Heaving, Stretching and Spicing Modes

Rui Xin Huang

Climate Variability in the Ocean

Heaving, Stretching and Spicing Modes

Rui Xin Huang

Heaving, Stretchingand Spicing ModesClimate Variability in the Ocean

123

Rui Xin HuangWoods Hole Oceanographic InstitutionWoods Hole, MA, USA

ISBN 978-981-15-2940-5 ISBN 978-981-15-2941-2 (eBook)https://doi.org/10.1007/978-981-15-2941-2

Co Publisher ISBN: 978-7-04-054255-4Jointly published with Higher Education PressThe print edition is not for sale in Chinese mainland. Customers from Chinese mainland pleaseorder the print book from: Higher Education Press.

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020This work is subject to copyright. All rights are reserved by the Publishers, whether the whole orpart of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way,and transmission or information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names areexempt from the relevant protective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information inthis book are believed to be true and accurate at the date of publication. Neither the publishersnor the authors or the editors give a warranty, express or implied, with respect to the materialcontained herein or for any errors or omissions that may have been made. The publishers remainneutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore189721, Singapore

For his pioneering and profound contributions to modernphysical oceanography

His 1962 paper opened up the field of spicity and its application

—Dedicated to the Centennial Anniversary ofHenry M. Stommel

Preface

This book is a collection of topics related to some fundamental aspects of thewind-driven circulation and climate change. We discuss methods used in thewater mass analysis and the climate study; in particular, we discuss theheaving modes induced by the adiabatic adjustment of the wind-driven cir-culation to wind stress perturbations. First, we can use simple reduced gravitymodels. Second, we can examine the climate variability in terms of verticalcoordinates based on water properties, such as the potential density, thepotential temperature and the salinity. These vertical coordinates based on thematerial properties of water parcels are Lagrangian coordinates. The com-bination of such material coordinates with the traditional fixed Euleriancoordinates in the horizontal directions gives rise to the Eulerian–Lagrangianhybrid coordinates, which can be used to identify heaving signals from cli-mate data generated from observations or computer models.

Some of the previous studies of climate based on isopycnal analysis werefocused on climate variability on isopycnal surfaces. The promise of thisapproach is that climate signals can be separated into two components:vertical movement of isopycnal layers and water property changes onisopycnal surfaces. However, it is more accurate to study the climate vari-ability associated with isopycnal layers; i.e., climate signals are examined interms of changes of the layer depth, the thickness, the spicity and others. Thismethod is called isopycnal layer analysis.

Water mass analysis has been the backbone of physical oceanography,and the most commonly used tools are the Theta-S diagram and the isopycnalanalysis. Recently, a potential spicity function whose contours are orthogonalto those of potential density has been defined. This opens up a new approachbased on the sigma–pi diagram, which can be used as an additional tool in thewater mass analysis. The sigma–pi diagram may provide new insights for thewater mass analysis, the ocean circulation and the climate change. In par-ticular, the introduction of an orthogonal coordinate system makes it possibleto define the distance in the parameter space. With the exact measure ofdistance, many aspects of the water mass analysis and the climate change canbe accurately quantified.

These basic materials have been presented through numerous seminars inmany institutions and discussed in some workshops, including “ClimateVariability in the World Oceans: Heaving and Isopycnal Analysis” (Instituteof Oceanography, CAS, Qingdao, China, March 2016) and “Workshop onAdiabatic Motions, Heaving Modes and Isopycnal Layer Analysis” (Xiamen

vii

University, Xiamen, China, April 2017). I am grateful for the encouragementfrom my colleagues and young friends. I have been benefited from discus-sions with my colleagues both in the USA and in China. Drs. XiangsanLiang, Jim Price, Bill Dewar, Bo Qiu, Ray Schmitt, Dezhou Yang andShengqi Zhou read parts of the draft and provided many critical suggestions.Dr. Xiaolin Zhang, Ms. Mary Zawoysky and Dr. Ruihe Huang read the earlyversion of the whole manuscript and pinpointed numerous mistakes. Drs. JoePedlosky, Quanan Zheng and Zijun Gan have been a continuous source ofscientific stimulation.

Woods Hole, USA Rui Xin HuangMarch 2020

viii Preface

Contents

1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Roles of Wind in Climate Variability . . . . . . . . . . . . . . . . . . 11.2 Main Thermocline in the World Oceans . . . . . . . . . . . . . . . . 21.3 Reduced Gravity Model, Advantage and Limitation . . . . . . . 6

1.3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 The Reduced Gravity in the World Oceans . . . . . . . . 11

1.4 Layer Outcropping: The Physics and the NumericalMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Climate Variability Diagnosed from the SphericalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Climate Variability Diagnosed in the z-Coordinate . . . . . . . . 172.2 External/Internal Modes in Meridional/Zonal Directions . . . . 28

2.2.1 Heat Content Anomaly . . . . . . . . . . . . . . . . . . . . . . . 282.2.2 Salinity Anomaly. . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.3 Density Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Adiabatic Signals in the Upper Ocean . . . . . . . . . . . . . . . . . 392.3.1 Adiabatic Adjustment in the Upper Ocean . . . . . . . . 412.3.2 Adiabatic Wave Adjustment in the Meridional

Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.4 The Regulation of MOC (MHF) by Wind Stress

and Buoyancy Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Surface Density Anomaly . . . . . . . . . . . . . . . . . . . . . 462.4.3 Correlation Between Surface Forces and MOC . . . . . 482.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5 Adiabatic Heaving Signals in the Deep Ocean . . . . . . . . . . . 532.6 Final Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Heaving, Stretching, Spicing and Isopycnal Analysis . . . . . . . . 613.1 Heaving, Stretching and Spicing Modes . . . . . . . . . . . . . . . . 61

3.1.1 Adiabatic and Isentropic Processes . . . . . . . . . . . . . . 613.1.2 Heaving, Stretching and Spicing Modes . . . . . . . . . . 623.1.3 External Heaving Modes Versus Internal

Heaving Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

ix

3.1.4 Wave Processes Related to Adiabatic InternalHeaving Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.1.5 Local Versus Global Heaving Modes . . . . . . . . . . . . 703.2 Potential Spicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.2 Define Potential Spicity by Line Integration . . . . . . . 743.2.3 Define Potential Spicity in the Least Square

Sense. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.2.4 Solve the Linearized Least Square Problem . . . . . . . 783.2.5 Potential Spicity Functions Based on UNESCO

EOS-80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2.6 Potential Spicity Functions Based on UNESCO

TEOS_10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.3 r–p Diagram and Its Application . . . . . . . . . . . . . . . . . . . . . 84

3.3.1 The Meaning of Spicity. . . . . . . . . . . . . . . . . . . . . . . 843.3.2 Density Ratio Inferred from the Density–Spicity

Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.3 The r–p Plane as a Metric Space . . . . . . . . . . . . . . . 110

3.4 Isopycnal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.4.1 The Lagrangian Coordinate . . . . . . . . . . . . . . . . . . . . 1383.4.2 Isopycnal Analysis in the Eulerian Coordinate . . . . . 1473.4.3 Isothermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 152

