**2. Methodology**

#### **2.1. Numerical experiments**

In order to clarify the multiscale mechanism for heavy rainfall, high-resolution numerical experiments were conducted with the NCAR/WRF mesoscale model [29]. The triply nested model domains (**Figure 1a**), i.e., the outermost domain (D1), the inner domain (D2), and the innermost domain (D3), are designed with horizontal spacing of 15 km, 3 km, and 600 m, respectively, with domain sizes of 281 × 281, 721 × 721, and 721 × 721 grid points, respectively. D1, which covers the northwest Pacific, eastern Tibetan plateau, Bengal Bay, and China, is employed to examine the large-scale environmental flow. D2, which covers east China, is one-way nested within D1 and is used to examine the major landfall processes. D3, which is fixed and one-way nested within D2, is used to analyze the detailed structure of rainfall system. A total of 37 vertical sigma levels are used for all the domains. The Kain-Fritsch cumulus parameterization scheme [30] with modification of convection trigger function [31] is used in D1. However, in D2 and D3, no cumulus parameterization scheme is considered to avoid its ambiguous application in high-resolution simulation. The WRF single-moment 6 (WSM6) class multiphase cloud scheme is employed in all domains to represent cloud physics. The Yonsei University (YSU) planetary boundary layer (PBL) scheme [32], using counter-gradient terms to represent nonlocal fluxes, is considered for PBL parameterization in D1 and D2. The YSU PBL scheme [33, 34] explicitly treats the entrainment layer at the PBL top with the surface buoyancy flux in line with results from large-eddy models. The PBL top is defined using a critical bulk Richardson number of zero. The turbulent kinetic energy (TKE) diffusion scheme is employed in D3 to deal with the PBL physics [35]. Furthermore, the rapid radiative transfer model (RRTM) scheme [36] and Dudhia scheme [37] are used for the parameterization of longwave and shortwave radiation, respectively.

**Figure 1b** indicates the phases for model integration in each domain. For domain D1, the time between 12UTC, 27 July, and 12UTC, 29 July, is chosen for an overall description of Fung-Wong's track and rainfall during landfall. In order to examine the detailed evolution of TC structure and rainfall, the time between 02UTC 28 and 12UTC 29 is selected for D2, and the time between 02UTC 28 and 12UTC 28 is selected for D3. The background field of D1 is interpolated from the analysis of NCEP Global Forecast System (GFS) whose horizontal resolution is 0.5°. The vortex initialization scheme developed by Ma and Tan [38] is employed to produce the initial analysis for TC simulation. In this scheme, sea level pressure (SLP) derived from satellite sea surface wind is used to generate the initial TC circulation. To ensure a reasonable simulation of Fung-Wong's track and intensity, two numerical experiments, e.g., Expt. CTRL (the one without vortex initialization) and Expt. VIRV (the one with vortex initialization), are conducted. The 3B-42 gridded rainfall datasets with the resolution of 0.25 × 0.25°, derived from the Tropical Rainfall Measuring Mission (TRMM), are employed for verification on rainfall simulation.

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⇀*ω*) is originally defined as the scalar product of velocity ( *<sup>V</sup>*

the twining structure of the vortex tubes [39]. Subsequent researches [20, 21] further examined

\_\_\_ ∂*v* <sup>∂</sup>*z*) <sup>+</sup> *<sup>v</sup>*(

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>v</sup>* \_\_\_ <sup>∂</sup>*<sup>θ</sup>*

Clearly, Eq. (2) indicates the equality between helicity and temperature advection for largescale flow, e.g., helicity should be positive (negative) for warm (cold) air advection. Therefore,

<sup>∂</sup>*<sup>y</sup>* <sup>−</sup> *<sup>v</sup>* \_\_\_ <sup>∂</sup>*<sup>w</sup>*

