The Whitham Equation. John D. Carter April 2, Based upon work supported by the NSF under grant DMS

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1 April 2, 2015 Based upon work supported by the NSF under grant DMS

2 Collaborators Harvey Segur, University of Colorado at Boulder Diane Henderson, Penn State University David George, USGS Vancouver Henrik Kalisch, University of Bergen Nathan Sanford, Seattle University (now at Northwestern U) Keri Kodama, Seattle University (now at George Mason U)

3 Outline I. Whitham s derivation II. Comparisons with experiments III. Stability of traveling-wave solutions

4 Whitham s derivation

5 Mathematical Model Assumptions Assume The fluid is incompressible The fluid is inviscid The motion of the fluid is irrotational There is zero surface tension The fluid rests on a horizontal impenetrable bed The flow is two dimensional

6 Variable Definitions z=ζ, water depth H z=h0 mean fluid level Λ h0 z x I I I I I I z=0 at the bottom g represents the acceleration due to gravity h0 represents the undisturbed fluid depth ζ = ζ(x, t) represents the local fluid depth φ = φ(x, z, t) represents the velocity potential of the fluid η = η(x, t) represents the free surface displacement Note: ζ(x, t) = h0 + η(x, t)

7 Euler Equations The Euler equations are φ xx + φ zz = 0 for 0 < z < ζ(x, t) φ z = 0 for z = 0 ζ t + φ x ζ x φ z = 0 for z = ζ(x, t) φ t + gζ + 2( 1 φ 2 x + φ 2 z) = 0 for z = ζ(x, t)

8 Euler Equations The Euler equations are dispersive with linear phase speed c 2 E ( ω ) 2 = g tanh(kh 0 ) = k k The Euler equations are complicated, so we use simplified models. Define ɛ = H h 0, a (dimensionless) measure of nonlinearity δ = h 0 λ, a (dimensionless) measure of shallowness

9 KdV The Korteweg-deVries (KdV) equation is derived from the Euler system by assuming δ 2 = O(ɛ) and truncating at O(ɛ 3 ). KdV is a model for small-amplitude, long waves.

10 KdV The dimensional KdV equation is given by η t + gh 0 η x + 3 gh0 ηη x + 1 2h 0 6 h2 0 gh0 η xxx = 0 KdV is dispersive and has phase speed c K = ω k = gh 0 ( h2 0k 2) These are the first two terms in the small-k series expansion of the positive root of the Euler phase speed.

11 Scaled phase speeds c g h Euler KdV k h 0

12 Whitham The dimensional Whitham equation is η t + 3 gk tanh(kh0 ) gh0 ηη x + e ik(x ξ) η 2h 0 2πk ξ dk dξ = 0

13 Whitham The dimensional Whitham equation is η t + 3 gk tanh(kh0 ) gh0 ηη x + e ik(x ξ) η 2h 0 2πk ξ dk dξ = 0

14 Whitham The dimensional Whitham equation is η t + 3 gk tanh(kh0 ) gh0 ηη x + e ik(x ξ) η 2h 0 2πk ξ dk dξ = 0 The Whitham equation is dispersive and has phase speed c W = ω k = g tanh(kh0 ) k This is exactly the positive root of the Euler phase speed. The Whitham equation is a model of small-amplitude waves.

15 Scaled phase speeds c g h Euler & Whitham KdV k h 0

16 Whitham The Whitham equation admits the following conserved quantities I 3 = I 1 = I 2 = ηdx η 2 dx ( ηk η η 3 ) dx These are used as checks on the numerics.

17 Comparisons with experiments

18 Experimental Setup Figure not to scale! x 61cm x 561cm x 1061cm x 1561cm 1 h 0 10cm Experiments conducted by Joe Hammack.

19 Experimental Initial Conditions Figure not to scale! x 61cm x 561cm x 1061cm x 1561cm 1 h 0 10cm 0 Experiments conducted by Joe Hammack.

20 Experimental J. L. Hammk Measurements: and H. Segur A 0 = 0.5cm me 0 II i X=t (g/h)f-x/h xperimental From wave J.L. systems: Hammack h = and 10 cm, H. Segur, L, = 122 Thecm, A, = 0.5 cm, M = , (a) z/h = O or Korteweg-deVries 37 = 0, (b) z/h equation = 50 or 37 and= water 25, (c) waves. s/h = 100 or 37 = 50, (d) z/h = 5, (e) z/h = Part 200 or Oscillatory = , waves, trajectory Journal baaed of Fluid on average wavenumber between and linear dispersion Mechanics, relation; 84: , - +, extrapolation of previous trajectory. ɛ = = 0.05 δ = = 0.08 ents

