Impact of subgrid-scale vertical turbulent mixing on eyewall asymmetric structures and mesovortices of hurricanes

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1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38, January B Impact of subgrid-scale vertical turbulent mixing on eyewall asymmetric structures and mesovortices of hurricanes Ping Zhu a *, Konstantinos Menelaou b and Zhenduo Zhu a a Department of Earth and Environment, Florida International University, Miami, FL, USA b Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada *Correspondence to: P. Zhu, Department of Earth and Environment, Florida International University, 1 SW 8th Street, Miami, FL 33199, USA. zhup@fiu.edu In this study, the multiple nested state-of-the-art Weather Research and Forecasting (WRF) model is used to investigate the impact of the subgridscale (SGS) vertical turbulent mixing parametrization on hurricane eyewall asymmetric structures and the formation of eyewall mesovortices. Hurricane Isabel (3) was simulated by a series of numerical experiments with different SGS vertical turbulent mixing parametrizations including the Yonsei University, Mellor Yamada Janjic, Mellor Yamada Nakanishi Niino. level and Mellor Yamada Nakanishi Niino 3 level schemes. The simulations show that the vertical turbulent mixing scheme not only substantially affects the SGS vertical transport of heat and moisture but also has an important bearing on the storm axisymmetric structure, eyewall mesovortices and other resolved asymmetric features in the vicinity of the hurricane eyewall. The analyses show that the vertical turbulent mixing processes provide a mechanism to affect the barotropic instability for generation of eyewall mesovortices through changing the vortex basic state potential vorticity (PV) field and generating eyewall disturbances with different frequencies. Our numerical experiments show that for given external conditions the magnitude and vertical distribution of the eddy exchange coefficients are the key factors that regulate the characteristics of eyewall disturbances. Such a modulation of eyewall structure by the eddy exchange coefficients is realized through a complicated interaction among SGS vertical turbulent mixing, mesoscale structures, diabatic heating, and barotropic instability. Key Words: turbulent mixing; hurricane; eyewall Received September ; Revised February 13; Accepted 1 March 13; Published online in Wiley Online Library June 13 Citation: Zhu P, Menelaou K, Zhu Z.. Impact of subgrid-scale vertical turbulent mixing on eyewall asymmetric structures and mesovortices of hurricanes. Q. J. R. Meteorol. Soc. : 38. DOI:1./qj.7 1. Introduction The inner core dynamics of a hurricane is believed to be one of the keys controlling hurricane structure and intensity change (e.g. Liu et al., 1999; Zhang et al.,, ; Wang and Wu, ). An intriguing part of the inner core dynamics is the eyewall mesovortices, which are frequently observed in the eyewall of an intense hurricane as vortical or swirling patterns with a variety of shapes, ranging from circles centred in the eye to convoluted structures distributed along the eye (Black and Marks, 1991; Willoughby and Black, 1996; Kossin et al., ; Knaff et al., 3). In the past decades our knowledge of mesovortices has been greatly advanced owing to the pioneer work by Schubert c 13 Royal Meteorological Society

2 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 17 et al. (1999). They argued that the cyclonic shear zone on the inner edge of the eyewall, which is in the vicinity of the radius of maximum winds, may be considered as an annular ring of high potential vorticity (PV) with large PV gradients pointing to both sides (i.e. PV increases and decreases with radius on the inner and outer edges of the annular PV ring, respectively). From the vortex Rossby wave point of view, opposite PV gradients on the two sides of the PV ring create counter-propagating PV waves: cyclonic on the inner edge and anticyclonic on the outer edge relative to the eyewall mean flow. If these waves become phase locked, each wave will make the other grow, leading to exponential instability. This barotropic instability argument was confirmed by their numerical simulations. Using an unforced -D barotropic non-divergent model, Schubert et al. (1999) illustrated that the eyewall PV ring is barotropically unstable and can be quickly broken down and pulled into discrete areas to create a number of mesovortices that result in a polygonal eyewall resembling that observed in intensifying hurricanes (e.g. Lewis and Hawkins, 198; Muramatsu, 1986; Hendricks et al., ). Using the same idealized framework, Kossin and Schubert (1) further showed that the generated mesovortices may undergo different merger processes, which are sensitive to the initial setting of the PV annulus: either relaxing to a vortex monopole for the smaller and radially thicker initial PV annulus, or forming a long-lived vortex lattice whose number may vary for the larger and radially thinner initial PV annulus. These results obtained from the highly simplified -D framework were confirmed by the idealized 3- D simulations without considering external forcing (Nolan and Montgomery, ; Nolan and Grasso, 3; Hendricks and Schubert, 1). In addition to providing a mechanism that well explains the formation of eyewall mesovortices, the -D barotropic non-divergent model further illustrated that the vorticity rearrangement (or mesovortex mixing) alone during the merger process can lead to a drop of central pressure as great as about hpa at the expense of reducing the maximum tangential winds. This may be understood from the simplest hurricane dynamics. The mixing of high PV from the eyewall into the eye by mesovortices reduces the maximum tangential winds, but at the same time it increases the quantity v t r (v t and r here represent the tangential wind speed and radius, respectively), which leads to the drop of central pressure constrained by the gradient wind balance. Thus, in a simple unforced dynamic framework, eyewall mesovortices provide a mechanism to affect hurricane development and intensity through changing the central pressure and maximum tangential winds. By comparing the results from a 3-D simulation with its axisymmetric -D version, Yang et al. (7) showed that the model-resolved asymmetric eddies in the form of vortex Rossby waves play important roles in modifying the axisymmetric structure of a hurricane. The resulting inward PV mixing from the eyewall into the eye results in a less-tilted eyewall, limits the drying and cooling effects of downdraughts and reduces the air sea entropy deficit under the eyewall, thereby reducing the strength of the primary circulation. Wang () further showed that the eyewall PV mixing can result in secondary eyewall formation. In a real hurricane, the barotropic instability, the eyewall PV structure and its evolution are expected to be further affected by the secondary circulations driven by moist diabatic heating and turbulent mixing processes. Using an idealized framework, Rozoff et al. (9) examined the potential impacts of diabatic heating on barotropic instability and mesovortices and demonstrated an internal mechanism of vorticity mixing that can interrupt the intensification process resulting from vorticity generation due to diabatic heating in the hurricane eyewall. Chen and Yau (1) showed that the PV anomalies in and at the top of the boundary layer can induce upward motion that gives rise to the inner cloud bands. In contrast to the vortex monopole or lattice in the -D barotropic model, a PV bowl instead tends to be formed and maintained by the continuous generation of high PV due to the latent heat release by the eyewall and rain band convection. They showed that the inward transport of continuously generated high PV due to Rossby wave and mesovortex mixing can intensify the hurricane. Their results seem to be opposite to Yang et al. (7), but we note that Chen and Yau (1) focused more on the vorticity transport of spiral bands outside the eyewall. While the importance of moist diabatic heating and turbulent mixing processes to eyewall dynamics has been established, how the diabatic heating and turbulent processes interact with the barotropic instability to affect the structure of the eyewall and the formation and development of mesovortices is still largely unknown. Although the diabatic heating of convection is important to eyewall structures, in this study we only focus on the vertical turbulent mixing processes (issues related to moist convection will be investigated in our future studies). Specifically, in this paper we attempt to address the following two issues. First, eyewall mesovortices do not tend to form all the time, despite the robust barotropic instability in the eyewall. Why then do mesovortices fail to form under certain circumstances? Does this have something to do with turbulent mixing processes? If so, how do turbulent mixing processes regulate the eyewall barotropic instability and affect the formation of mesovortices? Clarifying these issues will also help us to understand the role of turbulent mixing processes in conjunction with the dynamics of eyewall asymmetric structures in the storm-scale energetics. Recent studies (e.g. Bryan and Rotunno, 9; Rotunno et al., 9) show that subgrid-scale (SGS) horizontal mixing can have a substantial impact on hurricane development. To simply the problem, issues related to the SGS horizontal mixing will not be touched on in this study. Our focus is on the SGS vertical turbulent mixing and its impact on eyewall structure and mesovortices. Second, since the vertical turbulent mixing processes are not resolved in numerical models but are parametrized, it is not clear how the parametrized vertical turbulent mixing affects the hurricane simulations. In non-hurricane conditions, various vertical turbulent mixing schemes have been extensively evaluated (e.g. Holt and Raman, 1988; Holtslag et al., 199; Zhang and Zheng, ), but the sensitivity of simulated hurricanes to turbulent mixing schemes has not been thoroughly addressed. There are limited efforts on this (e.g. Li and Pu, 8), but they tended to focus on the overall effect of vertical turbulent mixing schemes on hurricane track and intensity. In two recent studies, Nolan et al. (9a, 9b) showed that highresolution hurricane simulations are extremely sensitive to the chosen vertical turbulent mixing scheme activated in the simulations. In particular, they showed that the vertical c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

