**5. Summary and discussion**

Using the conception of difference in total energy (DTE), Tan et al. (hereafter T04) [62] examined the impacts of initial small-scale disturbance on a "surprise" snowstorm through the analysis of energy cascading. Similar to the definition of DTE by T04, the total energy (*TE*) is

<sup>2</sup> + *Vi*,*j*,*<sup>k</sup>*

where *U*, *V*, and *T* are horizontal u-wind, v-wind components, and temperature, respectively.

A power spectrum of *TE*, averaged in the region of heavy rainfall, is analyzed. Wavenumbers 0, 1, and 2 are the TC scales following Krishnamurti et al. [63]. Meanwhile, the scales of the individual deep convective clouds reside around the azimuthal wave numbers 20–30. According to Saltzman [64], the TC scale is about several hundreds of kilometers, whereas the scale of convection, including updrafts and adjacent downdrafts, is only a few kilometers. It shows that a sizeable portion of the variance of *TE* is contributed by the first few harmonics (0–4) in the innermost region. The contribution from wavenumbers 3 to 55 (associated with medium- to small-scale processes) accounts for less than 10% of total *TE*, which agrees with the results from quasi-geostrophic models [65].

To better understand the energy transition during rainfall, this section further examines the relation between helicity, the magnitude of which is associated with kinematic energy, and CAPE, an indicator of potential energy. Krishnamurti et al. [63], by examining the scale interaction of hurricane inferred from the decomposition of the liquid water mixing ratio fields, found that nonlinear interaction of kinetic energy and available potential energy among cloud scales and the hurricane scale provide the energy to drive the hurricane. The generation of available potential energy and its transformation to kinetic energy takes place directly on the larger scales of the hurricane. Their results among hurricane scales and smaller scales show largely a cascade of energy, that is, hurricane scales lose energy when they interact with other scales. The evolution of CAPE and helicity of TC circulation during landfall (12UTC 27–12UTC 29 July) is investigated. It shows that an approximated negative correlation exists between *H*<sup>1</sup>

CAPE before the occurrence of rainfall, which is mainly featured by the decrease (increase) of

positive correlation, which decreases simultaneously. The decrease of CAPE should be associated with the significantly reduced heat flux from land surface and the large consumption

In other words, kinematic energy increases as potential energy is consumed. However, there

in more than 66.6% of the rainfall region. The maximum of CAPE is 3500 J kg−1 with *H*<sup>1</sup>

are greater than 400 m<sup>2</sup>

of 50–150 m<sup>2</sup>

of CAPE during the rainfall process. Scatter plot also shows that intensive *H*<sup>1</sup>

not directly fuel the growth of small-scale convection. Moreover, over the land, *H*<sup>1</sup>

is positive, with *H*<sup>1</sup>

corresponds to CAPE less than 500 J kg−1, while over the ocean, negative *H*<sup>1</sup>

low CAPE (e.g., CAPE of 3500 J kg−1 vs. *H*<sup>1</sup>

s−2. Five percent of *H*<sup>1</sup>

land, more than 95% of *H*<sup>1</sup>

CAPE as low as 1000 J kg−1.

is no clear correlation between CAPE and *H*<sup>2</sup>

). However after rainfall, the original negative relation is replaced by an approximate

<sup>2</sup> + *kTi*,*j*,*<sup>k</sup>* 2

) (5)

and

corresponds to

is positive

s−2. Over

corresponds to the

of

s−2) during the growth of convection.

, indicating that the energy from CAPE might

s−2, with the biggest of 800 m<sup>2</sup>

decreasing with CAPE. Most of the intensive *H*<sup>1</sup>

=287 K). I, j, and k are the numbers of x, y, and σ grid

<sup>2</sup>(*Ui*,*j*,*<sup>k</sup>*

defined here by considering kinetic and internal components:

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

*TE* = \_\_1

(the reference temperature *Tr*

*k* = *Cp* /*Tr*

CAPE (*H*<sup>1</sup>

50–150 m<sup>2</sup>

points, respectively.

High-resolution simulations are performed with nonhydrostatic WRF mesoscale numerical model to clarify the multiscale mechanisms leading to the heavy rainfall of TC Fung-Wong during landfall on southwestern coast of China. Numerical analysis shows that quasi-frontal structures are frequently generated at the boundary of warm LLJs associated with TC inflow and cold convective downdrafts, which favor the genesis of intensive rainfall. Some important features of the quasi-frontal structures, e.g., intensive vertical wind shear and small local Richardson number, exhibit similarity with that of K-H waves from supercell. The hodograph of LLJ which turns clockwise with height tends to produce positive helicity and favors the genesis of convection. An evident antiphase relationship between *H*<sup>1</sup> and CAPE during heavy rainfall suggests the energy transition from CAPE to kinematic energy.

For the future study, the structure, organization, and impact of convective rainfall systems in TCs during landfall remain a fruitful area for research. Convective cells are known to be favored downshear in TCs due to the shear-induced increase in convergence and upward motion downshear [68]. The variations of helicity and CAPE described in this paper should also connect with vertical shear. As has been stated by Molinari and Vollaro [28], the helicity and CAPE in the presence of large ambient shear exceeded those in storms with small ambient shear. The reduction in stability and increase in helicity might represent the positive influence of large vertical wind shear in offsetting the greater ventilation of the storm core. Many questions about heavy rainfall in landfalling TCs remain unanswered. How about the multiscale characteristics of helicity in different situation of shear? How do the multiscale helicity affect supercells of TCs? Observation (i.e., radar) and simulations are required to confirm the processes that such cells develop in landfalling TCs. Moreover, the detailed cascading process of rainfall should be carefully examined based on energy budget. The quantitative impact of systems in various scales on rainfall deserved to be examined with sensitive numerical experiments. Additional studies should be conducted to verify and expand upon the limited observation of cold pool associated with landfalling TCs. In particular, combination of numerical simulation and datasets including both onshore and offshore observations at various intensities and evolutionary stages across a spectrum of large-scale environments would improve understanding of cold pools and their various feedbacks on convection. To enhance the simulation on cold pools, improved representation of microphysics in models would also be beneficial. Furthermore, although the orographic effect is not addressed in this study as there is no clear evident of relationship between the terrain and the amplification of rainfall, the alteration, or reorganization of the convective clouds, frontal systems associated with TC when it encounters topographic features might be possible [69], which will be examined in a next study.

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