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Stability of Leap-Frog Constant-Coefficients
Semi-Implicit Schemes for the Fully Elastic System of Euler Equations. Case with Orography.
Pierre Bénard, Jan Masek, Petra Smolikova
To cite this version:
Pierre Bénard, Jan Masek, Petra Smolikova. Stability of Leap-Frog Constant-Coefficients Semi- Implicit Schemes for the Fully Elastic System of Euler Equations. Case with Orography.. 2004.
�hal-00003241�
Fully Elasti System of Euler Equations. Case with Orography.
P. Bénard
, J. Ma²ek +
, P. Smolíková y
CentreNationaldeReherhesMétéorologiques,Météo-Frane,Toulouse,Frane
+
SlovakHydro-MeteorologialInstitute, Bratislava,Slovakia
y
CzehHydro-MeteorologialInstitute,Prague,Czeh Republi
23June 2004
Corresponding address:
Pierre Bénard
CNRM/GMAP
42, Avenue G. Coriolis
F-31057TOULOUSE CEDEX
FRANCE
Telephone: +33 (0)5 610784 63
Fax: +33 (0)5 61 078453
e-mail: pierre.benardmeteo.fr
The stability of onstant-oeients semi-impliitshemes for the hydrostatiprimi-
tiveequationsandthefullyelastiEulerequationsinpreseneofexpliitlytreatedthermal
residuals has been theoretially examined in the earlier literature, but only for the ase
of a at terrain. This paper extends these analyses to the ase where an orography is
present, inthe shape of a uniformslope.
Itisshown, withmass-basedoordinates,thatfortheEuler equations,the preseneof
a slope redues furthermore the set of the prognosti variableswhih an be used in the
vertialmomentum equation tomaintain the robustness of the sheme, ompared to the
ase of a at terrain. The situationappears to be similarfor systems ast inmass-based
and height-basedvertial oordinates.
Still for mass-based vertial oordinates, an optimal prognosti variable is proposed,
and shown toresult ina robustnesssimilar tothe one observed for the hydrostatiprim-
itiveequations system.
Theprognostivariableswhihleadtorobustsemi-impliitshemessharetheproperty
of having umbersome evolution equations, and an alternative time-treatment of some
terms is then requiredto keep the evolution equationreasonably simple. This treatment
is shown not tomodify substantially the stability of the time-sheme.
This study nallyindiates that with a pertinent hoie for the prognosti variables,
mass-based vertial oordinates are equally suitable as height-based oordinates for e-
iently solving the ompressible Euler equationssystem.
The semi-impliit(SI)tehnique was introdued inmeteorology by Robert etal. (1972),
inorder toinrease the numerialeieny with respet toexpliitshemes, by allowing
largertime-steps. SI shemesare basedon anarbitraryseparationof the evolutionterms
between linear ontributions, treated impliitly, and non-linear (NL) residuals, treated
expliitly.
As disussed inBénard etal.,2004 (BLSV04 hereafter), several lasses of SI shemes
an be dened. In order to alleviate the problems linked to NL residuals, SI shemes
for whih the linear terms have non horizontally-homogeneous oeients (Thomas et
al. 1998), or even non-onstant oeients (Skamarok et al., 1997) an be designed.
This latter type of shemes is not examined here sine the fous of the present paper
is exlusively restrited to the so-alled lass of "onstant-oeients" SI shemes, that
is, to SI shemes in whih all the impliit linear terms have their oeients onstant
in time and horizontally homogeneous. For this lass of SI shemes, the relatively large
magnitude of NL residuals an result in instabilities, espeially for long time-step. Note
however that the stability an then be restored through an iteration of the sheme, as
shown in Bénard, 2003 (B03hereafter).
