Large Eddy Simulation of Non-Local Turbulence and Integral Measures of Atmospheric Boundary Layers

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(12) Dissertation for the Degree of Doctor of Philosophy in Meteorology presented at Uppsala University in 2003. Abstract Esau, I., 2003. Large eddy simulation of non-local turbulence and integral measures of atmospheric boundary layers. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 800. 30 pp. Uppsala. ISBN 91-554-5519-0 A new large eddy simulation (LES) code is developed and used to investigate non-local features of turbulent planetary boundary layers (PBLs). The LES code is based on filtered Navier-Stokes equations, which describe motions of incompressible, Boussinesq fluid at high Reynolds numbers. The code computes directly large-scale, non-universal turbulence in the PBL whereas small-scale, universal turbulence is parameterized by a dynamic mixed subgrid closure. The LES code is thoroughly tested against high quality laboratory and field data. This study addresses non-local properties of turbulence which emphasis on the stable stratification. Its basic results are as follows. The flow stability in PBLs is generally caused by two mechanisms: the negative buoyancy force (in the stable density stratification) and the Coriolis force (in the rotating system). The latter stabilizes the flow if the earth’s vorticity and the turbulent vorticity are antiparallel. The Coriolis force stability suppresses large-scale turbulence and makes large eddies asymmetric. The density stratification suppresses vertical scales of turbulence. Joint actions of the Coriolis and the buoyancy forces result in a more complex behavior of turbulence. Particularly, the layers of vigorous turbulence may appear in the course of development of low-level jets in baroclinic atmosphere. Non-local effects determine integral measures of PBLs, first of all the PBL depth. This study clearly demonstrates its pronounced dependences on the Coriolis parameter, the Kazanski-Monin internal stability parameter, and newly introduced imposed-stability and baroclinicity parameters. An LES database is created and used to validate an advanced PBL-depth formulation. LES support the idea that PBLs interact with the stably stratified free flow through the radiation of gravity waves, excited by large turbulent eddies at the interface. Igor Esau, Department of Earth Sciences, Meteorology, Villav. 16, 75236, Uppsala, Sweden. ©Igor Esau 2003 ISSN 1104-232X ISBN 91-554-5519-0 Printed in Sweden by Geotryckeriet, Uppsala, 2003.

(13) PAPERS INCLUDED IN THE THESIS This thesis is based on the following papers, which are referred to in the text by their Roman numerals:. I. Esau Igor. Large eddy simulation of planetary boundary layers with dynamic mixed subfilter closure. Environmental Fluid Mechanics (Submitted). II Esau Igor. Coriolis effect on coherent structures in planetary boundary layers. Journal of Turbulence (Accepted) III Esau Igor, Lyons Tom. (2002). Effect of sharp vegetation boundary on the convective atmospheric boundary layer. Agricultural and Forest Meteorology 114(1-2): 3-13 IV Zilitinkevich Sergej, Esau Igor. The effect of baroclinicity on the depth of neutral and stable planetary boundary layers. Quarterly Journal of the Royal Meteorological Society (Accepted) V Zilitinkevich Sergej, Esau Igor. (2002). On integral measures of the neutral barotropic planetary boundary layer. Boundary Layer Meteorology 104(3): 371-379 Comments: In paper III, the author made numerical experiments using a newly LES code developed by the author to reproduce data from buFex (1993) field experiment by Lyons et al. In Papers IV and V, the author is responsible for the LES validation of analytical equations, which derived by the supervisor Prof. S. Zilitinkevich..

(14) Contents. Introduction.....................................................................................................1 Large eddy simulation code............................................................................4 Equations....................................................................................................4 Turbulence closure .....................................................................................5 Numerical schemes ....................................................................................7 Boundary conditions ..................................................................................9 Quality of large eddy simulations.................................................................10 Non-local turbulence in planetary boundary layers ......................................13 Convective boundary layers .....................................................................13 Neutral boundary layers ...........................................................................15 Stable boundary layers .............................................................................17 Conclusions...................................................................................................22 Acknowledgements.......................................................................................24 References.....................................................................................................25.

(15) Abbreviations. PBL CBL SBL NBL LES DMM TKE NSE Log-law Re Rig Pr POD IO LLJ PC. Planetary boundary layer Convective boundary layer Stable boundary layer Neutral boundary layer Large eddy simulations Dynamic mixed closure Turbulent kinetic energy Navier-Stokes equations Logarithmic law of the wall Reynolds number Richardson number Prandtl number Proper orthogonal decomposition Inertial oscillation Low level jet Personal computer.

(16) Introduction. A large eddy simulation (LES) technique is gradually becoming a popular tool to investigate turbulent environmental flows. LES studies have several valuable advantages. They are (i) much cheaper and (ii) easier to handle than field measurements. They provide (iii) three-dimensional fields of flow characteristics under (iv) well-controlled external conditions. They are (v) more accessible in the sense that LES can be run in the conditions where field measurements would be difficult. Moreover, they are (vi) very helpful for conceptual understanding of environmental processes. Indeed, one or another physical process can be easily excluded from the consideration in LES. Therefore, the role of different processes can be clearly identified. Environmental sciences use the LES technique as a tool. Moreover, a number of studies (e.g. Khanna and Brasseur, 1998) used the LES technique as a "black box". It is possible since the LES technique is nothing but a numerical technique solving three-dimensional Navier-Stokes equations (NSE). The equations describe motions in a continues media disregarding particular kinds of fluid. This is especially true for a high Reynolds number flow. In such a flow, a molecular viscosity, which is a fluid specific feature, does not affect scales of practical importance. Successful LES of convective and neutral PBLs (e.g. Nieuwstadt et al, 1991; Andren et al, 1994; Moeng et al, 1996) have improved our knowledge of atmospheric turbulence. Particularly, these and other studies discovered the role of non-local turbulent processes in the maintenance of semiorganized turbulent motions in the atmosphere (Zilitinkevich et al, 1999). Turbulent processes in stably stratified boundary layers (SBLs) are more difficult to simulate (e.g. Mason and Derbyshire, 1990; Derbyshire, 1999). Firstly the stable stratification suppresses the vertical component of turbulent motions in SBLs. LES must be run at finer meshes to keep turbulence well resolved. Secondly the balance between generation of a turbulent kinetic energy (TKE) by the shear and dissipation of the TKE by the viscosity and the buoyancy must be simulated more carefully. Thus, there is a need in an accurate subgrid turbulence closure in the LES code. In spite of a few successful simulations of SBLs (e.g. Kosovic and Curry, 2000;. 1.

