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Technical note: Large-eddy simulation of cloudy boundary layer with the Advanced Research WRF model

Authors


Corresponding author: T. Yamaguchi, Earth System Research Laboratory, NOAA, 325 Broadway, Boulder, CO 80305, USA. (tak.yamaguchi@noaa.gov)

Abstract

[1] A thorough evaluation of the large-eddy simulation (LES) mode of the Advanced Research WRF model is performed with use of three cloudy boundary layer cases developed as LES intercomparison cases by the GEWEX Cloud System Study. Our evaluation reveals two problems that must be recognized and carefully addressed before proceeding with production runs. These are (i) sensitivity of results to the prescribed number of acoustic time steps per physical time step; and (ii) the assumption of saturation adjustment in the initial cloudy state. A temporary, but effective method of how to cope with these issues is suggested. With the proper treatment, the simulation results are comparable to the ensemble mean of the other LES models, and sometimes closer to the observational estimate than the ensemble mean. In order to ease the burden for configuration and post-processing, two new packages are developed and implemented. A detailed description of each package is presented. These packages are freely available to the public.

1. Introduction

[2] The large-eddy simulation (LES) mode of the Advanced Research WRF model (ARW) was implemented relatively recently. Since then, the number of studies using ARW as a LES model has gradually increased: for instance, Moeng et al. [2007], Wang et al. [2009], and Solomon et al. [2011] have used ARW with grid spacing of O(10 m) to perform LES of the atmospheric boundary layer. Although the capability of ARW as a LES model has been demonstrated in these previous studies, detailed evaluation of the LES version of ARW has not been well documented; Are the simulated turbulence and cloud fields satisfactory? How do the results with ARW compare with other LES models? Do any special procedures need to be followed?

[3] In this technical note, we study ARW in LES mode from various perspectives using the GEWEX Cloud System Study (GCSS) LES DYCOMS-II RF01 intercomparison case [Stevens et al. 2005, hereinafter DRF01]. We selected DRF01 since it is simple, and yet one of the more difficult cases to simulate amongst the GCSS intercomparison studies [Moeng et al., 1996; Siebesma et al., 2003; Ackerman et al., 2009; vanZanten et al., 2011]. DRF01 is simple because it is a nocturnal, non-precipitating stratocumulus case. However the sharp and large inversion jump make successful simulation difficult. The participating model results generally have a wide spread, e.g., the mean liquid water path (LWP) ranges from 10 to 60 g m−2, and the cloud fraction ranges from 20 to 100%. Our first attempt at simulating DRF01 with ARW resulted in cloud breakup within one hour. Two other GCSS cases, namely DYCOMS-II RF02 [Ackerman et al., 2009] and RICO [vanZanten et al., 2011], are also used to examine a particular issue.

[4] ARW has been developed as a community model; it offers various options for parameterization of microphysics, radiation, surface fluxes, land-surface, planetary boundary layer (PBL), and cumulus convection. In addition, it can perform one- and two-way nesting as well as data assimilation. Although these options are attractive, ARW was not necessarily developed with the LES user in mind. In order to configure ARW for a desired case (often idealized cases), one has to modify the source codes in numerous places, (e.g., specified surface flux, large-scale forcing), which can easily be a source of errors. The output format of the current distribution of ARW is a three-dimensional snapshot only. For LES, this presents a large obstacle since total output size can easily become very large, and post-processing can be a huge burden for analysis. For efficient workflow, we introduce two useful packages so that the ARW is a more user-friendly model in LES mode: one is a LES package for easy model setup, and the other is a statistics package, which outputs horizontal mean profiles at height coordinates during simulation. With these two packages, for instance, ARW can be easily configured to one of the GCSS cases, and compared with the GCSS ensembles. These two packages are available for ARW users at http://esrl.noaa.gov/csd/staff/tak.yamaguchi/code

[5] The outline of this technical note is as follows: ARW is briefly described in 2. A successful simulation of DRF01 with ARW in terms of the observation and GCSS ensemble is then presented in Section 3. Issues we encounter during this study are addressed in Section 4. The statistics and LES packages are described in Section 5. Summary and conclusions are presented in Section 6. For completeness, we briefly present the simulations for DYCOMS-II RF02 and RICO in Appendices A and B.

