a. Validation of TC secondary overturning circulation diagnosed by SEE and GSEE

Before applying the GSEE diagnostic tool developed in section 2 to analyze the simulated TC intensification, it is important to examine to the extent to which the TC second overturning circulation diagnosed by GSEE can represent the one directly simulated by HWRF. To do so, we carefully compared the simulated azimuthal-mean transverse circulation with those diagnosed by SEE and GSEE during the evolution of the simulated TC vortex. As an example, Fig. 3 shows the simulated azimuthal-mean vertical velocity and radial flow by EXP-SLOPE-1 averaged over the last 24 h compared with the corresponding SEE and GSEE diagnoses. Overall, there is a good agreement between the simulation and diagnoses in terms of the radial-height structure of vertical velocity and radial flow. In particular, the SEE and GSEE diagnoses well capture the location and vertical extension of the eyewall updrafts and the structure of inflow and outflow simulated by HWRF. However, the SEE diagnoses underestimate the magnitude of the eyewall updraft in the midtroposphere around 5–10 km and the outflow, particularly near the tropopause. In the meantime, the SEE diagnoses overestimate the eyewall updraft in the low troposphere and the radial inflow in the BL for both magnitude and radial extension.

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Comparison of overturning circulation between HWRF simulation and SEE/GSEE diagnoses averaged over the last 24 simulation hours from EXP-SLPOE-1. [a(1)] Azimuthal-mean vertical velocity simulated by HWRF; [a(2)],[a(3)] corresponding vertical velocity diagnosed by GSEE and SEE; [a(4)],[a(5)] bias errors of vertical velocity associated with GSEE and SEE (i.e., GSEE minus HWRF and SEE minus HWRF). [b(1)]–[b(5)] As in [a(1)]–[a(5)], but for radial flow. Note that the white color inside the blue shade in [b(3)] indicates the radial flow exceeds −45 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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The biases shown in the SEE analyses are reduced in the GSEE diagnoses to some extent owing to the inclusion of radial eddy forcing in the analysis. GSEE reduces the overestimated eyewall updraft in the low troposphere by SEE. In particular, GSEE nearly reproduces the magnitude and radial extent of radial inflow simulated by HWRF by cutting down the overestimated inflow by SEE substantially. In addition, GSEE slightly enhances the much-underestimated outflow near the tropopause by SEE. We also calculated the two-dimensional (2D) correlation between the radius–height structure of azimuthal-mean vertical velocity and radial flow simulated by HWRF and the corresponding diagnoses by GSEE and SEE. The results (provided in the supplemental file) show that the 2D correlation coefficients are close to 0.9 for most of the time throughout the simulation period, and GSEE diagnoses are better correlated to the HWRF simulated fields than SEE. However, there are noticeable differences between the simulated fields and SEE/GSEE diagnoses. For example, both SEE and GSEE underestimate the eyewall updraft in the midtroposphere and produce false weak subsidence around the radii of 35–50 km in the BL. These biases could be caused partially by the specified domain boundary conditions when numerically solving SEE/GSEE and partially by the limitation of 2D framework of SEE/GSEE in representing a 3D TC vortex. A detailed discussion on the cause of bias is provided in the supplemental file. Finally, we note that radial flow and vertical velocity affect TC intensification through terms $-\overline{u_r}\overline{\xi }$ (transport of absolute vorticity by radial flow) and $-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)$ (advection of tangential wind by vertical velocity), respectively, according to Eq. (1). The results of the tangential wind budget analyses (to be shown in the next section) show that $-\overline{u_r}\overline{\xi }$ dominates $-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)$ (Fig. 4). Thus, we argue that the biases in the azimuthal-mean vertical velocity field should not have a substantial impact on the GSEE diagnoses presented in the following sections.

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(first row) Azimuthal-mean SGS tangential eddy forcing $F_{{\text{sgs}}_{\lambda }}$ (m s⁻²) averaged over the last 24 h from EXP-SLOPE-1, BASELINE, and EXP-SLOPE-2, respectively. (second row) As in first row, but for the radial transport of absolute vorticity $-\overline{u_r}\overline{\xi }$ (m s⁻²). (third row) Vertical transport of tangential wind $-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)$ (m s⁻²). (fourth row) Tendencies from the resolved eddies. (bottom row) Net tendency of all terms on the RHS of Eq. (1). White contours in all panels indicate zero tendencies. Black contours in the second row are the absolute vorticity (s⁻¹). Black contours in the third and bottom rows are the tangential wind (m s⁻¹). Red solid and dashed contours in the second and bottom rows are the radial inflow and outflow (m s⁻¹), respectively. Red contours in the third row are the vertical velocity (m s⁻¹).

