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Aonnszsv LIBRARY TECHNICAL REPORT SECTION NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA 93940

APPROVED FOR PUBLIC RELEASE DISTRIBUTION UNLIMITED TECHNICAL NOTE NO. 72-3

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AN OPERATIONAL UPPER AIR ANALYSIS

USING THE VARIATIONAL METHOD

Technical Note No. 72-3

May 19 7 2

by

J. M. Lewis

FOREWORD

This technical note describes the Numerical Variational

Analysis (NVA) program used at FNWC Monterey to prepare upper

air analyses of wind and temperature for the Global Band Grid.

The grid has a mesh length of 2.5 degrees at the Equator and

extends around the world covering latitudes between 40S and

60N. (The tropical grid used at FWC Pearl Harbor is a sub-

set of this one.)

The program is unique to Naval Weather Service analysis

effort in the tropics in that it couples wind and temperature

through thermal wind constraints thereby automatically ensuring

vertical consistency. It is designed to make maximum use of

all source data — especially over sparsely sampled ocean areas.

Reviewed and approved on 20 May 19 72.

W. S. HOUSTON, Jr. Captain, U. S. Navy Commanding Officer Fleet Numerical Weather Central

li

TABLE OF CONTENTS

Page

TITLE PAGE i

FOREWORD ii

TABLE OF CONTENTS iii

LIST OF FIGURES AND TABLES iv

ABSTRACT vi

1. Introduction 1

2. Basic Ingredients for this Analysis, Observations and the General Thermal Wind Relation ...... 6

3. The Variational Formulation and Associated Euler Equations. . . . . .11

4. Method of Solution 19

5. Application 22

6. Concluding Remarks 29

REFERENCES. . . 33

APPENDIX 36

in

LIST OF FIGURES AND TABLES

Page

Figures

1 Schematic representation of "Ascent-Cascade" process of upper-air analysis 37

2 Sea level pressure at 00Z on 15 February 1972. . 38

3 Sea level pressure at 00Z on 15 February 1972. . 39

4 250 mb winds (knots) which serve as the upper boundary condition for construction of the intermediate levels 40

5 The wind observations (knots) used by the SCM scheme to construct the 250 mb analysis 41

6 Vector difference between the NVA velocity and

SCM velocity at 700 mb, W - \V . . 42

7 Vector difference between the NVA velocity and

SCM velocity at 400 mb, V - W. ........ 43

8 700 mb wind analysis (knots) by SCM scheme,

\V(700mb) 44

9 400 mb wind analysis (knots) by SCM scheme,

\V(400mb) 45

10 Wind shear (knots) between 400 and 700 mb, W(400) - \V(700), superimposed on the temperature field at 500 mb 46

11 Wind shear (knots) between 700 mb and the surface \V(700) - \V(SFC) , superimposed on the temperature field at 850 mb 47

12 Same as Fig. 11 except weights used in accord with Eq. (23) 48

Tables

1 Root-mean-square differences between adjusted fields (NVA) and SCM fields denoted by S(T), S(u) and S(v) where the differences are calculated at 500 mb and 700 mb for the temperature and wind, respectively 30

iv

Page Tables

2 Ratio between observational and dynamic weight as a function of cycle number . . . ... ... 31

3 The iterative solution to the Euler equations is accomplished by the accelerated Liebmann method. The number of iterations required for convergence is displayed as a function of the cycle. .... 32

ABSTRACT

An upper air objective analysis of wind and temperature

has been developed for operational use at Fleet Numerical

Weather Central. The primary feature of this analysis scheme

is the vertical and horizontal extrapolation of information

into the data-sparse regions from the data-rich surface and

jet aircraft level. The extrapolation is based on Sasaki's

recent extensions of the variational analysis method. The

hydrostatic equation and horizontal momentum equations are

used as dynamical constraints and these equations are combined

to relate temperature and wind through a generalized thermal

wind relationship. The governing analysis equations are

elliptic and amenable to solution by the relaxation process.

Vertical coupling of both wind and temperature is demonstrated

through the detailed examination of a case study in the North

Pacific.

VI

1. INTRODUCTION

The synoptic analysis of meteorological fields at

mid-tropospheric levels has been plagued by the scarcity

of conventional data. The data coverage over the densely

populated continents of Europe and North America has general-

ly been adequate for the definition of the large-scale or

extra-tropical cyclone features. For ocean operations,

however, the conventional data is strikingly inadequate to

define even these large-scale features. To augment the .

information at these mid-levels, the relatively rich sources

of data at the surface have been used. Yanai (1964) achieved

a. considerable improvement over standard methods when he

relied heavily upon the surface data,to build upper-air .

analyses of wind in the Caribbean. His success stemmed from

the ability to construct an accurate surface wind analysis

from relatively abundant ship and island reports. Adjoining

upper air charts were then built stepwise. by differential

analysis using the successive corrections method (SCM), an

operationally attractive scheme formulated by Bergthörsson

and Döös (1955) and modified by Cressman (1959).

Another approach that has been successful at the National

Hurricane Research Laboratory (NHRL) is the layer mean analy-

sis. In this analysis, the data base for the lower tropo-

sphere (1000 - 600 mb) and the upper troposphere (600 - 200 mb)

is effectively increased by amalgamating the information

from the individual mandatory levels. The apparent success

of this scheme is based on the two-layer structure of the

tropics during most of the year (Wise and Simpson, 1971).

