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Saltzman’s Model. Part I: Complete Characterization of Solution Properties

S. Lakshmivarahan School of Computer Science, University of Oklahoma, Norman, Oklahoma

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John M. Lewis National Severe Storms Laboratory, Norman, Oklahoma, and Desert Research Institute, Reno, Nevada

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Junjun Hu Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

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Print Publication:
01 Jun 2019
DOI:
https://doi.org/10.1175/JAS-D-17-0344.1
Page(s):
1587–1608
Full access

Abstract

In Saltzman’s seminal paper from 1962, the author developed a framework based on the spectral method for the analysis of the solution to the classical Rayleigh–Bénard convection problem using low-order models (LOMs), LOM (n) with n ≤ 52. By way of illustrating the power of these models, he singled out an LOM (7) and presented a very preliminary account of its numerical solution starting from one initial condition and for two values of the Rayleigh number, λ = 2 and 5. This paper provides a complete mathematical characterization of the solution of this LOM (7), herein called the Saltzman LOM (7) [S-LOM (7)]. Historically, Saltzman’s examination of the numerical solution of this low-order model contained two salient characteristics: 1) the periodic solution (in the physical 3D space and time) that expand on Rayleigh’s classical study and 2) a nonperiodic solution (in the temporal space dealing with the evolution of Fourier amplitude) that served Lorenz in his fundamental study of chaos in the early 1960s. Interestingly, the presence of this nonperiodic solution was left unmentioned in Saltzman’s study in 1962 but explained in detail in Lorenz’s scientific biography in 1993. Both of these fundamental aspects of Saltzman’s study are fully explored in this paper and bring a sense of completeness to the work.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: S. Lakshmivarahan, varahan@ou.edu

Abstract

In Saltzman’s seminal paper from 1962, the author developed a framework based on the spectral method for the analysis of the solution to the classical Rayleigh–Bénard convection problem using low-order models (LOMs), LOM (n) with n ≤ 52. By way of illustrating the power of these models, he singled out an LOM (7) and presented a very preliminary account of its numerical solution starting from one initial condition and for two values of the Rayleigh number, λ = 2 and 5. This paper provides a complete mathematical characterization of the solution of this LOM (7), herein called the Saltzman LOM (7) [S-LOM (7)]. Historically, Saltzman’s examination of the numerical solution of this low-order model contained two salient characteristics: 1) the periodic solution (in the physical 3D space and time) that expand on Rayleigh’s classical study and 2) a nonperiodic solution (in the temporal space dealing with the evolution of Fourier amplitude) that served Lorenz in his fundamental study of chaos in the early 1960s. Interestingly, the presence of this nonperiodic solution was left unmentioned in Saltzman’s study in 1962 but explained in detail in Lorenz’s scientific biography in 1993. Both of these fundamental aspects of Saltzman’s study are fully explored in this paper and bring a sense of completeness to the work.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: S. Lakshmivarahan, varahan@ou.edu
Keywords: Differential equations; Spectral analysis/models/distribution

1. Introduction

Thermally induced convection in response to a fluid heated from below has found many applications in meteorology. Among the events that stem from this process are buoyant plumes of air adjoining the ground or sea surface in response to solar heating of that surface, sea breezes that are generated through differential solar heating of sea and adjoining land surfaces, and the formation of stratus cloud over the ocean in the presence of warm sea surface temperature. The classical Oberbeck and Boussinesq approximation (Saltzman 1962) has provided the mathematical basis for the analysis of convection arising from thermal instability for well over a century. In addition to providing a historical account of the developments, Chandrasekhar (1961) contains a thorough analysis of the linear version of the thermally induced convection problem.

There are essentially two different routes to solve the system of nonlinear coupled partial differential equations (PDEs) arising from the above said approximation. The first is to numerically simulate convection using a suitable space–time grid. The second is to capture the spatial variations to any desired degree of accuracy using a finite number of Fourier modes and reduce the PDE to an initial value problem consisting of n coupled nonlinear ordinary differential equations (ODEs). This resulting system of coupled nonlinear equations is called a low-order model (LOM) of order n and is denoted by LOM (n). This latter approach is known as the Galerkin projection method or simply a spectral method (Canuto et al. 2007; Shen et al. 2011).

Early applications of both the space–time gridpoint method and the spectral method to the convection problem centered on linear dynamics and steady-state solutions. A nearly complete list of contributions to the problem up through the early 1970s is found in J. S. Turner’s classic treatise on buoyancy in fluids (Turner 1973, chapter 7). Prior to Saltzman’s (1962) contribution, a theoretical paper by Malkus and Veronis (1958) established the two-dimensional spectral framework that he followed.

Saltzman (1962) meticulously developed the structure of a family of LOM (n) for n ≤ 52. This family of models has been the basis for numerous subsequent studies. The well-known LOM (3), known as the Lorenz 1963 model (Lorenz 1963), is a member of this family of models as are the ones in Curry (1978).

By way of illustration, Saltzman (1962) concluded his paper with a preliminary and incomplete analysis of a seven-mode model, called the Saltzman LOM (7) [S-LOM (7)]. If Lorenz’s biography (Lorenz 1993) is any guide, using this LOM (7), Saltzman seems to have demonstrated the existence of a nonperiodic solution to Lorenz at the time when he visited Saltzman in 1961.

Despite the importance of this chaotic aspect of Saltzman’s LOM (7), it is surprising that he failed to mention it in his treatise. Yet we know from Lorenz’s scientific biography (Lorenz 1993, p. 137) that Saltzman was aware of this important feature. Although Lorenz (1993) gives full credit to Saltzman for showing him the preliminary nonperiodic results that were critical to his seminal paper, we the readership of Lorenz (1963) were unaware of this important interaction of the fellow classmates at MIT in the 1950s.

Thus, there was a strong motivation to conduct a full-scale examination of the properties of S-LOM (7). Our analysis reveals that S-LOM (7) exhibits an inherent competition between the amplification of energy resulting from the nonlinear interaction and dissipation resulting from the friction terms. For initial conditions (ICs) close to the origin, the amplification component dominates, and the solution X(t) grows in time. Once X(t) grows to a sufficient level, the dissipation part takes control leading to asymptotic convergence in one of the four (two in each of the invariant subspaces IS1 and IS2) 1D equilibrium manifolds described below. It turns out that the S-LOM (7) can be decomposed into a union of two copies of Lorenz-like LOM (3) and one copy of a linear LOM (3), which are interconnected by a set of nine nonlinear coupling terms. It turns out that each of these two Lorenz-like subsystems and the linear LOM (3) define three invariant subspaces for S-LOM (7). The solution X(t) ∈ R7 of the S-LOM (7), depending only on the distribution of energy in the initial condition and the value of the parameter λ, finds itself in one of the three invariant subspaces.

