1. Introduction
Tropical cyclone (TC) size is an important forecast metric as it directly and indirectly influences TC motion, intensity, track, and storm surge (e.g., Fiorino and Elsberry 1989; Fovell and Su 2007; Lin and Chavas 2012; Carrasco et al. 2014). There are a variety of metrics used to define the TC size, including the radius of the outermost closed isobar, the radius of vanishing wind, and the radius of 34-kt (about 17.5 m s−1) wind speed R34. In this study, R34 at 10 m above mean sea level (MSL) is used to define the storm size or width.
Bu et al. (2014) demonstrated that cloud-radiative forcing (CRF), the interaction of hydrometeors with longwave and shortwave radiation, has an important role in expanding the storm size. Averaged through a diurnal cycle, CRF consists of pronounced cooling along the anvil top and weak warming through the cloudy air. In particular, the within-cloud warming was relevant, enhancing convective activity in the TC outer core, leading to a wider eye, a broader tangential wind field, and a stronger secondary circulation. This forcing also functions as a positive feedback (Fovell et al. 2016), assisting in the development of a thicker and more radially extensive anvil than would otherwise have formed. CRF itself depends on the microphysics parameterization and Fovell et al. (2010) showed it is a major reason why simulations can be sensitive to microphysical assumptions.
Bu et al. (2014) also demonstrated that the GFDL-derived radiation scheme that was long employed operationally in the Hurricane Weather Research and Forecasting Model (HWRF) (cf. Tallapragada et al. 2014) did not handle CRF properly, resulting in deep clouds that were effectively transparent. Testing revealed, however, that implementing an ostensibly superior radiation scheme degraded model skill (L. Bernardet et al. 2014, personal communication). Analysis of those results led us to consider how the planetary boundary layer (PBL) influences storm size, in cooperation and competition with CRF, which is the subject of this study.
It is widely appreciated that boundary layer processes play an important role in TCs (e.g., Smith 1968; Ooyama 1969; Emanuel 1986; Van Sang et al. 2008). Among these processes are mixing acting on momentum and scalars such as temperature and moisture, the subgrid portion of which can be represented via diffusion coefficients 
There have been many papers addressing the sensitivity of simulated TCs to PBL parameterizations and assumptions (e.g., Braun and Tao 2000; Hill and Lackmann 2009; Nolan et al. 2009a,b; Smith and Thomsen 2010; Kepert 2012). Most previous studies of the PBL–TC relationship have focused on TC intensity, inner-core convection, and/or the TC PBL structure. A few studies, however, have explicitly examined the influence of the vertical diffusion on the storm structure, most of them reporting no significant influence. For example, Bryan (2012) found the 34-kt-wind radius to be only weakly sensitive (and the radius of maximum wind, or RMW, to be insensitive) to lυ, at least when reasonable values of the horizontal mixing length lh and the enthalpy–drag coefficient ratio 
Gopalakrishnan et al. (2013) demonstrated that eddy mixing strongly influences the intensity and depth of the TC low-level inflow and the GFS PBL parameterization was producing excessively large 
In this study, we focus on uncovering how and why the PBL vertical mixing impacts horizontal TC structure and size. The model and experimental design are discussed in section 2. Section 3 demonstrates how and why CRF and PBL mixing cooperate, and compete, to influence TC size. Section 4 presents the summary discussion.
2. Model and experimental design
The HWRF simulations in this study were carried out using the 2014 operational code. These experiments are “semi-idealized” in that we simplified the operational configuration by excluding land and decoupling the ocean model, employing a uniform and constant sea surface temperature (SST) of 302.5 K for the standard runs, and initializing with a horizontally homogeneous tropical sounding [modified from Jordan (1958); see Fovell et al. (2010)] without any mean flow. The Cao et al. (2011) “bubble” procedure was used to initiate the TC and all simulations spanned 4 full days with composite model fields being constructed for the final day in a vortex-following fashion, averaging over one full diurnal cycle. While changes in operational settings (primarily with respect to horizontal smoothing) from the 2013 version used by Bu et al. (2014) resulted in somewhat weaker storms for the same experimental design, all of the standard SST HWRF TCs attained major hurricane status (at least category 3 on the Saffir–Simpson scale) for the analysis period. Please note that most figures employ azimuthal averaging, thereby understating the maximum intensity of these asymmetric, beta-sheared storms (cf. Bender 1997; Bu et al. 2014).