In addition to increasing the data base for each

meteorological variable, the ability to couple variables

through dynamical or empirical relations is germane to

analysis in data-sparse regions. Thus, the observation of

a particular variable can indirectly produce information

about another variable. Sasaki (1958, 1969) has developed

a scheme based on variational calculus which couples the

meteorological fields by constraining the analysis to

satisfy a governing set of diagnostic or prognostic equations

This methodology has been used at Fleet Numerical Weather

Central (FNWC) to produce surface winds and pressure on a

global-band grid which encompasses the tropical belt (Lewis

and; Grayson, 1972). The work of Yanai (1964) also coupled

the fields at the surface by using a statistical-dynamical

model proposed for use at the Japan Meteorological Agency

(JMA) and reported by Masuda and Arakawa (1962).

The coupling of data in upper-air analysis has received

considerable attention in recent years. This interest has

been stimulated by the necessity to define observational

networks for numerical weather prediction (Döös, 1970). The

• 2

operational centers have traditionally used geostrophic

theory in conjunction with the successive corrections method.

With upsurge in global forecasting, however, the analysis

techniques have incorporated ageostrophic effects. The

National Meteorological Center (NMC) now analyzes heights

and winds by spectral methods which use Hough functions

(eigenfunctions of Laplace's tidal equation) for horizontal

modes and empirical orthogonal functions for the vertical

modes (Flattery, 1971). The results indicate that reasonable

height and wind fields are obtained, even at the equator.

Since the Hough functions are derived from equations which

neglect the nonlinear advection terms in the momentum equations,

there have been some problems when the analysis is used for

initializing the.primitive equation forecast model (Flattery,

1970). Gradient wind corrections have been made in an

attempt to account for the nonlinear imbalance.

The proposed upper-air analysis scheme at FNWC is designed

for operational use on a global band grid extending from

40S to 60N which has a mesh length of 150 n mi at the equator

and 75 n mi at 60N. The change of mesh size is a consequence

of the distortion inherent in the Mercator secant projection

(true at 22.5N and 22.5S) used for the analysis. The flight

support for turbo-prop aircraft has been a primary considera-

tion in the development of this scheme. These planes cruise

in the vicinity of the 400 mb surface and flight support

and guidance at this level are necessary. As mentioned

previously, the data-rich levels are the surface and the jet

aircraft level (vicinity of 250 mb). The extrapolation of

data from these data-abundant regions into the mid-troposphere

is accomplished by using Sasaki's numerical variational

analysis (NVA) method. This approach reduces the analysis

problem to a boundary value problem where the upper and lower

surface are specified and the interior is determined subject

to the governing elliptic analysis equations. Since there

are scattered observations between the surface and 250 mb,

the analysis scheme must have the capability of weighting

this local information as well as incorporating the extrapolated

information. The variational analysis scheme provides a

natural way to accommodate both the observations and the

dynamically extrapolated information.

The surface analysis which is used to provide boundary

values for the upper-air analysis is based on the operational

scheme reported by Lewis and Grayson (1972). This analysis

has been subjected to a two-dimensional adjustment by NVA.

The momentum equations used as constraints for this surface

analysis are consistent with the upper-air scheme except

that they contain the effects of surface friction. The

250 mb analysis is found by the SCM. A two-dimensional

adjustment could be performed at this level, but has been

postponed until the general features of the extrapolation

procedure are clarified. In order to incorporate dynamical

information at this level, the forecasted wind from the FNWC

primitive equation model (Kesel and Winninghoff, 1971) is

used as the guess field for the SCM between 20N - 60N.

Since aircraft rarely report geopotential (D-values) and,

moreover, because of the limited credibility of these observa-

tions, the coupling of temperature and wind has been used

to build the upper-air analyses. The subjective or manual

analysis of upper-air wind has heavily relied upon this

coupling or thermal wind relationship (see Saucier, 1955).

In order to couple these fields over the tropics and extra-

tropical regions, a more general form of the thermal wind

relation which includes ageostrophic effects must be employed.

Forsythe (1945) developed graphical techniques which enabled

the hand-analyst to account for ageostrophic effects between

the temperature gradient and wind shear. It is precisely

this traditional synoptic rule connecting temperature gradient

and wind shear that provides the vehicle for extrapolation

in the objective scheme.

2. BASIC INGREDIENTS FOR THIS ANALYSIS, OBSERVATIONS

AND THE GENERAL THERMAL WIND RELATION

The operational approach to numerical variational

analysis (referred to as NVA in future discussion) adopted

at FNWC is: (1) irregularly distributed observations of

each variable are independently interpolated to grid points

by the SCM, (2) a reliability is attached to the variable

at each grid point and is dependent upon both the number

of observations affecting this point and the distance of the

observations from the point, and (3) an adjustment of the

"observations" (the analysis resulting from the SCM) by the

NVA equations which have incorporated the model constraints.

