- Email: [email protected]
Validation of a non-hydrostatic numerical model to simulate stratified wind fields over complex topography
Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 273—282
Validation of a non-hydrostatic numerical model to simulate stratified wind fields over complex topography Christiane Montavon* Civil Engineering Department, Laboratory of Energy Systems, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland
Abstract A non-hydrostatic numerical model developed from a general purpose Navier—Stokes solver (CFDS-FLOW3D) has been proposed in order to simulate atmospheric flows over complex topography. In the validation tests presented here, the 3D numerical model has been run to simulate neutral and stratified wind fields over a 2D theoretical bell-shaped mountain. Model results have been compared to analytical solutions, obtained from linear mountain wave theory for the two approximations of (a) neutral flow and (b) stratified flow with atmospheric conditions leading to hydrostatic waves production. In both cases, the model results were in very good agreement with the analytical solution. For atmospheric conditions where nonhydrostatic effects become dominant, the model also proved to be able to reproduce trapped lee waves located downwind of the mountain. For highly non-linear atmospheric conditions, we have tried to reproduce the well-documented severe windstorm which was registered in January 1972 in the region of Boulder (Colorado), during which extreme horizontal wind values of more than 60 m/s were measured. To this purpose, the model was initialised with temperature and wind speed profiles from the meteorological soundings registered at Grand Junction station, located 300 km upwind of Boulder. A comparison of the model calculated potential temperature and horizontal wind component with the observations for this event shows the model ability to reproduce the very strong tropospheric air descent and flow acceleration over the mountain. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Stratified atmospheric flow; Complex terrain; Mountain waves; Numerical simulation; Full Navier—Stokes solver
1. Introduction Among the tools available for wind engineering applications, numerical models present an increasing interest. Several developments have already been proposed at *E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 2 4 - 5
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the Laboratory of Energy Systems for the prediction of local wind climates using numerical models [1,2]. For the purpose of simulating atmospheric processes at scales of the order of a few kilometers down to a hundred meters, we have chosen to use a numerical model solving the complete set of Navier—Stokes equations. The model presented here has been developed on the basis of the CFDS-FLOW3D solver. The standard version of the code works with the energy conservation equation and buoyancy term in the vertical velocity equation formulated in terms of the real temperature ¹. This makes it inappropriate for simulating stratified atmospheric flows with a vertical extent of several kilometres. The problem was circumvented by implementing the energy conservation equation and the buoyancy term using the potential temperature h and by defining a reference state for the atmosphere which corresponds to hydrostatic equilibrium. In this paper, we present results of simulations which have been carried out to test the validity of these modifications and the model’s ability to reproduce both hydrostatic and non-hydrostatic flow features which occur over complex topography. This work is part of an ongoing effort aiming at the simulation of the effect of real topography or obstacles on local wind conditions. Starting from a station with a known wind climate, such a model could help to establish relationships between wind conditions prevailing at different sites and be used to locate areas exhibiting the most favourable wind conditions, or areas which might be most exposed to extreme wind gusts.
2. Numerical model equations The numerical model equations are solved in the Boussinesq approximation, neglecting the effect of temperature variations on density, except in the buoyancy term. These equations read: f Continuity equation in the anelastic approximation and momentum conservation equation: L(o ) u ) ) i "0, Lx i L(o u ) L(o ) u u ) Lu Lp@ L Lu ) i # ) i j "! i# j d #(o !o)g ) d # k ij ) i3 %&& Lt Lx Lx Lx Lx Lx j j j j i
C A
(1)
BD
. (2)
f Modified energy conservation equation for the potential temperature h"¹(p /p)R@Cp: 0 L(oh) L(oh ) u ) L L i" # k h . (3) %&& Lx Lt Lx Lx i i i In the above k "k#k is the effective fluid viscosity (sum of the molecular and %&& T turbulent viscosity) and p@"p#2 o ) k!p is the total pressure p to which a contri3 ) ) bution associated with the turbulent kinetic energy k is added and the contribution of
C
D
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the hydrostatic pressure p is removed. The k—e model [3] provides closure for ) turbulence. A reference atmospheric state (denoted by the subscript ‘h’) is defined, which corresponds to an atmosphere in hydrostatic equilibrium. Starting from a real temperature profile ¹(z), the hydrostatic pressure profile p is obtained by numerically ) integrating dp "o ) g dz, assuming the perfect gas law o "p /R¹(z) is verified for ) ) ) ) each altitude z. The hydrostatic potential temperature profile is calculated from h "¹(p /p )R@Cp. ) 0 ) Assuming that the departure of the atmospheric state from the hydrostatic reference state is small and that the Mach number of the flow is low, an approximate form for the buoyancy term (o !o)g is derived which reads ) o (o !o)" ) (h!h ). ) ) h ) This expression is used in order to couple Eqs. (2) and (3). The complete set of equations is solved on a non-staggered grid by means of a finite-volume approach, all the variables being defined at the centre of the control volumes. Pressure and velocity are coupled through the iterative SIMPLE algorithm [4]. To avoid chequerboard oscillations between pressure and velocity, the normal velocity components at the control volume faces are interpolated by means of the Rhie and Chow algorithm [5]. More details on the numerical model characteristics can be found in the CFDS-FLOW3D user guide [6].
