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This chapter discusses double-diffusive convection, with a particular focus on the initial instability and eventual nonlinear evolution. It first considers the “salt-fingering” instability and then the “semiconvection” instability before discussing the possibility that the onsets of these instabilities at marginal stability have an amplitude that oscillates in time. The goal is to find the conditions that would result in a zero growth rate of the oscillation amplitude in order to determine the marginal stability constraint on the Rayleigh numbers for the onset of an oscillating instability. The chapter also shows how, after evolving beyond the onset of the instability, thermal diffusion between the moving parcel and the surroundings can alter the initial linear vertical profile of the horizontal-mean temperature into a “staircase” profile. This evolution of the temperature profile is investigated via nonlinear simulations.

This chapter looks at solving problems involving the thermal conductivity of matter using a technique developed by mathematicians in the late 18th and early 19th centuries. The key equation describes thermal diffusion, i.e., how heat appears to 'diffuse' from one place to the other, and much of the chapter presents techniques for solving this equation. The thermal diffusion equation for a sphere, Newton's law of cooling, the Prandtl number, sources of heat, and particle diffusion are discussed.

Diffusion is associated with the random, thermal motion of atoms that produces a change in the macroscopic concentration profile. This process occurs in gases, liquids, amorphous and crystalline solids of metals, ceramics, polymers, semiconductors, etc. The investigation of diffusion provides valuable information about the atomic structure of materials and the defects within them. Perhaps, most importantly, diffusion controls the rates of a wide range of kinetic processes associated with the synthesis of materials, processes by which we modify materials, and processes by which materials fail. The most common driving force for diffusion in a single-phase systems is associated with the entropy of mixing of its constituents (recall that we showed that the entropy of mixing of gases and the components of an ideal solution are always positive—see Sections 1.2.6 and 3.3). Since diffusional processes occur through the thermal motion of atoms (see below), it will not be surprising to learn that the rate of diffusion increases with increasing temperature. However, note that while the mechanisms of thermal motion in gases (random collision of atoms with each other) and liquids (e.g. Brownian motion) necessarily lead to mixing, the mechanisms of mixing within a solid are not as obvious. In solids, thermal motion corresponds to the vibrations of atoms near their equilibrium positions. Since the amplitude of such vibrations is much smaller than the nearest-neighbor separation, it would seem that such thermal motions cannot lead to mixing. Thus, the question ‘‘how do atoms migrate in solids’’ is not so simple. The equations describing diffusion were suggested by the physiologist Fick in 1855 as a generalization of the equations for heat transfer suggested by Fourier in 1824. Fick’s equations for diffusion can be obtained by analogy with Fourier’s equations for heat transfer by replacing heat with the number of atoms, temperature with concentration, and thermal conductivity with diffusivity. Fick’s first law provides a relationship between atomic currents and concentration gradients. As discussed above, this relationship can be understood by analogy with thermal conductivity or electrical conductivity.

Many physical properties depend on direction and the resulting anisotropy is best described with the use of tensors. Tensors are classified according to how they transform from one coordinate system to another. Therefore, we begin by describing transformations. There are several reasons why we want to do this: (1) transformations help us define tensors, (2) these tensors can be used to describe physical properties, (3) the effects of symmetry on physical properties can be determined by howthe tensor transforms under a symmetry operation, (4) the magnitude of a property in any arbitrary direction can be evaluated by transforming the tensor, (5) using these numbers, we can draw a geometric representation of the property, and (6) the transformation procedure provides a way of averaging the properties over direction. This is useful when relating the properties of polycrystalline materials to those of the single crystal. Mathematically, there is nothing fancy about these transformations. We are simply converting one set of orthogonal axes (Z₁, Z₂, Z₃) into another [math]. The two sets of axes are related to one another by nine direction cosines: a₁₁, a₁₂, a₁₃, a₂₁, a₂₂, a₂₃, a₃₁, a₃₂, and a₃₃. Collectively all nine can be written as aij where i, j = 1, 2, 3. The axes and direction cosines are illustrated in Fig. 2.1. It is important not to confuse the subscripts of the direction cosines. As defined in the drawing, a₁₂ is the cosine of the angle between [math] and Z₂, whereas a₂₁ is the cosine of the angle between [math] and Z₂. The first subscript always refers to the “new” or transformed axis. The second subscript is the “old” or original axis. The original or starting axes is usually a right-handed set, but it need not be. The transformed “new” axes may be either right- or left-handed, depending on the nature of the transformation. This will become clearer when we look at some transformations representing various symmetry operations. In any case, both the old and new axes are orthogonal.