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Exchange two equal fluid elements of volume $V$ without thermal diffusion in a Boussinesq approximation. Each retains its buoyancy. If the separation is $(\delta y,\delta z)$ and the background buoyancy difference is $\Delta b=B\delta y+N^2\delta z$, the pair's gravitational potential energy changes by ${\rho }_{0}V\delta z\Delta b$. For $B<0$, $N^2>0$ and poleward-rising slope $s=\delta z/\delta y$, this is negative exactly when $0<s<-B/N^2$. The minimum at fixed $\delta y$ is attained at half the isopycnal slope. This energetic test alone is not sufficient for dynamical baroclinic instability: rotation and constraints on parcel motion must also be considered.

Consider a steady, axisymmetric pure plume in a uniform ambient fluid, using the Boussinesq approximation and uniform top-hat cross-sectional profiles. If $r(z)$ is its radius, $U(z)>0$ its vertical velocity and $g^{\prime }(z)>0$ its reduced gravity, define the physical fluxesThey are respectively volumetric flow rate, momentum flux per reference mass density, and buoyancy flux. Only $B$ is conserved in this unstratified, source-free model. Fluid entrainment adds ambient volume to $Q$, and buoyancy increases $M$.

For constant entrainment coefficient $\alpha >0$, the Batchelor entrainment hypothesis gives an inward boundary speed $\alpha U$. Volume, vertical momentum and buoyancy balances therefore giveTo find a pure similarity solution, put $Q\propto z^p$ and $M\propto z^q$ relative to the plume virtual origin. The first balance gives $p-1=q/2$; the second gives $q-1=p-q$. Thus $p=5/3$, $q=4/3$, andIn particular $r=Q/\sqrt{\pi M}$ grows linearly while $U=M/Q$ decreases as $z^{-1/3}$. A finite source shifts the virtual-origin coordinate. Dividing all three fluxes by $\pi$ changes their balance coefficients, so pi-normalized and physical-flux formulas must not be mixed.

Vorticity is generated when density and pressure gradients are not parallel. Under the Boussinesq approximation, the source is $\nabla ({\rho }^{\prime }/{\rho }_0)\times \mathbf{g}$.

The Batchelor entrainment hypothesis takes the inward mean edge velocity of a turbulent plume to be $v_e=\alpha w$ in the Boussinesq approximation. Its density-corrected non-Boussinesq form is $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, where $\rho$ and ${\rho }_0$ are plume and ambient mass densities and $\alpha$ is the entrainment coefficient.

The Boussinesq approximation treats density as constant except where a small density variation multiplies gravity to produce buoyancy. The velocity remains incompressible.

A steady top-hat plume model supplied with pi-suppressed kinematic buoyancy flux $f_0=b^{2}w{g}^{\prime }>0$ but zero source volume flux and momentum flux hasThe full-cross-section buoyancy flux is $\pi f_0$. The Batchelor entrainment hypothesis and momentum balance yield these constants. Its Froude number satisfies $w^2/(g^{\prime }b)=5/(8\alpha )$. The Boussinesq approximation requires $g^{\prime }/g\ll 1$ and therefore fails at the ideal point source itself.

With common factors of $\pi$ removed, steady kinematic fluxes satisfy $Q_z=2\alpha \sqrt{M}$, $M_z=FQ/M$ and $F_z=-N^{2}Q$. The Batchelor entrainment hypothesis supplies volume, the integrated buoyancy force supplies momentum, and the environmental mass density gradient reduces buoyancy flux. This uses the Boussinesq approximation.

With mechanically and thermally equivalent upper and lower boundaries, the Boussinesq approximation permits reflection through the layer midplane together with reversal of the temperature perturbation and vertical velocity. In a reduced convection temperature equation this induces temperature inversion. A quadratic term which changes differently from the linear terms breaks this symmetry; a cubic term can preserve it. The Boussinesq approximation alone does not impose symmetry on unequal boundary conditions.

