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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.

Use the spatial internal-wave ray tracing model, with $dx/dz=\pm 16z$, and reflect the rays so that the frequency and tangential wave number are preserved. At a vertical wall the horizontal energy direction reverses; at a horizontal wall the vertical energy direction reverses. On the sloping wall the reflection depends on internal-wave slope criticality. The three-reflection orbit below encounters that wall at a height above $3/16$, so the wall reverses the horizontal energy direction there.

First take the packet to leave the bottom upward and to the left, with $0<x_0<6$. The four segments of the loop areThe left-wall reflection is at $z=\sqrt{x_0/8}$ and the top reflection at $x=8-x_0$. The slope intersection solves $8z_s^2+3z_s+x_0-10=0$, giving the internal-wave ray return mapThus the next formal bottom reflection is at $(R(x_0),0)$. On $0<x_0<6$, the slope heights are between approximately $0.544$ and $0.946$, consistent with the assumed reflection type.

An internal-wave attractor is a stable periodic ray, so solve $R(x_{\ast })=x_{\ast }$. This gives $z_s=2/3$, and thereforeThe successive vertices areThis is clockwise circulation in the $(x,z)$ plane. Differentiating the internal-wave ray return map showsNeighbouring bottom intersections converge to the periodic orbit, geometrically focusing the energy beam.

For completeness, the opposite upward launch direction has the reversed three-reflection itinerary: slope, top, left. In the interval where this itinerary is valid, putThe three-reflection reverse map is $R^{-1}$, with domain $R(0)<x_0<R(6)$. Its fixed point is the same, but $R_{\mathrm{rev}}^{\prime }(x_{\ast })=41/23>1$. Counterclockwise packets defocus from that orbit. Except for the exactly periodic reverse ray, they eventually leave this itinerary. In the full geometric reflection construction, omitted-side or additional-slope reflections redirect generic rays into the clockwise attracting branch. Tangency and exactly critical reflections are singular exceptions requiring finite-wavelength or dissipative treatment.

The bottom reflection is a formal spatial-ray idealization: part (b) shows that the WKB approximation itself fails near $z=0$, so the diagram does not imply arrival and reflection there in finite WKB travel time. In a real fluid, viscous attenuation of an internal-wave beam, finite wavelength, internal-wave breaking and ensuing turbulence and mixing limit the beam width and energy density. Unlimited focusing is a property of the ideal geometric model, not a prediction for the resolved physical wave.

Put $Z=ϵ\zeta$ and seek the slowly attenuating wave envelope $A(Z)$ inAt order ${ϵ}^0$, the equation gives the internal gravity wave dispersion relationAt order ${ϵ}^1$, retaining one derivative of the slowly varying amplitude givesUsing $\omega =N\sin \theta$ therefore yieldsHence the leading viscous attenuation of an internal-wave beam isThe wave energy density, being quadratic in the amplitude, decays twice as rapidly in the exponent.

The incident internal-wave ray has equation $z=-rx$. Intersecting it with the bottom $z=-H_0+sx$ givesIts distance to the slope is consequentlyThe subcritically reflected ray rises through the same vertical distance at angle $\theta$, so its next free-surface reflection is at

On the first leg, viscous attenuation of an internal-wave beam multiplies the energy density by $e^{-2k\ell /\mathrm{Re}}$. The bottom reflection multiplies it by ${\gamma }^2$ and changes the wavenumber to $\gamma k$. Since the corresponding Reynolds number is $\mathrm{Re}/{\gamma }^2$, attenuation on the second leg contributes $e^{-2{\gamma }^{3}k\ell /\mathrm{Re}}$. Ignoring boundary-layer enhancement as requested,