CORRIGENDUM

Tetsuya Takemi Department of Global Architecture, Osaka University, Osaka, Japan

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Richard Rotunno National Center for Atmospheric Research,* Boulder, Colorado

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Corresponding author address: Dr. Tetsuya Takemi, Dept. of Environmental Science and Technology, Tokyo Institute of Technology, G5-7, 4259 Nagatsuta, Yokohama, Kanagawa 226-8502, Japan. Email: takemi@depe.titech.ac.jp

Corresponding author address: Dr. Tetsuya Takemi, Dept. of Environmental Science and Technology, Tokyo Institute of Technology, G5-7, 4259 Nagatsuta, Yokohama, Kanagawa 226-8502, Japan. Email: takemi@depe.titech.ac.jp

1. Introduction

Takemi and Rotunno (2003, hereinafter TR03) examined the relative effects of subgrid-scale mixing and numerical filters in squall-line simulations in an idealized condition of a no-shear environment. TR03 found that simulations using common subgrid models with standard values for the model constants and without explicit numerical filters are characterized by poorly resolved grid-scale convective cells. They suggested that this problem should be controlled by setting eddy viscosity coefficients 1.5 to 2 times larger than the standard values. From a recent communication with Dr. G. Bryan of the National Center for Atmospheric Research, we have learned that the early version of the Weather Research and Forecasting (WRF) model (Skamarock et al. 2001) used in TR03 did not have the correct subgrid-scale mixing for some scalar quantities: vertical mixing of potential temperature and water vapor mixing ratio acted on the perturbation fields from the base state, not on the total fields. The purpose of this corrigendum is to reexamine the validity of the conclusions presented in TR03 by presenting some new simulations with corrected subgrid mixing.

2. Results and conclusions

The same experimental setup as that employed in TR03 is used in the present simulations. Here we examine only simulations using the subgrid parameterization scheme that predicts turbulent kinetic energy (TKE) and performs simulations of the evolution of an initial line thermal in a no-shear environment. In the no-shear environment, one expects the development of disorganized, scattered convective cells far behind the surface outflow boundary (Weisman and Rotunno 2004).

TR03 experimented with different values of the constants Ck and Ce, which appear, respectively, in the equations for the eddy viscosity Km,
i1520-0493-133-1-339-e1
and the subgrid-scale energy dissipation,
i1520-0493-133-1-339-e2
where e is the subgrid turbulent kinetic energy, and l is a length scale. The values (Ck, Ce) = (0.094, 0.93) were originally derived from theoretical estimates based on inertial-subrange, locally isotropic turbulence (Lilly 1967). Lilly (1966) noted that Ck = 0.124, if finite-difference effects are taken into account in the derivation. Based on theory and numerical experiments with stratified flow, Deardorff (1973) determined (Ck, Ce) = (0.065, 0.7) and later used the values (Ck, Ce) = (0.1, 0.7) (Deardorff 1980) for the large-eddy simulations of the planetary boundary layer. Moeng and Wyngaard (1988) revisited the issue and found that (Ck, Ce) = (0.1, 0.93) are more consistent with the available data. Schmidt and Schumann (1989) derived (Ck, Ce) = (0.0856, 0.845) based on the field measurement data, and Stevens et al. (1999) also found the same values in their derivation. Storm-scale and mesoscale cloud simulations have used a variety of values [e.g., Klemp and Wilhelmson (1978) used (Ck, Ce) = (0.2, 0.2)]; however, Ck = 0.1 seems to be commonly used in cloud simulations, and thus can be regarded as the standard value, although some variety in the choice of the Ck value is noted.

With the fixed value of Ce = 0.93, Figs. 1c and 5 of TR03 show the sensitivity of the simulated moist convection for Ck = 0.1, 0.15, 0.2, 0.25, 0.3. The simulations with Ck = 0.1 produced convective cells that were poorly resolved, while the simulations with Ck = 0.3 appeared to be more damped than necessary to control the noise in the solution.

With the correct physical mixing formulation now implemented in the WRF model (vertical mixing performed on the full scalar fields) we reexamine in Fig. 1 the behavior of the model-simulated moist convection for Ck = 0.1, 0.125, 0.15, 0.175, 0.2. Figure 1 shows that even with the corrected physical mixing, the solution with the standard value of Ck = 0.1 is still noisy and has many grid-scale structures in the no-shear case, as found in TR03. With Ck slightly increased to 0.125, there is a significant reduction in the number of grid-scale cells; however, there is still some small-scale noise in the solution. Increasing the constant Ck further to 0.15 results in smoother cellular structures and well-resolved features; however, with Ck = 0.2, the present simulations produce an overly damped solution, as few cells even survive.

As was done in TR03, the power spectral density of the vertical velocity at the 3-km level is calculated in the across-line direction in a 80 km × 80 km area around the rightward-moving surface outflow. Figure 2 shows the power spectral densities for the cases depicted in Fig. 1. Similar to the result of TR03, a significant energy buildup is seen at short wavelengths in the case using the standard Ck value. This energy buildup in short scales can be avoided by increasing the constant above the commonly used value. These results are basically the same as those presented in TR03, even though the Ck value suggested as optimal in TR03 produces an overly damped solution with the correct subgrid-turbulence formulation.

In summary, new simulations with corrected physical mixing using the standard subgrid model constants produces a noisy solution (as found in TR03), and some method to avoid (or control) the noise is still necessary. The experiments reported in Figs. 1 and 2 lead us to believe that TR03’s suggestion of adjusting the subgrid-model constants is still a viable method, but that TR03’s suggested values for the subgrid constants need to be revised downward. Apparently, the suggested values in TR03 (i.e., 1.5 to 2 times larger than the widely used standard values) are too large and provide overly strong damping when the correct subgrid mixing formulation is used. From the results of the present simulations, the suggested value for the constant should be approximately 1.5 times larger than the standard value.

