Description of the Book's Back Cover Figure:
Three-Dimensional Simulation of the Spiral Defect Chaos Convection State

Click on the image to play a movie (a 20 MB AVI file)
of a spiral defect chaos simulation in a Γ=30 cylindrical cell,
for Rayleigh number R=6000.

This particular dynamical state is called spiral defect chaos because of the spontaneous formation, motion, and annihilation of various spirals throughout the pattern (click on the above image to see a movie of the dynamics). Spiral defect chaos was first discovered experimentally by Stephen Morris, Eberhard Bodenschatz, David Cannell, and Guenter Ahlers in 1993 and the discovery was a bit of a shock because spiral defect chaos occurs for parameters such that uniform periodic stripe states are linearly stable (so scientists were expecting time-independent stripe states, perhaps somewhat distorted because of the lateral boundaries), and no reasons were known at the time why spirals should form near onset in a convecting flow. Experiments further showed that spiral defect chaos was found only in cells of sufficiently large aspect ratio (Γ at least about 25), which is one reason why earlier experiments and simulations had missed this state.

This simulation by O'Connor and Paul and related simulations by other researchers have provided several substantial scientific insights and illustrate an important theme of the book, which is that a combination of experiments, simulations, and theory is needed to understand nonequilibrium pattern formation. Indeed, the ability starting about the year 2000 to simulate three-dimensional fluid dynamics in domains that are comparable in size to those studied by experimentalists (aspect ratios from 30 to 80), for time scales comparable to what experimentalists can study (many horizontal thermal diffusion times), and with a variety of quantitatively accurate boundary conditions represented a major advance in fluid dynamics and more generally in understanding pattern formation. This particular simulation used a sophisticated spectral element code that was developed by Paul Fischer and other computational scientists over more than a decade of effort. The code achieves the exponentially rapid convergence (with respect to the number of numerical degrees of freedom) of a spectral code while retaining the flexibility to treat domains of arbitrary (smooth) geometry like a finite element code.

One insight provided by this simulation is that the experimentally observed spiral defect chaos state is plausibly a deterministic chaos and not some stochastic process, driven say by thermal fluctuations. That the dynamics is likely chaotic follows from two observations. One is that the numerical integration of the three-dimensional Boussinesq evolution equations is effectively noise-free and so deterministic and yet produces a state that seems identical visually and statistically to the experimental state. Second, a direct computation of the largest Lyapunov exponent λ₁ of the dynamics can be carried out using the computer code and shown to have a positive value, which establishes the existence of chaos empirically.

By using the ability of a computer code to implement periodic side walls (which is impossible to achieve experimentally), David Egolf and collaborators used related simulations to show that spiral defect chaos and stable stripe states (straight convection rolls) indeed coexist in that either state can be obtained in the same cell for the same physical parameters by choosing an appropriate initial condition. Keng-Hwee Chiam and collaborators used the spectral element code to set to zero the so-called mean flow (see Section 9.1.3 in the book), which analytical calculations had suggested to be important in wavenumber selection and in the formation of spirals in a convection pattern. Turning off the mean flow resulted in a state that no longer seemed to be chaotic. That the mean flow is important for spiral defect chaos helps to explain why this is a difficult state to understand mathematically since the mean flow provides a long-range interaction between different parts of the convecting patttern.

The parameters describing the fluid and the numerical calcuation are:

1. aspect ratio Γ = "cylinder radius" / "cylinder height" = 30.
2. Prandl number (ratio of kinematic viscosity ν to thermal diffusivity κ) σ = 0.86, which corresponds approximately to air at room temperature.
3. Rayleigh number R = 6000 ≈ 3.5 R_c.
4. lateral walls were perfect thermal conductors.
5. 768 spectral elements used.
6. a tensor product of 11th-order polynomials were used within each spectral element.
7. the numerical time step was Δt=0.0001 in units of a vertical thermal diffusion time.
8. the mathematical initial state was a motionless conducting state (zero velocity field) with small random perturbations in the linear temperature gradient. The image occurs 155.2 vertical diffusion times (about a fifth of a horizontal diffusion time) after this initial state.