DIAGNOSTIC WIND FIELD

Transcription

1 seminar DIAGNOSTIC WIND FIELD author: Matic Ivančič mentor: prof. dr. Jože Rakovec May 23, 2010 Abstract Diagnostic wind field procedure is calculation of wind on small scale and in complex terrain. Diagnostic wind filed is mass-consistent obteined by variational approach. The procedure is adjusting wind to terrain. Diagnostic wind field is used in air-pollution modeling, for planning wind turbines and for other studies in small scale.

2 Contents 1 INTRODUCTION 3 2 ATMOSPHERIC STABILITY Stable atmosphere Unstable atmosphere Connection between stability, dispersion classes and vertical temperature gradient Froude number THE VARIATIONAL TECHNIQUE Sasaki method Introduction stability in variational technique INITIALIZATION - FIRST GUESS Uniform wind Geostrophic wind Ground station data One ground station Two or more stations Wind profiles Mesoscale model CONCLUSION 14 2

3 1 INTRODUCTION The full three-dimensional wind field is needed for air pollution modelling. One way of calculating 3D wind filed is diagnostic wind model. Diagnostic wind field (DWF) is mass-consistent; there is no sources and no sinks of mass. Here we get the first condition for DWF: three-dimensional divergence must be equal to zero ( V = 0). This condition comes from continuity equation: 1 dρ ρ dt = V (1) Second condition is that there is no flow in or out of the ground. So the normal component of wind at the ground must be zero ( V n = 0). With this condition are included the boundary conditions that result from more or less complicated terrain. The additional condition is that the mass-consistent wind field should be as close too measurments, as possible. For this purpose a first guess wind field is constructed on the basis of the avaliable meteorological data. The calculation of diagnostic wind filed is achieved by the two step procedure: first guess for the wind field is constructed and then an adjustment is made to minimaze both - the wind divergence and the difference in respect to first guess wind field. The first step includes methods for horizontal and vertical interpolation of available wind data or calculation of first guess wind field. The second step includes a procedure for minimum possible modification to the first guess wind field so that the resulting (or final) wind field satisfies mass conservation. At the beginning we need consider some characteristics of atmosphere. We try to include data that influence wind: the most important thing is the stability of atmosphere. 3

4 2 ATMOSPHERIC STABILITY The stability of atmosphere is one of criteria for turbulence. There are three main stability classes: stable, neutral and unstable atmosphere. The hidrostatic stability depends on vertical temperature gradient. In neutral atmosphere vertical temperature gradient is the same as adiabatic lapse rate: T z = Γ a = g c p 10 K/km (2) If vertical temperature gradient is grater than Γ a we have stable atmosphere. atmosphere temperature decreases with height more then -10 K/km. 2.1 Stable atmosphere In unstable Air is more likely to flow around a terrain barrier rather than over it in stable atmosphere. Here we have more horizontal flow and almost no vertical flow (figure 1). The vertical temperature gradient is greater than adiabatic lapse rate. In this situation plume Figure 1: In stable atmosphere flow stops before reaching the top of the hill [8]. from stack does not sprayed and it travels in sheaf (figure 2). 2.2 Unstable atmosphere In unstable atmosphere the vertical movement is not repressed by the buoyancy effects. The atmosphere may be turbulent. The flow is more likely to go over a terrain barrier (figure 3). We have lower vertical temperature gradient as adiabatic lapse rate here. The plum from stack disperses fast in whole primary boundary layer (figure 4). 2.3 Connection between stability, dispersion classes and vertical temperature gradient As mentioned before the hydrostatic stability is connected with vertical temperature gradient. We have different classification about stabillity classes but Pasquill-Turner classification is most 4

5 Figure 2: Vertical temperature gradient and plume dispersion in stable atmosphere [4]. Figure 3: In unstable atmosphere flow goes under hills [8]. 5

6 Figure 4: Vertical temperature gradient and plume dispersion in unstable atmosphere [4]. often used. I have prepared table with connection betwen stability and vertical temperature gradient (figure 5). Figure 5: Connection betwen stability and vertical temperature gradient [9]. 2.4 Froude number Froude number is also one of characteristics of atmosphere. It is useful when we are dealing with flow over terrain barrier. We define Froud number with following equation: F r = U hn, (3) where U is horizontal wind velocity, h is the height of terrain barrier and N is Brunt-Vaisala frequency (connection with stability). We can also estimate Froude number with kinetic energy of flow (KE) and potenacial energy that flow needs to go over barrier: F r = KE P E 6 (4)

7 If Froude number is bigger than 1, KE is bigger than PE. In this case flow goes over hill and we can say that we have unstable atmosphere. For Froude number minor than 1 flow rather goes around hill and we have stable atmosphere. This is simulated in figure 6. Figure 6: Impact of different Froude number [3]. 7