As in Bu et al. (2014), our simulations employed three telescoping domains (with 27-, 9-, and 3-km horizontal grid spacings) along with some of the model physics used operationally during the 2014 season, such as the simplified Arakawa–Schubert (SAS) cumulus parameterization (remaining active in the 27- and 9-km domains after 24 h). In 2014, the operational configuration (cf. Tallapragada et al. 2014) also included the GFDL radiation scheme, the GFS PBL, and the tropical Ferrier microphysics parameterization (MP). These are compared to (or replaced by) RRTMG radiation (Iacono et al. 2008), the YSU PBL (Hong 2010), and Thompson MP (Thompson et al. 2008), respectively. Bu et al. (2014) showed that while RRTMG and GFDL generate nearly identical clear-sky radiative forcing profiles owing to longwave and shortwave radiation (see their Fig. 7a), CRF was not properly handled in the HWRF implementation of the latter. Therefore, owing to their strong similarity with RRTMG cases in which cloud-radiative forcing is deactivated (cf. Bu et al. 2014), HWRF simulations with GFDL radiation are also labeled “CRF-off” herein. Our work has shown that storm structure is significantly modulated by microphysical assumptions (cf. Fovell et al. 2016), but for simplicity we will focus solely on the Thompson scheme.
Also following Bu et al. (2014) we employ an axisymmetric version of Cloud Model 1 (CM1) (Bryan and Rotunno 2009) initialized with the Rotunno and Emanuel (1987) sounding. These simulations used 3-km radial grid spacing and 100-m resolution in the vertical (below 4 km MSL), were initialized with a weak vortex, and were integrated for 12 full days. As in Bu et al. (2014), Goddard radiation (Chou and Suarez 1994) and a version of Thompson microphysics were used for most experiments and the latitude was 20°N. Unless otherwise noted, the SST was 299 K as in Bryan and Rotunno (2009), the lower SST being motivated by this sounding’s cooler surface air temperature [see also Bryan (2012)]. All CM1 fields shown are averaged between days 10 and 12, inclusive, except in section 3d, in which an 8–12-day averaging interval was adopted for consistency with Bryan and Rotunno (2009).
3. Results
a. PBL cooperation with CRF
As reviewed above, Bu et al. (2014) demonstrated that CRF plays an important role in determining TC structure. This can be seen in HWRF simulations made using Thompson (“T”) microphysics and the GFS PBL scheme with either RRTMG (labeled CRF-on) or GFDL radiation (labeled CRF-off) for α = 0.7 and 0.25 in (1), representing the 2014 operational model setting and the recommendation of Gopalakrishnan et al. (2013), respectively. Enabling CRF can increase the storm size (as manifested by the 10-m R34) by a substantial (and MP-dependent) amount (cf. solid and dashed red, or solid and dashed blue contours in Fig. 1) because hydrometers interact with radiation to force gentle ascent, elevating the relative humidity through a deep layer mainly above the PBL, resulting in enhanced convective activity in the TC outer core. Although some details (including magnitude) are dependent on microphysics, resolution, and other factors, the expected pattern of net cooling along cloud top with warming through much of the cloudy area is seen in the T/RRTMG simulation (Fig. 2a), but not in its T/GFDL counterpart (Fig. 2b). The anvil cloud in the case with effective CRF is thicker and also wider, in part because cloud-radiative forcing itself acts as a positive feedback on anvil extent, as demonstrated in Fovell et al. (2016, see their Fig. 11.20).
