1. Introduction
The growth of ice crystals in nature is a complex phenomenon: atmospheric ice attains a variety of shapes that vary among relatively simple hexagonal plates and columns, dendritic structures, rosettes, and often irregular crystals that may be polycrystalline in form. Though crystal shapes vary, they can be characterized by two axis lengths, a and c, and their primary shapes by the aspect ratio (φ = c/a). For hexagonal prisms, the a dimension is half the distance across the basal (hexagonal) face, whereas the c dimension is half the length of the prism (rectangular) face. Less ideal crystals are characterized in terms of their maximum and minimum dimensions: for flat, platelike particles a is half the maximum dimension (referred to here as the maximum semidimension) and c is half the minimum dimension, with the opposite being the case for columnlike particles. Traditionally, the primary habit is characterized by aspect ratio variation with temperature, the classical picture of which is based on laboratory and in situ data and is arguably most valid for T > −22°C (e.g., Pruppacher and Klett 1997; Fukuta and Takahashi 1999; Bailey and Hallett 2009). These data suggest platelike (φ < 1) crystals for temperatures in the ranges of −1° to −3°C and −10° to −22°C, whereas columnlike crystals (φ > 1) occur in the ranges of −3° to −9°C and lower than −22°C. More recent data suggest important modifications to the habit diagram at temperatures below about −22°C (e.g., Bailey and Hallett 2009), specifically that plates tend to occur more often and that columns are rare. Some measurements support this (e.g., Bailey and Hallett 2002, 2004); however, these data are at odds with other laboratory studies of ice crystal formation and growth (e.g., Libbrecht 2003). The secondary habit of ice crystals depends on the degree of supersaturation, and this classification is most valid for T > −22°C. Laboratory measurements generally show that lower saturation states produce compact hexagonal crystals depending in part on nucleation. As the saturation increases toward liquid, crystals tend to hollow: columns become needlelike near −6°C, dendritic crystals appear around −15°C, with rosette and polycrystals forming at T < −22°C.
It is presently impossible for cloud models to capture the variety of ice crystal masses, shapes, sizes, and distributions that occur in real clouds. Nevertheless, all numerical models need some way to relate the mass of crystals to their shapes and sizes. Prior attempts have nearly universally involved using either equivalent volume (or diameter) spheres (e.g., Lin et al. 1983; Reisner et al. 1998; Thompson et al. 2004) or mass–size relationships (e.g., Mitchell et al. 1990; Harrington et al. 1995; Meyers et al. 1997; Woods et al. 2007; Thompson et al. 2008; Morrison and Grabowski 2008, 2010), though exceptions exist (e.g., Hashino and Tripoli 2007). These methods suffer from a number of deficiencies (Sulia and Harrington 2011, hereafter SH11). In comparison to wind tunnel data, the use of equivalent spheres underestimates ice growth at every temperature except around −10° and −22°C where growth is nearly isometric (e.g., Fukuta and Takahashi 1999). Mass–size relations allow the prediction of only a single size, and the coefficients in the mass–size power laws are derived from in situ data and are therefore tied to the observed particle growth histories. These methods are not linked to the growth mechanisms that determine the evolution of the two primary crystal axes (a and c) and therefore cannot evolve crystals in a natural fashion.
Bulk models are limited because they attempt to predict the evolution of only a few moments of the size spectrum, the shape of which is assumed a priori. Typically, only the ice mass mixing ratio is predicted in single-moment schemes, whereas in two-moment schemes the number mixing ratio is predicted as well. Moreover, most current bulk microphysical methods artificially categorize particles based on predefined classes and then use transfer functions to move particles among classes as they evolve. For instance, Woods et al. (2007) predicts numerous ice classes as a way to deal with the problem of habit evolution, but such methods are computationally costly because of the increased number of prognostic variables. In the method of Harrington et al. (1995), transfer functions across a somewhat arbitrary size boundary are used to evolve smaller pristine ice class particles into a larger snow class. A more recent trend in microphysical modeling is to predict different ice particle properties for a given ice class. For instance, the method of Morrison and Grabowski (2008) predicts a rime mass fraction for ice crystals that avoids the traditional approach of transferring ice to a separate graupel class, and Hashino and Tripoli (2007) use the Chen and Lamb (1994) approach for predicting habits in a hybrid-bin scheme. Such methods allow the use of fewer ice classes and provide for a more natural evolution of the ice particles based on their respective growth histories. To evolve crystal habit, a way to relate the evolution of the a- and c-axis distributions is required, and this has been a significant hurdle for numerical models to surmount.