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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b. TC vortex acceleration in the three experiments

The azimuth-mean tangential wind budget equation (Eq. (1)) provides an excellent way to understand how different dynamic processes contribute to the acceleration of the primary circulation of a TC. Using the model output, KZ-1 presented a detailed tangential wind budget analysis. Figure 4 summarizes the budget analyses of the three experiments. The SGS tangential turbulent eddy forcing $F_{{\text{sgs}}_{\lambda }}$ is negative in the inflow layer (Figs. 4a–c), which is the main momentum sink of a TC. The radius–height structure of $F_{{\text{sgs}}_{\lambda }}$ is consistent with that of Persing et al. (2013) and Zhu et al. (2019). On the other hand, the radial transport of absolute vorticity ($-\overline{u_r}\overline{\xi }$) generates large positive and negative tendencies in the inflow layer and outflow layer, respectively (Figs. 4d–f). This is easy to understand because $\overline{\xi }$ is positive everywhere, and thus the sign of $-\overline{u_r}\overline{\xi }$ is determined by the radial flow. The peak positive tendency of $-\overline{u_r}\overline{\xi }$ occurs somewhere between the strongest inflow and maximum $\overline{\xi }$ depending on the specific structure of a vortex. Similarly, since $\overline{w}$ is positive in the vicinity of radius of maximum wind (RMW), the sign of $-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)$ (Figs. 4g–i) is determined by the vertical gradient of tangential wind, which is positive and negative below and above the height of peak tangential wind, respectively. Therefore, $-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)$ has a sign opposite to $-\overline{u_r}\overline{\xi }$, causing the two terms to cancel each other. However, since the magnitude of $-\overline{w}(\partial {\overline{\upsilon }}_t/\partial z)$ is overwhelmed by $-\overline{u_r}\overline{\xi }$, the net tendency of $-\overline{u_r}\overline{\xi }-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)$ is to accelerate the primary circulation in the inflow layer near RMW. The tendencies resulting from the eddy correlation of model-resolved asymmetric eddies $-\overline{u_r^{\prime }{\zeta }^{\prime }}-\overline{{\upsilon }_t^{\prime }(\partial {\upsilon }_t^{\prime }/r\partial \lambda )}-\overline{w^{\prime }(\partial {\upsilon }_t^{\prime }/\partial z)}\hspace{0.17em}$ (Figs. 4j–l) are negligible compared to those of the advection terms by the mean flow $[-\overline{u_r}\overline{\xi }-\overline{w}(\partial \overline{{\upsilon }_t}/\partial z)]$. This is consistent with the argument by Nolan et al. (2007) that TC intensification is mainly driven by the TC symmetric response to the azimuthally averaged diabatic heating, rather than to the heating directly associated with individual asymmetries. However, we note that these idealized simulations were performed without ambient wind shear. Previous studies (e.g., Leighton et al. 2018) show that the asymmetric eddies play a larger role in TC intensification in a shear environment. Thus, we expect that $-\overline{u_r^{\prime }{\zeta }^{\prime }}-\overline{{\upsilon }_t^{\prime }(\partial {\upsilon }_t^{\prime }/r\partial \lambda )}-\overline{w^{\prime }(\partial {\upsilon }_t^{\prime }/\partial z)}$ would be larger if wind shear was prescribed in the idealized simulations.

The tendencies resulting from all terms on the RHS of Eq. (1) are plotted in Figs. 4m–o, which show that the maximum net positive tendency of tangential wind occurs just below the interface between the inflow and outflow near RMW, corresponding well with the peak tangential wind speed. The radius–height structures of the tendency terms are fairly similar among the three experiments. The main difference between them is the magnitude of individual tendency terms, in particular, the radial transport of absolute vorticity $-\overline{u_r}\overline{\xi }.$ KZ-1 analyzed how the larger sloping curvature of mixing length in EXP-SLOPE-1 generates the larger $-\overline{u_r}\overline{\xi }$ and kicks off a wind-induced surface heat exchange (WISHE)-like (Emanuel 1986) positive feedback among vortex acceleration via inward transport of absolute vorticity, surface evaporation, radial moisture convergence, and eyewall convection underlying TC intensification.

