Introduction to the Mathematics of Geophysical Waves in Fluids Course Notes. Paul Milewski

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1 Introduction to the Mathematics of Geophysical Waves in Fluids Course Notes Paul Milewski October 13, 2009

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3 Chapter 1 Introduction These notes are written to accompany an introductory level graduate course in wave propagation in fluids. They began as short course notes for 2-week classes taught at IMPA in Rio de Janeiro, Brazil and AIMS in Muizenberg, South Africa in Spring This course will present the main concepts and equations of linear and nonlinear waves using water waves and internal waves as a physical context. The goal is to give students an appreciation and intuition for the dynamics of waves and introduce them to the canonical equations appearing in the field. The course will cover the following topics, not necessarily in order: (i) Introduction/review of incompressible fluid dynamics, including free surface problems. (ii) Shallow water theory: hydraulic approximation, breaking waves and shocks. (iii) Weakly dispersive models: Boussinesq, Korteweg-de Vries and Su- Gardner. (iv) Dispersion - waves in deep water - phase and group speed and the kinematics of linear waves. (v) Dispersion and nonlinearity: the Nonlinear Schrodinger Equation. The notes are written informally, and cover a very selective range of topics. 3

4 4 CHAPTER 1. INTRODUCTION

5 Chapter 2 The Fluid Equations 2.1 Governing Equations In this section we present, without derivation, the equations which we shall use in the remnant of the notes. The Euler equations for ideal (frictionless) fluids are the statements of local conservation of mass, momentum and energy, written, ρ t + (ρu) = 0, (2.1) (ρu) t + (ρu u + pi) = ρg, (2.2) (ρe) t + (ρuh) = ρg u, (2.3) where x = (x, y, z) with z pointing in the direction opposite to gravity, u(x, t) = (u, v, w) is the fluid velocity, ρ is the density, p is the pressure, g is the external force per unit mass (usually if gravity alone is acting, g = ge z where e z is the unit vector in the z direction), E = e u2 is the specific total energy, e is the internal specific energy, H = E+ p ρ is the specific enthalpy. The term (ρu u) means i x i (ρu i u j ). To close the system (which has one more unknown than the number of equations) we have to add a thermodynamic relationship. In the simplest case this is a gas law, and most often one assumes isentropic gas dynamics which then replaces (2.3) and one obtains then p = p(ρ). These equations have been written in conservation form so that, if solutions are not smooth, one may write the appropriate weak form of the equations. These local conservation laws can be obtained from bulk laws of conservation through the Reynolds transport theorem (Leibnitz s integral rule in more dimensions): d dt Ω(t) F dv = Ω(t) F t dv + Ω(t) v n F da, where F is arbitrary (scalar or vector), Ω is a closed volume with boundary Ω that evolves with speed v and has the outward normal n. v is usually chosen to be equal to u the fluid velocity in order to have a Ω be a material volume. 5

6 6 CHAPTER 2. THE FLUID EQUATIONS There are 2 important incompressible cases, that is, cases where u = 0. One should view incompressibility as a consequence of the formula for expansion or contraction of an arbitrary material volume Ω(t) (the boundary of Ω follows fluid particles) d 1 dv = u dv (2.4) dt Ω(t) Ω(t) The first incompressible case is the case of constant density ρ = ρ 0. This applies to pure liquids under most physical situations, and is the approximation used for surface wave phenomena. In this case, the equations u = 0, (2.5) u t + (u u + p ρ 0 I) = g, (2.6) form a closed system, decoupled from the Energy equation. The second case is a little more subtle: equation (2.1) can be written ρ t + u ρ = ρ u, (2.7) Thus, with u = 0 we require that a nonconstant ρ satisfy ρ t + u ρ = 0. (2.8) This is called the stratified incompressible case. The interpretation of this equation is best seen in a Lagrangian framework. The particle paths X(t) of fluid parcels satisfies the ordinary differential equations dx dt = u(x(t), t). Thus, (2.8) implies that ρ(x(t), t) along such a path satisfies d dt ρ = 0. This means that the density of a fluid parcel does not change as it is carried by the flow, but, importantly, the density of nearby parcels need not be equal. There many physical situations in which this may apply and, in fact, most internal waves in the atmosphere and the ocean obey this physics. The closed equations of motion are then u = 0 (2.9) ρ t + u ρ = 0, (2.10) ρu t + ρ (u u + pi) = ρg, (2.11) The Lagrangian derivative of a quantity along a particle path t + u occurs often in fluid dynamics and is denoted d dt, the material derivative. In the incompressible case (u u) = u u and one often finds, say d ρ = 0, (2.12) dt d dt u = 1 p + g, (2.13) ρ

7 2.2. VORTICITY 7 replacing (2.10,2.11). Let us return to the physics of the stratified case: the simplest situation in which this occurs is in the ocean. Ocean waters have small but significant changes in density due to variations in salinity and/or temperature. The processes that reequilibrate these variations are molecular diffusion (of either salt in solution or heat) and can be assumed to happen on a timescale much longer than the wave motion of interest. (An exception is when this diffusion is enhanced by the small scales engendered by violent turbulence which mixes and uniformizes the fluid rapidly.) A formulation for the mixture of 2 components (say salt and water) including diffusive effects and its reduction to the above equations is beyond the scope of these notes. A common simplification made in the momentum equation of the stratified case is the Boussinesq approximation which replaces the variable density in the acceleration terms (ie the effect of the varying density on the inertia) with a constant, reference ρ 0 and leaves only the the variable density in the gravity term. Thus relative buoyancy due to variable density is preserved. The formal justification of this approximation will be discussed later. Lastly, we note that, if the internal energy is a constant, the energy equation is a consequence of mass and momentunm conservation when the solutions are differentiable. Weak solutions that includes shocks discontinuities in the variables or their derivatives satisfying the mass and momentum conservation may not satisfy the energy equation, even in a weak sense. We shall see examples of this when discussion breaking waves in shallow water EXERCISES 1. Show (2.4), the integral relation between the divergence and volume changes. 2. In the incompressible cases, show how the Energy equation is automatically satisfied by differentiable solutions satisfying mass and momentum conservation if one assumes that internal energy is constant. 3. Write separate equations for the conservation of mass for water and a salt in solution. The equation for the salt should include diffusive effects. Show how in the case of zero diffusivity the equations can be combined to our form of stratified mass conservation. 4. Show that (u u) = u u when u = Vorticity Vorticity plays an important role in fluid dynamics. As we shall see, irrotational (no vorticity) flows are much simpler than vortical ones. Vrticity is the local rate of rotation or the local angular momentum of a material fluid element. A physical motivation is to consider the circulation of the flow around a closed curve S bounding an area A on a plane with normal n defined according to the

