A discussion on nocturnal temperature profiles

Atmospherk Environment Vol. 23, No. 6, pp. 1177-1186, 1989. @X4-6981/89 S3.00+0.00 Printed in Great Britain. Q 1989 Person Rcu plc

A DISCUSSION ON NOCTURNAL TEMPERATURE PROFILES

DOMENICO ANFOSSI

Istituto di Cosmogeofisica del C.N.R., Cso Fiume 4, 10133 Torino, Italy

(First received 4 February 1987 and receivedfor publication 14 November 1988)

Abtmct-Observations of the nocturnal temperature profile up to 5OOm and of the surface micrometeorological variables were obtained at Philippsburg (F.R.G.) in the frame of the Mesoklip experiment. This data set forms the experimental basis for this work. Our main aims are: the validation of the analytical model due to Anfossi et al., 1976 (Q. JI R. met. Sot. 102, 173-180) with data collected in a complex terrain; the verification of a new method suggested by Surridge, 1986 (Atmospheric Environment 20, 803-806) to improve the forecast of the cited model; to state which measured temperature fall (at ground or at screen height) better fits the lower boundary condition imposed to the heat diffusion equation in order to derive our solution; to evaluate the ability of the model to estimate the nocturnal ground level temperatures.

Key word index: Nocturnal, temperature, inversion.

1. INTRODUCTION

In the past years a lot of work has been done on the problem of a better understanding of the mechanism of formation and development of ground based inversions and of forecasting the temperature profile in the first layers of the atmosphere in such conditions. That work was stimulated by the importance of these micrometeorological conditions for the dispersion of air pollution and for fog and frost formation.

Among the various studies, a series of simple analytical or empirical formulae for the prediction of nocturnal vertical temperature profiles under inversion conditions and of ground minimum temperature have been proposed.

The classical approach is due to Brunt (1934) who solved the heat diffusion equation considering:.(i) a semi-infinite medium, (ii) the radiative exchange as the main heat transfer mechanism, (iii) the diffusion coefficient constant with respect to time and space, (iv) the heat flux fat ground constant throughout the night.

The problem was reconsidered by Anfossi et al. (1976). They replaced the Brunt condition (i) setting an upper boundary moving upwards in time, across which vertical gradient of temperature must vanish. The time trend of the upper boundary is expressed by:

zi=,/4 Kt (1)

where K is the heat diffusion coefficient and zi is the height at which dT/dz= 0 (the height of the inversion top). Measurements performed in the PO Valley (Northern Italy) gave for K the value 0.34 mz s- l (Anfossi et al., 1974). For the vertical temperature profile they obtained:

T(z, t) = T(O, 0)-c T(O, 0) - T(O, t)] {exp[ -(z/zi)2]

- z/ziJ7t [erfc(z/zi) - erfc( l)] } (2)

and for the decrease of temperature at ground during the night:

2f t 1/Z T(O,O)- T(O,t)=--- -

() pcerf(1) Ka (3)

wheref(W me2) is the heat flux, p and care the density and specific heat of soil or of air. As Equation (3) is a boundary condition at the interface between air and

soil, the product pcfi for soil and for air should coincide even if temperature gradients in the ground are much smaller than those into the air. We recall (Brunt, 1934) that in the ground K represents a molecular diffusivity whereas in the air K refers to a radiative and/or eddy diffusivity. Anfossi et al. (1976) showed that Equation (2) gave a marked improvement over the Brunt solution.

Either in Brunt’s or in Anfossi et al.‘s analysis [ T(O,O) - T(0, t)], which is the fall in temperature at ground (z=O), is calculated by means of the corresponding fall in air temperature at screen height.

