Energy Balance Climate Models

 
 
Wiley-VCH (Verlag)
  • erschienen am 2. August 2017
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  • XVI, 369 Seiten
 
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978-3-527-68381-9 (ISBN)
 
Written by renowned experts in the field, this first book to focus exclusively on energy balance climate models provides a concise overview of the topic. It covers all major aspects, from the simplest zero-dimensional models, proceeding to horizontally and vertically resolved models.
The text begins with global average models, which are explored in terms of their elementary forms yielding the global average temperature, right up to the incorporation of feedback mechanisms and some analytical properties of interest. The effect of stochastic forcing is then used to introduce natural variability in the models before turning to the concept of stability theory. Other one dimensional or zonally averaged models are subsequently presented, along with various applications, including chapters on paleoclimatology, the inception of continental glaciations, detection of signals in the climate system, and optimal estimation of large scale quantities from point scale data. Throughout the book, the authors work on two mathematical levels: qualitative physical expositions of the subject material plus optional mathematical sections that include derivations and treatments of the equations along with some proofs of stability theorems.
A must-have introduction for policy makers, environmental agencies, and NGOs, as well as climatologists, molecular physicists, and meteorologists.
 
Eine prägnante Einführung in alle wichtigen Aspekte von Energiegleichgewichtsmodellen, von ganz einfachen Modellen (Nulldimension) bis hin zu horizontalen und vertikalen Modellen.
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978-3-527-68381-9 (9783527683819)
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Gerald R. North is University Distinguished Professor of Atmospheric Sciences Emeritus at Texas A&M University, having obtained his BS degree in physics from the University of Tennessee, PhD (1966) in theoretical physics from the University of Wisconsin, Madison. Among other positions he served eight years as research scientist at Goddard Space Flight Center before joining Texas A&M in 1986, where he served as department head 1995-2003. He is a fellow of AAAS, AGU, AMS, and recipient of several awards including the Jule G. Charney Award of the American Meteorology Society. He has served as Editor in Chief of the Reviews of Geophysics and Editor in Chief of the Encyclopedia of the Atmospheric Sciences, 2nd Edition. He has coauthored books on Paleoclimatology and Atmospheric Thermodynamics.

Kwang-Yul Kim is a professor in climatology and physical oceanography at Seoul National University. Upon graduation from Texas A&M with his Ph.D. degree in physical oceanography he was inducted into the Phi Kappa Phi Honor Society. He authored two books: Fundamentals of Fluid Dynamics and Cyclostationary EOF Analysis. He programmed several new energy balance models.
Preface

CLIMATE AND CLIMATE MODELS
Defining Climate
Elementary Climate System Anatomy
Radiation and Climate
Hiercharchy of Climate Models
Greenhouse Effect and Modern Climate Change
Reading this Book
Cautionary Note and Disclaimer
Notes on Further Reading

GLOBAL AVERAGE MODELS
Temperature and Heat Balance
Time Dependence
Spectral Analysis
Nonlinear Global Model
Summary

RADIATION AND VERTICAL STRUCTURE
Radiance and Radiation Flux Density
Equation of Transfer
Gray Atmosphere
Plane Parallel Atmosphere
Radiative Equilibrium
Simplified Model for Water Vapor Absorber
Cooling Rates
Solutions for Uniform-Slab Absorbers
Vertical Heat Conduction
Convective Adjustment Models
Lessons from Simple Radiation Models
Criticism of the Gray Spectrum
Aerosol Particles

GREENHOUSE EFFECT AND CLIMATE FEEDBACKS
Greenhouse Effect without Feedbacks
Infrared Spectra of Outgoing Radiation
Summary of Assumptions and Simplifications
Log Dependence of the CO2 Forcing
Runaway Greenhouse Effect
Climate Feedbacks and Climate Sensitivity
Water Vapor Feedback
Ice Feedback for the Global Model
Probability Density of Climate Sensitivity
Middle Atmosphere Temperature Profile
Conclusion
Notes for Further Reading

LATITUDE DEPENDENCE
Spherical Coordinates
Incoming Solar Radiation
Extreme Heat Transport Cases
Heat Transport Across Latitude Circles
Diffusive Heat Transport
Deriving the Legendre Polynomials
Solution of the Linear Model with Constant Coefficients
The Two-Mode Approximation
Poleward Transport of Heat
Budyko's Transport Model
Ring Heat Source
Advanced Topic: Formal Solution for More General Transports
Ice Feedback in the 2-Mode Model
Polar Amplification through Icecap Feedback
Chapter Summary