A theoretial framework for studying analytially the stability of these onstant-
oeients SI shemes in presene of thermal NL residuals and in simplied ontexts
has been proposed by Simmonsetal. (1978),Cté etal. (1983),and Simmonsand Tem-
perton (1997) for the hydrostatiprimitiveequation (HPE) system, and extended to the
fully elasti Euler equation (EE) system by B03. A more detailed review on the history
of these stability analyses for SI shemes an be found in this latter paper. Using the
same theoretial framework as in B03, a large sensitivity of the three-time levels (3-TL)
systeminBLVS04. Inpartiular,thehoieofthetwonewprognostivariablesappearing
in the EE system due tothe relaxationof the hydrostati hypothesis was shown to have
adramatiimpatonthe stability. In the aseof ageneralmass-basedoordinate(e.g.
Laprise, 1992),an optimalset for these twovariableswas proposed asfollows:
P =
p
(1)
d =
p
mR T w
(2)
whereT isthetemperature, wthevertialveloity,pthetruepressure, the hydrostati-
pressure,andm=(=)(inallthispaper,seeAppendixAforthedenitionofsymbols
whihare not dened in the text).
Finally, Bénard (2004) showed that an intrinsi instability of the two-time levels SI
sheme for the EE system ould be eliminated by hoosing a slightly modied SI linear
referene system, whih then an no longer be dened as the tangent-linear operator of
the omplete system aroundanexisting referene state.
Aommonpointtoalltheabovementionnedstudieswastoneglettheorographyasa
soure of nonlinear terms,and tofous mainlyon the instability indued by the thermal
nonlinearresiduals(asshowninBLVS04,thenonlineartermsinduedbythepressureeld
an be eliminated in mass-based oordinates by using appropriate oordinates, olumn-
integrated mass variables, and pressure variables). Conversely, in the ase where all NL
thermal residuals are negleted, Ikawa (1988) showed that expliitly treated orographi
terms ould make the evolutionof ompressible systems in height-basedoordinates un-
stableforthepartiular lassofprognostivariablesand SIshemesthathe examined. It
an thereforebe suspeted that aombinationofexpliitlytreatedthermalresidualsand
systems,thereisnoproofthatthehoie(1)(2)proposedaboveisstillthemostrelevant.
This latterstatementwasonrmedexperimentally,by showingthatthis hoieatu-
allyledtoinstabilitywhenanorography wasintrodued inthetwo-dimensional(vertial
plane)version of the ooperativemodel"Aladin-NH",desribed inBubnová etal.,1995,
(hereafter BHBG95). The instability ould be reprodued even for very simple ows,
inludinginitiallybalaned resting isothermal ows over anisolated mountain.
This experimentalfatprompted ustostudy moreindetailthe behaviourof the3-TL
SI EE system from the theoretial point of view in presene of orography, in order to
investigate the nature of the assoiated instability and to seek possible remedies. The
results of this study are reported in this paper.
ThestabilityanalysespresentedherearethusvalidfortheAladin-NHmodel(BHBG95)
as well asfor numerial models that would be based onthe same priniples (mass-based
oordinates, similar linearSI separation, et).
In setion 2, we will dene an aademi (simplied) framework whih allows alge-
braiallytratable stability analysesforthe 3-TL SIEEspae-ontinuous systeminpres-
eneofasimpleorographywhihonsistsina"uniformslopemountain";thenthestability
of the previously proposed hoie (1)(2)will be examined in setion3. In setion 4, an
alternative variable dlwill be proposed, and shown to result ina better stability inpres-
ene of orography. In setion 5, a numerial assessment of the validityof the theoretial
resultswillbepresented. Insetions6and 7someommentsonerningthe HPEsystem
and analternativetime-treatmentof the vertialmomentumequation willbe developed,
and the general onlusion willbe presented in setion 8.
The theoretial framework used for the analyses in this paper is basially the same as
in BLVS04, exept that an orography is introdued: the EE system is thus ast in the
pure unstrethed terrain-followinghydrostati-pressure-based =(=
s
) oordinateand
the owisassumed tobedry,adiabatiandinvisid,ina non-rotatingatmosphereand a
Cartesian framework.
