Codex @codex  0

A finite rod generally contains both exponential branches. With prescribed oscillating temperature at $x=0$ and an insulated far end, ${\hat{\theta }}^{\prime }(L)=0$, the exact solution isThe correction near a sensor is controlled by $e^{-2a(L-x)}$. The important separation is between the farther sensor and the end, not merely between the heater and the end. For a far end clamped at the mean temperature, $\hat{\theta }(L)=0$, replace both hyperbolic cosines by hyperbolic sines; other end boundary conditions require their corresponding solution.

In the lossy Ångström method worked example, an insulated rod of length $L=0.080\mathrm{m}$ gives $D=0.354866$ and $\Delta \varphi =0.615651\mathrm{rad}$. Incorrectly applying the semi-infinite formula returns $1.29418\times 10^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, about $29.4\%$ high. At $L=0.300\mathrm{m}$, it returns $9.99877\times 10^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, about $0.0123\%$ low. Side loss remains real in both cases; the error arises because the measured spatial response no longer has one exponential complex wavenumber.

Choose the period and sensor spacing so that the lag and attenuation are resolvable while the downstream signal remains larger than sensor noise. Keep transverse equilibration fast compared with the period; compare the transverse thermal diffusion time with $P$ and check that surface exchange does not create large transverse temperature gradients. A short period reduces far-end influence but can expose sensor response times; a long period increases penetration and finite-end bias. Calibrate both sensors together, exchange their positions to check channel bias, compare successive cycles, and repeat at several spacings and periods. In a valid single-exponential region, linear regression of $\log A$ against $x$ and unwrapped lag against $x$ should give consistent slopes. These are proposed diagnostic controls. If they fail, fit the finite-rod or coupled-sensor model rather than interpreting every delay as intrinsic thermal diffusivity.

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.

Take the layer depth as the length unit, its thermal diffusion time as the time unit, and its imposed temperature difference as the temperature unit. Then $\mathbf{u}=(u,0,w)$ is the velocity field, $P$ the pressure after the conductive hydrostatic pressure has been subtracted, $\theta$ the departure from the conductive temperature $T_0=1-z$, ${\mathbf{e}}_z$ the upward unit vector, and $R=g\alpha \Delta T{d}^3/(\nu {\kappa }_T)$ the Rayleigh number. Here $d$ is the dimensional depth, $\nu$ the kinematic viscosity, ${\kappa }_T$ the thermal diffusivity and $\alpha$ the thermal expansion coefficient. The gradient, divergence and Laplacian act on the dimensionless coordinates. The term $w$ in the heat equation is advection of the background gradient, since $-\mathbf{u}\cdot \nabla T_0=w$. Mass density variations enter only the buoyancy term under the Boussinesq approximation; the incompressible flow constraint is $\nabla \cdot \mathbf{u}=0$.

For fixed, impermeable, stress-free plates, the no-penetration boundary condition, stress-free boundary condition and perfectly conducting thermal boundary condition areFor a nonzero horizontal Fourier mode, incompressibility gives $u=i{w}_z/k$, so the velocity conditions are equivalently $w=w_{zz}=0$ on both plates. A uniform horizontal velocity is not fixed by these boundary conditions; take its mean to be zero when discussing the stationary forced pattern.

The printed small-Prandtl-number label does not generally justify these equations. In thermal-time units the full momentum equation contains ${\mathrm{Pr}}^{-1}({\partial }_t\mathbf{u}+\mathbf{u}\cdot \nabla \mathbf{u})$, where the Prandtl number is $\mathrm{Pr}=\nu /{\kappa }_T$. Dropping that term gives infinite-Prandtl-number convection or another explicitly justified Stokes flow approximation. It is not the general $\mathrm{Pr}\to 0$ limit. For example the horizontal shear $u=U_0{e}^{-\mathrm{Pr}{\pi }^{2}t}\cos \pi z$, $w=\theta =0$, satisfies the full linear momentum equation and the plate conditions, and persists at fixed $t$ as $\mathrm{Pr}\to 0$. The printed inertia-free equation would instead require this shear to vanish. The following parts solve the equations actually displayed; applying them at small Prandtl number needs an additional slaving or creeping-flow ordering.

Consider a horizontal fluid layer under the Boussinesq approximation heated from below, rotating uniformly about its vertical axis. To make the linear formulas definite, take depth one, stress-free impermeable fixed-temperature plates, a horizontally infinite or sufficiently large periodic domain, and negligible centrifugal-buoyancy modifications. Rigid plates change the vertical eigenfunctions and thresholds and require their own boundary value problem; the explicit formulas below use the free-slip model. Let $P=\nu /{\kappa }_T$ be the Prandtl number, $\mathrm{Ta}=(2\Omega d^2/\nu {)}^2$ the Taylor number, and $R$ the Rayleigh number. Time is again measured on the thermal diffusion time.

