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NCAR/TN-360+STRNCAR TECHNICAL NOTE
May 1991
The Energetics ofStochastic Continuum Equationsfor Fluid Systems
REXJ. FLEMING
CLIMATE AND GLOBAL DYNAMICS DIVISION
NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO
. . I I
TABLE OF CONTENTS
Preface . . . . . . . . . . . . . .
Acknowledgments . . . . . .
1. Introduction .........
2. Stochastic continuum equations
3. Energetics .......... .
4. Comments and conclusions
References . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . .
. . . . . . . . . . . . . .
4
.. . . . . . . . . . . .. 21
... . . . . . . . . . . . 34
... . . . . . . . . . . . 41
.. . . . . . . . .. . . . 43
iii
PREFACE
The numerical solution of the relevant prognostic equations for fluid systems involves
sources of uncertainty in the initial conditions and uncertainty in the external forces applied
to a physical system. The purpose of this note is to introduce stochastic continuum
equations for fluid systems that express this uncertainty dynamically. These equations,
written in analytical form, describe continuous field quantities which dynamically predict
the future and its believability. Beginning with the Navier-Stokes equations and express-
ing uncertainty as continuous field quantities, one avoids the serious shortcomings and
computational redundancy of previous methods using discrete amplitudes (gridpoints or
orthogonal functions). No assumptions are made concerning the original deterministic
equations, which predict the evolution of a single point in phase space. Rather, these are
a subset of the stochastic continuum equations, which predict an infinite cloud of points
in phase space. The amount of detailed structure in the shape of the cloud depends upon
the degree of derivative closure used in the continuum equations.
An examination of the dynamic growth of uncertainty was done "locally" by examining
the dominant terms in the field equations and "globally" by deriving the energetics of the
equations over the fluid volume. For those systems in which the energy expressions are
nonlinear cubic, the full energetics of these equations reveal the role of third moment
prognostic equations. The effects of derivative closure were investigated analytically.
The solution of the stochastic continuum equations offers a tremendous computational
improvement over previous fully stochastic dynamic methods. The equations are perfectly
suited to the new emerging parallel computer architecture. For large scale models like the
NCAR CCM, the continuum equations reduce the computational burden by a factor of 103
for second moments only, and by a factor of 106 for third moments. The computational
ratio for stochastic continuum to deterministic is still 0 (100). However, for beginning
research using second moments only, the ratio is 0 (12).
ACKNOWLEDGMENTS
This work was supported by the National Center for Atmospheric Research, sponsored
by the National Science Foundation, and by NSF Grant ATM-8805967. The author is
grateful to A. Kasahara and P.D. Thompson for their interest in the research. Drafting
of the figures was performed by the NCAR Graphics Department and the manuscript was
typeset by R. Bailey and E. Boettner.
v
1. INTRODUCTION
The formulation of the relevant prognostic equations for fluid systems (liquids and
gases) in geophysical applications involves a number of assumptions and hypotheses. The
numerical solution of these equations for realistic initial-value problems introduces an-
other level of uncertainty. The purpose of this note is to introduce stochastic continuum
equations for fluid systems that express this uncertainty dynamically.
These stochastic equations will be written in analytical form, describing continuous
field quantities, just as the deterministic hydrodynamic equations of Euler or those which
we have come to call "Navier-Stokes". In short, these equations dynamically predict
the future and its believability. These equations are applicable to the most complex
models and, in such cases, substantially reduce the computational burden over previous full
stochastic-treatments. With the new "fine-grain" parallel processing computer systems,
these stochastic continuum equations will be solvable in the very near future.
Major socio-economic benefits could accrue for users of environmental information if
dynamic predictions were accompanied by a dynamics-based estimate of how good those
projections were at the precise time in question. Deterministic predictions have been
providing only "half' the solution-the answer (or future state vector) as a function of
time. The "other half' of the problem is to determine the believability or variance of that
answer (based upon the same dynamics that produced the solution, but now recognizing
the uncertainties in the equations, the model and the initial conditions.) Better decisions
can be made when both pieces of information are available.
One methodology for providing this real-time predictability for low-order atmospheric
models has existed for some time. However, the computation required for the approach
(described below) has limited its use to a few academic exercises. The research described
here hopes to reverse this situation in two ways. First, the equations are in field form,
1
which more clearly show the dynamic relationships which cause uncertainty to grow in a
dynamic system. Second, the equations are formulated to take maximum advantage of
emerging parallel computer systems.
Equations which express the dependent variables in discrete form (in terms of a
regular mesh of points or in terms of orthogonal functions) and which subsequently pre-
dict covariance relationships for the resultant discrete amplitudes, were independently
derived by Epstein (1969) and Tatarskiy (1969). Epstein called his formulation of the
problem "stochastic dynamic prediction" and used Gleeson's (1968) continuity equation
for probability (similar to the Liouville equation for particles in phase space) as a basis. In
this stochastic dynamic approach the equations and physics were assumed to be perfect;
the uncertainty was in the initial conditions of the discrete amplitudes representing the
dependent variables. The method predicts the ensemble of possible initial states via a set
of coupled hierarchical moment equations.
A brief description of stochastic dynamic prediction is provided below. Given that a
numerical model has N dependent variable (grid points or spectral coefficients that are
functions of time), one can consider that these variables form an N-dimensional phase
space. A deterministic model predicts the evolution of a single point (the initial conditions
of the dependent variables) in this phase space. The stochastic dynamic equations predict
the evolution of an infinite ensemble of points over the space.
The stochastic equations for the expected values of the dependent variables (the
mean of the ensemble) contain the same terms as the original deterministic equations (the
deterministic equations can be considered a subset of the stochastic dynamic equations)
plus additional second moment (covariance) terms. The solution of the mean equations
is more accurate than the solution of the deterministic equations in a root-mean-square
sense.
2
Prognostic equations for the second moment terms involve third moments. Prognostic
equations for third moments involve fourth, etc. The system of equations is closed by some
approximation which assumes that moments of degree n + 1 in a given equation sum to
zero, or are expressible as combinations of moments of degree n. A further contribution
to the theory behind the stochastic dynamic equation set was established by considering
the energetics of the system, the closure of the equation set, and the impact of the closure
on the energetics and on predictability (Fleming, 1971a, 1971b).
The most serious limitation in the stochastic dynamic method has been the amount of
computation involved. This has hindered its implementation, and even frustrated research
efforts to make progress toward eventual implementation. If N dependent variables (grid
points or spectral coefficients) are predicted in the deterministic equations, then N(N+1)/2
covariance equations are required in the stochastic case. Thus, in a modest primitive
equation model with just 20,000 variables representing velocity, temperature, etc., in a
three-dimensional space, the number of covariance equations would be over 200 million.
It is the purpose of this note to introduce the stochastic continuum equations and
describe their energetics. In Section 2 we derive the stochastic continuum equations in
a general form-applicable to many problems and to different coordinate systems. The
analytical form of the stochastic continuum equations requires higher order derivative
expressions of the original deterministic equations. The important question of derivative
closure is discussed in Section 2, in terms of a Taylor series expansion, and later in Section 3,
in terms of conserved energy quantities. Section 3 derives the energetic relationships that
exist within the stochastic continuum equations. Conclusions are stated in Section 4, where
an estimate is provided for the amount of calculation required for particular applications
of the stochastic continuum equations. The method of solution, consistent with the new
generation of "fine-grain" parallel processors, is outlined.
3
2. STOCHASTIC CONTINUUM EQUATIONS
a. Derivation
We will see below that if the deterministic equations are nonlinear, then the mere
mathematical operation of taking the expected value of both sides of the prognostic
deterministic equations leads to a set of coupled hierarchical moment equations. Epstein
(1969) justified the use of the expected value operator, in view of the uncertainty in the
initial conditions of the discrete dependent variables. This justification is still valid and
allows us to use the same operator in the continuous equations. However, we should remind
ourselves that the stochastic nature of our prediction problem can be linked back to first
principles-even with "perfect" velocity measurements.
Our usual expression for the "convective derivative" is actually an approximation.
