Geostrophic Winds in Denmark:a Preliminary Study

Leif Kristensen and Gunnar Jensen

Ris@R-l145(EN) .

Ris@National Laboratory, Roskilde, DenmarkNovember 1999

DISCLAIMER

Portions of this document may be illegiblein electronic-image products. Images areproduced from the best available originaldocument.

MASTER

Geostrophic Winds in Denmark:a Preliminary Study

Leif Kristensen and Gunnar Jensen

Ris@National Laboratory, Roskilde, DenmarkNovember 1999

.

Abstract High-precision barometers have been deployed at six sites in Den-mark, four west and two east of the Great Belt. The purpose is to establish long

climatological records of the geostrophic wind as a supplement to the records oftens of years of duration of surface observations of wind, temperature, humidityetc., which have been obtained by Ris@ at many sites in Denmark. Three of thesesites are in principle sufficient to determine an average of the magnitude and di-rection of the geostrophic wind inside the triangle formed by the three sites. Ten,

out of twenty possible, triangles have been selected as suitable for studying thegeographical variations of the geostrophlc wind. A tentative conclusion from aboutone year of data is that statistically the geostrophic wind decrease in magnitudewhen going from west toward east. The data also showed that the largest mean val-ues of the geostrophic mean wind speed are in a direction sector from 285° to 315°.The Weibull parameters were calculated for all ten triangles. The curvature of theisobars were determined by using simultaneous pressure measurements at all sixsites and the geostrophic and gradient winds were calculated and compared to thegeostrophic wind based on three pressure measurements in one particular triangle.Combining the geostrophic wind with the surface wind measured at Tystofte insouthern Zealand, the two dimensionless constants A and B in the geostrophicdrag law were determined as functions of the surface friction velocity. These datasuggest that A = 0.5 and B = 3.5. The surface data at Tystofte and at BOrglumin Vendsyssel in northern Jutland were used to predict the geostrophic wind byapplying the geostrophlc drag law with these constants and the predictions were

compared to the observed geostrophic wind.

ISBN 87-550-2616-8ISSN 0106-2840

Information Service Department . Ris@ .1999

Contents

1

2

3

4

5

6

7

8

A

Introduction 5

Local Geometry 5

The

The

The

The

Geostrophic Wind 10

Thermal Wind 13

Gradient Wind 15

Surface Wind 18

Data and Data Analysis 19

7.1

7.2

7.3

7.4

Geostrophic Climatology .21

Gradient and Geostrophic Wind 26

Review of the Geostrophic Drag Law 29

Comparison with Surface Measurements 31

Concluding Remarks 35

Acknowledgements 37

References 38

Weibull Parameters A-1

IWO-R-1145(I3N)

—

1 Introduction

With the purpose of obtaining a climatology for the geostrophic wind climate,six stations with precision barometers of the type Vaisala PTB 200A have beenestablished in Denmark. At all the stations the air temperature is measured aswell since it is important to determine the pressure at same reference level. All

measurements are consecutive 10 minute averages.

The positions are shown in Fig. 1 and listed together with the barometer altitudesin Table 1.

58°

57°

56°

55°

54°6° 7° 8° 9° 10” 11” 12” 13° 14° 15° 16° 17”

Figure 1. The six barometer positions.

The barometers have a long-term accuracy of about 0.1 HPa which at sea sur-face corresponds the weight per unit area of a column of air of about 0.8 m. Tomatch this accuracy it is necessary to determine the altitudes better than about0.5 m. Professional surveyors from LE34 in Copenhagen determined the barometercoordinates of the barometer positions to this standard.

All the six stations were in operation on April 291998 at 13:45.

2 Local Geometry

We want to operate in a local coordinate system where the z-axis is normal tothe Earth’s surface and the two horizontal axes x and y point towards East andNorth, respectively.

The datum ED50 corresponds to approximating the surface of the Earth withellipsoid with the semi-axes a = 6378.3880000 km and b = 6356.9119461 km(Rasmussen 1996). This means that the eccentricity is ~ = 0.0820.

Figure 2 illustrates the situation.

RisO-R-l 145(EN)

Table 1. Positions and altitudes of barometers. Horizontal positions are given in

the datum ED50 and the altitudes in the vertical datum DNN (Danish normal

I Longitude

Ulborg 08°25’40.9040”

B@rglum 09°48’36.5570”

Kegmes 09°56’10.6492”

Bane 10°47’38.5100”

Gedser 11°56’34.2750”

RisO 12°05’22.1198”

Latitude

56°17’27.6974”

57°20’52.5701”

54°51’20.6491”

56°18’30.6509”

54°34’10.8587”

55°41’41.3339”

Altitude (m)

41.49

14.49

07.45

.38.47

02.28

08.04

Figure 2. Sketch showing the relation between the latitude p and the center angle@ for an ellipse with major semi-axis a and minor semi-axis b.

The equation for the ellipse is

(1)

as indicated in fig. 2, a point on the ellipse can be characterized by the angle@. However, the latitude p is defined as the complementary angle to the anglebetween the normal and the Earth’s axis. We want to determine the relationbetween p and @.

The point coordinates are

(2)

Ris@-R-l145(EN)6

and the equation for the tangent and the normal become

x Cos(p’ z sin q’—+—

b=1

a

and

x sin p’ Z Cospf {1a b— .—b a ‘coslf’sin~’ x–z ‘

respectively.

Referring to Fig. 2, simple geometry leads to

sin p’sin p =

/1 – # Coszp’ ‘

(3)

(4)

(5)

(6)

where

r6= 1–:

is the eccentricity of the ellipse.

It follows that

Jm Cos@

CoS’= ~“(7)

As a consequence of the smallness of ~ a first-order expansion in 62 of (5) and (7)should suffice. We get

C2cosp = (1 – ~ sin2p’) cosp’ (8)

and

~2sinq = (1 + ~ cos2p’) sinp’. (9)

Actually, we want @ expressed in terms of p and, to the same approximation, wehave

~2cos p’ = (1 + ~ sin2~) cos p (lo)

and

.s2sinp’ = (1 – ~ cos2q) sin~ (11)

Considering the Earth as a compressed, axisymmetric ellipsoid, we can specify aposition on the surface in the geocentric coordinate System (io, jO, ko) by

r=acosq’ cos~io+ acos~rsi n~jo+bsin~’ko, (12)

where A is the longitude

Rls@-R-l145(EN) 7

Using a as the length unit and applying the approximation

!l=J5=~_2,a 2

we may express r in terms of the latitude and the eccentricity as

(13)

r= (1+ ~sin2p)cospcos Ai0 + (1 + ~ sin2p) cosqsin~jo

e’+ (1 – ~[1 +cos’w]) sinpko. (14)

A differential distance 6s can now be expressed in terms of differential increments6P and 6A in latitude p and longitude A:

ds2 ~ dr . dr = {1 – 62(2 – 3sin2q)}6p2 + {l+c2sin2p} cos2~6A2. (15)

To define the local coordinate system at (w, A), we must determine the length ofthe vector r.

We find

lr12 = 1 – 62sin2p. (

The vertical unit vector k thus becomes

k= {1+ .52sin2p}cosqcos Ai0 + {1 +c2sin2q} cospsin~jo

+ {1 – c2cos2p} sinqko. (

In the tangent plane the unit vector i pointing East is

i=–sin~io+cos~jo. (18)

Finally, the unit vector pointing North in the tangent plane can be determined as

j = kxi

—— - {1 -e2cos2p} sinpcos Ai~ - {1 -c2cos2q} sinqsin Ajo

+ {1 + 62sin2p} cospko. (19)

We now define a the local reference coordinate systems in terms of the latitude

P.w and the longitude AM. The geocentric coordinate system is defined by theunit vectors i., j., and k.. The local reference system is then given by (18), (19)and (17) with p and ~ replaced by PM and AM:

i= —sin A~io+cOS AMjo, (20)

8 RIW-R-1145(EN)

(16)

(17)

j = – {1 – 62COS2qM} sinp~ cos AM iO

- {1 - .2 COS2pM} sin~~sin~~jo

+ {1 + 62sin2qiw} coswA4 ko,

and

k = {1 +,2cos2~M}cospM cosA~io

+ {1 + 62cos2~&f } Cos PM sin AM jo

+ {1 – C2sin2P&f} sin~MkO.

