Geophysical Convection Dynamics

By Jun-Ichi Yano

Geophysical Convection Dynamics, Volume Five provides a single source reference that enables researchers to go through the basics of geophysical convection. The book includes basics on the dynamics of convection, including linear stability analysis, weakly nonlinear theory, effect of rotation, and double diffusion. In addition, it includes detailed descriptions of fully developed turbulence in well-mixed boundary layers, a hypothesis of vertical homogeneity, effects of moisture, and the formation of clouds. The book focuses on the presentation of the theoretical methodologies for studying convection dynamics with an emphasis on geophysical application that is relevant to fields across the earth and environmental sciences, chemistry and engineering.

• Guides and prepares early-stage researchers to plunge directly into research
• Provides a synthesis of the existing literature on topics including linear stability analysis, weakly nonlinear theory, effect of rotation, double diffusion, description of fully developed turbulence in well-mixed boundary layers, hypothesis of vertical homogeneity, effects of moisture, formation of clouds at the top, and cloud-top entrainment instability
• Presents geophysical convection to readers as a common problem spanning the atmosphere, oceans, and the Earth's mantle

• Language: English
• Publisher: Elsevier Science
• Release date: May 8, 2023
• ISBN: 9780323998017

Jun-Ichi Yano

Dr. Jun-Ichi Yano has more than 30 years of research experience with various geophysical convection problems: those include the dynamics of atmospheric convection and its parameterization, interactions of convection and the large-scale dynamics in the tropical atmosphere, convection inside the giant planets and the Earth's core, and convection of self-gravitating systems in high rotation limit. He has also been extensively working on other problems of geophysical flows: theoretical studies of the vortex dynamics, and their applications to the Jovian atmospheres, oceans, and the tropical atmosphere; chaos theory and its applications to the atmospheric dynamics; wavelet analyses; tropical meteorology, microphysics, and numerical weather a hypothesis of vertical homogeneity, prediction problems.

Book preview

Part I: Stability analysis

Outline

Chapter 1. Introduction

Chapter 2. Rayleigh–Taylor instability

Chapter 3. Rayleigh–Bénard convection

Chapter 4. Weakly nonlinear theory

Chapter 5. Effect of rotation: Rayleigh–Bénard convection with rotation

Chapter 6. Double-diffusion convection

Chapter 7. Mantle convection

Chapter 8. Thermodynamics and dynamics

Chapter 9. Atmospheric thermodynamics

Chapter 10. Parcel stability analysis

Chapter 11. Pressure problem

Chapter 1: Introduction

Simplify, simplify, simplify, …

Henry David Thoreau

Abstract

Convection is a mode of fluid motion. At the same time, it is one of possible modes for transporting heat. All of us learn at school that there are three basic modes for this: convection, diffusion, radiation. Arguably, compared to the two other modes of heat transport, convection is fundamentally macroscopic. Diffusion is a process that heat is transported as a result of the kinetic energy of a given molecule transmitted to another to a certain preferred direction. Radiation is a process that heat is transmitted to another objects as an electro-magnetic wave (that is called radiation in this context: note that radiation has various different meanings in physics, unfortunately). Both processes are best understood in terms of the quantum mechanics, and in that sense, both are inherently microscopic processes. On the other hand, convection means a macroscopic movement of fluid transports heat in association with its movement. For understanding this, there is no obvious need to take into account of any quantum effects.

Keywords

simplification; convection; fluid mechanics; Navier–Stokes equation; stability; perturbation problem; stratification

1.1 What is convection?

Convection is a mode of fluid motion. At the same time, it is one of possible modes for transporting heat. All of us learn at school that there are three basic modes for this: convection, diffusion, radiation. Arguably, compared to the two other modes of heat transport, convection is fundamentally macroscopic. Diffusion is a process that heat is transported as a result of the kinetic energy of a given molecule transmitted to another to a certain preferred direction. Radiation is a process that heat is transmitted to another objects as an electro-magnetic wave (that is called radiation in this context: note that radiation has various different meanings in physics, unfortunately). Both processes are best understood in terms of the quantum mechanics, and in that sense, both are inherently microscopic processes. On the other hand, convection means a macroscopic movement of fluid transports heat in association with its movement. For understanding this, there is no obvious need to take into account of any quantum effects.

