Factors determining the decrease in surface wind speed following the evening transition

Transcription

1 Q. J. R. Meteorol. Soc. (2003), 129, pp doi: /qj Factors deterining the decrease in surface wind speed following the evening transition By A. LAPWORTH Met Of ce, Cardington, UK (Received 29 July 2002: revised 16 January 2003) SUMMARY Meteorological easureents were ade by surface instruentation at a at rural site over a period of several years. These are analysed to deterine any systeatic factors governing the decrease in surface wind speed during the evening. The 10 wind speed is found to depend on the surface cooling and gradient wind speed, with a larger decrease in the 10 wind speed with surface cooling when the gradient wind is stronger. The relationships are relatively unaffected by cooling rate. Moentu and heat uxes are also found to have well de ned relationships with surface cooling and gradient wind, with downward heat uxes reaching a axiu at a 2 degc cooling. These observations are copared with the predictions of a siple sei-analytical odel which gives qualitatively siilar results to the observations and shows that the greater 10 wind speed decrease at higher gradient wind speeds is due to a thicker boundary layer. The odel also predicts the foration of an upper inversion with an increase in 10 wind later in the evening and this prediction is copared with balloon and surface observations ade on a clear evening. A siple forecasting rule for the reduction in 10 wind during the evening is given. KEYWORDS: Forecasting rule Stable boundary layer 1. INTRODUCTION The reduction in surface wind speed and turbulence that coonly occurs during the evening is well known and any theoretical studies have been ade of the associated stable atospheric boundary layer (e.g. Nieuwstadt 1985; Derbyshire 1990). These studies have tended to concentrate on the fully stabilized boundary layer and relatively little experiental or theoretical work has been concerned with the ore practical question of how uch the surface wind and turbulence will have been reduced at a particular stage in the evening. Given the interittent and less predictable characteristics of stable layers copared with those of convective conditions, it is not iediately clear whether any useful guidance in answering such a question is, in fact, possible. The work described here is directed at deterining fro observations whether any systeatic relationships exist that ay give soe predictability to the evening variations in surface wind and to copare the relationships found with the results of a siple odel to give soe insight into the echaniss involved. In the rst section the surface site observations are described and an account is given of their analysis in order to deterine systeatic relationships. This is followed by a description of a sei-analytical odel of the growing nocturnal boundary layer and the results fro this are copared with the easureents and also with pro les fro a nite-difference odel. 2. FIELD MEASUREMENTS AND SITE DESCRIPTION The surface observations were ade over a period of ve years at an inland instruented eld site near Cardington, Bedfordshire in the east idlands of England (latitude 52 ± 06 0 N, longitude 00 ± 25 0 W). The surrounding country is fairly at arable land with relatively large elds and soe sall trees scattered along eld boundaries. Corresponding address: Met. Research Unit, Field Site, Cardington Air eld, Shortstown, Bedfordshire MK42 0SY, UK. e-ail: alan.lapworth@etof ce.co c Crown copyright,

2 1946 A. LAPWORTH There is a 100 high ridge running north-east to south-west passing 4 k to the southeast of the site. However, the ain obstructions to near-surface air ow are two large hangars 50 high situated 400 north of the site. Wind easureents were ade by an asyetric Gill Solent sonic aneoeter ounted at the top of a 10 ast, and screen teperatures were easured by an ASL platinu resistance theroeter ounted in an aspirated radiation shield at a height of 1.2. The instruent outputs were digitized and transitted at 4 Hz to a logging coputer where the data was blocked into 15-inute segents for purposes of calculating the ean and turbulent quantities. 3. ANALYSIS OF THE OBSERVATIONS For initial analysis, the data was quality controlled and observations that had been ade when the wind direction was in the sector fro 350 ± to 040 ± covered by the hangars were reoved. The data were then plotted in 24-hour segents and relationships were studied between the 10 wind and a nuber of eteorological variables, including teperatures at various heights, sensible-heat uxes, net radiation and wind direction. The strongest correlations were found between the 10 winds and near-surface teperatures, with the ean correlation coef cient for 15-inute eans of wind and teperature over 24 hours being around The correlation coef cient increased to around 0.7 if 30-inute eans were used. By far the ajority of variations contributing to the correlation were diurnal, although in a nuber of cases strong correlations were the result of frontal passages. If the observations were restricted to stronger daytie winds (axiu winds greater than 8 s 1 ) and greater diurnal teperature ranges (over 10 degc), then the average correlation coef cient rose to over 0.9. Figure 1 shows the diurnal variation of 10 wind and screen teperature on a nuber of days when the correlation coef cient was greater than This high correlation iplies a statistically near-linear relationship between wind and teperature and subsequent investigations were directed towards deterining the factors governing this relationship. Although the correlations iply a linear relationship for a particular day, the actual relationship (in particular the gradient of wind speed versus teperature) varies fro day to day and various dependencies were tested to deterine the cause of this variation. A study of 10 wind-speed screen-teperature plots con red that the relationship was fairly linear and gave the strong ipression that the gradient of the plot was higher on days when the geostrophic wind speed was high. Statistical plots using averages of all the data were ade which con red that the ain variable affecting the 10 wind-speed screen-teperature (U 10 T 1 ) relationship was the geostrophic wind speed which had the surprising effect of increasing the U 10 T 1 gradient as the geostrophic wind speed itself increased. This nding was con red by dividing all days into three ranges of axiu daytie wind speed U ax which radiosonde intercoparisons showed to be directly related to the geostrophic wind. Days were eliinated in which the correlation between teperature and sensibleheat ux was less than 0.3 as this was taken to indicate signi cant advection. Days were also eliinated on which the boundary layer was not convective at soe tie during the day these ainly occurred during the winter. For each wind-speed band an average plot of 10 wind speed versus screen teperature was constructed by referencing all daily screen teperatures to the evening transition teperature at which tie the atosphere is assued to be approxiately neutral with surface teperature T trans. Therefore, to plot the average curves, T 1 was replaced by 1T 1, where 1T 1 D T trans T 1. The transition teperature was taken as the teperature at which the sensible-heat ux at 10 becoes negative. All points were then binned at nite intervals along the teperature

