Wind Effects on Cable-Supported Bridges - You-Lin Xu - ebook

Wind Effects on Cable-Supported Bridges ebook

You-Lin Xu

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As an in-depth guide to understanding Wind Effects on Cable-Supported Bridges, this book uses analytical, numerical and experimental methods to give readers a fundamental and practical understanding of the subject matter. It is structured to systemically move from introductory areas through to advanced topics currently being developed from research work. The author concludes with the application of the theory covered to real-world examples, enabling readers to apply their knowledge. The author provides background material, covering areas such as wind climate, cable-supported bridges, wind-induced damage, and the history of bridge wind engineering. Wind characteristics in atmospheric boundary layer, mean wind load and aerostatic instability, wind-induced vibration and aerodynamic instability, and wind tunnel testing are then described as the fundamentals of the subject. State-of-the-art contributions include rain-wind-induced cable vibration, wind-vehicle-bridge interaction, wind-induced vibration control, wind and structural health monitoring, fatigue analysis, reliability analysis, typhoon wind simulation, non-stationary and nonlinear buffeting response. Lastly, the theory is applied to the actual long-span cable-supported bridges. * Structured in an easy-to-follow way, covering the topic from the fundamentals right through to the state-of-the-art * Describes advanced topics such as wind and structural health monitoring and non-stationary and nonlinear buffeting response * Gives a comprehensive description of various methods including CFD simulations of bridge and vehicle loading * Uses two projects with which the author has worked extensively, Stonecutters cable-stayed bridge and Tsing Ma suspension bridge, as worked examples, giving readers a practical understanding

