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Hamiltonian tools applied to non-hydrostatic modeling

Almut Gassmann

Max Planck Institute for Meteorology

Hamburg, Germany

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...

Hamiltonian form for the continuous moist turbulent equations

Spatial discretisation of Poisson brackets

Temporal discretisation

Contents • Hamiltonian form • Spatial discretisation • Temporal discretisation

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

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

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

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

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

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

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

Poisson bracket...

scalar triple product is

antisymmetric: A.(BxC) = -B.(AxC) = -C.(BxA)

Antisymmetry: swapping F and H only alters the sign, F=H 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

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

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

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: θv is not touched by the bracket philosophy, it is only

required 'somehow' at the interface position.

Contents • Hamiltonian form • Spatial discretisation • Temporal discretisation

prognostic equations

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 specialnearly hydrostatic state via the estimation of theta at the vertical interface.
- A combination of both requirements is also possible -> next slide.

Contents • Hamiltonian form • Spatial discretisation • Temporal discretisation

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

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

The nonlinear advection of momentum is split into two parts.

vorticity flux term

gradient of the kinetic energy

belongs to the 'divergent' part of the flow

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

Energy conserving time integration scheme...

Shallow water example

Energy budget equation

Contents • Hamiltonian form • Spatial discretisation • Temporal discretisation

Forward-backward time integration scheme for waves.

But kinetic energy term is implicit!

Goal: Explicit time stepping...

Contents • Hamiltonian form • Spatial discretisation • Temporal discretisation

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

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

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.

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.

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