The asymptotic behavior of X(t) in these invariant subspaces is essentially dictated by the rotation symmetry of the S-LOM (7) projected onto these invariance subspaces. Two of the invariant subspaces, labeled IS1 and IS2, each admit two branches of 1D equilibrium manifolds parameterized by λ. For the third invariant subspace IS3, the origin is the only stable attractor. Finally, we bring out the multifaceted behavior of S-LOM (7) using a deterministic ensemble experiment by starting the solution from the 128 corners of a seven-dimensional hypercube centered at the origin with sides of length 2α for α ∈ (0, 1].

There is a close connection between the class of energy conserving LOM (n) and the system of Volterra gyrostats and their generalization. Refer to Gluhovsky and Tong (1999), Lakshmivarahan and Wang (2008), and Tong (2009) for details.

In section 2, we identify the three invariant subspaces and the equilibria contained in them. The stability properties of these equilibria are developed in section 3. Section 4 contains an analysis of the global properties of the solution of S-LOM (7). Results of the deterministic ensemble and other experiments are contained in section 5. A concluding summary is given in section 6. Detailed analysis of the stability of the isolated equilibrium at the origin is contained in appendix A. Analysis of the boundedness of the solution is given in appendix B. Appendix C describes a simple encoding scheme to represent the 128 corners of the seven-dimensional hypercube used in the ensemble analysis given in section 5. Finally, in appendix D, we describe a linear invertible transformation, using which we prove the equivalence between the Lorenz-like LOM (3) in IS1 and the Lorenz (1963) model with and .

2. Analysis of Saltzman’s model

The equations of S-LOM (7) may be stated as follows. Let and , the S-LOM (7) is given by
e1
where is the vector field and is the Rayleigh number. From pages 336 and 340 in section 7 in Saltzman (1962), is a control parameter, and the components of this vector field (after relabeling: ) are given by
e2

It can be verified (Saltzman 1962) that the values of the coefficients in (2) depend on (i) the fundamental wavelength L in the (horizontal) x and 2H in the (vertical) z directions and (ii) the wavenumbers m and n in the x and z directions, respectively. Table 2 in Saltzman (1962) contains a listing of the numerical values of for low-order models of order n ≤ 52. For quick and easy reference, the subset of values of for the S-LOM (7) in (2) extracted from Saltzman (1962) are given in Table 1 of this paper. Notice that λ affects only three components, , , and , rather directly. Assuming the existence and uniqueness of the solution of (1), let denote the solution starting from the initial condition for a given . Clearly, defines the nonlinear flow that relates the initial condition to the solution at time . In other words, defines the flow induced by the vector field in (1).

  • Definition 2.1: A subset is an invariant set or subspace for the flow induced by S-LOM (7) in (1) and (2), if, for , for all . That is, the solution starting from S remains in S for all times.

    • Clearly, the whole space R7 and the origin are trivially invariant sets. Our interest is in finding nontrivial invariant sets.

  • Definition 2.2: A set of points, where the field vanishes, that is,
    eq1
    constitutes the equilibria for the S-LOM (7) (Hirsch and Smale 1973).

    • The equilibria may either be a collection of isolated points or a continuous curve or a manifold in R7 parameterized by λ.

Table 1.

Values of coefficients .

Table 1.

Notice that while an equilibrium is an invariant set, the converse is not true. It turns out that S-LOM (7) enjoys several equilibria and invariant sets.

a. Equilibrium E1

It can be verified by inspection that ; the origin of is an isolated equilibrium for S-LOM (7) for all . That is,
e3

b. Invariant subspace

It turns out that S-LOM (7) in (1) and (2) admits three invariant subspaces labeled as IS1, IS2, and IS3.

1) Invariant subspace IS1

It can be verified that the 3D subspace defined by
eq2
is an invariant subspace for S-LOM (7).
By projecting the S-LOM (7) onto this subspace, we obtain the reduced dynamics in IS1 given by
e4
where , with , where
e5
This closed subsystem resides in the 3D invariant subspace spanned by the original variables . Notice also that new variables are the surrogates for the original for in IS1.

2) Invariant subspace IS2

Proceeding likewise, it can be verified that
eq3
is an invariant subspace for the S-LOM (7) in (1) and (2). Again, by projecting S-LOM (7) onto this subspace, the resulting reduced dynamics in IS2 is given by
e6
where , with , where
e7
This closed subsystem resides in the 3D invariant subspace spanned by the original variables . Clearly, the new variables are the surrogates for the original for in IS2.

3) Invariant subspace IS3

The 3D subspace defined by
eq4
is an invariant subspace for the S-LOM (7). The projecting S-LOM (7) onto IS3 gives the following linear dynamics:
e8
where and given by
e9
This closed subsystem lies in the 3D invariant subspace spanned by , where are the surrogates for the original in IS3 for .

We hasten to add that a careful reading of the evolution of the solution of the S-LOM (7) given in Fig. 3, page 341 in Saltzman (1962) may suggest the presence of invariant subspaces IS1 and IS2.

4) Saltzman–Lorenz interactions: S-LOM (7)

We begin this discussion with the key quotation from Lorenz’s scientific biography The Essence of Chaos (Lorenz 1993, p. 137) that is most pertinent:

At the Tokyo meeting1 more than a decade earlier I had briefly mentioned the unexpected behavior of the twelve-variable model,2 but I felt that a full discussion of the relationship between lack of periodicity and growth of small disturbances, and its implications for long-range weather forecasting, belonged in a separate paper. For that paper I was anxious to use an even simpler system of equations as a principal illustrative example…. I tried to simplify the model still more without any luck…. My search came to an abrupt end one afternoon in 1961 when I was visiting Barry Saltzman at the Travelers Research Center…. Barry showed me a system of seven equations that he had been solving numerically. The equations were a bit like mine, but they modeled convective fluid motion driven by heating from below…. He was interested in periodic solutions and had obtained a number of them, but he showed me one solution that refused to settle down. I looked at it eagerly, and noted that four of the seven variables became very small. This suggested that the other three were keeping each other going, so that the system with only these three variables might exhibit the same behavior. Barry gave me the go-ahead signal, and back at M. I. T. the next morning I put the three equations on the computer and, sure enough, there was the same lack of periodicity that Barry had discovered. Here was the long-sought system whose existence I had begun to doubt.

