1. Introduction
Idealized numerical modeling studies of convective storms have yielded many valuable insights into their behavior over the past few decades. The idealized modeling approach typically initializes a horizontally homogeneous “storm environment” that is defined by a single vertical profile (usually taken from a real-world sounding, extracted from NWP model output, or from analytical functions). This environment is assumed to be representative of large-scale (i.e., mesoalpha to synoptic) conditions and is usually, but not always, assumed to be invariant in time (so that storm behavior within a given fixed environment can be studied). Convective development is then encouraged through a variety of artificial forcing mechanisms such as thermal bubbles (Klemp and Wilhelmson 1978), imposed low-level convergence (Loftus et al. 2008), and updraft nudging (Naylor and Gilmore 2012). These simplifications are reasonable if the simulation domain is on the order of 100 km in width and the simulation length is on the order of a few hours. They allow for precise control over the background environment, and thus better understanding of how different environments (in particular the vertical structure of the environment) modulate the simulated storm characteristics. Most studies of this type represent any forces operating on these large scales either by neglecting them entirely or by prescribing some initial balance condition: typically geostrophic balance.
In addition to the PPGF and the Coriolis force, friction is another force that is active at large scales. It is most important in the boundary layer, and particularly near the surface where the momentum flux from the surface dominates. In theory, when observed soundings are used to define the environment in idealized storm simulations, they should already be subject to the effects of friction. Under the aforementioned assumption of geostrophic balance (or no large-scale forces at all), however, the introduction of surface friction tends to modify the specified environmental sounding during the simulation (especially in the low levels), by reducing the near-surface wind speed (Adlerman and Droegemeier 2002; Wicker and Wilhelmson 1993). That is, the rhs of (1) now contains nonzero friction terms and the large-scale balance is upset. This problem has historically made it difficult to control the environmental wind profile while simultaneously incorporating the effects of surface friction through, for example, a drag parameterization or other imposition of no-slip or semislip lower boundary conditions. At least partly for this reason, the vast majority of idealized modeling studies have elected to use free-slip lower boundary conditions, thereby ignoring the direct effects of surface friction entirely.
However, it has long been recognized that the effects of surface friction are important for some aspects of severe convective storm behavior; a prime example is its role in tornado dynamics. Surface friction plays an important role in tornado structure and evolution through the formation of intense radial inflow in a shallow surface layer and subsequent rapid upward turning in the corner flow region (e.g., Lewellen and Lewellen 2007; Fiedler and Rotunno 1986; Davies-Jones 2014). Recent studies (Schenkman et al. 2014; Roberts et al. 2016; Markowski 2016; Mashiko 2016; Roberts and Xue 2017) have additionally found that surface friction may be implicated in processes leading to supercell tornadogenesis including as a potential direct source of horizontal vorticity that can be tilted into the vertical within the developing tornado. These reasons, among others, have motivated researchers to develop methods to incorporate the effects of surface friction while maintaining a specified low-level wind profile in simulations that include the tornado and its parent storm within the context of an idealized environment of the type discussed above.
Previous efforts to incorporate surface friction into idealized storm simulations have varied in their approach. Wicker and Wilhelmson (1993) were among the first to study the effects of surface friction on the development of a simulated tornado within the context of a full 3D supercell simulation. In their doubly nested grid approach, they applied surface friction to a restart of their inner (120-m grid spacing) tornado-resolving grid, 900 s before tornadogenesis in the original, free-slip inner domain simulation. The outer grid (600-m grid spacing) maintained a free-slip lower boundary condition throughout the simulation period. In this manner, they were able to show that their simulated tornado responded to the presence of surface friction by contracting in radius and intensifying. In their case, the nested simulation after the introduction of friction was likely too short for significant changes to occur in the environmental wind profile, which was additionally maintained by the lateral boundary conditions from the no-friction outer grid.
Other studies have attempted to remove or mitigate the modification of the environmental wind profile due to the action of surface friction through various approaches. Wilhelmson and Chen (1982, hereafter WC82) modified the surface drag formulation such that it was only applied to the perturbation wind (relative to the initial environmental wind profile). Adlerman and Droegemeier (2002) used the WC82 method in their idealized supercell simulations to study the effects of surface drag with different specified magnitudes of the drag coefficient on the behavior of the simulated mesocyclones. More recently, Roberts et al. (2016, hereafter R16) and Markowski (2016, hereafter M16) independently developed different methods by which surface drag could be incorporated into an idealized simulation framework while simultaneously maintaining a more-or-less unchanged environmental wind profile throughout a simulation. Coffer and Parker (2017) applied the R16 method to their idealized simulations of supercells in nontornadic and tornadic environments derived from composites of VORTEX-2 proximity soundings (Parker 2014).
The WC82, R16, and M16 methods can be cast within the context of a more general framework where friction is included in an (up to) three-way balance with the Coriolis and the PPGF. Johnson (1966) proposed the term geotriptic for this three-force balance (where, in addition to the surface friction itself, the internal turbulent eddy flux is also balanced).