Subsequent papers (Kloppel et al., 1978; Anfossi and Longhetto, 1985) extended the validity of Equa- tion (2) in different sites from the PO Valley. The first one reported a satisfactory agreement between calculated and measured nocturnal temperature profiles at Hamburg (F.R.G.) while the second one showed a good average agreement among Equation (2) and 136 nocturnal profiles (30 nights) measured at Hay (Aus- tralia) during the Wangara experiment (Clarke et al.,

1971). At this point it can be important to mention that

many different definitions of the nocturnal stable boundary layer (NSBL) height can be found in the literature. Yu (1978) defines it as the height at which turbulence disappears and above which shear stress and sensible heat flux become negligible. Other defini-

1177

1178 DOMENKC Abt OSSI

tions are as follows: height of the maximum wind profile (Clarke, 1970) height where the magnitude of the momentum flux reduces to 1% of its surface value (Businger and Arya, 1974), height to which significant cooling has extended (Melgarejo and Deardorff, 1974) height at which the vertical temperature flux falls to 5% of its surface value (Caughey et ul.. 1979), height where the temperature lapse rate changes to dry adiabatic, that is where d0/dz = 0 (Yamada, 1979). As mentioned above, we defined it (Anfossi et al., 1974, 1976) as the height at which d T/dz = 0. This was done because we developed our model (Anfossi et al., 1976) according to Brunt (1934) in the framework of a radiative diffusion process. As, in reality, both radiative and turbulent diffusion processes are responsible of the cooling of the NBSL, one could wonder whether one should consider the NSBL height to be coincident with the level at which dtI/dz =0 and not with that one at which dT/dz = 0 as we have done. Various estimates suggest that both processes are important even if different weights are given to the two processes. For instance Kondo and Haginoya (1985) evaluated that the ratio of the sensible heat flux H, to the net radiative heat flux R, in steady stable conditions at clear nights is: H,/R, = 1.4 u* (IL* expressed in m s- ‘). This means that H,/R,=0.7 if u,=O.Sms~ and =0.14 if u,=O.l rns-‘. Stull (1983) on the contrary estimated that 77% of the cooling of vertically integ- rated NSBL is due to the turbulence and 23% to the surface induced radiation divergence. In any case, near the top of the NSBL the turbulent heat flux is zero or approaching zero. Furthermore according to Kondo and Haginoya (1985) the radiational flux divergence exists from the surface to a level which is higher than that one where the turbulent heat flux divergence becomes negligible. Therefore it seems reasonable (at least from a practical point of view and also in consideration of the good results, mentioned above, obtained by our model) to assume the height zi of NBLS, where the heat flux is set equal to zero, coincident with the level at which dT/dz=O as we have done in previous papers (Anfossi et al., 1974, 1976; Anfossi and Longhetto. 1985) and in the present paper.

Yamada (1979) and Stull (1983) proposed two different empirical expressions for the nocturnal profiles of potential temperature. The first one (cubical in the height z) and the second one (exponential) were fitted, respectively, to the observations from 4 and 12 nights of the Wangara experiment. Anfossi and Longhetto (1985) compared Stull and Yamada expression to Equation (2) finding that: (i) the three formulas yield similar results for small values of the ratio z/zi, whereas for large values of zizi Yamada’s model departs from the other two; Stull and our model give almost the same results. It is to be noticed that Stull introduced an empirical constant which was adjusted to the Wangara data, whereas Equation (2) is an analytical solution of the diffusion equation which does not contain any empirical constant fitted to the Wangara observations.

Surridge (1986) proposed a method to extrapolate the vertical temperature profile from the measurements of the temperature at ground and at a second height (zz) near the ground. He calculated from Equa- tion (2) the following expression:

T(zzlt)- TE? = 1 __ exp[__(-_2 ,z,),

W’, 0) - WI t) +zz;z;Jn[erf(l)-erf(z,/zi)] (4)

from which it is easy to obtain:

zz& [erf(l)-erf(z,/z,)] zi = .___~.__-_-.__

T(z,, t) - T(O,O) --+exp[-(z,/zJ’].

7-(070) - T(O, r)

(5)

The iteration involved in Equation (5) is repeated until

the successive values of zi do not change by more than the required accuracy. The value zi is then substituted into Equation (2) from which the temperature profile can be computed. According to Surridge, this pro- cedure should be an improvement with respect to Equation (2) in that it does not assume a value of K.

