Fog prediction at Lisbon Airport using a one-dimensional boundary layer model

Transcription

1 Fog prediction at Lisbon Airport using a one-dimensional boundary layer model Meteorol. Appl. 8, (2001) João Teixeira 1 and Pedro M A Miranda, Centro de Geofisica and Department of Physics, University of Lisbon, Portugal 1 Present affiliation: UCAR Visiting Scientist at the Naval Research Laboratory, Monterey, California, USA Fog is the most important physical process that reduces surface visibility, having a major negative impact on air traffic. In this study, a one-dimensional boundary layer model was developed in order to predict the occurrence and intensity of fog. The model uses a selection of parameterisations, suggested by different authors, and uses the finite element method for the vertical discretisation of the equations. The 1977 Cabauw fog event is used to validate the model and eight fog events at Lisbon Airport are simulated. The validation shows that the model is able to predict realistically the fog s onset, growth and dissipation. The fog forecasts at Lisbon Airport are, in general, quite reasonable, and a simple simulation of a fog event, where advection played a fundamental role, also gives realistic results. 1. Introduction Reduced visibility at and near the surface has always been one of the main concerns in aeronautical meteorology. This problem seriously affects the taking-off and landing of aircraft. Reduced visibility at airports means that flights have to be delayed or diverted, which can result in heavy financial losses for the aviation industry. Among the different atmospheric processes that reduce visibility at the surface, fog is probably the most important. According to WMO (1966), fog is a suspension of water droplets or ice in the atmosphere that reduces horizontal visibility at the surface to less than 1 km. From a physical point of view, the reduction of visibility associated with fog is the result of the dispersion of visible radiation by water droplets or crystals and depends not only on the total amount of condensed water per volume, but also on the detailed distribution of drop sizes. Fog forecasting is usually performed by means of empirical rules or statistical methods (e.g. Tremant, 1989). Although new cloud schemes in global (Tiedtke, 1993) or regional (Zhao & Carr, 1997) Numerical Weather Prediction (NWP) models allow the existence of fog, these models are not yet able to forecast fog with a reasonable degree of accuracy, mainly due a lack of resolution (horizontal and vertical) and adequate parameterisations. Some new studies (Teixeira, 1999), however, seem to indicate the possibility of improvements in this respect. Reiff (1987) presents a review of operational models to forecast fog, and a review on numerical fog studies in general can be found in Bergot & Guedalia (1994). Most numerical studies of fog have used one-dimensional (1D) or two-dimensional (2D) boundary layer models, initially with a relatively simplified physics (Zdunkowksi & Barr, 1972; Brown & Roach, 1976), but with more elaborate physics in recent years (Musson-Genon, 1987; Turton & Brown, 1987; Bott et al., 1990). While there are new developments in threedimensional (3D) mesoscale studies of boundary layer clouds and fog (e.g. Golding, 1993; Ballard et al., 1991), 1D models and in particular one-column versions of climate or NWP models have been increasingly used to develop and validate fog and stratus parameterisations (e.g. Bretherton et al., 1999; Duynkerke et al., 1999; Teixeira, 1999). As with other boundary layer processes, which are generally assumed to be driven by the large-scale flow and thermodynamics, fog is an obvious candidate for parameterisation. Another good reason for the continuing interest in these simple models is the fact that the physical packages of global models are essentially 1D, making the development and validation of one-column models an important step in the improvement of weather and climate simulations. At Lisbon Airport fog is a relatively common and important event, with an average of 28 days per year of reported fog. According to the local forecasters, it tends to occur not only as pure radiation fog but also associated with particular advection patterns, justifying the hypothesis that, in some cases, it is generated over the Tagus estuary and then advected to the airport area. In order to better understand and predict this type of event, the Portuguese Institute of Meteorology installed two automatic weather stations for fog detection in These stations were positioned between 497

