The Lorenz System from Boussinesq Convection
This appendix proves Definition 36.62 of Nonlinear Dynamics and Chaos: that Equation (36.36), with the three parameters Equation (36.37), is what the equations of a convecting fluid layer become when they are truncated to one velocity mode and two temperature modes. The point of writing it out is stated in Remark 36.2: the Lorenz system is almost always presented as a dimensionless curiosity, and it is only by carrying the reduction through in SI variables that \(\sigma\), \(r\) and \(b\) can be read back as ratios of quantities a laboratory measures. Along the way the critical Rayleigh number \(\mathrm{Ra}_{\mathrm{c}}=27\pi^{4}/4\) and the critical wavenumber \(a^{2}=\tfrac{1}{2}\), both quoted in Definition 36.62 and in Equation (35.68), are derived.
Nothing is imported from outside the treatise. The starting equations are those of Fluid Dynamics — the incompressible Navier–Stokes equations Equation (35.32) and the advection–diffusion equation for heat — specialised by the Boussinesq approximation, which is that chapter's business and is discussed at Phenomenon 35.78; the linear-stability calculation and its comparison with measurement are Section 40.4. The attribution of the mode set is to Saltzman, whose 1962 study of finite convective amplitude in the Journal of the Atmospheric Sciences supplied the expansion Lorenz truncated; that paper has no entry in this bibliography, so the attribution is made in words, and the truncation and its consequences are Lorenz's [Lorenz:1963].
The layer, in SI units
A horizontal layer of fluid occupies \(0\le z\le d\), with \(z\) measured upwards against gravity \(\vect{g}=-g\hat{\vect{z}}\). The lower surface is held at temperature \(T_{0}+\Delta T\) and the upper at \(T_{0}\), so that \(\Delta T>0\) means heating from below. The fluid is characterised by four measured properties, all functions of the working substance and of its mean state:
the kinematic viscosity, the thermal diffusivity, the coefficient of thermal expansion and the reference density; \(d\) is in \(\mathrm{m}\), \(\Delta T\) in \(\mathrm{K}\) and \(g\) in \(\mathrm{m}/\mathrm{s}^{2}\). Rests on Theorem 35.37 and Definition 35.51.
Under the Boussinesq approximation — the density is treated as the constant \(\rho_{0}\) everywhere except in the buoyancy force, where it is \(\rho=\rho_{0}\left[1-\alpha_{T}\left(T-T_{0}\right)\right]\), and the flow is treated as incompressible — the motion of the layer of Definition A65.1 obeys
with \(p'\) the departure of the pressure from its hydrostatic value. The state of rest, \(\vect{u}=\vect{0}\), is a solution with the conducting temperature profile
Rests on Definition A65.1, Theorem 35.37 and Phenomenon 35.78.
Derives Proposition A65.2. Equation (A65.2) is Equation (35.32) of Theorem 35.37 with the body force \(\vect{g}\) retained and the density in that force alone allowed to vary. Writing \(\rho\vect{g}=-\rho_{0}g\hat{\vect{z}} +\rho_{0}g\alpha_{T}(T-T_{0})\hat{\vect{z}}\) and absorbing the first, constant, piece into the pressure — which is what \(p'\) means — leaves the buoyancy term displayed. Equation (A65.3) is the advection–diffusion equation for the temperature of an incompressible fluid, viscous heating being of order \(\nu U^{2}/(\kappa\Delta T)\) relative to the terms kept and negligible in every laboratory convection experiment. For the conducting state, put \(\vect{u}=\vect{0}\) and seek a steady \(T\): Equation (A65.3) becomes \(\nabla^{2}T=0\), whose solution with the imposed boundary values is the linear profile Equation (A65.4), and Equation (A65.2) is then satisfied by a \(p'\) depending on \(z\) alone.
∎The Boussinesq approximation is an approximation and not a limit theorem: it is justified when \(\alpha_{T}\Delta T\ll1\) and when the layer is thin against the density scale height, and the honest statement of its domain belongs with the convection material of Fluid Dynamics, not here. This is an internal dependency of Part III — Classical Mechanics on itself rather than a debt owed by Part II — Mathematical Methods, and it is the only thing in this section taken from elsewhere. For air at room temperature \(\alpha_{T}\approx3.4\times 10^{-3}\,/\mathrm{K}\), so a layer driven by \(\Delta T=10\,\mathrm{K}\) has \(\alpha_{T}\Delta T=0.034\): the approximation is good to a few per cent, which is the accuracy at which Section 40.4 tests the threshold.