(17) Figure 1: Left panel – streamwise turbulence energy spectra for various flows (after Chapman, 1979). Right panel – streamwise spectra of streamwise component of the velocity for various stability conditions in atmospheric boundary layers (after Larsen, 1986).. Saiki, Moeng and Sullivan, 2000), it is still a challenge to simulate turbulence in stably stratified layers. Problems with LES increase in the number and complicity when integral measures of the PBLs such as a PBL depth, a geostrophic drag coefficient, a geostrophic angle etc. are to study. Such measures are determined not only by small-scale turbulence but also by non-local effects due to large-scale gravity waves (Zilitinkevich and Mironov, 1996; Zilitinkevich et al, 2000; Zilitinkevich, 2000). Accordingly, the first aim of this study was to develop an improved LES code suitable for simulations of stably stratified layers at relatively coarse meshes. In its early days, the LES technique was considered as an application of the averaging operator over a finite volume. The idea was to separate largescale motions, u l , which are an exact solution of the discretized NSE from subgrid scale motions, u s , which are unknown. The computer LES code resolves the large-scale motions providing amplitudes and phases of every turbulent eddy. However, the subgrid scale motions must be parameterized involving one or another a priori assumption about the nature of small-scale motions. Using this approach, Deardorff (1972) conducted the first LES of a planetary boundary layer (PBL). He investigated neutral and unstable PBLs. The early LES technique was internally inconsistent. Firstly the averaging operator is not explicit. Its formulation depends on a numerical scheme in use. Secondly resolved and subgrid motions have inherently different features in this formulation. Leonard (1974) proposed a self-consistent, filtering, approach. The averaging approach follows from the filtering approach as its simplification. The Leonard's idea was to filter continues NSE with some filter operator. As 2.

(18) the result, continues filtered NSE for the large-scale motions, u l , incorporates a new term. The new term relates explicitly large and smallscale motions. It is important that in the continues case interactions between u l and u s can be exactly expressed through the large-scale motions u l (Yeo, 1987; Leonard, 1997; Carati, Winkelmans and Jeanmart, 2001). Thus, there is much less uncertainty in unknown small scale motions than it has been thought before. The progress in understanding of the nature of the incompressible NSE along with further development of the filtering approach have resulted in a new class of turbulence closures – dynamic turbulence closures (Germano et al, 1991; Germano, 1992). The dynamic turbulence closures assume that fluid motions consist of (i) small scale, universal, homogeneous turbulence and (ii) large scale, non-universal eddies. The former may be treated phenomenologically, for instance assuming the small scale turbulence has the Kolmogorov's inertial energy cascade (see Fig. 1). The later must be calculated explicitly. They depend on external parameters (boundary conditions), which are changing from flow to flow. Thus, the state-of-the-art LES technique is based on distinction between the universal and the non-universal turbulence in fluid. Any real LES code cannot distinct between these two kinds of turbulence clearly. Both discretization and numerical schemes make damage on the physical motions especially on the grid scale. It introduces a larger uncertainty in the LES technique. In spite of this, the dynamic turbulent closures are proved to be very useful in simulations for engineering flows (Piomelli, 1999). However, Mason (1994) doubted the applicability of such closures in PBL simulations. The author's LES code confirmed the robustness of the dynamic turbulent closures in studies of PBLs. Porte-Agel, Meneveau and Parlange (2000) and Porte-Agel et al (2001) come up with the same conclusion. The distinction between the universal and non-universal turbulence serves not only to ground the LES technique but also to encompass the area of the LES responsibility. That is an investigation of turbulence properties, which are determined by boundary conditions and external forces acting on the low. This summary attempts to highlight the author's view of the role of nonuniversal, non-local turbulence in PBLs. Section 1 presents the author's LES code. Section 2 demonstrates the quality of the LES. These two sections are mostly the content of Paper I. Section 3 (Papers II and III) addresses the nonlocal turbulence in convective PBLs. Section 4 (Papers II, IV, V) addresses the non-local turbulence in neutral PBLs. Section 5 (Paper IV) addresses the non-local turbulence in stable PBLs. Section 7 outlines conclusions of this study. 3.

(19) Large eddy simulation code Equations. The LES code is based on the following filtered equations. ∂uil ∂ =− (uil u lj + τ ij + p lδ ij ) − fω jδ ij − gβΘlδ i 3 ∂t ∂x j. (1). ∂uil =0 ∂xi. (2). ∂ ( Θl u lj + τ Θj ) ∂x j. (3). ∂Θl ∂t. =−. The superscript l denotes filtered (resolved) variables; the superscript L denotes variables filtered with a wider filter b. b. a. a. uil = ∫ ui ( x ′j )G∆ ( x j − x ′j )dx ′j , uiL = ∫ ui ( x ′j )Gα∆ ( x j − x ′j )dx ′j. (4). a = xj − ∆ j /2 , b = xj + ∆ j /2 superscript s denotes subfilter variables uis = ui − uil . Here x j = ( x, y , z ) are the axes of the Cartesian coordinate system directed to the East, the North and the Zenith, ui = (u, v, w) are the velocity components and ∆ j = ( ∆ x , ∆ y , ∆ z ) are the grid scales. The filtered potential temperature Θl and the filtered dynamic pressure p l are defined in the same way. Specific formulation of the filter function, G∆ ( x j − x ′j ) , is used only to calculate a subfilter turbulent stress, τ ij , and a subfilter heat diffusion, τ Θj . The author’s LES employs the Gaussian filter The. G∆ ( x j − x ′j ) =.  6( x j − x ′j ) 2  6  exp 2   π∆2j ∆ j  . (5). The filter works successively in x and y directions. Variables remain unfiltered in z direction. 4.

