There are two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces: thermal convection and internal gravity waves. The type depends on the thermal stratification within the fluid region. The Earth’s atmosphere and ocean, for example, are in most places convectively stable, which means that they support internal gravity waves, not (usually) convection (but see Chapter 7). On warm afternoons, however, the sun can heat the ground surface, which changes the vertical temperature gradient in the troposphere and makes the atmosphere convectively unstable; the appearance of cumulus clouds is an indication of...

Now we describe a numerical method for solving these equations on a computer. The vorticity-streamfunction formulation is introduced as a means of conserving mass. This formulation was used for this problem by Nigel Weiss and his collaborators (e.g., Moore et al., 1973; Weiss, 1981a,b). To introduce the reader to two very different spatial discretizations, the vertical derivatives are approximated with a local (finite-difference) method and the horizontal derivatives with a global (spectral) method. The nonlinear terms are computed in spectral space; a more efficient spectral-transform method is introduced in Chapter 10. The time integration is an explicit Adams-Bashforth scheme; an...

In this chapter we describe a linear stability analysis (i.e., solving for the critical Rayleigh number and mode) so readers can check their linear codes against the analytic solution. Dropping the nonlinear terms in Eqs. 2.10 and 2.11 not only simplifies the problem but also redefines the problem. For this linear analysis, each Fourier mode n can be considered a separate and independent problem since the linear terms in Eqs. 2.10–2.12 involve only a single value of n. The question being asked now is under what conditions, i.e., what values of Ra, Pr, and a, will the amplitude of...

Now the nonlinear terms are added to produce finite-amplitude simulations. Here we choose to calculate the nonlinear terms using a Galerkin method in spectral space; this is replaced with the more efficient spectral-transform method in Chapter 10. However, a Galerkin method provides a clear understanding for how nonlinear terms disperse energy among the available modes and what is meant by spectral aliasing. For a review of spectral methods see, for example, Canuto et al. (1988), Boyd (2001), and Peyret (2002).

The linear solution approximates only the initial growth of supercritical convection that begins with small temperature perturbations. Very quickly the...

The initial check of the nonlinear code can be done by monitoring the values of various $ {{T}_{n}} $ , $ {{\omega }_{n}} $ , and $ {{\psi }_{n}} $ at various depths as discussed in Chapter 4. However, to learn from these nonlinear simulations one needs to study them using computer graphics and analysis.

To produce a snapshot like those in Fig. 4.2, one needs to Fourier transform the spectral solution (n-space) to the grid (x-space). (Since the method we have chosen computes the solution in z-space, no transform in that direction is needed.) This is simply done according to Eqs. 2.8. One must, however, choose a...

In Section 1.1.1 we discuss how a mean (x-averaged) superadiabatic temperature gradient supports thermal convection (assuming a supercritical Rayleigh number) and how a mean subadiabatic temperature gradient supports internal gravity waves. However, so far we have focused on how to develop a numerical model for the former, prescribing a higher fixed temperature at the bottom boundary than at the top boundary. Recall that within the Boussinesq approximation the background is considered both isothermal and adiabatic, so where the mean temperature gradient, $ \partial {{T}_{0}}/\partial z $ , is negative the thermal stratification is superadiabatic and convectively unstable and where it is positive the thermal...

So far we have considered the diffusion of just one scalar quantity, temperature, and just one source of buoyancy, also temperature. Consider now a fluid composed of two constituents, a primary constituent and a small concentration of a secondary constituent with a different density, for example, salt in water (an ocean), MgO in a magmatic melt (a magma chamber), sulfur in liquid iron (a planetary core), or heavy elements in hydrogen (a giant planet or star). In these cases, buoyancy is partly thermal buoyancy, which is what we have considered so far, and partly compositional buoyancy due to variations in...

The explicit Adams-Bashforth scheme for integrating the equations in time, which is described (Eq. 2.18) and employed in Part 1, is relatively simple and efficient since at each time step the time derivatives are needed for only the current and previous time steps. This works fairly well and is second-order accurate, although the accumulated error is proportional to only $ \Delta {{t}^{2}} $ . Many more accurate time integration schemes do exist, which are higher order but computationally more expensive. As an example, in this chapter we describe fourth-order accurate Runge-Kutta and predictor-corrector schemes. However, the stability constraints on the size of the...

In Part 1 we chose to treat the horizontal direction with a spectral method and the vertical direction with a finite-difference method on a uniform grid. For some problems it is desirable to be able to employ a spatial resolution that varies with position. In this chapter we introduce two ways of doing this within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. We then outline how one can simulate the convection and gravity wave problems described in Part 1 either by using finite differences in both directions or by using a spectral method...

We begin this chapter by outlining how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. Then we focus on the side boundaries, outlining how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions. This permits fluid flow through these boundaries and nonzero mean flow, i.e., time-dependent horizontal flow that varies in the vertical direction but not in the horizontal direction. The model can then easily be converted from cartesian box geometry...

Magnetoconvection is the term usually used to describe thermal convection of an electrically conducting fluid within a background magnetic field maintained by some external mechanism (e.g., Roberts, 1967; Weiss, 1981a; Glatzmaier, 2005a; Rempel et al., 2009). For example, much of the solar magnetic field generated deep within the sun extends to the surface, where it is swept by the convection into intense small-scale magnetic flux concentrations called “sunspots,” which inhibit outgoing heat flux and so appear darker than the surrounding photosphere. Understanding this mechanism was the motivation for much of the magnetoconvection research that Nigel Weiss and colleagues pioneered in...

For studies of convection and/or gravity waves in atmospheres and interiors of stars and planets that span several density scale heights it is important to account for the effects of large variations in density with depth, i.e., density stratification. (Here we usually mean continuously stratified, i.e., no discontinuities in density with depth. An exception would be a localized phase transition within a planetary interior.) As mentioned in Section 1.2, when the modeled domain spans a density scale height or more the Boussinesq approximation (which we have employed up to this point) is not valid. A fully compressible model, however, would...

The effects of rotation on convection and gravity waves can be significant. Certainly flows in the atmospheres, oceans, and liquid cores of terrestrial planets are dominated by the Coriolis forces, as are the interiors of giant planets and stars. The sum of centrifugal and gravitational forces can go to zero at the top boundary of a rapidly rotating star or accretion disk. Poincaré forces arise due to the time rate of change of the planetary rotation rate.

We begin this chapter with the derivation and discussion of parts of the inertial term in the momentum equation that exist due to...