Internal-wave ray in uniform vertical shear ($m(t)=m_0-kst,z(t)=(\omega -\hat{\omega }(t))/(ks)$)

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.