(20) The vertical and the horizontal components of the Earth rotation are taken into account through the Coriolis parameter, f = 2Ω sin ϕ , where Ω = 7.45 ⋅10−5 s-1 is the constant angular velocity and ϕ is a latitude. The components of the vorticity are ω j = {(u2g − u2l − u3l ctgϕ ), ( −u1g + u1l ), u1l ctgϕ } . At the middle and high latitudes, the mean pressure gradient relates to the geostrophic wind as uig = 1 / f ( −∂p / ∂y , ∂p / ∂x,0) . The thermal expansion coefficient is β=0.003 K-1, g=9.81 m s-2 is the acceleration due to gravity. The Einstein's notation is used for summation. In this notation, δ ij = 1 at i=j and δ ij = 0 otherwise.. Turbulence closure Terms τ ij and τ Θj in Eq. (1) and (3) include unknown subfilter variables. They must be expressed only through the filtered variables. The exact form of the subfilter stress is (Leonard, 1974). τ ij = (uil u lj ) l − (uil ) l (u lj ) l + (uil u sj + uis u lj ) l + (uis u sj ) l = Lij + Cij + Rij. (6). The Reynolds term, Rij, is responsible for the energy dissipation. The cross term, Cij, is responsible for the energy backscatter from small scales to large scales. The net energy dissipation is a small difference between large terms Rij and Cij. A turbulent closure, which accounts for the Reynolds term only, is overdissipative. It dissipates about twice as much energy as it really occurs in the flow (Paper I). This fact explains why the Smagorinsky constant, Cs=0.1, suits boundary layer simulations much better than the analytically derived Cs=0.18-0.23. The problem becomes severe in a surface layer and in stably stratified PBLs. The Leonard term, Lij, used to be neglected even in backscatter subfilter closures (Mason and Thomson, 1992). However, it was recognized that Lij is responsible for up to 14% of the total energy transfer in the LES with the Gaussian or the top-hat filters (Leslie and Quarini, 1979). The author's LES employs a dynamic mixed subgrid closure (DMM) by Vreman et al (1994). The closure incorporates all three terms, Lij, Cij and Rij. It reads. τ ij = (uil u lj ) L − (uil ) L (u lj ) L − 144424443 scale similarity. part. 5. 2ls2 Sijl Sijl 1 424 3. eddy vis cos ity. (7) part.

(21) The scale similarity part parameterizes the Leonard term. The eddy-viscosity part parameterizes the sum of the Reynolds and the Cross terms. A principal feature of the DMM is that dissipation and diffusion in the LES cannot exist without turbulent activity on the smallest resolved scale. It prevents unphysical turbulent dissipation in the turbulence-free atmosphere with the mean wind shear. A possible energy backscatter and the effect of the flow anisotropy are taken into account through variations of a dissipative length scale, ls. The dissipative length scale can be calculated from resolved quantities in the course of simulations as L L L 1 ( Lij − H ij ) M ij l = , where 2 M ijL M ijL. (8). LLij = (uil u lj ) L − (uil ) L (u lj ) L. (6). M ijL = ( Sijl Sijl ) L − α 2 ( Sijl ) L ( Sijl ) L. (10). ) − (((u ) ) ) (((u ) ) ) − − (((u ) ) ((u ) ) − ((u ) (u ) ) ). (11). 2 s. (. H ijL = ((uil ) L (u lj ) L ). l L. l l L i. L l L l i. l l L j. l l i. L l L l j. l l L j. Here, the strain rate tensor is. S ij =. 1  ∂ui ∂u j  + 2  ∂x j ∂xi . (12). Vreman et al (1997) analytically found α2=52/3 = 2.92 for the Gaussian and the top-hat filters with the filter width ratio ∆Lx / ∆lx = ∆Ly / ∆l y = 2 , ∆Lz / ∆lz = 1 . The dynamical computation of ls is a mathematically ill-posed problem. To stabilize computations, it was prescribed that − 0.01 ≤ ∆ ≤ 0.8 . This is not too restrictive. The average value of ls / ∆ was about 0.18-0.23. This is in excellent agreement with Lesly and Quarini (1979) and Lilly (1966) analytical predictions, ls / ∆ =0.179-0.23. The dissipation length scale is much smaller in stably stratified PBLs. Figure 2.a shows typical nondimensional profiles of ls / ∆ in LES of conventionally neutral, convective and stable PBLs. The filter scale ∆ = ( ∆ x ∆ y ∆ z )1 / 3 is a constant here. A strong reduction of the dissipative length scale is obvious within a surface 6.

(22) layer as well as in stably stratified layers including an inversion layer at the top of PBLs. The turbulent diffusion of heat is parameterized in the following way −1 τ Θj = (Θil u lj ) L − ( Θil ) L (u lj ) L − PrLES l s2 Sijl. ∂Θl , ∂x j. (13). −1 where PrLES is a subgrid Prandtl number. Kondo et al (1978) proposed an empirical relation.  1 , 1 ≤ Rig  7 Rig  1  −1 , 0.01 < Rig ≤ 1 PrLES =  (14) 1  6.873Rig + 1 + 6.873Rig   1.5, Rig ≤ 0.01  where Rig = β∂Θ / ∂z /((∂u / ∂z ) 2 + (∂v / ∂z ) 2 ) is the gradient Richardson. number.. Numerical schemes The author's LES code is a finite difference code of the 2nd order of accuracy. The code employs a fully conservative central difference scheme (Morinishi et al, 1998) with the non-linear advection discretized in a skewsymmetric form. All variables are discretized on the staggered C-type grid with the pressure and the potential temperature at centers of cells. Although the scheme is not monotonic, no limiter is applied to correct the simulations. Non-monotonic schemes give rise to numerical wiggles around any physical fluctuation of the velocity or the temperature. The wiggles do not cause problems in simulations of convective, neutral for weakly stable PBLs. Physical turbulent fluxes are stronger than numerical fluxes under such conditions. Numerical fluxes complicate simulations of very stable layers. Limiters could reduce the numerical fluxes but they are computationally expensive. Sagaut and Grohens (1999) showed that the optimal discrete forms of the Gaussian and the top-hat filters coincide for the conservative central difference scheme of the 2nd order of accuracy. It makes the central-difference code independent of the choice of filters.. 7.

(23) 1.5. 1.5 a). b). 1. 1 z/H. z/H. analytical interval 0.5. 0. 0.5. 0. 0.1. 0.2. 0 −2. 0.3. −1. 0. 1. 2. Rig. Cs=ls/∆. Figure 2: Dynamically calculated profiles of the Smagorinsky constant C s = l s / ∆ (a) and the gradient Richardson number Rig (b) in author's LES: solid line – neutral PBL; dashed line – convective PBL; dashed-dotted line – stable PBL.. Thus, it makes the LES code self-consistent. The discrete compact filter in e.g. x direction is (15) u l ( X ) = a (u l ( X + ∆ ) + u l ( X − ∆ ) + bu l ( X )) i. i. x. i. x. i. where ui ( X ) is the velocity at the grid point X. The coefficients a, b are equal to 1/24, 22 for the basic filter G∆ , and 1/6, 4 for the wider filter Gα∆ . The Eq. (1) and (3) are integrated in time using the Runge-Kutta scheme in Jameson et al (1981) formulation. The time step is variable. It can be estimated (Dr. B-J. Boersma, personal communication) as.  ul  1 1 1  ∆ t = C  i + ls2 Sijl  2 + 2 + 2   ∆   ∆x  x ∆ y ∆z   i. −1. (16). Here, C is the Courant number. The fourth order Runge-Kutta scheme allows C=2.89. However, it is better to reduce C twice in convective and very stable PBLs. Incompressibility is enforced by a velocity correction through dynamic pressure calculations. The dynamic pressure is calculated by means of a fractional step method by Kim and Moin (1985) with modifications by Armfield and Street (1999). It has to be mentioned that the corrected velocity is not an exact discrete solution of the Navier-Stokes equation in this procedure. To satisfy both the continuity and the Navier-Stokes equations, the velocity field must be iterated several times at every time step. This LES code does not iterate since the improvement is rather small.. 8.