The rest of this paper is organized as follows: in section 2, we discuss these previous methods, introduce a new method that overcomes some of their limitations, and describe its implementation into a numerical cloud model. In section 3, we test the new method with simulations of a supercell in an environment derived from the 24 May 2011 Oklahoma tornado outbreak. The simulations compare traditional free-slip simulations with the new method, with the WC82 base state surface drag removal method, and with unbalanced surface drag. Section 3 also describes the results of corresponding simulations in which the prestorm environment includes resolved turbulent eddies. Section 4 describes and compares the structure and behavior of the simulated storm across the simulations, focusing on the structure of surface boundaries. Finally, section 5 summarizes the paper and discusses some potential modifications to the method and broader applications.
2. Description of the new method
a. Description of previous methods
During the column simulation these three forces together slowly modified the wind profile, eventually reaching a steady state in approximate geotriptic balance. R16 then used this final balanced wind profile to initialize the environment of their 3D simulations; as such, the wind profile would remain in a quasi-steady state away from storm-induced perturbations.
Similar to R16, though not running a completely separate simulation, M16 sought a balanced wind profile by first allowing friction to modify the wind profile in their “toy model” supercell tornado simulations prior to the activation of the artificial heat source to initiate convection. Then after the first hour of this “adjustment” period, they applied a nudging term to the model horizontal momentum equations that slowly relaxed the horizontal mean wind profile back toward the mean profile as it was at 1 h simulation time. The wind profile reached a near–steady state by 2 h, at which point they activated their heat source to generate the “toy” supercell. Moreover, they excluded the Coriolis force from their simulations entirely; the nudging terms thus “crudely mimic the influence of a large-scale horizontal pressure-gradient force and Coriolis force” (M16). The relaxation coefficient was chosen to minimize the impact on the storm-induced perturbations while still maintaining a constant “far-field” environmental wind profile.
Finally, instead of assuming an initial geostrophic balance as in R16, WC82 assumed that the surface drag was already in balance with an assumed PPGF (and possibly also Coriolis force, if it is active in the simulation). In a manner reminiscent of modifying the Coriolis term in (3) via the geostrophic balance relations (2), WC82 modified the surface drag formulation so that the drag was only applied to the perturbation wind relative to the initial environmental wind profile.
b. Description of the new method
Whereas R16 finds an adjusted wind profile that is consistent with a geotriptic balance between the model friction force, the Coriolis, and a PPGF specified via an initial geostrophic balance, the GWB method works the other way and computes a PPGF that is in geotriptic balance with Coriolis and friction for a given fixed wind profile. At least for applications that require or benefit from control over the environmental wind profile, this ability to compute the necessary PPGF for geotriptic balance [via (5)] for any given initial wind profile is the new method’s chief advantage over the R16 method. Moreover, the final adjusted wind profile in the R16 method will change as a function of details of the model friction parameterization, such as the value of the surface drag coefficient, whereas in the GWB method, the computed PPGF will change instead to compensate and maintain the same wind profile. In this regard, the GWB method is similar to the WC82 method, in that both effectively find a PPGF that counteracts the effects of surface friction on the environmental wind profile while still incorporating its effects on storm-induced perturbations. The chief advantage of the GWB method over WC82 is that it additionally balances the effects of friction above the surface, which may be substantial in some cases.
Finally, both the GWB and M16 methods work by adding an additional forcing term to the rhs of the model momentum equations. In M16, the forcing term is a Newtonian relaxation (or nudging) term that increases in magnitude the more the horizontally averaged wind field differs from the reference profile. In contrast, in the simplest configuration of the GWB method (after the initial 300-s adjustment period) this term is constant in time and there is no forced relaxation toward a reference wind profile, which simplifies interpretation of the results. In this regard, our method is reminiscent of that of Nowotarski et al. (2015) who also added artificial forcing terms to the model momentum equations (as well as the potential temperature and water vapor conservation equations). The forcing terms in their study were also only a function of height and were computed from the horizontal and time-averaged change in these variables over a 2-h period in a separate “stormless” simulation that included the growth of a realistic convective boundary layer. The goals of Nowotarski et al. (2015) and the current study are different, however, as they were concerned with the evolution of simulated storms in an evolving environment, whereas we again are concerned with how to maintain a fixed environment for simulated storms while still accounting for the effects of surface friction. Nevertheless, we adopt a very similar approach to initialize our supercell simulations with a turbulent boundary layer, where we additionally make use of a time-varying PPGF (see section 3d).