The spatial gradient of buoyancy determines the buoyancy anomaly produced by a small parcel displacement: $b^{\prime }=-\mathbf{\xi }\cdot \nabla \bar{b}$. Its vertical component is the squared buoyancy frequency in a stably stratified Boussinesq approximation. A horizontal component enters thermal wind and the propagation of a Boundary Rossby wave.

A buoyant thermal is a localized rising volume of fluid whose mass density is lower than its surroundings. It can form the head of a starting plume. A spherical mixed thermal of radius $R$ has volume $V=4\pi R^3/3$ and reduced gravity $g_t=g({\rho }_0-{\rho }_t)/{\rho }_0$ in the Boussinesq approximation.

The volumetric coefficient of thermal expansion is the fractional volume increase per temperature increase at fixed pressure and composition:The mass density form requires fixed mass $m$. Its units are ${\mathrm{K}}^{-1}$; ${\alpha }_T$ is another notation for this coefficient. For small temperature changes $\theta$ at fixed pressure,For a finite change in a single thermodynamic phase, integrate the possibly varying coefficient: $\ln (V_2/V_1)={\int }_{T_1}^{T_2}\alpha (T,P,N)dT$. A constant-coefficient approximation gives an exponential, while the displayed linear formula additionally requires $|\alpha \theta |\ll 1$.

For an ideal gas at fixed particle number,Integrating $1/T$ recovers the exact fixed-pressure thermal expansion ratio $V_2/V_1=T_2/T_1$. Combining expansion with isothermal compressibility gives the local response of any smooth simple compressible system of fixed composition,The ratio has units $\mathrm{Pa}{\mathrm{K}}^{-1}$. For the one-mole ideal gas state in isothermal compressibility, it is $P_0/T_0=333.33\mathrm{Pa}{\mathrm{K}}^{-1}$: a $1\mathrm{K}$ increase at fixed volume raises the pressure by about $333.33\mathrm{Pa}$. Maxwell relations identify this same ratio as the entropy slope with respect to volume and give $C_P-C_V=TV{\alpha }^2/{\kappa }_T$.

Under the Boussinesq approximation, the small fractional mass density change produces buoyancy while mass density is otherwise treated as constant. A positive temperature excess $\theta$ gives reduced gravity $g^{\prime }=g{\alpha }_T\theta$ in the convention used in natural ventilation. The coefficient therefore converts a heat budget into a buoyancy budget. The same measurable expansion response supplies entropy changes and heat capacity differences through Maxwell relations, connecting fluid motion to equilibrium thermodynamics.

The conductive state is the no-flow equilibrium in which heat conduction alone maintains a linear temperature profile across a layer with fixed boundary temperatures. Its hydrostatic pressure follows by integrating $P_b^{\prime }=-g{\rho }_b$, with mass density obtained from the equation of state. Under the Boussinesq approximation, deviations from this state are the variables used in the linear Rayleigh-Bénard convection problem.

A compressible intensification mechanism for a thin, nearly vertical flux tube in a superadiabatic atmosphere. An unstable downflow drains material and lowers internal gas pressure; approximate lateral balance $p_i+B^2/(2{\mu }_0)=p_e$ then permits the tube to contract and its field to strengthen at nearly conserved magnetic flux. It relies on compressibility and thermal evolution, which are absent from a strict Boussinesq approximation. Dynamic photospheric structures need not satisfy thin-tube equilibrium, so this mechanism does not explain every intense field concentration.

The density gradient is the gradient of a spatial mass density field. With an upward vertical coordinate in the Boussinesq approximation, the squared buoyancy frequency is $N^2=-(g/{\rho }_{\ast }){\partial }_z{\rho }_0$. A negative vertical density gradient therefore gives stable density stratification.

For a Boussinesq approximation, constant buoyancy frequency $N$, and buoyancy heating rate $R$, the quasi-geostrophic potential-vorticity equation becomes $D_{g}q/Dt=(f_0/N^2)R_z$, where $q={\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}+\beta y$. Heating changes the vertical gradient of buoyancy and therefore produces a potential-vorticity tendency proportional to its height derivative.

