The nature of small-scale non-turbulent variability in a mesoscale model

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1 ATMOSPHERIC SCIENCE LETTERS Atmos. Sci. Let. 13: (2012) Published online 11 May 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: /asl.382 The nature of small-scale non-turbulent variability in a mesoscale model Ivan Güttler 1 * and Danijel Belušić 2 1 Meteorological and Hydrological Service of Croatia, Zagreb, Croatia 2 Monash Weather and Climate, School of Mathematical Sciences, Monash University, Clayton, Victoria, Australia *Correspondence to: I. Güttler, Meteorological and Hydrological Service of Croatia, Grič 3, Zagreb, Croatia. ivan.guettler@cirus.dhz.hr Received: 24 October 2011 Revised: 26 March 2012 Accepted: 27 March 2012 Abstract Previous studies have shown that numerical diffusion plays a crucial role in the ability of mesoscale models to reproduce features similar to sub-meso motions found in observations, particularly in terms of spectral energy distribution. In this study, the impacts of surface heterogeneity and frequency of lateral boundary coupling are investigated as potential factors governing or modulating the occurrence of sub-meso motions. The spectral energy distribution after the reduction of model diffusion is analysed in detail. It is suggested that a combination of physical and numerical mechanisms is responsible for the final spectral shape. Copyright 2012 Royal Meteorological Society Keywords: sub-meso motions; WRF-ARW; surface heterogeneity; lateral boundary conditions; numerical diffusion 1. Introduction Belušić and Güttler (2010, hereafter BG10) have shown that the mesoscale model WRF-ARW can reproduce the magnitude of variability of small-scale non-turbulent motions with periods between 1 min and 1 h, termed sub-meso motions, for a stable weakwind CASES99 night. This was achieved by removing or reducing the horizontal numerical diffusion in the model. Seaman et al. (2012) reduced the imposed minimal TKE and length scale in WRF-ARW and obtained stronger variability of the plume behaviour in the presence of larger sub-meso motions (timescales larger than 12 min), which agrees well with the observations (Vickers et al., 2008, Hiscox et al., 2010). Therefore, the reduction of model diffusion, whether vertical or horizontal, seems to allow for greater variability that is in apparent accordance with the measurements. However, while the spectral energy, that is the variance, and the dispersion of the passive scalar plume in BG10 agreed well with the real world, the question remained whether the effect was due to physical or purely numerical reasons or a combination thereof. In the first case, the decrease of numerical diffusion might enable some previously filtered physical modes to appear. In the second case, the decrease of diffusion would foster the growth of numerical instabilities; however, these instabilities would have to distribute their energy in the frequency spectra in such a way to align well with the measured spectra. The third case, that is the combination of both physical and numerical mechanisms, would seem as the most probable: first of all, with removed or significantly reduced numerical diffusion, the appearance of numerical artefacts is eminent; hence, the physical modes surely cannot emerge uncluttered. Second, if the resolved variability is of purely numerical origin, it is not clear why it would follow the exact spectral shape of the measurements. The purpose of this study is to examine the nature of the variability introduced by the decrease of numerical diffusion. The following questions are in focus: 1. To what extent are the mechanisms that are responsible for this variability physical? 2. In what way is the energy redistributed between different time and space scales to yield the agreement with the spectra from measurements? The first question is studied by observing the response of the model to removing some of the forcings that could generate or modulate the physical modes. The second question is tackled by reducing the numerical diffusion at a certain moment of a simulation, rather than at the beginning, and by following the consequent development of the spectra. Section on Model and methods describes the details of these methodologies. 2. Model and methods Version 3.3 of the Weather Research and Forecasting (WRF) Advanced Research WRF (ARW) model (Skamarock et al., 2008) is used in this study to simulate the CASES99 period from 1800 UTC 18 October 1999 to 1200 UTC 19 October Four nested model domains with grid spacing of 9, 3, 1 and 1/3 km are centred at N, W. The grids consist of cells except for the 1/3 km domain that has cells. There are 81 vertical levels, with 9 levels in the first 500 m and with vertical layer depths increasing from 10 m near the Copyright 2012 Royal Meteorological Society