3 P. Zhu et al. turbulent mixing schemes can significantly alter the eyewall structure of the simulated hurricane. Some schemes tend to generate high-frequency perturbations along the eyewall, whereas others produce relatively smooth low frequency oscillations along the eyewall, but why different vertical turbulent mixing schemes lead to substantially different eyewall structure is still a question yet to be answered. It should be pointed out that most of the vertical turbulent mixing schemes used in models to simulate hurricanes were developed and verified in non-hurricane conditions. How these schemes perform and whether they can realistically represent the vertical turbulent mixing processes in hurricane conditions need to be extensively evaluated against observations. However, obtaining high temporal-resolution data that can be directly used to compute turbulent fluxes in harsh hurricane conditions is a difficult task. Recent advances in observations made a breakthrough in characterizing the boundary layer structure and turbulent mixing processes in hurricane conditions. For example, Zhang et al. (11b) and Zhang and Drennan () estimated the turbulent fluxes and vertical eddy diffusivity using the flight-level data collected by aircraft. Zhang and Montgomery () estimated the horizontal eddy diffusivity and mixing length in intense hurricanes using aircraft observations. Zhang et al. (11a) analysed the height scales of the hurricane boundary layer using GPS dropsondes. Using airborne Doppler measurements, Lorsolo et al. (1) estimated and mapped the turbulent kinetic energy (TKE) in hurricanes. With appropriate metrics, these observations may be used to evaluate the turbulent mixing schemes used for simulating hurricanes, which may enlighten various issues regarding the parametrization improvements. Based on recent observations and simulations, Kepert () provided a thorough summary of the methodologies for parametrizing the vertical turbulent fluxes in hurricane simulations. The focus of this study, however, is not on evaluating vertical turbulent mixing schemes using observations; rather, we design numerical experiments to examine the sensitivity of hurricane simulations to vertical turbulent mixing schemes. In particular, we investigate how different vertical turbulent mixing schemes affect the eyewall asymmetric structures and dynamics and the formation of eyewall mesovortices. Such a comparison between simulations not only allows us to look deeper into the underlying physical mechanisms of the impact of the vertical turbulent mixing processes on eyewall structure and mesovortices, but also may provide useful guidance for the improvement of the parametrization of vertical turbulent mixing processes in hurricane simulations. This paper is organized as follows. In section, we describe the baseline numerical experiments designed to study the impact of vertical turbulent mixing processes on eyewall asymmetries and mesovortices. The analyses and results from simulations are presented in section 3. In section, more sensitivity experiments are provided to investigate how the vertical turbulent mixing schemes affect the eyewall structure and the formation of mesovortices. Finally, the main findings of this study are summarized in section.. Numerical simulations of Hurricane Isabel (3) In this study, a series of numerical simulations of Hurricane Isabel (3) are conducted using the Advanced Research Weather Research and Forecasting (WRF-ARW) model (Skamarock et al., 8). To investigate the effects of vertical turbulent mixing processes on eyewall structure and mesovortices, four vertical turbulent mixing schemes are activated in the baseline sensitivity experiments. They are the Yonsei University scheme (; Hong et al., 6), the Mellor Yamada Janjic scheme (; Janjic, 199) and the two high-order turbulent mixing schemes: the Mellor Yamada Nakanishi Niino. level TKE scheme (MYNN-.) and 3-level TKE scheme (MYNN- 3.; Nakanishi and Niino, ). The and MYNN schemes are local turbulent mixing schemes based on the predicted TKE, whereas the scheme is a K-closure scheme but includes the non-local mixing effect. MYNN schemes are formulated using moist thermodynamics based on the variables conserved for moist reversible adiabatic processes, and thus are often called moist schemes. and, on the other hand, are dry schemes. These schemes are widely used in WRF simulations of many meteorological applications. Vertical turbulent mixing schemes are also known as planetary boundary layer (PBL) schemes in the literature. This is understandable since under normal conditions turbulent mixing is mainly in the PBL. However, when there is deep convection, intense turbulent mixing can exist in the convective clouds above the PBL. This is the case in hurricane conditions, particularly in the eyewall region and rain bands. Strictly speaking, representing turbulent mixing associated with the convective clouds is beyond the scope of PBL parametrization in a traditional sense. But in the WRF model, if there is indeed SGS vertical mixing in the free atmosphere, the mixing is handled by the same vertical turbulent mixing. For this reason, although the, and MYNN schemes are often referred to as PBL schemes in many studies, the vertical turbulent mixing scheme appears to be a more appropriate name for them in hurricane simulations since they are used to treat turbulent mixing both within and above the PBL. Although some of the schemes (e.g. ) do provide a way to diagnose the PBL height based on some empirical relationships, it is usually difficult to distinguish the turbulence directly affected by the surface processes from that induced by convective clouds in hurricane conditions since there is often no clean physical interface (or discontinuity), such as the strong inversion in some other cases, that separates the PBL from the free atmosphere. Thus, in this study, we make no effort to distinguish the turbulent mixing associated with the PBL processes and with the convective cloud processes. In the WRF model, there are SGS vertical and horizontal mixing parametrizations built into the WRF dynamic-core solver. When a vertical turbulent mixing scheme is activated, the vertical mixing parametrization associated with the WRF solver is overwritten. For this reason, the, and MYNN schemes are also called 1-D schemes. Activation of the 1-D vertical turbulent mixing scheme has no impact on WRF s SGS horizontal mixing parametrization associated with the WRF dynamic-core solver. In this study, the Smagorinsky first-order closure, which is recommended for real-case WRF simulations, is chosen to parametrize the SGS horizontal mixing. Since issues related to the SGS horizontal mixing are not investigated in this study, this option is kept the same in all sensitivity experiments. Also note that in the WRF model the available vertical turbulent mixing schemes usually work with their own c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

