Full text of DOI:10.1007/BF03183558

tained by the semi-experimental theoretical analysis
method are usually coarse and loose[3—7]. In order to hold
the original properties of the system, other constraint
methods need to be used. The energy constraint[8] is sim-
ply an effective method with clear physical sense. The
method, however, is only available for the adiabatic sys-
tem without friction.
Operator constraint principle
for simplifying atmospheric
dynamical equations
LI Jianping1, 2 & CHOU Jifan2
To make up for the lack of the scale analysis and get
consistently simplified dynamical equations, an operator
constraint method is proposed in this study. This method
is an extension of the energy constraint. The method is
based on the fact that the atmospheric system is an essen-
tially dissipative structure and is constructed on the quali-
1. National Key Laboratory of Atmospheric Sciences and Geophysical
Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese
Academy of Sciences, Beijing 100029, China;
2. Department of Atmospheric Sciences, Lanzhou University, Lanzhou
730000, China
Correspondence should be addressed to Li Jianping (email: ljp@lasg.
iap.ac.cn)
ˉ
19]1)
tative theory of atmospheric dynamical equations[9
,
and abides by the rule that the properties of corresponding
operators in the original and simplified equations should
be kept unchanged, thereby the simplified equations ob-
tained will not distort the essential properties of original
equations.
Abstract
Based on the qualitative theory of atmospheric
dynamical equations, a new method for simplifying equa-
tions, the operator constraint principle, is presented. The
general rule of the method and its mathematical strictness
are discussed. Moreover, the way that how to use the method
to simplify equations rationally and how to get the simplified
equations with harmonious and consistent dynamics is given.
1 General principle of the operator constraint
method
Keywords: simplification, operator equation, operator constraint
principle.
The full atmospheric dynamical equations can be
rewritten as an equivalent operator equation in Hilbert
space[17, 18] 1)
The atmospheric dynamical equations are very
complicated nonlinear, non-stationary and compressible
partial differential equations with dissipation and external
forcing[1—4]. They describe the various spatial-temporal
motions in the atmosphere. At present, many difficulties
cannot be overcome to solve the equations analytically
under the proper initial-boundary value conditions. How-
ever, the special spatial-temporal motion in the atmos-
phere has its own way, so some factors could be neglected
when we simplify equations for particular atmospheric
motion. This is not only easy for us to make mathematical
analysis, but we can also lay stress on the essential of mo-
tion and grasp the heart of problem.
The scale analysis developed since the 1940s has
been a main and broadly adoptive method with semi-
experimental property for simplifying equations[1—7]. The
method assumes that each term in the same equation does
not possess equal importance, thereby equations may be
simplified according to the rule that the minor terms are
left out by comparing the order of magnitude of every
term in the same equation. An apparent deficiency of the
scale analysis is that the method only compares the order
of magnitude of each term in the same equation and dis-
regards the links between equations in the original equa-
tions. Consequently, we probably get a set of inconsistent
simplified equations without consistent properties in the
original equations. Applying the method to simplifying
equations, sometimes we can find that the same term is
regarded as a secondary term and omitted in a certain
equation, but it is retained in another one. It is difficult to
explain this result in physical sense, therefore, results ob-
:
wM
-
°
ꢀ N(M)M ꢀ L(M)M [
,
(1)
®
wt
°
M |t 0 M0
¯
where the vector function
~
~
~ ~ ~
the sym-
c
M (u,v, w, U,T ) ,
~
*
*
*
~
~ ~ ~
bol c denotes transposition,
c
(u,v, w, U,T ) U (u ,v ,
*
*
*
*
w ,I ,T ) , u u 2 , v* v 2 , w* w 2 , I*
c
*
ˆ
,
,
, where u, v and
I
U*
U
T CV T CVT
w are zonal, meridional and vertical winds respectively,
Uꢁand T denote density of air and air temperature respec-
tively, Cv represents specific heat (for the concrete form of
the operators N(M) and L(M) see refs. [17, 18] and footnote
M
1)). The operator N( ) represents the nonlinear advection,
Coriolis force, pressure-gradient force, gravity, spherical
curvature, etc. L(M) embodies the effects of dissipative
terms. From the most universal sense, the abstract opera-
tor N(M) is an anti-adjoint and antisymmetrical operatorˈ
the L(M) is a self-adjoint and symmetrical operator,
namely
*
,
N(M) ꢂN (M) (N(M)M1,M2 ) ꢂ(M1, N(M)M2 ),
;
(N(M1)M,M) 0
*
,
L(M) L (M) (L(M)M1,M2 ) (M1, L(M)M2 ),
ı0.
(L(M)M,M)
1) Li Jianping, Qualitative theory of the dynamical equations of
atmospheric and oceanic motion and its applications, Ph. D. Dissertation
(in Chinese), Lanzhou University, 1997, 209.
