Nonlinear Waves: Woods Hole GFD Program 2009

Transcription

1 Nonlinear Waves: Woods Hole GFD Program 2009 Roger Grimshaw Loughborough University, UK July 13, 2009

2 Lecture 10: Wave-Mean Flow Interaction, Part II The Great Wave at Kanagawa 929 (JP1847) The Great Wave at Kanagawa (from a Series of Thirty-Six Views of Mount Fuji), Edo period ( ), ca Katsushika Hokusai (Japanese, ); Published by Eijudo Polychrome ink and color on paper

3 10.2 Lagrangian formulation Part 2: General theory Suppose that the physical system is governed by a Lagrangian L(φ, φ t, φ xi ; t, x) where the field variables are the vector-valued φ(t, x i ). Here t is time, and x i, i = 1, 2, are the spatial variables. The governing equations are then the Euler-Lagrange equations t ( L ) + ( L ) L φ t φ xi φ = 0. (1) Here we use the summation convention over the index i. From this formulation we can recover the conservation laws corresponding to the symmetries of the Lagrangian. Thus, for instance, energy conservation corresponds to time symmetry, that is when L/ t = 0, and momentum conservation corresponds to space symmetry, that is when L/ = 0. Wave action conservation corresponds to a phase symmetry.

4 10.2 Waves and mean flow We now suppose that the solution being sought consists of waves and a mean flow. To describe the waves, we introduce a phase parameter θ, such that φ(t, x i, θ + 2π) = φ(t, x i, θ). (2) For example, for small-amplitude sinusoidal waves, φ(t, x i ) a sin (k i x i ωt + θ), where k i is the wavenumber vector, ω is the wave frequency, a is the wave amplitude. Define a phase average by < >= 1 2π We then denote φ =< φ > and write 2π 0 ( )dθ. (3) φ = φ + ˆφ, (4) so that by definition φ is the mean flow and ˆφ is the wave field.

5 10.3 Wave action Since the Lagrangian has no intrinsic dependence on the parameter θ, it can be shown that the corresponding symmetry is A t + B i = 0, (5) A =< ˆφ θ L φ t >, B i =< ˆφ θ L φ xi >. (6) Then A is the wave action density, B i is the wave action flux. Importantly it is a wave quantity, being zero if there are no waves, and so is a good measure of wave activity. Equation (6) is a conservation law in all physical systems, assuming (as here) that there is no forcing and no dissipation. Note that formally this law is valid without restriction on amplitude and without restriction on the relative time and space scales of the waves vis-a-vis the mean flow. But we shall now proceed to implement it for small-amplitude waves in a slowly-varying medium.

6 10.4 Slowly varying medium We now suppose that the mean flow, the background medium and the wave parameters are slowly varying, and write ˆφ ˆφ(S(t, x i ) + θ; t, x i ), ω = S t, κ i = S. (7) Here ω is the local frequency, and κ i is the local wavenumber. Then the expressions (6) reduce to A L ω, B i L κ i. (8) Here L =< L > is the averaged Lagrangian and like ω, κ i is a slowly varying function of t, x i. But, recall that φ is rapidly varying in the phase itself. Note that (8) is formally valid without any amplitude restriction. Also, for slowly varying waves, we must add the equation for conservation of waves κ i t + ω = 0. (9)

7 10.5 Linearized waves Next we can decompose L as follows L = L 0 ( φ t, φ xi, φ; t, x i ) + L 1 ( ˆφ t, ˆφ xi, ˆφ; t, x i ), (10) where L 0 = L( φ, φ t, φ xi ; t, x i ) is the Lagrangian for the mean flow and L 1 is then Lagrangian for the wave field. The dependence of L 1 on the mean fields is here temporarily suppressed into the explicit t, x i dependence. In the small amplitude approximation the wave field is ˆφ(t, x i ) a(t, x i ) sin (S(t, x i ) + θ). (11) Then in this small amplitude approximation (11) L 1 D(ω, κ i ; t, x i )a 2, ω = ω U i κ i. (12) where we have extracted the dependence on the mean velocity field, U i and ω is the intrinsic frequency, while the remaining mean fields remain suppressed into the explicit x i, t dependence. In this linearized approximation, the mean fields are known. The result that L 1 depends only on the mean velocity U i through ω follows from Galilean invariance.

