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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.

Stable stratification is a constraint on motion, not an isolated parcel spring. A light-over-heavy density profile has $N^2=-(g/{\rho }_0){\bar{\rho }}_z>0$. An adiabatically displaced parcel has buoyancy anomaly $b=-N^2{\xi }_z$ relative to its surroundings. If one neglects the pressure response and movement of surrounding fluid, the vertical equation would be ${\ddot{\xi }}_z=-N^2{\xi }_z$. A finite spherical blob cannot satisfy that isolated-oscillator picture: incompressible flow requires surrounding fluid motion, and the pressure perturbation communicates its acceleration to the rest of the fluid.

This can be shown directly with the linear Boussinesq equations. For a uniform background, take a Fourier wavevector $({\mathbf{k}}_h,m)$. Pressure removes the component of the buoyancy force parallel to this wavevector. Taking the divergence of momentum to determine pressure and projecting the vertical component givessoThis pressure projection in a displaced stratified blob produces frequencies less than $N$ except for wavevectors with $m=0$. A localized spherical perturbation contains a range of directions and therefore a range of frequencies. It deforms and emits internal gravity waves, rather than remaining one sphere oscillating coherently at $N$. The added mass of a rigid sphere is a useful inertia analogy, but its numerical correction does not define an exact oscillation frequency for a deformable material blob in stratified fluid.

Internal waves and energy transport. The restoring force supports anisotropic internal gravity waves: their frequency depends on the direction, not just the length, of the wavevector. For a two-dimensional positive-frequency branch, $\omega =Nk/\sqrt{k^2+m^2}$ with $k>0$. Differentiation givesThe vertical group and phase velocities have opposite signs. Wave energy therefore travels along a direction different from the direction of phase propagation, a point essential in selecting radiation conditions for wave-generating obstacles.

For constant $N$, the linear perturbation energy density per reference mass isDot momentum with velocity and multiply buoyancy evolution by $b/N^2$; the $bw$ terms cancel, leaving ${\partial }_t\mathcal{E}+\nabla \cdot (\Pi \mathbf{u})=0$. The second term is the available potential energy stored by displacement against the stable density profile. In an ideal fluid, that energy is exchanged with kinetic energy or transported away, not irreversibly dissipated by buoyancy diffusion.

Steady flows, layers and topography. Stable stratification penalizes vertical displacement and favors flows with small vertical motion when the stratified Froude number $U/(NH)$ is small. The natural vertical displacement scale associated with kinetic energy $U^2$ is of order $U/N$; obstacles requiring much larger displacements can cause blocking and diversion instead of simple overpassing. In the ideal model fluid retains its buoyancy label, so motion tends to follow density surfaces. This inhibits vertical mixing but does not forbid horizontal rearrangement or turbulence powered by a shear flow.

A uniform current crossing topography generates stationary internal waves when its intrinsic frequency lies in the internal-wave range. For horizontal wavenumber $k$, stationarity gives $m^2=N^2/U^2-k^2$. Height-dependent wind and stratification replace this by the Scorer parameter relation derived in Question 1. A decrease of that parameter can produce a turning level, evanescence and vertical trapping, explaining lee-wave trains confined near a lower boundary. The exact channel superpositions in Question 1 also show that nonlinearity does not automatically destroy every wave solution.

Stability of the background density profile does not imply stability of every flow built on it. Sufficiently large waves can overturn density surfaces, and shear can supply energy to disturbances. Ideal advection can create fine structure, but irreversible homogenization of buoyancy in a real fluid additionally involves diffusion or dissipation. A finite-amplitude ideal solution is consequently distinct from a guaranteed stable or physically realizable state.

How rotation changes the picture. With the vertical Coriolis parameter $f$, the corresponding uniform-background inertia–gravity dispersion becomesWhen $N>|f|$, the propagating branch lies between $|f|$ and $N$. Rotation also allows a slow balanced sector: horizontal pressure and Coriolis forces balance, vertical pressure and buoyancy balance, and the circulation can be described through potential-vorticity inversion. Small Rossby number and small stratified Froude number lead to the quasi-geostrophic equations rather than to the fast internal-wave dynamics. Vertical shear is then tied to horizontal buoyancy gradients through thermal-wind balance.

