In Chapter 4 we showed that an arbitrary unitary operation U may be implemented on a quantum computer using a circuit consisting of single qubit and controlled-not gates. Such universality results are important because they ensure the equivalence of apparently different models of quantum computation. For example, the universality results ensure that a quantum computer programmer may design quantum circuits containing gates which have four input and output qubits, confident that such gates can be simulated by a constant number of controlled-not and single qubit unitary gates.

An unsatisfactory aspect of the universality of controlled-not and single qubit unitary gates is that the single qubit gates form a continuum, while the methods for fault-tolerant quantum computation described in Chapter 10 work only for a discrete set of gates. Fortunately, also in Chapter 4 we saw that any single qubit gate may be approximated to arbitrary accuracy using a finite set of gates, such as the controlled-not gate, Hadamard gate H, phase gate S, and π/8 gate. We also gave a heuristic argument that approximating the chosen single qubit gate to an accuracy ∈ required only Θ(1/∈) gates chosen from the finite set. Furthermore, in Chapter 10 we showed that the controlled-not, Hadamard, phase and π/8 gates may be implemented in a fault-tolerant manner.

OVERVIEW

As we emphasized back in Chapter 1, atmospheres are not static. The mass and composition of an atmosphere evolves over time, as a result of a great variety of chemical, physical, and biological processes. Now it is time to survey those processes in greater detail, and to put numbers on them to the extent possible in the limited space available in this chapter.

Throughout the following we will need to refer to some constituents of a planet as volatiles. These are “not rocks” – things that can become gases to a significant extent. The concept of a volatile is relative to the temperature of a planet. On Earth, water is a volatile but on Titan it is basically a rock, as is CO2, though N2 and CH4 remain as volatiles even at the low temperatures of Titan. On Earth, sand (SiO2) is a rock, but on a roaster – a hot extrasolar Jupiter in a close orbit – it could be a volatile.

For planets in which some atmospheric volatiles exchange with a condensed reservoir, as in the case of Earth's ocean and glaciers, the whole atmosphere–ocean–cryosphere system is best treated as a unit for many purposes, and we will refer to this as the volatile envelope. In other cases, the portion of the volatile envelope which resides in the atmosphere plays a distinguished role.

OVERVIEW

The atmospheres which are our principal objects of study are made of compressible gases. The compressibility has a profound effect on the vertical profile of temperature in these atmospheres. As things progress it will become clear that the vertical temperature variation in turn strongly influences the planet's climate. To deal with these effects it will be necessary to know some thermodynamics – though just a little. This chapter does not purport to be a complete course in thermodynamics. It can only provide a summary of the key thermodynamic concepts and formulae needed to treat the basic problems of planetary climate. It is assumed that the student has obtained (or will obtain) a more fundamental understanding of the general subject of thermodynamics elsewhere.

A FEW OBSERVATIONS

The temperature profile in Fig. 2.1, measured in the Earth's tropics, introduces most of the features that are of interest in the study of general planetary atmospheres. It was obtained by releasing an instrumented balloon (radiosonde) which floats upward from the ground, and sends back data on temperature and pressure as it rises. Pressure goes down monotonically with height, so the lower pressures represent greater altitudes. The units of pressure used in the figure are millibars (mb). One bar is very nearly the mean sea-level pressure on Earth, and there are 1000 mb in a bar.

OVERVIEW

This chapter will survey a few of the major questions raised by observed features of present and past Earth and planetary climates. Some of these questions have been answered to one extent or another, but many remain largely unresolved. This will not be a comprehensive synopsis of Earth and planetary climate evolution; we will be content to point out a few striking facts about climate that demand a physical explanation. Then, in subsequent chapters, we'll develop the physics necessary to think about these problems. Although we hope not to be too Earth-centric in this book, in the present chapter we will perforce talk at greater length about Earth's climate than about those of other planets, because so much more is known about Earth's past climate than is known about the past climates of other planets. A careful study of Earth history suggests generalities that may apply to other planets, and also raises interesting questions about how things might have happened differently elsewhere, and it is with this goal in mind that we begin our journey.

CLOSE TO HOME

When the young Carl Linnaeus set off on his journey of botanical discovery to Lapland in 1732, he left on foot from his home in Uppsala. He didn't wait until he reached his destination to start making observations, but found interesting things to think about all along the way, even in the plant life at his doorstep.

