1.Fundamental equations and concepts 2.Balanced flow and vortex motion 3.Waves 4.Instabilities 5.Nonlinear phenomena An Introduction to Geophysical Fluid Dynamics YODEN Shigeo Department of Geophysics, Kyoto University The 2nd KAGI-21 INTERNATIONAL SUMMER SCHOOL: Aug , 2005

1. Fundamental equations and concepts 1.1 Fundamental equations for Newtonian fluids

 Continuity equation conservation law of mass advection (Lagrange) form flux form  Momentum equation conservation law of momentum (Navie-Stokes Eq.) ~0

 advection vs. diffusion linear advection-diffusion equation (1D) nonlinear advection-diffusion equation: Burgers equation  Brownian motion Langevin equation for a particle stochastic motions of a huge ensemble of particles ?

 Energy equation conservation law of total energy (internal + kinetic) 1st law of thermodynamics (dU=ΔW ＋ ΔQ ) entropy conservation of entropy in adiabatic conditions

 Equation of state ideal gas Boussinesq fluid in the buoyancy term, and ρ=constant in the other terms incompressible fluid

 stress tensor momentum equation for continuum Cauchy’s 1st law of motion conservation law of angular momentum Cauchy’s 2nd law of motion for Newtonian fluids p : pressure

H L Advection of finite fluid elements in a steady 2D flow  deformation rate deformation rate tensor strain rate tensor  elongation-contraction rate  shear strain rate vorticity

1.2 Fundamental equations in a rotating frame  momentum equation in a rotating frame an orthogonal inertial frame with a set of unit vectors a rotating frame with which rotates with a constant angular velocity : position, velocity, and acceleration Coriolis force and centrifugal force CoCe

 inertial oscillation free motion of a particle on a rotating plane without Ce if we put inertial oscillation with a period of 2π/f (=π/Ω) inertial circle with a radius of |W 0 /f |

 momentum equation on the rotating Earth a spherical coordinate + rotation Ω ⇒ (longitude λ, latitude φ, radius r ) traditional approximation (r =a +z ; |z | << a ) conservation of angular momentum?

2. Balanced flow and vortex motion 2.1 Steady axisymmetric vortex in a rotating frame  constant density fluid a cylindrical coordinate (r, θ, z ) 2D motion independent of z + rotation Ω steady axisymmetric motion balance in centrifugal, Coriolis, and pressure grad. forces

 Rossby number Ro = Centrifugal force / Coriolis force = v / (2Ωr )  cyclostrophic-flow balance Ro >> 1 ex. tornado bathtub vortex  gradient-flow balance Ro ～ 1 ex. typhoon

 geostrophic-flow balance Ro << 1 ex. circumpolar vortex  inertial flow Pressure grad. force ～ 0

2.2 Circulation and vorticity  circulation theorem velocity, vorticity, and circulation Bjerknes’ circulation theorem Kelvin’s circulation theorem (by Stokes’s theorem) (solenoid term) B : baroclinic vector for barotropic fluid: B =0, or ρ = (p )

 vorticity equation momentum equation in a vector invariant form vorticity equation tilting term and stretching term  in a rotating system momentum equation and vorticity equation

2.3 Potential vorticity  conservation of potential vorticity –ideal gas potential vorticity potential vorticity equation  for inviscid fluid under conservative external forces PV thinking  invertibility: P ⇒ other dynamical quantities (u, T, ・・・ )  conserved in Lagrangian motion  combination with other conserved quantities: (Θ, q, ・・・ ) s : entropy or potential temperature Θ

 conservation of potential vorticity –shallow water shallow water equations potential vorticity equation

2.3 Vortex motions  Rankin vortex circular vortex patch  Contour dynamics evolution of vortex patch ex. elliptic vortex elongated one is unstable

 stratospheric polar vortex  point vortex Fujiwara effect Karman vortices Karman vortices around Robinson Crusoe Island DeFelice et al. (2000, BAMS ) McIntyre and Palmer (1983, Nature ) Fujii (2001)