This term is, of course, the second term on the r.h.s. of the sample equations below
dt A+(1)
dt t
where qf is a scalar and V is the velocity vector. In the formulation of this term (e.g.
see Prandtl and Tietjens, 1934) there is a truncation of a Taylor series expansion in the
expression for the velocity deformation tensor. It is usually assumed that there is a region
small enough surrounding a point pa in which the velocity is linearly dependent on the
distance from pa (thus eliminating second order and higher order terms in the Taylor series
expansion above). With this assumption of "homogeneous deformation," the convective
derivative reduces to V. VV, as in Eq. (1). This leads to a very small uncertainty when
this derivative is evaluated over distances greater than the calculus would dictate. Pielke
(1984) expresses his view that, strictly speaking, the equations for atmospheric motion
apply to space scales on the order of about a centimeter and time scales of a second or
4
so. Thus, over and above the usual errors associated with truncation and subgrid scale
dynamics, there is an element of uncertainty (however small) introduced by the nonlinear
term V V'V itself.
Formally, if one were to take the expected value of the Navier-Stokes equations
(considering only the x-component of velocity below)
Ou Ouat = -u- +... other terms...
one would haveAu Au / Au
%=-u -- cov yu, +... (2)
where bold-faced type indicates a mean quantity or expected value and where the
uncertainty introduced by the nonlinear term is given by the covariance of the product
of variables. Here the term cov(u^ |-) is a field quantity, like velocity.
This uncertainty and, hence, the covariance term in (2) is always assumed to be
zero in deterministic models. Thus, the substantial derivative in (2) reverts back to
the form in (1). The uncertainty associated with the neglect of higher order terms
in the Taylor series expansion for velocity deformation is minuscule compared to the
uncertainty associated with the velocity field as represented with real data. Such data
is contaminated with unwanted "real" signal, instrument errors, measurement noise, and
subsequent manipulation by analysis procedures.
If one uses a covariance term like that indicated in (2) in one's basic equations, then
it is prudent to have a prognostic equation for that covariance term. Unlike the previous
way of discretizing the field variables and then deriving prognostic covariance equations,
we will keep these covariance terms as field quantities and express their time derivatives
as equations in continuous form.
In the following derivation of the stochastic continuum equations, we will save space
and suffer no loss of generality if we neglect the vertical dimension and stress terms for the
5
moment. We will return to both subjects later. We further limit our beginning example
to the "shallow water" equations in Cartesian coordinates. Extension to the more general
primitive equations governing atmospheric motion, and to other coordinate systems is
straightforward.
The deterministic equations are:
Ou Ou Ohu == - -v - g - + fv
9v Ov Ah
vv= u Ov- g -fu (3)
Oh Oh O u 9 vh =-u- - v - hM + -
Ox ay \Ox y)
where u and v are velocity components in the x and y directions, h is the height of a free
surface of a single homogeneous frictionless fluid with "reduced" gravity g* (but referred
to here and later as just g), and f is the Coriolis parameter (f = f(y)).
Our dependent variables can be considered to be part of a larger vector x defined
over an N-dimensional phase space. Then
00
[()]= ] J f(V>(T ,t) dx (4)
-00
where E is the expected value operator, and where y ( z, t) is the N-dimensional proba-
bility density function satisfying:
JJ ... ( ,t)dxl, dx2 ... dxN = 1 .
We note that in general, if Z, = Z (~, t)
E (Za)= Za(Tt0 ) d z = Za ( )
E(Za ZB) = Za ZB + cov (Za ZB)
6
where the expected value operator gives the mean over the ensemble (in bold-faced type),
and where covy covariance. We also note that differentiation of (4) gives
dE[ f()] = E[df(zi)/dt] == Ji(7)> d (6)
where the continuity equation for probability justifies (6). Applying (6) to (5) we have
cov(ZaZ,?) = E [ZaZ +ZaZi] - Z Z - Z Z (7)
The above general probability expressions will be applied to the shallow water equations.
Using (6) on the set (3) yields the equations for the mean fields of u, v and h:
Ou ( Du\ Ou ( Ou
- -v -cov Kvy-
u =-ua - cov u -va -cov v
Ah (
vg v - fuv(8)
9( Oh\ O ( Oh\hi -_u--cov [u- - v _cov v¥
h -- cov h - - h -- cov h -Ox ox ay a Oyv
where we now have eight covariance terms over and above what the deterministic equation
set would have.
The covariances may be small or zero initially, but the variances of the dependent
variables, which reflect the uncertainty in the initial conditions (e.g., cov (uu) var (u),
cov (vv) = var (v), cov (hh) - var(h)) will grow with time and measurably impact the
covariance fields and mean quantities themselves.
Before deriving the general form, we take a particular covariance field in (8), deriving
the prognostic equation for cov(u a), in order to show how higher order derivatives
7
(derivative closure) enter into the picture. From the generic form of (7) we see that
/ ux ='.a ucov u- = E u -
k\ OxJ/ L9xau] . 9u
+u-\ -u-
We already have expressions for iu and ii from (3) and (8), respectively. It remains to
obtain sa and a4. We obtain the first by taking the partial derivative with respect to x
of both sides of the first equation in (3). This gives:
(O3Ocu Qu
Ox Ox
2 u- u-
av Ouax ay
02u- v
OxOy
02h-g Oz2
av+ fo'ax (10)
This equation is subject to the same uncertainty as present in the equation for iu. Therefore,
using (6) on (10) we have
au Ou- Ox -covax ax
Ou u \D-X x~)
02u- U
ax 2/ 02u
CV 9x 2
(av aOu- cov x- y
O9x Oy /
0 2u- v
OxOy- cov (v )
a xay)(11)
0 2 h Ovg x2 + fy x
Now, using (3), (8), (10) and (11) we have all the expressions to begin evaluating (9). This
gives:
cov (u T =
02hg Ox2
r Ou- -u - -- cc
9x
Ou OuOx Ox
(O9u\ 9Ou)v u- -v
(Ou Ou)- cov Ox Ox)
U( Ou- coy (v
02u- U -- (
ax2
Ov aOu- cov -- a
O9x Oy/)02U
- v 2y - covax9y
8
Ou-u
Ox (9)
Ov OuOx Oy
au Oh- V - - g-
9y axOu
-u Ox9x
Ou Ou+ u Ox Ox
02u-u Ox 2
av Ou 02u9x -y OxOy
Oh-9g' +
aufv ]-J Ox
(12)
Ov Ouax ay
02h- g Ox 2
(
- u
a2 U\
av.+ f (9x
Ou+ fv Ox
avl ,
d2 a \
In the first term in brackets in (12), we have products of three analytical functions.
Returning to our general notation,
E (Zc Zj Zy) =ZCaZ Z +. Za cov (Zf Z1)(13)
+ Z6 COv(ZaZ-) + Z y cov(ZaZf,) + (ZaZjZy)
where r is a third moment about the mean. Applying the above in (12) and (temporarily)
neglecting the third moment terms, we find:
cov (u =
02u OA9u I Ou\ IQuu Ou) (02u\a-O 2 cov(uu) - 3 .cov uy ) - u covy ax + cov uax)J
02u ( O vu\ 9u r ov + uv \a-- y cov (uV)-c -c covy u- +covOxay * ' 9xv c y x y + coy u Ozayx
F au Ov\ ( Ou\
-g cov -x -a + Cov U 2
[ (u a) (c v )]I x a x \^
(14)
Equation (14) above is the analytical expression for only one of the eight covariance terms
in equation set (8). We see that the r.h.s. of (14) will require prognostic equations for
other covariance fields (some involving even higher order derivatives). Before listing any
other equations let's consider several facts about (14).
The first term on the r.h.s. of (14) indicates that this covariance term will indeed
change, even if all the covariance fields are initially zero, for the variance of u will exist as
part of the uncertainty of the initial conditions. The second term on the r.h.s. of (14) is of
exponential form; however, the coefficient (8u/Ox) assures that the cov (u a9) term will
not exponentially amplify or decay, as the coefficient will change sign with passing wave
structure. The other covariance terms on the r.h.s. indicate that prognostic equations
9
for these will be required in order to close the system of equations. Indeed, all possible
covariance terms are required in principle to close the system. We will come back to this
point later.
b. General Form
We seek an easier way to express and derive these covariance fields. The basic
ingredients for deriving the prognostic equation for cov (ZaZ,) are prognostic deterministic
equations for Za and Zg,. In this paper we confine ourselves to deterministic equations
which are nonlinear quadratic. (The author has carried this analysis through for nonlinear
cubic systems, but these are not common in fluid systems.) These deterministic equations
(for the shallow water equations or the Navier-Stokes equations) can be written as Za =N
Z Zi,n Z 2,n + linear terms: where a is a dummy index, and where N is the number ofn=1
pairs of nonlinear terms. For example, in it, N = 2, u + v ). The expression for
Ou/Ox, for example, can be written in the general form
M
Z13 = >j Zl,m Z2, + linear termsm=1
where d is a dummy index and where M = 4 as we saw in (10).
