In barotropic model atmosphere some conditions exist throughout the motion
- Coincidence of pressure and temperature surfaces
- Absence of
- Vertical wind shear
- Vertical motions
- Horizontal velocity divergence
- Conservation of the vertical component of absolute vorticity
Classes of barotropic models:
- Non divergent barotropic model
- Simple non divergent barotropic model is based on the barotropic vorticity eqn
- Divergent barotropic model (shallow-water equations)
The barotropic (non-divergent) model assumptions:
- Absolute vorticity is conserved
The barotropic (non-divergent) model applies to
- Barotropic atmosphere or
- Level of non divergence
The barotropic (non-divergent) models are useful for:
- Tropical wind predictions
- Especially at levels where div. is small
Barotropic vorticity equation describes:
- The evolution of a
- Homogeneous (constant density)
- Non divergent
- Incompressible flow
- In which absolute vorticity is conserved following the motion
Barotropic model is obtained from:
- The barotropic vorticity equation
Finite difference analogs for the Jacobian do not conserve:
- Do not conserve:
- Kinetic energy
- Mean square vorticity over the model domain
Not suitable for use in model
Arakawa:
- Show that the linear combination of FD analogs for the Jacobian conserve both
- Kinetic energy
- Mean square vorticity over the model domain
What time derivative can be used in the barotropic model? How?
- The leap-frog time differencing scheme
- Two steps:
- Forward D then
- Central D
Assumptions and simplifications for the shallow water model:
- we have a layer of fluid with constant density
- (under hydrostatic assumptions) there is no vertical variation of the PGF if density is constant
What does the assumptions and simplifications for the shallow water model mean?
If initially the horizontal velocities are independent of height they will remain so during all times
What term doesn’t appear in the shallow water equations?
The vertical advection terms
Shallow water equations contain
- Slow moving Rossby waves
- High-frequency gravity waves
Requirements of handling high frequency gravity waves:
- Shorter time steps
- As demanded by CFL criterion
- Computationally expensive
- As demanded by CFL criterion
How to overcome the demanding requirements of handling high frequency gravity waves?
- Apply implicit time integration scheme
- Permits relatively longer time steps
Note that:
- Nonlinear part of the equation is dealt with explicitly
- Linear part of the equation is dealt with implicitly (known as: semi-implicit time-integration scheme)
Helmholtz equation:
- One equation and one unknown.
- Can be solved using relaxation techniques