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4 Heaving Modes in the World Oceans. . . . . . . . . . . . . . . . . . . . . 1614.1 Heaving Induced by Wind Stress Anomaly . . . . . . . . . . . . . 161

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.1.2 A Two-Hemisphere Model Ocean . . . . . . . . . . . . . . . 1654.1.3 A Southern Hemisphere Model Ocean . . . . . . . . . . . 1754.1.4 Adiabatic MOCs of the World Oceans

with Rectangular Basins . . . . . . . . . . . . . . . . . . . . . . 1844.1.5 MOC/MHF Simulated by a RGM in the World

Oceans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1904.2 Heaving Induced by Anomalous Freshwater Forcing . . . . . . 195

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.2.2 Model Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.2.3 Results from Numerical Experiments . . . . . . . . . . . . 1984.2.4 Experiment for 40 Year Continuing Freshening

of the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.3 Heaving Induced by Anomalous Wind, Freshening

and Warming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.3.2 A Simple Generalized Reduced Gravity Model . . . . . 2094.3.3 Numerical Experiments Based on This Reduced

Gravity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

x Contents

4.4 Heaving Induced by Convection Generated ReducedGravity Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.4.2 Model Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.4.3 Results from Numerical Experiments . . . . . . . . . . . . 2194.4.4 Numerical Experiments with Sinusoidal Reduced

Gravity Perturbations. . . . . . . . . . . . . . . . . . . . . . . . . 2304.5 Heaving Induced by Deep Convection Generated

Volume Loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.5.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.5.3 Results of Numerical Experiments. . . . . . . . . . . . . . . 234

4.6 ENSO Events and Heaving Modes . . . . . . . . . . . . . . . . . . . . 2414.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.6.2 Variability of Heat Content and Horizontal

Heat Fluxes Due to ENSO Diagnosedfrom the GODAS Data . . . . . . . . . . . . . . . . . . . . . . . 242

4.6.3 Meridional Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . 2454.6.4 Zonal Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2484.6.5 Vertical Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 2514.6.6 A Two-Hemisphere Model Ocean Simulating

ENSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

5 Heaving Signals in the Isopycnal Coordinate . . . . . . . . . . . . . . 2635.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.2 Casting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

5.2.1 FDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.2.2 MDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2665.2.3 Separating the Signals Into External and Internal

Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2675.2.4 Statistics in the Density Space. . . . . . . . . . . . . . . . . . 2695.2.5 External Signals in Terms of Layer Thickness . . . . . 270

5.3 Projecting Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2725.4 Difference Between the Casting Method and the Projecting

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2745.5 Isopycnal Layer Analysis for the World Oceans . . . . . . . . . . 276

5.5.1 External Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2765.5.2 Heaving Modes for r1 ¼ 30:9� 0:05 kg/m3 . . . . . . . 2805.5.3 Horizontal Distribution of Climate Variability for

r1 ¼ 30:9� 0:05 kg/m3 . . . . . . . . . . . . . . . . . . . . . . . 2835.5.4 The Heaving Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 2855.5.5 Regional Anomaly Patterns . . . . . . . . . . . . . . . . . . . . 2885.5.6 A Meridional Section Through 60.5° W . . . . . . . . . . 2915.5.7 A Zonal Section Along the Equator . . . . . . . . . . . . . 3005.5.8 A Zonal Section Along 45.17° N . . . . . . . . . . . . . . . 305

5.6 Isopycnal Layer Analysis Based on r0 . . . . . . . . . . . . . . . . . 308

Contents xi

5.7 Heaving Signals for the Shallow Waterin the Pacific-Indian Basin . . . . . . . . . . . . . . . . . . . . . . . . . . 3125.7.1 Application of the Casting Method to the

GODAS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3135.7.2 Isopycnal Layer Analysis of the Equatorial

Dynamics Based on Projecting Methods . . . . . . . . . . 3205.8 Heaving Signal Propagation Through the Equatorial

Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Appendix: Connection Between the MDC and the FDC . . . . . . . . 331References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

6 Heaving Signals in the Isothermal Coordinate . . . . . . . . . . . . . 3356.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3356.2 Casting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

6.2.1 FTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3366.3 Casting Method Applied to the GODAS Data . . . . . . . . . . . 338

6.3.1 The Choice of Temperature Scale . . . . . . . . . . . . . . . 3386.3.2 Statistics in the Temperature Space . . . . . . . . . . . . . . 339

6.4 Projecting Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3456.4.1 Isothermal Layer Analysis for the Layer

of h ¼ 20� 0:5 �C . . . . . . . . . . . . . . . . . . . . . . . . . . 3476.4.2 Structure in the Pacific Basin . . . . . . . . . . . . . . . . . . 352

6.5 Signals of Layer Depth and Zonal Velocityin the Pacific Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6.6 Z-Theta Diagram and Its Application to ClimateVariability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Appendix: Connection Between the MTC and the FTC . . . . . . . . 369References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

7 Climate Signals in the Isohaline Coordinate . . . . . . . . . . . . . . . 3737.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3737.2 Casting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

7.2.1 FSC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3757.2.2 MSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

7.3 Separating the Signals into External and Internal Modes . . . 3767.3.1 FSC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3777.3.2 MSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

7.4 Analysis Based on the GODAS Data . . . . . . . . . . . . . . . . . . 3777.5 Shallow Salty Water Sphere in the Atlantic Ocean . . . . . . . . 379References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

xii Contents

1Basic Concepts

1.1 Roles of Wind in ClimateVariability

The ocean is subjected to the following forces:wind, heat/freshwater fluxes and tidal dissipation.In the old paradigm the thermohaline circulation isdriven by surface thermohaline forcing. However,in the new paradigm the oceanic circulation is adissipation system, maintenance of which requiresexternal sources of the mechanical energy from thewind and the tidal dissipation. From anotherangle, the air-sea interaction can be classified intwo categories: mechanical and thermohaline.Mechanical interaction includes the atmosphericpressure and the momentum flux at the sea level.Thermohaline interaction includes different kindsof heat flux and freshwater flux through the air-seainterface. In a broad framework, it should alsoinclude other tracer (oxygen and other nutrients)fluxes through the air-sea interface. Therefore,from both angles wind forcing plays key roles inregulating oceanic stratification and circulation,and this is the focus in our study. In a quasi-steadystate, the wind stress plays two important roles.

First, the wind stress drives currents in theupper ocean. The circulation in the upper ocean ismostly due to the dynamical effect of wind stress.In physical oceanography, these currents are col-lectively called the wind-driven circulation(Fig. 1.1a). Wind-driven circulation has been dis-cussed in many classical papers and textbooks.

Second, the wind stress provides themechanical energy which is vitally important for

maintaining mixing in the ocean, including themixed layer and the subsurface ocean. Mixing inthe mixed layer affects the air-sea heat/freshwaterfluxes; thus, it is also a key factor regulating thethermohaline circulation (Fig. 1.1b).