3 EVENING SURFACE WIND SPEED Jul Aug Jan 2001 T U T U T U U,T U,T U,T UTC UTC UTC 10 Jun 1997 T U 04 Jun 1999 T U 20 Jun 1999 T U U,T U,T U,T UTC UTC UTC Figure wind (U) and teperature (T ) tie series for six days. axis. The standard deviation of the ean value of the points in each bin was calculated assuing a noral distribution and was plotted as error bars. The resultant average U 10 1T 1 curves for all three wind-speed bands are shown in Fig. 2. It should be noted that in this gure the points are well de ned because the large nuber of cases reduces the variance in the eans. The actual variance of the averaged points is of the order of 1 s 1. In the gure it can be seen that these ean plots are approxiately linear and that the slope of the curve of 10 wind speed versus screen teperature is greater for higher gradient winds. This is a surprising result as it indicates that surface cooling has a greater effect on higher gradient wind speeds. This does not necessarily iply that the surface wind will drop ore quickly with a higher geostrophic wind speed. As will be seen, the cooling rate is an independent variable which is usually lower at higher wind speeds, the increased turbulent uxes acting in opposition to the cooling of the radiative ux. This nding was con red by deterining the 10 =@1T 1 of linear ts to the U 10 1T 1 curve on the individual days in the study period and plotting these gradients against the axiu 15-inute daytie wind speeds U ax. The results are shown in Fig. 3 where the points have again been binned at nite intervals along the axis of the ordinate. This gure shows that over the range of wind speeds easured, there is an alost linear relation between the 10 =@1T 1 and daytie axiu wind, with increasing wind speed giving larger gradients. In this gure, larger errors are seen at higher wind speeds because there are fewer such cases. In order to deterine whether the U 10 1T 1 curves are cooling-rate dependent, the results were replotted with the days further divided into two bands of cooling rate. The cooling rates were deterined as the ean rates over the rst 4 hours of cooling after transition. The U 10 1T 1 plot for the iddle velocity band is shown for bands of the highest and lowest cooling rates in Fig. 4. It can be seen that the plot is very siilar for the two cooling rates, and this is also true for the lowest wind-speed band. For the

4 1948 A. LAPWORTH 8 6 U =9.5 s -1 U =6.4 s -1 U =3.9 s -1 4 U s T-T trans deg C Figure 2. Mean curve of 10 wind speed (U) versus screen-teperature fall after transition averaged over several years for three axiu daytie wind-speed bands (U, s 1 ). The error bars shown are for the variance of the ean values. 0.8 du/dt s -1 O C U ax s -1 Figure 3. Mean curve of 10 wind speed (U) versus screen-teperature gradients averaged over several years plotted against axiu daytie wind speed (U ax ).

5 EVENING SURFACE WIND SPEED O C h O C h -1 4 U s Figure T-T trans deg C Mean curve of 10 wind speed (U) versus screen-teperature fall after transition for one coon axiu daytie wind-speed band averaged over several years for two cooling-rate bands. highest wind-speed band, there appear slight differences, with the gradient decreasing by up to 10% for the highest cooling rates. These results con r that surface teperature is a far ore signi cant paraeter than tie in deterining the 10 wind speed during the evening. In effect, the boundary-layer wind appears to be in local equilibriu with the local teperature and geostrophic wind. Given the relatively well de ned relationship found to exist between the 10 wind and surface teperature during the evening cooling, plots were also ade of the turbulence uxes for oentu, heat and oisture against screen teperatures during this period. The plots are shown in Figs. 5, 6 and 7 for the sae three wind-speed bands used previously. Those for oentu and heat ux show well de ned curves which are well separated at different gradient wind speeds, as ight be expected. The oentu uxes fall away to a iniu as surface teperature falls, and if the friction velocity u, rather than oentu ux, is plotted against 1T 1 then the plot is linear and is very siilar to the U 10 1T 1 plot, falling to near zero at 1T 1 D 14 degc. The negative heat ux rises fairly sharply to a axiu at around 1T 1 D 2 degc and then falls. The initial rise is presuably related to the increase in teperature gradient and the associated initial rise in the surface teperature scale T, while the fall in surface friction velocity, u, accounts for the nal decrease in heat ux which is proportional to the square of u. The ux of oisture plotted against cooling has large statistical errors and the relationship is not well de ned in ters of wind speed, probably because oisture gradients are relatively independent of wind speed and teperature and so are unaffected by regularities forced by the dynaics and therodynaics. A signi cant upwards oisture ux appears to continue after the evening transition, which only becoes negative when the surface has cooled by ore than 8 degc. The surface Monin Obukhov length is also plotted against screen teperature in Fig. 8. This falls

6 1950 A. LAPWORTH Figure 5. Mean plots of oentu ux (u 0 w 0 ) versus screen-teperature fall after transition averaged over several years for three axiu daytie wind-speed bands (U, s 1 ). Figure 6. Mean plots of heat ux (w 0 T 0 ) versus screen-teperature fall after transition averaged over several years for three axiu daytie wind-speed bands (U, s 1 ).

