## 1. Introduction

Recent observations suggest that horizontal roll vortices, or rolls, frequently occur in the hurricane boundary layer (HBL). Most of these observations were obtained by Doppler radar from landfalling hurricanes (Wurman and Winslow 1998; Morrison et al. 2005; Lorsolo et al. 2008; Ellis and Businger 2010). In the radar observations, rolls are detected because they induce organized perturbations in the wind field. The wavelength of rolls estimated from the radar observations is highly variable, ranging from a few hundred meters to a few kilometers. For example, Lorsolo et al. (2008) observed rolls in Hurricanes Isabel (2003) and Frances (2004) with wavelengths ranging from 200 to 650 m. Morrison et al. (2005) found the majority of rolls during the landfalls of four hurricanes had wavelength in the range 1â€“2 km. Synthetic aperture radar observations identified rolls with wavelengths of 3â€“6 km (Katsaros et al. 2000). Zhang et al. (2008) provided the first in situ measurement of the roll-induced vertical fluxes in the HBL and showed that the rolls enhanced the total turbulent momentum and moisture fluxes in the HBL. However, the mixing effects of rolls are not represented in current hurricane models, which may limit their forecast accuracy. To represent the roll-induced vertical momentum and energy transports in hurricane numerical models, we must better understand their formation mechanism.

The formation mechanism and characteristics of rolls in the HBL were studied based on theoretical and numerical approaches. Foster (2005) applied linear and nonlinear analyses to the HBL momentum equations and argued that rolls are expected to be a common feature in the HBL. He demonstrated that the inflection points in the basic-state wind profiles cause the instability and lead to the generation of rolls. Such mechanism is analogous to the classical Ekman boundary layer instability (Faller 1965; Lilly 1966; Brown 1970, 1972). Since this type of instability is related to the inflection point in the boundary layer wind profile, it is commonly referred to as the inflection point instability. Wavelengths of the rolls estimated by Foster (2005) were similar to those observed by Morrison et al. (2005). Nolan (2005) used both a nonlinear axisymmetric model and a linear instability analysis approach to investigate the instability in the HBL. He also found that inflection points in the HBL wind profiles are responsible for the formation of rolls, which have wavelengths of 3â€“5 km. Zhu (2008) used the Large-Eddy Simulation version of the Weather Research and Forecasting model (WRF LES) nested within the WRF mesoscale model to simulate large eddies in a landfalling hurricane. He found roll-like perturbations existed in a statically stable boundary layer environment, suggesting these features were generated by the inflection point instability rather than the thermal instability. Nakanishi and Niino (2012) conducted an LES study at two different locations in the idealized HBL. By applying the EOF analysis to the model results, they identified rolls generated by the inflection point instability with spatial scales similar to those found by Foster (2005).

These previous studies investigated the rolls in neutral or near-neutral stratification (Foster 2005; Nolan 2005) or the rolls at specific locations within the HBL (Zhu 2008; Nakanishi and Niino 2012), but they did not systematically examine the conditions favorable for the formation of rolls by the inflection point instability and the impacts of the mean wind and stratification on the characteristics of rolls. In this study, we aim to gain a more comprehensive understanding of the formation of rolls under various HBL conditions and identify the important mean-flow factors affecting the characteristics of rolls. Specifically, this study addresses the following questions: What factors affect the spatial and temporal characteristics of the rolls generated by the inflection point instability? What are the spatial variations of the rolls in the HBL? What are the effects of stratification on the rolls? To answer these questions, we use a numerical approach that explicitly resolves the rolls in the HBL. We examine the formation and characteristics of rolls under various ambient environments by a set of idealized numerical experiments.

## 2. Method

### a. Modeling approach

We assume that rolls can be separated from the large-scale HBL flow because of their small spatial scale. Based on this assumption, we decompose the HBL flow variables into the mean and perturbations, as *a* can represent the velocity vector, potential temperature, etc. Similar flow decomposition approach was applied by Ginis et al. (2004) and Foster (2005) for studying boundary layer eddies or rolls under hurricane condition. In this study, the mean component represents the large-scale HBL flow, which consists of the primary cyclonic circulation and the secondary circulation induced by surface friction; the perturbations represent the rolls, which are the small-scale features generated by the inflection point instability of the mean HBL flow.