3. Comparison with solutions from linear mountain wave theory Linear mountain wave theory allows analytical solutions to be obtained for the perturbation of a horizontal flow by a bell-shaped mountain of small elevation [7]. The linearized equation for the vertical velocity component w can be transformed so that: L2w L2w 1# 1#l2(z) ) w "0 1 Lx2 Lz2
(4)
with the definitions w "Jo/o ) w(x,z), 1 0 1 LS 1 L2u gb S Lu 1 ! S2# ! , l2(z)" # 2 Lz u Lz2 u2 u Lz 4
(5) (6)
l(z) being the so-called Scorer parameter, bg"gL(ln(h)/Lz) is the square of the Brundt—Va> ysa> la> number and S"L(ln(o)/Lz) the heterogeneity of the atmosphere. The theoretical mountain profile h(x) used in this study is described by h(x)"h a2/(x2#a2), h being the maximum elevation and a the mountain half. . width. For this mountain profile, analytical solutions for the vertical velocity can be
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found for near-neutral flow and for hydrostatic flow, if the Scorer parameter is constant [8]. 3.1. Near-neutral flow For near-neutral flow (strong winds, neutral thermal stratification, small mountain) with al@1, the analytical solution for w becomes 1 !2x(a#z) w (x,z)"uN (0)h a . (7) 1 . [(a#z)2#x2]2 The calculated and theoretical predictions are plotted in Fig. 1. The solution for w obtained from the numerical model (plain line) for an incoming horizontal wind of 1 10 m/s compares very well with the analytical solution (dashed line). The solution was calculated with a"10 000 m, h "500 m. In the near-neutral case, the first term on . the r.h.s. of Eq. (6) becomes negligible. Moreover, with the choice of a constant velocity profile for the incoming flow, the dominant term in the Scorer parameter is the heterogeneity S. To meet the requirement of a constant value of l, which enables the derivation of a theoretical solution for the vertical velocity, the simulation for this particular case was done with a constant density profile. 3.2. Hydrostatic flow For hydrostatic flow (large mountain, strong atmospheric stability, weak winds, i.e. alA1), one obtains a solution for w exhibiting hydrostatic waves propagating 1
Fig. 1. Vertical velocity component (m/s) for the neutral atmospheric conditions obtained with u"10 m/s, a"10 000 m, h "500 m. Plain line: numerical model solution, dashed line: analytical solution. .
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Fig. 2. Vertical velocity component (m/s) for the hydrostatic mountain wave solution obtained with u"10 m/s, a"10 000 m, h "500 m, l"10~3 m~1. (a) Numerical and (b) analytical solution. Plain line: . positive value, dashed line: negative value.
vertically above the mountain, with the analytic form given by w (x,z)"uN (0)h a 1 .