We now derive the contribution to the general covariance term cov (ZAZp) from just
the first pair of nonlinear terms in Ze, and Zi. We temporarily withhold the linear terms
now as they do not contribute to the closure problem and will be added later. Note that
we now use a = cov.
10
a(Za,,Z) = E ZaZZg + ZaZ, - ZCZj - ZoZ,
= Zl,nZ2,nZ 3 + Zi,n a (Z2,nZf) + Z2,n a(Z 1 ,nZ1 )
+Z cr(Zi,nZ2,n) + T (Zl,nZ2,nZS)
+ZaZl,mZ2,m + Zca (Zl,mZ2,m) + Z1,mr (ZaZ2,m)
+Z2,m (ZaZl,m) + 7'(ZaZl,mZ2,m)
-Zl,nZ2,nZ, - Zo (Zl,nZ2,n)
-ZaZ1l,mZ2,m - Zca r(Z1,mZ2,m)
= Zl,n ca(Z2,nZ)3 ) + Z2,n O' (Zl,nZ,) + r (Zl,nZ2,nZ,)
+Zl,m 0 (Z2,mZa) + Z2,m O (Zl,mZa) + r (Zi,mZ2,mZa) (15)
Now applying the result of (15) to all the sums of pairs of nonlinear terms we have the
general prognostic equation for the analytic covariance fieldsN
a(ZZ#) = y [Zi,nC c (Z2,nZO) + Z2,n O (Z1 ,nZ/3 ) + 7(Zl,nZ2,nZ,3)]
(16)
+ S [Zl,m o'(Z2,mZa) + Z2,m O(Zl,mZa) + 7(Zl,mZ2 ,mZa)]
m=1
Here we have retained the third moment terms, instead of dropping them as in (14). It is
readily verified that the general expression (16) reduces to (14) for a(ZaZ,3) = a(uau).
A set of prognostic equations for third moment fields can be found using the same
procedure as above, and the relation
E[ZlZ2Z3Z4] = ZiZ2Z3Z4 + Zl T(Z2Z3Z4)
+ Z2 r(ZiZ3Z4) + Z3 T (ZlZ2Z4)
+Z 4 (ZiZ 2 Z 3 ) + Z 1Z 2 a (Z 3 Z4)
+Z1Z 3 r(Z2Z4) + ZiZ 4 a(Z2Z3) + Z 2 Z3 (ZlZ 4 )
+Z2Z4 o'(ZiZ3) + Z3Z4 a(Z 1 Z2) + A(ZlZ 2 Z3Z4)
11
(where A represents a fourth moment about the mean) to giveL
T(ZaZ Zy)= Z l,e 7 (Z2,Z/3Z.y) + Z2, t r(Zl1,tZ8Z.)£=1
-o (Z1,tZ2,e) a ( zz. ) + A (Zl,Z 2,tZ,3Z)]
M
+ [Zi,m r(Z 2 ,mZaZy) + Z 2,m (Zi,mZacZy)m=l (17)
-0 (Zi,mZ 2 ,m) (ZaZy) + A (Zi,mZ2,mZaZ-)]
N
+ [Z,1 n r(Z 2 ,nZaZ3) + Z 2,n 7(Zl,n ZjZ3)n=l
- a (Zl,nZ2,n) Cr(ZaZ3) + A (Z,nZ2,nZaZi)]
L M N
where > , Y , ~ are sums of pairs of nonlinear terms in Za, Z, and Z., respectively.t=1 m=1 n=l
To add the effects of the linear terms which are in the basic deterministic equations
(e.g., -g ah + fv in the equation for iu ) is straightforward. We now write the full
stochastic continuum equations in general form for the mean fields, second moment fields,
and third moment fields. [Wve have also added some general constants which may appear
in the nonlinear and linear terms of some deterministic equations.]
L L'
Z, = Y Cit [Zl,£Z2,t + -7(Zl,£Z2,£)] + y ba, Z (18)e=l t
L
a(Z'aZ/) = C~, [Zi,t ca(Z 2 Z,tZ1 ) + Z2 ,t a(Zi,tZ3) + r (Zi ,Z2,tZ8)]e=1
M
+ C,,m [Zl,m a (Z 2,mZa) + Z 2 ,m 0(Zi,mZa) + r (Zi,mZ 2,mZa)]m=l
L' M'
+E ba,e (ZeZ#) + E bi m d(ZmZa)
t=1 m=l(19)
12
T(Z^aZa3 Z)== E Cct, Zi,t r(Z2,eZZ,') + Z 2,- r(Zi,tZZ-y)e=1
~-((Zl,Z 2 ,e) 7(ZZ-y) + A(Zl,tZ2,tZ-y)]
Mr
+ E C',m [Zi,m T(Z 2,mZacZy) + Z2,m (Zi,mZaZY)m=l1
-a (Zi ,mZ 2 ,m) o (Zc Zy) + A (Zm Z 2 ,mZaZo)]
N (20)
+ A C cn ; Z, T (Z 2,nZczz3) + Z 2,n T(Z1 nZazz)n=l1
- (Zi,nZ2,n) 7(ZaZf) + A (Z1,nZ2,nZcoZfi)]
L' M'
+ b,t, r (ZZ,8ZY) + E b,jm 7 (ZmZaZy)£=1 m=l
N'
+ E b 'n7T(ZnZctZi)
n=1
L' M N'where the C's and b's are constants, and where >, E , E are sums of linear terms
'=l1 m'=1 n'=1
in Za, Z~ and Zy respectively.
Equations (18), (19), (20) are now in general form. The sums are of analytical
functions, not discrete amplitudes.
c. Derivative Closure
It remains for us to fully identify the analytical terms in the various Z. So far we have
only looked at i, iv, h of the shallow water equations and Oitu/x.
It is clear in (8) that we require several deterministic prognostic first order derivative
forms to derive prognostic equations for the covariance terms. We also saw that (14)
implied a need for prognostic second order derivative forms in order to derive prognostic
13
equations for terms like a (a) , ( y) and a (u h) It turns out that there is a
progression of all combinations of these derivative forms that is required. These are:
,x.Cx IOx'
ax'
ax'
OX2
02b
oh(X2 I
Oaiay'90oy'
9y'
02U
02x'
02hOxay '
y2 '"
ay2 '"
02hy2 ...
(21)
These derivative equations are obtained by forming the appropriate partial derivatives of
our original deterministic equation set (3). These equations are the general equations, Za,
needed to formulate the higher order moment equations. Consider one of these in (21):
0o2 uV 2 J
92u Ou
02v OuOx2 ay
03 h-g9 +
AX3
Ou 02 u aOu 02 03 u
Ox Ox2 Ox x2 - U
Ov 02u Ov 02u 03 u
x OaxOy x xOxOy Ox 2y
02vOx 2
(22)
We note that there are three third order derivatives on the r.h.s. of (22).
Before considering the importance of these higher order derivatives, it will be good
to recall how these equations are being used. The moment equations are describing a
"cloud" of points in phase space which surround the trajectory of the expected solution
in phase space. The moment equations (variances, covariances and third moments when
used) give us the shape of the infinite ensemble of points within the cloud. One eventually
faces a point of diminishing returns in keeping higher order derivatives in the Ze equations
14
while hoping to further refine the shape of the ensemble. (As a reminder, no assumptions
have been made in the dynamics of the original problem and the higher and higher order
derivatives of the Z, equations will have lesser and lesser effects on the mean solution.)
In a manner analogous to limiting the convective derivative to just the first order
terms (first order derivatives) from the Taylor series expansion of velocity deformation,
suppose one limited the Taylor series expansion of velocity covariance fields to second
order derivatives. (This is only analogous and not quite the same.) Were this done, the
Z., equations would be closed by the assumption that the net contribution of the third
partial derivatives in Za are negligible (e.g., third partial derivatives on the r.h.s. of (22)
set to zero). Similarly, one could maintain third partial derivatives and discard fourth. We
discuss such derivative closures below.
Since cov(u|9) is a field quantity, let's express it as a Taylor series expansion about
the point Po. W\e have then:
Au / A a a u Acov \u - = cov u a) + (x - o) -cov u )
aa a 9 (x-xo)2 a2 ( 9u>+(y - yo)-cov uy) + 2 ax 2 cov uy-}
(y-yo)2 a2 ( au au+ O-cov u- +...
2 y2a\ ax
Now since (u a(-) = - + u , applying the expected value operator to the
both sides yields the result:
a ( au' \Qu =u( ( a 2 u+cov [u W ) = covy ax + cov U a .