For the time-dependent problems, the windstress also plays a key role in regulating thegeneral circulation. In the ocean interior, changesin the wind stress can induce the basin-scaleadiabatic movement of the isopycnal layers. Forexample, if the wind stress is reduced, the mainthermocline in the basin interior becomes shal-lower and moves upward, and the correspondingisopycnal layers above the main thermoclinemove upward; such motions of isopycnal layersare called heaving. In general, the heavingmotions are indicated by the deformation of theisopycnal surfaces from the black curve to thered curve in Fig. 1.1c. Heave and heaving modecan induce large variability in terms of temper-ature and salinity at fixed points in the oceaninterior. Therefore, they are the dominant sourcesof climate variability and are also the primaryfocus of this book. In addition, wind stresschanges can lead to the shifting of the outcroplines, depicted as the transition from the blackcurve to the blue curve in Fig. 1.1c. The shiftingof the outcrop line and changes of the mixedlayer properties lead to different partitions of thewater mass formation/erosion in the density cat-egory, these processes are the most importantfactors inducing the thermohaline circulationvariability in the world oceans.

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020R. X. Huang, Heaving, Stretching and Spicing Modes,https://doi.org/10.1007/978-981-15-2941-2_1

1

In summary, the wind stress changes caninduce many aspects of changes in the oceaniccirculation; it is a great challenge to identify suchchanges from climate data. However, among manykinds of climate signals the heaving motionsinduced by the wind stress change alone are thedominating component; so that it is desirable toidentify such signals. The most clear-cut manifes-tation of heaving signals is the shifting of the mainthermocline in the ocean caused by wind stresschanges. Such signals often propagate in the formsof Rossby waves and Kelvin waves.

The common wisdom in the theory of theoceanic general circulation is that meridionaloverturning circulation (MOC) and the merid-ional heat flux (MHF) are directly linked to thediabatic heating/cooling across the air-sea inter-face and interior isopycnal/isothermal surfaces.Similarly, the thermohaline circulation in theocean also induces the zonal overturning circu-lation (ZOC), the zonal heat flux (ZHF), and thevertical heat flux (VHF). However, adiabaticmovements produced by wind stress anomalieson seasonal, annual, interannual and decadal timescales can generate anomalous MOC, MHF,ZOC, ZHF, and VHF.

As will be shown in this book, about 85%–

90% of the climate variability identified fromclimate data sets obtained from observations ornumerical simulations is heaving in nature. Thus,studying the role of heaving motions in the oceanand separating the contributions from heavingand non-heaving motions are vitally importantfor our understanding of the climate variability inthe real world.

1.2 Main Thermocline in the WorldOceans

The ocean is stratified; however, the stratificationin the ocean is not uniform. In fact, there is alayer of strong vertical temperature gradient inthe ocean, which is called the main thermocline.The suffix “cline” means a layer with a strongvertical gradient. For example, there are ther-moclines, pycnoclines, haloclines and lysoclines.The pycnocline is a layer with a maximumdensity gradient, and the halocline is a layerassociated with a maximum salinity gradient.The lysocline is a term used in geology, geo-chemistry and marine biology to denote the depthin the ocean below which the rate of dissolutionof calcite increases dramatically.

There are many types of thermoclines,including the diurnal thermocline, the seasonalthermocline and the permanent (main) thermo-cline. By definition, the main thermocline is alayer with a maximum temperature gradient. Inorder to identify the main thermocline, it is muchmore convenient to search the depth of the localmaximum of the vertical temperature gradient bystarting the search from the deep ocean andmoving upward.

Since density is the physical property directlylinked to dynamics, it is more appropriate to usethe term “pycnocline”. However, in most parts ofthe ocean, density in the upper ocean is primarilycontrolled by temperature, with the salinityplaying a minor secondary role. Hence, the mainthermocline and the main pycnocline are very

(c) Heaving/shifting of outcrop lines (a) Driving the currents

dniWdniWdniW

Surface mixing

Subsurface mixing

(b) Maintain mixing

Fig. 1.1 Roles of wind stress in the oceanic general circulation and climate change

2 1 Basic Concepts

close to each other in location; people often usesthe term “main thermocline” because it is directlylinked to heating/cooling and changes in theclimate system.

As shown in Fig. 1.2a, there is clearly astrong thermocline along the Equator, and it iscalled the equatorial thermocline. In the Pacificand Atlantic Oceans, it slopes down westward;but it slopes up westward in the Indian Ocean.Such patterns are primarily due to the fact thatthere are easterlies in both the Pacific andAtlantic basins; however, wind stress in theIndian Ocean has a very strong annual cycle andthe annual mean wind is westerly.

In the meridional sections, the main thermo-cline near the Equator appears in the form of adumbbell, i.e., the thermocline is shallow near theequator, but is deep in themiddle latitudes becauseof the bowl-shaped wind-driven subtropical gyresin both hemispheres (lower panels in Fig. 1.2).

The equatorial thermocline is asymmetric tothe equator, and this phenomenon is associatedwith the strength of the zonal equatorial currentsand the cross-equator flow in each basin. In theAtlantic basin, the strong meridional overturningcirculation associated with the thermohaline cir-culation manifests in the form of a northwardwarm current moving across the equator in theupper ocean. In the Pacific basin, wind stressover the latitude band of the Australian continentcreates the Indonesian Throughflow, whichmanifests in the form of a cross-equatorial cur-rent bringing the warm water from the mid-latitudes in the Southern Hemisphere into theNorthern Hemisphere. In fact, most of the watermasses entering the Equatorial Undercurrent inthe source region of the western Pacific basincome from the Southern Hemisphere. Zonal windstress in the equatorial Indian Ocean also givesrise to cross-equator flow.

Fig. 1.2 Typical annual mean temperature profiles in the world oceans, based on WOA09 climatology

1.2 Main Thermocline in the World Oceans 3

For examples, we show a case of using thevertical temperature gradient to identify the mainthermocline from WOA09 climatology(Fig. 1.3). The upper panel shows the tempera-ture contours. The corresponding vertical tem-perature gradient is shown in the lower panelsand the main thermocline obtained aftersmoothing is marked by the red curved line.

We also show a horizontal map of the mainthermocline depth in the world oceans diagnosedfrom WOA09 climatology (Fig. 1.4). The searchfor the main thermocline requires a slightlycomplicated code. First, we start the search fromthe 1500 m level and move upward, setting thebase of the main thermocline as the first gridwith a vertical temperature gradient larger thandT/dz = 0.01 °C/m. When the base of the ther-mocline is located, we search for the exact depthabove this grid where the vertical temperature

gradient is maximal. Using a single criterion ofdT/dz for the entire world oceans is, of course,not accurate, and this is modified as follows. Atthe equatorial band, the temperature gradient ismuch larger, so that we modify the criterion asdT=dz ¼ 0:01½1þ 4ð1� /j j=20Þ� for /j j\20,where / is the latitude in degrees. For mid-latitudes, this criterion is modified as dT=dz ¼0:01½1þ 0:025ð1� /j j=20Þ� for /j j[ 20. Sincethe main thermocline outcrops along strong zonalcurrents, such as the Gulf Stream, Kuroshio andACC, the main thermocline is confined to lowerand mid-latitudes only.