OVERVIEW

This results of this chapter are pertinent to a planet with a distinct surface, which may be defined as an interface across which the density increases substantially and discontinuously. The typical interface would be between a gaseous atmosphere and a solid or liquid surface. In the Solar System, there are only four examples of bodies having both a distinct surface and a thick enough atmosphere to significantly affect the surface temperature. These are Venus, Earth, Titan, and Mars; among these, the present Martian atmosphere is so thin that it only marginally affects the surface temperature, though this situation was probably different early in the planet's history when the atmosphere may have been thicker. Although thin atmospheres have little effect on the surface temperature, the atmosphere itself can still have interesting behavior, and the flux of energy from the surface to the atmosphere provides a crucial part of the forcing which drives the atmospheric circulation. This is the case, for example, for the thin nitrogen atmosphere of Neptune's moon Triton. Apart from the examples we know, it is worth thinking of the surface balance in general terms, because of the light it sheds on the possible nature of the climates of extrasolar planets already detected or awaiting discovery.

The exchange of energy between the surface and the overlying atmosphere determines the surface temperature relative to the air temperature.

OVERVIEW

So far, we have studiously avoided discussing the circulations of atmospheres or oceans, or indeed fluid mechanics of any type, with the exception of a brief foray into compressible one-dimensional hydrodynamics in Section 8.7.4. This is not because the subject is unimportant, but rather because the subject is too important to be relegated to the kind of superficial discussion we could accord it while doing justice to the rest of the physics governing the fluid envelopes of planets. This chapter provides a glimpse at what the reader has been missing. It highlights what the reader needs to keep in mind when learning atmosphere/ocean fluid dynamics, and, for the student who has already acquired some familiarity with that subject, connects fluid mechanical effects with the key planetary climate phenomena that have been the subject of this book. It is, in essence, a sampler of some of the many ways that large-scale fluid dynamics affects planetary climate.

This being the final chapter (for now) of a long journey, we will also take stock of how well we have done at coming to an understanding of the Big Questions introduced in Chapter 1. We wrap up with a reminder of the great breadth of largely unexplored problems the reader is already equipped to take on. The universe of problems becomes all the richer once planetary fluid dynamics is brought into the picture.

OVERVIEW

Why is the Earth generally hotter near the Equator than at the poles? Why is it generally hotter in summer than in winter, especially outside the tropics? Would this be true on other planets as well? How would the pattern change over time, as features of the planet's orbit vary? Would a very slowly rotating planet lose its atmosphere to condensation on the nightside? Would a planet whose rotation axis was steeply inclined relative to the normal to the plane of the orbit, or a planet in a highly elliptical orbit, have such an extreme seasonal cycle that it would be uninhabitable? The answers to these questions are to be found in the way the geographic and temporal pattern of illumination of the planet plays off against the thermal response time of the atmosphere, ocean, and solid surface of the planet. Generally speaking, in this section we seek to understand the features of a planet that determine the magnitude and pattern of geographic and seasonal variations in temperature.

Most of the discussion of temporal variability will focus on seasonal rather than diurnal variations, but much of the same considerations apply to both cycles, and so some remarks will be offered on the diurnal cycle as well. It should be kept in mind that the distinction between diurnal and seasonal cycle is meaningful only for bodies such as the Earth, Mars, or Titan whose rotation period is short compared with the period of orbit about the Sun.

When it comes to understanding the whys and wherefores of climate, there is an infinite amount one needs to know, but life affords only a finite time in which to learn it; the time available before one's fellowship runs out and a PhD thesis must be produced affords still less. Inevitably, the student who wishes to get launched on significant interdisciplinary problems must begin with a somewhat hazy sketch of the relevant physics, and fill in the gaps as time goes on. It is a lifelong process. This book is an attempt to provide the student with a sturdy scaffolding upon which a deeper understanding may be built later.

The climate system is made up of building blocks which in themselves are based on elementary physical principles, but which have surprising and profound collective behavior when allowed to interact on the planetary scale. In this sense, the “climate game” is rather like the game of Go, where interesting structure emerges from the interaction of simple rules on a big playing field, rather than complexity in the rules themselves. This book is intended to provide a rapid entrée into this fascinating universe of problems for the student who is already somewhat literate in physics and mathematics, but who has not had any previous experience with climate problems. The subject matter of each individual chapter could easily fill a textbook many times over, but even the abbreviated treatment given here provides enough core material for the student to begin treating original questions in the physics of climate.

OVERVIEW

Our objective is to understand the factors governing the climate of a planet. In this chapter we will be concerned with energy balance and planetary temperature. Certainly, there is more to climate than temperature, but equally certainly temperature is a major part of what is meant by “climate,” and greatly affects most of the other processes which come under that heading.