21 Experimental Measurements: A 0 = 1.5cm Korteweg-de Vries equation and water waves. Part z 0 I X = t (g/h)*-.y/h. Experimental wave systems: h = 10 cm, L* = 122 cm, A, = 1.5 cm, M = 8.36 x (a) From z/h = J.L. 0 or Hammack 37 = 0, and (a) H. z/h Segur, = 50 The or 37 = 25, (c) z/h = 100 or 37 = 50, 150 or 37 = 75, Korteweg-deVries (e) z/h = 200 or equation 37 = 100. and+, water trajectory waves. based on average wavenumber wo stations Part and linear 3. Oscillatory dispersion waves, relation Journal ;- +, of Fluid extrapolation of previous trajectory. Mechanics, 84: , wave necessarily appears leftward-running, i.e. the wave front appears at in each figure. Moreover, a point moves to John the D. left Carter or right The in Whitham succeeding Equation ɛ = = 0.15 δ = = 0.08

22 Whitham Expt #2: A 0 = 0.5cm, ɛ = 0.05, δ =

23 Whitham Expt #2, Larger t Interval

24 Whitham Expt #3: A 0 = 1.5cm, ɛ = 0.15, δ =

25 KdV

26 KdV Expt #2: A 0 = 0.5cm, ɛ = 0.05, δ =

27 KdV Expt #2, Larger t Interval

28 KdV Expt #3: A 0 = 1.5cm, ɛ = 0.15, δ =

29 St. Venant Equations

30 St. Venant Expt #2: A 0 = 0.5cm, ɛ = 0.05, δ = 0.08 h t + (hu) x = 0 (hu) t + ( 1 2 gh2 + hu 2) x = Expt

31 St. Venant Expt #3: A 0 = 1.5cm, ɛ = 0.15, δ = Expt St. Venant simulations computed by David George.

32 Serre/Green-Naghdi Equations

33 Serre Expt #2: A 0 = 0.5cm, ɛ = 0.05, δ = 0.08 h t + (hū) x = 0 ū t + ūū x + gh x 1 ( h 3( ū xt + ūū xx (ū x ) 2)) 3h = 0 x

34 Serre Expt #3: A 0 = 1.5cm, ɛ = 0.15, δ =

35 Experimental Comparison Summary Dispersion is important in these experiments. The Whitham equation does a good job modeling the evolution of these long waves of depression. Dissipation plays a role in the experiments, but Whitham is a conservative equation.

36 Stability

37 Dimensionless Whitham Equation The dimensionless Whitham equation is given by η t + 2ηη x + K η x = 0 where K(k) = tanh(k) k It admits family of 2π-periodic traveling-wave solutions of the form η(x, t) = φ(x Vt)

38 Whitham Solutions a b Π Π 2 5 Π 2 Π x Π Π Π 2 Π x c d Π Π Π 2 Π x Π Π Π 2 Π x Four representative 2π-periodic solutions of the Whitham equation. The wave speeds of these solutions are (a) V = , (b) V = , (c) V = , and (d) V =

39 Stability of Whitham Solutions The Whitham equation η t + 2ηη x + K η x = 0 has solutions of the form η(x, t) = φ(x Vt) In order to study the stability of these solutions 1. Change coordinates to a frame moving with speed V. y = x Vt τ = t

40 Stability of Whitham Solutions The Whitham equation becomes and the solution becomes η τ V η y + 2ηη y + K η y = 0 η(y, τ) = φ(y) 2. Consider perturbed solutions of the form η(y, τ) = φ(y) + ɛw(y, τ) + O(ɛ 2 ) 3. Linearization and separation of variables give w(y, τ) = W (y)e λτ Stability is determined by R(λ) > 0: unstable R(λ) < 0: stable R(λ) = 0: stable

41 Stability of Whitham Solutions 4. The resulting differential eigenvalue problem is 2η W + (V 2η)W K W = λw. 5. Floquet s theorem establishes that all bounded solutions have the form W (y) = e iµy l= Ŵ l e ily, µ [ π, π) 6. This gives L Ŵ = λŵ where {ˆf 0,n m + i ( µ + m )ˆf 1,n m if m n L nm = ˆf 0,0 + i ( µ + n )ˆf 1,0 + i(µ + n) tanh(µ+n) µ+n if m = n.

42 Stability of Whitham Solutions a Λ b Λ Λ Λ 500 c Λ d Λ Λ Λ Spectra corresponding to the four Whitham solutions.

43 Stability of Whitham Solutions 03 a Λ 03 b Λ Μ Μ c Λ d Λ Μ Μ Plots of R(λ) versus µ, the Floquet parameter.

44 Nearly Peaked Whitham Solution a Π Π Π 2 Π x b Λ Λ Μ c Λ Plots of (a) the solution, (b) its stability spectra, (c) R(λ) vs µ.

45 Whitham Stability Summary Small-amplitude traveling-wave solutions of the Whitham equation with period 2π are stable. Large-amplitude traveling-wave solutions of the Whitham equation with period 2π are unstable with respect to a modulational instability. Periodic solutions with kh 0 < always exhibit modulational instability (for proof see Hur & Johnson). These results are qualitatively different than the KdV result where all traveling-wave solutions are stable.

46 Summary The Whitham equation provides a good model for the evolution of long waves of depression. Some Whitham solutions are unstable with respect to a modulational instability. The Whitham equation is an interesting equation that needs more attention, especially is dissipative generalizations.

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