(x2!a2) sin lz!2xa cos lz . (a2#x2)2
(8)
Fig. 2a presents the results for w"JoN /oN w (x,z) from a numerical simulation with 0 1 u"10 m/s, a"10 000 m, h "500 m, and a hydrostatic density profile. The Scorer . parameter was set so that l"10~3 m~1 for the lowest 10 km, and the stability was thereby increased. The analytical solution for l"10~3 m~1 is given in Fig. 2b. The comparison of both wave patterns below 10 km shows the model ability to reproduce the vertical wavelength expected from theory. The effect of the stronger atmospheric stability above this level can also be observed as a reduction of the vertical wavelength above 10 km. The amplitude of the hydrostatic waves are also well reproduced, as well as the amplitude decrease with altitude due to the density reduction. Both for the neutral and hydrostatic cases, the model results are in very good agreement with the analytical solutions. 3.3. Non-hydrostatic flow For an intermediate condition with al"1, no analytical solution exists for the whole atmosphere, against which a validation could be done. Different studies [9,10] show, however, that for such situations, trapped lee waves develop downwind of the mountain crest. Such a situation has been simulated with u"20 m/s, a"2000 m, h "1000 m, l"5]10~4 m~1 and with a hydrostatic density profile. The results for .
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Fig. 3. Solution for the non-hydrostatic situation, calculated with u"20 m/s, a"2000 m, h "1000 m, . l"5]10~4 m~1. (a) Potential temperature, (b) vertical velocity component (m/s). Plain line: positive value, dashed line: negative value.
Table 1 Comparison of flow features obtained with FLOW3D, TVM and RAMS for the simulation of non-hydrostatic waves calculated with a constant density profile and the same set of parameters as in Fig. 3 Amplitude and location of the first lobe of positive vertical velocity in the lee side FLOW3D TVM RAMS
3.16 m/s &3.25 m/s &2.75 m/s
x"7.6 km, z"6.2 km x+7.0 km, z+8.0 km x+7—8 km, z+8.0 km
Horizontal lee wavelength 10—12 km &11 km &13 km
the potential temperature and for the vertical velocity are presented in Fig. 3. It is apparent that the model produces trapped lee waves as expected. A comparison with results obtained by the mesoscale models RAMS and TVM [8] was performed for non-hydrostatic conditions and constant density profile (presented in Ref. [11]). The comparison is summarised in Table 1, where a few flow features were reported for the three models. From the latter, it can be concluded that FLOW3D prediction lies in the range of variation of mesoscale models.
4. Comparison with observations recorded during the severe Boulder windstorm of January 1972 A severe wind storm, with horizontal wind speed higher than 60 m/s, was reported in the region of Boulder (Colorado) during January 1972. This event is very well
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Fig. 4. Observations (from Ref. [13]) of (a) the horizontal velocity and (b) potential temperature field for the Boulder windstorm of January 1972.
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documented [12,13]. Vertical slices of the horizontal wind velocity and potential temperature obtained from aeroplane observation for this event are given in Fig. 4. A very strong descent of air originating from the higher troposphere was observed, together with an important flow acceleration near the ground downwind of the mountain. Waves also developed behind the mountain. To test the model ability to reproduce this highly non-linear situation, we initialised the model with the temperature and horizontal velocity soundings registered at Grand Junction, a station 300 km upwind of the Boulder area [14]. Fig. 5 presents the model results at time t"4800 s for the potential temperature field as well as for the horizontal wind component for an idealised mountain shape (h"2000 m, a"10000 m). With a vertical displacement of some 5000 m for the isolines of potential temperature (see the isolines for 305 and 310 K), the strong air descent above the mountain is well reproduced. The flow is strongly accelerated leading to velocities of about 60 m/s close to the ground behind the mountain crest. The wave formation downwind is also reproduced and can best be observed in the potential temperature isolines. Wave breaking is also simulated near the tropopause level in a zone where an overturning of the potential temperature isolines occurs. This results in a flow-reversal zone (negative horizontal velocity), corresponding quite well with the observations of Fig. 4a. Here again, the model results compare very well with solutions obtained from other mesoscale models. For example, a solution given by Peltier and Clark [14] for the same flow conditions presents, after an integration time of 8000 s, a maximum value for the wind speed in excess of 60 m/s located at some 8—10 km behind the mountain crest at an altitude of about 3 km. For the waves forming in the lee side, both FLOW3D and the model of Peltier and Clark give a wavelength of 25 km. And finally, both models present a flow reversal zone and a deflection of the tropopause with an amplitude slightly bigger than 5000 m.
Fig. 5. Numerical results for the (a) potential temperature and (b) horizontal velocity field for the Boulder windstorm of January 1972. Plain line: positive value, dashed line: negative value.
