Approximations to full equations as long as use in a GCM Primitive Equations Plus other equations How to solve equations Linear Advection Equation
Mannis, David, Co-Publisher/President has reference to this Academic Journal, PHwiki organized this Journal N.B. the material derivative, rate of change following the motion:
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Approximations to full equations as long as use in a GCM r = a + z (a=radius of Earth) z a Coriolis in addition to metric terms proportional to can be ignored 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, in addition to mass.
How to solve equations Few analytical solutions to full Navier-Stokes equations, in addition to only as long as 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) x, then Re-arrange to give approximation as long as 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 in addition to time Centered time in addition to space scheme
Numerical Stability in addition to numerical solutions Schemes may be accurate but unstable: e.g. simple centred difference scheme as long as 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 in addition to /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 ( in addition to ocean) governed by straight as long as ward physics Discretisation has problems but generally can be understood in addition to quantified. NO tuneable parameters so far NO need as long as knowledge of past in addition to only need present to initialise models. BUT
Mannis, David Co-Publisher/President
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