Abstract

Hybrid bulk–bin microphysics schemes discretize particle size distributions into bins for calculating microphysical process rates, while retaining a limited number of bulk prognostic quantities and assuming an underlying analytic functional form for the particle size distributions as in traditional bulk microphysics schemes. In this paper, the treatment of sedimentation in two-moment bulk and hybrid schemes is compared using different numerical methods. Using the first-order upwind method for calculating sedimentation in conjunction with a widely used, two-step, time-splitting approach that updates model fields after transport by air motion followed by calculation of sedimentation, it is shown analytically that despite using a spectrum of fall speeds corresponding to different particle sizes, hybrid schemes converge with increasing bin resolution toward bulk schemes that utilize only characteristic moment-weighted particle fall speeds. While not strictly convergent, it is also shown that solutions using bulk and hybrid schemes are often similar for other numerical methods and approaches. Noticeable improvement using the hybrid scheme occurs in a few circumstances: when the Courant number associated with falling precipitation is large (>>1), requiring substepping, semi-implicit, or Lagrangian-type methods for numerical stability; or when a one-step approach is employed that calculates hydrometeor transport in a single step using a velocity that combines both vertical air motion and particle fall speed. Thus, it is concluded that the use of hybrid rather than bulk schemes is justified for some, but not all, applications, and care should be taken to determine the appropriateness of hybrid schemes for specific applications.

Abstract

This paper compares simple theoretical expressions relating vertical velocity, perturbation pressure, updraft size, and dimensionality for cumulus convection, derived in Part I, with numerical solutions of the anelastic buoyant perturbation pressure Poisson equation and vertical velocity w. A range of thermal buoyancy profiles representing shallow to deep moist convection are tested for both two-dimensional (2D) and three-dimensional (3D) updrafts. The theoretical expressions give similar results for w and perturbation pressure difference from updraft top to base Δp compared to the numerical solutions over a wide range of updraft radius R. The theoretical expressions are also consistent with 2D and 3D fully dynamical updraft simulations initiated by warm bubbles of varying width.

Implications for nonhydrostatic modeling in the “gray zone,” with a horizontal grid spacing Δx of O(1–10) km where convection is generally underresolved, are discussed. The theoretical and numerical solutions give a scaling of updraft velocity with R (~Δx) consistent with fully dynamical 2D and 3D simulations in the gray zone, with a rapid decrease of maximum w at relatively small R and a slower decrease at large R. These results suggest that an incorrect representation of perturbation pressure may be an important contributor to biases in convective strength at these resolutions. The theoretical solutions also provide a concise physical interpretation of the “virtual mass” coefficient in convection parameterizations and can be easily incorporated into these schemes to provide a consistent scaling of perturbation pressure effects with R, updraft height, and the buoyancy profile.

Abstract

This study investigates relationships between vertical velocity, perturbation pressure, updraft size, and dimensionality for cumulus convection. Generalized theoretical expressions are derived from approximate analytic solutions of the governing momentum and mass continuity equations for both two-dimensional (2D) and axisymmetric quasi-three-dimensional (3D) steady-state updrafts. These expressions relate perturbation pressure and vertical velocity to updraft radius R, height H, and thermal buoyancy. They suggest that the vertical velocity at the level of neutral buoyancy is reduced from perturbation pressure effects by factors of and in 2D and 3D, respectively, where is a nondimensional length, with somewhat different scalings lower in the updraft (α is a parameter equal to the ratio of vertical velocity horizontally averaged across the updraft to that at the updraft center). They also indicate that updrafts are weaker in 2D than 3D, all else being equal, with a difference of up to a factor of 2 in vertical velocity for as a direct result of differences in mass continuity between 2D and axisymmetric 3D flow. Differences between these expressions and other analytic solutions, including those derived from single normal mode Fourier/Fourier–Bessel expansion of the buoyant perturbation pressure Poisson equation, are discussed. Part II of this study compares the theoretical expressions with numerical solutions of the buoyant perturbation pressure Poisson equation for a wide range of thermal buoyancy profiles representing shallow-to-deep moist convection and also with fully dynamical 2D and 3D updraft simulations.