(24) Boundary conditions As in a number of LES (e.g. Andren et al, 1994), the author's LES employs a logarithmic law (log-law) to match the high Re surface conditions. The 2nd order central difference scheme along with the C-type staggered grid require only tangential subfilter stresses, τ i 3 and τ Θ 3 , to be determined at the nonslip surface. For simplicity, the influence of pressure gradients, buoyancy and nonlinear accelerations are neglected below the first grid level. It gives the relation τ k 3 = u*2 , k=1,2. The surface stress velocity, u* , can be determined from the log-law for a rough surface and the velocity at the first computational level ∆ z / 2 as. 1 ui ( x, y , ∆ z / 2) (17)  ∆z   1 / κ ln  2 z0  where z0 is surface roughness and κ=0.41 is the von Karman constant. u* =. Finally, the subfilter surface stress is nothing but. τ k 3 ( z = 0) = u*2. uk ( x , y , ∆ z / 2 ) , k=1,2 uk ( x , y , ∆ z / 2 ). (18). The subfilter surface heat flux, τ Θ 3 , is prescribed explicitly. The von Neumann boundary conditions set up at the upper boundary. ∂uk ( x, y , z = L3 ) ∂Θ( x, y, z = L3 ) = = 0, k=1,2 ∂z ∂z. (19). The Dirichlet condition must be used for the vertical component of velocity, w (x, y, z=L3) = 0. Lateral boundary conditions are periodic. Periodicity reduces code run time. Moreover, it would be difficult to impose correct turbulent inflow conditions on the lateral boundaries.. 9.

(25) Quality of large eddy simulations. Our interest is to obtain Re-independent statistics rather than to conduct truly infinite Re LES. Two major factors limit the Reynolds number in LES. The first one is the grid resolution. The second one is an ability of a subgrid closure to support velocity fluctuations on the grid scale. It has been already found experimentally (Wei and Wilmarht, 1989), numerically and analytically (Coleman et al, 1990) that low order turbulent statistics become Re-independent at a rather small Re≈5000. This fact facilitates simulations of the PBLs. Many important characteristics of the PBLs can be reproduced with a high quality even in coarse resolution runs. Figures 3, 4 support this thesis (see also Fig. in Paper I). These Figures show LES runs along with laboratory data. Figures reveal clearly a good agreement between all LES runs and laboratory data in the core of the boundary layer. Generally, the laboratory data possess less scatter than the atmospheric data. Therefore, they are more suitable to the LES validation. The quality of simulations improves with the resolution refinement. 2 However, the coarsest run with the resolution 0.05h still reproduces σ uu and 2 σ ww in a reasonable agreement with the data. Hence, the conclusion is that the second order turbulent statistics are fairly insensitive to the resolution of the LES code. This is somewhat expected. Indeed, motions on the scale of h 2 2 give major contribution to σ uu and σ ww . Such motions are well resolved in all runs. This follows from the analysis of the velocity spectra (Fig. 5). The spectral maximum of the TKE is seen in all runs. The grid scale is the smallest scale of motions in the LES. The small-scale motions considerably contribute to higher order statistics. Hence, the higher order statistics must reveal a stronger Re-dependence. Atmospheric measurements reveal a rather large scatter. This is mainly due to a complex interplay of different forces in real atmosphere. Moreover, PBLs are never exactly neutral. In addition, they are often baroclinic. Nevertheless, the agreement between LES of neutral PBLs and the atmospheric data is rather good. It is clearly seen in Fig. 6. The truly neutral LES agree better with the Pennell and LeMone (1974) data in Fig. 6.b. They obtained the data in a very deep (h>600 m.), neutrally stratified atmospheric. 10.

(26) 1. 1 a). 0.8. b). 0.8. z/H. 0.6. z/H. 0.6 0.4. 0.4. 0.2. 0.2. 0. 0. 2. 4. 0. 6. −2. 0. σ3uuu/u3. 2. σuu/u2. 2. 4. *. *. Figure 3: Profiles of the non-dimensional variance, σ uu / u* (a), and the third order 3 3 statistics, σ uuu / u* (b) of the streamwise component, u, of the velocity: solid line LES resolution 0.0125h; dashed line - LES resolution 0.025h; dash-dotted line - LES resolution 0.05h; □ – data from Andreopoulos and Bradshaw (1981), ○ – data from Krogstad et al (1992). 2. 1. 2. 1 a). 0.8. b). 0.8. z/H. 0.6. z/H. 0.6 0.4. 0.4. 0.2. 0.2. 0. 0. 0.5. 2 2 /u ww *. σ. 1. 0 −0.2. 1.5. 0. 3 3 /u www *. σ. 0.2. 0.4. Figure 4: The same as Fig 3, but for the variance of the vertical component of the velocity.. PBL. It makes their data the most relevant to the conditions of the numerical experiment. Other authors obtained data in relatively shallow (h<400 m.) PBLs. Stable stratification of the free atmosphere does not allow the PBL to reach the equilibrium Ekman depth h = 0.5u* / | f | . It results in a number of effects, which are considered in Section 5. Among these effects is an enhancement of the v -fluctuations. Such a systematic enhancement is clearly seen in Fig. 6.b. The conventionally neutral LES agree better with Tjernström and Smedman (1993) data than the truly neutral ones. Both runs underpredict small-scale fluctuations in the lower part of PBLs (Fig. 6.e). We can conclude that a coarse resolution LES code with the DMM closure realistically produces Re-independent turbulent statistics. Possible inaccuracy in the atmospheric measurements is larger than the inaccuracy of the LES with the resolution ∆ z = 0.025h .. 11.

