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Begin with the Boussinesq approximation primitive equations on a beta plane, decompose every field into a zonal mean and a disturbance, and average over longitude. The zonal momentum equation then contains the divergence of the eddy momentum flux $\overline{u^{\prime }{v}^{\prime }}$, while the mean density equation contains the divergence of the eddy density flux $\overline{{\rho }^{\prime }{v}^{\prime }}$. At small Rossby number, use geostrophic balance, hydrostatic pressure, thermal-wind balance, and the leading eddy equations to combine those fluxes.

Defineand introduce the residual mean circulationThe added eddy-induced velocity is nondivergent, soIt absorbs the eddy density-flux divergence into advection by the transformed circulation, giving

The mean zonal momentum equation becomeswhere the zonally averaged Eliassen–Palm flux in the meridional-vertical plane isThus the transformed Eulerian mean gathers the wave forcing into one flux divergence and makes density evolve under one residual circulation.

Write the density as a hydrostatic reference profile $\hat{\rho }(z)$ plus a small perturbation ${\rho }^{\prime }$, and define buoyancy and buoyancy frequency byThe Boussinesq approximation to the Navier-Stokes equation, together with mass conservation, iswhere $D/Dt$ is the material derivative, $\nu$ is kinematic viscosity, and $\kappa$ is mass diffusivity. Linearizing about rest and eliminating pressure and buoyancy givesFor a plane wave proportional to $e^{i(kx+mz-\omega t)}$, with $K^2=k^2+m^2$, the viscous-diffusive dispersion relation isIn the inviscid limit this becomesFor weak diffusion the two oscillatory roots areso a freely evolving Fourier mode decays.

A single plane wave is also an exact solution of the nonlinear equations. Every field depends only on its phase $\Theta =\mathbf{k}\cdot \mathbf{x}-\omega t$, while incompressible flow gives $\mathbf{u}\cdot \mathbf{k}=0$. Therefore $\mathbf{u}\cdot \nabla$ annihilates both $\mathbf{u}(\Theta )$ and $b(\Theta )$, and all nonlinear advection terms vanish.

For a boundary-forced wave with real $\omega$, nonzero $\nu$ or $\kappa$ instead makes the bulk vertical wavenumber complex, attenuating the propagating beam. Because diffusion raises the spatial order of the equations, additional short vertical-wavenumber roots form viscous and scalar boundary layers; they allow a no-slip velocity condition and a scalar no-flux condition to accompany impermeability. These layers and bulk attenuation become essential near critical internal-wave reflection, where the inviscid reflected wavelength collapses.

The Boussinesq approximation replaces density by a common reference value ${\rho }_0$ in inertia, mass conservation, and pressure acceleration, while retaining small density differences in the gravitational buoyancy term. It gives incompressible flow and is appropriate here wheneven though those small differences drive the room-scale motion. It would fail for order-one thermal density contrasts or strongly compressible ventilation.

The Batchelor entrainment hypothesis sets the mean inflow speed across a turbulent plume edge to $\alpha$ times a representative axial plume speed, where $\alpha$ is the entrainment coefficient. It closes integral plume balances by relating plume growth to its speed. Applied here, it produces an entraining axisymmetric warm plume above the floor source and a one-sided cold wall line plume below the vent. Treating both as turbulent top-hat plume models neglects source regions, detailed profiles, wall friction, interaction between the two plumes, and the finite thickness of the density interface; these are the principal modelling assumptions.

The quasi-geostrophic approximation requires small Rossby number, nearly horizontal geostrophic balance, hydrostatic vertical balance, small interface or density displacements, stable background stratification, and horizontal scales much larger than the vertical scale. The Boussinesq approximation and a beta plane are used, ageostrophic motion enters only at the order needed to evolve potential vorticity, and here the buoyancy frequency $N$ is constant. Under these assumptions the materially conserved three-dimensional quasi-geostrophic potential vorticity is

Let the velocity perturbation be $(u,w)$, pressure perturbation be $p$, and density perturbation be ${\rho }^{\prime }$ about the hydrostatic background $\hat{\rho }(z)$. The linearized inviscid Boussinesq approximation givesStable stratification means ${\hat{\rho }}_z<0$, and the buoyancy frequency isDifferentiate the momentum equations to eliminate $p$, use incompressibility, and then use the density equation to eliminate ${\rho }^{\prime }$. This yieldsFor $w\propto e^{i(kx+mz-\omega t)}$, the internal gravity wave dispersion relation is ${\omega }^2=N^2{k}^2/(k^2+m^2)$.

The Boussinesq approximation uses a constant reference density in inertia and continuity while retaining the small temperature-dependent density deficit in buoyancy. Here $({\rho }_0-\rho )/{\rho }_0=(T-T_0)/T_0$, so the plume remains incompressible but rises under $g(T-T_0)/T_0$.

The Batchelor entrainment hypothesis sets the ambient inflow speed at the exposed plume edge to $EW$, where $E$ is a dimensionless entrainment coefficient. For this one-sided heated wall plume, it gives $dQ/dz=EW$.

The variables $u,v,w$ are eastward, northward, and upward velocity, $\varphi =p^{\prime }/{\rho }_0$ is the pressure perturbation divided by reference density, $\sigma$ is buoyancy, $N$ is the constant buoyancy frequency, and $\beta y$ is the equatorial approximation to the Coriolis parameter. The equations express, respectively:

Together they are the hydrostatic Boussinesq approximation for long equatorial waves.

A monochromatic internal gravity wave in uniform stratification has a spatially linear phase. In the nonrotating inviscid Boussinesq approximation, its dispersion relation is ${\omega }^2=N^2|{\mathbf{k}}_h{|}^2/(|{\mathbf{k}}_h{|}^2+m^2)$. The wave vector is normal to its phase planes, while the group velocity lies along them.

In the Boussinesq approximation, a spherical buoyant thermal with conserved total buoyancy $B_0$ obeys $\dot{b}=\alpha u$ and $d(Vu)/dt=B_0$, with $V=4\pi b^3/3$. A zero-volume, zero-impulse point source gives $b=\alpha z$, $K{z}^3\dot{z}=B_{0}t$, where $K=4\pi {\alpha }^3/3$. Hence $z^4=2B_0{t}^2/K$ and $u=\sqrt{B_0/(2K)}/z$. The zero-time singularity represents an ideal source limit, not a finite initially regular sphere.

Ignoring end boundaries, the buoyancy-driven turbulent density diffusion model admits $\bar{\rho }={\rho }_0[1+A(t)C{z}^5]$ for $C>0$. Since $({\partial }_z\bar{\rho }{)}^{3/2}\propto A^{3/2}{z}^6$, the amplitude obeys $A^{\prime }=65^{3/2}K{W}^2\sqrt{gC}{A}^{3/2}$. With $A(0)=1$, $A=(1-t/t_{\ast }{)}^{-2}$ and $t_{\ast }=[35^{3/2}K{W}^2\sqrt{gC}{]}^{-1}$. Its formal growth is not a physical finite-time singularity of a closed mixing tube: ignored end fluxes, loss of the Boussinesq approximation and the finite energy budget invalidate the continuation.

In the constant-property convection convention, the strong Boussinesq approximation supplements small relative mass density changes with constant dynamic viscosity, thermal diffusivity, specific heat capacity and thermal expansion coefficient throughout the modeled temperature interval. A uniform reference density replaces density in inertia and mass conservation, while a linear density anomaly is retained in gravitational buoyancy. The temperature obeys an advection-diffusion equation with constant coefficients. The term “strong” is not uniform across all fluid-mechanics literature: some stratified-flow treatments instead emphasize the use of a globally constant reference density. The constant-property convection usage is explained in courses.physics.ucsd.edu/2021/Spring/physics218c/Spiegel%20Convection.pdf .

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.

In the Boussinesq approximation, density-weighted fluxes are $Q={\rho }_{0}bW$, $M=(2/3){\rho }_{0}b{W}^2$ and $F=(2/3)gbW({\rho }_0-\rho )$. With two-sided Batchelor entrainment $e=\alpha W$, the integral balances areThe factors $4/3$ and $3$ depend on the triangular profile and the definition of the entrainment coefficient.

In the Boussinesq approximation, downward excess cell weight gives $\mu w^{\prime \prime }=G+g^{\prime }n(x)$. The no-slip boundary condition at both walls and zero total fluid flux determine the excess axial pressure gradient $G$, with $b=a\beta$. In particular, $w=C[e^{\beta x}-\cosh b-(x/a)\sinh b]+G(x^2-a^2)/(2\mu )$, $C=g^{\prime }{n}_{0}a\beta /(\mu {\beta }^2\sinh b)$. For large $b$, writing $\xi =x/a$ and $W=g^{\prime }{n}_0{a}^2/(\mu b)$ gives the outer profile $w/W=(1+\xi )(1-3\xi )/2$. A thin layer restores no slip at the illuminated wall. Cells descend there and fluid returns upward on the other side. At $b=0$ the limiting solution is $G=-g^{\prime }{n}_0$, $w=0$.