Chapter 11 - Pressure problem

Abstract

In concluding Part I of this book, we finally consider the problem of pressure. There are two major reasons to consider this problem in standalone manner. First, because the pressure is a major force that acts against buoyancy. As emphasized many times already, the buoyancy is a primary force that drives convection. This is essentially the definition of convection. Convection is considered a kind of instability, and the most common approach in atmospheric science for inferring a possibility of convective instability is to perform a parcel stability analysis, as discussed in the last chapter. The procedure is simple enough that it can be applied to any types of convection problems. However, this approach is fundamentally limited by considering a particular fluid particle in its own isolation. In reality, a fluid particle can move around within a fluid only by pushing round other part of fluid. Those displaced fluids, in turn, displace other volumes of fluid around, and this process propagates to a full domain of fluid in concern. Such interactions between fluid particles are primarily performed by pressure forces, thus a close examination on a role of pressure elucidates the limits of the traditional parcel-based approach in the most vivid manner. The second reason is rather technical, but practically very important: how do we compute the pressure in a give fluid-dynamics problem? This question may appear to be, in the first sight, totally trivial, but actually not. To see this point, let's count the equations involved in a problem of fluid mechanics: the Navier–Stokes equation is a main equation to be solved to define time evolution of a fluid flow. This equation is essentially interpreted as a conservation law of momentum for a fluid, and its main role is to predict an evolution of fluid velocity with time, as clear from a way it is written. Time evolution of buoyancy is evaluated by solving a thermodynamics equation, effectively a conservation law of entropy. Additionally, there is an equation for mass continuity. However, we don't have any equation for the pressure somehow at our hand. We have to somehow work out certain relations among those equations for deriving an equation for the pressure. This chapter derives this equation for the pressure. To get a feeling on a role of pressure in convection problems, in turn, we are going to explicitly solve a problem of pressure under a given buoyancy distribution. As a simple example, we consider a problem with a spherically homogeneous buoyancy anomaly.