Thermal convection

Thermal convection is fluid motion driven by temperature-dependent buoyancy. It transports heat by bulk motion in addition to thermal conduction.

Horizontal convection is buoyancy-driven circulation forced by lateral variations of temperature along one horizontal boundary. Hot and cold surface regions drive a circulation even though all thermal forcing is applied at the same height. An insulated bottom and sides require zero net heat exchange through the remaining surface in steady state, but permit heat to enter its warmer region and leave its colder region.

For downward depth $z=ϵy$, take $(u,w)=({\psi }_z,-{\psi }_x)$, surface temperature $T=x$, a flat impermeable stress-free upper boundary and a no-slip insulated bottom. In the steady infinite-Prandtl number thin-layer interior, ${\psi }_{zzzz}=-\mathrm{Ra}$ with $\psi ={\psi }_{zz}=0$ at the surface and $\psi ={\psi }_z=0$ at the bottom. The displayed profile follows. Sidewall end layers close the circulation; the interior profile must not be imposed unchanged at those walls.

The interior first temperature correction obeys ${\Theta }_{zz}=u={\psi }_z$, $\Theta (0)=0$ and ${\Theta }_z(ϵ)=0$. Hence ${\Theta }_z=\psi$, which integrates to the displayed polynomial. Its contribution to depth-integrated horizontal advective heat flux is $\int u\Theta dz=-\int {\psi }^{2}dz=-19{\mathrm{Ra}}^2{ϵ}^9/1451520$. The leading conductive flux is $-ϵ$. These statements apply away from the sidewall end layers at fixed Rayleigh number as the layer becomes shallow.

Heat-carrying fluid motion dominated by an externally imposed flow or shear. Buoyancy may still influence it; forced and free contributions can coexist. In surface-layer scaling, $z\ll |{\ell }_O|$ is predominantly shear-driven, while $z\gg |{\ell }_O|$ is predominantly buoyancy-driven for lower-boundary heating.

Convection driven primarily by buoyancy generated by heating or cooling, rather than an externally imposed mean flow. In a constant-flux unstable surface layer, the local speed scales as $(B_{0}z{)}^{1/3}$.

For a positive approximately constant vertical turbulent buoyancy flux, dimensional turnover scaling gives $K_b\sim B_0^{1/3}{z}^{4/3}$ and $N^2\sim -B_0^{2/3}{z}^{-4/3}$. The negative squared frequency describes unstable overturning, not a real restoring oscillation. The leading dissipation is height-independent, unlike neutral wall turbulence.

A finite-wavenumber convection instability with a resonant three-wave triad can yield three coupled amplitude equations. For the gradient nonlinearity $\nabla \cdot (|\nabla \Theta {|}^2\nabla \Theta )$ and weak quadratic asymmetry $-ϵ\gamma \nabla \cdot (\Theta \nabla \Theta )$, projection onto a triad of length squared $K$ gives the stated equation and its cyclic copies. The quadratic coefficient is $+\gamma K$, while both self and cross cubic coefficients are $-3K^2$ for angles of $120$ degrees. The finite critical mode requires positive linear damping in the underlying long-wave equation.

For a resonant three-wave triad with rotational and translational symmetry but no spatial reflection, the cubic amplitude equations have the cyclic form $\dot{A}=\mu A+\overline{B}\overline{C}-{\nu }_1|A{|}^{2}A-({\nu }_2+\delta )|B{|}^{2}A-({\nu }_2-\delta )|C{|}^{2}A$. A half-turn induces complex conjugation, so all coefficients are real. The coefficient $\delta$ distinguishes the two cyclic cross-couplings that a reflection would interchange.

For the rotational hexagon amplitude equations, put $d={\nu }_2-{\nu }_1$ and $N={\nu }_1+2{\nu }_2$. A roll with $r^2=\mu /{\nu }_1$ has transverse eigenvalues $-d{r}^2\pm \sqrt{{\delta }^2{r}^4+r^2}$, each twice, radial eigenvalue $-2\mu$, and one neutral translation. A hexagon with $\mu =N{h}^2-h$ has real-amplitude eigenvalues $h-2N{h}^2$ and $-2h+2d{h}^2\pm 2i\sqrt{3}\delta h^2$; its imaginary-amplitude eigenvalues are $-3h,0,0$. These follow by diagonalizing the cyclic Jacobian matrix. Stability is understood modulo translations. If ${\nu }_2>{\nu }_1>0$ and $\delta \ne 0$, the upper positive hexagon branch reaches a Hopf bifurcation at $h=1/d$, $\mu =(2{\nu }_1+{\nu }_2)/d^2$.