The origin of the local reference coordinate system is

We find

r~.i=()

(21)

(22)

(23)

(24)

and

r~. j =0(64) (25)

so we will ignore that rm is not quite perpendicular to the horizontal plane atthe origin of the local coordinate system.

A position (14) given by latitude p and longitude A can now be expressed inCartesian coordinates in the local reference coordinate system. We find

g=(r–r~). j=r. j=

( )1 – ~ {2c0s2~M – sin2$o} (cosp~ sinp – sinPM COS9COS(J– ~M)) . (27)

Sometimes we will need to determine the latitude (fM from the components of theunit vector k given by (22).

Ris@-R-l 145(EN) 9

Rewriting (22) as follows

k=zoi13+zljo+z2h3, (28)

where

(29)

we define

Z. = .Z~ + z? – z; = c0s(2~M) + ~2sin2(2ff&f) + 0(64) (30)

Z. = 222 Z:+ Z: = sin(2p~) – 62cos(2qA4) sin(2p~) + 0(64). (31)

Excluding terms of higher order in e than 4, we get

COS(29M) = 2. – 622;

and

sin(2ffM) = Zs + ~2ZcZs.

(32)

(33)

3 The Geostrophic Wind

(%Y1,P1)

(X3, !J3,P3)

[L2,Ap,]

G- Y2, P2)

Figure .3. Triangle for determination of the geostrophic wind velocity.

10 Rk@-R-l145(EN)

The geostrophic wind velocity is given by (see e.g., Dutton (1986))

G=; kxV2p,

where

(34)

(35)

is the horizontal gradient, p and p are the density of air and the pressure, both atthe ground, and

~ = Q sin(p~) = 1.45810-4 S-l sin(p~) (36)

is the Coriolis parameter.

We assume that we can determine G by means of the pressure measured at threedifferent places

7’i-?’M=zii+gij, i= 1,2,3 (37)

which are not on the same straight line in the tangent plane defined by i and j.This means that, inside the triangle determined by the three points, the pressurevaries linearly with the position coordinates. In other words,

As (34) shows, we must find the magnitude and the direction of

v2p=p; i+p; j

and we have the three equations

{:::} X{;}=

with the three unknowns p., p; and p;.

We find

PI

P2P3 }

P: = ;{P1(Y2 ‘Y3) +p2(g3 ‘?41) +P3(Y1 ‘!42)}

and

1Pj = ~{pl(~3 ‘~2)+p2($l ‘z3)+p3($2–~1)},

where D is the determinant:

1 z~ g~D= 1 X2 y2 .

1 Z3 y3

Ris@-R-l145(EN)

(38)

(39)

(40)

(41)

(42)

(43)

11

We see immediately that the direction of the geostrophic wind is given by

kx{PLi+Pjj} =–Pji+Pij=

+[{P1(Z2 -X3) +P2(~3 -Z1)+P3(Z1 -&?)} i

+ {P1(Y2 ‘Y3) +p2(y3 ‘!h) +P3(V1 ‘Y2)} d .

The direction of the geostrophic wind vector is then

~G = arctan2(pl(xz – x3) +p2(X3 – Xl) +p3(XI – X2),

pl(~2 – ~3) +p2(y3 – !h) +p3(!/1 – ~2)) .

(44)

(45)

The magnitude of Vzp can be expressed in terms of the lengths of the sides in thetriangle

el =

tz =

[3 =

/(x3 - X2)2 + (Y3 - Y2)2

J(X1 - X3)2+ (!41- !43)2 (46)

~(z2 - Z1)2+ (U2 - !/1)2

and the three pressures.

(V,p)z = -&—

(~;{P; +p2p3} + ~i{P; + P3P1:

where

A = :/{+/1 +/2 +t’3}{+I?l +I!2 – 13}{+11 –/2 +/3}{ –/1 +/2 +/3}, (48)

according to Heron’s formula, is the area of a plane triangle with the side-lengths/1, /2 and /3.

It is possible to estimate how the random errors c@] on the measurements prop-agate to V2p. Assuming that the measuring errors on the pressures are the sameand equal to d~], we may use the equation

‘2’(v’p)21=’2@12{&(v2~)2r (49)

We have

&(v2p)2 = & {2@, + (/; -1:- 4)p3 + (/: - f?;- l?;)p2} . (50)

12 ~k@-R-l145(EN)

Also, we have the identity

so that

(51)

Similar expressions for d/8pz (Vzp)2 and 8/8p3(VZP)2 are easily obtained by uti-lizing the symmetry and changing the indices in a cyclic manner.

Introducing the definitions

{1}’{!:!!:!}

and

{5}={::;}

(Note that Apl + Apz + Ap3 = O.)

(53)

(54)

we get

I&$n5’~

Thus,

{

–A2Ap2 + A3Ap3(Vzp)’ = &

}

–A3AP3 + AI API .–Al ApI + A2Apz

(55)

g{&(v2P)2}2 = & {A~Ap12 + 4AP22 + 4A1132

– A2/i3Ap2Ap3 – -’43-41Ap3ApI – A~.42Ap~Ap2 } . (56)

The error of the magnitude G = IGI of the geostrophic wind speed becomes

(5[G]=2/$v2P&{&(v2p)2r

(57)

With the purpose of reducing systematic errors when dealing with observation, allthe equations are made symmetric in (ZI, X2, $3), (W, Y2, u3), and (PI, P2,P3).

4 The Thermal Wind

The geostrophic wind velocity given by (34) is under steady, barotropic conditionsand the geostrophic wind becomes almost equal to the constant wind velocity aloft.

Ris@-R-l145(EN) 13

However, if there are horizontal temperature variations G is not constant withheight. A good discussion of this subject has been given by Dutton (1986) who

shows that

8G 1 c9Tg kxV2T+F8z—= —

a,z fT— G,

where T is the air temperature in “K and g is the acceleration of gravity.

The vertical gradient of G is what is called the thermal wind.

Applying (34) and the hydrostatic equation

we may rewrite (58) in the form

aG 1

{

13p 1—k X ~v~p– ~V2T .8Z = Tpf

(58)

(59)

(60)

Let us for a moment generalize (37) and include the vertical coordinate z, i.e.

P= P(~, Y,z), (61)

and consider the pressure variation around the point (z., y., Zo).

Locally, the variation c!p is given by

dp =p; c!x +p:Jy +pjaz, (62)

where the derivatives are taken at the point (ZO,yO,ZO) and where dz = x —xo,

anddy=y —yo, and bz=z –zo.

The two-dimensional, constant-pressure surface through (xo, YO,ZO) is defined by

o = p;dz +p;l$y +p:dz. (63)

On this surface the level line is defined by C$Z= O, i.e.

P;~x +P;~Y = o (64)

The unit tangent vector to this line is

‘=-%%”

(65)

The normal n to t in the tangent plane is the principal normal. It is tangent tothe line of steepest descend and given by

/! .1/.

“=%=x+%$”The normal to the surface, the binormal

/. {./

‘=’’”===vector, becomes

(66)

(67)

Ris@-R-l145(EN)14

We can define the analogue unit vectors for the temperature field

‘T = Z’(z, y,z)

as follows

‘=-w+‘=-R=%%+

and

‘=’xn=KH=&%”

(68)

(69)

(70)

(71)

The necessary and sufficient condition that the two constant-surfaces for p and Tcoincide locally in the neighborhood of (ZO,y., Z. ) is that b and B are parallel.According to (67) and (71) this means that in this case

Comparing this equation and (60), we see that the condition that the thermal windis zero is that the constant-pressure surface and the constant-temperature surfacecoincide. When (72) is fulfilled we say that we have barotropic stratification and

baroclinic stratification when this is not the case.

