The Solar Tachocline

Cover......Page 1
Half -title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Contributors......Page 9
Preface......Page 11
Part I Setting the scene......Page 15
1.1 Preamble......Page 17
1.2 Some basic properties of the Sun......Page 18
1.3 Solar spin-down......Page 21
1.4 The rotation of the Sun today......Page 23
1.5 The solar tachocline......Page 26
1.6 On the basic dynamics of the tachocline......Page 28
1.7 Tachocline instability and the solar dynamo......Page 39
Acknowledgments......Page 44
2.1 The maculate Sun......Page 45
2.2 Braking the Sun\'s rotation......Page 46
2.3 Solar spin-down......Page 49
2.4 Solar intermittency......Page 52
2.5 Into the 1990s......Page 56
2.6 From recollections to reflections......Page 58
References......Page 61
Part II Observations......Page 65
3.1 Introduction......Page 67
3.2 Helioseismic techniques......Page 69
3.3 Solar internal structure and rotation......Page 75
3.4 Dynamical properties of the tachocline......Page 83
3.5 The structure of the tachocline region......Page 88
3.6 Outlook......Page 93
References......Page 95
Part III Hydrodynamic models......Page 101
4.1 Introduction......Page 103
4.2 Setting the stage......Page 104
4.3 The tachocline: governing equations and simplifying assumptions......Page 107
4.4 Laminar penetration......Page 109
4.5 A hydrodynamic remedy against the spread of the tachocline: anisotropic turbulence......Page 112
4.6 Which turbulence?......Page 113
4.7 Ventilation time and mixing......Page 116
4.8 Internal gravity waves......Page 117
4.9 Conclusion......Page 118
References......Page 119
5.1 Introduction......Page 123
5.2 The upper tachocline: penetrative convection......Page 124
5.3.1 Overview......Page 132
5.4 Summary......Page 139
5.4.1 Acknowledgments......Page 140
References......Page 141
6.1 Introduction......Page 143
6.2 The …-effect......Page 144
6.3 Turbulent heat transport......Page 147
6.4 Solar models......Page 150
6.5 Stellar models......Page 153
6.6 Antisolar rotation?......Page 156
References......Page 157
Part IV Hydromagnetic properties......Page 159
7.1.1 Magnetic fields in the tachocline......Page 161
7.2.1 First models......Page 162
7.2.2 Towards a self-consistent model: the governing equations......Page 166
7.2.3 The Gough & McIntyre boundary layer analysis......Page 169
7.2.4 Numerical solutions of the Gough & McIntyre model......Page 172
7.2.4.1 Steady-state calculations......Page 173
7.2.4.2 Time-dependent calculations......Page 177
7.2.5.1 Stability of the slow tachocline models......Page 180
7.2.5.3 Boundary conditions......Page 182
7.3 Dynamo field confinement: the fast tachocline......Page 184
7.3.2 Discussions and prospect for the dynamo confinement model......Page 189
7.4 Discussion and prospects......Page 191
References......Page 193
8 Magnetic confinement and the sharp tachopause......Page 197
8.1 Introduction......Page 198
8.2 Gyroscopic pumping and HSZ burrowing......Page 203
8.3 The nearly laminar magnetic interior......Page 209
8.4 The high-latitude tachocline and its invisible shear......Page 211
8.5 The effects of compositional stratification Nµ......Page 220
8.6 Concluding remarks......Page 223
References......Page 225
9.1 Introduction......Page 227
9.2 Some basic aspects of two-dimensional MHD turbulence on a β-plane......Page 232
9.3 The Rhines scale for MHD turbulence on a β-plane......Page 236
9.4 Spectral transfer and turbulent dissipation in β-plane MHD......Page 241
9.5 Discussion and conclusions......Page 249
References......Page 252
Part V Instabilities......Page 255
10.1 Introduction......Page 257
10.2 Classical shear and magneto-shear instabilities......Page 262
10.3.1 Basic equations: nonlinear and linear......Page 265
10.3.2 Ideal fluid linear MHD instabilities......Page 266
10.3.3 Ideal fluid nonlinear MHD instabilities......Page 269
10.3.4 Effects of drag......Page 272
10.4 Towards three dimensions......Page 274
10.4.1.1 Equations......Page 275
10.4.1.2 Reference states......Page 277
10.4.1.3 Hydrodynamic instabilities......Page 278
10.4.1.4 Magnetohydrodynamic instabilities......Page 279
10.4.2 Fine radial structure instabilities......Page 282
10.5 Conclusions: implications for the Sun......Page 284
References......Page 287
11.1 Introduction......Page 289
11.2.1.1 The instability mechanism......Page 291
11.2.1.2 The role of diffusion......Page 294
11.2.1.3 The influence of rotation......Page 296
11.2.1.4 The interaction with a shear flow......Page 298
11.2.2 Nonlinear evolution......Page 301
11.3 The instability of isolated flux tubes......Page 304
11.4 The buoyant rise of isolated flux tubes......Page 306
11.5 Implications for the tachocline......Page 308
References......Page 310
12.1 Introduction......Page 313
12.2 Angular momentum transport in general......Page 314
12.3 The magnetorotational instability......Page 315
12.4 Statistical models of anisotropic MHD turbulence......Page 319
12.5.1 Introduction......Page 321
12.5.2 Local dispersion relation......Page 323
12.5.3 Case of zero magnetic field......Page 324
12.5.4 Limit of a weak magnetic field......Page 325
12.5.6 Application to the tachocline......Page 326
12.5.7 General remarks......Page 327
References......Page 328
Part VI Dynamo action......Page 331
13.1 Observations......Page 333
13.2.1 Differential rotation......Page 338
13.3 Where is the solar dynamo?......Page 345
13.4.1 The form of the magnetic field......Page 349
13.5 Modulation......Page 354
13.6 Structure of the tachocline......Page 358
Part VII Overview......Page 365
14.1 Afterthoughts......Page 367
14.2.1 How is the tachocline defined, and where is it?......Page 368
14.3 Comments on modelling and simulations......Page 374
14.4 The (possible) role of experiments......Page 378
References......Page 380
Index......Page 383