wave attractors: linear yet nonlinear - Semantic Scholar

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International Journal of Bifurcation and Chaos, Vol. 15, No. 9 (2005) 2757–2782 c World Scientific Publishing Company


WAVE ATTRACTORS: LINEAR YET NONLINEAR LEO R. M. MAAS Royal Netherlands Institute for Sea Research (NIOZ ) P.O. Box 59, 1790 AB Den Burg, The Netherlands [email protected] Received February 22, 2005

A number of physical mechanisms give rise to confined linear wave systems whose spatial structure is governed by a hyperbolic equation. These lack the discrete set of regular eigenmodes that are found in classical wave systems governed by an elliptic equation. In most 2D hyperbolic cases the discrete eigenmodes are replaced by a continuous spectrum of wave fields that possess a self-similar spatial structure and have a (point, line or planar) singularity in the interior. These singularities are called wave attractors because they form the attracting limit set of an iterated nonlinear map, which is employed in constructing exact solutions of this hyperbolic equation. While this is an inviscid, ideal fluid result, observations support the physical relevance of wave attractors by showing localization of wave energy onto their predicted locations. It is shown that in 3D, wave attractors may co-exist with a regular kind of trapped wave. Wave attractors are argued to be of potential relevance to fluids that are density-stratified, rotating, or subject to a magnetic field (or a combination of these) all of which apply to geophysical media. Keywords: Internal waves; wave focusing; wave attractors; linear hyperbolic BVPs (boundary value problems); self similarity; wave chaos.

1. Introduction The first part of the title, “wave attractors”, joins together two concepts usually not encountered in conjunction. “Wave” here refers to waves in fluids, with an emphasis on those encountered in the ocean, but with an application to other large-scale fluid systems such as the atmosphere, the liquid outer core of the earth, or the stellar interior. These waves arise due to the presence of a restoring force, such as gravity, or the Coriolis force due to the rotation of the earth or star. Waves form the response to a perturbation of a stable equilibrium state with which the fluid is stratified. Waves are able to “act at a distance”: they propagate energy and momentum over large distances and deposit these quantities elsewhere, locally affecting the state of the fluid and adjacent bottom. “Attractor” is a term borrowed from the field of dynamical systems, where it is used to designate the asymptotic state to which a continuous

or discrete system evolves. This usually concerns a contraction of phase space onto a lower dimensional subspace, which seems to betray the presence of a hidden force that attracts all initial states onto the asymptotic state. A well-known example of an attractor is the one that appears upon iteration of the quadratic map [Hofstadter, 1981]. This map associates to any preimage x0 ∈ [0, 1], an image x1 = λx0 (1 − x0 ), which is again in the interval [0, 1], provided 0 ≤ λ ≤ 4. Iteration implies one uses x1 as new preimage, whose image in turn provides the second iterate x2 , which process is repeated indefinitely. For values of λ ≤ 3 the iterates converge onto an attractor consisting of a single fixed point x∞ , see Fig. 1(a). For higher values of λ, the attractor consists of a sequence of 2n points (n ∈ N), a so-called periodic attractor of period n. But, from a finite value of λ < 4 upwards, n → ∞ and the attractor becomes chaotic, except in some finite-sized parameter intervals, see Fig. 1(b). The

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1.0 0.8

λ/4

0.6

x

0.4

0.2

x0 0.2

x1

x2 x 0.4

0.6

0.8

1

(a) Fig. 1.

0.0

2.8

4.0 (b)

(a) Iteration of the quadratic map x → λx(1 − x) for λ = 2.9. (b) Asymptotic state x∞ as a function of parameter λ.

asymptotic state is again a reduced state space, but one which in the chaotic state is visited irregularly, and which is characterized by diverging trajectories of initially neighboring states. But what is a wave attractor? Surprisingly, most of us do have an intuitive notion as we have encountered one from early childhood on: the beach, Fig. 2. The peculiar phenomenon that wind waves of wavelengths from centimeters up to tens of meters always appear to just approach the beach, is well known, in particular when the beach manifests itself as a gentle slope. The reason why we do not see

(a) Fig. 2.

these waves reflecting is because they often break, or dissipate energy in some other way. Thus, while we know that physically this “attraction” is due to a combination of wave refraction (wave speed decreasing upon decrease of depth) and nonlinear wave breaking (due to steepening of the wave, following the simultaneous decrease in wavelength), the beach does seem to draw in waves from the neighboring sea or ocean, which has big consequences. Not only is the energy that is being deposited used to swirl up sediments, but also does the associated onshore transport of fluid necessitate the formation of

(b)

A beach seems to attract surface waves due to (a) refraction and (b) breaking.

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along-shore currents and so-called rip currents that return the water seawards. However, while the beach is instructive to build the notion of a surface wave attractor, it is clear that it is not “attractive” under all circumstances. The ability of the beach to attract waves depends on the wave’s steepness compared to that of the bottom. For, if we assume that the beach slopes over some finite distance L only, the depth being uniform seaward from this point, then from the perspective of longer, less steep and lower-frequency waves, the relative extent of the sloping beach will be smaller and smaller. For the longest waves the sloping beach approaches a vertical coast, which is fully reflective, and for which it no longer acts as an attractor. This paper will discuss a class of waves that are quite common in geophysical fluids for which the appearance of wave attractors, however, does seem to be generic (Secs. 3–7). This relates to the fact that these waves are very different from classical waves encountered e.g. at the surface of the sea, or in the form of acoustic and electromagnetic waves. For this reason, Sec. 2 first recalls some of the properties of these classical waves by briefly considering those appearing at the surface of a lake or sea. A striking aspect of the new class of waves prompted the last part of the title. Despite linearity of governing equations and boundary conditions certain properties of the waves are reminiscent of phenomena encountered otherwise in nonlinear dynamical systems. These will be discussed where appropriate.

2. Surface Waves Surface waves derive their name from the fact that they obtain their largest excursions at the surface to

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which they are trapped and along which they propagate horizontally, see Fig. 3(a). This intensification takes place at the surface because the spatial structure of these waves is governed by an elliptic partial differential equation satisfying a maximum principle which states that solutions of these equations take on extreme values on the boundary of their domain of validity [John, 1975]. An example is the Laplace equation, ∇2 φ = 0, where ∇ stands for the gradient operator. This equation is obeyed by the velocity potential φ that for short waves determines the velocity field u = ∇φ in an incompressible (∇ · u = 0), irrotational (∇ × u = 0) fluid. Surface boundary conditions determine a relation between the wave’s frequency ω and horizontal wavenumber vector k, the dispersion relation. A peculiarity of surface waves is that the frequency relates to wavenumber magnitude (or wavelength) only, not to its direction: ω = ω(|k|). This property is responsible for the fact that regardless of the precise frequency-wavelength dependence surface wave energy propagates (with group velocity vector cg ≡ ∇k ω) in the same direction as the wave crests and troughs, determined by phase velocity vector c = ωk/|k|2 , see Fig. 3(b). This property is so familiar to us that we may well turn a blind eye to its special nature. There are two consequences to the fact that the surface wave frequency relates only to wavelength. First, since a wave’s frequency does not change upon reflection from a plane surface, neither can its wavelength. For short waves, reflecting from a slowly varying boundary having a radius of curvature that is everywhere large compared to the wavelength, each reflection is from a locally plane surface. The invariance of the wavelength identifies them as mono-longitudal (“single-length”) waves.

θI

cg // c

y

θr

θI

x θr

Snell’s law: θr= θI (a)

(b)

(c)

Fig. 3. (a) Orbital motion (circles) and surface trapping of a surface gravity wave propagating to the right. (b) Surface wave energy propagates horizontally in the same direction as the crests and troughs. (c) Top view of incident parallel rays reflecting from a curved wall.