3. Application of the GWB method to idealized supercell simulations
a. Overview
Next, we explore the utility of the GWB method through a series of idealized simulations of a strongly tornadic supercell and its environment using the Bryan Cloud Model, release 18.2 (CM1; Bryan and Fritsch 2002). We initialized the simulations using a sounding derived from the RUC model that was representative of the inflow environment of the tornadic supercells of the 24 May 2011 central Oklahoma tornado outbreak (Fig. 1; Orf et al. 2017; L. J. Wicker 2017, personal communication). The model domain is 200 × 200 × 20 km3 with a horizontal grid spacing of 250 m in an inner 100 × 100 km2 region (hereafter the “inner domain”) and gradually stretched to 1 km at the lateral boundaries. The surrounding outer stretched region of grid points is intended to provide a larger area to minimize undesired lateral boundary condition effects, as well as to serve as a zone where turbulent eddies can be introduced (see section 3d) before entering the inner high-resolution domain. A stretched vertical grid is imposed with 53 levels with spacing from 20 m near the surface to 800 m at the model top. The domain translates with a constant [u, υ] = [14.2, 12.1] m s−1 to keep the simulated storm near the center of the domain (though the Coriolis and frictional forces still act on the original ground-relative winds). To initiate convection, we use the updraft nudging technique of Naylor and Gilmore (2012) within an ellipsoidal region of radii 10 km × 10 km × 1.5 km, centered at x = 100 km, y = 100 km, and z = 1.5 km. The maximum magnitude of the nudging is 10 m s−1 and is applied over the first 900 s of the model integration. We employed the triple-moment version of the NSSL microphysics scheme (Mansell 2010; Dawson et al. 2014) and a 1.5-order prognostic TKE subgrid turbulence closure method based on Deardorff (1980). All simulations included the horizontal components of the Coriolis force with a constant Coriolis parameter f = 1 × 10−4 s−1. We divide our experiments into sets with the following naming convention: [S, NS]-[T, NT]-[D, ND]-[GEO, GWB, WC], where we apply all combinations of each term in brackets for a total of 20 simulations. We describe each of the terms in the brackets in Table 1 and in detail in the following sections.
Skew T–logp and hodograph for the RUC-derived 24 May 2011 inflow sounding used in the simulation experiments.
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
Simulation configuration. Experiment naming key: S = storm, NS = no storm, T = resolved BL turbulence, NT = no resolved BL turbulence, D = surface drag, ND = no surface drag.
b. Simulation of the environmental wind profile evolution in the absence of convection
We show the temporal evolution of the low-level wind profile (horizontally averaged over the inner domain) for each of the NS-NT- experiments in Fig. 2. Turning to the experiments without surface drag first, NS-NT-ND-GEO (Fig. 2d) assumes an initial large-scale geostrophic balance [i.e. computes the PPGF using (2)], while the PPGF in NS-NT-ND-GWB (Fig. 2e) is computed from (5) and thus additionally accounts for the effects of internal mixing on the profile. NS-NT-ND-GEO shows only small changes in the wind profile over time, mainly in a slight reduction in the wind speed near the surface (see circles in Fig. 2d) and in a straightening of the hodograph between 500 and 1000 m AGL (hodograph segment between × and star markers in Fig. 2d), while changes in NS-NT-ND-GWB are barely noticeable (Fig. 2e). The minor changes to the hodograph in NS-NT-ND-GEO reflect the subtle action of internal subgrid-scale turbulent mixing on the wind profile; the fact that these changes are relatively small confirms that the internal mixing by itself has minimal effect, at least for this particular wind profile and subgrid-scale turbulence parameterization.
Hodographs for each of the no-storm runs without resolved boundary layer turbulence (NS-NT-). For clarity, only the lower portions of the hodographs are plotted. The hodographs were computed from a weighted (by gridcell area) horizontal average of the entire domain. Initial hodographs are shown in black, while the 1-, 2-, and 3-h forecast times are shown in blue with changing line styles (see legend). Red markers (see legend) indicate specific heights (m AGL).
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
The experiments with surface drag, in contrast, show substantial differences in the evolution of the near-surface wind profile. The hodograph evolution in NS-NT-D-GEO (Fig. 2a) differs substantially from that in NS-NT-D-GWB (Fig. 2b). The near-surface hodograph (i.e., below 1 km AGL) in NS-NT-D-GEO lengthens substantially over time owing to a reduction in the near-surface wind speed by the action of the surface drag. A similar effect is seen in the adjustment to the 3 May 1999 hodograph in R16 when they ran the original wind profile through a 1-D simulation that creates a steady-state balanced sounding (their Fig. 1). It can be inferred that the wind profile that a simulated storm “sees” will differ in potentially important ways from the initial environment if friction is not included in the large-scale force balance with the PPGF computed appropriately. On the other hand, in NS-NT-D-GWB, the wind profile remains nearly constant in time, by design (Fig. 2b). Finally, the NS-NT-D-WC hodograph evolution is very similar to that of NS-NT-D-GWB, except again for a slight reduction in the wind speeds just above the surface (Fig. 2c, circles) and a straightening of the hodograph between 500 and 1000 m AGL (Fig. 2c, hodograph segment between × and star markers), This represents a slight weakening of the shear in the lowest couple hundred meters relative to the initial profile. This behavior is nearly identical to that in NS-NT-ND-GEO (Fig. 2d, circles) and is due to the WC82 method only “balancing” the effects of the drag at the first level above the surface; the GWB method computes a PPGF that also accounts for the effects of surface drag and turbulent mixing throughout the entire profile.