Dutton (1986) discussed (58) and found that the second term is about two ordersof magnitude smaller than the first. Consequently, we will use the approximation

(73)

henceforth.

5 The Gradient Wind

The previous considerations have been based on the pressure gradients and not onthe second derivatives. This implies that all isobars of p are considered straight.The geostrophic balance means in this case that the vector sum of the horizontalpressure gradient and the Coriolis force is zero. If we want to consider the curvatureof the isobars and include the second derivatives, the centrifugalin the balance. In order to study this we generalize (38):

P(~>Y) = Po + PL~ + P;Y + ; {P:z~2 + 2P&xY + P;YY2} .

Ris@-&l145(EN)

force also enters

(74)

15

The surface described by (74) is a so-called quadric surface. The tangent plane isin general horizontal in exactly one point (sO, yo) and this point will be either a

maximum point, ‘a high’, a minimum point, ‘a low’, or a saddle point surrounded

by two high and two low pressure domains. This is determined by signs of theeigenvalues p$ and p!! of the symmetric tensor

“’={%2} (75)

When are p: and p!! are both positive/negative the surface has a minimum/maximum.

When p~ and p!! have opposite signs the surface has a saddle point. There is ofcourse the possibility that one, p!! say, is zero. In this case the surface is a ‘trough’(low pressure, p! > O) or a ‘ridge’ (high pressure). When both p! and p!l are zerothe surface is a plane.

The determinant of the tensor (75) is equal to the product of p~ and p!!, i.e.

(76)

and can be used as a diagnostic tool.

The position (z., go) of the extremum is determined by setting the first derivativesof (74) equal to zero. This leads to the linear equations

{2%}{:}=-{2}with the solution

{1

Xo 1

{

P;yP; – P:vP;Yo = p;zpj’y – p&2 P:yP; – P:zP;

The differential equation for the isobars is

p;dx + p;dy = 0.

(77)

(78)

(79)

Let us follow a Lagrangian particle along the isobar going through the center. Itsposition (x(t), y(t)) in parametric form with time t as parameter must obey

()(~>ti) = :,: = ; (-P;) P;) ~

We really want to determine the radius of curvature

~ = (22 +@2

I

Xti”XY

The second derivatives become

x = ;: (–P;) = ; {–P:y~ – P;gY} = * {P:VP; – P;yP:

(80)

(81)

(82)

16 Ris@-R-l145(EN)

LoH

H

Figure ~. Symbolic representation, by means of isobars of a low-pressure system, a

high-pressure system and a saddle point. When there is no horizontal density gra-dient the air moves along the isobars, clockwise around a low and counterclockwisearound a high.

and, similarly,

Y = --&{P;VA ‘}– PLPy

Inserting in (81) we obtain

( )3/2

PL2 +P;2R=

P;yPi 2 – 2P;yPiPj + P&Pj2

(83)

(84)

If R is not infinite the isobars have curvature, positive around a low pressureand negative around a high pressure. When R is infinite (large compared to thelinear dimensions of Denmark) we consider the isobars straight and the pressuresurface a plane. In this case geostrophic balance means that the wind aloft has

the magnitude and the direction of the geostrophic wind. When the isobars arecurved the magnitude of the wind speed aloft is equal to the gradient wind ~. Therelation between Q and G is given by (Dutton 1986)

()GG2

jEz +:–1=0. (85)

This equation represents a balance between the centrifugal force Q2/R, the Coriolisforce f~,and the pressure force fG.

Riso-R-l145(EN) 17

We see that the important parameter is the dimensionless quantity

(86)

As (85) shows, this parameter, the so-called Gradient Rossby Number (Dutton1986), determines whether or not ~ and G are approximately equal. Solving (85)for ~/G, we get only one meaningful solution (see Fig. 5):

G –1 + @ + 4G/(fR)—=G 2G/(~R) “

(87)

2.0

1.5

~

G’ lo

0.5

0.0 ,–~ 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure 5. The solution to (87).

If G/(fR) is small compared to one we get, by expanding the square root

G G

({-1)

G2 G

E ‘~-~+o fR ‘l-fin(88)

The balance (85) cannot be maintained when G/(f R) < –1/4.

6 The Surface Wind

At the Earth’s surface the Coriolis force is of no importance and the resistance forcefrom the surface becomes dominant. When the terrain is horizontally homogeneouswith a roughness length Z. the variation close to the surface of the wind speedwith height z is under neutral conditions given by

(89)

18 Ris@-R-l145(EN)

where K H 0.4 is the von K6.rm5n constant and U* the friction velocity.

In this idealized situation it is possible to establish a connection between the

magnitude and direction of the geostrophic wind and those of the surface wind.An elegant derivation of this relation, the geostrophic drag law, is given by Tennekes

& Lumley (1978). If the angle from the direction of the geostrophic wind to thatof the surface wind is a then

‘cOs”=~{ln(a-Al (90)

Gsina = ~B, (91)

where A and B are dimensionless constants. There is a considerable uncertaintyabout what the best values of these constants. Since they are based on the loga-rithmic wind profile (89), pertaining to a neutral stratification of the surface layer,

an experimental determination of A and B must be carried out under conditionswith strong surface winds. Even then there does not seem to be a general agree-ment as to what values should be used. Tennekes & Lumley (1978) and Mortensenet al. (1993) and others use A = 1.8 and B = 4.5 whereas Panofsky & Dutton(1984) suggest A m O and B ~ 5 for strong winds.

The two equations (90) and (91) may be recast in the form

and

(92)

a=arctan’(+%)-AB) (93)

We see that if the geostrophic wind velocity G is known, i.e. both magnitude anddirection, then the magnitude and direction of the surface wind can be calculatedat a given latitude (for the determination of ~) and for a given roughness length Z.by first solving (92) for U* and inserting in (89) and, subsequently, by using (93)to calculate the turning of the wind a from the geostrophic wind to the surfacewind.

Figure 6 shows schematically the relation between G, U and a.

7 Data and Data Analysis

The data from each station is stored year by year as an ASCII file with a namewhich includes information about the name of the station and the year. The formis “pnam.ye.dat”, where “p” shows that it is a pressure file, where %am” standsfor the first three letter of the name of the stationt, and where “ye” is the last twodigits of the year. For example the pressure data from RIs@,from 1998 are storedin the file “p~is_98.dat”.

The files are format ted as records, where numbers are separated by space, andwhere “

tTh~D~ni~hl~tt~r~f,=,,, “@”,and %“ willherebe replacedby “a”, “o”, and“aa”, respectively.

Ris@-R-l145(EN) 19

~— “-”-‘“--— —

I ‘T

-..

L.—

~:”’’-”’=’ —“-J - -.. .... .

~-

Figure 6. Schematics of the relations between the geostrophic wind G, the surfacewind U, and the angle a between them.

the first holds the latitude in degrees, minutes, and seconds,

the second record the longitude in degrees, minutes, and seconds,

the third the altitudes in m, and

each of the following the pressure at the observation altitude, the air tem-perature in 0C, the year with all four digits, and the month, day, hour, andminute in the form “MODAHOMI”.

The pressure records used in the following analysis cover the periods given inTable 2.

Table 2, Timetable for pressure measurements.

Ulborg

B@rglum

Kegn=s

Bane

Gedser

RlsO

Start

1998 0429 13:45

1998 0429 13:45

1998 0429 13:45

1998 0429 13:45

1998 0429 13:45

1998 0429 13:45

stop

1999 0606 23:55

1999 0606 23:55

1999 0606 23:55

1999 0323 10:25

1999 0627 09:55

1999 0606 23:55

As explained in section 3 we may calculate the geostrophic wind speed and direc-tion from three barometer stations.