The approach developed here predicts the evolution of ice particle habit by tracking the evolution of ice aspect ratio following Chen and Lamb (1994, hereafter CL94) and SH11. Similarities exist in the approach to that presented in Hashino and Tripoli (2007), with the exception that the method presented here is derived for a bulk cloud model using four prognostic variables. The method provides a way to relate the a- and c-axis evolution through the use of laboratory-determined parameters, making the method amenable to improvements through new measurements. The approach eliminates thresholds between habits and follows the recent paradigm of predicting particle properties instead of multiple particle classes. The parameterization is developed here with testing of the method described in Harrington et al. (2013, hereafter Part II).
2. Single-particle evolution
The Fickian-distribution method
The nonlinear impact of crystal shape on growth is embodied by an evolving capacitance in the Fickian-distribution method (C; Fig. 1): For the same volume particle, C increases nonlinearly as the aspect ratio deviates from unity. This is due to the tightening of the vapor density gradients near crystal edges (Libbrecht 2005, their Fig. 5, and see SH11 for a full explanation). While the capacitance contains this characteristic feature of the vapor field (Fig. 2), the aspect ratio is constant in the traditional capacitance model. The consequences of keeping aspect ratio constant are illustrated in Fig. 1: A single crystal is grown for 5 min at liquid saturation and at T = −6°C either with a constant or evolving aspect ratio. When φ is constant, the nonlinear feedbacks in growth are not captured, and so the crystal grows to a smaller final capacitance, and therefore mass, than when φ evolves freely. Consequently, predicting crystal aspect ratio is critical for capturing nonlinear changes in vapor diffusion.
Capacitance as a function of aspect ratio for oblate (plate) and prolate (column) crystals with volume equivalent to that of a sphere (req) of radius 50 (solid), 25 (dashed), and 10 (dotted–dashed) μm. Evolution of an ice particle at T = −6°C, liquid saturation, an initial equivalent volume spherical radius of 10 μm, and initial φ = 10 for the capacitance model with φ constant (solid black arrow) and Fickian distribution with evolving aspect ratio (solid gray arrow).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Capacitance as a function of aspect ratio for oblate (plate) and prolate (column) crystals with volume equivalent to that of a sphere (req) of radius 50 (solid), 25 (dashed), and 10 (dotted–dashed) μm. Evolution of an ice particle at T = −6°C, liquid saturation, an initial equivalent volume spherical radius of 10 μm, and initial φ = 10 for the capacitance model with φ constant (solid black arrow) and Fickian distribution with evolving aspect ratio (solid gray arrow).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Capacitance as a function of aspect ratio for oblate (plate) and prolate (column) crystals with volume equivalent to that of a sphere (req) of radius 50 (solid), 25 (dashed), and 10 (dotted–dashed) μm. Evolution of an ice particle at T = −6°C, liquid saturation, an initial equivalent volume spherical radius of 10 μm, and initial φ = 10 for the capacitance model with φ constant (solid black arrow) and Fickian distribution with evolving aspect ratio (solid gray arrow).