2 170 I Güttler and D Belušić surface to 350 m at 20 km. The initial conditions for all domains and 6 hourly boundary conditions for the 9 km domain are obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) operational analyses. The nesting is applied depending on the experiment, either in one-way mode between successive domains or in one-way mode for the 3- and 1 km domains, followed by offline nesting of the 1/3 km domain using 1 h outputs from the 1 km domain. This setup is similar to BG10. The default (here labelled full ) horizontal diffusion in WRF-ARW consists of two separate options: implicit and explicit (for details, see BG10). The control simulation here uses only the constant explicit horizontal diffusion with the horizontal diffusivity K H = 2m 2 s 1 for all domains. Several experiments were performed to examine the sensitivity of the results to the value of K H, and it was found that for this particular value the agreement between the model and measurements spectra is the best (BG10). The choice of various parameterizations and model options is as in BG10 (e.g. their simulation givendiff). The control simulation agrees well with the CASES99 (Poulos et al., 2002) wind measurements in terms of frequency spectra and is used as the reference for all other experiments. The experiments are summarized in Table I and explained in the following sub-sections. All the results are shown only for the 1/3 km domain 1 min outputs from the first model half-level that is at 5 m above the surface. The frequency and wavenumber spectra are calculated as in BG10. Specific details of the wavenumber spectra computations can be found in Skamarock (2004) Origins of the variability Two test simulations are performed in order to examine the physical nature of the modelled variability. If the physical sub-meso motions are in any way related to the surface properties, the homogenization of the surface should reduce the variability. Therefore, the first test simulation has flat terrain with a single landuse category, grassland, over the entire domain, and is labelled the homogene simulation. The second test simulation tends to examine the role of boundary conditions. In the usual online 1-way nesting, a parent domain prescribes at each time step the boundary conditions to its child domain. The time step of the 1 km domain is 4 s, which means that the boundary conditions could affect the variability at timescales that are in focus here, that is larger than 1 min, in the 1/3 km domain. This effect is avoided if the 1/3 km domain obtains the initial and boundary conditions from the 1 km domain s 1 h outputs. The latter is accomplished by using the ndown program that is a part of the WRF-ARW modelling system designed to enable 1- way offline nesting. This simulation is labelled the ndown simulation. Not all possible physical mechanisms are taken into account by these two test simulations. These are, however, the most obvious ones, since they should remove the physical effects of lower and lateral boundaries on the boundary-layer variability Spectral shape In order to examine the spectral energy redistribution, the simulation is first run with full diffusion until 04 UTC 19 October (22 h local time 18 October) and then restarted with the constant explicit horizontal diffusion as in the control simulation (labelled the restart1 simulation). In this way, the changes in the spectra are influenced only by the decrease of diffusion and not by the domain initialization. The restart1 wavenumber spectra are compared to the wavenumber spectra from the restart2 simulation, which has full diffusion both before and after the restart. Specifically, we examine the time evolution of power spectral densities for several wavenumbers. This approach enables us to follow the development of variability in the spectra and to determine how it attains its final shape. 3. Results The control simulation is first compared to the measurements and the full diffusion simulation as given in WRF-ARW by default. The results are equivalent to BG10 by showing the increase of variability at all temporal scales, with the shape of power spectra resembling the measurements (cf. Figure 1) Origins of the variability Figure 1 depicts the wind components time variability for the homogene and ndown simulations compared Table I. Simulations summary. Simulation Horizontal diffusion on 9, 3 and 1 km domains Horizontal diffusion on 1/3 km domain Nesting Surface full Full Full 1-way all Realistic control Given Given 1-way all Realistic homogene Given Given 1-way all Uniform ndown Given Given ndown to 1/3 km Realistic restart1 Full 18 UTC 04 UTC full ndown to 1/3 km Realistic 04 UTC 12 UTC given restart2 Full 18 UTC 04 UTC full ndown to 1/3 km Realistic 04 UTC 12 UTC full Horizontal diffusion full includes both explicit and implicit diffusion, while given includes no implicit diffusion and has constant K H = 2m 2 s 1.