2. Advanced Research WRF

[6] ARW is a non-hydrostatic model, which solves the compressible system of equations in a flux form governed by a mass vertical coordinate, i.e., hydrostatic-pressure vertical coordinate, defined as

display math

where pdh is the hydrostatic component of pressure of dry air, and

display math

is the dry air mass for a column. Here the subscripts t and s denote the domain top and surface. The basic prognostic variables are the covariant velocities (u, v, w), dry air mass, geopotential, potential temperature (θ), and subgrid-scale (SGS) turbulence kinetic energy (TKE). Depending on the microphysics scheme, mass mixing ratios and number concentrations (per unit mass of air) are also predicted. The equation for the momentum, potential temperature, SGS TKE and other scalars has a form coupled with the dry air mass, for instance,

display math

where F represents the summation of turbulence mixing and any other forcings, and

display math

is the coordinate (or contravariant) vertical velocity.

[7] Temporal discretization is done with a time-split integration scheme described in Klemp et al. [2007]. The scheme separates the high frequency mode, i.e., acoustic and gravity waves, from the low frequency mode (or physical mode). ARW uses a third-order Runge-Kutta scheme, and during each Runge-Kutta step, the horizontally propagating high frequency mode is integrated with a forward-backward scheme with the acoustic time step, which is typically one order of magnitude smaller than the physical time step, while an implicit scheme is used for the vertically propagating high frequency mode. ARW employs the staggered Arakawa C grid for its spatial discretization. Geopotential is located at the vertical velocity point, while pressure and density are located at the scalar point. Advection is computed with a scheme described in Wicker and Skamarock [2002]. As recommended in the ARW user's guide, the fifth-order scheme for the horizontal advection and the third-order scheme for the vertical advection are used. For scalar advection, the monotonic limiter tested by Wang et al. [2009] is used. The diffusion due to turbulence mixing is formulated with eddy diffusivity (i.e., K-theory) for the height coordinate coupled with the dry air mass; this involves a coordinate transformation from the mass coordinate to the height coordinate for the horizontal derivatives. A 1.5-order TKE closure [Klemp and Wilhelmson, 1978; Deardorff, 1980] is used as a SGS turbulence parameterization.

[8] For the microphysics parameterization, a bulk two-moment scheme [Feingold et al., 1998] is used in this study. Particular attributes of the scheme are that it uses a hybrid bin-bulk approach that attempts to retain the accuracy of a bin-scheme without tracking a large number of prognostic scalars. It also predicts cloud supersaturation following the semi-analytical approach of Clark [1973] and activates the ambient aerosol accordingly. The scheme has been successfully applied in the RAMS [Feingold et al., 1998] and WRF models [Wang et al., 2009; Wang and Feingold, 2009a, 2009b; Kazil et al., 2011].

3. DYCOMS-II RF01

[9] ARW is configured following the case specification listed in DRF01 [Stevens et al., 2005]. The horizontal resolution is 96×96 grid points with 35 m grid spacing, and the vertical resolution is 300 levels with 1.5 km domain depth, which gives roughly 5 m grid spacing for the initial mass levels. The physical time step is 0.1 seconds and the number of acoustic time steps per physical time step is 10, i.e., the acoustic time step is 0.01 seconds. A small physical time step has to be used due to the sensitivity related to the acoustic time step (or acoustic Courant number), which is discussed in the next section. The simulation is of 4-hour duration. The horizontal mean statistics are saved every simulated minute, without time averaging, while the full three-dimensional snapshot is saved every ten simulated minutes.

[10] The turbulence is initiated with pseudo-random perturbations, which are applied to potential temperature and SGS TKE in the lowest five layers. The potential temperature perturbation is a random perturbation between ±0.02 K, while the SGS TKE is initialized with 0.04(5−k) m2 s−2, where k = {1,2,3,4,5} is a vertical level index.

[11] The following values are used for the two-moment microphysics scheme: the aerosol number concentration is 100 mg−1 (100 cm−3 at an air density of 1 kg m−3) which produces little drizzle; the mean and geometric standard deviation of the aerosol size distribution are 0.2 microns and 1.5, respectively; the aerosol composition is ammonium sulfate; the geometric standard deviation of both the cloud water and rain water size distributions is 1.2. Even though DRF01 is a non-precipitating stratocumulus case, precipitation is allowed to form after the first hour, i.e., once turbulence has developed.

[12] Snapshots of cloud fields at the last time step are characterized by closed cellular convection with large cloud cover (Figure 1). The distribution of the LWP field is similar to the other simulations of DRF01 [e.g., Stevens et al., 2005; Yamaguchi and Randall, 2012]. A cross-section of cloud water shows that the downdrafts are located below cloud holes, and updrafts are broader and have smaller magnitudes on average than the downdrafts. This is consistent with the aforementioned studies as well as observations, e.g., Gerber et al. [2005].

