A more comprehensive model for stratospheric chemistry and transport theory was developed for the purpose of aiding predictions of changes in the stratospheric ozone content as a consequence of natural and anthropogenic processes. This new and more advanced stratospheric model is time dependent and the dependent variables are zonal means of the relevant meteorological quantities which are functions of latitude and height. The model was constructed by the best mathematical approach on a large IBM S360 in American National Standard FORTRAN. It will be both a scientific tool and an assessment device used to evaluate other models. The interactions of dynamics, photochemistry and radiation in the stratosphere can be governed by a set of fundamental dynamical equations

Chen, H. C.

English

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https://ntrs.nasa.gov/search.jsp?R=19780013749 2020-03-22T04:00:50+00:00Z
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(NASA-CR-156722) SOFTWARE DEVELOPMENT: 	 M78-21692
STRATOSPHERE MODELING Final Report (Systems
and Applied sciences Corp.) 14 p
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SOFTWARE DEVELOPMENT
FOR
STRATOSPHERIC MODELING
(Final Report)
1
R-SAD-12 /77- 34
Software Development
for
Stratospheric Modeling
(Final Report)
Prepared for:
John Campbell
NASA Code 624
Goddard Space Flight Center
Greenbelt, Maryland 20771
December 197T
Systems and Applied Sciences Corporation
6811 Kenilworth Avenue, Suite
Riverdale, Maryland 20840
(301) 699-5400
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1. Report No.
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Oov,,nment Accession No. 3. Recipient's Catalog No.
4. Title an	 uolit a
Software Development -Stratosphere Modeling
I
5. Repair	 ate
December 9, 1977
6. Performing Orgonixotion Code
7. Authors)
I	 Dr. H. C. Chen
B. P rfa	 r	 0	 an' a tare Report No.
I"9WI3-if71V'" 34
9• Performing Organisation Nome and Address
Systems & Applied Sciences Corporation
6811 Kenilworth Ave., Suite 610, 111verdale, MI
10. Work Unit No.
11
	 Contract
13. Type of Report and Period Covered
Final Project ReportU. Sponsoring Agency Name and Address
Goddard Space Flight Center
Greenbelt, Maryland 20771 	 114. Sponsoring Agency Code
626
15. Supplementary Nob,
16. Abstract
17• Key Words (S. bcted by Author(,)) 10. Distribution Statement
19. Security Clm,t l. (04 this report) 20. Security C10ee11. (of 161s pope) 21. No. of Page, 22. Price•
FOREWORD
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This report describes work performed by Systems and Applied
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Sciences Corporation (SAS) under NASA Contract No. NAS5-24255. The work
involves development of an improved two-dimensional model of chemical
k
and physical processes in the stratosphere.
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TABLE OF CONTENTS
Page
FOREWORD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
SECTION 1 — INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1-1
SECTION 2 — BASIC EQUATIONS . . . . . . . . . . . . . . . . . . . 2-1
SECTION 3 — NUMERICAL METHOD . . . . . . . . . . . . . . . . . 3-1
SECTION 4 — PRELIMINARY RESULTS AND DISCUSSION . . . . . . . . 4-1
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SECTION 1 - INTRODUCTION
A more comprehensive model for stratospheric chemistry and transport
theory is being developed for the purpose of aiding predictions of charges in the
stratospheric ozone content as a consequence of natural and anthropogenic
processes. This new and more advanced stratospheric model is time dependent
and the dependent variables are zonal means of the relevant meteorological
quantities which are functions of latitude and height. The model is constructed
by the best mathematical approach on a large IBM 5360 in American National
Standard FORTRAN. It will be both a scientific tool and an assessment device
used to evaluate other models. Program blocks are kept under rt maximum of
100 statements per routine and each routine has but one function. The report
sets out the detailed formulation of a numerical model both in physics and
mathematics. The interactions of dynamics, photochemistry and radiation
in the stratosphere can be governed by a set of fundamental dynamical equations
which will be described in Section 2. Some physical ideal is obtained from ar_
existing FORTRAN computer source deck of Crutzen's two-dimensional model
of the stratosphere which is subsequently applied in building new and more
comprehensive models. Section 3 describes the numerical method used in the
integration. Some preliminary results and discussions are presented in Section 4.
ORIGINAL PAGE IS
OF POOR QUALITY1-1
nL
2-1
SECTION 2 - BASIC EQUATIONS
The model is an integration of an extensive set of fimdamental equations
which can be used to predict the future state of the atmospheric circulation from
lmowledge of its present state. In an (x, y,e) coordinate system in which x =
distance eastwards, y = distance northwards, and = log (p o/p); with po = 1000 mb
and p = pressure, u, v, and w are the three corresponding velocity components
In
	
	
respectively. In this log-pressure system the eastward component of the
momentum equation is:
	