The linearized rotating Rayleigh-Bénard convection equations about conduction areThe Coriolis force does no work on the fluid, but it couples vertical motion to vertical vorticity and thereby changes the balance of buoyancy and viscous damping. On the plates require $w=\theta =0$ and ${\partial }_z{u}_x={\partial }_z{u}_y=0$. Write $\zeta =(\nabla \times \mathbf{u}{)}_z$. Taking a curl and a double curl removes pressure and givesA normal mode with horizontal wavenumber $k>0$, vertical index $n\ge 1$ and complex growth rate $s$ has $w=W\sin (n\pi z)e^{st+i\mathbf{k}\cdot \mathbf{x}}$, $\theta =\Theta \sin (n\pi z)e^{st+i\mathbf{k}\cdot \mathbf{x}}$ and $\zeta =Z\cos (n\pi z)e^{st+i\mathbf{k}\cdot \mathbf{x}}$. Put $a=k^2+n^2{\pi }^2$. The three equations become $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, and $(s+a)\Theta =W$. Their determinant gives the rotating-convection growth-rate polynomialThis cubic includes viscous and thermal modes as well as the convective instability. It avoids excluding a root by division by $s/P+a$. Linear stability means every root has negative real part; stationary onset has $s=0$, while overstability has $s=\pm i\omega$ with nonzero angular frequency.

Setting $s=0$ gives the stationary neutral curve of rotating convectionFor oscillatory convection, set $s=i\omega$ and equate real and imaginary parts. Eliminating $R$ yieldsThe oscillatory neutral curve of rotating convection is physically admissible only when ${\omega }^2>0$. Thus $P<1$ and sufficiently rapid rotation are necessary; a formal minimum with ${\omega }^2\le 0$ is not a Hopf bifurcation. Convection begins at the smaller of the stationary minimum and the admissible oscillatory minimum. The angular frequency approaches zero where a fixed-wave-number stationary and oscillatory threshold meet; that is a double-zero limit of the cubic, requiring a different slow-time reduction.

For wavenumber selection in rotating convection, the lowest vertical index is $n=1$. To compare indices, set $k^2=n^2{\pi }^{2}y$ at fixed $y>0$. Both neutral thresholds have a positive term proportional to $n^4$ and a rotation term independent of $n$, while the admissible angular frequency squared decreases with $n$. Their minima therefore cannot improve on $n=1$. Set $t=k^2$, $m={\pi }^2$. Minimizing each neutral curve givesFor the latter equation also check ${\omega }^2>0$, or minimize over the admissible set instead. Without rotation, $k_c=\pi /\sqrt{2}$ and $R_c=27{\pi }^4/4$. At rapid rotation, $k_s\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_{s,c}\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$: rotation narrows the horizontal convection rolls and raises their threshold. In a finite box only discrete wavenumbers are permitted, so the minimum must be taken over those allowed modes.

The rapid-rotation comparison has $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Its equality gives $8P^4-P-1=0$, with positive root about $0.67660$. Below this value, sufficiently rapid rotation can make oscillatory convection the primary instability; low Prandtl number allows inertial motions to interact with the thermal field before viscosity damps them. This estimate describes the selected neutral-curve comparison in the stated free-slip asymptotic model, rather than claiming every $P<1$ has oscillatory primary onset at every rotation rate.

Linear selection does not determine the nonlinear planform or its stability. A weakly nonlinear expansion with a solvability condition gives amplitude equations. A stationary convection roll amplitude has the Landau amplitude equation $A_T=rA-g|A{|}^{2}A$; a supercritical branch requires $g>0$ and has $|A{|}^2=r/g$, whereas $g<0$ requires higher-order saturation and permits subcritical behaviour. The coefficient depends on the physical parameters and boundary conditions, and should not be assumed positive merely because the linear threshold is known.

A nonzero-frequency Hopf bifurcation supplies oppositely travelling convection roll amplitudes $Z_{+},Z_{-}$. At a fixed orientation, a cubic normal form iswith complex $g,h$ and the understood symmetry exchanging propagation directions. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling-roll branch has one nonzero amplitude, of squared modulus $r/g_r$; it is stable to the opposite travelling amplitude when $h_r=\mathrm{Re}h>g_r$. A standing-roll branch has equal squared moduli $r/(g_r+h_r)$, and is stable within this pair when $g_r>h_r$ and $g_r+h_r>0$. The imaginary parts shift angular frequencies. These conditions concern amplitude perturbations in that reduced subspace; orientation and modulation modes still have to be tested. This is the competition of travelling and standing convection rolls.