The analysis is performed every 6 hr and the natural

candidate for the guess field of SCM is the previous analysis

or, equivalently stated, the 6 hr persistence field. In

addition to the persistence field, however, the FNWC primitive

equation model produces a forecast every 12 hr over the

Northern Hemisphere. The guess field for the wind is 6 hr

persistence south of 20N. Between 20N - 60N, the 6 hr and

12 hr forecasts are alternately used. That is, the 06 GMT

and 18 GMT global-band analyses use the 6 hr forecasts from

00 GMT and 12 GMT, respectively. Similarly, the 00 GMT

and 12 GMT analyses use 12 hr forecasts from 12 GMT (previous

day) and 00 GMT, respectively. The temperature guess field,

however, is strictly the 6 hr persistence field. The feed-

back from wind to temperature occurs in the NVA scheme.

Only when the "observations" (SCM analyses) are independent

can the variational adjustment process guarantee maximum

likelihood estimates (Cramlr, 1955; Lewis and Grayson, 1972).

Four successive scans are made to correct the guess

field. For the analysis of temperature, the radii are 600,

300, 200 and 100 n mi, respectively. Grid points which are

600 n mi or further from a report are unmodified, i.e.,

the 6 hr persistence becomes the current analysis. It is

precisely this kind of event that often causes the SCM

to produce "unmeteorological" fields, i.e., the data-sparse

areas adjoining the data-rich areas show discontinuities

which are characteristic of the mathematical weighting func-

tions of the SCM and are not attributable to meteorological

events.

In a similar fashion, three scans are used by the SCM

for wind analysis. Here the two component directions are

analyzed together. In addition to speeding the analysis

procedure, the decision to accept or reject wind data is

based on the vector rather than the separate sealer components.

The radii for the wind analysis are 450, 300 and 150 n mi,

respectively. The successively decreasing scan radii are

designed to capture the larger scale features on the preliminary

scan and then include the smaller scale features on the

following scans. Stephens (1967) has discussed the relation-

ship between characteristic meteorological scale and the scan

radii under the assumption of uniformly distributed data.

The second ingredient to NVA is the model dynamics

which will be used to couple the variables and provide the

path to extrapolation from the data-dense to data-sparse

regions. The governing equations are modified in a manner

consistent with the NVA at the surface (Lewis and Grayson,

1972). Namely, the nonlinear advection terms are approximated

by the distributions resulting from the SCM. The primary

advantage is the simplification of the Euler equations and

the guarantee of an elliptic differential equation system

regardless of the weights used in the variational formulation.

The guaranteed ellipticity leads to rapid convergence when

the relaxation process is used. The governing equations in

Mercator coordinates are:

It * ^vü " fv " "m H w

|I + W-Vv + fu - -m |i (2)

|± = RT C3)

a = In (po/p) (4)

where: t

x

y

a

V

time

W

T

£

m

e

R

east-west coordinate (positive eastward)

north-south coordinate (positive northward)

vertical coordinate (positive upwards)

ui + vj A.

i

J

Ul + Vj

temperature

horizontal velocity

unit vector along positive x

unit vector along positive y

velocity from SCM

geopotential

Coriolis parameter

: image scale factor for Mercator

projection true at 22.5N. and 22.5S

cose sec0 o

latitude (9 =22.5 degrees)

reference pressure (1013 mb)

gas constant for dry air

= m 9C ) I + m 3Q • !JT— i + m ^ j ^7 gradient operator

To simplify the notation, the nonlinear advection terms

are denoted by:

A = \V*Vu

B = \V-Vv

The general thermal wind relation is derived by eliminating

the geopotential from the momentum equations, (1) and (2),

(5)

C6)

by substituting from the hydrostatic equation, (3). In

component form, this relation is

8 r8u. 3 A ^ r 3v D 3T c ,-, Jt for> = " JO A + £ 3^ • mR a* = F <7>

It $) = - la 5 - £ "IF - mR ly - G C8)

where the separation of terms by functions F and G will prove

useful in the discussion on minimization.

10

3. THE VARIATIONAL FORMULATION AND ASSOCIATED EULER EQUATIONS

The timewise localized version of NVA (Sasaki, 1970) is

adopted for the upper-air analysis and the functional to be

minimized assumes the form:

I = \U{[a(u-u)2 + S(v-v)2]

+ [ß(T-T)2]

+ [a(Jt C3o"^ + a(9t C9^ 1} J t-xTy-) dV

(9.)

where

a

s a

xe

o xe,ye

df :

Gauss' precision modulus for wind

Gauss' precision modulus for temperature

Dynamical weight

east-west coordinate on earth

north-south coordinate on earth

analysis resulting from the SCM

Jacobian of transformation from earth to

Mercator coordinates

incremental volume (dxdyda)

The relationship between earth and map coordinates is

dxedye ■ J(^5£) dxd^ • (10)

Consequently, (9) implies that equal weight is given to

11

equal earth areas, not equal map areas. For the Mercator

prbj ection,

(.11)

By virtue of (7) and (8), the functional can be rewritten

I =

r , 2 2 ^ a[(u-u)z + (v-v)z]

♦ 3[(T-T)2]

+ a[F2 + G2]

> j(vii) dy x,y (12)

J

The first set of bracketed terms force the analysis toward

the SCM velocity in proportion to 5. Similarly, the second

bracketed term forces the temperature toward the SCM

temperature in proportion to 3. The weights a and 3 are

called observational weights or Gauss' precision moduli

and specified beforehand, i.e., before the minimization

process. They are a measure of the confidence in the

respective SCM fields. Following the discussion of these

weights in the paper by Lewis and Grayson (1972), let

''S - l/ow , 3 = l/crT (13)

where a is the mean-square difference between the separate

wind components of NVA and SCM and aT is the mean-square

difference between the NVA and SCM temperatures.