The software has been developed and Table 3 shows a sample output. The topconsists of various household data, such as the name and version of the program

20 Ris@-R-l145(EN)

used, ellipsoid data, geographical positions in decimal degrees of the three stationsand that of their center of mass, the altitudes of the three stations, and the Coriolisparameter pertaining to the center of mass position. In the main body of the table

are the pressures, reduced to sea level, the temperatures in ‘C, the geostrophicwind speed G, the error 6[G] according to (57), the geostrophic wind direction,

the magnitude and direction of the thermal wind, and the observation time.

Table 3. Sample output from the stations Ris@, Kegnm, and Gedser.

#Source code: C:\LKDOC\GSTR\TYST.A.B\G-WINDOS.PAS. Time St=p: 3 November1999, 15:29.##Pressure and temperature##Semimajor axis (datumED50): 637S. 388 km.#Eccentricity (datumED50): 0.08199189045.##Site Latitude Longitude Altitude#===================================#Kegnaes 54.8557 9.9363 7.45#Gedser 54.5697 11.9429 2.28#Risoe 55.6948 12.0895 8.04#meanpos 55.0445 11.3190#COriOlis parameter: 0.0001195319##Year: 1998.##Distances:#Gedser - Risoe : 125.611 km#RisOe - Kegnaes : 165.696 km#Kegnaes - Cedser : 133.192 km## Kegnaes Gedser Risoe G errG dirG ThW dirThWYEARMODAHOMI# mb c mb c mb c mls mls deg S-(-l) deg#=====================================.========..=...=...=..===..= ..==.= . . . ..==..===.=== . ...=*

1009.81 10.1 1008.76 S.3 1009.95 14.4 7.69 0.83 60.4 0.014 88.1 1998 04 29 13 451009.81 10.2 1008.70 9.3 1010.04 14.6 8.48 0.84 62.5 0.012 94.9 1998 04 29 13 551009.71 10.2 1008.62 9.3 1010.31 14.4 9.91 0.83 70.2 0.012 94.6 1998 04 29 14 051009.71 10.2 1008.54 9.1 1010.23 14.3 10.08 0.83 68.1 0.012 92.6 1998 04 29 14 151009.71 10.2 1008.48 9.0 1010.07 14.7 9.83 0.83 64.7 0.013 92.7 1998 04 29 14 251009.71 10.2 1008.41 8.7 1009.95 15.3 9.80 0.84 61.8 0.015 91. S 1998 04 29 14 35

7.1 Geostrophic Climatology

We can now determine the geostrophic wind by means of three stations as shownin section 3. We have chosen 10 such triangles of stations to investigate a pos-sible geographical variation of the geostrophic wind climate. Figure 7 shows thepositions of the centers of these triangles and Table 4 their geographical positions.

We have calculated time series of the geostrophic wind speed G and direction DGas 10 minute averages for all ten centers. These records in their the entire lengthshave been used to determine for each center the mean (G), the mean square (Gz)and histograms of G in twelve 30° sectors, centered around 000°,030°,...,330°and for all directions. Under the assumption that the probabilityy density function

Ris@-R-l145(EN) 21

58°

57”

56°

55°

54”

I -7” w /’!1

.I I h 1 I I 1 I

6° 7° 8° 9° 10° 11° 12” 13” 14” 15” 16° 17”

Figure 7. Center positions for the 10 triangles, numbered from O to 9.

Table J. Positions of the ten centers. The positions are given in the datum ED50.

No

o

1

2

3

4

5

6

7

8

9

Center I Triangle

BOULKE B@rglum Ulborg Kegmes

BOULBA B@rglum Ulborg Bane

ULKEBA Ulborg Kegn~s Bane

BOULRI B@-glum Ulborg RlsO

ULKERI Ulborg Kegnm Ris@

BOKEBA B@rglum Kegn=s Bane

BAKEGE Bane Kegrws Gedser

IBAKERI Bane Kegnax Rkw

‘

KEGERI Kegnax Gedser Ris@

BAGERI I Bane Gedser RlsO

Longitude

09°23’40”

09°40’34”

09°43’19”

10°13’20”

10°09’30”

10°10’50”

10°53’41”

10°56’01”

11°19’08”

11°36’58”

Latitude

56°10’05”

56°39’14”

55°49’24”

56°27’18”

55°37’26”

56°10’22”

55°14’55”

55°37’26”

55°02’42”

55°31’36”

22 Ris@-R-l145(EN)

of G is well represented by the Weibull distribution

p(G)=j(~)k-’exP(-[~]k) (94)

(G) and (G’) have been used to determine the parameters d and k.

At all the centers, except at ULKERI (4) and at BAGERI (9), the highest valueof G was found in the 300° sector. The mean (G3000) for this sector and for alldirections are given in Table 5.

Table 5. Maximum, mean and mean in the 300° sector of G.

No

o

1

2

3

4

5

6

7

8

9

Center

BOULKE

BOULBA

ULKEBA

BOULRI

ULKERI

BOKEBA

BAKEGE

BAKERI

KEGERI

BAGERI

max G

59.8 m/s

52.8 m/s

46.9 m/s

54.4 m/s

44.6 m/s

47.8 m/s

46.3 m/s

53.1 m/s

39.9 m/s

45.5 mfs

DG

196°

295°

219°

289°

230”

310°

273°

256°

215°

355°

Time

1998 0714 10:25

1999 0125 15:35

1998 0714 10:25

1999 0125 15:35

1998 0714 10:25

1998 1025 23:55

1998 1025 23:15

1998 1025 23:15

1998 1028 07:05

1998 1106 11:05

(G)

11.9+ 0.1 m/s

12.0+ 0.1 m/s

11.5+0.1 m/s

11.7+0.1 m/s

10.6+ 0.1 m/s

12.1+0.

11.2+0.

11.7+0.

m/s

m/s

m/s

10.6+ 0.1 m/s

11.7+ 0.1 m/s

(G300° )

14.9+ 0.2 m/s

15.8+ 0.2 m/s

14.1+0.2 m/s

15.1+ 0.2 m/s

12.3+ 0.2 m/s

14.2+ 0.2 mfs

13.4+ 0.2 m/s

13.2+ 0.2 m/s

12.9+ 0.2 m/s

13.2+ 0.2 m/s

Table 6 is a list of the Weibull parameters, (k, A) for all directions and (k300., A3000)for DG = 300° and Figure 8 shows as an example the frequency distributions of

G for all directions and for the 300° sector. The complete information about allthe Weibull parameters can be found in the appendix.

At each of the centers listed in Table 4 the maximum value of G was selected,together the corresponding direction and the time of occurrence, for a preliminaryinvestigation of extreme events. Only data from before March 19, 1999, wherethere was a major interruption in the data recording in Bane, were used.

Each of the ten maximum values of G were compared to the simultaneous valuesof G at the nine other centers. Several of the maximum values of G were simul-taneous. Further, the data from Ulborg were periodically interrupted for ratherlong durations. The result is that, with this selection criteria, we have found onlythree events where all the centers provide relevant data. The results are listed inTable 7 and shown in Figs. 9, 10, and 11.

Ris@-R-l145(EN) 23

Table 6. Weibull parameters.

No

o

1

2

3

4

5

6

7

8

9

0.1

0.08

0.06

0.04

0.02

0

Center

BOULKE

BOULBA

ULKEBA

BOULRI

ULKERI

BOKEBA

BAKEGE

BAKERI

KEGERI

BAGERI

k

1.86

1.90

1.90

1.87

1.92

2.03

1.90

1.91

1.88

1.93

d

13.4 m/s

13.5 m/s

13.0 m/s

13.2 m/s

11.9 m/s

13.6 m/s

12.7 m/s

13.2 m/s

11.9 m/s

13.2 m/s

0.1

0.08

0.06

0.04

0.02

0

k3000

2.05

2.18

2.16

2.09

2.16

2.15

2.21

2.27

2.00

2.09

I

d3000

16.8 m/s

17.9 m/s

15.9 m/s

17.0 m/s

13.9 m/s

16.0 m/s

15.1 m/s

14.9 m/s

14.6 m/s

14.9 m/s

I I 7I

-1

0 10 20 30 40 50 0 10 20 30 40 50

G (m/s) G3000 (m/s)

Figure 8. Histograms and corresponding Weibull distributions for all directions(left frame) and for the 300° sector (right frame) at center number O, BOULKE.

