The Energy Method, Stability, and Nonlinear Convection

Before considering the nonlinear partial differential equations encountered in convection we approach the problem of stability or instability for the diffusion equation.

For many flows it is sufficient to regard the fluid as incompressible. In this situation, with a constant temperature throughout, the velocity field, v, and pressure, p, are determined from the Navier-Stokes equations, .

In this section we explain an important idea in the theory of energy stability. This is the idea of coupling parameters which was briefly introduced in the last chapter and was originally developed by Professor D.D. Joseph.

It is often useful to define a convection problem on a half space. For example, (Hurle et al., 1982) use the velocity in a phase change problem to transform their stability analysis to one on a half-space; also (Hurle et al., 1967) have heat conducting half-spaces bounding a fluid layer to investigate the effects of finite conductivity at the boundary. While it may offer some simplicity to deal with a half-space configuration, from the mathematical point of view it does introduce new complications. In particular, (Galdi and Rionero, 1985) derive a very sharp result on the asymptotic behaviour of the base solution for which the energy maximum problem for R _E admits a maximizing solution. Roughly speaking, either the base solution must decay at least linearly at infinity, or the gradient of the base solution must decay at least like 1/z ² (if z > 0 is the half-space.) To describe this result and related ones in geophysics it is convenient to return to the general equations for a heat conducting linearly viscous fluid.

Thus far in convection studies we have explored uses of the energy method that have concentrated on employing some form of kinetic-like energy, involving combinations of L ² integrals of perturbation quantities. While this is fine and yields strong results for a large class of problem, there are many situations where such an approach leads only to weak results if it works at all, and an alternative device must be sought. In fact, recent work has often employed a variety of integrals rather than just the squares of velocity or temperature perturbations. (Drazin and Reid, 1981), p. 431, point out that this natural extension of the energy method is essentially the method advocated by Lyapunov for the stability of systems of ordinary differential equations some 60 years or so ago. In this chapter we indicate where a variety of different generalized energies (or Lyapunov functionals) have been employed to achieve several different effects.

The topic of this section is historically important since it is now believed that in the original experiments of (Bénard, 1900) (the convection problem now bears his name), the driving mechanism was one of surface tension variation due to temperature. We point out that convective instability seems to have been first described by James Thomson, the elder brother of Lord Kelvin, (Thomson, 1882), a very clear account of this is given in (Drazin and Reid, 1981) , p. 32.

If we allow the Newtonian constitutive equation for the stress to be replaced by something for a more exotic fluid then a variety of interesting convection problems arise. Nonlinear energy stability analyses for these fluids are only relatively recent. In the first three sections we concentrate on thermal convection in a micropolar fluid. Other effects are considered in section 9.4 and section 9.5 reviews work in various other classes of generalized fluids.

A physically important class of problem involves convection in a layer when the basic temperature is not simply a linear function of z, but where it depends also directly on time. For example, any attempt to grow semiconductor crystals in space will be faced with the problem of a gravity field approximately 10^-5 times as strong as that on Earth, and hence slight movement of the spacecraft will lead to a basic state that changes with time. Also, any geophysical convection problem driven by solar radiation comes into the category under consideration here since the Sun’s heating effect follows either a diurnal cycle or a yearly one, depending on the timescale involved.

As (Rosensweig, 1985) points out, the interaction of electromagnetic fields and fluids has been attracting increasing attention due to applications in many diverse areas. He writes that the subject may be divided into three main categories:

Ferrohydrodynamics (FHD) is of great interest because the fluids of concern possess a giant magnetic response. This gives rise to several striking phenomena with important applications. Among these are the spontaneous formation of a labyrinthine pattern in thin layers, the self-levitation of an immersed magnet, and of particular interest here, the enhanced convective cooling in a ferrofluid that has a temperature-dependent magnetic moment. The very well written book by (Rosensweig, 1985) is a perfect introduction to this fascinating subject. He very briefly refers to thermo-convective instability in FHD, which is what we concentrate on here. Another, more general, but again very readable account of ferromagnetism may by found in (Landau et al., 1984) . We now present the relevant equations for FHD, in the forms appropriate to this chapter. Then a brief account is given of a striking convective-like instability, before embarking on the thermo-ferro convection problem.