The generalset of equations (3)-(9) of BLVS04is thus still valid, even in presene of
orography,and will beused here asa startingpoint.
The generalmethodfor the analyses is alsosimilar toB03 and BLVS04: an "atual"
steadybasi-stateX andaSI-referenestateX
are hosen,andtheanalysisisperformed
for smallperturbationsaroundX (see B03for deeper explanations and notations). Both
X and X
are assumed isothermal, resting and hydrostatially-balaned. The perturba-
tions around X are assumed to remain small, hene the soure terms of the system an
be linearized around X, and symbolially noted L. Similarly, the soure terms of the
linear referene system assoiated tothe SI sheme are noted L
. The SI sheme is then
implemented aording to Eq. (12) of BLVS04, and the residual (L L
) is treated ex-
pliitely. The domain is taken as a two-dimensional vertial plane along the x diretion.
The temperature of the basi atual state T deviates from the one of the SI-referene
state T
, thusgenerating expliitlytreated thermalresiduals.
As outlinedabove, the major dierene between the framework used in BLVS04 and
the present one is that a uniform slopes is nowassumed for the surfae height z
s
of the
bottom boundary:
z
s
(x)=sx; (3)
s
unbounded in thex diretioninorderto laterallowsimplernormal mode analyses,some
are must betaken to assess the physialrelevane of this framework.
The olumn-integrated mass variationsare desribed through the prognosti variable
q=ln(
s
). The stationarity of X for the omplete system imposes the pressure-gradient
fore to vanish in the horizontal momentum equation. Hene (using the above assump-
tions):
R Trq+r
s
=0 (4)
or:
rq= gs
R T
(5)
whereq(x)istheqeldinthe atualstateX andristhe(=x)operatoralongonstant
surfaes. Under all mentionned assumptions, it an be heked easily that X is then
a stationary state for the omplete system (3)(9) of BLSV04, and onsequently for the
linearized system assoiated toL as well.
It appears that both elds
s
and q are unbounded when x beomes unbounded.
However, this mathematial feature does not indue any partiular problem from the
physial point of view, sine only spatial or temporal variationsof these elds appear in
the equations. In otherwords, even if the height ofthe orographyis unbounded alongx,
all orographi soure terms always remain bounded sine the slope s is a nite number.
From the physial point of view, the proposed framework is thus perfetly relevant for
desribing the ow overa slanted unbounded orography.
As outlined in B03, due to the eliminationof upper and lower boundary onditions
whihwillbeperformedinalltheseanalyses,amore"loal"pointofviewanbeadopted
small-saleperturbations inside alimited regionof the atmospherewhen the larger sale
environment is given by X. Using this loal point of view, the slopes must therefore be
understood as the mean slope of the orography at a sale muh larger than the sale of
the perturbations onsidered inthe analysis. In this paper, we willexamine atmospheri
perturbationsatthekilometrihorizontalsale,onsistentlywithfuturetargetsforNWP
appliations, hene,the slopesretained in the analyses willbeonsistent with terrestrial
slopes at an horizontal sale of 50-100 km, that is, s 2 [ 0:05;+0:05℄. For smaller (i.e.
hetometri)perturbationsales, steeperslopes would have to be onsidered.
The smallperturbation f
X aroundthe X state is dened by:
e
T = T T (6)
e
q = q q: (7)
Sinetheotherprognostivariableshaveazeroreferene-value,thetildesymbolisomitted
for them. For the speiation of q in the X
state, we follow the approah usually
adopted in pratial NWP appliations, that is, q
= ln(
00
), where
00
is an arbitrary
onstant. The omplete set ofequationsisthen linearizedaroundthis q
value, assuming
noorography.