15
In a similar way, other derivatives of other covariances can be replaced and the Taylor
series expansion becomes:
cov u ) = cov (u
, _ (ou QuN (O2u)]+(x-xo) covy ^ a ) + coy U j
[ (au au (( 2u\1+(y-o) cov ( ) + cov u )]
(X [ \ 9y Qx) \ Qxy\
+ ( 2-- o 3 cov - &2) + cov u 3)]
(y- yo) 2 (&)u 192u\ (u 92u ( _u)
+ 2 ) [cov - y2) +2 cov -(9 &xcy) + covu ua 2
It is evident that by retaining all second order derivations, one is maintaining all first
order terms in the Taylor series expansion and some second order terms in the expansion.
This suggests that some covariance terms might be dropped, or alternatively, that only a
few need be added to gain the full effect of the next higher order in the expansion.
In Section 3 on energetics, we will see that eliminating third or higher order derivatives
does not affect the energy conservation properties of the stochastic continuum equations.
After viewing the energetics, we will be able to say more about the impact of derivative
closure on the shape of the ensemble.
As a final point on derivative closure, we consider the impact on the usually small
stress terms. In the Navier-Stokes equations we neglected the stress terms listed below
(for the x-component only)
Au 1 (Ou av aw) 0a2u 02u 02u)
_ ='"5 + -V + - + a~) + V ax~2 + "ay + az16
16
These terms are linear in form (assuming that the viscosity, Iz, is a known constant)
and their inclusion in the general form of the stochastic continuum Eqs. (18)-(20) is
straightforward. In the equation for a(uu), the full effect of the stress terms is felt.
The equation for U(u Iu) will only partially contain the effects of the stress terms
if third order derivatives are dropped in the Za equations. However, there should ensue
only a slightly optimistic result in underestimating the growth of uncertainty and this will
be quite small in most fluid applications. In the applications where viscosity has a larger
role; numerical experiments will have to be performed to determine the proper derivative
closure for the stress terms.
d. Variance Fields
The growth of uncertainty in the velocity field is assured if there is the least bit of
uncertainty present in the initial conditions, var (v) 7 0, and the velocity field is not
uniformly constant. The usual way of expressing uncertainty is to provide the variance of
each quantity one is trying to predict.
We can examine the growth of uncertainty from two quite different perspectives. One
way is to look at the "local" growth at a point within a field by the dynamics affecting
that point. Another way is to look at the error growth from a "global" perspective, i.e., to
look at the energetics of the system as a whole. We will consider both methods, discussing
the energetics of uncertainty in Section 3.
In this section we take a cursory look at the local error growth at a point by examining
the influence of various terms of the velocity deformation tensor. For brevity in this
discussion, we neglect third moments and third derivatives in the predictions of three
stochastic fields (u, v and h in the shallow water equations). Then there are 171 covariance
fields. In the general Eq. (19) for the prognostic equations for each of these 171 covariance
17
fields, there would be on the order of ten analytical expressions on the r.h.s. of each of
those equations. We clearly cannot afford to write out the full equation set here. We
have shown how to attain them. For brevity, we can see some qualitative influences in the
growth of uncertainty by just inspecting the variance fields of u, v and h. These are:
a(uu) = -2+[< (uu) + u a u ) + v a u
+ g( )) + ( - a(uv) (23)
+ v go. u 9v) + 9v)a(vv) = -2 (vv) + va (- +uCT (v
+ga VT) +(f f+f ) (uv)] (24)
a(hh) =-2[(± +0L a)(hh)+uu(h a )
( ( Ou'\ ( Ov\ (h Oh+h(a (h- )+a (h- )+v y h )
h Ox h+ X (uh) + -Ž a(vh) (25)
Ox Qy
Let us now look at the isolated effects of divergence or convergence (depending upon
whether + a) is positive or negative). On the r.h.s. of (23), the dominant term will
beOu
a(uu) = -2- a(uu)+...
since, in the early stages of the time integration, the variance terms are relatively larger
than other covariance terms. Thus it is seen from this component that local convergence
(Iaa negative) will tend to increase the local variance or uncertainty, while local divergence
will tend to decrease the local uncertainty.
Inspection of Eqs. (24) and (25) of the other component of divergence also reveals that
local convergence (9 negative) will tend to increase the local uncertainty of v and h. Thus,
18
from the sum of these components we have a net convergence leading to a net increase
in the local uncertainty and a net divergence leading to a net decrease in uncertainty. A
similar inspection (not outlined here, as it involves a(uv) and a consideration of all possible
cases) reveals that relative vorticity maximum (positive or negative) lead to a more rapid
growth of uncertainty.
From the above inspection of terms in the equations we have a general picture of
the local growth of uncertainty as follows. As long as there is initial uncertainty in the
velocity field, the uncertainty in the system will grow with time. Uncertainty can be simply
advected from place to place, but will always grow more rapidly in areas of relative vorticity
maximum. The effects of divergence (convergence) are to slow the growth in regions of
divergence and to increase the growth in regions of convergence.
The above results are illustrative and informative, but only strictly apply to the early
stages in the growth of uncertainty and the above assumptions. As time progresses, the
full nonlinearity of the equations requires numerical prediction to evaluate the uncertainty.
Indeed, as perhaps each dynamic situation in nature has no exact analogue, each stochastic
model of a dynamic situation will give real-time predictability that is uniquely a function
of the dynamics of that event, the uncertainties of that particular data set that described
the initial conditions, and the uncertainties of the external forces on the system at that
moment. Each of these can change from hour to hour and this is what makes stochastic
dynamic prediction so interesting.
19
3. ENERGETICS
The energetics of the stochastic continuum equations will depend upon the energetics
of the original deterministic problem to which they are being applied. In some physical
systems the energy quantities are conserved. In those systems in which energy is not
conserved because of energy generation and dissipation, the energy relationships can still
be useful for gaining insight into the dynamics of a model and can serve as useful diagnostic
tools. We will use the concepts of Lorenz (1955) in describing the energy of the system in
terms of kinetic and available potential energy.
In some physical systems, the energy terms are quadratic in dependent variables, and
in others they are cubic in dependent variables. The cubic energy relationships are more
difficult to deal with. However, the advantage of the stochastic continuum equations being
expressed in analytical form will allow us to address the more challenging case of cubic
energy expressions.
The total kinetic energy K for the shallow water equations, the primitive atmospheric
equations in the a-system, and primitive equations in the 0-system are given by the cubic
expressions:
K = ( 2 + 2) hds shallow water
sS
K = (u2+ v2) Psdv P.E. sigma
Vor
K= 1 1 (u+2) -dve P.E. isentropic
where p is density, p, is surface pressure, 0 is potential temperature. We will derive the
stochastic energy exchange relationships for the shallow water equations and the other
applications will be similar.
20
In the deterministic version of the shallow water equations it can be shown that the
sum of kinetic (K) and available potential energy (A) is conserved. Here K is defined by
K = pI khds and A = h2 ds (26)
S 8
where k = 2 (U2 + V2 ), and s is surface area. The area can be an unbounded plane or the
area of an infinite channel (periodic in x and bounded by walls through which there is no
flow), or the area can be expressed as a surface of a sphere (if we add appropriate metric
coefficients to the definition of K). It can be shown that
dK g h 2VV (27)
and
dA ~=_ [ h2V V (28)dt 2 J
Thus, we see that K and A are conserved. Using the convention that in the conservation
of two general quantities A and B, if d = f ( )ds and d equal the same integral
but with opposite sign, then C (A, B) read "the conversion of A to B" will be written as
C (A, B) = - ( ) ds. Thus, we see that in the deterministic case, (27) and (28) lead to.9
C(A,K)= PJ h2V. (29)
When a deterministic model conserves A and K, then a stochastic dynamic model
will conserve the sum of these quantities plus their "uncertain" counterparts, UA and UK.
The expected values of the deterministic energy quantities yield the following results and
21
definitions:
A= p h2ds2
S
UA= g p a (hh) ds
S
K= p h(u2+v2) d
s (30)
UK = 2 h [a (uu) + o (vv)]
+2[ua (uh)+vo (vh)]
+ T (uuh) + r (vvh) }ds
where A and K are formed from the means of the deterministic variables, and where UA
and UK are formed from the remaining terms.
It is perhaps useful to try to add a "quasi-physical" meaning to the concept of
"uncertain" energy-a nomenclature adopted by Epstein (1969). As a gross simplification,
consider that all realizations of the atmospheric wind field fall into one of three discrete
values (8, 10 or 12 ms-1) for u and v within unit areas. Whereas the actual observations
made over the globe are subject to all the sources of error with which we are familiar,
consider that the analyzed wind field is smoothed to be 10 ms- 1 for each wind component
per unit area. Since in general zP > (5)P where p is a positive power, the real energy will be
greater than the analyzed energy. The real energy per unit area is K = (area) - u2 + v 2 =
3 * [82 + 82 + 102 + 102 + 122 + 122] = 1022 m 2 s- 2 The analyzed wind field, hence the
initial conditions of the model, yields K = 100 m 2 s- 2.
A deterministic modeler and a stochastic modeler would both begin with initial
conditions of K = 100 m 2 s- 2 and K = 100 m 2 s- 2 respectively. Both begin with their
best knowledge of the initial state of the atmosphere. The stochastic modeler goes on
22
to estimate the initial uncertainty (unless a previous stochastic prediction and analyses
sequence exists, which would be better) and expresses that uncertainty in the form of
initial variances of a (uu) and a (vv), such that the uncertain energy, UK, is about 2 or 3
percent of K.
Another way to view the uncertainty is in terms of an energy spectrum of wavenum-
bers. Then the "certain" energy spectrum is formed by known mean quantities, the
amplitudes of the sine and cosine of each wave form, and the precise position of each
wave is determined by its phase. The "uncertain" energy spectrum tells us that waves
exist of a given scale size in the system, we have a measure of their range of amplitudes via
the variance, but we cannot determine their phase-their precise position in the system
is uncertain. In a stochastic model, we predict the dynamically expected answer and the
gradual decay of certain knowledge into uncertainty-or the transition of certain energy
into uncertain energy.
We will prove that the sum of the four quantities in (30) is conserved and derive the
energy conversion terms between the four quantities. Since third moment quantities are a
part of the definition of UK, it is clear that we will require prognostic equations for third
moment terms to complete the proof.
In an earlier work on stochastic energetics, Fleming (1971a) examined the complete
energy exchanges for a quasi-geostrophic system using Lorenz' (1955) energy definitions.
The problem with the earlier work was that the generation, conversion and dissipation
of energy quantities were expressed in terms of the spectral orthogonal functions used to
discretize the dependent variables. In what follows, we will express the energetics of the
stochastic continuum equations in analytical terms. One now gains a better illustration
of the role of dynamics in contributing to energy exchange between certain and uncertain
energy forms without regard to how the dependent variables are expressed.
23
We begin by looking at the time rate of change of A. (In the remainder of this paper
we shall simply drop the constant density factor from the energy conversion terms.) We
have:
dAt = 2 h2ds = g h-ds
=-g h u- +v-+h --+ )x 9y O9x Oy
[0h Qh( ((u+a\(uO)+7(vx- )+<(h)\+OyJ(h- )\ds
This can be simplified by noting that:
hu ds= u (h2)ds
0 u1h) ds - (1h)i ds
The first integral above reduces to zero by Gauss' theorem.
Performing the same operation on the corresponding y-component term and combining
terms we have:dA g / D v
dt 2 ( Ox )
/h [a (u) + ¢(a )
+h [ (V) +C (h)] }ds .
But the first term in the second integral can be written as:
A0 =]ha7 a(uh)]h [a uL ) +a (h-)] h /h [(uh)
=J {O L[ha(uh)] -a (uh ) - h
= -uOx
(31)
24
The second bracket in the second integral of (31) reduces similarly. Thus, (31) becomes:
h 2 (Ou\9x
+g a(uh) +JOx
+ -) dsOy
(h )}
We see that the first integral is equivalent to the negative of the deterministic C (A, K).
Now let us look at how UA changes with time. We will retain third and fourth moment
terms in all cases where they should appear. We have:
d(UA)dt
g -a(hh)
-2{ OuOx
+v+ -y a(hh)+ h a h Ou
(Ox +a (h )K ou
+ua hO- )
+vaO (h- h
+ h a(uh)
+ -- a(vh)+rOy
+ r vh- yK y + ( hh) jdskOY
The above integral can be reduced by noting that since:
+a (Oh \+~ K Ox J
=2a( hx-(oh)7
sf Ohds = U ux (hh)ds
2 l
= J a [ua(hh)]ds-f-2 (hhOu ds
= Ox
1 o u2J Ox
25
dAdt
(32)
]
( Oh
then
0-o(hh)Ox
( h)Ox
+ 1Uhh)
The above is true by Gauss' theorem and the fact that the covariance fields are symmetric.
Using this expression and the similar one for the y-component reduces d (UA) /dt to:
d[(UA) ( af )u 9v )
= r (d or hh+ r
+ r uh- ) +r vh-y +r hh- +7' hhyh ds
~-gj = Oh( ^hhds
This can further be reduced by noting that
Ox 2Ox +\) O ) + x )
+ ( h)+=r (ahh +2r uh (h
Therefore:
j r(uh ds= (uhh) -r hh ds
( hh Ids
Using the above expressions, we have:
d(UA) _ g Atfu OvN 'h--- = i- + -a hh)+2hhh- )+ )+a / O
+7I-hh I+7- -hhii ds (33)aOx / \9y /JJ
f Oh '1-g] Tpo (uh) + -o- (vh) ds
We see that the second integral above is just the negative of the second integral in dA/dt,
Eq. (32). Thus, this integral represents the conversion of A to UA or C (A, UA). We will
soon show how the first integral in (33) is related to UK.
26
First, let us look at dK/dt. We begin by defining k = (u 2 + v 2 ). We then have:
dK a()dsdt == / - (kh) ds =dt J Qt ' /{k!
=-|{kV.Vh+kh.VV+hV Vk+gV V (lh2)
( Oh
\Y/ +o (h-)ay y
+hu a(lu)au
+hv [ (u v[ V )
/ au
+a v(ay)] ds
Now expressing all the covariance terms temporarily as "cov" we have
_/{v= -v .
V(kh) + khV V + gV V
(khV ) +g V (h2)
(lh2)+ "cov" } ds
+ "cov" } ds
=-9- ~ V(h2)ds- "cov" ds2
= -2 {. (Vh2)-h2V .V }ds- "cov" ds
h 2V Vds - J "cov" ds
h2 (0u + -) ds - "cov" dsQ9y)
We see that the first integral above is the negative of the first integral in (32) and thus is
C (A, K). Therefore:
27
+k [a (u
dKdt
g2
+ h d
(9x
dt =C (AK)- {(U2 V2) [a Ua +a va) +a (h +a ha j
K r 2 ( ha am+hu o7 U- +or Iv
1 Qax' ay}
+hv [auy7) +0a (v) ;ds
(34)
It now remains for us to look at the very long integral of d (UK) /dt. This is:
dt ) == 2 J \ h(a(uu) + a (vv)) +2[ua(uh)+va (vh)] + r(uuh) + r (vvh) ds
(35)
The nine prognostic equations on the r.h.s. of (35) together involve over 130 terms. These
are written in Appendix A and it is shown there that (35) reduces to the following two
integrals:
dt(UK) _ g f f( u v a0 (hh\9h[a(h u\ ( 9v)]
+r( - hh) +T (7-hh)}ds9 2) (9y } }
+ {2 +v [a ( a) a a ) + (ha) + a h d 36)
[ ( [ )\ ( u )]+hu [U + V
+hv u- ) +a (v )] }ds
We note that the first integral in (36) is the negative of the first term in (33). We note that
the second integral in (36) is the negative of second integral in (34).Thus we can write:
d UK) =C(UA,UK)+C(K,UK). (37)dt
We summarize the summarize the results of (32), (33), (34) and (37) below. (We now drop the bold
faced type for K and A since we are now in the stochastic system of equations and K and
28
A refer to certain kinetic and certain available potential energy.)
dA'=t -C (A, K) -C (A, UA)at
= C (A, K) -C (K, UK)
(38)dUA
d = C (A, UA) - C (UA, UK)
dUKd = C (K, UK) + C (UA, UK)dt
We summarize the definitions of these conversion terms as:
C(AK)- =g/ h2 (u + v)s (39)
C (A,UA)=-g |h a (uh) + oa(vh)ds (40)
+ (8 hh I + T ( -hh) }ds (41)aOx / \ 9y
C(KITUK)= j (UT2"+ V2)2 [(Ohu o(Oh V(h )orO+a N ]
+uh [r (u-) + ( v )]
+vh[a(u +a(v )]}ds (42)
We see that the C (A, K) in the stochastic continuum equations is the same as in the
deterministic case and involves the integral product of h2 and divergence. The uncertainty
in the initial conditions leads to a growth of uncertain energy via C (A, UA) and C (K, UK).