a) u+uax u+vay u+w L u =ua tan co+2Ov sin 0- a—Z 	 (1)
where a = radius of the earth, St = rate of rotation of the earth, cp = latitude,
Z is geopotential. The thermodynamic equation is:
ag+u a g+v a g + wa6 =q	 (2)
at	 ax	 ay	 3 
where B = T (p o/p)k , T = temperature, q is the rate of change of potential
temperature, K is the ratio of gas constant for dry air R to specific heat of air
at constant pressure C . The equation for conservation of matter is:
10
Since our aim is to deal explicitly with zonal mean quantities only.
Accordingly let a over bar denote the zonal mean and a dash denote the departure
therefrom, then equations (1) to (3) yield, after taking an average around a
latitude circle:
att (T a' COS c^1) + ay(V T) + a (W T) _ - a N^ T ^) - a (WIT )	 (4)
at (De COS c0)+ a+ (W B) =qe ^COS (P- ay(^/)-a (77r-OT)
(5)
a P 
-F -^- 'v = oay a
where T = (D a cos cp + u) a cos cp
V=ve ^ coscp
W=we ^ coscp
Note that the continuity equation (3) takes the simple form (6) which is the
advantage of using the perssure coordinates.
Equations (4) to (6) may become a complete set by adding the y
component momentum and hydrostatic equations. For our own purpose of not
introducing the gravity waves we introduce an approximation to the y component
2-2
ORIGINAL PAGE IS
OF POOR QUALITY
w
(6)
2-3
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momentum equation that the zonal mean wind is in geostrophic equilibrium with
the zonal mean geopotential field. Combining this with the hydrostatic equation
leads to the thermal wind relationship:
3_ :r __Rexp(-k^)	 aB
aj	 2Q tancO	 ay
Equations (4) to (7) can be regarded as a closed set which completely
determines the relationships among the dependent variables r, B, V, W. The
other variables can be written in terms of the relevant quantities and are held
as functions of time. latitude and altitude.
I.
(7)
SECTION 3 - NUMERICAL METHOD
It is convenient to make the equations dimensionless. The scaling quantities
is given a subscript s. Equations (4) to (7) become:
C) t (
T a^s^oos(p)+ (TV) +a (TW) =H 	 (8)
C)
 
t ( B e 
^s^cos (p) + app (8 V) + a (8 ^^ = q el sx cos cp+ Q	 (9)
d -
	
e-k s	 00-
a	 T = - tan ep
	 09	 (10)
a^	 w=
 09
	
(11)
where H and Q are the divergence of eddy momentum and eddy heat flux,
respectively. * is a stream function which is introduced to rewrite equation (6)
in the form of equation (11). The method of solution is to eliminate the time
derivatives from (8) and (9) using (10). Substituting for V and W from (11)
gives an equation for >Ir .
_	
0 e k ks	 [Es _	 0V e	 l +sing(p)1
-'V^^^ + 2 TC ^'W + tann W ^^P^P  	 Tfp +
	sin 9	 J q'9 +
L 
s T  + e k soC + kts e k sOcp/tart cO
1	
(p = HE + t;s
 H +
eke*	 e-44 scos  (p q^0+ ((p + Q tan 91 /tan tp	 (12)
ORIGINAL PAGE
OF PAR QUAtITY
3-1
w
1
The ^ and op used as subscripts denote differentiation with respect to that
.luantity. It may be solved with suitable boundary conditions for * which may
be used in (8) and (9) to get new values of T, 6 at the advanced time step.
Equation (12) is then solved at the new time value and the process repeated.
A Crank-Nicholson implicit finite difference scheme is used in the
numerical calculations. The Alternate Direction Implicit method is adopted so
that the finite difference equation can be solved by simply inverting a tridiagonal
matrix in each of the two dimensions sequentially. The nonlinear coefficients
in the equations can be taken into account by using a predictor - corrector
procedure. A more detailed description is followed.
Eq. (12) is an elliptic type, second order partial differential equation which
can be solved by running a marching method for the time-dependent equation out
to a sufficiently large tim-. Equations (8) and (9) are time-dependent first order
partial differential equations. A general form for the set of equations can be
written as:
St = A Sig + B No + C Soo + D Sg + E So + F	 (13)
The ADI scheme for equation (13) is the following:
S n+2 =S n +AtIA6 (Sn+)+BSlo+C6o(Sn)
+ D 6, (Sn+112 ) + E 6(P (Sn ) + F1	(14)
3-2
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Sn+1 = Sn + Ot r A 6gg2
 (Sn+z ) + B S n + C 6^ (Sn+l)
+ D 6 (Sn+ Z) + E 6(, (Sn+1 ) + F^	 (15)
where:
6	 Sjk = (Sj+l,k - 2 Sj,k + Sj-1,k )/A^
26^	 Sjk = (Sj,k+1 - 2S j k +S J k-1 )/Q4)2
6^ Sjk - (Sj+l , k - Sj-1,k OA2
60 SJ.k = (Sj.k+1 - Sj.k-1 )/2A(p
A simple tridiagonal matrix inversion will solve equations (14) and (15) to give
the solution of equation (13).
3-3
EN
E
N
0
SECTION 4 - PRELIZiIINARY RESULTS AND DISCUSSIONS
Equation (12) is an elliptic type, boundary value problem so that the crossing
term 
'^o 
is not a dominant term which can be approximated at the old time step
as shown in equation (14) and (15). This enables us to use the Crank-Nicholson
'cheme and the ADI method to solve equation ( 12) without changing the coordinates
for getting rid of the crossing term. Equation (12) has been solved without over
100 total time steps before it reached the steady state solution.
We have taken the average for each two consecutive values of IP in the
process of getting fast convergence. The values of ^/ are then used to determine
V and W which are substituted in equations (8) and (9) for solving r and 0 at the
advanced time step. This process has been repeated many times.
The terms in the right hand side of equation (12) are source functions which
drive the rate of heating due to absorption of solar radiation and the rate of cooling
by thermal radiation. Eddy fluxes of heat and momentum are treated by large-
scale diffusion coefficients. For these fixed driving terms the solutions of the
set of equations do not converge very well. It may be due primarily to the poor
initial data available. Therefore, we have to find the steady state solution first
before seeking any time-dependent solutions.
M
Y
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11
Q
4-1
ORIGINAL PAGE IS
OF POOR QUALITZ


Software development: Stratosphere modeling

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