1

A Framework for The Analysis of Physics-Dynamics Coupling Strategies

Andrew Staniforth, Nigel Wood

(Met Office Dynamics Research Group)

and Jean Côté

(Meteorological Service of Canada)

2

Outline

What do we mean by physics? Extending the framework of Caya et al

(1998) Some coupling strategies Analysis of the coupling strategies Summary

3

What do we mean by physics?

212 ( ) mol

Dp

Dt

uΩ u Ω Ω r u

2, ,

,mol i j i j

i j

DK R L C

Dt

2

,i

mol i i jj

DqK q C

Dt

. 0D

Dt

u

Molecular effectsRadiationMicrophysics (conversion terms, latent heating)

Underpinning fluid dynamical equations:

Physics = Boundary Conditions+RHS:

4

Filtered equation set1

2 ( ) . ' 'D

PDt

UΩ U Ω Ω r u u

, ,,

. ' ' i j i ji j

DR L C

Dt

u

,. ' 'ii j

j

DQq C

Dt u

. 0D

Dt

U

Added sub-filter physics (neglect molecular effects):

Modified Boundary ConditionsTurbulenceConvectionGravity-wave stress

5

Physics Physics split into two parts:

– True physics (radiation, microphysics)

(albeit filtered)

– Sub-filter physics = function of filter scale (NOT a function of grid scale though often assumed to be! Convergence issues Mason and Callen ‘86)

» CRMs, LES etc = turbulence

» NWP/GCMs = turbulence + convection + GWD

» Mesoscale = turbulence + ?

Boundary conditions also!

6

What do we mean by physics-dynamics coupling?

For explicit models with small t no issue - all terms

(physics/dynamics) handled in the same way (ie most CRMs, LES etc) and even if not then at converged limit

Real issue only when t large compared to time scale of processes, then have to decide how to discretize terms - but 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 + viscid physics corrector (boundary conditions corrupted)

7

Holy grail of couplingLarge scale modelling (t large): SISL schemes allow increased t and therefore a

balancing of the spatial and temporal errors whilst retaining stability and accuracy (for dynamics at least)

If physics is not handled properly then the coupling will introduce O(t ) errors and the advantage of SISL will be negated

Aim: Couple with O(t2) accuracy + stability

8

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)

( )( )

dF tF t G

dt

9

CLZ98’s model represents either a damping term (if real and

> 0) or oscillatory term (dynamics) if imaginary

G = const. forcing (diabatic forcing in CLZ98) Model useful (CLZ98 diagnosed problem in

their model) but:– neglects advection (and therefore cannot analyse

eg SL orographic resonance)

– neglects spatio-temporal forcing terms

10

Extending CLZ98’s model These effects can be easily included whilst

retaining the analytical tractability of CLZ98:

Since Eq is linear can Fourier decompose

( , )j j jj j j

Di i R x t

Dt

F

F F F

DU

Dt t x

( )ki kx tj j j k

j j j

DFi F F i F R e

Dt

where

is the kth Fourier mode ofF FAssumed only one frequency of oscillation

11

Exact Regular Solution Consider only 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

12

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:

13

Application to Coupling Discretizations

Assume a semi-Lagrangian advection scheme Apply a 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

14

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

15

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

16

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 correctorThird = Implicit Physics only corrector

** *

** *1 0dd

Fi F

t

FF

17

Analysis

Each of the 4 schemes has been analysed in terms of its: – Stability

– Accuracy

– Steady state forced response

– Occurrence of spurious resonance

18

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

19

Accuracy

2 2

1 ...2

iexact t i tE i te

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

20

Forced Regular Response

As for the free mode, the forced response can be determined by seeking solutions of the form:

Accuracy of the forced response can again be determined by comparison with the exact analytical result.

kkx tiforcedtkF eF

21

Steady State Response of the Forced Solution

A key aspect of a parametrization scheme (and often the only fully understood aspect) is its steady state response when k=0 and >0.

Accuracy of the steady-state forced response can again be determined by comparison with the exact analytical result:

kikx

steady

i kU

R eF

22

Forced Resonant Solution

Resonance occurs when the denominator of the Forced Response vanishes

0kkU

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

23

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)

24

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)

25

Summary Numerics of Physics-Dynamics coupling key to

continued improvement of numerical accuracy of models

Caya et al (1998) has been 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