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