Write the translation group action as $(A,B,C)\mapsto (e^{i{\theta }_1}A,e^{i{\theta }_2}B,e^{-i({\theta }_1+{\theta }_2)}C)$. Modulo lattice translations, the stabilizer subgroup of $(h,h,h)$, with nonzero real $h$, is the sixfold rotation group. Its fixed-point subspace of a group action is the real line $(a,a,a)$. The stabilizer subgroup of $(r,0,0)$, with nonzero real $r$, consists of the circle ${\theta }_1=0$ together with a half-turn; its fixed-point subspace of a group action is the real line $(a,0,0)$. The equivariant branching lemma therefore supplies generic branches of both axial types.

In the hexagonal convection amplitude equations, a pure convection roll has $|A{|}^2={\mu }_2/(3K)$ and $B=C=0$. The transverse diagonal linear growth cancels because self and cross cubic coefficients coincide. Broken up-down symmetry leaves the off-diagonal coupling $\gamma KA$, giving eigenvalues $\pm |\gamma |K|A|$ and an unstable orientation perturbation whenever $\gamma \ne 0$. With zero quadratic coupling these directions are neutral at the retained order, so that degenerate case cannot be declared strictly unstable by this calculation.

A convection roll is a circulating cell pattern whose ideal velocity and temperature fields are periodic across a chosen horizontal direction and independent of the coordinate along the convection roll axis. In a two-dimensional stream function representation, its velocity field is $(-{\psi }_z,0,{\psi }_x)$. Rayleigh-Bénard convection can select stationary or oscillatory convection rolls; a two-dimensional convection roll may be unstable to three-dimensional disturbances, a different convection roll orientation, or slow amplitude and phase modulations. Existence of a saturated Landau amplitude equation branch does not alone establish its stability.

oscillatory convection is time-dependent convection emerging through nonzero-frequency overstability or a Hopf bifurcation. At onset a complex-conjugate growth-rate pair crosses the imaginary axis. Travelling and standing convection rolls are distinct nonlinear possibilities, and their stability cannot be decided from the linear angular frequency alone.

At fixed orientation, opposite travelling amplitudes can obey ${\dot{Z}}_{\pm }=(r+i{\omega }_0)Z_{\pm }-(g|Z_{\pm }{|}^2+h|Z_{\mp }{|}^2)Z_{\pm }$. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling branch has one nonzero intensity $r/g_r$ and is stable to its counter-propagating amplitude when $h_r>g_r$. A standing branch has both intensities $r/(g_r+h_r)$ and is stable within the pair when $g_r>h_r$ and $g_r+h_r>0$. These follow by linearizing the real intensity equations. Spatial modulation and other orientations remain separate stability tests.

stationary convection has a time-independent velocity and temperature field, in the chosen fixed frame. Its onset from a conduction state corresponds to a real growth rate passing through zero. Stability of a resulting steady convection roll is a separate nonlinear question.

A stable stratification gives a restoring buoyancy force and positive squared buoyancy frequency for an adiabatically displaced parcel. An adverse specific entropy gradient can instead drive convection; other forces such as rotation can modify the full stability criterion.

Convective overshoot carries material from an unstable convecting region into an adjacent stably stratified region because its momentum does not vanish at the stability boundary. In stars this mixes material beyond the boundary determined by a local stellar convective stability criterion, changing stellar core masses and nuclear lifetimes.

Double-diffusive convection occurs when two properties, such as temperature and composition, affect buoyancy but diffuse at different rates. A stabilizing composition gradient can oppose a destabilizing thermal gradient and change the form of heat transport.

Fingering convection is a form of double-diffusive convection in which a rapidly diffusing stabilizing contribution to buoyancy competes with a more slowly diffusing destabilizing contribution. The combined mass density stratification can resist an ordinary vertical displacement, yet unequal diffusion permits narrow rising and sinking structures to grow.

A displaced parcel initially carries both contributions. If the restoring anomaly diffuses away faster than the destabilizing anomaly, the remaining buoyancy reinforces its motion. In the common thermal-compositional case, thermal diffusivity exceeds solutal diffusivity; the initial stability of the combined mass density profile is therefore insufficient to decide stability once diffusion acts.^[ref]

In seawater, salt fingering realizes this competition with temperature and salinity: the thermal contribution stabilizes while salt-rich material above fresher material supplies the destabilizing contribution. Its laboratory examples make the different transport rates visible.^[ref]

In a star, chemical composition can play the slower role. A larger mean molecular weight tends to increase mass density at fixed pressure and temperature, so a molecular-weight inversion can oppose the thermal restoring effect. Rapid thermal transport can then permit fingering even when the total local stratification is stable.^[ref]

The density ratio measures the competition, and the salt-finger instability criterion shows why there is also an upper limit to the stabilizing contribution: if it is too strong, unequal diffusion no longer produces growing modes. The elevator mode gives a particularly simple local pattern with vertical motion and no height dependence.