And, of course, exploration of this system led to Lorenz’s seminal contribution, deterministic nonperiodic flow (Lorenz 1963).

A number of observations are in order:

  1. Lorenz’s system of three equations are given by
    e10
    using the spectral expansion (refer to Saltzman 1962, section 7)
    e11
    Comparing (10) with the reduced system (5) in IS1 and (7) in IS2, it turns out that both of these are Lorenz-like systems.

  2. The three mode models in (5) and (7) differ from the Lorenz model in (10) only in the scaling of the variables. The parameter in (10) corresponds to in (5) and (7). In the analysis of the Lorenz model (10), the aspect ratio parameter is set at 8/3 and the Prandtl number is set at 10. A careful look at the derivation of S-LOM (7) in Saltzman (1962) reveals that he has already incorporated and in the numerical computation of the values of the coefficients .

  3. While our intent is not in analyzing the comparative powers of different low-order models to explain the convection phenomenon, an observation relating to the S-LOM (7) is worth mentioning here.

    The second sentence in Saltzman’s (1962) conclusions (section 8 of his paper) reads as follows:

    …in spite of its simplicity the system treated does, in fact, appear to contain a good deal of the real physical content of the problem [Bénard’s laboratory experiment].

    To further amplify on this statement, consider water at 20°C with the follow physical parameters: coefficient of thermal diffusivity , kinematic viscosity3 , and coefficient of volume expansion , and where the depth of the water H is 2 mm in accord with order of magnitude fluid depth in experiments reviewed by Chandrasekhar (1961, chapter 2).

    In this case, the Rayleigh number given by
    eq5
    where g is the acceleration of gravity and is the temperature difference over the depth of water. If λ = 2, the adverse temperature gradient is , with .

    Under these circumstances, it is interesting and reasonable to consider the favored convective regime as a function of the Rayleigh number—one with a three-wave pattern over the horizontal distance of or four waves over this horizontal distance . The regime at equilibrium for λ = 2 is the 3-wave regime. If λ is decreased by 5%, λ = 1.9 instead of λ = 2, the three-wave pattern remains with slight changes in magnitude of the spectral components; however, if λ is increased by 5%, from λ = 2 to λ = 2.1, the regime changes to a four-wave pattern. In short, if is increased from 3.7° to 3.9°C, there is a regime change, whereas if it is decreased to 3.5°C, there is no regime change. In essence, this was the type of question Rayleigh considered in development of theory underlying Bénard’s experiments (Rayleigh 1916, 537–539)—the size of cells in the presence of fluid instability.

  4. S-LOM (7) in (1) and (2) is the union of the three reduced systems in IS1, IS2, and IS3 and also contains an additional nine nonlinear interaction terms. Despite these additional nonlinear interaction terms, the following analysis shows that asymptotically the solution of S-LOM (7) finds itself in either IS1 or IS2. Further, depending only on the I.C and , after entering into IS1 or IS2, the solutions approach stable equilibrium in the 1D manifold parameterized by . These stable values of the Fourier amplitudes correspond to the periodic solution of the Oberbeck–Boussinesq equation in Saltzman (1962). This property partially captured in Fig. 1. For values of , the solution exhibits chaotic behavior in IS1 and IS2.

Fig. 1.
Fig. 1.

A pictorial view of the asymptotic behavior of S-LOM (7). While the invariant subspaces IS1, IS2, and IS3 have a common dimension , for clarity these invariant subspaces are shown as distinct sets. and are the loci of the stable equilibria embedded in IS1 and IS2, respectively.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

3. Invariant subspaces and equilibria

Against the backdrop of the existence of equilibria and three invariant subspaces, we now describe the stability of the equilibrium at E1 and the asymptotic properties of the three reduced dynamics in ISi, i = 1, 2, 3.

a. Stability of the equilibrium E1

To further characterize the stability properties of E1, we need to compute the eigenvalues of the Jacobian, of the field in (2) at given by
e12

Let be the eigenvalue–vector pair for the matrix in (12), where it is assumed without loss of generality that . In appendix A, it is shown that this computation of the eigensystem for (12) reduces to that of four separable, simple systems. An immediate consequence of this separability is that it greatly simplifies the problem of computing eigenvectors of (12). It follows from appendix A that exactly two eigenvectors of (12) reside in each of the three invariant subspaces IS1, IS2, and IS3 and the seventh one lies along the seventh unit vector .

The entries of eigenvalues of (12) in Table 2 as a function of are obtained from the computation in appendix A by sorting the [ is defined in (A2)–(A6) in appendix A]. For example, and .

Table 2.

Eigenvalues of at for different λ.

Table 2.

Referring to Table 2, the origin undergoes bifurcation twice: first at and again when . It can be verified that the unstable eigenvector at is given by
eq6
At , the two unstable eigenvectors are given by
eq7
Thus, for λ < 1.0004, the origin is a stable attractor. For , the origin becomes a saddle point. For , solutions starting close to the origin leaves the origin along the unstable direction given above. But for , the solutions leave the neighborhood of the origin along or depending on the IC and . Also notice that the energy in and are concentrated along dimensions and but that for are concentrated along and . It turns out, as shown below, that and lie in two complementary invariant subspaces.

We now move on to the analysis of the three reduced subsystems. Since it is easy to analyze the linear subsystem in IS3, we take it up first.

b. Analysis of the reduced linear dynamics in IS3

Referring to (8) and (9), is decoupled from and . Besides, since , it turns out that
e13
which tends to 0 exponentially fast. The dynamics of is given by
e14
The eigenvalues of the 2 × 2 matrix in (14) are the roots of
eq8
It can be verified that the two roots and of are real, distinct, and negative for all . Hence, the origin of the invariant subspace IS3 is a stable attractor in IS3 and for all . Refer to the illustration in Fig. 1.

An immediate import of this analysis is that, if any part of the energy in the solution of S-LOM (7) starting from initial conditions spills over to IS3, then that portion of the energy will dissipate to zero.

c. Analysis of stability of the reduced nonlinear dynamics in IS1

Solving the system of three nonlinear equations in the three variables in
e15
where is given in (5), we obtain the locus of the equilibrium in IS1 (see Fig. 1). To this end, define a new set of parameters:
eq9
The values of are given in Table 3.
Table 3.

Values of parameters ai and bi, with i = 1, 2, 3, 4.