Figure 1.

The instantaneous cloud-field snapshot at four hours. (top) LWP and (bottom) a cross-section of cloud water mixing ratio at the light green dashed line on the LWP plot. The vertical velocity is superimposed as line contours on the cloud water plot. Downdrafts (updrafts) are colored red (light green), and the black contour is the zero isoline. The contour lines are drawn every 0.5 m s−1.

[13] The simulated turbulence is satisfactory for the 35 m horizontal grid spacing in terms of the power spectrum of the vertical velocity. According to the energy cascade theory, the power is expected to decrease as wavenumber, math formula, with the slope of math formula beyond the inertial subrange for developed turbulence. Bryan et al. [2003] have shown that for a finite-difference model, the power spectrum falls off faster than math formula for wavelengths shorter than 6Δx. Following Moeng et al. [2010], we compute the power spectrum of the vertical velocity with a two-dimensional FFT over a horizontal plane at the 160th mass level (approximately 800 m and in the cloud layer), and then average with respect to the horizontal wavenumber, math formula. The time average over the last hour is taken to get a smooth curve. The hourly mean power spectrum is shown in Figure 2. The inertial subrange exists at approximately 400 m and smaller wavelengths, and the spectral slopes for scales less than 6Δx, are steeper than math formula, as expected.

Figure 2.

Hourly mean power spectrum of the vertical velocity at the ∼ 800 m level for the fourth simulated hour. The power, Ew, is multiplied by wavenumber on the vertical axis. A reference energy cascade line with the slope of math formula is shown with the dashed line. The vertical dotted line is located at the 6Δx wavelength.

[14] Comparison with the GCSS ensemble is presented in Figures 3 and 4. Overall, ARW produces results closer to the observational estimate than the GCSS ensemble mean. For some variables, ARW is a relative minimum or maximum, e.g., liquid water potential temperature and cloud water mixing ratio. ARW maintains larger LWP and a thicker cloud layer than the GCSS mean by maintaining a lower cloud base and slightly lower entrainment rate than the GCSS mean. The moist conservative variables, i.e., liquid water potential temperature and total water mixing ratio, are very well mixed below the inversion, and they are cooler and moister than the GCSS mean, especially in the upper half of the mixed layer; this results in a cloud water mixing ratio close to the observation. The resolved-scale turbulence of ARW is stronger than the GCSS mean, and the second and third moments of the vertical velocity are well matched to the observational estimate, especially in the cloud layer. The results presented here are for the two moment microphysics of Feingold et al. [1998], and may not reflect the results produced by other microphysics schemes available in ARW.

Figure 3.

Comparison with the GCSS ensemble members for DRF01: (a) time series of liquid water path and vertically integrated TKE, (b) hourly-mean vertical profiles of the fourth hour. θl is liquid water potential temperature, r is total water mixing ratio, rl is cloud water mixing ratio. The blue lines are the ARW results, and black lines are the GCSS ensemble mean. The dark shading covers the first and third quartile, and the light shading covers the entire ensemble range. The black-filled circles show the observational values.

Figure 4.

Same as Figure 3b but for the turbulence statistics. w is vertical velocity.

4. Issues

4.1. Acoustic Time Steps

[15] ARW integrates the acoustic and gravity-wave modes explicitly with the acoustic time step during each Runge-Kutta step. The acoustic mode is meteorologically insignificant and gravity waves are less important compared with the other physical mode for stratocumulus clouds, so that the numerical representation of this high frequency mode should have insignificant influence on the simulation. We found, however, that the choice of the number of acoustic time steps per physical time step, NAT, or acoustic Courant number defined below, has significant impact on the simulated cloud and turbulence. For the worst case, rapid cloud dissipation takes place.

[16] The forward-backward time integration scheme used for the acoustic mode is stable if

display math

where CrA is acoustic Courant number, cs≈300 m s−1 is the speed of sound, Δτ = Δt/NAT is the acoustic time step, Δt is the physical time step, and Δx is the horizontal grid spacing. The maximum Courant number on the far right-hand side has already been made smaller by replacing math formula with 0.5. For instance, NAT≥4 gives stable solution for Δt = 0.2 seconds and Δx = 35 m. Based on empiricism it is recommended that NAT≤12 (W. Skamarock, personal communication, 2011). Consideration of the acoustic Courant number in the vertical direction is not required since ARW uses an implicit scheme for the vertically propagating high frequency waves.

