Download talk – document - PDC – Physics Dynamics Coupling

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
Outline
 Physics-Dynamics & their coupling
 Extending the framework of Caya et al
(1998)
 Some coupling strategies
 Analysis of the coupling strategies
 Summary
© Crown copyright Met Office
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
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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)
© Crown copyright Met Office
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
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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:
dF (t )
  F (t )  G
dt
 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
© Crown copyright Met Office
CLZ98’s model
dF (t )
  F (t )  G
dt
•  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
© Crown copyright Met Office
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:
DF
 i F    F  Rk ei ( kxk t )
Dt
 Solution = sum of free and forced solution:
F  F free  x, t   F forced  x, t 
 Fk
© Crown copyright Met Office
free
e
 kU i 
i  kx 

t 

i kx k t 
Rk e

  i   kU   k 
Exact Resonant Solution
 Resonance occurs when denominator of forced
solution vanishes, when:
  i   kU  k   0
which, as all terms are real, reduces to:
    kU  k  0
 Solution = sum of free and resonant forced solution:
F  F free  x, t   F forced  x, t 

 Fk
© Crown copyright Met Office
free

i  kx   kU t 
 Rk t e 

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
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Fully Explicit/Implicit
F t t  Fdt
t
 i  F t t  1    Fdt 
     F t t  1    Fdt 

 Rk  e
i  kx k  t t  
 1   e 
i kxd k t 
 Time-weights:  dynamics,  physics,  forcing
 =0 Explicit physics - simple but stability limited
 =1 Implicit physics - stable but expensive
© Crown copyright Met Office

Split-Implicit
F *  Fdt
t
 i  F *  1    Fdt   0
F t t  F *
t

i  kx k  t t  
   F t t  Rk  e 
 1   e 
Two step predictor corrector approach:
 First = Dynamics only predictor (advection + GW)
 Second = Physics only corrector
© Crown copyright Met Office
i kxd k t 

Symmetrized Split-Implicit
F*  Ft
  1     F t  1   Rk e 
i kx k t 
t
F **  Fd*
 i  F **  1    Fd*   0
t
F t t  F **
t
i  kx k  t t  
   F t t  Rk e 
Three step predictor-corrector approach:
 First = Explicit Physics only predictor
 Second = Semi-implicit Dynamics only corrector
 Third = Implicit Physics only corrector
© Crown copyright Met Office
Analysis
 Each scheme analysed in terms of its:
Stability
Accuracy
Steady state forced response
Occurrence of spurious resonance
© Crown copyright Met Office
Stability
 Stability can be examined by solving for the free mode
by seeking solutions of the form:
F  Fk
t
free i kxt 
e
and requiring the response function
F t t
i kU t
E  t e
Fd
to have modulus  1
© Crown copyright Met Office
Accuracy
 Accuracy of free mode determined by expanding E in
powers of t and comparing with expansion of
analytical result:
E exact  e
i   t
© Crown copyright Met Office
i   

 1   i    t 
2
2
t 2
 ...
Forced Regular Response
 Forced response determined by seeking solutions of
form:
F  Fk
t
forced i kxk t 
e
 Accuracy of forced response again determined by
comparing with exact analytical result.
© Crown copyright Met Office
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:
F
© Crown copyright Met Office
steady

Rk eikx
  i   kU 
Forced Resonant Solution
 Resonance occurs when the denominator of the
Forced Response vanishes
    kU  k  0
 For semi-Lagrangian, semi-implicit scheme there can
occur spurious resonances in addition to the physical
(analytical) one
© Crown copyright Met Office
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)
© Crown copyright Met Office
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)
© Crown copyright Met Office
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
© Crown copyright Met Office
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 27792799
 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)
© Crown copyright Met Office
Thank you!
Questions?