Numerics of Parameterization

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Numerics of Parameterization Powered By Docstoc
					Numerical methods IV (time stepping)
by Nils Wedi (room 007; ext. 2657)

In part based on previous material by Mariano Hortal and Agathe Untch
Governing Equations 3 Slide 1

ECMWF

What is the basis for a stable numerical implementation ?
 A: Removal of fast - supposedly insignificant - external and/or internal acoustic modes (relaxed or eliminated), making use of infinite sound speed (cs) and/or the hydrostatic approximation from the governing equations BEFORE numerics is introduced.  B: Use of the full equations WITH a semi-implicit numerical framework, reducing the propagation speed (cs 0) of fast acoustic and buoyancy disturbances, retaining the slow convective-advective component (ideally) undistorted.

 C: Split-explicit integration of the full equations, since explicit NOT practical (~100 times slower)
 Determines the choice of the numerical scheme

Governing Equations 3

Slide 2

ECMWF

Choices for numerical implementation
 Avoiding the solution of an elliptic equation fractional step methods; Skamrock and Klemp (1992); Durran (1999)

 Solving an elliptic equation
Projection method; Durran (1999) Semi-implicit; Durran (1999); Cullen et. al.(1994); Benard et al. (2004); Benard (2004)

Preconditioned conjugate-residual solvers or multigrid methods for solving the resulting Poisson or Helmholtz equations; Skamarock et. al. (1997)

Governing Equations 3

Slide 3

ECMWF

Split-explicit integration
Skamarock and Klemp (1992); Durran (1999); Doms and Schättler (1999);
„Slow‟ part of solution

„Fast‟ part of solution

e.g. implemented in popular limited-area models: Deutschland Modell, WRF model
Governing Equations 3 Slide 4

ECMWF

Semi-implicit schemes
linearised term, treated implicit

non-linear term, treated explicit (i) coefficients constant in time and horizontally (hydrostatic models Robert et al. (1972), Benard et al. (2004), Benard (2004) ECMWF/Arpege/Aladin NH) (ii) coefficients constant in time Thomas (1998); Qian, (1998); see references in Bénard (2004) (iii) non-constant coefficients Skamarock et. al. (1997), (UK Met Office NH model, EULAG model)

Governing Equations 3

Slide 5

ECMWF

Design of semi-implicit methods
 Treat all terms involving the fastest propagation speeds implicitly (acoustic waves, gravity waves).  Assume that the energy in those components is negligible.  Consider the solvability of the resulting implicit system, which is typically an elliptic equation.

Governing Equations 3

Slide 6

ECMWF

Example: Shallow water equations
u h  u  U0 g 0  t  x x   h  U h  H u  0 0  t x x 

Linearized:

Linear analytic solution:

u( x, t )  u0eit eikx
  U 0  gH k
H denotes here a mean state depth.

Phase speed:

c

Governing Equations 3

Slide 7

ECMWF

Shallow water equations
 u u u  u v  fv  0  x y x  t  v v v   u v  fu  0  x y y  t    u v     u v  (   )   0 x y  t  x y  

: advection
: gravity-wave (or sometimes called „adjustment‟) term

In the linear version:

t  2

s
2 U0

 V02

t 

s 2 

t 

s 2 U 02  V02  

advection In the atmosphere

adjustment

combined
2   U 0  V02

gH    300 m / s

5 in synoptic-scale models s  10 m

==> Δt≤ 236 sec ~ 4 min
ECMWF

Governing Equations 3

Slide 8

Explicit time-stepping

• Leap-frog explicit scheme
  t  n  n 1 n 1 n n u j  u j  tV j u j  s  x j    t  n  n 1 n 1 n n v j  v j  tV j v j   y j s    t    n 1 n 1 n n  j   j  tV j  j     V jn  s 

  gH x   y   s   V j  (u j , v j )    ( x ,  y )  x A j  A j x  A j x  y A j  A j y  Aj y
x j+Δy x x x j-Δx x x

Stability:

t 

s 2 

x j x j-Δy

x j+Δx

If we treat implicitly the advection terms we do not get a Helmholtz equation
Governing Equations 3 Slide 9

ECMWF

Increasing the allowed timestep

• Forward-backward scheme
 u   t  x  0       u  0  t x 
von Neumann gives

 t n  n1  j  n  (u j 1  u n1 ) j j   2x  u n1  u n  t ( n1   n1 ) j j 1 j 1  j 2x 
t x  sin(kx )  2

forward

backward

doubles the leapfrog timestep

• Pressure averaging
u n1  u n1 j j 2t  1 1 1  1 {(1  2)(n1  n1 )  [(n1  n1 )  (n1  n1 )]} j j j j j 1 j 2x
t x
doubles the leapfrog  sin(kx )  2 timestep

if ε=0 ------> leapfrog
if ε=1/4 we get
Governing Equations 3 Slide 10

ECMWF

Split-explicit time-stepping
  s n 1 u  u  t V n u n j s j j  j    s n 1 n n v j  v j  ts V j v j    s n 1 n n  j   j  ts V j  j 
t f  n  n 1 s  x j u j  u j  s   n 1 s t f  n  y j v j  v j  s  t f    n 1 s j j    V jn  s 

t s  M t f

slow
ts  s 2 U V
2 0 2 0

fast
Stability as before but M times a simpler problem.

t f 

s 2 

Potential drawbacks: splitting errors, conservation. However recent advances for NH NWP suggested in (Klemp et. al. 2007)
Note: The fast solution may be computed implicitly.
Governing Equations 3 Slide 11

ECMWF

Semi-implicit time-stepping
  t  n 1 n 1  n 1 n 1 n n u j  u j  tV j u j  2s  x ( j   j )    t   n 1 n 1 n n v j  v j  tV j v j   y ( n 1   n 1 )  j j 2s      t  1   n 1 n 1 n n n n 1  j   j  tV j  j   2s   (V j  V j ) 
x  y  s   V j  (u j , v j )    ( x ,  y )  x Aj  Aj x  Aj x  y Aj  Aj y  Aj y
x j+Δy x x

Solve:
 2 n 1  j 4( s )  n 1  F n , n 1 j  ( t ) 2
2

j-Δx x x

x j x j-Δy

x j+Δx
x

Helmholtz equation !