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Second, as the wave vector direction is not present in the dispersion relation, the wave is free to change its propagation direction upon reflection, and indeed, will do so in order to be able to satisfy the impermeability condition on the reflecting surface. This condition is satisfied when the wave reflects specularly, i.e. according to Snell’s law. This law states that the angle of reflection equals the angle of incidence, measured with respect to a line perpendicular to the wall at the point of reflection (dashed in Fig. 3(c)). When applied to subsequent parts of a curved boundary, for an incoming wave propagating in a fixed direction, it reveals the sensitive dependence of the angle of the reflected waves upon the shape of the boundary and, consequently, the deformation of a wave front. This sensitivity is borne out by the generically chaotic spreading of wave rays in most container shapes (exceptions are formed by the simplest container shapes, such as circles, ellipses and rectangles, for which mostly periodic ray paths are found). This property is intensely studied in the field of quantum chaos [Gutzwiller, 1990]. But, despite its discovery in quantum mechanical context, this is really a property of the Helmholtz equation proper and this equation governs many phenomena in physics, including the aforementioned surface waves. This can be readily seen by describing their decay from the surface downwards by an exp(kz), or sinh(kz) vertical z-dependence. When inserted in the Laplace equation, it is obvious that the horizontal structure of the wave field is determined by the Helmholtz equation (∇2h + k2 )φ = 0, where ∇2h = ∂ 2 /∂x2 + ∂ 2 /∂y 2 denotes the horizontal Laplacian. For short waves (k  1, assuming k

(a)

is scaled with basin scale L) a WKB-ansatz is often made, φ = A exp(ikS), and eikonal and amplitude equations are derived for the wave’s phase S and amplitude A. This gives approximate solutions (as a perturbation series in k−1 ) for high wavenumber eigenmodes. The eikonal equation yields straight ray paths that reflect specularly from the boundary. This reduces the description to a billiard problem, in which rays follow the path of a ball on a frictionless billiard which has the shape of the container to which the waves are confined. An example of such a chaotic path in a stadium is shown in Fig. 4(a). The ray dynamics can be seen as a nonlinear map of the circumference onto itself, in which the location of a reflection point and the angle of reflection are determined by initial boundary point and ray direction. For certain container shapes, Berry [1981] gives explicit conditions for this map to become chaotic. With a chaotic spread of most of the rays, wave energy propagating along them spreads out ergodically over the container. This sets the background noise level. Any large-scale structure of the high-wavenumber eigenmode energy distribution is obtained from the few exceptional periodic orbits that persist. These orbits are unstable, but the time needed for waves on neighboring rays to diverge is long enough that the amplitudes on these stand out above the noise level. For this reason, these periodic orbits were termed scars [Heller, 1996]. See for instance the localization of energy along a diamond-shaped orbit in the stadium, Fig. 4(b). The localization of wave energy on particular orbits is exploited in today’s most powerful microlasers [Gmachl et al., 1998]. In this figure, also note the remarkably dotted character of the periodic orbit’s amplitude, which

(b)

Fig. 4. (a) Pattern of a single ray in a stadium, from [Berry, 1987]. (b) High wavenumber mode in a stadium with a single, diamond-shaped periodic ray (solid line). Taken from [Heller, 1996].

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bears testimony of the monolongitudal nature of the wave field. It is interesting to speculate that similar scarring as that of quantum eigenstates may perhaps be present on the surface of lakes or inland seas, which might be visible with satellite altimetry or other remote sensing devices. However, in order for scarring to take place, the waves should both be short enough for WKB-theory to apply, but on the other hand also be long enough, compared to the range over which the bottom shoals, for the coast to be reflective. It remains to be seen whether these conditions are not mutually exclusive and thus whether scarring might actually take place in such geophysical circumstances, or not. To summarize: the spatial structure of surface waves is governed by elliptic differential equations that lead to surface trapping, and therefore to horizontal wave propagation (in arbitrary direction). In a confined container, the wave field is described by eigenmodes that are determined by a discrete spectrum of well-separated eigenfrequencies. The higher modes are determined by waves that are short compared to the basin scale. For these modes an approximate description can be obtained, based on plane wave theory. For surface waves these are governed by a dispersion relation that implies monolongitudal waves that propagate energy parallel to their phase. As a consequence, in most containers ray patterns will generally be chaotic and thus disperse energy uniformly over the container. The exceptions are formed by scars: high-energy eigenstates that are formed by periodic orbits along which waves retrace and stand out above the noisy background. Nearly all of these properties will be lost for waves encountered in continuously stratified or rotating fluids that will be considered next. Interestingly, the only exceptional property that survives is actually quite relevant physically: the localization of wave energy on periodic orbits.

3. Internal Waves The ocean is stratified, both in its density as well as in its angular momentum field. The ocean’s density field is determined mainly by the amount of dissolved salts and heat, although in some turbid coastal environments dissolved sediments contribute to a fluid’s stratification. Fluxes of heat and fresh water through the ocean’s boundaries shape these salinity and temperature fields. To first approximation the ocean can

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be considered stratified in the direction of gravity (vertical) only. This stratification is capped by a shallow, 50–100 m deep, homogeneous wind-mixed layer near the surface. Below this there is a thin layer in which the density increases rapidly: the seasonal pycnocline. This pycnocline supports and traps so-called interfacial gravity waves, which are very akin to the surface gravity waves, except for a reduction of the restoring force of gravity by a factor proportional to the relative density contrast over the pycnocline. Below this interfacial layer, the density field increases in a gradual fashion, which provides a stable stratification. Seen as a collection of an increasing number of homogeneous layers, separated by smaller and smaller density jumps, it is evident that each of their interfaces is able to support internal gravity waves too. Indeed, in the limit of a continuous stratification, it is clear that the whole sea becomes penetrable to internal gravity waves. Because of earth rotation (characterized by rotation vector Ω) the ocean is also stratified in terms of angular momentum, even in the absence of density stratification. For a solid body rotation this stratification is stable. It is therefore again able to support waves, termed inertial (or gyroscopic) waves. Within the corotating frame of reference the restoring force is identified as the Coriolis force [Tolstoy, 1973]. In the real ocean, gravitational and Coriolis restoring mechanisms appear, of course, together. For methodological reasons they will here first be treated separately, their combined effect being briefly discussed in Sec. 6. We refer to both types of waves as internal waves, because both have their maximum displacement in the interior of the fluid. Both also have a spatial structure that is governed by a hyperbolic equation and, as will be shown, in general lack eigenmodes. And, finally, both waves satisfy a dispersion relation ω = ω(α) that relates wave frequency to wave number direction α. Here π/2 − α is the angle between wave vector and direction of gravity ∠(k, g) or rotation axis ∠(k, Ω). This leads to monochromatic waves of fixed inclination, so-called monoclinical (single-angled) waves. These propagate energy obliquely downwards or upwards, parallel to the wave crests and troughs, so that energy propagation is perpendicular to phase propagation: cg ⊥ c. As in this dispersion relation the wavelength is unconstrained, waves do change their wavelength and get focused or defocused upon reflection from an inclined boundary. In confined areas, focusing

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usually dominates and in 2D the waves will then approach a limiting orbit: a wave attractor along which in an inviscid description, the wave field develops a spatial singularity. Here, emphasis will be on the internal gravity waves for which 2D linearized inviscid results are exact, owing to the fact that particle paths are rectilinear and stay in the vertical plane in which wave propagation takes place. This implies that in a real container the presence of vertical front and end walls, parallel to the wave propagation plane, do not require any inviscid adjustment (the motion being parallel to these walls). In contrast, inertial waves have circular particle paths. Therefore they have a velocity component perpendicular to the energy propagation plane, and hence do require some adaptation to such front and end walls. Inertial waves impose an intrinsically 3D problem, that can only approximately (and far from these walls) be modeled by a 2D description. Attempts to generalize these results to 3D are discussed in Sec. 7 and provide an interesting connection to the elliptic problems discussed before.

3.1. Internal gravity wave laboratory observations Experimental evidence of the oblique nature of internal gravity wave propagation was first given by G¨ ortler [1943] and Mowbray and Rarity [1967]. Figure 5, kindly provided by Frans-Peter Lam, exemplifies some of the unusual properties of internal waves. It shows a side view of a 38 cm high tank, filled with a uniformly, salt-stratified fluid (characterized by stability frequency N = 1.5 rad/s, defined in the next section). Internal waves of particular frequency are generated by vertical oscillation of the cylinder of 22 mm radius in the upper left-hand corner. The oscillation amplitude is 4 mm and the oscillation has a frequency of 0.924 rad/s. Visualization of the internal waves is by means of the pseudo-schlieren method [Dalziel et al., 1998; Sutherland et al., 1999] which uses the fact that the amount of deflection of light rays depends on the stratification. Comparing to a deflection pattern observed when the cylinder is still at rest, perturbations are sensed and show up as bright bands, denoting lines of equal vertical displacements. The cylinder emits four beams. The two upward propagating beams are apparently absorbed at the free surface. The one propagating downward to the left is immediately reflected, thus joining the

?

