Analysis of the planetary boundary layer with a database of large-eddy simulation experiments

Analysis of the planetary boundary layer with a database of large-eddy simulation experiments I. Esau a,b a G.C. Rieber Climate Institute at the Nansen Environmental and Remote Sensing Center, Thormohlensgt.
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Analysis of the planetary boundary layer with a database of large-eddy simulation experiments I. Esau a,b a G.C. Rieber Climate Institute at the Nansen Environmental and Remote Sensing Center, Thormohlensgt. 47, 5006, Bergen, Norway b Bjerknes Center for Climate Research, Bergen, Norway Corresponding author: Tel , address: (Igor Esau) Keywords: Planetary boundary layer, arge eddy simulations, Atmospheric turbulence, Parameterizations. Abstract Observational studies of a planetary boundary layer (PB) are difficult. Ground-born measurements usually characterize only a small portion of the PB immediately above the surface. Air-born measurements cannot be obtained close to the surface and therefore cannot capture any significant portion of the PB interior. Moreover, observations are limited in choice of instrumentation, time, duration, location of measurements and occasional weather conditions. Although turbulence-resolving simulations with a large-eddy simulation (ES) code do not supplant observational studies, they provide valuable complementary information on different aspect of the PB dynamics, which otherwise difficult to acquire. These circumstances motivated development of a medium-resolution database (DATABASE64) of turbulence-resolving simulations, which is available on DATABASE64 covers a range of physical parameters typical for the barotropic SB over a homogeneous rough surface. ES runs in DATABASE64 simulate 16 hours evolution of the PB turbulence. They are utilized to study both transition and equilibrium SB cases as well as to calibrate turbulence parameterizations of meteorological models. The data can be also used to falsify theoretical constructions with regards to the PB. 1 Introduction The turbulent planetary boundary layer (PB) is often stably stratified, e.g. at high latitudes and in night time at all latitudes. A stably stratified PB (SB) is characterized by a lower 1 surface temperature relative to temperature of air above it. The principal SB feature is the downward, toward the surface, direction of the mean vertical sensible heat flux. There are at least three physical mechanisms leading to the SB development, namely, the warm air advection, the radiative heat loss due to larger thermal emissivity of the surface, and the elevated latent heat release in clouds. Those mechanisms often interplay to create quite complicated structures of the SB. The exact re-stratification mechanisms are however out of focus in this work. We consider the PB re-stratification as an external factor. Hence we focus on the SB structure consistent with given external parameters within a typically observed range. Numerical large-eddy simulation (ES) technique is a method to generate the SB structure consistent with specified and controlled external parameters. The external parameters are: the time-independent downward temperature flux, F = w' θ ' = H s ρc at the surface ( z = 0 ), θ / p 1 and the initially imposed temperature stratification, ( ) 1/ N = gθ 0 θ / z. The former, F θ 0 in SB, is a prescribed constant, which is more typical for the surface with large heat capacity, e.g. water surfaces. The later, N or the Brunt-Vaisala frequency, is modified within the SB by the turbulent mixing. There is an interest to follow its evolution. Here, ρ is the air density, cp is the air specific heat at the constant pressure, H s is the sensible heat flux, z is the height above the impenetrable surface, g is the gravity acceleration, θ 0 is the reference temperature characterizing the air thermal expansion, θ is the potential temperature. Preparing the SB simulations, it is perhaps the most difficult to compose a range of its characteristic external parameters. The SB is scarcely known. Esau and Sorokina (009) composed the SB climatology for the Arctic on the basis of literature review and data processing. This climatology is used to select relevant ES runs for the database. Additional runs were motivated by the needs to improve SB parameterisations (Mahrt, 1998; Tjernstroem et al., 005; Cuxart et al., 006; Beare et al., 006; Holtslag et al., 007; Steeneveld et al., 008; Byrkjedal et al., 008). Mauritzen et al. (007) have already used this database for such a crucial improvement. To understand the added value of the ES database, one should consider the best traditional data obtained during short field campaigns. The SB observations are difficult. On the one side, ground-born measurements usually characterize only a small portion of the SB immediately above the surface and therefore strongly affected by local surface features. The typical height of a meteorological mast is about 10 m to 30 m, whereas even the shallowest SB are of 50 