4 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries Best track MYNN. MYNN Latitude ( ) UTC, Sept UTC, Sept Longitude ( ) Figure 1. Domain configuration of WRF simulations of Hurricane Isabel (3). The background shades indicate the simulated 1 m wind speed at UTC September 3 from the run. White, blue, red, green, and black curves with small circles indicates the best hurricane track and simulated track with,, MYNN-., and MYNN-3. schemes, respectively. Square boxes indicate the WRF two-way nested domains. This figure is available in colour online at wileyonlinelibrary.com/journal/qj surface layer schemes. In the real world the surface fluxes and the vertical transport in the PBL are all controlled by the turbulent mixing processes and cannot be separated as the parametrization schemes do in models. For this reason, in this study when we say SGS vertical turbulent mixing parametrization it refers to both the WRF 1-D vertical turbulent mixing scheme and its associated surface layer scheme. It is a whole package for handling SGS vertical turbulent processes, and we do not separate them in this study. How the surface layer parametrization affects the eyewall asymmetric structures will be investigated in our future studies. In addition to the previously listed baseline experiments with four 1-D SGS vertical turbulent mixing schemes, more numerical tests are performed to investigate the sensitivity of eyewall structure, particularly the eyewall vortex Rossby waves and mesovortices, to the details of a SGS vertical turbulent mixing scheme. These additional sensitivity experiments and the related findings and discussions will be presented later in section. Other major model physics include the Rapid Radiative Transfer Model (RRTM; Mlawer et al., 1997) for longwave radiation, the Dudhia scheme for short-wave radiation (Dudhia, 1989), the Thompson scheme for microphysics (Thompson et al., 8), the Kain Fritsch scheme for deep convection (Kain and Fritsch, 1993), which is only activated in the coarse WRF parent domain (see below) and the Noah land-surface model (Chen and Dudhia, 1). These schemes were kept the same in all numerical experiments. In all simulations, vortex-following two-way nests are used to resolve the complicated mesoscale processes in the eyewall region. Figure 1 shows the domain configuration. The parent domain with a horizontal grid-mesh of and grid spacing of 8.1 km is sufficiently large to cover a large part of the tropical west Atlantic so that the large-scale flow pattern can be realistically simulated. Two nests following the vortex are activated in the simulations with a grid-mesh of 1 1 and 1 1, respectively. The nesting ratio is 1:3, so that the innermost domain has a fine resolution of.9 km. A total of 6 levels are configured in the vertical with 13 and 7 layers below km and 1 km, respectively. The initial and boundary conditions are supplied with Geophysical Fluid Dynamics Laboratory (GFDL) model data, which contains a bogus vortex. The bogus vortex, which is generated from an axisymmetric hurricane simulation by a cylindrical coordinate version of the GFDL model, is forced to have approximately the same size and intensity as reported in real time by the National Hurricane Center (NHC). The GFDL analyses are available every 6 h and can be directly processed by the WRF Preprocessesing System (WPS) to generate the desired initial and boundary conditions for the configured WRF simulations. All simulations are executed starting from UTC September 3 and run for 8 h. Since this study focuses on eyewall structure and mesovortices and their relationship to the vertical turbulent mixing processes, to obtain the realistic background flow pattern and to make an easy and meaningful comparison between numerical experiments the WRF simulations were nudged towards the GFDL analyses in the WRF parent domain. This forced agreement in storm track (Figure 1) will not have a substantial impact on the intensity and eyewall structure since nudging is not activated in the nested domains. During the simulations, the sea-surface temperature (SST) is updated every 6 h from the GFDL analysis data. All simulations are performed using WRF version 3.1. Table 1 summarizes the major model configurations and baseline numerical experiments. c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

5 P. Zhu et al. Table 1. Model configurations and numerical experiments. Model configurations Grid mesh Horizontal resolution Vertical levels Nudging Cumulus parametrization 37X89 (d1) 81 m 6 (13 below km) On On 1X1 (d) 7 m 6 (13 below km) Off Off 1X1 (d3) 9 m 6 (13 below km) Off Off Numerical experiments Experiment name Vertical turbulent mixing scheme Simulation period run Yonsei University scheme 8 h run Mellor Yamada Janjic scheme 8 h MYNN-. run Mellor Yamada Nakanishi Niino. TKE scheme 8 h MYNN-3. run Mellor Yamada Nakanishi Niino 3. TKE scheme 8 h Storm center pressure (hpa) MYNN. MYNN 3. NHC best track (b) 9 8 Maximum surface winds (m s 1 ) MYNN. MYNN 3. NHC best track Time (UTC), Sept. Sept., 3 Figure. Time series of hurricane centre sea-level pressure from the four baseline simulations with different vertical turbulent mixing schemes compared with the NHC best track data. (b) The same as except for maximum 1 m wind speeds. This figure is available in colour online at wileyonlinelibrary.com/journal/qj 3. Simulation results In this section, we will first present how different SGS vertical turbulent mixing schemes affect hurricane intensity and axisymmetric eyewall structure, and then focus on the impact of vertical turbulent mixing parametrization on eyewall asymmetric structures and the formation of mesovortices Simulated hurricane intensity and axisymmetric eyewall structures Figure compares the time series of simulated storm centre sea-level pressure and maximum surface winds from the four baseline simulations with observations, where observed hurricane centre pressures and maximum winds were obtained from the hurricane report issued by the NHC. Since the NHC data are obtained using a specific data-processing method, a point-to-point comparison with the WRF simulations is not adequate. Nevertheless, the comparison provides a way to qualitatively evaluate the WRF s performance in simulating hurricane intensity and the sensitivity of simulated storm intensity to the vertical turbulent mixing parametrization. Among all simulations, the storm centre pressure and maximum surface winds predicted by the run are closest to the observations, followed by the run. The performance of the MYNN-. and MYNN-3. runs is quite similar, but they both substantially underestimate the observed hurricane intensity. In addition to the different simulated storm intensities, the numerical experiments with different vertical turbulent mixing schemes also generate different axisymmetric eyewall structures. Figure 3 shows the radial-height structure of azimuthal-mean vertical velocity, PV, hydrometeor mixing ratio and radial/tangential components of horizontal winds averaged from 6 UTC September to UTC September 3. Consistent with its predicted strongest storm intensity, the run also generates the largest eyewall updraughts, PV, hydrometeor content and radial/tangential winds, which are several times stronger than those of the two MYNN runs. It is not clear, though, c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

6 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 1 w (m s 1 ) and V r (m s 1 ) PV (PVU) and V t (m s 1 ) Hydrometeor q c+r+i+s+g (g kg 1 ) 1 1 (b) 1 (c) (d) 1 (e) 1 (f) (g) 1 (h) 1 (i) MYNN (j) 1 (k) 1 (l) MYNN radius (km) radius (km) radius (km) Figure 3. Left column: height radius variations of azimuthal-mean vertical velocity (color shades) and radial component of horizontal winds (contours) from the four baseline simulations. White contours denote radial inflow ( 1, 6, 11,, 1 m s 1 ) and black contours denote radial outflow (1, 6, 11 m s 1 ). Middle column: the same as the left column except for PV (colour shades) and tangential component of horizontal winds (contours). The unit of PV is PVU (1 PVU = 1 6 kg 1 Km s 1 ). The contour interval is 1 m s 1. Right column: the same as the left column except for hydrometeor mixing ratio including cloud, rain, ice, snow and graupel. Note that the variables shown here are averaged from 6 UTC September to UTC September 3. This figure is available in colour online at wileyonlinelibrary.com/journal/qj if these differences in axisymmetric storm structures are the causes for or the outcome of the different strength of the simulated storm intensity (Figure ). Nonetheless, these results suggest that the hurricane intensity and axisymmetric storm structure are sensitive to the vertical turbulent mixing parametrization. How vertical turbulent mixing affects the storm intensity is an important but complicated question. But since this study focuses on eyewall asymmetric disturbances and mesovortices, we will leave this question for future research. The role of vertical turbulent mixing is to transport the energy obtained from the ocean surface upward to the atmosphere aloft. The tendency generated by the vertical turbulent mixing is an important term in the heat and moisture budgets, which may be represented as θ t =... w θ =...+ z z (K θ θ ), (1) z q t =... w q =...+ z z (K q q ), () z where w, θ and q denote vertical velocity, potential temperature and water vapour mixing ratio, respectively. c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