3 The nature of small-scale non-turbulent variability u (m s 1 ) t (UTC) PSD (m 2 s 2 Hz 1 ) 10 2 CONTROL HOMOGENE NDOWN FULL CASES99 f (Hz) v (m s 1 ) PSD (m 2 s 2 Hz 1 ) t (UTC) f (Hz) Figure 1. Zonal (up) and meridional (down) wind component 1 min time series (left) and power spectra (right) comparisons between the control, homogene, ndown and full simulations at the first model half-level (5 m) in the central grid point, and the CASES99 measurements at 5 m. Vertical error bar denotes the 95% confidence interval. with the control simulation and the measurements. Rather surprisingly, the main result is that there is no significant change in the time variability of wind components, neither with the homogeneous surface nor with different boundary conditions. Additional insight can be gained from the spatial structure of the flow, as revealed by the wavenumber spectra. Figure 2 shows the comparison between the wavenumber spectra of the homogene, ndown, control and full simulations. Contrary to the time domain, the differences between different experiments are pronounced in the spatial domain. The homogenization of the surface in the homogene simulation leads to smaller energy at the large spatial scales when compared to the control simulation, which is primarily due to the removal of orography and consequently the flow features associated with the orography. The ndown simulation has the largest energy at all scales when compared to the other simulations. This can be attributed to the fact that the system is less perturbed at the boundaries in the ndown experiment; hence, the entire system is more stationary, which enables the spatial stationary modes to build up more efficiently. It is, therefore, obvious that the changes in physical forcings do induce significant changes in the flow field; however, these changes are not reflected to the time domain. The latter is mostly due to the relative stationarity of spatial modes. Since the obvious physical mechanisms are not responsible for the temporal variability, the cause of temporal variability might predominantly lie in the numerical domain. The legitimate question in that case is, what is the reason for the spectral agreement with the measurements? The inclusion of full diffusivity clearly separates the full simulation from the control simulation in the high wavenumber part of the spectra (Figure 2) and is consistent with the analogous reduction found in the frequency spectra (Figure 1). Therefore, there seems to be a relationship between the spatial and temporal domains when the variability is numerically induced. This observation can be used for the investigation of the evolution of spectra The spectral shape Figure 3 depicts the time evolution of the spatial power spectra for several wavenumbers for the restart1 and restart2 simulations. After reducing the horizontal diffusion in the restart1 simulation at 04 UTC, the results are seemingly twofold. On the one hand, there is a sudden increase of energy at wavenumbers

4 172 I Güttler and D Belušić PSD ((m 2 s 2 )*km) PSD ((m 2 s 2 )*km) 10 1 CONTROL HOMOGENE NDOWN FULL wavenumber (km 1 ) 10 1 wavenumber (km 1 ) Figure 2. Wavenumber power spectra for the zonal (top) and meridional (bottom) wind component at the first model half-level (5 m) in the control, homogene, ndown and full experiments. The spectra are averaged over the period 00 UTC 12 UTC 19 October between 0.79 and 1.5 km 1, which starts at the exact time of the step decrease of diffusion. This apparent lack of a time lag of the energy response to the reduction of diffusion when passing to lower wavenumbers does not seem to imply that energy is transferred across scales. It rather suggests the build-up of energy at each individual scale, which certainly is the case due to the abrupt decrease of the horizontal diffusivity and the finite spectral width of the diffusion filter. On the other hand, the fastest increase of the spectral energy occurs for the two highest wavenumbers shown (1.19 and 1.5 km 1 ). The energy at the next lower wavenumber of 0.79 km 1 increases more slowly, while for the wavenumber of 0.5 km 1,there is even a time lag present, particularly evident for the meridional component. This can be explained as follows. As the increasing scales approach the cut-off scale of the diffusion filter (Langhans et al., 2012), they become less influenced directly by the diffusion. The wavenumber of 0.5 km 1 corresponds to 6 x, which is approximately at the lower limit of scales that are not significantly directly filtered by the diffusion (Skamarock, 2004; Langhans et al., 2012). Therefore, the bulk of the energy increase at this scale is not a direct consequence of the decrease of diffusion at this scale, but rather occurs because of the transfer of energy from smaller scales. The noticed time lag corroborates such conclusion. It follows that for scales smaller than 0.5 km 1 but larger than the Nyquist wavenumber, the upscale energy transfer is superimposed on the direct energy increase at individual scales. The heuristic conclusion is that the energy is redistributed from the smallest spatial scales by the model 10 1 restart2 vs restart1: W-E wind component 10 1 restart2 vs restart1: S-N wind component PSD (m 2 s 2 km) of u 2D field 10 1 PSD (m 2 s 2 km) of v 2D field time (UTC) time (UTC) 0.12 km km km km km km 1 Figure 3. Time evolution of the spatial spectra for several wavenumbers (from 0.12 to 1.50 km 1 ) in the restart1 (thick lines) and restart2 (dashed lines) experiments for the zonal (left) and meridional (right) wind component.