*y*axis is parallel to the direction in which the rolls are aligned. We will refer to the

*y*axis (

*x*axis) as the along-roll axis (cross-roll axis). The along-roll variations of the perturbations are assumed negligible; that is,

*x*â€“

*z*plane. To distinguish the velocity components projected onto the two different coordinate systems, we use uppercase letters

*y*and the azimuthal direction

*Î»*is defined as

The complete sets of equations for the mean flow and rolls are presented in the appendix. For the purpose of this study, the governing equations for the mean flow and rolls are further simplified. We consider the evolution of the rolls to take place in two phases: the linear phase and the nonlinear phase. During the linear phase, rolls are formed by the instability of the mean flow and have the following characteristics: 1) roll velocities grow exponentially with time, but remain at least one order smaller than the mean winds, and thus the nonlinear terms in their governing equations are negligible; and 2) the averaged fluxes induced by rolls are too weak to cause any significant modifications to the mean flow, and therefore we can assume the mean flow remains unchanged during the linear phase. The evolution of rolls enters the nonlinear phase after roll velocities reach sufficiently large magnitude (comparable to the mean winds), during which the nonlinear effects and the roll-induced fluxes are important. In this paper, we focus on the generation and initial evolution of rolls and therefore only consider the linear phase. Thus some terms in the governing equations of the mean flow and rolls can be neglected, as discussed below.

### b. The basic-state HBL flow

The basic-state HBL flow is used as the background environment for the formation of rolls. The basic-state HBL wind profiles are obtained by resolving the steady-state mean wind equations, neglecting time dependency and roll-induced forcing terms in (A4)â€“(A6). While the basic-state potential temperature profiles are prescribed analytically. We assume that the basic-state HBL wind and potential temperature profiles remain unchanged over time during the linear phase of rolls.

*f*plane centered at 20Â°N. At the upper boundary (

*H*is the vertical extent of the atmospheric layer that we consider), we assume the wind is under gradient wind balance; that is,

^{âˆ’1}, a radius of maximum wind (RMW) of 40 km, and a parameter

*B*of 1.3 (parameter

*B*controls how rapidly

The model described above is called the basic-state HBL model. Figure 3 shows an example of the basic-state HBL wind solution with the choice of *r* = 40 km), 2RMW (*r* = 80 km), and 3RMW (*r* = 120 km), are shown in Fig. 4. Both the radial and azimuthal wind profiles have inflection points (only the inflection points in the radial wind profiles are marked in Fig. 4). The vertical wind shear reaches extreme value at the inflection point. Compared with the inflection point in the azimuthal wind profile, the inflection point in the radial wind profile has considerably larger wind shear.

### c. Equations for rolls in the linear phase

Periodic conditions are applied at the lateral boundaries. At the upper and lower boundaries, we set

### d. Experimental design

Three groups of experiments are designed to investigate the impacts of the basic-state wind and stratification on the formation and characteristics of rolls. In the numerical experiments presented below, the SRM is embedded at every grid point of the basic-state HBL model from 1RMW to 3RMW. The horizontal and vertical resolutions and the domain sizes for both models are listed in Table 1. The calculations in each experiment are done in two steps. During the first step, the basic-state HBL model resolves (1)â€“(3) to derive the basic-state wind distribution. During the second step, the SRM resolves (7)â€“(10) based on the basic-state wind profiles and the prescribed potential temperature profiles. Both theoretical analysis (Foster 2005) and three-dimensional LES study (Nakanishi and Niino 2012) suggested that the angle between the along-roll axis and the azimuthal wind is relatively small. Therefore, for simplicity we assume that

Numerical parameters for the basic-state HBL model and the SRM.

Group N is designed to investigate the effect of basic-state HBL winds. In this group of experiments, we neglect the effect of stratification on the rolls by assuming neutral stratification (i.e.,

Values of asymptotic mixing length used in group N. Group N is designed to investigate the effect of basic-state HBL wind distribution on rolls. Neutral stratification is applied in group N and distribution of the basic-state HBL winds is varied by changing

Groups S and M are designed to investigate the effect of stratification on roll formation and characteristics. The basic-state HBL winds used in groups S and M are as in experiment N3 (^{âˆ’1} and the mixed-layer height

Values of lapse rate used in group S. Group S is used to investigate the effect of stable stratification on rolls. The basic-state HBL winds used in group S are as in experiment N3 (

Values of mixed-layer height used in group M. Group M is used to investigate the effect of the mixed layer on rolls. The basic-state HBL winds used in group M are as in experiment N3 (^{âˆ’1}) is used in group M. The same potential temperature profile is applied at all locations in each experiment.

## 3. Effect of mean wind on rolls

### a. Basic structure of rolls

Rolls are formed in all experiments of group N owing to the inflection point instability. They have similar structures, but their wavelengths and growth rates vary with the changes of the mean wind profiles. The typical structure of rolls is shown in Fig. 7. Since rolls in the linear phase grow exponentially with time, the variables shown in Fig. 7 are normalized to remove the time-dependent part. The length scale *K* is the turbulent viscosity. Here we use the maximum value in the

### b. Factors affecting growth rate and wavelength of rolls

We next explore the characteristics of rolls, specifically their wavelength and growth rate, and identify mean wind factors that affect these characteristics. The SRM solutions indicate roll variables are in the form of normal mode. For example, the vertical velocity can be written as *k* is the horizontal wavenumber, *e*â€²ã€‰ is kinetic energy averaged over the entire SRM domain.

Figure 9 shows the wavelength of rolls and

Figure 10 shows the growth rate of rolls and the bulk wind shear from all experiments in group N. The bulk wind shear represents the magnitude of mean wind shear and is defined as

## 4. Effects of stratification on rolls

### a. Effect of the stable stratification

The growth rates of rolls in the group S experiments are shown in Fig. 11a. The major effect caused by the stable stratification is the reduction the growth rate of rolls. At a given location, the growth rate decreases with increasing lapse rate. When the stratification is sufficiently strong, the growth rate can be reduced to zero, which means the inflection point instability is completely suppressed. We find that the classic Ri criterion^{1} (Miles 1961) can be used to determine whether the inflection point instability can operate under stable stratification. Ri is defined as

The effect of stable stratification can be explained by the buoyancy work in the kinetic energy budget equation [see (11)]. The correlation between

### b. Effect of the mixed layer

Figure 14a shows the growth rate of rolls in group M. Overall, the mixed layer relaxes the suppressing effect of stratification on the inflection point instability. In the presence of a mixed layer, rolls can be formed at the locations where the inflection point instability is completely suppressed without the mixed layer (e.g., experiment S5 in group S). The roll growth rate increases with increasing

The typical structure of rolls in the group M experiments is shown in Fig. 15. The near-surface part of the rolls appears to be similar to those in groups N (Fig. 7) and S (Fig. 12). However, one noticeable change appears in the upper part where the roll streamlines become inclined from the vertical axis (

*D*is the ratio of the vertical

*e*-folding length to the vertical wavelength. Following Sutherland et al. (1994),

*D*is referred to as the penetration ratio. Larger

*D*suggests that the internal waves have less vertically decaying property and more vertically propagating property. For example, if

*D*is infinitely large, the internal waves propagate vertically without any decay. After introducing

*x*direction). So we only keep

*D*of the internal waves generated by rolls with known

Figure 16 shows *D*, derived based on (16) and (18), as functions of the mixed-layer height at RMW. As the mixed-layer height increases, the inclination angle of the internal wave phase lines also increases. This implies that the phase lines of internal waves are more inclined from the vertical direction when the mixed layer gets higher. The penetration ratio also increases with the increasing mixed-layer height, suggesting that the internal waves generated by the rolls have less vertically decaying property and more vertically propagating property under nonuniform stratification with a higher mixed layer.

