SYDNEY CHAPMAN. National Center for Atmospheric Research, Boulder, Colo., and Geophysical Institute, University of Alaska, College, Alaska, U.S.A.

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1 ATMOSPHERIC TIDES

2 SYDNEY CHAPMAN National Center for Atmospheric Research, Boulder, Colo., and Geophysical Institute, University of Alaska, College, Alaska, U.S.A. RICHARD S. LIND ZEN Dept. of Geophysical Sciences, University of Chicago, Chicago, Ill. U.S.A. ATMOSPHERIC TIDES Thermal and Gravitational D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND

3 ISBN-13: e-isbn-13: DOl: / D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1970 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher

4 PREFACE Everyone is familiar with the daily changes of air temperature. The barometer shows that these are accompanied by daily changes of mass distribution of the atmosphere, and consequently with daily motions of the air. In the tropics the daily pressure change is evident on the barographs; in temperate and higher latitudes it is not noticeable, being overwhelmed by cyclonic and anticyclonic pressure variations. There too, however, the daily change can be found by averaging the variations over many days; and the same process suffices to show that there is a still smaller lunar tide in the atmosphere, first sought by Laplace. Throughout nearly two centuries these 'tides', thermal and gravitational, have been extensively discussed in the periodical literature of science, although they are very minor phenomena at ground level. This monograph summarizes our present knowledge and theoretical understanding of them. It is more than twenty years since the appearance of the one previous monograph on them - by Wilkes - and nearly a decade since they were last comprehensively reviewed, by Siebert. The intervening years have seen many additions to our knowledge of the state of the upper atmosphere, and of the tides there, on the basis of measurements by radio, rockets and satellites. Combined with progress in the dynamical theory, this has led to the definite abandonment of Kelvin's proposed resonance explanation of why the half-daily change of the barometer is so much more prominent than the 24-hourly component variation, although the latter is dominant in the air temperature changes that cause both these pressure variations. The main cause is the absorption of solar radiation by ozone in the stratosphere, aided by water vapor. The dynamical theory indicates that the strength of the tides increases with height in the atmosphere, so that in the ionospheric regions they are no longer minor; there they are major aeronomic features, involving very strong winds, and tidal changes of height reckoned in kilometers. This has been well confirmed by observation. As now developed, the theory solves some (though not yet all) of the major traditional problems relating to the oscillations observed at ground level. The refinement of the theory involves a new generation of problems, ranging from the nature of the daily variations in the lowest few meters of the air near the ground, to the influence of factors of the environment in the atmosphere above 50 km - such as hydro magnetic forces, molecular viscosity and conduction, and the modified composition at high levels. The theory is even being extended to deal with thermal tides in the atmosphere of Mars. The shortcomings of the theory at present are indicated, and the likely directions of future progress are pointed out.

5 VI SYDNEY CHAPMAN AND RICHARD S.LlNDZEN Chapter I gives a historical account of the subject, which has something of the flavor of a detective story in which the whodunit was the source of the semidiurnal pressure variation: when Kelvin proposed his hypothesis, the existence of the ozone layer, the chief cause, was unknown. Chapters 2S and 2L describe the main known facts about the various components of the solar thermal tide and the lunar gravitational tide, as revealed by the thermometer, barometer and anemometer, and as discussed and represented, graphically and mathematically, in various ways and by different authors. Chapter 3 summarizes the present state of the dynamical theory; it is by R. Lindzen, who also contributed the last three sections of Chapter 1; apart from this, Chapters 1 and 2 are by S. Chapman. It is hoped that readers will benefit from the full subject index, author index, place index, and the Guide to and Classification of the Figures and Tables, on which much effort has been spent. The typing assistance of Mrs. D. G. Fisher, Mrs. C. MacBride and Mrs. L. Reiffel is gratefully acknowledged. SYDNEY CHAPMAN RICHARD S. LIND ZEN July 15, 1969

