 Solution

Equations and conventions. Let ${\rho }_0$ be a constant reference density and define total buoyancy acceleration by $\sigma =-g(\rho -{\rho }_0)/{\rho }_0$. With reduced pressure $\Pi =p/{\rho }_0+gz$, the nonrotating ideal Boussinesq equations areDensity departures are small compared with ${\rho }_0$, so they can be neglected in inertia and mass continuity but retained in the gravitational force. The reference stratification must also change density only weakly over the depth of interest. incompressible flow is already assumed; the Boussinesq approximation additionally restricts the size of the density variations, rather than following automatically from incompressible flow. An ideal fluid here has neither viscosity nor buoyancy diffusion.

A stationary reference profile $\bar{\sigma }(z)$ has buoyancy frequencyStable stratification has $N^2>0$: an upward-displaced parcel retains its old buoyancy and has negative buoyancy relative to its new surroundings.

Vorticity equation and linearization. For motion in the $(x,z)$ plane, choose the streamfunction by $u={\psi }_z$, $w=-{\psi }_x$. The $y$-component of vorticity is then $u_z-w_x={\nabla }^2\psi$, where ${\nabla }^2={\partial }_x^2+{\partial }_z^2$. Taking the $y$-component of the curl of momentum givesThe minus sign in the buoyancy curl follows from $(\nabla \times \sigma \hat{z}{)}_y=-{\sigma }_x$.

Write $\psi =\bar{\psi }(z)+{\psi }^{\prime }$ with ${\bar{\psi }}_z=\bar{u}(z)$ and $\sigma =\bar{\sigma }(z)+{\sigma }^{\prime }$. The linear material derivative is $D_t={\partial }_t+\bar{u}{\partial }_x$. The perturbation equations areThe shear-curvature term arises because $w^{\prime }$ advects the basic vorticity ${\bar{u}}_z$. Apply $D_t$ to the first equation and use the second; $D_t$ commutes with ${\partial }_x$ and does not differentiate the height-only coefficients. Therefore

For a stationary nonzero horizontal wavenumber $k$, the buoyancy equation gives $\hat{\sigma }=N^2\hat{\psi }/\bar{u}$. Substitution into the vorticity equation givesHere ${\ell }^2$ is the Scorer parameter. Where $m^2>0$, the vertical structure is oscillatory; where $m^2<0$, it is evanescent. A rigid flat lower boundary gives $\hat{\psi }(0)=0$ for $k\ne 0$. In an upper evanescent region the growing solution is excluded by boundedness. When propagation persists to infinity, an appropriate radiation condition distinguishes outgoing from incoming energy for a forced problem.

If $\bar{u}=U_0+Sz$ with $S>0$, its curvature term vanishes. Bounded $N^2$ implies ${\ell }^2=N^2/{\bar{u}}^2\to 0$ aloft. Thus every fixed nonzero $k$ eventually lies in an evanescent region. If stratification is sufficiently strong near the bottom, a lower oscillatory region is separated from the upper evanescent region by a turning level of a stationary internal wave. Waves reflect at that level and at the rigid lower boundary, giving vertically trapped modes for the horizontal wavenumbers that satisfy both conditions. In a slowly varying example the phase condition is approximately ${\int }_0^{z_t}mdz=(n-1/4)\pi$, from a lower node and the turning-point connection; the key point is trapping by the decrease in the Scorer parameter, not a physical lid.

Rigid channel modes. With constant $U=\bar{u}>0$ and $N$, the vertical nodes require $m_n=n\pi /H$, $n=1,2,\dots$. Stationarity then requiresThe specified strict interval admits positive $k_n^2$ for exactly $n=1,2$. Choosing $k_1,k_2>0$, every real superposition of these nonzero-wavenumber modes isThe two independent complex constants encode the amplitudes and horizontal phases; negative wavenumbers are supplied by complex conjugation, not additional independent constants.

Exact finite amplitude. Put $\lambda =N^2/U^2$. Both modes and every superposition satisfy ${\nabla }^2{\psi }^{\prime }=-\lambda {\psi }^{\prime }$. For the total streamfunction $\Psi =Uz+{\psi }^{\prime }$, the total buoyancy is $\sigma =N^{2}z+(N^2/U){\psi }^{\prime }=(N^2/U)\Psi$. A stationary velocity $({\Psi }_z,-{\Psi }_x)$ is tangent to its streamfunction contours, so $D\sigma /Dt=0$ exactly. Moreover the nonlinear self-advection of ${\nabla }^2{\psi }^{\prime }$ vanishes because it is proportional to ${\psi }^{\prime }$ itself. The remaining vorticity equation iswhich is an identity. Both rigid boundaries remain impermeable, and the curl-free residual in momentum determines a pressure. Thus the displayed two-mode family consists of exact finite-amplitude Boussinesq solutions, with no small-amplitude restriction on the constants in these equations. Very large amplitudes can overturn the stratification or leave the physical Boussinesq regime; exact existence does not establish stability.

The two-constant statement concerns wave disturbances with $k\ne 0$. If $k=0$ is literally included, arbitrary stationary horizontal shear adjustments ${\psi }^{\prime }=F(z)$ and height-only buoyancy changes are also possible. They are changes of the basic state and are not constrained by the wave equation obtained after division by $k$. Excluding that mean-flow sector is necessary for the intended count.