These conversions, (40) and (42), only involve second moment terms. The subsequent
conversion of UA to UK (41) involves second and third moment terms. The fourth moment
terms on the r.h.s. of third moment equations integrated to zero.
29
What has been shown above is that, just as the sum of A and K are conserved in
the deterministic set (3), the sum of A, K, UA, and UK are conserved in the stochastic
continuum equations corresponding to (3). Moreover, since the above four quantities,
defined in (30), contain no moment terms with derivatives, the conservation over the
domain is apparently unaffected by neglecting higher-order derivatives in other prognostic
equations for covariance fields involving derivative terms. However, as stated earlier, the
fidelity of fluid quantities at a point in the domain or the accuracy of energy conversion
between energy expressions will be affected by the derivative closures employed. For
example, noting that( 9u\ 1 09
a u ) 2 a- x(uu)
we can modify our previous expansion of a (u ) into:
a (ou) = 2 a- (uu)o + 2 a:-(uu)o
+ 1 (uu)O +x6y 02 6x2 aQ3
+ 42 a--ya (uu)0 + au4 xay'( uu)0
bX3M
4
+12 Oxya (uu) 0 +' "
Thus, the elimination of third derivatives in the covariance equation leads to neglecting
a term like -T4a(uu). This implies a smoothing of the variance field. Initially, the
velocity variance field might be uniform. Various sources of error, from external forcing
and from the nonlinear dynamic situation, evolve the variance field into one with some
structure. Thus, maintaining nth order derivatives restricts the evolution of details in the
field structure of such quantities as a (uu), a (vv), a (uv), a (uh), a (vh), and a (hh) as
derivatives of those field quantities are maintained only up to degree n + 1.
A summary of these energy conversions is shown in Fig. 1. In addition, we have
completed the diagram under the assumption that additional physics has been added to
30
the models: heating to generate (G) available potential energy, and friction to dissipate
(D) kinetic energy. Under the assumption that such physics is perfectly known and perfectly
parameterized, the arrows on the G (UA) and D (UK) are generally as indicated; i.e., the
effects of perfect physics will tend to retard the growth of uncertainty, so that the entire
system is more predictable (cf. Fleming, 1972).
31
gC (A, K) =2
J h2 ( + )ds
C( c(A,UA) =-g |-a (uh) + a a (vh)ds
G o(f.,,---/{.,(,)(o )[u [v( ). )U v h, ..hff1 2 (+h) [ (0/2i O h( + h (+ r vN1hh ( ) }
C(KUK)= (u2 +v2) a u +v xv ) + a (h-) +o ha)]
+uh[a (u a (v vh [a (U o +a o(v v ds
FIG. 1. Stochastic energy diagram for continuum equations.
32
0
4. COMMENTS AND CONCLUSIONS
a. Moment Closure; More general equations
Several additional comments on the stochastic continuum equation are provided below.
It is clear that a moment closure scheme is required in the general form of the equations (18)
- (20). Fleming (1971a) showed that third moments were important for energy transfer
in a barotropic model, and we saw above that they are similarly a part of the energetics
of primitive equation models. It is beyond the purpose of this paper to delve deeply into
the closure question, so we shall simply say that evidence exists that an eddy-damped
quasi-normal closure will more than adequately suffice for the length of time integrations
anticipated for the use of the stochastic continuum equations. As one integrates sufficiently
far into the future, the uncertain energy becomes so great that it is folly to continue.
The eddy-damped quasi-normal closure replaces fourth moments with products of
second moments (A12 3 4 = 012034 + 0'13024 + c14o 2 3 ) in the third moment prognostic equa-
tions and, further, adds a damping term in such equations to account for the nonlinear
mixing that naturally occurs. Fleming (1971a) analyzed this closure in a stochastic
dynamic barotropic model and found that it gave good results out to 21 days (when
compared against Monte Carlo calculations of large sample size). In another quite dif-
ferent application, Fleming (1973) performed a stochastic calculation on a maximum
simplification of the spectral form of the shallow water equations to isolate gravity waves
superimposed upon a basic flow. This highly skewed system was able to reproduce a third
moment quantity (skewness in a gravitational mode) extremely well out to a few hundred
wave periods, and to obtain comparable results in an adjustment parameter out to three
weeks. Moreover, Fleming (1991) has shown that in chaotic systems there is an optimal set
of damping coefficients that can be found to match Monte Carlo calculations of extremely
large sample size.
33
Solving the shallow water equations on a sphere involves additional linear terms and
replacing the derivatives with respect to x and y with derivatives with respect to A and
X (longitude and latitude). Similarly, any model with quadratic nonlinear terms in any
coordinate system can be expressed in the form of (18) and the subsequent second and
third prognostic moment equations formed from (19) and (20).
b. Computational Estimates
The stochastic continuum equations have been introduced as a superset of the original
deterministic equations for any given model. An approximate ratio (R) of the stochastic
calculation to the deterministic calculation can be obtained by considering only the non-
linear terms. We could calculate an exact ratio, except for the fact that we have not
yet discussed the concept of uncertainty in the external forcing terms of a model. This
inclusion of uncertainty in some kinds of external forcing terms may be relatively trivial
(e.g., as in Fleming, 1972) or as computationally complex as the nonlinear terms. (An
example of the latter occurs when the forcing is of the form exemplified by
where a and b are processes (ranging from simple constants to complex algorithms) which
contain uncertainty - this subject is discussed in a separate paper). Thus, we may consider
the value of R computed below to be an upper bound.
The general equation set (18) - (20) is a closed system when the eddy-damped quasi-
normal closure is used; i.e., fourth moments on the r.h.s. of (20) are replaced with products
of second moments, and a damping term -Krijk is added to the r.h.s. of (20). The set
is solved simultaneously as one would solve the deterministic set, equation (18) without
covariance fields; however, it is informative to consider three distinct steps within a given
time step.
34
The first step is to formulate and to store the various derivatives of the dependent
variables (from the initial conditions of the variables and from the subsequently predicted
values of the variables at later time steps). Whether these derivatives are calculated by
finite differences in physical space or by the spectral transform method (physical space to
spectral space to physical space) we are left with stored fields of derivatives at a mesh of
points in physical space.
The second step is to solve the mean equations, the general form given by Eq. (18). The
solution of these equations can be accomplished by finite difference methods, finite element
methods, or the spectral transform method. For global models, the spectral transform
method (cf. Orszag, 1970 or Eliasen et al., 1970) is the most popular, and we will use this
for a comparison later.
The third component of the algorithm is to evaluate the r.h.s. of equations (19)
and (20). There are many equations and many more terms to be multiplied and added.
However, these terms are just products of derivative fields and moment fields. (Consider
the non-differentiated mean variables (e.g., u, v, h) as zeroth order derivatives as these also
appear.) The moment fields are represented by a mesh of points over the space domain.
This is the same mesh as used to form the derivatives. Since we have already evaluated the
derivative fields at these same points, the evaluation of the r.h.s. of (19) and (20) is simply
multiplication and addition of terms at the same point. This completes the algorithm for
a given time step.
It can be shown that, for a full three-dimensional calculation with five dependent
variables (e.g., as in the NCAR Community Climate Model), with all third moments
included and with second order (but not higher) derivatives included in the za euqations,
the ratio is given by:
4 x 106 + 90N45N
35
where N is the number of modes (e.g., wavenumber in the east-west direction) in the
spectral transform method. For N = 180, we find R = 0 (500). If third moments were
neglected, we would have R = 0 (12). There are a number of reasonable assumptions one
could make between the above extremes.
Terms like r(uuu), r(vvv), etc. indicate the degree of non-Gaussian structure that
exists in phase space as a result of the struggle between coherent non-Gaussian forcing of
variables and the coherence that is being destroyed by nonlinear mixing among the waves
and eddies of the fluid system. Thus, while r(uuu) may be interesting, it will be small
and not contribute much to the shape of the ensemble. Will r (uu ax) 3= 3 i rt(uuu) be
significant? Perhaps. Will r (uu ) = r(uuu) - 2r (u a ) be significant? We
suspect not.
Ignoring the effects of many such third moment quantities leads to an estimate of R
of order 100. There are a number of other tricks and reductions that we could mention
here, but let's leave the ratio as R = 0 (100).