The phenomenon of double-diffusive convection in a fluid layer, where two scalar fields (such as heat and salinity concentration) affect the density distribution in a fluid, has become increasingly important in recent years. The behaviour in the double-diffusive case is much more diverse than for the Bénard problem. In particular, linear stability theory, cf. (Baines and Gill, 1969), predicts that the first occurrence of instability may be via oscillatory rather than stationary convection if the component with the smaller diffusivity is stably stratified. Finite amplitude convection in the doublediffusive context was investigated by (Veronis, 1965; Veronis, 1968a) whose results suggested steady finite amplitude motion could occur at critical values of a Rayleigh number much less than that predicted by linearized theory. Several later papers confirmed this, usually by weakly nonlinear theory, see e.g., (Proctor, 1981) and the references therein. The boundary layer analysis of (Proctor, 1981) is an interesting one and provides some explanation for the energy results of (Shir and Joseph, 1968). The phenomenon of double diffusive convection and even multi-diffusive convection is examined in detail in the next chapter.

In the standard Bénard problem the instability is driven by a density difference caused by a temperature difference between the upper and lower planes bounding the fluid. If the fluid layer additionally has salt dissolved in it then there are potentially two destabilizing sources for the density difference, the temperature field and the salt field. A similar scenario could be witnessed in isothermal conditions but with two dissolved salts such as sodium and potassium chloride. When there are two effects such as this the phenomenon of convection which arises is called double diffusive convection. For the specific case involving a temperature field and sodium chloride it is frequently referred to as thermohaline convection. There are many recent studies involving three or more fields, such as temperature and two salts such as NaCl, KCl. For the three or greater field case we shall refer to multi-component convection.

In section 3.2 we have discussed the Boussinesq approximation for the equations of thermal convection in a linear viscous fluid. The equations arising from the Boussinesq approximation, (3.41), have been extensively employed in this book. However, there are situations in which compressibility effects are important, or one wishes to study convection in a very deep layer. In these circumstances, the Boussinesq approximation is generally believed to be inappropriate. In this chapter we examine some of the models which have been employed to take account of compressibility effects or convection in a deep layer. We begin in this section with an energy analysis of two models for thermal convection in a deep layer.

The instability of the thermal conduction solution when a layer of fluid is heated from below and convection cells form is treated extensively in this book. Up to this point we have mostly assumed the properties of the fluid are constant. It may, however, be argued that viscosity should always be treated as a function of temperature in the Bénard problem, since it is one of the fluid properties which does exhibit a considerable change with varying temperature. To appreciate this variation we simply look at values for mundane fluids. For example, (Rossby, 1969) quotes that the kinematic viscosity of water varies from 0.01008 cm²sec^-1 at 20°C to 0.00896 cm²sec^-1 at 25°C, this being approximately a 10% change. Over the same temperature range the thermal conductivity only exhibits a 1.5% change. (Rossby, 1969) also quotes values for the viscosity of a 20 cSt silicone oil, of 0.2137 cm²sec^-1 at 20°C and 0.1904 cm²sec^-1 at 25°C, i.e. approximately a 20% variation. Over this temperature range the thermal conductivity of the same silicone oil is constant. The viscosity values just quoted are for typical room temperatures over a range in which the Boussinesq approximation may be expected to hold. If the temperature range is greater the viscosity variation is also typically much larger. For example (Lide, 1991) states that the viscosity of olive oil varies from 138.0 centipoise at 10°C to 12.4 centipoise at 70°C.

A pioneering piece of work on penetrative convection is the beautiful paper of (Veronis, 1963). Our description of penetrative convection relies much on his paper.

The purpose of this chapter is to review and examine work on nonlinear energy stability theory applied to models for circulation within the oceans. This is a highly applicable subject and one which may lead to bounds which could be of use in environmental engineering. Since it is believed thermohaline circulation in the ocean may have an effect on climate change and even global warming, cf. (Clark et al., 2002), any useful thresholds energy stability bounds can yield will be welcome.

The purpose of this chapter is to describe two very efficient methods for solving eigenvalue problems of the type encountered in linear and energy stability convection problems. The techniques referred to are the compound matrix method, which is simple to implement, and the Chebyshev tau technique. The chapter is intended to be a practical guide as to how to solve relevant eigenvalue problems. Several examples from fluid mechanics and porous convection are included. First we briefly describe a standard shooting method.