[17] Sensitivity tests are performed by changing NAT and Δt for DRF01 as well as two additional GCSS cases: a drizzling stratocumulus case (DYCOMS-II RF02 [Ackerman et al., 2009, hereinafter DRF02]) and a precipitating shallow cumulus case (RICO [vanZanten et al., 2011]). Table 1 lists the acoustic Courant number tested for each case. All cases are numerically stable based on (5). For the sensitivity test with DRF01, the acoustic Courant number is the same for (NATt) = (6,0.1) and (12, 0.2), and (4, 0.1) and (8, 0.2), respectively. The duration of each simulation is 2 hours for DRF01 and DRF02, and 24 hours for RICO. The comparison results of ARW with the GCSS ensemble for DRF02 and RICO are presented in Appendices A and B.

Table 1. Physical Time Step, Number of Acoustic Timesteps per Physical Time Step, and Acoustic Courant Number for the Various Cases
DYCOMS-II RF01DYCOMS-II RF02RICO
Δt (s)</emph>NATCrAΔt (s)</emph>NATCrAΔt (s)</emph>NATCrA
0.260.2860.580.375180.375
0.280.2140.5100.31100.3
0.2100.1710.5120.251120.25
0.2120.1430.260.20.580.1875
0.140.2140.280.150.5100.15
0.160.1430.2100.12   
0.180.107      
0.1100.086      
0.1120.071      

[18] The time series of LWP and vertically integrated TKE for DRF01 and DRF02 are presented in Figure 5. First, the sharp drop in LWP and sharp increase in TKE in the first hour are due to spin-up of the resolved scale turbulence through the initial perturbation and radiative cooling [Moeng et al., 1996]. For both variables in the figure and for both cases, the larger NAT reduces to a larger value for the same physical time step. For the same acoustic Courant number, the results are comparable. Ultimately, the results converge for smaller acoustic Courant numbers. For DRF01, the results converge with NAT = 12 and 10 for Δt = 0.1 seconds, which means that the simulated high frequency mode is insensitive to the simulated cloud and turbulence only for an acoustic Courant number less than 0.086. The convergence acoustic Courant number for DRF02 is 0.15 with (NATt) = (8,0.2), which is larger than for DRF01.

Figure 5.

Time series of LWP and vertically integrated TKE for simulations with various NAT and physical time step: (a) for DYCOMS-II RF01 and (b) for DYCOMS-II RF02.

[19] An additional two simulations with DRF01, restarted with NAT = 4 at 60 and 90 minutes of the run with (NATt) = (10,0.1), are performed to see if this sensitivity is related to the spin-up of turbulence. As shown in Figure 6, the spin-up of turbulence is not responsible for this sensitivity. The LWP decreases immediately after restarting with a larger acoustic Courant number.

Figure 6.

Change in the LWP after restarting with NAT = 4, 60 and 90 minutes after the beginning of the simulation with NAT = 10. The physical time step is 0.1 seconds for these runs.

[20] The RICO case does not exhibit much sensitivity (Figure 7) to the choice of acoustic Courant number. For this case, convergence is not clear; depending on one's convergence criterion (absolute or relative), one could state that the results have already converged with the largest acoustic Courant number or one could argue that convergence is achieved only for the two smallest acoustic Courant numbers.

Figure 7.

Results for the acoustic-Courant-number-convergence test for the RICO case. (a) A one-hour running mean is used for smoothing. SHF denotes the surface sensible heat flux, and LHF denotes the surface latent heat flux. (b) The profiles are time-averaged over the last four hours.

[21] At this point, it is advisable to check if the results converge sufficiently well compared with a smaller acoustic Courant number, so that the results are insensitive to the simulated high frequency mode. We recommend that one should obtain a convergence for both time series and selected profiles, depending on one's convergence criterion. Omitting this convergence test may produce misleading results, or results that appear to perform well, but for the wrong reasons; for instance, Figure 8 shows that the results with (NATt) = (12,0.2) are closer to the GCSS ensemble mean than those simulated with (NATt) = (10,0.1).

Figure 8.

Same as Figure 3 but for the results with two different NAT and Δt. The red lines are results with (NATt) = (12,0.2), the blue lines are for (NATt) = (10,0.1), and the black lines are the GCSS mean.

[22] A separate research effort to ascertain and eliminate the cause of this sensitivity has been undertaken.

