Non-hydrostatic Modelling- Basic Equations -- Simplifications -

- Vertical Coordinates -

Hans-Joachim Herzog

St.Petersburg Summer SchoolSestroretsk , 11.-17.6.2006

DWD - Potsdam , Germany

1

1. BASIC EQUATIONS - physical basis

Budget Equations for Momentum , Mass , Heat , Water Components

constitute a model describing the impact of gravity and Earth rotationover an enormously wide spectral range of internal processes caused by heat, mass, momentum, radiation transfer and phase changes of

water essentially determined by turbulence.

2

1.1 Coordinate-free basic equations

( ) ( )kk

k

ee

IJtd

qd

RJtdpd

tdhdRJvp

tded

vtd

d

Tvptdvd

+⋅−∇=

++⋅∇−=⇔++⋅∇−⋅∇−=

⋅∇−=

⋅∇−×Ω−∇−−∇=

r

rrrrr

r

rrr

ρ

ερερ

ρρ

ρφρρ 2

vr - barycentric velocity

p - pressure

∑=k

kρρ - total density

ρρ k

kq =- mass fraction (specific content)

of constituent k

( )he - specific internal energy ( enthalpy)with ( h = e + p / )

φΩr

T ε

Rr

eJr ( )kJ

r

kI

- molecular stress tensor - molecular dissipation

- gravitational potential

- Earth rotation vector

- radiation flux vector

- diffusion flux of heat (of kq )

- source / sink of kq

Doms, G. et al.,2002 :LM documentation

Part I: Dynamics andNumerics

(www.cosmo-model.org)

ρ

3

; k = v , l , f

( k = d , v , l , f )

1.2 Choice of basic equations dependent on a followingnumerical scheme

momentum equationstotal mass equation (continuity equation)enthalpy equationwater constituent equations

+ equation of state

elimination of densitymain-stream approach

Prognostic equation for pressureinstead of total density

continuity equation is hidden !

4

1.3 Common physical approximations

- original budget equations formulated for mean flow by Reynolds averaging approach ( van Mieghem 1973 )

- molecular fluxes, dissipation and almost all moleculardiffusion fluxes are neglected compared to turbulent flux terms

- latent heat of vaporation and sublimation assumed constant- specific heats of moist air are replaced by specific heat of dry air

5

turbulence averaging symbols dropped

1.3 Common physical approximations

( )

( )

( )

( ) 1

1

,,,,

11

1

2

−

−

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+=

++⋅∇−=

+−⋅∇−=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+⋅∇−=

+=

′′′′⋅∇−×Ω−∇+∇−=

vdflv

d

vd

flflflfl

flvv

mvd

pdh

vd

pd

vd

pd

hpd

TRpTqqqRRRp

IFPtd

qd

IIFtd

qd

Qcc

Qcc

vpcc

tdpd

Qtdpd

tdTdc

vvvptdvd

ρ

ρ

ρ

ρ

ρρφρρ

rr

r

r

rrrrr

we follow this line(e.g. LM and MM5)