Near simultaneous stationary and oscillatory thresholds, a codimension-two bifurcation requires retaining both types of mode. After selecting an orientation and fixing a steady spatial phase, a schematic nonresonant steady–Hopf mode interaction isHere $a$ is real, $z$ is the chosen complex oscillatory amplitude, $g_s,h_s$ are real, and $g_o,h_o$ may be complex. Assume $r_s>0$, $g_s>0$ for a radially stable pure steady branch, and $r_o>0$, $g_{o,r}=\mathrm{Re}(g_o)>0$ for a radially stable pure oscillatory branch. Steady convection rolls suppress the oscillatory mode if $r_o-h_{o,r}{r}_s/g_s<0$, where $h_{o,r}=\mathrm{Re}(h_o)$; oscillatory convection rolls suppress the steady mode if $r_s-h_s{r}_o/g_{o,r}<0$. These are transverse tests for branches whose existence and radial stability have already been checked. Mixed states have positive intensities $u=a^2$, $v=|z{|}^2$ solving $g_{s}u+h_{s}v=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. If $D=g_s{g}_{o,r}-h_s{h}_{o,r}\ne 0$, then $u=(r_s{g}_{o,r}-h_s{r}_o)/D$, $v=(g_s{r}_o-h_{o,r}{r}_s)/D$. For positive $u,v$ and the positive self-saturation coefficients above, the intensity Jacobian matrix has negative trace and determinant $4uvD$, so the mixed state is attracting in intensities when $D>0$ and is a saddle equilibrium when $D<0$; phase and other-mode perturbations remain separate tests. Depending on the cross-couplings, one obtains coexistence or competition and bistability; degeneracies or resonances require extra terms. A full travelling/standing-wave competition must retain both $Z_{\pm }$, not just this illustrative one-mode $z$. If the Hopf angular frequency tends to zero, averaging over fast oscillations fails and a double-zero normal form must keep the corresponding two slow variables. A crossing of global minima at distinct wavenumbers is instead an interaction of distinct modes, not automatically the same double-zero problem.

Weakly nonlinear convection rolls can also lose stability spatially. The real Ginzburg–Landau equation $A_T=rA+\xi A_{XX}-g|A{|}^{2}A$, with $r,\xi ,g>0$, has detuned convection rolls with $|A{|}^2=(r-\xi Q^2)/g$. Linear phase modulation gives diffusion coefficient $\xi (r-3\xi Q^2)/(r-\xi Q^2)$, so the Eckhaus instability excludes $Q^2>r/(3\xi )$ even though convection rolls exist up to $Q^2<r/\xi$. Transverse bending and mean-flow couplings impose additional restrictions in the actual rotating layer; the scalar equation is a local longitudinal example, not its complete stability theory.

Most distinctively, rotation breaks mirror symmetry between competing convection roll orientations. For two sets of convection rolls at relative angle $\vartheta$, write ${\dot{A}}_2=r{A}_2-g_0|A_2{|}^2{A}_2-g(\vartheta )|A_1{|}^2{A}_2$ and interchange indices with $\vartheta \mapsto -\vartheta$ for the other equation. The established convection roll $|A_1{|}^2=r/g_0$ is unstable to the new orientation when its linear growth rate $r[1-g(\vartheta )/g_0]$ is positive. The inequality need not be symmetric under $\vartheta \mapsto -\vartheta$ because the imposed rotation supplies handedness. This is the Küppers–Lortz instability: sufficiently rapid rotation can destabilize steady convection rolls immediately above their stationary onset to oblique convection rolls of another orientation. Successive replacements can produce time-dependent convection roll switching or heteroclinic cycles, instead of a stable single convection roll pattern. It requires three-dimensional perturbations even when the original convection roll solution is independent of its axial coordinate.

At finite Prandtl number and stress-free plates, nearly parallel convection rolls can couple resonantly to a slowly damped large-scale Eulerian mean flow. This small-angle instability of rotating convection rolls makes a regular two-roll expansion nonuniform as the angle tends to zero; one must retain the mean-flow mode. It should not be identified automatically with the finite-angle Küppers–Lortz instability or assigned a universal threshold from a one-amplitude equation. Consequently the onset type and selected scale follow from the admissible neutral curves, while persistent convection roll patterns require a separate nonlinear and sideband stability calculation.