It is clear that the main thermocline in the fivesubtropical basins appears as the local bowl-shaped maximum. As will be discussed shortly,the zonal slope of the squared thermocline depthis linearly proportional to the Ekman pumpingrate divided by the reduced gravity. This simple

Fig. 1.3 Identifying the main thermocline along 179.5° E from WOA09; the red curve indicates the position of themaximum vertical temperature gradient, i.e., the main thermocline

4 1 Basic Concepts

relation gives rise to some interesting phenomenaidentifiable from Fig. 1.4. In fact, the mainthermocline in the North Atlantic Ocean is muchdeeper than that in the North Pacific Ocean. Thisis primarily due to the high salinity in the NorthAtlantic Ocean; consequently, water in the upperocean is dense. Consequently, the reducedgravity is relatively small; as a result, the ther-mocline in this basin is deep.

The thermocline in the Indian Ocean isdeepest in the world oceans, with a maximaldepth of more than 900 m (Figs. 1.2b and 1.4).Wind stress perturbation induced adiabaticmotions (The adiabatic motions induced by thewind stress perturbation) can penetrate to thebase of the main thermocline; as a result, thewind stress perturbations in the world oceans cangenerate adiabatic signals below 1000 m. Fur-thermore, as will be shown in Chap. 2, adiabaticmotions may penetrate to the whole depth of thewater column in the world oceans. Such adiabaticmotions in the deep ocean may be linked to the

variability of the Antarctic Circumpolar Currents,the strength of which is primarily regulated bythe Southern Westerly.

Figure 1.4 also shows that the thermoclinedepth along the western coast of Australia is muchdeeper than that along the other eastern bound-aries of the world oceans. As an example, thetemperature and potential density distributionsalong 29.5° S are shown in Fig. 1.5. It is readilyseen that the main thermocline depth along theeastern boundary of the Indian basin is about750 m, much deeper than the corresponding depthof 100 m in the South Pacific and 150 m in theSouth Atlantic. This may be linked to the unusu-ally strong poleward eastern boundary current inthe South Indian Ocean, the Leeuwin Current.When the strength of the Leeuwin Current chan-ges, the thermocline depth along the easternboundary of the South Indian Ocean may changein response. Even if the wind stress and otherparameters for the wind-driven circulation in theSouth Indian Ocean remain unchanged, the depth

Fig. 1.4 Main thermocline depth in the world oceans, based on WOA09

1.2 Main Thermocline in the World Oceans 5

of the main thermocline must change as well.Therefore, the wind-driven circulation in theSouth Indian Ocean is intimately linked to thestrength of the Leeuwin current.

1.3 Reduced Gravity Model,Advantage and Limitation

For the wind-driven circulation in the ocean, thedepth of the main thermocline can be describedby a simple reduced gravity model as discussedbelow; while the detailed dynamical descriptionof wind-driven circulation is referred to Huang(2010).

1.3.1 Model Formulation

A reduced gravity model is formulated as fol-lows. The oceanic stratification is simplified as atwo-layer fluid environment. The upper andlower layers have the density q0 and q0 þDq; theupper layer thickness is denoted as h, and thelower layer is assumed to be infinitely thick, andthus motionless. In the ocean interior the fric-tional and inertial terms are negligible. For sim-plicity, we discuss a model formulated on themid-latitude beta plane, and we will assume windstress in a simple form: sx ¼ sxðyÞ; sy ¼ 0. For asteady state, the momentum equations arereduced to

Fig. 1.5 Zonal maps of temperature (a) and potential density (b) along 29.5° S, based on WOA09 climatology

6 1 Basic Concepts

�fhv ¼ �g0hhx þ sx=q0 ð1:1aÞfhu ¼ �g0hhy ð1:1bÞ

where g0 ¼ gDq=q0 is the reduced gravity. Inaddition, the model satisfies the continuityequation

huð Þx þ hvð Þy¼ 0 ð1:1cÞ

Note that a major assumption made in thetraditional level model based on the quasi-geostrophic approximation is that the change instratification in the horizontal direction remains asmall fraction. As shown above, a major featureof the reduced gravity model is to include thefinite amplitude variability of the layer thicknessin both the continuity equation and the momen-tum equation. As such, the reduced gravity modelcan simulate the nonlinear mechanism associatedwith finite layer depth change, which is one of themost important dynamical features of basin-scalewind-driven circulation in the ocean.

Cross-differentiating and subtracting (1.1a)and (1.1b) lead to the vorticity equation

bhv ¼ �sxy=q0 ð1:2Þ

This equation is called the Sverdrup relation.Substituting (1.2) into (1.1a) gives rise to a first-order ordinary differential equation

hhx ¼ � f 2

g0q0bsx

f

� �y

ð1:3Þ

This equation is a key to understanding theadjustment of wind-driven circulation: in asteady state the slope of the squared thermoclinedepth is proportional to the Ekman pumping rateand inversely proportional to the reduced gravity.As climate conditions change, the wind stressinduced Ekman pumping rate and the buoyancyof the upper ocean vary accordingly. As a result,thermocline slope changes, warm water in theupper ocean is redistributed, and the circulation

and sea water properties in the upper ocean gothrough a three-dimensional adjustment, which isthe major focus of this book.

Integrating Eq. (1.3) leads to the equationregulating the layer thickness, which is valid inthe basin interior. In order to balance the vorticityover the whole basin, the integration should bestarted from the eastern boundary, as discussed inHuang (2010). The zonal integration leads to thesquared layer thickness for the basin interior

h2 ¼ h2e þ2f 2

g0q0bsx

f

� �y

xe � xð Þ ð1:4Þ

This equation can also be written in terms of theEkman pumping in the following form

h2 ¼ h2e �2f 2we

g0bxe � xð Þ; we ¼ � sx

fq0

� �y

ð1:5Þ

For the given wind stress profile, one can usethese two equations to calculate the layer thick-ness in the basin interior. However, there are twoapproaches.The first approach is to specify theupper layer depth along the eastern boundary andthen integrate westward. This approach has beenwidely used in many textbooks. This approach isso popular that many young people entering ourfield would think that this is the only approach.

The second approach is quite interesting andstrongly relevant to climate study. This approachis to assume that the total amount of warm waterabove the main thermocline is constant, and wewant to find the solution which satisfies theconstraints in Eq. (1.4) (or Eq. 1.5) andEq. (1.1c).

In fact, Eq. (1.1c) is the steady version of themore general form of the continuity equation

ht þ huð Þx þ hvð Þy¼ 0 ð1:1c0Þ

Integration of Eq. (1.1c′) over the whole basinleads to the conservation law

1.3 Reduced Gravity Model, Advantage and Limitation 7

ZZhðtÞ dxdy ¼ const: ð1:1c00Þ

Namely, the total amount of warm water in theupper layer remains constant in time.

In this case, we can no longer specify the layerthickness along the eastern boundary apriori;instead, we need to find a solution that satisfiesthe integral constraint of a constant volume ofwarm water above the main thermocline.

In fact, most numerical models are based onsuch a constraint: we start the model for an initialstate of rest, with a given mean layer thicknessover the whole basin. Since the wind-drivennumerical models conserve the total volume ofthe warm water, when the model reaches a finalstate, we have a solution which satisfies thevolume conservation constraint.

