Chapter 1 Dynamics

• Vortical motions


  Rotational flow that is almost non-divergent
• Weather systems

  • Associated with strong winds around vortices, such as extratropical cyclones or hurricanes
• Helmholtz decomposition


  V = V non-divergent + V irrotational
• Streamfunction


  Describes the rotational part of the flow
• Vorticity


  Measure of the local spin about a point in a fluid, defined as the curl of velocity
• Absolute vorticity


  Sum of planetary vorticity and the vorticity of the fluid flow relative to the Earth's surface
• Cyclonic flow


  Relative vorticity has the same sign as the Coriolis parameter f
• Geostrophic balance


  Pressure gradient force is equal and opposite to the Coriolis force
• Geostrophic streamfunction


  Defined as p'/ρ₀f₀
• Geostrophic vorticity


  Defined as V × Vg
• Hydrostatic balance


  Gravitational force is balanced by the vertical pressure gradient force
• Anelastic approximation


  Variations in density and pressure are dominated by the decrease with height associated with gravity and the weight of fluid above in the column, in hydrostatic balance at all levels
• The anelastic approximation is often expressed in terms of the corresponding equation for mass conservation: V·(ρru) = 0
• Replacing p' in the vertical momentum equation (22)


  1. Some manipulation

  2. Showing that

  3. Dw/Dt = (1/p) * (dp/dz) * Pr + g * p'/p
• N²


  Static stability, can be written as g * (d(ln(p))/dz + 1/p * dp/dz)
• It is a typically a good approximation to assume that H₂ » H, and also that the height scale He is much greater than characteristic vertical scales of the velocity, H. In this case the final term related to N2 x 1/He can be neglected
• In practice it is also a good approximation to take ρ₀ ≈ ρ₀ in this equation, where ρ₀ is a constant
• Anelastic approximation


  Filtering out sound waves, flow speeds of interest are very much less than the speed of sound
• Anelastic expression for hydrostatic balance

  1. Combining the anelastic form of the vertical momentum equation

  2. Yielding the anelastic form of hydrostatic balance: dp'/dz = -ρ₀g
• Derivation of anelastic horizontal momentum equation


  1. Dt/Dt + fk x V = -∇h(p')

  2. Since p_r = p_r(z) is constant in the horizontal, the second term is 0

  3. Resulting in -f₀√(p'/p_r)∇hψ_g
• Derivation of anelastic thermodynamic equation


  Db'/Dt + N²w = 0
• Summarising the anelastic equations

  • Horizontal momentum equation
  • Vertical momentum equation
  • Thermodynamic equation
  • Mass conservation or continuity equation
• ψ_g


  Geostrophic streamfunction, given by p'/ρ₀f₀
• θ'


  Perturbation potential temperature
• b'


  Buoyancy, given by g * θ'/θ₀
• N²


  Static stability parameter, N is the Brunt-Väisälä frequency
• Recall lines of constant streamfunction are streamlines. Streamlines are lines that are parallel to the velocity vector at each point in a fluid.
• Recall N is the frequency of oscillation of an air parcel displaced vertically in a statically stable atmosphere related to how easy or difficult it is to push air up or down.
• Boussinesq approximation


  • Simplifies the equations of motion by assuming the density height scale is large relative to the height scales of the motion
  • Density can be taken as constant in mass continuity, so velocity is non-divergent
• In the troposphere, the density height scale, Hp, is typically 6 km. This means that the Boussinesq approximation is a poor approximation for weather systems where the scales of the motion span the troposphere with a characteristic height scale of about 5 km.
• In the majority of the lecture notes, the equations will use the Boussinesq approximation because it simplifies the analysis and equations presented.
• Thermal wind


  Relates the vertical shear of the geostrophic wind to the horizontal temperature gradient
• Deriving the equation for thermal wind balance

  1. Combining the equations for hydrostatic and geostrophic balance

  2. Resulting in (∂u_g/∂z, ∂v_g/∂z) = (-1/f₀ρ₀) k x ∇_h b'
• The thermal wind balance relationship can also be written in terms of the geostrophic relative vorticity: (∂ξ_g/∂z) = -(f₀/N²) ∇²_h b'
• Ageostrophic wind


  Represents the departure of the wind from geostrophic balance
• The horizontal divergence and vertical motion are associated with the ageostrophic wind
• The evolution of the flow is mainly due to the ageostrophic wind, as the geostrophic flow is defined by a diagnostic relationship with the pressure field
• Potential vorticity (PV)


  A key variable that provides the clearest route to understanding dynamics or the evolution of a balanced flow dominated by rotation
• PV is conserved following fluid parcels for adiabatic, frictionless flow
• If the distribution of PV is known throughout the atmosphere, it can in principal be inverted to obtain knowledge of all other variables relevant to the dynamics such as winds, temperature and pressure