List of Figures ............................................................. ix List of Tables ............................................................. xii Preface ................................................................... xiii Acknowledgements ........................................................... xiv 1. Atmospheric Convection with Analogies in Astrophysics and the Laboratory .......................................................... 15 Robert M. Kerr 1.1 Introduction ........................................................ 15 1.2 Reynolds number and modeling ........................................ 17 1.3 Dry convective scaling .............................................. 20 1.4 Precipitating convection ............................................ 23 1.5 Hierarchy of scales ................................................. 27 1.6 Improving LES ....................................................... 30 1.7 Conclusion .......................................................... 32 2. Solar and Stellar Convection: A Perspective for Geophysical Fluid Dynamicists ......................... 37 Peter A. Gilman 2.1 Introduction ........................................................ 38 2.2 Solar motions ....................................................... 39 2.3 Global features of convection zone .................................. 43 2.3.1 Structure with radius ......................................... 43 2.3.2 Influence of rotation ......................................... 45 2.3.3 Upper boundary layer .......................................... 45 2.3.4 Lower boundary layer .......................................... 46 2.3.5 Waves and instabilities in and near the convection zone ....... 47 2.4 Simulating solar convection ......................................... 48 2.5 Summary of methods and results for compressible convection .......... 49 2.6 Convection as driver of differential rotation ....................... 51 2.7 Interaction of the convection zone with the solar surfaces and the shear layer at the base ......................................... 52 2.8 Concluding remarks .................................................. 54 3. Unsteady Non-Penetrative Thermal Convection From Non-Uniform Surfaces ... 59 Richard D. Keane, Noboyuki Fujisawa and Ronald J. Adrian 3.1 Introduction ........................................................ 59 3.2 Experimental apparatus and procedure ................................ 62 3.3 Results ............................................................. 65 3.3.1 Heat transfer characteristics ................................. 65 3.3.2 Existence of horizontal mean flow ............................. 67 3.3.3 Patterns of convection ........................................ 67 3.4 Summary and conclusions ............................................. 74 4. Astrophysical Convection and Dynamos .................................... 85 Axel Brandenburg, Ake Nordlund and Robert F. Stein 4.1 Introduction ........................................................ 85 4.2 Deep solar convection ............................................... 87 4.3 Low Prandtl number effects .......................................... 88 4.4 The entropy gradient ................................................ 90 4.5 The thermal time scale problem ...................................... 94 4.6 The formation of magnetic structures ................................ 97 4.7 Magnetic dynamo action .............................................. 99 4.8 Downward pumping ................................................... 100 4.9 Outstanding problems ............................................... 101 5. Dynamics of Cumulus Entrainment ........................................ 107 Wojciech W. Grabowski 5.1 Introduction ....................................................... 107 5.2 Turbulent entrainment in cumulus clouds and in buoyancy-driven flows .............................................................. 111 5.3 Entrainment as a result of interfacial instabilities ............... 113 5.4 Entrainment and buoyancy reversal .................................. 121 5.5 Conclusions ........................................................ 123 6. The 2/7 Law in Turbulent Thermal Convection ............................ 129 Stephane Zaleski 6.1 Introduction ....................................................... 129 6.2 Problem definition ................................................. 130 6.3 Simple approaches to scaling ....................................... 131 6.3.1 Similarity arguments based on dimensional analysis ........... 131 6.3.2 Marginal stability and boundary layer similarity ............. 132 6.4 Mechanistic approaches to scaling .................................. 133 6.4.1 Inviscid interior scaling .................................... 133 6.4.2 Plume theories with a single length scale .................... 134 6.4.3 Plume theory with several length scales ...................... 135 6.4.4 2/7 scaling: Shraiman-Siggia theory .......................... 136 6.4.5 Range of validity ............................................ 136 6.5 Comparison with experiments ........................................ 138 6.6 Critique ........................................................... 139 6.7 Conclusion ......................................................... 140 7. Organization of Atmospheric Convection over the Tropical Oceans: The Role of Vertical Shear and Buoyancy ................................ 145 Margaret A. LeMone 7.1 Introduction ....................................................... 145 7.2 Convection in the fair weather mixed layer ......................... 146 7.2.1 Larger aspect-ratio mixed-layer banded structures ............ 149 7.3 Precipitating convection ........................................... 152 7.3.1 Buoyancy ..................................................... 152 7.3.2 Shear ........................................................ 154 7.4 Conclusions ........................................................ 161 8. Images of Hard Turbulence: Buoyant Plumes in a Crosswind ............... 165 Andrew Belmonte and Albert Libchaber 8.1 Introduction ....................................................... 165 8.2 Hard Turbulence .................................................... 167 8.3 Experimental techniques ............................................ 169 8.3.1 The convection cell .......................................... 169 8.3.2 Visualization ................................................ 170 8.3.3 Image processing ............................................. 172 8.4 Shadowgraph images ................................................. 172 8.5 Intensity correlation measurements ................................. 177 8.6 Discussion ......................................................... 179 9. Convection in Cloud-Topped Atmospheric Boundary Layers ................. 185 Christopher S. Bretherton 9.1 Introduction ....................................................... 185 9.2 Global distribution and importance of boundary layer cloud ......... 186 9.3 Convective dynamics of CTBLs ....................................... 190 9.4 Further observations and conclusions ............................... 195 10. Solar Granulation: A Surface Phenomenon ................................ 199 Mark Peter Rast 10.1 Introduction ...................................................... 200 10.2 Granular dynamics ................................................. 201 10.3 Heat transport .................................................... 207 10.4 Flow stability .................................................... 211 10.5 Conclusion ........................................................ 216 11. Turbulent Convection: What has Rotation Taught Us? ..................... 221 Joseph Werne 11.1 Introduction ...................................................... 221 11.2 Nonrotating Rayleigh-Benard convection ............................ 222 11.3 Turbulent convection theories ..................................... 223 11.3.1 Priestley's theory ......................................... 223 11.3.2 Kadanoff, Zaleski and Zanetti's theory ..................... 223 11.3.3 Shraiman and Siggia's theory ............................... 224 11.3.4 Cautionary comment on scaling theories ..................... 224 11.3.5 She's theory ............................................... 225 11.3.6 Yakhot's theory ............................................ 225 11.4 Rotating Rayleigh-Benard convection ............................... 226 11.5 Numerical simulation of rotating convection ....................... 226 11.5.1 Intermittent flow fields ................................... 227 11.5.2 Cyclonic plumes ............................................ 227 11.5.3 Ekman pumping .............................................. 227 11.5.4 Linear thermal Ekman layer ................................. 230 11.5.5 Nonlinear Ekman spirals .................................... 232 11.5.6 Plume-plume interactions ................................... 234 11.5.7 Rotating hard turbulence ................................... 235 11.6 Conclusions ....................................................... 236 12. Helical Buoyant Convection ............................................. 241 Douglas Lilly 12.1 Rotating thunderstorms and tornadoes .............................. 241 12.2 Analysis and illustrations ........................................ 244 12.3 Further discussion ................................................ 250 13. Modeling Mantle Convection: A Significant Challenge in Geophysical Fluid Dynamics ......................................................... 257 David A. Yuen, S.Balachandar and U.Hansen 13.1 Introduction ...................................................... 258 13.2 Model and numerical techniques .................................... 259 13.2.1 Anelastic liquid model ..................................... 260 13.2.2 Internal solid-state phase transitions ..................... 261 13.2.3 Thermal-chemical convection ................................ 262 13.2.4 Mantle rheology ............................................ 263 13.2.5 Numerical methodologies .................................... 264 13.3 Past achievements and computational challenges .................... 264 13.3.1 Sample past results ........................................ 265 13.3.2 Computational requirements ................................. 268 13.4 Results ........................................................... 269 13.4.1 Viscous heating in mantle convection ....................... 269 13.4.2 High Rayleigh number thermal-chemical convection ........... 271 13.5 Perspectives and future directions ................................ 273 14. Turbulent Transport in Rotating Compressible Convection ................ 295 Nicholas H. Brummell 14.1 Introduction ...................................................... 295 14.2 Local modelling of rotating compressible convection ............... 297 14.2.1 Turbulent transport of convective energy ................... 299 14.2.2 Turbulent transport of (angular) momentum .................. 302 14.3 Conclusions ....................................................... 306 15. Potential Vorticity, Resonance and Dissipation in Rotating Convective Turbulence ............................................................. 309 Peter Bartello 15.1 Background ........................................................ 310 15.2 Normal mode equations, conservation and resonance ................. 312 15.2.1 The <GGG> interactions ..................................... 314 15.2.2 The <AAA> interactions ..................................... 314 15.2.3 The <GAA> interactions ..................................... 314 15.2.4 The <GGA> interactions ..................................... 314 15.3 Numerical Results ................................................. 315 15.3.1 Metais et al. (1994) revisited ............................. 315 15.3.2 Simulations with large vertical dissipation ................ 316 15.4 Conclusions ....................................................... 320 16. Numerical Simulations of Convection in Protostellar Accretion Disks .... 323 William Cabot 16.1 Introduction ...................................................... 323 16.1.1 What are protostellar accretion disks? ..................... 323 16.1.2 Why is convection (potentially) important? ................. 324 16.2 Properties of protostellar disks .................................. 326 16.2.1 General disk properties .................................... 326 16.2.2 Under what conditions do disks become convective? .......... 326 16.2.3 Simplifying assumptions .................................... 327 16.3 Numerical hydrodynamic simulations ................................ 327 16.3.1 Further simplifying assumptions ............................ 327 16.3.2 Governing equations ........................................ 329 16.3.3 Boundary conditions ........................................ 330 16.3.4 Parameters ................................................. 331 16.4 Simulation results ................................................ 332 16.4.1 Incompressible simulations ................................. 332 16.4.2 Compressible simulations ................................... 332 16.5 Discussion ........................................................ 336 16.5.1 Why does disk convection generate inward transport of angular momentum? .......................................... 339 16.5.2 What are the consequences? ................................. 340 16.5.3 What more needs to be done? ................................ 341 16.5.4 Conclusions ................................................ 342 17. A New Model for Turbulence: Convection Rotation and 2D ................. 345 V.M.Canuto, M.S.Dubovikov, A.Dienstfrey and D.J.Wielaard 17.1 Turbulent convection .............................................. 345 17.1.1 New stochastic equations ................................... 345 17.1.2 Numerical results ......................................... 347 17.1.3 Conclusions ................................................ 347 17.2 Rotating turbulence ............................................... 347 17.2.1 Basic results .............................................. 347 17.2.2 2D-3D states in rotating turbulence ........................ 352 17.2.3 Decaying turbulence ........................................ 353 17.2.4 Conclusions ................................................ 354 17.3 2D Turbulence ..................................................... 354 17.3.1 Basic features ............................................. 354 17.3.2 Basic equations. Time evolution of the energy spectrum ..... 354 17.3.3 Numerical results .......................................... 355 17.3.4 Conclusions ................................................ 357 18. Transport Using Transilient Matrices ................................... 363 Roland B. Stull and Jerzy Bartnicki 18.1 Introduction ...................................................... 363 18.2 A transilient turbulence parameterization ......................... 365 18.2.1 Mixing potential, Y, first estimate ........................ 366 18.2.2 Influences of nonlocal static stability .................... 366 18.2.3 Convective overturning and subgrid turbulence .............. 367 18.2.4 Unequal grid spacing ....................................... 368 18.2.5 Transilient matrices ....................................... 368 18.2.6 Use of transilient matrices ................................ 369 18.2.7 Turbulent flux and mixed-layer depth ....................... 369 18.3 Split time step and the destabilization problem ................... 369 18.4 Calibration ....................................................... 371 18.5 Illustrative model ................................................ 372 18.6 Simulation results for idealized scenarios ........................ 373 18.6.1 Neutral boundary layer ..................................... 374 18.6.2 Unstable (free-convective) mixed layer ..................... 376 18.6.3 Mechanically mixed layer ................................... 377 18.6.4 Both buoyant and mechanically mixed layer .................. 378 18.6.5 Stable boundary layer ...................................... 378 18.6.6 Diurnal cycles of boundary layer forcings (including pollution dispersion) ............................ 380 18.6.7 Discussion ................................................. 382 18.7 Conclusions ....................................................... 384 Index ...................................................................... 389