NPS ARCHIVE1966FERRENTINO, P.

;;dfflBnnOOORi""MM

WW,

A TWO-DIMENSIONAL OMEGA EQUATION FORTHE 1000-700 MB LAYER WITH DIABATIC HEATING

PETER S, FERRENTINO

$3iibSBffl

ismmmi Bff»

mim

1

tfffifi

matinm GEMS

msMmsam

BuWMsHHbw1i'v

"',-

''I

i

m mw

BOOK&$Be

HI: ''!.''

'

:

- ''

'

;

WfflH

Wlttl tnUifi

|jj»:

i

:

:;'.?;''v7i'''''

'

rWP'''''; ''," ',* JK]B$|| . \

Bfi";'','' V 'Nf'^U.: :

: .. .'

mJtHViJwfn

H itnwi

11V 1

'

TJffiW'ii'i

:

L'' !.''''...

.

'

''''• ';-::' ;';',/

NHEK B5Kffissv

SdCoct

LIBRARYNAVAL POSTGRADUATE SCHOOL

MONTEREY, CALIF. 93940

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY CA 93943-5101

This document has been approved for publicrelease and rale; its distribution is unlimited.

6 &3

NAVAL POSTGRADUATE SCHOOt

A TWO-DIMENSIONAL OMEGA EQUATION FOR THE

1000-700 MB LAYER WITH DIABATIC HEATING

by

Peter S. FerrentinoLieutenant, United States Navy

Submitted in partial fulfillmentfor the degree of

MASTER OP SCIENCE IN METEOROLOGY

from the

UNITED STATES NAVAL POSTGRADUATE SCHOOL

May 1966

ABSTRACT

A two-dimensional omega equation is derived by

combination of the vorticity and thermodynamic equations.

The desired omega is then taken to be the logarithmic average

in the 1000-700 nib layer. A diabatic term, after Laevastu,

for oceanic areas only is included to deduce the empirical

temperature and vapor-pressure changes associated with

sensible and latent heating in the maritime layers. Over

both continental and oceanic areas a frictional vorticity

sink is included in order that excessive energy cannot be

generated over the ocean. Among other novel features is the

use of the Holl static-stability parameter which affords

vertical consistency with analyses prepared by Fleet

Numerical Weather Facility.

TABLE OP CONTENTS

Section Pag©

1. Introduction 11

2. Th© Basic Omega Equation 11

3. Vortical Distribution of Pressure and 13Omega

k. Vertical Integration of the Omega 15Equation

5. The Lower Boundary Condition 19

6. The Diabatic Heating Term 21

7. State-Change Parameters for the 25Convective Case

8. Numerical Procedures 30

9. The Computer Program 36

10, Results and Conclusions 38

11, A^mowledgei-nents ';.l

12, Bibliography 57

LIST OP ILLUSTRATIONS

Figure Page

1. Pressure Distribution in the Vertical 13

2. The Omega Profile 15

3. Three Cases of the Lower Boundary 20

1|. Model of the Convective-atmosphereModification 22

5. AT^p as a Function of ATD 26

6. 00Z 28 April 1966 8£0 rab D Field i+2

7. 00Z 28 April 1966 Diabatic Heating [[3

8. 00Z 28 April 1966 Stability kk

9. 00Z 28 April 1966 8£0 mb Adiabatic 1^5Omega

10. 00Z 28 April 1966 FNWF 8£0 mb Omega I|6

11. 00Z 28 April 1966 8£0 mb Omega l±7

12. 12Z 28 April 1966 8^0 mb D Field 1+8

13. 12Z 28 April 1966 8£0 mb Omega 1+9

ll|. 00Z 29 April 1966 8£0 mb D Field £0

15. 00Z 29 April 1966 8£0 mb Omega 5l

16. 12Z 29 April 1966 8£0 mb D Field $2

17. 12Z 29 April 1966 8£0 mb Omega 53

18. 00Z 28 April 1966 Omega Lower Boundary Sh

19. 00Z 28 April 1966 FNWF Omega Lower ^Boundary

LIST OP SYMBOLS

Scalar Quantities

temperature

Ta surface air temperature

Id surface dew-point temperature

hv sea water temperature (surface)

|ivb wet -bulb temperature

VC. condensation level temperature

Q potential temperature

Sa surface vapor pressure

Qs surface saturation vapor pressure

^v vapor pressure over water

Oc condensation level vapor pressure

P pressure

IA surface pressure

re condensation level pressure

H height above mean sea level

H.T terrain height

D Z (actual)- Z (standard atmosphere)

I

J

absolute vorticity

relative vorticity

coriolis force

C<J vertical motion of a pressure surface

UJlq vertical motion at the lower boundary

LJp vertical motion due to frictional effects

LIST OP SYMBOLS (continued)

LOr vertioal motion due to terrain effects

(J* stability parameter

G"Ji Holl stability parameter

V wind speed

VI© 10-meter wind speed

YY\ mass

P density

t time

Cd coefficient of drag•

Q heating rate per unit mass

Qc heat flux for the inversion case

O^ cross isobar angle

Oft moist adiabatic lapse rate

Constants

04* o/Cp dry adiabatic lapse rate

QD dew-point lapse rate

R gas constant for dry air

Cp specific heat at constant pressure

Ilatent heat of vaporization

^ acceleration of gravity

8

LIST OP SYMBOLS (continued)

Vector-Dealer, operators

\y velocity

\^o geostrophic wind velocity

WA gradient of A

\7 A Laplacian of A

vJ(A>d) Jacobian of A and B

1. Introduction

Little practical us© has been mad© of vertical motions,

CJ« dp/</t, in short-range forecasting due to the complicated

and lengthy computations required. Vertical motion computa-

tions are usually the by-product of multilevel baroclinic

models and single values must be extracted only after going

through the entire three-dimensional procedure. In this

paper a two-dimensional vertical motion equation is developed

using vertically integrated parameters with an assumed vertical

motion profile. The layer-mean omega is especially useful

for short-range thickness forecasts and the 1000-700 mb layer

has been selected for study. Derivation of the CJ -equation

is similar to that by Thompson|_18J

except a diabatic heating

term providing a mechanism for development over oceanic

areas has been included. Due to the empiricisms employed

the non-elliptic conditions mentioned by Pedersen [l6J do

not arise.

2. The Basic Omega Equation

The diagnostic omega equation is derived by standard

means from a vorticity equation and a thermodynamic equation

which retains the diabatic term. Equation (1) is the time

derivative of the first law of thermodynamics in (x, y, p, t)

coordinates.

ii

Substituting, T =- | $? , which comes from the hydrostatic

assumption and the ideal gas law; then dividing by, -g/R,

yields equation (2).

•pfitlno- on enuation ( ?) with the Laolacian gives

(2)

Operating on equation (2) with the Laplacian gives the final

form of the thermodynamic equation as shown by equation (3).

Here, O^s— Rp3*^, is the Holl stability parameter to be

further discussed in section (i|).

The vorticity equation in pressure coordinates is shown

by equation (1|)

.

As presented by Thompson llSjand numerous other writers

the last term of the left side of (I4) is approximately equal

to that on the right side, and the two terms are henceforth

deleted. Then, a useful form of the vorticity equation is

arrived at (equation 5) by making the geostrophic assumption

for vorticity and for velocity, C - JLV* > W,*$.WZ fand by

- f * ftaking the logarithmic pressure derivative.

