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

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

Jan 14, 2024 10 likes •243 views

A well-balanced scheme for a 2D finite difference, non-hydrostatic atmospheric model Andreas Dobler 22/09/06 Thanks to: Christoph Schr, Jrg Schmidli, Stefan Wunderlich 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Outline

  • Introduction
  • Governing equations
  • Model features
  • Reduction of local

truncation errors (LTEs)

  • LTEs in atmosphere
  • ver steep topography
  • Analysis of LTEs in

terrain following coordinates

  • Cut-cell approach
  • Conclusions
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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Problems with nearly hydrostatic flows

  • The gravity term and the vertical pressure

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

−z

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Governing equations

  • 2D non-hydrostatic Euler equations in a dry,

non-rotating atmosphere:

  • Gravitational potential: = gz
  • Sum of kinetic and internal energy: e

∂ ∂t[  u  w e]  ∂ ∂ x[ u u

2 p

uw ue p]  ∂ ∂ z[ w uw w

2 p

we p] =−[ ∂ /∂ x ∂/∂ z 0 ]

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

How to reduce the LTEs

  • Subtract a global, constant, hydrostatic

background state

  • Subtract a local, time-dependent, hydrostatic

background state that matches the actual state

  • f the atmosphere in case of hydrostatic balance

exactly -> no LTEs -> well-balanced method

  • Cut-cells

∇ p

∗=− ∗∇ 

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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)

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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
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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Simulation of an atmosphere at rest over steep topography

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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 :

E pz=z

2

6 pzzzO  z

4

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Analysis of LTEs in terrain following coordinates – Horizontal part

  • Invertible coordinate transformation s = s(x,z)
  • Error in horizontal pressure gradient:
  • > For a horizontal uniform pressure distribution

this would be zero in an ideal cut-cell approach

E px= x

2

6

 pxxxs 

s x s

2

6 psssps   sx−s xO  x

4O  s 4

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Analysis of LTEs in terrain following coordinates – Horizontal part, continued

  • Assumptions: (e.g., Gal-Chen),

appropriate discretization of metric terms and horizontal uniform pressure distribution

  • >

E px= x

2

6 pzzz z x

313 pzz z xx

pzzz z x

2 − 2 z s 2

z x

2O  x 4

z ss=0,s= x

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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
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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Cut-cells

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Results for atmosphere at rest

[m s

−1]

[m s

−1]

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Possible reasons for unexpected big LTEs

  • Finite volume implementation details
  • Cut-cell approach
  • Bugs, etc.
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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

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

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Thank you for your attention !

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Linear, non-hydrostatic flow

  • u = 10 m/s
  • N = 0.01 1/s
  • Time: 5000 s
  • dz ~ 300 m
  • dx = 400 m
  • Gaussian hill
  • Height: 1 m
  • Halfwidth: 1000 m
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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Linear, non-hydrostatic flow – exact solution (contour interval = 0.001 m/s)

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Linear, non-hydrostatic flow – simulation results (contour interval = 0.001 m/s)

Well-balanced Full equations

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Linear, non-hydrostatic flow – simulation results (contour interval = 0.001 m/s)

Well-balanced Full equations

Without Rayleigh damping

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Andreas Dobler / Institute for Atmospheric and Climate Science, ETH Zürich / andreas.dobler@env.ethz.ch

Thank you again for your attention !