8 8 CHAPTER 2. THE FLUID EQUATIONS right-handed rule with the direction of integration along S, 1 ζ = u dr. Area(A) Defining the limit as A 0 as ζ n we have, by Stokes theorem, that ζ = u. The governing equation for the vorticity is obtained by taking the curl of the momentum equation. For the constant density case and if g = 0 S ζ t + u ζ = ζ u. This left hand side is just the material derivative of vorticity. The right side is the vortex stretching term (note that it is ξ times the derivative of u in the direction of ξ) which strengthens the local vorticity following fluid particles if there is local fluid convergence in a plane perpendicular to the vorticity vector. It reflects conservation of angular momentum which means that fluid will spin faster as it converges (the spinning skater effect). To understand this term in the equations consider locally that the vorticity is in the positive e 3 direction, then, ζ u = ζ (u z e 1 + v z e 2 + w z e 3 ) = ζ (w x e 1 + w y e 2 + w z e 3 ) which means that the change in the magnitude of vorticity vector is a maximum if w is aligned with e 3 (the other two components to leading order tilt the vorticity vector). Thus, if w x = w y = 0, then ζ u = (u x + v y )e 3. In two dimensions this term is identically zero (since the vorticity vector is normal to the plane of the flow and the fluid in the plane is divergence free). In any case, in standard ideal flows (see below for exceptions), if the flow is initially irrotational (vorticity zero everywhere), it will remain irrotational. Shear another name for vorticity used mostly for two-dimensional flows has to be there from the start. Clearly there are ways that vorticity can be generated by fluid flows, and we mention three. The standard mechanism is the shedding of vorticity from thin regions near boundaries where it is generated by the viscous no-slip condition. Vorticity can also be generated by the barotropic mechanism: the vorticity equation for variable density has a generation term on the right hand side equal to ρ 2 ρ p. Lastly, topological changes in fluid domains can generate vorticity: an example is a breaking wave. If the flow is irrotational, then there is a velocity potential φ such that u = φ which simplifies the momentum equation (2.6) considerably and yields, through an integration, Bernoulli s equation ρ 0 (φ t ( φ)2 + gz) + p = 0. Furthermore, because of incompressibility, the velocity potential satisfies Laplace s equation φ = 0. Since the velocity potential is harmonic, complex variable methods have been used for many classic problems in 2 dimensional flows.

9 2.3. BOUNDARY CONDITIONS EXERCISES 1. Derive the barotropic vorticity generation term. 2. Fill in the steps of the argument that shows that the convergence in the plane perpendicular to the vorticity vector intensifies vorticity. 2.3 Boundary Conditions There are three main types of boundary conditions that we need to discuss: the conditions at rigid walls, the conditions in the far field, and the conditions at free boundaries. At rigid nonporous walls the standard condition is the so-called slip condition: that the fluid slides along the wall without penetrating it. If Γ is a rigid boundary with normal vector e Γ then u e Γ = 0 on Γ Generally, at walls, the fluid pressure is a priori unknown. Also, walls are material particle paths a sometimes useful fact for constructing solutions and placing walls a posteriori. The boundary condition for moving walls is also easy to treat (see exercises). The free surface boundary conditions between two fluids has 2 distinct parts: that the free surface follow material particle paths (or as often stated that the free surface moves with the velocity of the fluid at the surface) and that the sum of the stresses on a thin material volume including the free surface does not lead to infinite accelerations as this volume shrinks to include only the free surface. The former is called the kinematic boundary condition (KBC) and the latter the dynamic boundary condition (DBC). The KBC is best written in terms of a level set: suppose that F(x, t) = 0 defines the evolution of the fluid interface. Then, we need to state that, at the interface F = 0, F evolves with the fluid velocity there F t + u F = 0, at F = 0. Often, one assumes that the free surface is a graph z = η(x, t). (Here a change of notation has occurred: x has ceased to mean (x, y, z) and now means the horizontal variables (x, y) only. In these notes, it should be clear from context whether the vertical variable has been separated out of the problem.) Then one can take F = η z and the KBC becomes η t + u η = w, at z = η. Ideal fluids do not support tangential stresses so the DBC arises out of only the normal stress (pressure) balance at the interface. This is written p + p = γκ, at F = 0. Here p ± are the pressures in each fluid as one approaches the interface from each side, and γκ is the pressure jump at the interface due to surface tension

10 10 CHAPTER 2. THE FLUID EQUATIONS effects: κ is the mean curvature of the interface and γ is the coefficient of surface tension. The sign of the curvature is defined such that κ > 0 when the domain with the fluid with pressure p is locally convex at the surface. Note that in the absence of surface tension, the pressure is continuous at the surface a requirement to avoid infinite accelerations of the massless surface. The far field conditions are usually dependent on the physical problem at hand, but the most common is the assumption of appropriate decay in all variables to an equilibrium state. Often far field conditions are avoided alltogether by considering periodic boundary conditions (or equivalently a periodic physical domain) EXERCISES 1. Write down the slip boundary condition for a (flexible) wall given by the equation z = b(x, t).