Since convection is a matter of a movement of fluid, it would be best understood as a problem of the fluid mechanics. Thus, this book also follows this line.

1.2 Examples of geophysical convection

An example of geophysical convection that we experience in our daily life is a heavy afternoon shower on summer day, with rain falling from heavy clouds. These clouds are products of atmospheric convection. We can observe how these clouds evolve with time by standing at a spot with a good view of a long distance. We see that many popcorn like clouds gradually develop from the surface in early afternoon. This initial phase of development would be more like a gradual increase in number of clouds, rather than growth of individual clouds. However, at a certain point, we notice that some of those clouds begin to grow upwards rather dramatically. We also begin to notice that high clouds begin to spread horizontally, and also begin to cover immediately above ourselves. That would be the best moment to look for a shelter, if we don't carry any proper rain jacket, because a shower would come relatively quick. An upward growth of these clouds is an indication that heat is also transported upwards associated with them.

If you have ever tried to observe various celestial objects by your home telescope, one of the easiest objects to observe are the blackish spots found on the surface of Sun, called sun spots. This is also a consequence of convection. Sun spots appear to be dark and black, because they are colder than the surrounding, being corresponding to descending brunches of convection. Number of sun spots that we find on the surface of the Sun changes from day to day. It would be rather a meticulous effort count their number every day. However, it turns out to be a very useful record for understanding the evolution of the activity of the whole Sun over years.

1.3 Methodologies for studying convection

Convection in general, but those in geophysical systems more specifically, can be studied in various different manners. The most basic approach would be to observe them carefully. A basic idea for such careful observations is sketched above for two examples. The initial step of scientific observations would be inevitably qualitative, as we observe how convective clouds develop. However, to make your observations scientifically useful, it is important to be quantitative: it is important to report a number of sun spots, rather than just reporting that you have observed spots on the surface of Sun. Second important methodology is the laboratory experiments. It was Henri Bénard, who studied convection carefully for the first time in history by his laboratory experiment. We may even argue that he discovered convection by his experiment. Of course, it is not possible to confine geophysical convection inside your laboratory. However, it is more than often possible to conceive an experiment that reproduces an essential process going on with a given type of geophysical convection. In short, you may be able to produce an afternoon shower inside of your water tank!

From those observations and laboratory experiments, we can induce various conclusions about the nature of convection: both something general for all convections as well as those more particular for a given type of convection. However, those inductions always remain kinds of inference. We ultimately need something else to deduce our conclusion in a more solid manner. That is a role of theory.

As Galileo Galilei said effectively, the nature is written in the words of mathematics. As Issac Newton demonstrated explicitly for the mechanics, mathematics describing the nature takes the form of differential equations. Thus, from this perspective, the problem of understanding the nature reduces to that of solving the differential equations. We may say that the role of theory of physics is to identify a right differential equation and solve it. This book is throughout written from this perspective. The basic differential equation that describes the evolution of the fluid motions is the Navier–Stokes equation, thus we review this equation next.

1.4 Fluid mechanics

This book assumes that readers already have some basic knowledge of the fluid mechanics, especially what the Navier–Stokes equation is. In short, the basic equation that govern the evolution of the fluid flow, given in terms of the fluid velocity, v, is the Navier–Stokes equation:

(1.1)

Here, t is the time, p the pressure, ρ the density, ν the molecular viscosity, and F represents an additional external force acting on the fluid such as the gravity, the Coriolis force (when the system under a rotational framework as often the case with geophysical flows).

Eq. (1.1) may be understood as a straight extension of Newton's second law, i.e., the conservation law of momentum, to a continuous medium: the acceleration term is identified as the first in the left hand side, which may be balanced by an external force, F, as the last term in the right hand side. However, more terms appear by considering a continuous medium: the momentum (or velocity) can be moved around by the movement itself (i.e., advection) as given by the second term in the left hand side; the pressure force (first in the right hand side) further pushes the fluid; and finally, the velocity may be dissipated into heat by viscosity (second term in the right hand side).

In many geophysical applications, as any naive order-of-the-magnitude estimate can demonstrate, contribution of the molecular viscosity is very small. Thus, we may wish to neglect this term, and work instead with the following equation:

(1.2)

This is the Euler equation. In the first sight, the Euler equation is a good approximation for most of the geophysical applications. For this reason, in this book, the molecular term is often dropped, and simply the Euler equation is adopted for the analysis.