Salt fingering is double-diffusive convection in which narrow rising and sinking fluid structures develop because heat and dissolved salt diffuse at different rates. Warm, salty water above colder, fresher water can have a stable total mass density stratification: the stabilizing temperature difference outweighs the destabilizing salinity difference.^[ref]

A small upper parcel displaced downward enters colder surroundings. It loses heat relatively quickly while retaining more of its salt, becoming denser and continuing downward. A fresher parcel displaced upward can gain heat while remaining relatively fresh and light. The distinction between thermal diffusivity and solutal diffusivity supplies the feedback; stability of the initial combined mass density profile does not exclude this mechanism. The density ratio records the competing contributions, while the salt-finger instability criterion specifies the parameter regime.

A laboratory analog can replace heat and salt by two dissolved substances. David Osmond describes a 1820-by-80-by-120 millimetre tank with salt and sugar solutions entering at opposite ends and an overflow at the centre. The supplied solutions had matching mass density, but salt diffused faster than sugar. The salt-rich plume could lose salt faster than it gained sugar, producing a lighter core and denser surroundings. Fingering and diffusive layers appeared in different regions, showing that identical source densities need not remain dynamically equivalent when composition diffuses.^[ref]

In this analog, salt plays the faster-diffusing role played by heat in the ocean, while sugar plays the slower-diffusing role of oceanic salt. Keeping that correspondence explicit prevents reversing the physical explanation.

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.

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.

Thermosolutal convection is double-diffusive convection driven by temperature and concentration gradients. Their buoyancy contributions can oppose one another, while unequal thermal diffusivity and solutal diffusivity allow steady, oscillatory and finite-amplitude states. For a layer heated from below with a stabilizing concentration gradient, thermal buoyancy is destabilizing and solutal buoyancy restoring.

In double-diffusive convection, a density ratio compares the buoyancy contributions from two stratifying quantities. A convention must be stated. For vertical coordinate $z$ increasing upward and warm salty water above cooler fresher water, take $R_{\rho }=\alpha T_z/(\beta S_z)$, with positive thermal expansion coefficient $\alpha$ and haline contraction coefficient $\beta$.

The linear equation of state gives ${\rho }_z={\rho }_0(-\alpha T_z+\beta S_z)$. Thus $R_{\rho }>1$ makes the combined density stratification stable, even though the salt contribution alone is destabilizing. Faster heat diffusion can still enable salt fingering. The reciprocal ratio is also used in the literature, so a numerical density ratio is meaningless without its convention.^[ref]

The haline contraction coefficient measures the fractional increase of mass density with salinity at fixed temperature and pressure:Its units depend on the salinity variable $S$: a dimensionless mass fraction and a value in grams per kilogram require different numerical coefficients.

For small changes, $\delta \rho /\rho \simeq -\alpha \delta T+\beta \delta S$, with thermal expansion coefficient $\alpha$. This formula separates two competing buoyancy contributions. In thermosolutal convection, their combination determines the static density gradient, while unequal diffusion rates determine whether that apparently stable gradient can nevertheless become unstable.

Let $T_i,S_i$ be the temperature and salinity of the original supplied liquid, and $T_{\infty },S_{\infty }$ those of the surroundings. For reservoir values $T,S$, define the dimensionless cooling and freshening measureswith positive reference differences. Here $\alpha$ is the thermal expansion coefficient, $\beta$ the haline contraction coefficient, $A$ the interface area and $R_{\rho }=\beta (S-S_{\infty })/[\alpha (T-T_{\infty })]$ the density ratio. Increasing $\theta$ describes cooling toward the surroundings; increasing $\eta$ describes freshening. The flux law below is an assumed double-diffusive convection closure.

The steady subscript denotes balance between supply, overflow and interfacial transfer. A model of this kind separates the stabilizing bulk salinity contrast from the faster transfer of heat across the interface.