11 Chapter 3 Linear Interfacial flows We start with the simplest, yet already quite rich problem of the evolution of small disturbances on the interface between two fluids. Here we will see emerge information about a variety of wave phenomena that are observed in real fluids: the linear wave equation in the limit of small depth and dispersive effects as the depth (relative to wavelength) is increased, the effects of surface tension and of a possible shear. Consider the flows between 2 rigid horizontal walls at z = H 2 > 0, z = H 1 < 0 of immiscible fluids of constant density ρ 2 < ρ 1, which, in equilibrium, have an interface at z = 0. The mathematical formulation of this problem for the 2 fluid potentials and the interface η(x, t) is: φ 1 + φ 1 zz = 0, H 1 < z < η φ 1 z = 0, z = H 1 η t + φ 1 η = φ 1 z, z = η ρ 1 (φ 1 t ( φ1 ) (φ1 z) 2 + gη) = ρ 2 (φ 2 t ( φ2 ) (φ2 z) 2 + gη), z = η η t + φ 2 η = φ 2 z, z = η φ 2 + φ 2 zz = 0, η < z < H 2 φ 2 z = 0, z = H 2 This problem is clearly difficult: it has nonlinear boundary conditions and is a free-boundary problem: the domain of the solution is part of the problem! We shall simplify it by linearizing the equations and assuming that the deviations from z = 0 are small. Thus all terms of the form G(x, z, t) z=η one writes G(x, 0, t). The result is to replace the boundary conditions at z = η by η t = φ 1 z, z = 0 ρ 1 (φ 1 t + gη) = ρ 2(φ 2 t + gη), z = 0 η t = φ 2 z, z = 0 11

12 12 CHAPTER 3. LINEAR INTERFACIAL FLOWS We consider periodic solutions in an unbounded domain in x and write the solution as a Fourier integral. That is, η = R 2 η(k, t)e ik x dx and φ 1,2 = R 2 φ 1,2 (k, z, t)e ik x dx. The result is that solutions to the Laplace s equations are φ 1 = A(k, t)cosh[(z + H 1 ) k ] φ 2 = B(k, t)cosh[(z H 2 ) k ] Substituting these into the boundary condition, together with η = C(k, t) will lead to a second order differential equation d 2 C dt 2 + g(ρ 1 ρ 2 ) k sinh( k H 1 )sinh( k H 2 ) ρ 2 sinh( k H 1 )cosh( k H 2 ) + ρ 1 cosh( k H 1 )sinh( k H 2 ) C = 0, which shows that the Fourier modes of wavenumber k oscillate at frequency ω(k) given by the dispersion relation ω 2 = g(ρ 1 ρ 2 ) k sinh( k H 1 )sinh( k H 2 ) ρ 2 sinh( k H 1 )cosh( k H 2 ) + ρ 1 cosh( k H 1 )sinh( k H 2 ). Note that often these steps are bypassed by assuming a solution to the equations from the start in the form of a monochromatic wave φ 1 = Acosh[(z + H 1 ) k ]e i(k x ωt) φ 2 = B cosh[(z H 2 ) k ]e i(k x ωt) η = Ce i(k x ωt) and obtaining the dispersion relation from the condition that the resulting 3 equation for A, B, C have a nontrivial solution. We have used complex solutions for φ and η to simplify the calculations. This is appropriate since the problem is linear and has real coefficients, which allows to conclude that the real and imaginary part of a complex solution are solutions to the problem. Once the dispersion relation is obtained, the general solution to the linear problem may be written as the Fourier integral η = η + (k)e ik x ω + (k)t + η (k)e ik x ω (k)t dk, R 2 where ω ± are the two branches of the dispersion relation (which, in one dimension may correspond, for example with right- and left- travelling waves) and η ± corresponds to the projection of the initial data (with 2 degrees of freedom for a second order system in time) onto the Fourier coefficients. Explicit computations of this integral are unwieldy although it is very useful in long time/space asymptotics where stationary phase methods can be used (see the section on the Kelvin ship wake).

13 13 Before discussing some of the details of this dispersion relation we point out a few characteristics of this type of solution. The wave crests are perpendicular to k and the wavelength is 2π/k. The crests propagate at the phase velocity c p = ω k e k. If ω is a function of k as is the case here, then the dispersion relation is isotropic: all directions are equivalent. If ω/ k is a constant, then the waves are nondispersive, that is, all wavelengths travel at the same speed. Dispersive equations have the property that initially coherent or localized data will disperse into an oscillatory wave field as time increases since Fourier components travels at different speeds. Likewise one can construct, for dispersive equations, initial data that is disperse and that will dispersively focus into a coherent localized state at a chosen time. There are several limits to discuss. Note that, when we take H 1,2 below, although formally one can take the limit directly in the dispersion relation, one should modify the boundary condition to a decay of Φ 1,2 at infinity, yielding Φ 1,2 exp[± k z] instead of the form we used above. 1. Deep Water. H 1,2 Defining ρ 0 = (ρ 1 + ρ 2 )/2 and δρ = (ρ 1 ρ 2 )/2 we obtain ω 2 = g δρ ρ 0 k. (3.1) In this case the waves are dispersive and there is the so called reduced gravity factor g δρ ρ 0 slowing the time scale of the waves. Note that c p 1/ k and therefore longer waves travel faster. In this case the velocity field decays exponentially with depth: u z=±2π/ k = e 2π u z=0. Particle paths can be shown to be almost circular. 2. Finite Depth Surface Waves. ρ 2 = 0 ω 2 = g k tanh( k H 1 ). (3.2) This is the standard free surface waves dispersion relation valid for water of arbitrary depth. Note that the density is irrelevant! 3. a. Shallow Water. The limit k H 1 = k H 2 1. Taking only the leading order terms in the dispersion relation, one obtains ω 2 = g(ρ 1 ρ 2 )H 1 H 2 ρ 2 H 1 + ρ 1 H 2 k 2.

14 14 CHAPTER 3. LINEAR INTERFACIAL FLOWS This means that the phase velocity is constant and the dynamics evolves according to the wave equation u tt c 2 u = 0, c 2 = g(ρ 1 ρ 2 )H 1 H 2 ρ 2 H 1 + ρ 1 H b. Shallow Water with finite depth effects. ρ 2 = 0, k H 1 1 For clarity we restrict to one horizontal dimension, one fluid and expanding the dispersion relation in a Taylor series, obtain: ω ± gh 1 k(1 1 6 k2 H 2 1 ). The signs correspond to left and right travelling waves. It is interesting here to consider an equivalent partial differential equation (for η by abuse of notation!) with exactly the plus branch of this dispersion relation (rescaled) η t + η x + µ η xxx = 0. The η x term may be eliminated by changing to a frame moving at speed 1 to the right with ξ = x t, then the µ 2 factor may be absorbed into the time scale with τ = µ 2 t. The result is the linear Korteweg-de Vries (KdV) equation - also called the Airy or Stokes equation η τ η ξξξ = 0. These transformations mean that the leading behavior is a travelling pulse but that in times of O(µ 2 ), initially coherent data for this equation will disperse into an oscillatory wavefield. In this shallow water limit the velocity field is almost uniform with depth: u z= H1 = (1 1 2 k 2 H 2 1) u z=0. Note that the definition shallow water depends on the ratio of wavelength to depth (kh is the dimensionless number). Thus a tsunami over 4km of ocean water is shallow (typical wavelengths are 200km) whereas capillary ripples in 5cm of water are deep water waves (typical wavelength 2cm). As the water becomes shallower the particle paths become elongated quasi-ellipses. 4. Internal Shallow Water. H 2, k H 1 1 Consider the same situation as in 3, but with a fluid of infinite depth above. Such scenarios (or their equivalent upside down versions) exist widely in nature. This is sometimes called the one and a half layer model. ω g δρ ρ 1 H 1 k(1 ρ 2 2ρ 1 k H 1 ).