12

Here, 7*5^1 > is the absolute vorticity. Next subtract

(5) from (3), and the result is the omega equation:

-T8^'^))(6)

3. Vertical Distribution of Pressure and Omega

Instead of the distribution of pressure indexes normally

considered, that employed here is given by,

where V) is an arbitrary pressure level. The 1000-500 mb

layer is divided into sub-layers as shown by Figure (1).

The resultant pressures, while essentially logarithmic, bear

values nearly identical to those of the mandatory levels.

k ——— - P4

3 ___ ?2>

2 ——— Pz

1 __-_- P|

Pa

po 2-l

_2- Pd 2 H =

- P^2 x

- P*2"$ =

Po2

500.0 rab

59l[.6 mb

707.1 mb

81;0.9 mb

1000.0 mb

Figure 1. Pressure Distribution in the Vertical

13

//

The values of OO- dp/dt, the "vertical velocity in pres-

sure coordinates , La assumed to vary parabolically in the

vertical with a profile defined by equation (7) which

extends nearly to the 5>00 mb level.

u>=A(j<ALt+bfA2.\ (7)

For boundary conditions it is assumed that^OOat p=p

(the kinematic boundary condition at the assumed surface of

the earth, p ). Also for convenience it is assumed that at p

C/C^a , i.e. that the 3>00-mb divergence is zero. Then,

upon differentiation of equation (7) with the 500-mb diver-

gence taken to be zero, equation (8) results. It should be

noted that if the value ( ^%yp)«0 is assumed zero at p=p 3

the value of B is altered by only 5$.

or.

Therefore, B=1.3863A, and the omega profile in terms of A is

given by equation (9).

cb-mA LA + i.m3^ (9)

Since, CO =o, the ratio between 60,, and ^)^, defines the

omega profile (equation 10) for the 0-2 layer (see Figure 2)

Ik

CO, ^AAz(- £/**+/. 3863)(10)

, ^ P1*

p v'&J | pn

^W6 p<UJ-*

f6

co^= 1.9/44 <*>i

Figure 2. The Omega Profile

For the pressure-range considered (see Figure 2), the layer-

logarithmic-mean, UJ =0.950dO|,so that U>, , will be used f or U) .

Note finally, the profile assumed here applies only to the

large-scale adiabatic, frictionlcss component of the vertical

velocity (that is terrain irregularities are considered

absent )

.

l\. Vertical Integration of the Omega Equation

The Holl stability parameter Lllj is used since it is one

of the vertical—consistency requirements used by FNWF (Fleet

Numerical Weather Facility) and this study uses FNWF processed

data. The Holl parameter is a modification of the standard

stability parameter used here, as shown by equations (11), (12),

and (13).

15

&dx> r\Qp? dp Jc?f ap(id

Letting, RT= - <3j ^|L gives,a S£Tp

(12)

Equation (12) may now be finite differenced over the 0-2 layer.

<r« §-(**-*•) -(To-Tl) (13)

Equation (13) represents the Holl stability parameter in

finite-difference form. In effect, Oh , by (13) gives the

i,logarithmic-pressure\'average' over the 0-2 layer. This

-

use of O^ is compatible with the assumed logarithmic pressure

distribution and will be used throughout the analysis which

follows.

The omega equation (6) will now be integrated term by

term over the isobaric layer (Po, ?z) • Prom (6), one may

rewrite the QJ -equation as:

,/(D)

/ v\(E) X'

whose parts (A),...,, (E) will now be discussed individually.

16

(A) The first term is approximated by substituting

layer mean values: ^^60,) , 60, = U)

(B) With the mathematical relation, p* £> <~Q - & Q h<*>r

the second term becomes, . J \ X*

whose center-finite-differenced form over the 0-2 layer

becomes:

(ii+)

2

Then, using the specified vertical motion profile the rela-

tion^2.= 1.711+3^\ is substituted. However, it is desired

to introduce lower boundary effects due to friction and

terrain by letting &Jaȣ0|.o , the U)uo term then being pro-

determined and placed on the forcing function side of the CO-

equation. Expression (15) below is the final form of (11+)

including effects of the lower boundary.

Term (B) i +% ( ^M2JSi |A?/V3*>, \

j.fW/_ai ^o ^(15)

The lower boundary, LOUO , will be discussed in section (5).

(C) Term (C) in the geostrophic-diagnostic model repre-

sents geostrophic advection of absolute vorticity. Averaging

in the vertical with respect to the logarithm of pressure leads

to equation ( 16)

:

17

j- MjM)i^pi m (ajU^Iky (i6)

The derivation of equation (16) makes use of the mean value

theorem, so that p = p- is represented by p and may be re-

moved from within the integral. Equation (16) then becomes,

M.l(*,ty .iil^^-Jil^l) d7)

Lty.

(D) Term (D) involves the geostrophic advection of

thickness within the logarthmic pressure integral. Its

value is well approximated by equation (18):

ftv##'ih»'the right side of which gives equation (19).

h£vzJ(z> ,a*) « Ii£ ^J(e<, zz-so) d9)

Equation (19) shows that the geostrophic advecting wind may bo

taken arbitrarily as the level (1) wind.

(E) The diabatic heating term will be vertically

integrated in section (6).

The integrated form of the resulting omega equation, subject

to the lower and upper boundary conditions (atpfcandp^_)

becomes,

18

+t =

Equation (20) is a Helmholtz-type equation in the variable,

5L60, . This grouping, 0*^60, , precludes making the usual

simplifying approximation of most three-dimensional models.

(see for example, Haltiner et al 9 [I.

F2^ to) - <rv*a)-i-

6o\vV + z«/(r- W6o & cnvio

The final solution of equation (20) is to be divided by O^

leaving U)t= ^O , where ^ is the mean vertical motion which

approximates the 8£0 mb vertical motion.

5>. The Lower Boundary Condition

Vertical motion at the lower boundary (Muo , is the sum of

a terrain and a friction term, Lduv a ^T-\- ^F •

Here, CO-f , the terrain-effected vertical motion may be de-

scribed by equation (21) from Berkofski and Bertoni[2

[.

Cor « -ftft/r • IP^t (21)

Terrain height,^, is a smoothed field used operationally

by FNWF. The wind velocity at terrain level, IV-p , will be

arrived at by an objective procedure to be described below.

19

The friction term L0? , as shown by equation (22) is a

simplified version of Gressman's formula used by Haltiner

et al|"9] .

CO?=-f>T 3L

Ct>Vt$t (22)

Q> is the geostrophic drag coefficient derived by Cressmanj 7 [>

the field of which is in regular use at PNWF, The field of

C~£ is a set of constants, one for eachAj] grid-point in the

Northern Hemisphere.

Since the J" subscripts infer application at terrain height,

values are used that approximate the particular terrain heights

involved. This is done by dividing the 0-2 layer into three

terrain-height dependent cases.

_ A _700_ m

1 J00_m_

?z Case 3: 2200 m £ 2-r

-2200 m use Pz values

Case 2: 900 m^Z/2200 m

900 m use Pi values

Case 1: sfc^&r^OO m

use P values

p

Figure 3. Three Cases of the Lower Boundary

Division of the 0-2 layer as shown by figure (3) is

dependent upon the gradient level above terrain, since \\/T is

approximated by some geostrophic wind throughout. As a basis

for the selection of the particular geostrophic wind, \\Aj , note

that the gradient level in a neutral atmosphere occurs at

2|

approximately 700 m above terrain level for a wide range of

surface roughness, as shown by Blackadar J 3 . Therefore,

vertical velocity at the lower boundary is case (1) if the

terrain height is 900 m or lower. This values corresponds to

a "standard atmosphere" height which is 700m or more below

the (1) level, and (0) level parameters are then used to

compute the lower boundary. Case (2) is to be used if the

terrain height j?T lies within a gradient-level range of p f

but not P^ . Case (3) occurs for all terrain heights within

700 m of pt ,

Equation (23) is the terrain-induced vertical velocity at

the lower boundary after application of the geostrophic

assumption.