Gandia et al. (1985) suggested to use a model of exponential decrease of temperature at ground to predict the nocturnal minimum temperature at ground which reads:

W4 t) - TK4 t,)

T(O, 0) - T@, L,) = exp(A t + B) (6)

where t, is the time at which the minimum temperature occurs, A and B are two constants evaluated on the basis of best fit analysis. In Equation (6), B should be equal to zero to satisfy the initial condition. How- ever the authors (Gandia, pers. comm., 1987) included the B term for the following reason. Their equipment for measuring temperature was automatic and operated every 2-h. As a consequence generally their first measurement did not coincide with sunset. They took the first measurement available after sunset as the one corresponding to t = 0. Therefore B was inserted to take into account this time lag.

The purposes of this paper are the following: (i) to test the accuracy of Equation (2) in in predicting the temperature profiles in a complex terrain; (ii) to compare the performance of the Surridge’s method (Equa- tion (5)) with our original method (Equation (2)); (iii) to establish which temporal decrease of temperature [ T(O,O) - T(0, t)], (z =0 or screen height) is the more appropriate for Equation (2); (iv) to compare the performances of Equations (3) and (6) in predicting the temporal trend of temperature at ground.

2. DATA

To achieve the above results we utilized the nocturnal observations of temperature profiles measured at Philippsburg (in the neighbourhood of Karlsruhe, F.R.G.) during the Mesoklip Experiment (Fiedler and Prenosil, 1980). To try to be more general in the

Discussion on nocturnal temperature profiles 1179

comparison involved in item (ii), the Wangara expet- iment data were also used.

Mesoklip was a big field experiment in the Rhein Valley executed in cooperation between numerous German Institutes and coordinated by the Institute of Meteorology of Karlsruhe. The Rhein Valley at Karls- ruhe is about 40 km wide. On both sides it is sur- rounded by hills as high as 300400 m. During the Mesoklip Experiment, 10 measuring stations were simultaneously operated across the valley from 17 to 28 September 1979. In each station the vertical profiles of air temperature and wind speed and direction were measured at a rate of one profile h-i during the subperiods of measurement. Data were recorded with a vertical resolution of 25 m. Furthermore, in the Philippsburg station (which is almost in the middle of the valley), a complete series of micrometeorological measurements have also been performed at 10 min intervals during the whole experiment. They include: radiation balance at ground, dry and wet temperature and wind speed at 0.5,2 and 6 m above ground; wind direction at 6 m; pressure at ground level; temperature in the ground at several levels, i.e. 0, 2,4, 8, 16, 32, 64, 128 and 256 cm. Surface temperature (z = 0) was measured by a radiation thermometer. Temperatures near the ground were affected by an error c 0.1 K whereas radiosonde temperatures had an error < 1 K. For further details on the experiment the reader is referred to the report by Fiedler and Prenosil (1980).

In the present study only the observations taken at Philippsburg were considered as they include the measurement of the net longwave radiation (RN) and both the ground level (To) and screen level (T,) temperatures. Further they allow the evaluation of soil heat flux (G,). This last can be evaluated (Georg, 1971) as the sum

G,=(pcAT,fAt), *AZ, -t(pcAT/At),-AZ,+ . * -

+(pcAT/At);Az, (7)

where the subscripts refer to the various decreasing soil slabs and p and c are, respectively, the soil density and specific heat. The last term in the sum of Equation (7) refers to the slab at which no change in the soil tem~rature occurs. We firstly evaluated ATjAt in each slab and then obtained the values of G, and of the various pc in each layer by means of a method of optimization (numerical best fit) designed to solve non-linear estimation problems.

As regards Wangara data, we considered two data sets: a larger one, containing 136 profiles from 30 nights, referred to as ‘all data’ and a smaller one, consisting of 35 profiles from 12 nights, referred to as ‘Stull data’. As indicated in Anfossi and Longhetto (1985), for the first data set the nights were selected when no rain or thunderstorm occurred in the prece- ding afternoon. The second set, which is a subset of the previous one as it is contained in it, consists of a data set selected by Stull(1983) in order to examine profiles free from the influence of low-pressure systems or fronts and from probable measuring errors.