It is not safe to draw too definite conclusions from these time limited records ofthe geostrophic wind velocities. Further, not all the ten centers are equally wellsuited for analyzing the geostrophic wind field. In fact, by comparing Figs. 7 and10, we see that three of them, BOULKE (0), BOKEBA (5), and BAGERI (9), arerather ‘shallow’.

Even then the results above seem to indicate that:

24 Rk@-R.-ll45(EN)

Table 7. Three events where at least one center has its maximum value of G.

No

o

1

2

3

4

5

6

7

8

9

Center

BOULKE

BOULBA

ULKEBA

BOULRI

ULKERI

BOKEBA

BAKEGE

BAKERI

KEGERI

BAGERI

1998071410:25

G

59.8 m/s

36.0 m/s

46.9 m/s

37.6 m/s

44.6 m/s

19.1 m/s

27.9 m/s

33.4 m/s

24.7 m/s

33.2 m/s

DG

196°

183°

219°

184°

230°

258°

236°

229°

227°

214°

1998102807:05 1999012515:35

G

08.7 m/s

10.8 m/s

05.6 m/s

12.4 m/s

14.0 m/s

20.9 m/s

37.2 m/s

17.0 m/s

39.9 m/s

20.7 m/s

DG G

347° 53.1 m/s

253° 52.8 m/s

123° 34.3 m/s

248° 54.4 m/s

140° 18.9 m/s

183° 34.6 m/s

191° 33.1 m/s

180° 35.2 m/s

215° 28.5 mjs

271° 30.3 m/s

DG

318°

295°

311°

289°

295°

264°

273°

263°

271°

253°

58°

57°

56°

55°

54°6° 7° 8° 9° 10° 11° 12° 13° 14° 15° 16° 17°

Figure 9. Extreme geostrophic wind speed on July 1~, 1998. Centers O, 21 and ihave their maximum values on this date.

1. the largest mean values of G are in the 300° direction sector,

2. both (G) and (G300. ) increase in magnitude when going from west to east,and

Rkw-R-l145(EN) 25

58°

57°

56°

55°

54°

A

6° 7° 8° 9° 10° 11° 12° 13° 14° 15° 16° 17°

Figure 10. Extreme geostrophic wind speed on October 28, 1998. Center 8 has its

maximum value on this date.

58°

57°

56°

55°

54°

A

6° 7° 8° 9° 10° 11° 12° 13° 14° 15° 16° 17°

Figure 11. Extreme geostrophic wind speed on January ,25, 1999. Centers 1 and 3have their maximum values on this date.

3. when G is at maximum in the western part of the country the simultaneousvalues in the eastern part are considerably smaller and not corresponding tothe maximum values there.

7.2 Gradient and Geostrophic Wind

In section 5 we discussed the gradient wind ~ where the curvature of the isobarsis taken into account by including the centrifugal force in the balance with thehorizontal pressure gradient and the Coriolis force. We are actually able to deter-mine this curvature from the six pressure observations since there are exactly sixunknown constants in the quadric surface described by (74).

26 Rls@-R-l145(EN)

The curvature of isobars may also result in a different estimate of the geostrophicwind speed G6 compared to that obtained from three pressure observations G3

where the isobars are assumed to be straight. The directions corresponding to G6and ~ however, are the same. We have carried out a comparison between G3, G6,and Q and center number 8 (KEGERI). Figure 12 shows these three speeds, theGradient Rossby Number, and the directions for a short period of three days andFigs. 13, 14, 15 more completi comparisons between G6 and G3, ~ and G3, and~GG and ~G~, respectively.

360°

~~ —--”-180°

0° ‘ I I

30 m/sI

15 m/s

Oct. 10, 98 Oct. 12, 98 Oct. 14, 98 Oct. 16, 98

Figure 12. Lower frame: G6 (thick line), G3 (thin line), and ~ (dashed line). Mid-dle frame: G/(fR). Top frame: direction of the geostrophic wind from three pres-sure measurements (thick line) and from six pressure measurements (thin line).Data from center number 8 (KEGERI).

r=nl30 1-

.. ...”...-.“5 .?.,:,s... ........,.. -1

05 10 15 20 25 30 35 40 45

G3 (m/s)

Figure 13. Comparison between the geostrophic wind speed based on six simultane-ous pressure observations and that based on three at center number 8 (KEGERI).

Rls@-R-l145(EN) 27

G (m/s)

45 I I I I I I I I

40

. . . “.. . ...”.

35

30 [

25

20

15

10

5

005 10 15 20 25 30 35 40 45

G3 (m/s)

Figure 14. Comparison between the gradient wind speed and the geostrophic windat center number 8 (KEGERI) based on three simultaneous pressure observations.

180”. .f I I I

. . . .. .“.. ... . .

.. .. .... .. . ..

.-.”. . 4l.”~“..

.. . ..090° ... . . .

..”. . . :. :.....“.. -” ~.c. . “.. . ,. --..:”...%

“ !.,.:“..:.”,,..0“..“”.-,~... .:... ....... .. ......’.”.... 0 .“........,.. . >,...Lti..:vt%:,-:*&;:* ..“:’.“. <:=l”.“d:. ~..: ...U..*,..>,.,i,y.

DGG – DGS,#’&” “’“*- ...

000° ,.:‘**:. .;?., “..

. .“. . ~::..“.‘“:7?”?.:,,~~i’,.::“.ff::i.+-Y.-”... .. ~J. .::. .-” ..”. -. .:”::....... ............. .. .... ..... . . “.~.. ..... ...... -.. .-.... . .. .–090°

.. . .. .“.. .”“.

.. .

l.. I. .

–180° “ I I I I

000” 090” 180° 270° 360°

Figure 15. The difference between the direction of the geostrophic wind based on

six stations and that based on three stations at center number 8 (KEGERI).

Figure 12 seems to show that G6 tracks Gz quite well and also that there is goodagreement between the directions DGC and DG~. The gradient wind ~ is usuallysomewhat lower than the geostrophic wind which of course is no surprise (see Fig.5) when observing that the middle frame shows that the Gradient Rossby Numberis nearly always positive, corresponding to low-pressure situations. The few timesthis number is negative the gradient wind is larger than the geostrophic wind.

Figures 13, 14, and 15 is an indication of how large deviations one must expectwhen comparing velocities and directions obtained in different ways.

28 Ris@-R-l145(EN)

7.3 Review of the Geostrophic Drag Law

It is possible to compare the geostrophic wind and the measured surface wind at

Tystofte. Ten minute averages of wind speed and direction has been contiguouslyrecorded at the top of a 39.3 m mast since May 25, 1982. Its geographical co-ordinates are (A, p) == (11°19’48”, 55°14’24” ), i.e. about 23 km straight north ofthe center KEGERI (8). Kristensen et al. (1999) discuss these data and show howWAsP (Mortensen et al. 1993) can be used to determine, for a given measured windspeed, the friction velocity u. for any direction of the wind. Consequently, we havesimultaneous records of the geostrophic wind and the surface wind and should beable to determine the dimensionless constants A and B in the geostrophic draglaw (90) and (91).