8 3 THE VARIATIONAL TECHNIQUE As mentioned in first section the calculation of diagnostic wind field has two parts: initialization of wind filed and application of the varitional technique. Firstly I will present second part because it has influence to first part. The mathematical formulation of the mass-consistent models uses a calculus-of-variations approach originally developed by Sasaki [1] and Sherman [2]. 3.1 Sasaki method Firstly we need to define variables which appear in our problem. Let use standard cartesian coordinate system x, y, z and u, v, w are the wind-components in x, y and z direction. Now we can write wind vector as V = (u, v, w) and the initial field (first guess) as V 0 = (u 0, v 0, w 0 ). We can also define the modelling domain as Ω. How to prepare initial wind field V 0 will be explained in section 4. Now we describe the varitional problem of minimizing the difference between the final wind field and it s first guess. In mathematical language that means we must find the minimum of the following functional: E(u, v, w) = Ω [ α 2 1(u u 0 ) 2 + α 2 1(v v 0 ) 2 + α 2 2(w w 0 ) 2] dxdydz (5) where α 1 and α 2 are the weights with name Gaussian precision moduli. With different weights for horizontal and for vertical wind component we can intruduce stability in variational problem (discussion will be in next subsection 3.2). We also have one strong condition: 3D divergence must be zero over all computing domain: V = u x + v y + w z = 0 (6) We can include condition 6 into equation 5 with Lagrange multiplier λ(x, y, z) and get new functional: [ F (u, v, w, λ) = α1(u 2 u 0 ) 2 + α1(v 2 v 0 ) 2 + α2(w 2 w 0 ) 2 + λ V ] dxdydz (7) Ω Searching for the extreme (minimum) of this functional over the whole domain Ω leads to the Euler-Lagrange equations: 2α 2 1(u u 0 ) λ x = 0 (8) 2α 2 1(v v 0 ) λ y = 0 (9) 2α2(w 2 w 0 ) λ z = 0 (10) u x + v y + w z = 0 (11) 8

9 We must also consider the boundary condition that the normal component of wind at the solid border of domain Ω must be zero: V n = 0 at the ground (12) With this condition we make wind adjustment to terrain. Assuming that α 1 and α 2 are constants throughout the domain and substituting u, v, w from equations 8, 9, 10 and 11 we get the following equation: 2 ( ) λ x λ 2 y 2 + α1 2 ( λ α 2 z 2 = u0 2α2 1 x + v 0 y + w ) 0 z This equation is hardly solved analytically. So we must find λ with some numerical method. Usually there are two options: - SOR (Successive Over-Relaxation) method - ADI (Alternating Direction Implicit) method Once we get λ we can find the solution for wind field from 8, 9 and 10: (13) u = u λ 2α1 2 x v = v λ 2α1 2 y w = w λ 2α2 2 z (14) (15) (16) 3.2 Introduction stability in variational technique As have seen in section 2 there typically is horizontal motion in stable atmosphere and air flows around hills. In unstable atmosphere vertical motion is not suppressed and air is likely to travel also over terrain barriers. In section 3.1 we have mentioned that we can include the stability with effects of Gaussian precision moduli α 1 and α 2. Actually the most important is their ratio. So we can write: ( ) 2 α1 α = (17) The value of α governs the relative adjustment between the horizontal and the vertical wind component. We use α >> 1 (α 1 > α 2 ) for unstable atmosphere and adjustment of the vertical motion predominates in this case. In stable conditions we have α << 1 (α 1 < α 2 ) and flow adjustment occurs primarly in the horizontal plane. For neutral stratification we use value α = 1 (α 1 = α 2 ). We can also connect the Gaussian precision moduli with the measured wind variances: α 2 α i = 1 2σ 2 i (18) 9

10 Variance of wind components is, on the other hand,connected with the degree of turbulence. Great value of σh 2 and small value of σ2 z is typical for stable atmosphere and inversely for unstable atmosphere (here σh 2 means variance of the horizontal wind component and σ2 z means variance of the vertical wind component. So this way of connection is logical. We can rewrite equation 17: ( ) 2 σz α = (19) Variance of the horizontal wind (σh 2 ) also depends on horizontal velocity U: bigger velocity usually means bigger variance. And the same is with vertical variance. So we can also estimate value of paramater α with following equation: σ h ( ) w 2 α (20) u In literature one may found the table connecting the stability classes with Gaussian precision moduli. Table is on figure 7. Figure 7: Connection between Gaussian precision moduli and stability of atmosphere. This connection is used in model WINDS [5]. 10