15 15 The same exercise as in the previous case yields the equivalent equation η t + η x µ 1 6 Hη xx = 0. This equation is the linear Benjamin-Ono (BO) equation. The decay in fluid velocity is exponential in the deep layer and the velocity is almost uniform in shallow water. The Hilbert transform H has been introduced through its Fourier symbol: F[Hη] = isign(k)f[η] It is usually introduced as an integral operator Hη = 1 π η(t) x t dt. We shall return to the Hilbert transform: this nonlocal operator appears often in wave models. The combination H x is the Dirichelet to Neumann map (DtN) of Laplace s equation for the half-plane. Thus it is an approximation of the map between surface potential and surface vertical velocity in fluids of infinite depth. 5. Shear layer. Adding a mean flow with φ 1,2 ±(U/2)x + φ 1,2 and H 1,2 yields the anisotropic dispersion relation (k = (k 1, k 2 )). ω 2 = g δρ ρ 0 k 1 4 U2 k 2 1. One notices here a common effect of shear flows: the possibility of hydrodynamic instability evidenced by ω complex. Indeed if g δρ k ρ 0 U 2 k1 2 < 1 4 the linear disturbances grow exponentially. Note that since k provides an inverse length scale we can see that this criterion is exactly related to the Richardson number criterion for instability when Ri g δρ (z) ρ 0 U (z) 2 < Surface Tension Effects. The effect of surface tension changes the high k behavior (short waves). When we include the effects of surface tension we introduce a length scale in the problem (in addition to H 1, H 2 ). For reasonable situations this scale tends to be small (less than 10cm.) In view of this, for simplicity, we consider here that k H 2. ω 2 = (g(ρ 1 ρ 2 ) + τ k 2 ) k sinh( k H 1 ). ρ 2 sinh( k H 1 ) + ρ 1 cosh( k H 1 ) In geophysics surface tension effects are important in the generation of ripples by wind which is a deep water phenomenon and thus simplifies the relation above to ω 2 = (g(ρ 1 ρ 2 ) + γ k 2 ) k /(ρ 2 + ρ 1 ).

16 16 CHAPTER 3. LINEAR INTERFACIAL FLOWS 3.1 EXERCISES 1. Make the appropriate rescalings and arguments to obtain the linear KdV and BO equations from the dispersion relation. What is the dimensionless parameter µ? 2. Obtain the dispersion relation which includes the effect of surface tension. In order to find the expression for the curvature of the surface, use that κ = n. 3. [NUMERICAL] Particle paths for linear waves. Consider the following way to write the solution to the water wave problem for a one-dimensional freesurface: φ(x, z, t) η(x, t) ω 2 = iωc = Ce i(kx ωt) = g k tanh( k H) cosh[(z + H) k ] e i(kx ωt) sinh( k H) The particle paths satisfy the differential equations ẋ = u(x, t), where x = (x, z), u = (u, w)(x, z, t) = (φ x, φ z ). Since the problem is linear C is arbitrary. We can choose C = 0.05, but experiment with values so that the plots come out nicely. In shallow water the appropriate nondimensional measure of amplitude is C/H, whereas in deep water it is Ck. (a) Progressive Waves. For k = 1, plot a sample of particle paths with ω > 0, for 3 cases H = 0.2, H = 0.7, H = 2. In each case and on the same graph plot 6 paths initiating at x = 0 and z = 0, 0.2H, 0.4H, 0.6H, 0.8H, H and plot η(x, 0). Describe your results. Do particle paths close? If the horizontal displacement in one period equal to the wavelength 2π/k? (b) Standing Waves. For a superposition of two waves, both with k = 1, one with ω > 0 and one with ω < 0, plot a sample of particle paths for 3 cases H = 0.2, H = 0.7, H = 2. In each case plot paths initiating at x = 0 and at x = π/2 for z = 0, 0.2H, 0.4H, 0.6H, 0.8H, H and plot η(x, t) at various times. Show that the free surface displacement is a standing wave. (c) In deep water (H ), and for C small, show that the particle paths are approximately circles, centered at (x 0, z 0 ) and of exponentially decaying radii kc exp[ k z 0 ].

17 Chapter 4 Conservation Laws Before continuing the discussion of real physical problems, we digress briefly to the problem of a conservation laws. All physical process at some level can be written in this fashion, and the mathematical results and intuition will serve us well later. Conservation laws are also intimately related to hyperbolic PDEs: in fact we shall restrict ourselves to first order conservation laws where the flux law (see below) is a function of the dependent variables alone, not their derivarives. In physically reasonable situations this usually gives rise to hyperbolic PDEs. Flux laws can also depend on derivatives of the dependent variables (such as Fick s law of diffusion) and then the result is usually not wave-like, but diffusivelike. 4.1 Scalar Conservation Laws Consider the conservation law for the arbitrary quantity ρ(x, t) ρ t + [Q(ρ)] x = 0. This is called a conservation law because d dt x2 x 1 ρ dx = Q(ρ(x 1, t)) Q(ρ(x 2, t)). Thus, the rate of change of the total amount of ρ in [x 1, x 2 ] is equal to the difference in Q between the two endpoints: giving Q the property that it is the flux of ρ. In other words Q is the amount of ρ transported in the positive x direction per unit time. Defining U(ρ) through Q(ρ) = ρu(ρ), we have that ρ t + [U(ρ)ρ] x = 0, and that U(ρ) is the transport velocity of ρ. Scalar laws are precisely those where U is a function of the quantity ρ alone. Lastly, we can, assuming solutions ρ(x, t) are smooth, define c(ρ) = Q (ρ) and write ρ t + c(ρ)ρ x = 0. 17