7 P. Zhu et al. Time variations of SGS heating rates (HR) (K h 1 ) 3 1 MYNN 3. MYNN. (no sst updating) 6 6 Time (UTC), Sept. Sept., 3 (b) mean SGS HR (K h 1 ) (c) mean SGS HR (K h 1 ) (d) MYNN. mean SGS HR (K h 1 ) (e) MYNN 3. mean SGS HR (K h 1 ) Figure. SGS heating rates vertically averaged over m from the four baseline simulations with different vertical turbulent mixing schemes. The dotted line represents the experiment with the scheme but without 6 h SST updating. Note that the heating rate includes both sensible and latent heating and has been averaged over the entire innermost domain D3. (b e) Height radius variations of azimuthal-mean SGS heating rates averaged over the time periods from 6 UTC September to UTC September 3. This figure is available in colour online at wileyonlinelibrary.com/journal/qj Prime and overbar represent perturbation and mean. K θ and K q are the turbulent eddy exchange coefficients for heat and moisture, respectively. WRF assumes K θ = K q.in the WRF model, the SGS heating rate and moistening rate ( z (K θ θ z ), z (K q q z )) can be directly output from a vertical turbulent mixing scheme. The total SGS heating rate (R sgs ), which combines both sensible and latent heating, may then be written as [ ] R sgs = 1 ρc p ρc p z (K θ θ z ) + ρl z (K q q z ) = z (K θ θ z ) + L C p z (K q q z ), (3) where ρ, C p,andl denote the air density, specific heat at constant pressure and specific latent heat of evaporation, respectively. Thus the magnitude of the total SGS heating rate (R sgs ) may be used as a measure to examine the direct effect of an SGS vertical turbulent mixing scheme on hurricane simulations. Using Eq. (3), the total SGS heating rate (R sgs ) can be readily calculated from the tendencies generated by the WRF vertical turbulent mixing scheme saved in the model output. Figure shows the time variations of domainmean (D3) total SGS heating rates averaged over m and the height radius plots of azimuthal-mean SGS heating rates averaged over the time period from 6 UTC September to UTC September. The time series of SGS heating rates due to the vertical turbulent mixing from the four baseline simulations (Figure ) show an apparent discontinuity every 6 h. The detailed analyses show that this discontinuity is caused by the 6 h SST updating from the GFDL analyses during the simulations. The GFDL SST field has been adjusted to the bogus vortex; thus it contains a cold SST pool induced by the storm, which changes its location and magnitude every 6 h as the storm evolves. Since the surface fluxes are calculated by the difference between SST and air temperature at the lowest model level, the changes in SST lead to the discontinuity in SGS heating rate. To confirm this, we performed exactly the same simulations but without SST updating. As an illustration, the total SGS heating rate from the run without SST updating is also shown in Figure. The discontinuities are completely removed in the run without SST updating. This is true for all simulations with different vertical turbulent mixing schemes. Note that this discontinuity in SST updating does not appear to have a substantial impact on the storm intensity (Figure ), possibly due to the fact that the SGS heating induced by the vertical turbulent mixing is only one term in the heat and moisture budgets. Other important terms in the budgets do not have such a discontinuity. The simulations show that the storm intensity is a little stronger without SST updating, probably because of warmer SST at initial time for this particular case. Despite the discontinuities caused by SST updating, Figure clearly shows that the run generates the largest SGS heating rates. The large amount of energy (mostly the latent heat as indicated by Figure ) transported by the parametrized vertical turbulent mixing is one of the important energy sources to allow the simulated storm vortex to reach the observed intensity. The height radius variation of SGS heating clearly shows that the enhanced SGS heating is mainly in the eyewall region. Compared with the run, the total SGS heating rates due to vertical c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

8 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 3 (W m ) (W m ) 1 1 (b) 3 1 MYNN. MYNN 3. Surface latent heat flux MYNN. MYNN 3. Surface sensible heat flux Figure. Simulated azimuthal-mean radial profiles of surface latent heat fluxes and sensible heat fluxes (b) averaged over the time period from 6 UTC September to UTC September 3. This figure is available in colour online at wileyonlinelibrary.com/journal/qj turbulent mixing generated by the MYNN runs are very weak, particularly in the eyewall region, where they are several times weaker than that of the run. As stated previously, in the WRF model different vertical turbulent mixing schemes usually work with their own surface layer parametrizations. Although they are all formulated based on the Monin Obukhov similarity theory, the WRF surface layer schemes are not exactly the same. Thus it is important to examine whether the different SGS heating rates due to vertical turbulent mixing are related to the different surface layer parametrizations. Figure shows the radial profiles of azimuthal-mean surface latent and sensible heat fluxes averaged from 6 UTC September to UTC September. As expected, the run generates larger surface latent heat fluxes than other runs, particularly in the vicinity of the eyewall. Note that the four simulations generate almost identical heat fluxes in the outer region (i.e. radius approximately greater than 7 km). This is consistent with the large SGS heating rates concentrated in the eyewall region (Figure ). However, it does not appear that the SGS heating rate is solely determined by the surface heat fluxes. The run also generates a larger SGS heating rate than the two MYNN runs. However, as indicated by Figure, the surface latent heat fluxes generated in the run are smaller than those in the MYNN runs. Note that the run does generate larger sensible heat fluxes, but they are nearly three times smaller than the latent heat fluxes in these simulations. Thus a large surface heat flux does not necessarily mean a large SGS heating rate, although it is an important factor in the SGS heating. In simulations for a given surface heat flux, the vertical distribution of SGS turbulent heat transport also depends on how the vertical mixing processes are parametrized. 3.. Eyewall asymmetric structure and mesovortices In this study, the output from the innermost domain allows us to examine the eyewall evolution and activity of eyewall asymmetric disturbances, including mesovortices, in detail during the -day WRF simulations. Our analyses show that the four baseline experiments with different vertical turbulent mixing schemes generate substantially different eyewall structures. Figure 6 shows the mean PV field from to 8 m averaged over a h period along with the azimuthal-mean PV profiles. A striking difference among these experiments is that the eyewall PV field in the and runs exhibits high-frequency perturbations along the eyewall even after a long time period average, whereas in the MYNN runs the eyewall PV field shows relatively smooth low-frequency variations along the eyewall. In a recent study, Fierro et al. (9) showed that as horizontal resolution increases smaller-scale eddies develop and tend to dominate over coherent low-wavenumber variations. However, based on the fixed high horizontal resolution simulations in our study, variations in the strength of highfrequency perturbations can also be generated by different vertical turbulent mixing schemes. To analyse the detailed structure of eyewall asymmetric disturbances, we examined the eyewall structure at an instant of time. Figure 7 shows the PV field at 19 UTC September from the four baseline experiments, where the PV field has been averaged over a depth from to 8 m. The instantaneous PV field shares similar characteristics to that of the mean PV field (Figure 6) in the sense that highfrequency perturbations appear to dominate the eyewall PV field in the and simulations, in contrast to the relatively low-frequency eyewall PV variations in the MYNN runs. High-frequency eyewall disturbances in the and runs were also reported in previous numerical simulations by Nolan et al. (9a, 9b). They argued that these smaller-scale variations resemble those known as misocyclones or eyewall vorticity maxima (EVMs) identified from radar data and flight-level data (Abserson et al., 6; Marks et al., 8). They showed that these EVMs consist of significant updraught and downdraught couplets. We shall show later in the section (Figure 11) that these high-frequency eyewall disturbances can result in strong vertical transport of heat and moisture. In contrast to the and runs, the high-frequency perturbations are suppressed in the MYNN runs. Instead, the eyewall perturbations appear to be dominated by lowfrequency variations. In this case, the existence of eyewall mesovortices is easier to identify from the PV fields. For example, in the MYNN-. run (Figure 7(c)), the somewhat spiral shell-shaped feature on the southeast side of the eyewall PV ring (indicated by the black arrow) may be considered as the signature of an eyewall mesovortex. In the MYNN-3. run (Figure 7(d)), the apparent elliptical eyewall PV ring may have resulted from stretching by a pair of mesovortices. To clearly show the eyewall mesovortices in Figure 7, we decomposed the PV field in terms of wavenumbers using the Fourier transform. Figure 8 shows the PV perturbations of low wavenumbers from 1 to (i.e. perturbations associated with wavenumbers greater than have been filtered out) along with the azimuthal-mean radial PV profile (i.e. wavenumber zero) and the normalized power spectra of PV perturbations for radius 3 8 km. The PV field without high-frequency perturbations clearly shows that the asymmetries in the and runs at this moment are most likely associated with the wavenumber 1 perturbations, whereas a wavenumber perturbation appears to dominate the low-frequency PV field in the MYNN runs. Since the total perturbation energy is conserved under the Fourier c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