2 J Teixeira and P M A Miranda the Tagus estuary and the airport and were kept in continuous operation between June 1992 and December Some of those data are analysed in this paper. Prior to this work, the problem of forecasting fog at Lisbon Airport had already been tried with statistical methods. It was the main purpose of this work to develop and apply a 1D boundary layer model to simulate and study fog at Lisbon Airport. A version of this model has already been used as the atmospheric boundary layer component of a wind-wave prediction system (Teixeira et al., 1995). In section 2 the model is described. A simple validation of the model is presented in section 3. The results from the simulation of some fog events at the Lisbon Airport are presented in section 4. In section 5 a simulation with a simple advection pattern is presented in order to try to simulate a fog event that was generated over the Tagus estuary and then advected to the airport area. 2. Model description The 1D boundary layer model used in the present study has prognostic equations for the following mean variables: horizontal wind components, liquid water potential temperature and total specific humidity. A prognostic equation for the turbulent kinetic energy is also used for the turbulence closure. Some 1D studies of fog (e.g. Musson-Genon, 1987) use models with rather elaborate short-wave radiation schemes. One of the main objectives of this work is to show that a simpler parameterisation of the effects of solar radiation can be used with a good degree of accuracy. The interaction between fog droplets and solar radiation is, in the present case, parameterised using flux profiles of a simple prescribed shape proposed by Hanson & Derr (1987). The other main difference between the present model and other published material is the use of linear finite elements for the vertical discretisation of the prognostic equations Model equations For a moist atmosphere it is convenient to use thermodynamic variables that are conserved during phase transitions, such as the liquid water potential temperature (θ 1 ) and the total specific water content (q t ) given by: θ L v = θ 1 ct q p 1 1 q t = q + q l content. The reader is referred to Betts (1973) and Deardorff (1976) for details. Under horizontally homogeneous conditions, and assuming incompressibility, the momentum, water conservation and thermodynamic equations can be written in the standard form: u = ( ) t z wu + f ( v V g ) v = ( ) t z wv f ( u U g ) qt t z wq G t z = ( ) + + = ( ) 1 ρ θ1 t z w θ θ 1 Tc F + L G z z + A where the bar denotes mean quantities and the prime refers to the fluctuations in the quantities, u and v are the horizontal wind components, w is the vertical wind component, U g and V g are the geostrophic wind components, f is the Coriolis parameter, F is the radiation flux and G is the drop settling flux. For convenience, let ψ stand for any of the prognostic variables (u, v, q t or θ 1 ), then (w ψ ) is the vertical component of the turbulent flux of property ψ, and A ψ is the horizontal advection of property ψ. As can be noted in the previous equation set, horizontal homogeneity is only strictly assumed for all turbulent terms and for the advective terms in the momentum equations. A horizontal pressure gradient is implicit in the geostrophic wind terms and some room is given for horizontal advection of temperature and humidity, although those terms must be given, or computed from the known variables. As a consequence, 1D models can only deal with either radiation fogs or with very simple advection fogs, where the advective terms can be parameterised. The model includes a fifth prognostic equation for the mean turbulent kinetic energy (TKE), which is written as (e.g. Stull, 1989): e = + t z we wp wu u wv v z z + g ρ T w θ v ε o o where θ v is the fluctuation of the virtual potential temperature, p is the pressure and ε represents the TKE dissipation. p A qt θ1 where L v is the latent heat of vaporization, c p is the specific heat of air at constant pressure, T is the temperature, q is the specific humidity and q l is the liquid water 498

3 2.2. Parameterisation schemes (a) Turbulence The model follows the K-diffusion approach to relate the sub-grid turbulent fluxes and the dissipation ε with the mean variables of the model. The crucial point in the closure scheme is the establishment of a relationship between the diffusivity coefficient and the TKE, in what is sometimes referred as a 1.5 order closure (e.g. Therry & Lacarrère, 1983). The mixing length and the dissipation length are computed using the diagnostic relations proposed by Therry & Lacarrère (1983), as used in Musson-Genon (1987). The height of the boundary layer is considered to be the level where the value of TKE is 0.01 m 2 s 2. (b) Radiation The broad band flux emissivity method (Stephens, 1984) is followed for the parameterisation of long-wave radiation. Inside the fog the emissivity includes contributions from water vapour and liquid water droplets. For the liquid water emissivity a relation proposed by Stephens (1978) is used. Fog prediction using a one-dimensional boundary layer model The model uses a statistical sub-grid scale condensation scheme based on ideas initially developed by Sommeria & Deardorff (1977) and Mellor (1977). The cloud cover and the liquid water are diagnosed from the mean values of the conserved variables total water and liquid water potential temperature, the variance of these