>(coj)^ = _a, £. T(b* ,2T) -^£4 1'i<* t>» v\ < 23

Here, the V\ subscript refers to the levels 0, 1, and 2.

Densities are standard atmosphere values for the respective

levels,

6. The Diabatic Heating Term

Diabatic heating in this model results from an exchange

of heat between the sea surface and the atmosphere. The ex-

change is divided into three cases; neutral, convective, and

inversion, the division being dependent upon the sea-air

temperature difference, T ~ T .* w a

The neutral case (0^ C £ T. -TA <3°C) results in no net

transport of heat across the air- ocean interface and the

21

diabatic heating term is arbitrarily taken as zero when this

condition prevails.

^fe_^g_gYe-C-^Ye_Jlig£ (

i

^v"ta- 3°C) results in a net heating

of the atmosphere. The oonvective atmosphere is assumed to

consist of a dry adiabatic lapse rate to the condensation

level and a moist adiabatic lapse rate above. This particular

model of the convective case is discussed by Burke [5 J in his

paper on the transformation of cP to mP air. Heat gain by the

atmosphere is manifested by a layer-mean temperature gain

induced by a one-hour trajectory of surface air over a warmer

sea-surface, and the layer temperature change is shown by

figure (1|).

-K\ V

V- ATwe> -\st/ilvw

\ \"Piw "^^"« \'

\-Pcj.

V^-ATi>->\&fo\

A

U

-P.

Figure k» Model of the Convective-Atmosphere Modification

Heating is composed of: (1) sensible heat from the sur-

face to the condensation level and (2) latent heating above

this level. This result follows by an analysis of the Burke

model and its implications reguarding heat transport. Martin

I lljj has shown that the large-scale transports of heat have

22

already been included In those terras of equation (6) in

which Q does not specifically appear. Thus Qi/Cp , for use

in equation (6) is given by the local temperature change.

Equation {2$) is the diabatic heating ten (£) referred

to ' hat ion (6)..

d ? w^A --Rpv* ATa\ _ /ATwa\At /

(25)

Integrating logarithmically in the vertical yields the

final form of the diabatic term as shown by equation (26).

(26)

In each of equations 2fy, 25, and 26, the operator-^ represents

the finite-difference version of the local time derivitive,

while, P^ , represents a convective-condensation level

pressure.

According to the model depicted in figure (1|), the modi-

fication is largely determined by the static stability which

in turn largely depends on OJ . As an assumed infinite source

of heat, the ocean will first modify a thin surface layer of

air and then, due to convective activity, will distribute the

modification (temperature change) throughout the 0-2 layer.

An overestimation of heat exchange should be expected with

23

this particular process since convection is not instantaneous

This overestimation may be partially balanced by heat lost

from the top of the 0-2 layer. However, the non-inclusion of

a moisture-continuity equation may be more serious in regions

of upper ridges.

The inversion case (T^ - TA< 0) or c_cjolin£ o^sej. where

heat is lost from the atmosphere to the ocean is defined

by an empiricism from Laevastu [ 12J.

Qc = 3,0 Vic (Tk; -Ta) gm-cal/cmx-2L

thrs (27)

Equation (27) represents the heat flux across the 10-meter

level. This is an empiricism requiring the 10-meter wind,

V|o, which will be approximated by making a frictional

correction to the 1000 mb geostrophic wind as shown by

equation (28)

:

V)a « a.sJL IkxvDo (28)

fThe factor 0.5 is due to surface frictional effects and will

be discussed in section (7).

Before integrating equation (27) over the 0-2 layer

the initial distribution within the layer must be determined.

Normally cooling effects due to inversion conditions are con-

fined to a surface layer only several hundred meters thick,

however, since the layer mean cooling effect is desired over

the entire 0-2 layer it will be assumed that cooling is dis-

tributed throughout the 0-2 layer. Thus, dividing equation

2h

(27) by the mass of a column of air extending from the surface

pressure PA tc Pa , with total mass,M = (PA - PjJ/g, gives the

mean temperature change in a vertical column extending from

PA to Pt as,

A. -. ®*3L (29)

Finally we have, ?a,

_ RaV*£ = - El^lF^ ^, - £*-V"q (30)

which is the final result for the inversion case assuming

neglible heat release from fog-droplet condensation,

7. State-Change Parameters for the Convective Case

Air-temperature change and wet-bulb temperature change

equations must be derived for use in equation (26). The

state-change equations (31) and (32) were initially developed

by Mosby [l5L Amot IJ, and more recently by BoyumjijJ.

However, in a FNWF study by Carstensen and Laevastu[6 J the

following best-fit equations were found to give hourly changes

with a high degree of skill:

£Ja =.aiz.(%j-T^-o.io 1'vio'B^ °cAr (31)At

£* * o./5{eui-eA)- 0J6 - W/VFe* mt/hr (32)

it

Equation (3D is substituted directly into the sensible-

heat term of equation (26) while the wet-bulb temperature

change contributes directly to the latent-heat term. In

25

computing AJ^l it is necessary to Include the small IncrementAt

of temperature found by proceeding along a mixing ratio isopleth

from P^, to ?c^ then down a moist adiabat to Pc ,(see figure

5). Equation (33) shows this relationship.

ATwa = &Ti>-t(f*-Xn)AZ}*- , &&* J.406-2L (33)

Here A^pc is the one-hour height change of the condensation

heights determined by the height of intersection of the dry

adiabat and the mixing ratio isopleth based upon the tempera-

ture - dew-point spread at the surface.

A2pc = 2>ci -2

Figure 5. ^TL,o as a Function of /IVTk

(31+)

Substituting equation (3k) into (33) gives the local rate of

change of the wet-bulb temperature as a function of dew-point

and a ir - 1

o

nrnor n tnro ohnnro,

A"U = AT* , //d-W )/aTa _ A/bAt At ^t>-tfd /I At At

(35)

26

The dew-point temperature change /$i is found by taking

the time differential of equation (36) below 10 1,

TA-Th = a^L(e s -ga) 06)

Here, es is the saturated vapor pressure and e^ the observed

value, both at 10-meters. The finite-difference form of the

time differential of dew-point resulting from equation (36)

then follows:

^j^^Alji . £x^zLAe* a^4^^LAeg (37)

At. At- 4>$ At <t?> &<

The vapor pressure change, &&A t is given by equation (32).

The time rate of change of saturated vaoor pressure, ^5 ,

is calculated using the Cla&sius- Ol'afjfyron equation:

^^ KTa^ At (38)

fir 6JO?<fXp $<//*,£ SV/?,o (39)

293. \(* 7>Equation (39) is an integrated form of the Clausius - Clapyron

equation.

There are still some unsolved parameters in the system.

The moist adiabatic lapse rate, ^ , cannot be considered a

constant with respect to pressure even in the limited opera-

tional range of this study. At the condensation level, Pc ,

the moist adiabat is calculated from the following |"lo"j :

21

£>,4>ZZ L<2

(1(0)

All subscripts G refer to the condensation level.