Doms et al. 2002Dudhia 1993, MWR 121

pd

dcR

ppp

TT

⎟⎟⎠

⎞⎜⎜⎝

⎛=→

=→

00

π

πθ

Bryan and Fritsch2002, MWR 130

- overview article -

6

Definition of heat source term

hpd

T Qc

Qρ

1:=

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++⋅∇−= f

pd

sl

pd

v

pdT S

cLS

cLRH

cQ

rr

ρ1

ρ/ll IS =ρ/ff IS =⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

≈′′⋅∇−≈

′′⋅∇−=⋅∇−

zK

zvc

c

Tvcc

Hc

hpdpd

pdpdpd

θρπρ

θρπρ

ρρρ

11

1:1

r

rr

Turbulent heat flux Radiation fluxDiabatic heating due to

cloud microphysicalsources per unit mass

1.3 Common physical approximations 7

Definition of moisture source term

mpd

M Qc

Qρ

1:=

( ) ( ) ( )[ ]flfl

pd

dfl

pd

vv

d

v

pd

dM PPFF

cTRSS

cTRF

RR

cTRQ

rrrrr+⋅∇++⋅∇++−⋅∇⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

ρρ1

flvx

kz

qKkqwqvFx

hxxx

,,=∂∂

−≈′′≈′′=rrrr

ρρρ

kvqP flT

flflrrr

,,, −=

Turbulent flux forwater constituents

Precipitation (gravitational diffusion) fluxes

Cloud heat sources

1.3 Common physical approximations 8

1.4 Mass-consistent formulation of T - and p – prognosticequations

Mpdvd

pdTpd

vd

pd

vd

pd

Tpd

Qccc

Qccc

vpcc

tdpd

Qtdpd

ctdTd

ρρ

ρ

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+⋅∇−=

+=

1

1

r

vpcc

tdpd

Qvcp

tdTd

vd

pd

Tvd

r

r

⋅∇−=

+⋅∇−=ρ ( )

Mpdvd

pdTpd

vd

pd

vd

pd

MTvd

pd

vd

Qccc

Qccc

vpcc

tdpd

QQcc

vcp

tdTd

ρρ

ρ

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+⋅∇−=

++⋅∇−=

1r

r

mass-consistent approachsource terms in prognostic p-equation often neglected

Dudhia 1993 , Doms et al. 2002

up to here coordinate-free –parameterisation problem dropped in the lecture

9

1.5 Spherical coordinates formulation ofadiabatic dry model part

Equation system to start from:

vptdvd rrr

×Ω−∇−∇−= 21 φρ

vc

ptdTd

v

r⋅∇−=

ρ

vpcc

tdpd

v

p r⋅∇−=

TRp d ρ=

vtd

d r⋅∇−= ρρ

pcR

oo

ppT ⎟⎟

⎠

⎞⎜⎜⎝

⎛=θ

0=td

dθ

introducing spherical coordinates

λϕr

longitude

latitude

Radial distancefrom Earth centrehow to do that ?

10

Deep-Atmospheric Non-hydrostatic Equationsfor a Rotating Spherical Atmosphere

Lagrangian formalism from Theoretical Mechanics applied

kkk qp

qL

qL

tdd

∂∂

−=∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

ρ1

&

( ) 01=

∂∂

+k

k

qq

tdd &D

Dρ

ρ 3,2,1=k

0=∂∂

+∂∂

kk q

qt

θθ&

Newton‘s second law of motion

continuity equation

first law of thermodynamics

kk qq &,

NTL φ−=2

2

21

21

⎟⎟⎠

⎞⎜⎜⎝

⎛==

tdrdvTr

r

( )kNN qφφ =

( )2rd rkj qq

T&& ∂∂

∂=

22D- generalised coordinates

- Lagrangian function

- kinetic energy

- Newtonian gravitational potential

functional determinantsquared

- metric form

11

Specifications

rqrq kk &&&& ,,;,, ϕλϕλ Ω+==

( )drdrdrrd ,,cos ϕλϕ=r

( )( )2222222

cos21

21 rrr

tdrdT &&&r

++Ω+=⎟⎟⎠

⎞⎜⎜⎝

⎛= ϕλϕ

( )( ) ⎟⎠⎞

⎜⎝⎛ Ω−−++Ω++= 222222222 cos

212cos

21 ϕφϕλλϕ rrrrL N&&&&

222 cos21

Ω−= ϕφφ rN

ϕϕ

λ24

2

2

2

2

2

2

cos

00

00

00

r

rT

T

T

=

∂∂

∂∂

∂∂

=&

&

2D

geopotential = gravity potential =Newtonian potential + centrifugal potential

we define : rwrvru &&& === :,:,cos: ϕλϕ

12

rrp

rvuu

tddw

rp

rrwv

ruu

tddv

rp

rrwu

rvuwv

tddu

∂∂

−∂∂

−=+

−Ω−

∂∂

−∂∂

−=++Ω+

∂∂

−∂∂

−=++Ω+Ω−

φρ

ϕ

ϕφ

ϕρϕϕ

λφ

ϕλϕρϕϕϕ

1cos2

11tansin2

cos1

cos1tancos2sin2

22

2

momentum equations

continuity equation

( ) 02coscos

11=+

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+ wrr

wvurtd

d ϕϕλϕ

ρρ

first law of thermodynamics

00cos

≠∪=∂∂

+∂∂

+∂∂

+∂∂

=r

wrv

ru

ttdd θ

ϕθ

λθ

ϕθθ

Literature:Hinkelmann,K.H.: Primitiveequations. WMO Training Seminar.Moscow 1965, pp.306-375.White,A.A. et al.: Consistent approximate models of the global atmosphere...Q.J.R.Meteorol.Soc. 2005, 131,pp.2081-2107.relevant for ‘Unified Model’ of UK Met Office!