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Graphical illustration of vapor flux enhancement due to nonsphericity in the capacitance model. Each particle has the same volume, with vapor density contours of the same values surrounding the particle and computed for an oblate spheroid. Spherical particles have isometric vapor fields, and hence the vapor flux is uniform over the surface. However, as the aspect ratio φ varies away from one, decreasing in this case, the vapor flux increases over the axis of greatest curvature (end of the a axis) and decreases over the other axis (c axis). This process leads to an overall increase in the diffusive flux toward the particle.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Graphical illustration of vapor flux enhancement due to nonsphericity in the capacitance model. Each particle has the same volume, with vapor density contours of the same values surrounding the particle and computed for an oblate spheroid. Spherical particles have isometric vapor fields, and hence the vapor flux is uniform over the surface. However, as the aspect ratio φ varies away from one, decreasing in this case, the vapor flux increases over the axis of greatest curvature (end of the a axis) and decreases over the other axis (c axis). This process leads to an overall increase in the diffusive flux toward the particle.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Graphical illustration of vapor flux enhancement due to nonsphericity in the capacitance model. Each particle has the same volume, with vapor density contours of the same values surrounding the particle and computed for an oblate spheroid. Spherical particles have isometric vapor fields, and hence the vapor flux is uniform over the surface. However, as the aspect ratio φ varies away from one, decreasing in this case, the vapor flux increases over the axis of greatest curvature (end of the a axis) and decreases over the other axis (c axis). This process leads to an overall increase in the diffusive flux toward the particle.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
The equations above constitute the core of adaptive habit evolution using the Fickian-distribution method. The equations are modified for ventilation effects that require computation of crystal fall speeds. The method of SH11 is followed, and the main equations are discussed in appendix A. In Part II, laboratory data are used to critique the above equations and prior mass–size methods. It will be shown that the above method is robust and relatively accurate, whereas most mass–size relations produce crystal masses, sizes, and fall speeds that are generally in error.
3. Bulk adaptive habit parameterization
Prediction of bulk ice habit requires the particle size distribution-integrated forms of the mass growth equation [Eqs. (1) and (4)] and the mass distribution hypothesis [Eq. (2)]. In forming these equations, a quantitative link between the distributions of the a and c axes is needed. Two-dimensional size spectra have been used for this purpose (e.g., Chen and Lamb 1999) but are computationally costly. Here, δ is used to relate the a- and c-axis distributions, so that a single distribution can be used, making the method numerically efficient. The development presented here assumes the evolution of a single habit type, and so mixes of particle habits (plates and needles) and complex crystal types (e.g., rosettes and polycrystals) are not considered, but could be added by allowing for at least two particle classes and following a method similar to Hashino and Tripoli (2008). Frequently used coefficients from the derivations below are listed in Table 1.
Coefficients resulting from derivations presented in the main text and appendix.
a. Axis length history
Because particles are initially assumed to be isometric, δ* is initially equal to one. However, the inherent growth ratio δ is typically not unity and varies with temperature, and so δ* will also vary in time. This is an important departure from traditional bulk methods that implicitly fix δ* at a particular value for a given habit type. In the method proposed here, the relation between c and a will also evolve. Since δ* is a weighted time average of δ, it is naturally constrained to lie within the limits of δ derived from laboratory data (about 0.27 and 2.1). Note the distinction between δ and δ*: δ is a measure of the mass distributed over the a and c axes during growth (one time step), whereas δ* relates the two axis lengths over time and allows for the use of a single size spectrum in either the c- or a- axis length. It is worth noting that δ* is not a prognostic variable, but is diagnosed from separate prediction of the c- and a-axis lengths. The parameterization depends on the exactness of this historical relation (δ*), which is demonstrated in Part II.
b. Axis distributions
To define the aspect ratio requires the c-axis distribution while the equivalent volume sphere distribution (r) is needed for vapor growth calculations (see section 2). Therefore, it is necessary to change variables from a to that of r or c in Eq. (9), which can be done analytically by using δ*. The resulting distributions are similar to Eq. (9). The general form of the distribution conversion and the distributions of the c and r axes [Eqs. (B6) and (B5), respectively] are given in section a of appendix B. Each has characteristic lengths (cn and rn) related to the mean axis length that characterize the mean particle shape.
c. Mass mixing ratio growth and mass distribution relationship for ice particle distributions
Knowing the particle distributions [Eqs. (9), (B5), and (B6)] and the historical relation between the axis lengths [Eq. (6)] makes it possible to derive equations for the evolution of the mass mixing ratio and the axis length mixing ratios. Because the axis mixing ratios are proportional to an and cn, the characteristic sizes are used throughout to represent mean habit evolution. Evolving qi follows a procedure similar to that for single particles (section 2a): The mass growth equation is combined with an equation describing the change in characteristic equivalent volume radius. Once the change in rn is known, aspect ratio evolution can then be computed using relationships between the different axis lengths.
d. Fall speed and ventilation
e. Procedure for evolving habits
Predicting changes in mass mixing ratio, axis lengths, fall speeds, and density can now be accomplished using the equations above when solved over a time step, the details of which are provided in the appendices. The derivation of the method is complicated, so an overview of the equations used to evolve bulk habits is warranted. Moreover, the equations are cast in a form suitable for use in Eulerian models since the method is designed ultimately for this purpose. The main equations used are also summarized in Tables 2 and 3.
Summary of variables: Beginning of time step. Only the mixing ratios (Ni/ρa, qi, A, and C) are stored externally to the growth calculations. To compute the change in qi, A, and C over a time step, initial values of various quantities must be derived from these quantities. The initial variables at the beginning of a time step used in the growth model are given here, and a description in section 3e.
Summary of prognostic variables: End of time step. The growth method evolves the ice mass mixing ratio (qi), and the a- and c-axis mixing ratios (A and C) over a time step. This table contains the main equations in the order that they are used. Note that initial values of the variables from Table 2 are assumed. At the end of the calculations, only the three mixing ratios are passed back and stored.
Time stepping of the variables can now be done, and the main equations are summarized in Table 3. For reasons given in section 3c and in SH11, particle size and mass changes are computed by finding the change in rn using Eq. (B23) giving rn(t + Δt). Knowing rn at the end of a time step allows an(t + Δt) to be determined using Eq. (B26). Including axis ratio-dependent ventilation is then accomplished by replacing δ in Eq. (B26) with δυb [Eq. (20)]. The change in cn can also be computed, but to do so requires knowing δ*(t + Δt), which is diagnosed from Eq. (B27) since rn(t + Δt) and an(t + Δt) are known. For columnlike crystals, cn(t + Δt) is determined from the relation between cn and an in Eq. (B6). For platelike crystals, the change in mean aspect ratio is first computed, and then cn(t + Δt) is found from Eq. (B32). At this point, the axis mixing ratios are recomputed from Eq. (10).
Finally, the mass mixing ratio at the end of a time step is computed. To do so requires knowing how the mean ice density has changed over time. The new density is a volume-weighted average between the density at the prior time step and the density added during growth given by Eq. (B25). Once 
In the parcel model tests described in Part II of this paper, our goal is to illustrate that the average effects of adaptive habit evolution can be captured with the procedure described above. Other issues emerge when the method is implemented in an Eulerian framework. For example, separate advection of the a- and c-axis length mixing ratios could potentially lead to errors in the evolution of axis ratio. However, tests indicate that such errors are small; this is a subject of a forthcoming paper on the Eulerian implementation of the method. Other issues also arise: Ice can be nucleated at multiple locations within the cloud, and regions of differing ice classes can be mixed together. Furthermore, the adaptive habit method needs to be connected in a physically accurate way to both riming and aggregation processes. These issues are beyond the scope of the present work, but are discussed briefly in Part II and in more depth in our forthcoming paper on the Eulerian implementation and testing.
4. Summary, remarks, and future development
Prior methods of parameterizing ice mass growth from vapor have used the traditional capacitance model; however, the capacitance depends on the major semiaxis length (either a or c) and the aspect ratio 
Acknowledgments
We are grateful for the detailed comments provided by three anonymous reviewers, which substantially improved this manuscript. K. Sulia and J. Harrington would like to thank the National Science Foundation for support under Grants ATM-0639542 and AGS-0951807. In addition, J. Harrington was supported in part by the Department of Energy under Grant DE-FG02-05ER64058. K. Sulia was supported in part by an award from the Department of Energy (DOE) Office of Science Graduate Fellowship Program (DOE SCGF). The DOE SCGF Program was made possible in part by the American Recovery and Reinvestment Act of 2009. The DOE SCGF program is administered by the Oak Ridge Institute for Science and Education for the DOE. ORISE is managed by Oak Ridge Associated Universities (ORAU) under DOE Contract Number DE-AC05-06OR23100. All opinions expressed in this paper are the author’s and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE. H. Morrison was partially supported by the NOAA Grant NA08OAR4310543; U.S. DOE ARM DE-FG02-08ER64574; U.S. DOE ASR DE-SC0005336, sub-awarded through NASA NNX12AH90G; and the NSF Science and Technology Center for Multiscale Modeling of Atmospheric Processes (CMMAP), managed by Colorado State University under Cooperative Agreement ATM-0425247.