We expect A and B to depend on the stability of the atmospheric surface layer(Tennekes 1982). We are mostly interested in strong winds where the flux of specificmomentum is dominating, i.e. where the thermal stratification can be consideredclose to neutral. We have consequently excluded, somewhat arbitrarily, situationswhere U* is smaller than 0.4 m/s. According to (89), this value of u. correspondsto a wind speed of about 5 m/s at 10 m over a terrain with the roughness lengthZ. = 0.05 m/s. The values A and B, obtained from the data, showed a considerablescatter. Figure 16 shows a histogram and a Gaussian fit to the data.

400

350

300

250

200

150

100

10

400

350300

250

200

150

100

:0

r II

20

20

Figure 16. Two-dimensional histogram and Gaussian fit to A and B, obtained fromthe Tystofte wind data and the simultaneous pressure observations at Kegnax,Gedser, and Risti. The mean values, the standard deviations, and the correlationcoefficient are ((A), (B)) = (0.8, 4.1), (aA, aB) = (2.9, 2.9), and pAE = –0.16,respectively.

To test to what extent A and B depend on U* we have also calculated averagevalues of A and B by successively excluding data where U* is smaller than a certainvalue. the result is shown in Fig. 17.

As mentioned in 6 recommended values are O~ A ~ 1.8 and 4.5 ~ B ~ 5. We findthat A = 0.5 and B = 3.5 are in reasonable agreement with our observations, inparticular for higher wind speeds.

Rk@-R-l145(EN) 29

5

4

A,1332

1

0

–1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

u. (m/s)

Figure 17. The mean values of A (closed circles) and B (open circles) as functionsof the lower value of the friction velocity.

There seems to be a large discrepancy between our findings and those used rou-tinely in e.g. WAsP. However, in some applications this is of limited consequence.For example, if for a given geostrophic wind we want to determine how the frictionvelocity changes with changing roughness length there is the simple approximaterelation (Kristensen et al. 1999)

()z; 9

U:=U* — ,I@

(95)

where the primed variables are the new values of the friction velocity and theroughness length. The exponent q is about 0.069 for (A, l?) = (1.8, 4.5) and 0.067 =1/15 for (A, B) = (0.5, 3.5). In the first case the friction velocity changes by 17.2%when the roughness length changes by a factor of 10. In the second case thecorresponding change i U* is 16.770.

To estimate the influence on the relation between the surface wind and the geostrophicwind we rewrite (92) and (93) as

KG— = J(L– A)2+IP,U*

and

tan(a) = ~,

where

()‘=ln g

(96)

(97)

(98)

characterizes the magnitude of the surface wind speed in a given terrain with theroughness length Z. at a given latitude. The range

10< L<15 (99)

30 Rk@-R-l145@N)

covers all situations of practical importance.

If the changes in A and B are dA and 6B, respectively, the change in a becomes

Since L will then be somewhat larger in magnitude than both a and B, we ap-

proximate (96) by a second-order expansion in A/L and B/L.

KG— = {(L-A)’ +B2}’12u*

{

A AZ + B2 112= L 1–27+ Lz 1{

~L~_ A+ A2+B2 12

L zL2 – ~( )]

–2;

z L–A+~. (101)

Roughly we have

(5G 6A B 6B—%G –—+17”L

(102)

The approximate relations (100) and (102) show that a change in A is moreimport ant for G than the same change in B. The opposite is the case for a.

7.4 Comparison with Surface Measurements

We have compared surface wind data with the geostrophic wind at two differentsites, using the geostrophic drag laws (92) and (93) with A = 0.5 and B = 3.5,derived by comparing Tystofte data and data from center number 8 (KEGERI).

First we used the Tystofte data to predict the direction and magnitude of the

geostrophic wind and compared with the geostrophic wind at center 8 (KEGERI)which is, within 2°, approximate y 22 km due south of Tystofte. Figure 18 showstime series over 10 days of the predicted and the observed direction and magnitudeof the geostrophic wind.

It is obvious that the tracking is far from perfect. For example, there seems to bea significant discrepancy between the geostrophic wind speeds on May 131998.The directions, however, do not contradict one another. That particular day thereis, according to the weather surface map, a high pressure of 1035 HPa about800 km north of Denmark, where the surface pressure is about 1025 HPa. A

possible explanation for the discrepancy is therefore that the gradient wind whichis indeed larger than the geostrophic wind should have been compared to theprediction.

Figure 18 also shows that changes in the predicted and the observed geostrophicwind speed do not always occur simultaneously. This is illustrated quite well bythe record: late on May 91998 the observed geostrophic wind speed precedes thepredicted by a couple of hours and 24 hours later the situation is the opposite.

Rk@-R-l145(EN) 31

In the first case the geostrophic wind direction is about 180° and in the secondbetween 0° and 90°.

We have used the entire Tystofte and KEGERI records to compare predicted andobserved geostrophic wind. The result is shown in Figs. 19 and 20. In both plots

there is a large scatter and Fig. 19 shows that the predicted geostrophic wind speedGpred. in the mean is smaller than the observed GOb~. In fact, a least square fitgives the result Gpred. % 0.7GOb~ + 3 m/s.

360°

270°

180°

090°

000°

25mis , I I I I I I I i I I20m/s

15mjs

10m/s

5 mjs

Omis

May 81998 Ivfay131998 May 181998

Figure 18. Comparison between predicted (thin line) and observed (thick line)Geostrophic wind at center number 8 (KEGERI). Upper frame: directions. Lowerframe: geostrophic wind speeds. The prediction is based on the Tystofte data.

35

30 1 ““.1.. ... .

0 5 10 15 20 25

Gobs(m/s)

Figure 19. Comparison between the predicted geostrophic

30 35 40

wind speed at Tystofleand the observed geostrophic wind at center number 8 (KEGERI).

We have also carried out a more independent test of the geostrophic drag lawby using the values of A and B derived from the Tystofte and KEGERI data to

32 Risg-R-l145(EN)

I

180° I I I I . . f

.1

..

–180° “. .

I I I

000° 090° 180° 270° 360°

Dobs,

Figure 20. The difference between the predicted and the observed direction of thegeostrophic wind based on data from Tystofte and center number 8 (KEGERI).

compare the geostrophic wind at center number 1 (B OULBA) (see Fig. 7) and thatpredicted by the surface data at B@rglum about 78 km to the north (geographicaldirection 6°). A ten-day record of the prediction and the observation is shown inFig. 21. This figure shows good agreement at times and also direct disagreement forrather long periods. Using the entire record, we compare magnitudes and directionsin Figs. 22 and 23. The scatter is larger than in the corresponding plots Figs. 19and 20, but here GPred. does not underpredict Gobs, to the same extent (Gpred. ~

().8Gobs. + 3 m/s).

360°

270°

180°

090°

000”

25mis I I I I I I I .Lm I I I 120mfs

15 m/s

10 mfs

5 m/s

Om/s

May 131998 May 181998 May 231998

Figure 21. Comparison between predicted (thin line) and observed (thick line)Geostrophic wind at center number 1 (BO ULBA). Upper frame: directions. Lowerframe: geostrophic wind speeds. The prediction is based on the B@rglum data withA and B derived from the Tystofie-KEGERI data.

There are several possible explanations for the less than perfect agreement between

Ris@-R-l145(EN) 33

40

35

30

25

Gpre&@) 2“

15

10

5

0

0 5 10 15 20 25 30 35 40

Gobs.(ds)

Figure .22’. Comparison between the predicted geostrophic wind speed at Bgrglum,

using A and B from the Tystofte-KEGERI data, and the observed geostrophic windat center number 1 (BO ULBA).

180° . . . . . ... .. .. . . . . ... .“, ,..,

,“. . ::..“ ----

090°

Dpreci. – Dobs.

000°

–090”

–180° t . .I I I. .. . I

000° 090° 180°

Dobs.

Figure 23. The difference between the predicted and thegeostrophic wind, based on data from Bflrglum with ATystofte-KEGERI data, and data from center number 8

270° 360°

observed direction of theand B derived from the(BOULBA).

predicted and observer geostrophic wind.

1.

2.