18 18 CHAPTER 4. CONSERVATION LAWS The quantity c has the same units as U (and is thus a speed also) but is in general different from U - unless the equation is linear. It is the speed of propagation of disturbances or of information. It is called the characteristic velocity of the equation. One often cites the example of modelling traffic flow: if ρ is the density of cars along a single lane road, and if one can associate a function that gives the car s velocities U as a function of the traffic density (a strong modeling assumption indeed!) then one obtains the conservation of cars with the flux Q = ρu(ρ). In this case what is the interpretation of thevelocity c? Take the example of cars arriving at a traffic jam: although the cars may be moving in the positive x-direction, the jam may be increasing in length as new cars arrive so that the disturbance (the edge of the jam) is propagating to the left. We shall see soon another conservation law model when ρ represents the density of waves, otherwise known as the wavenumber k! This form of the equation is solvable by the method of characteristics. Consider the initial value problem where ρ(x, 0) = ρ 0 (x). Define the characteristic curves in the x, t plane, x = ζ(t), defined by d dt ζ = c(ρ). Along these curves, the value of ρ, ρ(ζ(t), t) = q(t) satisfies d dt q = ρ t + ρ x ζ t = ρ t + cρ x = 0. Thus ρ is constant along characteristics and the characteristics are therefore straight lines ξ(t) = x 0 + c(ρ 0 (x 0 ))t. The solution for arbitrary values of x, t is then constructed by propagating characteristic curves from every point x 0 at t = 0. This solution is sometimes written in the implicit way ρ(x, t) = ρ 0 (x c(ρ 0 )t). An important feature of this solution is that at the time t = min x R,t=0 {( c (ρ)ρ x ) 1 } there is the first crossing of characteristics, and the solution no longer makes sense (at least in terms of the PDE). This crossing of characteristics corresponds to ρ x and the formation of a shock. We shall discuss in later sections how to reconcile shocks with the conservation law. A more general form of a scalar conservation law PDE with a source term is ρ t + c(ρ, x)ρ x = f(ρ, x).

19 4.2. SYSTEMS OF CONSERVATION LAWS 19 For this equation the method of characteristics yields the equations d ζ dt = c(q, ζ), (4.1) d q dt = f(q, ζ). (4.2) Although we have reduced the problem to solving a system of 2 ODEs, in general, the characteristics are not straight lines and q is not constant on them. 4.2 Systems of Conservation Laws Systems of conservation laws take the form v t + [Q(v)] x = 0, for a vector v = [v 1,... v N ] T of conserved quantities (for example, mass, momentum and energy) with corresponding fluxes Q = [Q 1,..., Q N ] T. For smooth solutions one can rewrite this as v t + A(v)v x = 0, A = Q i v j. One can now consider the dynamics along a space-time curve x = ζ(t) dv dt = ζv x Av x, and one may linearly combine these equations using the vector of weights l(v) by writing l T dv dt = lt ( ζi A)v x. Choosing ζ = λ(v) where λ,l are such that the linear combination of derivatives of v is zero along characteristics gives l T dv dt = 0, det(λi A) = 0, (λi A)T l = 0. A system is defined as hyperbolic if there are N linearly independent vectors l satisfying this. Together with these left-eigenvectors are N characteristic speeds λ. What is the special significance of these speeds? They are the speeds at which information is carried by the system: if one considers the dependence of a solution at a point x and time t on the solution at a previous time t t, one can draw line segments of slope λ and express the solution as combinations of the variables at this previous time. It is also along these curves that weak discontinuities may propagate: along those curves the local evolution of v is independent of v x. From this one can infer that along these curves discontinuities in v x can exist. (These should not be confused with shocks which

20 20 CHAPTER 4. CONSERVATION LAWS require conservation laws to interpret.) Suppose v is continuous across a curve (not yet assumed to be characteristic) but v x is not. By geometry, locally, [v t ] = dx dt [v x], where dx dt is the local derivative of the curve and [ ] indicates the jump in the quantity. Computing the derivatives on either side of this curve with the equation v t = A(v)v x and subtracting, we get dx dt [v x] = A(v)[v x ], which requires dx dt = λ and that the jump in v x be proportional to the righteigenvector of A: (λi A)r = 0, [v x ] = αr. One can sometimes go further in solving hyperbolic systems. Suppose one can find a function R(v) such that dr = l T dv, or, if that is not possible dr = µ(v)l T dv, for an arbitrary integrating factor µ. Then, clearly, the evolution along the corresponding characteristic λ diagonalizes to R t + λr x = 0. The new dependent variable R is called the Riemann invariant of the system. This form of the equation makes explicit the concept that the characteristic speeds are the speeds at which information is carried. If R s can be found for all the linear combinations l T dv, then the system becomes completely diagonal R t + DR x = 0, where D(R) is a diagonal matrix with the characteristic speeds along the diagonal and R = [R 1,...,R N ] T. Unfortunately any diagonalization is in general impossible for N > 2, but, fortunately,a complete diagonalization is always possible (in principle, perhaps not explicitly) for N = 2. This fact follows from integrability conditions of differential forms. 4.3 Shocks 4.4 Problems 1. There are many different ways to compute the time at which shocks form. Here are two of them. (i) Starting with the fact that the characteristics are straight lines, compute directly the time by finding the first crossing of (infinitesimally) nearby characteristics. (ii) Write an evolution equation for the slope of the solution s(x, t) = ρ x, solve it by the method of characteristics and show that it develops a singularity in finite time. 2. Propose a reasonable Flux law for traffic flow. Explain why it is reasonable.

21 4.4. PROBLEMS Phase plane analysis. The equations for the solution of a single hyperbolic PDE (4.2), given that they are a system of 2 autonomous ODEs, are amenable to phase plane analysis. Consider the PDE u t + uu x = sin(x), u(x, 0) = u 0 (x). Interpret the equations and their solutions in the phase plane.