APPENDIX A
Fall Speed and Ventilation Effects
APPENDIX B
Bulk Parameterization Details
a. Distribution relations
b. Integration of mass growth equation
Power-law fit coefficients for the shape factor [fs; Eq. (B9)] as a function of aspect ratio.
Shape factor as a function of aspect ratio. The exact expression for oblate and prolate spheroids along with the polynomial fit are shown.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Shape factor as a function of aspect ratio. The exact expression for oblate and prolate spheroids along with the polynomial fit are shown.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
Shape factor as a function of aspect ratio. The exact expression for oblate and prolate spheroids along with the polynomial fit are shown.
Citation: Journal of the Atmospheric Sciences 70, 2; 10.1175/JAS-D-12-040.1
c. Mass distribution relations for particle distributions
d. Mass mixing ratio, size, and aspect ratio evolution
e. Fall speeds and bulk ventilation coefficients
REFERENCES
Avramov, A., and J. Y. Harrington, 2010: Influence of parameterized ice habit on simulated mixed phase Arctic clouds. J. Geophys. Res., 115, D03205, doi:10.1029/2009JD012108.
Avramov, A., and Coauthors, 2011: Towards ice formation closure in Arctic mixed-phase boundary layer clouds during ISDAC. J. Geophys. Res., 116, D00T08, doi:10.1029/2011JD015910.
Bailey, M., and J. Hallett, 2002: Nucleation effects on the habit of vapour grown ice crystals from −18° to −42°C. Quart. J. Roy. Meteor. Soc., 128, 1461–1483.
Bailey, M., and J. Hallett, 2004: Growth rate and habits of ice crystals between −20° and −70°C. J. Atmos. Sci., 61, 514–554.
Bailey, M., and J. Hallett, 2009: A comprehensive habit diagram for atmospheric ice crystals: Confirmation from the laboratory, AIRS II, and other field studies. J. Atmos. Sci., 66, 2888–2899.
Bohm, J., 1992: A general hydrodynamic theory for mixed-phase microphysics. Part I: Drag and fall speed of hydrometeors. Atmos. Res., 27, 253–274.
Chen, J.-P., and D. Lamb, 1994: The theoretical basis for the parameterization of ice crystal habits: Growth by vapor deposition. J. Atmos. Sci., 51, 1206–1221.
Chen, J.-P., and D. Lamb, 1999: Simulation of cloud microphysical and chemical processes using a multicomponent framework. Part II: Microphysical evolution of a wintertime orographic cloud. J. Atmos. Sci., 56, 2293–2312.
Fridlind, A., A. Ackerman, G. McFarquhar, G. Zhang, M. Poellot, P. DeMott, A. Prenni, and A. Heymsfield, 2007: Ice properties of single-layer stratocumulus during the Mixed-Phase Arctic Cloud Experiment: 2. Model results. J. Geophys. Res., 112, D24202, doi:10.1029/2007JD008646.
Fukuta, N., and T. Takahashi, 1999: The growth of atmospheric ice crystals: A summary of findings in vertical supercooled cloud tunnel studies. J. Atmos. Sci., 56, 1963–1979.
Harrington, J. Y., M. P. Meyers, R. L. Walko, and W. R. Cotton, 1995: Parameterization of ice crystal conversion processes due to vapor deposition for mesoscale models using double-moment basis functions. Part I: Basic formulation and parcel model results. J. Atmos. Sci., 52, 4344–4366.
Harrington, J. Y., K. Sulia, and H. Morrison, 2013: A method for adaptive habit prediction in bulk microphysical models. Part II: Parcel model corroboration. J. Atmos. Sci., 70, 365–376.
Hashino, T., and G. J. Tripoli, 2007: The Spectral Ice Habit Prediction System (SHIPS). Part I: Model description and simulation of the vapor deposition process. J. Atmos. Sci., 64, 2210–2237.
Hashino, T., and G. J. Tripoli, 2008: The Spectral Ice Habit Prediction System (SHIPS). Part II: Simulation of nucleation and depositional growth of polycrystals. J. Atmos. Sci., 65, 3071–3094.