9 P. Zhu et al. 6 (b) Y (km) Y (km) X (km) X (km) (c) 6 MYNN. (d) 6 MYNN Y (km) Y (km) X (km) X (km) PV (PVU) (e) 3 Basic state PV profile MYNN. MYNN Figure 6. (a d) Mean PV field from to 8 m from the four baseline simulations averaged over the period from 6 UTC September to 6 UTC 13 September 3. The unit of PV is PVU (1 PVU = 1 6 kg 1 Km s 1 ). (e) Azimuthal-mean PV profiles, where the small circle on the profile indicates the minimum of the inner edge PV of the eyewall PV ring and two vertical bars indicate the midpoint between the inner-edge PV minimum and the PV peak on both sides of the PV slope. This figure is available in colour online at wileyonlinelibrary.com/journal/qj transform, one can decompose the variance of the PV perturbations in terms of wavenumbers represented by N 1 ση = c(n), () n=1 where ση is the variance of the PV perturbations, n is the wavenumber, c(n) is the Fourier transform coefficient and N is total number of points being analysed. Thus we may define a normalized power spectra as E(n) = c(n) ση, N 1 E(n) = 1. () n=1 Clearly, E(n) provides a way to quantify the contribution of a certain wavenumber to the total perturbation energy as a percentage. Figure 8(f) shows the normalized power spectra of the eyewall PV perturbations averaged over the annulus c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

10 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 19: UTC, Sept. (b) 19: UTC, Sept (c) 1 MYNN. 19: UTC, Sept. (d) 6 1 MYNN 3. 19: UTC, Sept Figure 7. Examples of PV field at 19 UTC September 3 from the run, run (b), MYNN-. run (c) and MYNN-3. run (d), where the unit of PV is PVU (1 PVU = 1 6 kg 1 K m s 1 ). Note that the PV field has been averaged from to 8 m. This figure is available in colour online at wileyonlinelibrary.com/journal/qj from 3 to 8 km, where the spectra for wavenumbers greater than 1 have been added together for a clear illustration. The power spectra clearly show that the low-frequency PV perturbations are dominated by wavenumber 1 in the and simulations, but wavenumber has the largest contribution to the PV perturbations in the MYNN runs. As expected, high-wavenumber perturbations contribute more to the total energy in the and runs than that in the MYNN runs. In addition to their impact on the eyewall asymmetric disturbances, the four vertical turbulent mixing schemes also lead to different structures of azimuthal-mean PV radial profiles (Figure 8), similar to that seen for the mean PV profile averaged over a time period (Figure 6). Here we would like to emphasize two important differences of the azimuthal-mean PV radial profiles between simulations. First, although all four experiments show that there is an elevated PV in the inner edge of the eyewall PV maximum, the ratio of PV in the eye to the mean PV (averaged from the storm centre to the outer edge of PV) in the run is smaller than that in the and MYNN runs. Second, the width of the eyewall PV ring is quite different in these simulations. As illustrated by Figure 6(e), the width of the c 13 Royal Meteorological Society eyewall PV ring here is defined as follows: a midpoint between the PV peak and the minimum of inner edge PV of the PV ring may be found on the both sides of the PV slope. Then, the radial distance between the midpoints may be defined as the width of the eyewall PV ring. Although such defined width of the eyewall PV ring is somewhat arbitrary, it qualitatively distinguishes the different characteristics of the basic state eyewall PV field between simulations. As indicated in Figures 8 and 6(e), the width of the eyewall PV ring in the run is much thicker than that in the and MYNN runs. To understand why these parameters, such as the ratio of inner edge PV to the mean PV of a vortex and the width of the eyewall PV ring, are important to the instability of the eyewall asymmetric disturbances, it is helpful to revisit Schubert s barotropic instability theory of eyewall vortex Rossby waves in the absence of diabatic heating and boundary layer forcing. Considering a basic state relative vorticity field ζ (r) represented by ξ 1 + ξ ξ ζ (r) = < r < r1 r 1 < r < r r < r <, Q. J. R. Meteorol. Soc. : 38 () (6)

11 6 P. Zhu et al. PV (PVU) Azimuth mean radial PV profile (b), WN 1 (PVU) (c) MYNN. MYNN Distance (km) Distance (km) 1 1 Distance (km), WN 1 (PVU) Distance (km) Distance (km) (d) MYNN., WN 1 (PVU) (e) MYNN 3., WN 1 (PVU) (f) Distance (km) Distance (km) Distance (km) NPS of PV for R=3 8km MYNN. MYNN >1 Wavenumber Figure 8. Decomposed PV field at 19 UTC September 3 (shown in Figure 7) using the Fourier transform. Basic state radial PV profile (i.e. wavenumber ). (b e) PV perturbations of wavenumbers 1 from the,, MYNN-. and MYNN-3. runs, respectively. (f) Normalized power spectra of PV averaged over the radius from 3 to 8 km. Note that the contributions from perturbations with wavenumber greater than 1 have been added together for a clearer illustration. This figure is available in colour online at wileyonlinelibrary.com/journal/qj Figure 9. Isolines of the dimensionless frequency ν as a function of δ and γ for wavenumbers m = 3,,, 6, 7, 8. Positive growth rate (instability) ζav occurs only in the shaded areas. The isolines are ν i =.1,.,.3,... The largest growth rates occur in the lower-right corner of each panel (adapted ζav from Schubert et al., 1999). where constants ξ 1 and ξ are the vorticity jumps at the radius r 1 and r respectively, Schubert et al. (1999) was able to derive an analytical solution of the linearized barotropic non-divergent vorticity equation for a small disturbance to an elevated vorticity ring (ξ 1 < andξ > ). They showed that the barotropic instability of eyewall vortex Rossby waves depends on δ = r 1 /r and γ = (ξ 1 + ξ )/ζ av, the two parameters that describe the basic state of the vorticity ring, where ζ av is the averaged vorticity over the region r r. Their analytical solution allows them to calculate the dimensionless complex frequency ν ζ av for various wavenumbers m as a function of δ and γ. Figure 9 shows the result for wavenumbers m = 3,,, 6, 7, and 8, where the unstable area has been shaded with grey. The figure clearly indicates two important results of vortex Rossby wave instability: (i) the thinner the annular vorticity ring (i.e. the larger the value of δ = r 1 /r ) and the smaller inner-region vorticity (i.e. the smaller the value of γ ), the greater the instability growth rate; and (ii) the larger the wavenumber m, the thinner the vorticity ring (i.e. a larger δ) is required for vortex Rossby waves to be unstable. In the previously summarized Schubert s barotropic instability theory, the parameters δ and γ that describe the basic state vorticity ring are treated as the external c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