5 The nature of small-scale non-turbulent variability 173 dynamics, therefore resulting in the physically consistent spectral shape in the frequency domain. The magnitude of the energy input at the smallest scales is in our case governed by the specified K H. The prospect that the artificial small-scale perturbations are consistently redistributed by the model to attain the realistic spectra could be used to introduce particular stochastic forcing to the model at limited time and space scales and still yield realistic energy content in the simulations. This might be used to introduce in a controlled way the variability that is currently missing in the models. 4. Summary and conclusions The small-scale non-turbulent motions in the mesoscale model WRF-ARW are shown not to be substantially governed by the surface heterogeneity and lateral boundary forcing, regardless of the changes in the wavenumber spectra of velocity components that result from different surface characteristics and nesting procedures. In the case of homogeneous surface, spatial power spectra are reduced for the largest spatial scales as expected, while for the case with less frequent lateral coupling, energy build-up is detected at all scales. However, this does not correspond to a significant change in temporal variability. On the other hand, activity on small spatial and temporal scales depends significantly on the strength of the horizontal diffusion. A decreased diffusion enables the model to closely reproduce the strength of variability in measurements, as evidenced through the agreement of the power spectra. A test simulation with a step reduction of the numerical diffusion long after the model spin-up enabled the inspection of the time evolution of the wavenumber spectra. The goal of this exercise was to determine the reasons for the agreement of spectral shapes, having known that the most likely origin of variability in the model is numerical instability. The highest wavenumbers exhibited fastest increase of variability after the diffusion reduction, which is consistent with the possible upscale energy transfer to intermediate spatial scales. The newly introduced variability became a stationary feature within about 30 min after the diffusion decrease. In general, changes to any part of the spectrum can be due to direct energy input at those scales or from upscale or downscale energy transfer from the neighbouring scales. Our results make a combination of physical and numerical mechanisms the most probable explanation for the final shape of the spectra, where physical modes mostly govern large scales, numerical modes small scales, and the model upscale energy transfer feeds the intermediate scales. However, details of the processes involved need more understanding. Since it is not yet fully clear why experiments containing numerical instabilities show the level of realism that is comparable to measurements, a setup using higher resolution, for example LES, or more idealistic and controllable experiments should be used in order to make progress on the remaining open questions. In addition, other details of the lateral boundary forcing still wait to be systematically explored. These include the impact of the spatial variation of the vertical profiles imposed to a nested domain in its buffer zone. Acknowledgements The work was partially supported by the Croatian Ministry of Science, Education and Sports project Nos and The simulations were performed on the computer cluster of the Faculty of Science, University of Zagreb. Two anonymous reviewers are gratefully acknowledged for their constructive comments. References Belušić D, Güttler I Can mesoscale models reproduce meandering motions? Quarterly Journal of the Royal Meteorological Society 136: DOI: /qj.606. Hiscox AL, Miller DR, Nappo CJ Plume meander and dispersion in a stable boundary layer. Journal of Geophysical Research 115: D DOI: /2010JD Langhans W, Schmidli J, Schär C Mesoscale impacts of explicit numerical diffusion in a convection-permitting model. Monthly Weather Review 140: Poulos GS, Blumen W, Fritts DC, Lundquist JK, Sun J, Burns SP, Nappo C, Banta R, Newsom R, Cuxart J, Terradellas E, Balsley B, Jensen M CASES-99: a comprehensive investigation of the stable nocturnal boundary layer. Bulletin of the American Meteorological Society 83: Seaman NL, Gaudet BJ, Stauffer DR, Mahrt L, Richardson SJ, Zielonka JR, Wyngaard JC Numerical prediction of sub-mesoscale flow in the nocturnal stable boundary layer over complex terrain. Monthly Weather Review 140: DOI: /MWR-D Skamarock WC Evaluating mesoscale NWP models using kinetic energy spectra. Monthly Weather Review 132: Skamarock WC, Klemp JB, Dudhia J, Gill DO, Barker DM, Duda MG, Huang X-Y, Wang W, Powers JG A description of the Advanced Research WRF Version 3. NCAR/TN-475+STR. Vickers D, Mahrt L, Belušić D Particle simulations of dispersion using observed meandering and turbulence. Acta Geophysica 56: