Department of Hydraulic Engineering and Environment,
Polytechnical University of Valencia, Ap. de Correos 22012,
46080 Valencia, Spain
ABSTRACT
We have elaborated an interpolation numerical wind model to be used as inputdata in regional air pollution dispersion studies. This model evaluates theprincipal components for the wind observations and the wind patterns for thegrid nodes in a complex terrain by solving a partial differencial equationapplying the Finite Element Method. The wind field for every measurementtime is obtained as a linear combination of the wind patterns, with coefficientsevaluated by means of the simultaneous observations and the principalcomponents. This is part of a study to evaluate the dispersion capacity ofair pollution in the Comunidad Valenciana (Spain) related with land use plans.
1.- INTRODUCTION
The study of the air pollution dispersion in a regional scale requires theknowledge of the wind field over a large area, generally with irregular andcomplex terrain; these features make those studies more difficult. The use ofGaussian models, which are easy to implement in a computer, is in those casesinappropiate, because these models are based on the hypothesis that the terrainaround the emission source is uniform; besides these models suppose uniformityin wind direction and velocity.
In a regional scale the topography causes over the air movement effectsof deviation, sloping, channeling, etc; those effects must be known to elaborateair pollution dispersion models which cover large areas. Besides, we should
consider that such effects can be different depending on the predominant winddirection, due to the irregular topographic relief of the zone we are studying.
By these reasons, the knowledge of the wind field, which determine thepollutant dispersion, is a relatively complex task, mainly if we want to use theclassical equations of turbulent fluid flow, namely continuity, Navier-Stokes andenergy equations. This problem formulation gives rise to a non linear set ofpartial differential equations, whose solution requires by one side initial andboundary conditions extremly difficult to know, and by the other therequirements of computer time and memory are so big that make impossibleits application in normal air pollution dispersion studies.
Those complications have led to the interpolation models, which use windmeasurements in several meteorological stations inside the study zone; thosemodels try to determine the best wind fields that match to the measurementsand can be used as input data in air pollution calculations. The model we haveused can be included in this type of model
2.- DATA TO BE USED IN THE MODEL
The type of data which the numerical model requires to make calculations isvery important for its operation. By one side it needs meteorologicalmeasurements and by the other it needs information about terrain topography.With regard to meteorological data, we have used simultaneous windmeasurements (velocity and direction) in several meteorological stations locatedinside or close to the study area. We have collected data from those stationsfor a determinated period, and we have extrapolated them to a height of 10 mover the terrain. To carry out this extrapolation we have used the power lawexpression:
V(z) ,z" (1)
where V^ is the measured wind speed at height Zj, and the exponent n is givenby Justus and Mikhail [5]:
037 - 0/0881 InV, ,_n =
1 - 0/0881 ln(z/10)
In this way we have elaborated a data base of wind speed componentsto be used in our numerical model.
The study zone is going to be a rectangular area of 220 Km in X (Est)direction and 330 Km in Y (North) direction, which includes the zone we areinterested to study, that is the territory of the Comunidad Valenciana (Spain).In order to handle the terrain topography we have divided the study zone in44x66 elements of 5x5 Km and we have incorporated to the model as a database the average height over the sea level for every one of the 45x67 nodesincluded in the grid.
The study zone, the profile of the Comunidad Valenciana and the locationof the meteorological stations used in this study are shown in Fig. 1.
Fig. 1.- Study zone and location of meteorological stations.
The numerical model has been developed following the next phases:
3.1.- Calculation of principal components.The principal components method is based on the hypothesis that every set ofsimultaneous wind observations can be represented as a linear combination ofthe mean wind and these principal components.
We call V(r) to the column vector formed with the (u,v) windcomponents which are observed simultaneously in N meteorological stationsfor the instant r; directly you can write this vector as:
2N
where: V^ = Mean wind component vector.Vj = Vectors corresponding to the wind principal
components. They form a base independent oftime.
N = • Number of stations where we have measurements.a (r)= Time scalar functions. They are calculated in the
following way:
a;(f) = [V(r) - VJ *V. (4)
Using Lagrangian multipliers we deduce that the Vj vectors, which we arelooking for and which give the best adjust of the wind observations are theeigenvectors of the covariance matrix A, which is formed as A = MxM\M being the matrix constructed with all the wind measurements along the timeperiod that has been considered, and whose columns are the differences
On the other hand the eigenvalues of A represent the relativeeigenvectors weight in terms of decreasing significance in representing thedata, and we will consider how many principal components we can take torepresent the original observed data in a rather approximated way.