(27) 4. 3. 10. 10 a). b). 3. 2. 2. 10. (kz) Suu/u*. (kz) Suu/u2*. 10. 2. 10. 1. 10. 1. 0. 10. 0. 1. 10. 10. 2. 10. 0. 10. 1. 10 kz. kz. 10. 2. 10. c) *. (kz) Sww/u2. *. (kz) Sww/u2. d). 1. 10. 1. 10. 0. 10. 0. 10. 0. 1. 10. 2. 10. 0. 10. 1. 10 kz. kz. 10. Figure 5: Non-dimensional streamwise spectra of the velocity components u (a, b) and w (c, d) plotted versus non-dimensional wave number kz: solid line – LES resolution 0.0125h; dashed line – LES resolution 0.025h; dashed-dotted line – LES −5 / 3 resolution 0.05h; the straight line is ( kz ) ⋅ k Panels (a) and (c) correspond to the PBL core z=0.5h, panels (b) and (d) corresponds to the surface layer z=0.05h. 1. 1 a). 1. b). 0.9. 1. c). 0.9. 1 d). 0.9. 1 e). 0.9. 0.9. 0.8. 0.8. 0.7. 0.7. 0.7. 0.7. 0.7. 0.7. 0.6. 0.6. 0.6. 0.6. 0.6. 0.6. 0.5. 0.5. 0.5. 0.5. z/H. 0.8. z/H. 0.8. z/H. 0.8. z/H. 0.8. z/H. z/H. 0.9. 0.5. 0.5. 0.4. 0.4. 0.4. 0.4. 0.4. 0.4. 0.3. 0.3. 0.3. 0.3. 0.3. 0.3. 0.2. 0.2. 0.2. 0.2. 0.2. 0.2. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0. 0. 2. 4 σ2 /u2*. 6. 0. 0. 2. 4. σ2vv/u2. 0. 0. 0. 1. 2. σ2ww/u2. −1. −0.5 2. 0 2. σuw/u2* , σvw/u2*. 0. 0. 0.5 3. σwe/u3*. 1. f). 0 −0.4. −0.2 σ3wp/u3*. 0. 0.2. 2 / u*2 (a), σ vv22 / u*22 (b),2 2 Figure 6: Profiles of non-dimensional variances, σ uu 2 2 σ ww / u* (c), non-dimensional components of the turbulent stress σ uw / u* , σ vw / u* 3 3 (d) and the third order statistics, σ we / u*3 (f): solid line – truly / u*3 (e), σ wp neutral LES with resolution 0.0125h; dashed line – conventionally neutral LES with resolution 0.025h; □ – data from Andreopoulos and Bradshaw (1981), ○ – data from Krogstad et al (1992); * - data from Pennell and LeMone (1974). uu. *. *. 12.

(28) Non-local turbulence in planetary boundary layers. Convective boundary layers Convective boundary layers (CBLs) are inherently unstable. Small scale, stochastic perturbations of infinitesimal initial amplitudes evolve into large scale, persistent coherent structures often visible as convective clouds. Individual non-local eddies in CBLs are known in literature as puffs, thermals and plumes (Hunt, 1998). Multivariate statistical analyses gives a possibility to estimate the importance of the non-local turbulence in the atmospheric convection. Wilson (1996) obtained that the largest scale structures in the CBL comprises more than 65% of the total TKE. Author's calculations gave a close value (>55%) of the total TKE in the largest scale eddies. Thus, LES quantitatively confirmed a common opinion that the CBLs are dominated by coherent structures of a very large scale. Moreover, a proper orthogonal decomposition (POD) – a modification of the multivariate statistical analysis – provides a typical shape of the coherent structures in three dimensions (Paper III). This result is absolutely impossible to derive from ground- or air-born meteorological measurements. Measurements in three dimensions, the prerequest for POD analysis, would be extraordinary expensive and difficult to achieve. The typical shape of the coherent eddy is essentially a vortex ring in the shear-free CBL. Plate 1 shows the coherent eddy. The wind shear and the Coriolis force modify the shape of the coherent structures. The atmospheric PBLs are usually heated too intensively to demonstrate the Coriolis effect on the coherent structures. However, the Coriolis effect is important in deep ocean convection. The wind shear modifies the vortex ring profoundly. It aggravates the eddy vorticity in the mean wind direction. It also stretches the vortex ring. Plate 2 shows the typical shape of the coherent eddy in the sheared CBL at the latitude ϕ = 45o North. 13.

(29) The discussed coherent eddies represent only the first level of coherent structures in CBLs. Cloud cells and cloud streets represent the next level (Atkinson and Zang, 1996). They are also known from laboratory experiments as rolls and convective cells. Such structures have scales of 550 km in the horizontal. It makes their simulation rather difficult due to capacity restrictions of modern computers. Nevertheless convective cells and rolls were reproduced in the author's LES. Plate 5 shows a three dimensional instant view of the convective cells in the shear-free CBL. Plate 6 shows a view of the rolls in the sheared CBL without Coriolis force. These LES contradict the widely accepted explanation that the rolls (cloud streets) result from the action of the Coriolis force on the CBL (Brown, 1991). At the same time, these LES are supported by a number of satellite pictures in which cloud streets over equatorial areas are clearly identified. Moreover, these LES support a new theory by Elperin et al (2003), which explains the roll formation through the action of the geostrophic wind on coherent eddies in CBLs. The coherent structures on both levels have a random phase with respect to a homogeneous, featureless surface. However, abrupt change of the surface properties may affect the structures significantly. The buFex field experiment revealed the role of a sharp vegetation boundary on coherent structures in the CBL. The buFex (1993) field experiment (Lyons, 1993) was conducted in the southwestern Australia. The terrain is a vast flat plain. The western part of the plain is under agricultural vegetation (mostly wheat). The eastern part is under native vegetation (mostly local shrub and bushes). Different types of vegetation result in different turbulent heat fluxes as well as in different roughness of these two areas. Moreover, the heat flux and the roughness change abruptly. Under such conditions, the CBL turbulence develops a convective roll, which is attached to the sharp vegetation boundary. LES disclosed that the upward branch of the roll is attached to the warmer, native vegetation surface and the downward branch to the colder, agricultural vegetation surface. Therefore, convective eddies are strongly suppressed over the agricultural area over 3-5 km from the boundary. Development of such boundary-attached rolls would unavoidably dry out a strip of agriculture close to the boundary. At the same time, it could increase rainfall behind the boundary. Quantitatively, about 20-30% of evaporated water is transported from the agricultural area into the native vegetation area. This explains climatologically significant increasing of rainfalls over the native vegetation, which has been registered in this area for the last 40 years. Plate 7 presents a three dimensional instant view of convective structures in this LES. The strongest upward and downward motions were simulated in the direct vicinity of the vegetation boundary. 14.