However, there is a rather subtle issue in neglecting the viscosity term. Most fundamentally, the basic nature of the partial differential equation changes by dropping the viscosity term. Mathematically speaking, the Navier–Stokes equation is parabolic, whereas the Euler equation is hyperbolic: see e.g., Sommerfeld (1949) for the definitions.

Along the same line, as a more practical problem, by dropping the viscosity term, the order of the differential equation reduces. As a result, we need only a single condition for the velocity at the surface (such as a vanishing of the normal component) for solving the Euler equation, whereas more conditions must be imposed for solving the Navier–Stokes equation. When we use the Euler equation as an approximation of the Navier–Stokes equation, thus, we may still wish to construct a solution in such manner that the original boundary conditions imposed for the Navier–Stokes equation are satisfied. Obviously, this is generally not possible due to a lower order of the Euler equation. It transpires that, although the Euler equation may be a good approximation in most part of the fluid, close to the boundary, we still have to apply the Navier–Stokes equation for solving the problem in full. In other words, a region close the boundary of a system requires a special treatment. This is called the boundary-layer problem. This is a beginning of a long story to understand why we also have a zone called a boundary layer geophysical flows: see, e.g., Sorbjan (1989). Fortunately, we will not encounter much of boundary-layer problems in this book, but see part II.

[Ex]

By following the scale analysis in a later part of the book, show explicitly that the molecular viscosity term is negligible in most of the geophysical applications.

1.5 Stability

An essence of the convection problem may be considered a question of the stability of the fluid. The basic notion of the stability may be understood by considering the problem of boiling water in a kettle: you pour water into the kettle, and place it on an oven. At first nothing visibly happens. The oven heat the bottom of the kettle, and the temperature of water becomes higher at the bottom than at the top. This tendency is compensated by molecular diffusive transport of heat upwards. Thus, the kettle remains quiet for while. However, at a certain point, as the kettle is heated more, we begin to hear a noise inside the kettle: water begins to make a movement inside. We may call this transition as onset of convection.

Thus, a first goal of convection investigation is to determine a point of onset of convection as a state of the system changes. A basic mathematical procedure for identifying such a point, and also resulting behavior of the system is called the stability problem: in words, the original state (a purely diffusive state in the case of the kettle problem above) becomes unstable, and a new state (i.e., convection) arises.

1.6 Perturbation problem

When we focus the problem of the behavior of the fluid close to the onset, the convective movement would still be weak, and we can consider a problem by a linearization of the system around the original basic state (e.g., purely diffusive), say, . We may designate a small deviation from this basic state as . In technical term, this small added quantity is called perturbation, and the problem to be solved is called a perturbation problem.

This problem can symbolically be stated as:

(1.3)

where L is a linear operator arising from this linearization. The problem can be solved by assuming an exponential tendency for a time evolution of the system, thus . Here, σ is the growth rate. There are three possibilities for the solution:

(i)   : (asymptotically) stable

(ii)   : unstable

(iii)   : marginally stable, neutral.

1.6.1 Example

To see a working of the perturbation problem mathematically, let us consider a simple system given by:

(1.4)

Here, R is a positive constant. This type of ordinary differential equations is called a dynamical system, which defines its evolution by a temporal tendency of a dependent variable, x. Here, the equation contains only a first derivative of x.

We first seek steady solutions by setting in the left hand side, thus the problem reduces to:

(1.5)

Possible solutions are

(1.6)

Here, is added to suggest steady solutions above.

The perturbation analysis can be performed by adding a small deviation from the above, designated by , thus

Assuming the perturbation is small, we find:

By substituting this expression into the full Eq. (1.4), we obtain a perturbation equation to solve:

(1.7)

As outlined above, we set , then we find the growth rate to be:

(1.8)

Thus, and , respectively, when and : the first and the second steady solutions are, respectively, unstable and stable.

[Ex]

Repeat the same perturbation analysis as above for the case with . How would you treat with imaginary steady solutions for , which would be unphysical, if x is a physical variable?

[Ex]

Perform the linear stability analysis of the following system:

(1.9a)

(1.9b)

[Ex]

Show that the system (1.9a, b) reduces to the system (1.4) by the following transformation:

(1.10a)

(1.10b)

Chs. 2 and 3 will focus on this linear stability analysis, i.e., linear perturbation