Condensation parameters of temperature, Tc , and pressure,

Pc , are readily calculable from equation (l[l) and (lj.2) after

Edson j^8]:

to- Si

Pc =K*)%

(III)

(1*2)

For equation (1|0), the vapor pressure at the condensa-

tion level, ec , must be calculated. Equation (k3) » the inte-

grated form of the Clausius -Clapeyron equation may be used

since air at the condensation level is saturated, and Pc and

Tt are known from (1|1) and (1|2),

£<. J&.lofetp272.1(0 T,

(1*3)

In like manner the vapor pressure of the sea surface, e ,

(from equation 32) is calculated since the sea surface is

saturated (equation l\)\) , Also, a correction factor of

0.98 is required for salinity effects.

UJ = (b.«?)(4/o?)s><f>Z93./4

28

(kk)

Finally, the advecting wind at lO-meters,^^, must be

derived for use in equations (31) and (32). The geostrophio

wind at the surface is taken as equal to the 1000 mb geo-

strophio wind. However, due to frictional interaction with

the ocean surface, the geostrophic wind must be reduced in

magnitude and rotated to the left, to yield the advecting

frictional wind,!J/, . For the convective case the observed

cross-isobar angle at mid-latitudes is approximately lf> L10!*

Reduction of the 10-raeter wind speed as expressed by

the ratio, V/V* , where V^ is the 10-meter geostrophic wind

speed, may be obtained from a study by Lettau|

13"I. By

assuming a roughness length of £ = 0.1 cm for oceanic areas,

the 10-meter geostrophic wind ratio for neutral conditions

is, V/V* = 0,6(i Lottau then allows for stability criteria,

and for the convective case a value of V/V« = 0.7^' is arrived

at. Similarly, V/V*g= 0.J> for the inversion case.

Advection by the frictional wind is derived by first

decomposing the vector advection as shown by equation (1|5>):

Then multiplying by O.1

^ and rotating the field through the

angle K = l£ by a rotational change of coordinates leaves

equation ( Ij.6 )*

29

Assuming the geostrophic conditions, Ufo* —£ S^> ^--zr^equation (i|6) may be written in ita final form aa given by

equation (i|7).

(1*7)

Similarly, equation (!j8) is derived for the adveotion of

vapor pressure by tie 10-meter wind.

V>*'We* = 0.1 l_<u*ttaT(*<>£a)

f

8. Numerical Procedurea

All finite-difference operators used are standard cent-

ered-differencea of the type,

V^A = 2aIVM J(A,b) = JkL.J(a tb)

with five-point grida having a mesh distance, d = 3^1 km

at 60 latitude. !Iere,M is the map factor for polar stereo-

graph projections and d is the grid spacing.

The variables needed for input into the diagnostic omega

equation x^ere scaled as follows:

30

A Al!m =>Vt .2 p =p. .2 rab

)9 *y) .2" sec" 1 € =C .2** mb

^O =60.2 mb/sec CJJ =0J.2 crnVsec*-A

? / — 4 *°cm^Jto = ^.2 mb/sec ^T = :

2t* 2

}> =b .2/?

cm ^ =<$ .2" gm-cm7sec*

T =| .2 C -T = l.lj £6142x10" sin^sec-'

The following physical constants were also needed in the

computations of Oj . All values of those constants have

been expressed in cgs units.

d = 38,1000,000 cm

* — 980.0 cm/sec 1

R s .0287 X 10 7 cmVsec* - °K

K as 0.2+61 X 107

cmx/secl - *K

c* = 1.003 X 107

crnVsec1- - *K

L = 2500 X 107

crnVsec*o T/

-

- ft.

*d = 0.9771 X 10_t/

°C/crn

rfc= 0.1600 X 10 C/cm

/*= 1.213 X 10" 3 gm/cm3

c>ss 1.0££ X 10" 3 gm/cm*

r*= 0.919 X 10"* gm/cm3

The scaled Helmholtz equation, which follows from

equation (20), expressed in units of, mb - cm 2/sec 3, is 1

shown on next pace.

31

vt^A)- H.^osilhi^cb)'^ \<rH»

x&^-kX,/^ .1-/A

"2

'2 -p^^.Z-42049)

All height values have been replaced by D-values, the

deviation of the height field from a standard atmosphere

height.

The scaled stability parameter is given by equation

(5>0) and is in units of cm I/sec^

Z'taooo35^Pt^xr (5o

For computation of the stability parameter D is re-

scaled to D = D .2 to prevent overflow when adding the

standard height,

2

>6~^ /000 290,000 cm. Stability calcu-

lations are based on 1000-700 mb differences and throughout

the derivation all gradients and differences at the P, and

Pz levels are assumed equal to those at 8£0 mb and 700 mb

respectively.

To prevent computational instability during the numerical

Helmholtz solution a minimum value of stability is required.

32

This value corresponds to a maximum lapse of ~%X(\

:

(fo-7.) *"$*£ [(M'}?" + 21°°° c

"

Equation (51) is the scaled lower boundary condition

in units of mb/sec.

(5D

Here, the subscript Y\ refers to the terrain-height dependent

cases, 1, 2, and 3 explained in section (5). The factor 10""

converts dynes/cm 2- to millibars.

Equations for the diabatic term are computed in terms

of heating rates, Q/Cp , which are assigned to grid points

according to the existing grid condition: convective,

inversion, or neutral.

Equation (52) shows the scaled heating rate for the convective

i*(52)

case.

A

A -

The factor, 3600, converts hours to second and heating rate

is now in units of mb -aC/sec.

Equation (53) shows the scaled heating rate for the

inversion case (a cooling process) also in mb - °C/sec.

33

JA) = 4Afa^^^^?^^-^ ,53Cp f2fl( ICCf

The factor (i|.l81| X 10) converts gm-cal/sec to dyne-cm/sec;

K = 2\\ X 3600 which converts 2l|-hours to seconds; 0.03 re-

places 3.0, and permits velocity computation to be made in

cm/sec.

Land area grid points are masked and receive heating

rate values of zero as do oceanic grid points which satisfy

the neutral conditions. The remaining oceanic grid points

receive either heating or cooling values depending on the

prevailing condition at the point. The diabatic term is then

computed and the scaled form in units of mb-cm /sec 3 is shown

by equation (ft)

.

(ft)

Here, P approximates P for ease of computation.

Other equations involved in the computation of the dia-

batic term are scaled as follows:

(1) The air temperature change is units of °C/hr.

At ' \ /

-q

-0.7M1

^d1CX>StS^(4aJ^^^\S

m(^J^ +A^tf*\

(55)

»*

3k

(2) The dew-point temperature change in units of °C/hr.

— 0.7A2H- d(&i%"T(ia , Ga) - siw«*(4$»A$+

+ ^l A^)]l^ Co.&Z2-L?'£.s) AJfiAi

.X

(56)

(3) The wet-bulb temperature change in units of *C/hr.

where,

(58)

^-0.497* ^±!±^i!±^W

(l±) The condensation level temperature in units of C.

(5) The dew-point temperature in units of C.

(6) The condensation level pressure in units of

(59)

millibars, Pc =&"&

and(60)

Hero Tai is computed using values of Tft

and T* in enuation 58.

R* = Pa

35

A A AHere T~ is calculated using values of T

A| +AT*/At and

ir+A^/At in equation?* ( £6 ) , (59.), (55), and (56).

9. The Computer Program

The Control Data Corporation l60l| digital computer

was used in this study. It has a core capacity of 32,768

words of I48 bits each. An operational field of 1,977

grid points forming a 5l X \\7 octagon inscribed within the

9°N latitude circle was employed. The grid-mesh is 38I km

true at 60°N latitude.