6 TABLE OF CONTENTS PREFACE v CHAPTER 1. INTRODUCTORY AND HISTORICAL 1.1. Introduction: Pytheas, Bacon, Newton and Laplace 1.2. The Barometric and Other Daily Variations I.2A. True or Apparent Time, and Mean Time I.2B. The Harmonic Dial 1.3. Thermal Tides and Kelvin's Resonance Theory 1.4. More Realistic Atmospheric Models 1.5. The Phase of S2(P) 1.6. Doubts as to the Resonance Theory 1.7. Renewed Hope in the Resonance Theory 1.8. Atmospheric Oscillations as Studied by Weekes and Wilkes 1.9. Rockets Exclude Resonance Ozone Absorption of Radiation the Main Cause of S2(P) Upper Air Data Theoretical Calculations of the Diurnal Thermal Tide Other Features of Atmospheric Oscillations CHAPTER 2s. THE SOLAR DAILY ATMOSPHERIC OSCILLATIONS AS REVEALED BY METEOROLOGICAL DATA 2S.I. The Material Studied; Ground Level Data 2S.2. Harmonic Analysis of S; The Non-Cyclic Variation 2S.3. The Seasonal Variation of S 2S.3A. Daily Seasonal Integers (J (Sigma) or SN (Bartels, 1954) 2S.4. The World-Wide Distribution of S, Particularly of S(p) 2S.4A. S2(P) 2S.4A.I. Types of Associated Legendre Functions 2S.4A.2. The Spherical Harmonic Expression of S2 (p) 2S.4B. Sl (p) 2S.4C. S3(P) 2S.4D. S4(P) 2S.5. The Daily Variation of Air Temperature T

7 VIII SYDNEY CHAPMAN AND RICHARD S. LINDZEN 2S.6. The Daily Wind Variation S(V) 2S. 7. Atmospheric Daily Changes above Ground Level 2S. 7 A. Daily Variations between the Ground and 30 km 2S.7B. Daily Variations from 30 km-60 km 2S.7C. Daily Variations from km 2S.7D. Daily Variations in the Thermosphere 2S.7E. Analysis of Data Covering Only a Fraction of a Day CHAPTER 2L. THE LUNAR ATMOSPHERIC TIDE AS REVEALED BY METEOROLOGICAL DA TA 66 2L.l. Introduction 66 2L.2. The Tropical Lunar Air Tide 67 2L.3. The Lunar Air Tide Outside the Tropics 67 2L.4. The Month and the Lunar Day 69 2L.4A. The Main Harmonic Components of the Lunar Tidal Potential 72 2L.5. Methods of Computation of L from Observed Data; Early Methods Based on Apparent Lunar Time 74 2L.6. The Chapman-Miller (or C-M) Method for Meteorological Variables 78 2L.6A. Use of the Integers Mu (or J1) instead of the Integers Nu or W~~ 2L.6B. The Components Sp 81 2L.7. Vector Probable Errors 81 2L.8. The Determination of L z from Only a Few Meteorological Readings ~~ ~ 2L.9. The Lunar Semidiurnal Barometric Tide Lz(P) 84 2L.I0. The Expression of L z (p) in Spherical Harmonic Functions 86 2L.l1. The Asymmetry of L z (P) Relative to the Equator, and its Seasonal Variation 91 2L.12. Comparison of Lz(p) and Sz(p) 93 2L.13. The Lunar Tidal Wind Variation 95 2L.14. The Lunar Tidal Variation of Air Temperature 100 2L.15. The Lunar Tidal Changes of Height of Various Pressure Levels 103 2L.16. Brief Mention of the Lunar Geomagnetic Tide 104 W CHAPTER 3. QUANTITATIVE THEORY OF ATMOSPHERIC TIDES AND THERMAL TIDES 3.1. Introduction 3.2. Equations 3.3. Methods of Solution 3.3A. Laplace's Tidal Equation 3.3B. Vertical Structure Equation

8 ATMOSPHERIC TIDES IX 3.3C. Outline of Overall Procedure Sources of Excitation A. Gravitational Excitation B. Thermal Excitation Due to Exchange of Heat with the Ground C. Thermal Excitation Due to Direct Atmospheric Absorption of Insolation D. Summary Explicit Solutions A. The Migrating Solar Semi diurnal Thermal Tide B. The Solar Diurnal Thermal Tide C. The Lunar Semidiurnal Tide D. Other Components Shortcomings of Present Calculations A. Surface Topography B. Dissipation B.1. Infrared Cooling B.2. Molecular Viscosity and Conductivity B.3. Ion Drag and Thermal Tides in the Ionosphere C. Non-Linear Effects D. Neglect of Mean Winds and Horizontal Temperature Gradients E. Additional Remarks Comparison of Theory with Data 169 List of Symbols for Chapter GUIDE TO THE FIGURES AND TABLES 175 REFERENCES 179 INDEX OF NAMES 188 INDEX OF SUBJECTS 191 INDEX OF PLACES 199