8 10.6 Wave action for linearized waves Then, from the expressions (8) we get that A = L 1 ω = D ω a2, B i = L 1 = D a 2. (13) κ i κ i But we also have that L 1 / a = 0, so that we get the dispersion relation D(ω, κ i ; t, x i ) = 0. (14) This defines ω = U i (t, x i )κ i + ω, ω = Ω(κ i ; x i, t). Then substitution into (14) and differentiation with respect to the wavenumber κ i yields D ω c gi + D = 0, (15) κ i where c gi = ω = U i + cgi, cgi = ω, (16) κ i κ i defines the group velocity. Hence we finally get that B i = c gi A and the wave action equation (5) is A t + (c gia) = 0. (17)

9 10.7 Energy and momentum We now need to provide a more physical interpretation of wave action, and for this we shall consider the conservation laws for energy and momentum for the full Lagrangian system (1), obtained from the Lagrangian L(φ, φ xs ; x s ), s = 0, 1, 2, 3 where x 0 = t. It is useful to note the general expression, valid for any ψ = ψ(x s ), Then putting ψ = φ xr T rs x s (ψ L ) = ψ xs x s φ xs L φ xr generates the conservation laws + ψ L φ. (18) = L x s, where T rs = φ xr L φ xs Lδ rs (19) T rs is the energy-momentum tensor of classical physics. We can identify T 00 with energy density, T 0i with energy flux, T i0 with momentum density, and T ij with momentum flux. We can now apply the averaging operator to the conservation laws (19) and so obtain the equations governing the averaged total energy < T 00 > and averaged total momentum < T i0 >. But these are not particularly useful as they contain both the mean fields and the waves.

10 10.8 Pseudoenergy and pseudomomentum Instead of the total energy and momentum, we define the pseudoenergy and pseudomomentum (see Andrews and McIntyre (1978)) by putting ψ = ˆφ xs in (18) and then averaging, so that T rs x s = L 1 x r, where T rs =< ˆφ xr L 1 ˆφ xs L 1 δ rs > (20) Here recall L 1 is the Lagrangian (10), defined by L = L 0 + L 1 where L 0 is the mean Lagrangian which depends only on the mean flow. Hence T rs is an O(a 2 ) wave property. We can identify T 00 with pseudoenergy density, T 0i with pseudoenergy flux, T i0 with pseudomomentum density, and T ij with pseudomomentum flux. But (20) is not a conservation law unless L 1 is independent of x s, and this inter alia requires that the mean flow is independent of x s. Note that putting ψ = ˆφ θ yields the wave action equation (6). Thus, if phase averaging corresponds to averaging over a particular coordinate, θ = x s, then T s0 = A and T si = B i, noting that in this case the diagonal term T ss is then absent from the conservation law (20). Thus the wave action is pseudoenergy for time averaging, and is pseudomomentum for space averaging.

11 10.9 Wave energy In general wave energy is not as useful a quantity as wave action, as in general it is not a conserved quantity. Suppose that the mean flow consists of a mean velocity U i, and a vector-valued mean field Λ (mean depth, etc.), which for convenience will be required to satisfy the equation dλ dt + Λ ijλ U i x j = 0. (21) Then, we follow Bretherton and Garrett (1968) and define the wave energy E as the pseudoenergy in a reference frame moving with the mean flow. E = T 00 + U i T i0 =< d ˆφ L 1 dt ˆφ L 1 >, t The corresponding wave energy flux is F i =< d ˆφ dt d dt = t + U i. (22) L 1 ˆφ U i L 1 > (23) xi Here L 1 = L 1 ( ˆφ, ˆφ t, ˆφ xi ; U i, λ; x i, t). Further we suppose that the dependence on ˆφ t is only through d ˆφ/dt = ˆφ t + U i ˆφxi.