Table 3.
Solving (15), it follows that the set of all equilibria of the reduced subsystem (5) in IS1 is given by
eq10
eq11
and
e16
From the third equation, either or
e17
Since (see Table 3), it follows that
eq12
Hence, for , is given by
e18
Substituting (18) in the first two equations in (16), we get a total of three equilibria:
e19
A remark on the symmetry of the dynamics (4) and (5) is in order here.
It can be verified that the Lorenz-like system (4) and (5) has an intrinsic symmetry with respect to the rotation by π radians about the axis. Define
eq13
which represents a rotation operator by π radians with respect to the axis. If , then , and , the identity matrix. Also, it can be verified that
eq14
That is
eq15
Stated in other words, (4) and (5) are said to be equivariant (Gilmore and Letellier 2007) under the symmetry matrix . Hence, it is not surprising that there is a system of two families of equilibria with signs given by and , which are rotations by π radians of each other with respect to . For future reference, we denote these two branches of the equilibrium manifold as and , respectively.
To get an idea of the structure of this equilibria, from (18) and (19), consider one of the branches given by
e20
A plot of versus and a 3D view of this branch of equilibria for are given in Fig. 2 and Fig. 3.
Fig. 2.
Fig. 2.

Plots of the steady states in as a function of λ.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Fig. 3.
Fig. 3.

A 3D plot of the steady states in as λ is varied from 1.1 to 30.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

A sampling of the steady-state values as a function of are given in Table 4.

Table 4.

A sampling of the values of the steady states in as a function of λ.

Table 4.

To characterize the stability of these equilibria, first consider the Jacobian of (4) and (5) given by
e21

1) Analysis of the equilibrium at

At this equilibrium, takes the form
e22
whose eigenvalues are given by and the two roots of
e23
It can be verified the two roots of (23) are real and negative only for . Hence, this equilibrium in is unstable for .

2) Analysis of the equilibrium at

The variation of the equilibrium and the eigenvalues of the Jacobian at the equilibria of the Lorenz-like system in (4) and (5) for are given in Table 4. For , this equilibrium is an attractor and the solution Y(t) spiral toward and settle down on Y listed in Table 4. Recall that , , and are the three amplitudes of the 2D Fourier expansion of the solution of the Oberbeck–Boussinesq approximation (Saltzman 1962). Thus, for in this range, the stable asymptotic solution naturally corresponds to the periodic solution in the physical space. However, for , the solution of (4) and (5) undergoes Hopf bifurcation, the equilibria becomes unstable, and the solution in IS1 exhibits chaotic behavior. Stated in other words, the Lorenz-like system in (4) and (5) exhibits characteristics quite similar to the Lorenz (1963) model in (10). Refer to chapter 32 in Lewis et al. (2006) for details.

A sample of this chaotic behavior for and IC , , and is illustrated in Fig. 4a. We further quantify the key signatures of this chaotic behavior by computing the Lyapunov exponent (Wolf et al. 1985) and the fractal dimension (Grassberger and Procaccia, 1983) of the strange attractor for the system (4) and (5) in IS1. Variation of the Lyapunov index with is given in Fig. 4b. Clearly, becomes positive when , indicating the onset of instability leading to the chaotic behavior. Further, the sum for all confirms that the phase volume shrinks even in this chaotic regime. A comparison of the fractal dimension of the strange attractor for the Lorenz model in (10), the Lorenz-like system (4) and (5) in IS1, and (6) and (7) in IS2 are given in Table 5.

Fig. 4.
Fig. 4.

(a) Phase plot for chaotic behavior in IS1 [IC, , , , and ]. (b) Variation of the Lyapunov exponent with λ for the Lorenz-like dynamics (4) and (5) in IS1 starting from the same initial condition as in Fig. 4a, refers to the ith exponent and sum is .

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Table 5.

Comparison of the fractal dimension of Lorenz attractor with those of the three sets of attractors arising from S-LOM (7).

Table 5.

d. Analysis of asymptotic stability of the reduced nonlinear dynamics in IS2

Solving the system of three nonlinear equations in the three variables in in
e24
where is given in (7), we obtain the locus of the equilibria in IS2 (see Fig. 1). To this end, define control parameters
e25
The values of are given in Table 3.
Solving (24), it follows that the set of all equilibria for the reduced dynamics in (6) and (7) in IS2 are given by
e26
From the third equation in (26), either or
e27
Since b1b2b3 is negative (refer to Table 3), it is immediate that
e28
Hence, for , is given by
e29
Substituting (28) in the first two equations in (26), we obtain a total of three equilibria given by
e30
For future reference, we denote these two branches of the equilibrium manifold in IS2 as and , respectively.
To analyze the stability of these equilibria, consider the Jacobian of (6) and (7) given by
e31

1) Analysis of the equilibrium at

At this equilibrium, takes the form
e32
whose eigenvalues are given by and the two roots of
e33
It can be verified the two roots of (33) are real and negative only for . Hence, this equilibrium is unstable for .

Since (6) and (7) are also a Lorenz-like system, it enjoys symmetry under rotation by π radians. Hence there are two branches of the equilibria with signs and .

2) Analysis of the equilibrium at

For definiteness, consider the branch of equilibria given by
e34
A plot of the steady states versus are given in Fig. 5. Figure 6 contains a 3D plot of the steady states, which in turn defines periodic solution in the physical space. A sampling of the values of the steady states in IS2 and the eigenvalues of the Jacobian along the steady states for are given in Table 6. Again, the solution of (6) and (7) undergoes a Hopf bifurcation at , and for , the equilibria become unstable. In this unstable regime, the model exhibits chaotic behavior. Fractal dimension of the resulting attractor is given in Table 5. A sample of this behavior for and IC , , and is illustrated in Fig. 7.
Fig. 5.
Fig. 5.

Plots of the steady states in as a function of λ.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Fig. 6.
Fig. 6.

A 3D plot of the steady states in as λ is varied from 1.2 to 30.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Table 6.

A sampling of the values of the steady states in as λ is varied.

Table 6.
Fig. 7.
Fig. 7.

Time evolution and phase plot for chaotic behavior in IS2 [IC, , , and ].

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

4. A global perspective

In this section, we address two important global properties of the S-LOM (7) in (1). These relate to the rate of change of phase volume and the boundedness of the trajectories of (1).