<sup>∂</sup>*z*( \_\_ *u*

<sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *<sup>w</sup>*(

<sup>∂</sup>*<sup>y</sup>* <sup>=</sup> <sup>−</sup> *<sup>V</sup>*

According to Tan and Wu [20], for large-scale motion, helicity (hereafter *<sup>H</sup>* <sup>1</sup>

⇀), which represents the rotational characteristics in the motion direction and

\_\_\_ ∂*u*

) can be rewritten as

\_\_\_ ∂*v* <sup>∂</sup>*<sup>x</sup>* <sup>−</sup> \_\_\_ <sup>∂</sup>*<sup>u</sup>*

*<sup>v</sup>*) <sup>=</sup> <sup>−</sup>|⇀*<sup>v</sup>* |2 \_\_\_ <sup>∂</sup>*<sup>α</sup>*

⇀) and vorticity

) can be approxi-

<sup>∂</sup>*z*) (1)

⇀ <sup>⋅</sup> <sup>∇</sup> *<sup>θ</sup>* (2)

<sup>∂</sup>*y*) (3)

<sup>∂</sup>*<sup>z</sup>* (4)

) and its component

can be further expressed as

**2.2. The multiscale conception of helicity**

*H*<sup>1</sup> ≈ −*u*(

*<sup>H</sup>*<sup>1</sup> <sup>≈</sup> <sup>−</sup>*<sup>u</sup>* \_\_\_ <sup>∂</sup>*<sup>θ</sup>*

For small-scale motion, helicity (hereafter *H*<sup>2</sup>

*<sup>H</sup>*<sup>2</sup> <sup>≈</sup> *<sup>u</sup>* \_\_\_ <sup>∂</sup>*<sup>w</sup>*

*<sup>H</sup>*<sup>1</sup> <sup>≈</sup> *<sup>v</sup>* <sup>2</sup> \_\_\_<sup>∂</sup>

*u*), then

where *θ* is the potential temperature.

Based on the assumption of thermal wind balance, *H*<sup>1</sup>

very large gradient of helicity is associated with frontogenesis.

Assuming *α* = *α*(*z*) the angle between velocity vector ⇀*v* (|⇀*v*|2 = *u*<sup>2</sup> + *v* <sup>2</sup>

⇀⋅

Helicity (*H* = *V*

mated as

*<sup>u</sup>* (i.e., tan*α* = \_\_*<sup>v</sup>*

vector (⇀*ω* = ∇ ∧ *V*

its multiscale conception.

**Figure 1.** (a) The triply nested model domains for numerical simulation (D1, the outermost domain; D2, the inner domain; D3, the innermost domain) and (b) the phases of model integration for each model domain.

**Figure 1b** indicates the phases for model integration in each domain. For domain D1, the time between 12UTC, 27 July, and 12UTC, 29 July, is chosen for an overall description of Fung-Wong's track and rainfall during landfall. In order to examine the detailed evolution of TC structure and rainfall, the time between 02UTC 28 and 12UTC 29 is selected for D2, and the time between 02UTC 28 and 12UTC 28 is selected for D3. The background field of D1 is interpolated from the analysis of NCEP Global Forecast System (GFS) whose horizontal resolution is 0.5°. The vortex initialization scheme developed by Ma and Tan [38] is employed to produce the initial analysis for TC simulation. In this scheme, sea level pressure (SLP) derived from satellite sea surface wind is used to generate the initial TC circulation. To ensure a reasonable simulation of Fung-Wong's track and intensity, two numerical experiments, e.g., Expt. CTRL (the one without vortex initialization) and Expt. VIRV (the one with vortex initialization), are conducted. The 3B-42 gridded rainfall datasets with the resolution of 0.25 × 0.25°, derived from the Tropical Rainfall Measuring Mission (TRMM), are employed for verification on rainfall simulation.

