n.b. the material derivative, rate of change following the motion:
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xu
tDt
DtDt
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.u
N.B. the ‘material derivative’, rate of change following the motion:
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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:
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“Primitive Equations”
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Plus other equations
• Continuity equation
• Ideal Gas Law
• First Law of Thermodynamics
These equations can be shown to conserve energy, angular momentum, and mass.
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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
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Re-arrange to give approximation for derivatives
• First order accurate:
• Second order accurate
• Fourth order accurate
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Linear Advection Equation
Differential equation becomes following difference equation
Second order accurate in both space and time
Centered time and space scheme
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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)
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Examples of Numerical Schemes
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More solutions
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Staggered Grids
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Grids on Sphere
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Vertical Grid/Coordinates
Hybrid coordinates
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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…..