For example, we run a simple model subjectedto these two constraints. Assume the Ekmanpumping rate over the subtropical basin (from/s ¼ 15�N to /n ¼ 40�N) is we ¼ �Epf�

10�6 sin p /� /sð Þ= /n � /sð Þ½ � m/sð Þ, (Epf is theEkman pumping factor, which can vary from0 to 2.5), and g0 ¼ 0:01 ðm/s2Þ. To get moreaccurate solutions the calculation is based onthe corresponding equations in the sphericalcoordinate.

In Fig. 1.6, layer depths under two differentamplitudes of forcing for these two experimentsare displayed side by side. Under the sameamplitude of wind forcing, the maximum layerdepth (near the western boundary) and the min-imum layer depth (along the eastern boundary)for these two model constraints are different.

If we fix the upper layer thickness along theeastern boundary at the value of he ¼ 300 m, thezonal slope of the thermocline and the layerthickness in the basin interior increase with theenhancement of Ekman pumping, as shown bythe transition from the dashed blue curves to thesolid blue curve in Fig. 1.7c. In particular, thelayer thickness maximum along the westernboundary increases rapidly, as shown by the

(a) h (m), Epf=0.5, he=300 m (b) h (m), Epf=2.375, he=300 m

(c) h (m), Epf=0.5, V=V0 (d) h (m), Epf=2.375, V=V0

0 10E 20E 30E 40E 50E 60E15N

20N

25N

30N

35N

40N

300

350

400

450

0 10E 20E 30E 40E 50E 60E15N

20N

25N

30N

35N

40N

300

400

500

600

700

800

0 10E 20E 30E 40E 50E 60E15N

20N

25N

30N

35N

40N

350

400

450

500

0 10E 20E 30E 40E 50E 60E15N

20N

25N

30N

35N

40N

100

200

300

400

500

600

700

800

Fig. 1.6 Layer thickness (in m) in the case of constant layer thickness along the eastern boundary (upper panels) andthe case of constant volume of warm water (lower panels)

8 1 Basic Concepts

solid blue curve in Fig. 1.7b; while the layerthickness along the eastern boundary (dashedblue curve) remains unchanged, as required bythe model constraint. As a result, the totalamount of warm water in the ocean increasesgradually, the blue curve in Fig. 1.7a. In thiscase, the change in the total volume of warmwater in the upper layer implies a source of warmwater; it could come from the lateral boundariesor the bottom boundary of the model. In anycase, the model run discussed above cannot beconsidered as adiabatic or isolated from otherparts of the ocean.

If the total volume of warm water in the upperlayer is fixed (V ¼ V0, black curves in Fig. 1.7),the situation can be considered to be adiabatic. Inthis case, the zonal slope of the thermocline inthe basin interior increases with the enhancementof Ekman pumping (the black curves inFig. 1.7c). The layer thickness maximum alongthe western boundary also increases; however,the layer thickness along the eastern boundarydeclines quickly (the solid and dashed blackcurves in Fig. 1.7b).

When wind forcing is very strong (Epf !2.5), the layer thickness along the easternboundary vanishes. If wind forcing is strongerthan this critical value, the upper layer thicknessnear the eastern boundary is zero. By definition,layer thickness cannot be negative, so that theupper layer should outcrop near the eastern

boundary. Outcropping is a very importantphysical phenomenon in the ocean, and thesimple reduced gravity model described byEqs. (1.1a, b, c′, c′′–1.5) is no longer valid. Infact, dealing with outcropping requires specialmethods in analytical and numerical models, aswill be discussed shortly.

In the case of a fixed amount of warm water,enhancement of Ekman pumping leads to thethree-dimensional redistribution of warm waterin the upper ocean, as shown in Figs. 1.8 and 1.9.Figure 1.8 shows that warm water is transportedfrom both the low and high latitudes to themiddle latitudes (Fig. 1.8a); warm water ismoved from the eastern boundary to the westernboundary (Fig. 1.8b); and warm water isremoved from the upper ocean and pumpeddown to the deep part of the ocean, Fig. 1.8c.

Such a three-dimensional shifting of watermass can be clearly demonstrated in terms of theshifting of the mass center (Fig. 1.9). WhenEkman pumping is enhanced, the center of thewarm water mass in the upper ocean moveswestward (Fig. 1.9a), northward (Fig. 1.9b) anddownward (Fig. 1.9c).

From the volumetric distribution of warmwater, one can infer the volumetric anomalyprofiles in the meridional, zonal and verticaldirections, defined as the deviation from thepivotal case of Epf = 1.0 (Fig. 1.10). For exam-ple, if the Ekman pumping factor is reduced to

Fig. 1.7 a Total volume of the upper layer; b eastern boundary layer thickness as a function of the Ekman pumpingfactor; c layer thickness along the mid-latitude (30° N)

1.3 Reduced Gravity Model, Advantage and Limitation 9

Epf = 0.5, the volumetric anomaly in themeridional direction is shown as the blue curvein Fig. 1.10a. In such a case, warm water volume

at middle latitudes is reduced, while it isincreased at high/low latitudes. On the otherhand, if Ekman pumping is enhanced to Epf =

Fig. 1.8 Warm water mass shifting due to the enhancement of Ekman pumping subjected to the fixed volume.aMeridional distribution of warm water volume (1015 m3/0.25°); b Zonal distribution of warm water volume (1015 m3/0.3°); c Vertical distribution of warm water volume (1015 m3/5 m)

0 1 230E

29E

28E

27E

26E

25E

24E

(a) Xc (degreee)

27.0N

27.5N

28.0N

28.5N(b) Yc (degree)

260

240

220

200

Dep

th (m

)

(c) Zc (m)

Ekman pumping factor Ekman pumping factor Ekman pumping factor0.5 1.5 2.5 0 1 20.5 1.5 2.5 0 1 20.5 1.5 2.5

Fig. 1.9 Migration of mass center due to the enhancement of Ekman pumping

Fig. 1.10 Modes of warm water volumetric anomaly, relative to the pivotal case of Epf = 1.0

10 1 Basic Concepts

1.5, 2.0 and 2.5, the warm water volume isincreased at middle latitudes, but it is reduced athigh/low latitudes.

For the zonal profiles, when Ekman pumpingis reduced to Epf = 0.5, the warm water volumeanomaly is negative in the western basin, but it ispositive in the eastern basin, as shown by theblue curve in Fig. 1.10b. On the other hand,when Ekman pumping is enhanced, warm wateris pushed towards the western basin, leading topositive anomaly there and negative anomaly inthe eastern basin.

Among these features, the most important oneis the downward push of the warm water asso-ciated with the enhancement of the Ekmanpumping rate. Since the adjustment process isadiabatic, the total heat content (HC) is con-served during the adjustment; therefore, in thecase of a simple reduced gravity model, the HCanomaly profile must be in the form of a firstbaroclinic mode (Fig. 1.10c).

As shown by the blue curve in Fig. 1.10c, ifEkman pumping is reduced to Epf = 0.5, there ismore warm water between 300 and 400 m, butthere is less warm water between 400 and 600 m.The reduction of the warm water volume at agiven depth implies that there is cooling at thislevel. Accordingly, the baroclinic modes of thewarm water volume could be interpreted as thewarming/cooling signals diagnosed at the corre-sponding geopotential levels.