In analogy with the observational weights 5 and 3, the

dynamical weight a is expressed as

12

a = l/at (14)

where ot is the mean-square value of local time rate of

change of wind shear as calculated from (7) and (8). Alter-

natively stated, a is a measure of the steadiness or degree

with which u, v, and T satisfy F = G = 0. As a increases,

the analyzed fields are forced closer to the steady state

relationships.

The observational weights 5 and 3 should be a function

of space (and time) and generally distinguish between (1) grid

points that benefited from nearby observations and (2) grid

points unaffected by observations and consequently reflecting

6 hr persistence after the SCM analysis. A method for speci-

fication of a and ß in terms of the distance-dependent weighting

implicit in the SCM has been discussed by Lewis and Grayson

(1972). During the development stages of the upper air analy-

sis, the weights have not varied as a function of space but

have been chosen to reflect only typical variance of the

wind and temperature.

As mentioned in the Lewis and Grayson (1972) paper, the

separately specified values of o , aT and a. are not preserved

in the minimization process. However, the ratios between

these values are preserved. This fact will be obvious when

the Euler equations (Eqs. 17-19) are examined. If a, b and

c are chosen for a , aT and ö., the final analysis produces

13

values d, e and f, respectively, where a:b:c = d:e:f. Rather

than specifying a , a„ and a , it is sufficient to specify

Qy/°t and cJT/at.

A necessary and sufficient condition for minimization

of (12) is the vanishing of the first variation, denoted by

51. The sufficiency of this condition is directly attributa-

ble to the quadratic nature of the integrand (Lanczos, 1970,

p. 41). Using integration by parts and the commutative

properties of the 6-process (Lanczos, 1970, p. 56), the

functional takes the form

0 = 61 = (P6u ♦ Q6v ♦ W6T) äV ♦ ilZT^tllZlttilV .• C»)

The expressions P, Q and W are differential expressions which

will be explicitly written later.

The boundary terms indicated on the right-hand side of

(15) involve integrals over the bounding surfaces, viz.,

the walls at 40S and 60N, the 250 mb level and the earth's

surface. The east-west boundaries are cyclic because of the

360° longitudinal span of the analysis region. The conditions

necessary to cause these integrals to vanish are called the

natural boundary conditions. There is generally more than

one set of acceptable boundary conditions. The simplest

boundary conditions are: 6T = 0 on the north and south

walls and 6u = 6v = 0 along the upper and lower surfaces.

14

These are Dirichlet conditions and allow no adjustment of

temperature and velocity along the boundaries where the

respective variations vanish. Alternative boundary condi-

tions can be found by examination of the integrals given

in the appendix. The simplicity of the Dirichlet conditions,

especially in regard to the programming of numerical methods

for solution, has led to the adoption of this set of boundary

conditions.

Having disposed of the terms evaluated on the surface

boundaries and because of the independence of the variations

of u, v and T, the solution to (15) is

p = Q = W = 0 (16)

where

P =

Q =

i1 4 - C|) (U-Ü) da

2~ + mfR 3? C*y> + f ^7

f2 4 . (|j (v.-3

► = 0

do'

- mfR Ja ^3x (£) -f 82A

3a

' = 0

(17)

(18)

15

® ©

w =

V.(m"2VT) - (mR)~2 (§)(T-T)

L7x (^^) + VfJLl m R

m

= 0 (19)

® and

©

c - & * ♦!! J 3a 3a (20)

The local vertical unit vector is k and the dot and cross-

products have been incorporated in (19). Equations (17)

and (18) can be combined to produce the vector representation:

© © ^ VV)

r f2 ^ - (|)(V

3a'

+ fRkx 4~ VT da fkx i- c

= 0 (21)

® © The equations, (17)-(19), or equivalently, (19) and (21), are

called The Euler equations. Together with the Dirichlet

boundary conditions, they constitute the governing equations

for the upper-air analysis.

The derivation of (17)-(19) has assumed that a is invariant

in space. A variable a can easily be included in the equations,

but the applicability of the model constraints over the entire

analysis region dictates the constancy. The ratios (3/a and

16

a/a are the controlling factors in (19) and (21), respectively.

As the reliabilities of the SCM wind and temperature increase,

the ratios a/a and ß/a increase, respectively. Consequently,

term © in both (19) and (21) forces the analysis toward the

SCM analysis in proportion to these ratios.