34

The derivation of the geostrophic drag law is based on the assumption thatwe have steady-state wind situation where the temporal derivatives in theequation of motion can be neglected. This assumption is cert airily not alwaysfulfilled.

It is also assumed that the geostrophic wind is constant with height, i.e.that we have a barotropic stratification. However, there are many situationswhere this is not the case and the implication then is that G does not in

Ris@-R-l 145(EN)

an unambiguous way represents the dn”ving wind aloft to be related to thesurface wind.

The isobars may have strong curvature. This means that the geostrophic

wind calculation from three pressure stations does not represent the realgeostrophic wind inside the triangle, defined by the three stations. Conse-quently, it will be difficult to determine to which point in the triangle the

calculated geostrophic wind actually pertains. This problem is particularlypronounced when to comparing surface winds at a station which is close to

the edge of the triangle.

A strong curvature of the isobars may also introduce a centrifugal force in abalance with the pressure force and the Coriolis force. As we saw in section5 this three-force balance implies that the wind aloft is better represented bythe gradient wind than by the geostrophic wind. This means that in the draglaw in these cases should be based on the surface wind and the gradient wind.

The temperature stratification in the atmospheric surface layer has a stronginfluence on the parameters A and B. According to the discussion and reviewby Jensen et al. (1984) the variation of these parameters as functions ofU*/~/L, where L is the Monin-Obukhov length, is particularly strong nearneutral stratification ([L[ +- co).

Concluding Remarks

It has been the purpose of this report to describe an installation of six pressurestations in Denmark and to point out how the records often-minute averages fromt~ese six stations can be exploited to study the climatology of the free wind andhow it might vary geographically in Denmark.

The barometers are accurate and long-term stable within 0.1 HPa which at theEarth’s surface corresponds to the weight per unit area of a column of air with aheight of less than one meter. The positions and heights of the instruments haveconsequently been determined with a matching accuracy. We have even appliedgeometric equations pertaining to the local ellipsoid with the datum ED50 but,comparing to calculations using ordinary spherical geometry, this refinement seemsof very little importance as long as the temperature correction of the pressure toa common reference height is carefully carried out. A test showed that distancesof about 150 km can be off by 300 m. However, in this test the largest error was0.9 m/s in a situation where the geostrophlc wind speed was about 35 m/s andthe direction error never exceeded O.1°.

All six pressure stations were in operation on April 291998 and the recordingcontinues. There has been some major interruptions at Bane and at Ulborg (seeFig. 1), mostly due to power failure and the present preliminary data analysisdoes not include data after March 23, 1999.

Using the equations from section 3 we have studied the geostrophic wind usingten triangles with centers distributed over a large part of Denmark (Fig. 7). (Sixpressure-station positions can actually be corners in 20 different triangles, butmany of them are rather flat.)

For each triangle we have calculated the mean of the geostrophicirrespective of direction, and also in 12 direction sectors. Further,

Ris@-R-l145(EN)

wind speed,the Weibull

35

3.

4.

5.

8

parameters have been determined sectorwise and for all directions. Apparentlythe means are in general larger the further we go west and it also seems that thelargest means are found in the 300° sector.

We have also selected all the situations where at least one of the centers has itslargest value of the geostrophic wind. There were three such situations, two wherethe largest values were in Jutland (Figs. 9 and 11) and one where it was justsouth of Zealand (Fig. 10). The first two indicate that the directions of the largegeostrophic wind speeds were from SW to NW and that the magnitude of thespeeds decrease with increasing longitude. The third case seems more inconclusivewith small geostrophic wind speeds in Jutland and with no geographical uniformityin the directions.

If we want to take into account the curvature of the isobars we must include morethan three pressure stations. As discussed in section 5, the six stations are justsufficient to determine the six coefficients if we assume that the surface pressureis a second order polynomial in the horizontal coordinates. The data have beenanalyzed from this point of view and the geostrophic wind speed, calculated fromsix pressure records G6, has been compared to the geostrophic wind speed G3

based on the pressure records at the three stations Kegnas, Gedser, and Rkt$.There is a general agreement between G6 and G3 and between the directions aswell. Also the gradient ~ wind has been determined from the six pressure records.

We find, as expected, that most of the time G deviates from GG and Gs more thatthese deviate from one another. This of course is no real surprise, but it may beimportant for future analyses since it could be more reasonable to use ~ ratherthan G as the external forcing of the surface wind.

The surface data at Tystofte has been used together with the pressure measure-ments at Kegmes, Gedser, and Rk@ for reviewing the dimensionless constants Aand B in the geostrophic drag law. We found that when the friction velocity isgreater than 0.6 m/s, corresponding to a wind speed greater than about 8 m/s

at the height 10 m over a uniform terrain with the roughness length 5 cm, thenA = 0.5 and B = 3.5 are in good agreement with data. This should be compared tothe values A = 1.8 and B = 4.5 used routinely for neutral stratification in WAsP.Taking this discrepancy as a measure of uncertainty, (100) and (102) imply thatthe uncertainties in predictions of the ageostrophic angle cr and the geostrophicwind speed are about 5° and 1070, respectively.

With the values A = 0.5 and B = 3.5 we predicted the geostrophic wind fromthe Tystofte data and from the B@rglum data. The first prediction is much betterthan the second, primarily for two reasons: the values of A and B were found onbasis of the data we are using for the prediction in the Tystofte case, and B@rglumis at the corner of the very triangle we use to determine the geostrophic wind.

Ultimately, we want to carry out an extreme-wind analysis on the geostrophicwind and the geographical variation of the 50-year event, similar to the analysis

by Kristensen et al. (1999). However, the records we have so far are too short induration to make such an analysis meaningful. From that point of view, we hopethat the measurements will continue for at least 10 years.

We suggest that more pressure stations are put into operation. If we added pressurestations at say Helgoland in the North Sea, on the west side of Oslo Fjord, atBornholm south of Sweden in the Baltic Sea, we would eventually “be able todetermine the geostrophic wind climate all the way to the borders of Denmark ascan be seen by inspecting Figs. 1 and 7. Even with the records we have now and

in the near future we find that the quality of the data is so good that it might berewarding to let it form the basis of a Ph.D. study.

36 Ris@-R-l145(EN)

Acknowledgements

This project was funded by the EFP-project ENS–1363/97–O04. Our colleagues

Peter Kirkegaard, Jakob Mann, and Ole Rathmann have been helpful reviewingthe equations. The last also provided the WAsP terrain description at Tystofte

and B@rglum. Arent Hansen has constructed the very efficient pressure stationsand is to a large extent responsible for their smooth operation.

Rk@-R-l145(EN) 37

References

Dutton, J. A. (1986). The Ceaseless Wind: An Introduction to the Theorg ofAtmospheric Motion, Dover Publications, Inc, Mineola, NY 11501.

Jensen, N. O., Petersen, E. L. & Tkoen, I. (1984). Extrapolation of MeanWind Statistics with Special Regard to Wind Energy Applications, WCP-86, WiVIO/TD-15, World Meteorological Organization.

Kristensen, L., Rathmann, O. & Hansen, S. O. (1999). Exstreme Winds in Den-

mark, R–1 068(EN,,, Ris@ National Laboratory.

Mortensen, N. G., Landberg, L., Troen, I. & Petersen, E. L. (1993). Wind Analysisand Application Program WAsP, I–666(EN), Ris@ National Laboratory. Vol2: User’s Guide.

Panofsky, H. A. & Dutton, J. A. (1984). Atmospheric Turbulence: Models andMethods for Engineering Applications, John Wiley & Sons, Inc., New York.

Rasmussen, L. M. (1996). Introduction til Grafiske Informationssystemer (In-troduction to Geographical Information Systems, FOFT F-9/1996, DanishDefence Research Establishment. In Danish.