m to 100 m deep. Moreover, the SB is often capped with temperature inversion. The inversion in potential temperature is observed in more than 95% of the cases in Northern high latitudes (Serreze et al., 199; iu et al., 006). On the other side, the air-born measurements usually cannot be done sufficiently close to the surface to capture any significant portion of the SB. Flights below 50 m level are problematic, although development of unmanned aircrafts can facilitate the low-level data sampling (Reuder et al., 008). Recent advances in remote sensing techniques (Bösenberg and inné, 00; Emeis et al., 004; Cooper et al., 006; Frehlich et al., 008), dropsonde and tethered balloon techniques can provide high quality data (Holden et al., 000) but require considerable improvement of resolution, especially near the surface. In these circumstances, ES can provide a valuable supplement data for complementary and independent studies of the SB structure and dynamics (Stevens and enshow, 001). In some ways, ES data are indispensible as they can provide information, which is difficult if possible to acquire by other approaches. In particular, ES is useful due to easy accessibility, controllability, accuracy and repeatability of its data. The ES can be rerun at any request to check for impact of different physical and numerical factors as well as to obtain any kind of dynamical quantity of interest. The major ES drawbacks are related to unavoidable idealizations in model experiments. Therefore, it is important not to overstate the limits of the ES data validity. As it is recognized now (Mason, 1994; Muschinski, 1995; Esau, 004), the ES data characterise dynamics of a ES-fluid flow, which in many aspects resembles the atmospheric flow, but may differ from it in aspects ultimately sensitive to the properties of the smallest and the largest resolved-scale motions in the model. Hence, the ES data must be always falsified against observations where the observations are available. This paper describes a database (DATABASE64) of turbulence-resolving simulations. DATABASE64 covers the physical parameters characterizing the barotropic SB over a homogeneous aerodynamically rough surface. Each ES run in DATABASE64 simulates 16 hours evolution of the SB. The ES produces three dimensional fields of fluctuations of the meteorological variables, namely, the three components of velocity and the potential temperature. These fluctuations are further processed to obtain turbulence statistics as well as horizontally and time averaged profiles of the required quantities. DATABASE64 can be used for both transition and equilibrium SB studies as well as for calibration of the turbulence 3 parameterizations. DATABASE64 is available on and it has been already used in several research studies. Esau and Zilitinkevich (006) utilized DATABASE64 to characterize the SB parameter space in order to develop analytical resistance laws. Zilitinkevich et al. (007; 008) used DATABASE64 to support a total turbulent energy theory. Mauritzen et al. (007) and Canuto et al. (008) utilized DATABASE64 to calibrate new parameterizations. Several other research groups have also expressed their interest in DATABASE64 (e.g. Frehlich et al., 008; Basu et al., 008; Jeričević et al., 009; Perez et al., 009; Grisogono and Rajak, 009). This study aims to demonstrate several new ways to work with DATABASE64. The examples make contribution to selected problems of the boundary layer theory, parameterisations and understanding. The paper is organized in the following way. The next Section describes the numerics of the large-eddy simulation code ESNIC (Esau, 004). The attention is paid to treatment of fluid static stability. This Section is given to facilitate the understanding of the ES advantages and limitations. The reader may omit it if those details are of no interest for him. Section 3 describes the numerical experiments and results of sensitivity tests. Section 4 gives several examples of DATABASE64 applications. Section 5 summarizes the presentation. The large-eddy simulation code ESNIC The present database has been obtained with the large-eddy simulation code ESNIC developed by the author at Uppsala University and Nansen Environmental and Remote Sensing Center (Esau, 004). The code numerically solves Navier-Stokes equations of motions for incompressible Boussinesq fluid and the transport equations for the potential temperature and passive scalars. The equations for passive scalars are identical to that for the potential temperature and will not be described here. et us define the Cartesian coordinate system ( x, y, z) with the axis directed to East, North and Zenith correspondingly. The components of velocity in this coordinate system are u i = ( u, v, w). The potential temperature is defined as θ = T ( p p / ) where T is the absolute temperature and p is the pressure with p ( z = 0) = p0. The layer of neutral static stability would have θ / z = 0. Equations of motions, continuity and the scalar transport are u / t = / x ( u u + τ + pδ ) + F, (1) i j i j u 4 u / x = 0, () i i θ t ) + F. (3) / = / x j ( θu j + τ θ j b Here we use the Einstein rule of summation and the Kroneker delta, which is δ = 0 if i j and δ = 1otherwise. In Eq. (1)-(3), the Boussinesq approximation has been applied (Zeytounian, 003). The pressure p must be understood as the deviation from the hydrostatic pressure. The continuity equation reduces to the non-divergence equation by this approximation. The external forces specified in this set of simulations are the three dimensional Coriolis and reduced buoyancy forces. The forces are F coriolis F = F + F where u coriolis buoyancy r r = ( Ω u) and 1 Fbuoyancy = gθ 0 θδ i3. We take and Fθ = 0 in DATABASE64. The terms, τ and τ, are of special interest. They are responsible for the energy θj dissipation and the temperature fluctuations diffusion in the ES code. Therefore, modeling of τ and τ θj is critical for the ES performance and accuracy. In ESNIC, τ and τ θj are not dissipative terms as it is adopted in turbulence parameterizations in the large-scale meteorological models. The terms τ and resolved transport terms, u iu j and θ u j τ θj have the same mathematical properties as the, but describe the spectral energy (temperature fluctuation) transport across the smallest resolved scale (the mesh scale) in the code. They are traceless tensors with 6 independent components for It should to be clearly understood that τ and τ and 3 components for τ θj. τ θj cannot be constructed by analogy with the molecular dissipation/diffusion as it is done in the meteorological parameterizations. To construct τ and τ, ESNIC utilizes an analytical solution of a simplified variational θj optimization problem for the spectral energy transport in the inertial interval of scales. The detailed derivation, tests and proper references can be found in Esau (004). The essence of this variational problem (Pope, 004) is to find such values of the term τ that balance as accurately as possible the amount of energy cascading through the mesh scale, Δ, with the amount of energy cascading through some larger resolved scale, Δ Δ. The latter value can be explicitly computed, as the term (see Eq. (6) below), up to the accuracy of the 5 numerical scheme. In a nearly laminar low Reynolds number (Re) flow, where the direct interaction hypothesis 1 is justified, invoking of into the formulation for τ provides almost exact spectral turbulence closure. Unfortunately, in the high Re flow such as the atmospheric SB, a large fraction of energy is cascading indirectly through interactions between motions with significantly different scales. In this case, is not sufficient to describe the magnitude of the spectral energy transport. saturates at about 50% of the total turbulent stress magnitude on a fine resolution mesh (Sullivan et al., 003). More sophisticated constructions for τ are required (Vreman et al., 1997). Depending on the choice of optimization methods and amount of information extracted from the resolved flow fluctuations different turbulence models were proposed. ESNIC employs relatively unsophisticated and therefore computationally inexpensive model. It is a reduced version of a dynamic-mixed model that reads τ = l S S, (4) s l s = 1 ( H ) M M M, (5) H ( u u ) ( u ) ( u ) =, (6) = i j i l ( ( ui ) ( u j ) ) ) ( ui ) j l l ( ) ) ( u j ) ) l l l l ( ) [( u ) ) ( ) ) ( ) ( ) ) ] i u j ui u j, (7) M ( S S ) α ( S ) ( S ) =, (8) S ( u / x + u x ) 1 = i j j / i. (9) 1 The most energy exchange is due to the interaction between eddies with the closest wave lengths. Following Germano (1986), Carati et al. (001) showed that the exact expression of τ through the resolvedscale fluctuations of the velocity could be obtained with the infinite Taylor series expansion. The more terms of the expansion are retained the more accurate and complete τ is obtained. The computational cost of this approach increases rapidly, rendering it of being mostly of the academic interest. 6 Here A A is the scalar product, A = ( A A ) 1/, the superscripts l and denote filtering i j i i i with the mesh length scale and the twice mesh length scale filters. The filters squared aspect ratio is α =. 9 for the Gaussian and the top-hat filters discretized with the nd order of accuracy central-difference schemes. The reader should observe that the formulation in Eq. (5) for the mixing length scale is given in quadratic form. It implies imagery values for the mixing length in certain flow conditions that physically means the turbulence energy backscatter from the small to large scales. The turbulence model for the temperature fluctuations is j s ( θ / x ) 1 τ θ = Pr l S, where (10) Pr = Pr 0 + ari, Pr = lim Pr 0 j Ri 0 ( u ) 1 0 θ / z i / = 0.8, a = 5, and (11) Ri = gθ z. (1) The numerical discritization in the ESNIC code is the nd order fully conservative finitedifference skew-symmetric scheme on the uniform staggered C-type mesh with the explicit Runge-Kutta 4 th order time scheme. The schemes and the relevant references are given in Esau (004). The mesh scales are Δ, Δ and x y Δ z with the 1 st computational level for u,v, θ is placed at z1 = Δ z / and for w at Δ z. The lateral boundary conditions are periodic. It allows implementation of exact (up to the computer accuracy) and fast direct algorithms to solve the continuity Eq () with a pressure correction. But it also limits the size of resolved perturbations and spans of trajectories to ½ of the size of the computational domain and may cause an artificial energy accumulation at the longest resolved scales. 