Acknowledgments

We would like to thank Dr. George Bryan for his helpful suggestions and discussions.

REFERENCES

  • Deardorff, J. W., 1973: The use of subgrid transport equations in a three-dimensional model of atmospheric turbulence. J. Fluids Eng., 95 , 429438.

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  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18 , 495527.

  • Klemp, J. B., and R. B. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35 , 10701096.

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  • Lilly, D. K., 1966: On the application of the eddy viscosity concept in the inertial subrange of turbulence. Manuscript 123, National Center for Atmospheric Research, Boulder, CO, 19 pp.

  • Lilly, D. K., 1967: The representation of small-scale turbulence in numerical simulations. Proc. IBM Scientific Computing Symp. on Environmental Sciences, Yorktown Heights, NY, 195–210.

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  • Moeng, C-H., and J. C. Wyngaard, 1988: Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci., 45 , 35733587.

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  • Schmidt, H., and U. Schumann, 1989: Coherent structure of the convective boundary layer derived from large-eddy simulations. J. Fluid Mech., 200 , 511562.

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    • Export Citation
  • Skamarock, W. C., J. B. Klemp, and J. Dudhia, 2001: Prototypes for the WRF (Weather Research and Forecasting) model. Preprints,. Ninth Conf. on Mesoscale Processes, Fort Lauderdale, FL, Amer. Meteor. Soc., J11–J15.

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  • Stevens, B., C-H. Moeng, and P. P. Sullivan, 1999: Large-eddy simulations of radiatively driven convection: Sensitivities to the representation of small scales. J. Atmos. Sci., 56 , 39633984.

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  • Takemi, T., and R. Rotunno, 2003: The effects of subgrid model mixing and numerical filtering in simulations of mesoscale cloud systems. Mon. Wea. Rev., 131 , 20852101.

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    • Export Citation
  • Weisman, M. L., and R. Rotunno, 2004: “A theory for strong long-lived squall lines” revisited. J. Atmos. Sci., 61 , 361382.

Fig. 1.
Fig. 1.

Horizontal cross section of vertical velocity at the height of 3 km, contoured at a 1 m s−1 interval at 4 h for the cases of (a) Ck = 0.1, (b) Ck = 0.125, (c) Ck = 0.15, (d) Ck = 0.175, and (e) Ck = 0.2. The gust front is also indicated by long-dashed lines. A 40 km by 40 km portion of the entire domain is shown.

Citation: Monthly Weather Review 133, 1; 10.1175/MWR-2847.1

Fig. 2.
Fig. 2.

Power spectral density of vertical velocity at the 3-km level for across-line (x) direction obtained using the TKE scheme with various Ck values. A k−5/3 line is also plotted for reference.

Citation: Monthly Weather Review 133, 1; 10.1175/MWR-2847.1

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Save
  • Deardorff, J. W., 1973: The use of subgrid transport equations in a three-dimensional model of atmospheric turbulence. J. Fluids Eng., 95 , 429438.

    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18 , 495527.

  • Klemp, J. B., and R. B. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35 , 10701096.

    • Search Google Scholar
    • Export Citation
  • Lilly, D. K., 1966: On the application of the eddy viscosity concept in the inertial subrange of turbulence. Manuscript 123, National Center for Atmospheric Research, Boulder, CO, 19 pp.

  • Lilly, D. K., 1967: The representation of small-scale turbulence in numerical simulations. Proc. IBM Scientific Computing Symp. on Environmental Sciences, Yorktown Heights, NY, 195–210.

    • Search Google Scholar
    • Export Citation
  • Moeng, C-H., and J. C. Wyngaard, 1988: Spectral analysis of large-eddy simulations of the convective boundary layer. J. Atmos. Sci., 45 , 35733587.

    • Search Google Scholar
    • Export Citation
  • Schmidt, H., and U. Schumann, 1989: Coherent structure of the convective boundary layer derived from large-eddy simulations. J. Fluid Mech., 200 , 511562.

    • Search Google Scholar
    • Export Citation
  • Skamarock, W. C., J. B. Klemp, and J. Dudhia, 2001: Prototypes for the WRF (Weather Research and Forecasting) model. Preprints,. Ninth Conf. on Mesoscale Processes, Fort Lauderdale, FL, Amer. Meteor. Soc., J11–J15.

    • Search Google Scholar
    • Export Citation
  • Stevens, B., C-H. Moeng, and P. P. Sullivan, 1999: Large-eddy simulations of radiatively driven convection: Sensitivities to the representation of small scales. J. Atmos. Sci., 56 , 39633984.

    • Search Google Scholar
    • Export Citation
  • Takemi, T., and R. Rotunno, 2003: The effects of subgrid model mixing and numerical filtering in simulations of mesoscale cloud systems. Mon. Wea. Rev., 131 , 20852101.

    • Search Google Scholar
    • Export Citation
  • Weisman, M. L., and R. Rotunno, 2004: “A theory for strong long-lived squall lines” revisited. J. Atmos. Sci., 61 , 361382.

  • Fig. 1.

    Horizontal cross section of vertical velocity at the height of 3 km, contoured at a 1 m s−1 interval at 4 h for the cases of (a) Ck = 0.1, (b) Ck = 0.125, (c) Ck = 0.15, (d) Ck = 0.175, and (e) Ck = 0.2. The gust front is also indicated by long-dashed lines. A 40 km by 40 km portion of the entire domain is shown.

  • Fig. 2.

    Power spectral density of vertical velocity at the 3-km level for across-line (x) direction obtained using the TKE scheme with various Ck values. A k−5/3 line is also plotted for reference.

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