Fig. 5. Side view of a laboratory observation of generation and reflection of internal wave beams in a uniformly stratified fluid. Arrows indicate energy propagation direction. (Courtesy of Frans-Peter Lam.)

initially downward-moving beam in generating one broad beam that propagates downward to the right. The energy propagating away from the source is indicated by white arrows. Starting from rest, a movie shows that the waves first fan in onto the asymptotic direction (which represents the broad spectrum of waves covering the Heaviside process involved in turning on the oscillation). Afterwards, wave energy slowly penetrates into the fluid, parallel to the bright phase lines. The unique inclination α of these lines (and the beam they constitute) is evident and closely approximates the angle cos−1 ω/N , predicted by the dispersion relation. The phase lines themselves, however, propagate perpendicular to their own orientation. For the broad beam phase lines move upwards to the right. This verifies the orthogonality of group and phase velocity. Upon reflection from the sloping side the beam brightens up, which is an increase in the amplitude of vertical displacement, due to a velocity increase accompanying compression of the beam’s width due to focusing. The following reflections from bottom and left side wall are only faintly visible, and the subsequent one at the surface is completely invisible. Inferred energy propagation is denoted by dashed arrows. Yet, the next focusing reflection from the upper slope is visible again (upper right).

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In order to model this process in the simplest possible terms, the perturbations of a weakly stratified motionless fluid will be considered next.

3.2. Internal wave equations in a 2D plane In an external field of gravity g, consider a fluid that is initially at rest, and which is weakly but stably stratified in density around an average mean density ρ∗ ; thus density ρ = ρ∗ + ρ0 (z). The pres
0 sure field z ρg dz is in hydrostatic balance with this density field. Then, any perturbations of this state ρ (x, z, t), p (x, z, t) will propagate as internal gravity waves having small velocities u (x, z, t) such that nonlinear advective terms can be neglected. The complete density field is thus described by ρ = ρ∗ + ρ0 (z) + ρ (x, z, t) and the background stratification by the stability 1/2 , assumed confrequency N = (−gρ−1 ∗ dρ0 /dz) stant here. Define buoyancy as b = −gρ /ρ∗ and perturbation pressure p as the reduced pressure p divided by ρ∗ , where the reduced pressure is the total minus the hydrostatic pressure field associated with the mean and static density fields. Thus, in a linearized, inviscid description the perturbations are governed by the following reduction of the Euler equations: ∂u ∂p =− ∂t ∂x

(1)

∂p ∂w =− +b ∂t ∂z

(2)

∂b + wN 2 = 0 ∂t

(3)

∂u ∂w + = 0. ∂x ∂z

(4)

The incompressibility condition (4) allows the introduction of a streamfunction u=−

∂ψ , ∂z

w=

∂ψ . ∂x

We will look for monochromatic (single-frequency) waves and assume [ψ, p, b] = [Ψ(x, z), −iωΠ(x, z), B(x, z)] exp(−iωt). Inserting this in the buoyancy Eq. (3) shows that b = −iwN 2 /ω. Hence, upon substituting

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these relations, the momentum equations (1), (2) reduce to ∂Ψ ∂Π = ∂z ∂x  2  ∂Π N ∂Ψ = . −1 ω2 ∂x ∂z By cross-differentiation and subtraction these can be combined into a single equation for the spatial structure of the streamfunction field ∂ 2 Ψ (N 2 − ω 2 ) ∂ 2 Ψ − = 0. ∂z 2 ω2 ∂x2 For plane waves Ψ ∝ exp i(kx + mz) one obtains the dispersion relation ω2 k2 = = cos2 α. N2 k 2 + m2 The latter identity follows from a polar description k = (k, m) = κ(cos α, sin α). It shows that the wave frequency is independent of the wave number magnitude κ = |k| and depends only on its angle α. This is expressed succinctly as ω = ω(α). For waves having this property, regardless of the actual frequencyangle dependence, group and phase velocity vectors are perpendicular. For the given dispersion relation this is verified by mN ω (m, −k) ⊥ 2 k = c. 3 κ κ We nondimensionalize the spatial coordinates, x = LX, z = DZ, where L is the basin half-width and ωL D=√ N 2 − ω2 provides a stretching of the vertical coordinate such that the governing equation takes on its canonical form ∂2Ψ ∂2Ψ − = 0. (5) ∂Z 2 ∂X 2 This is the well-known wave equation, now cast in spatial coordinates solely. The latter fact makes the problem unusual. The wave equation is normally solved as an initial value problem only, as one is not inclined to prescribe future behavior. In the present context, however, the presence of a solid boundary to which the fluid is confined makes a prescription of the streamfunction over the whole boundary perfectly natural (Dirichlet boundary conditions). Boundary forcing may be prescribed by a function of the along-boundary coordinate s : Ψ = Ψb (s). In its absence impermeability simply requires Ψ = 0 cg = ∇k ω =

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at the boundary. For the Laplace equation, such a vanishing of the field at the boundary implies that only the trivial (zero) solution exists. In Sec. 4 we will see that this is quite different for the hyperbolic equation, and in this sense poses the simplest possible nontrivial second-order PDE with homogeneous boundary conditions. Not only does one find nontrivial solutions, but there are actually infinitely many of them and the problem is left underdetermined, making it formally ill-posed. In the following section the conditions under which the problem becomes well-posed will be discussed. Note that on employing a scaling of the pressure field, Π = (L/D)P , the momentum equations reveal that streamfunction Ψ and pressure P are “hyperbolically conjugate” fields:

The virtual depth τ in (6) is a lumped parameter that contains specifications of the medium (H0 , L, N ) and the wave field (ω). The choice for symbol τ stems from its proportionality to the wave period in the limit ω  N . Typical values can be estimated as follows. On a plane rotating with angular frequency f /2 (for the earth this −5 −1 is 7.252 × 10  rad s ), ω in the numerator is replaced by ω 2 − f 2 . For a hundred meter deep lake, that is 20 km long, which is stratified with N = 10−2 s−1 , low-frequency oscillations having  ω 2 − f 2 = 10−4 s−1 yield τ = O(1). For a deep ocean basin of 5 km depth, being 5000 km long, with aweaker stratification N = 10−3 s−1 , waves having ω 2 − f 2 = 10−5 s−1 yields τ = O(0.2).

3.3.1. Rectangle

∂P ∂Ψ = ∂Z ∂X ∂P ∂Ψ = , ∂X ∂Z equivalent to the Cauchy–Riemann conditions for a 2D Laplace problem. Therefore the pressure field satisfies the same hyperbolic equation. Only its boundary condition is more complicated. It is of oblique-derivative type, relating the pressure’s along-boundary variation to its normal derivative.

In a rectangular domain, −1 ≤ X ≤ 1, −τ ≤ Z ≤ 0, that contains a uniformly stratified fluid, the hyperbolic equation can be solved explicitly by separation of variables:     Z 1+X sin mπ . Ψ = An,m sin nπ 2 τ This vanishes at the boundaries for integer n and m and satisfies the hyperbolic equation provided τ =±

3.3. Simple-shaped domains Here the hyperbolic equation will be solved on some simple-shaped domains and conditions for uniqueness will be discussed. With the adopted stretching of the vertical coordinate, the bottom at z = −H(x) ≡ −H0 h(x/ L) is nondimensionally given by Z = −τ h(X), see Fig. 6. Here the basin’s deepest point H0 and depth scale D serve to define the nondimensional depth √ H0 N 2 − ω 2 H0 = . (6) τ= D L ω The nondimensional bottom shape satisfies 0 ≤ h(X) ≤ 1 and is confined to the interval |X| ≤ 1.

-L -H0

x

L z=-H(x)

Fig. 6. Sketch of a uniformly stratified basin with a rigid lid surface.

2m , n

see Fig. 7. Superficially one may still view these classical, sloshing type of regular solutions as eigenfunctions and τ as corresponding eigenvalues, but the fact that τ is rational indicates its pathology. It is dense (any irrational number can be approximated by a rational to any desired accuracy) and it

Streamlines 0

isopycnals

Z

Z=Z=-τ −1

X

1

Fig. 7. Rectangle with a sketch of two streamlines (one of which coincides with the boundary) and two states of the central isopycnal, depicted one half period apart, for the (m, n) = (1, 1) mode. Arrows indicate their oscillation.

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is infinitely degenerate. That is, replacing (m, n) by j(m, n) leaves τ = 2jm/(jn) = 2m/n unchanged, satisfying the boundary conditions for j integer, but produces a j 2 -fold version of the original streamfunction field. The pathology indicated by these solutions becomes clear if the rectangle is tilted relative to the direction of gravity. Separation of variables no longer provides solutions that vanish at the boundaries, and other methods of solution have to be considered, such as will be presented below.

3.3.2. Wedge The pathology of the rectangle is also visible in exact solutions in a wedge shaped domain, first obtained by inspection in [Wunsch, 1968]. To explicitly construct such solutions in a wedge one may exploit an analog of the conformal mapping method, employed for solving boundary value problems for the Laplace equation (see Appendix). Here it suffices to simply point at the qualitative properties of the wave field. For, if the waves have an angle of propagation that is steeper than that of the bottom, then, upon reflection from the bottom, the waves do not alter their horizontal propagation direction and hence, in the end, turn up in the apex where these waves will break. This happens regardless of the precise value of that angle of propagation (and hence of the related frequency). Again the beach acts as an attractor, but now for internal gravity waves, see Fig. 8.