Stability: now only limited by the advection terms
Note: if we also treat the advection terms implicitly we do not get a Helmholtz equation!
Governing Equations 3 Slide 12

ECMWF

Compressible Euler equations

Davies et al. (2003)

Governing Equations 3

Slide 13

ECMWF

Compressible Euler equations

Governing Equations 3

Slide 14

ECMWF

A semi-Lagrangian semi-implicit solution procedure

(not as implemented, Davies et al. (2005) for details)

Davies et al. (1998,2005)
Governing Equations 3 Slide 15

ECMWF

A semi-Lagrangian semi-implicit solution procedure

Governing Equations 3

Slide 16

ECMWF

A semi-Lagrangian semi-implicit solution procedure
Non-constantcoefficient approach!

Helmholtz equation
(solutions see e.g. Skamarock et al. 1997, Smolarkiewicz et al. 2000)

Governing Equations 3

Slide 17

ECMWF

Semi-implicit time integration in IFS
Choice of which terms in RHS to treat implicitly is guided by the knowledge of which waves cause instability because they are too fast (violate the CFL condition) and need to be slowed down with an implicit treatment. In a hydrostatic model, fastest waves are horizontally propagating external gravity waves (long surface gravity waves), Lamb waves (acoustic wave not filtered out by the hydrostatic approximation) and long internal gravity waves. => implicit treatment of the adjustment terms. L= linearization of part of RHS (i.e. terms supporting the fast modes) => good chance of obtaining a system of equations in the variables at “+” that can be solved almost analytically in a spectral model.
Governing Equations 3 Slide 18

ECMWF

Two-time-level semi-Lagrangian semiimplicit time integration in the hydrostatic IFS DX
Dt  RHSx
X   0.5tL  X 0  0.5tL0  tRHS x1/2  tL1/2  X *

For compact notation define:  tt L  0.5 ( L  L0 )   L1/2 “semi-implicit correction term”

DX  RHS 1/2  tt L => x Dt

Notations: X : advected variable RHS: right-hand side of the equation L: part of RHS treated implicitly Superscripts: “0” indicates value at dep. point (t) “1/2” indicates value at mid-point (t+0.5Δt) “+” indicates value at arrival point (t+Δt)

L=RHS => implicit scheme L= part of RHS => semi-implicit (β=1) L=0 => explicit (β=0)
Governing Equations 3 Slide 19

ECMWF

Semi-implicit time integration in IFS
   Dvh  f k  v h     Rd Tv ln p  Pv  K v Dt  Tv DT   PT  K T Dt 1  (  1)q  p
1

 1  p (ln ps )      ( v h )d t ps 0 
 Dvh  RHSv   tt γ  T  Rd Tr  ln ps Dt DT  RHST   tt  D  Dt  (ln ps )  RHS p   tt  D  t
Governing Equations 3 Slide 20



semi-implicit corrections



semi-implicit equations
ECMWF

 Dvh  RHSv   tt γ  T  Rd Tr  ln ps Dt DT  RHST   tt  D  Dt  (ln ps )  RHS p   tt  D  t

Semi-implicit time integration in IFS





semi-implicit equations
Reference state for linearization: Tr ref. temperature

  Where:
γX



Rd X dpr   d  pr d  1 Rd Tr  c pd pr




psr ref. surf. pressure
=> lin. geopotential for X=T

dpr   X   X d  d  0 => lin. energy conv. term for X=D 1 dpr 1   X   p  X d  d  sr 0 Governing Equations 3 Slide 21 ECMWF

Linear system to be solved
D   0.5t 2 ( T   Rd Tr log( ps  ))  D* T   0.5t D   T * log( ps  )  0.5t  D   P*

Eliminate all variables to find also a Helmholtz equation for D+ :
(1  0.25 2t 2 (  RdTr )2 )D  D*  0.5t2 ( T *  RdTr P* )


Governing Equations 3

~ I    D  D
2 



  γ  Rd Tr
Slide 22

operator acting only on the vertical I = unity operator ECMWF

Semi-implicit time integration in IFS



~ I    D  D
2 



Vertically coupled set of Helmholtz equations. Coupling through

  γ  Rd Tr
Uncouple by transforming to the eigenspace of this matrix gamma (i.e. diagonalise gamma). Unity matrix “I” stays diagonal. =>



~ 1  i  D  D
2 



One equation for each 1  i  N Lev

In spectral space (spherical harmonics space): n(n  1)  m  ~ m because 2Ynm   n(n  1) Ynm   1  i  Dn  Dn a2 2 a   Once D+ has been computed, it is easy to compute the other variables at “+”. Governing Equations 3 Slide 23 ECMWF


				
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