Either for the Mesoklip or for the Wangara data, pro&s in which no inversion appeared were discarded. Profiles in which an inversion was present at an height >z2 but in which T(z,) was less than T(z,) were discarded too as they would give completely wrong results with Surridge’s technique. In all the analyzed cases zi was >z2 (25 m for Mesoklip and 50 m for Wangara).

3. RESULTS AND DISCUSSION

Figures 1, 2 and 3 refer to the Philippsburg data. Heights above ground are indicated as ordinate axis. The difference between calculated and measured temperatures, averaged over all the sounding, are reported in the abscissa axis of Figs la, 2a and 3a, whereas the corresponding standard deviations are shown in Figs lb, 2b and 3b. Crosses refer to the temperature computed by means of our model (Equation 2), whereas asterisks refer to Surridge’s method (Equations 5 and 2). Figures 1, 2 and 3 differ only in the choice of the starting time to and in the estimate of AT, = [ T(O,O) - T(0, t)]. In the calculations of Fig. 1 (case I), t, is chosen as the time in the afternoon at which the net radiation (RF’) changes sign, and AT, is the fall in ground temperature (z =O). In Fig. 2 (case II) t, is chosen as before but AT, is the fall in air temperature at the screen level (2 m). In Fig. 3 (case III), t, is the time at which the diurnal wave, at the screen height, reaches its maximum value and AT, refers again to screen level temperatures.

It was stated in the original paper (Anfossi et al., 1976) that the results are only theoretically valid up to height zi. Therefore Figs 1, 2 and 3 report only data obtained under measured zi. In a subsequent paper (Anfossi and Longhetto, 1985) the comparison was tried up to 500 m, bearing in mind the possibility of using Equation (2) for climatological purposes. We noticed that data above zi generally agreed well with our model even if there is no sound theoretical basis for the extension,

Table 1 shows the results of the comparison with all the data, i.e. without excluding those data above zi. Tables 2 and 3 refer to the Wangara experiment. Table 2 reports the ‘all data’ and Table 3 the ‘Stull data’. In both Tables 2 and 3 t, is the time of the wave maximum in the afternoon and AT, represents screen height temperatures. They are referred to as case III.1 and 111.2, respectively, as they are comparable to the data denoted as case III in Fig. 3 and Table 1. Also Wangara data are subdivided into two groups: ‘heights under zi or ‘all heights’ (0 <z < 500 m).

Inspection of Figs 1, 2, 3 and Tables 1, 2 and 3 suggests the following considerations.

(a) In all the comparisons the first point of the profile is exactly zero by definition. When considering Surridge’s method, even the second point (25 m for Mesoklip and 50 m for Wan- gara) is exactly zero by definition.

(b) Either considering the average values or the

1180 DOMFNK o ANPOSSI

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Fig. 1. Comparison among Philippsburg night-time temperature profiles and the ones estimated by means of our model (Equation (2)) and Surridge’s method (Equation (5)) for heights under I,. Asterisks refer to Surridge’s method and crosses to our model. The average differences between calculated and measured temperature at each height are indicated in Fig. la, whereas the corresponding standard deviations are denoted in Fig. lb. In this graph t, is the time in the afternoon at which R, changes sign and AT, is the temperature decrease during the night at

ground level (case I).

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Fig. 3. As in Fig. 1 except that t, is the time in the afternoon at which the temperature wave at screen height exhibits the maximum value and AT, is the

screen level temperature fall (case III).

Table 1. Comparison among Philip~burg night-time tem~rature profiles and the ones estimated by means of Surridge’s method (Equation (5)) and our model (Equation (2)) for ‘all heights’. The grouping of data in cases I, II and III is the same as that presented in Figs 1, 2 and 3, respectively. Numbers in columns 2, 4, 6, 8, 10 and 12 represent, for the indicated heights, the average differences between calculated and measured temperatures;

numbers in the adjacent columns represent the corresponding standard deviations.