In this balanced sector, a planetary or topographic PV gradient supports Rossby waves, whose restoring mechanism is conservation of PV under displacement across that gradient. Stable stratification couples horizontal levels through the ${\partial }_z[(f_0^2/N^2){\psi }_z]$ term, and the mountain-wave calculation in Question 3 illustrates its vertical propagation filter. Thus stratification can support rapid buoyancy-restored waves, constrain slowly evolving balanced circulation, and shape interactions with boundaries; which description applies depends on the velocity, length, time and rotation scales.

Take $U_0,N_0,k>0$ and assume an initially propagating mode, $k<N_0/U_0$. Put $s=N_0/U_0$ and $\delta (z)=1+\gamma z$. Neglecting $U^{\prime \prime }/U$ as stipulated, the Scorer parameter and stationary internal-wave WKB solution areThe positive root $m>0$ selects upward group velocity for the negative intrinsic frequency branch $\hat{\omega }=-kU$. The WKB approximation needs a slowly varying background relative to the local vertical wavelength, in particular $|m^{\prime }|\ll m^2$ and $|m^{\prime \prime }|\ll |m{|}^3$, away from a turning level or singular mean profile.

The local dispersion relation is $\omega =kU-Nk/(k^2+m^2{)}^{1/2}=0$. Its intrinsic and laboratory group velocities are, with $K=(k^2+m^2{)}^{1/2}$,Thus intrinsic energy propagation is upstream and upward, while laboratory energy propagation is downstream and upward, parallel to $(k,m)$. Constant-phase lines of an internal gravity wave have slope $dz/dx=-k/m$ and normal spacing $2\pi /K$; the horizontal spacing remains $2\pi /k$.

If $\gamma <0$, $m$ grows with height. The wavevector and laboratory energy ray become more vertical; phase lines become flatter and closer together. For finite nonzero $N$, the heightis a critical level of an internal gravity wave: $U\to 0$ and the vertical wavenumber diverges. The laboratory group velocity tends to zero. Importantly, $|m^{\prime }|/m^2\to |\gamma |/s$, so the formal WKB condition need not deteriorate before this level if that ratio is small. Nevertheless the stationary equation is singular at the level, the phase has no finite limit, and the horizontal perturbation velocity grows as $m^{1/2}$; the inviscid linear wave cannot be continued uniformly through it without additional physics.

If $\gamma >0$, $m$ decreases to zero atThe wavevector and laboratory ray become horizontal, the phase lines become vertical, and the normal spacing increases. Above $z_t$, $m^2<0$ and the disturbance is an evanescent wave. Here $|m^{\prime }|/m^2$ diverges, so the WKB approximation fails in a turning region and must be replaced by a local connection solution.

For the normal modethe linear material derivative becomesSubstitution into the equation from part b and division by $-k^2(\overline{u}-c{)}^2$ gives the Taylor–Goldstein equationwhereand the Scorer parameter is

For a ridge-fixed disturbance, $c=0$. If the Scorer parameter varies on a height scale much longer than the vertical wavelength, the local WKB approximation giveswhere $m^2=l^2-k^2>0$. The disturbance is vertically oscillatory there and can carry wave activity upward. Where $l^2<k^2$, $m$ is imaginary and the solution is vertically evanescent.

A rigid ridge supplies the lower boundary condition and an upper radiation or decay condition selects the physical solution. If $l^2$ decreases through $k^2$, the crossing is a turning level: the wave is reflected or decays above it and is trapped beneath it. Increasing $\overline{u}$ normally reduces $N^2/{\overline{u}}^2$, while decreasing $N^2$ reduces it directly; subject to the curvature term $-{\overline{u}}_{zz}/\overline{u}$, either change therefore promotes vertical trapping of an atmospheric gravity wave.

An atmospheric internal gravity wave is vertically trapped when its squared vertical wavenumber changes from positive to negative with altitude. A decrease of the Scorer parameter, caused for example by increasing wind speed or decreasing buoyancy frequency, creates a turning level and an evanescent upper region.