Lamb, D., and W. Scott, 1972: Linear growth rates of ice crystals grown from the vapor phase. J. Cryst. Growth, 12, 21–31.
Lamb, D., and J. Verlinde, 2011: The Physics and Chemistry of Clouds. Cambridge University Press, 584 pp.
Libbrecht, K., 2003: Growth rates of the principal facets of ice between −10°C and −40°C. J. Cryst. Growth, 247, 530–540.
Libbrecht, K., 2005: The physics of snow crystals. Rep. Prog. Phys., 65, 855–895.
Lin, Y., R. Farley, and H. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Climate Appl. Meteor., 22, 1065–1092.
López, M. L., and E. E. Ávila, 2012: Deformation of frozen droplets formed at −40°C. Geophys. Res. Lett., 39, L01805, doi:10.1029/2011GL050185.
Meyers, M. P., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud microphysics parameterization. Part II: The two-moment scheme. Atmos. Res., 45, 3–39.
Milbrandt, J. A., and M. K. Yau, 2005: A multimoment bulk microphysics parameterization. Part II: A proposed three-moment closure and scheme description. J. Atmos. Sci., 62, 3065–3081.
Mitchell, D. L., 1996: Use of mass- and area-dimensional power laws for determining precipitation particle terminal velocities. J. Atmos. Sci., 53, 1710–1723.
Mitchell, D. L., R. Zhang, and R. L. Pitter, 1990: Mass-dimensional relationships for ice particles and the influence of riming on snowfall rates. J. Appl. Meteor., 29, 153–163.
Morrison, H., and W. Grabowski, 2008: A novel approach for representing ice microphysics in models: Description and tests using a kinematic framework. J. Atmos. Sci., 65, 1528–1548.
Morrison, H., and W. Grabowski, 2010: An improved representation of rimed snow and conversion to graupel in a mutlicomponent bin microphysics scheme. J. Atmos. Sci., 67, 1337–1360.
Nelson, J., 1994: A theoretical study of ice crystal growth in the atmosphere. Ph.D. dissertation, University of Washington, 183 pp.
Nelson, J., and C. Knight, 1998: Snow crystal habit changes explained by layer nucleation. J. Atmos. Sci., 55, 1452–1465.
Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. Kluwer Academic Publishers, 954 pp.
Reisin, T., Z. Levin, and S. Tzivion, 1996: Rain production in convective clouds as simulated in an axisymmetric model with detailed microphysics. Part I: Description of the model. J. Atmos. Sci., 53, 497–519.
Reisner, J., R. Rasmussen, and R. Bruintjes, 1998: Explicit forecasting of supercooled liquid in winter storms using the MM5 mesoscale model. Quart. J. Roy. Meteor. Soc., 124, 1071–1107.
Sheridan, L. M., J. Y. Harrington, D. Lamb, and K. Sulia, 2009: Influence of ice crystal aspect ratio on the evolution of ice size spectra during vapor depositional growth. J. Atmos. Sci., 66, 3732–3743.
Sulia, K. J., and J. Y. Harrington, 2011: Ice aspect ratio influences on mixed-phase clouds: Impacts on phase partitioning in parcel models. J. Geophys. Res., 116, D21309, doi:10.1029/2011JD016298.
Thompson, G., R. Rasmussen, and K. Manning, 2004: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part I: Description and sensitivity analysis. Mon. Wea. Rev., 132, 519–542.
Thompson, G., P. Field, R. Rasmussen, and W. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 5095–5115.
Walko, R. L., W. R. Cotton, M. P. Meyers, and J. Y. Harrington, 1995: New RAMS cloud microphysics parameterization. Part I: The single moment scheme. Atmos. Res., 38, 29–62.
Walko, R. L., W. R. Cotton, G. Feingold, and B. Stevens, 2000: Efficient computation of vapor and heat diffusion between hydrometeors in a numerical model. Atmos. Res., 53, 171–183.
Woods, C., M. Stoelinga, and J. Locatelli, 2007: The IMPROVE-1 storm of 1–2 February 2001. Part III: Sensitivity of a mesoscale model simulation to the representation of snow particle types and testing of a bulk microphysical scheme with snow habit prediction. J. Atmos. Sci., 64, 3927–3948.