Boundary conditions around the octagonal grid were

determined by the standard subroutines used. Laplacian,

Jacobian, and all other five-point center-difference

operations set the grid boundary to zero. The Helmholtz

relaxation operation set the edge and the next interior

border point to zero.

Equation {\\% was solved by a two-dimensional Liebmann

relaxation technique wherein the^l +l)-iterate for any point

A

is givenby.

A*+l

/ " V^,;-(^b).i-C,-,-

2- lB;,- + /

Here, n is the over-relaxation coefficient and the residual

at any step, R n , is,

4V^.i-(AB)ij-£'i

-1 n

-it>r, + 1

36

The over-relaxation coefficient used was, }\ = 1.1*H[,

which allowed convergence in approximately 30 scans over

the 1977 point grid using an initial guess field of zero. The

Convergence criterion is that the iteration ceases when €(n) (v\±t\ (*) z.. a

defined by % sA -A falls below 3000 mb-cry sec .

This value corresponds to a vertical velocity of 0.3 X 10

mb/sec for a stability of 83 X 10 cm /sec (the standard

atmosphere stability).

Total computation time was approximately 1 minute and

30 seconds with 23 seconds required for the heating term

and 26 seconds for the boundary condition.

Due to the abrupt cut-off criteria used for delineating

the three cases of diabatic heating (inversion, convective,

neutral) the final Q/e field was smoothed with a five-point

smoother of the form,

A = A + KSPA

The smoothing coefficient k is a constant value of (1/8)

and the field was smoothed twice. The smoothing operation

removed small scale irregularities caused by the cut-off

criteria of the heating term but also reduced peak values.

A standard FITWF filtering process followed all operations

involving the Laplacian. Tills process removes all small

scale features with wave numbers greater than 15 at latitude

37

10. Results and Conclusions

A series of four successive 12-hourly data-sets be-

ginning with the 00Z maps of 28 April 1966 and ending with

12Z, 29 April 1966 were used. Since the only diabatic

heating mechanism introduced into this study was that aris-

ing by "conduction" from the underlying oceanic surface,

the results have been depicted only over the North Atlantic

Ocean. This region has a large density of reporting ships

and thus the reason for its selection.

The computational omegas and their associated parameters

for 00Z 28 April 1966 are contained in figures (6) through

(11) which are appended. Of these, not only the diabatic

omegas are shown, but also, the additive effect of the dia-

batic influence as contrasted with adiabatic computations.

In figures (12) through (1?) only the final-product omegas

of this study are shown together with the corresponding

FNWF analyzed 8^0 mb D-fields to servo as identifdor's",:

that is, to indicate qualitative coherency between the verti-

cal motions and the associated motion systems.

In the sequence of figures (6) through (11), it is of

interest to note the pattern similarity between comparative

adiabatic gSJgJiAfi&l motion computationsrthose. produced here,

and, for the same timrs, thosejproduced tyy F1F./F, which are based

upon the procedure described by Haltiner et alf"9l. The

magnitude and position of the updraf t-centers show a strong

similarity with those of FITV/F, however, the model described

by this paper shows larger downdraft areas extending east-

38

ward from Newfoundland. This apparent discrepancy may be

attributed to the two differing treatments in L̂O(figures

18 and 19). The^c^at the continent-edge essentially becomes

the boundary condition for the oceanic computations.

On the other hand, however, one of the major objectives

of this study was to test a simplified ^>l» which uses only

parameters pertaining to the standard levels of analysis

(equation 23). In this connection, the greatest time-

consuming aspect of the FNWF <-o -computation is that of

deriving U)uo, and associated parameters at terrain height re-

quiring pressure extrapolation. The general similarity in

the fields of ^(.©by the two operational procedures which

have been discussed is evident by an inspection of figures

(18) and (19), and this can perhaps be a justification for

continuing the simpler ^©computations.

The diabatic effects, which were obtained for the 00Z

April 28 synoptic time, depict values of Q/up, so that the

large positive center southeast of Newfoundland is realistic

with regard to the northerly flow passing over the Gulf

Stream. The maximum heating value for the four map periods

over the Gulf Strewn Is h/^v = O.O67 mb - °C/sec, which

corresponds to a layer mean heating rate of 0.3 °C/hr for the

1000-700 mb layer. The equivalent thickness chaiige for this

layer is approximately 3 meters/hr.

For comparison, Petterssen»s values \yf\ for the

1000-500 mb thickness change during a similar synoptic

situation in late March show a maximum value of 6 to 8

39

meters/hr in the vicinity of the Gulf Stream. These values

for sensible and latent heating were computed using flux

calculations at the surface based on empiricisms similar

to those of Laevastu. It should be noted that heating rates

for this time period in late March are indeed greater than

late April values.

Finally, with regard to the diabatic term, note that

in the area of qualitative verification no significant dia-

batic cooling values were observed. These areas are limit-

ed in extent in winter and spring but could be of greater

synoptic consequence in the summer and fall seasons.

The effect of the diabatic term as it appears on the

forcing function side of equation (21;) gives rise to figure

(11), applicable at approximately 850 mb, but actually rep-

resenting the layer-mean l3. As could be anticipated from

the standpoint of heat injection into moving parcels, the

change (inclusion of the diabatic term) has been such as

to expand the southern rim of the trough which extends to-

wards the southwest from Iceland, Similar aspects of coher-

ency between the resultant co of this study and the trough

movement and development may be traced out, feature by

feature as one proceeds through the synoptic sequences.

For example, the pronounced trough over the Atlantic has,

by 00Z of 29 April, become oriented North-South just east

of Greenland. At the same time the updraft cell has taken

on this same orientation just west of the same trough, so

that the Li> -field gives some confirmation of the dynamic

!;.0

processes involved in these map changes.

The model for vertical motion, as presented by this

paper, appears to be quite sucessful and future plans involve

inclusion of a more realistic frictional term. These more

realistically computed omegas (extended to 500 mb) are then

to be used for feedback to yield a prognostic z-field, hour-

by-hour. The ultimate aim, of course, Is to realize a small-

er R.H.S. error as the motion systems progress over the ocean,

11. Acknowledgements

The original research for this paper was by Professor

Prank L. Martin of the United States Naval Postgraduate

School. His assistance ano encouragement are gratefully

acknowledged. Also, the personnel of Fleet Numerical Weather

Facility, especially Mr. Leo C. Clarke, are acknowledged

for their invaluable assistance in adapting the model to a

computer program.

kl

cCD

0)

h:bo

%

-v • •^.

>*?

~<P o-.

(1)

V.

IT

U3

31

cCO

00

**>;

<>%&

-V

-v

-V o*-^^>*»'.

.A* ps

o w<s.

? \: •'£ ^k^.j. .

"I ", 1

"

, " —

'• .^

j5?o'^

wD

Li-

C

•*.

.+ •. / \ NCO ^

o-<?v

+-V

ro

o>

£6^-4f

+ w

\

vV

#.o-V

*

X

'

. -V

O &-V

e5o\

X

-V

f .

^' (A- . -TAs • .^^' .-AV. >>

**&

$K^oSj

Q> fe

<"o-

v

X.Sj

'*

A

p\

+

V kvo.

+

S\T&\

§oexdn

*o,uo

<£o:

-V

6"

aV

-V

*8

CD

X

om-+

X •£&£

"f

X

..?a_>-

f

-v?

