x (u + 2c) + (c)" x (u + 2c) = g" x h, (u # 2c) + u" x (u # 2c) # (c)" x (u # 2c) = g" x h. Along dx dt = u + c, dx dt = u " c, d(u + 2c) dt d(u " 2c) dt = g dh dx. = g dh dx. If h = const or h(x) = mx + b,! Riemann invariants Apparently this method does not generalize to {x,y,t}.

21 Math structure #3: Linearize eq ns # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. # + " x (uh) + " y (vh) = 0, u + g" x # = 0, v + g" y # = 0.

22 Math structure #3: Linearize eq ns # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. " $ $ $ # # + " x (uh) + " y (vh) = 0, g% (expect ' h '( u + g" x # = 0, c(x, y) = gh h ' & v + g" y # = 0. )! (# g) + " x (u h $ gh) + " y (v h $ gh) = 0, (u h) + gh $" x (# g) = 0, (v h) + gh $" y (# g) = 0. Good form for linearized equations: h = h(x,y)

23 Structure #3: Linearized eq ns (# g) + " x (u h $ gh) + " y (v h $ gh) = 0, (u h) + gh $" x (# g) = 0, (v h) + gh $" y (# g) = 0. Consequence (a): Eliminate {u h,v h}

24 Structure #3: Linearized eq ns (# g) + " x (u h $ gh) + " y (v h $ gh) = 0, (u h) + gh $" x (# g) = 0, (v h) + gh $" y (# g) = 0. Consequence (a): Eliminate {u h,v h}! 2 (# g) = $ % {gh % $(# g)} Linear wave equation in 2-D, with variable speed (c 2 = gh). (Come back to use this.)

25 Structure #3: Linearized eq ns (# g) + " x (u h $ gh) + " y (v h $ gh) = 0, (u h) + gh $" x (# g) = 0, (v h) + gh $" y (# g) = 0. Consequence (a): Eliminate {u h,v h} 2 (# g) = $ % {gh % $(# g)} Linear wave equation in 2-D, with variable speed (c 2 = gh). Consequence (b): Wave eq n drops one t-derivative. Define vorticity: "(x, y,t) = # y u $# x v! # = 0

26 Applications: Linearized wave eq n 2 (# g) = $ % {gh % $(# g)} Tsunamis A good model for the propagation of a tsunami across the open ocean (away from shore, where the wave compresses horizontally, and grows vertically). In the open ocean, the local speed of propagation, in any direction, is c = gh(x, y)

27 Applications: Linearized wave eq n 2 (# g) = $ % {gh % $(# g)} Tsunamis A good model for the propagation of a tsunami across the open ocean (away from shore, where the wave compresses horizontally, and grows vertically). In the open ocean, the local speed of propagation, in any direction, is c = gh(x, y) Wave shoaling (see lecture 20)

28 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (1) (2) (3) Define " = # y u $# x v Compute, from (2), (3) (#) + u" x (#)+ v" y (#) + (#)(" x u + " y v) = 0 (4)

29 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (1) (2) (3) Define " = # y u $# x v Compute, from (2), (3) (#) + u" x (#)+ v" y (#) + (#)(" x u + " y v) = 0 (4)! # + " x (u#)+ " y (v#) = 0 (5)! Total (integrated) vorticity is conserved.

30 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (1) (2) (3) Define " = # y u $# x v Compute, from (2), (3) (#) + u" x (#)+ v" y (#) + (#)(" x u + " y v) = 0 # + " x (u#)+ " y (v#) = 0 Total (integrated) vorticity is conserved. From (1) (# + h) + u" x (# + h)+ v" y (# + h) + (# + h)(" x u + " y v) = 0 (4) (5) (6)

31 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (#) + u" x (#)+ v" y (#) + (#)(" x u + " y v) = 0 (1) (2) (3) (4) (# + h) + u" x (# + h)+ v" y (# + h) + (# + h)(" x u + " y v) = 0 # $ + h ) + u" # x( $ + h )+ v" # y( $ + h ) = 0! ( (7) (6)