Rolls, the stream function, and the elimination of the pressure
Attention is restricted to motions independent of the horizontal coordinate \(y\) — rolls with axes along \(y\). The velocity is then written in terms of a stream function \(\psi(x,z,t)\) of SI dimension \(\mathrm{m}^{2}/\mathrm{s}\),
and the temperature by its departure from the conducting profile,
For any pair of functions the Jacobian is written
so that \(\left(\vect{u}\cdot\vect{\nabla}\right)h=J(\psi,h)\) for the field Equation (A65.5). Rests on Proposition A65.2 and Equation (35.11).
Definition A65.4 restricts attention to solutions of the three-dimensional equations Equation (A65.2) and Equation (A65.3) that happen not to depend on one of the three spatial coordinates. It is a symmetry ansatz inside the observed \(3+1\) spacetime, exactly like the axisymmetric ansatz used for Stokes flow, and it is not a model of a two-dimensional universe; the fluid, the layer and the rolls are all three-dimensional objects. The ansatz is a genuine restriction — real convection at large \(\mathrm{Ra}\) is not roll-like — and that restriction, not the dimensionality of space, is what has to be watched: it is one of the approximations listed in Remark A65.12.
For motions of the form Equation (A65.5), the system Equation (A65.2)–Equation (A65.3) is equivalent to the pair
in which the pressure has disappeared and \(\nabla^{2}\), \(\nabla^{4}\) act in the \((x,z)\) plane. Rests on Definition A65.4 and Proposition A65.2.
Derives Proposition A65.6. Incompressibility. \(\vect{\nabla}\cdot\vect{u} =-\pp_{x}\pp_{z}\psi+\pp_{z}\pp_{x}\psi=0\) identically, so Equation (A65.5) solves the constraint of Equation (A65.3) and no Lagrange multiplier is needed.
Vorticity. For a \(y\)-independent flow the vorticity has only a \(y\) component,
Taking the \(y\) component of the curl of Equation (A65.2) annihilates the pressure gradient, since the curl of a gradient vanishes; the buoyancy term \(g\alpha_{T}\theta\hat{\vect{z}}\) — the \(z\)-dependent part of \(T-T_{0}\) having been absorbed into \(p'\) along with the hydrostatic piece — contributes \(-g\alpha_{T}\pp_{x}\theta\); and the advective term contributes \(\left(\vect{u}\cdot\vect{\nabla}\right)\omega_{y}=J(\psi,\omega_{y})\), the vortex-stretching term \(\left(\vect{\omega}\cdot\vect{\nabla} \right)\vect{u}\) vanishing because \(\vect{\omega}\) points along \(y\) and nothing depends on \(y\). Hence
and substituting Equation (A65.10) and multiplying through by \(-1\) gives Equation (A65.8).
Temperature. Substituting \(T=T_{\mathrm{c}}+\theta\) into Equation (A65.3), and using \(\nabla^{2}T_{\mathrm{c}}=0\) because Equation (A65.4) is linear in \(z\),
and \(u_{z}=\pp_{x}\psi\) with \(\dd T_{\mathrm{c}}/\dd z=-\Delta T/d\) turns the third term into \(-\left(\Delta T/d\right)\pp_{x}\psi\), which moved to the right is Equation (A65.9). The physical content of that term is the whole instability: fluid rising through the mean gradient (\(\pp_{x}\psi>0\)) arrives warmer than its surroundings, which increases \(\theta\), which by Equation (A65.8) drives more rising motion.
∎The scaling, and the two dimensionless groups
Scale lengths by \(d\), time by \(d^{2}/\kappa\), the stream function by \(\kappa\) and the temperature departure by \(\nu\kappa/\left(g\alpha_{T}d^{3}\right)\),
Then Equation (A65.8)–Equation (A65.9) become, with the tildes dropped,
and exactly two dimensionless groups survive, the Prandtl and Rayleigh numbers
which are Equation (36.37) and Equation (36.38). Rests on Proposition A65.6 and Definition 35.51.
Derives Proposition A65.7. Under Equation (A65.13), \(\nabla^{2}=d^{-2}\tilde{\nabla}^{2}\), \(\pp_{t}=\left(\kappa/d^{2}\right)\pp_{\tilde{t}}\) and \(J=d^{-2}\tilde{J}\) acting on the scaled fields. The four terms of Equation (A65.8) then carry the prefactors
so dividing by \(\kappa^{2}/d^{4}\) leaves the factor \(\nu/\kappa=\sigma\) on the right-hand pair and gives Equation (A65.14). The four terms of Equation (A65.9) carry
and dividing by the first leaves the third with the factor
which is Equation (A65.15).
The dimension check. It is worth carrying out in full, because it is what allows \(\sigma\) and \(r\) to be read back as measurements:
and \(\sigma=\nu/\kappa\) is a ratio of two quantities in \(\mathrm{m}^{2}/\mathrm{s}\), so both are pure numbers, as Definition 35.51 requires. That no third group appears is the substance of the reduction: the four material properties Equation (A65.1) and the three imposed quantities \(g,\Delta T,d\) enter the scaled equations only through \(\sigma\) and \(\mathrm{Ra}\), and \(\rho_{0}\) has dropped out entirely, having been absorbed into \(p'\).
∎Linear stability and the critical Rayleigh number
Let the boundaries \(z=0,1\) be stress-free and perfectly conducting, so that \(\psi=\pp_{z}^{2}\psi=\theta=0\) there. Then the conducting state is marginally stable to the roll disturbance of dimensionless horizontal wavenumber \(\pi a\) precisely when
and the minimum of Equation (A65.21) over \(a\) is attained at \(a^{2}=\tfrac{1}{2}\), where
the value quoted in Definition 36.62 and in Equation (35.68). Rests on Proposition A65.7 and Phenomenon 35.78.
Derives Theorem A65.8. Linearisation. Drop the Jacobians from Equation (A65.14)–Equation (A65.15); they are quadratic in the disturbance and cannot affect the threshold.
The modes. Take
which satisfy every boundary condition: \(\sin(\pi z)\) and its second derivative vanish at \(z=0,1\). Both are eigenfunctions of the Laplacian, \(\nabla^{2}\to-k^{2}\) with
the sum of the squared horizontal wavenumber \(\pi a\) and the squared vertical wavenumber \(\pi\). Marginal stability means a steady disturbance, so set \(\pp_{t}=0\).
The two-by-two system. With \(\pp_{x}\theta=-\pi a\Theta\sin(\pi ax)\sin(\pi z)\) and \(\pp_{x}\psi=\pi a\Psi\cos(\pi ax)\sin(\pi z)\), the two linearised equations reduce to
A nonzero solution exists exactly when the determinant vanishes, \(k^{4}\cdot k^{2}=\mathrm{Ra}\,\pi^{2}a^{2}\), that is
which is Equation (A65.21).
The minimum. Write \(X=a^{2}>0\) and \(F(X)=\left(1+X\right)^{3}/X\). Then
which is negative for \(X<\tfrac{1}{2}\) and positive for \(X>\tfrac{1}{2}\): the unique minimum is at \(X=a^{2}=\tfrac{1}{2}\), and substituting gives Equation (A65.22). Numerically \(27\pi^{4}/4=657.51\). The corresponding horizontal wavelength is \(2\pi/(\pi a)=2/a=2\sqrt{2}\) in units of \(d\), which is the \(\lambda_{\mathrm{c}}=2\sqrt{2}\,d\) of Equation (35.68): convection cells are about \(1.4\) times as wide as the layer is deep, a prediction with no free parameter that Section 40.4 tests.
∎Fluid Dynamics writes the marginal curve as \(\mathrm{Ra}(a)=\left(\pi^{2}+a^{2}\right)^{3}/a^{2}\) and Nonlinear Dynamics and Chaos writes it as Equation (A65.21); the two are the same curve in different variables, and a reader comparing them will otherwise think one of them wrong. In Fluid Dynamics the symbol \(a\) is the horizontal wavenumber itself, in units of \(1/d\); here and in Definition 36.62 it is that wavenumber in units of \(\pi/d\), so that \(a_{\text{ch.\,}14}=\pi a\). Substituting turns one expression into the other, and the minimum sits at \(a^{2}=\pi^{2}/2\) in the first convention and \(a^{2}=\tfrac{1}{2}\) in the second. The second is used here because it is the one in which Equation (36.37) reads \(b=4/(1+a^{2})\), giving Lorenz's \(b=8/3\). Rests on Theorem A65.8 and Equation (35.68).