0> X X X X^0

+

%

. \

X

Q

,«fco

>c

Xi H-

:uj • x

8-V

x

do'- 1<

XX ^

+ •

-V

£*

XL \

y.'.

. -A

*o •

x•

A-

irO*>:

*

*

<$b'

*f

*

k

r; ".-f

ri

-/. f

£<h. Wx

K+

*o0V

tf

/Vf

[jjv. %&,

ro.'«**%'«***«,

mmmm;';^.<$> .

* •

-v•

'

*-A

X

+

S

+

.%<• v .. . rib- . J3v • •

/^,-r ^/

(2/ V§& Cfo?

j©/b O-o, 'HP A"

$ro-

'-b• .^

• 6-

a-

BIBLIOGRAPHY

1. Amot, Audvin. On the Temperature Difference Betweenthe Air and the Sea Surface and Its Applicability inthe Practical Weather Analysis. MeteorolgiskeAnnaler, BD. 1, HE. 19, 19UU

•

2. Berkofski, L. and E. A. Bertoni. Mean TopographicCharts for the Entire Earth. Bulletin of the AmericanMeteorological Society, 36, 1955, 350-354.

3. Blackadar, A. K. The Vertical Distribution of Windand Turbulent Exchange in a Neutral Atmosphere.Journal of Geophysical Research, Vol. 67, ITo. 8,

July, 1962.

4. Boyum, G. A Study of Evaporation and Heat ExchangeBetween the Sea Surface and the Atmosphere.Geofysisko T^ublikas joner, Vol. XXII, Ho. 7, January, 1962

5. Burke, C. J. Transformation of Polar Continental Airto Polar Maritime Air. Journal of Meteorology,Vol. 2, Ho. 2, June, 1945

6. Carstensen, L. P. and T. Laevastu. FBWF Technicalllote Ho. 17, April, 1966.

7. Cressman, G. P. Improved Terrain Effects in BarotropicForecasts. Monthly Weather Review, 88, pp. 327-342,1960.

8. Edson, II. Numerical Cloud and Icing Forecasts.Scientific Services Technical Note No. 13, Headquarters3D Weather Wing, Scientific Services.

9. Haltiner, G. J., L. C. Clarke, and G. E. Lawniczak.Computation of the Large Scale Vertical Velocity.Journal of Applied Meteorology, Vol. 2, No. 2,

pp. 242-259, April, 1963.

10. Haltiner, G. J. and F. L. Martin. Physical and Dynam-ical Meteorology. New York, McGraw-Hill, 1957.

11. Holl, M. M. J. P. Bibbo, and J. R. Clark. LinearTransforms for State-Parameter Structure, editiontwo, Meteorology International Technical Memorandum,No. 1, 1 October, 1963.

12. Laevastu, T. Factors Affecting the Temperature of theSurface Layer of the Sea. Societas Scientiarum Fennica,Helsinki, i960

57

13. Lettau, H. H. Wind Profile, Surface Stress and Geo-strophic Drag Coefficients in the Atmospheric SurfaceLayer. Advances in Geophysics, Vol. 6, pp. 2l].l-257>

Academic Press, Hew York, 1959.

111. Martin, P. L. Derivation of the Diabatic Heating Term.Unpublished Paper, 1966.

15. Mosby, H. The Sea-Surface and the Air. ScientificResults of the Norwegian Antarctic Expeditions 1927-1928, Ho. 10, Oslo, 1933.

16. Pedersen, K. An Experiment in Quantative PrecipitationForecasting with a Quasi-Geoatrophic Model. Universityof Chicago Department of the Geophysical Sciences,Report Ho. 1, October, 1962.

17. Petterssen, S., D. L. Bradbury, and K. Pedersen.Heat Studies in Weather Analysis and Forecasting.Dept. of the Geophysical Sciences, University ofChicago, September, 1963.

18. Thompson, P. D. numerical Weather Analysis and Pre-diction. Hew York, MacMillan, 1961.

58

INITIAL DISTRIBUTION LIST

No. Copies

1. Lt. P. S. Perrentino 51807B WithingtonChina Lake, Calif.

2. P. L. Martin 6

Environ. SciencesTJSIT Postgraduate SchoolMonterey, Calif.

3

.

Library 2TJ. S. Naval Postgraduate SchoolMonterey, California 9391*0

1+. Dept. of Meteorology & Oceanography 1

U. S. Naval Postgraduate SchoolMonterey, California 9391+0

5. Defense Documentation Center 20Cameron StationAlexandria, Virginia 223 H+

6. Office of the U. S. Naval Weather Service 1

IT. S. Naval Station (Washington NavyYard Annex)

Washington, D. C. 20390

7. Chief of Naval Operations 1OP-09B7Washington, D. C. 20350

8. Officer in Charge 1Naval Weather Research FacilityU. S. Naval Air Station, Bldg. R-I48Norfolk, Virginia 23511

9. Commanding Officer 1FWC/JTWCC0MNAVMARFP0 San Francisco, Calif. 96630

10. Commanding Officer 1U. S. Fleet Weather Central, KodiakPPO Seattle, Washington 98790

11. Commanding Officer 1U. S.. Fleet Weather Central, PearlHarbor

FPO San Francisco, California 96610

59

INITIAL DISTRIBUTION LIST (CONT. )

12, Commanding OfficerU. S. Fleet Weather Center. RotaFPO ITow York, Hew York 09540

13. Commanding OfficerFleet Weather Central, SuitlandNavy DepartmentWashington, D. C. 20390

Ul« Commanding OfficerFleet Weather CentralU. S. Naval Air StationAlameda, California 9^501

15. Commanding Officer and DirectorNavy Electronics LaboratoryAttn: Code 2230San Diego, California 92152

16. Officer in ChargeU. S, Fleet Weather Facility, ArgentiaFP0 New York, New York 09597

17. Officer in ChargeU. S. Fleet Weather Facility, KeflevikFPO New York, New York 09571

18. Officer in ChargeTJ. S. Fleet Weather Facility, YokosukaFPO San Francisco, California 96662

19. Officer in ChargeU. S. Fleet Weather Facility, Sangley

PointFPO San Francisco, California 96652

20. Officer in ChargeFleet Weater FacilityTJ. S. Naval Air StationSan Diego, California 92135

21. Officer in ChargeFleet Weather FacilityU. S. NavalAir Station^uonset Point, Rhode Island 02819

22. Officer in ChargeFleet Weather FacilityBox 85Naval Air StationJacksonville, Florida 32212

60

INITIAL DISTRIBUTION LIST (CONT.

)

23. Officer in Charge 2Fleet Numerical Weather FacilityIT. S. Haval Postgraduate SchoolT ~onterey, California 9391+0

2l|. I". S. Naval War College 1

Newport, Rhode Island O28I4I+

25. Director, Naval Research Laboratory 1

Attn: Tech. Services Info. OfficerWashington, D. C. 20390

26. Office of Chief Signal Officer 1Research and Development DivisionDepartment of the ArmyWashington, D. C.

27. Commander 1

Air Force Cambridge Research CenterDepartment of the ArmyWashington, D. C.

28. Geophysics Research Directorate 1Air Force Cambridge Research CenterCambridge, Massachusetts

29. Naval Air Technical Training Unit 1IT. S. Naval Air StationLakehurst, New Jersey 08733

30. Program Director for Meteorology 1National Science FoundationWashington, D. C.

31. Headquarters 2nd Weather Wing (MAC) 1United States Air ForceAPO vr633New York, New York

32. American Meteorological Society 11+5 Beacon StreetBoston, Massachusetts

33. Commander, Air Weather Service 2Military Airlift CommandU. S. Air ForceScott Air Force Base, Illinois 62226

3k* U. S. Department of Commerce 2Weather BureauWashington, D. C.