Terms (T) and (3) in (21) represent the geostrophic

coupling between wind and temperature. In fact, except for

the differentiation with respect to a, this is the geostrophic

thermal-wind relationship. Interpreted physically, the verti-

cal change of horizontal temperature gradient contributes

r> L (If

to the curvature, —J- , of the wind profile. Term @ 3az

brings the nonlinear advection of momentum into the process

and is generally an important contributer in the regions of

strong horizontal shear such as in the area neighboring the

jet stream. Although this term is approximated by the SCM

analysis, the approximation is improved by a "re-cycling"

technique explained in the next chapter. 2

The horizontal curvature of temperature, V T, and the

vertical variation of vorticity, •*— (£»V>< \V), are related by

terms (l) and (5) in (19). Again, these terms stem from a

strict hydrostatic-geostrophic balance. The nonlinear

advective effects are contained in (5) and are associated

with vertical variation of streamline curvature as expressed

3 3 u by u T- (•—-y), etc. These nonlinear terms are subject to ^ 3x

17

improved approximation by the re-cycling method mentioned

above.

In the development of the finite-difference analogs of

the Euler equations, a departure from the exact form has

been made. As shown by Forsythe and Wasow (1960, p. 182),

the construction of symmetric operators follows directly

from the finite-difference form of the functional, Eq. (12)

in this case. Assuming that a centered difference is used 3 \V to approximate -r-—- in the functional, e.g.,

3 W, _ \V250 " V700 fn~. "5F~J400 mb " ln(7ÖÖ/25Ö) ' CZZJ

3 \V then the finite-difference analog to —7 should include 3a

values at both the second level above and the second level

below the one in question. Because of the limited vertical

extend of the discretized analysis space, only the first

level above and the first level below are used in calculating

the second derivative. This departure from rigorous form

introduces some discrepancy between the exact minimum for

(12) and the value obtained by using the solution to the

assumed finite-difference form of (17)-(19). However, the

basic concept of coupling the variables and extrapolation

remains intact.

18

4. METHOD OF SOLUTION

The set of Euler equations are coupled elliptic

differential equations (Helmholtz type) subject to

Dirichlet conditions. The form of the equation lends

itself to a vertically staggered grid arrangement, viz.,

an alternating sequence of temperature and wind fields

along the a-axis. Equation (19) associates the vertical

gradient of vorticity, term (T), and vertical gradient of

momentum advection, term (4J, with the horizontal tempera-

ture distribution. Consequently, the finite-difference

8 \V evaluation of ■*-— can be constructed using only the levels

adjacent to the temperature. A similar evaluation is possible

in connection with (21) and the finite-difference form of

9T ^—. A principal advantage of the staggering scheme is the

reduction in both computer memory and time requirements.

The staggered system with the assignments for temperature

and wind is shown in Fig. 1. The operational sequence of

events in the analysis package is (1) the surface wind and

pressure field are analyzed by the two-dimensional NVA

scheme (Lewis and Grayson, 1972), (2) the SCM upper-air

analysis is used for temperature and wind and (3) the varia-

tional analysis is accomplished through the solution of

(17)-(19). The solution of these Euler equations is schematical

ly represented in Fig. 1.

19

An elaboration upon the cycling process is instructive:

STEP 1. The S50 mb temperature is found by solving (19)

by the accelerated Liebmann method (Miyakoda, 1961).

On the first cycle, the available wind field at the

adjacent upper level is the SCM analysis.

STEP 2. The 700 mb wind field is found by solving (17) and

(18), again by the accelerated Liebmann method.

The temperature field at 850 mb found in STEP 1 is

now used in conjunction with the SCM temperature

3T analysis at 500 mb to calculate TT—. The wind fields dO

at 400 mb (SCM on this first cycle) and the surface

2 are used to estimate 3" V

T7~ STEP 3. The temperature at 500 mb is found by solving (19),

which uses NVA fields below this level and SCM

fields above.

STEP 4. The wind field at 400 mb is found by solving (17)

and (18), which use NVA fields below and SCM fields

above.

STEP 5. The temperature at 300 mb is found by solving (19),

which uses the NVA fields at all levels except 250

mb where the wind analysis is derived by the SCM

scheme.

With the completion of this complete cycle through the

a-layers, a re-cycling process is performed. Now, the adjusted

fields from the first cycle are used as forcing functions in

20

(17)-(19). As expected, the convergence of the Liebmann

method is much faster during the re-cycling (see Table 3).

The nonlinear advection terms, A and B, are also re-calculated

and. these improved estimations are used in (17)-(19). The

process of re-cycling could be continued indefinitely but

obviously at the expense of computer time.

One re-cycle through the levels accounts for most of

the adjustment which is fortunate from an operational

viewpoint. A more rigorous criterion for termination could

be based upon the ratios of observational to dynamical weight.

Thus, having specified 0 /o*t and o^/o beforehand, the final

analysis should approach these ratios as the number of cycles

increases.

21

5. APPLICATION

The analysis scheme has been tested using data collected

on 15 February 1972. Although the objective analysis was

performed over the entire grid (~7000 points at each level),

the synoptic discussion will concentrate on the Pacific Ocean

region between 20 and 60N. The jet-aircraft flights between

the Pacific islands and mainland (Asia and North America)

provide credible winds for the upper boundary condition.

Additionally, the ship traffic and island density anchor

the surface analysis.

Figure 2 depicts the sea level pressure analysis via

two-dimensional NVA (Lewis and Grayson, 1972). The data

coverage is uniformly dense and both pressure and wind

reports have been superimposed over the pressure pattern.

The surface wind analysis obtained simultaneously by the

NVA scheme is shown in Fig. 3. This analysis serves as the

lower boundary condition for the construction of the upper

air analysis. The wind has been expressed in knots to

conform with the standard operational format.