In short, the results of KZ-1 and Fig. 4 confirm that the inward transport of absolute vorticity is the main driving force for the vortex acceleration. Thus, the goal of this study is to decompose the total $-\overline{u_r}\overline{\xi }$ calculated directly from model output into different components induced by individual processes using GSEE. This decomposition of $-\overline{u_r}\overline{\xi }$ may shed a new light on how these processes contribute to the vortex intensification. While turbulence can affect TC intensification both dynamically and thermodynamically as illustrated in Fig. 1, our focus in this paper will be on the dynamic aspect of how radial and tangential turbulent eddy forcing affects TC intensification.

c. Role of unbalanced radial turbulent eddy forcing in TC intensification

The gradient-wind balance provides a concise description of the fundamental TC dynamics. A real TC, however, deviates from the gradient-wind balance due to the turbulent processes, particularly in the BL. To quantitatively measure how far the tangential wind deviates from the balanced gradient wind, we define a quantity known as the normalized net radial forcing (NNRF) as

$\text{NNRF}=\frac{C-\left|-\frac{1}{\overline{\rho }}\frac{\partial \overline{p}}{\partial r}\right|}{\left|-\frac{1}{\overline{\rho }}\frac{\partial \overline{p}}{\partial r}\right|}.$(9)

Using the model output, we calculated NNRF. Fig. 5 shows the radius–height distribution of NNRF averaged over the last 24 h from EXP-SLOPE-1, EXP-BASELINE, and EXP-SLOPE-2, respectively. The structure of NNRF from the three experiments shares similar characteristics. The lowest part of the inflow layer is subgradient reflecting the vortex slowing down by the frictional force induced by turbulence. In the outflow core just above the inflow layer, the tangential wind of the vortex is supergradient. The difference among the three experiments is mainly on the magnitude of NNRF. EXP-SLOPE-1 produces much stronger supergradient winds than the other two experiments. The maximum NNRF reaches 0.5. The radial inflow and outflow induced by turbulence and convection are further affected by the radial turbulent eddy forcing $F_{{\text{sgs}}_r}$. Figure 6 shows the azimuthal-mean $F_{{\text{sgs}}_r}$ from the three experiments. The radius–height structures of $F_{{\text{sgs}}_r}$ from the three experiments are similar. In the inflow and outflow layers, $F_{{\text{sgs}}_r}$ is mainly positive and negative, respectively, reflecting that the SGS turbulence acts as a frictional force to slow down the motion. How radial turbulent eddy forcing affects TC intensification will be discussed shortly.

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Radius–height structure of azimuthal-mean NNRF (color shades) and tangential wind (black contours) averaged over the last 24-h simulations from EXP-SLOPE-1, EXP-BASEINE, and EXP-SLOPE-2, respectively. White contour indicates zero NNRF.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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Azimuthal-mean radial eddy forcing resulting from the parameterized SGS turbulent processes $F_{{\text{sgs}}_r}$ (m s⁻²) averaged over the last 24 h from EXP-SLOPE-1, BASELINE, and EXP-SLOPE-2, respectively. White contour indicates zero tendencies. Red solid and dashed contours are the radial inflow and outflow, respectively.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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It is worth noting that the core of positive NNRF in Fig. 5 matches the peak tangential wind well, implying that the supergradient wind may have something to do with the vortex intensification. To further illustrate the relationship between the supergradient wind and vortex tangential wind, Fig. 7 shows the NNRF at the location of the peak tangential winds as a function of the peak tangential wind for the simulation period of 60–120 h from the three experiments. It clearly shows that the peak tangential wind speed is somewhat correlated to its NNRF particularly for the BASELINE and EXP-SLOPE-1 experiments. The poorer correlation between peak tangential wind and NNRF in EXP-SLOPE-2 suggests that the mechanisms governing the vortex intensification in EXP-SLOPE-2 may differ from those in the other experiments. KZ-1 presents a detailed analysis of how the vortex in the three experiments evolves into different intensification pathways from the same initial and environmental conditions. Here, we are looking into this issue from another perspective of how turbulent eddy forcing contributes to the tangential wind tendency using GSEE diagnoses (presented in the following sections).