c. Supercell storm simulations
We then performed a corresponding set of simulations, this time with early updraft forcing included but again with no resolved boundary layer turbulence (the S-NT- simulations). The updraft forcing was applied starting at 300 s—just after the initial PPGF profile adjustment period to avoid contaminating the PPGF profile with the effects of storm-induced perturbations—and ending at 1200 s into the simulation. In each simulation, an intense supercell storm formed with a maximum updraft near 80 m s−1 (Fig. 3a) and persisted for 3+ h, producing multiple intense tornado-strength vortices with maximum surface vorticity and wind speeds exceeding 1 s−1 and 100 m s−1 in some cases (Figs. 3b and 3c, respectively). (Transient updrafts reaching 140+ m s−1 in Fig. 3a are associated with intensification of the vortices either at the surface or aloft). However, after 3 h, there was a tendency for the storm in each simulation to begin to lose supercell characteristics, as other cells that developed nearby began to merge their outflow with the primary cell, leading to upscale growth (not shown). For this reason we will only concern ourselves with the model output out to 3 h. As will be discussed in section 3d, the structure and behavior of the simulated storms, particularly near the surface, were substantially different between the simulations.
Time series of domain maximum (a) vertical velocity (m s−1), (b) surface vertical vorticity (s−1), and (c) surface wind speed (m s−1) for each of the storm simulations without resolved boundary layer turbulence (the S-NT- simulations). The black vertical lines denote the start and end times of the temporal composites described in section 4.
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
Similar to the NS-NT- experiments, we plot the hodograph evolution for the set of storm simulations in Fig. 4. In this case, however, we computed the hodographs from a horizontal average of the horizontal wind components in a 20 × 20 km2 area at the southeast corner of the inner 250-m mesh, considered to be representative of the “far-field” inflow environment outside of the region most affected by storm-induced perturbations. If the GWB method is working well, we would expect this region far from the storm to show very limited modification over the period of the simulation similar to the corresponding “no-storm” case. As a comparison of Figs. 2 and 4 shows, there are still impacts from the storm visible in the far-field profile evolution, especially at the later times for S-NT-ND-GEO and S-NT-ND-GWB (Figs. 4d and 4e, respectively). Nevertheless, these are minor, and the GWB method clearly maintains a quasi-steady far-inflow wind profile for out to 3 h (Figs. 4b and 4e), as desired. The S-NT-D-WC experiment also exhibits a relatively steady wind profile (Fig. 4c), though not as steady as S-NT-D-GWB (Fig. 4b). The fact that the GWB method can preserve the initial wind profile with only limited modification in the far inflow environment while still allowing the environment closer to the storm to evolve appropriately is clearly an advantage for studies that wish to systematically investigate the behavior of storms in a prespecified/user-defined shear environment.
As in Fig. 2, but for the storm simulations without resolved boundary layer turbulence (S-NT-).
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
d. Resolved versus subgrid-scale turbulence with the GWB method
Markowski and Bryan (2016) compared LES-scale simulations of atmospheric boundary layers with and without resolved-scale turbulent structures. They demonstrated that when the initial environmental flow is laminar (i.e., does not contain appreciable turbulent structures that are resolved on the grid), common LES turbulent closure assumptions tend to overestimate the amount of shear in the surface layer. In their simulations where resolved turbulence was activated via the introduction of random perturbations to the initial state that were then recycled through the periodic boundaries, the near-surface shear was reduced, as was the fraction of TKE partitioned into the subgrid-scale parameterization. This issue potentially has implications for the interpretation of the effects of surface drag on a wide range of simulated storm behavior, and we refer the reader to Markowski and Bryan (2016) for details. For our purposes, when resolved turbulent structures are present in the prestorm boundary layer environment, the environmental wind profile is clearly no longer even approximately horizontally homogeneous. Yet, there still may be a desire to maintain the horizontal mean profile in the same way as when the prestorm environment is laminar. To demonstrate the utility of the GWB method for this situation, we perform and analyze two additional sets of experiments in which the boundary layer contains resolved turbulent structures.