Abstract

New theoretical analytic expressions are derived for the evolution of a passive scalar, buoyancy, and vertical velocity in growing, entraining moist deep convective updrafts. These expressions are a function of updraft radius, height, convective available potential energy (CAPE), and environmental relative humidity R _H . They are quantitatively consistent with idealized three-dimensional moist updraft simulations with varying updraft sizes and in environments with differing R _H . In particular, the analytic expressions capture the rapid decrease of buoyancy with height due to entrainment for narrow updrafts in a dry environment despite large CAPE. In contrast to the standard entraining-plume model, the theoretical expressions also describe the effects of engulfment of environmental air between the level of free convection (LFC) and height of maximum buoyancy (HMB) required by mass continuity to balance upward acceleration of updraft air (i.e., dynamic entrainment). This organized inflow sharpens horizontal gradients, thereby enhancing smaller-scale lateral turbulent mixing below the HMB. For narrow updrafts in a dry environment, this enhanced mixing leads to a negatively buoyant region between the LFC and HMB, effectively cutting off the region of positive buoyancy at the HMB from below so that the updraft structure resembles a rising thermal rather than a plume. Thus, it is proposed that a transition from plume-like to thermal-like structure is driven by dynamic entrainment and depends on updraft width (relative to height) and environmental R _H . These results help to bridge the entraining-plume and rising-thermal conceptual models of moist convection.

Abstract

Idealized three-dimensional supercell simulations were performed using the two-moment bulk microphysics schemes of Morrison and Milbrandt––Yau in the Weather Research and Forecasting (WRF) model. Despite general similarities in these schemes, the simulations were found to produce distinct differences in storm structure, precipitation, and cold pool strength. In particular, the Morrison scheme produced much higher surface precipitation rates and a stronger cold pool, especially in the early stages of storm development. A series of sensitivity experiments was conducted to identify the primary differences between the two schemes that resulted in the large discrepancies in the simulations.

Different approaches in treating graupel and hail were found to be responsible for many of the key differences between the baseline simulations. The inclusion of hail in the baseline simulation using the Milbrant––Yau scheme with two rimed-ice categories (graupel and hail) had little impact, and therefore resulted in a much different storm than the baseline run with the single-category (hail) Morrison scheme. With graupel as the choice of the single rimed-ice category, the simulated storms had considerably more frozen condensate in the anvil region, a weaker cold pool, and reduced surface precipitation compared to the runs with only hail, whose higher terminal fall velocity inhibited lofting. The cold pool strength was also found to be sensitive to the parameterization of raindrop breakup, particularly for the Morrison scheme, because of the effects on the drop size distributions and the corresponding evaporative cooling rates. The use of a more aggressive implicit treatment of drop breakup in the baseline Morrison scheme, by limiting the mean––mass raindrop diameter to a maximum of 0.9 mm, opposed the tendency of this scheme to otherwise produce large mean drop sizes and a weaker cold pool compared to the hail-only run using the Milbrandt––Yau scheme.

Abstract

A new two-moment stratiform cloud microphysics scheme in a general circulation model is described. Prognostic variables include cloud droplet and cloud ice mass mixing ratios and number concentrations. The scheme treats several microphysical processes, including hydrometeor collection, condensation/evaporation, freezing, melting, and sedimentation. The activation of droplets on aerosol is physically based and coupled to a subgrid vertical velocity. Unique aspects of the scheme, relative to existing two-moment schemes developed for general circulation models, are the diagnostic treatment of rain and snow number concentration and mixing ratio and the explicit treatment of subgrid cloud water variability for calculation of the microphysical process rates.

Numerical aspects of the scheme are described in detail using idealized one-dimensional offline tests of the microphysics. Sensitivity of the scheme to time step, vertical resolution, and numerical method for diagnostic precipitation is investigated over a range of conditions. It is found that, in general, two substeps are required for numerical stability and reasonably small time truncation errors using a time step of 20 min; however, substepping is only required for the precipitation microphysical processes rather than the entire scheme. A new numerical approach for the diagnostic rain and snow produces reasonable results compared to a benchmark simulation, especially at low vertical resolution. Part II of this study details results of the scheme in single-column and global simulations, including comparison with observations.

Abstract

Simulations of a squall line observed on 20 May 2011 during the Midlatitude Continental Convective Clouds Experiment (MC3E) using 750- and 250-m horizontal grid spacing are performed. The higher-resolution simulation has less upshear-tilted deep convection and a more elevated rear inflow jet than the coarser-resolution simulation in better agreement with radar observations. A stronger cold pool eventually develops in the 250-m run; however, the more elevated rear inflow counteracts the cold pool circulation to produce more upright convective cores relative to the 750-m run. The differing structure in the 750-m run produces excessive midlevel front-to-rear detrainment, reinforcing excessive latent cooling and rear inflow descent at the rear of the stratiform region in a positive feedback. The contrasting mesoscale circulations are connected to early stage deep convective draft differences in the two simulations. Convective downdraft condensate mass, latent cooling, and downward motion all increase with downdraft area similarly in both simulations. However, the 750-m run has a relatively greater number of wide and fewer narrow downdrafts than the 250-m run averaged to the same 750-m grid, a consequence of downdrafts being under-resolved in the 750-m run. Under-resolved downdrafts in the 750-m run are associated with under-resolved updrafts and transport mid–upper-level zonal momentum downward to low levels too efficiently in the early stage deep convection. These results imply that under-resolved convective drafts in simulations may vertically transport air too efficiently and too far vertically, potentially biasing buoyancy and momentum distributions that impact mesoscale convective system evolution.

