© Crown copyright Met Office A Framework for The Analysis of Physics-Dynamics Coupling Strategies Andrew Staniforth, Nigel Wood (Met O Dynamics Research)

© Crown copyright Met Office

A Framework for The Analysis of Physics-Dynamics Coupling Strategies

Andrew Staniforth, Nigel Wood (Met O Dynamics Research)and Jean Côté (Met Service of Canada)

© Crown copyright Met Office© Crown copyright Met Office

Outline

Physics-Dynamics & their coupling

Extending the framework of Caya et al (1998)

Some coupling strategies

Analysis of the coupling strategies

Summary

What is dynamics and physics?

Dynamics =

Resolved scale fluid dynamical processes:

Advection/transport, rotation, pressure gradient

Physics =

Non-fluid dynamical processes:

Radiation, microphysics (albeit filtered)

Sub-grid/filter fluid processes:

Turbulence + convection + GWD


What do we mean by physics-dynamics coupling? Small t (how small?) no issue:

All terms handled in the same way (ie most CRMs, LES etc)

Even if not then at converged limit

Large t (cf. time scale of processes) have to decide how to discretize terms

In principle no different to issues of dynamical terms (split is arbitrary - historical?)

BUT many large scale models have completely separated physics from dynamics

inviscid predictor + viscous physics corrector (Note: boundary conditions corrupted)


Aim of coupling

Large scale modelling (t large):

SISL schemes allow increased t and hence balancing of spatial and temporal errors

Whilst retaining stability and accuracy (for dynamics at least)

If physics not handled properly then coupling introduces O(t ) errors & advantage of SISL will be negated

Aim: Couple with O(t2) accuracy + stability


Framework for analysing coupling strategies

Numerical analysis of dynamics well established

Some particular physics aspects well understood (eg diffusion) but largely in isolation

Caya, Laprise and Zwack (1998) simple model of coupling:

Regard as either a simple paradigm or F(t) is amplitude of linear normal mode (Daley 1991)

CLZ98 used this to diagnose problem in their model

( )( )

dF tF t G

dt


CLZ98’s model

represents:

Damping term (if real and > 0)

Oscillatory term (dynamics) if imaginary

• G = constant forcing (diabatic forcing in CLZ98)

• Model useful but:

Neglects advection (& therefore orographic resonance)

Neglects spatio-temporal forcing terms


( )( )

dF tF t G

dt

Extending CLZ98’s model

Add in advection, and allow more than 1 -type process

In particular, consider 1 dynamics oscillatory process, 1 (damping) physics process:

( )ki kx tk

DFi F F R e

Dt

Solution = sum of free and forced solution:

, ,

k

free forced

i kx ti kx kU i tfree k

kk

F F x t F x t

R eF e

i kU


Exact Resonant Solution

Resonance occurs when denominator of forced solution vanishes, when:

, ,free forced

i kx kU tfreek k

F F x t F x t

F R t e

Solution = sum of free and resonant forced solution:

0ki kU

0kkU

which, as all terms are real, reduces to:


Application to Coupling Discretizations

Apply semi-Lagrangian advection scheme

Apply semi-implicit scheme to the dynamical terms (e.g. gravity modes)

Consider 4 different coupling schemes for the physics:

Fully Explicit/Implicit

Split-implicit

Symmetrized split-implicit

Apply analysis to each


Fully Explicit/Implicit

1t t t

t t tdd

Fi F

t

FF

1t t tdFF

1k d ki kx t t i kx tkR e e

Time-weights: dynamics, physics, forcing

=0 Explicit physics - simple but stability limited

=1 Implicit physics - stable but expensive


Split-Implicit

*

* 1 0t

tdd

Fi F

t

FF

*

1k d k

t ti kx t t i kx tt t

k

FR e e

t

FF

Two step predictor corrector approach:

First = Dynamics only predictor (advection + GW)

Second = Physics only corrector


Symmetrized Split-Implicit

*

1 1 k

ti kx tt

k

FR e

t

FF

**

k

t ti kx t tt t

k

FR e

t

FF

Three step predictor-corrector approach:

First = Explicit Physics only predictor

Second = Semi-implicit Dynamics only corrector

Third = Implicit Physics only corrector

** *

** *1 0dd

Fi F

t

FF


Analysis

Each scheme analysed in terms of its:

Stability

Accuracy

Steady state forced response

Occurrence of spurious resonance


Stability

i kx tfreetkF eF

Stability can be examined by solving for the free mode by seeking solutions of the form:

and requiring the response function

to have modulus 1

t tkU

td

i tFF

E e


Accuracy

2 2

1 ...2

iexact t i tE i te

Accuracy of free mode determined by expanding E in powers of t and comparing with expansion of analytical result:


Forced Regular Response

Forced response determined by seeking solutions of form:

Accuracy of forced response again determined by comparing with exact analytical result.

kkx tiforcedtkF eF


Steady State Response of the Forced Solution

Key aspect of parametrization scheme is its steady state response when k=0 and >0

Accuracy of steady-state forced response again determined by comparing with exact analytical result:

kikx

steady

i kU

R eF


Forced Resonant Solution

Resonance occurs when the denominator of the Forced Response vanishes

0kkU

For semi-Lagrangian, semi-implicit scheme there can occur spurious resonances in addition to the physical (analytical) one


Results I

Stability:

Centring or overweighting the Dynamics and Physics ensures the Implicit, Split-Implicit and Symmetrized Split-Implicit schemes are unconditionally stable

Accuracy of response:

All schemes are O(t) accurate

By centring the Dynamics and Physics the Implicit and Symmetrized Split-Implicit schemes alone, are O(t2)


Results II

Steady state response:

Implicit/Explicit give exact response independent of centring

Split-implicit spuriously amplifies/decays steady-state

Symmetrized Split-Implicit exact only if centred

Spurious resonance:

All schemes have same conditions for resonance

Resonance can be avoided by:

• Applying some diffusion ( >0) or

• Overweighting the dynamics (at the expense of removing physical resonance)


Summary

Numerics of Physics-Dynamics coupling key to continued improvement of numerical accuracy of models

Caya et al (1998) extended to include:

Advection (and therefore spurious resonance)

Spatio-temporal forcing

Four (idealised) coupling strategies analysed in terms of:

Stability, Accuracy, Steady-state Forced Response, Spurious Resonance


Application of this analysis

A simple comparison of four physics-dynamics coupling schemes Andrew Staniforth, Nigel Wood and Jean Côté (2002) Mon. Wea. Rev. 130, 3129-3135

Analysis of the numerics of physics-dynamics coupling Andrew Staniforth, Nigel Wood and Jean Côté (2002) Q. J. Roy. Met. Soc. 128 2779-2799

Analysis of parallel vs. sequential splitting for time-stepping physical parameterizations Mark Dubal, Nigel Wood and Andrew Staniforth (2004) Mon. Wea. Rev. 132, 121-132

Mixed parallel-sequential split schemes for time-stepping multiple physical parameterizations Mark Dubal, Nigel Wood and Andrew Staniforth (2005) Mon. Wea. Rev. 133, 989-1002

Some numerical properties of approaches to physics-dynamics coupling for NWP Mark Dubal, Nigel Wood and Andrew Staniforth (2006) Q. J. Roy. Met. Soc. 132, 27-42 (Detailed comparison of Met Office scheme with those of NCAR CCM3, ECMWF and HIRLAM)


Thank you!Questions?

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© Crown copyright Met Office A Framework for The Analysis of Physics-Dynamics Coupling Strategies Andrew Staniforth, Nigel Wood (Met O Dynamics Research)