3 The experiment set up and quality assessment DATABASE64 is a collection of ESNIC runs each computed independently from slightly perturbed laminar flow under a fixed set of external control parameters. The surface boundary conditions for temperature are prescribed in each run through a constant temperature flux τ ( t, z = 0) const (13) θj = The constant flux conditions have some drawbacks. Derbyshire (1999) and Basu et al. (008) argued that the prescribed heat flux boundary conditions should be avoided in the SB as there is a possibility to exceed the maximum physical heat flux consistent with the intensity of the flow turbulence. In this case, there is a possibility for runaway surface cooling in the ES. 7 Indeed, the simulations run into troubles under some parameter combinations crossing probably into unphysical areas of the parameter space (see examples in the next Section). Moreover, the bifurcations in the SB turbulence dynamics cannot be simulated 3. The surface boundary conditions for momentum are given through the log-layer formulation for the friction velocity u = κ ui ( z1) / ln( z1 / z0 ), (14) τ z = 0) = δ u u ( z ) / u ( ). (15) ( i3 i 1 i z1 The log-linear Monin-Obukhov stability functions are not used in ESNIC. Nevertheless, the consistent log-linear behaviour is recovered in the simulated fields. The ES runs were integrated for 16 hours. The three dimensional data were sampled every 600 s and processed during the simulations. DATABASE64 contains the horizontally and time averaged data and turbulent statistics. The averaging was done over each half an hour of simulations that is over 3 subsequent samples. At the end of each run, the instant three dimensional fields of u, v, w, θ were stored in the database. The turbulence statistics were obtained in the post-processing as follows. Consider for example the mean resolved vertical temperature flux profile w 'θ '. et n 1 φ x = φ( xi, y, z, t) be an averaging operator in the direction x where n = x / Δ x is the n i= 1 total number of samples (grid nodes) along x. Then the flux is defined as w (16) ' θ '( z, t ) = w( x, y, z, t) ( θ ( x, y, z, t) θ ( x, y, z, t) x y ) x y t [ ti + δt ] where δt is the 3 subsequent instant data samples. 3 Possibilities for sharp transitions in the intensity of the vertical turbulent mixing in the SB were theoretically discovered by McNider et al. (1995). The transitions were linked to the bifurcations and hysteresis found in the SB parameter space where two different states with low and high levels of the turbulent mixing are possible for the same range of control parameters. Derbyshire (1999) demonstrated the bifurcations in the SB parameterizations. This kind of the behaviour has been supposedly reported by yons and Steedman (1981) for the Australian nocturnal SB. It remains to be seen whether the ES also reproduce the SB bifurcation dynamics. 8 The mesh size of 64 by 64 by 64 grid points is only marginally sufficient for the SB studies (Esau and Zilitinkevich, 006). The larger mesh size however would seriously increase the demand for the computational resources. For instance, the use of 18 by 18 by 18 mesh will increase the computer time per run by more than an order of magnitude. To assess the database quality several additional runs were conducted. The sensitivity of the ES runs to the mesh resolution has been shown in Esau and Zilitinkevich (006). We will not repeat it here. More results on the ES sensitivity to the mesh resolution could be found in Beare and McVean (001). A lack of resolution severely damages the inertial interval (the large wave numbers) in the turbulence energy spectrum. The model partially absorbs the damage reducing the effective Smagorinsky constant C s = l s / Δ and hence the sub-grid scale flux τ. The averaged C in the SB core (from z = h / 3 to z = h / 3 ) drops from its nearly theoretical value (eslie s and Quarini, 1979) of to C s = 0. in the well resolved runs with 3 48 levels within the SB C s = 0.1 for the runs with 1 4 levels within the SB. The similar reduction of the gridscale dissipation, and C s, has been documented in the Horizontal Array Turbulence Study (HATS) atmospheric experiment (Kleissl et al., 006). The dynamic adjustment. It should not draw the average Cs reduction has limits of Cs below ~0.1. Hence, more stable runs must be produced in smaller size domains. Here, the domain height varies from z ~ 4000 m in the EB to z ~ 80 m in the most stable SB. Without a care
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