The divergence of the vector field in (1) using the values of the coefficients in Table 1 is given by
e35
Hence, by Liouville’s theorem in Arnold (1992, section 6, article 27, chapter 3), the phase volume decreases at an exponential rate proportional to the term . Consequently, the equilibrium has to be a manifold of zero phase volume. This is corroborated by our analysis in sections 2 and 3, where it is shown that the equilibrium is either the origin or the 1D manifolds in IS1 and IS2.
To examine the boundedness of the solution of (1), consider a generalized energy functional given by
e36
where is a diagonal matrix given by
e37
with for .
It is shown in appendix B that for large time t,
e38
Since the vector field in (1) is smooth in time t, the solution cannot become unbounded in finite time. Combining these, it is immediate that the solution remains bounded for all time .

5. An ensemble analysis

To further understand the dependence of the behavior of the solution of (1) on the initial conditions and the parameter , in this section, we describe the results of a deterministic ensemble experiment. We choose a set of 27 = 128 initial conditions corresponding to the coordinates of the vertices/corners of the seven-dimensional hypercube centered at the origin with the length of the sides given by 2α for and varying . Thus, is one vertex as is . Using the one-to-one encoding scheme described in appendix C, we compress the seven coordinates into an integer. Accordingly, the vertex with coordinates is denoted by the number zero and the vector with coordinates is denoted by the number 127. Likewise, the vector with coordinate is denoted by the integer 83. See appendix C for details.

a. Experiment 5.1

By keeping fixed and varying the initial conditions over the 128 vertices of the hypercube labeled from 0 through 127 with the model solutions computed using the Runge–Kutta routine in MATLAB are plotted in Figs. 8 and 9. A record of the equilibria to which the solutions converge are given in Table 7. Recall from our earlier discussion in section 3 that each of the steady-state solutions of S-LOM (7) defines a periodic behavior in the physical space.

Fig. 8.
Fig. 8.

Plot of the components of the solution of (1) for converging to equilibria in IS1.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Fig. 9.
Fig. 9.

Plot of the solution of (1) for converging to equilibria in IS2.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Table 7.

A record of the equilibria to which the solutions converge when λ = 2.

Table 7.

Figure 8 contains the plot of the solution corresponding to 96 initial conditions that lead to convergence to the equilibria in IS1. Of these, the solutions corresponding to initial conditions converging to the equilibria in are shown in the left column of Fig. 8, and the solutions from the rest of the initial conditions converging to its image equilibria in are shown in the right column of Fig. 8.

Figure 9, likewise, contain the plot of the solutions from the rest of 32 initial conditions converging to the equilibria in . Of these, exactly solutions converge to , as shown in the left column of Fig. 9, and the rest solutions converge to the equilibria in , as given in the right column of Fig. 9.

Stated in other words, the entries in the Table 7 and the plots in Figs. 8 and 9 together provide a complete picture of the behavior of the ensemble for .

b. Experiment 5.2

By keeping as in experiment 5.1 and increasing , we repeated the analog of experiment 5.1. It turns out in this case, the solution from all the 128 points converge to an equilibrium in IS2 with exactly converging to and the rest converging to as shown in Table 8 and Fig. 10.

Table 8.

As in Table 7, but for λ = 2.5.

Table 8.
Fig. 10.
Fig. 10.

Plot of the solution of (1) for converging to equilibria in IS2.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

c. Experiment 5.3

In this experiment, we explore the behavior of the solution of the S-LOM (7) in (1) and (2) starting from the same initial condition as in Saltzman 1962 paper, namely, and and when . This initial condition is such that there is nonzero initial energy in each of the subspaces IS1, IS2, and IS3. Since , the energy within the subspaces IS1 and IS2 grow and become chaotic, and the solution in the three subspaces interact through the common dimension . Phase plots versus for for this case are given in Fig. 11. From the figure, we can easily identify the chaotic behavior in the form of a double attractor, one in each of the subspaces IS1 and IS2 and their interaction with IS3. Variation of the seven Lyapunov exponents for the S-LOM (7) in (1) and (2) with starting from the same initial condition as in Saltzman (1962) is given in Fig. 12. Positive values of and further confirm the simultaneous onset of chaos in IS1 and IS2. Also notice that the overall sum of all the seven Lyapunov exponents are negative. It should be interesting to identify the different basins of attraction for various values of .

Fig. 11.
Fig. 11.

Phase plot of solution to S-LOM (7) with and —same initial condition as in Saltzman (1962).

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

Fig. 12.
Fig. 12.

Variation of the Lyapunov exponent with λ for the S-LOM (7) in (1) and (2) starting from the same initial condition as in Fig. 11, refers to the ith exponent and sum = .

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

6. Summary and conclusions

While our analysis does not add much to the physical aspects of convection that is already contained in Saltzman (1962), it brings completeness to Saltzman’s profound contribution to the theory of convective motion in fluids, which is concisely summarized as follows:

  1. The state space for the S-LOM (7) in (1) is the union of three invariant subspaces IS1, IS2, and IS3. The presence of the three invariant subspaces is an artifact resulting from the structure of S-LOM (7) in (1) and (2).

  2. The phase volume decreases exponentially in time and the solutions to S-LOM (7) remain uniformly bounded for all initial conditions and values of .

  3. The S-LOM (7) when projected onto IS3 reduces to an asymptotic stable linear dynamics with the origin as the unique global attractor in IS3.

  4. There are two branches of the equilibria—one-dimensional manifold parameterized by , in IS1 dented by and , which are reflections of each other with respect to axis. Similarly, there are two branches of equilibria—one-dimensional manifold parameterized by , in IS2 dented by and , which are also reflections of each other with respect to axis.

  5. Chaotic regime: For values of , the solutions in IS1 and IS2 exhibit chaotic behavior resulting from a Hopf bifurcation. The resulting instability is characterized by the existence of a positive Lyapunov exponent and the fractal dimension of the strange attractor.

  6. Periodic solution of Lorenz-like LOM (3) in (4) and (5) and (6) and (7).

    • Our analysis of the Lorenz-like LOM (3) given by (4) and (5) in IS1 and that by (6) and (7) in IS2 thus far concentrated on establishing a set of qualitative similarity measures between these two systems and the Lorenz 1963 model (Lewis et al. 2006, chapter 32) in (10). This list includes (i) the existence of bifurcation of the equilibrium at the origin for values of close to 1, (ii) the presence of the 1D equilibrium manifolds in IS1 and IS2 parameterized by , (iii) the existence of Hopf bifurcation for , and (iv) and computation and comparison of Lyapunov exponents and the fractal dimension of the resulting chaotic attractor.

    • However, a careful review of the classical literature (Sparrow 2012) relating to the analysis of the Lorenz 1963 model reveals that this latter model also exhibits a rich variety of periodic behavior. This observation calls for the analysis of the periodic solutions of the Lorenz-like LOM (3) in (4) and (5) and (6) and (7).