22 22 CHAPTER 4. CONSERVATION LAWS

23 Chapter 5 Kinematics of Linear Waves 5.1 The Eikonal and Amplitude Equations Quite a lot of information about wave the wave field can be obtained by understanding the kinematics of the waves, where the information is completely encapsulated in the dispersion relation. In order to simplify things, consider a one (horizontal) dimension problem with ω = Ω(k). That is, we shall deal here with an evolution equation of the form η t + Lη = 0, where the linear spatial operator L yields the required dispersion relation. Of course, for such a linear problem, we could just write the general solution η(x, t) = η(k)e i(kx Ω(k)t) dk, η(k) = 1 η(x, 0)e ikx dx. 2π R Here, we take another approach consider single phase solutions of the type η(x, t) = A(X, T)e iθ(x,t)/ǫ, (X, T) = ǫ(x, t), ǫ 1. If the second derivatives of Θ are order one, in the O(ǫ 1 ) vicinity of any point this solution is a monochromatic wave (single wavenumber) with k = Θ X, ω = Θ T. We retain these as global definitions of the slowly varying wavenumbers and frequency k(x, T), ω(x, T). Substituting the single-phase solution into the governing equation will yield, at leading order in ǫ, the dispersion relation ω(x, T) = Ω(k(X, T)). It follows from the definition of k and ω that k T + ω X = 0, or k T + (Ω(k)) X = 0, or k T + (c p k) X = 0. (5.1) This is the equation for the conservation of waves. Ω is the flux function for k(x, T) which the local density of wave-crests per unit (2π) length in X. This 23 R

24 24 CHAPTER 5. KINEMATICS OF LINEAR WAVES equation is often called the Eikonal equation or the equation of goemetrical optics. It is our first nonlinear equation, governing the dynamics of linear waves! From the third version of the equation we see that the transport speed of wave crests is the phase velocity Ω/k. Differentiating in the middle form above k T + c g k X = 0, c g Ω (k). Where we have now defined the group velocity c g which is the characteristic speed of the wave transport equation. Thus, along the characteristic curves Ẋ = c g (k(x)) the wave density k is a constant. For problems in more dimensions the equations would read k T + c g X k = 0, c g = k Ω(k), k = 0. There is also a similar equation for the frequency ω. Since ω T = Ω (k)k T = c g ω X we can write ω T + c g ω X = 0. We thus come to the following conclusion: individual wavecrests move at the phase velocity but the density of waves travels at the local group velocity. How is this possible? The answer is that a crest that belonged to a wave with a certain wavelength at initial times will change to belong to a wave with different wavelength at later times. A simple exercise is to take the deep water gravity surface wave dispersion relation (3.1) with ρ 2 = 0: ω = g k. [We digress here to discuss one-way models and dispersion relations. A one-way model should have an odd dispersion relation so that c p has a given sign and thus to factor the dispersion relation ω 2 = g k into 2 one-way models, one should write ω = ±sign(k) k. However, since we are restricting ourselves to linear waves and taking real parts of complex solutions, we can simply consider k to be positive. When solving nonlinear problems we will have to be more careful with this.] Returning to the problem at hand: c p = 1 1, c g = g k 2 g k. So for initial data concentrated at the origin, waves of wavenumber k will be found along the family of lines X/T = c g, which, in turn means that a given crest will have the trajectory x(t) given by dx dt = c p = 2c g = 2 x t, or, x = αt2. As time advances a particular crest experiences an increasing wavelength. Consider now the O(ǫ) terms resulting from the substitution of the slowly varying single phase solution into our model equation. These yield A T + c g A X = 1 2 (c g) k k X A,

25 5.2. EFFECTS OF VARIABLE TOPOGRAPHY 25 or, in a conservative form ( A 2 ) T + ( c g A 2) X = 0. This is the equation for conservation of wave energy also called the Amplitude equation or the equation of Physical Optics. It is a linear equation with nonconstant coefficients since in general c g will vary with X, T. The interpretation of the equation is that the quantity x2(t) x 1(t) A 2 dx is constant in time between any two space-time curves [x 1, x 2 ] satisfying ẋ i (t) = c g (k(x i )). The generalization to more dimensions is immediate. ( In fact ) the following general conservation law for arbitrary a(k) is true: a(k) A 2 T + ( c g a(k) A 2) = 0. The law for the conservation of wave action X is the choice ( ) A 2 A + (c 2 ) g = 0. ω ω T This is the more appropriate conservation law for a physical system (not our simple one) in the presence of mean flows. 5.2 Effects of Variable Topography We now turn to the possible meaning of ǫ. There are two possibilities: the initial data is such as to define ǫ or, more interestingly ǫ is given by a physical scale of the problem, such as slowly varying topography. Suppose, for example in the finite depth surface waves relation (3.2) that H 1 = H(ǫx). In other words suppose waves are propagating into a topography that varies slowly with respect to wavelength (a common occurrence in waves). Then the dispersion relation is ω 2 = g k tanh( k H(X)), or ω = Ω(k, X). The equation for conservation of waves (5.1) becomes X k T + c g k X = Ω X. Suppose, most generally that ω = Ω(k, X, T). Then, the above equation remains unchanged, but, ω T + c g ω X = Ω T. The conclusion is that if the medium is spatially dependent, but not timedependent, the frequency of waves following characteristics is constant but the wave density is not. The opposite is true for a time-only dependent medium. The equation for energy (or wave-action) should be derived in the particular application.