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NNRF at the peak tangential wind as a function of peak tangential wind speed for the simulation period of 60–120 h along with the correlation coefficients between NNRF and peak tangential wind speed from EXP-SLOPE-1, EXP-BASELINE, and EXP-SLOPE-2, respectively.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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The somewhat correlated positive NNRF to the peak tangential wind shown in Figs. 5 and 7 raises a question of how the unbalanced dynamics associated with the supergradient and subgradient winds resulting from the radial turbulent eddy forcing affects TC intensification. With GSEE, this question may be adequately addressed by calculating $-\overline{u_r}\overline{\xi }$ induced by the radial eddy forcing. However, as indicated by Eq. (8), the radial eddy forcing $(F_r+F_{{\text{sgs}}_r})$ does not appear as an independent forcing term but is entangled with the diabatic heating as $(\partial /\partial z)[(F_r+F_{{\text{sgs}}_r}){\overline{\chi }}^{2}Q]$. Thus, GSEE cannot be forced by $(F_r+F_{{\text{sgs}}_r})$ alone without diabatic heating. Alternatively, we can solve GSEE and SEE with full forcing (i.e., diabatic heating and tangential/radial eddy forcing for GSEE and diabatic heating and tangential eddy forcing for SEE); then, their difference should tell us the contribution of radial turbulent eddy forcing to the tangential wind tendency via Eq. (1). Moreover, we can further decompose the radial eddy forcing $F_{{\text{sgs}}_r}$ (Fig. 6) into the supergradient and subgradient components based on the positive and negative NNRF (Fig. 5). Using the model output, we performed one set of SEE diagnosis and two sets of GSEE diagnoses (Sup-GW and Sub-GW with supergradient and subgradient components of radial eddy forcing $F_{{\text{sgs}}_r}$, respectively). Table 2 lists the SEE/GSEE diagnoses along with their forcings. The difference of $-\overline{u_r}\overline{\xi }$ derived from GSEE and SEE diagnoses, then, can be used to identify individually how the radial eddy forcing associated with the supergradient wind and subgradient wind affects the TC intensification.

Table 2.

SEE and GSEE diagnoses with their forcings.

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Figure 8 shows the NNRF and the difference of $-\overline{u_r}\overline{\xi }$ between GSEE (Sup-GW, Sub-GW) and SEE diagnoses averaged over the different time periods throughout the simulation of EXP-SLOPE-1. It clearly shows that the peak tangential wind (indicated by *) falls in the positive regime of $-\overline{u_r}\overline{\xi }$ induced by the radial eddy forcing component associated with the supergradient wind (Figs. 8b.1–b.4) throughout the simulation period, suggesting that the unbalanced dynamics associated with the supergradient wind does contribute positively to TC intensification. The radial eddy forcing component associated with the subgradient wind (Figs. 8c.1–c.4), on the other hand, creates positive and negative $-\overline{u_r}\overline{\xi }$ below and above the peak tangential wind, indicating that while accelerating the peak tangential wind, it also tends to lower the height of peak tangential wind. In short, GSEE diagnoses confirm that the unbalanced dynamics associated with supergradient wind and subgradient wind plays a nonnegligible role in TC intensification.

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(a) Azimuthal-mean NNRF; (b),(c) $-\overline{u_r}\overline{\xi }$ derived from GSEE forced by the supergradient and subgradient components of radial eddy forcing subtracted by that from SEE, respectively, averaged over the periods of 37–48, 49–72, 73–96, and 97–120 simulation hours from EXP-SLOPE-1. White contours are the zero tendencies. Black contours are the tangential wind speeds and * indicates the location of peak tangential wind.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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d. Role of tangential turbulent forcing in TC intensification