The Galerkin truncation
Fix a horizontal wavenumber \(a\). Write \(\tilde{x}\) for the dimensionless horizontal coordinate and \(z\) for the dimensionless vertical one, reserving \(x(s)\), \(y(s)\) and \(z_{\ast}(s)\) for the three amplitudes — these last are the \(x\), \(y\) and \(z\) of Equation (36.36), decorated here only so that no symbol does two jobs. Substitute into Equation (A65.14)–Equation (A65.15) the three-mode ansatz
with the dimensionless time
and project the resulting equations onto the three modes. Discarding the single term that leaves the retained set, the amplitudes obey
with the dot denoting \(\dd/\dd s\) and
These are Equation (36.36) and Equation (36.37); at the critical wavenumber \(a^{2}=\tfrac{1}{2}\) they give \(b=8/3\). Rests on Proposition A65.7, Theorem A65.8 and Definition 36.62.
Derives Theorem A65.10. Write \(\tilde{x}\) for the horizontal coordinate throughout, to keep it apart from the amplitude \(x(s)\), and \(z_{\ast}\) for the third amplitude, to keep it apart from the vertical coordinate \(z\). Put
with constants \(A,B,C\) to be determined and the abbreviations
All three are mutually orthogonal over one horizontal period and over \(0\le z\le1\), and all satisfy the boundary conditions of Theorem A65.8.
Step 1: the vorticity equation has no nonlinearity. Since \(\nabla^{2}\psi=-k^{2}\psi\) with \(k^{2}\) from Equation (A65.24), the Jacobian \(J\left(\psi,\nabla^{2}\psi\right)=-k^{2}J(\psi,\psi)=0\) identically: a single roll does not advect its own vorticity. Using \(\pp_{\tilde{t}}=k^{2}\dd/\dd s\) from Equation (A65.30), Equation (A65.14) becomes
because \(\pp_{\tilde{x}}C_{1}=-\pi aS_{1}\) and the \(S_{2}\) term of \(\theta\) is independent of \(\tilde{x}\). Dividing by \(-k^{4}A/\sigma\),
which is the first of Equation (A65.31) provided
Step 2: the Jacobian in the temperature equation, in full. This is the step that decides whether three modes close, and it cannot be seen without doing it. From Equation (A65.33),
Assembling \(J(\psi,\theta)=\pp_{\tilde{x}}\psi\,\pp_{z}\theta -\pp_{z}\psi\,\pp_{\tilde{x}}\theta\) gives three products. The two involving \(y\) combine, since
using \(\sin\phi\cos\phi=\tfrac{1}{2}\sin2\phi\): the horizontal dependence cancels completely, and what is left is a mode with twice the vertical wavenumber and no horizontal structure. That is the physical origin of the third mode — a roll carrying warm fluid up on one side and cool fluid down on the other distorts the horizontally averaged temperature profile — and it is why \(z_{\ast}\) must be kept. The remaining product is
by \(\sin\phi\cos2\phi=\tfrac{1}{2} \left[\sin3\phi-\sin\phi\right]\). Its \(\sin(\pi z)\) half is \(+\pi^{2}aAC\,xz_{\ast}C_{1}\), a term in the retained set; its \(\sin(3\pi z)\) half is not, and it is the only term produced anywhere in the calculation that leaves the three-mode set. Discarding it is the entire content of the truncation.
Step 3: projection. Insert Equation (A65.39), the retained half of Equation (A65.40), and \(\pp_{\tilde{x}}\psi=\pi aA\,x\,C_{1}\) into Equation (A65.15), use \(\nabla^{2}C_{1}=-k^{2}C_{1}\) and \(\nabla^{2}S_{2}=-4\pi^{2}S_{2}\), and collect the coefficients of the two orthogonal modes. The \(C_{1}\) component gives
and the \(S_{2}\) component, remembering the minus sign carried by \(z_{\ast}\) in Equation (A65.33), gives
Step 4: fixing the constants. Dividing Equation (A65.41) by \(k^{2}B\) and Equation (A65.42) by \(-k^{2}C\),
Comparison with Equation (A65.31) imposes three conditions, to be read together with Equation (A65.37):
The last two give \(C=k^{2}B/\left(\pi^{2}aA\right)\) and \(C=\pi^{2}aAB/\left(2k^{2}\right)\); equating them yields \(\pi^{4}a^{2}A^{2}=2k^{4}\), hence
by Equation (A65.24), which is the coefficient in Equation (A65.28). Then Equation (A65.37) gives
the last equality being Equation (A65.26), and \(C=k^{2}B/(\pi^{2}aA)=B/\sqrt{2} =\mathrm{Ra}_{\mathrm{c}}(a)/\pi\) after substituting Equation (A65.45). With \(\theta=By\,C_{1}-Cz_{\ast}S_{2}\) this is exactly Equation (A65.29). Finally the first of Equation (A65.44) reads, using Equation (A65.37),
and the last term of Equation (A65.43) gives
which completes Equation (A65.32). At \(a^{2}=\tfrac{1}{2}\), \(b=4/(3/2)=8/3\).
