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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.

Raffaele Marino is a physicist whose research includes turbulence in space plasmas and fluids with density stratification. A first-authored study investigated intense vertical drafts using direct numerical simulation of equations with the Boussinesq approximation. In some simulated regimes, roughly $10\%$ of the volume accounted for as much as $50\%$ of the total dissipation.^[ref]

These figures concern the modeled flows, rather than a universal measured fraction of the ocean. They illustrate why the average can hide strong spatial concentration. Dividing the two fractions gives $0.50/0.10=5$: the active region's mean dissipation density is five times the whole-volume mean. The remaining $90\%$ supplies the other half, so its mean is only $0.50/0.90$ times the whole-volume mean. The contrast between the two regions is consequently ninefold.

The result gives an illuminating connection between intermittency and physical mixing: a relatively small region can dominate energy conversion. The simulations test a proposed mechanism for variability; identifying the same mechanism in a particular environment also requires observations of that environment.

In an ideal unbounded fluid governed by the Boussinesq approximation, take uniform background temperature and salinity gradients $T_z,S_z>0$, with $z$ upward. Let $\alpha ,\beta >0$ be the thermal expansion coefficient and haline contraction coefficient, and let the thermal diffusivity ${\kappa }_T$ exceed the solutal diffusivity ${\kappa }_S>0$. With the conventiona strictly stable total mass density gradient has $R_{\rho }>1$. The band admitting exponentially growing salt fingering modes isThe upper equality is the marginal long-wavelength limit. This local linear stability result assumes no rotation and constant positive kinematic viscosity; finite boundaries can restrict the available wavelengths.^[ref]

A dimensional derivation makes the signs explicit. Put $A=g\alpha T_z$ and $B=g\beta S_z$, where $g>0$ is the gravitational acceleration. Consider a normal mode with growth rate $\sigma$, squared wavenumber $K=|\mathbf{k}{|}^2>0$ and horizontal fraction $q=k_h^2/|\mathbf{k}{|}^2$, where $\mathbf{k}$ is the wave vector and $k_h$ its horizontal magnitude. Eliminating the temperature, salinity and pressure perturbations from the linearized equations gives the characteristic polynomialwhere $\nu >0$ is the kinematic viscosity.

For an elevator mode, $q=1$ and the perturbation is independent of $z$. Its constant term isIf $R_{\rho }=A/B<{\kappa }_T/{\kappa }_S$, choose a sufficiently small positive $K$ to make $P(0)<0$. Since $P(\sigma )$ tends to positive infinity along the positive real axis, the intermediate value theorem supplies a positive real growth rate.

For the converse, write $P(\sigma )={\sigma }^3+a_1{\sigma }^2+a_2\sigma +a_3$. When $R_{\rho }\ge {\kappa }_T/{\kappa }_S$, all three coefficients are positive, andThe Routh-Hurwitz criterion therefore puts every root in the left half-plane. This covers oblique modes too, since $0\le q\le 1$. If $R_{\rho }<1$, ordinary density-driven overturning is already available; that is a different regime from fingering within a stably stratified background.

A stratified incompressible shearing sheet retains vertical buoyancy while treating mass density as constant in inertial terms, as in the Boussinesq approximation. Let $\theta$ be a vertical displacement, with $D\theta /Dt=u_z$, so stable stratification supplies acceleration $-N^2\theta {\mathbf{e}}_z$ for positive squared buoyancy frequency $N^2$.

For uniform local coefficients, no vertical shear, and axisymmetric plane waves with wavevector $(k_x,0,k_z)\ne 0$, the linear incompressible equations giveHere $q$ is the orbital shear parameter and ${\kappa }_r$ the radial epicyclic frequency. In a Keplerian shearing sheet, $q=3/2$ and ${\kappa }_r^2={\Omega }^2$, recovering the previously displayed form. The dispersion relation interpolates between epicyclic restoration for a vertical wavevector and buoyancy restoration for a radial wavevector. If both squared restoring frequencies are nonnegative, this branch has no exponential growth; negative $N^2$ permits convective orientations.

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 $G=-g^{\prime }{n}_0$, $w=0$.