32 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (#) + u" x (#)+ v" y (#) + (#)(" x u + " y v) = 0 (# + h) + u" x (# + h)+ v" y (# + h) + (# + h)(" x u + " y v) = 0 # ( $ + h ) + u" # x( $ + h )+ v" # y( $ + h ) = 0 (1) (2) (3) (4) (6) (7) " # + h ) ( is called the potential vorticity (7) is called Ertel s theorem (1942) Potential vorticity is carried (without change) by each fluid particle (i.e., by each water column)

33 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. # ( $ + h ) + u" ( # x $ + h )+ v" ( # y $ + h ) = 0 (1) (2) (3) (7) Let G(") be any differentiable function! Then G(#) = dg d# $% Any smooth function of etc. # {G( $ + h )} + u" # x{g( $ + h )}+ v" # y{g( $ + h )} = 0 " ( # + h ) (8) is carried by each water column

34 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. # + " x (u#)+ " y (v#) = 0 # {G( $ + h )} + u" {G( # x $ + h )}+ v" {G( # y $ + h )} = 0 (1) (2) (3) (5) (8)

35 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. # + " x (u#)+ " y (v#) = 0 # {G( $ + h )} + u" {G( # x $ + h )}+ v" {G( # y $ + h )} = 0 (1) (2) (3) (5) (8) # {# $ G( % + h )} + " {u# $ G( # x % + h )}+ " {v# $ G( # y % + h )} = 0! (9) Infinitely many conservation laws!

36 Math structure #4: Track vorticity in 2-D # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (1) (2) (3) # {# $ G( % + h )} + u" {# $ G( # x % + h )}+ v" {# $ G( # y % + h )} = 0 (9) The potential vorticity is very constrained. Q: Can we split the motion from these equations into 2 pieces: 2 irrotational pressure waves, with speed a rotational wave, constrained by (9)? g(" + h)

37 Last topic: How do waves break? # + " x {(u)(# + h)} + " y {(v)(# + h)} = 0, u + u" x u + v" y u + g" x # = 0, v + u" x v + v" y v + g" y # = 0. (1) (2) (3) The shallow water equations are hyperbolic, so waves can break in shallow water, and they do. Q: What mathematical model(s) of wave breaking should be added to these equations to describe wave breaking in shallow water? dissipative shock waves? dispersive collisionless shocks? other? (please specify)

38 How do waves break in shallow water? Choice 1: A plunging breaker - dissipative (CLAWPACK probably uses this)

39 How do waves break in shallow water? Recall Hammack s experiments in shallow water Undular bore - dispersive

40 How do waves break in shallow water? Front of 2004 tsunami reaches the shore of Thailand Note two breaking wave fronts (photos from Constantin & Johnson, 2008)

41 How do waves break in shallow water? Front of 2004 tsunami reaches the shore of Thailand Train of oscillatory waves behind front (Constantin & Johnson)

42 How do waves break in shallow water? Photo due to Clark Little, SWNS Summary: The shallow water equations are similar to the equations of gas dynamics in 2-D. But breaking water waves seem to be more complicated than ordinary shock waves in gas dynamics. Q: How to model wave breaking properly?

43 Thank you for your attention (photo due to Clark Little, SWNS)

44 Wave shoaling in shallow water Q: Why do waves crests in shallow water often line up parallel to the beach? Lima, Peru 2004 Duck, NC 1991

45 Wave shoaling in shallow water 2 # = $ %{gh(x, y)$#} If h = h(x) near shore, then beach c(x) = gh(x) x y deep ocean

46 Wave shoaling in shallow water Q: Why do waves crests in shallow water often line up parallel to the beach? Jones Beach Long Island, NY

Lecture 7: Oceanographic Applications.

Lecture 7: Oceanographic Applications. Lecturer: Harvey Segur. Write-up: Daisuke Takagi June 18, 2009 1 Introduction Nonlinear waves can be studied by a number of models, which include the Korteweg de