deep equations 13

Consistent shallowness-approximation via the Lagrange route

spherical geometry : a - constant Earth radius

zar += z( - variable )

shallowness : za >>

0=∂∂

=∂∂

ϕφ

λφ .constg

zr≈⇒

∂∂

≡∂∂ φφ

,

( )

( )( )zwavauaTL

azaaT

dzdadardzqzaq

N

kk

&&&

&&&

r&&&&

====−=

Ω−=++Ω+=

==+=

:,:,cos:,cos,

cos21,cos

21

,,cos,,,,,

2

222222222

ϕλϕϕφ

ϕφφϕλϕ

ϕλϕϕλϕλ

D

14

shallow spherical equation

momentum equations

gzp

tdwd

pa

ua

utdvd

pa

va

utdud

−∂∂

−=

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+Ω+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+Ω−

ρ

ϕρϕ

ϕ

λϕρϕ

ϕ

1

1coscos

2

cos1sin

cos2

continuity equation( ) 0cos

cos11

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

+zwvu

atdd

ϕϕ

λϕρ

ρ

first law of thermodynamics

00cos

≠∪=∂∂

+∂∂

+∂∂

+∂∂

=z

wav

au

ttdd θ

ϕθ

λθ

ϕθθ

15

Comparison with deep equations indicates that the shallownessapproximation , consisting of

- ‘traditional approximation’ ( neglecting the -terms Eckart 1960 )

ϕcos2Ω

- ‘shallow - approximation’ ( )

- ‘small-curvature approximation’ ( neglecting

in momentum equations, and in the continuity equation ) ,

is achieved dynamically consistent as one package with the Lagrangianapproach .

., constz

za =∂∂

>>φ

rvu

rwv

rwu 22

,, +

rw2

16

Basic Equationsconstituted by physical laws

having always simplifications down to parameterisations. Choice of prognostic variables/equations

anticipates the way of numerical integration process

Spherical coordinates

Shallowness-approximationand/or

Hydrostatic approximation

Projection-planeequations

by mapping

Unified ModelPhilosophy

Rotated sphericalcoordinates Global Model

Limited AreaModel

Linear Mode Analysisfor understanding dynamics

of the model core

Some simplified overview 17

2. Linear Mode Analysis

Importance and meaning of a linear mode analysis is to recogniseessential properties of the compressible non-hydrostatic model equations and understand simplifications due to filtering. Why are the full equations important and interesting enough to become increasingly standard model equations?

The modes involved are- Rossby-, or advective mode- Internal gravity (buoyancy)-inertial modes- Acoustic modes- Lamb mode

Literature: Miller, M. , 2002 : Atmospheric Waves.Meteorological Training Course Lecture Series, ECMWF.

Changnon, J. M. , P. R. Bannon , 2005 : Wave Response during Hydrostatic and Geostrophic Adjustment. Part I : Transient Dynamics. JAS, 62 , May 2005 , 1311-1329.

Thuburn, J. , N. Wood, A. Staniforth , Normal modes of deep atmospheres.2002 a : Spherical geometry. Q.J.R.M.S. 128 , pp. 1771-1792.2002 b : f - F - plane geometry. Q.J.M.S. 128 , pp. 1793-1806.

18

2.1 Linear model framework

Linear Model frameworkCompressible , dry , inviscid , unforced , ideal gas ,Cartesian coordinates , f - plane , invariant in y-direction ,Linearisation on f - plane about constant basic currentand height-variable basic state temperature with constantstability.