For dry atmospheric thermodynamics, choose potential temperature as the materially conserved scalar in Ertel's theorem. The displayed density-normalized scalar has units $\mathrm{K}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-1}$. Its conservation in adiabatic inviscid motion depends on conservative body forces and a composition-fixed equation of state. It differs in normalization from the Boussinesq approximation convention using buoyancy without the variable density factor. Large-scale balanced weather can be organized by its distribution, with potential-vorticity inversion additionally requiring balance and boundary information.

The local meridional eddy buoyancy flux is the covariance of disturbance meridional velocity with disturbance buoyancy. In a Boussinesq approximation, $b^{\prime }=-g{\rho }^{\prime }/{\rho }_0$. This is a local transport density, distinct from a plume's cross-section-integrated buoyancy flux. In the transformed Eulerian mean, division by the background buoyancy frequency squared gives an eddy-induced meridional overturning streamfunction.

An elevator mode is a normal mode whose fluid motion is purely vertical while its velocity, temperature and composition perturbations are independent of height. Neighbouring columns can move upward and downward without a vertical return-flow structure, hence the name. Such modes describe the fastest-growing disturbances in the ideal local salt fingering problem.^[ref]

Take $z$ upward and write the velocity field and scalar perturbations asThe incompressibility condition is automatic because $\nabla \cdot \mathbf{u}={\partial }_{z}w=0$. More surprisingly, the perturbation's own advection of every one of these fields vanishes:The background need not be uniform: if its temperature and salinity gradients are constants $T_z,S_z$, vertical motion still advects those backgrounds. This supplies the terms $T_{z}w$ and $S_{z}w$ below.

In a nonrotating local model with the Boussinesq approximation, constant kinematic viscosity $\nu$ and constant thermal diffusivity ${\kappa }_T$ and solutal diffusivity ${\kappa }_S$, the ansatz reduces the equations toHere $g>0$ is gravitational acceleration, $\alpha$ the thermal expansion coefficient, $\beta$ the haline contraction coefficient, and ${\nabla }_h^2={\partial }_x^2+{\partial }_y^2$ the horizontal Laplace operator. A constant pressure perturbation is compatible with the absent horizontal motion.

For example, let all three fields be proportional to $e^{\sigma t}\cos (kx)$, with $k>0$. The upward and downward columns alternate with wavelength $2\pi /k$. Their amplitudes satisfy a three-variable linear differential equation; eliminating the scalar amplitudes gives the $q=1$ characteristic polynomial used in the salt-finger instability criterion. Although that criterion is a linear stability result, the vanishing self-advection means that this particular ansatz also solves the corresponding nonlinear local equations exactly within the stated ideal model.

That exactness does not imply that unlimited growth describes a laboratory tank. An impermeable horizontal wall requires the vertical velocity to vanish there, excluding a nonzero height-independent pattern. Other perturbations can also grow on the rising and falling columns and disrupt them. Elevator modes are therefore useful ideal building blocks, with boundary conditions and additional disturbances determining how faithfully a real flow follows them.

Ertel potential vorticity is the scalar product of absolute vorticity with the gradient of a materially conserved stratifying scalar. In Boussinesq flow with buoyancy $b$,It is materially conserved by inviscid, adiabatic motion.

For nonrotating convection under the Boussinesq approximation with fixed-temperature impermeable stress-free boundary conditions, a marginal vertical mode $\sin (n\pi z)$ has neutral Rayleigh number $(k^2+n^2{\pi }^2{)}^3/k^2$. The minimum is at $n=1$, $k=\pi /\sqrt{2}$, and $R_c=27{\pi }^4/4$. These follow by eliminating the temperature from $(D^2-k^2{)}^{2}w=R{k}^2\theta$ and $(D^2-k^2)\theta =-w$, then minimizing over $k$.