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5. Conclusions A modified version of the CFDS-FLOW3D numerical model has been tested to simulate atmospheric flow processes over complex terrain. Simulations were performed over a two-dimensional theoretical mountain shape for various atmospheric conditions. For small mountain elevations, the model results were compared to analytical solutions obtained from the theory of linear mountain waves. For both the near-neutral and hydrostatic approximations, the simulation results were in close agreement with theory. The model’s ability to reproduce highly non-linear atmospheric processes leading to violent windstorms such as the one observed at Boulder was also verified. From the presented results, it is expected that such a numerical model, used over real three-dimensional topography, will become a very interesting tool to study the effect of topography or obstacles on local wind conditions.
Acknowledgements The present work is part of a Ph.D research project funded by the Swiss Federal Institute of Technology in Lausanne, and has benefited from previous research work, aiming at selecting appropriate tools to simulate atmospheric flows over complex topography, and undertaken with the funding of the Swiss Federal Bureau of Water Management (CRUEX project) and the Swiss National Science Foundation (Grant No. 2000 287 27).
References [1] C. Montavon, C. Alexandrou, J.-A. Hertig, Wind Potential Evaluation Over Complex Terrain By Means Of A Three-Dimensional Atmospheric Numerical Model, Proc. European Union Wind Energy Conf., Go¨teborg, 1996, pp. 526—529. [2] C. Montavon, J.-A. Hertig, C. Alexandrou, Analyse des conditions d’exposition aux vents forts des ponts sur la valle´e de la Mentue, Rapport LASEN, EPF Lausanne, Switzerland, 1995. [3] B.E. Launder, D.B. Spalding, Mathematical models of turbulence, Academic Press, New York, 1972. [4] J.P. Van Doormal, G.D. Raithby, Enhancement of the SIMPLE method for predicting incompressible fluid flows, Numer. Heat Transfer 7 (1984) 147—163. [5] C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1527—1532. [6] CFDS-FLOW3D, Release 3.3: User Manual, Computational Fluid Dynamics Services, Harwell Laboratory, Oxfordshire OX11 0RA, UK, 1994. [7] P. Queney, G.A. Corby, N. Gerbier, H. Koschmieder, J. Zierep, Technical Note No. 34: The Airflow over mountains, WMO-No. 98. TP.43, Geneva, Switzerland, 1960. [8] P. Thunis, Formulation and evaluation of a non-hydrostatic vorticity-mode mesoscale model, Report EUR 16141 EN, Joint Research Centre, European Commission, Environm. Institute, 1995. [9] P. Queney, The problem of airflow over mountains: a summary of theoretical studies, Bull. Amer. Meteor. Soc. 29 (1948) 16—26. [10] R.R. Long, Some aspects of the flow of stratified fluids, I: a theoretical investigation, Tellus 5 (1953) 42—58.
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[11] C. Montavon, Validation du mode`le nume´rique non-hydrostatique FLOW3D pour la simulation d’e´coulements atmosphe´riques stratifie´s en terrain accidente´, Diploma thesis, Cours 3e`me cycle, IMHEF, EPF Lausanne, Switzerland, 1996. [12] D.K. Lilly, E.J. Zisper, The front range windstorm of 11 January 1972 — a meteorological narrative, Weatherwise 25 (2) (1972) 56—63. [13] J.B. Klemp, D.K. Lilly, Numerical simulation of hydrostatic mountain waves, J. Atmos. Sci. 35 (1978) 78—107. [14] W.R. Peltier, T.L. Clark, The evolution and stability of finite-amplitude mountain waves. Part II: Surface wave drag and severe downslope windstorms, J. Atmos. Sci. 36 (1979) 1498—1529.
