Most physically significant large-scale atmospheric circulations
have time scales on the order of Rossby waves but much larger
than the time scales of gravity waves. The maximum stable time
step determined from the CFL condition for gravity waves is much
less than that to accurately simulate the relevant phenomena.
Process
Speed (m/s)
Stability
Condition
Max ∆t
∆x=20km
Max ∆t
∆x=200km
Sound, air
340
∆x/cs
1 min
10 min
Sound, water
1500
∆x/cs
15 sec
2 min
Gravity wave
√gH
∆x/ √gH
10 min-1 hr
2 min (water)
2 hr-10 hr
15 min(water)
Internal wave NH ~ 3
∆x/NH
2 hr
18 hr
Jets
∆x/U
3 hr
1 dy
Rossby Wave U~0.05-1
∆x/U
6 hr - 5 dy
2dy - 7wk
Interior Flow
∆x/U
2 dy
3 wk
U~2
U~0.1
Ways to Circumvent Time Constraints
• Approximate the full governing equations with
a “filtered” set of equations.
– Primitive Equations
– Euler Equations
– Vorticity Eqns with Geostrophic Approx.
• Use numerical techniques to stabilize the
fast-moving waves. Generally, accuracy of
the fast waves is sacrificed for efficiency.
Implicit Schemes
Discretize: Center the time-derivative and choose level for RHS.
Explicit:
The same equation with 2x the time step
gives the more common form of an
explicit scheme.
Implicit:
Most often use some combination of
the two schemes => Semi-Implicit
Example: Diffusion
Assume that the diffusion coefficient is constant.
Explicit:
von Neumann =>
Stability Constraint =>
Example: Diffusion
Implicit:
This system can be solved by inverting the tridiagonal matrix. Using
the boundary conditions to solve backwards in time! Often fully
implicit schemes are too difficult to invert efficiently.
von Neumann Stability =>
=> Unconditionally stable A1 for all ∆t>0.
=> Only first order accurate.
Example: Diffusion
Semi-Implicit Scheme: A combination of the implicit and explicit.
Crank-Nicholson Scheme
von Neumann Stability =>
=> Unconditionally Stable AND accurate.
Notes on Using Implicit Schemes
• Higher order implicit schemes are not
necessarily more stable than the related
explicit methods.
– Example: the 3rd and 4th order Adams-Moulton
schemes (Backward and Trapezoid are the 1st
and 2nd order of this family) generally amplify
solutions for any choice of time step.
• Generally, only use implicit schemes for those
terms that are crucial to fast waves. Use
explicit methods on all other terms.
Example: Shallow Water
Equations and Gravity Waves
Split into Baroclinic and
Barotropic modes.
Rossby wave = Low Frequency
Gravity wave = High Frequency
Mixed Schemes
Trapezoid method. Stable when the mean flow, U, < gravity wave speed, the
fluid depth, H, > the wave perturbations and the CFL condition for the Rossby
wave is satisfied. These conditions are not always satisfied in the polar
regions of the Earth’s atmosphere.
Here, the averaging can occur in a couple different ways:
The trapezoid method as above.
The longer time step can be subdivided into M smaller steps ∆t/M, iterated
only for the gravity wave term and then averaged over the longer time step.
Note: the gravity wave term is slowed down for this calculation.
LeVeque: Finite Difference Methods for Ordinary
and Partial Differential Equations; SIAM, 2007.