∎Reading the parameters back
Undoing the scaling Equation (A65.13) makes every symbol of Equation (36.36) a statement about the layer. The variables. \(x\) is the amplitude of the convective roll: the physical stream function is \(\kappa\,A\,x\,S_{1}\), so the fluid speed is of order \(\left(\kappa/d\right)\left(1+a^{2}\right)\abs{x}\) — \(2\times 10^{-5}\,\mathrm{m}^{2}/\mathrm{s}\) divided by a centimetre, of order \(1\,\mathrm{mm}/\mathrm{s}\) for air at \(\abs{x}\) of order unity. \(y\) measures the horizontal temperature variation at the roll's own wavenumber, and \(z_{\ast}\) the horizontally averaged distortion of the vertical temperature profile. Their common temperature scale is not arbitrary: substituting Equation (A65.46) into Equation (A65.13) gives
where \(\Delta T_{\mathrm{c}}\) is the temperature difference at which that wavenumber first becomes unstable — so \(y\) and \(z_{\ast}\) are temperature departures measured in units of the critical driving difference divided by \(\pi\). The parameters. \(\sigma=\nu/\kappa\) is the Prandtl number of the working fluid alone: about \(0.71\) for air, \(7\) for water at \(20\,\mathrm{^\circ\mathrm{C}}\), of order \(0.02\) for liquid sodium and \(10^{2}\) or more for oils. Lorenz's \(\sigma=10\) is therefore a plausible fluid and not a dial setting. \(r=\mathrm{Ra}/ \mathrm{Ra}_{\mathrm{c}}\) is the imposed temperature difference in units of the one that starts convection, so \(r=28\) means a layer driven \(28\) times past onset. \(b=4/(1+a^{2})\) is a pure function of the horizontal wavenumber of the roll and of nothing else at all. Rests on Theorem A65.10 and Remark 36.2.
Five approximations were made, and they are of quite different standing. One is the Boussinesq approximation (Remark A65.3), good to a few per cent in ordinary laboratory convection. Two is the restriction to rolls (Remark A65.5), which is a fact about the observed flow only near onset. Three is stress-free boundaries, chosen because they make \(\sin\pi z\) an exact eigenfunction; the laboratory case of two rigid plates gives \(\mathrm{Ra}_{\mathrm{c}}\approx1708\) instead of \(657.5\) (Phenomenon 35.78) and no closed-form modes at all, so the tidy arithmetic above is bought with a boundary condition no experiment has. Four is the fixing of a single horizontal wavenumber \(a\), which suppresses every interaction between cells of different sizes. Five, and the serious one, is the discarding of the \(\sin\left(3\pi z\right)\) term at Equation (A65.40). That discarded term is small only while the convection is weak. At Lorenz's \(r=28\) it is not small, the neglected modes are not small, and Lorenz said so plainly in the paper that introduced the system [Lorenz:1963]. The projection is exact on the linear terms — which is why Theorem A65.8 is a genuine result about fluids — and an approximation on the nonlinear ones, which is why Equation (36.36) beyond onset is a model of convection and not convection. Remark 36.63 makes the same point in the chapter, and the two statements are meant to stand together: what Equation (36.36) is good for is the study of a low-dimensional dissipative flow that happens to have been derived from a fluid, not the prediction of a heat flux. Rests on Theorem A65.10 and Remark 36.63.
The Lorenz System from Boussinesq Convection discharges the derivation owed at Definition 36.62 of Nonlinear Dynamics and Chaos. The four stages announced there are Rolls, the stream function, and the elimination of the pressure (the stream function, which eliminates the pressure), The scaling, and the two dimensionless groups (the explicit scaling and the two dimensionless groups Equation (36.38)), Linear stability and the critical Rayleigh number (the linear stability calculation giving \(\mathrm{Ra}_{\mathrm{c}}=27\pi^{4}/4\) at \(a^{2}=\tfrac{1}{2}\) for stress-free boundaries [Rayleigh:1916]), and The Galerkin truncation (the Galerkin truncation producing Equation (36.36) with the parameters Equation (36.37)). Everything the chapter builds on Equation (36.36) — the volume contraction, the fixed points, the strange attractor — is derived there and does not depend on this section; what this section supplies is the licence to call \(\sigma\), \(r\) and \(b\) measurements rather than dials, which is the demand Remark 36.2 makes of every model in that chapter.