Introduction of ‘ field variables ‘ in the sense of Eckart 1960( cf. Gassmann and Herzog 2006 )*

( ) ( ) ppTcwuTwu stps

t ′=′′′=ρρρρ :,:

* Gassmann and Herzog : A consistent Time-Split Numerical Schemeapplied to the Nonhydrostatic Compressible Equations.- accepted to be published in: Monthly Wea.Rev. 2006

19

2.1 Linear model framework

0

01

0

0

0

2

2

=+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=−⎟⎟⎠

⎞⎜⎜⎝

⎛Γ+

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

wNcg

xU

t

xuw

zp

xU

tc

cgp

zw

xU

t

ufvx

Ut

xpvfu

xU

t

p

s

p

θθ

θθµ

with the definitions

221

scg

H+−=Γ

2

21

scg

gN

H+=

zddgN θ

θ=2

- Eckart - coeffient

- reciprocal scale-height -Brunt-Väisälä frequencysquared

µ - tracer

20

General case :nonhydrostatic

compressible

1=µ∞<2

sc21

Identical structural wave equation for both andp w

( ) ( ) ( ) 0~1~~ 2222

2222

2

222 =⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+−

∂∂

++⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−

∂∂

+wp

fDcx

NDwp

zfD

s

µ

with the individual tendency operator ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

≡x

Ut

D~

variable separation ansatz: ( ) ( )( ) ( ) ( )( )ztxppztxww p

nnw

nn ψψ ,~,,~

( ) ( ) ( ) 0~1~~12

222

22222

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡∂∂

−+++⎟⎟⎠

⎞⎜⎜⎝

⎛+

n

n

sn

n

n pw

xfD

cND

pw

fDh

µ( )

( ) 0,22

,2

=+ wpnn

wpn

zdd ψνψ

horizontal, time-dependent structuralequation

vertical structuralequation

2.2 Vertical structural solution

with vertical boundary condition : 0=w for ∞<=∩= Tzzz 0( )

( ) 022

2

=+ wnn

wn

zdd ψνψ

( ) 0=wnψBC : BC :

( )( ) 0=Γ+ pn

pn

zdd ψψ

( )( ) 02

2

2

=+ pnn

pn

zdd ψνψ

02

222 >=

Tn z

n πν

( )( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= z

znz

zn

znz

TTT

pn

πππψ sincos ( )( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−= z

znz

T

wn

πψ sin

...,3,2,1=n- vertical wavenumber squared

222 Γ+=−nnh ν - separation constant in horizontal structural equation

( ) ( )zp Γ−=exp0ψ( ) 00 =wψ

220 Γ−=ν 02 =−

nh

0=n

22

2.3 Horizontal structural equation

( ) ( )txktxw nn ω−sin~, ( ) ( )txktxp nn ω−cos~,wave solution

frequency equation

( )[ ] ( )( )[ ]( )[ ] =

−−−−+−−

22

222222 /fUk

cUkfkUkN

n

snn

ωωωµ

Gossard and Hook , 1975 : Waves in the Atmosphere. p. 112, eq. (23.7)

All possible wave frequencies of the given linear model are contained in the frequency equation above. They are going to be discussed in the following.

2

1

nh22 Γ+nν

2

2

22

41

sn c

NH

−+ν

23

04

14

1 22

2222

2

2222

2

4

=⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛++− f

HNk

cf

Hk

c ns

nns

n νµνµσσµ

( )22 : nn Uk ωσ −= - intrinsic frequency

∞<= 21 scµ

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++≈+⎟⎟

⎠

⎞⎜⎜⎝

⎛++≈ 2

22222

2222

41

41

Hkcf

Hkc nsnsan ννσ

( )2

2

222

22

222

2

414

1

sn

n

gn

cf

Hk

fH

Nk

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

≈ν

νσ

pairs of internalacoustic mode frequencies

pairs of internal gravity mode frequencies

1=µ

Nonhydrostatic- compressible/elastic

∞→

=2

1

scµ

24

Nonhydrostatic- incompressible/anelastic ∞→= 21 scµ

04

14

1 22

2222

2

2222

2

4

=⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛++− f

HNk

cf

Hk

c ns

nns

n νµνµσσµ

∞→2sc 1=µ

( )2

22

22

222

2

41

41

Hk

fH

Nk

n

n

gn

++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

≈ν

νσ

acoustic mode filtered out due to ∞→2sc

internal gravity modesremain untouched

∞<

=2

0

scµ

25

Hydrostatic approximation- compressible/elastic ∞<= 20 scµ

04

14

1 22

2222

2

2222

2

4

=⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛++− f

HNk

cf

Hk

c ns

nns

n νµνµσσµ

( ) 2

22

222

41 f

H

Nk

n

gn ++

≈ν

σ

∞<2sc0=µ

internal gravity modeswith limited validity

222

41H

k n +<<ν

22zx LL >>

- acoustic mode filtered out due to hydrostaticapproximation

- hydrostatic filtering, however, is insufficient to representgravity waves with high-resolving models

- Motivation for nonhydrostatic modelling!