The upper boundary condition is the SCM analysis of wind

at 250 mb. This field is displayed in Fig. 4 and wind

observations incorporated into this analysis are shown in

Fig. 5. The primitive equation forecast (12 hr) is used as

the guess and accounts for the definition of features in

data-void areas. For example, the pronounced ridge near

22

160-170E and extending from 30-45N is virtually the

unadulterated guess field.

With the specification of the boundary conditions and the

assignment of the weight ratios a/a and 3/a, the Euler

equations can be solved. The solution is accomplished by

using the accelerated Liebmann method in the level-by-level

sequence discussed in Chapter 4. The convergence rate of

this iterative method is strongly dependent upon the weight

ratios. As ß/a and a/a increase, the rate of convergence

increases. Essentially, term (?) in both (19) and (21)

becomes increasingly important and the coupling between the

wind and temperature is weakened. Operationally, the range

of values for these ratios must be restricted to prevent

the usage of unreasonable amounts of computer time. The

chosen ratios, however, should guarantee a feedback or coupling

between the wind and temperature. The results discussed in

this paper have been obtained with the following ratios:

a/a = o^/o = 1.5

ß/a = <Jt/aT = 250

(23)

where the implied space, time and temperature units are n mi,

hr and °C, respectively.

In order to attach physical meaning to these weights,

the root-mean-square difference between the SCM fields and

23

the adjusted fields must be examined. Representative values

at 500 mb and 700 mb are displayed in Table 1. It is signifi-

cant from an operational viewpoint that the major adjustment

occurs within two complete cycles through the ö-layers. This

table also depicts the root-mean-square value of the shear

tendency, S(F) and S(G). Since Aa ~ 0.9 for the layer between

the surface and 400 mb, values of S(F) and S(G) can be

roughly interpreted as the 400-1000 mb wind shear change over

a one hr period.

A measure of the overall convergence (distinguished from

the convergence at a particular level on a given cycle) can

be formulated in terms of the difference between calculated

and hypothetical ratios. From the form of (12) and the

definitions for the weights given in (13), the following

relationship should hold:

2-1. ~-l. ~-l. -1. -1 3 : a : a : a : a

= S2(T):S2(u):S2(v):S2(F):S2(G)

(24)

where S is the root-mean-square designator. From (24) we

get c 2

S/o <

= [S(F)/S(T)J

= [S(G)/S(T)]

V

(25)

24

and r

a/a \

V.

= [S(F)/S(u)]

= [S(F)/S(v)]

= [S(G)/S(u)]

= [S(G)/S(v)]

(26)

Average values of a/a and ß/a based on (25) and (26)

are displayed in Table 2. The marked improvement within

two cycles is evident. However, the theoretical values can

never be obtained because of the truncation error introduced .

by the finite difference analogs to (17)- (19). Additionally,

the license used to formulate the second derivative in terms

of adjacent levels (see end of Chapter 3) ultimately causes

some discrepancy.

As expected a priori, the amount of computer time necessary

to solve for u, v or T at a particular level decreases with

every cycle through the a-layers. Table 3 shows the number

of iterations for both temperature and wind at 500 and 700 mb,

respectively. The convergence criterion was established to

yield solutions accurate to 10~2 - 10"loC and 10"1 - 1 kt

for temperature and wind, respectively. These values are

substantially below the accuracy of the observations.

In order to reveal the magnitude of adjustment in the

wind field, a vector difference between the NVA and SCM

25

analyses is displayed. Figures 6-7 show the vector difference

at 700 and 400 mb, respectively, after two complete cycles

through the a-layers. The SCM analysis at 700 and 400 mb

is shown in Figs. 8 and 9, respectively. An encouraging

result that appears on both Figs. 6 and 7 is the organization

of the adjustment. That is, the scale of the adjustment is

the same order of magnitude as the synoptic scale disturbances

and does not exhibit two or three mesh-length oscillations.

The maximum adjustment at 700 mb occurred in connection

with the cyclone at approximately 40N, 175W. The SCM

analysis shows definite cyclonic circulation above the surface

low pressure center, but the adjustment basically increases

the intensity of this vorticity center. Whereas the SCM

analysis indicates wind speeds the order of 10 kt to the

west of the center, the NVA scheme has speeds around 30 kt.

There were no 700 mb observations in this area to verify the

adjustment, but the strength of the surface system lends

support to both the sense and magnitude of the modification.

Another interesting adjustment occurred east of the Japanese

Islands in the vicinity of the col in the sea level pressure.

The magnitude of 700 mb wind above this area is decreased by

NVA and the cyclonic vorticity is weakened. The cyclonic

vorticity is increased, however, directly above the two

low pressure systems. Generally, the circulations at the

surface have been incorporated into the 700 mb chart.

26

The scale of adjustment at 400 mb appears to be larger

than at 700 mb as expected. For example, the upper-air

circulation accompanying the mid-Pacific low is adjusted

in the cyclonic sense but over a much larger east-west

extent. A conspicuous amplification of the ridge along 145W

can be seen by examination of the 400 mb adjustment field.

Again, this seems to reflect the feedback of information

from 250 mb.