Ym)

Case I Case II Case III Equation (5) Equation (2) Equation (5) Equation (2) Equation (5) Equation (3)

T,-T,,, u T,--T,,, IJ T,-T, u T,--T,,, Q T,-T, n T,-T,,, u

W WI K) W) W) (K) W 6) (W W W) W

2: 0 0 0 0 0 1.3 0 1.2 0 0 -0.2 0 0 0.5 0 0 0 0 -0.2 0

50 -0.5 0.9 0.9 1.0 0.1 0.5 -0.2 0.9 0.0 0.4 -0.4 75 -0.3 1.3 0.7 0.9 0.1 0.8 -0.2 1.1 -0.1 0.7 -0.7

100 0.2 1.5 z: 1.0 0.1 0.9 -0.2 1.2 -0.3 0.8 -0.9 125 0.9 1.8 1.1 0.1 1.1 -0.2 1.3 -0.5 1.0 -1.1 150 1.7 2.1 0:s 1.2 0.2 1.2 -0.2 1.4 -0.5 1.2 - 1.2 175 2.5 2.4

:; 1.4 0.3 1.4 0.0 1.5 -0.5 1.4 - 1.2

200 3.3 2.8 1.5 0.4 1.5 0.1 1.5 -0.5 1.5 - 1.1 225 4.1 3.2 1.2 1.5 0.6 1.6 0.4 1.5 -0.3 1.7 -0.9 2.50 4.9 3.7 1.6 1.5 0.8 1.7 0.6 1.5 -0.2 1.8 -0.7 275 5.6 4.1 1.9 1.6 1.0 1.9 0.8 1.5 -0.1 2.0 -0.5 300 6.3 4.6 2.2 1.8 1.1 2.0 1.1 1.6 0.1 2.2 -0.3 325 1.0 5.1 2.5 1.9 1.3 2.2 1.3 1.7 0.2 2.5 -0.1 350 7.7 5.6 2.8 2.1 1.5 2.4 1.5 1.8 0.4 2.7 0.1 375 8.4 6.0 3.2 2.2 1.6 1.7 1.8 0.6 2.9 0.3 400 9.0 6.5 3.5 2.4 1.8 % 2.0 1.9 0.7 3.2 0.6 425 9.7 7.0 3.8 2.6 2.0 3:0 2.2 2.0 0.9 3.3 0.8 450 10.3 7.5 4.1 2.8 2.1 3.2 2.3 2.1 1.0 3.5 0.9 475 11.0 7.9 4.4 3.0 2.3 3.4 2.5 2.2 1.1 3.7 1.1 500 11.6 8.4 4.6 3.3 2.4 3.6 2.6 2.3 1.3 3.9 1.2

:6 l:o 1.1 1.1 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.4 1.5 1.6 1.6 1.7 1.8 1.9 1.9 2.0

standard deviations, the average agreement between our model and the Philippsburg data (Figs 1, 2, 3 and Table 1) is noticeable. Case II appears particularly good. The agreement with

Wangara data is good too, as already stated in a previous paper (Anfossi and Longhetto, 1985). As regards this second data set the agreement is worse in those cases in which only the heights

1182

Table 2. As in Table 1 but for the Wangara ‘all data’. Results related to ‘all heights’ are reported in the 1.h.s. of the table while those related to ‘heights

under 3,‘ are in the r.h.s. of the table

All heights Heights<:, Equation (5) Equation (2) Equation (5) Equation t2)

7; - T W) m Ch

r, -’ ‘T, (i T,-T,,, (T 7, -~ T”, v (K) 6) (K) !K) (KI WI

0 0 0 0 0 0 0 0 0 SO 0 0 0.5 I .o 0 0 0.5 I .(I

100 0.1 0.5 0.9 1.2 0.1 0.5 0.9 1.2 150 0.3 0.7 1.0 l.i 0.2 0.8 1.2 0.8 200 0.4 1.0 1.: I.1 0.” 1.0 I .4 1.3 250 0.6 1.3 1.0 1.3 0.1 1.5 1.7 I.8 300 0.7 1.7 0.9 1.4 -0.1 I.8 1.8 1.9 400 0.8 2.5 0.5 1.6 0.4 15 0.0 1.2 500 0.9 3.2 0. I 1.9

