lec11 (2) - R. G. Prinn, 12.806/10.571 Atmospheric Physics...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
R. G. Prinn, 12.806/10.571 Atmospheric Physics & Chemistry, March 14, 2006 Solving the Basic Equations for the Atmosphere in 3-D ∂ρ d ( u ρ ) d ( v ) d ( w ) Mass = − Continuity t dx dy dz u d ( uu ) d ( vu ) d ( wu ) = − + ... + pressure gradient t dx dy dz Equations of Motion v d ( uv ) d ( vv ) d ( wv ) +Coriolis = − + ... force (momentum t dx dy dz continuity) +gravity w d ( uw ) d ( vw ) d ( ww ) = − + ... t dx dy dz +friction Thermodynamic T d ( uT ) d ( vT ) d ( wT ) 1 D (1/ ) Equation = − + J p v (energy continuity) t dx dy dz c Dt J : radiation, conduction, latent heat release, etc D(1/ρ) / Dt : conversion between thermal and mechanical energy in fluid system Chemical ∂χ d ( u χ ) d ( v ) d ( w ) + Chemical Production Continuity = − –Chemical Loss Equation t dx dy dz
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
3. Spatial representations a. Finite difference schemes (truncated Taylor expansion at J grid-points) b. Spectral techniques (express variables using truncated series of N orthogonal harmonic functions and solve for N coefficients of expansion;) see c. Interpolation schemes (interpolates between grid points e.g. using a polynomial) d. Finite element schemes (minimizes error between actual and approximate solutions using a “basis function”, good for irregular geometries, c.f. (b) above which is good for regular geometries) 4. Explicit and Implicit time stepping Explicit: () x t t = f ...., x * t ,... Implicit: x t t = f ..., x t , x t t ,.... (Implicit methods more stable (but often less accurate) than explicit methods for longer time steps)
Background image of page 2
Time stepping and stability Time stepping and stability In the numerical model, time is treated in discrete units and the time intervals chosen depend on the size of the model grid boxes. Intuitively, don’t want to transport across more than a grid cell over a time step. General Rule for stability: the CFL condition ut 1 x e.g. Typically in the atmosphere, max u = 100m/s & grid spacing = 200 km, so constraint is Δ t < 2000 seconds (33min) x u t
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
5. Example Finite Difference Schemes (A) Advection (a) Forward / Upstream (explicit, conditionally* stable) (time) (space) n1 n j () j + (first order accurate) t Δ t n n ⎫⎫ u n j j 1 ( u j > 0 ) j Δ x ⎪⎩ u x n n (first order) n j1 + j u j ( u j 0 ) Δ x ⎭⎭ ut Δ *For stability need Courant No. 1 Δ x (b) Centered / Centered or Leap-frog (explicit, neutrally** stable) n j j + 1 (second order) t 2 Δ t n n j1 u u n j
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 10

lec11 (2) - R. G. Prinn, 12.806/10.571 Atmospheric Physics...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online