## 5. Discussion

In the three groups of experiments described above, we assume that the angle between the along-roll axis and the azimuthal direction

In group N, we vary the basic-state HBL wind distribution by changing

According to our analysis, the spatial scale of rolls is proportional to *I* and

Previous theoretical studies (Foster 2005; Nolan 2005) primarily focused on the effect of the mean HBL wind structure on the rolls, but the effect of stratification has not been fully investigated. Nolan (2005) used neutral stratification throughout the atmospheric layer that he considered. Foster (2005) included the near-surface unstably stratified layer but he did not consider the stably stratified layer above the mixed layer. The unstably stratified layer Foster (2005) considered did not appear to affect roll characteristics significantly. This study emphasizes the effects of stratification on rolls. We find that the stable stratification in the HBL may suppress the inflection point instability under some conditions. This may explain why rolls do not always exist in the HBL observations (e.g., Morrison et al. 2005) even though the inflection point in the radial wind profile is always present. Moreover, we find the stably stratified layer provides the environment for rolls to couple with internal waves. The characteristics of rolls and internal waves are affected by the mixed-layer height, suggesting the mixed-layer height is a critical length scale.

Spatial resolutions of the majority of current hurricane models are too coarse to resolve rolls and therefore their effects are not explicitly included. To properly parameterize the effects of rolls, process-oriented studies need to be conducted to study their dynamics, such as their formation and nonlinear evolution. One possible limitation of our numerical approach is the assumption of two-dimensional structure of rolls. Although this assumption can be justified because rolls tend to be elongated along the mean wind direction within the HBL, a high-resolution three-dimensional LES model would be the best tool for studying rolls in hurricanes. However, it is not practical at present because of computer limitations. Utilization of the SRM imbedded into a HBL model allows us to simplify the problem and dramatically decrease the amount of computation. This numerical approach can serve as a useful tool to study both the thermal and inflection point instabilities in the HBL and the effects of the environmental factors such as the mean wind and stratification. Because the equations describing the HBL mean flow and the SRM are coupled, this approach can be further extended to study the nonlinear phase of rolls, their interactions with the internal waves, as well as their interactions with the mean HBL flow.

## 6. Conclusions

We have investigated the role of the mean wind (large-scale wind) and the stratification on the inflection point instability in the hurricane boundary layer (HBL). The study was performed using a two-dimensional Single-Grid Roll-Resolving Model (SRM) to resolve linear-phase rolls under various HBL conditions. The key findings in this study are summarized as follows. Rolls generated by the inflection point instability are characterized by tilted streamlines in the vicinity of the inflection point of the radial wind profile. Kinetic energy budget consideration reveals that the tilted streamlines are critical for the rolls to extract kinetic energy from the mean wind. We have identified two important factors of the mean HBL winds that affect the characteristics of rolls: the dynamical HBL height affects the wavelength of rolls, and the magnitude of the mean wind shear affects the growth rate of rolls. Therefore, under neutrally stratified HBL, the wavelength of rolls increases with the distance from the storm center (outside of the RMW), while their growth rate decreases. The stable stratification in the HBL can suppress the growth of rolls, and rolls can be generated only if the minimum gradient Richardson number is less than 0.25. The nonuniform stratification with a mixed layer has less suppressing effect on rolls. If the mixed layer is sufficiently high, the stably stratified layer above has minor effect on the growth of rolls.

Rolls generated by the inflection point instability can trigger internal waves in the stably stratified layer, which have the same horizontal wavelength, growth rate, and angular frequency as rolls. These internal waves have both vertically propagating and decaying properties. We derived analytical solutions for internal waves, which relate the properties of the internal waves to the properties of boundary layer rolls. We find as the mixed-layer height increases, the phase lines of the internal waves are more inclined from the vertical direction, and the internal waves have less vertically decaying property and more vertically propagating property.

Since there is a growing interest in parameterizing the roll-induced mixing effect in hurricane models, we think it is necessary to outline the important environmental factors affecting the formation of rolls and their characteristics. We believe that this work improves the current understanding on rolls in the HBL and provides important guidelines for the parameterization of rolls.

## Acknowledgments

We acknowledge Ralph Foster for his valuable input throughout this study and Jun Zhang for providing the composite hurricane boundary layer observations. We thank Jun Zhang and one anonymous reviewer for providing the insightful comments that significantly improved this work. This research was funded by the Office of Navy Research through Award N000141210447.

## APPENDIX

### Equations for the Mean Flow and Roll Vortices in the Hurricane Boundary Layer

**v**is the velocity vector and

#### a. Equations for the mean flow

#### b. Equations for rolls

*y*axis and the along-roll variations are negligible. The governing equations for rolls in the local Cartesian coordinates at radius

*r*=

*R*are

*x*and

*y*directions, respectively.

The numerical methods applied for solving the mean-flow and roll equations are described in Ginis et al. (2004).

## REFERENCES

Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in a neutral atmosphere.

,*J. Geophys. Res.***67**, 3095â€“3102, doi:10.1029/JZ067i008p03095.Brown, R. A., 1970: A secondary flow model of the planetary boundary layer.

,*J. Atmos. Sci.***27**, 742â€“757, doi:10.1175/1520-0469(1970)027<0742:ASFMFT>2.0.CO;2.Brown, R. A., 1972: On the inflection point instability of a stratified Ekman boundary layer.

,*J. Atmos. Sci.***29**, 850â€“859, doi:10.1175/1520-0469(1972)029<0850:OTIPIO>2.0.CO;2.Donelan, M. A., B. K. Haus, N. Reul, W. J. Plant, M. Stiassnie, H. C. Graber, O. B. Brown, and E. S. Saltzman, 2004: On the limiting aerodynamic roughness of the ocean in very strong winds.

,*Geophys. Res. Lett.***31**, L18306, doi:10.1029/2004GL019460.Ellis, R., and S. Businger, 2010: Helical circulation in the typhoon boundary layer.

,*J. Geophys. Res.***115**, D06205, doi:10.1029/2009JD011819.Faller, A. J., 1965: Large eddies in the atmospheric boundary layer and their possible role in the formation of cloud rows.

,*J. Atmos. Sci.***22**, 176â€“184, doi:10.1175/1520-0469(1965)022<0176:LEITAB>2.0.CO;2.Foster, R. C., 2005: Why rolls are prevalent in the hurricane boundary layer.

,*J. Atmos. Sci.***62**, 2647â€“2661, doi:10.1175/JAS3475.1.Foster, R. C., 2009: Boundary-layer similarity under an axisymmetric, gradient wind vortex.

,*Bound.-Layer Meteor.***131**, 321â€“344, doi:10.1007/s10546-009-9379-1.Ginis, I., A. P. Khain, and E. Morozovsky, 2004: Effects of large eddies on the structure of the marine boundary layer under strong wind conditions.

,*J. Atmos. Sci.***61**, 3049â€“3064, doi:10.1175/JAS-3342.1.Holland, G. J., 1980: An analytic model of the wind and pressure profiles in hurricanes.

,*Mon. Wea. Rev.***108**, 1212â€“1218, doi:10.1175/1520-0493(1980)108<1212:AAMOTW>2.0.CO;2.Katsaros, K. B., P. W. Vachon, P. G. Black, P. P. Dodge, and E. W. Uhlhorn, 2000: Wind fields from SAR: Could they improve our understanding of storm dynamics?

,*Johns Hopkins APL Tech. Dig.***21**, 86â€“93.Kepert, J. D., 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part I: Linear theory.

,*J. Atmos. Sci.***58**, 2469â€“2484, doi:10.1175/1520-0469(2001)058<2469:TDOBLJ>2.0.CO;2.Kepert, J. D., 2012: Choosing a boundary layer parameterization for tropical cyclone modeling.