From the preceding chapter, we know that the temperature of a chunk of matter provides a measure of its energy content. Suppose that the planet receives energy at a certain rate. If uncompensated by loss, energy will accumulate and the temperature of some part of the planet will increase without bound. Now suppose that the planet loses energy at a rate that increases with temperature. Then, the temperature will increase until the rate of energy loss equals the rate of gain. It is this principle of energy balance that determines a planet's temperature. To quantify the functional dependence of the two rates, one must know the nature of both energy loss and energy gain.

The most familiar source of energy warming a planet is the absorption of light from the planet's star. This is the dominant mechanism for rocky planets like Venus, Earth, and Mars. It is also possible for energy to be supplied to the surface by heat transport from the deep interior, fed by radioactive decay, tidal dissipation, or high temperature material left over from the formation of the planet.

OVERVIEW

Our objective in this chapter is to treat the computation of a planet's energy loss by infrared emission in sufficient detail that the energy loss can be quantitatively linked to the actual concentration of specific greenhouse gases in the atmosphere. Unlike the simple model of the greenhouse effect described in the preceding chapter, the infrared radiation in a real atmosphere does not all come from a single level; rather, a bit of emission is contributed from each level (each having its own temperature), and a bit of this is absorbed at each intervening level of the atmosphere. The radiation comes out in all directions, and the rate of emission and absorption is strongly dependent on frequency. Dealing with all these complexities may seem daunting, but in fact it can all be boiled down to a conceptually simple set of equations which suffice for a vast range of problems in planetary climate.

It was shown in Chapter 3 that there is almost invariably an order of magnitude separation in wavelengths between the shortwave spectrum at which a planet receives stellar radiation and the longwave (generally infrared) spectrum at which energy is radiated to space. This is true throughout the Solar System, for cold bodies like Titan and hot bodies like Venus, as well as for bodies like Earth that are habitable for creatures like ourselves. The separation calls for distinct sets of approximations in dealing with the two kinds of radiation.

Introduction

Of all the standard clinical imaging techniques ultrasound is by far the least expensive and most portable (including handheld units smaller than a laptop computer), and can acquire continuous images at a real-time frame rate with few or no safety concerns. In addition to morphological and structural information, ultrasound can also measure blood flow in real-time, and produce detailed maps of blood velocity within a given vessel. Ultrasound finds very wide use in obstetrics and gynaecology, due to the lack of ionizing radiation or strong magnetic fields. The real-time nature of the imaging is also important in measuring parameters such as foetal heart function. Ultrasound is used in many cardiovascular applications, being able to detect mitral valve and septal insufficiencies. General imaging applications include liver cysts, aortic aneurysms, and obstructive atherosclerosis in the carotids. Ultrasound imaging is also used very often to guide the path and positioning of a needle in tissue biopsies.

Ultrasound is a mechanical wave, with a frequency for clinical use between 1 and 15 MHz. The speed of sound in tissue is ∼1540 m/s, and so the range of wavelengths of ultrasound in tissue is between ∼0.1 and 1.5 mm. The ultrasound waves are produced by a transducer, as shown in Figure 4.1, which typically has an array of up to 512 individual active sources. In the simplest image acquisition scheme, small subgroups of these elements are fired sequentially to produce parallel ultrasound beams.

Introduction

Of the four major clinical imaging modalities, magnetic resonance imaging (MRI) is the one developed most recently. The first images were acquired in 1973 by Paul Lauterbur, who shared the Nobel Prize for Medicine in 2003 with Peter Mansfield for their shared contribution to the invention and development of MRI. Over 10 million MRI scans are prescribed ever year, and there are more than 4000 scanners currently operational in 2010.

MRI provides a spatial map of the hydrogen nuclei (water and lipid) in different tissues. The image intensity depends upon the number of protons in any spatial location, as well as physical properties of the tissue such as viscosity, stiffness and protein content. In comparison to other imaging modalities, the main advantages of MRI are: (i) no ionizing radiation is required, (ii) the images can be acquired in any two- or three-dimensional plane, (iii) there is excellent soft-tissue contrast, (iv) a spatial resolution of the order of 1 mm or less can be readily achieved, and (v) images are produced with negligible penetration effects. Pathologies in all parts of the body can be diagnosed, with neurological, cardiological, hepatic, nephrological and musculoskeletal applications all being widely used in the clinic. In addition to anatomical information, MR images can be made sensitive to blood flow (angiography) and blood perfusion, water diffusion, and localized functional brain activation.