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4. Arbitrarily-Shaped Channel While the conformal-map type of transformation of a domain described in the appendix is useful in obtaining solutions in a few exactly solvable domains, it is not helpful in general. For this reason we retrace steps and note that conventional methods aim at obtaining closed-form solutions (possibly in the form of infinite series) that satisfy boundary conditions at one single stroke over a whole boundary segment, e.g. by choosing the boundary to coincide with a coordinate line in some transformed coordinate frame. The price for this “ambitious” approach is that this is possible only for a very limited class of geometries, tractable by separation of variables [Moon & Spencer, 1988]. When searching for solutions in arbitrary 2D domains, however, it might be fruitful to adopt a more modest approach which, at first, requires the boundary condition to be satisfied in a single point of the boundary only. To elaborate on this, consider the general solution of (5), Ψ(X, Z) = f (X + Z) − g(X − Z), for arbitrary functions f (X + Z) and g(X − Z). Assume one knows the value of f at the given characteristic X + Z = c1 , f1 = f (c1 ) say. The intersection of this down-sloping characteristic and the bottom defines the particular up-sloping characteristic X − Z = c2 emanating from this intersection. Then, requiring the vanishing of the streamfunction at this single point of reflection one infers that the value of g on the reflected characteristic, g2 = g(c2 ), should equal f1 , see Fig. 9. As this statement is true regardless of which intersection one considers, it applies to all boundary reflections of characteristics of this web. We conclude that f1 is an invariant of the entire web. Questions that remain are, first: is one such a web covering the entire domain, or if not, how

X-Z=c2 X+Z=c1 f1 Z Fig. 8. Wedge showing the approach of the apex for all waves whose oblique angle of propagation is steeper than that of the bottom. Arrows indicate energy propagation direction.

g2 Ψ=0 → g2=f1

X Fig. 9. Example of characteristics reflecting from a piece of boundary (heavy solid line) and inference that can be made.

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“many” webs are needed (in any measurable sense)? And, second: what is the physical meaning of this invariant? The first is basically a geometric question, concerned with the covering of space. Characteristic webs exist regardless of whether waves propagate on them or not, much like drainage systems exist whether or not any water is transported through them. The second is a dynamic question, and is important for the actual visibility of the waves. The meaning of these invariants can be inferred by noting that the conjugate pressure field is described in terms of the same functions by P (X, Z) = f (X + Z) + g(X − Z). Hence, the pressure at any point is the summation of the two invariants carried on the two unique characteristics through that point. For this reason, one may refer to each such invariant as the web’s partial pressure (without implying any relation to similar terminology used in gas dynamics). Determining the partial pressure field is referred to as the dressing of the characteristics. In particular, we note that at the flat surface (Z = 0) the partial pressure is half the actual pressure. Hence the construction of this type of solution shows that the Dirichlet boundary condition (Ψ = 0) is incomplete. It also requires that the pressure is specified over some part of the boundary. Or, in other words, that the along-boundary pressure gradient is specified, which at the flat surface, Z = 0, is equivalent to the normal gradient of the streamfunction field, ∂Ψ/∂Z. But, the previous remarks concerning the underlying characteristic webs show that one cannot also specify the pressure over the whole boundary (Cauchy data). This would easily lead to an overdetermined problem when two different pressures are specified on two different boundary points of one and the same web. Part of the problem is thus to find out over which socalled fundamental intervals the pressure can indeed be freely specified. For a particular geometry this is tackled below.

4.1. Example: Parabolic channel Consider as example a parabolic channel whose bottom is at Z = −τ (1 − X 2 ) discussed in [Maas & Lam, 1995]. For a particular “depth”, τ = 0.9, characteristics are launched at an arbitrary initial point x0 . Its web contracts onto a limit cycle, see Fig. 10. This same limit cycle is reached regardless of the

Fig. 10. Example of a single web for τ = 0.9. Characteristics emanating from an arbitrary initial point x0 reveal their approach to a limit cycle (the two rectangles): the wave attractor. Adapted from [Maas & Lam, 1995].

initial source location. Since wave energy propagates along these characteristics, this implies that waves of frequency commensurate to this particular τ -value are drawn to this limit cycle from all over the basin. For this reason this limit cycle acts as a wave attractor and its high degree of predictability make for a singular phenomenon. The presence of the wave attractor is also stable to perturbations of the depth H0 or, what is equivalent because of the lumped character of parameter τ , perturbations of stratification N and frequency ω. This is evident by taking a bird’s eye view of the surface reflections of the attractor. Formally speaking, if we label successive surface reflections of the web as x−1 , x−2 , . . . when following the web leftwards from x0 and x1 , x2 , . . . when going rightwards, then the surface reflections of the attractor are given by a limit set, consisting of points that are visited periodically. These are collectively referred to as x∞ and are displayed over some range of τ in Fig. 11. For τ < 1/2 the bottom is subcritical everywhere, and the waves are trapped at the beaches (corners). For τ > 1 waves penetrate deeper, and reflect more times from the sides, prior to surfacing again and are not further considered here. Note that the attractor is first of all present over continuous τ (frequency) intervals. This denies the presence of any particular set of eigenfrequencies and corresponding eigenmodes. This implies the breakdown of the usual machinery employed in computing the response in elliptic problems. The latter consists in projecting the forcing onto a set of eigenfunctions, yielding a near-resonance when forcing frequency is close to an eigenfrequency. Its spatial structure is dominated by the corresponding

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Fig. 11. Bird’s eye view of attractor for 1/2 ≤ τ ≤ 1. Adapted from [Maas & Lam, 1995].

eigenmode. For internal waves in a continuously stratified fluid eigenmodes are absent, and any applied forcing frequency directly responds with a spatial singularity. Yet, at certain distinct values of τ successive bands are separated from each other by explosive (tangent) bifurcations, such that complicated attractors (having many surface and bottom reflections) replace simple attractors. In the laboratory experiments, to be discussed next, the goal was to observe the simplest attractors. The idea is that there will be more interior shear zones when the attractors are more complicated, so that the latter may therefore be erased by viscous effects. A simple diagnostic measure for the amount of shear along an attractor would be formed by the length of the limiting orbit. Moreover, the roll-up of the web onto the wave attractor, together with the invariance of the partial pressure per web, implies that any pressure differences between two neighboring webs, will be squeezed onto smaller and smaller scales on approaching the attractor. These pressure gradients drive the internal wave currents, which therefore increase without bound. In an inviscid description a velocity singularity develops (that will in reality be checked by nonlinear and viscous effects).

4.1.1. Self-similarity of the Lyapunov exponent The Lyapunov exponent measures the asymptotic rate at which two neighboring characteristics converge very close to the attractor (see Fig. 12). It

Fig. 12. Lyapunov exponent λ+ for a parabolic channel in the range 1/2 ≤ τ ≤ 1, together with two subsequent zooms indicated by rectangular boxes. Adapted from [Maas & Lam, 1995].

mimics the complexity of the underlying attractor intervals and shapes. The simplest attractors have the strongest convergence rate. The two enlargements shown in the zooms of Fig. 12 reveal the self-similarity of the Lyapunov exponent in parameter space. It will appear in the next section, that a similar self-similarity is surprisingly also obtained in physical space. In a smooth convex fluid domain (of roundedoff triangular shape) the self-similarity of the Lyapunov exponent was found to be related to the presence of a devil’s staircase in the rotation number. The latter represents the progression of the boundary reflections of characteristics in an

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orientation preserving map of the boundary onto itself. It is also shown that under smooth deformations of the boundary, attractors (large convergence rates) appear in tongue-like structures in parameter space, so-called Arnol’d tongues [Manders et al., 2003].