Abstract

Idealized simulations of the 15 May 2009 squall line from the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2) are evaluated in this study. Four different microphysical setups are used, with either single-moment (1M) or double-moment (2M) microphysics, and either hail or graupel as the dense (rimed) ice species. Three different horizontal grid spacings are used: Δx = 4, 1, or 0.25 km (with identical vertical grids). Overall, results show that simulated squall lines are sensitive to both microphysical setup and horizontal resolution, although some quantities (i.e., surface rainfall) are more sensitive to Δx in this study. Simulations with larger Δx are slower to develop, produce more precipitation, and have higher cloud tops, all of which are attributable to larger convective cells that do not entrain midlevel air. The highest-resolution simulations have substantially more cloud water evaporation, which is partly attributable to the development of resolved turbulence. For a given Δx, the 1M simulations produce less rain, more intense cold pools, and do not have trailing stratiform precipitation at the surface, owing to excessive rainwater evaporation. The simulations with graupel as the dense ice species have unrealistically wide convective regions. Comparison against analyses from VORTEX2 data shows that the 2M setup with hail and Δx = 0.25 km produces the most realistic simulation because (i) this simulation produces realistic distributions of reflectivity associated with convective, transition, and trailing stratiform regions, (ii) the cold pool properties are reasonably close to analyses from VORTEX2, and (iii) relative humidity in the cold pool is closest to observations.

Abstract

The dynamical effects of increased aerosol loading on the strength and structure of numerically simulated squall lines are explored. Results are explained in the context of Rotunno–Klemp–Weisman (RKW) theory. Changes in aerosol loading lead to changes in raindrop size and number that ultimately affect the strength of the cold pool via changes in evaporation. Thus, the balance between cold pool and low-level wind shear–induced vorticities can be changed by an aerosol perturbation. Simulations covering a wide range of low-level wind shears are performed to study the sensitivity to aerosols in different environments and provide more general conclusions. Simulations with relatively weak low-level environmental wind shear (0.0024 s⁻¹) have a relatively strong cold pool circulation compared to the environmental shear. An increase in aerosol loading leads to a weakening of the cold pool and, hence, a more optimal balance between the cold pool– and environmental shear–induced circulations according to RKW theory. Consequently, there is an increase in the convective mass flux of nearly 20% in polluted conditions relative to pristine. This strengthening coincides with more upright convective updrafts and a significant increase (nearly 20%) in cumulative precipitation. An increase in aerosol loading in a strong wind shear environment (0.0064 s⁻¹) leads to less optimal storms and a suppression of the convective mass flux and precipitation. This occurs because the cold pool circulation is weak relative to the environmental shear when the shear is strong, and further weakening of the cold pool with high aerosol loading leads to an even less optimal storm structure (i.e., convective updrafts begin to tilt downshear).

Abstract

The study of evolution characteristics of initial perturbations is an important subject in four-dimensional variational data assimilation (4DVAR) and mesoscale predictability research. This paper evaluates the impact of microphysical scheme complexity on the propagation of the perturbations in initial conditions for warm-season convections over the central United States. The Weather Research and Forecasting Model (WRF), in conjunction with four schemes of the Morrison microphysics parameterization with varying complexity, was used to simulate convective cases using grids nested to 5-km horizontal grid spacing. Results indicate that, on average, the four schemes show similar perturbation evolution in amplitude and spatial pattern during the first 2 h. After that, the simplified schemes introduce significant error in amplitude and spatial pattern. The simplest (liquid only) and most complex schemes show almost the same growth rate of initial perturbations with different amplitudes during 6-h forecast, suggesting that the simplest scheme does not reduce the nonlinearity in the most complex scheme. The evolution of vertical velocity and total condensates is more nonlinear than horizontal wind, temperature, and humidity, which suggest that the observations of cloud variables and vertical velocity should have a shorter time window (less than 1 h) compared to horizontal wind, temperature, and humidity observations. The simplified liquid-only microphysics scheme can be used as an acceptable substitute for the more complex one with a short time window (less than 1 h).