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1 climate dynamics concepts scaling and multiple equilibria by gerrit lohmann alfred wegener institute helmholtz centre for polar and marine research bremerhaven germany department of physics university of bremen bremen ...

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                                             1
                    Climate Dynamics:
             Concepts, Scaling and Multiple Equilibria
                    by Gerrit Lohmann
       Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research,
                     Bremerhaven, Germany.
          Department of Physics, University of Bremen, Bremen, Germany.
                    Lecture Notes 2020
       version of April 13, 2020
                     Contents
                     I      First part: Fluid Dynamics                                                                                                                          9
                     1     Basics of Fluid Dynamics                                                                                                                           10
                           1.1      Material laws             .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     11
                           1.2      Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                           14
                           1.3      Someexercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . .                                                           17
                           1.4      Integral and differential formulation* . . . . . . . . . . . . . . . . . . . . . .                                                        21
                           1.5      Elimination of the pressure term                        .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     24
                           1.6      Non-dimensional parameters: The Reynolds number                                          .  .  .   .  .  .   .  .  .   .  .  .  .   .     25
                           1.7      Characterising flows by dimensionless numbers . . . . . . . . . . . . . . . . .                                                            28
                           1.8      Dynamicsimilarity: Application in engineering*                                    .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     29
                     2     Fluid-dynamical Examples                                                                                                                           35
                           2.1      Convection in the Rayleigh-Bénard system . . . . . . . . . . . . . . . . . . .                                                            39
                                    2.1.1        Elimination of pressure and vorticity dynamics                                 .  .   .  .  .   .  .  .   .  .  .  .   .     41
                                    2.1.2        Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . .                                                      49
                                    2.1.3        Galerkin approximation: Obtaining the Lorenz system . . . . . . . . . .                                                      50
                           2.2      Bernoulli flow*               .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     52
                           2.3      Couette flow* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                            61
                                                                                                   2
                     CONTENTS                                                                                                                                                   3
                     II       Secondpart: Dynamical systems                                                                                                                  65
                     3     Preparation and tools                                                                                                                              64
                           3.1      Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                            67
                           3.2      Fourier transform                .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     76
                           3.3      Covariance and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                           84
                           3.4      Transport phenomena                    .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     90
                           3.5      General form of wave equations                          .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .     92
                     4     General concepts                                                                                                                                   96
                           4.1      ProgrammingwithR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                                              96
                           4.2      RMarkdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
                           4.3      Netcdf and climate data operators . . . . . . . . . . . . . . . . . . . . . . . . 106
                                    4.3.1        TheBash,apopularUNIX-Shell . . . . . . . . . . . . . . . . . . . . 111
                                    4.3.2        Reducing data sets with CDO . . . . . . . . . . . . . . . . . . . . . . 114
                                    4.3.3        Asimplemodelofsealevelrise                              .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   118
                           4.4      Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
                                    4.4.1        Linear stability analysis                  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   127
                                    4.4.2        Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 135
                                    4.4.3        Lorenz system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
                     5     Statistical Mechanics and Fluid Dynamics                                                                                                         149
                           5.1      Mesoscopic dynamics* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
                           5.2      TheBoltzmannEquation* . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
                           5.3      H-Theoremandapproximation of the Boltzmann equation* . . . . . . . . . . . 159
                           5.4      Application: Lattice Boltzmann Dynamics                                 .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   164
                                    5.4.1        Lattice Boltzmann Methods* . . . . . . . . . . . . . . . . . . . . . . 164
                                    5.4.2        Simulation set-up of the Rayleigh-Bénard convection                                      .  .   .  .  .   .  .  .  .   .   169
                                    5.4.3        System preparations and running a simulation . . . . . . . . . . . . . . 172
                     4                                                                                                                                       CONTENTS
                     III        Thirdpart: Dynamics of the climate system                                                                                                  178
                     6     AtmosphereandOceanDynamics                                                                                                                       179
                           6.1      Pseudo forces and the Coriolis effect . . . . . . . . . . . . . . . . . . . . . . 179
                           6.2      Scaling of the dynamical equations . . . . . . . . . . . . . . . . . . . . . . . 182
                           6.3      Thecoordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
                           6.4      Geostrophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
                           6.5      Conservation of vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
                                    6.5.1        Potential vorticity equation (ζ + f)/h . . . . . . . . . . . . . . . . . 201
                                    6.5.2        Taylor-Proudman Theorem . . . . . . . . . . . . . . . . . . . . . . . 210
                           6.6      Wind-driven ocean circulation                        .  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   213
                                    6.6.1        Sverdrup relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
                                    6.6.2        EkmanPumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
                                    6.6.3        Ekmanspiral* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
                                    6.6.4        Western Boundary Currents . . . . . . . . . . . . . . . . . . . . . . . 234
                           6.7      Thermohaline ocean circulation                          .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   241
                                    6.7.1        Conceptual model of the ocean circulation: Stommel’s box model . . . . 252
                                    6.7.2        Non-normal dynamics of the ocean box model                                     .  .   .  .  .   .  .  .   .  .  .  .   .   260
                     7     Simple Climate Models                                                                                                                            265
                           7.1      Engery balance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
                           7.2      Moist atmospheric energy balance model*                                 .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   288
                           7.3      Interhemispheric box model                       .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .  .  .   .  .  .   .  .  .  .   .   293
                                    7.3.1        Modeldescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
                                    7.3.2        Runthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
                                    7.3.3        Modelscenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
                           7.4      Weather and climate: Stochastic climate model . . . . . . . . . . . . . . . . . 304
                           7.5      Projection methods: coarse graining* . . . . . . . . . . . . . . . . . . . . . . 328
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...Climate dynamics concepts scaling and multiple equilibria by gerrit lohmann alfred wegener institute helmholtz centre for polar marine research bremerhaven germany department of physics university bremen lecture notes version april contents i first part fluid basics material laws navier stokes equations someexercises chapter integral differential formulation elimination the pressure term non dimensional parameters reynolds number characterising ows dimensionless numbers dynamicsimilarity application in engineering dynamical examples convection rayleigh benard system vorticity boundary conditions galerkin approximation obtaining lorenz bernoulli ow couette ii secondpart systems preparation tools pendulum fourier transform covariance spectrum transport phenomena general form wave programmingwithr rmarkdown netcdf data operators thebash apopularunix shell reducing sets with cdo asimplemodelofsealevelrise bifurcations linear stability analysis population statistical mechanics mesoscopic th...

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