12 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 7 Table. Parameters of basic state PV of different experiments. Parameter MYNN-. MYNN-3. δ γ controlling factors. Their analysis itself is unable to provide further information on how these parameters change in response to the change of external forcings. In reality, the basic state vorticity is affected by the diabatic heating and turbulent mixing processes. The numerical experiments performed in this study allow us to investigate how the SGS vertical turbulent mixing parametrizations affect these important parameters that largely determine the barotropic instability of eyewall vortex Rossby waves. As shown in Figures 8 and 6(e), the basic state radial PV profiles are substantially sensitive to the vertical turbulent mixing parametrizations. Based on the width of the eyewall PV ring defined previously, one may provide a rough estimate of these important parameters. As an example, Table lists the estimated values of δ and γ of the h mean PV radial profiles (Figure 6(e)) from the four baseline experiments. In these estimates, r 1 and r in Eq. (6) are defined at the midpoints on the inner and outer slope of the PV profile as illustrated by Figure 6(e). ξ 1 + ξ and ξ are then taken as the mean PV of < r < r 1 and r 1 < r < r, respectively. The results show that the influence of the vertical turbulent mixing on the barotropic instability of vortex Rossby waves appears to be complicated according to Schubert s theory. For example, in the run, the thicker annular PV ring seems unfavourable for barotropic instability, particularly for high-wavenumber perturbations. However, the smaller ratio of the inner edge PV to the vortex mean PV implies a greater instability growth rate once the instability is initiated. In contrast, the relatively thinner annular PV ring in the and MYNN runs may support instability for a wider range of wavenumbers, but the large ratio of the inner-edge PV to the vortex mean PV suggests a relatively slow instability growth rate. Such a dependence of the basic state PV on the SGS vertical turbulent mixing is likely due to the fact that the diabatic heating is sensitive to the SGS scheme as inferred from Figures 3 and Figure 11 shown later, and the SGS scheme, in turn, responds to the diabatic heating. As a result, different diabatic heating rates/distributions result in different basic state PV profiles during the course of each model integration since diabatic heating is an important source of generation of PV. In short, the above analyses (Figures 6, 7 and 8) suggest that the eyewall asymmetries are affected by the vertical turbulent mixing schemes through two different ways. First, different parametrizations tend to generate different frequency eyewall disturbances. We note that some extremely high-frequency perturbations shown in the and simulations are likely not vortex Rossby waves. As discussed previously, these small-scale eddy circulations more likely resemble misocyclones or EVMs shown in radar and aircraft measurements. Their formation mechanism may be complicated and will not be discussed in the paper, but we will show later that these small-scale eddy circulations can create strong horizontal and vertical mixing to affect the vortex axisymmetric structures. Second, the vertical turbulent mixing parametrization plays an important role in modulating the basic state PV radial profile, likely via diabatic heating, which in turn can strongly affect the barotropic instability of eyewall vortex Rossby waves and the development of misocyclones. The altered axisymmetric structure of the storm further induces changes in the vertical turbulent mixing. These mechanisms interact with each other to result in a really complicated impact of vertical turbulent mixing parametrization on eyewall asymmetries. As shown previously, the basic state PV radial profile can change substantially due to different vertical turbulent mixing schemes. One of the likely causes of the different widths of the annular PV ring may be the resolved eyewall disturbances, since their induced horizontal mixing provides a mechanism to dilute the eyewall PV concentration. This may be inferred from the horizontal fluxes induced by the resolved structures. In the cylindrical coordinate, the resolved horizontal fluxes of PV by the radial component of the horizontal winds at a given height z and radius r may be written as v r η (z, r) = 1 N 1 [v r,n (z, r) v r (z, r)][η n (z, r) η(z, r)], N n= where v r and η represent the radial component of horizontal winds and PV, respectively. The wide overhat indicates the azimuthal average along the circle at the radius r. The prime denotes the perturbation with respect to the average. N is the total number of grid points in the circle at radius r. In the cylindrical coordinate, positive and negative v r η represent outward and inward transport of PV, respectively. Using the WRF output, the resolved horizontal fluxes of PV in the radial direction, v r η (z, r), can be calculated. Figure 1 shows the azimuthal-mean resolved horizontal PV fluxes in the radial direction averaged from 6 UTC September to UTC September. All simulations show negative (inward) PV flux inside the PV ring (maxima) and positive (outward) PV flux outside the PV ring (maxima) above the inflow layer. This result is expected from a down-gradient transport perspective. As illustrated by the figure, the resolved processes in the run generate larger horizontal fluxes of PV in the radial direction than those in the MYNN runs. The different strength in radial fluxes of PV by the resolved structures may be one of the reasons for the relatively concentrated PV ring in the MYNN runs compared with that in the run. In addition to generating horizontal mixing, the resolved organized structures can also efficiently transport energy, moisture and momentum vertically through the non-local mixing mechanism (Zhu, 8). From the model output, we can directly compute the vertical fluxes at a given height induced by the resolved structures using the eddy correlation method similar to that for calculating horizontal PV fluxes. Figure 11 shows the height radius variation of resolved vertical buoyancy fluxes and moisture fluxes, ŵ θ v (z, r) and ŵ q (z, r) averaged from 6 UTC September to UTC September. To show the vertical structure clearly, the vertical profiles of resolved fluxes at an arbitrary radius in the vicinity of the eyewall (here 7.7 km is chosen) are also shown in the figure. A couple of things can be seen from the figure. First, the strongest resolved vertical fluxes are in the eyewall region, but its influence extends far to the outer region of the hurricane. This is understandable (7) c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

13 8 P. Zhu et al. Horizontal fluxes of PV by the radial component of horizontal winds ( 1 ms ) 1 (b) (c) MYNN. 1 1 (d) MYNN Figure 1. Height radius variation of azimuthal-mean resolved horizontal fluxes of PV in the radial direction ( v r η ) averaged from 6 UTC September to UTC September 3. Black contour indicates the zero value of PV fluxes. White contours indicate the eyewall PV field. The outer and inner contours represent 19 PVU and 1 PVU, respectively (1 PVU = 1 6 kg 1 Km s 1 ). This figure is available in colour online at wileyonlinelibrary.com/journal/qj since the resolved mesoscale processes, which often consist of well-defined updraughts and downdraughts that can effectively promote vertical transport, can exist not only in the eyewall but also in rain bands in the outer regions of a hurricane. Second, the resolved vertical fluxes generated in the run are much stronger than other simulations, which is consistent with the stronger storm intensity in this run since resolved vertical flux is an important mechanism for the upward energy transport. To quantify the individual contribution of different frequency eyewall disturbances to the resolved vertical and horizontal transport, we decomposed the vertical buoyancy and moisture fluxes, and radial PV fluxes using the Fourier transform. Figure shows the normalized power spectra of these fluxes averaged over the height from to 8 m, radius from 3 to 8 km and time period from 6 UTC September to UTC September, where wavenumbers higher than are grouped together for a clear illustration. The decomposition clearly shows the differences between the / and MYNN simulations. In the and simulations, the resolved eyewall perturbations with wavenumbers greater than contribute roughly one third of the total energy of vertical and horizontal fluxes, whereas the disturbances with low wavenumber 1 collectively contribute about % of the total energy for buoyancy fluxes and about 3% for moisture fluxes. The wavenumber 1 perturbation, however, appears to be more important to horizontal PV transport, as it alone contributes more than % of the total energy of radial PV fluxes. On the other hand, the perturbations with wavenumbers greater than in the MYNN simulations only contribute roughly 1 1% of the total energy of vertical and horizontal fluxes. In this case, low-wavenumber perturbations are more important to the transport. As indicated by the figure, wavenumber 1 or alone contributes about 1 % of the total energy for the vertical fluxes, whereas for horizontal transport wavenumber 1 contributes more than 3% of the total energy. Previous analyses indicate that the SGS vertical turbulent mixing parametrization exerts a substantial impact on the dynamics of the eyewall through the regulation of the basic state PV distribution in the eyewall vicinity and the generation of eyewall disturbances with different wavenumbers. These two mechanisms also interact with each other due to the fact that high-frequency disturbances can generate strong horizontal mixing to change the vortex basic state axisymmetric structure, which in turn changes the barotropic instability of eyewall vortex Rossby waves. The formation of eyewall mesovortices is thus a delicate process that can be easily changed or destroyed due to the interaction among the SGS vertical turbulent mixing, diabatic heating and resolved eyewall structures. Lastly, it is important to point out that this argument is purely from the barotropic instability perspective. We do not exclude the possibility that mesovortex-like circulations may form due to other mechanisms, which will be further investigated in our future studies.. Sensitivity of eyewall asymmetries to vertical turbulent mixing parametrization The previous results raise an important question that needs to be further addressed: why do the and schemes tend to produce high-frequency perturbations along the eyewall whereas the MYNN schemes tend to produce stronger low-frequency variations? This phenomenon was also observed and discussed by Nolan et al. (9a, 9b), but the underlying reasons for this phenomenon have not been clarified. A further question that may help address this c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