5. Saturation Adjustment for the Initial Profiles

[23] The initial sounding of DRF01 is designed to generate cloud water, in other words it is supersaturated. Models that use a microphysics parameterization based on saturation-adjustment (i.e., all-or-nothing approach), commonly diagnose cloud water before time integration to bring the fields to saturation-adjustment. Without this procedure, a large amount of latent heat is released by condensation at the first time step, which creates undesirable gravity waves. The current ARW initialization code does not treat this problem. This portion of the code has to be corrected in case the microphysics parameterization is based on the all-or-nothing approach. This is, however, not applied to the two-moment scheme used in this study, because it predicts supersaturation, which condenses water rather slowly as shown below.

[24] In order to demonstrate the above issue, two simple simulations are performed: one with a saturation-adjustment scheme [Kessler, 1969], and the other with the two-moment scheme [Feingold et al., 1998]. No forcings or initial perturbations are applied, so that condensation is responsible for any excited waves. This test is performed on a 4×4 horizontal grid, because without initial perturbation the simulated fields in one column are exactly the same as other columns. The vertical resolution, horizontal and vertical grid spacings, and physical and acoustic time steps are the same as the specification described in Section 3. The statistics data are saved every time step, up to one simulated minute, and then every minute for later processing. The duration is two simulated hours.

[25] For the saturation-adjustment scheme, the LWP oscillates immediately after the first time step (Figure 9a). The vertical motion reaches over 1 m s−1, and then decreases with oscillations. On the other hand, the two-moment scheme condenses liquid water gradually and reaches the maximum LWP after about 20 seconds. Although the two-moment scheme generates vertical motion, it is much weaker. The oscillations of the LWP for the saturation-adjustment scheme persist over two hours (Figure 9b). The vertical motion becomes weaker but does not dissipate, while the vertical motion for the two-moment scheme quickly dissipates and the LWP remains steady.

Figure 9.

A comparison test between the Kessler scheme and the Feingold scheme. The maximum math formula means the largest magnitude of the vertical velocity.

6. Technical Development for ARW Toward a User-Friendly LES Model

6.1. Statistics Package

[26] The numerical algorithm of the statistics package is based on the statistics output code of System for Atmospheric Modeling (SAM) [Khairoutdinov and Randall, 2003]. Due to the fact that ARW uses a mass vertical coordinate, the horizontal mean of each level is not at a height coordinate, as is common in other LES and GCSS studies. In order to output the horizontal average on the height coordinate, the local column profile is vertically interpolated to the output level, which is the mass level of the base state. The monotonic cubic interpolation of Steffen [1990] is used as a default interpolation method: it is third-order accurate and does not create new minima or maxima between two levels (Figure 10). The horizontal averaging should be performed with mass-weighting for the compressible system [Canuto, 1997]:

display math

where the double overbar represents the mass-weighted horizontal mean, the overbar denotes horizontal mean, and ρ is air density. Note that the horizontal mean density and pressure are given without mass-weighting [Canuto, 1997]. It is also important to note that for compressible turbulence,

display math

where the prime denotes perturbation from the mass-weighted horizontal mean. From (6) and (7), one can see math formula and math formula. In the statistics package, the interpolated value at an output level is combined horizontally with mass-weighting for averaging:

display math

where, the angle brackets represent interpolation to the output level.

Figure 10.

An example of a profile constructed with a monotonic cubic interpolation with Steffen's method (red line). The filled circles represent the local value at discrete levels. The green line is the profile constructed with the linear interpolation.

[27] Resolved scale and SGS fluxes are the horizontal mean of the vertical advective and diffusive fluxes computed by the advection and diffusion schemes of the ARW dynamical core, respectively. All of these fluxes are collected for output and diagnosis. For the height coordinate, advective fluxes can be written as ρwf. Decomposing to the horizontal mean and its perturbation, followed by horizontal averaging yields

display math

where we have used math formula and math formula. As seen in (3), the vertical advective flux is computed with μdω, so a velocity transformation to w must be performed. The hydrostatic relationship, which is a diagnostic relation for the non-hydrostatic system and part of the coordinate definition can be written for the ARW dynamics as

display math

where z is height, ρd is dry air density, and g is the gravitational acceleration. With the definition of ω given in (4) and the hydrostatic relationship, one can transform from μdω to w using

display math

Thus, the transformation of the vertical advective flux is performed by

display math

This transformation is performed before interpolation. The vertical diffusive flux is computed at the height coordinate, thus no transformation is necessary. After vertical interpolation, the air density is applied to obtain the mass-weighted horizontal mean; thus the flux is obtained as

display math

Examples of flux for diagnostic variables are discussed in Appendix C.

[28] Higher-order moments, (e.g., variance), as well as the resolved scale TKE and its tendency terms, discussed in Appendix D, involve the horizontal mean of certain quantities. For these output variables, the mean profile is constructed at a high-resolution height level, and then the high-resolution mean profile is interpolated to the local mass level for diagnosis. For instance, variance is diagnosed as

display math

where the tilde represents the interpolation of the mass-weighted high-resolution horizontal mean from the high-resolution level to the local level. The use of the high-resolution vertical level minimizes the bias appearing in the perturbation and gradient due to interpolation of the mean profile to the mass level. An example is shown in Figure 11. Let us assume that 1) the local column profile happens to be the same as the mean, 2) the output height level is located at the half level of the mass level, and 3) a high-resolution level is not used. The perturbation in this example should be zero at the mass level. In order to compute the perturbation, first the local column will be interpolated to the output height for the horizontal mean. Next, the mean profile is interpolated back to the mass level. It is clearly seen that the procedure creates a bias, associated with the difference between the cross and filled circle marks. The resulting gradient also includes bias. The high-resolution mean profile helps reduce the bias. Our experiment shows that the use of the monotone cubic interpolation also reduces bias compared with the linear interpolation. The resolution of the high-resolution level is a user-specified parameter in the name list; Other user specified parameters are the timing of the output, number of samples over the output period for time average, and conditional averages, e.g., in-cloud statistics. In this technical note, twenty levels are added between two output levels.

Figure 11.

An example showing how bias forms with interpolation when the perturbation is computed at the mass level. The filled circles represent the local value at the mass level, the open circles represent the mean value at the output height, and the crosses represent the interpolated mean value from the output level to the mass level where the bias appears. Linear interpolation is considered here. In this example, the perturbation at any level should be zero since the local and mean profiles are set to be the same. The bias, which is the difference between the cross and filled circles, becomes smaller as the resolution of the output level increases.

6.2. LES Package

[29] Our LES package offers a straightforward model setup by simply setting the name list parameters and supplying forcings and soundings in a simple text file. Like the statistics package, the algorithmic strategy is based on SAM. The list of options that can be used with the current LES package is summarized in Table 2. The functionality of unused options in this technical note have been tested by the authors. The package accepts a time-varying forcing and soundings, and the input soundings are specified on the height coordinate.

Table 2. List of the Available Options for the Current Version of the LES Package, and the Case Name, Which Uses the Listed Option
OptionsCase
Specified surface temperatureDRF01, DRF02, RICO
Specified surface sensible heat fluxDRF01, DRF02
Specified surface latent heat fluxDRF01, DRF02
Specified surface friction velocityDRF01, DRF02
Specified large scale subsidence (applied to horizontal velocity, potential temperature, mixing ratios, and aerosol/cloud number concentrations)DRF01, DRF02, RICO
Specified large scale advective tendency for potential temperatureRICO
Specified large scale advective tendency for water vapor mixing ratioRICO
Specified radiative forcing 
Mean-profile nudging for horizontal velocity 
Mean-profile nudging for potential temperature 
Mean-profile nudging for water vapor mixing ratio 
Geostrophic wind update 
Specified initial perturbationDRF02, RICO
Specified codes for the surface calculationRICO

[30] For the specified surface parameters, subsidence, and advective and radiative forcings, linear interpolation is used to calculate the value or profile at the current time step. Then, monotonic cubic interpolation is used to calculate the value at each local level for the time-interpolated profile. The tendency due to subsidence is calculated using the first-order upwind scheme in the advective form in the height coordinate, with the subsidence vertically interpolated to the local level.

[31] Nudging is applied to the mass-weighted horizontal mean:

display math

where τ is the user specified nudging time scale. The nudging tendency is computed at a high-resolution height coordinate, then interpolated back to the local level. The default number of high resolution levels is 10 times as many as the model resolution. Prior to the nudging tendency calculation, the horizontally averaged profile is computed at a high-resolution level. For efficiency, the vertical interpolation of the input soundings to the high-resolution level is performed at the beginning of the simulation and stored for later use. and linear interpolation is used to build a profile at the current time step. This high-resolution sounding is also used for the geostrophic wind update.

[32] There are two subroutines in the LES-package module file in which a user can add special code for the initial perturbation and surface fluxes. This is handy since it obviates the need to modify several source files.

