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 U~2 ∆x/U 3 hr 1 dy

Rossby Wave

U~0.05-1 ∆x/U 6 hr - 5 dy 2dy - 7wk

Interior Flow U~0.1 ∆x/U 2 dy 3 wk

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: DiffusionImplicit:

von Neumann Stability =>

=> Unconditionally stable A1 for all ∆t>0.

=> Only first order accurate.

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.

Example: Diffusion

Semi-Implicit Scheme: A combination of the implicit and explicit.

von Neumann Stability =>

Crank-Nicholson Scheme

=> Unconditionally Stable AND accurate.

• 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.

Notes on Using Implicit Schemes

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 Ordinaryand Partial Differential Equations; SIAM, 2007.