14 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries (c) (b) 1 1 Resolved moisture flux w q (m s 1 g kg 1 ) (d) (e) (f) (g) (h) Resolved buoyancy flux w q v (m s 1 K) MYNN. MYNN 3. (i) Resolved buoyancy flux (m s 1 K) at radius 7.7 km MYNN. MYNN MYNN. MYNN 3. (j) Resolved moisture flux (m s 1 g kg 1 ) at radius 7.7 km MYNN. MYNN Figure 11. (a h) Height radius variations of azimuthal-mean resolved vertical buoyancy fluxes and moisture fluxes (ŵ θ v and ŵ q ) averaged from 6 UTC September to UTC September 3 from the four baseline simulations. Black contours indicate the hydrometeor mixing ratio. The outer and inner contours represent 1. and 3 g kg 1, respectively. (i, j) Vertical profiles of resolved buoyancy fluxes and moisture fluxes at a radius of 7.7 km. This figure is available in colour online at wileyonlinelibrary.com/journal/qj issue would be: among various parts of a scheme, what is the key component of the scheme that is fundamentally responsible for such a difference? The answers to this question will provide critical guidance for developing physically robust parametrizations that can realistically represent the SGS vertical turbulent mixing processes in hurricane conditions. However, diagnosing a cause among several complicated schemes is like seeking a needle in a haystack. To do so, we have looked into the details of schemes and performed numerous experiments to test possible causes responsible for generating high- or lowfrequency disturbances along the eyewall. In this section, we c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

15 3 P. Zhu et al. Normalized power spectra of resolved vertical buoyancy fluxes, w q v (%) MYNN. MYNN > (b) Normalized power spectra of resolved vertical total water fluxes, w q t (%) MYNN. MYNN > (c) Normalized power spectra of resolved horizontal radial fluxes of PV, n r h (%) MYNN. MYNN > Wavenumber Figure. Normalized power spectra of resolved vertical buoyancy fluxes w θ v averaged over the radius from 3 to 8 km, height from to 8 m and time period from 6 UTC September to UTC September 3. (b, c) The same as except for vertical moisture fluxes w q, and radial PV fluxes v r η, respectively. This figure is available in colour online at wileyonlinelibrary.com/journal/qj present our findings and provide physical explanations for them. Our investigations show that the magnitude and vertical distribution of the eddy exchange coefficient mainly control the characteristics of eyewall asymmetric disturbances in our simulations. Figure 13 shows the height radius variation of the azimuthal-mean eddy exchange coefficients for heat and momentum from the four baseline simulations. To clearly show the vertical structure, the profiles of eddy exchange coefficients at a radius of 7.7 km (an arbitrary radius in the vicinity of the eyewall) are also shown in the figure. Clearly, the and runs the two that generate high-frequency perturbations along the eyewall share the same characteristics of eddy exchange coefficient, which is substantially smaller in magnitude and in depth than that in the MYNN runs. At the first glance, the large SGSheatingratesintheandruns(Figure) seem to be in conflict with their small eddy exchange coefficients (Figure 13). But this paradox can be readily clarified from the K-theory of turbulence closure. As stated earlier, the sensible heating rate, moistening rate and total heating rate due to the SGS vertical turbulent mixing may be written as z (K θ θ z ), z (K q q z ), z (K θ θ z ) + L C p z (K q q z ), respectively. From these expressions, it is clear that the SGS heating rates do not directly depend on the eddy exchange coefficients; rather they are determined by a complicated combination of the first derivative of the eddy exchange coefficient and the second derivative of mean variables. Fundamentally, K-theory is a local closure that assumes down-gradient diffusion in which vertical transport (or vertical flux) is determined by the vertical gradient of a mean variable and the eddy exchange coefficient, a parameter that measures eddy mixing ability. Thus strong mixing (large eddy exchange coefficient) does not necessarily lead to large SGS heating rates and vice versa. For example, in a well-mixed condition (the strongest mixing that one c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

16 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 31 K m (m s 1 ) K h (m s 1 ) 1 3 (b) (c) (d) (e) MYNN (f) MYNN (g) 1 MYNN 3. 3 (h) 1 MYNN (i) 1 1 K m (m s 1 ) at radius 7.7 km MYNN. MYNN 3. (j) 1 1 K h (m s 1 ) at radius 7.7 km MYNN. MYNN Figure 13. (a h) Height radius variations of azimuthal-mean eddy exchange coefficients for momentum and heat averaged from 6 UTC September to UTC September 3 from the four baseline simulations. (i, j) Vertical profiles of eddy exchange coefficient at a radius of 7.7 km. This figure is available in colour online at wileyonlinelibrary.com/journal/qj could find), in order to obtain finite vertical fluxes the eddy exchange coefficient has to approach infinity. This singularity is regarded as the major limitation of K-theory when vertical mixing is dominated by strong non-local mixing. Regardless of the deficiency associated with K- theory, the paradox between Figures and 13 indicates nothing but a highly nonlinear relationship between SGS heating and the method of parametrizing vertical turbulent mixing via the eddy exchange coefficients. Eddy exchange coefficients are the key parameters in a vertical turbulent mixing scheme. The substantial difference in eddy exchange coefficients between schemes (Figure 13) indicates that these important parameters are not well parametrized and constrained under hurricane conditions. Recently observational analyses on the eddy exchange coefficients (e.g. Zhang et al., 11b) may shed new light on how to evaluate the eddy exchange coefficient parametrization under hurricane conditions. However, c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

