Numerical Experiments on Two-dimensional Turbulence on a Rotating Sphere

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The 2nd KAGI-21 INTERNATIONAL SUMMER SCHOOL Aug.
15-27, 2005
An Introduction to Geophysical Fluid Dynamics
YODEN Shigeo Department of Geophysics, Kyoto
University

• Fundamental equations and concepts
• Balanced flow and vortex motion
• Waves
• Instabilities
• Nonlinear phenomena

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1. Fundamental equations and concepts

• 1.1 Fundamental equations for Newtonian fluids

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

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• 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

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• Energy equation
• conservation law of total energy (internal
  kinetic)
• 1st law of thermodynamics (dU?W?Q )
• entropy
• conservation of entropy in adiabatic
  conditions

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• Equation of state
• ideal gas
• Boussinesq fluid
• in the buoyancy term,
• and ?constant in the other terms
• incompressible fluid

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• stress tensor
• momentum equation for continuum
• Cauchys 1st law of motion
• conservation law of angular momentum
• Cauchys 2nd law of motion
• for Newtonian fluids
• p pressure

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• deformation rate
• deformation rate tensor
• strain rate tensor
• elongation-contraction rate
• shear strain rate
• vorticity

Advection of finite fluid elements in a steady 2D
flow
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• 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

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• inertial oscillation
• free motion of a particle on a rotating plane
  without Ce
• if we put
• inertial oscillation with a period of 2p/f
  (p/O)
• inertial circle with a radius of W0/f

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• momentum equation on the rotating Earth
• a spherical coordinate
• rotation O ? (longitude ?, latitude f, radius
  r )
• traditional approximation (r a z z ltlt a )

conservation of angular momentum?
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2. Balanced flow and vortex motion

• 2.1 Steady axisymmetric vortex in a rotating
  frame
• constant density fluid
• a cylindrical coordinate (r , ?, z )
• 2D motion independent of z rotation O
• steady axisymmetric motion
• balance in centrifugal, Coriolis, and pressure
  grad. forces

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• Rossby number
• Ro Centrifugal force / Coriolis force v /
  (2Or )
• cyclostrophic-flow balance
• Ro gtgt 1
• ex. tornado
• bathtub vortex
• gradient-flow balance
• Ro 1
• ex. typhoon

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• geostrophic-flow balance
• Ro ltlt 1
• ex. circumpolar vortex
• inertial flow
• Pressure grad. force 0

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• 2.2 Circulation and vorticity
• circulation theorem
• velocity, vorticity, and circulation
• Bjerknes circulation theorem
• Kelvins circulation theorem

(by Stokess theorem)
(solenoid term) B
baroclinic vector
for barotropic fluid B 0, or ? (p )
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• 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

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• 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 (T,
  q, )

s entropy or potential temperature T
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• conservation of potential vorticity shallow
  water
• shallow water equations
• potential vorticity equation

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

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• stratospheric polar vortex
• point vortex
• Fujiwara effect
• Karman vortices

Fujii (2001)