U.S. DEPARTMENT OF COMMERCENATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION

NATIONAL WEATHER SERVICENATIONAL METEOROLOGICAL CENTER

OFFICE NOTE 222

A Noteon Tuning the Gradient Wind RelationCurrently:.Used in the LFM

Edward A. O'LenicDevelopment Division

AUGUST 1980

This is an unreviewed manuscript, primarilyintended for informal exchange of informationamong NMC staff members..

Introduction

An effort to choose a wind relation to calculate winds from heightsfor use in the Optimum Interpolation system (O/I) in regions of sparsewind observations, has prompted the evaluation of various wind relationsduring the past year. The simplest wind relation to incorporate curvatureeffects is the gradient wind equation, a modified version of which isnow used in the analysis portion of the LFM (Brown, 1971).

Background

The gradient wind relation under investigation may be derived fromthe invariant form of the equations of motion in component form, inpressure coordinates,

au. a ; 0 - -.- u ' fa V to +u2, yv)2 (c-if)v +. '.patp + ax($ + 2...a a -'- V2 P ()=

at + .SP~(p + u +~ 2)1 + (0C+f)u + a 2 2

-HP~~~~~~~~~~-Dv ~ ~ ~ ~ ~ ~ ~ ~~~rs =' 0

at + 7-7+ ¥+ + (~'+flu + DP ~2 p

where 0 = gZ

g = 9.8 ms - 2

Z = height of pressure surface

u = eastward component of wind

v = northward component of wind

= relative vorticity of wind

= av -- :au

Dx ayy

f = coriolis parameter

= dp/dt

Assuming W -terms are negligible and letting / W =X fvg, and$/DY= -fug, where ug and vg are the geostrophic wind components,

_2 + v z2.

andK = 2 ,we get

au + fVg + (aK - v) - fv = 0DXIt~~ 0 ax ~~~~~(2)

vx- fug + (UK + uO) + fu = 09 y.

Du , vAssuming that the horizontal accelerations At, a t,may be absorbed into

a constant factor, kl, applied to the non-linear term, and further, that

vorticity and kinetic energy may be approximated by their geostrophiccounterparts, we get

~'= Ug- 1 Z( + g g)

(3)klk 2 aKg

V*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .

= Vg + f (X Vg.g)

where u*, v* = gradient wind components,

Sin L - Sin 20k2 = Sin 30 - Sin 20, 0 < k2 < 1 for 20° < L < 30 °

*

k2 is a latitude weighting parameter which has been inserted in equations(3) to facilitate a gradual transition from gradient winds northward of300N, where k2 = 1, to guess winds south of 20°N, where k2 = 0. The

parameter kl, is the subject of the efforts described in the nextsections. In vector form, equations (3) become

+klkk+ k xVKg - v ) (4)

0:~~~~~~ :Procedure

* Seven synoptic cases were chosen from which to compute gradient

winds (Appendix A). For each case, 10 sets of winds, each using adifferent value of kl, were calculated and verified against observationsusing the SUMAC verification code. Verification statistics were compiledfor the 110 station NAllO area and the 96 station EUR96 area (Figure 1).

Results

Graphs of RMS vector wind deviation and bias (gradient versus observed)versus k1 (Appendix B) show that, while the RMS generally reached aminimum for 0.10 < k1 < 0.25, the gradient winds exhibited a negative

bias which decreases monotonically with increasing k1. Further, thisbias has its largest magnitude an most rapil rate of decrease at 250 mb,where it varies from about -1 ms to -3 ms for 0.05 < k1 < 0.33. Thisbehavior of the bias indicates that the non-linear term, Xk x VK- gg,

was negative below 100 mb for all values of k1 for all 7 cases in thetest. Thus, the effects of cyclonic curvature apparently dominated the

* NOTE: Hough height analyses were used.

KAr"' . .

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..... ~.- .....

: · f, . 0 t . .- . : ; . ' : f X 0 t: DD~ ~ ~ ~ ~ ~ ~~~. '.* .... ... , . , , ..': \.'t0 ,0 ti00 : ;0 .- ,0S;f; :0f :f 0 : fffEd

Filg u re 1 .S UM AC ve rif ic at i on ar~e as N AllO a nd EURPs9 6.N . 0 . .

results of this test. Ideally one would want to choose cases in whichcyclonic and anticyclonic curvature were distributed in equal amountsamong troughs and ridges, respectively. Equation 4 implies that, forsuch cases in the northern hemisphere, the gradient wind exceeds thegeostrophic wind in ridges to the north of the jet, and lags the geo-strophic wind in troughs to the south of the jet (Figure 2). Graphs ofgradient wind RMS error and bias were also plotted (Appendix B) for the96 station EUR96 area (Figure 1). The upper air height gradients inthis area during May were weak in contrast to those over NAl110. The RMSvector wind deviation and the bias of the gradient winds were considerablylower over EUR96, and the rapid decrease in bias with increasing k, observedover NA110, is only weakly present.

In view of the tendency for the RMS vector wind deviation (gradientvs observed) to reach a minimum for 0.10 < k1 < 0.25, along with the large

increase in negative bias for increasing values of k1, the value k1 = 0.15at all mandatory levels from 850 to 100 mb was chosen as the optimumvalue of k1. As a test, winds were computed from a 12Z 21 October

1979 Optimum Interpolation (O/I) height analysis. To verify that thischange in k1 improves the details of the wind fields, winds computedusing the old values, k1 = 0.2, 0.2, 0.3, 0.3 at 850, 500, 250, 100

mb, respectively, are shown in Figures 3 and 4, while winds computedusing k1 = 0.15 at each level are displayed in Figures 5 and 6. Observa-tions are shown in Figure 7 and 8.

Comparing the gradient winds (Figures 3-6) with observations(Figures 7, 8) reveals that the gradient winds reproduce the overalldetails of the observed wind field quite well. Further comparison how-ever, reveals that winds computed using k1 = 0.30 (at 250 mb) were consis-tently weaker than those observed in troughs by from 20 to 110 percent,and stronger than observations in ridges by up to 40 percent. Thesecharacteristics are most marked in the trough-ridge-trough pattern extend-ing from 45N, 145W into the western United States (Figure 3), in thebroad trough over north central Europe (Figure 4). Notable differencesbetween gradient winds and observations in the opposite sense occur inthe ridge over the eastern U. S. (Figure 3), in the short-wave ridgenear Iceland (Figure 4), where gradient winds are slightly weaker thanobserved, and in the trough over Tripoli (Figure 4), where gradientwinds are stronger than observed. Gradient winds calculated using k1

= 0.15(Figures 5, 6) improve upon those calculated using k = 0.30 mostnotably in the trough-ridge-trough pattern west of 100W, in the troughnorth of Cuba (Figure 5), and in the trough along 30W, southeast ofGreenland (Figure 6). Changing k1 from 0.30 to 0.15 also reduces thewinds in the ridges, but only 5 to 10 percent in most cases, and thenonly in sharp ridges. Large scale features, such as the broad troughover north central Europe (Figures 4, 6) remain unchanged by this changein k1.

Vcsas' \ 4VRAb I

6LV b

Figure 2. Physical interpretation of equation 4.

j:6r80

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12Z 21 October' 197 A 46A~-4U

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E-wr op e,,

12Z ~21 October, 1979

Summary and Conclusions

A version of the gradient wind law currently in operational use atNMC has been studied in order to assess the usefulness of modifying oneof its parameters, and its possible usefulness for providing winds tothe 0/I. A series of tests indicated that the RMS vector wind deviation(gradient vs. observed) for winds computed from this relation reached aminimum for values of the parameter k1 of 0.1 < k1 < 0.25. The 250 mb

mean speed error, or bias, was negative for all cases in both the NA110and EUR96 areas. Further, this bias becam¥ increasingly negative withinc easing kl, varying from about -1.0 ms for k = 0.05, to -3.0ms for k = 0.33. Considering these RMS and bias variations, itwas decided that a value of k1 = 0.15, instead of the value k1 = 0.30,

currently used at 250 mb in the LFM, might result in improved gradientwinds, especially with respect to their mean speed bias. Comparisons ofwinds calculated using both values of k1 with each other, and withobservations, confirmed that winds calculated using the updated k1brought wind speeds in most troughs into better agreement with observa-tions, with only slight decreases in wind speeds in ridges. This certainlycan be regarded as an improvement, at least for the 12Z 21 Oct 79 case.

Table 1 lists the value of k1 currently used in the LFM and gives asuggested list of updated values for all standard levels based on theresults of this study.

Table 1. Current and Suggested Values of Parameter k1 in LFM

Pressure level (mb)

1000

850

700

500

400

300

250

200

150

100

70

50

Current Value

0.0

Suggested Value

0.0

0.1

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.3

0.1

0.1

0.15

0.15

0.15

0.15

0.15

0.15

0.1

0.1

0.1

References

Brown, John, A., 1971:Numerical PredictionAugust 1971.

New Initialization Procedure for the 6-Layer (PE)Model. Technical Procedures Bulletin No. 65,

Acknowledgements

I wish to express appreciation to A. J. Desmarais for providingboth a coded version of the gradient wind equation, and the impetus tocarry out this study. Conversations with Dr. John A. Brown clarifiedmany questions. John R. Ward supplied the height-analysis for the 12Z21 October 1979 case.

APPENDIX A

250 mb Height-Isotach Analyses

N

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APPENDIX B

RMS vector wind deviation and bias from SUMAC verifications(gradient winds vs. observed winds)

- - . ~~~~~~~~~~~~~~~~ I K ~~~~~~~~~~~~~, ....... ~~~~~~~~~~~~~~~~~~~~~.....-

JAI_: -NA.

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