Validation of a non-hydrostatic numerical model to simulate stratified wind fields over complex topography Christiane Montavon* Civil Engineering Department, Laboratory of Energy Systems, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland
Abstract A non-hydrostatic numerical model developed from a general purpose Navier—Stokes solver (CFDS-FLOW3D) has been proposed in order to simulate atmospheric flows over complex topography. In the validation tests presented here, the 3D numerical model has been run to simulate neutral and stratified wind fields over a 2D theoretical bell-shaped mountain. Model results have been compared to analytical solutions, obtained from linear mountain wave theory for the two approximations of (a) neutral flow and (b) stratified flow with atmospheric conditions leading to hydrostatic waves production. In both cases, the model results were in very good agreement with the analytical solution. For atmospheric conditions where nonhydrostatic effects become dominant, the model also proved to be able to reproduce trapped lee waves located downwind of the mountain. For highly non-linear atmospheric conditions, we have tried to reproduce the well-documented severe windstorm which was registered in January 1972 in the region of Boulder (Colorado), during which extreme horizontal wind values of more than 60 m/s were measured. To this purpose, the model was initialised with temperature and wind speed profiles from the meteorological soundings registered at Grand Junction station, located 300 km upwind of Boulder. A comparison of the model calculated potential temperature and horizontal wind component with the observations for this event shows the model ability to reproduce the very strong tropospheric air descent and flow acceleration over the mountain. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Stratified atmospheric flow; Complex terrain; Mountain waves; Numerical simulation; Full Navier—Stokes solver
1. Introduction Among the tools available for wind engineering applications, numerical models present an increasing interest. Several developments have already been proposed at *E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 2 4 - 5
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C. Montavon/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 273–282
the Laboratory of Energy Systems for the prediction of local wind climates using numerical models [1,2]. For the purpose of simulating atmospheric processes at scales of the order of a few kilometers down to a hundred meters, we have chosen to use a numerical model solving the complete set of Navier—Stokes equations. The model presented here has been developed on the basis of the CFDS-FLOW3D solver. The standard version of the code works with the energy conservation equation and buoyancy term in the vertical velocity equation formulated in terms of the real temperature ¹. This makes it inappropriate for simulating stratified atmospheric flows with a vertical extent of several kilometres. The problem was circumvented by implementing the energy conservation equation and the buoyancy term using the potential temperature h and by defining a reference state for the atmosphere which corresponds to hydrostatic equilibrium. In this paper, we present results of simulations which have been carried out to test the validity of these modifications and the model’s ability to reproduce both hydrostatic and non-hydrostatic flow features which occur over complex topography. This work is part of an ongoing effort aiming at the simulation of the effect of real topography or obstacles on local wind conditions. Starting from a station with a known wind climate, such a model could help to establish relationships between wind conditions prevailing at different sites and be used to locate areas exhibiting the most favourable wind conditions, or areas which might be most exposed to extreme wind gusts.
2. Numerical model equations The numerical model equations are solved in the Boussinesq approximation, neglecting the effect of temperature variations on density, except in the buoyancy term. These equations read: f Continuity equation in the anelastic approximation and momentum conservation equation: L(o ) u ) ) i "0, Lx i L(o u ) L(o ) u u ) Lu Lp@ L Lu ) i # ) i j "! i# j d #(o !o)g ) d # k ij ) i3 %&& Lt Lx Lx Lx Lx Lx j j j j i
C A
(1)
BD
. (2)
f Modified energy conservation equation for the potential temperature h"¹(p /p)R@Cp: 0 L(oh) L(oh ) u ) L L i" # k h . (3) %&& Lx Lt Lx Lx i i i In the above k "k#k is the effective fluid viscosity (sum of the molecular and %&& T turbulent viscosity) and p@"p#2 o ) k!p is the total pressure p to which a contri3 ) ) bution associated with the turbulent kinetic energy k is added and the contribution of
C
D
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275
the hydrostatic pressure p is removed. The k—e model [3] provides closure for ) turbulence. A reference atmospheric state (denoted by the subscript ‘h’) is defined, which corresponds to an atmosphere in hydrostatic equilibrium. Starting from a real temperature profile ¹(z), the hydrostatic pressure profile p is obtained by numerically ) integrating dp "o ) g dz, assuming the perfect gas law o "p /R¹(z) is verified for ) ) ) ) each altitude z. The hydrostatic potential temperature profile is calculated from h "¹(p /p )R@Cp. ) 0 ) Assuming that the departure of the atmospheric state from the hydrostatic reference state is small and that the Mach number of the flow is low, an approximate form for the buoyancy term (o !o)g is derived which reads ) o (o !o)" ) (h!h ). ) ) h ) This expression is used in order to couple Eqs. (2) and (3). The complete set of equations is solved on a non-staggered grid by means of a finite-volume approach, all the variables being defined at the centre of the control volumes. Pressure and velocity are coupled through the iterative SIMPLE algorithm [4]. To avoid chequerboard oscillations between pressure and velocity, the normal velocity components at the control volume faces are interpolated by means of the Rhie and Chow algorithm [5]. More details on the numerical model characteristics can be found in the CFDS-FLOW3D user guide [6].