26

2.4 Analytical solution

Nonstationary solutions for linear cases considered above

( )( )

( ) ( )( )ztxkUkf

fktzxv pnn

n

ψωω

−−−

− sin~,, 22

( ) ( )( )

( ) ( )( )ztxkUkf

Ukktzxu pnn

n

n ψωω

ω−

−−− cos~,, 22

( ) ( ) ( )( )ztxktzxp pnn ψω−cos~,,

( )( ) ( ) ( )( )ztxk

UkN

ztzx

cg w

nnnp

ψωωθ

θ−

−cos~,, 2

( ) ( ) ( )( )ztxktzxw wnn ψω−sin~,,

The vertical structural function is sufficient to representthe vertical structure of the variables .

( )( )zpnψ ( )( )( )zw

nψ( )θ,,, wpvu

This finding suggests for a vertical difference approximation the application ofa Charney-Phillips ( CP- ) grid instead of a Lorenz ( L- ) grid!

27

( ) 0≠− nUk ω

2.4 Analytical solution

( ) ( )zp Γ−=exp0ψ ( ) 00 =wψ0=n22

0 Γ−=ν 02 =−nh

The case

with

defines the Lamb mode which is a particular solution of the verticalstructural equations with boundary conditions included. Itsfrequency is

indicating a horizontally propagating acoustic wave evanescent invertical direction. For a pure Lamb wave

and is valid.

( ) 2220

2so ckUk ≈−= ωσ

0=w 0=θ

28

0~

0~1

0~

0~

0~

2

2

=+⎟⎟⎠

⎞⎜⎜⎝

⎛

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−

∂∂

+

=−⎟⎟⎠

⎞⎜⎜⎝

⎛Γ+

∂∂

+

=+

=∂∂

+−

wNcgD

xuw

zpD

c

cgp

zwD

ufvD

xpvfuD

p

s

p

θθ

θθµ

0~

0

0~

0~

0~

2 =+⎟⎟⎠

⎞⎜⎜⎝

⎛

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−

∂∂

=−⎟⎟⎠

⎞⎜⎜⎝

⎛Γ+

∂∂

+

=+

=∂∂

+−

wNcgD

xuw

z

cgp

zwD

ufvD

xpvfuD

p

p

θθ

θθ

0~

0~1

0

0~

0~

2

2

=+⎟⎟⎠

⎞⎜⎜⎝

⎛

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛Γ−

∂∂

+

=−⎟⎟⎠

⎞⎜⎜⎝

⎛Γ+

∂∂

=+

=∂∂

+−

wNcgD

xuw

zpD

c

cgp

z

ufvD

xpvfuD

p

s

p

θθ

θθ

full equations anelastic equationshydrostatic equations

0~ ⇒wD 0~ ⇒pD

04

1~

0~

0~

222

2

2

2

=∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

=∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=∂∂

+∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

xuNp

HzD

xuf

xvD

xp

xvf

xuD

0~ ⇒⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

xuD

04

1~22

2

2

2

2

2

=⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

+∂∂ p

HzNf

xpD

geostrophic potential vorticity equation- advective mode only

2.5 Simplified scheme towards filtered equations29

04

122

2

2

2

2

2

=−∂∂

+∂∂ w

Hzw

xw

fNwith

Nonhydrostatic model equationsHydrostatic ( primitive )

equationsCompressible/elastic Incompressible/anelastic *

equations with fullphysical structure

full set of prognosticvariables

0=∂∂

+⎟⎠⎞

⎜⎝⎛ Γ−∂∂

xuw

z

filter condition:

0=td

dp

reduction of equation set due to reduction to hydrostatic equation

0=td

dw0=−⎟⎟

⎠

⎞⎜⎜⎝

⎛Γ+

∂∂

θθ

pcgp

z

( ) tzxFppzx

∀=Γ−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂ ;,2

2

2

2

2

boundary value problem constitutes adiagnostic relation for p

filter condition:

boundary value problem constitutes a diagnostic relation for w

( ) tzxGwHz

w∀=−

∂∂ ;,

41

22

2

Internal acoustic andgravity waves are com-pletely contained.