4.1.2. Dressing the webs Complete streamfunction (or pressure) fields can be constructed by dressing the webs. This requires first, to find the fundamental intervals, i.e. the parts of the boundary over which the (partial) pressure can be supplied; second, to choose a pressure field on those boundaries, and third, to deduce the streamfunction (or pressure) field in any arbitrarily chosen point within the fluid domain by determining on which two webs this point lies, and what the partial pressures of these two webs are. The fundamental intervals are determined by the channel’s critical points. These are the points where the bottom slope is parallel to one of the two characteristics. For the parabolic channel, shown in Fig. 10, and for the τ -value adopted, there are two fundamental intervals at the surface. The channel’s two acute corners are degenerate versions of critical points (buried in them). Hence, one fundamental interval is bounded by the corner and the nearest surface reflection of the web coming from the other corner, see Fig. 13(a). The other fundamental interval is bounded by the surface reflections of the characteristics emanating from the two critical points at the sloping bottom [Fig. 13(b)]. The attractor in this example is symmetric with respect to the midplane. For other frequencies, there exist two asymmetric attractors, which are each other’s mirror images. In that case there are three fundamental intervals [Maas & Lam, 1995]. Fundamental intervals are used as regions from which a number of webs are launched, see Figs. 13(a) and 13(b). The areas that are influenced are complementary in the sense that each point of the fluid domain has exactly two characteristics passing through it. For some points these come from different fundamental intervals, for others from the same. By specifying the pressure at the surface in the fundamental intervals (here two displaced half-sinusoids) the streamfunction field is constructed by subtracting the partial pressure on the characteristic sloping upward to the right, from that on the other, see Fig. 14. To facilitate this, one may first determine

(a)

(b) Fig. 13. (a) and (b) Fundamental intervals in a parabolic uniformly-stratified channel for τ = 0.9. Edges of these intervals are indicated at the surface by dashed bars and are used as regions from which a number of webs are launched. Adapted from [Maas & Lam, 1995].

the pressure over the whole surface by using the fact that the partial pressure is conserved on a web. The surface pressure reveals compressed and alternatingly mirrored repetitions of the original halfsinusoids. The streamfunction field shows a cellular pattern, whose cells get compressed on approaching the wave attractor. Figure 15 represents a similar solution for a half-trapezoidal basin with a square attractor. In Fig. 15(b) the enlargement of the triangular region above the attractor’s focusing location (where it reflects from the slope) reveals the self-similarity of the streamfunction field in physical space.

4.2. Experimental observation of internal gravity wave attractor In Sec. 3.1 an experimental observation was shown of internal waves in a uniformly-stratified fluid forced by vertical oscillation of a horizontallyoriented cylinder. While two focusing reflections

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1 0.5 0 -0.5 -1 -1

0

1

(a)

(b)

Fig. 14. (a) Example of a streamfunction field obtained by prescribing the partial pressure in the two fundamental intervals (b), indicated by dashes on the top. Adapted from [Maas & Lam, 1995].

Fig. 15. (a) Example of a streamfunction field in the vertical cross-section of a half-trapezoidal uniformly-stratified channel. (b) Zoom of the area that is indicated by dashed lines. Adapted from [Maas et al., 1997].

could be discerned near the sloping side, the final approach of the waves onto a wave attractor was apparently suppressed by viscous damping. In another experiment these waves were generated in a similar configuration by another forcing mechanism, namely by oscillation of the supporting table (with frequency 2ω), see [Maas et al., 1997]. This oscillation can be viewed as a periodic perturbation of the gravitational acceleration, one of the parameters of the problem. The waves are thus parametrically excited. This can be made manifest by separating the temporal from the spatial dependence. The former is governed by a Mathieu equation, containing a parameter that depends sinusoidally on time. This equation has instability regions, the first and most important of which occurs at the subharmonic of the driving frequency [Abramowitz & Stegun, 1964]. It is this

subharmonic of frequency ω which grows in time, and which is chosen from the wave attractor’s most dominant frequency domain having the largest wave attractor. Figure 16 shows the set-up of the container with sloping side wall. It is filled with a uniformly stratified fluid (N = 1.89 s−1 ) and waves of frequency ω(= 1.44 s−1 ) propagate energy obliquely along straight lines, making an angle cos−1 ω/N with the vertical. The visualization employs fluorescine dye, which is injected in alternating layers. A laser shines from above on the dye layers, and a camera observes their displacement from the side. It takes about 11 min (150 wave periods) for the attractor to completely develop. Then it shows oscillations around the predicted wave attractor location, see Fig. 17. In a reanalysis, the originally horizontal dye layers are subtracted from their

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Laser

g

perturbed versions, amplifying the visibility of the waves, see Fig. 17(b) [Maas et al., 1997]. In order to capture the evolution of the wave’s vertical displacement field the amplitude of the subharmonic and its phase (modulo π) are computed by averaging over a number of characteristics that are all perpendicular to the two long sides of the wave attractor, see Fig. 18. The averaging is performed to remove the

Fluorescine dye

Camera

g→g(1+ε sin 2ω t)

Fig. 16. Laboratory set-up employing parametric excitation adopting a uniform stratification.

(a)

(a) (b)

(c) (b) Fig. 17. (a) Maximum vertical displacement of fluorescine dye lines observed about 11 min after starting the oscillation of the supporting table. (b) Amplitude of the subharmonic of the oscillation frequency of the residual color intensity field in any bin upon subtracting the initial dye field (dominated by horizontal lines).

Fig. 18. (a) Beam of rays used to average the observed amplitude and phase of the subharmonic displacement field. Solid lines indicate two long sides of attractor that cross the beam. (b) Amplitude and (c) phase of the subharmonic of the oscillation frequency averaged over the beam width, as a function of time t measured in terms of wave periods T . The abscissa indicates the along-beam distance in pixels (slope at the right-hand side). Taken from [Lam & Maas, 2005].

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presence of artificial minima and maxima in these fields in the along-attractor direction which are due to the finite width of the original, horizontal dye bands [Lam & Maas, 2005]. The result confirms that the wave attractor appears “out of the blue”. Some weak precursors are already visible after five minutes. The attractor’s two branches are clearly visible, and are strongest immediately upon creation, but nevertheless persist in a somewhat blurred fashion over much longer times. The attractor’s branch that is closest to the (focusing) slope is always the sharpest. The phase is nearly uniform over each branch of the attractor (which are 180◦ out of phase with each other), particularly at onset. This suggests it grows as a standing wave. But the persistent weak deviations that develop later on reveal the wave’s asymptotically propagating character, which is in fact captured more clearly in terms of propagating nodal lines in a movie [Maas et al., 1997]. Note that while the observations do confirm a localization of internal waves around the attractor, and thus its physical relevance, they do not provide any indication of the fine structure as present in the hypothetical streamfunction field. A few comments can be made. First, the experiments provide observations on vertical displacement of dye lines. A fair test should therefore compare computed displacements of (initially horizontal) isopycnals with observed dye line displacements. This is performed in [Lam & Maas, 2005], where it is assumed that a weak surface sloshing mode is first excited which then acts to force the internal wave field. This requires one to address the effect of a forcing that extends beyond fundamental intervals. Second, one needs to incorporate viscous and nonlinear effects. In the above inviscid approach the two fundamental intervals are isolated. Yet, the observations show that these neglected effects provide a clear synchronization across the attractor which needs further elucidation.

5. Inertial Waves A uniform-density, uniformly-rotating fluid supports inertial waves [Greenspan, 1968]. A fluid that is rotating as a solid body has elastic properties because it is stratified in angular momentum, in a radial direction. In this balanced state the outwarddirected centrifugal force is balanced by an inwarddirected radial pressure gradient. When a particle is displaced adiabatically outwards or inwards

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(retaining its original angular momentum) it is driven backwards by the imbalance between local pressure gradient and the centrifugal force. Free inertial waves can be described by considering the linearized momentum equation in a corotating frame of reference. For steady rotation (angular frequency Ω), within this frame the centrifugal force is dynamically present as the Coriolis force (the second terms in the first two equations below) ∂p ∂u − 2Ωv = − ∂t ∂x ∂p ∂v + 2Ωu = − ∂t ∂y ∂w ∂p =− ∂t ∂z ∂u ∂v ∂w + + = 0. ∂x ∂y ∂z For monochromatic waves of frequency ω < 2Ω the spatial structure of the inertial wave field is thus determined by  2  2 4Ω ∂ p ∂2p ∂2p + 2− −1 = 0, (7) 2 2 ∂x ∂y ω ∂z 2 the Poincar´e equation [Cartan, 1922]. In some laboratory experiments waves were excited by superimposing a slight (  1) modulation (of frequency ω < 2Ω0 ) on the steady rotation: Ω = Ω0 (1 +  sin ωt) [Beardsley, 1970; Maas, 2001; Manders & Maas, 2003]. The Euler force, also derived from the centrifugal force, provides a body force in the momentum equations [Tolstoy, 1973]. Due to conservation of vorticity this implies accelerating and decelerating horizontal flows. When the tank has a sloping wall, this flow, however, is forced also upwards and downwards, triggering inertial waves in the bulk of the fluid domain. When considered in a tank that is relatively long in the along-slope direction, a quasi 2D treatment in the vertical cross-slope plane is warranted (∂y ≈ 0). Thus its spatial structure is again determined approximately by the 2D spatial hyperbolic equation, and predicts the occurrence of wave attractors. The appearance of these attractors is confirmed by amplitude distributions of the in-plane (u, w) velocity fields of harmonic waves for different forcing (modulation) frequencies (see Fig. 19). The predicted locations of the wave attractor are superimposed as a solid line. The energy, while

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0.5

1

1.5

2

1 0.75 z 0.5 0.25 0 0

0.5

1

1.5

2

x

Fig. 19. Velocity amplitude of inertial waves obtained from particle image velocimetry in a rotating, uniform-density fluid. Taken from [Manders & Maas, 2003].

localized around the wave attractor, is not as uniformly present as for the internal gravity wave experiment as the waves also propagate in the along-slope direction (out of the plane).

Fig. 20. Dye patterns observed six modulation periods apart, seen from above in a half-trapezoidal tank. Adapted from [Maas, 2001].

background rotation. This is visualized by a remarkably strong leftward dye displacement midway over the slope that matches estimates from Particle Image Velocimetry.

5.1. Generation of a mean flow Due to wave focusing the fluid is mixed owing to unresolved processes such as Kelvin–Helmholtz instabilities. As this focusing principally takes place upon the wave’s reflection from the slope, it is at this location that we expect mixing to occur preferentially. “Mixing” in this case obviously refers to a mixing of angular momentum (the fluid being homogeneous in density). By considering a mixture of two parcels that are brought together at an intermediate radial position, it can be shown that the angular velocity of this mixture is higher than the solid body angular velocity at that location. Thus the mixture results in a cyclonic mean flow (i.e. moving in the same direction as the tank). An experimental confirmation of the occurrence of such a mean flow is visible in Fig. 20. The slope is in the upper half of the figures. The modulation excites inertial waves that focus onto a squareshaped attractor over the middle of the slope where they mix. Over that location this mixing accelerates fluid into the same direction as the (mean)

6. Geophysical Application Rotating stars and planets as the earth often possess homogeneous fluid layers that result from convection. The earth e.g. contains a liquid outer core whose density is constant. Also the earth’s atmosphere and ocean can be modeled to some approximation as a homogeneous fluid contained in a spherical shell. The spherical shell is a geometry where one may directly anticipate the occurrence of wave attractors. On the one hand it supports inertial waves, that orient themselves with respect to the rotation axis, yet on the other hand its boundaries are clearly neither parallel nor perpendicular to this axis, thus, in principle, favoring focusing and defocusing reflections. Bretherton [1964] first pointed at the geometric peculiarity that such a domain supports exceptional periodic orbits of inertial waves, see Fig. 21(a). These periodic orbits were actually found to be attracting [Stewartson, 1971], demonstrating

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inertial oscillations 0

N

Ω N=0

Z

h 1

1

2

3

S

1

Y

N=const

0

inertial oscillations

Fig. 21. Inertial wave rays approach a wave attractor (periodic orbit) in a (a) homogeneous or (b) stratified spherical shell. In the homogeneous case they approach one that is symmetric with respect to the equator. In the stratified case they approach the point where the hyperboloidal turning surface intersects the bottom (a little North of the local inertial latitude indicated by the straight solid line). Adapted from [Bretherton, 1964] and [Maas, 2001].

the dominance of focusing over defocusing. On comparing this to the laboratory experiment in the previous section, one may anticipate that mixing accompanying the focusing of inertial waves, may drive prograde (eastward) zonal currents. This mixing takes place at locations where the attractor’s focusing reflections occur, and this mixture (eastward currents, depicted by crosses that indicate velocity arrow-tails) can then spread along a cylinder that is parallel to the rotation axis which forms an isosurface of angular momentum. Outside the frequency intervals over which waves are trapped onto an attractor waves propagate poleward. However, fluid envelopes such as the ocean are frequently also stratified in density. This additional radial gravitational restoring mechanism has two effects [Friedlander & Siegmann, 1982]. First, the characteristics become curved and second, the fluid domain that is accessible to the waves becomes bounded by turning surfaces over which the governing differential equation changes from hyperbolic to elliptic. For the usual oceanographic case, stratification is the larger of the two (2Ω < N ). The wave attractors are as a consequence concentrated on an equatorial band bounded by hyperboloidal turning surfaces, see Fig. 21(b). Interestingly, these may coexist with other wave attractors that form the counterpart of the waves that were propagating poleward in the homogeneous rotating spherical shell [Dintrans et al., 1999]. These are point attractors, which are located at the intersection of turning surfaces and the bottom of the spherical shell. Since the turning surfaces are very close to the

Y 0

0

N S

-1 1

2

3

-1

Scaled frequency

Fig. 22. (a) Wave attractors in equatorial basin. (b) Domains of attraction, see text.

so-called critical latitudes ±φ, where the adopted frequency is very nearly ±2Ω sin φ, these waves are locally referred to as (near) inertial waves and the attractors as inertial wave attractors. In an equatorial basin the fluid domain is confined between bottom (Z = 0), surface (Z = h) and (inertial) turning latitudes (Y = ±1), see Fig. 22(a). This case shows the coexistence of an equatorial wave attractor, approached by solid rays, and (northern) inertial wave attractor, approached by dashed rays. On mapping the domains of attraction of these two types of attractors as a function of scaled frequency (related to dimensionless depth h) and surface launching position Y [Fig. 22(b)] it appears that equatorial wave attractors exist for a discrete set of continuous frequency intervals (gray), while the inertial wave attractors whether of a southern (white, label S) or northern (black, label N) kind exist for any frequency considered [Maas & Harlander, 2005]. This may explain the observed ubiquitous presence of (near) inertial waves in the oceans [Fu, 1981].

7. Internal Waves in 3D The spatial structure of the perturbation pressure field of monochromatic internal gravity waves in a three-dimensional basin is again governed by the Poincar´e equation (7) except for a different factor multiplying the z-derivatives, and except for satisfying simpler boundary conditions. Thus we will confine ourselves to internal gravity waves here. Except for some symmetric, nonfocusing geometries, exact solutions of this equation, satisfying the boundary conditions, are hard to find. The problem is that

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at any interior point the pressure is no longer the sum of just two partial pressures, but instead an integral over the cone of fixed angle with respect to gravity (the angle determined through the dispersion relation by the wave’s frequency). Here we will not resolve the full pressure field, but infer some qualitative properties by means of ray tracing. Restricting one’s attention to just the path of an internal wave beam is motivated by the fact that in two-dimensional problems the distribution of these paths (in 2D equivalent to the characteristics) dominates the actual spatial energy distribution and in particular the occurrence of wave attractors. Since the wave’s frequency is fixed, also the beam’s angle with respect to the vertical will be fixed. But the beam might change its horizontal angle of propagation (neglecting the possibility of diffraction: spreading into multiple directions). The wave beam is considered to consist of only short waves, as these propagate their energy along straight lines. Thus we will only need to establish the locations where the wave beam reflects, whether this is up or downwards, and into what horizontal direction.

(a)

7.1. Ray tracing in paraboloidal basin As an example, we here apply this to a particular geometry: a paraboloidal basin that is filled with a uniformly stratified fluid. Here we assume that similar rescaling has been performed as in the two-dimensional problem. That is, we assume the paraboloid has surface radius 1 and depth τ . Now consider plane, short waves of definite frequency that are “launched” at the surface, at position (x0 , 0), along horizontal direction φ0 measured with respect to the x-axis. Note that because of axial symmetry of basin and equations it is sufficient to consider launching at the arbitrarily located x-axis solely. We then consider a single reflection at the sloping bottom, which, as the incident and reflected waves, is treated as locally planar. Requiring the incident and reflected wave to obey the same dispersion relation (and thus to satisfy the same governing equations), the condition of vanishing normal flow at the point of reflection implies that the wave is subject to instantaneous refraction, see Fig. 23. While the wave vector component in the along-slope, tangential direction is unchanged, the wave vector in the cross-slope direction changes due to a focusing or defocusing reflection. The horizontal direction of the reflected wave φr is determined

(b) Fig. 23. (a) Side and (b) top view of a short internal wave packet that reflects from a sloping bottom.

by the local bottom slope s = |∇H| and horizontal direction of the incoming wave, φi (which equals φ0 plus the angle that the bottom gradient vector makes with the x-direction). It yields sin φr =

s2 − 1 sin φi 1 + 2s cos φi + s2

[Phillips, 1963; Eriksen, 1982; Gilbert & Garrett, 1989; Manders & Maas, 2004]. Applying the same reflection law (in appropriately rotated coordinate frames) on subsequent reflections, the wave’s ray path can thus be followed as it bounces through the paraboloid. We find that upon a large number of reflections, rays either converge onto a vertical plane through the basin’s axis of symmetry of unique orientation φ∞ , or do not converge at all and wrap around the basin again and again, see Fig. 24. However, when rays do converge, within that axial plane they further converge onto limit cycles, which are the same as the attractors found for the 2D channel.