Chinese Science Bulletin Vol. 46 No. 1 January 2001
1053

NOTES
ꢃM,M1,M2 
H
0 (:)
, where H
0 (:)
is a complete Hilbert
anti-symmetric property of
. Otherwise, there must
N(M)
space with the following inner product and norm
be false “sources” or “sinks”, thereby elementary proper-
ties of the original system are lost. Considering the system
without dissipations and forcings, i.e. the adiabatic non-
friction system, eq. (1) becomes
c
ꢄM1,M2
ꢅ
M1M2d:
³
:
ꢀ
2ʌ
ʌ
rf
2
c
M1M2r sinT dr dT dO,
wM
³ ³ ³
.
(4)
0
0
rs
ꢀ N(M)M 0
wt
Making H inner production with , the energy conserva-
tion follows immediately, namely
(M ,M)1 2
.
M
0
M
0
The physical senses implied by the properties of N(M) and
L(M) as mentioned above are that N(M) represents the
various kinds of reversible processes of energy conserva-
tion and L(M) shows the irreversible processes of energy
dissipation. For a simplified equation, its corresponding
operators should hold the properties of N(M) and L(M) so
that the essential physical properties of the original equa-
tions will not be broken down. This is simply the operator
constraint principle for simplifying equations. Let the
simplified equation be
.
(5)
M(t) M0
For any simplified equation of (4),
~
wM
~
~ ~
ꢀ N(M)M 0
,
(6)
wt
still holds the anti-adjoint properties of
~
~
if
N(M )
N(M),
it results immediately by making H0 inner production with
M that there is also the energy conservation in the simpli-
fied equation (6), i.e.
~
~
wM
~
~
.
(7)
This indicates that eq. (6) does not break down the energy
M (t) M0
.
(2)
ꢀ N(M)M ꢀ L(M)M [
wt
According to the rule mentioned above, the simplified
~
conservation in the original equation. If
does not
N
~
operator
~
should be an anti-adjoint operator and
N(M)
possess the anti-adjoint property of N, it can be proved
that the simplification destroys the energy conservation
property of eq. (4). There are many examples such as the
barotropic non-divergent model, the linear barotropic
model, the shallow water model, and the p-coordinate
primitive equation model, which satisfy the principle as
indicated earlier. Additionally, two improper examples
from the scale analysis are also given because they do not
have the above properties.
It can be concluded from the above analysis that the
energy constraint method is a special case of the operator
constraint method and in the process of simplification the
operator constraint method is more concise and much
clearer than the energy method.
should be self-adjoint. We can prove that there is
L(M)
the same asymptotic behavior in eqs. (1) and (2). This
shows that there is no false source or sink and there is still
a global attractor in the simplified equation obtained by
the principle. The simplified equation is still a “dissipative
structure”, which does not destroy the asymptotic behav-
ior of the solution of original equations and holds the
whole properties and physical laws of original equations.
Hence, the simplified equations obtained by the operator
constraint principle are still very strict and their dynamical
relations are harmonious and consistent. For example, the
large-scale atmospheric equations can be written as fol-
lows[9—11, 18]
:
3
Simplification of the operator L
wBM
wt
.
(3)
ꢀ Nl (M)M ꢀ LlM [
The operator L(M) is an asymmetric and positive
~
The operators
properties of
and in eq. (3) do not lose the
Ll
Nl (M)
operator, so is the simplified operator
.
L(M)
and L(M)[9—11, 18], so there is the same
Let lij be a certain term in L(M) and lji the asymmetric
term of lij. If lij is left out, lji must be omitted. Otherwise,
the asymmetric and non-negative property of L(M) will be
lost, thereby physical properties of the original system are
distorted. In the following two simplifications used are
discussed. The boundary conditions may be either with or
N(M)
long-time behavior between eqs. (3) and (1) and the sim-
plification is reasonable.
2
Simplification of the operator N
According to the discussions stated above,
is
N(M)
without topography[17, 18] 1)
.
an anti-symmetric operator, so is the simplified operator
~
Because the turbulent viscosity force is much greater
than the molecule viscosity force, the molecule viscosity
in the motion equations is usually neglected while dissipa-
tion is taken account of. After the molecule viscosity is
.
N(M)
Let nij be a term in
and nji the anti-symmetric
N(M)
term of nij. To the problem considered and properties of
the objective studied, if nij is regarded as minor and can be
neglected, nji should also be omitted in order to hold the
1) See the footnote on page of .
1054
Chinese Science Bulletin Vol. 46 No. 1 January 2001

~
omitted, the operator
can be rewritten as follows:
L(M)
that is to say,
prove whether
is asymmetric. Then, we need to
L(M)
ꢂ l
0
ꢂ l
0
0
0
0
0
0
0
0
0
0
ª
º
»
»
»
»
»
~
«
«
«
«
«
ı0.