Table 3. As in Table 2 but for the Wangara ‘Stult data’

All heights Heights < z, Equation (5) Equation (2) Equation (5) Equation (2)

T,-T,,, D T,-T,,, d T,-T,, u T,-T,,, o WI W (W w WI (K) (K) w

0 0 0 0 0 0 0 0 0 50 0 0 0.1 1.2 0 0 0.2 1.1

100 0.0 0.5 0.4 1.1 0.0 0.5 0.4 I.2 150 0.1 0.8 0.6 0.9 0.1 0.9 0.7 0.9

:: -0.1 0.0 1.2 1.6 0.5 0.4 0.9 1.1 0.2 0.3 1.1 1.5 0.8 0.9 0.9 1.4 300 -0.2 2.0 0.1 1.1 0.8 1.5 1.3 0.8 400 -0.7 2.9 -0.6 1.2 0.4 1.5 0.0 1.2 500 -1.2 3.8 - 1.3 1.4 _ _

under zi are considered. This means that our model reproduces better the upper part of the average shape of measured Wangara profiles than the lower one.

(c) Surridge’s method fits the data well. As regards Philippsburg data it gives results comparable to ours for cases II and III and presents a lack of agreement in case I for z> 150m. As far as Wangara data are concerned, particularly in the case of ‘Stull data’, Surridge’s approach gives better agreement. We must recall that in this case &ridge’s method fixes exactly the value of the temperature at 50 m. This height can be a significant fraction of zi in many profiles, thus improving the general agreement. Had we included in the current analysis those profiles having zI > z2 but T(z,) d T(z,) we should have found an average agreement between data and Surridge’s estimates worse than that found with our method.

(d) Figure 2a shows the best agreement between our model or, likewise Surridge’s approach, and the Mesoklip data. Figure la shows, on the contrary, a worse agreement for &ridge’s approach, particularly above 150m. It can be

concluded that it is better to estimate Al”, by the fall in air temperature at the screen level and to fix the starting time when R, becomes negative.

(e) An interesting point that arises from these conclusions is why AT’,, must be estimated from the screen height temperatures and not from the soil temperatures. The same problem was also present in the classical derivation due to Brunt as clearly underlined by Srinivasan and Narasimha (1983). An explanation cannot be found in the difference between the temperature values measured at screen height and those evaluated at the first point of each sounding which could, in principle, be different as different instruments measure the two quantities and the averaging times. In fact the average agreement between the two evaluations is good. It must be taken into account that it is quite difficult to measure exactly the temperature of the ground (I =O). Furthermore, even if one is almost sure to have measured such a quantity with a satisfactory de- gree of accuracy, it is to be remembered that the ground surface is the interface between two adjacent media, soil and air, and thus it should reflect the properties of both media. Therefore, it


(f)

can be said that probably the screen height (about 2 m) is near enough to the ground surface to behave as the lower boundary of the heat diffusion and is far enough from the ground to reflect the atmospheric properties which effect the diffusion. In fact, Equation (2) is a solution of the heat diffusion equation describing a heat transfer process in the atmosphere which is either radiative or turbulent (or both together), while in the first atmospheric sublayer adjacent to the ground the molecular conduction is likely to play an important role (Monteith, 1957). As a consequence, the location at which we must set the lower boundary condition should be at a height which is just above this thin layer in order to exclude the layer in which the physical phe- nomenon not taken into consideration by the model can occur. Wind velocities have not been presented and discussed in the present analysis as, in our previous studies, it was found that the correlation coefficient between wind speed at about 100 m and K was approaching zero, i.e. < 0.1 for the PO Valley (Anfossi et al., 1974) and 0.2 to 0.3 for Hay (unpublished data), independently of the average wind speeds at the same height (-2ms - ’ in the PO Valley and 2 10 m s- ’ for the Hay data).