    • The latter analysis can be done in one of two ways. First is to use a numerical method described in appendix E of Sparrow (2012) for locating a periodic orbit. Clearly, this involves detailed exploration of the state space and the parameter space, which in principle could be time consuming. A second alternative is to construct an invertible transformation, using which Lorenz-like systems in (4) and (5) and in (6) and (7) can be reduced to the Lorenz system in (10). Indeed, by rescaling the state variables and the time variable t in appendix D, we construct a linear invertible transformation that proves the equivalence between LOM (3) in (4) and (5) and a specific version of (10) with , , where in (10) is related to the in (4) and (5) by the relation . A similar transformation between (6) and (7) and (10) can be likewise obtained.

  7. For general initial conditions with nonzero energy in IS1 and IS2, the S-LOM (7) in (1) and (2) exhibits simultaneously similar behavior in IS1 and IS2. The presence of the double attractor is noteworthy.

Finally, Saltzman (1962) has had great appeal to readership in fluid mechanics and meteorology. This work has brought some sense of completeness to Saltzman (1962) and should fundamentally be viewed as a tribute to him and a means to expand the usefulness of the family of low-order models for convection.

Acknowledgments

We wish to record our sincere thanks to Qin Xu, National Severe Storms Laboratory, for his questions related to the proof of the boundedness of the solution in appendix A and Alex Gluhosky, Purdue University, for his interest and encouragement. We are very grateful to the two anonymous reviewers, whose multifaceted questions helped to improve the overall flow and the readability of the paper.

APPENDIX A

Stability of Equilibrium at the Origin

Analysis of the stability of the equilibrium E1 at the origin relates to quantifying the eigenstructure of the sparse, banded Jacobian matrix in (12), which takes the general form
ea1
where for , , , , , , and , whose numerical values are given in Table 1.
Let and be an eigen pair of in (A1), that is, . By direct substitution, it can be verified that the sparse, banded structure of allows us to naturally decompose the seven-dimensional linear system
ea2
into a collection of four disjoint subsystems each of whose dimension is no more than two, as shown below:
ea3
ea4
ea5
and
ea6
Indeed, by solving these subsystems, we can recover the full eigenstructure in (A1).
  • Solution of : Trivially, the seventh eigenvalue given by is independent of . Its corresponding eigenvector is
    ea7
    which is the seventh standard unit vector in .
  • Solution of : The two eigenvalues, say, and arising from solving the 2 × 2 linear system in (A3) are given by the solution of the characteristic polynomial
    ea8
    Let . The variation of as a function of is given in Fig. A1a. It can be easily verified that the two (distinct) eigenvectors corresponding to and take the general form as
    ea9
    where . From the definition of the invariant subspace IS1, it is immediate that these two eigenvector are in IS1.
    Fig. A1.
    Fig. A1.

    Plot of the variation of eigenvalues and of as a function of λ: (a) only, (b) only, and (c) and .

    Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

  • Solution of : Following similar reasoning, the next two eigenvalues and are obtained by solving the characteristic polynomial
    ea10
    Let . The variation of as a function of is given in Fig. A1b. The two (distinct) eigenvectors corresponding to and take the general form as
    ea11
    where . These two eigenvectors belong to the subspace IS2.
  • Solution of : The last two remaining eigenvalues and are the roots of the characteristic polynomial
    ea12
    of the solution in (A5). It is a simple exercise to verify the eigenvectors for and taking the general form
    ea13
    where . These two eigenvectors belong to the subspace IS3.

  • Summary of the stability analysis: From the variation of the seven eigenvalues of as a function of , it is immediate that the first bifurcation occurs at . Below this value, the origin is a hyperbolic attractor. Above this value, the origin becomes a saddle. More specifically, becomes positive for and the corresponding unstable eigen direction carries the solution away from the origin. As we increase , around , the eigenvalue of the system becomes positive, creating another unstable eigen direction . However, from Fig. A1c, it is clear that remains the dominant eigen value for all , but becomes dominant for . Hence, for all , any solution starting close to the origin moves away from it along the eigenvector in IS1 depending on and settles down in an equilibrium in IS1. For , solution close to the origin moves away from it either along or along , depending on both the initial condition and value of

For , the system becomes chaotic but remains within IS1 or IS2, depending on the initial condition.

APPENDIX B

Boundedness of the Solution of S-LOM (7)

Consider a quadratic energy function given by
eb1
where is a diagonal matrix:
eb2
with positive diagonal entries. Since the S-LOM (7) in (2) is a forced, dissipative system, cannot be a constant along its trajectory.
Our goal is to choose the diagonal elements of such that the sign of can be conveniently and analytically evaluated. Computing the time derivative of along the trajectory of (2) and collecting the like terms, we obtain
eb3
which is the sum of 6 cubic and 10 quadratic terms.
Since
eb4
setting
eb5
the coefficients of the five cubic terms except that of vanish. Now setting
eq16
or
eb6
that is,
eb7
we force the coefficient of the remaining cubic terms also to zero. Clearly, there are infinitely many choices for K1, K2, and K3 satisfying (B7).
Assuming that the values of are chosen to satisfy (B5) and (B7), we now regroup the remaining 10 quadratic terms into three groups consistent with the properties of the invariant subspaces IS1, IS2, and IS3 in section 2. Accordingly,
eb8
where
eq17
eq18
eq19
and
eb9

From Table 1, since , in (B9) is negative definite. To analyze the negative definiteness of the rest of the three quadratic forms in (B9), we invoke two basic facts from the theory of quadratic forms.

  • Fact 1: If , then , where the symmetric part and the skew-symmetric part are given by and .

  • Fact 2: From , it is immediate that .

  • Sign definiteness of : Using these facts,
    eb10
    with , , and . Hence, is negative definite when
    eb11
    Substituting for and using the values of from Table 1, since , (B11) reduces to
    eb12
  • Sign definiteness of : Again, from
    eb13
    with , , and , it follows that is negative definite if . Substituting for and using the values of from Table 1, since , is negative definite when
    eb14
  • Sign definiteness of : By similar arguments
    eb15
    with , , and , is negative definite if . Substituting for and using the values of from Table 1, since , is negative definite when
    eb16
    Stated in other words, all the four quadratic forms in (B9) are simultaneously negative definite if there exist constants , , and that satisfy (B7), (B12), (B14), and (B16) simultaneously.
    Setting in (B7), it follows that and . Using the standard majorizationB1 argument, inequalities in (B12), (B14), and (B16) hold if
    eb17
    Clearly, (B17) holds for all . Hence, and the solution of S-LOM (7) in (2) is bounded for all

APPENDIX C

An Encoding of the Ensemble Members

In this appendix, we develop a succinct encoding scheme to represent the set of all 128 initial conditions, which correspond to the corners of the seven-dimensional hypercube that is centered at the origin with the length of the sides equal to 2α for some . An example of such a hypercube in 2D is given in Fig. C1.

