1National Severe Storms Laboratory, Norman, Oklahoma, U.S.A.

ABSTRACT: This paper highlights the results of our efforts to incorporate parameterized expressions for non-inductive and inductive charging, lightning discharges, and formation of screening layers into a three-dimensional dynamic cloud model (ie. "cloud model"). The cloud model is time dependent and involves simultaneous solution of the equations of atmospheric motion with equations for mass, heat, and water substance continuity. Electrification is modeled by solving additional continuity equations for liquid and ice hydrometeor space charge densities and a Poisson equation for scalar electric potential. The present dynamical cloud simulation approach extends earlier kinematic modeling studies of tornadic supercell and New Mexico mountain thunderstorms. At present we are engaged in efforts to validate the preliminary simulations versus observations of morphologically similar storms and also to investigate the impact of the storm environment on the simulated electrical and lightning morphology.

INTRODUCTION

In recent modeling studies, we have evaluated the role of non-inductive graupel-ice charging in the early electrification of a mountain thunderstorm (Ziegler et al. 1991) and have explored the relationships between space charge, electric fields, and lightning morphology in a tornadic storm (Ziegler and MacGorman 1994). These studies employed a kinematic model with parameterized expressions for non-inductive and inductive charging, lightning discharges, and formation of screening layers. However, large or rapidly moving storms and storm systems are not usually observed with Doppler radars to the extent required by the time-varying kinematic model. Hence, it is desirable to develop a capability to simulate electrifying storms to complement the kinematic modeling method.

In the present study, we report results of our efforts to incorporate these electrification mechanisms into a time dependent, three-dimensional dynamic cloud model. To demonstrate the new model’s capabilities, we simulate the development of a supercell thunderstorm, terminating the cloud model integration after the first few simulated intra-cloud lightning discharges to focus on the early storm electrification phase.

MODEL FRAMEWORK

a. Airflow dynamics

The numerical simulations are made using a non-hydrostatic, fully compressible, three-dimensional dynamic cloud model (Straka and Anderson 1993). Prognostic equations are included for three momentum components, pressure, potential temperature, and turbulent kinetic energy. There are also conservation equations for the number concentrations (when required) and mixing ratios of aerosols, water vapor, and hydrometeors.

The advection and diffusion numerics in the simulation model all include a conservation principle. For momentum advection, the model uses a time-centered, quadratic-conserving differencing scheme or a sixth-order local spectral scheme (Straka and Anderson 1993). Choices for scalar advection include a forward in time, sixth-order, flux divergence-corrected, Crowley scheme, and a time-centered, second-order flux scheme. The diffusion parameterization is based on K-theory, with the mixing coefficient derived from the variable turbulent kinetic energy.

b. Microphysics

The model employs a general microphysical package that was designed for cloud and mesoscale models in which new information about microphysical processes could be added without rewriting significant portions of the source code. Optional features include: (1) prediction of N habits and size categories of hydrometeor, aerosol, and chemical species, including more than 12 distinct ice crystal habits; (2) prediction of microphysical processes with either a number of parameterizations or with explicit microphysics, the latter incorporating logarithmically spaced mass categories to represent the size distribution; (3) hybrid applications with some species using parameterizations and some using explicit microphysics. The options desired for a particular simulation are easily chosen via namelist parameters input to the model during compilation.

In its present configuration, the model solves prognostic equations for the following hydrometeor habits: liquid drops (cloud and rain), frozen raindrops (900 kg m^-3), bullets, solid and hollow columns, needles, sheaths, thick and thin plates, planar and spatial dendrites, sectors, stellars, columns with plates, plates with dendrites, snow aggregates, three density categories of graupel (ie. 200 kg m^-3, 450 kg m^-3, 700 kg m^-3), and two size categories of (high density) hail (ie. 5-20 mm, > 20 mm, respectively). Hydrometeor formation and growth rates include the nucleation of water drops, primary and secondary ice nucleation, vapor diffusion, stochastic coalescence, continuous collection of cloud droplets, wet and dry growth of graupel and hail, melting and freezing, breakup of large drops and aggregates, conversions from one hydrometeor specie to another, variable hail and graupel particle density, ice particle temperature, and sedimentation.

c. Electrification

The formulation of electrification processes is essentially identical to that described by Ziegler et al. (1991), though generalized to incorporate optional treatments of charging. Prognostic equations are integrated for space charge on cloud ice, cloud droplets, rain, snow, graupel, and hail. The vector electric field is derived from the scalar electric potential, which in turn is derived by inverting a Poisson equation forced by net space charge density.

The space charge continuity equation for hydrometeors is given by

where r _i is the space charge (C m^-3) of hydrometeor habit l, V_t,l is terminal velocity (ms^-1) for habit l, S_l is sources and sinks of space charge density (C m^-3 s^-1) for habit l, r ₀ is air density (Kg m^-3), x_i are the Cartesian coordinates (m), u_i are the Cartesian wind components (ms^-1), and K_h is the sub-grid eddy mixing coefficient (m² s^-1). The equations for the source and sink terms for the space charge that are included are non-inductive charging, inductive charging, screening layer charging, and mass transfer. The definitions for total space charge, potential, and electric field components are

where r _t is the total charge density on hydrometeors (C m^-3), f is scalar electric potential (V), and e ₀ = 8.8592x10^-12 N^-1 m^-2 C² is the electrical permittivity of air.

d. Charge source terms

Four options for the non-inductive charging rates per collision are available in the model: 1) Helsdon and Farley (1987), wherein various levels of fixed separated charge are assumed per rebounding collision between graupel/hail and snow particles; 2) Ziegler et al. (1991), wherein collisional charging is a function of size of the graupel/hail and snow particles; 3) Saunders et al. (1991), utilizing a lookup table approach to specify charge per collision as a function of particle size, cloud liquid water content, temperature, and other parameters (total charging determined by quadrature over modeled particle size distribution); 4) Saunders and Peck (1998), which follows the Saunders et al. (1991) method, with functional replacement of liquid water content by rime accretion rate. (Since the ICAE, Takahashi noninductive charging has been added as a fifth option.)