variables and a distribution function. The distribution function used is the one proposed by Bougeault (1981). This distribution function has already been used successfully by Musson-Genon (1987) in a 1D boundary layer model of fog Boundary conditions and numerical schemes At the model s lower boundary, the mean variables and the TKE are imposed at z 0. At the upper boundary (z = 2 km) the fluxes of the mean variables and of TKE are set to zero, except for liquid water potential temperature, for which a vertical gradient of K m 1 is imposed, a typical stratification above the boundary layer. It is important to note that in this study the surface boundary conditions are not predicted. It is clear that to produce operational fog forecasts, the surface conditions must be predicted as well. In any case, as a first approach to the problem, we think that isolating the atmospheric modelling problem from the surface and soil, by imposing the observed values of surface parameters in our simulations, is an adequate way to pursue this study of fog. As mentioned before, we follow a simple procedure suggested by Stull (1989) to parameterise the interaction between fog droplets and solar radiation, in contrast with other models that use more detailed schemes. The solar radiation flux profile inside the fog is prescribed, following Hanson & Derr (1987), as a combination of exponential functions. This profile is ultimately based on the values of the cloud albedo, cloud absorption and the downward solar flux at the top of the fog. We follow the relations suggested by Manton (1980) between the cloud albedo, absorption and the liquid water path. (c) Drop settling In spite of the fact that fog droplets rarely exceed 20 µm, Brown & Roach (1976) found that drop settling can play a significant role in the evolution of fog. This is due to the small updraft vertical velocity inside radiation fog. In order to represent drop settling, the parameterisation used by Brown & Roach (1976) is followed; the flux G is parameterised as G = W g q l where W g is the mean settling vertical velocity that is linearly related to the liquid water. (d) Sub-grid scale condensation The spatial discretisation of the equations uses a finite elements method, with linear base functions. This method has shown good results in previous studies with 1D boundary layer models (e.g. Mailhot & Benoit, 1982). The finite elements method presents some general advantages, when compared with the finite differences method (for a review, see Temperton, 1991). Two sets of base functions are used in a staggered grid. The first set includes the mean variables and the forcing functions (geostrophic wind, advection), whereas the second set includes the turbulent fluxes and the TKE. In all experiments shown, both grids are regularly spaced in the vertical. The time discretisation of the equations is done implicitly, following Richtmeyer & Morton (1967). 3. Validation experiment In order to validate the model, a fog event that occurred at Cabauw in the Netherlands, on 3 August 1977, was simulated. This particular fog situation has already been studied using 1D boundary layer models developed by Grandin (1983) and Musson-Genon (1987) (hereafter referred to as G83 and M87, respectively). All the data used in this validation study was taken from G83 and M87. The data is described in G83 and M87, and consists of the following: vertical profiles of the mean variables, taken as initial conditions at UTC, 3 August 1977; 499

4 J Teixeira and P M A Miranda hourly series of surface values of the mean thermodynamic variables; and hourly series of the surface geostrophic wind and hourly series of the temperature wind in the layer m. The initialisation of TKE is done using a spin-up procedure (e.g. Burk & Thompson, 1982). The geostrophic wind in the layer m is calculated using the thermal wind relations and, above this layer, the geostrophic wind is considered constant. The temperature advection is also determined, for the layer m, using the thermal wind values. The observed temperature at 0.45 m height is taken as the model surface temperature. It starts with a value of about 15 C at UTC, decreasing to a minimum value of about 11 C just after UTC. After that, the surface temperature, increases almost linearly to just over 16 C at UTC. The simulation uses a time step of 60 s and a vertical grid spacing of 20 m. To relate the simulated liquid water content with visibility, a diagnostic relation suggested by Kunkel (1984) is used: where VIS is the visibility and β = 144.7(ρq - 1) The time evolution of the simulated visibility profile is presented in Figure 1(a), which can be compared with Figure 1(b) showing observed visibilities. The analysis of the figures shows that the fog s generation and initial growth (until around UTC) is reasonably well simulated. However, the minimum of visibility (maximum of liquid water) observed around UTC and the weakening of the fog between and UTC are not well simulated, either by the present model or by M87 and G83. On