3. Comparison with solutions from linear mountain wave theory Linear mountain wave theory allows analytical solutions to be obtained for the perturbation of a horizontal flow by a bell-shaped mountain of small elevation [7]. The linearized equation for the vertical velocity component w can be transformed so that: L2w L2w 1# 1#l2(z) ) w "0 1 Lx2 Lz2
(4)
with the definitions w "Jo/o ) w(x,z), 1 0 1 LS 1 L2u gb S Lu 1 ! S2# ! , l2(z)" # 2 Lz u Lz2 u2 u Lz 4
(5) (6)
l(z) being the so-called Scorer parameter, bg"gL(ln(h)/Lz) is the square of the Brundt—Va> ysa> la> number and S"L(ln(o)/Lz) the heterogeneity of the atmosphere. The theoretical mountain profile h(x) used in this study is described by h(x)"h a2/(x2#a2), h being the maximum elevation and a the mountain half. . width. For this mountain profile, analytical solutions for the vertical velocity can be
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found for near-neutral flow and for hydrostatic flow, if the Scorer parameter is constant [8]. 3.1. Near-neutral flow For near-neutral flow (strong winds, neutral thermal stratification, small mountain) with al@1, the analytical solution for w becomes 1 !2x(a#z) w (x,z)"uN (0)h a . (7) 1 . [(a#z)2#x2]2 The calculated and theoretical predictions are plotted in Fig. 1. The solution for w obtained from the numerical model (plain line) for an incoming horizontal wind of 1 10 m/s compares very well with the analytical solution (dashed line). The solution was calculated with a"10 000 m, h "500 m. In the near-neutral case, the first term on . the r.h.s. of Eq. (6) becomes negligible. Moreover, with the choice of a constant velocity profile for the incoming flow, the dominant term in the Scorer parameter is the heterogeneity S. To meet the requirement of a constant value of l, which enables the derivation of a theoretical solution for the vertical velocity, the simulation for this particular case was done with a constant density profile. 3.2. Hydrostatic flow For hydrostatic flow (large mountain, strong atmospheric stability, weak winds, i.e. alA1), one obtains a solution for w exhibiting hydrostatic waves propagating 1
Fig. 1. Vertical velocity component (m/s) for the neutral atmospheric conditions obtained with u"10 m/s, a"10 000 m, h "500 m. Plain line: numerical model solution, dashed line: analytical solution. .
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277
Fig. 2. Vertical velocity component (m/s) for the hydrostatic mountain wave solution obtained with u"10 m/s, a"10 000 m, h "500 m, l"10~3 m~1. (a) Numerical and (b) analytical solution. Plain line: . positive value, dashed line: negative value.
vertically above the mountain, with the analytic form given by w (x,z)"uN (0)h a 1 .
(x2!a2) sin lz!2xa cos lz . (a2#x2)2
(8)
Fig. 2a presents the results for w"JoN /oN w (x,z) from a numerical simulation with 0 1 u"10 m/s, a"10 000 m, h "500 m, and a hydrostatic density profile. The Scorer . parameter was set so that l"10~3 m~1 for the lowest 10 km, and the stability was thereby increased. The analytical solution for l"10~3 m~1 is given in Fig. 2b. The comparison of both wave patterns below 10 km shows the model ability to reproduce the vertical wavelength expected from theory. The effect of the stronger atmospheric stability above this level can also be observed as a reduction of the vertical wavelength above 10 km. The amplitude of the hydrostatic waves are also well reproduced, as well as the amplitude decrease with altitude due to the density reduction. Both for the neutral and hydrostatic cases, the model results are in very good agreement with the analytical solutions. 3.3. Non-hydrostatic flow For an intermediate condition with al"1, no analytical solution exists for the whole atmosphere, against which a validation could be done. Different studies [9,10] show, however, that for such situations, trapped lee waves develop downwind of the mountain crest. Such a situation has been simulated with u"20 m/s, a"2000 m, h "1000 m, l"5]10~4 m~1 and with a hydrostatic density profile. The results for .
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Fig. 3. Solution for the non-hydrostatic situation, calculated with u"20 m/s, a"2000 m, h "1000 m, . l"5]10~4 m~1. (a) Potential temperature, (b) vertical velocity component (m/s). Plain line: positive value, dashed line: negative value.