By this means, starting with the observations in a number ofmeteorological stations, we calculate the arithmetic mean of the (u,v)components and the principal components in these stations. We haveconsidered seven stations (Almazora, Murcia, Alicante, Manises, San Javier,Tortosa and El Altet, Fig. 1) because in all of them 4 daily wind observations
are available during the period 1979-1985, and so 14 principal componentshave been evaluated. Besides the eigenvalues of the matrix A indicate that thesix former principal components represent more than the 80 % of windobservations; by this reason only these six ones are going to be used inapproximated studies.
3.2.- Calculation of wind patterns.Now, we pretend to obtain approximated values of mean wind and principalcomponents in every node of the grid by means of those obtained inmeteorological stations. We are going to do this by using an interpolationmodel based on the continuity equation.
formWe assume that the wind vector can be separated into two parts in the
V = V, + V/ (5)
where VQ is an initial wind field, which is only function of wind observationsin different stations and V modifies the initial wind field in order to satisfy thecontinuity equation.
In the process to obtain the model formulation a variational functionalmust be minimize. This variational functional is:
E(u,v,w,X) = (w-w/
(6)> duX _
axdv dw T , , ,_ + _] dx dy dzdy dz *
where X is a Lagrangian multiplier used to take into account the continuityequation. After some mathematical considerations we arrive to the followingpartial differential equation:
dxi ax 2_ax
\CL* dy dzi ax
2aJ az ay az,
(7)
whose solution enables us to determine the wind field from the expressions:
To solve the equation (7) we have used the Finite Element Method. Thisapplication has been done taking into account the following considerations:
4.1.- Integration domain.It is constituted by an hexaedron of 220x330 Km, with a vertical projectionshown in Fig. 1; and includes all the territory of the Comunidad Valenciana.The lower boundary is formed by the terrain topography, and the upperboundary is defined by the expression:
H(x,y) = H, + kh(x,y) + (1-k) h^ (11)
where: H(x,y) = Height of the upper boundary over the point (x,y).h(x,y) = Height of the point (x,y) above the sea level.H* = Parameter representing the height of the upper boundary over the
ground.k = Parameter to define the relative gradient of the upper boundary.hg = Mean height of the terrain above the sea level.
If k = 0, H(x,y) = H. + hg = constant : upper boundary is horizontal.If k = 1, H(x,y) = H* 4- h(x,y): upper and lower boundary are parallel.
The integration domain is divided into 44 elements in X direction, 66elements in Y direction, and 6 elements in Z direction. Every element has avertical projection of 5x5 Km, and its height changes according to its relativeposition; those in first level (directly over the ground) are 1 m height, thosein second level, 9 m, and those in third level, 40 m. Elements in the threeupper levels have a height of one third of the distance between the upper andthe lower boundary, decreased in 50 m which correspond to the three lower
levels. In this way we perform a better study of the lower zone, which has thehighest wind velocity gradient.
4.2.- Coefficient values.After carrying out a study of the magnitude order for the coefficients inexpression (7), we have used the values a, = 0.000224 s /m and#2 = 7.071 s*'Vm. These values give a rate (ai/aj? in the neighborhood of10"* and, according to Dickerson [3], correspond to the relative magnitudes ofthe vertical and horizontal fluxes. For the upper limit done by expression (11),H, = 1000 m and k = 0.75 have been used.
4.3.- Initial wind Field.It is constituted by the mean value of the wind components and the fourteenprincipal components interpolated to all the nodes in the integration domain;to do this the following expressions have been used:
E "A/ E v/r/«o(*,y) = 4 - ; v,,(x,y) = - (12)
E W E i/
where: Ug(x,y), VQ(X,V) = Wind components extrapolatedto the point (x,y), at a height of 10 m over theground.Uj, Vj = Wind components in the station j, at aheight of 10 m over the ground.TJ = Distance between the j station and the point
4.4.- Boundary conditions.It is possible to perform several combinations of boundary conditions to solvethe equation (7), but not all of them give wind patterns which can beconsidered acceptable. We have tested different combinations of the conditionsX = constant or dX/dn = 0 on the boundaries using a reduced model whichhas elements of 10x10 Km. This new grid contains 22 elements in X direction,33 elements in Y direction and 5 elements in Z direction. For each one of thepossible combination we have calculated mean wind speed and the first sixwind patterns (which represent more than 80 % of the observations). In somecases very distorted patterns and unreasonable values of wind speed in a lot ofstudy nodes have been obtained.