(g) To evaluate Equation (2) it was assumed a value of K equal to 0.34 m2 s- 1 (Anfossi et al., 1974). We tested this choice with an independent estimate of the average air diffusion coefficient K. This was obtained from Equation (3) inserting the measured values of AT,, and starting time t, as above discussed. For the heat flux f in Equa- tion (3) we used Go as stated in a previous paper (Anfossi and Longhetto, 1985). Table 4 (third column) shows the results. It comes out that K assumes a value of about 0.3-0.4 mz s-t, which is in good agreement with the one estimated measuring in the PO Valley the time evolution of the radiative inversion top (Equation (1)). Repea- ting the same computation for the soil (inserting the average value of pc evaluated through Equa- tion (7) in the first 2 cm layer, that is 2.2

x lo6 Jme3K-’ we obtain K=1.07 x 10-7 mz s- ‘. It is interesting to notice that this value is quite similar to that one, K = 0.8 x 10e7 m2 s- ‘, obtained in a different German

site, precisely at Bremen, by Mast et al. (1984). (h) We compared the evaluation of zi by Equations

(1) and (5) and averaged over all profiles obtain- ing the figures quoted in the fourth and fifth columns of Table 4. These figures indicate that, apart from case II, Surridge’s method does not estimate zi in a satisfactory way (this is particularly true for case 111.1). This fact is difficult to understand. In fact, as the method gives reliable estimates of temperature profiles and forces the profiles to coincide with the first two measured points, thus deriving an estimate of zi which varies from one observation to the other one (whereas our method uses always the same parametrization of zi, i.e. Equation (1) with K =0.34 m2 s- ‘) one could expect a better agreement between the experimental zi values and those computed by means of Surridge’s method. This means that even a small inaccuracy of the model (Equation (2)) in representing actual profiles near the surface or limitations in the measurement precision can cause incorrect estimates of zi.

Figures 4 and 5 refer to the prediction of temperature fall and of minimum temperature at ground. In both graphs, on the abscissae axis we indicated the values of the ratio appearing in 1.h.s. of Equation (6) measured each 10 min at Philippsburg. On the ordinate axis we reported the values of the same ratio computed: (a) in Fig. 4, by means of Equation (6) with A= -0.225 and 8=0.187 (Gandia et al., 1985); (b) in Fig. 5, by means of the following relationship

T(O, t) - T(O, L)

T(0, 0) - T(f), fm) = 1 - Jt/t, (8)

which is easily obtained from Equation (3). It appears that Equation (8) and therefore our model (Equa- tion (3)) gives the better agreement. The same conclu- sion comes out from the statistics of the comparison: the correlation coefficient, slope and intercept are:

Table 4. Average values, relative to the data sets considered, of two relevant physical quantities: the heat diffusion coefficient in atmosphere evaluated through Equation (3) and the ratio between calculated (through Equation (1) and Equation (S), respectively) and measured zi. I, II, III, III.1 and III.2

correspond to the same cases of Figs 1, 2, 3 and Tables 1, 2 and 3

Site Type K z;/zy

(m* s-i) Equation (1) Equation (5)

Philippsburg I 0.31 0.83 0.52 (F.R.G.) II 0.44 0.83 1.04

III 0.41 1.07 1.40

Hay III.1 0.39 1.10 3.31 (Australia) III.2 0.39 0.95 1.27

1184

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tit+ *tf + it+ t+ t

t tttt ti t t t tt tt

;;;_

t

.9 1 + tt + +++ + + ++ tt ttt

H t+ t ++ tt t*t

‘1

f +

, t *+++ + +

t t + +

Fig. 4. Scatter diagram of { [ Z’(0, t) - 7’(0, t&l/[: T(0, 0) - 7’(0, tm)] ). In the abscissae we reported the values measured at Philippsburg during the nights every IO min and in the ordinates the corresponding ones esti-

mated by means of Equation (6).

0.89, 0.74 and -0.01 for Fig. 4 and 0.93, 1.02 and -0.02 for Fig. 5. As pointed out in section 1,B should be zero in Equation (6). As a consequence we performed a best fit analysis on our data (setting B=O) finding A= -0.305. However the agreement between measured and calculated values did not improve significantly. As a matter of fact we found the same figure (0.89) for the correlation coefhcient, 0.87 for the slope. and 0.07 for the intercept. We also add that Equation (6) was empirically established. In the original paper (Gandia et nl., 1985) A and B values are annual averages obtained by a best fit analysis of data taken at a particular site (located near the Valentian Coast in Spain). On the contrary Equation (8) was not obtained for a specific site and does not contain any empirical constant.

4 Conclusions

This paper presents and discusses the applications of an analytical model (Equation 2) for the forecasting of nocturnal temperatures in the first layers of the

atmosphere to the data collected during the Mesoklip Experiment in the Rhein ValIey. Previous papers (Anfossi er al., 1976; Kloppel et al., 1978; Anfossi and Longhetto, 1985) dealt with a similar topic but the climatologies of those sites were different from each other and’from the present one, which is typical of a complex terrain. The average agreement was found to be good.

The application of a method suggested by Surridge (1986) to improve the forecastings of Equation (2) is discussed. The comparison of this last method with the Mesoklip temperature profiles did not demonstrate a significant improvement over the original method, rather a worsening of the results in some cases. On the contrary Surridge’s approach fitted the Wangara profiles slightly better than our model.

An interesting result of this study was related to the lower boundary condition of our model. It was found that the more appropriate temperature fall to intro- duce into Eq~tion (2) was that one measured at the screen height and not at z = 0. Disregarding the diffi- culties in meaningfuRy measuring the temperature at z=O, it is suggested that this fact could be related ta the presence of a thin laminar sublayer just above the


I.? -

1.f -

1.0 -

*9 -

.8 -

s ii .7 -

2 i .6 - _ .5 -

.4 -

.3 -

+

.I ,2 .3 .4 .5 ,F; .? .8 .9 1.0 1.1 1.2 1.3

~~~E~

Fig. 5. As in Fig. 3 but the values on the ordinates have been calculated by means of Equation (8).

ground within which heat is transferred by molecular Anfossi D. and Longhetto A. (1985) Cooling processes and

conduction. The capability of Equation (2) to predict the ground

temperatures during the night was checked. It was found that these forecasts are more accurate than those obtained by an empirical formula proposed by Gandia et al. (1985).

Acknowledgements-This work was performed while the author was visiting the Institut fiir Meteorologie und Klima- forschung in Karlsruhe (F.R.G.); he would like to thank Prof. F. Fiedler, Director of the Institute, for his kind hospitality, stimulating discussions and for the permission to use the data from the Mesoklip Experiment, and his colleagues for their friendfy company and discussions. The stay at Karlsruhe was supported by a grant by Progetto Strategico Clima e Am- biente dell’Area Mediterranea del Consiglio Nazionale delle Ricerche.

Caughey S. J., Wyngaard J. C. and Kaimal J. C. (1979) Turbulence in the evolving stable boundary layer. J. ntmos. Sci. 36, 1041-1052.

Clarke R. H. (1970) Observational studies in the atmospheric boundary layer. Q. J1 Roy. met. Sot. %, 91-114.

Clarke R. H., Dyer 3. J., B&ok R. R., Reid D. G. and Troup A. S. (1971) The Wangara experiment: boundary layer data. CSIRO Div. Meteor. Physics. Tech. Rep. No. 19.

Fiedler F. and Prenosil T. (1980) Das Mesoklip Experiment. Mesoskaliges Klimaprogramm in Oberrheintal. Wissen-

schafUiche Berichte des Meteorologischen Instituts det CJni- versitat Karlsruhe 1.

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1186 DOMENICO ANFOSS

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A discussion on nocturnal temperature profiles