3 Stability analysis with the fP;dg set of variables
In this setionthe stability of the 3-TLSI EE system isexaminedfor the set ofvariables
fP;dgwhihwasproposed inEqs. (22) and(65) ofBLVS04foreliminatingthe problems
linked witha disrepany between T and T
inthe ase of at terrain. The linearization
of the system is performed in the same way as in BLVS04, but now retaining all linear
s
divergene D
3 is:
D
3
=D+d+ gs
R T
~
u (8)
where uis the horizontalwindomponent,D=ru,and
~
=(=). Additionally,the
expression of (=)_ is requiredfor the equationof P sine:
dP
dt
=(1+P) _ p
p _
!
(9)
The linearized version of (=)_ writes:
_
= SD gs
R T
(I S)u (10)
and the linearized L system with orography beomes:
D
t
= R Trr 0
"
G e
T
T
(G I)P
#
R Tr 2
e
q (11)
d
t
= g
2
R T
~
(
~
+I)P (12)
e
T
t
=
R T
C
v (r
0
u+d) (13)
P
t
= Sr 0
u C
p
C
v (r
0
u+d) (14)
e
q
t
= ND+ gs
R T
Nu (15)
where the operator r 0
isdened by:
r 0
=r+ s
H
~
(16)
and H = (R T=g) (see Appendix A for other notations). The SI linear system L
is
obtained diretly from the above system (11)(15) by substituting T ! T
and s ! 0
(and onsequently r !r); itisof oursenot modiedwith respet tothe ase without
orography. Asaonsequene, alltermsassoiatedtosare treatedexpliitlyin(11)(15).
The method of analysis then follows the one proposed in B03, and the reader is invited
torefer tothis paper for the details of notations and the algebraidevelopments.
First, in this paragraph, the above system is shown to fullthe fouronditions [C1℄
[C4℄ dened in B03, and whih are required for making possible the spae ontinuous
analyses with the proposed method. The numberof prognosti variablesis P =4 in the
sense of B03, and the spae ontinuous state-vetor is X = (X
1
;:::;X
4
) = (D;d;
e
T;P).
The linear operator l in [C1℄ involves l
1
=
~
, l
3
= r, and l
4
= (
~
+I)r, respetively
appliedto(11),(13),and(14),asinsetion7ofBLVS04. The"unbounded"linearsystem
then beomes:
~
D
t
= R Trr 0
"
e
T
T (
~
+I)P
#
(17)
d
t
= g
2
R T
~
(
~
+I)P (18)
r
e
T
t
=
R T
C
v (r
0
D+rd) (19)
(
~
+I)r P
t
= r
0
D C
p
C
v (
~
+I)(r 0
D+rd) (20)
The struture equation, whihallows todetermine the time-ontinuousnormal modes of
this linear system writes:
1
2
4
t 4
+
2
t 2
r 02
+
~
(
~
+I)
H 2
!
+N 2
r 02
=0 (21)
where:
2
= C
p
C
v
R T (22)
N 2
= g
C
p T
(23)
This struture equationappears to beformallysimilar tothe struture equation without
orographyinBLVS04, exeptthat theoriginalhorizontalgradientoperator risreplaed
by r 0
. The determination of the ontinuous normal modes thus follows diretly from
the ase without orography: the ondition [C'2℄ requires T > 0 and the struture of the
normal modes of the ontinuous system is then given by:
X
j
(x;)=
X
j exp
(ik
r +
s
2H )x
(i 1=2)
forj 2(1;:::;4) (24)
where
X
j
is the omplex magnitude for the onsidered variable, and (k
r
;) 2 IR. Note
that is a non-dimensional vertial wave-number, and =2 represents a mode with a
vertial wavelength equal to the harateristi height H of the atmosphere (here and in
later similarusesrelatedtowaves geometry, isof ourse3.1415...). Thereal value =k
r
represents the horizontalhalf-wavelengthof the mode, that is, the distane between two
onseutive zeros of the real part of the omplex mode along the x diretion. For suh
a linear pertubation X
j
(x;), the energy density of the perturbation is proportional to
X 2
j
where=(=R T)(see e.g. Bannon,1995). This energydensity deomposesitselfin