When the Ekman pumping rate is enhanced,the warm water is pushed downward, inducingwarming at deeper levels and cooling at shal-lower level, as indicated by the black, red andgreen curves in Fig. 1.10c.

Baroclinic modes of the HC anomaly inducedby changes in the Ekman pumping rate (associ-ated with wind stress perturbations) under theadiabatic conditions are the most important fea-tures often diagnosed from climate datasets.

It is important to notice that for the secondmodel, the assumption of constant volume isequivalent to assuming that the adjustment of themodel is under the adiabatic constraint. In theocean, for a relatively short time scale, theamount of warm water in the upper ocean can betreated as nearly constant. As such, wind stress

perturbations would induce changes in the slopeof the main thermocline. When Ekman pumpingis enhanced, the layer thickness near the easternboundary declines, and the layer thickness alongthe western boundary increases, as shown by theblack curves in Fig. 1.7b. Such an adjustmentprocess is carried out by the anomalous currentsand waves, and during the adjustment of thewind-driven gyre, the warm water in the upperocean goes through a three-dimensional redistri-bution. Although such motions are adiabatic,they can carry substantial amounts of the HCanomaly with them; as a result, such adiabaticmotions can cause temporal heat transport inthree dimensional space. These motions are theprototype of heaving motions in the ocean andthey are the primary subject of this book.

1.3.2 The Reduced Gravityin the World Oceans

The common way of formulating a reducedgravity model is to assume that the reducedgravity is a constant in space and time. By defi-nition, g0 ¼ gDq=q0, where q0 and q1 ¼ q0 þDqare the mean density of the upper and lower lay-ers. There are two important issues. First, in thecalculation one should use the potential density,not the in situ density. In the following example,we use r1, i.e., the potential density using 1000 dbas the reference pressure. The reason for using r1,instead of r0, is as follows. In the Atlantic basin,r0 is not monotonic in the deep ocean; thus, thecalculation of reduced gravity might be somewhatinaccurate. Second, by carrying out the calcula-tion of reduced gravity for each water column, onerealizes that the reduced gravity is a horizontallydistributed function.

As an example, we show the result of using459 m as the interface to calculate the meanpotential density for the upper and lower parts ofthe water column and the inferred reducedgravity for the world oceans. The calculation isbased on climatology obtained from the GODASdata (Behringer and Xue 2004) as shown inFig. 1.11. It is clear that in the Pacific basin, thereduced gravity varies within the range of 0.015–

1.3 Reduced Gravity Model, Advantage and Limitation 11

0.03 m/s2; on the other hand, it is smaller in theAtlantic basin, within the range of 0.01–0.02 m/s2. In the Indian basin, the reducedgravity is in the middle range.

We can also estimate the reduced gravitymodel, using an isopycnal surface as the interfaceto calculate the mean potential density for theupper and lower parts of the water column andthe inferred reduced gravity for the world oceans.For example, one can use r0 ¼ 26:8 ðkg/m3Þ asthe interface, and the result is quite similar to thatshown in Fig. 1.11.

In the commonly used reduced gravity model,the reduced gravity is assumed to be constant intime and space; the typical value used in thesemodels varies over the range of 0.01–0.02 m/s2.In most cases, people specify this value as moreor less arbitrary. However, as discussed above,the reduced gravity can be calculated from theclimatological data in an accurate way.

It is important to emphasize that the reducedgravity model is a highly truncated model in thedensity coordinate. As such, it cannot describe thecirculation very accurately. However, the reducedgravity model excludes the complicated thermo-haline processes and the fast signals associated

with the external gravity waves; hence, this typeof model is much easier to study either analyti-cally or numerically. Since the high speed baro-tropic waves, such as the barotropic Rossbywaves, are excluded, the reduced gravity modelsare not suitable for the study of short time scales.

It is possible to extend the commonly usedreduced gravity model into a model including thethermodynamics and the horizontally non-constant reduced gravity. The relevant informa-tion can be found in Huang (1991, 2010),McCreary and Yu (1992). The modified model iscalled the generalized reduced gravity model, inwhich the reduced gravity is a function of hori-zontal coordinates and time. In Chap. 4, suchmodels will be used in the study of the circulationvariability in the connection with density anomalyinduced by heating/cooling or freshening.

1.4 Layer Outcropping: The Physicsand the Numerical Method

As shown in Fig. 1.2, most isothermal layersoutcrop at high latitudes; similarly, most isopy-cnal layers also outcrop at high latitudes. Layer

Fig. 1.11 Reduced gravity (in cm/s2) for the world oceans, inferred from the GODAS data

12 1 Basic Concepts

outcropping is a phenomenon that happens in thestratified ocean, and our models should simulatesuch phenomenon accurately. However, in theearly stages of model development in history,layer models or isopycnal models did not takelayer outcrop into consideration. The nonlinear-ity associated with layer outcropping is of criticalimportance. For a simple layer model, the con-tinuity equation is

ht þðhuÞx þðhvÞy ¼ 0 ð1:6Þ

Using the scale thickness H and velocity scale U,we define the following non-dimensional variables

h ¼ Hð1þ dh0Þ; ðu; vÞ ¼ Uðu0; v0Þ; t ¼ t0H=U

ð1:7Þ

The non-dimensional continuity equation is asfollows

h0t0 þ ðu0x þ v0yÞþ dðu0h0x þ v0h0yÞ ¼ 0 ð1:8Þ

where the non-dimensional layer thicknessparameter is

d ¼ DH=H�Oð1Þ ð1:9Þ

Hence, the nonlinearity of layer thickness changeis O (1). In comparison, the nonlinearity associ-ated with advection terms in momentum equa-tions is Ro � O (1). Therefore, for large-scalewind-driven circulation associated with largelayer thickness change or outcropping the non-linearity in the continuity equation is the mostimportant dynamical issue and should not beignored.

For technical convenience, many modelers seta lower bound of layer thickness on the order of10 m. Such models ignored the criticallyimportant roles of density fronts in the oceancirculation; thus, the model results were dra-matically different from reality.

The importance of simulating the density frontaccurately was recognized in the 1980s, andsubsequent development in isopycnal models hasmade special effort in dealing the density front(or isopycnal outcropping). In terms of numerical

calculation, to simulate the outcrop line the so-called positive definite scheme should be used inthe layer depth calculation. Most numericalmodels are based on the so-called central finitedifference scheme in calculating the layer thick-ness. However, such a scheme may lead to anegative layer thickness near the outcrop line;hence, it is not suitable for a case withoutcropping.

To deal with the outcrop line, one has to usethe so-called positive-definite finite differenceschemes; the essential feature of such scheme isto guarantee that layer thickness is never nega-tive. A simple choice is the well-known upwindscheme, which can guarantee the layer is nevernegative; many other higher-order accuracyschemes are also available.

A layer model or isopycnal coordinate modelincluding the outcropping line can provide muchmore accurate information related to densityfronts, such as the Gulf Stream and KuroshioCurrent; accordingly, they can simulate theoceanic circulation and climate changes moreaccurately.