the other hand, the model predicts local minimum values of visibility (<60 m) a few metres below the fog s top, in agreement with observations, but not necessarily at the right time. There is a slight overestimation of the fog s depth. However, this overestimation rarely exceeds 20 m, which is the value of the model s vertical resolution. So we do not consider this to be a serious problem of the model. The fog dissipation period is again reasonably well simulated by the model. This result shows that the simplified parameterisation scheme, which was used for the short-wave radiation effects, is able to generate a realistic simulation of the fog s dissipation stage. It can be said that, in general, the model performs reasonably well in simulating this particular fog event. It is interesting to note that these results, although far 500 VIS = ln ( 002. ) β Figure 1. Vertical profile of visibility for the fog event at Cabauw. (a) Numerical simulation of evolution; contours are 60 and 300 m. (b) Observed evolution; contours are 30, 60 and 300 m. Courtesy of Luc Musson-Genon. from perfect, seem to compare better with the observations, in terms of fog onset and evolution, than LES (Large Eddy Simulation) model results from a study of the same case (Nakanishi, 2000). It is not our intention, however, to claim that a 1D model can represent the general properties of fog in a more accurate way than LES models. But the fact that the 1D model s visibility looks more realistic than the LES visibility, shows the difficulty and complexity of fog simulation and prediction. It also suggests that 1D fog prediction can still be a very viable option. In order to assess the role of the different processes on the fog s evolution, the temperature tendencies were analysed. The horizontal advection tendency is constant during the simulation and equal to 0.18 K day 1. Until the onset of fog the cooling tendencies close to the surface from radiation and turbulent mixing are almost constant in time and with values of the same magnitude of the advection values. After the fog s onset, the liquid water increases and as a consequence the long-wave cooling starts to play a major role, with large values close to the fog s top as is typical of stratiform boundary layer clouds. After sunrise, the surface temperature increases and the solar radiation starts to penetrate into the cloud, warming the fog. These effects together with the long-wave cooling at the top produce a more intense turbulent mixing that contributes to an intensification of the fog event for a couple of hours, until it finally dissipates the fog. Figure 2 shows the evolution of the profiles of temperature and specific humidity. These profiles seem to be related, as expected, since close to the surface the atmosphere is in a quasi-saturated state. At UTC

5 a pronounced surface inversion can be seen on the temperature profile, characteristic of the onset of fog. At UTC, a 120 m mixed layer, in both T and q, has been formed. If this is compared with the results shown in Figure 1 it is possible to identify this mixed layer as the fog itself. The mixed layer is a characteristic of deep radiation fog (e.g. Brown, 1987; Stull, 1989). The inversion is now located at the fog s top, where a strong radiative cooling process is active. At UTC the temperature profile becomes unstable due to the solar heating of the surface and the consequent generation of turbulence. In general, the simulated structure of both T and q is quite realistic when compared with the observations. The overall behaviour of the model in predicting boundary layer properties and the interaction with the surface are discussed in Teixeira et al. (1995). The validation result shown here is just an example of what the model is able to achieve in terms of fog simulation. A much more detailed validation of the model is discussed in Teixeira & Miranda (2000), where the model s sensitivity to the parameterisation of some different physical processes is also studied. Fog prediction using a one-dimensional boundary layer model 4. Fog events at Lisbon Airport After the validation, the model is used to simulate eight cases of fog that occurred at Lisbon Airport, which correspond to all cases of dense fog observed in the chosen winter. The data used in these simulations consisted of the following: vertical profiles of the initial conditions for the mean variables (θ, u, v, q ) and initial surface pressure, p, at UTC for each one of the eight cases, taken from radiosondes launched at Lisbon Airport; hourly series of the surface values of the variables mentioned above; and geostrophic wind at the surface, 850 hpa and 700 hpa estimated from synoptic charts at and UTC and linearly interpolated in the vertical and in time. The synoptic charts were also used to evaluate the advection of temperature, using the thermal wind relation. All simulations use a time step of t = 60 s and a vertical grid spacing of z = 20 m. For each case, three different sets of simulations were performed, in order to