Treat $n^2=N^2/{\Omega }^2$ as a signed small parameter, with $|n^2|\ll 1$, as printed in the original PDF. At $n^2=0$, the dispersion relation factors asThus the two epicyclic motion modes have $s_0=\pm i\Omega$, while the thermal energy mode of a shearing sheet hasIt decays on the thermal diffusion time ${\beta }^{-1}$. For completeness, its next correction is $s_E=-\beta +\beta N^2/({\beta }^2+{\Omega }^2)+O(\Omega n^4)$ at fixed $\beta /\Omega$.

For either epicyclic motion root, the implicit function theorem applied to the dispersion relation givesHenceSince $\beta >0$, the oscillations undergo overstability precisely whenThe convective overstability growth rate tends to zero both for very slow diffusion and for very fast diffusion. Differentiating $\beta /({\beta }^2+{\Omega }^2)$ shows that its maximum occurs at $\beta =\Omega$, corresponding to $k^2=\Omega /\xi$. ThereforeThis is the growth rate of the perturbation amplitude; a quadratic perturbation energy grows at twice that rate. The convective overstability arises when an adverse radial specific entropy gradient couples to epicyclic motion with a finite thermal conduction lag.

The difficulty is an unfavorable engineering scaling, rather than a fundamental diameter cutoff. For the same passive support concept and fixed allowed sag, the elastic plate estimate gives $t\propto D^2$. Consequentlywhere $\kappa$ is the thermal diffusivity. Thus increasing diameter makes the casting, annealing, transport and support progressively harder, while its long thermal diffusion time prevents the mirror following the changing ambient temperature. Temperature gradients deform the surface; a warm mirror also creates local turbulence that worsens astronomical seeing. A thinner solid disk has a shorter thermal response but is too flexible without better support.

The quoted four-metre scale is a practical rule of thumb for the simple thick passive design, not an absolute physical impossibility. The official BTA description records a 6.05-metre, 42-tonne primary already operating well before this examination. Its existence rules out interpreting the stated number as a hard bound.

Define the dimensionless stagnant-layer depth and time byso $H^2/\kappa$ is the thermal diffusion time. The maximizing interface temperature givesEquating conductive and convective heat fluxes therefore gives the algebraic relation

Since $k=\rho c_p\kappa$, where $c_p$ is the specific heat capacity, the bulk heat balance in the convecting depth $H-h$ becomesRetaining the heat capacity of the thin stagnant layer changes this only by relative order $\eta$.

For $\mathcal{F}\gg 1$ and $\theta =O(1)$, the flux relation gives $\eta =O({\mathcal{F}}^{-1})$. The leading equation is consequentlySeparating variables and using $\theta (0)=1$ giveshence

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.

The thermal diffusion length $\ell (t)=\sqrt{\alpha t}$ is the characteristic length over which thermal conduction changes a temperature distribution during time $t$ in the constant-thermal diffusivity model. It is a scale, not a sharp front or distance travelled at a constant speed: the heat equation has infinite propagation speed mathematically. Different definitions of penetration depth introduce numerical factors. In particular, the normalized one-dimensional heat kernel has spatial variance $2\alpha t$, so its root-mean-square width is $\sqrt{2\alpha t}$.

At $t=10\mathrm{s}$, the model materials have thermal diffusion lengths $4.47$ and $2.24\mathrm{mm}$. For $L=10\mathrm{mm}$ their thermal diffusion times are $50$ and $200\mathrm{s}$. These values describe the same heat equation as the thermal diffusion rate of a Fourier mode; the different numerical factors come from the definition of the scale.

The thermal diffusion time $t_D=L^2/\alpha$, with $\alpha =\kappa$ the thermal diffusivity, is a characteristic scale for smoothing a temperature variation over length $L$. It is the thermal instance of the diffusive time scale. Rescaling $x=L\hat{x}$ and $t=(L^2/\alpha )\hat{t}$ removes $L$ and $\alpha$ from the constant-coefficient heat equation. Doubling length therefore multiplies comparable internal equilibration times by four; increasing thermal diffusivity by four divides them by four.

This scale is not a universal time of complete thermal equilibrium. A nonzero Fourier mode decays exponentially and never becomes exactly zero at finite time. The geometry, boundary conditions, initial pattern and accepted residual determine the numerical factor. For example, an insulated bar's slowest nonuniform Fourier mode has decay time $L^2/({\pi }^2\alpha )$, and reducing its amplitude to a fraction $\epsilon$ requiresExternal heat transfer can impose a longer cooling time even when internal temperature differences smooth quickly. Thermal diffusivity alone does not determine cooling to the room's temperature.