We first discuss the NS-T- experiments–analogous to the NS-NT- experiments but including a field of resolved turbulent structures in the boundary layer. Our main purpose is to analyze the impact of resolved turbulence on the average environmental wind profile for the same experimental setup as the original NS-NT- experiments (i.e., without storm-induced perturbations). Our method for encouraging the development of turbulence is similar to the “point” method of Muñoz-Esparza et al. (2014). We apply random uniformly distributed potential temperature perturbations of ±0.25 K to the initial conditions, as well as every 200 s during the model integration to a 20-gridpoint-wide zone positioned 40 grid points from each inflow boundary. We apply the perturbations from the surface up to 1 km AGL, the approximate depth of the boundary layer in this case. This mechanism leads to well-developed turbulent structures in the boundary layer by approximately 1–2 h, which continue to grow in intensity until the end of the simulation (Fig. 5). That is, the turbulence never reaches a statistically steady state. We do not, however, consider this to be detrimental to our goal, which is to understand the impact on the wind profile in a qualitative sense. Future work may focus on developing a more robust methodology for generating a turbulent boundary layer in a statistically steady state with a given mean wind profile, perhaps by using an “eddy injection” method similar to Bryan et al. (2017). Because the resolved turbulence is changing in intensity with time, for the two experiments where we applied the GWB method—NS-D-GWBT- and NS-ND-GWBT—we update the PPGF profile every time step throughout the simulation, instead of only the first 300 s as in the NS-T- experiments. Again, we use the grid-area-weighted domain horizontal average of the horizontal momentum time tendencies to compute the PPGF profile, which effectively removes the effect of small-scale fluctuations owing to the turbulent eddies.
As in Fig. 3a, but for the no-storm simulations with resolved boundary layer turbulence (NS-T-).
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
In Fig. 6, we show vertical profiles valid at 2 h of the subgrid-scale versus resolved turbulence kinetic energy (TKE), horizontally averaged over the inner domain, for each corresponding pair of experiments without (NS-NT-) and with (NS-T-) resolved boundary layer (BL) turbulence (left and right columns, respectively). Resolved TKE was computed according to
Vertical profiles valid at 2 h of inner domain horizontal mean subgrid-scale (solid), resolved (dashed), and total (dotted) TKE for each of the (left) NS-NT- and (right) NS-T- simulations. Red markers indicate the same heights AGL as in Figs. 2 and 4.
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
As in Fig. 2, but for the no-storm simulations with turbulent boundary layer (NS-T-) runs.
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
When the GWB method is not active (i.e., in the GEO and WC experiments), the presence or absence of boundary layer turbulence has only modest effects on the overall evolution of the wind profile, despite the large differences in the vertical profiles of TKE between corresponding pairs (cf. left and right columns of Fig. 6). However, there is slightly more modification of the hodograph between ~500 and 1000 m in the non-GWB turbulent BL experiments (Figs. 7a,c,d) than in the corresponding nonturbulent BL experiments (Figs. 2a,c,d). This additional modification can be explained by the presence of substantial resolved TKE maximized near 500 m AGL in the turbulent-BL runs (Fig. 6 right column), but with the subgrid-scale TKE remaining nearly the same as in their nonturbulent BL counterparts (Fig. 6 left column).
When the GWB method is active and surface drag is present, the surface drag is offset by a strong PPGF (Fig. 8: red line near the surface). Above this shallow surface layer (or throughout the profile for the runs without surface drag; blue lines), the magnitude of the PPGF is much lower and its profile differs between the turbulent and nonturbulent BL simulations (Fig. 8a). In particular, the turbulent-BL simulations show less southward-directed (or more northward-directed) PPGF in the turbulent runs (dashed lines) than the nonturbulent runs (solid lines), indicating a tendency for the resolved turbulent mixing in this portion of the profile to weaken the meridional winds more than in the nonturbulent simulations (Fig. 8b). In the nonturbulent simulations, the PPGF and Coriolis are nearly balanced above approximately 200 m AGL (cf. solid lines with dotted black line in Fig. 8a). Between about 100–200 m, both subgrid-scale and resolved mixing are instantaneously acting to weaken the shear in this layer by increasing the meridional winds from 50 to 100 m and decreasing them above (Fig. 8b). The PPGF counters these tendencies in order to maintain the wind profile (and thus the shear) in this layer (Fig. 8a).
Vertical profiles of the meridional component of (a) the PPGF acceleration and (b) the frictional force for the four no-storm GWB experiments. Red and blue curves indicate the experiments with and without surface drag, respectively. Solid and dashed curves are for the experiments without and with resolved BL turbulence, respectively. For the resolved-BL-turbulence experiments, the PPGF profile is shown for a model time of 2 h. The profile of the average Coriolis force (which is nearly identical between the four experiments since it depends only on the nearly constant vertical wind profile) is also shown (blacked dotted line) for reference in (a). Note the symmetric logarithmic scale for the abscissa.
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
Otherwise, the fact that the differences are relatively minor between the turbulent and non-turbulent-BL simulations can be explained by the overall small forcing from the TKE (either resolved or subgrid) on the wind profile above the shallow surface layer and the fact that the surface layer is completely dominated by the subgrid-scale TKE, as seen in other LES-type simulations with similar resolutions (see Markowski and Bryan 2016, and references therein). We think it is plausible, however, that different wind profiles may exhibit somewhat different sensitivities, particularly when considering the effects of the thermodynamic profile. For example, turbulence would be enhanced for large virtual temperature lapse rates and suppressed for small lapse rates.
Finally, we also performed a series of storm experiments–analogous to the S-NT- experiments–but with a turbulent boundary layer (the S-T- experiments). The initialization of these simulations as well as the application of the GWB method is somewhat more complicated than in the corresponding S-NT- simulations. We desire to initiate storms in an environment where the boundary layer already has appreciable resolved turbulent structures. Therefore, we took the model output valid at 2 h from the no-storm GWB simulations (NS-T-D-GWB and NS-T-ND-GWB) and used them as the initial conditions for the appropriate sets of storm simulations (i.e., with and without surface drag, respectively). Since as already discussed the GWB method ensured that the average wind profile was preserved in NS-T-D-GWB and NS-T-ND-GWB, this means that the new set of S-T- simulations all start with the same average environmental wind profile as those storm simulations without resolved BL turbulence (i.e., the S-NT- simulations). This approach is very similar to that of Nowotarski et al. (2015) who likewise initialized their supercell storm simulations from a previous set of simulations of convective boundary layers with fully developed turbulence. In their simulations, however, it was difficult to maintain the average low-level vertical wind profile over the course of their storm simulations because of the lack of a large-scale PGF to balance the turbulent mixing (Nowotarski et al. 2014); the GWB method provides this as follows. Recall that the PPGF profile was continuously updated for the two no-storm GWB experiments (NS-T-D-GWB and NS-T-ND-GWB), such that the average wind profile remained constant even while the boundary layer turbulence continuously evolved (cf. Fig. 5). For the two GWB storm experiments (S-T-D-GWB and S-T-ND-GWB) we used the PPGF profile valid at 2 h in NS-T-D-GWB and NS-T-ND-GWB, respectively, and then held it fixed for the duration of the two storm simulations. This ensures that in the horizontal mean, the PPGF profile balances the frictional force profile at least at the initial time. While we could have continued to update the PPGF profile in the face of the changing horizontal average of the horizontal momentum time-tendencies, as in the NS-T- experiments, in this case we would have been also compensating for the average of the storm-induced perturbations to the wind profile, which is undesirable for the present purposes. We show the wind profile evolution for the S-T- experiments in Fig. 9. Here, as in the storm simulations without resolved turbulence (the S-NT- experiments), the profiles are taken as a horizontal average of a 20 × 20 km2 area at the southeast corner of the inner 250-m mesh, to represent the far-inflow region (the 20 × 20 km2 is sufficient to average out the effects of the resolved turbulence, not shown). The same basic evolution is seen when comparing the hodographs between pairs of experiments from the NS-T- (Fig. 7) and S-T- (Fig. 9) sets indicating that the presence of the storm does not substantially affect the far-inflow wind profile, and that the GWB method reliably maintains it.
As in Fig. 7, but for the storm simulations with turbulent boundary layer (S-T-) runs.
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
4. Impact of surface drag on simulated storm structure
Finally, motivated by the aforementioned recent work on this problem, we turn to a brief analysis of the impact of surface drag on simulated near-surface storm structure and behavior. As noted earlier, each simulation produced several intense tornado-strength vortices, with an overall increase in activity after ~2 h (cf. Fig. 3). As an initial foray, we wish to reveal any obvious differences in near-surface kinematic structure that persist over time across the simulations, since the near-surface wind field associated with the storm and attendant tornadoes would be expected to be most directly affected by the presence or absence of surface drag. To this end, we constructed temporal composites by tracking the location of maximum vertical vorticity within a 12 × 12 km2 moving box at 500 m AGL every 60 s during the second hour of each simulation and rejecting those times when the vertical vorticity decreased below 0.1 s−1. Additionally, we rejected those times when the position of the vortex deviated more than 3 km from its median position of all the previous times; we found this to be necessary to avoid centering on spurious nontornadic or vortices that had significant storm-relative rearward motion (i.e., “occluding” vortices). We then aligned the model fields in the horizontal at the individual times relative to the surface vortex location and performed a simple average across the times.
We show the resulting composites for surface horizontal wind speed (color fill), asymptotic contraction rate (ACR; green contours),1 and surface simulated radar reflectivity (dBZ; black contours) in Fig. 10. While we performed these composites for both the S-NT- and S-T- sets of experiments, the results were not substantially different in a qualitative sense, and thus we show only the S-NT- results for the sake of brevity. The presence or absence of surface drag profoundly alters the near-surface kinematic structure of the storm. First, the two experiments without surface drag (S-NT-ND-GEO and S-NT-ND-GWB; Figs. 10d and 10e) exhibit a large area of strong surface winds >20 m s−1 up to 10 km from the vortex. In contrast, the three experiments with surface drag (S-NT-D-GEO, S-NT-D-GWB, and S-NT-D-WC; Figs. 10a–c) all show the strongest winds in a much smaller area close to the simulated tornado.