(14)
ꢄ
L(M)M,M
ꢅ
0
0
0
0
~
The answer is positive. However, to prove formula (14)
~
,
(8)
L(M)
ꢂ l
0
0
we need to introduce a new Hilbert
H1(:)
complete space with the following norm
, which is a
0
0
«
¬
»
¼
0
0
ꢂ l5
1 2
2
1
~
2
2
2
2
§
©
·
¸
¹
~
~
~
~
where l and l5 are:
,
(15)
M
u ꢀ v 1 ꢀ w1 ꢀ U ꢀ T
¨
1 0
1
1
w
kO
w
1
U*
l
U*r sinT
where |g|1 takes the norm in the Hilbert space
.
V (:)
wO r sinT wO
is a complete space with the following norm
V (:)
1
w kT sinT w
1
U*
ꢀ
1/ 2
U*r sinT
1
U*r2
1
2
wT
r
wT
§
¨
·
¸
ª
º
wf
wr
§
¨
·
¸
2
.
(16)
f
« f
ꢀ
d:»
w
wr
w 1
wr
1
³
¨
¸
:
,
«
©
¹
»
ꢀ
krr2
¬
¼
U*
©
¹
Using the new space, we can prove formula (14). This
indicates that simplification can only alter the state space
but not the properties of operators, and consequently, it is
clear that the simplified equation does not distort the
evolution law of the original equation in the qualitative
sense.
w
KO
w
1
U*
l5
U*r sinT
wO r sinT wO
1
w KT sinT w
1
U*
ꢀ
U*r sinT
wT
r
wT
2
1
w
wr
w 1
wr
U
~
Kr r 2
ꢂ CKDS TS 2T
4
Summary
,
ꢀ
*
*
U r2
In the theoretical study of atmospheric dynamics the
respectively; kO, kT and kr represent turbulent viscosity
coefficients in horizontal and vertical directions respec-
tively; KO, KT, Kr are turbulent conductivity in horizontal
primitive equations often need to be simplified. Simplifi-
cation changes state space, but properties of simplified
operators should hold those of original operators. The rule
can assure that simplification does not distort elementary
physical laws and global properties of the original system.
This is just the operator constraint principle presented by
this note for simplifying atmospheric dynamics equations.
According to our discussions, this method has a strict
mathematical basis and definite physical sense. Simplified
equations with harmonious and consistent dynamics can
be obtained by applying the combined method of the scale
analysis and the operator constraint principle. It should be
noticed that, however, if we study the phenomenon related
to long-term process of the atmosphere, according to the
operator principle the simplified system should still be a
forced dissipative nonlinear system, which should neither
be an adiabatic system without friction nor a linear sys-
tem[11—19]. Only in this way do the simplified equations
not distort the long-time behavior of the original system
and also can better results be obtained.
and vertical directions, respectively. It is easy to prove
~
that there are the same properties between
and
L(M)
L(M). Therefore, the simplification is appropriate.
After the molecule viscosity is left out, it is common
to think that the turbulent viscosity in horizontal direction
is much less than that in vertical direction. As a result, the
turbulent viscosity in horizontal direction is also omitted.
Is this simplification true? The analysis shows that after
neglecting the turbulent viscosity in horizontal direction,
L(M) becomes
~
ª
«
«
«
«
«
º
»
»
»
»
»
ꢂ l
0
0
0
0
0
0
0
0
0
0
0
0
0
~
ꢂ l
0
~
~
,
(9)
L(M)
0
ꢂ l
0
0
0
~
«
¬
»
¼
0
0
0
ꢂ l5
Acknowledgements This work was jointly supported by the Knowl-
edge Innovation Key Project of Chinese Academy of Sciences (Grant No.
KZCX2-203), the National Key Basic Research Project (G1998040900),
the National Natural Science Foundation of China (Grant Nos. 40023001,
49905007 and 49805006), the Innovation Project of IAP/CAS (8-1301)
and the President Foundation of the Chinese Academy of Sciences.
where
~
1
w
w
1
*
,
(10)
(11)
l
kr
*
U r2 wr wr U
~
1
w
wr
w
wr
1
U*
~
.
l5
Kr
ꢂ CKDs Ts 2T 2
U*r2
It may be proved that
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~
~
,
(12)
ꢄ
L(M)M1,M2
ꢅ
ꢄ
L(M)M2 ,M1
ꢅ
Chinese Science Bulletin Vol. 46 No. 1 January 2001
1055

NOTES
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(Received February 5, 2001)
1056
Chinese Science Bulletin Vol. 46 No. 1 January 2001