Formulating these equations in continuum form over the discrete grid form has meant
an improvement by a factor of 1000 in the second-moment only case, and a factor of
a million in the third moment eddy-damped closure case. The gain comes from three
reasons: (i) the correlation of grid points widely separated in space are virtually zero
anyway (especially third moment quantities) and do not enter into the continuum set,
(ii) second and third moment terms of grid points reasonably close together (but tightly
packed) are similar in value (and thus represent redundant information), and (iii) any
realistic finely detailed structure in the phase space cloud which could be captured by a
grid-system is lost by a smoothing of the continuum equation approach when higher order
derivatives are dropped (this represents the only real loss of these equations).
36
In practice, an impossible computational problem has now become possible with such
continuum equations. Since no method of predicting the uncertainty dynamically is being
used today (at least for the complex geophysical codes), it would seem appropriate to begin
even with the simple "second-moment only, second-derivative only" closures as a means of
providing at least an estimate of the dynamically evolving uncertainty.
c. Other Computational Aspects
"Fine-grained" parallel computing systems (cf. Willis, 1985) can be put to maximum
use in solving the stochastic continuum equations. These new systems can have thousands
of parallel processors (each with a local memory of several thousand addresses). The
stochastic continuum equations are solvable with such a highly parallel system. Moreover,
because the computationally bound part of the kernel is the r.h.s. of the prognostic
moment equations where we have only products of expressions at the same gridpoint,
these equations involve no delay in memory access. Only several hundred field variables
and a few hundred other fields for constants and work space are required per gridpoint
(per processor). Therefore, sufficient local memory exists to solve third moment equation
versions of the stochastic continuum equations. Thus, these equations are solved with such
machines running at near peak performance. Adding higher order derivatives to the z
equations or adding higher moment equations only adds to the parallelism.
Several steps may be required for practical implementation. One of these may be
a light damping term applied to the moment equations to control high wavenumber
aliasing when multiplying field A times field B over the same lattice structure (gridpoint
representation). Another practical consideration for the mesoscale is the use of the semi-
Lagrangian time integration scheme (cf. Sawyer, 1963; Robert, 1981, 1982 and many recent
papers). This removes the nonlinear advection terms from the equations but transfers the
nonlinear problem to obtaining the "proper" trajectory departure point (requiring high-
37
order interpolation). The stochastic approach to this method is more properly the subject
of a separate discussion.
d. Conclusion
The purpose of this note is to introduce the stochastic continuum equations. These
equations predict both the future and its believability dynamically. The general form of
these equations is expressed analytically. Their solution is achieved by closing the moment
equations at some level and by closing the higher order derivatives at some order.
An examination of the dynamic growth of uncertainty is done "locally" by noting
the dominant terms in the field equations and "globally" by deriving the energetics of
the equations over the fluid volume. The local effects are shown to be dominated by
relative vorticity and divergence. The energetics of the stochastic continuum equations
are shown to be energy conserving (when the corresponding deterministic equations are
energy conserving). The nonlinear cubic energy form for the deterministic equations, used
as an example in the paper, require the inclusion of prognostic third moment equations.
It is shown that the conversion of C(A, K) is the same integral expression in both the
deterministic and stochastic versions of the system equations. The conversions C(A, UA)
and C(K, UK) involve second moment quantities. The C(UA, UK) involve third moment
terms.
The effects of the derivative closure in the zQ equations are investigated analytically.
The expression of covariance fields in a Taylor series expansion reveals that dropping
derivatives of order n is equivalent to ignoring derivatives of order n + 1 of covariance
quantities. The practical aspect of this is that one will lose some ability to depict the
detailed structure of the ensemble evolution in phase space. For example, example, expanding the
field cov(u au) revealed that ignoring third derivatives in the z, equations led to ignoring
the fourth derivatives of the variance field cov(uu). Since the continuum equations can
38
provide our first real-time predictability of the horizontal velocity field (through cov(uu)
and cov(vv), as well as its first three derivatives), this appears to be a reasonable derivative
closure. Although no derivative closure is particularly advocated until further calculations
in a complex model are performed, this closure ignoring third derivatives is used in
computational estimates. The ratio of stochastic calculations to deterministic calculations
is 0 (100). Such calculations may be possible in the not-too-distant future.
In summary, we see no major limitations in implementing the stochastic continuum
equations. They are perfectly suited for the new emerging parallel computer architecture.
The advantages of improved predictions, and more meaningful information content for the
users, suggests that they may prove beneficial for a variety of applications.
39
REFERENCES
Browning, G.L., J.J. Hack and P.N. Swarztrauber, 1989: A comparison of three numerical
methods for solving differential equations on a sphere. Mon. Weather Rev., 117,
1058-1075.
Eliason, E., B. Machenauer, and E. Rasmusson, 1970: On a numerical method for integra-
tion of the hydrodynamical equations with a spectral representation of the horizontal
fields. Report No. 2, Institut for Teoretick Meteorologi, Univ. of Copenhangen.
Epstein, E.S., 1969: Stochastic dynamic prediction. Tellus, 21, 737-757.
Fleming, R.J., 1971a: On stochastic dynamic prediction: I. The energetics of uncertainty
and the question of closure. Mon. Weather Rev., , 99, 851-872.
, 1971b: On stochastic dynamic prediction: II. Predictability and utility. Mon.
Weather Rev., 99, 927-938.
, 1972: Predictability with and without the influences of random external forces. J.
Appl. Meteor., 11, 1155-1163.
,1973: Gravitational adjustment in phase space. J. Appl. Meteor., 12, 1114-1122.
, 1991: Chaos and stochastic dynamic closure for low-order geophysical systems.
(Submitted for publication).
Gleeson, T.A., 1968: A modern physical basis for meteorological prediction. Proc. of First
Nat'l Conf. on Statistical Meteorology, Amer. Meteorol. Soc., 1-10.
Lorenz, E.N., 1955: Available potential energy and the general circulation. Tellus, 7,
157-167.
40
Orszag, S.A., 1970: Transform method for calculation of vector coupled sums: Application
to the spectral form of the vorticity equation. J. Atmos. Sci., 27, 890-895.
Pielke, R.A., 1984: Mesoscale Meteorological Modeling. Academic Press, Orlando, Fla.,
612 pp.
Prandtl, L., and O.G. Tietjens, 1934: Fundamentals of Hydro- and Aeromechanics. Dover,
New York, N.Y., 270 pp.
Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological
equations. Atmos. Ocean, 19, 35-46.
, 1982: A semi-Lagrangian and semi-implicit numerical integration scheme for the
primitive meteorological equations. J. Meteor. Soc. Japan, 60, 319-325.
Sawyer, J.S., 1963: A semi-Lagrangian method of solving the vorticity advection equation.
Tellus, 15, 330-342.
Tatarskiy, V.I., 1969: The use of dynamic equations in the probability prediction of the
pressure fields. Izvestia, Atmospheric and Oceanic Physics, 5, 162-164.
Thompson, P. D., 1986: A simple approximate method of stochastic-dynamic prediction
for small initial errors and short range. Mon. Weather Rev., 114, 1709-1715.
Willis, W.D., 1985: The Connection Machine. MIT Press, Cambridge, Mass., 190 pp.
41
APPENDIX A
Here we reduce the extended integral expression for d(UK)/dt. Using set (3) and the
general stochastic continuum equations (18)-(20), we rewrite (35):
d(UK)dt
12
J {h [a(uu) + a(vv)] +2 [u(uh) + va(vh)]+ T(uuh)+ r(vvh) ds (A1)
Writing each prognostic equation in order, term by term, this becomes:
d(UK) 1dt 2 J
2h [u (u u )Ouxu-/ au
+ T7 UU |\ 9xj
u+ -a(uu) + v7
ox
( Ou+T*"a
( Ou
+ g (u )h+gv KE)
uv+ -vT(uv) +'
Ox(D v-) + -- a(vv)
+ga v )+ f o(uv)~~~Y-~
+ [o(uu) + o(vv)]
+/ (h+a u x+< dx
+2u uo7( 7h)
- + h+gc Kh
Oh, 9x
Oh (Ou+v-+h -+
Oy O9x
( Oh
u va+ o a(uh) + vaOxu h)
-fa(vh) +rU a h(Oxu
Ov
Qy)
+ ~ a(vh)
+ Ou h)+'rWy
Oh+ cT(UU)'
au+ - a(uh)
+r UV-\ Qy,
+ ha au
i u )h\^r~n~x
Oh+ , a(uv)
Oy
+ ~ a(uh)
42
au+ -7-c(uv)
+ 2h ucr( av)\ Ox
-fa(uv)]
+ u (u
+ ha uOu\ax)
]
(9 x
A+ 7 UU ax
(9x
O9h ,(9u\+g--fv+a7 u-x
9x ( Qx u+ o(uh) u 9u+ 2o(uh) u T + v' / [ x
(v h)
+ h'\Oy^/
/ ah)
+ua v-.