17 3 P. Zhu et al. (b) 1.h (c) h (d) h (e) K m (m s 1 ) (f) K h (m s 1 ) MYNN. 1.h.h h 6 MYNN. 1.h.h h Figure. PV field from the baseline run -1.h run (b), -.h run (c) and -h (d) at 9 UTC 13 September 3, where the unit of PV is PVU (1 PVU = 1 6 kg 1 Km s 1 ). Note that the PV field has been averaged from to 8 m. The domain-averaged vertical profiles of eddy exchange coefficient of momentum and heat at this time are shown in (e) and (f), respectively. This figure is available in colour online at wileyonlinelibrary.com/journal/qj since environmental conditions and vertical turbulent mixing can be substantially different for different hurricanes and different regions of a hurricane, how to appropriately use observations for parametrization validation is a complicated scientific issue that needs to be further addressed. In this paper, we made no attempt to judge which scheme or what value of the eddy exchange coefficients is right or wrong. Our major objective is to demonstrate how different vertical turbulent mixing schemes affect the evolution and structure of eyewall asymmetries. In the following paragraphs, we will use the scheme as an example to illustrate how the eddy exchange coefficient modulates the frequency of eyewall disturbances and the development of eyewall mesovortices. In the scheme, the eddy exchange coefficients for momentum (K m ) and heat (K h ) are parametrized differently below and above the boundary layer as follows: for z h K m = kw s z(1 z h ), K h = K m /P r, for z > h K m = l f m (R i ) ( U z ) + ( V z ), K h = l f h (R i ) ( U z ) + ( V z ), (8) (9) where U, V, z and h denote x-direction and y-direction mean wind speeds, height and boundary layer depth, respectively. k is the von Karman constant, w s the mixed layer velocity scale, P r the Prandtl number, l the mixing length, R i the Richardson number, and f m and f h the stability function for momentum and heat, respectively. For detailed information of these parameters, please refer to Hong et al. (6). From the equations, it is easy to see that for certain surface fluxes and boundary layer stability the vertical distribution and the magnitude of the eddy exchange coefficients are determined by the boundary layer height, which is diagnosed based on the critical bulk Richardson number and virtual potential temperature at the lowest model level using an iteration method. Thus we may change the shape and magnitude of the eddy exchange coefficients by simply adjusting the boundary layer height. Note that these experiments are done only for demonstration purposes but not as any means for tuning of the scheme. Here, we present three additional experiments with the scheme in which everything else is the same except that we increased the boundary layer depth by a factor of 1.,. and. These experiments were named -1.h, -.h and -h, respectively. Note that an accurate determination of boundary layer height is difficult under hurricane conditions, particularly in the eyewall region, since there is no clean physical interface, such as a strong inversion that exists in some other conditions, to separate the turbulence directly linked to the surface processes from that generated aloft due to convective clouds. In such conditions, the turbulent mixing layer is not determined by the boundary layer height but by the physical processes that generate turbulence. The boundary layer height in the scheme is only a diagnostic variable and is used to determine the magnitude and vertical structure of the turbulent eddy exchange coefficients. Whether such a parametrization in terms of the diagnosed boundary layer height can appropriately represent the turbulent eddy c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

18 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 33 1 UTC Sept. (b) 11 UTC Sept (c) UTC Sept (d) 13 UTC Sept. (e) UTC Sept (f) 1 UTC Sept (g) UTC Sept. (h) 17 UTC Sept (i) UTC Sept Figure 1. Evolution of eyewall mesovortices in the -h run on September 3. This figure is available in colour online at wileyonlinelibrary.com/journal/qj exchange coefficients under hurricane conditions requires thorough evaluations against observations. Nonetheless, through adjusting the boundary layer height, one can obtain different magnitude and vertical structure of the turbulent eddy exchange coefficients. In other schemes, such as and MYNN, the turbulent eddy exchange coefficients are determined based on the predicted TKE, mixing length and stability. In this case, one can adjust the mixing length to obtain different eddy exchange coefficients. The experiments show that the change in turbulent eddy exchange coefficients significantly alters the structure and dynamics of eyewall asymmetries and the formation of mesovortices. As an example, Figure shows the eyewall PV field at an instant time (9 UTC 13 September) from the four experiments along with the domain mean vertical profiles of the eddy exchange coefficients for momentum and heat at this time. For comparison, the eddy exchange coefficient profiles from the run and MYNN-. run are also shown in the figure. The modulation of eyewall asymmetric PV structure by the eddy exchange coefficients is clearly shown in the figure. As the eddy exchange coefficients increase, the frequency of PV disturbances decreases and the eyewall PV field becomes more organized. When the eddy exchange coefficients reach a magnitude close to those in the MYNN-. run, the PV fields in the run exhibit similar features to those shown in the MYNN runs. In the case of the -h run in which the eddy exchange coefficients become exceptionally large, robust swirling eyewall mesovortices form along the eyewall and can be easily identified from the PV field throughout the simulation. Figure 1 tracks the simulated eyewall mesovortices evolving with time on September from the -h run. It should be noted that the change of the eddy exchange coefficients in the simulations not only alters the structure of the eyewall asymmetries but also leads to different storm intensity (not shown here). How the interaction among the vertical turbulent mixing, diabatic heating and eyewall structure leads to storm intensification is an important question, but it is beyond the scope of this paper. Further examination shows that the characteristics of eyewall disturbances in the simulations appears to be established early in the spin-up stage. Figure shows the simulated water vapour mixing ratio at a height of 7 m at 1 UTC September (i.e. 1 h right after the start of the simulations) from the seven numerical experiments c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

19 3 P. Zhu et al. 1 MYNN. 1 (b) MYNN (c) 1 (d) (e) 1.h 1 (f).h (g) 1 h (h) K h (m s 1 ) MYNN. MYNN 3. 1.h.h h Figure. Water vapour mixing ratio (g kg 1 ) at 7 m from various experiments at simulation hour 1 after the start of simulations along with the domain-mean vertical profiles of eddy exchange coefficient of heat. This figure is available in colour online at wileyonlinelibrary.com/journal/qj along with the domain mean vertical profiles of the eddy exchange coefficients. In the two MYNN runs, the variation of moisture is rather smooth. On the other hand, the moisture field in the and runs revealed small-scale fluctuations in the eyewall region. With the increase of eddy exchange coefficients in the sensitivity runs, the smallscale fluctuations shown in the moisture field are suppressed. The spatial variation of moisture in the -.h and - h runs shows a smooth variation pattern similar to that in the MYNN runs. The initiated perturbations with different frequencies evolve as time goes on. At simulation hour 1 (Figure 17), for example, the small-scale fluctuations along the eyewall have developed into small-scale eddy circulations resembling misocyclones or EVMs discussed previously, whereas the smooth variations shown in the MYNN and -.h and -h runs gradually evolve into mesovortices. As indicated by the figure, the signature of eyewall mesovortices is so clear that it can be easily seen in the moisture fields. Note that the eddy exchange coefficients barely change throughout the simulations (Figures (h) and 17(h)). Because of this, the eyewall disturbances can maintain their frequencies that were established in the early stage. This result suggests that although model spin-up is normally considered to be not physical important flow c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()

20 Impact of SGS Vertical Turbulent Mixing on Eyewall Asymmetries 3 1 MYNN. 1 (b) MYNN (c) 1 (d) (e) 1.h 1 (f).h (g) 1 h (h) K h (m s 1 ) MYNN. MYNN 3. 1.h.h h Figure 17. The same as Figure except for simulation hour 1 after the start of simulations. This figure is available in colour online at wileyonlinelibrary.com/journal/qj features can be established in this stage, which can have important impacts on the simulations in the later stages. The transition from small-scale eddy circulations to low-frequency vortex Rossby waves as the eddy exchange coefficients are increased in the experiments is also shown in the experiments when we adjusted the eddy exchange coefficients to large values (not shown). Similarly, when the eddy exchange coefficients are reduced in experiments with the MYNN schemes, high-frequency perturbations develop along the eyewall. The modulation of eyewall horizontal structure by the eddy exchange coefficients is so robust that it can be easily identified from either PV or moisture fields, as shown in Figures and 17. Why then do small eddy exchange coefficients favour the development of small-scale eddy circulations, whereas large eddy exchange coefficients tend to foster low-frequency perturbations (which may be considered to be vortex Rossby waves) leading to eyewall mesovortices? The answer may lie in the complex interaction among vertical turbulent mixing, mesoscale structures, diabatic heating and barotropic instability of vortex Rossby waves. In the and simulations, when the eddy exchange coefficients are sufficiently small the weak turbulent mixing cannot efficiently transport the energy obtained from the ocean surface upward. As a result, the energy has to be released locally to induce the development of high-frequency disturbances. These resolved high-frequency disturbances are very efficient at vertical and horizontal mixing since they contain intense updraught and downdraught couplets (Nolan et al., 9a) and thus serve as an important c 13 Royal Meteorological Society Q. J. R. Meteorol. Soc. : 38 ()