Table 1 Comparison of flow features obtained with FLOW3D, TVM and RAMS for the simulation of non-hydrostatic waves calculated with a constant density profile and the same set of parameters as in Fig. 3 Amplitude and location of the first lobe of positive vertical velocity in the lee side FLOW3D TVM RAMS
3.16 m/s &3.25 m/s &2.75 m/s
x"7.6 km, z"6.2 km x+7.0 km, z+8.0 km x+7—8 km, z+8.0 km
Horizontal lee wavelength 10—12 km &11 km &13 km
the potential temperature and for the vertical velocity are presented in Fig. 3. It is apparent that the model produces trapped lee waves as expected. A comparison with results obtained by the mesoscale models RAMS and TVM [8] was performed for non-hydrostatic conditions and constant density profile (presented in Ref. [11]). The comparison is summarised in Table 1, where a few flow features were reported for the three models. From the latter, it can be concluded that FLOW3D prediction lies in the range of variation of mesoscale models.
4. Comparison with observations recorded during the severe Boulder windstorm of January 1972 A severe wind storm, with horizontal wind speed higher than 60 m/s, was reported in the region of Boulder (Colorado) during January 1972. This event is very well
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Fig. 4. Observations (from Ref. [13]) of (a) the horizontal velocity and (b) potential temperature field for the Boulder windstorm of January 1972.
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documented [12,13]. Vertical slices of the horizontal wind velocity and potential temperature obtained from aeroplane observation for this event are given in Fig. 4. A very strong descent of air originating from the higher troposphere was observed, together with an important flow acceleration near the ground downwind of the mountain. Waves also developed behind the mountain. To test the model ability to reproduce this highly non-linear situation, we initialised the model with the temperature and horizontal velocity soundings registered at Grand Junction, a station 300 km upwind of the Boulder area [14]. Fig. 5 presents the model results at time t"4800 s for the potential temperature field as well as for the horizontal wind component for an idealised mountain shape (h"2000 m, a"10000 m). With a vertical displacement of some 5000 m for the isolines of potential temperature (see the isolines for 305 and 310 K), the strong air descent above the mountain is well reproduced. The flow is strongly accelerated leading to velocities of about 60 m/s close to the ground behind the mountain crest. The wave formation downwind is also reproduced and can best be observed in the potential temperature isolines. Wave breaking is also simulated near the tropopause level in a zone where an overturning of the potential temperature isolines occurs. This results in a flow-reversal zone (negative horizontal velocity), corresponding quite well with the observations of Fig. 4a. Here again, the model results compare very well with solutions obtained from other mesoscale models. For example, a solution given by Peltier and Clark [14] for the same flow conditions presents, after an integration time of 8000 s, a maximum value for the wind speed in excess of 60 m/s located at some 8—10 km behind the mountain crest at an altitude of about 3 km. For the waves forming in the lee side, both FLOW3D and the model of Peltier and Clark give a wavelength of 25 km. And finally, both models present a flow reversal zone and a deflection of the tropopause with an amplitude slightly bigger than 5000 m.
Fig. 5. Numerical results for the (a) potential temperature and (b) horizontal velocity field for the Boulder windstorm of January 1972. Plain line: positive value, dashed line: negative value.
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5. Conclusions A modified version of the CFDS-FLOW3D numerical model has been tested to simulate atmospheric flow processes over complex terrain. Simulations were performed over a two-dimensional theoretical mountain shape for various atmospheric conditions. For small mountain elevations, the model results were compared to analytical solutions obtained from the theory of linear mountain waves. For both the near-neutral and hydrostatic approximations, the simulation results were in close agreement with theory. The model’s ability to reproduce highly non-linear atmospheric processes leading to violent windstorms such as the one observed at Boulder was also verified. From the presented results, it is expected that such a numerical model, used over real three-dimensional topography, will become a very interesting tool to study the effect of topography or obstacles on local wind conditions.
Acknowledgements The present work is part of a Ph.D research project funded by the Swiss Federal Institute of Technology in Lausanne, and has benefited from previous research work, aiming at selecting appropriate tools to simulate atmospheric flows over complex topography, and undertaken with the funding of the Swiss Federal Bureau of Water Management (CRUEX project) and the Swiss National Science Foundation (Grant No. 2000 287 27).
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