The first rigorous treatment of the outcrop linecan be traced back to the pioneering study ofParsons (1969), who pointed out the criticallyimportant physical constraint related to the out-crop line: in a reduced gravity model, the outcropline should be both the zero thickness line and astreamline.

It is clear that the lowest order dynamicsregulating the wind-driven circulation in theocean interior cannot satisfy both constraints. Inorder to satisfy both dynamical constraints, thereshould be a thin boundary layer in which thehigher-order dynamical terms playing the role ofsatisfying both constraints along the outcrop line.A condensed description of Parson’s solution canbe found in Huang (2010).

One example is shown in Fig. 1.12, where anoutcrop window appears in the northwest cornerof the basin; next to the outcrop line, there is aninterior boundary layer. Under the assumption ofa fixed amount of warm water in the upper layer,if the wind stress forcing is further increased, theoutcrop window will extend further. The

1.4 Layer Outcropping: The Physics and the Numerical Method 13

expansion of the outcrop window implies that thewarm water in the upper ocean is redistributed.The redistribution of the warm water in the oceanimplies a three-dimensional redistribution ofmass, heat and salt; thus, the wind stress anomalycan give rise to an anomalous transport of mass,heat and salt in the meridional, zonal, and verti-cal directions. In Chap. 4 we will discuss thisissue in details, using several simple reducedgravity models for an idealized two-hemispheremodel ocean, an idealized model for the SouthernHemisphere, including a periodical channelmimicking the Antarctic Circumpolar Current,and a reduced gravity model for the worldoceans.

In the traditional view, the meridional heatflux is directly tied to the thermohaline circula-tion and the heat exchange across the air-seainterface or isothermal surfaces. Heaving modesdiscussed in this book may be the other mecha-nism giving rise to the sizeable three dimensional

transport of mass, heat and freshwater on theinterannual and decadal time scales.

Parson’s work can be easily extended into thecase of double gyres in a single hemisphere.When there is enough warm water, the upperlayer covers the entire two-gyre basin. However,when the amount of warm water is not enough tocover the whole basin, the outcropping windowfirst appears in the middle of the subpolar basinnear the western boundary, as shown in thesketch in Fig. 1.13. As the wind stress isenhanced further, the outcrop window expandsand eventually extends into the subtropical basin.In the case of strong wind forcing and a smallamount of warm water, the upper layer is con-fined to a rectangular region in the southwestcorner of the basin, very much like the warmpool in the Pacific basin.

Parson’s model is much easier to be solved bynumerical integration. As an example, we showone set of solutions obtained by a reduced gravity

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

Outcropwindow

0 0.2

0.60.8

1.4

1.4

1.2

1.2

1.4

1.2

1

0.40.8

0.6

0

0.2 0.4 0.6

0.81

1

1

0.8

0.8

1

1.2

Fig. 1.12 A solution of thesubtropical gyre with anoutcrop window, thehorizontal/vertical axes andlayer depth are in non-dimensional unit

14 1 Basic Concepts

model based on spherical coordinates with arealistic coastal geometry of 1° � 1° resolutionand forced by the climatological annual meanwind stress obtained from the GODAS data.These cases were obtained by running the modelfor 200 years by starting from an initial state ofrest. Two cases with the initial layer thickness of

250 and 75 m were run. In the first case, there isenough warm water, so that most of the sub-tropical basins are covered by the upper layer(Fig. 1.14). On the other hand, the upper layeroutcrops in the most part of the ACC and a largepart of the subpolar gyre in the Pacific andAtlantic basins.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1X

Y

= 0.2 = 0.4

= 10 = 8.0

= 2.0

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Fig. 1.13 The non-dimensional position of theoutcropping window for anidealized two-gyre basin,where k is a non-dimensionalnumber indicating thenonlinearity of the model

Fig. 1.14 The upper layer thickness for the world oceans forced by the climatological annual mean wind stress of theGODAS data, with the initially uniform thickness of 250 m

1.4 Layer Outcropping: The Physics and the Numerical Method 15

However, as the amount of warm water isreduced, the upper layer can no longer cover thesubtropical basins. As a result, there are outcropwindows in the subtropical basins which gradu-ally expand (Fig. 1.15). Although these experi-ments were run under the assumption of differentinitial upper layer thickness and forced by thesame wind stress, one can also run the modelwith the same amount of warm water in the upperlayer, but gradually change the strength of windforcing. As discussed above, under such anassumption, the outcrop window will graduallyexpand, very much like the situations shown inFigs. 1.14 and 1.15.

References

Antonov JI, Seidov D, Boyer TP, Locarnini RA, Mis-honov AV, Garcia HE, Baranova OK, Zweng M,Johnson DR (2010) World Ocean Atlas 2009, Volume

2: Salinity. In: Levitus S (ed) NOAA Atlas NESDIS69. U.S. Government Printing Office, Washington,DC, p 184

Behringer DW, Xue Y (2004) Evaluation of the globalocean data assimilation system at NCEP: The PacificOcean. In: Eighth symposium on integrated observingand assimilation systems for atmosphere, oceans, andland surface, AMS 84th Annual Meeting, WashingtonState Convention and Trade Center, Seattle, Wash-ington, pp 11–15

GODAS data, Provided by the NOAA-ESRL PhysicalSciences Division, Boulder Colorado from their Website at https://www.esrl.noaa.gov/psd/

Huang RX (1991) A note on combining wind andbuoyancy forcing in a simple one-layer ocean model.Dyn Atmos Oceans 15:535–540

Huang RX (2010) Ocean circulation, wind-driven andthermohaline processes. Cambridge Press, Cambridge,791 pp

McCreary JP, Yu Z (1992) Equatorial dynamics in a 2½-layer model. Prog Oceanogr 29:61–132

Parsons AT (1969) A two-layer model of Gulf Streamseparation. J Fluid Mech 39:511–528

Fig. 1.15 The upper layer thickness for the world oceans forced by the climatological annual mean wind stress of theGODAS data, with the initially uniform thickness of 75 m

16 1 Basic Concepts

2Climate Variability Diagnosedfrom the Spherical Coordinates

The spherical coordinates are commonly used inoceanography; in particular, most climate studiesand datasets are based on the z-coordinate.Therefore, in the first part of this chapter weexamine the climate variability using thez-coordinate; at the end of this chapter, we willalso explore using longitudinal and latitudinalcoordinates to diagnose the climate variability. Inthese coordinates climate signals can be sepa-rated into the external and internal modes. Theexternal modes indicate the net change of heatcontent (or salt/density content) integrated overthe world oceans; these modes are directly linkedto anomalies in net external forcing, such as heatflux (or freshwater/density flux). On the otherhand, the internal modes represent the internalexchanges of heat, salt and density between dif-ferent layers (latitude/longitude bands). Theseprocesses may involve internal diabatic compo-nents, which give rise to changes of heat(salt/density) in individual layers; however, theglobal net contribution of these internal diabaticprocesses must be zero by definition. Therefore,separating the climate signals into external andinternal modes in these coordinates may revealinteresting phenomena hidden in climate chan-ges. The dynamical/thermodynamic processesleading to such internal modes of variability maytake place primarily within isopycnal layers,which are slanted in the traditional sphericalcoordinates. Hence, to explore the cause of cli-mate variability observed in the ocean, oneshould also look at climate variability from

different angles. Isopycnal/isothermal analysis isone of such tools, and it will be examined indetails in this book.