Radial profiles of temporally and azimuthally averaged 10-m wind speed from semi-idealized HWRF simulations using the GFS PBL and Thompson (“T”) MP varying the α parameter (0.7 in red, 0.25 in blue) and cloud-radiative forcing (CRF-on solid, CRF-off dashed). The CRF-on and CRF-off simulations utilized RRTMG and GFDL radiation, respectively; see text. Gray dashed line indicates 34-kt (17.5 m s−1) speed threshold R34.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Temporally and azimuthally averaged total condensation (shaded, note logarithmic scale) and net radiation [negative (dashed) and positive (solid), contour interval 0.1 K h−1] for Thompson/GFS storms using α = 0.7 with (a) RRTMG (labeled CRF-on) and (b) GFDL (labeled CRF-off) radiation.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
After we identified the GFDL scheme’s lack of cloud-radiative forcing for deep clouds, the Developmental Testbed Center (DTC) and the HWRF team evaluated the RRTMG scheme along with the Thompson MP for adoption in the operational HWRF (see introduction). Their analyses of retrospective simulations demonstrated that the HWRF forecast skill was generally degraded when the new physics was included and, as a consequence, neither package was adopted for the 2014 TC season. The T/RRTMG model storms developed a positive size bias, which was especially pronounced among the Atlantic cases. In the east Pacific subset, the T/RRTMG cases tended to exhibit positive biases early on, but encountered colder SSTs sooner, resulting in negative size biases at longer forecast lead times.
Our working hypothesis was that excessive mixing associated with the GFS scheme with 
Consistent with this interpretation, Fig. 1 reveals that CRF and α have qualitatively similar influences on horizontal storm size. Note that varying α (for fixed CRF) causes the 34-kt-wind radius to increase significantly independent of the radiation scheme employed (cf. red and blue contour pairs). For instance, with GFDL radiation, increasing α from 0.25 (blue dashed) to 0.7 (red dashed) shifts R34 from 90 to 150 km. The narrowest storm used the α = 0.25 value suggested by Gopalakrishnan et al. (2013) with GFDL radiation, while the widest employed the 2014/15 operational setting (0.7) with RRTMG. Thus, it is seen that the physics interplay between CRF and mixing can alter the 34-kt-wind radius by factor of 2, and there is a material impact on the eye size as well.
We note that the range of R34 found in the experiments above (e.g., 90–205 km) is consistent with observations of TC size derived from ships, buoys, aircraft reconnaissance, and satellite-derived algorithms [see review in Knaff et al. (2016)]. These include an interquartile range of 138–277 km in the Atlantic basin extended best-track (Kimball and Mulekar 2004), 1.8°-mean (1° standard deviation) satellite-derived 34-kt radius of Wu et al. (2015) in the western North Pacific TCs, and range of 90–300 km in Atlantic basin storms with concurrent Hurricane Wind Analysis System (H*Wind) and QuikSCAT data (Chavas et al. 2015).
b. Vertical eddy mixing influence on storm size
1) Sensitivity to α
The expansion of the 34-kt-wind radius seen in Fig. 1 occurs because the PBL mixing acts in a very similar manner as CRF in expanding storm size, as illustrated by the temporally and spatially averaged microphysical diabatic tendency and tangential wind fields shown in Fig. 3. Implementing CRF for fixed α (right column) and varying α with CRF active (left column) both result in a radially more extended heating field, causing the wind field (as illustrated by the tangential wind differences in the bottom row) to expand outward in qualitatively similar manners, for the reasons discussed in Bu et al. (2014). The impact on the eye size is also obvious in the difference plots. Note that while the GFDL/α = 0.7 and RRTMG/α = 0.25 runs possessed nearly identical 10-m-wind profiles (Fig. 1), their tangential winds differed more substantially aloft. We are focusing on the 10-m winds because these are used in skill assessments. However, these results serve as a reminder that the near-surface winds alone may not be sufficient to accurately determine actual storm size.