26 26 CHAPTER 5. KINEMATICS OF LINEAR WAVES An example is particularly interesting: assume that ω = gh(x)k, the shallow water dispersion relation. One can wonder whether these theories applies to such a nondispersive case, but, in fact, these theories were first developed for such situations! Then, along dx dt = c dk g, dt = Hx 2H c gk. This can be rearranged to read dk dh = k 2H, and similarly for the Energy equation (for a derivation of the energy equation, see the exercises in the section on shallow water theory), and therefore k H = constant, A 2 H = constant. This effect of decreasing wavelength and increasing amplitude as the water becomes shallow is called shoaling. 5.3 The Linear Schrödinger Equation Consider now the inclusion of next order terms (O(ǫ 2 )). The result is a modification of the amplitude equation to read, (in the case where k X = 0 reflecting a monochromatic wave in a constant medium), [ ] 1 A T + c g A X = iǫ 2 Ω A XX. The waves are allowed to be modulated in amplitude. Following the waves at their group velocity with the transformation χ = X c g T i 1 ǫ A T = 1 2 Ω A χχ, which is the Schrödinger equation for a free particle where ǫ takes the role of Planck s constant h (the constant Ω /2 can be scaled out). It is usual to define the very slow timescale τ = ǫt = ǫ 2 t and write ia τ = 1 2 Ω A χχ, We now discuss a curiosity: if one takes the Schrödinger equation for a particle of mass m in a potential V (x): i hψ t = h2 2m Ψ xx + V (x)ψ, and performs the high-frequency analysis choosing Ψ = a(x, t)e iφ(x,t)/ h, whence the Eikonal equation is k = φ x, ω = φ t = 1 2m k2 + V k t + 1 m kk x = V x (x),

27 5.4. STATIONARY PHASE 27 whose solution by the method of characteristics satisfies: d dt x = k m, (5.2) d dt k = V x(x). (5.3) This is of course the classical Hamilton s equations for a particle in a potential, where k is the momentum of the particle! This is due, of course, to the fact that Schrödinger wrote his equation precisely because its high frequency asymptotics gives classical mechanics (ie, he worked backwards from what we have done). What is curious is that the large scale motion of classical objects like waves obey the same equation as the small scale (quantum) limits of classical particles. We make a last comment on the kinematic equation for waves. Since these are hyperbolic PDEs, the immediate question is whether they can form shocks? The answer is yes, and the physical interpretation is clear: suppose initial data is such that faster waves (in group velocity) are initially behind slower waves. Then these faster waves will overtake the slower ones creating a region where we have a superposition of waves (remember that since the problem is linear that is allowed). At the leading edge of this region is a wavenumber shock where the solution changes wavelength abruptly. Within this region, one should now write the equations for each phase independently, that is, η = j A j(x, T)e iθj(x,t)/ǫ. Another question arising at these shocks is the amplitude behavior. This requires a nonlinear correction to these equations which we address in the special topics section. 5.4 Stationary Phase 5.5 EXERCISES 1. Show that the Eikonal equation is an equation for the conservation of wavecrests by integrating over a domain [x 1 (t), x 2 (t)] and choosing ẋ 1,2 so that the resulting equation is d x2 dt x 1 k dx = 0 and interpreting x 2 x 1 k dx as a multiple of the number of waves in the interval [x 1 (t), x 2 (t)]. 2. Show that for (3.2) with H 1 = H(X), Ω X = Hx H (kc g 1 2ω) > 0 when H(X) < 0 and k, ω > 0. Conclude that the wavenumber increases (and therefore the wavelength decreases) as right-travelling waves enter shallower water. 3. Consider the kinematic of waves governed by k T + c g (k)k X = 0 with initial data k(x, 0) = k 0 (X). This is a hyperbolic PDE solvable in closed form by the method of characteristics. Show that two nearby characteristics first intersect at a time equal to the minimum over the initial data of the quantity ( c g(k 0 )(k 0 ) X ) 1. This is when a wavenumber shock is created. For the Klein Gordon equation u tt = u + u xx, with initial data having a slowly varying wavenumber k 0 (X) = 2 + sin(x/2), find the time at which the wavenumber shock occurs.

28 28 CHAPTER 5. KINEMATICS OF LINEAR WAVES 4. A good example of a scalar conservation law are traffic flow models. In these ρ is the density of cars on a single lane road, and Q is the flux of cars. For such a problem, propose reasonable properties for Q, U and deduce the meaning of U and c. 5. [NUMERICAL] Linear wavepacket dynamics for the KdV. Consider the initial value problem for the linear KdV equation u t +3u x +u xxx = 0, with initial data u 0 (x) = exp[ (x/10) 2 ]cos(x). You shall solve this in two ways and compare the solutions: (a) Solve the linear KdV directly with periodic boundary conditions on x [ 50, 50] up to t = 10. Comment on why you think the equation was chosen to have the 3u x term. (b) Solve the linear Schrödinger equation ia τ = 3A χχ, where τ = ǫ 2 t and χ = ǫ(x c g t) with appropriate initial conditions and up to the appropriate time so as to attempt to reconstruct the solution obtained in (a). Compare the two solutions. (You have to decide what to choose for ǫ and therefore how to rescale the variabled to compare the solutions.) (c) Extra Credit. Derive the linear Schrödinger equation from the linear KdV by using an appropriate form for the solution. 5. [NUMERICAL] Wavenumber shocks for u t u xxx = The Eikonal equation for nondispersive waves. Consider the propagation of nondispersive waves accorsing to u tt = c 2 u. For a fixed frequency ω write the Eikonal equation and show that in any number of dimensions, the level sets of the solutions are equidistant sets (ie. the distance between 2 level sets is a constant irrespective where you are on the set). Interpret this in terms of, say, the propagation of light. Derive the amplitude equation and interpret it geometrically.

29 Chapter 6 Shallow Water Theory We have seen that the definition of shallow water depends on the problem. One alternative definition which we have seen holds true is that shallow water waves are waves whose velocity field has not decayed appreciably at the bottom: in other words, the horizontal fluid velocity is almost constant in the water column. There are two main ways to derive nonlinear shallow water models: making an assumption, a priori, of the structure of the velocity field. This is the more traditional hydraulic approach that was used by St. Venant, Serre, and more recently Green and Naghdi. Alternatively, one scales the full Euler equations appropriately and uses a small parameter ( shallowness ) that appears to write an asymptotic expansion of the solution. Each has its merits, and we present both. We shall limit our discussions to one horizontal dimension - the extension to two dimensions is straightforward. We shall also include topographical effects (variable bottom) - by definition they are most important in shallow water! Equations that include topographical effects which either vary slowly or on the same scale as the waves are fairly straightforward to write - at least at leading order. If the topography varies on a faster scale than the wavelength of the waves (rough topography), then new formulations must be used. 6.1 Hydraulic Derivation Consider a fluid layer bounded below by a bottom at y = b(x) and above by a free surface y = η(x, t). The total fluid height h η + b. We now assume the following: (i) The flow is uniform in depth, that is, the horizontal fluid velocity does not depend on the vertical variable u(x, t). (ii) The pressure is hydrostatic and constant (zero) at the free surface. (Note that we shall not mention the vertical velocity of the fluid.) Considering an area of fluid between two fixed vertical lines at x = x 1, x 2. The integral form of the conservation of volume or 29