### **2.2. The multiscale conception of helicity**

**2. Methodology**

**2.1. Numerical experiments**

280 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

In order to clarify the multiscale mechanism for heavy rainfall, high-resolution numerical experiments were conducted with the NCAR/WRF mesoscale model [29]. The triply nested model domains (**Figure 1a**), i.e., the outermost domain (D1), the inner domain (D2), and the innermost domain (D3), are designed with horizontal spacing of 15 km, 3 km, and 600 m, respectively, with domain sizes of 281 × 281, 721 × 721, and 721 × 721 grid points, respectively. D1, which covers the northwest Pacific, eastern Tibetan plateau, Bengal Bay, and China, is employed to examine the large-scale environmental flow. D2, which covers east China, is one-way nested within D1 and is used to examine the major landfall processes. D3, which is fixed and one-way nested within D2, is used to analyze the detailed structure of rainfall system. A total of 37 vertical sigma levels are used for all the domains. The Kain-Fritsch cumulus parameterization scheme [30] with modification of convection trigger function [31] is used in D1. However, in D2 and D3, no cumulus parameterization scheme is considered to avoid its ambiguous application in high-resolution simulation. The WRF single-moment 6 (WSM6) class multiphase cloud scheme is employed in all domains to represent cloud physics. The Yonsei University (YSU) planetary boundary layer (PBL) scheme [32], using counter-gradient terms to represent nonlocal fluxes, is considered for PBL parameterization in D1 and D2. The YSU PBL scheme [33, 34] explicitly treats the entrainment layer at the PBL top with the surface buoyancy flux in line with results from large-eddy models. The PBL top is defined using a critical bulk Richardson number of zero. The turbulent kinetic energy (TKE) diffusion scheme is employed in D3 to deal with the PBL physics [35]. Furthermore, the rapid radiative transfer model (RRTM) scheme [36] and Dudhia scheme [37]

are used for the parameterization of longwave and shortwave radiation, respectively.

**Figure 1.** (a) The triply nested model domains for numerical simulation (D1, the outermost domain; D2, the inner

domain; D3, the innermost domain) and (b) the phases of model integration for each model domain.

Helicity (*H* = *V* ⇀⋅ ⇀*ω*) is originally defined as the scalar product of velocity ( *<sup>V</sup>* ⇀) and vorticity vector (⇀*ω* = ∇ ∧ *V* ⇀), which represents the rotational characteristics in the motion direction and the twining structure of the vortex tubes [39]. Subsequent researches [20, 21] further examined its multiscale conception.

According to Tan and Wu [20], for large-scale motion, helicity (hereafter *<sup>H</sup>* <sup>1</sup> ) can be approximated as

$$H\_{\rm i} \approx -\mu \left(\frac{\partial v}{\partial z}\right) + \nu \left(\frac{\partial u}{\partial z}\right) \tag{1}$$

Based on the assumption of thermal wind balance, *H*<sup>1</sup> can be further expressed as

$$H\_1 \approx -\mu \frac{\partial \theta}{\partial x} + \nu \frac{\partial \theta}{\partial y} = -\overrightarrow{V} \cdot \nabla \cdot \Theta \tag{2}$$

where *θ* is the potential temperature.

Clearly, Eq. (2) indicates the equality between helicity and temperature advection for largescale flow, e.g., helicity should be positive (negative) for warm (cold) air advection. Therefore, very large gradient of helicity is associated with frontogenesis.

For small-scale motion, helicity (hereafter *H*<sup>2</sup> ) can be rewritten as

$$H\_{\mathbf{z}} \approx \ln \frac{\partial w}{\partial y} - \mathbf{v} \frac{\partial w}{\partial \mathbf{x}} + \mathcal{W} \left(\frac{\partial v}{\partial \mathbf{x}} - \frac{\partial u}{\partial y}\right) \tag{3}$$

Assuming *α* = *α*(*z*) the angle between velocity vector ⇀*v* (|⇀*v*|2 = *u*<sup>2</sup> + *v* <sup>2</sup> ) and its component *<sup>u</sup>* (i.e., tan*α* = \_\_*<sup>v</sup> u*), then

$$H\_1 \approx \sqrt{2\frac{\partial}{\partial z}} \left(\frac{\mu}{\upsilon}\right) = -|\left|\overrightarrow{\nabla}\right|^2 \frac{\partial a}{\partial z} \tag{4}$$

The multiscale conception of helicity discussed above is fundamental for the understanding of energy cascade, which occurs either by Taylor's mechanism of stretching and spin-up of small-scale vortices due to large-scale strain or twisting of small-scale vortex filaments due to a large-scale screw.