Radius vs height cross sections showing the temporally averaged symmetric components of microphysics diabatic forcing (shaded) and tangential wind (10 m s−1 contours) from Thompson/RRTMG simulations using the GFS PBL with (a) α = 0.7 and (b) α = 0.25. (c) The α = 0.7 minus α = 0.25 difference fields; the superposed field is tangential velocity difference (1 m s−1 contours). Also shown are Thompson/GFS simulations with α = 0.7 for the (d) RRTMG radiation and (e) GFDL radiation cases. (f) The RRTMG minus GFDL difference fields. To facilitate comparisons, (a) and (d) are identical.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
As noted above, the α parameter was added to (1) to control eddy mixing in the TC inner core with the GFS PBL scheme. Figure 4a shows vertical profiles of 
Profiles of vertical eddy diffusivity (m2 s−1) averaged in time and through an annulus extending from 30 to 200 km from the storm center, for T/RRTMG simulations. (a) Momentum and scalar diffusivity (Km = Kh) from runs with α = 1.0, 0.7, 0.4, and 0.25. (b) As in (a), but showing scalar eddy diffusivity Kh from YSU simulations using the default and modified Ribcr values superimposed.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
As in Fig. 1, but for (a) GFS PBL simulations using various values of α; (b) as in (a), but after nondimensionalization with respect to RMW and maximum velocity; and (c) as in (a), but with default and Ribcr = 0.25 YSU simulations superimposed.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Similar to Fig. 3, but showing water vapor (shaded) and eddy diffusivity applied to vapor Kh (10 m2 s−1 contours), for HWRF Thompson simulations using the GFS PBL. (a) α = 0.7, (b) α = 0.25, and (c) difference fields. (d) CRF-on, (e) CRF-off, and (f) difference fields. Note that, to facilitate comparisons, (a) and (d) are identical.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
In the atmosphere, the water vapor concentration decreases quasi linearly with height and, as a consequence of the parabolic vertical shape of 
Figures 6d–f extend the comparison to two simulations varying CRF for fixed α and illustrate two important points. First, CRF itself induces a change in the PBL mixing. This is not surprising as the parameterized mixing responds to the circulation changes induced by cloud–radiation interaction. Second, the effect of altered mixing on the vapor field in this experiment is dominated by the CRF influence, which is sizable and not confined to the boundary layer. As a consequence, we will explicitly control the PBL mixing in some subsequent sensitivity tests in order to separate these two effects.
2) Influence of SST on α sensitivity
Examination of DTC’s HWRF retrospective cases from their initial Thompson and RRTMG tests described above suggested to us that the impact of α could vary from case to case, and even from region to region, with some TCs being quite insensitive to the value employed. From these cases, we surmised that the less convectively favorable the environment, the less influence eddy mixing of moisture would, or could, have. Within the semi-idealized framework, we can establish a more unfavorable environment by simply lowering the SST from its standard value of 302.5 K. In this subsection, we explore how SST modulates the impact of α on the storm size, selecting values of 300 and 298 K to examine.
Colder SSTs result in smaller and weaker storms, other factors being equal (Fig. 7), consistent with Holland (1997), Lin et al. (2015), and Chavas et al. (2016), and the disparity between the larger and smaller α diminishes as well. While the water vapor and 
As in Fig. 1, but for T/RRTMG/GFS simulations with α = 0.7 (black) and α = 0.25 (red) for SSTs of (a) 302.5, (b) 300, and (c) 298 K. Black dots indicate 34-kt-wind radii.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
As in Fig. 6c, but showing vapor and Kh difference fields between α = 0.7 and α = 0.25 simulations with SSTs of (a) 300 and (b) 298 K.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
In this experiment, there are two convolved factors: the diminished entropy supply from the sea surface directly reduces the storm intensity but also indirectly decreases the eddy mixing since 
Figure 9 shows temporally averaged 10-m-wind profiles for four different SSTs for these “fixed K” simulations using both the original and decreased amounts of mixing. For the latter, 
The 10-m wind speeds (m s−1) vs radius averaged between days 8 and 12 for “fixed K” CM1 axisymmetric simulations imposed with Km and Kh profiles from HWRF’s α = 1 control run (black) and with the same profiles divided by three (red) for simulations with SSTs of (a) 300, (b) 299, (c) 297, and (d) 295 K. Dashed curves indicate simulations in which either Km or Kh is reduced. Black dots indicate 34-kt-wind radii; in (a) the control run’s 34-kt-wind radius is beyond the range depicted.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Thus, it appears that TC size can be directly modulated via water vapor transport in the boundary layer, with the sensitivity to α values gradually disappearing as the entropy supply from the sea surface declines. This reveals that the inclusion of lower-sensitivity cases could serve to partially obscure the influence of PBL mixing in ensemble statistics incorporating a large number of events.