30 30 CHAPTER 6. SHALLOW WATER THEORY mass reads; d x2 h(x, t) dx = u(x 1, t)h(x 1, t) u(x 2, t)h(x 2, t). dt x 1 The right hand side can be written x 2 x 1 (uh) x dx and, therefore, for arbitrary, fixed, x 1, x 2, and if the functions h and hu are differentiable in [x 1, x 2 ]: x2 x 1 h t + (uh) x dx = 0. In order to write the integral conservation of momentum, we first compute total force due to the hydrostatic pressure p = ρg(η y) of the fluid on one side of a vertical line on the fluid of the other side: F f = η b p dy = ρgh2 /2. Second we compute the force on the fluid by the bottom between the two vertical lines x = x 1, x 2 : F b = x 2 x 1 ρghb x dx. Note that this force is zero if the bottom is flat. The conservation of momentum now reads d dt x2 ρhu dx = ρu(hu) x2 + ρg h2 x 1 x 1 2 x2 x2 + x 1 x 1 ρghb x dx. The density is irrelevant and the we can rewrite the conservation law as x2 ) (hu) t + (hu 2 + g h2 ghb x dx = 0. x 1 2 x Finally, we can say that since the integral is zero on arbitrary intervals, that the integrand must be zero everywhere, giving the St. Venant or Shallow Water eqations (SWE) h t + (hu) x = 0, (6.1) ) (hu) t + (hu 2 + g h2 = ghb x, (6.2) 2 This is the so-called conservation form of the equations where each equation is of the form U t + Q x = F. This form allows one to see an integral conservation law immediately and therefore to enlarge the possibility of solutions to ones involving discontinuities. If we intend to consider smooth solutions, we can rewrite the system as x h t + (hu) x = 0, (6.3) u t + uu x + gh x = gb x, (6.4) Note that clearly there is not a unique conservation form for a given set of first order hyperbolic wave equations (hyperbolic will be defined below), and that physics must be used to discern between useful and useless conservation versions of the equations.

31 6.1. HYDRAULIC DERIVATION 31 One can also wonder about further conservation laws, such as the energy. Conservation of energy can be derived from first principles: ( ) ( ) hu gh2 hu gh2 u = 0. t This equation is consistent in smooth parts with the equations of mass and momentum (ie it is trivially satisfied). However, this ceases to be true if interpreted as an integral conservation law for shock solutions! Then, integral conservation of mass and momentum do not imply conservation of energy. In fact they imply strict dissipation Linear Waves The linearized equations are obtained by taking h = h 0 (x) + η(x, t) (h 0 (x) is the mean depth of the fluid) and dropping all quantities quadratic in u and η: which can be rewritten as the single wave equation x η t + (h 0 u) x = 0, (6.5) u t + gη x = 0 (6.6) η tt g(h 0 η x ) x = 0. If h 0 is constant then the wave speed is fixed at ± gh 0. If h 0 varies slowly compared to the variations of η (say h 0 (X)) where X = ǫx, then, the waves have a dispersion relation ω 2 = h 0 (X)k 2, as we used in prior considerations Weakly Nonlinear Waves For simplicity, let us assume b x = 0 (flat bottom), h 0 = g = 1. Then the nonlinear equations read η t + u x = (ηu) x, (6.7) u t + η x = uu x (6.8) We can symmetrize these equations by considering the linear combinations R 1 (η + u), 2 L 1 (η u). 2 Which, as we shall see, stand for mostly right- and left-moving disturbances. (In the nonlinear section below we will see how to make this split exact through Riemann invarients.) R t + R x = 3 2 RR x LL x (RL) x, (6.9) L t L x = 1 2 RR x LL x 1 2 (RL) x (6.10)

32 32 CHAPTER 6. SHALLOW WATER THEORY If one could say here that L = 0, for example, then we would have immediately that R satisfies R t + R x RR x = 0. Setting L = 0 however is inconsistent with the equations but, if we say that R is small (say order ǫ) and L is initially zero then L will remain O(ǫ 2 ) for long times. Thus, we can set R = ǫ R, and a correct approximation to the system is R τ R R ξ = 0, where the variable τ = ǫt is a slow time in a frame of reference following the wave at its linear speed: ξ = x t. This equation often is called the Inviscid Burgers Equation or the Hopf Equation. This equation, as we have seen before, is completely solvable up to the time at which a shock forms (and shocks will form in this case for any nonconstant initial data). In this case, the interpretation is that nonlinearity is leading to the steepening of the wave, and a shock corresponds to a breaking wave. In the regime of this equation the shocks would be called weak shocks or weak hydraulic jumps of the original shallow water system since we have made an a priori assumption on the smallness of R Fully Nonlinear Waves - Riemann Invarients Shocks EXERCISES 1. For the equation η tt g(h 0 (ǫx)η x ) x = 0, show by direct substitution of a wave of the form η(x, t) = A(X, T)e iθ(x,t)/ǫ, where (X, T) = ǫ(x, t), that the conservation of wave energy ( A 2) T + ( c g A 2) X = 0 holds. 2. Fill in the steps that start with the transformation to R, L and finishes with the Hopf equation R τ R R x = 0, 3. [NUMERICAL] Shoaling. Write a program to integrate numerically the shallow water equation η tt g(h 0 (x)η x ) x = 0, on x [ 20, 100] with periodic boundary conditions, taking g = 1 and h 0 = 1 x/200 for x [0, 100] and h 0 = 1 for x [ 20, 0], and initial data η(x, 0) = exp( (x + 10) 2 ) and η t (x, 0) appropriate such that the wave motion is to the right. Compute the solution until the wave has propagated up to x 90. Confirm of contradict the shoaling predictions that k H = constant, c = H and A 2 H = constant. Note that you will need fairly high resolution for this problem. 6.2 From Euler to Shallow Water We have now seen how to obtain the linear KdV equation (dispersive) and the nonlinear Hopf equation (nondispersive ) from shallow water limits. It is tempting to put them together and get the full nonlinear Korteweg-de Vries Equation. In order to do that we consider the Euler equations for (u, v)(x, y, t),