Fig. C1.
Fig. C1.

A 2D hypercube of side length = 2 centered at the origin.

Citation: Journal of the Atmospheric Sciences 76, 6; 10.1175/JAS-D-17-0344.1

The seven-dimensional hypercube of interest has 27 = 128 corners. The coordinates of each of these corners corresponds to a string of ±1 of length 7. The encoding scheme that enumerates these 128 corners is given below:

  1. Let i be an integer in the range 0 to 127.

  2. Let be the binary representation of the integer i.

  3. Define an encoding function as follows:
    eq20
  4. Then associate the label i with the new string of ±1 obtained as follows:
    eq21
    where , depending on bj = 1 or 0.

Accordingly, the four corners of the 2D hypercube in Fig. C1 is given in Table C1. That is, the four nodes A, B, C, and D are encoded as 0, 1, 2, and 3 in Table C1. We can likewise develop a table for the 128 nodes of the seven-dimensional hypercube of interest. As an example, consider . In binary, is 1010011. Since , the corner of the hypercube with these coordinates is denoted by the integer .

Table C1.

The four corners of the 2D hypercube in Fig. C1.

Table C1.

APPENDIX D

Equivalence between the Projected S-LOM (7) onto IS1 Given by (4) and (5) and the Lorenz 1963 Model Given in (10)

Define a set of new Fourier amplitude variables
ed1
and a new time variable
ed2
where , , and are the Fourier amplitude–scale factors and is the time-scale factor.
Using (D1) and (D2) in (4) and (5), after simplification, we get the transformed version of the reduced dynamics in IS1 as
ed3
and
ed4
where
ed5

Recall that the values of the coefficients extracted from Saltzman (1962) are given in Table 1. Now setting , our goal is to choose the four scaling factors , , , and such that the vector field of (D4) and (D5) matches that of Lorenz model given in (10).

Equating the coefficients of the like terms on the right-hand side of (D5) with that of the Lorenz model in (10), we obtain the following set of seven equations relating the above four scaling parameters and the three parameters , , and in the Lorenz model:
ed6
Substituting the value of from Table 1 in (D6), we readily obtain the following:
eq22

Stated in other words, there exists an invertible linear scaling of the Fourier amplitudes and the time given in (D1) and (D2), using which we can transform the reduced S-LOM (3) given in (4) and (5) [obtained by projecting S-LOM (7) onto the invariant subspace IS1] to the Lorenz 1963 model with a specific value of the Prandtl number , aspect ratio , and the Rayleigh parameters related by .

By using a similar line of argument, we can show that the S-LOM (3) in (6) and (7) is indeed equivalent to the Lorenz model in (10).

REFERENCES

  • Arnold, V. I., 1992: Ordinary Differential Equations. Springer, 334 pp.

  • Canuto, C., A. Quarteroni, M. Y. Hussaini, and T. A. Zang, 2007: Spectral Methods: Fundamentals in Single Domain. Springer, 581 pp.

    • Crossref
    • Export Citation

  • Chandrasekhar, S., 1961: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, 654 pp.

  • Curry, J. H., 1978: A generalized Lorenz system. Commun. Math. Phys., 60, 193–204, https://doi.org/10.1007/BF01612888.

  • Gilmore, R., and C. Letellier, 2007: The Symmetry of Chaos. Oxford University Press, 545 pp.

  • Gluhovsky, A., and C. Tong, 1999: The structure of energy conserving low-order models. Phys. Fluids, 11, 334–343, https://doi.org/10.1063/1.869883.

  • Grassberger, P., and I. Procaccia, 1983: Measuring the strangeness of strange attractors. Physica D, 9, 189–208, https://doi.org/10.1016/0167-2789(83)90298-1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hirsch, M. W., and S. Smale, 1973: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, 358 pp.

  • Lakshmivarahan, S., and Y. Wang, 2008: On the relation between energy-conserving low-order models and a system of coupled generalized Volterra gyrostats with nonlinear feedback. J. Nonlinear Sci., 18, 75–97, https://doi.org/10.1007/s00332-007-9006-6.

    • Crossref
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  • Lewis, J. M., S. Lakshmivarahan, and S. Dhall, 2006: Dynamic Data Assimilation: A Least Squares Approach. Vol. 13. Cambridge University Press, 654 pp.

    • Crossref
    • Export Citation

  • Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141, https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lorenz, E. N., 1993: The Essence of Chaos. University Of London, 227 pp.

    • Crossref
    • Export Citation

  • Malkus, W. V. R., and G. Veronis, 1958: Finite amplitude cellular convection. J. Fluid Mech., 4, 225–260, https://doi.org/10.1017/S0022112058000410.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rayleigh, L., 1916: On convection currents in a horizontal layer of fluid, when higher temperature is on the under side. London Edinburgh Dublin Philos. Mag. J. Sci. 32, 529–546, https://doi.org/10.1080/14786441608635602.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Saltzman, B., 1962: Finite amplitude free convection as an initial value problem—I. J. Atmos. Sci., 19, 329–341, https://doi.org/10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shen, J., T. Tang, and L. L. Wang, 2011: Spectral Methods: Algorithms, Analysis and Applications. Springer, 472 pp.

    • Crossref
    • Export Citation

  • Sparrow, C., 2012: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Vol. 41. Springer, 270 pp.

  • Tong, C., 2009: Lord Kelvin’s gyrostat and its analogs in physics, including the Lorenz model. Amer. J. Phys., 77, 526–537, https://doi.org/10.1119/1.3095813.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turner, J. S., 1973: Buoyancy Effects in Fluids. Cambridge University Press, 367 pp.

    • Crossref
    • Export Citation

  • Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985: Determining Lyapunov exponents from a time series. Physica D, 16, 285–317, https://doi.org/10.1016/0167-2789(85)90011-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

The International Symposium on Numerical Weather Prediction, Tokyo, Japan, 1960.

2

A simplified form of the filtered equations of numerical weather forecasting.

3

In keeping with Saltzman’s model, we keep the Prandtl number (Pr) = ν/κ = 10 and accordingly change the value of ν from its true value of 1.0 × 10−2 cm2 s−1 to the value shown above.