12 10.10 Radiation Stress The wave energy equation then becomes E t + F U i = R ij ( d L 1 x j dt ) e, (24) R ij = T ij + U j T i0 Λ ij λ L 1 λ, (25) is the radiation stress tensor. Here the subscript e denotes the explicit derivative with respect to t, x i when the wave field and the mean fields U i, λ are held fixed. Finally we need an equation for the mean flow, which is obtained by variation of the mean fields, subject to the constraint (21). The mean momentum equation is t ( L 0 U i ) + x j (U j L 0 U i Λ ij λ L 0 λ + L 0δ ij ) ( L 0 ) e = R ij + ( L 1 ) e. (26) x j When Λ ij = Mδ ij is isotropic, there is a mean pressure Q such that R ij x j = T i0 t (U jt i0 ) x j + Q. (27)

13 10.11 Slowly varying waves ˆφ ˆφ(S(t, x i ) + θ; t, x i ), In this case we get the reductions ω = S t, κ i = S. Pseudoenergy: T 00 ωa L 1, T 0i ωb i, (28) Pseudomomemtum: T i0 κ i A,, T ij κ i B j L 1 δ ij, (29) Wave energy: E ω A L 1, F ω (B i U i A), (30) Radiation Stress: R ij κ i (B j U j A) + L 1 δ ij Λ ij λ L 1 λ. (31) Recall that ω = ω U i κ i is the intrinsic frequency.

14 10.12 Slowly varying linearized waves Now, we have a dispersion relation (14) so that ω = Ω(κ i ; λ; x i, t), ω = κ i U i + ω, and the wave action equation is (17) A t + (c gia) = 0, where c gi = U i + Ω/ κ i is the group velocity. For linearized waves L 1 = 0 so wave energy E = ω A, and the pseudoenergy T 00 = ωa. The wave energy equation (24) and the radiation stress (25) become E t + ([U i + c U i x gi]e) = R ij i x j + E ω (dω dt ) e, (32) R ij = A(κ i c gj + Λ ij λ Ω λ ). (33)

15 10.13 Extensions Modal waves: Often waves are confined to waveguides such that there is propagation in only a reduced set of spatial dimensions, and a modal structure in the remaining spatial dimensions. Examples are water waves and internal waves. This general theory can still be used, but we need to combine the Lagrangian averaging with integration across the waveguide. Generalized Lagrangian mean (GLM) theory: For applications to fluid flows we need to identify a suitable Lagrangian. Although these can sometimes be found in the Eulerian formulation, it is usually best for our present purposes to use a Lagrangian formulation of the equations of motion. But to be able to consider finite amplitude waves, we define the particle displacements from a mean position that moves with the mean velocity, U i. This is the GLM theory developed by Andrews and McIntyre (1978). Thus, x i are Lagrangian variables moving with the Lagrangian mean velocity U i, relative to which we define the particle displacements ξ i. The Eulerian variables are then x i = x i + ξ i, and the Eulerian velocity is u i = U i + dξ i /dt (d/dt = / t + U i / ) and < ξ i >= 0.

16 REFERENCES An exact theory of nonlinear waves on a Lagrangian mean flow Andrews, D.G. and McIntyre, M.E J. Fluid Mech., 89, On wave-action and its relatives Andrews, D.G. and McIntyre, M.E J. Fluid Mech., 89, Wavetrains in inhomogeneous moving media Bretherton, F.P. and Garrett, C. J. R. 1968, Proc. Roy. Soc., A302, Wave Action and Wave-Mean Flow Interaction, with Application to Stratified Shear Flows Grimshaw, R. H. J Ann. Rev. Fluid Mech., 16, Linear and Nonlinear Waves Whitham, G. B Wiley, New York.