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The butterfly effect is the intuitive idea that an arbitrarily small change in an initial condition can eventually change the observed future substantially. In dynamical systems, sensitive dependence makes this precise for a map $F:X\to X$ on a metric space $(X,d)$: there is one separation scale $\delta >0$ such thatThe same $\delta$ works at every point, whereas $y$ and the separation time $n$ may depend on the point and the requested closeness. For a flow map, replace the iteration of a map $F^n$ by ${\Phi }_t$ with some $t\ge 0$. The state space matters: restricting to a single fixed point removes the alternative nearby states needed for sensitivity.

Sensitivity is compatible with deterministic evolution. Given an exact initial condition, there is still one orbit; uncertainty concerns which exact state a measurement represents. It also allows accurate short-term prediction. For a continuous function $F$, each fixed finite iterated function $F^n$ is continuous, so sufficiently close initial states remain close at any specified finite step. What fails is a single initial accuracy that controls every future step.

The logistic map illustrates both sides. At $r=2.8$, nearby orbits in the basin of attraction of its attracting fixed point approach one another. At $r=4$, arbitrarily close states can eventually separate by a macroscopic fraction of $[0,1]$. The definition asserts the existence of separating neighbours, not that every nearby pair must separate: $x$ and $1-x$ have exactly the same next state under the logistic map and hence the same subsequent orbit. Nor must separation increase at every step.

A positive Lyapunov exponent describes average exponential growth of an infinitesimal displacement along an orbit, when the relevant limit exists. It is related to sensitivity but is not a substitute for its uniform definition. For a differentiable interval map, the chain rule givesAt $r=4$, the logistic map has $\lambda =\log 2$ for Lebesgue measure-almost every initial condition, while exceptional orbits can have different exponents. Finite-separation growth is only approximately exponential while the linear approximation remains useful. In a bounded state space, a separation cannot grow exponentially forever.

Sensitivity alone is also weaker than Devaney chaos, which additionally requires topological transitivity and dense periodic points. For example, $F(x)=2x$ on the real line has sensitive dependence and a positive Lyapunov exponent, but its only periodic point is zero, so it is not Devaney-chaotic. This distinguishes growth of uncertainty from the richer organization of recurrent orbits.

Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963) studied a thermal convection model with three ordinary differential equations. Its numerical orbits exhibited persistent nonperiodic motion and instability to small changes of the initial condition. Lorenz discussed the consequences for very-long-range weather forecasting: a deterministic law need not make a detailed distant future reliably recoverable from approximate observations. The historically relevant lesson of the Lorenz system is this growth of uncertainty under deterministic evolution.^[ref]

Convection transports properties such as energy or composition through bulk fluid motion. Thermal convection is driven by temperature-dependent buoyancy.

Darcy-Bénard convection is thermal convection in a porous layer heated from below, with velocity governed by Darcy's law. With fixed temperatures and impermeable boundaries, a normalized two-dimensional model uses streamfunction $\psi$ and temperature $\theta$:The control parameter is a porous-medium Rayleigh number. On $0<z<1$ with temperature values $0,-1$, the conductive state is $\psi =0$, $\theta =-z$.

The study of dynamical systems asks how a state changes in time: which states persist, which orbits repeat, and how a small perturbation changes the future. A dynamical system combines a state space with an evolution rule. For a seasonal population, the state might be its abundance and one step one generation; for a harmonic oscillator, a state vector includes both position and velocity. Choosing a sufficient state is essential: position alone does not determine where an oscillator moves next.

The two basic descriptions are a discrete dynamical system, $x_{n+1}=F(x_n)$, and an ordinary differential equation, $\dot{x}=f(x)$. Repeated iteration of a map gives $x_n=F^n(x_0)$; an autonomous differential equation with unique solutions gives a flow map $x(t)={\Phi }_t(x_0)$. In either case, evolving for two successive durations agrees with evolving for their sum. The logistic map provides a particularly simple discrete dynamical system; the Lorenz system supplies a continuous-time example with a very different geometry.

Deterministic evolution says that the exact rule and exact initial condition specify one future. Predicting that future from measurements is a separate question. An instrument supplies a range of possible initial conditions, and a computation adds small errors. Near an asymptotically stable fixed point, nearby orbits converge and the long-term state can become easier to predict. With sensitive dependence on initial conditions, arbitrarily close starting states can eventually lead to a visible disagreement, even though each follows the same rule.

Much of the usefulness of dynamical systems comes from describing outcomes without writing every solution explicitly. Fixed points identify possible resting states; periodic orbits identify repeating behaviour; stability theory asks whether those behaviours survive small disturbances. A bifurcation diagram shows how changing a bifurcation parameter changes the possible outcomes. This is useful in population dynamics, where a stable abundance and an alternating abundance have different consequences, and in thermal convection, where a steady flow and a changing flow transport heat differently.