61

INITIAL DISTRIBUTION LIST (COITT. )

35. Commandant of the Marine CorpsNavy Department (Code DP)Washington, D. C. 203 80

36. Office of Naval ResearchDepartment of the NavyWashington, D. C. 20360

37. U. S. Naval Oceanographic OfficeAttn: Division of OceanographyWashington, D. C. 20390

38. SuperintendentUnited States Naval AcademyAnnapolis, Maryland 211+02

39. DirectorCoast and Geodetic SurveyTJ. S. Department of CommerceAttn: Office of OceanographyWashington, D. C.

I4O. Office of Naval ResearchDepartment of the NavyAttn: Geophysics Branch (Code 1+16

)

Washington, D. C. 20360

1+1, Office of Naval ResearchDepartment of the NavyAttn: Director, Surface and Amphibious

Programs (Code 1+63)

1+2. Program Director OceanographyNational Science FoundationWashington, D. C.

1+3. Council on Wave ResearchDepartment of Civil EngineeringUniversity of CaliforniaBerkeley, California

kk» DirectorNational Oceanographic Data CenterWashington, D. C,

1+5. DirectorWoods Hole Oceanographic InstitutionWoods Hole, Massachusetts 0251+3

1+6. ChairmanDepartment of Meteorology & OceanographyNew York UniversityUniversity Heights, BronxNew York, New York

52

INITIAL DISTRIBUTION" LIST (CONT.

)

if7. DirectorScripps Institution of OceanographyUniversity of California, San DiegoLa Jolla, California

!|8. Bingham Oceanographic LaboratoriesYale UniversityNew Haven, Connecticut

1|9. Director, Institute of Marine ScienceUniversity of Miami#1 Rickenbacker CausewayVirginia KeyN1«ral f Florida

50. Department of Meteorology & OceanographyChairmanUniversity of HawaiiHonolulu, Hawaii

51. DirectorLamont Geological ObservatoryTorrey CliffPalisades, New York

52. Chairman, Department of OceanographyOregon State UniversityCorvallis, Oregon 97331

53. Chairman, Department of OceanographyUniversity of Rhode IslandKingston, Rhode Island

£lf, Texas A & M UniversityChairman, Department of OceanographyCollege Station, Texas 778I4.3

55. Executive OfficerDepartment of OceanographyUniversity of WashingtonSeattle, Washington 98105

56. National Research Council2101 Constitution AvenueWashington, D. C.Attn: Committee on Undersea Warfare

57. ChairmanDepartment of OceanographyThe Johns Hopkins UniversityBaltimore, Maryland

63

INITIAL DISTRIBUTION LIST (COIIT.)

58 . LibraryFlorida Atlantic UniversityBoca Raton, Florida

59. Department of MeteorologyUniversity of CaliforniaLos Angeles, California

60. Department of the Geophysical SciencesUniversity of ChicagoChicago, Illinois

61. Department of Atmospheric ScienceColorado State UniversityFort Collins, Colorado

62. Department of Engineering MechanicsUniversity of MichiganAnn Arbor, Michigan

63. School of PhysicsUniversity of MinnesotaMinneapolis, Minnesota

61*. Department of MeteorologyUniversity of Utah.Salt La!:o City, Utah

65. national Center for Atmospheric ResearchBoulderColorado

66. Department of Meteorology and ClimatologyUniversity of WashingtonSeattle, Washington 98105

70. Department of MeteorologyUniversity of WisconsinMadison, Wisconsin

71. Department of MeteorologyFlorida State UniversityTallahassee, Florida

72. Department of MoteorologyMassachusetts Institute of TechnologyCambridge, Massachusetts 02139

73. Department of MeteorologyPennsylvania State UniversityUniversity Park, Pennsylvania

61*

INITIAL )IS£RIBUTIOH LIST (COHT.)

7l|. Hawaii Institute of Geophysics 1University of HawaiiHonolulu, Hawaii

75. University of Oklahoma 1Research IntituteNorman, Oklahoma

76. Atmospheric Science Branch 1

Science Research InstituteOregon State CollegeCorvallis, Oregon

77. The University of Texas 1Electrical Engineering Research LaboratoryEngineering ScienceBldg. 631AUniversity StationAustin, Texas 78712

78. Department of lleteorology 1Texas A & 1-1 UniversityCollege Station, Texas 77o)j3

79. Lamont Geological Observatory ±Columbia UniversityPalisades, Hew York

80. Division of Engineering and Applied Physics 1Room 206, Pierce HallHarvard UniversityCambridge, I'lassachusetts

81. Department of '"ochanics 1The Johns Hopkins UniversityBait imore , I ^ryiand

82. University of California 1E. 0. Lawrence Radiation LaboratoryLivermore, California

83. Department of Astrophsics and 1Atmospheric Physics

University of ColoradoBoulder, Colorado

8fy. Weather Dynamics Group 1Aerophysics LaboratoryStanford Research InstituteMenlo Park, California

65

INITIAL DISTRIBUTION LIST (COIIT. )

85. Meteorology International, Inc.P. 0. Box 1361].

Monterey, California 939U0

86. The Travelers Research Center, Inc.6^0 Main StreetHartford, Connecticut

87. United Air LinesDirector of MeteorologyP. 0. Box 8800Chicago, Illinois

88. Department of MeteorologyUniversity of MelbourneGrattan StreetParkville, VictoriaAustralia

89. Bureau of MeteorologyDepartment of the InteriorVictoria and Drummond StreetsCarlton, VictoriaAustralia

90. International Antarctic Analysis CentreI468 Lonsdale StreetMelbourne, VictoriaAustralia

91. Department of MeteorologyMcGill UniversityMontreal, Canada

92. Central Analysis OfficeMeteorological BranchRegional Adm. BuildingInter. AirportDorval, Quebec, Canada

93. Meteorological Office315 Bloor Street WestToronto 5, Ontario, Canada

n';

. >a: fcment of MeteorologyUnivorsity of CopenhagenCopenhagen, Denmark

95. Institute of MeteorologyUniversity of HelsinkiHelsinki - Port^ania, Finland

66

INITIAL DISTRIBUTION LIST (CONT.

)

96. Institut fur Theoretische MeteorologieFreie Universitat BerlinBerlin-DahlemThiel-allee l\9

Federal Republic of Germany

97. Meteorological InstituteUniversity of ThessalonikiThessaloniki, Greece

98. Meteorological Service1|1|, Upper O'Connell StreetDublin 1, Ireland

99. Department of MeteorologyThe Hebrew UniversityJerusalem, Isreal

100. Geophysical InstituteTokyo UniversityBunkyo-kuTokyo, Japan

101. Meteorological Research InstituteKyoto UniversityKyoto, Japan

102. Department of Astronomy and MeteorologyCollege of Liberal Arts and SciencesSeoul national UniversityPong Soong Dong, Chong ITo KuSeoul, Korea

103. Central Meteorological OfficeI Song Wul Dong, Sudaemon KuSeoul, Korea

I0I4. Department of Motoorolo^-yInstituto de GeofisicaUniversidad Nacional de MexicoMexico 20, D. P., Mexico

105. New Zealand Meteorological ServiceP. 0. Box 722Wellington, G. E. New Zealand

106. Institutt for Teoretisk MeteorologiUniversity of OsloBlindern, Oslo, Norway

67

INITIAL DISTRIBUTION LIST (CONT.