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Contents

Cover

Title Page

Copyright

Dedication

Foreword

Foreword

Preface

Acknowledgements

Chapter 1: Wind Storms and Cable-Supported Bridges

1.1 Preview

1.2 Basic Notions of Meteorology

1.3 Basic Types of Wind Storms

1.4 Basic Types of Cable-Supported Bridges

1.5 Wind Damage to Cable-Supported Bridges

1.6 History of Bridge Aerodynamics

1.7 Organization of this Book

1.8 Notations

References

Chapter 2: Wind Characteristics in Atmospheric Boundary Layer

2.1 Preview

2.2 Turbulent Winds in Atmospheric Boundary Layer

2.3 Mean Wind Speed Profiles

2.4 Wind Turbulence

2.5 Terrain and Topographic Effects

2.6 Design Wind Speeds

2.7 Directional Preference of High Winds

2.8 Case Study: Tsing Ma Bridge Site

2.9 Notations

References

Chapter 3: Mean Wind Load and Aerostatic Instability

3.1 Preview

3.2 Mean Wind Load and Force Coefficients

3.3 Torsional Divergence

3.4 3-D Aerostatic Instability Analysis

3.5 Finite Element Modeling of Long-Span Cable-Supported Bridges

3.6 Mean Wind Response Analysis

3.7 Case Study: Stonecutters Bridge

3.8 Notations

References

Chapter 4: Wind-Induced Vibration and Aerodynamic Instability

4.1 Preview

4.2 Vortex-Induced Vibration

4.3 Galloping Instability

4.4 Flutter Analysis

4.5 Buffeting Analysis in the Frequency Domain

4.6 Simulation of Stationary Wind Field

4.7 Buffeting Analysis in the Time Domain

4.8 Effective Static Loading Distributions

4.9 Case Study: Stonecutters Bridge

4.10 Notations

References

Chapter 5: Wind-Induced Vibration of Stay Cables

5.1 Preview

5.2 Fundamentals of Cable Dynamics

5.3 Wind-Induced Cable Vibrations

5.4 Mechanism of Rain-Wind-Induced Cable Vibration

5.5 Prediction of Rain-Wind-Induced Cable Vibration

5.6 Occurrence Probability of Rain-Wind-Induced Cable Vibration

5.7 Case Study: Stonecutters Bridge

5.8 Notations

References

Chapter 6: Wind-Vehicle-Bridge Interaction

6.1 Preview

6.2 Wind-Road Vehicle Interaction

6.3 Formulation of Wind-Road Vehicle-Bridge Interaction

6.4 Safety Analysis of Road Vehicles on Ting Kau Bridge under Crosswind

6.5 Formulation of Wind-Railway Vehicle Interaction

6.6 Safety and Ride Comfort of Ground Railway Vehicle under Crosswind

6.7 Wind-Railway Vehicle-Bridge Interaction

6.8 Notations

References

Chapter 7: Wind Tunnel Studies

7.1 Preview

7.2 Boundary-Layer Wind Tunnels

7.3 Model Scaling Requirements

7.4 Boundary Wind Simulation

7.5 Section Model Tests

7.6 Taut Strip Model Tests

7.7 Full Aeroelastic Model Tests

7.8 Identification of Flutter Derivatives

7.9 Identification of Aerodynamic Admittance

7.10 Cable Model Tests

7.11 Vehicle-Bridge Model Tests

7.12 Notations

References

Chapter 8: Computational Wind Engineering

8.1 Preview

8.2 Governing Equations of Fluid Flow

8.3 Turbulence and its Modeling

8.4 Numerical Considerations

8.5 CFD for Force Coefficients of Bridge Deck

8.6 CFD for Vehicle Aerodynamics

8.7 CFD for Aerodynamics of Coupled Vehicle-Bridge Deck System

8.8 CFD for Flutter Derivatives of Bridge Deck

8.9 CFD for Non-Linear Aerodynamic Forces on Bridge Deck

8.10 Notations

References

Chapter 9: Wind and Structural Health Monitoring

9.1 Preview

9.2 Design of Wind and Structural Health Monitoring Systems

9.3 Sensors and Sensing Technology

9.4 Data Acquisition and Transmission System (DATS)

9.5 Data Processing and Control System

9.6 Data Management System

9.7 Structural Health Monitoring System of Tsing Ma Bridge

9.8 Monitoring Results of Tsing Ma Bridge during Typhoon Victor

9.9 System Identification of Tsing Ma Bridge during Typhoon Victor

9.10 Notations

References

Chapter 10: Buffeting Response to Skew Winds

10.1 Preview

10.2 Formulation in the Frequency Domain

10.3 Formulation in the Time Domain

10.4 Aerodynamic Coefficients of Bridge Deck under Skew Winds

10.5 Flutter Derivatives of Bridge Deck under Skew Winds

10.6 Aerodynamic Coefficients of Bridge Tower Under Skew Winds

10.7 Comparison with Field Measurement Results of Tsing Ma Bridge

10.8 Notations

References

Chapter 11: Multiple Loading-Induced Fatigue Analysis

11.1 Preview

11.2 SHM-oriented Finite Element Modeling

11.3 Framework for Buffeting-Induced Stress Analysis

11.4 Comparison with Field Measurement Results of Tsing Ma Bridge

11.5 Buffeting-Induced Fatigue Damage Assessment

11.6 Framework for Multiple Loading-Induced Stress Analysis

11.7 Verification by Case Study: Tsing Ma Bridge

11.8 Fatigue Analysis of Long-Span Suspension Bridges under Multiple Loading

11.9 Notations

References

Chapter 12: Wind-Induced Vibration Control

12.1 Preview

12.2 Control Methods for Wind-Induced Vibration

12.3 Aerodynamic Measures for Flutter Control

12.4 Aerodynamic Measures for Vortex-Induced Vibration Control

12.5 Aerodynamic Measures for Rain-Wind-Induced Cable Vibration Control

12.6 Mechanical Measures for Vortex-Induced Vibration Control

12.7 Mechanical Measures for Flutter Control

12.8 Mechanical Measures for Buffeting Control

12.9 Mechanical Measures for Rain-Wind-Induced Cable Vibration Control

12.10 Case Study: Damping Stay Cables in a Cable-Stayed Bridge

12.11 Notations

References

Chapter 13: Typhoon Wind Field Simulation

13.1 Preview

13.2 Refined Typhoon Wind Field Model

13.3 Model Solutions

13.4 Model Validation

13.5 Monte Carlo Simulation

13.6 Extreme Wind Analysis

13.7 Simulation of Typhoon Wind Field over Complex Terrain

13.8 Case Study: Stonecutters Bridge Site

13.9 Notations

References

Chapter 14: Reliability Analysis of Wind-Excited Bridges

14.1 Preview

14.2 Fundamentals of Reliability Analysis

14.3 Reliability Analysis of Aerostatic Instability

14.4 Flutter Reliability Analysis

14.5 Buffeting Reliability Analysis

14.6 Reliability Analysis of Vortex-Induced Vibration

14.7 Fatigue Reliability Analysis based on Miner's Rule for Tsing Ma Bridge

14.8 Fatigue Reliability Analysis based on Continuum Damage Mechanics

14.9 Notations

References

Chapter 15: Non-Stationary and Non-Linear Buffeting Response

15.1 Preview

15.2 Non-Stationary Wind Model I

15.3 Non-Stationary Wind Model II

15.4 Buffeting Response to Non-Stationary Wind

15.5 Extreme Value of Non-Stationary Response

15.6 Unconditional Simulation of Non-Stationary Wind

15.7 Conditional Simulation of Non-Stationary Wind

15.8 Non-Linear Buffeting Response

15.9 Notations

References

Chapter 16: Epilogue: Challenges and Prospects

16.1 Challenges

16.2 Prospects

Index

This edition first published 2013

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Library of Congress Cataloging-in-Publication Data

Xu, You-Lin, 1952-

Wind effects on cable-supported bridges / You-Lin Xu.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-18828-6 (cloth)

1. Cable-stayed bridges. 2. Wind-pressure. I. Title.

TG405.X885 2013

624.2′52–dc23

2012038933

ISBN: 9781118188286

To

Wei-Jian, my wife; Anna, my daughter

–Y.L. Xu

Foreword

by Ahsan Kareem

Economic growth and modernization has resulted in greater demand for cable-supported bridges that are designed to carry large volumes of vehicle traffic and railways over a long span. At the end of year 2010, there were at least ten suspension bridges with a main span over 1,200 m and ten cable-stayed bridges with a main span over 700 m.

Ambitious cable-supported bridge projects are often constructed along coastal areas and in or near to cities that are vulnerable to typhoons and high winds; therefore, wind effects on cable-supported bridges are key issues in their design, construction, operation and maintenance. However, a good knowledge and understanding of bridge aerodynamics is not widespread amongst university students and practicing engineers, mainly because of the multi-disciplinary nature of the subject. Another reason is that there is no book that addresses this topic comprehensively.

In anticipation that more cable-supported bridges and some super-long-span cable-supported bridges will be designed and built in the 21 century throughout the world, the demand for such a book will increase as this subject shall be taught at major universities and colleges at least at the postgraduate level. A comprehensive book like this one on this subject, covering not only the fundamental knowledge but also state-of-the-art developments, will certainly facilitate learning and preparation of students to face the challenges posed by the bridges of tomorrow.

Dr. You-Lin Xu has conducted teaching, research and consultancy work in the field of wind engineering and bridge engineering for almost 30 years. He was engaged in wind tunnel studies and wind-induced vibration control in Australia from 1989 to 1995. Together with his students and colleagues, he has worked extensively on wind loading and effects on the Tsing Ma suspension bridge in Hong Kong since 1995 and on the Stonecutters cable-stayed bridge in Hong Kong since 2003. In recognition of his contribution, he was awarded the Robert H. Scanlan Medal by the American Society of Civil Engineers in 2012. This medal is awarded to those who make extraordinary contributions to mechanics and its applications, with special reference to bridge aeroelasticity. He has taught the subject “Wind Engineering” to MSc students at The Hong Kong Polytechnic University since 1998.

Essentially, the fundamentals presented in the book are drawn from his lecture notes, and most of the state-of-the-art developments presented in the book have resulted from their relevant publications in international journals and conference proceedings. Dr. Xu, in my opinion, is at the right stage of his career, with the distinguished academic and profession background, to synthesize this interdisciplinary work into a comprehensive and exhaustive book. I am confident that it will be very well received, both in academia and in design practice.

The history and latest developments of bridge wind engineering both demonstrate that any progress made in this subject evolves from the synergy between research and practice. This book does bridge the gap between the theoretical research and practical application. Covering a comprehensive range of topics and the most up to date information on the subject, it will also inspire researchers and academics to pursue new methodologies and innovative technologies for the design, construction, operation and maintenance of wind-excited cable-supported bridges.

Ahsan Kareem, NAE, Dist. Mem. ASCERobert M Moran Professor of EngineeringNatHaz Modeling LaboratoryUniversity of Notre Dame, Indiana

Foreword

by Hai-Fan Xiang

To meet the social and economic needs for efficient transportation systems, many cable-supported bridges have been built throughout the world. The Nanpu cable-stayed bridge, with a main span of 423 m, which was successfully built in Shanghai in 1991, marks a milestone in the history of the construction of cable-supported bridges in China. Since then, a high tide of construction of long-span cable-supported bridges has emerged in China. As of 2012, among the ten longest suspension bridges of a main span over 1200 m in the world, five are in China. Similarly, there are five cable-stayed bridges in China among the world's ten longest cable-stayed bridges with a main span over 700 m.

With accumulated experience and advanced technology, the construction of super-long-span cable-supported bridges to cross straits has also been planned around the world, such as Messina Strait in Italy, Qiongzhou Strait in China, Sunda Strait in Indonesia and Tsugaru Strait in Japan. However, as span length increases, cable-supported bridges are becoming lighter in weight, more slender in stiffness, lower in damping and more sensitive to wind-induced vibration. The requirements of functionality, safety, and sustainability of the bridges against wind hazards have presented new challenges to our wind engineering community. A comprehensive book like this one on this subject, covering not only the fundamental knowledge but also state-of-the-art developments, will definitely help the learning and preparation of our students and engineers who face these challenges.

Dr You-Lin Xu graduated from Tongji University, where I have been working for about 60 years. Dr Xu and his research team at The Hong Kong Polytechnic University have worked extensively on wind loading and effect on the Tsing Ma suspension bridge in Hong Kong since 1995 and on the Stonecutters cable-stayed bridge in Hong Kong since 2003. This book is structured to systemically move from introductory areas through to advanced topics with real-world examples. It should serve well to advance the research and practice in the field of wind engineering in general, and wind effects on cable-supported bridges in particular.

This book is actually a summary of the work they have done in the past 17 years. I would give my warm congratulation to Dr Xu for this excellent work.

Hai-Fan Xiang, Professor Emeritus,MCAE, Advisory DeanCollege of Civil EngineeringTongji University, Shanghai, China

Preface

The well-known collapse of the original Tacoma Narrows Bridge in 1940 identified the importance of understanding wind effects on long-span cable-supported bridges. Extensive research and practice have been carried out since then. The pioneering work of Professors Robert Scanlan and Alan Davenport, among others in the 1960s and 1970s, laid down a foundation for the subject of bridge wind engineering. The advanced theories and modern technologies developed in the past 40 years have made it possible to construct suspension bridges with a main span over 1990 m and cable-stayed bridges with a main span over 1000 m.

The need to construct super-long-span cable-supported bridges to cross straits and to bring people together to live and work has become more obvious in this century. However, the great increase in destructive wind storms due to global climate change has affected many parts of the world, and the vulnerability of long-span cable-supported bridges to strong winds has been increased significantly. The demand imposed by this on the functionality, safety and sustainability of super-long-span cable-supported bridges against wind hazard has presented new challenges to our wind engineering community. This was the original incentive of the author in writing this book: to provide the fundamental knowledge from which modern bridge wind engineering has evolved for our graduate students, and to present the state-of-the-art development from the past 40 years in the field to both graduate students and practicing engineers, so that we are better prepared for new challenges in bridge aerodynamics.

Chapter 1 of this book presents the background materials, including basic notions of meteorology, basic types of wind storms, basic types of cable-supported bridges, wind damage to cable-supported bridges and history of bridge aerodynamics. Chapter 2 introduces wind characteristics in atmospheric boundary layer. Chapters 3 to 6 respectively describe mean wind load and aerostatic instability of bridges, wind-induced vibration and aerodynamic instability of bridges, wind-induced vibration of stay cables and wind-vehicle-bridge interaction. These contents cover the fundamentals of bridge aerodynamics, which are suitable as an elective subject for final-year undergraduate students. As three important tools in the studies of bridge aerodynamics in addition to theoretical analysis discussed in the previous chapters, Chapters 7 to 9 respectively introduce wind tunnel technique, computational wind engineering simulation and wind/structural health monitoring technology.

The materials presented in the first nine chapters are appropriate for graduate student courses. Special topics, such as buffeting response to skew winds, multiple loading-induced fatigue analysis, wind-induced vibration control and typhoon wind field simulation, are introduced in Chapters 10 to 13, respectively. These chapters are rather independent of the others and can be used individually. In Chapter 14, reliability analysis of wind-excited bridges is described, laying down a foundation for the probabilistic wind-resistant design of long-span cable-supported bridges. As a frontier in this field, Chapter 15 presents the preliminary study results on non-stationary and nonlinear buffeting responses. Finally, challenges and prospects of bridge aerodynamics as a scientific but practical subject are highlighted in Chapter 16.

The history and latest developments in bridge wind engineering both demonstrate that any progress made in this subject stems from the synergy between research and practice. Case studies on real long-span cable-supported bridges are provided within almost every chapter of this book.

I embarked on the field of wind engineering in 1983 as a master student at Tongji University, China, and continued my study in this field as a PhD student at University of Sydney in 1989 in Australia. I have, fortunately, been involved in long-term collaborative research and practice with the Hong Kong Highways Department since 1995 for wind effects on, and structural health monitoring of, the Tsing Ma suspension bridge and the Stonecutters cable-stayed bridge in Hong Kong. I have taught the subject “Wind Engineering” to Master of Science students since 1998 at The Hong Kong Polytechnic University. Most importantly, I have been inspired by the work of many outstanding scholars and engineers in the past 30 years, and I would like to dedicate this book to them.

In writing the book, I am always reminded that it mainly serves as a textbook for graduate students and practicing engineers to understand bridge aerodynamics and straddle the gap between theoretical research and practical application. Its readers are assumed to have some background in structural analysis, structural dynamics, probability theory, and random vibration.

I would be very happy to receive constructive comments and suggestions from readers.

You-Lin XuThe Hong Kong Polytechnic [email protected] 2012

Acknowledgements

The writing of this book has been a challenging and time-consuming task that could not have been completed without the help of many individuals. I am grateful to the many people who helped in the preparation of this book.

A few PhD students, former and present, at The Hong Kong Polytechnic University participated in some research works presented in this book: Prof. Le-Dong Zhu, Prof. Wen-Hua Guo, Prof. Wen-Shou Zhang, Dr. Zhi-Wei Chen, Dr. Wen-Feng Huang, Dr. Wai-Shan Chan, Dr. Ling-Yun Wang, Dr. Kei-Man Shum, Dr. Chi-Lun Ng, Mr. Qing Zhu, and Mr. Bin Wang.

Several colleagues and research staff, former and present, at The Hong Kong Polytechnic University made contributions to some research works described in this book: Prof. He Xia, Prof. Jia-Hao Lin, Prof. Jun Chen, Prof. Yong-Le Li, Prof. Nan Zhang, Dr. Yong Xia, Dr. Quan-Shun Ding, Dr. Liang Hu, Dr. Hai-Jun Zhou, Dr. Guo Liu, Dr. Lin Huang, Dr. Ting-Ting Liu, Dr. Zhi-Gi Li, and Mr. Shan Zhan.

In particular, Mr. Cao-Dong Zhang was involved in the writing of Chapter 1; Dr. Wen-Feng Huang was involved in the writing of Chapters 2, 12 and 13; Mr. Qing Zhu participated in the writing of Chapters 3, 4, and 7; Miss Yi-Xin Peng drew some figures in Chapter 7; Mr. Bin Wang participated in the writing of Chapter 8; Dr. Yong Xia was engaged in the writing of Chapter 9; Dr. Zhi-Wei Chen prepared the first draft of Chapter 14; Dr Liang Hu prepared the first draft of Chapter 15; Mr Xiao-Liang Meng proof-read the book and prepared the index; and Mr Jian-Fu Lin examined all the figures and tables, for which I am most grateful.

I wish to thank Dr. Chi-Ho Hui and Dr. Kai-Yuen Wong, the chief engineer and the senior engineer respectively, from the Hong Kong Highways Department for their strong support when we were working on the relevant projects of the Stonecutters Bridge and the Tsing Ma Bridge in Hong Kong.

I wish to express my appreciation to my former professors who provided a foundation of knowledge on which I could build. I would also like to acknowledge the anonymous reviewers whose comments on the proposal of the book led to a much better presentation of the materials. I have been influenced by the work of many outstanding scholars and researchers in this field, and most of their names will be found in the reference list attached to each chapter.

My research work in this area has been largely supported by the Hong Kong Research Grants Council, The Hong Kong Polytechnic University, The Natural Science Foundation of China, and the Hong Kong Highways Department over many years. All the support is gratefully acknowledged.

