Reflection of an internal-wave ray

At a stationary slope, an internal wave preserves frequency and tangential wavenumber. Unlike specular reflection, the reflected energy ray preserves its angle to the vertical; a supercritical slope can reverse its horizontal propagation direction.

Let an internal gravity wave energy ray make angle $\theta$ with the horizontal, so that $\sin \theta =\omega /N$. A boundary of local slope magnitude $s$ is subcritical, critical, or supercritical according as $s$ is smaller than, equal to, or larger than $\tan \theta$. At criticality the inviscid reflected wavelength tends to zero.

In subcritical internal-wave reflection, the boundary is everywhere less steep than the energy ray. The reflected ray leaves the boundary without the singular shortening associated with a critical slope.

At critical internal-wave reflection, the reflected group velocity is tangent to the boundary and the inviscid reflected wavenumber diverges. Kinematic viscosity, mass diffusivity, nonlinear steepening, and wave breaking regularize the ideal singularity.

Reflection from periodic topography couples an incident horizontal wavenumber $k$ to sidebands $k+j{k}_T$. Expanding the impermeability condition on a boundary of small amplitude $h_0$ produces the $j=\pm 1$ sidebands at order $h_0$ and the $j=0,\pm 2$ corrections at order $h_0^2$.

Internal-wave ray tracing follows the group velocity while preserving the wave frequency. In a uniformly stratified fluid, each straight ray keeps the angle ${\sin }^{-1}(\omega /N)$ to the horizontal until reflection or a change in the medium.

For a uniform vertical shear flow $U=sz$ with $s>0$, constant buoyancy frequency $N>0$ and horizontal wavenumber $k>0$, choose the initially rising branch $m_0<0$. The ground-frame angular frequency is the conserved ray HamiltonianThe internal-wave ray tracing equations give $\dot{k}=0$ and $\dot{m}=-ks$, hence $m=m_0-kst$ and $z=(\omega -\hat{\omega })/(ks)$. As $t\to \infty$, the vertical wavenumber tends to negative infinity, the intrinsic frequency tends to zero and the height approaches the critical level $z_c=\omega /(ks)$. For $\theta =\arctan (|m|/k)$, the laboratory ray slope isThe second term is mean advection divided by vertical group velocity. The zero-shear case has no finite $z_c$ and must instead be treated as propagation in a uniform current.

For a stationary internal gravity wave in constant buoyancy frequency $N$ and wind $U=U_0(1+z/H)$, set $q_0=k{U}_0/N<1$. Its upward group velocity has ratio $dx/dz=k/m=q/\sqrt{1-q^2}$, where $q=q_0(1+z/H)$. Integrating from the origin gives a circular ray with center $(H\sqrt{1-q_0^2}/q_0,-H)$ and radius $H/q_0$. The ascending ray becomes horizontal at the turning level of a stationary internal wave; its reflected branch continues downwind along the same circle.

A periodic spatial internal-wave ray toward which neighbouring ray paths converge after reflection in a confined stratified basin. A fixed point of an internal-wave ray return map is attracting when its derivative has magnitude below one. Viscous attenuation of an internal-wave beam and internal-wave breaking regularize the ideal concentration of energy. A spatial ray construction does not remove turning-level limitations of the WKB approximation.

Map one boundary intersection of an internal-wave ray to the next intersection with the same boundary after a specified reflection itinerary. A periodic ray is a fixed point. Linear contraction, $|R^{\prime }|<1$, gives an internal-wave attractor; the reversed itinerary, when invertible, has reciprocal multiplier and is locally repelling.

A level where the squared vertical wavenumber of an internal gravity wave touches zero quadratically without changing sign. The ordinary WKB approximation fails there; unlike a simple sign-changing turning point, this does not itself provide an evanescent region on one side. Finite-wavelength analysis determines how solutions connect across or meet a boundary at the level.

In uniform positive flow $U$, upward stationary internal gravity waves have laboratory group velocity $U({\cos }^2\theta ,\sin \theta \cos \theta )$. Packets emitted by a localized source at startup lie on $(x-Ut/2{)}^2+z^2=(Ut/2{)}^2$, with later emissions inside. This envelope belongs to the permanent stationary component, not every possible startup transient.

A lower ray reaches $H$ after ${\tau }_1=H/[U_1\sin {\theta }_1\cos {\theta }_1]$, at $x_H=H\cot {\theta }_1$. Conserved horizontal wavenumber gives $\cos {\theta }_2=(U_2{N}_1/U_1{N}_2)\cos {\theta }_1$. The transmitted front is $(x_H,H)+(t-{\tau }_1){\mathbf{c}}_{g2}$ on propagating branches; reflection reverses the vertical group velocity.

A broad-spectrum stationary source below height $H$ first sends an upward packet there after $2H/U$, at $x=H$. This uses the maximum vertical group velocity $U/2$ at $\theta =\pi /4$. A source lacking the corresponding wavenumber has a later actual arrival.

For a nonrotating internal gravity wave of fixed angular frequency in uniform buoyancy frequency, the dispersion relation fixes the ray angle $\theta$ to the horizontal. Consider a planar slope at angle $\alpha$ in the focusing orientation, with $0<\alpha <\theta <\pi /2$. Conservation of tangential wavenumber at reflection gives the ratio of total wavenumber magnitudesThe reflected wavelength and beam width each shrink by ${\gamma }^{-1}$. At fixed angular frequency, the magnitude of group velocity also shrinks by ${\gamma }^{-1}$; conservation of beam-integrated energy flux therefore multiplies the wavelength-averaged energy density by ${\gamma }^2$.

The orientation matters. Reversing the ray itinerary gives the reciprocal, defocusing factor; a horizontal boundary gives $\gamma =1$. Thus subcritical slope magnitude alone does not imply amplification for every incident direction. The divergence as $\alpha \to \theta$ signals the failure of the ideal inviscid reflection model near internal-wave slope criticality.