study the sensitivity of the results to (a) the inclusion of horizontal advection of temperature and (b) to the lower boundary condition for moisture. The results are summarized in Table 1. In general, the results are reasonable. In the default simulations (no advection of temperature, no surface saturation Figure 2. Numerical simulation of the Cabauw fog event. Vertical profiles of (a) temperature and (b) specific humidity at 00.00, 03.00, and UTC. imposed) and when the advection of temperature is included, the model is able to predict four of the eight cases of observed fog. When the saturation of the surface moisture is imposed, the model is able to predict seven of the eight cases of fog. The results are not very sensitive to the imposed temperature advection. It could be argued that this is the result of the method used for the evaluation of thermal advection, which disregards mesoscale features, but it is consistent with the hypothesis that all those cases correspond to events of radiation fog. In Figure 3 the evolution of the simulated and observed visibility are shown for the two best cases, occurring on 21 January and 2 December The model results are quite reasonable, when compared with the observations. In Figure 3(a), the evolution of the simulated visibility is quite remarkable, when compared with the observations. This is true for most of the time, but after 501

6 J Teixeira and P M A Miranda Table 1. Summary of the results of the fog prediction experiments at Lisbon Airport (see text for details). Yes means that fog was predicted by the model in that run. Case Control simulation Temperature advection Saturated surface 15 November1989 No No No 2 January 1989 No No Yes 4 January 1989 Yes No Yes 21 January 1989 No No Yes 1 December 1989 Yes Yes Yes 2 December 1989 Yes Yes Yes 5 January 1989 Yes Yes Yes 7 February 1990 No Yes Yes that, and during the fog evolution, the model results are quite realistic. Even the dissipation stage is simulated in a reasonable way, although with a certain delay. These good results suggest that, in some situations, the use of a 1D model might be enough to achieve a good fog forecast at Lisbon Airport. A more detailed analysis of all simulations emphasises the importance of the schemes used to deal with water and temperature fluxes into the model s domain. It is clear that to achieve a good fog forecast, a model to forecast the surface conditions must be introduced and that horizontal inhomogeneities must be taken into account. A simple example is shown in the following section. 5. Simulation with a simple local advection pattern It has been known for some time that the Tagus estuary is a possible source for many cases of fog observed at Lisbon Airport (see Figure 4). In order to study this situation, and to be able to forecast (nowcast) fog at Lisbon Airport, the Portuguese Meteorological Institute installed in the winter of 1992/93 a couple of automatic fog detection stations between Lisbon Airport and the Tagus estuary. Figure 3. Comparison between observed and simulated visibility values for (a) case 1 (21 January 1989) and (b) case 2 (2 December 1989) of fog at Lisbon Airport UTC the model is not able to simulate the dissipation of fog in a realistic way. In Figure 3(b) there is a discrepancy between the model and the observations at UTC, because the observed fog started already before midnight, while there is no fog in the initial state of the model. After 502 A good example of the output of these stations can be found in Figure 5; this shows the time evolution of the measured visibility at both stations between UTC and UTC for 9 January In both stations the visibility changes rather abruptly and fog is observed at Station 1 more or less one hour earlier than at Station 2. Horizontal advection is thus the simpler explanation for the observed behaviour. The values measured at the stations provide some helpful information in order to incorporate a simple horizontal advection term for the total water content in the 1D model: q t = C t adv q t x

7 Fog prediction using a one-dimensional boundary layer model Figure 6. Comparison between observations (smoothed with a moving average filter) and model results at Station 1 for the advection fog case (9 January 1993), from to UTC. Figure 4. Location of Lisbon Airport and the Tagus estuary. In this case, the introduction of a simplified water advection helps to solve the problem of estimating the horizontal inhomogeneities of the thermodynamic variables. It should be noted, however, that this is a very crude way of estimating the horizontal advection terms and that it has been used here only as a test. 6. Conclusions Figure 5. Observed visibility at the two automatic fog stations near Lisbon Airport in the advection fog case (9 January 1993), from to UTC. where x is the distance between the stations, q t = 1g kg 1 and C = x/ t adv is the advection velocity (with t adv 1 h, which is the approximate time lag between the arrival of fog at Station 1 and Station 2). This advection term