7. Summary and Conclusions

[33] This technical note presents an evaluation of the LES mode of ARW from various view points, in addition to providing a description of the LES and statistics packages implemented in ARW. The new packages facilitate setup and post-processing and should be beneficial to every user running ARW as a LES model.

[34] ARW is capable of producing results comparable to those from other LES models. Aside from the saturation adjustment for the initial condition, the current ARW exhibits sensitivity to the acoustic Courant number as illustrated in our simulations of the GCSS DYCOMS-II RF01. The sensitivity varies among the cases tested. At this point it is recommended that a convergence test be performed every time the model configuration is changed. The current ARW tends to be numerically expensive due to this sensitivity, which may limit the physical time step to a value much smaller than the stability criterion for the physical mode. An ongoing study will attempt to relieve ARW of the sensitivity.

Appendix A:: DYCOMS-II RF02

[35] ARW is configured following the case specification listed in Ackerman et al. [2009]. The domain size is 6.4×6.4×1.5 km3 with horizontal (vertical) grid spacing of 50 (approximately 12) m. The physical (acoustic) time step is 0.2 (0.025) seconds. The duration is 6 hours.

[36] The case specifies a cloud droplet number concentration Nd of 55 cm−3 for single-moment microphysics schemes, while our two-moment scheme predicts Nd. For this reason, two simulations are performed: one with an initial aerosol concentration of 120 cm−3 and the other with 95 cm−3. The former produces a similar LWP time series to the GCSS ensemble mean while the latter produces a mean droplet number concentration of 56 cm−3 over the last 4 hours.

[37] Results are presented in Figure A1. The run with 120 cm−3 initial aerosol concentration produces the quasi-steady LWP and steady increase of TKE. The entrainment rate and vertical profiles are similar to the GCSS ensemble mean without sedimentation, due to lack of drizzle. The run with 95 cm−3 initial aerosol concentration depletes LWP due to precipitation. Without a source, the aerosol is depleted by collision-coalescence and wet removal, which promotes further precipitation. The profiles are somewhat similar to the GCSS ensemble mean with the sedimentation calculation, even though the cloud field of ARW is transient under the precipitation feedback. An ad hoc addition of an aerosol source could possibly help ARW match results with the GCSS mean.

Figure A1.

Comparison of ARW with the GCSS ensemble members for the DYCOMS-II RF02 case. The red (blue) line is ARW with an aerosol concentration of 120 (95) cm−3, and the black (dashed) line is the GCSS ensemble mean with (without) sedimentation. The shading is created only with the simulations with sedimentation. Entrainment rate is smoothed with a 20-minute running mean, after being computed with the smoothed PBL height. PBL height is smoothed with the same running mean.

Appendix B:: RICO

[38] ARW is configured following the case specification listed in vanZanten et al. [2011]. The domain size is 12.8×12.8×4 km3 with horizontal (vertical) grid spacing of 100 (approximately 40) m. The physical (acoustic) time step is 0.5 (0.05) seconds. The duration of the simulation is 24 hours.

[39] Results are presented in Figure B1. The differences between ARW and the GCSS ensemble mean mainly come from the lack of precipitation below the cloud base for ARW. The two-moment microphysics scheme of Feingold et al. [1998] creates rainwater as much as the GCSS mean, but evaporates rainwater in the cloud layer. Comparing ARW with Figure 4 of vanZanten et al. [2011], ARW agrees quite well with the GCSS mean without precipitation.

Figure B1.

Comparison of ARW with the GCSS ensemble members for the RICO case. The blue line is ARW, and the black line is the GCSS ensemble mean.

Appendix C:: Flux for Diagnostic Variables

[40] Some of the thermodynamic fluxes have to be diagnosed with the advective and diffusive fluxes collected from the ARW dynamical core. For instance, liquid water potential temperature flux is diagnosed with

display math

where qc is the summation of mixing ratios in the liquid phase; math formula is the Exner function; and wq for each of the liquid-phase moist variables are collected from the ARW dynamical core; cp is the isobaric specific heat of dry air; Lv is the latent heat of vaporization.

[41] The flux involving virtual temperature contains triple moments so that deriving a diagnostic formula requires additional consideration. Here we derive the virtual potential temperature flux. The virtual potential temperature is defined as

display math

where qv is water vapor mixing ratio and δ = (RvRa)/Ra≈0.608. Ra and Rv are the gas constants for dry air and water vapor, respectively. The vertical flux can be written as

display math

This is exact and holds even with math formula.