Temporal composites (see text for details on the construction of the composites) centered on the location of maximum surface vorticity for each of the S-NT- runs. Shown in each composite is ground-relative surface wind speed (HWS; color fill, m s−1), surface asymptotic contraction rate (ACR; green contour; 0.0075 s−1 shown), and radar reflectivity (black contours, 10 dBZ increment, starting at 20 dBZ).
Citation: Monthly Weather Review 147, 11; 10.1175/MWR-D-18-0462.1
The two simulations without surface drag also exhibit a much more prominent rear-flank gust front (RFGF) as well as a prominent boundary extending off to the northeast of the surface vortex into the forward flank of the storm that resembles the “left flank convergence boundary” (LFCB) identified in the simulations of Beck and Weiss (2013). In contrast, the simulations with surface drag have a less prominent boundary in the mean in the forward flank region. Moreover, the structure of the RFGF and its orientation relative to the surface vortex is quite different between the simulations with and without surface drag. In the no-drag simulations, the RFGF appears as a single entity (at least in the mean, as defined by the compositing procedure) that appears nearly contiguous with the LFCB/FFGF, both having a roughly SSW to NNE orientation (green contours in Figs. 10d and 10e). The simulated tornado in these simulations is located very near the intersection of these two boundaries with less arcing of the RFGF. This structure is qualitatively similar to that seen in several other published simulations of supercells that used free-slip lower boundary conditions (e.g., Adlerman and Droegemeier 2002; Dahl et al. 2012; Beck and Weiss 2013). In contrast, in each of the three drag simulations, the RFGF forms a tightening cyclonic arc or spiral through at least 180°, that wraps around the simulated tornado such that the latter is “tucked” into the tighter, western portion of the spiral or even kinematically “detached” from it (green contours in Figs. 10a–c). Except for perhaps S-NT-D-GWB, there is little evidence for a prominent forward flank boundary in the mean, although inspection of individual times (especially later in the simulation) indicates that the drag simulations do form several transient forward flank boundaries (not shown). This general forward- and rear-flank boundary structure is consistent with that found in several recent published simulations that included surface drag parameterizations (e.g., Schenkman et al. 2014, 2016; Dawson et al. 2015; Roberts et al. 2016; Coffer and Parker 2017; Coffer et al. 2017) and in observation-based studies (e.g., Skinner et al. 2011; Lee et al. 2012; Marquis et al. 2012; Kosiba et al. 2013; Skinner et al. 2014). Additionally, the drag experiments all show another persistent boundary trailing SSW from the vortex itself. This boundary is roughly parallel with the eastern portion of the RFGF and separated from it by ~5 km, and bears striking resemblance to the so-called secondary rear flank gust front or internal momentum surge identified in several recent modeling and observational analysis studies of tornadic supercells (e.g., Skinner et al. 2011; Lee et al. 2012; Schenkman et al. 2016; Marquis et al. 2012; Kosiba et al. 2013; Skinner et al. 2014). While the no-drag simulations also exhibit these secondary gust fronts, they are much more transient and thus do not show up in the composite fields.
Figure 10 suggests that the overall kinematic structure of the storm at the surface, at least during times when the surface vortex is most intense, is not appreciably different between the GWB/WC and non-GWB simulations. However, the two experiments that used a PPGF that offset surface drag (S-NT-D-GWB and S-NT-D-WC) have stronger southeasterly flow in the inflow region and weaker northwesterly flow in the outflow region north of the surface vortex than is the case in the experiment that included both surface drag and assumed initial geostrophic balance via (2) (S-NT-D-GEO). This can be seen by comparing wind vectors in the corresponding regions in Figs. 10b and 10c with Fig. 10a). The weaker southeasterly inflow into the storm in S-NT-D-GEO is clearly due to the surface drag weakening the near-surface flow (cf. hodograph Fig. 4a). The stronger environmental or background southeasterly flow in S-NT-D-GWB and S-NT-D-WC apparently cancel out some of the northwesterly storm-induced outflow northwest of the low-level vortex center to produce weaker total ground relative winds there (cf. wind vectors in the upper-left quadrant of Figs. 10b and 10c with Fig. 10a). Additionally, the winds in the composite vortex in S-NT-D-WC (color filled region in Fig. 10c) are noticeably weaker than in the other two drag simulations (Figs. 10a,b); this is also evident in the time series of maximum horizontal wind speed (Fig. 3, blue line). The reason for this difference is unclear, but may be related to the somewhat weaker shear that develops with time in the lowest ~500 m in the far-field inflow wind profile for this experiment (Fig. 4c, circle and × markers) as compared to S-NT-D-GEO and S-NT-D-GWB (Figs. 4a and 4b, respectively). A complete investigation into the reasons for these differences and a thorough comparison with previous simulations and observations is beyond the scope of this study but will be pursued in future work.