TIME DEPENDENCE IN THE 1-D MODEL
Differential Equation for Time Dependence
Decay of Anomalies
Seasonal Cycle on a Homogeneous Planet
Spread of Diffused Heat
Random Winds and Diffusion
Numerical Methods
Spectral Methods
Chapter Summary
Appendix: Solar Heating Distribution

NONLINEAR PHENOMENA IN EBMS
Formulation of the Nonlinear Feedback Model
Stürm-Liouville Modes
Linear Stability Analysis
Finite Perturbation Analysis and Potential Function
Small Ice Cap Instability
Snow Caps and the Seasonal Cycle
Mengel's Land Cap Model
Chapter Summary

TWO HORIZONTAL DIMENSIONS AND SEASONALITY
Beach Ball Seasonal Cycle
Eigenfunctions in the Bounded Plane
Eigenfunctions on the Sphere
Spherical Harmonics
Solutions of the EBM with Constant Coefficients
Introducing Geography
Global Sinusoidal Forcing
Two Dimensional Linear Seasonal Model
Present Seasonal Cycle Comparison
Chapter Summary

PERTURBATION BY NOISE
Time-Independent Case for Uniform Planet
Time-Dependent Noise Forcing for Uniform Planet
Green's Function on the Sphere: f=0
Apportionment of Variance at a Point
Stochastic Model with Realistic Geography
Thermal Decay Modes with Geography

TIME-DEPENDENT RESPONSE AND THE OCEAN
Single-Slab Ocean
Penetration of a Periodic Heating at the Surface
Two-Slab Ocean
Box-Diffusion Ocean Model
Steady State of Upwelling-Diffusion Ocean
Upwelling Diffusion with (and without) Geography
Influence of Initial Conditions
Response to Periodic Forcing with Upwelling Diffusion Ocean
Summary and Conclusions

APPLICATIONS OF EBSM: OPTIMAL ESTIMATION
Introduction
Independet Estimators
Estimating Global Average Temperature
Deterministic Signals in the Climate System

APPLICATIONS OF EBMS: PALEOCLIMATE
Paleoclimatology
Precambrian Earth
Glaciations in the Permian
Glacial Inception on Antarctica
Clacial Inception on Greenland
Pleistocene Glaciations and Milankovic

Chapter 1
Climate and Climate Models


The global climate system consists of a large number of interacting parts. The material components and their sub-members include the following:

  1. 1. the atmosphere and its constituents such as free molecules and radicals of different chemical species, aerosol particles, and clouds;
  2. 2. the ocean waters and their members such as floating ice, dissolved species including electrolytes and gases as well as undissolved matter such as of biological origin and dust;
  3. 3. the land components with characteristics such as snow and ice cover, permafrost, moisture, topographical features and vegetation with all its ramifications.

The space-time configuration of abstract fields that are used to characterize properties of interest (such as temperature, density, and momentum) attributed to these components and their sub-members vary with time and position and each exhibits its own spectrum of time and length scales. Heat (or more formally, enthalpy) fluxes, moisture, and momentum fluxes pass from one of these material components to another, sometimes through subtle mechanisms. Determination of whether and how these constituent parts combine to establish a statistical equilibrium may seem challenge enough, but the climate dynamicist also seeks to understand how the system responds to time-dependent changes in certain control parameters such as the Sun's brightness, or the chemical composition of the atmosphere. Although we have been at it for many decades now, the grand problem is still far too complicated to solve at the desired level of accuracy (no bias) and precision (error variance) even though preliminary engineering-like calculations are being used routinely in scenario/impact studies because policymakers must (should!) make use of even tentative information in their deliberations (IPCC, 2007, 2013).

Serious attempts at quantitative climate theories can be said to have begun in the late 1960s, although some very clever attempts predate that by decades (see Weart, 2008). The theory of global climate is emerging from its infancy but it hardly constitutes a set of principles that can be converted into reliable numerical forecasts of climate decades ahead or that can be unequivocally used in explaining the paleoclimatic record. However, some valuable insights have been gained and many problems can be cast into the form of conceptual frameworks that can be understood. We now have an idea of which of the components are important for solving certain idealized problems, and indeed, in some cases, it appears that the problems can be made comprehensible (but not strictly quantitative) with models employing only a few variables.