A steady well-mixed hot salty reservoir receives volume flux $Q$ and loses the same flux over a weir. Heat and salt balances are $Q(T_i-T)=A{F}_T$ and $Q(S_i-S)=A{F}_S$. If $\beta F_S/(\alpha F_T)=R_F$ and $\alpha F_T=b(\alpha \Delta T{)}^{4/3}/R_{\rho }^2$, define $R_0=\beta (S_i-S_{\infty })/[\alpha (T_i-T_{\infty })]$. Then ${\eta }_s=(R_F/R_0){\theta }_s$ and $C=Ab[\alpha (T_i-T_{\infty }){]}^{1/3}/(Q{R}_0^2)$. For $R_0>1$ and $0\le R_F<1$ the interface is statically stable while preferential heat transfer can drive double-diffusive convection.

Let $T_i,S_i$ be the temperature and salinity of the original supplied liquid, and $T_{\infty },S_{\infty }$ those of the surroundings. For reservoir values $T,S$, define the dimensionless cooling and freshening measureswith positive reference differences. Here $\alpha$ is the thermal expansion coefficient, $\beta$ the haline contraction coefficient, $A$ the interface area and $R_{\rho }=\beta (S-S_{\infty })/[\alpha (T-T_{\infty })]$ the density ratio. Increasing $\theta$ describes cooling toward the surroundings; increasing $\eta$ describes freshening. The flux law below is an assumed double-diffusive convection closure.

With a constant buoyancy-flux ratio and the preceding steady initial state, the heat and salt balance equations give $d\eta /d\theta =q$ and ${\eta }_s=q{\theta }_s$, yielding $\eta =q\theta$ after shutdown.

After inflow and overflow cease, a fixed-volume reservoir initially in the preceding steady state obeys $\eta =q\theta$ and $\dot{\theta }=k(1-\theta {)}^{10/3}/(1-q\theta {)}^2$, with $k=Ab[\alpha (T_i-T_{\infty }){]}^{1/3}/(V{R}_0^2)$. Integrating gives $k(t-t_0)=G(\theta )-G({\theta }_s)$, where $G=3(1-q{)}^2(1-\theta {)}^{-7/3}/7+3q(1-q)(1-\theta {)}^{-4/3}/2+3q^2(1-\theta {)}^{-1/3}$. For $q<1$, heat contrast tends to zero only at infinite time, while a nonzero salt contrast remains.

A five-mode Galerkin method model retains the roll velocity $a$, its correlated temperature and concentration modes $b,d$, and the mean-profile distortions $c,e$. Its parameters include the Prandtl number $\sigma$, diffusivity ratio $\tau$, thermal forcing $r$, stabilizing solutal forcing $r_s$, and a positive geometric decay ratio $\varpi$. The velocity equation is $\dot{a}=\sigma (-a+rb-r_{s}d)$, while the scalar equations are $\dot{b}=-b+a(1-c)$, $\dot{c}=\varpi (-c+ab)$, $\dot{d}=-\tau d+a(1-e)$ and $\dot{e}=\varpi (-\tau e+ad)$. The solutal forcing is normalized using the thermal diffusion scale, so its steady stabilizing effect is $r_s/\tau$.

Linearization of the Lorenz model of thermosolutal convection at conduction yields $(\lambda +\sigma )(\lambda +1)(\lambda +\tau )-\sigma r(\lambda +\tau )+\sigma r_s(\lambda +1)=0$. A Hopf bifurcation requires the cubic coefficient identity $a_3=a_1{a}_2$ and positive frequency squared. Its threshold is the stated $r_H$, with ${\omega }^2=\sigma r_s(1-\tau )/(\sigma +1)-{\tau }^2>0$. If $r_s={\beta }^2\tau$ and $\delta =\sigma /(1+\sigma )$, then $r_H=1+\tau /\delta +{\beta }^2\tau \delta +O({\tau }^2)$. Comparing with the steady fold of a thermosolutal Lorenz model gives a leading nonnegative difference $\tau (\beta \delta -1{)}^2/\delta$ for $\beta \ge 0$.

A nonzero steady Lorenz model of thermosolutal convection branch obeys $r(x)=(1+x)+r_s\tau (1+x)/({\tau }^2+x)$, with $x=a^2$. For $0<\tau <1$, it has an interior saddle-node bifurcation precisely when $r_s>{\tau }^3/(1-{\tau }^2)$. Differentiating gives $x_{\ast }=-{\tau }^2+\sqrt{r_s\tau (1-{\tau }^2)}$ and the stated minimum. The weaker-versus-stronger threshold distinction matters: a sufficient bound need not be the sharp subcriticality condition. Steady-branch existence alone does not decide oscillatory stability.

Layered convection consists of mixed convecting layers separated by more weakly transporting interfaces. In a giant planet, composition gradients can support such interfaces and reduce heat loss, contributing to delayed cooling of an inflated giant planet.