**4. Numerical analysis on the multiscale systems associated with the** 

**4.1. Quasi-frontal structure viewed from helicity, low-level jet (LLJ), and cold pool**

which produces a zone of elevated conditional instability favorable for rainfall genesis.

tive warm and moist LLJs associated with TC inflow frequently appear and move toward the TC core region and finally meet with the strong cold convective downdrafts which was induced by the convective detrainment from the middle-to-upper troposphere (PBL, **Figure 3**). As a result, quasi-frontal structure is generated at the boundary between the warm LLJ and cold downdraft. The LLJ intensifies as the front gradually sharpens. The large curvature near the quasifront should serve to accelerate the buoyant air and the growth of convection. According to Eq. (4), the vertical shear of the angle between vector ⇀*v* and *u* component is equivalent to the ratio

should be generated when the LLJ turns clockwise with height. It is interesting to find that before the occurrence of heavy rainfall, the shear vector of LLJ generally turns counterclockwise with

∂*z* is positive). This relation indicates the existence of negative helicity and cold advection. However, as heavy rainfall occurs, the shear vector turns clockwise with height, showing a positive helicity. It is also noticed that high potential vorticity (PV, >2.5 PVU) associated with the mesoscale disturbances mainly occupies 700–950 hPa before the genesis of heavy rainfall. These PV disturbances then grow rapidly and extend to the whole troposphere, in companion with the genesis of heavy rainfall. Moreover, to evaluate the influence of the frontogenetical forcing on the growth of heavy rainfall, the Sawyer-Eliassen equation is further calculated. It is found that evident frontogenetical forcing (stream function value is −10.1 hPa m s−1) is formed at the quasifrontal area to the southeastern part of heavy rainfall. The forcing is intensified (−12.2 hPa m s−1) and extended from 400 hPa to a lower level (700 hPa) at the peak time of rainfall (04UTC 28 Jul).

) and the square of total horizontal velocity. In this regard, positive helicity

during TC landfall, this study also identifies that rela-

Numerous studies have highlighted the role of preexisting boundaries intersecting the primary convective system where cyclonic-only mesovortices were observed to form at the intersection point [40, 41]. More detailed analysis also indicated shearing instability [42] as the genesis mechanism for cyclonic-only low-level vortices formed along mesoscale boundaries such as gust fronts [43, 44]. These mesoscale boundaries are usually associated with frontal structures, which are required to be examined to clarify the mechanism of heavy rainfall. While it has long been recognized that the low-level jet (LLJ) is an efficient moisture transport mechanism [45] and a source of large-scale destabilization through warm advection [46, 47], the frontogenetical character of the boundary of LLJ can be important for the genesis of MCSs [48]. Therefore, the frontogenesis process similar to that of Augustine and Caracena [49] is investigated here to understand the characteristics of boundaries associated with MCSs. In particular, a timeaveraged composite vertical cross section at 6 h preceding the mature stage shows that the LLJ in the plane of the cross section ascending the northeastward sloping frontal surface. Trier et al. [50] argued that long-lived MCSs are aided by the frontogenetical lifting of air by the LLJ,

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**torrential rainfall**

Through the analysis on the evolution of *H*<sup>1</sup>

between helicity (*H*<sup>1</sup>

height (\_\_\_ <sup>∂</sup>*<sup>α</sup>*