Internal gravity waves are com-pletely contained, acoustic wavesare filtered out.

Internal gravity waves are contained, but insufficientlypresented acoustic waves excluded.

( )22zx LL >>

Hydrostatic and geostrophic adaptation

Hydrostatic and geostrophic adaptationGeostrophic adaptation

Damping/filtering of acoustic wavesdue to an appropriate numericalscheme ( split-explicit , semi-implicit-semi-Lagrange )Models: MM5, LM, WRF-NCAR,

UK-Unified Model, etc. ...WK78 , Cullen, Gassmann

Numerical treatment of elliptic pressureequation is difficult and needs to be done

with care (terrain-following coordinate).An excellent research model is EULAG(Grabowski, W.W.,P.K.Smolarkiewicz,2002,MWR 130, 939-952.)

Most operational global and limitedarea models are hydrostatic.

* refinements of anelastic approximation: P.B.Bannon,1996, J.Atm.Sc. 53, No.23, 3618-3628.

30

3. Vertical coordinates

The choice of an appropriate vertical coordinate is always to aim at improvingsimulations over mountainous terrain

Introduction of terrain-following vertical coordinate in meteorological modelling by Phillips 1957 , Gal-Chen and Somerville 1975 , applying a terrain-followingnormalisation with surface pressure ( time-dependent coordinate -> deformable )or surface-height ( time-independent coordinate -> nondeformable )

- Phillips’ sigma-coordinate -> larger-scale hydrostatic modelling- Gal-Chen’s coordinate -> small-scale nonhydrostatic modelling

.

There are problems in computing the pressure gradient term with pressurecoordinate in hydrostatic models in case of steeper mountains which has led tothe introduction of step-terrain orography ( Mesinger 1984, Mesinger et al. 1985,1988 -> NCEP Regional Eta Model ), which seems however not appropriate innonhydrostatic modelling ( Gallus and Klemp 2000 ).

.

31

Revival of Z-coordinate as shaved-cell approach in combination withfinite-volume method – a possible breakthrough ?

.

step topographyin Z- coordinate model

terrain-following coordinate model

Z-coordinate modelwith piecewise constant slopes

(shaved-cell approach)

3.1 Representation of mountains in nonhydrostatic models

Revival of Z-coordinate in nonhydrostatic modelling !

Bonaventura, L. , 2000:J. of Comput. Phys. 158, 186-213.

Gallus, A. W., J. B. Klemp, 2000:MWR 128 , 1153-1164.

Adcroft, A. et al. , 1997:MWR 125 , 2293-2315.( for Ocean Modelling )

Steppeler, J. et al. , 2002:MWR 130 , 2143-2149.( application in LM )

research state ! research state !

Applied in several non-hydrostatic models even

in operational mode( LM , MM5 , UK , etc. )

smooth slope

smooth slope

32

3.2 Introduction of a time-independent terrain-following coordinate

Vertical coordinate may be any monotonic function of geometrical height.ζThis -system is fixed in physical space, and is non-orthogonal.ζIt is of Gal-Chen type.

The lowest coordinate surface of constant coincides with the smooth model orography.

The lower boundary condition seems easy but should be formulatedcarefully and consistent which is actually not simple to be done.

.

.

.

.ζ

How to transform the basic equations into such a non-orthogonal - system?ζ

Method : Start from the basic equations in covariant vector form

33

.

3.3 Transformation method

Literature: Dutton, J.A. , 1986 : The ceaseless wind. pp. 129-144 , 248-251Pielke, R.A. , 1984 : Mesoscale Meteorological Modeling. pp.102-127Gal-Chen, T. , R.C.J. Somerville , 1975 : On the use of a coordinatetransformation for the solution of the Navier-Stokes equations.J.Comput.Phys. 17, 209-229

Zdunkowski, W., A.Bott : Dynamics of the Atmosphere. A course inTheoretical Meteorology. Cambridge Univ.Press 2003.