11 4 INITIALIZATION - FIRST GUESS As mentioned in section 3.1 we need a basic information of the 3D wind filed - the first guess ( V 0 ). There are different possibilities to prepare first guess. With better first guess we can expect beter final results. It is good that maximum possible avaliable data will be included in calculation. There are several different diagnostic wind models: NOABL, AIOLOS, TALdia, WINDS, CAL- MET, MATHEW,... All of this models have almost the same variational technique but they are differ in inicialization. All equtions from this section we can find in [5]. 4.1 Uniform wind In this case we pripare the constant wind for all modeling domain. Horizontal wind components u 0 and v 0 are independent of space and we put zero for vertical wind component (w 0 = 0). This kind of initialization is usefull for test simulations because in that situations the topografic influence can be seen very well but this initialization has no practical meaning for real calculations. 4.2 Geostrophic wind The input for this kind of intialization is the wind free atmosphere velocity in the top of modeling domain. Here we also need stability parameter and surface rougnes (z 0 ). Again initial vertical wind velocity w 0 has value 0. In this case the initial wind is not constant in space. We must interpolate the wind from the top of the domain to each vertical layer. We can use a sort of equations that are close to the Ekman wind profile in PBL; for example: u 0 (z) = u [ ln z ( ) ] z z 0 z 2 + b µ + b z0 µ (21) k z 0 h h v 0 (z) = u k [ a µ z z 0 h + a µ ( ) ] z 2 z0 where u is friction velocity, k is von Karman konstant (k=0,40) and h is the PBL-height. Stability is imported with parameters a µ, a µ, b µ and b µ. Friction velocity we can calculate with following: k u = G ) 2 (23) (ln h b z0 µ + a 2 µ h (22) where G is geostrophic wind speed. Equations 21, 21 and 23 can not be solved analiticaly but we must calculate iterativitly because there is not enought avaliable data. This kind of inicializations is used in models AIOLOS and TALdia (part of model AUSTAL2000). 11

12 4.3 Ground station data When we use data from ground stations we must separate case with only one ground station and case with more than one station One ground station The principle for this inicialization is similar to inicialization with geostrophic wind (section 4.2). Difference here is that in previous section we use wind in the top of domain and in this case we use the wind measurement at the ground. We again need to interpolate wind to all modeling layers. We can use the equations 21 and 22 and different equation for friction velocity u : u = kv obs ( ln z obs z 0 + b µ ζ + b muζ 2 ) 2 + (aµ ζ + a µζ 2 ) 2, (24) where V obs is wind speed at station at hight z obs and ζ = z obs z 0 h. Again stability is imported with parameters a µ, a µ, b µ and b µ. With such approach I have prepared some calculation of wind profiles. It will be represented on section??. We presume that u 0 and v 0 are independent on x and y coordinate and vertical component w 0 is equal to zero Two or more stations Similiar situation is with two or more surface stations. But here wind components u 0 and v 0 are dependant on x and y direction. Firstly we need to make vertical profile for each station with equations 21 and 22. And than we start horizontal interpolation for each vertical layer with the following equation: (u 0, v 0 ) i,j = Ls l=1 (u st, v st ) l f(r l ) Ls l=1 f(r l) (25) where u 0 and v 0 are the interpolated components in location (i,j), L s is number of surface stations, u st and v st are velocity components over surface station and f(r l ) is the weighting factor for each station being dependant on distance r l from station to point (x,y). We can use inverse square of distance for weighting factors: f(r l ) = 1 r 2 l (26) In intialization we can also include which stations are more representative. Equation for weighting factors are then in the folloving form: f(r l ) = s l r 2 l (27) where s l have values between zero and one (more representative station has bigger value). 12

13 4.4 Wind profiles In previous section we made vertical profiles with theoretical equations as a function of stability. If we have measurements of vertical profiles we inlude this data instead of theoretical equations and additional calculations are the same as in section 4.3. We can also include one vertical measurment and one or more ground stations in inicialization. In this case we must calculate theoretical vertical profiles for ground stations and use bigger weighting factor s l for vertical measurment. 4.5 Mesoscale model The operational mesoscale models have typical horizontal resolution of few kilometers to few tens of kilometers. We can not make wind fields to finer scale with operational prognostic mesoscale models because the calculation costs too much time. The solution for wind fields in smaller scale are combinations of prognostic and diagnostic wind models. We take the results of mesoscale model as first guess in initialization and then make the diagnostic procedure to finer scale. 13

14 5 CONCLUSION Wind field is essentially important in air-quality modeling. Only if we have a good approximation for the 3D wind field we can expect good final results. But we must know that the diagnostic wind field is only an approximation for wind field and the real wind is usually much more complicated. As I mentioned we can combine the diagnostic wind filed procedure with information from the prognostic model for use in fine sptial resolution. In this case we can expect better results because the prognostic wind models already discribe reliable wind in a mesoscale. 14

15 References [1] Sasaki, Y. (1970): Some basic formalism in numerical variational analysis. Monthly Weather Review. [2] Sherman, C.A. (1978): A mass-consistent model for wind fields over complex terrain. J. Appl. Meteor. [3] Stull, R.B.: An introduction to boundary layer meteorology. Kluwer Academic Publishers, Dordercht. [4] Gilbert M. Masters, INTRODUCTION TO ENVIRONMENTAL ENGINEERING AND SCIENCE. [5] E. Georgieva, E. Canepa, A.Mazzino: WINDS, Release 4.2, Users Guide (August 2002). [6] AUSTAL2000, Program Documentation of Version 2.4, Appendix D. [7] (May 23, 2010) [8] (May 23, 2010) [9] (May 23, 2010) 15