was introduced into the 1D model, in the first 100 m above the surface and until the fog s onset, in an experiment that was set up to simulate the fog evolution at Station 1. In this study a time step of t = 30 s and a grid spacing of z = 4 m were used. Figure 6 shows the time evolution of the measured and simulated visibility at Station 1. The patterns of variation of the simulated and measured visibility are remarkably similar. The most striking feature is the arrival of fog, almost at the same time, both in the observations and in the simulations. Also, after the arrival of fog at the station, the simulated values of visibility are very close to the observed ones. In fact, the difference between the simulated and measured values of visibility, after the fog arrival, is typically never greater than 30 m. In this study, a 1D boundary layer model was developed, in order to study fog at Lisbon Airport. The 1D model has prognostic equations for the mean quantities (horizontal wind components, liquid water potential temperature and total specific water) and for the mean turbulent kinetic energy, which is used to parameterise the turbulent fluxes of the mean variables. The model uses simple parameterisations of the effects of solar and long-wave radiation, with a good degree of accuracy. The equations are discretised using the linear finite elements, in a staggered grid. The validation study showed that the fog s onset and initial growth is reasonably well simulated. The model predicted minimum values of visibility a few metres below the fog top in agreement with observations. The fog dissipation stage was again reasonably well simulated by the model. The main conclusion from the validation is that the use of a simple solar radiation parameterisation scheme is probably good enough for the forecast of fog dissipation. The model was used to simulate eight cases of fog that occurred at Lisbon Airport. The fog forecast results were reasonable and not very sensitive to the imposed temperature advection. The analysis of the time evolution of the simulated and observed visibility for the two best simulations suggests that, in some situations, the use of a 1D model might be enough to achieve a good fog forecast. In order to study fog events that are generated over the 503

8 J Teixeira and P M A Miranda Tagus estuary and then advected to the airport area, a further case was analysed, using data from two automatic fog detection stations located between the airport and the estuary. In both stations the visibility was found to change abruptly, allowing for the evaluation of a mean propagation speed of the fog front, which was used to incorporate a simple horizontal advection term for the total water content in the 1D model. An analysis of these 1D fog simulations at Station 1 showed a remarkable similarity between the simulated and measured visibility. The methodology followed in the present study is rather simple and cannot account for all features of the process of onset and dissipation of fog. In many cases, namely in the area of Lisbon, mesoscale processes are dominant in the definition of the horizontal advection of atmospheric properties and it would be impossible to address the problem without a full 3D mesoscale model. While these models are available, they are still much more difficult to use for operational boundary layer prediction, both for computational and data initialisation reasons, leaving some space for the practical use of simpler models, like the one presented in this paper. In any case, in order to produce operational fog forecasts with the 1D model presented in this paper, we will need to be able to predict the surface boundary conditions, by coupling the model with a surface scheme, and the large scale forcing. This can be achieved by coupling the 1D model with the output of a global or limited-area NWP model, much in the same way as done by Musson-Genon (1989). Acknowledgements The authors acknowledge the contribution of the Portuguese Institute of Meteorology, and in particular of João Jacinto Ferreira and Fernanda Tavares, for providing the data used in this study and for their support. We also would like to thank A. Beljaars, H. Wessels and L. Musson-Genon for providing information concerning the validation fog case. Comments made by two anonymous referees are also gratefully acknowledged. This study was accomplished with the financial support of the Portuguese Foundation for Science and Technology under Grant 3/3.2/EMG/1968/95. References Ballard, S. P., Golding, B. W. & Smith, R. N. B. (1991). Mesoscale model experimental forecasts of the Haar of Northeast Scotland. Mon. Wea. Rev., 119: Bergot, T. & Guedalia, D. (1994). Numerical forecasting of radiation fog. Part I: Numerical model and sensitivity tests. Mon. Wea. Rev., 122: Betts, A. K. (1973). Non-precipitation cumulus convection and its parameterization. Q. J. R. Meteorol. Soc., 99: Bott, A., Sievers, U. & Zdunkowski, W. (1990). A radiation fog model with a detailed treatment of the interaction