We note in passing that in both the HWRF and CM1 mixing experiments (Figs. 7 and 9), the intensity difference between greater and lesser mixing is not a simple function of SST. TC intensity is a complex function of many factors, including available energy from the sea surface as well as the competition between inner- and outer-core convective activity (e.g., May and Holland 1999; Wang 2009; DeMaria et al. 2012). Although decreasing the water vapor diffusion through the whole domain may suppress the convection in the eyewall region somewhat, the outer convection may be reduced even more. As a consequence, the net influence may be to actually intensify the TC; this deserves further study.
3) Influence of α on scalars and momentum
We have shown that the unmodified GFS PBL parameterization produces vigorous mixing and reducing this via α results in the model storms becoming smaller. Since 
The experiment reveals that the mixing applied to scalars (being water vapor, temperature, and nonprecipitating condensate for GFS) has a greater influence on storm size than that applied to momentum. Reducing the mixing applied to these fields by two-thirds (without explicitly modifying 
(a) Temporally and azimuthally averaged 10-m wind speeds (m s−1) from HWRF T/RRTMG/GFS simulations using α = 1 with vertical eddy diffusivity unmodified (control; black), reduced momentum mixing (Km/3; red), and reduced scalar mixing (Km/3) applied either to all applicable scalars (green) or just to water vapor (blue). (b) Vertical profiles of wind speed (m s−1) averaged in time and through an annulus extending from 100 to 250 km from the storm center for cases as in (a).
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Microphysics diabatic heating and tangential winds as in Fig. 3, but for HWRF T/RRTMG/GFS α = 1 runs using (a) Kh/3 (applied to all scalars), (b) unmodified eddy mixing (control), and (c) Km/3. See also Fig. 10.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Whatever the explanation, it is clear that though the vertical shear clearly varies inversely with α, the normalized wind profiles from the GFS 
Vertical profiles of the horizontal wind, averaged in time and through an annulus extending from 100 to 250 km from the storm center, and normalized with respect to the wind speed at about 300 m MSL, from GFS runs with α = 1, 0.75, 0.5, and 0.25; the Km/3 and Kh/3 (all scalars) tests; and simulations with the YSU, MYJ, and QNSE PBL schemes. Black squares on the GFS α =1 profile indicate model levels. The Kh/3 (vapor only) case is indistinguishable from the Kh/3 (all scalars) profile shown.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Since modifying the mixing applied to some fields can alter the entire circulation, the diffusion not being directly manipulated is also affected to some degree. As a consequence, we also consider a version of the last subsection’s fixed-K CM1 experiment in which 
c. Comparison with YSU scheme
Since its inception, HWRF has used some version of the GFS PBL scheme, while the YSU parameterization is a popular choice with the Advanced Research version of WRF (ARW) core. As these nonlocal schemes evolved from a common ancestor, they unsurprisingly retain many similarities, including the same prescribed parabolic shape function for eddy mixing below the boundary layer depth h. In our tests, however, YSU tends to produce shallower boundary layers that, owing to (1), make the eddy mixing magnitudes smaller and shift the level of maximum mixing closer to the surface (solid black curve on Fig. 4b). One can now anticipate this has an impact on vertical moisture transport and, thus, storm radial extent. At 10-m MSL, the YSU storm’s R34 is about 182 km, comparable to the GFS simulation with α = 0.4 (Fig. 5c).
There are several potentially influential differences between the current YSU and GFS implementations, including their handling of the turbulent Prandtl number (as noted earlier) and the free atmosphere above the PBL, as well as the present need to employ different surface-layer parameterizations. However, in the present study, by far the most important factor involves the specification of the critical bulk-Richardson number, Ribcr, which influences the PBL height h, with larger Ribcr resulting in greater boundary layer depths. In the original MRF scheme (Hong and Pan 1996), the Ribcr value of 0.5 suggested by Troen and Mahrt (1986) was adopted, while recent practice with the GFS scheme in HWRF has been to set Ribcr = 0.25 over water, with optional modification based on the surface Rossby number (cf. Vickers and Mahrt 2004). Although originally set to 0.5 (Hong et al. 2004), YSU evolved to employ different Ribcr values for unstable and stable conditions, and currently uses Ribcr = 0.0 for the unstable PBL (Hong 2010).
YSU’s stability-dependent handling of Ribcr results in the relatively shallow boundary layer depth seen in Fig. 4b. When the default YSU scheme is altered to adopt an Ribcr of 0.25, as in the current GFS parameterization, the fields and storm structures more closely resemble the GFS results seen earlier (Figs. 4b and 5c). There is more substantial mixing over a greater depth (cf. Figs. 13a,b), producing a larger vertical transport of water vapor above about 700 m MSL (Fig. 13c).3 Averaged through the 30–200-km annulus, the modified YSU mixing (dashed black curve on Fig. 4b) closely resembles that produced by the GFS scheme when α = 0.7. The wind field at 10-m MSL is also expanded somewhat (Fig. 5c) and comparable to GFS with α = 0.7.
Water vapor and scalar mixing fields similar to Fig. 6, but for T/RRTMG simulations using YSU with (a) Ribcr = 0.25 and (b) the default setup. (c) Difference fields.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
As the GFS and YSU parameterizations possess other differences that can impact the hurricane circulation, we consider yet another fixed-K experiment in which the effect of mixing depth is explored (Fig. 14) in a more controlled fashion. For this experiment, the GFS-supplied 
Temporally averaged 10-m winds similar to Fig. 9, but for CM1 simulations using HWRF-derived vertical eddy mixing profiles with vertical extents that have been unmodified (control; solid black), halved for both Km and Kh (solid red), halved for Km only (dashed black), and halved for Kh only (dashed red). The 34-kt-wind threshold is indicated (dashed gray).
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Up to this point, we have focused on azimuthally averaged fields, partly for simplicity. However, this disguises the differences among the storms with respect to asymmetric structures, especially beyond the TC inner core. Figure 15 presents mass-weighted mean vertical velocity between the surface and 500 hPa from the HWRF simulations, again temporally averaged over the simulations’ final diurnal cycle. The YSU storm (Fig. 15a) is compact, with a narrow and asymmetric eye, and relatively little outer-rainband activity. Raising Ribcr to 0.25 (Fig. 15b) results in an enhanced primary rainband structure (cf. Houze 2010), more closely resembling what the GFS scheme produces with α = 0.4 (Fig. 15c). This feature is most prominent when the GFS eddy mixing is even less constrained (Fig. 15d). As these are semi-idealized experiments, there is no correct answer, but it remains that PBL mixing is clearly influential in modulating outer-storm structure.
Mass-weighted mean vertical velocity (m s−1) from HWRF T/RRTMG simulations using (a) the default YSU scheme with Ribcr = 0.0, (b) the YSU scheme with Ribcr = 0.25, (c) the GFS scheme with (c) α = 0.4, and (d) the GFS scheme with α = 0.7. Range rings of 50 and 150 km are depicted, and tops of plots represent north.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
d. Comparison with selected axisymmetric studies
In contrast to our study, the axisymmetric studies of Bryan and Rotunno (2009), Bryan (2012), Chavas and Emanuel (2014), and Frisius (2015) reported little sensitivity of TC size to vertical mixing, which can be manipulated via the vertical mixing length 
In this subsection, we employ axisymmetric CM1 simulations configured similarly to Bryan and Rotunno (2009), albeit with somewhat coarser (3 km) radial grid spacing and Thompson microphysics. Vertical mixing lengths of 100 and 25 m are examined with 
Owing to friction, a storm’s fastest winds are typically located hundreds of meters above the surface, and both R-E relaxation cases attain temporally averaged peak tangential velocities [computed in the manner of Bryan and Rotunno (2009)] of about 90 m s−1 (not shown), consistent with the values provided in Fig. 2 of Bryan and Rotunno (2009). At 10 m MSL, however, both are quite compact relative to the HWRF simulations examined previously, with R34 of only about 65 km (blue curves in Fig. 16a). The narrowness persists through the 12-day simulation period following maturity (Fig. 17a) and extends through the troposphere, associated with a very clear absence of outer-convective activity (shown for lυ = 100 m in Fig. 18a). The vertical eddy mixing field (shown for lυ = 100 m in Fig. 19b) and applied equally to momentum and scalars bears some resemblance to the HWRF runs using GFS (Fig. 6a) in that both have a parabolic vertical shape with a maximum at about 0.5–0.6 km MSL beyond the RMW. While fairly large values of mixing extend vertically into the eyewall, a characteristic explained by Kepert (2012), note the mixing strength tapers off much more rapidly in the radial direction.