5. Summary and discussion
We have developed a new method to compute a large-scale pressure gradient for use within atmospheric simulation models with idealized background environments. The method is particularly useful for idealized simulations of convection and other phenomena where it is desirable to precisely control and maintain the large-scale kinematic environment while simultaneously including the effects of friction. A prime example are high-resolution supercell tornado simulations initialized from a single environmental sounding where satisfying both of these conditions has historically proven difficult. This difficulty arises due to the tendency for surface friction in particular to modify the low-level environmental wind profile over time if the large-scale PGF is not calculated appropriately to account for the frictional force in the large-scale force balance. The primary appeal of the new method is its applicability to an arbitrary wind profile coupled with any given surface drag and turbulence formulation, including in the presence of resolved turbulent structures that evolve over time. The method is conceptually simple and easy to implement in existing models, and works by determining the horizontal force as a function of height that is needed to cancel out the horizontal frictional and Coriolis forces. The method estimates this force profile as the negation of the time tendency of the horizontal momentum equations early in the simulation at each grid level in a suitable (i.e., perturbation free) column (or average of columns). This vertical force profile is then included as an additional term in the horizontal momentum equations for the duration of the simulation.
We demonstrated the method by applying it to a series of idealized simulations of a supercell using a sounding representative of the inflow environment of the storms in the 24 May 2011 Oklahoma tornado outbreak. With the Geotriptic Wind Balance method, the far-field low-level wind profile remains essentially unchanged throughout the 3–4 h model integration, even with the presence of surface drag and/or resolved turbulent eddies. The simulations with resolved boundary layer turbulent eddies were (somewhat surprisingly) very similar in their overall behavior with their nonturbulent counterparts, though subtle differences in the evolution of the wind profiles did exist. Namely, there was a tendency for slightly more modification of the wind profile to occur over time when comparing corresponding experiments in the turbulent versus the nonturbulent boundary layer sets in which the GWB method was not employed. In ongoing work we are investigating the sensitivity of simulated supercells and their attendant tornadoes to the presence of surface drag and resolved versus subgrid-scale turbulence; as part of this effort we will continue to assess the utility of the GWB method for other wind profiles and higher grid resolutions in which turbulent structures would be better resolved. Results of these investigations will be reported in a future paper.
While in this study we focused on the goal of controlling (via maintaining) a fixed background wind profile, the emphasis should be on the word “control” rather than “fixed”. The GWB method can easily be generalized to control other aspects of the environmental evolution, and has similarity to some methods found in the published literature. Nolan and Rappin (2008) applied a forcing term to the model horizontal momentum equations to maintain a time-invariant background vertical wind profile in order to investigate the effects of vertical wind shear in idealized simulations of tropical cyclones. Bryan et al. (2017) applied a similar method to maintain a time-invariant background flow that was in gradient wind balance in limited-area LES experiments to explore the wind profile in the tropical cyclone boundary layer. That is, they explicitly computed the required PPGF that would satisfy gradient wind balance rather than geostrophic as is more commonly used, and added this term to the model momentum equations in much the same manner as in the present study.
Other researchers have sought to examine in a controlled manner the impact of a changing background wind profile on simulated convective storm evolution. Letkewicz et al. (2013) used a method they dubbed “base-state substitution” where a simulation was performed with one kinematic and thermodynamic profile defining the horizontally homogeneous base state, which was then replaced with another at some time during the simulation. A limitation of this approach is seemingly the abrupt nature in which the background profile changes. Using an approach more similar to our own, Kost and Richardson (2004) applied a time-varying force profile to the momentum equations that slowly modified the background wind profile from one state to another with stronger wind shear. Our method can easily be modified to perform this function while simultaneously including the effects of surface friction. Another potential application in this vein would be for studying the effects of changing surface roughness within a single simulation, by interpolating over a short period the PPGF profiles that balance the frictional forces associated with two or more different values of the drag coefficient that are “activated” at specific times in the simulation. Furthermore, our method is not limited to convective storm or tropical cyclone simulations; it can be applied to, for example, idealized orographic flow simulations that include surface frictional effects. The same procedure can be used to help maintain the upstream wind profile that eventually flows over the mountain.
Acknowledgments
The authors wish to thank Dr. Chris Nowotarski and two other anonymous reviewers for their comments, which greatly improved the quality of the manuscript. We are grateful for the many fruitful discussions with colleagues regarding this work, including Drs. Johannes Dahl, Paul Markowski, Yvette Richardson, Lou Wicker, Ted Mansell, George Bryan, Brice Coffer, Patrick Skinner, Robin Tanamachi, and Corey Potvin, among many others. We also acknowledge Dr. George Bryan for his continued support of the CM1 model. Most of the postprocessing and plotting of the simulation output was carried out using the Python programming language and leveraged numerous open-source modules, most notably the MetPy (May et al. 2017) and matplotlib (Hunter 2007) packages. Support for this work came from the first author’s institution in the form of start-up funding.
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This purely kinematic quantity is a measure of the long-term rate at which adjacent air parcels approach each other given a steady kinematic flow field, and is more useful for identifying airstream boundaries than simple divergence (Cohen and Schultz 2005; Betten et al. 2018).
