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7 Linearization of Fluid Equations: Waves and Instabilities

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7 Linearization of Fluid Equations: Waves and Instabilities D.G.Korycansky Astro204notes November1,2017 37 7 Linearization of fluid equations: waves and instabilities The fluid-dynamical equations are difficult to solve due to their non-linearity. In general, closed- form solutions to non-linear equations are difficult or impossible to find. There are special cases, of course, but in general we must resort to numerical computation for solutions, which technique brings its own complications. The most typical form of nonlinearity the we encounter is the quadratic nonlinearity due to the advection terms of the equations: u ∇u. In such cases, one possible line of· analysis is to consider perturbation techniques, that is, the analysis of “small” disturbances of a simpler basic flow. Formally one can write the fluid variables in the form of an expansion in powers of a small parameter ε: 2 2 ρ = ρ0 + ερ1 + ε ρ2 ... u = u0 + εu1 + ε u2 ..., (7.1) etc. The zero subscripts denote a basic state, such as steady-state equilibrium or a uniform flow. The series expansions are substituted into the fluid equations and the resulting terms are grouped order by order. As noted, the zero-order terms will describe a basic state. First-order equations will yield linear equations describing the evolution of small perturbations. Second and higher orders will contain non-linearities; in some cases the nonlinear terms may be amenable to further analysis, but for our purposes here we will take the attitude that ε denotes an “infinitesimal” perturbation for which higher order effects may be neglected. Such a treatment has its limitations in the sense that growing solutions, if found, will only be valid for short times until the non-linear/higher-order terms can no longer be neglected. Nonetheless, analysis of linearized equations is a valuble tool that has yielded many insights. An example of how the order separation plays out for a sample set of terms in the quadratic non-linearity u ∇u, where the series is substituted in, yielding · 2 u ∇u = u0 ∇u0 + ε(u0 ∇u1 + u1 ∇u0) + ε (u0 ∇u2 + u1 ∇u1 + u2 ∇u0) + ... (7.2) · · · · · · · The order-ε equations are linear; depending on the base state they may have simple (constant) coefficients in time or space, thus giving solutions that are sinusoidal or exponential in the relevant dimensions. Most typically the base state is at least constant in time, so that the perturbations have sinusoidal or exponential (growing/decaying) solutions in time. Dimensions in time or space with constant-coefficient equations allow sinusoidal/exponential variation, thus reducing the di- mensionality of the linear differential equations. Analyses typically seek to reduce the equations to ordinary differential type or algebraic relations among fequencies, wavenumbers, and physical parameters. In many cases, boundary conditions give rise to eigenvalue problems with restricted spectra of distinct solutions. It is probably simplest to proceed from here by way of examples. 7.1 Sound waves (acoustic waves) (Regev 6.2.1–6.2.2, pp 310–312, 315, Landau and Lifshitz 63, pp. 245–249) § § Start with the one-dimensional equations of mass and momentum conservation for compressible inviscid flow; we finesse the energy equation by writing the pressure as a function of density (the D.G.Korycansky Astro204notes November1,2017 38 adiabatic equation of state): ∂ρ ∂ + (ρu) = 0, ∂t ∂x ∂u ∂u 1 ∂ p γ + u = , P = Kρ . (7.3) ∂t ∂x −ρ ∂x ρ ργ We assume that 0 and U0 are constant (independent of x), and it follows that p0 = K 0 is constant as well. Writing ρ = ρ0 + ερ1, u = U0 + εu1, p = p0 + ε p1, we have to linear order ∂ρ ∂ρ ∂u 1 +U 1 + ρ 1 = 0, ∂t 0 ∂x 0 ∂x ∂ ∂ 2 ∂ρ u1 u1 cs 1 +U0 = , (7.4) ∂t ∂x −ρ0 ∂x 1/2 where cs =(γP0/ρ0) is the adiabatic sound speed and we have substituted for the perturbation 2 pressure p1 = c ρ1. The coefficients are constants, so we can write ρ1 = ρ1 exp[i(kx ωt)] and s − similarly for u1, ρ ωρ ρ ρ ω 2 1 i 1 + ikU0 1 + ik 0u1 = 0, i u1 + ikU0u1 + ikcs = 0. (7.5) − − ρ0 Write this system in matrix form ω ρ i( + kU0) ik 0 ρ − 2 1 cs = 0. (7.6) ik i( ω + kU0) u1 ρ − 0 In order for there to be non-trivial solutions, the determinant of the matrix has to vanish, giving us the dispersion relation 2 2 2 ( ω + kU0) k c = 0, or ω = k(U0 cs). (7.7) − − s ± (Note that we have two possible waves, depending on the sign chosen for the kcs term.) 7.2 Dispersive and non-dispersive waves, phase function, wave vector, phase and group velocities The relation ω = k(U0 cs) is an example of a non-dispersive wave, that is all, all Fourier compo- nents (i.e. wavenumbers± k) travel at the same speed. This can be seen by comparing the relation- ship with the outcome of doing the same analysis for the linear advection equation in which we substitute the same dependence exp[i(kx ωt)]: − ∂ f ∂ f + c = 0 iω f + ikcf = 0 ω = ck. (7.8) ∂t ∂x → − → All modes propagate at the same speed c = ω/k. We can introduce the idea of phase speed cp = ω(k)/k. D.G.Korycansky Astro204notes November1,2017 39 If ω is not a linear function of k, then cp will be a function of k, meaning that different modes will propagate a different speeds, and the wave in question will be dispersive. A “packet” of waves that occupies a relatively small region of space (and will hence require a spectrum of modes in k-space) will spread out or disperse as it travels. An example of a dispersive wave is one that propagates along the surface of water of depth h in gravity g with surface tension T . The dispersion relation (which I won’t derive here, yet, at least) is T ω2 = gk tanhkh + k3. (7.9) ρ0 1/2 Considering first the case of infinite depth h ∞ and T = 0, we have ω = (gk) or cp = 1/2 1/2 → g k− . That is, long waves (small k) propagate faster than short ones (large k). For the case including surface tension T , sufficiently short waves will propagate fastest (ω/k ∝ k1/2). We can also introduce the additional concept of the group velocity vg = dω/dk, which gives the propagation speed of a packet as a whole and the speed at which the energy of the wave propagates. For multiple space dimensions, we have a wave vector k that describes the collection of wave numbers in the multiple dimensions e.g. k =(kx,ky,kz). Sometimes it is useful to think in terms of a phase function φ(x,t) for which the sinusoidal variation of the wave ∝ exp(iφ); crests of the wave can be marked by values of φ = 0, 2π,4π, etc. In that case we have the relation between the wave numbers k, the frequency ω and φ: ∂φ k = ∇φ, ω = . (7.10) − ∂t 7.3 Two-dimensional internal waves in a stratified atmosphere (Regev et al. 4.4.2-4.4.3 pp. 215–221) § In this case we look at two-dimensional incompressible waves in the Boussinesq approxima- tion. The equilibrium state is a constant density ρ0 but there is a background temperature gradient dT0/dz > 0 (hotter on top, a dynamically stable situation). The linearized equations for perturba- tions (ρ,u,w, p,T) are ∂u ∂w + = 0 ∂x ∂z ∂u 1 ∂ p + = 0 ∂t ρ0 ∂x ∂w 1 ∂ p + + gρ = 0 ∂t ρ0 ∂z ∂T dT + w 0 = 0. (7.11) ∂t dz with the Boussinesq EOS ρ/ρ0 = αT . − Substituting our paradigm q = qexp[i(kxx+kzz ωt)] for quantities q, and eliminating ρ in favor of T we have − ikx p ikz p dT0 ikxu + ikzw = 0, iωu + = 0, iωw + gαT = 0, iωT + w = 0. (7.12) − ρ0 − ρ0 − − dz D.G.Korycansky Astro204notes November1,2017 40 Note that the imcompressibility condition tells that k u = 0; perturbation velocities are perpen- · dicular to the wave vector. From that condition we have u = kzw/kx, and substituting into the 2 − equation for u, we get p = ρ0ωw/k , so that the w-equation is − x 2 ω kz ω α w + 2 w ig T = 0. (7.13) kx − Substituting for iT =(dT0/dz)w/ω, we get 2 kz ω2 α dT0 1 + 2 g w = 0. (7.14) kx − dz For non-trivial solutions the term in brackets vanishes giving us the dispersion relation 2 2 1/2 ω2 α dT0 kx ω kx = g 2 2 , or = N 2 2 , (7.15) dz kx + kz kx + kz 1/2 where N =(gαdT0/dz) is the Brunt-V¨ais¨al¨afrequency. The Brunt-V¨ais¨al¨ais (naively, the way I think) the frequency at which a parcel of fluid in a stratified medium would oscillate vertically. Some things to note: 1) unlike sound waves, these internal waves are dispersive: the ω is not a linear function of the wave number (or the wave vector denoted by k =(kx,kz)). For multidimen- sional waves we write ω ω kx kz vp = kˆ = , vg = ∇kω, (7.16) k (k2 + k2)1/2 (k2 + k2)1/2 (k2 + k2)1/2 | | x z x z x z where the subscripted nabla symbol indicates derivatives with respect to the wave vector compo- nents k. For two-dimensional sound waves we have c ω = c(k2 + k2)1/2, v = v = (k ,k ) (7.17) x z p g 2 2 1/2 x z (kx + kz ) Thus k u = k vg = ω (k vp = ω is true by definition).
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