(30) Neutral boundary layers In spite of the fact that the atmospheric neutral boundary layers (NBLs) are never exactly neutral, the NBL was thought to be the simplest object to study and to simulate. These studies (e.g. Andren et al, 1994) neglect the fact that atmospheric NBLs always develop against more or less stably stratified turbulence-free atmosphere. Zilitinkevich and Esau (Paper III) proposed to distinguish between the truly neutral PBL – developing in a neutrally stratified fluid, and the conventionally neutral PBL – developing against the background stable stratification. The former PBL has been well studied experimentally, theoretically and numerically. These studies agree on values of key integral parameters and scaling. For instance, different studies (Rossby and Montgomery, 1935; Caldwell, van Atta and Heland, 1972; Mason and Thomson, 1987; Coleman, 1999) obtained the same value of the constant Ch=0.5 in Rossby and Montgomery formula: (20) h = Ch u* / | f |, for the depth, h, of an equilibrium, barotropic NBL. The same value of Ch=0.495 has been obtained from the author's LES of the truly neutral PBL. In atmospheric NBLs, Ch=0.1-0.2 (e.g. Taylor, 1969; Tjernström and Smedman, 1993; Hess and Garratt, 2002). Contrary to the classical view, Zilitinkevich et al (2002) and Zilitinkevich and Baklanov (2002) suggested that Ch is not a constant. It depends on an imposed-stability parameter, µ N = N / | f | , where N is the free-flow BruntVäsälä frequency and f is the Coriolis parameter. Zilitinkevich and Baklanov derived a diagnostic equation:.  C 2C  Ch = C R 1 + R 2uN µ N  CS  . −1 / 2. ,. (21). where CR, CS and CuN are dimensionless constants to be determined empirically. These constant were estimated in the author's LES as CR=0.5, CS=1.0 and CuN=0.56, – in a fairly good agreement with earlier estimations from available atmospheric data. Figure 7 shows the theoretical curve after Eq. (21), along with LES data.. 15.

(31) hE|f|/u*. 0.6. 0.4. 0.2. 0. 0. 200. 400. 600. 800. 1000. 1200. µN=N/|f|. Figure 7: The dependence of the dimensionless depth, | f | hLES / u* , of the barotropic, truly and conventionally neutral PBLs on the imposed-stability parameter, µ N . Black circles, squares and diamonds represent author's LES. Crosses represent earlier LES (Mason and Thomson, 1987; Lin et al, 1997) and DNS – direct numerical simulation (Coleman, 1999). The solid line is calculated after Eq. 2 (20) with Eq. (21) where CR=0.5 and CuN / CS =0.56. Ri 1.8. hLES/hE (Eq. 6). 6.25. 1.56. 0.69. 0.39. 1.2. 1.6. 1.6 typical atmospheric interval of the free flow Ri. 1.4. 1.2. 1. 0. 0.4. 0.8. µ =Γ/N Γ. Figure 8: The dependence of the dimensionless depth, hLES /h[Eq.(20) with Eq.(23)], of the simulated baroclinic conventionally neutral PBL on the baroclinicity parameter, µ Γ . The upper scale correspond the values of the free flow Richardson number −2 Ri= µ Γ . Black squares represent author's LES. The solid line is calculated after Eq. 2 (20) with Eq. (23) where C0=0.67, CR=0.5 and CuN / C S =0.56. LES is a very useful tool to study baroclinic boundary layers. In the baroclinic atmosphere, Ch depends not only on the imposed-stability parameter but also on a baroclinicity parameter, µ Γ = Γ / N . Here, Γ is the baroclinic shear:.  ∂u g Γ =   ∂z. 2 2   ∂v g    +      ∂z  . 1/ 2. g = | f | Θ0 16.  ∂Θ  2  ∂Θ  2      +   ∂x   ∂y  . 1/ 2. (22).

(32) where (ug, vg) are components of the geostrophic wind, and Θ is the potential temperature. On the assumption that Γ=constant, scaling analysis of the Ekman equations gives.  C 2C  C h = C R (1 + С0 µ Γ ) 1 + R 2uN µ N  CS   1/ 2. −1 / 2. (23). Figure 8 shows a reasonably good agreement between the theoretical curve Chbaroclinic / Chbarotropic (after Paper IV) and LES data. Zilitinkevich (2002) suggested that the radiation of internal gravity waves is a physical mechanism responsible for decreasing of the equilibrium NBL depth. Small scale, universal turbulent eddies cannot radiate gravity waves. Frequencies of such waves would be larger than the typical atmospheric N. Such waves would be evanescent. Thus, the radiation process requires the existence of large-scale non-universal eddies. Indeed, non-universal turbulence is important in NBLs. The POD analysis of the author's LES attributes about 30% of the total TKE to the largest coherent structures in the NBL. Finnigan and Shaw (2001) obtained about the same value in their wind tunnel studies of a high Re boundary layer. Plate 3 presents the typical three-dimensional structure of coherent eddies in the NBL. This structure closely resembles horseshoe vortices. The horseshoe vortices are well recognized in laboratory flows too (Robinson, 1991). They results in low- and high-speed velocity streaks in an snapshot of the boundary layer flow. In agreement with the Ekman analysis, the Coriolis force strongly modifies the large eddies in the NBL. The horizontal component of the Coriolis force makes the structure in Plate 3 asymmetric. The vertical component of the Coriolis force suppresses the structure as whole. This is clearly seen in Plate 4.. Stable boundary layers It is generally recognized that turbulent eddies become smaller and less intensive under the action of stable stratification. Garg et al (2000) suggested to distinct three regimes of stability: (i) a buoyancy-affected regime in which turbulence is suppressed but still active, (ii) a buoyancy-controlled regime in which turbulence ceases during the transition and restores again in a steadystate and (iii) a buoyancy-dominated regime in which turbulence ceases completely during the transition and exists only intermittently in sporadic spots in an almost laminar, steady-state flow. 17.