The coupling between the temperature and wild fields

is displayed in Fig. 10. If the model constraints were

strictly geostrophic-hydrostatic, the direction of the

shear vector would be along the isotherms and its magnitude

would be proportional to the isotherm gradient. There are

some noticeably ageostrophic regions especially in the

vicinity of weak thermal gradients. The correlation between

the shear and the thermal pattern is evident in the baro-

clinic regions, e.g., the extensive SW-NE flow from mid-

Pacific to the Canadian border and the SW-NE flow just east

of Japan.

With the weight ratios given by (23), the influence

of the wind field on the temperature field is not as con-

spicuous as the reverse coupling, i.e., the influence of

temperature on wind. This statement is further substantiated

by the results in Table 1 indicating that the NVA temperature

27

is very close, in an rms sense, to the SCM temperature.

Stated mathematically, term (Y) in (19) dominates the other

terms in this equation. In order to isolate the influence

of wind on temperature, an experiment was conducted in which

ß=0. Thus, term (Y) vanishes in (19) and the temperature

at any level is a function of the wind field and the boundary

values of temperature on the north and south walls.

As expected, the rate of convergence is slowed down

considerably in this case; in fact, the number of iterations

increased by a factor of 7. The 850 mb temperatures for

the case 3=0 and the control case, i.e., weights given by

(23), are shown in Figs. 11 and 12, respectively. The

maximum or minimum values or "central" values are in obvious

disagreement. Effectively, the (3 = 0 case has no anchoring

information on the interior. It is very encouraging, however,

to see the coupling between isotherm curvature and the wind

shear vector. Especially interesting is the cut-off feature

in mid-Pacific.

28

6. CONCLUDING REMARKS

The primary results based on this study are (1) the

variational method can incorporate detail into the 400-700 mb

wind fields via vertical coupling and (2) the iterative

technique for solving the analysis equations is efficient

enough to withstand the rigid time requirements of operational

analysis and prediction. Based on information from the case

study, the estimated time for solution to the Euler equations

is 15 min [Central Processor Unit (CPU)] on the CDC 6500

computer. The SCM analysis takes approximately 15 min also,

yielding a combined time of 30 min for the entire analysis.

The analysis will be routinely transmitted (every 6 hr)

to the outlying weather centers in June, 1972.

Considerable effort is currently concentrated on increasing

the data-base for the analysis. In particular: (1) the

incorporation of pibals (pilot balloons) information will

ultimately strengthen the low-level wind field; (2) the

winds derived from satellite photographs will provide support

at both the upper-level (~250 mb) and low-level (-700 mb);

and (3) temperature information from SIRS soundings will be

added to the conventional temperature reports.

29

TABLE 1. Root-mean-square differences between adjusted

fields (NVA) and SCM fields denoted by S(T), S(u) and S(v)

where the differences are calculated at 500 mb and 700 mb

for the temperature and wind, respectively. The mean-square

values of F and G (see Eqs. 7 and 8) at 500 mb are denoted

by S(F) and S(G).

Cycle S(T) °C

S(u) kt

S(v) kt

S(F) i kt hr 1

S(G) , kt hr i

1 0.324 3.83 4.34 7.68 5.75

2 0.325 3.82 4.02 6.39 5.12

3 0.323 3.85 4.01 6.20 5.02

4 0.322 3.88 4.03 6.16 5.07

30

TABLE 2. Ratio between observational and dynamic weight as

a function of cycle number. The hypothetical ratios are shown

in parentheses and the calculations based on values from

TABLE 1.

r - - (1.5) £ (250) Cycle a *• J a v J

1 2.78 424

2 2.18 327

3 2.06 305

4 2.04 308

31

TABLE 3. The iterative solution to the Euler equations is

accomplished by the accelerated Liebmann method. The number

of iterations required for convergence is displayed as a

function of the cycle. These numbers represent the calcula-

tions at 700 mb and 500 mb for wind and temperature, respectively

Cycle Temperature Wind

1 11 9

2 6 8

3 5 6

4 4 5

32

REFERENCES

Bergthörsson, P., and B. Döös, 1955: Numerical weather map analysis. Tellus, 7_, 329-340.

Cramer, H., 1955: The Elements of Probability Theory. New York, Wiley, 281 pp.

Cressman, G. P., 1959: An operational objective analysis system. Mon. Wea. Rev., 94, 367-374.

Döös, B., 1970: Numerical experimentation related to GARP. Global Atmospheric Research Programme (GARP), WMO-ICSU Joint Organizing Committee, Pub. Ser. No. 6, 32 pp.

Flattery, T. W., 1970: Spectral analysis and forecasting. Numerical Weather Prediction Activities, National Meteorological Center (NMC), Second-half 1969, 51 pp.

Flattery, T. W., 1971: Spectral models for global analysis and forecasting. Proc. 6th AWS Tech. Exch. Conf., U. S. Naval Acad., Air Weather Service Rept. 242, 42-54. ~

Forsythe, G. E., 1945: A generalization of the thermal wind equation to arbitrary horizontal flow. Bull, of Amer. Meteor. Soc., 26, 371-375.

Forsythe, G. E., and W. R. Wasow, 1960: Finite-Difference Methods for Partial Differential Equations, New York, Wiley, 444 pp.