So, we discarded the combinations of boundary conditions that suppliedno acceptable results and accepted the unique case for which logical windpatterns had been obtained. The boundary conditions corresponding to this casewere X = 0 on the North, West and lower boundary and d\/dn = 0 on theEast, South and upper boundary.
The boundary condition dX/dn = 0 indicate that in the correspondingboundary there is no correction of the initial wind field. It seems quite logicalon the East and part of the South boundary because they are taken over sea.It is worthwhile to denote that the boundary conditions used by Dickerson(1978), those are X = 0 on the lateral boundaries and d\/dn = 0 on the lowerboundary, gave distorted wind patterns with our model.
5.- RESULTS OBTAINED
We have used our model to calculate wind patterns to 10 m height over theterrain for the mean wind (VpJ and the fourteen principal components (VpJcorresponding to the period 1979-1985. Fig. 2 to 5 show the results obtainedfor the mean wind and the first three principal components.
Fig. 2.- Pattern for mean wind. Fig. 3.- Pattern for the firstprincipal component.
Fig. 4.- Pattern for the secondprincipal component.
Fig. 5.- Pattern for the thirdprincipal component.
The wind field (Vf) corresponding to each observation will be evaluatedas a linear combination of the mean wind and the fourteen patterns, by usingthe expression:
2N= V (13)
for each node in the grid, where a,(r) are calculated using the expression (4).Note that the expression (13) is an extension of the expression (3) applied tothe study nodes and using the wind pattern instead of the wind measurements.By this way the equation (7) is only to be solved 2N + 1 times.
In order to check the validity of the model we have compared the resultswith the experimental data on the nine meteorological stations shown in Fig.1, those are the seven stations used to evaluate the principal components plustwo others in Valencia and Teruel (these two ones with only three daily windobservations). This validation is referred to compare the mean calculated andobserved wind velocity modulus and also to evaluate the mean deviation ofdirection between calculated and observed wind. The averaging time is theperiod 1979-1985. The results are shown in Fig. 6 for the mean velocities andFig. 7 for the mean deviation of direction.
We have elaborated an interpolation mathemetical wind model to predictregional wind fields using as data the wind observations in severalmeteorological stations and the terrain topography. This model evaluates theprincipal components for the wind observations and the wind patterns for thegrid nodes by solving a partial differential equation applying the FiniteElement Method. The wind field for every measurement time is obtained as alinear combination of the wind patterns, with coefficients evaluated by meansof the simultaneous observations and the principal components.
The model has been applied to the territory of the ComunidadValenciana, using four daily simultaneous wind measurements in sevenmeteorological stations in the period 1979-1985. After comparing wind resultswith wind measurements in nine meteorological stations we can make thefollowing observations:
The situations visualized by the model don't deviate much fromreality, specialty on the location of the seven meteorological stationsused to evaluate the principal components.
We must be conscious of the generalizations and simplifications wehave made with the purpose of modelling the wind behaviour. Atany rate the results of the model are rather satisfactory.
- It is observed that level of error is higher than in other models ofbetter quantification related with wind phenomena, if we considerthe results on Valencia and Teruel, the two stations not usedpreviously. This occurs because of simplifications made in modelformulation and due to the fact that some of the stations are locatedin urban zone (i.e., Valencia). Beside, we must take into accountthat the model is predicting wind fields in 3015 nodes from windobservations collected in only seven stations.
We want to point out that the present model can be considered asan easy using tool to simulate the wind behaviour over large areasof terrain. More complicated models could adjust better to reality,but their cost in time and computer memory would be much higher.
This model doesn't consider atmospheric stratification nor mixingheight due to the fact that measurements related with thesephenomena are not available over the territory of the ComunidadValenciana. Land and sea breeze effects can be considered asincluded in the wind measurements.
Results obtained with this model are being used as input data in studiesabout air pollution dispersion capacity in the Comunidad Valenciana.
ACKNOWLEDGMENTS. The authors wish to thank Prof. Carlos Lopez,from the Institute de Mecanica de Fluidos e Ingenieria Ambiental deMontevideo (Uruguay), for his many helpful suggestions and discussionsduring the development and implementation of this model. This work hasbeen supported in part by the Programa de Cooperacion Cientifica conIberoamerica 1992 del Gobierno Espanol, and by the Conselleria d'ObresPubliques, Urbanisme i Transports de la Generalitat Valenciana.
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