The budget analyses of tangential wind tendency shown in Fig. 4 suggest that dynamically tangential turbulent eddy forcing $F_{{\text{sgs}}_{\lambda }}$ can have both negative and positive impacts on TC intensification. Directly, $F_{{\text{sgs}}_{\lambda }}$ serves as a frictional force to slow down the primary circulation of a TC as shown in Figs. 4a–8c. This is the well-known effect of $F_{{\text{sgs}}_{\lambda }}$. Indirectly, the weakened tangential winds by $F_{{\text{sgs}}_{\lambda }}$ break down the gradient-wind balance and induce the radial inflow, which transports absolute vorticity inward to accelerate the TC vortex. It remains unclear as to which dynamic effect (positive or negative) tends to dominate the other under what conditions. Note that turbulence also transports energy and moisture upward to foster convection. This thermodynamic effect of turbulence on TC intensification will not be discussed in this paper; rather, we focus only on the dynamic effect of turbulence. With GSEE, this issue can be adequately addressed. As shown in Eq. (8), $F_{{\text{sgs}}_{\lambda }}$ comes into the equation as an independent forcing term $-(\partial /\partial z)(\overline{\chi }\eta F_{{\text{sgs}}_{\lambda }})$; thus, we can force GSEE only with $F_{{\text{sgs}}_{\lambda }}$ while setting other forcings to be zero. This will allow us to isolate the component of secondary circulation solely forced by $F_{{\text{sgs}}_{\lambda }}$. Once the radial flow induced by $F_{{\text{sgs}}_{\lambda }}$ is diagnosed by Eq. (8), the resultant tangential wind tendency can be calculated by $-\overline{u_r}\overline{\xi }$. Then, the net dynamic effect of $F_{{\text{sgs}}_{\lambda }}$ on vortex intensification can be assessed by $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ Figs. 9 and 10 show the radius–height structures of $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ averaged over the different time periods throughout the simulations of EXP-SLOPE-1 and EXP-SLOPE-2, respectively. In both experiments, $F_{{\text{sgs}}_{\lambda }}$ dominates in the lowest part of the BL and causes the net tendency of $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ to be negative, consistent with our current knowledge that $F_{{\text{sgs}}_{\lambda }}$ is the main momentum sink during the development of a TC. However, there is an interesting difference between the two experiments. In EXP-SLOPE-1, aloft at the height of peak tangential wind (indicated by *), the net tendency of $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ is in fact positive throughout the simulation (Fig. 9), suggesting that $F_{{\text{sgs}}_{\lambda }}$ does contribute positively to the increase of peak tangential wind and favors vortex intensification if considering the indirect dynamic effect of $F_{{\text{sgs}}_{\lambda }}$ on tangential wind tendency via $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})$.

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Net tangential wind tendency induced by the turbulent eddy forcing $F_{{\text{sgs}}_{\lambda }}$ only $[-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}]$ averaged over the time periods of 25–36, 37–48, 49–60, 61–72, 73–84, 85–96, 97–108, and 109–120 simulation hours from EXP-SLOPE-1. White contours are the zero tendencies. Black contours are the tangential wind speeds, and black star * is the location of peak tangential wind.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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As in Fig. 9, but for 49, 50, 51–60, 61–72, 73–84, 85–96, 97–108, and 109–120 simulation hours from EXP-SLOPE-2.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-23-0243.1

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EXP-SLOPE-2, on the other hand, shows a different story. The radius–height plots (Fig. 10) show that the location of peak tangential wind (indicated by *) falls in the negative regime $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ from the very beginning when the vortex starts to develop to the end of the simulation, suggesting that the net dynamic effect of $F_{{\text{sgs}}_{\lambda }}$ in this case overall is to slow down the tangential wind including the peak tangential wind. The different radius–height structures of $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ with respect to the peak tangential wind provides an additional reason for why the vortex in EXP-SLOPE-2 behaves differently and fails to intensify like that in EXP-SLOPE-1.

The results shown in Figs. 9 and 10 suggest that the role of tangential turbulent eddy forcing in TC intensification is complex and depends on the details of turbulence parameterization. For the TKE scheme developed from Mellor–Yamada (MY) Level-2 turbulence model (Mellor and Yamada 1982) used in this study, the large sloping curvature of mixing length in EXP-SLOPE-1, which produces the large TKE in the eyewall (KZ-1), causes $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ to be positive near the location of peak tangential wind, suggesting that the net effect of tangential turbulent eddy forcing plays a role in facilitating vortex intensification in this case. In contrast, in EXP-SLOPE-2 in which the small sloping curvature of mixing length produces the small TKE in the eyewall (KZ-1), the negative effect of tangential turbulent eddy forcing appears to dominate, causing $-\overline{u_r}\overline{\xi }(F_{{\text{sgs}}_{\lambda }})+F_{{\text{sgs}}_{\lambda }}$ to be negative near the location of peak tangential wind, which is unfavorable to TC intensification. This result may have important implications in real TCs. For intensifying storms, the net effect tangential turbulent eddy forcing could strengthen peak tangential wind if TKEs in the BL and in the eyewall are large, which helps kick off the WISHE-like positive feedback, leading to the rapid intensification of TCs like the case of EXP-SLOPE-1. On the other hand, when TKEs are small like the case of EXP-SLOPE-2, the overall net negative effect of tangential turbulent eddy forcing could be a barrier for TC intensification.