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(b)

φ0

y

φ∞ x0

x

(c)

(d)

Fig. 24. Perspective (a), (b) and top views (c), (d) of ray paths in a uniformly stratified basin (τ = 0.65) launched at surface positions (x0 , 0) for x0 = 0.3 and 0.8, and angle φ0 = π/2.

When rays do not converge, they cluster around the critical line in an endless succession of focusing and defocusing reflections. The critical line (dashed in Fig. 24) connects the critical points at which the bottom slope equals characteristic slope. These rays represent a kind of internal edge waves. One way to classify the wave’s behavior is by considering the asymptotic horizontal angle of the axial plane to which the rays converge, φ∞ , as a function of launching position 0 ≤ x0 ≤ 1 and horizontal launching direction, φ0 , restricted to the interval (−π, 0) because of symmetry. An example is plotted for τ = 0.65 in Fig. 25, where a certain color represents the asymptotic direction according

to the color table on its right-hand side. White (which is outside the color table’s range) represents the absence of any convergence, hence it refers to the internal edge waves. For the two examples in Fig. 24, starting at x0 = 0.3 and 0.8 respectively, the rays are launched perpendicular to the x-axis, φ0 = π/2. One verifies the presence and absence of an asymptotic plane respectively, predicted by Fig. 25. Note the very intricate, lobed structure of the white, internal edge wave domains. Also note that even when only attracting planes exist, the asymptotic direction reached is distributed quite unevenly, see for instance the dominance of the green color, which represents φ∞ ≈ 1, for

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Fig. 25. Asymptotic horizontal direction φ∞ (indicated by a color) reached by rays in a uniformly stratified basin (τ = 0.65) launched at (x0 , 0) into direction φ0 . White corresponds to lack of convergence (edge waves).

Fig. 26. As Fig. 25 for rays launched perpendicular to the x-axis, i.e. φ0 = π/2, as a function of virtual depth τ and radial position x0 .

0.3 < x0 < 0.6. It is interesting that for some launching locations and virtual depths, φ∞ is nearly constant. Regardless of launching direction one is able to predict quite accurately the asymptotic plane. The example shown suggests that for this geometry internal edge waves are not as common as

wave attractors. To probe this we varied the virtual depth τ . In order to show the result in a similar color graph, we restrict ourselves here to perpendicular launching only, φ0 = π/2. Then, variation of radial launching position x0 demonstrates that for other values of τ internal edge waves may be

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much more common, see Fig. 26. We recognize the earlier choice τ = 0.65 as being at the border of a τ -region where internal edge waves are more common (at least, when launching perpendicularly). The bands in Fig. 26 reflect those in the 2D context, Fig. 11. At the same time, the presence of internal edge wave regimes within those bands, provides for a still richer parameter plane. Internal edge waves will stand out energetically for those rays that are strictly periodic. Otherwise, they ergodically fill a band in between two caustics (two circles acting as turning surfaces). Interestingly, internal edge waves trapped at the foot of the continental slope have probably been observed on several occasions [Horn & Meincke, 1976; Huthnance & Baines, 1982; Lerczak et al., 2003], see Fig. 27. The presence of a critical line, that divides an up-slope region where the bottom is supercritical from a down-slope region where it is subcritical, seems to be a crucial ingredient for its existence. Indeed, previous theories have looked in vain for trapped (regular) internal waves in a singleangled wedge, capped by a rigid lid surface, that lacks such a critical line [Wunsch, 1969; Muzylev & Odulo, 1980; Sherwin, 1991; Lien & Gregg, 2001; Llewellyn Smith, 2004]. However, by choosing an appropriate shape of the bottom containing a critical line it does seem possible to construct analytical solutions of trapped internal edge waves (unpublished results both by S. Llewellyn Smith as well as by the present author).

Fig. 27. Observed distribution of internal tidal energy in the Eastern Atlantic. The vertical coordinate has been scaled such that the internal wave rays are straight. Note the clustering around the location where the slope is critical. From [Horn & Meincke, 1976].

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8. Discussion Classical confined wave systems, such as surface gravity or acoustic waves, are dominated by regular eigenmodes, associated with a discrete set of eigenfrequencies. For short waves, these eigenmodes have a localized character: scars. Short waves are described by a dispersion relation which states that upon reflection from the boundary only the wavelength of such waves is fixed, not their direction. The waves can succinctly be described by their ray paths. Rays bounce back and forth across the basin, reflecting from a boundary according to Snell’s law, as on a classical billiard. Each bounce can be seen as the iteration of a two parameter map that relates initial and subsequent circumferential location plus horizontal propagation direction. While this map is in general chaotic, the localization is determined by the periodic rays that are allowed by the shape of the fluid domain. Thus, the periodic rays, while being unstable to perturbations, are the least-repelling orbits, for which reason wave energy lingers in their vicinity. When such a classical wave system is continuously forced, the steady response is determined by a projection of the forcing on the complete set of eigenmodes. The response is usually inversely proportional to the difference of forcing frequency to the nearest eigenfrequency. An infinite response is predicted when this difference vanishes (a resonance), which, in steady conditions, will be regularized by viscous effects (damping). When considering the response to a nearly-resonant forcing that is turned on at some initial moment, the amplitude of the response grows slowly in time, until either the previous steady response is reached, or the damping is unable to check the forcing and a global, explosive instability sets in. For the geophysically relevant internal waves, considered here, the behavior of a confined wave system is completely different, at least in two dimensions (a vertical plane). The waves are dominated by the presence of wave attractors, which exist over continuous frequency bands and attract waves for any frequency considered, regardless of where they are forced. This is because these waves, relative to the vertical (gravity or rotation axis), have a fixed direction, not wavelength. They have a spatial singularity, where wave energy increases without bound, until checked by viscous [Rieutord et al., 2001] and nonlinear processes. Experiments in uniformly stratified, and rotating homogeneous

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fluids confirm the existence of these wave attractors. Field observations from a small lake [Fricker & Nepf, 2000] present a first indication of their existence in nature and the ubiquitous presence of inertial oscillations in the oceans might perhaps be due to them [Maas, 2001]. These attractors may play a decisive role in geophysical media as the ocean, atmosphere, liquid outer core of the earth, other planets or stars [Ogilvie & Lin, 2004], because the waves that approach them localize energy onto small scales where they contribute to mixing and drive mean flows. These processes may for the same reason be relevant in industrial applications, like turbomachinery. An important difference between both wave systems lies in the nature of the equation that governs their spatial structure. Surface (and acoustic) waves are governed by an elliptic equation, internal waves by a hyperbolic equation. Are there any other physical systems whose spatial structure is governed by a hyperbolic equation? Supersonic waves come to mind. However, it is hard to envisage that these waves propagate in a container that is comoving at the same speed. A more likely candidate is formed by electron–cyclotron waves that arise in a magnetic field when considered in the limit of an electron fluid amidst a frozen ion lattice [Tolstoy, 1973]. The transverse Lorentz force experienced by moving electrons in the magnetic field, is completely analogous to the Coriolis force acting on a moving water parcel in a rotating fluid. It is interesting to speculate that these electron density waves might actually be focused by an appropriately reflecting boundary (either hard, or soft, as the edge of the magnetic field), and to consider what macroscopic relevance such a wave attractor may have. In the introduction, a well-known example of a wave attractor was presented in the form of a beach. However, with hindsight this is quite surprising, as this example was drawn from the class of surface waves. Recall, however, that the absence of wave attractors (and the presence of eigenmodes) in Sec. 3 followed from the coast being reflective, in other words, from a consideration of long waves. Indeed, surface wave literature acknowledges the presence of a continuous spectrum of short waves, which are those that were subject to breaking. This leaves us with the challenge to investigate how one might formulate also for elliptic problems a map such that upon iteration the beach attractor appears as the map’s fixed point.

The construction of approximative high wavenumber modes in the elliptic case, and exact (approximative) wave attractors in the 2D (3D) hyperbolic case relies to a large degree on condensing the linear dynamics to the iteration of a nonlinear map of the boundary onto itself. This map was 2D and in general chaotic in the former case, and 1D and (mathematically) dissipative in the 2D latter case. Iteration of such maps gives rise to self-similar bifurcation patterns in parameter space and self-similar patterns in physical space. These properties are usually reminiscent of nonlinear dynamics. However, they here arise in strictly linear PDEs, so that solutions retain their superposability. This tension led to the subtitle of this paper: “linear yet nonlinear”, which applies also to the concept “quantum chaos” to which the wave attractor is found to be complementary. This subtitle paraphrases the personal motto of Simon Stevin (1548–1620), a self-taught Belgian–Dutch mathematician, physicist, civil engineer and counseler of the Prince of Orange, which reads “magic and yet no magic” (in ancient Dutch: wonder en is gheen wonder). This not only describes Stevin’s positive attitude when delving into the unknown, but may perhaps also express the sentiment felt when observing the rich dynamical structure of wave attractors.

