= Solution
\b[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 gives
$$
w_t=\frac{k_h^2}{k_h^2+m^2}b,\qquad b_t=-N^2w,
$$
so
$$
\boxed{\omega^2=N^2\frac{k_h^2}{k_h^2+m^2}.}
$$
This <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.
\b[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 gives
$$
c_{gx}=\frac{Nm^2}{(k^2+m^2)^{3/2}},\qquad
c_{gz}=-\frac{Nkm}{(k^2+m^2)^{3/2}}.
$$
The 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 is
$$
\mathcal E=\frac12|\mathbf u|^2+\frac{b^2}{2N^2}.
$$
Dot 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.
\b[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.
\b[How rotation changes the picture.] With the vertical Coriolis parameter $f$, the corresponding uniform-background inertia–gravity dispersion becomes
$$
\omega^2=\frac{N^2k_h^2+f^2m^2}{k_h^2+m^2}.
$$
When $N>|f|$, 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.
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