Boundary-forced internal gravity wave

A moving boundary excites an internal gravity wave through its kinematic boundary condition. For wavenumber $k$ and angular frequency $\omega$, the vertical velocity field satisfies $W^{\prime \prime }+k^2(N^2/{\omega }^2-1)W=0$. The radiation condition selects outward group velocity; when $\omega >N$, the bounded response is an evanescent wave.

For a plane internal gravity wave in uniform buoyancy frequency, substitute $\zeta =A(\mu t,\mu z)e^{i(kx+mz-\omega t)}$ into the displacement equation. The leading order gives ${\omega }^2=N^2{k}^2/(k^2+m^2)$; the next order gives the displayed transport equation with $c_{gz}=-\omega m/(k^2+m^2)$. A boundary amplitude $a(T)$ radiates as $A=a(T-Z/c_{gz})$, choosing the sign of $m$ so the group velocity points into the fluid. The wavelength-averaged energy is $N^2|A{|}^2/2$ and its vertical flux is $c_{gz}$ times that energy. A causal ramp therefore creates a translating front, not an instantaneous monochromatic field throughout the half-space.

A right-going, boundary-forced internal gravity wave has horizontal momentum per unit inertial density $P=(k/\omega )\overline{E}$. Its envelope-induced Eulerian mean acceleration is ${\overline{u}}_t=P_t$ to leading slow order. In the frame moving at horizontal phase velocity $c=\omega /k$, the change of mean kinetic energy relative to the initial current $-c$ is $-{\rho }_{0}c\int \overline{u}dz$ at quadratic wave order. Its time derivative is minus the boundary work $\overline{W}={\rho }_{0}d\int \overline{E}dz/dt$. The lab-frame mean kinetic energy alone is fourth order in wave amplitude; retaining the moving-frame cross term is essential.

For a nonrotating uniform-stratification fluid under the Boussinesq approximation driven at $0<\omega <N$ by a localized vertical-wall velocity, positive vertical wavenumbers radiate down and right and negative ones up and right. The real velocity separates into functions of $Z_{+}$ and $Z_{-}$. Its in-phase component is half the imposed boundary profile on each beam; the quadrature component is its Hilbert transform partner. Thus compact localization at maximum boundary speed does not imply compact localization of all components at every phase. A finite-duration source contains several frequencies and spreads energy over their different ray angles and group speeds.

A fixed terrain obstacle forces a zero laboratory-frequency internal gravity wave through the kinematic boundary condition $w=U{h}_x$. Its intrinsic frequency is $-kU$. Propagation, radiation and the transient arrival of its permanent component are distinct questions.

Define intrinsic positive frequency $\omega$ and ground frequency $\hat{\omega }=Uk+\omega$ for a nonrotating internal gravity wave. Stationarity with $U>0$ selects $k<0$ and $\sqrt{k^2+m^2}=N/U$. Differentiating the full dispersion before imposing that condition gives the displayed ground-frame group velocity. Upward energy propagation requires $m<0$ and $|k|<N/U$. Its ray inclination is ${\cos }^{-1}(U|k|/N)$. The ground-frame group velocity is opposite the normal intrinsic phase velocity; the intrinsic group velocity is perpendicular to that phase velocity. A strictly stationary ground-frame phase pattern has zero phase velocity, so mixing frames can produce a false angle claim.

For a sinusoidal small-amplitude bed, constant current $U>0$ and buoyancy frequency $N$, put $m=\sqrt{N^2/U^2-k^2}>0$. The linear vertical velocity amplitude is $W(z)=ikU{\eta }_0\sin [m(H-z)]/\sin (mH)$. The upward-wave coefficient is $A_{+}=-kU{\eta }_0{e}^{-imH}/[2\sin (mH)]$. At $mH=n\pi$, the prescribed nonzero bed motion and homogeneous rigid-lid condition are incompatible with a bounded stationary inviscid solution. With angle $\theta$ between intrinsic group velocity and vertical, $\tan \theta =m/k$, giving $kH\tan \theta =n\pi$.

For a localized terrain source in uniform flow, the stationary phase method applied to $kx+m(k)z$ gives phase $\Phi =(N/U)\sqrt{x^2+z^2}$ on the downstream branch. Permanent phase lines are circular arcs, whereas an individual monochromatic component has straight phase lines. Reflected arcs use an image source; transmitted phase is continuous across the velocity interface.

At a step in buoyancy frequency without a jump in background mass density, vertical velocity and fluid pressure are continuous. For a fixed horizontal wavenumber and angular frequency, these are continuity of $W$ and $W^{\prime }$. They determine reflection and transmission between propagating internal gravity waves and evanescent waves.