(33) 7. E (m2 s−2). 6 5 4 3 2. 0. 1. 2. 3 time (hours). Lz. 4. 5. 6. ∫. Figure 9: Time evolution of the total TKE, E = 1 / Lz E ( z )dz , in nocturnal SBLs. Symbols represent different runs of the author's LES. Here, Lz is the height of the 0 domain, E(z) is the TKE.. Evolution of E/u2*. 8 7. height (meters). 250. 6 5. 200. 4 150. 3 SBL top. 2. 100. 1 50 1. 2. 3 4 time (hours after sunset). 5. 6. Evolution of |u|/u. *. Development of a short time elevated turbulent layer related to break of a low level jet 50 45. height (meters). 250. 40. 200. 35 150. 30. SBL top 100. 25 20. 50 1. 2. 3 4 time (hours after sunset). 5. 6. Figure 10: Development of an elevated turbulent layer in the author's LES of the baroclinic SBL. The upper panel shows time evolution of the TKE. The lower panel shows the time evolution of the wind speed.. 18.

(34) The buoyancy-affected turbulence resembles very much turbulence in NBLs. Since the buoyancy force suppresses most of all the largest, coherent structures, the buoyancy-affected regime is dominated by the small-scale universal turbulence. The buoyancy-controlled regime is a typical case of nocturnal boundary layers. Figure 9 shows the evolution of the integral TKE in a nocturnal stably stratified boundary layer (SBL) simulated with the author's LES code. The initial decay and the late night restoration of the turbulence are clearly seen. The mechanism of the turbulence decay is well understood in the frameworks of the classical theory of turbulence. The classical theory accounts for the shear instability as the only source of the TKE in SBLs. The dissipation along with the work of the buoyancy force destroy turbulence. Isotropic turbulence in a linearly stratified media can only decay in a flow with the Richardson number, Rig>0.25. Large-scale models use this value of Rig to distinct between turbulent and laminar flows. Such a simplification causes a number of problems and breakdown of turbulent parameterizations in large-scale models (e.g. Marht, 1998; Derbyshire, 1999). As the matter of fact, active turbulence exist in nocturnal SBLs (Marht, 1985) and in long-lived SBLs (Zilitinkevich and Calanca, 2000) after the period of the initial transition. Author's LES confirm these observations. The classical theory does not account for many important but essentially nonlocal mechanisms supporting the turbulence. Among them is KelvinHelmholtz instability, the elliptic instability and the inflection point instability. Two latter mechanisms can be very efficient under the action of the Coriolis force (Miyazaki, 1993). Figure 10 shows development of an elevated turbulent layer at a level of the inflection of the wind profile in the LES of baroclinic SBLs. The buoyancy-dominated regime also occurs in the PBLs. The SHEBA and CASES99 field experiments have registered such cases. Hopfinger (1987) called patchy turbulence of the buoyancy-dominated regime as fossilized turbulence. This turbulence can exist only inside fluid volumes, which have been already mixed up by some non-local process. Surprisingly, the role of non-local turbulence being minimal in the buoyancy-effected regime grows in the buoyancy-controlled regime and becomes essential in the buoyancy-dominated regime. The turbulent Prandtl number Pr = u ' w' / w' Θ' (∇ z Θ / ∇ z u ) is a good indicator of this role. At large Ri, the kinetic energy is transported largely by gravity waves, which cannot transport the heat. Hence, Pr(Rig) must saturate at very large Rig. However, the wave transport of the heat flux still takes place through the wave breaking. A visible saturation of the Prandtl number characterizes the 19.

(35) buoyancy-dominated regime. The behavior of Pr in the LES is in excellent agreement with the behavior of Pr found in new atmospheric measurements (Pordyjak, Monti and Fernando, 2002). Plate 8 presents an instant threedimensional picture of the SBL with active gravity waves. Radiation of gravity waves affects integral measures of the SBLs (Zilitinkevich and Calanca, 2000). Zilitinkevich et al (2002) and Zilitinkevich and Baklanov (2002) demonstrated a PBL-depth dependence on both the imposed-stability parameter, µ N = N / | f | , and the KazanskiMonin internal-stability parameter, µ = u* / | f | L , where L is the MoninObukhov length scale. The extension of the Eq. (23) to essentially stable boundary layers reads.  C 2C C2  C h = C R (1 + С0 µ Γ ) 1 + R 2uN µ N + R2 µ  CS CS   1/ 2. −1 / 2. (24). Figure 11 shows a general agreement between the theoretical curve and the author's LES. Another important non-local process is the development of an inertial oscillation (IO) in the atmospheric SBLs. Figure 12 shows a time evolution of the wind speed and the wind direction in LES of a long-lived SBL. The IO is also clearly seen in Fig. 10. The oscillation periods are | f | in the turbulence-free atmosphere and f + ν t / h in the SBL. Here, ν t is an eddy viscosity. The difference in oscillation periods within and above of the SBL leads to the formation of a thin layer of a pronounced wind maximum. Such a wind maximum is known as a low-level jet (LLJ). There are no publications reporting dependence of LLJ parameters on small-scale turbulence in SBLs. Just opposite, a number of authors (e.g. Högström, Hunt and Smedman, 2002) have reported a strong correlation between development of the LLJ and intensification of turbulence in SBLs. It is probable that the late night restore of turbulence follows the development of the LLJ. This problem, however, needs additional investigation.. 20.

(36) |f|hLES/u*. 0.1. 0.05. 0. 0. 500. 1000. 1500. 2000. µ. Figure 11: The dependence of the dimensionless depth, | f | hLES / u* , of the. simulated baroclinic stably stratified PBL ( µ N =340, Γ / N =0.44) on the internal stability parameter, µ . Black squares represent author's LES. The solid line is calculated after Eq. (20) with Eq. (24) where C0=0.67, CR=0.5, CS=1.0 and CuN / CS2 =0.56. Inertial Oscillation profile. 40 35 30. m. 25 20 15 10 5 0 6 14. 4 12. 2. 10 8. 0 6 m/s. −2. 4 m/s. Figure 12: Time evolution of the wind speed and the wind direction in the author's LES of a long-lived SBL. The evolution orbits represent the inertial oscillation on different layers within and above the SBL. The time interval between knots on each orbit is one hour.. 21.