Brief survey of the method

jj xr~∂∂

=r

τ

jj x~∇=η

ij

ij δητ =

two differentbasis vectors

covariant:

contravariant:

orthogonality relation:

jjj

j uuv ητ ~~ ==r

ljlji

i

jji

liljj

ij

i

ugxx

xpG

uxuu

tu

~~~2~~

~1~

~~~~~~

3 Ω−∂∂

−∂∂

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛Γ+

∂∂

+∂∂

ερ

( )ii

sisii

i

uGxG

uxuv ~~

~~1~~

~~

, ∂∂

=Γ+∂∂

=⋅∇r

Momentum equations in contravariant form:

Divergence of wind vector:

34

3.3 Transformation method

further specifications

l

j

l

iji

xx

xxG

∂∂

∂∂

=~~~

j

l

i

l

ji xx

xxG ~~

~∂∂

∂∂

=j

i

xxG ~

~∂∂

= zxxz

ljljilj ∂

∂∂∂

∂=Γ=Γ

ζ~~

~~ 23

ii

xxuw ~

~3

∂∂

=

transformation

( )zyx ,, ( )ζ,, yx

( )ζ,,,,,, 321 yxzyxxxx =

( )zyxyxxxx ,,,,~,~,~ 321 ζ=

old set

new set monotonic in z !

ζ&,,~,~,~ 321 vuuuu = and

35

3.3 Transformation method

with ( ) ( ) ( )tzyxpzptzyxp ,,,,,, 0 ′+= ( )ζ,,, yxzz =

and the definition 0~~: >

∂∂

−=∂∂

−=−=ζz

xxGG j

i

(left-handed system!)

We arrive finally at the nonhydrostatic equations in the - systemassuming a dry-adiabatic, unforced, inviscid model atmosphere, herefor the sake of simplicity, in a - system . A peculiarity of most models is to use a prognostic variable in the verticalmomentum equation which is not the contravariant vertical wind component

but the common covariant variable instead .

ζ

ζ,, yx

ζ& zw &=

Havingζ

ζζζ ∂

∂+

∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

= &y

vx

utz

wy

vx

uttd

d

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

+∂

∂=⋅∇

ζζ&r G

yvG

xuG

Gv 1

hydrostatic vertical distribution of basic state pressure !( )zp0

flux-form very importantfor model implementation!

36

vfpxz

Gxp

tdud

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂′∂

∂∂

+∂′∂

−=ζρ ζζ

11

ufpyz

Gyp

tdvd

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂′∂

∂∂

+∂′∂

−=ζρ ζζ

11

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−′

+∂

′∂=

0

0011pp

TT

TTgp

Gtdwd

ρρ

ζρ

vpcc

wgtdpd

vd

pd r⋅∇−=−

′0ρ

,vc

ptdTd

vd

r⋅∇−=

ρ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

∂∂

+∂∂

= wyzv

xzu

G ζζ

ζ 1&

TRp d ρ=

3.4 Transformed equations

Nonhydrostatic equationswritten with generalisedterrain-following - coordinate

Further introduction of sphericalcoordinates - with shallowness-approximation - leads to the dynamicalcore equations of the Lokal-Modell (LM)

ζ

relation between contravariantand physical vertical motion

37

Mind the pressure gradient terms !

3.5 Lower boundary condition

… has an important practical implication. It should be formulated as consistent as possible within a given numerical scheme, where the schemeis also necessary to be adapted to this problem. A successful approach has been developed by Almut Gassmann*.

At terrain-following lower boundary (LB) free-slip condition :

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

+∂∂

= wxzv

xzu

G1ζ&

tLB ∀= ,0ζ&

LBLB x

zvxzu

Gw ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=1

*Gassmann, A. , 2004 : Formulation of the LM’s Dynamical Lower Boundary Condition.COSMO-Newsletter No.4, Febr. 2004, 155-158. (www.cosmo-model.org)

Gassmann, A. , H.-J. Herzog , 2006 : A consistent time-split numerical scheme applied to thenonhydrostatic compressible equations. MWR, accepted to be published.