[42] The following relationship can be derived with math formula:

display math

Thus, the resolved scale virtual potential temperature flux can be expressed as

display math

With the advective flux, (i.e., and wq), from the ARW dynamical core, the advective flux for θq for each moist species can be expressed as

display math

The horizontal mean resolved scale virtual potential temperature flux is diagnosed in the statistics package as

display math

where wadv is the vertical velocity transformed from μdω, which is the vertical advective velocity.

[43] For the SGS flux diagnosis, consider that the double overbar represents the grid volume mean in (C3), and thus prime denotes the SGS-turbulence quantity deviating from the resolved scale motion. The vertical diffusive flux collected from the ARW dynamical core has the form of

display math

where the double bracket represents the grid volume average; the double prime represents SGS turbulence; K is eddy diffusivity computed from the SGS turbulence scheme. Since K is defined at the scalar level, it has to be interpolated to the flux level. With this formula, one can compute the diffusive flux for Tq and then the SGS virtual temperature flux can be obtained with the third line of (C3) as

display math

Finally the horizontal mean SGS virtual temperature flux is diagnosed in the statistics package as

display math

At the surface, we use the following formula by assuming the triple moment to be zero in the second line of (C3):

display math

Appendix D:: TKE and Its Tendency Terms

[44] Resolved scale TKE and its tendency terms are given as a summation of terms defined at different staggered levels. In the statistics package, each term is interpolated to the output level and then combined. For instance, the resolved scale TKE is computed as

display math

where tensor notation has been used on the l.h.s. and the subscripts s, u, v, w, uw, and vw for the right angle bracket denote interpolation to the output level from the level at scalar, u, v, w, half level of the u level, and half level of the v level.

[45] The resolved scale TKE equation for the compressible system can be written as

display math

where σij is the stress tensor, i.e., SGS momentum flux. Note that p′ as well as ρ′ are perturbations from the horizontal mean without mass-weighting as discussed in Section 5.1. A detailed derivation can be found in Canuto [1997]. The pressure redistribution term does not disappear for the compressible system. Since the quantities with an overbar are horizontally averaged, they are only a function of height:

display math

where math formula has been used, the pressure transport and redistribution terms have been combined, and the hydrostatic relationship has been used for the buoyant production:

display math

[46] Turbulence transport can be diagnosed with

display math

The quantities on the r.h.s. can be obtained from the WRF dynamical core except the mean value, which is computed as a high-resolution mean profile. The density in the first term on the r.h.s. is interpolated to the ui level. For i = 3, i.e., vertical velocity, this formula reduces to

display math

[47] In order to derive the above formula, one starts with

display math

[48] Manipulation yields

display math

Applying the average and then using the relationship math formula one obtains

display math

[49] Using the relationship math formula and some manipulation reduces (D9) to (D5). Alternatively, one can obtain the turbulence transport starting by multiplying the total advective tendency of the momentum equation, math formula, by math formula, and then subtracting the shear production. We decided not to take this path because this involves the conversion from math formula to ρ as well as the coordinate transformation for the horizontal gradients: for instance,

display math

[50] The total pressure transport is computed as

display math

Instead of performing the coordinate transformation for the horizontal gradients, we interpolate p to the u (v) level to compute the gradients for the x (y) direction on the constant local height.

[51] Diffusion is computed as

display math

where the term in the second angle brackets on the r.h.s. of the first line is the dissipation, which is the negative of the shear production of the SGS TKE, S.

[55] Shear production is computed as

display math

while buoyant production is computed as

display math

[56] As discussed for diffusion, dissipation is computed as

display math

[57] The horizontally averaged SGS TKE equation can be written as

display math

where e is SGS TKE, S is shear production, B is buoyant production, D is dissipation, and math formula is the vertical SGS flux available from the ARW dynamical core as well as the production and dissipation terms without density (i.e., S, B, and D). The total transport term is computed as

display math

Acknowledgments

[52] The authors thank Marat Khairoutdinov for allowing them to use the SAM's program codes for the development of the two packages discussed in the text. The authors thank Bill Skamarock for insightful discussions on the acoustic time stepping, and Jan Kazil, Seoung-Soo Lee, and Hailong Wang for technical discussions. This study is supported by the NOAA Climate Program Office through the Climate Process Team, Cloud Macrophysical Parameterization and its Application to Aerosol Indirect Effects (PI V. Larson).

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