9. Appendix: Internal Gravity Waves in a Wedge The wave equation (5) that one aims to solve in a particular domain can sometimes be solved by employing a coordinate transform that leaves this equation unaltered, but changes the shape of the domain. This analog of the conformal mapping method, often employed in solving Laplace’s equation, enables one to derive solutions for internal gravity waves in a wedge. The coordinate transform is obtained by employing the fact that real ξ and imaginary ζ parts of any function F (σ) of a complex variable σ = x + iy are harmonic, ξ + iζ = F (x + iy). That is, each satisfies a Laplace equation, ∆ξ = ∆ζ = 0, where ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 . By simply substituting y = −iz in any pair of harmonically conjugate functions, as well as in the Laplace equation, the new coordinates satisfy the wave equation: ∂ 2 ξ/∂x2 − ∂ 2 ξ/∂z 2 = 0, and similarly for ζ [Maas & Lam, 1995]. We refer to ξ and ζ (now as functions of x and z) as “hyperbolically conjugate” functions,

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as they satisfy ξx = ζz , ζx = ξz .

(8) (9)

Subjecting (5) to such a “hyperbolically conformal” coordinate transformation: (x, z) → (ξ(x, z), ζ(x, z)), we obtain, by working out partial derivatives, Ψξξ (ξx2 − ξz2 ) + Ψζζ (ζx2 − ζz2 ) + 2Ψξζ (ξx ζx − ξz ζz ) + Ψξ (ξxx − ξzz ) + Ψζ (ζxx − ζzz ) = 0.

(10)

In view of Eqs. (8) and (9), the “hyperbolic CauchyRiemann” conditions, (10) simplifies to J(ξ, ζ)(Ψξξ − Ψζζ ) = 0,

(11)

which is again the wave equation, but now in the transformed plane, except along lines where the Jacobian of transformation J(ξ, ζ) ≡ ξx ζz − ξz ζx = ξx2 − ξz2 = ζz2 − ζx2

(12)

vanishes, J(ξ, ζ) = 0. Since the boundary conditions are unaffected, the freedom introduced with the coordinate transform can be used to let the boundary take on a simple form, for instance by reducing it to a coordinate line ζ = constant. Applied to a wedge, we assume vanishing of the streamfunction Ψ = 0 at surface z = 0 and sloping bottom z = sx. For this wedge-shaped geometry Wunsch [1969] obtained explicit solutions by inspection, that can be retrieved by first mapping the wedge domain onto a channel, and then using the well-known internal wave channel modes. The aforementioned map is obtained by choosing F (σ) = 2 ln σ. Written in polar form, with σ = r exp(iθ), where, as usual, r = (x2 + y 2 )1/2 and 2θ = 2 tan−1 (y/x) = ln((x−iy)/(x+iy))1/2 , the real and imaginary parts of F (σ), 2 ln r and 2θ, represent two independent solutions of Laplace’s equation. Thus, replacing y = −iz, one obtains two independent solutions of the wave equation ξ = ln(x2 − z 2 ) = ln(x + z) + ln(x − z),

(13)

and ζ = ln

x+z = ln(x + z) − ln(x − z), x−z

(14)

that are clearly “hyperbolically conjugate”, see (8) and (9), which are useful as new coordinates. For the moment we assume that the bottom is subcritical (−1 < s < 0), so that for x > 0 singularities along the critical characteristic (z = −x) coming

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from the apex do not lie within the fluid domain and the Jacobian of transformation (12) never vanishes. The surface z = 0 is mapped onto ζ = 0, the bottom z = sx, on ζ = ln[(1 + s)/(1 − s)] ≡ −h, a constant. The apex, (x, z) = (0, 0), is mapped to the “point” at −∞. The wedge is thus mapped onto an infinitely long channel, −∞ < ξ < ∞ for −h ≤ ζ ≤ 0. The equation for internal waves is unchanged in the new coordinates, see (11). Hence, the standard solutions of internal waves, obtained by separation of variables, can be employed in the transformed frame. These are wave modes, standing between surface and bottom in the transformed vertical direction, ζ, which propagate in the transformed horizontal direction, ξ: Ψ=

∞ 

An eiµn ξ(x,z) sin µn ζ(x, z),

n=−∞

where µn = nπ/h. These are the solutions obtained by inspection in [Wunsch, 1968, 1969]. For n positive (negative), the waves propagate into the positive (negative) ξ direction, that is, waves propagate out of (into) the apex. The amplitudes An are determined by the boundary condition (specified at certain x, say). When choosing A−n = An one may construct standing waves [Wunsch, 1968], but as Wunsch (1969) argues, it is physically more likely that waves only enter the wedge (obtained by choosing An = 0, n > 0). Waves reflecting from the bottom are accompanied by an intensification of the currents. As also the current shear intensifies, this will lead to instabilities of this shear flow, to breaking and eventually, to mixing (processes not described by the wave equation above). The propagating nature of the internal waves in a wedge of subcritical slope was experimentally confirmed in [Wunsch, 1969; Cacchione & Wunsch, 1974].

Acknowledgments The author is grateful to the ENOC2005 conference organizers for their kind invitation to present the topic covered in this paper. He is much indebted to Jo¨el Sommeria, Dominique Benielli, Stuart Dalziel, Astrid Manders and Frans-Peter Lam for kindly allowing the use of experimental facilities, results and reanalysis thereof obtained at the Ecole Normale et Superieure de Lyon, Coriolis Laboratory in Grenoble, and the G. K. Batchelor Laboratory of the University of Cambridge. Thanks are also due to Margriet Hiehle, Nelleke Krijgsman

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and Lucas Merckelbach for support in preparing figures and text, and Uwe Harlander and Arno Swart for comments on the first version of the manuscript.

Glossary of Symbols Symbolic expression α

Angle of wave vector with horizontal, or group-velocity vector with vertical.

λ+

Lyapunov exponent; + indicates characteristics are launched towards x > 0.

µ

Wave number.

N

Brunt–Vais¨ al¨ a (stability) frequency; stratification of rest state 1/2 . (−gρ−1 ∗ dρ0 /dz)

p (x, t)

Reduced pressure field, i.e. actual pressure minus hydrostatic pressure field.

A, An

Amplitude of wave field, or of the nth mode.

b(x, t), B(x, z)

Buoyancy field b = −gρ (x, t)/ρ∗ and its spatial amplitude.

p, P (x, z)

Perturbation pressure, p /ρ∗ , and its spatial structure.

c, cg

Phase velocity vector ωk/κ2 and group velocity vector ∇k ω.

Π(x, z)

Scaled spatial structure pressure field P (x, z)(L/D).

ρ, ρ∗

D

Depth scale, in uniformlystratified nonrotating fluid: √ ωL/ N 2 − ω 2 .

Density field, constant reference density.

ρ0 (z), ρ (x, t)

Density stratification of state of rest, density perturbation.



Small dimensionless perturbation.

r

Radial coordinate.

φ

Potential function, or latitude. Horizontal launching/ asymptotic direction of oblique internal waves.

σ s

Complex function. Along perimeter distance. Also designates bottom slope |∇H|.

S

Phase function.

Horizontal propagation angle of incident/reflected waves.

t

Time.

τ

Lumped parameter H0 /D; interpreted as scaled depth or scaled wave period.

ψ(x, z, t), Ψ(x, z)

Streamfunction field and its spatial structure.

φ0,∞

φi,r

f

Coriolis parameter f = 2Ω sin φ. In laboratory: f = 2Ω.

f, g

Partial pressure.

g

u = (u, v, w)

Velocity vector field.

Gravitational acceleration vector.

x = (x, y, z)

Cartesian coordinate frame.

H(x), H0 , h(X)

Topography, topography depth scale, and dimensionless scaled topography.

x0,1,2,...,∞

Initial, subsequent and asymptotic IW surface reflection points.

X = (X, Y, Z)

Stretched Cartesian coordinate frame, x = LX, z = DZ.

ξ, ζ

Transformed coordinates.

ω

Wave frequency.

Ω0 , Ω

Constant (and sometimes weakly time-dependent) frame rotation rate.

k = (k, l, m)

Wave number vector. In 2D: k = κ(cos α, sin α).

κ = |k|

Wave number magnitude.

L

Horizontal length scale of fluid domain.

λ

Parameter in 1D map.

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