PR

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Hamiltonian tools applied to non-hydrostatic
modeling
Almut Gassmann Max Planck Institute for
Meteorology Hamburg, Germany
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Motivation...
Max Planck Institute for Meteorology and German
Weather Service (DWD) are developing a new
global model system in a joint project
ICON Members and collaborators present at IPAM
Marco Giorgetta, Peter Korn, Luis Kornblueh,
Leonidas Linardakis, Stephan Lorenz, Almut
Gassmann, Werner Bauer, Florian Rauser, Hui Wan,
Peter Dueben, Tobias Hundertmark, Luca
Bonaventura My part non-hydrostatic atmospheric
model
Contents...

1. Hamiltonian form for the continuous moist
   turbulent equations
2. Spatial discretisation of Poisson brackets
3. Temporal discretisation

Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Governing equations for NWP and climate
simulations
Is there really a need to reconsider that? No!
But As soon as moisture and turbulence
averaging come into play, things become
ugly ? Approximations to thermodynamics may
distort local mass and/or energy
consistency. Unfortunately, we do not know the
longer term impact of such small, but systematic
errors.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Starting from general equation set...
Momentum equation
Conservation of total (moist) mass
First law of thermodynamics
Conservation of tracer mass q specific quantities
Approximations that do not change mass or energy
balance - neglect the molecular heat flux
against the turbulent one W ? R - neglect the
molecular dissipation against the turbulent
one Make sure that the diffusion fluxes of the
constituents and the conversion terms sum up to
0.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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The energy budget must be closed ...
shear production buoyancy production
dissipation
rudimentary mean turbulent kinetic energy equation
This suggests adding
in the heat equation
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Specifics for a moist atmosphere...
The internal energy u is not a suitable
variable. We have to unveil the phase changes of
water or chemical reactions, and thus consider
rather the enthalpy h instead of the internal
energy u.
The ideal equation of state is assumed to be
valid also for averaged quantities
virtual increment
Finally, a prognostic temperature equation is
obtained.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Towards flux form equations...
viewpoints of fluid dynamics

• Particle view (Lagrange)

• Field view (Euler)
• Flux form for scalars
• specific moisture quantities
• some form of entropy variable
• Because the actual entropy s including all
  moisture quantities is impractical to handle, we
  decide for a compromise the virtual potential
  temperature.
• We write the wind advection in Lamb form to
  unveil the vorticity (reason particle
  relabelling symmetry).

The views are equally valid and suitable
for building a numerical model. In our ICON
project, we decide for the Eulerian standpoint.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Equation set that unveils the entropy
production...
entropy production
virtual potential temparture
Next step Poisson bracket form for the
non-dissipative adiabatic limit case...
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Hamiltonian dynamics ...
A suitable Hamiltonian functional at least covers
the adiabatic part of the dynamics
The Hamiltonian is a function of the density, a
suitable thermodynamic variable and the velocity.
With the choice of the virtual potential
temperature density we obtain ''dynamic''
and ''thermodynamic'' functional derivatives
independently.
Hamiltonian dynamics (at least for the adiabatic
part)
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Poisson bracket...
scalar triple product is antisymmetric A.(BxC)
-B.(AxC) -C.(BxA)
Antisymmetry swapping F and H only alters the
sign, FH gives a zero bracket result Note
only the divergence operator appears, not
the gradient operator, duality of the div and
grad operators is automatically given Background
integration by parts rule
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Summary so far...

• Exact Hamiltonian form with Poission brackets
  seems to be only practicable to write down for
  idealized (dry, non-turbuent) flows.
• This is no contradiction to the conservation of
  the total energy, because friction contributes
  correctly also to the internal enery by
  dissipative heating, and phase changes do not
  change the total energy.
• The structure of the Poisson bracket guarantees
  for correct energy conversions and thus energy
  conservation comes as a by product.
• Mass conservation is automatically given.
• The virtual potential temperature enters the
  equations as a passive tracer as expected. In
  the dry adiabatic non-dissipative limit case, the
  entropy is conserved.
• Prognostic variables might be chosen freely.
  Nice prognostic variables are the density and
  virtual potential temperture density (also the
  Exner pressure).

Next step Discretize brackets instead of single
terms in the equations...
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Numerics with brackets...
R. Salmon (2005, etc.) ...From the standpoint
of differential equations, conservation laws
arise from manipulations that typically include
the product rule for derivatives. Unfortunately,
the product rule does not generally carry over to
discrete systems try as we might, we will never
get digital computers to respect it. However, in
the strategy adopted here, conservation laws are
converted to antisymmetry properties that
transfer easily to the discrete case digital
computers understand antisymmetry as well!
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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C-grid discretisations...
Poisson brackets convert easily the Arakawa C
grid (also in the vertical ? Lorenz grid)
Requirements - divergence via Gauss theorem -
Laplacian-consistent inner product
Note ?v is not touched by the bracket
philosophy, it is only required 'somehow' at the
interface position.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Role of the potential temperature as a 'tracer'...

• Higher order advection scheme is interpreted as
  to give an interface value for theta.
• Well balancing approach (for terrain
  following-coordinates) might be interpreted as to
  give a special nearly hydrostatic state via the
  estimation of theta at the vertical interface.
• A combination of both requirements is also
  possible -gt next slide.

Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Well balancing...

• Atmospheric motions are nearly hydrostatic. The
  local truncation error of the pressure gradient
  term might violate the nearly hydrostatic state.
  Workaround locally well balanced reconstruction
  with the help of a local hydrostatic background
  state (Botta et al., 2002)

contrib. to covariant horizontal equation
interface value in flux divergence term
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Vorticity flux term...
absolute potential vorticity
gives the mass flux

• Same type of vector reconstruction.
• In case of an (irregular) hexagonal grid, further
  consistency requirements are required, which
  determine the stencil and method for the vector
  reconstruction.
• The PV takes the same role as theta in the
  previous considerations it is the tracer
  quantity in the vorticity equation and might be
  subject to further conditions (anticipated
  vorticity flux method etc.).

Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Time integration scheme...
State of the art nonhydrostatic models

• fully implicit (expensive and global)

• vertically implict
• horizontally explicit (forward-backward wave
  solver in combination with a Runge-Kutta type
  scheme for advection split-explicit)

How does the time integration scheme look like,
if we have discretized Poisson brackets?
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Energy conserving time integration scheme...
Shallow water example
Energy budget equation
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Goal Explicit time stepping...
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Predictor step philosophy...

• similar to RK2 procedure
• proposed prediction step

Linear implicit method behaves similarly to RK2
split explicit scheme and is unstable.
Linear implicit method for the predictor
step behaves similarly to the proposed explicit
scheme and is stable. Linear stability analysis
reveals an unstable behaviour, which is not found
in numerical experiments, presumably because the
whole scheme is nonlinear.
The predictor step plays the role of divergence
damping in traditional split-explicit methods.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Pressure gradient term...
pressure gradient term 'unfortunately implicit'
Trick to make it explicit
relabeling of time levels gives explicit scheme
Remark This approach is known with empirical
weights as acoustic mode filtering (Klemp et al,
2007). We can shed different light on this
procedure. Our new weigths are physically based.
Contents Hamiltonian form Spatial
discretisation Temporal discretisation
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Summary on the temporal discretisation...

• Product rule for derivatives in the context of
  time integration.
• Strict splitting between 'wave solver' and
  'advection' becomes questionable.
• New light is shed on split-explicit schemes
  Alternative explanations for
• divergence damping
• acoustic mode filtering
• The vorticity flux term is still an outsider
  here. Because it should be energetically neutral,
  the mass flux therein must be consistent with the
  continuity equation also in the time level
  choice. The time level of the PV itself is not
  constrained.

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Baroclinic wave test case Non-hydrostatic ICON
model on the hexagonal grid (dx
240km) including some of the numerical issues
disussed in the talk. The run is without
additional diffusion.