(37) Conclusions. One of the main result from this work is successful simulation of the environmental turbulence by means of the LES code. The code itself is based on mathematically self-consistent, filtering approach which allows to distinct between large-scale (resolved) and small-scale (parameterized) turbulence. Distinguishing between universal and non-universal turbulence has shown to be a fruitful idea. It resulted in accurate LES of a range of practically important planetary boundary layers including convective, conventionally neutral and stable layers. An accurate LES code on coarse meshes helps to transfer the LES technology from a small number of research groups at supercomputer centers to a large number of researchers working with desktop PCs. Another main achievement of this study is better understanding of the role of non-local processes and non-universal turbulence in PBLs. Previous studies paid the main attention either to visualization of instant threedimensional snapshots of the flow or to statistical analysis of one- or twodimensional sampling series. On the contrary, this study paid the main attention to multivariate statistical analysis in three dimensions. Moreover, the study did not perform a three-dimensional analysis of a single LES run. It addressed a physical problem, the effect of the Earth rotation, in this threedimensional analysis. In order to go through this problem, a new technology was applied. First, the LES code had generated three-dimensional turbulent fields of the velocity and temperature. Second, typical non-local eddies were separated in the PBLs. Third, the role of external parameters was understood by comparing the shapes of non-local eddies. This technology for the first time explicitly visualized how the Coriolis force changes the typical largestscale eddies in the PBL. The horizontal component of the Coriolis force makes the eddies asymmetric and the vertical component suppresses the eddies as whole. Three-dimensional studies of non-local turbulence are, however, still computationally expensive and time consuming. The third achievement is the LES validation of the new theory of nonlocal turbulence developing by S. Zilitinkevich. It is rather difficult to systematically verify this theory using atmospheric measurements. The latter provide a complex interplay of baroclinicity, stratification and the Coriolis 22.

(38) force. They are often unsteady. LES allows to separated effects with due regard to every force. This study clearly showed that the PBL depth, h, depends on three non-dimensional parameters which relate h to external forces. The parameters are (i) the Kazanski-Monin internal-stability parameter, µ = u* / | f | L , and two new parameters: (ii) the baroclinicity parameter, µ Γ = Γ / N , and (iii) the imposed stability parameter, µN = N / | f | . Finally, this study has significantly improved our understanding of SBLs. The author's LES reproduced nocturnal and long-lived SBLs with the timeconstant surface heat flux, w' Θ' , as large as 0.17 K m s-1 , whereas previous LES were able to work with w' Θ' <0.1 K m s-1. The author's LES also reproduced initial decay and restore of turbulence in the SBL. They also reproduced successfully the inertial oscillations. The latter are very sensitive to accurate simulation of the turbulent energy cascade due to very long period of the oscillations. The inertial oscillation is a concern of many others LES codes as it has been recently disclosed at the 15th Symposium on Turbulent Boundary Layers.. 23.

(39) Acknowledgements. I would like to thank my supervisor Prof. Sergej Zilitinkevich whose support I have been always feeling. Hours of our discussion resulted in my better understanding of fundamental physics as well as in my better vision of practical demands from atmospheric sciences. He contributed not only directly to my research but also indirectly organizing my short-time scholarships at the best fluid dynamics centers in Europe. I would like also to thank Academician, Prof. V. Dymnikov, Prof. A. Filatov and Sir Prof. J.C.N. Hunt who took care on me and spent their time answering my questions during my visits. I express my gratitude to Prof. R. Harwood and Dr. Ch. Merchant who were my hosts at Edinburgh University. I also express my gratitude to Prof. G. Djolov who was my host during my visit to the University of Venda, South Africa. I am grateful to my teachers: Profs. M. Tjernström, Ch. Kiselman, A.-S. Smedman, Drs. B. Grisogono, L. Enger and C. Nappo who passed to me sand-grains of knowledge. My especial gratitude is to my colleagues and friends: Academician, Prof. V. Lykossov, Drs. M. Tolstykh, V. Alexeev, E. Kazantzev, B.-J. Boersma, M. Pourque, J. Rimshans and P. Samuelsson who helped me with development of the LES code. I would like to mention guests of our department: Profs. T. Lyons and L. Marht, Drs. V. Grjanik, N. Kleeorin and I. Rogachevsky with whom we have initiated a fruitful collaboration. I appreciate the financial support from the following projects and funds: the SIDA Project "Non-local turbulent transport in weather prediction and air pollution modeling" – SRP-2000-036, the ARO Project "Improved Parameterization of Stably Stratified Boundary Layer Turbulence in Atmospheric Models" – DAAD 19-01-1-0816, the European Community – Access to Research Infrastructure action of the Improving Human Potential Programme (contract No~HPRI-CT-1999-00026). Some LES runs were done at the Institute for Numerical Mathematics, Russia. Personally, many thanks to my wife, Oxana, and our friend Marina Rung who corrected my English. I am grateful to my colleagues at MIUU and LUVA, especially to Maj-Britt Johannesson, Dr. Hans Bergström and Prof. Sven Israelsson. Finally, I would like to thank all my friends and relatives in Russia, Germany and worldwide, with whom, thank to the Internet, I always keep in touch. 24.

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(44) P lB aL te.3:T hesam easinP late1butinP late4:T hesam easinP late2butinN B L N. Plate 1: Three-dimensional structure of the first characteristic eddy in CBL without the Coriolis effect. The plot contains streamlines of the flow in; (xz) – black, and (yz) – blue, plains and isosurface of normalized streamwise component of the coherent vorticity, 2 / 3 max ω z. Plate 2:The same as in Plate 1 but with the Coriolis effect at latitude 45 degrees North.. 29.

(45) Plate 5: Convective cells simulated by the author’s LES. The color surface shows anomalies (blue – colder, red – warmer) of the surface temperature. Red isosurfaces show upward motions with vertical speed >0.5 m/s. In this LES run. Plate 6: Convective rolls simulated by the author’s LES. The color surface shows anomalies (blue – colder, red – warmer) of the surface temperature. Red isosurfaces show upward motions with vertical speed >1.5 m/s. In this LES run. w' Θ' |z =0 =0.1 K m/s, ug=0 m/s.. w' Θ' |z =0 =0.4 K m/s, ug=10 m/s.. Plate 7: Three-dimensional snapshot of the CBL in the LES run over heterogeneous surface. Red isosurfaces show upward motions with the vertical speed >1 m/s, blue isosurfaces show downward motions with vertical speed <-1 m/s. The solid line shows the location of the sharp vegetation boundary in the LES domain. The arrow shows the direction of the geostrophic wind.. Plate 8: Three-dimensional snapshot of wave-like structures in the LES run in the SBL. Red isosurfaces show upward motions with the vertical speed >0.1 m/s, blue isosurfaces show downward motions with vertical speed <-0.1 m/s. The solid line shows the location of the sharp vegetation boundary in the LES domain. The arrow shows the direction of the geostrophic wind.. 30.

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