A vote of thanks must go to Mr. James Murphy, Publisher at John Wiley & Sons, for his patience and encouragement from the beginning and during the preparation of this book, and to Ms. Clarissa Lim and Ms. Shelley Chow, Project Editors at John Wiley & Sons, for their patience and scrutiny in the editing of this book.

Finally, I am grateful to my family for their help, encouragement and endurance.

1

Wind Storms and Cable-Supported Bridges

1.1 Preview

Because of their competence for long spans, many cable-supported bridges, including both cable-stayed bridges and suspension bridges, have been built throughout the world. These long-span cable-supported bridges are often remarkably flexible, low in damping and light in weight. Therefore, they can also be susceptible to the action of wind. For example, the Tacoma Narrows suspension bridge, which had a main span of 853 m and was built to link the Olympic Peninsula with the rest of the state of Washington, oscillated through large displacements at a wind speed of about 19 m/s and collapsed on November 7, 1940, only four months and six days after the bridge was opened to the public.

Modern long-span cable-supported bridges carry a large volume of vehicles and may experience considerable vibration due both to moving vehicles and to turbulent winds. The considerable vibration of the bridge and the crosswinds may, in turn, affect the running safety of vehicles. Therefore, adequate treatment of wind effects in design is essential to the safety and functionality of both long-span cable-supported bridges and vehicles running on the bridges.

In this introductory chapter, the meteorology of wind storms is first introduced to provide information on the basic features of wind storms. The basic configuration, structural systems, and aerodynamic characteristics of both cable-stayed and suspension bridges are then described to facilitate understanding of aerodynamic phenomena and performance of the bridges discussed in the subsequent chapters. Wind-induced excessive vibration and damage to long-span cable-supported bridges are discussed, focusing on the lessons learned from them by the engineering profession. Finally, the history of bridge wind engineering, particularly for cable-supported bridges is outlined to look back to the past and look forward to the future.

1.2 Basic Notions of Meteorology

1.2.1 Global Wind Circulations

Wind, or the motion of air with respect to the surface of the earth, is fundamentally due to differences in the amount of solar heat received by the atmosphere over various areas of the earth's surface because of the shape of the earth and its position relative to the sun [1]. The differences in solar radiation between the poles and the equator produce temperature and pressure differences. These, together with the effects of the earth's rotation, the curvature of the path of motion and friction at the earth's surface, initiate and break down the air movement into six district circulations, as shown in Figure 1.1.

Figure 1.1 Idealized global circulation.

There are three in each hemisphere: easterly trade wind, westerly wind in the temperate zone, and polar easterly wind [2]:

Easterly trade wind: the permanent subtropical high-pressure zone in the Northern Hemisphere initiates a flow towards the equator with low pressure at ground level. The rotation of the earth creates a virtual force which is known as the Coriolis force. The Coriolis force is perpendicular to the wind direction and to the right in the Northern Hemisphere. Thus, the flow is bent westwards and forms a trade wind. This is called the easterly trade wind, as it comes from the east. The subtropical high-pressure zone in the Northern Hemisphere is at approximately 30° latitude, because the flow away from the equator at high altitude cannot penetrate further north due to the Coriolis force.Westerly wind in temperate zone: in the layers near the ground of the temperate zone, flow is directed towards the north by the subtropical high-pressure zone and the polar front low-pressure zone. Due to the Coriolis force, it is then diverted eastwards and becomes a westerly wind in the temperate zone. The polar front is a surface where a warm westerly wind flows to the south of the polar front and a cold easterly wind flows to the north of the front. The equilibrium at this front is very sensitive to changes in the temperature, velocity and humidity of the two air masses.Easterly polar wind: from the high pressure at the North Pole, the air flows to the south at low altitudes. The flow is then diverted to the west and becomes the cold easterly polar wind. The two main flows – the west wind in the temperate zone and the easterly polar wind – pass along each other at 50°–60° northern latitude, thereby forming the low pressure polar front.

1.2.2 Pressure Gradient Force

The most important forces acting on a particle of air are pressure gradient force, Coriolis force and frictional force [3]. The first two of these are particularly important to the upper level air, where the effect of frictional force is insignificant. Nevertheless, the frictional force must be taken into account for air near the earth's surface.

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