Since neither observations nor laboratory experiments explicitly provide information to accommodate charging versus ice crystal habit, the same parameterizations are used for all crystal types in the model. Depending on what option is employed, the model evaluates the parameterization either analytically or by quadrature from look-up tables, and can consider partial and complete distributions.

The model also includes expressions for inductive charging, screening layer formation, and charge transfers associated with (microphysical) mass transfer (Ziegler et al. 1991). A new lightning parameterization, which can simulate either IC or CG flashes, has recently been added to the model (MacGorman et al. 1998) and is activated in the present run. (Since the ICAE, a discrete breakdown channel model has been added as a new optional treatment for lightning discharges.) An option to include explicit ion processes following Helsdon and Farley (1987) is under development. The model also does not treat inductive charging owing to any ice — ice, ice — drizzle or ice — rain rebounding collisions; from any liquid — liquid rebounding collisions; or from shedding of cloud, drizzle, or rain owing to melting or wet growth.

RESULTS

The simulated supercell is initialized in a horizontally homogeneous base state environment with 2200 J kg^-1 of convective available potential energy and a 1/2 circle hodograph with an arc of 50 m s^-1 (ie. 0-5 km shear = 31.8 m s^-1). The horizontal and vertical grid spacing are set at 1 km and 0.5 km, respectively, and the domain size is 40 km x 40 km x 20 km. Vertical motion is initiated by introducing a warm thermal bubble with an amplitude of 1 ° C, horizontal and vertical radii of 10 and 1.4 km respectively, and an elevation of 1.4 km AGL. For this test, electrification mechanisms are treated as by Ziegler et al. (1991). A reversal temperature of —10 ° C for non-inductive graupel-ice charging is assumed. The model is integrated for a period of 35 min.

Early supercell electrification displays a distinctly exponential character (Fig. 1). As also shown by Ziegler et al. (1991) for a New Mexican mountain thunderstorm, the exponential character of the early electrification is caused by graupel-ice collision rates and mean separated charge per collision increasing exponentially. After the storm achieves a maximum field of breakdown intensity (~ 150 kV m^-1; see MacGorman et al. 1998) just after 31 min, IC discharges commence at a rate of 2-3 flashes min^-1 (Fig. 1, inset). The later, mature stage of electrification, including the individual interflash recoveries, is more linear in character due to a relatively slowly varying microphysical charging current in the storm after ice contents become established.

The simulated supercell morphology (Fig. 2) bears many similarities to earlier kinematic model results for the Binger, Oklahoma tornadic storm (Ziegler and MacGorman 1994). Just prior to the first IC flash, the storm has achieved an altitude of 14 km and a maximum updraft of 40 m s^-1. The net space charge density ranges from roughly —1 to 1 nC m^-3, and the corresponding maximum electric field magnitude is 135 kV m^-1. The intense convection elevates the main charge region in the updraft, and sloping main negative and upper positive charge regions gradually separate with downstream distance into the anvil due to the differing fallspeeds of cloud ice, snow particles, graupel, and hail.

CONCLUSIONS

We have developed a versatile cloud model which may be used to investigate a wide range of questions concerning storm electrification. The model may be configured to simulate both isolated storms, such as illustrated here, and larger Mesoscale Convective Systems (MCSs). In addition to studying early electrification, we are presently using the new model to investigate lightning parameterizations and the variations of supercell electrical and lightning morphology across a spectrum of convective instability and wind shear values representing typical storm environments.

ACKNOWLEDGEMENTS: Model runs were performed by JMS on the CRAY J-90 computer at the University of Oklahoma through sponsorship of the OU Graduate College and NSF Grant EAR-9512145. Additional support for this project was provided under NSF Grants ATM-9311911, ATM-9120009, ATM-9617318, and EAR-9512145.

REFERENCES

Helsdon, J. H., and R. D. Farley, 1987: A numerical study of a Montana thunderstorm: 2. Model results versus observations involving electrical aspects. J. Geophys. Res., 92, 5661-5675.

Saunders, C. P. R., W. D. Keith, and R. P. Mitzeva, 1991: The effect of liquid water on thunderstorm charging. J. Geophys. Res., 96, 11,007-11,017.

Saunders, C. P. R., and S. L. Peck, 1998: Laboratory studies of the influence of the rime accretion rate on charge transfer during crystal/graupel collisions. J. Geophys. Res., 103, 13,949-13,956.

Straka, J. M., and J. R. Anderson, 1993: Numerical simulations of microburst-producing storms: Some results from storms observed during COHMEX. J. Atmos. Sci., 50, 1329-1348.

MacGorman, D. R., J. M. Straka, and C. L. Ziegler, 1998: A lightning parameterization for numerical cloud models. Preprints, 19^th Conference on Severe Local Storms, AMS, pp. 348-351.

Ziegler, C. L., D. R. MacGorman, J. E. Dye, and P. S. Ray, 1991: A model evaluation of non-inductive graupel-ice charging in the early electrification of a mountain thunderstorm. J. Geophys. Res., 96, 12,833-12,855.

Ziegler, C. L., and D. R. MacGorman, 1994: Observed lightning morphology relative to modeled space charge and electric field distributions in a tornadic storm. J. Atmos. Sci., 51, 833-851.