The beauty of the subject is that a small formula can organize a large variety of behaviour. The logistic map moves from an attracting fixed point to attracting periodic orbits and then to parameter values supporting sensitive motion. Its period-doubling cascade has a recognizable structure across many unimodal maps. Nevertheless, Devaney chaos, Glendinning chaos, and a positive Lyapunov exponent express different properties; a claim of chaos should specify its definition, its state space, and the parameter values under consideration.

A lava lake is a persistent exposed body of molten or partly molten magma. Surface radiative cooling, thermal convection, latent heat, and internal radiogenic heating can jointly control formation and melting of its crust.

The Lorenz system is a three-variable autonomous differential equation modeling an idealized thermal convection flow. In Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963), published in the Journal of the Atmospheric Sciences 20, pp. 130–141, numerical solutions of the model illustrated nonrepeating motion and instability to changes in the initial condition. Lorenz used this example to examine obstacles to very-long-range weather forecasting.^[ref]

Introduce specific heat capacity $c_p$, latent heat per unit mass $L_f$, and $\Delta T=T_m-T_s>0$. Write the prescribed convective heat flux as $F=K(\overline{T}-T_m)$, whereThere is a dimensional convention to specify: with $Q$ measured in $\mathrm{W}\mathrm{k}{\mathrm{g}}^{-1}$ and dimensionless $\lambda$, this printed expression makes $K$ a heat transfer coefficient only if $\mu$ is kinematic viscosity. If $\mu$ denotes dynamic viscosity, the numerator needs ${\rho }_0^{2}g\alpha Q$ instead. The question says only “viscosity”; use the consistent kinematic interpretation, or equivalently the corrected dynamic formula. The following balances apply to either convention once $K$ has units $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$.

For a thin lava lake crust with a linear temperature profile, Fourier's law gives the outward conductive flux $k\Delta T/a$. The Stefan condition balances it against upward thermal convection and freezing, while the well-mixed liquid loses the convective flux and gains radiogenic heating. To leading order in $a/H$ the radiogenically heated lava-lake crust model isThe fixed surface temperature already incorporates radiative cooling, so no extra radiative loss is added to the interior balance. The linear conductive profile is interpreted as a quasi-steady approximation, so sensible-heat storage within the crust is not evolved. Thinness alone does not establish this approximation: rapid thermal adjustment in the crust is also needed, for example a small Stefan number during the initial conduction-driven growth. Replacing the liquid depth by $H-a$ and retaining the moving liquid-volume enthalpy would introduce thin-crust corrections: the requested constant-temperature branch refers to the leading model above. A transient thermal profile in a thick or newly nucleating crust would require the full Stefan problem, rather than this linear-profile closure.

Using the thermal diffusion time $H^2/\kappa$, defineThe three independent dimensionless parameters are the convective-conductive coefficient $B$, the heating parameter $R$, and the Stefan number $S$. For positive $B,R,S$ the evolution equations becomewhere prime denotes $d/d\tau$. Temperature has the exact solutionConsequently the unique constant-temperature initial condition isAt this temperature, the convective flux equals the column's radiogenic heat production. Starting with no crust, the constant-temperature lava-crust growth law is obtained by separating $h^{\prime }=S(1/h-R)$:The crust grows towards $h_{\ast }=1/R$, or $a_{\ast }=k\Delta T/({\rho }_{0}QH)$, where conduction carries all the radiogenic input. The logarithm makes reaching this equilibrium take infinite time. For an initial thickness $h_0>0$, subtract $[-R{h}_0-\log (1-R{h}_0)]/(S{R}^2)$ on the same side of the equilibrium; the corresponding antiderivative uses $\log |1-Rh|$ on either branch. The thin-crust description through equilibrium requires $R\gg 1$.

For general initial temperature, the full early and intermediate history is specified by the exact temperature above and one scalar ordinary differential equation. A particularly convenient integral formulation for an initially bare surface isThis removes the infinite initial derivative of $h$. The right-hand side points into $y>0$ at zero, and is decreasing in $y$ when ${\theta }_0\ge 0$; two nonnegative solutions therefore cannot separate, which gives uniqueness. Its solution, together with the temperature formula, determines the entire model evolution and can be computed without a phase-boundary search. The early expansion follows by substitution:The dimensional leading crust thickness is $a\sim \sqrt{2k\Delta Tt/({\rho }_0{L}_f)}$. Thus hotter liquid reduces the first correction to conductive crust growth, whereas colder liquid permits more rapid thickening.