between radiative transfer and fog microphysics. J. Atmos. Sci., 47: Bougeault, P. (1981). Modeling the trade-wind cumulus boundary layer. Part I: Testing the ensemble cloud relations against numerical data. J. Atmos. Sci., 38: Bretherton, C. S., Krueger, S. K., Wyant, M. C., Bechtold, P., van Meijgaard, E., Stevens, B. & Teixeira, J. (1999). A GCSS boundary layer model inter-comparison study of the first ASTEX Lagrangian experiment. Bound. Layer Meteorol., 93: Brown, R. & Roach, W. T. (1976). The physics of radiation fog. II. A numerical study. Q. J. R. Meteorol. Soc., 102: Burk, S. D. & Thompson, W. T. (1982). Operational evaluation of a turbulence closure model forecast system. Mon. Wea. Rev., 110: Deardorff, J. W. (1976). Usefulness of liquid-water potential temperature in a shallow cloud model. J. Appl. Meteorol., 15: Duynkerke, P. G., Jonker, P. J., Chlond, A., van Zanten, M. C., Cuxart, J., Clark, P., Sanchez, E., Martin, G., Lenderink, G. & Teixeira, J. (1999). Intercomparison of three- and one-dimensional model simulations and aircraft observations of stratocumulus. Bound. Layer Meteorol., 92: Golding, B. W. (1993). A study of the influence of terrain on fog development. Mon. Wea. Rev., 121: Grandin, G. (1983). A one-dimensional PBL model with a sub-grid scale condensation scheme for stratiform clouds and fogs. Report 72, Uppsala University, Uppsala, Sweden. Hanson, H. P. & Derr, V. E. (1987). Parameterization of radiative flux profiles within layer clouds. J. Clim. Appl. Meteorol., 26: Kunkel, B. A. (1984). Parameterization of droplet terminal velocity and extinction coefficient in fog models. J. Climate Appl. Meteorol., 23: Mailhot, J. & Benoit, R. (1982). A finite-element model of the atmospheric boundary layer suitable for use with numerical weather prediction models. J. Atmos. Sci., 39: Manton, M. J. (1980). Computations of the effect of cloud properties on solar radiation. J. Rech. Atmos., 14: Mellor, G. L. (1977). The gaussian cloud model relations. J. Atmos. Sci., 34: Musson-Genon, L. (1987). Numerical simulation of a fog event with a one-dimensional boundary layer model. Mon. Wea. Rev., 115: Musson-Genon, L. (1989). Forecasting in the vertical with a local dynamical interpretation method. Mon. Wea. Rev., 117: Nakanishi, M. (2000). Large-eddy simulation of radiation fog. Bound. Layer Meteorol., 94: Oliver, D. A., Lewellen, W. S. & Williamson, G. G. (1978). The interaction between turbulent and radiative transport in the development of fog and low-level stratus. J. Atmos. Sci., 35: Pilie, R. J., Mack, E. J., Rogers, C. W., Katz, U. & Kocmond, W. C. (1979). The formation of marine fog and the development of fog-stratus systems along the California coast. J. Appl. Meteorol., 18: Reiff, J. (1987). Forecasting boundary layer clouds and fog: a review on operational numerical models. In Proc. of the International Symposium on Mesoscale Analysis and 504

9 Fog prediction using a one-dimensional boundary layer model Forecasting, August 1987, Vancouver, Canada. Published by ESA, ESTEC, Noordwijk, The Netherlands, Richtmyer, R. D. & Morton, K. W. (1967). Difference Methods for Initial-Value Problems. Wiley-Interscience, New York, 405 pp. Sommeria, G & Deardorff, J. W. (1977). Subgrid-scale condensation in models of nonprecipitating clouds. J.Atmos. Sci., 34: Stephens, G. L. (1978). Radiation profiles in extended water clouds. II: Parameterization schemes. J. Atmos. Sci., 35: Stephens, G. L. (1984). The parameterization of radiation for numerical weather prediction and climate models. Mon. Wea. Rev., 112: Stull, R.B. (1989). An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, The Netherlands, 666 pp. Teixeira, J. (1999). Simulation of fog with the ECMWF prognostic cloud scheme. Q. J. R. Meteorol. Soc., 125: Teixeira, J., Abreu, M. P. & Soares, C. G. (1995). Uncertainty of ocean wave hindcasts due to wind modelling. J. Offshore Mechanics and Arctic Engineering, 117: Teixeira, J. & Miranda, P. M. A. (2000). Simulation of fog: model validation, sensitivity and case studies at Lisbon Airport. CGUL Technical Report No 2000/01, Lisboa, Portugal, 36 pp. Temperton, C. (1991). Finite elements. In Proc. ECMWF Seminar in Numerical Methods in Atmospheric Models. ECMWF, Reading, United Kingdom. Therry, G. & Lacarrère, P. (1983). Improving the eddy kinetic energy model for planetary boundary layer description. Bound. Layer Meteorol., 25: Tiedtke, M. (1993). Representation of clouds in large-scale models. Mon. Wea. Rev., 121: Tremant, M. (1989). The forecasting of sea fog. Meteorol. Mag., 118: Turton, J. D. & Brown, R. (1987). A comparison of a numerical model of radiation fog with detailed observations. Q. J. R. Meteorol. Soc., 113: WMO (1966). International Meteorological Vocabulary. World Meteorological Organization, Geneva, 275 pp. Zdunkowski, W. G. & Barr, A. E. (1972). A radiative conductive model for the prediction of radiation fog. Bound. Layer Meteorol., 2: Zhao, Q. & F. H. Carr (1997). A prognostic cloud scheme for operational NWP models. Mon. Wea. Rev., 125: