xu

tDt

DtDt

D

x

.u

N.B. the ‘material derivative’, rate of change following the motion:

Approximations to full equations for use in a GCM

1. r = a + z (a=radius of Earth) z << a

2. Coriolis and metric terms proportional to

can be ignored

3. For large scales, vertical acceleration is small, hence vertical component becomes:

“Primitive Equations”

Plus other equations

• Continuity equation

• Ideal Gas Law

• First Law of Thermodynamics

These equations can be shown to conserve energy, angular momentum, and mass.

How to solve equations?• Few analytical solutions to full Navier-Stokes

equations, and only for fairly idealised problems. Hence need to solve numerically.

• At heart of all numerical schemes is a Taylor series expansion:– Suppose we have an interval L, covered by N equally

spaced grid points, xj=(j-1) Δx, then

Re-arrange to give approximation for derivatives

• First order accurate:

• Second order accurate

• Fourth order accurate

Linear Advection Equation

Differential equation becomes following difference equation

Second order accurate in both space and time

Centered time and space scheme

Numerical Stability and numerical solutions

• Schemes may be accurate but unstable:– e.g. simple centred difference scheme for linear

advection scheme will be stable only if Courant-Friedrichs-Levy number less than 1.

• Many schemes can have artificial (computational mode)

• All schemes distort true solution (e.g. change phase and/or group speed)

• Some schemes fail to conserve properties of system (e.g. energy)

Examples of Numerical Schemes

More solutions

Staggered Grids

Grids on Sphere

Vertical Grid/Coordinates

Hybrid coordinates

Summary so far

• Dynamics of atmosphere (and ocean) governed by straight forward physics

• Discretisation has problems but generally can be understood and quantified.

• NO tuneable parameters so far

• NO need for knowledge of past and only need present to initialise models.

• BUT…..