B1

The maximum of the lhs is less than the minimum of the rhs.

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Arnold, V. I., 1992: Ordinary Differential Equations. Springer, 334 pp.

  • Canuto, C., A. Quarteroni, M. Y. Hussaini, and T. A. Zang, 2007: Spectral Methods: Fundamentals in Single Domain. Springer, 581 pp.

    • Crossref
    • Export Citation

  • Chandrasekhar, S., 1961: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, 654 pp.

  • Curry, J. H., 1978: A generalized Lorenz system. Commun. Math. Phys., 60, 193–204, https://doi.org/10.1007/BF01612888.

  • Gilmore, R., and C. Letellier, 2007: The Symmetry of Chaos. Oxford University Press, 545 pp.

  • Gluhovsky, A., and C. Tong, 1999: The structure of energy conserving low-order models. Phys. Fluids, 11, 334–343, https://doi.org/10.1063/1.869883.

  • Grassberger, P., and I. Procaccia, 1983: Measuring the strangeness of strange attractors. Physica D, 9, 189–208, https://doi.org/10.1016/0167-2789(83)90298-1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hirsch, M. W., and S. Smale, 1973: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, 358 pp.

  • Lakshmivarahan, S., and Y. Wang, 2008: On the relation between energy-conserving low-order models and a system of coupled generalized Volterra gyrostats with nonlinear feedback. J. Nonlinear Sci., 18, 75–97, https://doi.org/10.1007/s00332-007-9006-6.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lewis, J. M., S. Lakshmivarahan, and S. Dhall, 2006: Dynamic Data Assimilation: A Least Squares Approach. Vol. 13. Cambridge University Press, 654 pp.

    • Crossref
    • Export Citation

  • Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141, https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lorenz, E. N., 1993: The Essence of Chaos. University Of London, 227 pp.

    • Crossref
    • Export Citation

  • Malkus, W. V. R., and G. Veronis, 1958: Finite amplitude cellular convection. J. Fluid Mech., 4, 225–260, https://doi.org/10.1017/S0022112058000410.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rayleigh, L., 1916: On convection currents in a horizontal layer of fluid, when higher temperature is on the under side. London Edinburgh Dublin Philos. Mag. J. Sci. 32, 529–546, https://doi.org/10.1080/14786441608635602.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Saltzman, B., 1962: Finite amplitude free convection as an initial value problem—I. J. Atmos. Sci., 19, 329–341, https://doi.org/10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shen, J., T. Tang, and L. L. Wang, 2011: Spectral Methods: Algorithms, Analysis and Applications. Springer, 472 pp.

    • Crossref
    • Export Citation

  • Sparrow, C., 2012: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Vol. 41. Springer, 270 pp.

  • Tong, C., 2009: Lord Kelvin’s gyrostat and its analogs in physics, including the Lorenz model. Amer. J. Phys., 77, 526–537, https://doi.org/10.1119/1.3095813.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turner, J. S., 1973: Buoyancy Effects in Fluids. Cambridge University Press, 367 pp.

    • Crossref
    • Export Citation

  • Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985: Determining Lyapunov exponents from a time series. Physica D, 16, 285–317, https://doi.org/10.1016/0167-2789(85)90011-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A pictorial view of the asymptotic behavior of S-LOM (7). While the invariant subspaces IS1, IS2, and IS3 have a common dimension , for clarity these invariant subspaces are shown as distinct sets. and are the loci of the stable equilibria embedded in IS1 and IS2, respectively.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Plots of the steady states in as a function of λ.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    A 3D plot of the steady states in as λ is varied from 1.1 to 30.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Phase plot for chaotic behavior in IS1 [IC, , , , and ]. (b) Variation of the Lyapunov exponent with λ for the Lorenz-like dynamics (4) and (5) in IS1 starting from the same initial condition as in Fig. 4a, refers to the ith exponent and sum is .

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Plots of the steady states in as a function of λ.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    A 3D plot of the steady states in as λ is varied from 1.2 to 30.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Time evolution and phase plot for chaotic behavior in IS2 [IC, , , and ].

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Plot of the components of the solution of (1) for converging to equilibria in IS1.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Plot of the solution of (1) for converging to equilibria in IS2.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Plot of the solution of (1) for converging to equilibria in IS2.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Phase plot of solution to S-LOM (7) with and —same initial condition as in Saltzman (1962).

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Variation of the Lyapunov exponent with λ for the S-LOM (7) in (1) and (2) starting from the same initial condition as in Fig. 11, refers to the ith exponent and sum = .

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Plot of the variation of eigenvalues and of as a function of λ: (a) only, (b) only, and (c) and .

  • <sc>Fig</sc>. C1.
    View in gallery
    Fig. C1.

    A 2D hypercube of side length = 2 centered at the origin.

Fig. 1.

A pictorial view of the asymptotic behavior of S-LOM (7). While the invariant subspaces IS1, IS2, and IS3 have a common dimension , for clarity these invariant subspaces are shown as distinct sets. and are the loci of the stable equilibria embedded in IS1 and IS2, respectively.
Fig. 2.

Plots of the steady states in as a function of λ.
Fig. 3.

A 3D plot of the steady states in as λ is varied from 1.1 to 30.
Fig. 4.

(a) Phase plot for chaotic behavior in IS1 [IC, , , , and ]. (b) Variation of the Lyapunov exponent with λ for the Lorenz-like dynamics (4) and (5) in IS1 starting from the same initial condition as in Fig. 4a, refers to the ith exponent and sum is .
Fig. 5.

Plots of the steady states in as a function of λ.
Fig. 6.

A 3D plot of the steady states in as λ is varied from 1.2 to 30.
Fig. 7.

Time evolution and phase plot for chaotic behavior in IS2 [IC, , , and ].
Fig. 8.

Plot of the components of the solution of (1) for converging to equilibria in IS1.
Fig. 9.

Plot of the solution of (1) for converging to equilibria in IS2.
Fig. 10.

Plot of the solution of (1) for converging to equilibria in IS2.
Fig. 11.

Phase plot of solution to S-LOM (7) with and —same initial condition as in Saltzman (1962).
Fig. 12.

Variation of the Lyapunov exponent with λ for the S-LOM (7) in (1) and (2) starting from the same initial condition as in Fig. 11, refers to the ith exponent and sum = .
Fig. A1.

Plot of the variation of eigenvalues and of as a function of λ: (a) only, (b) only, and (c) and .
Fig. C1.

A 2D hypercube of side length = 2 centered at the origin.
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