Tennekes, H. (1982). Similarity relations, scaling laws and spectral dynamics,in F. T. M. Nieuwstadt & H. van Dop (eds), Atmospheric Turbulence andAir Pollution ModeZling, D. Reidel Publishing Company, Dordrecht, Holland,Boston, U.S.A., London, England, chapter 2, 37-68.

Tennekes, H. & Lumley, J. L. (1978). A First Course in Turbulence, The MITPress, Cambridge, Massachusetts, and London, England. Fifth printing.

38 Rls@-R-l145(EN)

A Weibull Parameters

Here we present the means (G) and the YVeibu]l parameters k and A for each 30°sector and all directions at the ten centers.

Table 8. Center O, BO ULKE.

DG

000°030°060°

090°120°150”180°210”240°270°

300°330°

000° – 330°

Table 9. Center 1, BOULBA.

DG

000°030°060°090°120”150°180°210°240°270°300°330°

000° – 330°

{G)

8.6127.840

9.3388.7648.171

10.67412.59313.09612.60212.27514.92512.766

11.927

(G)

9.5428.8288.9279.347

9.38610.79010.73311.54012.27113.36415.82211.53612.003

k

2.098921.913682.02168

1.999022.304832.090301.710611.851022.429482.205002.053961.894491.86121

k

2.068591.702262.389662.318672.406221.813281.851022.047592.149862.117912.180101.811381.90086

A

9.7238.837

10.539

9.8899.223

12.05114.12014.74514.21213.86016.84814.385

13.431

A

10.7729.895

10.07110.550

10.58712.13812.08413.02613.85615.08917.86612.97613.527

Rk@-R-l145(EN) A-1

Table 10. Center 2, ULKEBA.

DG

000°

030°060°090°120°150°

180°210°240°270°300°330°

I 000° – 330°

Table 11. Center 3, BOULRI.

DG

000°030°060°090°120°150°180°210°

240°270°300°330°

000° – 330°

(G)

8.672

8.1647.920

9.9198.2909.857

10.79712.06013.14712.51114.117

11.67111.548

(G)

9.6428.4948.5698.9669.184

10.46910.08111.120

12.45713.56915.05110.41211.713

k

1.65015

1.776971.754722.02088

2.106991.997121.816261.831192.190872.252222.155171.936841.89547

k

1.968401.954392.448842.406222.436131.972021.753412.05894

2.230832.008962.087901.587901.87250

A

9.698

9.1748.894

11.194

9.36011.12212.14613.57214.84514.12515.940

13.16013.013

A

10.8769.5809.662

10.11410.35711.81011.32012.553

14.06515.31216.99311.60513.193

A-2 Rk@-R.-ll45(EN)

Table 12. Center ~, ULKERI.

1--’-000°030°060°090°120°150°180°210°240°270”300°330°

000° – 330°

Table 13. Center 5, BOKEBA.

DG

000°030°060°090°120°150°180°210°240°270°300°330°

000° – 330°

(G)

8.8127.2507.6969.7408.6778.354

9.44410.92312.52111.56412.34810.72710.590

(G)

13.3019.7067.633

9.99510.48310.1079.943

11.04312.63413.50214.17513.77112.071

k

1.71892

1.627612.012391.994732.134062.037621.907361.971002.094642.285032.158951.806071.92241

k

1.768771.932262.026682.627702.286402.200242.030151.980792.330112.315962.148622.026012.02597

d

9.883

8.0998.685

10.9909.7989.429

10.64412.32214.13713.05413.94312.06411.938

A

14.94310.9438.615

11.24911.83411.41211.22212.45814.25915.23916.00615.54213.624

Ris@-R-l145(EN) A-3

Table 1,/. Center 6, BAKEGE.

DG

000°030°060°090°120°150°180°210”240°270°

300°330°

000° – 330°

Table 15. Center 7, BAKERI.

DG

000°030°060°090°120°150°180°210”240°270°

300°330°

000° – 330°

(G)

9.2177.4167.323

9.4579.9609.4239.3989.838

12.88113.38913.403

9.84211.226

(G)

10.679

8.1567.524

10.01410.339

8.7549.221

10.91514.32213.35513.20411.05511.681

k

1.720741.586751.817902.212222.186462.030661.778801.732292.203392.128562.209731.844311.90040

k

2.000321.769481.858002.386312.397292.100511.814511.835542.068632.048202.268201.814971.90779

A

10.338

8.2658.239

10.67811.24610.63510.56211.04014.54415.11815.13411.07912.651

A

12.0509.1638.472

11.29711.6639.884

10.37312.28516.16815.07514.90612.43613.165

A-4 RW-R-1145(EN)

Table 16. Center 8, KEGERI.

DG

000°030”060°090°120°150°180°210°240°270°300°

330°000° – 330°

Table 17. Center 9, BAGERI.

D~

000°030°060°090°120°150°180°210°240°270°300°330°

000° – 330”

(G)

8.811

6.9697.8079.1779.4649.0308.9679.730

11.55112.52112.947

9.94110.556

(G)

10.8328.5647.747

9.72510.395

9.6629.592

13.47213.10712.40913.18912.82411.726

k

1.60263

1.521061.599632.058322.333312.161991.850861.775872.157392.159712.00338

1.847771.88320

k

1.779121.990111.99874

2.528002.391812.113141.812361.945132.022312.242552.086941.727511.92551

d

9.8297.7328.707

10.36010.68110.19610.09610.93413.04314.13814.61011.19211.892

A

12.1739.6638.741

10.95811.727

10.90910.79015.19214.79214.01014.89014.38813.220

Rk@-R-l145(EN) A-5

Bibliographic Data Sheet Ris@–R–l 145( EN)

Title and author(s)

Geostrophlc Winds in Denmark: a preliminary study

Leif Kristensen and Gunnm Jensen

ISBN ISSN

87-550-2616-8 0106-2840

Dept. or group Date

Department of Wind Energy and Atmospheric Physics November 22, 1999

Groups own reg. number(s) Project/contract No.

1105021-04 ENS-1363 /97-OO04

Pages Tables Illustrations References

43 17 23 8

Abstract (Max. 2000 char.)

High-precision barometers have been deployed at six sites in Denmark, four west

and two east of the Great Belt. The purpose is to establish long climatological

records of the geostrophic wind as a supplement to the records of tens of years

of duration of surface observations of wind, temperature, humidity etc., which

have been obtained by Rls@ at many sites in Denma-k. Three of these sites are in

principle sufficient to determine an average of the magnitude and direction of the

geostrophic wind inside the triangle formed by the three sites. Ten, out of twenty

possible, triangles have been selected as suitable for studying the geographical

variations of the geostrophic wind. A tentative conclusion from about one year

of data is that statistically the geostrophlc wind decrease in magnitude when go-

ing from west toward east. The data also showed that the largest mean values

of the geostrophlc mean wind speed are in a direction sector from 285° to 315°.

The Weibull parameters were calculated for all ten triangles. The curvature of the

isobars were determined by using simultaneous pressure measurements at all six

sites and the geostrophic and gradient winds were calculated and compared to the

geostrophic wind based on three pressure measurements in one particular triangle.

Combining the geostrophic wind with the surface wind measured at Tystofte in

southern Zealand, the two dimensionless constants A and B in the geostrophicdrag law were determined as functions of the surface friction velocity. These datasuggest that A = 0.5 and B = 3.5. The surface data at Tystofte and at B@rglumin Vendsyssel in northern Jutland were used to predict the geostrophic wind byapplying the geostrophic drag law with these constants and the predictions werecompared to the observed geostrophic wind.

Descriptors INIS/EDB

ATMOSPHERIC PRESSURE; BAROMETERS; DENMARK; DR.AG; GEODESY;

VELOCITY; WIND

Available on request from:Information Service Department, Ris@National Laboratory(Afdelingen for Informationsservice, Forskningscenter Ris@)P.O. Box 49, DK–4000 Roskilde, DenmarkPhone (+45) 46774677, ext. 4004/4005 . Fax (+45) 46774013