threeparts: kineti,availablepotential,andavailableelastienergy density. Thenormal-
modes(24)haveanhorizontalvariationalongonstantsurfaeswhihisonsistentwith
the growth of the modes with height due to the Boussinesq eet and with the elevation
of the terrain along the x axis. The normal modes X
j
thus have a uniform amplitude
along true horizon tal surfaes only. However, the energy density of these normal modes
is spatially uniform, as in the at-terrain ase, due to the ompensating variation of the
mass density :
r(X
j )=
~
( X
j
)=0 (25)
Foran eigenmode haraterized by (k
r
;),the eigenvalues of derivative operators are:
r =
ik
r +
s
2H
=ik (26)
~
=
i 1
2
(27)
r 0
= i
k
r +
s
H
=ik 0
: (28)
The veriation of [C3℄[C4℄ proeeds easily, asin the ase withoutorography. It should
benotedthatforthereferenesystemlL
,theroperatorisusedeverywhere, thusleading
tothe eigenvalue found in(26). For [C3℄, wehave:
1
= (i 1=2) (29)
3
= ik (30)
4
= ik(i +1=2); (31)
and for [C4℄:
13
= R kk 0
(32)
13
= R k 2
(33)
14
= R Tkk 0
(i +1=2) (34)
14
= R T
k 2
(i +1=2) (35)
24
= g
2
R T (
2
+1=4) (36)
24
= g
2
R T
( 2
+1=4) (37)
31
=
C
v ik
0
(38)
32
=
R T
C
v
ik (39)
31
=
32
= R T
C
v
ik (40)
41
= ik 0
1 C
p
C
v
(i +1=2)
(41)
41
= ik
1 C
p
C
v
(i +1=2)
(42)
42
= =
42
= ik C
p
C
v
(i+1=2): (43)
All other
ij
oeients are zero, and [C1℄[C4℄ are nally fullled.
The analysis then proeeds as for the at-terrain ase in BLVS04: For the 3-TL SI
sheme, a numerialgrowth-rate is introdued through:
X(t) = X(t t) (44)
X(t+t) = 2
X(t t); (45)
(46)
and the stability equationan beexpressed as inBLVS04:
Det(M)=0; (47)
where M is given by (46)(49) of BLVS04, used with the above values of
j
and
ij .
This eighth-degree stability polynomial equation in an be solved numerially: for
any geometrial struture dened by a pair (k
r
, ), the modulus of the eight roots
1 (k
r
;);:::;
8 (k
r
;) gives the growth rate of the eight orresponding eigenmodes. The
growth-rateof thegeometrialstruture(k
r
,)isthendened by themaximummodulus
of the eight roots:
r 1 r 8 r
If one of the roots has a modulus larger than one, then the orresponding geometrial
struture is unstable.
In this paper, the asymptoti growth-rate for a given geometrial struture (k
r , ) is
denedbythevalueoftheabovegrowth-rateinthelimitoflargetime-steps. Asdisussed
inB03,examinationofasymptotigrowth-ratesisrelevantsineSIshemesareusedwith
long time-steps in pratie, and thus, asymptotigrowth-rates provide agoodindiation
of the robustness of a sheme independently of the partiular value of the time-step.
For onveniene, aparameter for the thermalnonlinearityan beintroduedthrough
=
T T
T
: (49)
As an illustration of the results, the asymptoti growth-rate for the kilometri sale
horizontal mode with k
r
= 0:00157 m 1
and for T = 280 K is depited in Fig. 1 as
a funtion of and s. The growth-rate whih is plotted is the maximum growth-rate
obtainedwhenrepeatingtheaboveanalysisfordisretevaluesof desribingthe interval
[2 , 100℄. This interval represents vertial wavelenths varying between 500 m and H.
In pratie however the maximum growth-rate for this gure is reahed for the shortest
vertial mode = 100 (not shown). The domain where the growth-rate is smaller than
1.1 is restrited to a very small area along the axes, whih means that for ows with
signiantvaluesofthe slopeandthe thermalnonlinearity,the shemeishighlyunstable.
