A well-balanced scheme for a 2D finite difference, non-hydrostatic - PowerPoint PPT Presentation

A well-balanced scheme for a 2D finite difference, non-hydrostatic atmospheric model Andreas Dobler 22/09/06 Thanks to: Christoph Schär, Jürg Schmidli, Stefan Wunderlich 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 1 / 18

Outline  Introduction  LTEs in atmosphere over steep topography  Governing equations  Analysis of LTEs in  Model features terrain following  Reduction of local coordinates truncation errors  Cut-cell approach (LTEs)  Conclusions 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 2 / 18

Problems with nearly hydrostatic flows  The gravity term and the vertical pressure − z gradient are almost balanced but relatively big  Local truncation errors may introduce large non- physical vertical accelerations  With terrain following coordinates, metric terms appear in the horizontal derivatives, introducing local truncation errors there too 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 3 / 18

Governing equations  2D non-hydrostatic Euler equations in a dry, non-rotating atmosphere: ∂ x [ u  e  p  ] ∂ z [ w  e  p  ] ∂ t [  e  ] =− [ 0 ]  u  w  0 2  p  uw ∂ /∂ x  u ∂  ∂  u  ∂ 2  p ∂/∂ z  w  uw  w  Gravitational potential: = gz   Sum of kinetic and internal energy: e 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 4 / 18

Model features  Centred finite differences, non-staggered grid  Time discretization: Leapfrog or Runge-Kutta  Divergence filter  Numerical diffusion / computational mixing  Rayleigh damping at top boundary  Co-existing finite volume version with cut-cells approach 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 5 / 18

How to reduce the LTEs  Subtract a global, constant, hydrostatic background state ∗ =− ∗ ∇  ∇ p  Subtract a local, time-dependent, hydrostatic background state that matches the actual state of the atmosphere in case of hydrostatic balance exactly -> no LTEs -> well-balanced method  Cut-cells 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 6 / 18

Local, time-dependent background state for every grid cell  Must fulfil the hydrostatic relation  Must interpolate state variables at cell centre  Must fulfil the equation of state for ideal gases -> ODE. Solvable with assumption on potential temperature (e.g., piecewise constant or linear) 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 7 / 18

Simulation of an atmosphere at rest over steep topography  Gaussian hill, height: 1500 m, halfwidth: 5000 m  Zero initial wind speed  Constantly stratified atmosphere, initial bottom potential temperature 288 K, initial Brunt Väisälä frequency 0.01 s -1  6 hours simulation, dt = 0.3 s  dx = 1000 m, dz ~ 300 m 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 8 / 18

Simulation of an atmosphere at rest over steep topography 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 9 / 18

Analysis of LTEs in terrain following coordinates – Vertical part  Density is given exactly at grid points  Error in vertical pressure gradient using centred finite differences : 2 E p z = z p zzz  O   z 4  6 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 10 / 18

Analysis of LTEs in terrain following coordinates – Horizontal part  Invertible coordinate transformation s = s(x,z)  Error in horizontal pressure gradient: 2 2 E p x = x  s s x − s x   O   x 4   O   s 4   p xxx  s   p sss  p s   s x 6 6 -> For a horizontal uniform pressure distribution this would be zero in an ideal cut-cell approach 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 11 / 18

Analysis of LTEs in terrain following coordinates – Horizontal part, continued  Assumptions: (e.g., Gal-Chen), z ss = 0,  s = x appropriate discretization of metric terms and horizontal uniform pressure distribution 3  1  3 p zz z xx 2   O   x 2 2 E p x = x 2 z s 4  -> 2 − p zzz z x 6 p zzz z x z x 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 12 / 18

Finite Volume cut-cell approach  Simple cut-cell approach implemented, but only in (old) finite volume version  Topography boundary behaves like a symmetry line -> Momenta are reflected at boundary  Cut-cells are treated as whole cells -> no stability problems due to small cell size  Method is not strictly conservative 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 13 / 18

Cut-cells 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 14 / 18

Results for atmosphere at rest − 1 ] − 1 ] [ m s [ m s 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 15 / 18

Possible reasons for unexpected big LTEs  Finite volume implementation details  Cut-cell approach  Bugs, etc. 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 16 / 18

Conclusions  The well-balanced method reduces the LTEs associated with the pressure gradient force  An ideal cut-cell approach eliminates the LTEs associated with metric terms in the horizontal pressure gradient force  Therefore, an easy implementable cut-cell approach for testing purposes in our FD well- balanced model would be desirable 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 17 / 18

Thank you for your attention ! 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 18 / 18

Linear, non-hydrostatic flow  u = 10 m/s  Gaussian hill  N = 0.01 1/s  Height: 1 m  Time: 5000 s  Halfwidth: 1000 m  dz ~ 300 m  dx = 400 m 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 19 / 18

Linear, non-hydrostatic flow – exact solution (contour interval = 0.001 m/s) 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 20 / 18

Linear, non-hydrostatic flow – simulation results (contour interval = 0.001 m/s) Well-balanced Full equations 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 21 / 18

Linear, non-hydrostatic flow – simulation results (contour interval = 0.001 m/s) Without Rayleigh damping Well-balanced Full equations 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 22 / 18

Thank you again for your attention ! 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch 23 / 18