Kesel, P. G., and F. J. Winninghoff, 1971: Fleet Numerical Weather Central's four processor primitive equation model Proc. 6th AWS Tech. Exch. Conf.,U., S. Naval Acad., Air Weather Service Rept. 242, 17-41.

Lanczos, C, 1970: The Variational Principles of Mechanics, 4th ed. University of Toronto Press, Math Expositions No. 4, 375 pp.

Lewis, J. M., and T. H. Grayson, 1972: The adjustment of surface wind and pressure by Sasaki's variational matching technique. Accepted for publication in J. Ap'pl. Meteor.

Masuda, Y., and A. Arakawa, 1962: On the objective analysis for surface and upper level maps. Proc. Intern. Symp. Numerical Weather Prediction Tokyo, 1960, Meteor. Soc. Japan, 55-66.

33

Miyakoda, K., 1961: Contribution to numerical weather prediction - computation with finite difference. Geophysical Institute, Tokyo University, 190 pp. (see p. 96).

Sasaki, Y., 1958: An objective analysis based on the variational method. J. Meteor. Soc. Japan, 36, 77-88.

1969: Proposed inclusion of time variation terms, observational and theoretical, in numerical variational objective analysis. J. Meteor. Soc. Japan, 47, 115-124. * ~ ——

_, 1970: Some basic formalisms in numerical variational analysis. Mon. Wea. Rev., 98, 875-910.

Saucier, W. M., 1955: Principles of Meteorological Analysis, University o± Chicago Press, 438 pp.

Stephens, J. J., 1967: Filtering responses of selected distance dependent weight functions. Mon. Wea. Rev., 95, 45-46.

Wise, C. W., and R. H. Simpson, 1971: The tropical analysis program of the National Hurricane Center, Weatherwise, 24, 164-173.

Yanai, M., 1964: An experimental objective analysis in the tropics. Tech. Paper No. 22, Dept. of Atmos. Sei., Colorado State University, 21 pp.

34

APPENDIX: NATURAL BOUNDARY CONDITIONS

The right hand side of (15) has categorically grouped the

terms involving integrals over the boundary surfaces. These

integrals originate from the last set of bracketed terms in

(12). The mechanics of extracting the surface integrals

from the functional is explained in Lanczos (1970). Before

writing the explicit form of these integrals, it is convenient

to adopt the following notation:

x', xM : boundaries of analysis space in east-west direction

y', y" : boundaries of analysis space in north-south direction

a1, a" : boundaries of analysis space in vertical direction

Realizing that the global-band is cyclic in the longitudinal

or east-west direction, the relationship

4>(x») = 4>(x") (Al)

holds, where <J> represents any of the variables, u, v or T or

derivatives of these variables. The explicit form is:

35

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+ £^ © rr ©

w • &?> ^* % [«• |Äjx: dyd0

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O. ff 0 ^ of II r. 3B,a" , , af l \ ,, 3A,a" , ,

i? J I ^ °' " ? ' ' ' TO "' x"

where [4>(x,y,a)] , = <J>(x",y,a) " «Kx1»/»0") in conformity with

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UNCLASSIFIED Security Classification

DOCUMENT CONTROL DATA -R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified)

I. ORIGINATING ACTIVITY (Corporate author)

Fleet Numerical Weather Central

2a. REPORT SECURITY CLASSIFICATION

Unclassified 2b. GROUP

3. REPORT Tl TLE

An Operational Upper Air Analysis Using the Variational Method

4. DESCRIPTIVE NOTE.S (Type of report and inclusive dates)

Technical Note No. 72-3, May 1972 8- AUTHOR(S) (First name, middle initial, last name)

John M. Lewis

6- REPORT DATE

May 1972 7a. TOTAL NO. OF PAGES

50 ...- 76. NO. OF REFS

20 8a. CONTRACT OR GRANT NO.

b. PROJEC T NO.

3a. ORIGINATOR'S REPORT NUMBER(S)

9b. OTHER REPORT NOISI (Any other numbers that may be assigned this report)

10. DISTRIBUTION STATEMENT

Approved for public release; distribution unlimited,

It. SUPPLEMENTARY NOTES

13. ABSTR AC T

An upper air objective analysis of wind and temperature has been developed for operational use at Fleet Numerical Weather Central. The primary feature of this analysis scheme is the vertical and horizontal extrapolation of information into the data-sparse regions from the data-rich surface and jet aircraft level. The extrapolation is based on Sasaki's recent extensions of the variational analysis method. The hydrostatic equation and horizontal momentum equations are used as dynamical constraints and these equations are combined to relate temperature and wind through a generalized thermal wind relationship. The governing analysis equations are elliptic and amenable to solution by the relaxation process. Vertical coupling of both wind and temperature is demonstrated through the detailed examination of a case study in the North Pacific.

DD """ 1473 1 NO V 65 I *T / W

S/N 0101-807-6801.

(PAGE 1) 49 UNCLASSIFIED

Security Classification


KEY WORDS LINK C

Numerical Variational Analysis (NVA)

Upper-Air Analysis

Global-band grid

Objective Analysis

Tropical Analysis

General thermal wind constraint

Coupling between wind and temperature analysis

DD ,',r..1473 «BACK) (PAGE 2) 50


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