7 EVENING SURFACE WIND SPEED 1951 Figure 7. Mean plots of oisture ux (w 0 q 0 ) versus screen-teperature fall after transition averaged over several years for three axiu daytie wind-speed bands (U, s 1 ). 800 U =9.5 s -1 U =6.4 s -1 U =3.9 s -1 LMO T-T trans deg C Figure 8. Mean plots of Monin Obukhov length (LMO) versus screen-teperature fall after transition averaged over several years for three axiu daytie wind-speed bands (U, s 1 ).

8 1952 A. LAPWORTH draatically with initial cooling after the evening transition. The statistical errors are large near neutrality but are reduced as cooling increases. As there was a fairly well de ned evening relationship between surface wind speed and teperature, an investigation was also ade of the effects of surface waring during the orning. It was found that a very siilar relationship existed between 10 wind speed and screen teperature to that found during the evening, but with slightly different gradients. However, this relationship could only be obtained if the screen teperature was referenced to the evening transition teperature, as de ned by the surface ux change criterion, rather than the orning transition teperature. The orning ux teperature relationships differ copletely fro those found during the evening as a surface convective layer fors under the stable layer at an early stage of waring. 4. MODEL In order to investigate the echaniss underlying the ain qualitative features of these results, a siple two-layer odel of the evening boundary layer after transition was constructed. This odel is dynaical and does not consider radiative effects. A description of the concepts on which the odel is constructed will rst be given, and this is followed by detailed derivation of the relevant equations. During the late afternoon and evening of a convective day, the atosphere is no longer heated by the surface but starts to cool. Initially the boundary layer becoes neutral with an equilibriu height scale H e given by: H e D 0:2 u f where u is the friction velocity, and f is the Coriolis paraeter. As the surface cools further, transition occurs and an inversion fors close to the surface and with further cooling its top starts to rise as shown in Fig. 9. The equilibriu value of this inversion top height h e is shown by Yaada (1979) and Nieuwstadt (1980) to be: (1) h e D 2.w0 µ 0 / s : s =@t In this equation µ is the potential teperature, t the tie, w the vertical coponent of wind velocity and the subscript s is used to denote the values of variables at the height of the surface. The pried values are turbulent deviations fro the ean values and the overbar indicates an average over tie. The iportance of this scale is that downward heat ux w 0 µ 0 can only take place within it as it contains the necessary vertical teperature gradient. The boundary layer now consists of two layers, a lower stable layer which is increasing in thickness and an upper neutral layer as shown in Fig. 9. Soon after transition both layers are turbulent, with different turbulent height-scales in each. The botto stable layer has an associated turbulent height-scale that Zilitinkevich (1972) showed is given by: H e D 0:4 s u L s f where L s is the Monin Obukov length-scale in the surface layer given by: (3) L s D µ 0u 2 kgµ (4)

9 EVENING SURFACE WIND SPEED 1953 Figure 9. Scheatic diagra of odel pro les. See text for further explanation. where µ 0 is the ean potential teperature of the layer, g is the acceleration due to gravity, k is von Karan s constant and µ is the turbulent potential-teperature scale given by: µ D w0 µ 0 u : (5) The iportance of the scale H e is that the boundary layer tends to be turbulent with non-zero u 0 w 0 below it and cal above. It will be seen that H e is initially greater than h e, but that subsequently, as indicated in Fig. 9, h e tends to grow as the downward heat ux increases while H e decreases with increasing stability in the lower layer. Eventually the two heights will becoe equal and at this point the inversion top is no longer able to grow because there will be no turbulent conduction available to transport heat downwards fro above the inversion. Instead, the inversion strength will increase at the top and as will be described later, an increase in surface wind and hence u stops the height-scale H e fro decreasing further and a sharp inversion fors at this height as shown in Fig. 10. During the initial period after transition the ain wind shear will occur near the surface as the ixing length is liited there by the sall height-scale. As the surface inversion grows and strengthens, the ixing-length scales will start to decrease with the local Monin Obukov scale L at increasingly greater heights above the surface. Therefore, there will be increasing wind shear at these levels, decreasing the nearsurface wind values. It will be shown fro the odel results in section 5 that this process of surface wind reduction is ore effective at higher geostrophic wind speeds because a thicker inversion and hence greater total wind shear fors in these cases which accords with the observations. When the upper inversion fors, the wind shear in the lower boundary layer becoes constant or decreases as described in section 5 and the surface wind becoes constant or increases as shown in Fig. 10. To ipleent the odel, local siilarity with assued pro les of u and µ will be used to deterine values of wind speed U and potential teperature µ within each of the two layers. The pro les of u and µ are chosen to give continuity at the top of the surface inversion. The pro les of the associated uxes are shown in Fig. 9. For the case

10 1954 A. LAPWORTH Figure 10. Scheatic diagra of odel pro les during foration of a sharp upper inversion. See text for further explanation. that h becoes equal to H, conservation of heat is used to deterine the teperature discontinuity at the inversion top. Forulae for the non-equilibriu values of the surface inversion top h and stable-layer height-scale H will be quoted fro past results. U and µ will then be atched at the interface to give forulae for surface values of u and µ and pro les of all variables can be derived fro these. This nal set of equations is solved iteratively. In the rst stage expressions will be derived for U and µ. During evening cooling of the surface the assuption will be ade that the atosphere is in a state of quasi equilibriu so that siilarity-based ixing-length relations can be applied. Using functions fro standard ixing-length theory, velocity and teperature gradients for the surface inversion D u 1 C 5z (6) D µ kz 1 C 5z L where z is the height above the surface. Equations (6) and (7) are local equations that is they use local values of u, µ and L. Near the surface (z < L=5) they express surface siilarity. However, at heights z À L=5 it can easily be shown that they siplify to Nieuwstadt s (1985) expressions giving the ux Richardson nuber and gradient Richardson nuber as constants (D 0.2) in the upper boundary layer. Therefore, in this odel these equations will be taken to apply over the whole depth of the boundary layer. In the top neutral layer: D u =k.z h d/ D 0: (9) Here d is the displaceent height relative to the inversion top and is expected to lie between 0 and h. Tests have shown that the results are very insensitive to the actual value of this ixing length over a very large range and for the present results a value

11 EVENING SURFACE WIND SPEED 1955 equal to h=4 was used. In this layer a linear pro le will be assued for u, falling away to zero at z g, where the subscript g denotes the value of the variable at the height of the geostrophic wind. Again the results are very insensitive to the pro le chosen. In order to integrate these functions to deterine U and µ, assuptions ust be ade about the variation of u and µ with height. Both of these should decrease with height, and in order for µ to be continuous across the top of the surface inversion, it will becoe zero there as µ is zero in the layer above. u should tend to zero at the boundary-layer scale height appropriate to the layer concerned. Initially this height will be very uch greater than the height of the top of the surface inversion. For siplicity the vertical variation of both these quantities will be taken to be linear. As will be seen below, pro les taken fro a one-diensional nite-difference odel show that these assuptions are not unreasonable and it is unlikely that oderate departures fro linearity will affect the essentially qualitative nature of the results obtained. A scheatic diagra of the wind-speed, teperature and ux pro les of the odel is shown in Fig. 9. We then have: ± ² u D u s 1 z ± H µ D µ s 1 z ² h and the subscript s is again used to denote the values of variables at the surface. Integration of Eqs. (6) and (7) with respect to z, after substituting for L, u and µ fro Eqs. (4), (10), and (11) now gives: U D u s k µ D µ s C µ s k µ z ln zh 1 C 5Ls zhh Hh C H ln " z ln z 0T z 0 zh C 5Ls ( z.1 H= h/ 2.1 z=h / C zh 2 h 2 C 2H 2 h (10) (11) ± 1 z H ² ¼ (12) H ± h 1 ln 1 z ² ¼ (13) H where z 0 and z 0T are the oentu and teperature roughness lengths, respectively. The Richardson nuber in this layer is an iportant quantity used below and is given by: ± Ri D z L s 1 z ² 2 ± 1 z ² ¼ C 5z : (14) H h In the top neutral layer, integrating Eq. (8) for z g À h: U D U g u s 1 h n ± zg ² o ln 1 (15) k H d µ D µ g : (16) These equations (12) and (13) in the botto layer, and (15) and (16) in the top layer for U and µ will now be atched across the interface to give values for u and µ as follows. Continuity of U and µ at the top of the surface inversion give: hzg u s D ku g µln dz 0 1 h H ln ± zg d ² C 5 Ls H C H H h 1 ln ¼ 1 h H (17)

12 ¼ ¼ 1956 A. LAPWORTH µ h µ s D k.µ g µ s / ln 1 z 0T C 5 2H 2 L s h H C 2H 2 h H h 1 ln 1 h ¼ : (18) H The above equations apply to the case where the boundary-layer height H is greater than the inversion height h. The corresponding equations for the case when H equals h will now be derived. As the surface cools, then eventually the turbulent top of the boundary layer ay drop below the inversion top. In this case µ, which for large z is proportional to u and p.@µ=@z/, ust go to zero at the boundary-layer top H rather than the inversion top h. Because the turbulent ux above height H is zero, the inversion top is then unable to grow to the height h required by Eq. (27) below. This equation is based on heat conservation and in order to continue to conserve heat the teperature pro le assued in the derivation of Eq. (27) ust be altered. As the ascending inversion top is attepting to grow into a descending non-conducting region a relatively sharp teperature discontinuity ay for at the top of the boundary layer if H is changing slowly. As will be discussed below, at the tie of foration of this discontinuity the surface wind speed and friction velocity u are increasing slightly. By Eqs. (3) and (4) this will have the effect of increasing H e and so the boundary-layer height H, which follows H e, will eventually stop falling and the teperature discontinuity will be con ned to a fairly narrow layer. For siplicity, a teperature pro le that is linear with a capping discontinuity will be assued. Conservation of heat then shows that the size of the discontinuity ±µ is given by: ±µ D 1µ.h H / : (19) H Then using the stable-layer Eqs. (12) and (13) for the case h D H : U D u s k µ D µ s C µ s k z ln z z 0 H C 5z L s z ln z z 0T H C 5z L s (20) : (21) At the interface, fro Eqs. (15), (16) and (19), µ D µ g ±µ and U D U g and hence u and µ can be obtained: ¼ H u s D ku g ln 1 C 5HLs (22) z 0 ¼ H µ s D k.µ g µ s / ln 1 C 5HLs : (23) The Richardson nuber in this case is given by: ± Ri D z.nl s 1 z ² o C 5z : (24) H In order to solve the above equations, expressions ust now be derived for the two height-scales h and H. z 0T

13 EVENING SURFACE WIND SPEED 1957 Following the evening transition, an inversion fors in the layer adjacent to the surface, and as surface cooling continues the top of this layer rises. Yaada (1979) and Nieuwstadt (1980) have shown that the one-diensional D µ can be integrated across the boundary layer if assuptions are ade about the teperature pro le to give a rate equation for the rise of the top of the s=@t.h h e /.µ h µ s / where h e is given in Eq. (2), w is the vertical coponent of wind velocity and the subscript h is used to denote the values of variables at the height of the top of the inversion. This equation shows that the height h will either rise or fall fro its current value towards its equilibriu value h e. Equation (26) can be integrated using an integrating factor to give the height to the inversion top as: h D 1 Z 1µs h e d (27) 1µ s 0 where 1µ s is the surface cooling below the teperature at transition. The other scale height H is then deterined as follows. Nieuwstadt and Tennekes (1981) have shown that a rate equation for the decrease in the nocturnal boundary-layer height H with cooling is D 4 3 (25) H e / µ h µ s : (28) As with Eq. (26), this equation shows that the value of H always tends towards its equilibriu value H e. This can be integrated using an integrating factor to give the height H as: H D 41µ s 4=3 3 Z 1µs 0 1=3 H e d : (29) As noted above, the equilibriu value of H, H e is given in Eq. (3) but this can be odi ed in the near-neutral liit as suggested by, for exaple, Derbyshire (1990) to be the solution of s u L H e D 0:4 f.1=l C 0:8=H e / : (30) The equations for h and H (which require the equations for L, h e and H e ) and the equations for U, µ, u s and µ s for either the case before the upper inversion fors or afterwards can now be solved iteratively to obtain the variation of U 10, the turbulent uxes and L s with the screen-teperature depression 1µ (D 1T 1 ) at z D 10. Because observations and nite-difference odels show that the wind velocity in the upper layers of a stable boundary layer undergoes an inertial oscillation, this was included in the upper-boundary value of U g in the present odel. The aplitude of the oscillation was taken to be 20% of the ean wind, this being a coonly observed value. There is a discontinuity in the gradient of all the curves where H becoes less than h. The position of this discontinuity is very sensitive to the exact cooling rate

14 1958 A. LAPWORTH 8 U =9.5 s -1 U =6.4 s -1 U =3.9 s -1 U s T-T trans deg C Figure 11. Model curves of 10 wind speed (U) versus screen-teperature fall after transition for three axiu daytie wind speeds (U, s 1 ). Dashed lines indicate that an upper inversion has fored. and the assued initial conditions for the stable boundary-layer height, and in practice the averaging used to obtain ean values fro the observations would sooth out the U 10 1T 1 curve into a continuous line. It is noticeable that individual U 10 1T 1 curves often show axia although there could be any causes for these. 5. INTERCOMPARISON OF RESULTS Plots showing the variation of U 10, the uxes and L s with 1T 1 for three different values of U g at a cooling rate of 1.5 degc hour 1 are given in Figs. 11, 12, 13 and 14. Figure 15 shows the variation of H and h with 1T 1 in the odel. The cooling rate is in the range observed and was chosen to best illustrate the effects of foration of the upper inversion, which, as noted above, is sensitive to the initial conditions used in the odel. For intercoparison purposes odel geostrophic wind speeds were chosen using a U g =U ax ratio of 1.7 as indicated by a series of sonde ascents. It can be seen that there are qualitative siilarities with the observations plotted in Figs. 2, 5, 6 and 8. The plot of U 10 against 1T 1 in the odel (Fig. 11) shows an alost linear variation (apart fro the rise where the boundary-layer height crosses the inversion top) and in particular, the gradient of the line increases with the geostrophic wind as observed in practice. The reason for this is related to the increase in the height of the top of the surface inversion h with geostrophic wind U g which can be seen in Fig. 15. It can be shown fro the relevant equations above that both h and H increase with wind speed. This increase in h (or H ) with U g then affects the 10 wind speed as follows. The vertical shear in the stable boundary layer is greatest adjacent to the surface where the Richardson nuber becoes vanishingly sall with the ixing length which is of the order of z. In this situation the velocity at a given height above the surface then tends to be controlled by the relatively sall axiu shear possible between that level and the top of the

15 EVENING SURFACE WIND SPEED 1959 Figure 12. Model plots of oentu ux (u 0 w 0 ) versus screen-teperature fall after transition for three axiu daytie wind speeds (U, s 1 ). Dashed lines indicate that an upper inversion has fored. Figure 13. Model plots of heat ux (w 0 T 0 ) versus screen-teperature fall after transition for three axiu daytie wind speeds (U, s 1 ). Dashed lines indicate that an upper inversion has fored.

16 1960 A. LAPWORTH 800 U =9.5 s -1 U =6.4 s -1 U =3.9 s -1 LMO T-T trans deg C Figure 14. Model plots of Monin Obukhov length (LMO) versus screen-teperature fall after transition for three axiu daytie wind speeds (U, s 1 ). Dashed lines indicate that an upper inversion has fored. 800 U =9.5 s -1 U =6.4 s -1 U =3.9 s -1 Zil, Zi T-T trans deg C Figure 15. Model plots of the height of the top of the surface inversion (z i ) and Zilitinkevich height (Zil) versus screen-teperature fall after transition for three axiu daytie wind speeds (U, s 1 ). The three inversiontop height plots all start fro zero height at T D 0. Dashed lines indicate that an upper inversion has fored.

17 EVENING SURFACE WIND SPEED 1961 boundary layer which is liited by the bulk Richardson nuber of the upper part of the layer. Using the Richardson nuber de nition the shear U g U 10 is given by: s g1µ.h 10/ U g U 10 D (31) µ 0 Ri b where Ri b is an average bulk Richardson nuber. Hence: D g.h 10/ 4T 0 Ri b 1µ : (32) It can be shown fro Eq. (14) that as L decreases, the ean Ri in the layer approaches a constant value. Therefore, fro Eq. (32) the 10 =@1µ increases as h (or H ), which is as observed. So it can be seen that the underlying reason for the apparently anoalous greater susceptibility of wind speed to surface cooling at higher geostrophic winds is a result of the thicker boundary layer that fors in these conditions for all stages during the transition. As the upper inversion fors the near-surface wind speed and u are both starting to increase which is an iportant factor in aintaining its sharpness because H stops decreasing. This increase is ainly associated with the increase in geostrophic wind speed as the upper wind starts its inertial oscillation. There are other possible in uences. One is that the change in teperature pro le as the inversion fors, itself contributes to the near-surface wind-speed increase. This echanis operates only after the inversion has fored. Provided that the surface cooling rate is not too great, the foration of the inversion reduces the teperature gradient in the boundary layer below the inversion as can be seen in Fig. 10. This causes a reduction in the wind shear in the upper part of this layer because the Richardson nuber in this region is aintained near its critical value while shear is concentrated in the surface layer with its low ixing length. The net effect is to increase the 10 wind speed. There is an increased wind shear at the growing upper inversion itself but because this layer is thin, this is not enough to copensate for the reduced wind shear in the thicker layer below. Another in uence is the interittent nature of the stable boundary layer which is associated with sei-periodic increases in activity. The ain qualitative features of the turbulence quantity variations with 1T 1 are also reproduced by the odel. The odel surface oentu ux u 0 w 0 curves shown in Fig. 12 decrease with increasing 1T 1, as is expected because U also decreases with 1T 1. However, the downward heat ux w 0 T 0 shown in Fig. 13 initially increases with increasing 1T 1 as the vertical teperature and hence µ increase, and then decreases as the continuously decreasing oentu ux predoinates. The for of the curve is governed by the constancy of the ux Richardson nuber at increasing stability. The axiu downward heat ux occurs at around 1T D 2 degc, which is siilar to the observed 1T 1 value observed experientally. The surface Monin Obukhov length L shown in Fig. 14 initially decreases extreely sharply with increasing 1T 1 and continues to fall at a reduced rate as 1T 1 increases further. The variation of the U 10 1T 1 curve with cooling rate for U ax D 6:4 s 1 in the odel is shown in Fig. 16. It can be seen that there are variations between the average 10 =@1T 1 for the different cooling rates but the differences are fairly sall. The foration of the upper inversion signi cantly affects the overall slope of the slow cooling-rate curve which would otherwise have a uch steeper gradient than the other

18 1962 A. LAPWORTH 1. 9 O C h O C h O C h -1 U s T-T trans deg C Figure 16. Model plot of curve of 10 wind speed (U) versus screen-teperature fall after transition at one coon axiu daytie wind speed for three different cooling rates. Dashed lines indicate that an upper inversion has fored. two curves. As noted above, the averaging process would sooth out the hups of individual cases for this cooling rate. The odel clearly predicts that foration of a sharp upper inversion with a concoitant increase in surface wind is ore likely at lower cooling rates. The large increase in surface uxes for lower screen teperatures shown by the odel when the upper inversion fors is not seen in the averaged observations, although a sall increase ay be indicated by the overall reduced slopes of the experiental curves relative to the odel. This ay indicate that a sharp upper inversion does not coonly for. In this context it should be noted that the Nieuwstadt odel which is applicable to higher stabilities also has a sharp upper inversion and gives higher uxes than those observed experientally for the coldest screen teperatures. The sei-analytical odel described above was used to give soe insight into the uid dynaics of the evening transition but inevitably contains approxiations. For coparison, the relevant equations were integrated in full as a nite-difference gradient-transfer odel to deterine whether the ain features of the analytic results were retained. The nite-difference odel was one-diensional and integrated the standard Navier Stokes equations with the inclusion of a Coriolis force ter. An expanding grid was used in the vertical. The equations were closed using a ixing-length schee based on local siilarity forulations. Radiative effects were not included. Tie stepping was perfored using the neutrally stable leapfrog schee for the Coriolis ter, but with an explicit forward tie-stepping schee for the viscous ters to avoid instability. The odel was run in double precision arithetic and the tie step was continuously adjusted to eet the viscous instability criterion. At the surface a no-slip condition was iposed and teperature cooling was prescribed during the evening transition starting fro a convective daytie boundary layer. At the top boundary a constant teperature

19 EVENING SURFACE WIND SPEED 1963 Figure 17. Graphical results fro nite-difference odel. See text for further explanation. was prescribed together with a constant geostrophic wind. The odel was run with a series of geostrophic wind speeds siilar to those observed. The results fro this odel con red that the pro les of u and µ within the stable layer were approxiately linear which was one of the ain assuptions ade in ipleenting the sei-analytic odel. Figure 17 shows the ain output plots fro this odel. The rst four graphs in this gure show the variation of surface wind speed, oentu ux, heat ux and Monin Obukov length with surface teperature and can be copared with Figs. 11, 12, 13 and 14, respectively, fro the sei-analytic odel. The two geostrophic wind

20 1964 A. LAPWORTH speeds used in these plots are equivalent to the highest (thick line) and lowest (thin line) wind speeds used in the sei-analytic odel. The fth plot (botto left) shows the variation of surface wind speed with teperature for the highest (thick line) and lowest (thin line) cooling rates and the sae geostrophic wind speed shown in Fig. 16. There is reasonably good qualitative agreeent between the plots obtained fro the two odels and they share any coon features, although the nite-difference odel tends to round off the sharpness of the transition seen in the analytic odel when an upper inversion fors. The variation of surface wind-speed teperature gradient with geostrophic wind speed shown in Fig. 17 is observable although not as well arked as in the sei-analytic odel output, and in both odels the surface wind increases with cooling after the upper inversion fors. However, the wind-speed teperature gradients in Fig. 17 are higher and less linear than in Fig. 11. The fth graph shows that this plot is less dependent on cooling rate than geostrophic wind speed, and as in Fig. 16 the slowest cooling rate gives a ore oscillatory curve. The oentu uxes in both odels initially fall fairly quickly as the surface cools. The negative heat uxes in the nite-difference odel rise sharply with cooling and reach a axiu at around 2 degc cooling below neutral. After this the negative heat uxes initially reduce with further cooling, although not to such an extent as observed or calculated in the seianalytic odel, before rising again as the upper inversion fors. The axiu at a surface cooling of around 2 degc is, therefore, a feature both observed and predicted by both odels. The size of the axiu is less well arked in Fig. 17 than in Fig. 13. The Monin Obukov length variation agrees very well between the odels with its initial sharp fall off with cooling. The sharp elevated inversion predicted by the sei-analytic odel when its two vertical length-scales interact is observed to for in the nite-difference odel and this is shown in the nal plot of Fig. 17 in which the Zilitinkevich heights are arked by dashed lines. Two pro les are shown, one before the upper inversion has fored, with the higher associated Zilitinkevich height, and one after. This odel shows a siilar variation of Zilitinkevich height with cooling after the upper inversion fors to that inferred fro the analytic odel it stops falling sharply and starts rising towards the upper inversion. The stabilization of this height is the ajor factor deterining the foration of a sharp upper inversion. The height at which the upper inversion fors in the nite-difference odel is greater for stronger geostrophic winds, as expected fro the analytic odel behaviour. The height of upper-inversion foration is also greater when the nite-difference odel is run at slower surface cooling rates. This is expected fro Eq. (2) because the surface inversion grows faster in height at slower surface cooling rates while the Zilitinkevich height-scale of the stable layer is relatively unaffected. Therefore, the two height-scales will becoe equal higher up in the boundary layer. 6. CASE-STUDY OF UPPER-INVERSION FORMATION The experiental results so far presented have been of surface observations but it is hard to validate the odel results for upper-inversion foration solely by using these. A case-study was ade at Cardington on the night of 22/23 May 2001 using the balloonborne turbulence probe syste described by Lapworth and Mason (1988), and Lapworth (1995). The night was clear until near the end of the observing period. Figure 18 shows the U 10 1T 1 curve for this night and it will be seen that the 10 wind dies away steadily as the surface cools until at a 1T 1 value of 5 degc the wind increases with further cooling. This curve resebles the odel prediction of Fig. 11.

21 EVENING SURFACE WIND SPEED /23 May U s :10 00:50 23:00 21:30 19: T-T trans deg C Figure 18. Measured curve of 10 wind speed (U) versus screen-teperature fall after transition with soe ties indicated by arrows for the night of 22/23 May 2001 at Cardington. Figure 19 shows the potential-teperature pro le as easured by a nuber of balloonounted probes at the ve 1T 1 values indicated by arrows and labelled with ties on Fig. 18. It should be noted that in soe cases the balloon is at a fairly constant height while in others it is rising or descending. It can be seen that an upper inversion has fored at around the 1T 1 value that the surface wind increases with cooling. The upper inversion appears sharp in the easureents ade with the balloon at constant height. This is partly because gravity waves on the inversion ove the isothers up and down past the probe which has a relatively constant height. The inversion appears after the top of the boundary layer, as easured by the u pro le, has dropped below the top of the surface inversion as easured by the w 0 T 0 pro le. It was noticeable that the upper inversion started to for as the inertial jet becae proinent at the inversion level, which enhanced shears and hence turbulent uxes at this level. It would be of value to show that the inversion was fored by the turbulent echanis indicated rather than by other echaniss such as radiation, subsidence or advection but this is dif cult. A pro le ade by a radioeter as the upper inversion started to for showed no signi cant radiative divergence at the inversion level, and no fog or cloud was noticed during the following few hours. The ean potential teperature above the upper inversion did not increase signi cantly during these hours. These observations tend to eliinate radiative and subsidence echaniss, but leave advection as a possibility. One possible type of advective phenoenon would be a gravity current, but none of the eteorological variables showed the characteristic horizontal discontinuities. Pro les of turbulent heat ux ade during the rst 2 hours that the upper inversion was growing show a good correlation between the cooling rate below the inversion inferred fro the easured uxes and the cooling rate that was observed. However, the ux spectra showed that a signi cant part of the turbulent cooling was at the longest wavelengths

22 1966 A. LAPWORTH Figure 19. Five potential-teperature pro les at different ties for the night of 22/23 May 2001 at Cardington at surface teperatures corresponding to the arrows on Fig. 18. which results in such large statistical errors as to ake quantitative intercoparison dif cult. So although there are indications that the inversion is fored by a onediensional turbulent echanis, the effects of advection cannot be ruled out. Further such case-studies would be needed to give fuller validation of the odel results. 7. FORECASTING RULE The observations above have shown that during the evening transition the wind speed over land has a statistical relationship to the screen teperature and geostrophic wind. This relationship can be used as the basis of a siple forecasting rule, providing the screen teperature is rst forecast for the evening using a rule such as Barthra s (Barthra 1964). The procedure then is as follows: (i) Deterine whether the evening is suitable for application of the rule. In particular there should be no frontal passages or other advective changes expected and a steady geostrophic wind. The upwind fetch should be over land with no signi cant orography. (ii) Deterine the wind speed U trans ( s 1 ) and teperature T trans ( ± C) at transition usually about 1.5 to 2 hours before sunset. (iii) Deterine the value U ax of the axiu 15-inute-ean wind during the day. If this is not available substitute a value of 0.6 ties the geostrophic wind speed. (iv) Deterine the curve of tie versus screen teperature for the evening fro, for exaple, Barthra s rule to deterine the teperature T at the tie required.

23 (v) EVENING SURFACE WIND SPEED 1967 Use the following relation to deterine the expected 10 wind speed ( s 1 ) for U ax > 2:5 s 1 : U D.0:18 0:07U ax /.T trans T / C U trans (33) where T is the screen teperature in degrees Celsius. (vi) The errors have a standard deviation of 1 s 1 for perfect-teperature prognosis, and vary depending on how uch advection effects interfere. However, this rule does give soe coparative feel for how strong the effect of nocturnal stabilization is in reducing the near-surface wind speed in relation to other factors such as a varying pressure gradient. (vii) In any applications the wind gust is ore iportant than the ean. Because, as noted previously, the friction velocity u behaves siilarly with cooling to the wind speed U, the predicted gust during the evening can be found by ultiplying the predicted wind speed U by a constant factor, usually of the order of 1.5 over land. The above rule is derived fro a fairly at rural inland site and so ay not necessarily apply to other sites. In particular it ay not apply to coastal sites especially where the sea land air-teperature difference is arked. However, four weeks observations taken fro a site at Coleshill which is a few iles downwind fro the Biringha conurbation give results that appear very siilar to those obtained at Cardington. The odel does not indicate any great sensitivity to variation of either oentu or theral roughness lengths. It will be seen that siilar forecasting rules for the oentu and heat uxes can be deterined fro Figs. 5 and 6. Equation (33) can be put in non-diensional for by altering the cooling ter.t trans T / to a buoyancy.t trans T /=T 0 which is appropriate because the cooling affects the dynaics through changes in stability. 8. CONCLUSIONS Observations ade over a ve-year period fro a at inland surface site were analysed to deterine whether any statistical relationships existed that could be used to predict the reduction in surface wind during evening cooling of the surface. The surface wind was found to depend statistically on the decrease in teperature as the surface cooled in a well de ned and linear way which is dependent ainly on the gradient wind and is relatively unaffected by the rate at which the cooling takes place. Surprisingly, the decrease in wind with teperature was found to be greatest for the strongest gradient winds. Moentu and heat uxes were also found to depend in a well de ned way on the screen-teperature decrease and gradient wind but there is no siilar relationship with oisture uxes. The downward heat ux was observed and odelled to reach a axiu at a surface cooling of around 2 degc. A siple sei-analytical odel shows a qualitatively siilar behaviour to the observations and in the odel results it can be seen that the stronger effect of cooling on the surface wind at higher gradient wind speeds is due to a thicker boundary layer. The odel predicts an increase in surface wind later in the evening if an upper inversion fors. Observations fro a tethered balloon and surface easureents show that during a clear night the surface wind did increase when an upper inversion fored although it was not possible to deterine whether the echanis of inversion foration was related to that occurring in the odel. A onediensional nite-difference odel gave results that could be interpreted in ters of the sei-analytic odel.

24 1968 A. LAPWORTH A siple forecasting rule for the evening surface wind was derived fro the observations. ACKNOWLEDGEMENTS I a grateful to colleagues at the Meteorological Research Unit at Cardington for their help and to Mr T. Jones and Mr J. McGregor in particular for installing, aintaining and calibrating the surface instruents. I also wish to thank Mr M. Sart MBE and Mr I. Walton for operating the balloon and Dr B. Claxton for logging the data. Finally thanks are due to the Grafha Water sailing club for causing the question to be raised. REFERENCES Barthra, J. A A ethod of forecasting a radiation night cooling curve. Meteorol. Mag., 93, Derbyshire, S. H Nieuwstadt s stable boundary layer revisited. Q. J. R. Meteorol. Soc., 116, Lapworth, A. J Probles in the use of inclinoeters and agnetoeters to deterine translation and orientation of a turbulence probe. J. Atos. Oceanic Technol., 12, Lapworth, A. J. and Mason, P. J The new Cardington balloon-borne turbulence probe syste. J. Atos. Oceanic Technol., 5, Nieuwstadt, F. T. M A rate equation for the inversion height in a nocturnal boundary layer. J. Appl. Meteorol., 19, A odel for the stationary, stable boundary layer. Pp in Proceedings of the IMA conference on turbulence and diffusion in stable environents, Cabridge, Ed. J. C. R. Hunt. Oxford University Press Nieuwstadt, F. T. M. and Tennekes, H A rate equation for the nocturnal boundary-layer height. J. Atos. Sci., 38, Yaada, T Prediction of the nocturnal surface inversion height. J. Appl. Meteorol., 18, Zilitinkevich, S. S On the deterination of the height of the Ekan boundary layer. Bound-Layer Meteorol., 3,