If ${\theta }_0>{\theta }_{\ast }$, the temperature decreases and the instantaneous stationary lava-crust thickness $h_e(\tau )=1/[B\theta (\tau )]$ increases towards $h_{\ast }$. Starting from zero, the crust always remains below this rising curve: at a first contact $h^{\prime }=0$ but $h_e^{\prime }>0$, so it cannot cross from below. Hence hot-start crust grows monotonically. It also stays thinner than the constant-temperature solution: the early expansion puts it below, and at a proposed first crossing their derivative difference is $-SB({\theta }_0-{\theta }_{\ast })e^{-B\tau }<0$, preventing crossing from below. If $0\le {\theta }_0<{\theta }_{\ast }$, the temperature increases and the instantaneous stationary lava-crust thickness decreases. The crust stays thicker than the constant-temperature solution by scalar comparison, but it may cross the decreasing $h_e$, reach a maximum and subsequently melt back towards $h_{\ast }$. Such overshoot is possible, not inevitable; the two relevant relaxation rates are $B$ and $S{R}^2$. The numerical plot uses $B=4$, $R=20$, $S=0.1$, so crust adjustment is faster than temperature relaxation and the cool-start example does overshoot:

The curves use ${\theta }_0/{\theta }_{\ast }=0.5,1,1.5$ and initially $h=0$. They solve the complete reduced time-dependent equations, rather than replacing the moving stationary thickness by the actual crust. At late times, set $h=h_{\ast }+v$, $\delta ={\theta }_0-{\theta }_{\ast }$; linearization gives $v^{\prime }=-S{R}^{2}v-SB\delta e^{-B\tau }$. For $S{R}^2\ne B$,At equal rates the forced term is $-SB\delta \tau e^{-B\tau }$. In particular, slow warming with $B<S{R}^2$ approaches equilibrium from a thicker crust, explaining the plotted remelting. These long-time expressions describe the linearized response; nonlinear corrections can have their own faster exponential rates. All predictions are restricted to $a/H\ll 1$, nonnegative liquid superheat, and the assumed vigorous-convection flux law. If crust growth violates those conditions, the reduced model must be replaced rather than extrapolated to total freezing.

The density anomaly of water makes mass density increase with temperature up to $T_m$ and decrease above $T_m$. Since temperature increases with depth in the stated layer, the density profile first increases to its maximum ${\rho }_m$ at the $T=T_m$ isotherm and then decreases toward the warm bottom.

The upper region has denser water below lighter water and hence stable density stratification. Below the density maximum, density decreases downward, so lighter water lies beneath denser water and gives unstable density stratification. The lower region therefore overturns by thermal convection, while the cold upper region remains a stagnant cap. This coexistence is penetrative convection.

The approximation in part v requires the dimensionless stagnant depthto remain small. Since $0<\theta \le 1$ makes the numerator order one, this condition isUsing the solution from part v, the equivalent time range isor, at the level of asymptotic equivalence,The cooling becomes appreciable on the shorter scale $\tau =O({\mathcal{F}}^{-1})$, so these ranges overlap widely when $\mathcal{F}\gg 1$.

The Rayleigh number based on the convecting depth isThe definition of $\mathcal{F}$ implicit in part iii givesIt follows thatIf $1-\eta =O(1)$ and $\theta \gg {\mathcal{F}}^{-3/5}$, then $\mathrm{Ra}\gg O({\mathcal{F}}^{9/5})$. The thermal convection therefore remains strongly supercritical throughout the range in which the thin stagnant-layer approximation is valid.

For a well-mixed lava lake of depth $H$ and a thin conductive crust of thickness $a$, put $\Delta T=T_m-T_s$, and use the convective flux $K(\overline{T}-T_m)$ with constant heat transfer coefficient $K>0$. The Stefan condition and liquid heat balance give, to leading order in $a/H$,Here latent heat $L_f$ is per unit mass and $k=\rho c_p\kappa$. Crust heating and changes of liquid depth are neglected. The linear conductive profile uses a quasi-steady approximation; thinness alone does not justify neglecting crust thermal storage. During initial growth, a small Stefan number supplies a sufficient rapid-adjustment regime. Define $h=a/H$, $\theta =(\overline{T}-T_m)/\Delta T$, and $\tau =\kappa t/H^2$. Scaling by the thermal diffusion time $H^2/\kappa$ and $\Delta T$ gives $h^{\prime }=S(1/h-B\theta )$ and ${\theta }^{\prime }=R-B\theta$, where $B=KH/k$, $R=Q{H}^2/(c_p\kappa \Delta T)$, and $S=c_p\Delta T/L_f$ is the Stefan number. Thus $\theta =R/B+({\theta }_0-R/B)e^{-B\tau }$ exactly. The reduction applies while $a/H\ll 1$, the conductive profile is approximately linear, and the thermal convection closure remains valid; it does not model complete freezing of a deep lake.

A fluid has unstable density stratification when a vertical displacement amplifies itself. Denser fluid above lighter fluid can overturn by Rayleigh-Taylor instability or develop thermal convection when the density difference is thermal.