,*Mon. Wea. Rev.***140**, 1427â€“1445, doi:10.1175/MWR-D-11-00217.1.Kepert, J. D., and Y. Wang, 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part II: Nonlinear enhancement.

,*J. Atmos. Sci.***58**, 2485â€“2501, doi:10.1175/1520-0469(2001)058<2485:TDOBLJ>2.0.CO;2.Lilly, D., 1966: On the instability of Ekman boundary flow.

,*J. Atmos. Sci.***23**, 481â€“494, doi:10.1175/1520-0469(1966)023<0481:OTIOEB>2.0.CO;2.Lorsolo, S., J. L. Schroeder, P. Dodge, and F. Marks, 2008: An observational study of hurricane boundary layer small-scale coherent structures.

,*Mon. Wea. Rev.***136**, 2871â€“2893, doi:10.1175/2008MWR2273.1.Miles, J. W., 1961: On the stability of heterogenous shear flows.

,*J. Fluid Mech.***10**, 496â€“508, doi:10.1017/S0022112061000305.Moon, I.-J., I. Ginis, T. Hara, and B. Thomas, 2007: A physics-based parameterization of airâ€“sea momentum flux at high wind speeds and its impact on hurricane intensity predictions.

,*Mon. Wea. Rev.***135**, 2869â€“2878, doi:10.1175/MWR3432.1.Morrison, I., S. Businger, F. Marks, P. Dodge, and J. Businger, 2005: An observational case for the prevalence of roll vortices in the hurricane boundary layer.

,*J. Atmos. Sci.***62**, 2662â€“2673, doi:10.1175/JAS3508.1.Nakanishi, M., and H. Niino, 2012: Large-eddy simulation of roll vortices in a hurricane boundary layer.

,*J. Atmos. Sci.***69**, 3558â€“3575, doi:10.1175/JAS-D-11-0237.1.Nolan, D. S., 2005: Instabilities in hurricane-like boundary layers.

,*Dyn. Atmos. Oceans***40**, 209â€“236, doi:10.1016/j.dynatmoce.2005.03.002.Powell, P., J. Vickery, and T. A. Reinhold, 2003: Reduced drag coefficient for high wind speeds in tropical cyclone.

,*Nature***422**, 279â€“283, doi:10.1038/nature01481.Sutherland, B. R., C. P. Caulfield, and W. R. Peltier, 1994: Internal gravity wave generation and hydrodynamic instability.

,*J. Atmos. Sci.***51**, 3261â€“3280, doi:10.1175/1520-0469(1994)051<3261:IGWGAH>2.0.CO;2.Wurman, J., and J. Winslow, 1998: Intense sub-kilometer boundary layer rolls in Hurricane Fran.

,*Science***280**, 555â€“557, doi:10.1126/science.280.5363.555.Zhang, J. A., and W. M. Drennan, 2012: An observational study of vertical eddy diffusivity in the hurricane boundary layer.

,*J. Atmos. Sci.***69**, 3223â€“3236, doi:10.1175/JAS-D-11-0348.1.Zhang, J. A., K. B. Katsaros, P. G. Black, S. Lehner, J. R. French, and W. M. Drennan, 2008: Effects of roll vortices on turbulent fluxes in the hurricane boundary layer.

,*Bound.-Layer Meteor.***128**, 173â€“189, doi:10.1007/s10546-008-9281-2.Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks Jr., 2011: On the characteristic height scales of the hurricane boundary layer.

,*Mon. Wea. Rev.***139**, 2523â€“2535, doi:10.1175/MWR-D-10-05017.1.Zhu, P., 2008: Simulation and parameterization of the turbulent transport in the hurricane boundary layer by large eddies.

,*J. Geophys. Res.***113**, D17104, doi:10.1029/2007JD009643.

^{1}

The Richardson number criterion states that for a stratified shear flow, the necessary (but not sufficient) condition for the flow to be unstable is that the minimum Ri is less than 0.25, and the sufficient (but not necessary) condition for the flow to become stable is that the minimum Ri is greater than 0.25.