The 10-m wind speed from CM1 simulations, averaged between days 8 and 12. (a) Simulations using R-E relaxation (blue curves) and Goddard radiation with CRF on (black curves) with lυ = 100 (solid) and 50 m (dashed). (b) Simulations using Goddard radiation with CRF off and lυ = 100 m (black curve) and a fixed-K run using the R-E (lυ = 100 m) simulation’s eddy mixing field (red curve). The lυ = 100-m profiles from (a) are included in gray for reference.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Hovmöller (time vs radius) diagrams of tangential wind speed at 10 m MSL, with 20 m s−1 contour bolded, for CM1 simulations with lυ = 100 m and using (a) R-E relaxation, (b) Goddard radiation with CRF on, (c) Goddard radiation with CRF off, and (d) Goddard with CRF on, but with fixed eddy mixing from the R-E simulation in (a). In (a) and (b), 20 m s−1 contours from corresponding simulations using lυ = 25 m are shown in green. In (d), 20 m s−1 contours for CRF-off version shown in green.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
Radius vs height cross sections showing the temporally averaged microphysics diabatic forcing (shaded) and tangential wind (m s−1) from CM1 simulations using (a) R-E relaxation, (b) Goddard radiation with CRF on, (c) Goddard radiation with CRF off, and (d) Goddard with CRF on, but with fixed eddy mixing from the R-E simulation in (a). The 20 m s−1 contours are bolded. All simulations used lυ = 100 m and were averaged between days 8 and 12.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
As in Fig. 6, but for CM1 simulations using (a) Goddard with CRF on and (b) R-E relaxation. (c) The difference fields. Both simulations used lυ = 100 m and were averaged between days 8 and 12.
Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0231.1
In contrast, the CM1 simulations employing Goddard radiation (with CRF active) yield much wider storms (black curves in Fig. 16a), which are more comparable to the HWRF simulations and expand progressively with time (Fig. 17b). These results are consistent with the findings of Hakim (2011). There is a suggestion of more outer convective activity (shown for lυ = 100 m in Fig. 18b) and greater sensitivity to the vertical mixing length (Figs. 16a and 17b). In line with the broader and stronger storm circulation, the vertical eddy mixing field is larger in both magnitude and radial extent (Fig. 19a). As anticipated from prior results, the more vigorous mixing contributes to increasing the water vapor content in the upper portion of the boundary layer relative to the R-E case (Fig. 19c).
The three principal differences between the R-E relaxation and Goddard/CRF runs for a given 
Second, the effect of eddy mixing in isolation can be tested via a final fixed-K experiment, this time utilizing the R-E case’s temporally averaged Km (=Kh) field (Fig. 19b). Again, this is externally applied from the start of the simulation, and Goddard radiation is employed with CRF active. Despite the relatively weak and restricted mixing at outer radii, the storm is still able to organize rapidly (Fig. 17d), develop radially extensive convective activity (Fig. 18d), and attain a 10-m-wind profile comparable to the standard CRF example (red curve in Fig. 16b). That this is mostly due to CRF is demonstrated by the much slower expansion of this experiment’s CRF-off version (20 m s−1 contour superposed in green in Fig. 17d), which more closely resembles the other CRF-off storm (Fig. 17c) with respect to size and expansion rate. In agreement with our previous fixed-K experiments, these results suggest that a vertical mixing field that evolves with time to become radially extensive can assist in the progressive expansion of a tropical cyclone but is not absolutely necessary, particularly when clouds are permitted to interact with radiation.
4. Discussion and summary
Bu et al. (2014) demonstrated that cloud-radiative forcing (CRF) can exert a substantial influence on numerically simulated tropical cyclones (TCs), especially with respect to the storm’s horizontal scale. Specifically, it is the within-cloud longwave warming component of CRF that indirectly enhances convective activity in the TC outer core, thereby generating the diabatic heating that broadens the wind field. They further established that the radiation scheme employed by the operational HWRF, which derived from the old GFDL parameterization, was very deficient in handling CRF, to the point that it was essentially absent. However, as mentioned in the introduction, when HWRF was applied to historical cases using a more realistic radiation package, model skill with respect to important storm characteristics, such as intensity, position, and size, was degraded. In particular, most storms developed a positive size bias with respect to R34, the radius of the 34-kt (17.5 m s−1) wind at 10 m above the surface, one of the important metrics used in model verification.