Ou
Ov (Ov\+ - a(uh)+v (- h
+ f(uh)+ ( U h}-
+ h a(uv)
u (vh)+x
+va vv-y
+ ha (vOvay/9v
+ - a(vh)
+ r v- yhOh)
Oh '+ T a(vv)+ (
O yr
( h+T -uv )
\ 9X
+ la( [v+ 2a(vh) uO[ox
(u h)09 h
Ou
V ax/
(u vh+ T' -x v h
av O-h + / y)av\+v- +g -+ fu+ tu-
Qy ay \ x)
9(Uh)+ V ( uh)+ O -(uuh) +vr -uh
+-x h \9ya)
a(uh)-oa (v (oua(uh) + A
Ov h- h I9Yy
+T (
(u+ A (v-uh)( Oy
(Oh )+UT t UU
+ h7 uu O\
+ g ( uh) - f(uvh)ICyX j \
Oh+ - r(uuu) + vr
u++- r(uuh) + hr
(Qh \[~Y~ u u
tuu (9 )
( Oh )+A U uu 1
(\ Oh
+A v-uu 4vu -Y
(h u+ k hOx)
A h - uu
+ h -Oy I
) +A (h7uu)
43
+ 2v [u
]]
au+ 5-yr(vuh)
9y
(u-uh)u auh
+h (uuv)
+ -,r(uuh)+^(U/
aul
Oh\
Dv
+ 2 w
- a(uu) a U
2[u (Ox h)-(x v )
+>(v vh)+ A yy y h(oy
+ ar(uvh) + vr
o(vh) -x (av )
(v h) + -r(vvh)
(Oh N
+hr9 T VV
@X
+ - r(uvv) + VT
Ou+ -~ r(hvv) + hr
( vv)
O9v
Oh+ a -(VVV)
rhv v+ -r(hvv)
+ a ( h OvO9y /
+A (h vv)( ux )
+A h-vvOy
We work on (A2) in parts. First we isolate those terms involving g and leave the other
terms in the integral as "other." (Also note that the Coriolis terms cancel.)
d(UK) _dt
ha (u h)
ha v yy + -(Oh i
+ r uh 1
- | ("others") ds
This can be reduced using the relation
0ao-(uh) =
Oh Oh\+Ox ( uh)o+ h- +x Ox0~
O + (h +a y/
+ (vh )} ds
( u OhOxJ +a h )(Ouxin the first term and combining with the second term to give
02g5- [ha(uh)]- 2gha (h O
44
a(vh) + A (u vh(or)X+ g ( vh) + fr(uvh)]TY~~~~
- (vv) cr(u OhOx--
+ (uOh hOx vv + (v A V(o )y }ds
(A2)
(A3)
(9xA·
I g
Using the same relation for corresponding y-terms, integrating, and using Gauss' theorem,
we are left with
-2gh (h ,) and - 2gha h y )
The remaining second moment terms in (A3) can be rewritten as
2gu [ ay (hh) =g 9 [ua (hh)]- g(hh) .
Rewriting the corresponding y-term in a similar fashion and integrating yields
O--ga(hh)-x and
The third moment term in x can be written as
2gr (uh ah)( o x
a= 9 -g-r(uhh)- grox
Upon integration, this will reduce to -g (Oj hh). The y-form becomes -gt (A hh)
Adding these changes to (A3) we have:
d(UKT)dt 2
(2h [r (h )
+ ( hh+ Ox-1/
+ ) a (hh)
(v h)+( -hh ds/ J
(A4)
- | J("other form") ds
However, we see that this first integral in (A4) is the negative of the first term in (33).
Thus we can write
d(UK) = C(UA,UK) - ("other forms")dsdt 2
(A5)
Now let us isolate the fourth moment quantities in "other terms" in (A5). These are:
45
Ov- ga(hh)-y-
au .ax h
+ a' Ov
3A uu h)Ox J
+ A uuu 7 +' IO x
2A uv a h(ou)2A uv-ax h)
3A (v Nh)3 (vv-hO y /
+ A uuv A )
+ A huV x)
+ A (vvv O)+A(^^)19
+A (uu h)+
+A vvh +(ax V J
Since for a general fourth moment A(ZiZ2 Z3 Z4 ), we have
0XAZ( I Z2 Z3 Z4 ) -= A
Ox ( "OZ1 N
'X Z2Z3Z 4 ) + (Zl Z3Z4aZ3
Z1 Z2 - Z4 ) +(A ZIZ2Z3- )) \ u~x
and similarly for a A(ziZ2 3 Z4 ), then all the fourth moment terms will integate to zero by
Gauss' theorem.
Now isolating the third moment terms from "other terms" we have (deliberately
grouping terms in rows):
3hr (uu T) +( 9u)Ohy r(uuu)+
2hr uu v) + hr (uu
2hr u v + hr (vv(o\ 9Xu)
Oh+ -T r(uuv)+
+ - r(uvv)+Ox
+ h TVVV)++ - 7(vvV)+
ay
U ( h) + 2 ( Oh)4ur uxh +2ur uu xxu) (ox.
2u [r ( vh) + T (uI . \y \
r(uuh)++2-7r(Uuh)+
Ox
hv )
9y /
46
3h (vv a)(9 y)
Ou+2 -u r(uvh)+ay
I
1�
A '+,r UV (9y .
2v vh) +r (u h) + Tr (uv )+2 T r(uvh)+
+ 4vr (v h) + 2vr (VV ) +2 (Vvh)+
+ 2ur - uh + ur (uu - + - r-(uuh)+
(Ou ( Oh Ov+ 2v7 uh) + vT- uu - ) + - r(uuh)+
( 9v , ( Oh a Ou+ 2uT V ~ h + ur vv d + T r(hvv)+
Ov A ( Ov+ 2VT V h) + vT VV )- + -aT(vvh).v-y) +- (vvh).
Using
O (Ozi 0 2 z( 0Z39X r(Z 1 2 Z 3 )= T Z2Z 3 ) +7 x Z 3 ) +T IZlZ2 x)
and Gauss' theorem, each of the above rows integrates to zero.
We have just seen that all of the fourth moment terms and all of the third moment
terms (except the two that contribute to C(UA, UK)) integrate to zero in the equation for
d(UK)/dt. Now looking at the remaining second moment terms in (A5) or (A2) we have
47
(deliberately grouping terms in rows):
-21{ 2uu2/2vv a
a ( )Oux[(7 Ox^~)
(vh)
+ a (v h
Oh)]
+ 6u - a(uh) + 2u a(vh)+
+6v - a(vh) + 2v - a(uh)+
Oh Oua(uu) 3u + 3h
o [ u) x Ox]
a(uv) [2u -y + 2h ]
a(vv) 3v 5 + 3h ayj
a Oh Oav]7(uv) 2v- +2h-
. r 9.J
a(uh) [2u O
a v
Ou]+2v- a
+2v-Ox
c(uu) h +hv]
v)u Oh +Oua(vv) u T + h -L x 9x
+4uha (u) +
+2uha y(u)+
+ 4vha (v - +
+ 2vh7a v O- +\ 9x}
+ 2uv [a
+ 2uv [aL
+ 2vha (u
Ou Ny h)
(v h)O9x
Ou)ay +
+ uho7 (v } ds+ Qx } )
It can be readily verified, using the same covariance relationships that we have used,
e.g.,
(Oxh)( h\+a UOx a(uh),
and using Gauss' theorem that the first six rows reduce to the terms in the following
integral, and that the last four rows each reduce individually to zero.
48
d(UK)dt
(A6)
(Oh+a U-Ox +
+ oa v )] +
Thus, we can write (A6) as:
d(UK) _dt
C(UA, UK')
f 1 ( 2 O2h ( ah ( 8Ohs2 (~u ~ 2)'[9x + {ay)
Ou
/ OvOux)
+( h-iTOX,(A7)
+ a ay)-
+ a v )
Noting that the second integral in (A7) is the negative of the second integral in (34), we
can write
d(U) = C(UA, UK) + C(K, UK)dt
49
+ hu [
+ hv [o-
+ah 9