)

107. Institute of GeophysicsUniversity of BergenBergen, Norway

108. Pakistan Meteorological DepartmentInstitute of Meteorology and GeophysicsKarachi, Pakistan

109. Royal Swedish Air ForceM. V. C.Stockholm 80, Sweden

110. Department of MeteorologyImperial College of ScienceSouth KensingtonLondon S, H. 7, United Kingdom

111. Meteorological OfficeLondon R.BracknellBerkshire, United Kingdom

112. national Research. Institute forMathematical Sciences

. S . I. R,P. 0. Box 395Pretoria, Union of South Africa

113. Commonwealth Scientific and IndustrialResearch Organization

3H+ Albert StreetEast Melbourne, C. 2, Victoria

111). DirectorPacific Oceanographic GroupITanaimo, British ColumbiaCanada

115. Ocean Research InstituteUniversity of TokyoTokyo, Japan

68

IMlMaaiEIEDSecurity Classification

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

I. ORIGINATING ACTIVITY (Corporate author)

U. S. NAVAL POSTGRADUATE SCHOOL

2a. REPORT SECURITY CLASSIFICATION

UNCLASSIFIED2 6 GROUP

3. REPORT TITLE

A TWO-DIMENSIONAL OMEGA EQUATION FOR THE1000-700 MB LAYER WITH DIABATIC HEATING

4. DESCRIPTIVE NOTES (Type ot report and inclusive dates)

MASTER OP SCIENCE THESIS (METEOROLOGY)5 AUTHORfS.) (Last name, ti rat name, initial)

FERRENTINO, PETER S.

LIEUTENANT, U. S. NAVY

6- REPORT DATE

1 Hay 19667a. TOTAL NO. OF PAGES

',. 66

7b. NO. OF REFS

188a. CONTRACT OR GRANT NO.

b. PROJECT NO.

9a. ORIGINATOR'S REPORT NUMBERfSJ

9b. OTHER REPORT HO(S) (A ny other numbers that may be assignedthis report)

M^\410. AVAILABILITY/LIMITATION NOTICES

This document has been a^'-^c-e-c1 f~* mblicBBlaaafl and ralp; its d-.at.rin,.;.nm is nnHtnl ed.

11 SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Chief of Naval Operations (OP-09B7"Department of the NavyWashington, D. C. 20^60

13. ABSTRACT

A two-dimensional omega equation is derived by combinationof the vorticity and thermodynamic equations. The desiredomega is then taken to be the logarithmic average in the1000-700 tub layer. A diabatic term, after Laevastu, foroceanic areas only is included to deduce the empiricaltemperature and vapor-ores sure changes associated withsensible and latent '\etit: ' \ the maritime layers. Over -

both continental and oceanic areas a frictional vorticitysink is included in order that excessive energy cannot begenerated over the ocean. Among other novel features isthe use of the Holl static-stability parameter whichaffords vertical consistency in the analyses prepared byFleet Numerical Weather Facility.

DD FORM 1473 69 UNO LASS TETEDSecurity Classification

UNCLASSIFIED

Security Classification

14-KEY WORDS

LINK A

ROLELINK B

ROLELINK C

ROLE

LAYER-MEAN OMEGA

01EGA EQUATION

VERTICAL MOTION

DIABATIC HEATING

INSTRUCTIONS

1. ORIGINATING ACTIVITY: Enter the name and addressof the contractor, subcontractor, grantee, Department of De-fense activity or other organization (corporate author) issuingthe report.

2a. REPORT SECURITY CLASSIFICATION: Enter the over-all security classification of the report. Indicate whether"Restricted Data" is included. Marking is to be in accord-ance with appropriate security regulations.

26. GROUP: Automatic downgrading is specified in DoD Di-

rective 5200. 10 and Armed Forces Industrial Manual. Enterthe group number. Also, when applicable, show that optionalmarkings have been used for Group 3 and Group 4 as author-ized.

3. REPORT TITLE: Enter the complete report title in all

capital letters. Titles in all cases should be unclassified.If a meaningful title cannot be selected without classifica-

tion, show title classification in all capitals in parenthesisimmediately following the title.

4. DESCRIPTIVE NOTES: If appropriate, enter the type ofreport, e.g., interim, progress, summary, annual, or final.

Give the inclusive dates when a specific reporting period is

covered.

5. AUTHOR(S): Enter the name(s) of authors) as shown onor in the report. Enter last name, first name, middle initial.

If military, show rank and branch of service. The name ofthe principal author is an absolute minimum requirement.

6. REPORT DATE; Enter the date of the report as day,

month, year, or month, year. If more than one date appearson the report, use date of publication.

7a. TOTAL NUMBER OF PAGES: The total page countshould follow normal pagination procedures, Le. , enter thenumber of pages containing information.

76. NUMBER OF REFERENCES: Enter the total number of

references cited in the report.

8a. CONTRACT OR GRANT NUMBER: If appropriate, enterthe applicable number of the contract or grant under whichthe report was written.

8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriatemilitary department identification, such as project number,subproject number, system numbers, task number, etc.

9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi-

cial report number by which the document will be identifiedand controlled by the originating activity. This number mustbe unique to this report.

9b. OTHER REPORT NUMBER(S): If the report has beenassigned any other report numbers (either by the originatoror by the sponsor), also enter this number(s).

10. AVAILABILITY/LIMITATION NOTICES: Enter any lim-

itations on further dissemination of the report, other than those

imposed by security classification, using standard statementssuch as:

(1) "Qualified requesters may obtain copies of thisreport from DDC"

(2) "Foreign announcement and dissemination of thisreport by DDC is not authorized.

"

(3) "U. S. Government agencies may obtain copies ofthis report directly from DDC. Other qualified DDCusers shall request through

(4) "U. S. military agencies may obtain copies of thisreport directly from DDC Other qualified usersshall request through

(5) "All distribution of this report is controlled. Qual-ified DDC users shall request through

If the report has been furnished to the Office of TechnicalServices, Department of Commerce, for sale to the public, Indi-

cate this fact and enter the price, if known.

1L SUPPLEMENTARY NOTES: Use for additional explana-tory notes.

12. SPONSORING MILITARY ACTIVITY: Enter the name ofthe departmental project office or laboratory sponsoring (pay-ing for) the research and development Include address.

13- ABSTRACT: Enter an abstract giving a brief and factualsummary of the document indicative of the report, even thoughit may also appear elsewhere in the body of the technical re-

port. If additional space ia required, a continuation sheet shallbe attached.

It is highly desirable that the abstract of classified reportsbe unclassified. Each paragraph of the abstract ahall end withan indication of the military security classification of the in-formation in the paragraph, represented as (TS). (S). (C), or (V).

There is no limitation on the length of the abstract. How-ever, the suggested length is from 150 to 225 words.

14. KEY WORDS: Key words are technically meaningful termsor short phrases that characterize a report and may be used asindex entries for cataloging the report. Key words must beselected so that no security classification is required. Identi-fiers, such as equipment model designation, trade name, militaryproject code name, geographic location, may be used as keywords but will be followed by an indication of technical con-text. The assignment of links, roles, and weights ia optional.

DD FORM1 JAN 04 1473 (BACK)

70UNCLASSIFIED

Security Classification

thesF268

A two-dimensional omega equation for the

3 2768 002 06566 6DUDLEY KNOX LIBRARY