38

LBLB xz

tv

xz

tu

Gtw

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ 10=

∂∂

tLBζ&

wfpp

TT

TTgp

Gtw

+⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−′

+∂

′∂−=

∂∂

0

0011ρρ

ζρelimination of by use ofLBt

w⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎥⎦

⎤⎢⎣

⎡−′

−′

+∂∂

∂∂

+∂∂

∂∂

=∂

′∂wfT

Tgpp

TTg

tv

yz

tu

xzGp

ρρ

ρρρ

ζ0

0

00

elimination of

in u- and v-momentumequation

ζ∂′∂ p elimination of

by use of u- and v-momentumequation

tv

tu

∂∂

∂∂ ,

3.5 Lower boundary condition 39

3.6 Upper boundary condition

. Non-penetrative boundary condition at 0=ζ

- rigid lid ( flat )

- no mass flux across the boundary ( ) 0...,,, =∂∂ Tvuζ

Danger of wave reflection without additional absorbing remedies !Sponge technique ( philosophy of Davies + Kallberg 1976, 1983 )

. Radiative upper boundary condition

It is possible to be implemented in a nonlinear nonhydrostatic model (LM) for real-data integrations, although resting on limited assumptions ( linear, hydrostatic, incompressible Klemp, Durran 1983, and also Bougeault 1983 )

demonstrated in Almut Gassmann’s lecture

wkNp ˆˆ ρ

=′

0==wζ&

41

22

2

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

=∂∂

yz

xz

BxzF

yz

xzF

yz

tu

vu

22

2

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

=∂∂

yz

xz

ByzF

yz

xzF

xz

tv

uv

ypvffF

xpvffF

vv

uu

∂′∂

−−=

∂′∂

−+=

ρ

ρ1

1

wfTTg

pp

TTgB −

′−

′=

ρρ

ρρ 0

0

00

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

+∂∂

+∂∂

=∂

′∂22

1yz

xz

BFyzF

xz

Gp vu

ρζ

3.5 Lower boundary condition

Neumann boundarycondition at

relevant forimplementation

LBζ

40

V.BJERKNES , who was the first advocat of NWP, wrote 1904 :‘ If it is true, as every scientist believes, that subsequent atmospheric statesdevelop from the preceding ones to physical law, then it is apparent that the sufficient and necessary conditions for rational solution of forecastingproblems are the following:. . . A sufficiently accurate knowledge of the laws according to which onestate of the atmosphere develops from another . . . The problem is of hugedimensions. Its solution can only be the result of a long development . . .’

1998, when A. ARAKAWA retired , he has reflected Bjerknes‘ famousnote such : ‘ I will not be able to see the completion of the ‘great challenge’, but I am happy to see at least its beginning .’

Where we are now ?

4. Some reflections – all the problems are already thought over 42

5. Outlook

.Outline of main points :

Increasing tendency towards unified global non-hydrostatic modelsTendency towards global gridpoint models Introduction of quasi-homogeneous, quasi-isotropic grids

( geodesic grid : icosahedron polyhedron ) . New formulation principles in view of a Vortex-Energy Theory( Nambu-bracket theory) from P. Nevir ( 1993, 1998 ) opens the way for the construction of new spatial difference schemes generalisingthe classic ideas from Arakawa for the Jacobian operator up to the general adiabatic nonhydrostatic compressible equations connecting both global and local accuracy . Recent suggestions for such newdifference constructions have been made by R.Salmon (2005).

Literature: Heikes,R., D.A.Randall,1995: MWR 123,1881-1887.Majewski,D. et al.,2002: MWR 130,319-338.Salmon, R.,2005: Nonlinearity 18,R1-R16.Nevir,P.,R.Blender,1993:J.Physics A 26,L1189-L1193.Nevir,P. 1998: Habilitationsschrift, FU Berlin,317p.

43

( )∫∫∫ ++= φρρρτ uvdH ~2/~ 2r

θlnpcs =

vcp

tdTd

vtd

d

vptdvd

v

r

r

rrr

⋅∇−=

⋅∇−=

×Ω−∇−∇−=

ρ

ρρ

ρφρρ 2

( ) HSs

ts

HMt

HSvHMvHhvtv

a

~,~,

~,~,

~,~,~,~,~,~,

ρρ

ρρ

=∂

∂

=∂∂

++=∂∂ rrrr

( )∫∫∫= sdS ρτ~

∫∫∫= ρτdM~

∫∫∫ ⋅= aa vdh ξτrr~

Entropy

Mass

Helicity

Nevir‘s Nambu formulation of hydro-thermodynamicbasic equations

!vrr

×∇= 3ξ

A Nambu-bracket approachto construct conserving algorithms should be possible

44

Energy

I invite you to take place in this boat !. . . and in any case, thank youfor your attention and your patience thatyou followedme !

45

Cartoon of Norman Phillips, from: Dynamic Meteorology (ed. P.Morel) 1973