That result motivated a study of how and why the planetary boundary layer (PBL) and its parameterization affect storm size, in cooperation and competition with CRF. Our principal finding is that vertical mixing of scalars in the boundary layer, primarily water vapor, also influences storm size via modulating outer-core convective activity. In this case, it is eddy mixing that helps transport water vapor to the top of the boundary layer, elevating the relative humidity there, and making the outer core more favorable for convection. Thus, a compelling parallel is seen with respect to CRF, the difference being that the heating associated with in-cloud warming was focused above the boundary layer, while the PBL influence is essentially “bottom up” from the sea surface.
As a consequence, we see why TC structure is sensitive to the PBL parameterization, particularly with respect to the magnitude and shape of the eddy diffusion distributions they generate. This was demonstrated using HWRF’s operational boundary layer code, the GFS PBL scheme, which was modified in the past to incorporate a tuning parameter α to throttle mixing in the TC inner core (Gopalakrishnan et al. 2013) because observations (e.g., Zhang et al. 2011a) suggested the model was too diffusive. We showed that α has a profound influence on R34. Indeed, we believe that it was the excessive diffusion produced by the GFS scheme that was previously compensating for the lack of cloud-radiative forcing in HWRF such that when the radiation issue was fixed, the simulated TCs developed a positive size bias, leading to poorer wind structure, intensity, and position forecasts.
Eddy mixing applied to momentum also appears to influence storm size, at least relatively close to the surface. However, what changed the most was the vertical shear, as the winds farther aloft were less impacted. Since the breadth of the 10-m-MSL wind field is one of the parameters used to judge model forecast skill, and information from farther aloft is often absent, this result raises the possibility that available storm size and/or intensity information can be skewed or misinterpreted. That said, examination of retrospective cases made using HWRF suggested that TCs are not always responsive to variations in the eddy mixing and the sensitivity is diminished when the TC environment is generally less favorable, which we demonstrated via experiments in which the sea surface temperature (SST) was lowered.
The GFS and YSU PBL schemes share a common ancestor and represent parameterizations that determine PBL height based on near-surface vertical stability and wind shear (cf. Vickers and Mahrt 2004). We demonstrated that how the critical Richardson number, Ribcr, was specified in these schemes had a major impact on the magnitude and depth of the eddy mixing, indirectly influencing TC size through vertical diffusion of boundary layer scalars. This finding motivates closer study in the future as it appears to provide a more physically defensible way of modulating mixing, especially because there is uncertainty with respect to the structure of the hurricane inner core and how the PBL depth should be defined (cf. Zhang et al. 2011b). Finally, through a comparison with recent axisymmetric studies, we appreciate that vertical mixing can indeed influence the progressive expansion of a TC in this restricted physical framework, but it does so most efficiently when acting with the assistance of cloud-radiative forcing.
Acknowledgments
The authors gratefully acknowledge the assistance of Drs. Ligia Bernardet, Mrinal Biswas, Gregory Thompson, Christina Holt, and three anonymous reviewers. This work was supported by the National Atmospheric and Oceanic Administration’s Hurricane Forecast Improvement Program (HFIP) under Grant NA12NWS4680009, by the National Aeronautics and Space Administration’s Hurricane Science Research Program under Grant NNX12AJ83G, and by the Developmental Testbed Center (DTC) visitor program. The DTC Visitor Program is funded by the National Oceanic and Atmospheric Administration, the National Center for Atmospheric Research, and the National Science Foundation.
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Additionally, according to the WRF code, version 3.7.1, YSU applies a separate mixing coefficient Kq to moisture, which in the mixed layer utilizes a somewhat modified turbulent Prandtl number formulation from that employed for Kh. This results in slightly different mixing being applied to moisture and heat.
It is noted that for the 2015 HWRF (Tallapragada et al. 2015), the α parameter was replaced by a strategy that does not require an externally set free parameter. See Bu and Fovell (2015) for more information.
This comparison involves Kh as the YSU scheme permits Pr to become smaller than unity, resulting in larger mixing being applied to scalars than to momentum.