Although most studies of climate changes arefocused on the upper ocean, with the limitedinformation available about the deep ocean, onecan also infer the climate variability in the deepocean, from climate datasets generated fromeither observations or numerical simulations.Due to the server limits of data availability forthe deep ocean, any result related to the deepocean is at best speculation only. Nevertheless,such studies may reveal some importantdynamics related to climate changes. Our dis-cussion below is based on the GODAS data(Behringer and Xue 2004).

2.1 Climate Variability Diagnosedin the z-Coordinate

In the following discussion, we calculate the totalheat content anomaly for the world oceans, withthe mean annual cycle removed. Note that theGODAS data is a monthly mean climatologywhich began in 1980. The horizontal resolutionis 1� � 0:333�; there are 40 non-uniform layersin the vertical direction, and the center of thelowest layer is at the depth of 4478 m. Since thevertical grid in the GODAS data is non-uniform,we will use heat content per unit thickness, inunits of J/m. In addition, figures are plotted in a

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020R. X. Huang, Heaving, Stretching and Spicing Modes,https://doi.org/10.1007/978-981-15-2941-2_2

17

stretched vertical coordinate, with fine resolutionfor the upper one kilometer.

The heat content anomaly is calculated asfollows:

dhc ¼ q0Cp h i; j; k;mð Þ � �h i; j; k;mað Þ� �dxðjÞdydzðkÞ

ð2:1Þ

where q0Cp ¼ 1035� 4184 J/m3=�C is the unitconversion constant, h is the potential tempera-ture, m = 1, …, 420 is the time in month, ma isthe corresponding month in the mean annualcycle. The heat content anomaly for each levelcan be calculated by horizontally integrating dhcdefined in Eq. (2.1) over the world oceans

Hc k;mð Þ ¼ZZ

Adhc dxdy ð2:2Þ

This variable can be further separated into thebarotropic (external) and baroclinic (internal)modes as follows

Hc k;mð Þ ¼ HBTc ðmÞþHBC

c k;mð Þ� �dzðkÞ ð2:3Þ

The barotropic mode is defined as the verticalmean

HBTc ðmÞ ¼

X40

k¼1

Hc k;mð Þ=D ð2:4Þ

where D = 4736 m is the effective depth of themodel ocean in the GODAS data. Therefore, thebarotropic mode is the net change of the verticalmean heat content anomaly integrated over theworld oceans. The baroclinic modes are definedas

HBCc k;mð Þ ¼ Hc k;mð Þ=dzðkÞ � HBT

c ðmÞ ð2:5Þ

Both the barotropic and baroclinic modes aredefined as the net heat content anomaly permeter.

The global sum of the external mode in thez-coordinate should be the same as the globalintegration of the external mode in the densitycoordinate. The current modeling practice is to

use the thermal isolating boundary conditionsaround the lateral and bottom surfaces; so thatheating/cooling of the model ocean must gothrough the air-sea interface. The time evolutionof the corresponding summation reflects thegeneral warming/cooling of the world oceans dueto the anomalous air-sea thermal interaction.

On the other hand, the internal modes in thedensity coordinate and the baroclinic modes inthe z-coordinate diagnosed from data may havequite different patterns. These differences reflectthe dynamic nature of the internal modes in thesetwo different coordinates; such differences canprovide useful information regarding the natureof climate variability in the world oceans, andthis is the reason to use different coordinates formapping out the climate variability.

As shown in Fig. 2.1, patterns of the timeevolution of total signals and the baroclinic modesignals are quite similar. Since the total signalsare the sum of barotropic and baroclinic signals,the similarity in the patterns of total signals andbaroclinic signals indicates that the contributiondue to the barotropic signals is a small fractiononly. There are clearly high frequency oscillatorycomponents in the upper ocean. For the deepocean, the signals are predominantly on decadaltime scales. The Root-Mean Square (RMS) of thebaroclinic mode is shown by the dashed red linein Fig. 2.1c, and its magnitude (0.1 � 1020 J/m)is much smaller than that of the baroclinicmodes.

The time evolution of the barotropic mode isshown in Fig. 2.1d. As noted above, the ampli-tude of this mode is much smaller than thebaroclinic modes. This mode reflects the totalheat content change in the world oceans, andsuch change is entirely due to the external heatflux variability associated with the air-seainteraction.

In order to reveal the relation between thesesignals, we plot the time evolution of these sig-nals in Fig. 2.2. It is clearly seen that theamplitude of the barotropic mode is muchsmaller than the baroclinic modes, especiallynear the surface. For example, the RMS ampli-tude of the barotropic mode is 0.115 � 1020 J/m,

18 2 Climate Variability Diagnosed from the Spherical Coordinates

while the RMS amplitude of the baroclinic modesignal in the top layer is 1.51 � 1020 J/m, about13 times larger than that of the barotropic mode.

In the surface layer (at 5 m depth), the heatcontent anomaly (the red curve in Fig. 2.2a) wentthrough the period of increase before 1996;afterward, there was a period of oscillationwithout much increase. Similarly, the baroclinicmode at 105 m depth also shows a period of nosystematic increase between 2005 and 2015.A period of no temperature increase is oftencalled a hiatus, which has been widely discussedin recent literatures.

Conceptually, the barotropic mode is con-trolled by the air-sea heat exchange anomaly; onthe other hand, the baroclinic modes are con-trolled by internal processes. In the surface layerthe baroclinic mode is ten times stronger than thebarotropic mode; in fact, the heat contentanomaly in the surface layer is primarily regu-lated by internal dynamical processes, and theanomalous air-sea heat flux and solar insolationplay minor roles only. Therefore, although theheat content anomaly of the surface layer, such asthat associated with the recent hiatus, can be

important to human society, it does not neces-sarily reflect the overall change of heat content inthe world oceans; one should also examine thestatus of the barotropic mode.

In the subsurface layers, the heat contentanomaly trend is quite different. The heat contentanomaly over the depth range of 949–1193 mincreased over the whole period of the datarecord, as shown in Fig. 2.2b. In fact, the heatcontent increase rate was high before 1994. Itslowed down during the period of 1995–2007;afterwards, the heat content of these subsurfacelayers increased with a speed doubled. It is par-ticularly interesting to note that the heat contentof the entire depth of the world oceans wasactually negative for the period of 1997–2006,i.e., the heat content anomaly over these middledepth layers has signs opposite to that of thedepth-integrated heat content of the worldoceans. Such baroclinic modes of the heat con-tent anomaly are due to the adiabatic adjustmentof isothermal layers in the ocean, and this will bediscussed in detail in the following chapters.

On the other hand, in the deep ocean, espe-cially near the sea floor, the amplitude of the

Fig. 2.1 Heat content anomaly for the world oceans inferred from the GODAS data, in 1020 J/m. a The total heatcontent; b the baroclinic modes; c the RMS of the baroclinic modes (black curve) and barotropic mode (red dashedline); d the barotropic mode

2.1 Climate Variability Diagnosed in the z-Coordinate 19