The field of climate dynamics is vast, embracing virtually every subfield of the geosciences (even "pure" physics, chemistry, and biology) from the quantum mechanics of photons being scattered, absorbed, and emitted by/from atmospheric molecules in radiative transfer processes to the study of proxies such as tree-ring widths and isotopic evidence based on fossilized species deposited and buried long ago in sediments deep below the ocean's floor. The in-depth coverage of these subfields is generally presented in the traditionally separate course offerings of curricula in the geosciences. This book is concerned with the integration of this array of material into a composite picture of the global climate system through simplified phenomenological models. The approach will be to pose and examine some problems that can be solved or analyzed with the classical techniques of mathematical physics. Throughout we attempt to use these analytical methods, but will introduce and use numerical methods and simulations when necessary. However, our main strategy will be to idealize the physical problem in such a way as to render it solvable or at least approachable, then compare or draw analogies either to the real world or to the results of solving a more believable model - hardly a foolproof procedure but likely to be instructive. In short, we hope to get at the heart of some climate problems in such a way that the reader's intuition for the composite system can be developed and more informed approaches can be taken toward the solution of specific problems.

The energy balance climate models (EBCMs) generally deal with an equation or a set of coupled equations whose solution yields a space-time average of the surface temperature field. Unfortunately, the solutions cannot usually describe the temperature field above the boundary layer of the atmosphere except in rare circumstances. This is a severe limitation, leaving us with only partial answers to many questions we would like to pose. On the other hand, we are blessed with many reasons supporting the importance of the surface temperature field:

  1. 1. Space-time averages of surface temperature are easily estimated and many instrumental records provide good data, not just contemporarily, but over the last century.
  2. 2. Space-time averages of surface temperature data are close to being normally distributed, making them easy to understand and treat. This is not so for precipitation and some other variables. Moreover, the larger the space-time scale, the more information from point sources can be combined into the average, resulting in a reduction in the random measurement errors on the mean estimates.
  3. 3. The time series of space-time averages of surface temperature is particularly simple, resulting in applicability of autoregressive behavior of order unity in many cases.
  4. 4. Nearly all paleoclimate indicators provide information about the surface temperature, extending the data base that can be used in testing. There are never enough data to check and adjust models, especially complex numerical models. Paleoclimatology can potentially provide more data that can be used to understand climate models.
  5. 5. As we will show, the surface temperature is also the easiest variable to model, especially for large area and time averages. It becomes more difficult as the space-time scales in the problem decrease. In this book, we will start with the largest space and time scales and find that there is a natural progression of estimates from the largest to the smallest space-time scales. Moreover, averaging over large scales reduces some errors in models as well as in measurements.
  6. 6. Most of the externally applied perturbations to the climate system that are of interest are directed at large spatial and temporal scales. This happens to be the case for the four best known perturbations: greenhouse gases, volcanic dust veils, anthropogenic aerosols, and solar brightness. It is intuitively appealing (as well as motivated by physics, as we shall see) that the large space-time scale perturbations result primarily in the same large space-time scales of thermal response patterns in the climate system.
  7. 7. The study of energy balance models is cheap. This can be a factor when questions are posed from paleoclimatology, for example. Big models are simply too expensive to experiment with in the first trials. With the speed up of modern computers, many paleoclimate problems can be examined with general circulation models (GCMs), but not every one of them.
  8. 8. The study of exoplanets has become important in recent years. The habitable zone of a planet's orbital and atmospheric/oceanic dynamical/chemical parameters may fall into the purview of energy balance models.
  9. 9. Finally, the surface temperature is important for societal well-being and it is easily grasped, although the idea of large space-time scales is less easily identifiable and appreciated by the average person.

Unfortunately, as soon as we go above the near-surface environment, the mathematical difficulties of solving the climate problem even for the temperature becomes orders of magnitude more difficult. Also, for all its importance, precipitation cannot be solved by simple models because it depends too sensitively on the circulation of the atmosphere (and the ocean).

1.1 Defining Climate


Before proceeding, we must define what we mean by climate. As an illustrative example, we restrict ourselves at first to the global average surface temperature. Our definition is abstract and not strictly an operational one unless certain (reasonable but, unfortunately, unverifiable) conditions are fulfilled. When we examine records of globally averaged temperature at the Earth's surface we find that it fluctuates in time. Figure 1.1 shows a century-long record of both annual and global averages (estimates of these, to be more precise) and, except for a possible upward slope, we find departures from the mean linear trend that persist over a few years or even decades.

Figure 1.1 Time series of thermometer-based global average temperatures from the website of Goddard Institute for Space Studies: www.giss.nasa.gov. The units are in Kelvin and the temperature values are "anomalies" or deviations from a long-term mean (1951-1980).

(Goddard Institute for Space Studies (NASA) (2017).)

Consider an abstraction of the real system. We borrow from the discipline of time series analysis (which may have originated in the subdiscipline of theoretical physics called statistical mechanics) the concept of an ensemble.1 By this, we mean to consider a segment of a record of some quantity versus time (e.g., the record of estimates of annual-mean and global-average temperatures...

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