# Investigation on the Intensification of Supertyphoon Yutu (2018) Based on Symmetric Vortex Dynamics Using the Sawyer–Eliassen Equation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulations

#### 2.1. Model Configuration and Numerical Experiments

_{max}) increasing to 150 kt, according to the JTWC. Yutu then mainly moved west-northwestward over the WNP.

_{max}of the TC (one-minute average maximum wind speed), to determine the TC track and intensity.

#### 2.2. HWRF Simulations

_{max}for CTL occurred between 18 and 42 h; the V

_{max}increased to approximately 60 kt. All the ocean temperature sensitivity experiments (Tp1 and Tm1) obtained comparable typhoon intensities on the first day when the simulated Yutu did not yet move into the region with the modified ocean temperature. The strongest intensity for the simulated Yutu was found in Tp1, and the strongest V

_{max}reached approximately 150 kt, at 90 h. Moreover, Tm1 still produces a RI, but the intensification rate was weaker than CTL, indicating the more dominant effect of the ocean temperature reduction. Tp1 and Tm1 obtained comparably fast intensifications in their first-day forecasts, whereas their later developments, before reaching their peak intensities, is closely related to the initial increases or reductions in ocean temperature, respectively.

## 3. The Sawyer–Eliassen Equation in Height Coordinates

#### 3.1. The Unbalanced SE Formulation

#### 3.2. Solution of the SE Equation

^{−20}. The domain for the SE equation is 0–25 km in the vertical direction, with a vertical resolution of 200 m and 0–3 degrees in the radial direction, with a radial resolution of 0.025 degrees. The boundary conditions for the vortex when solving Equation (26) are $\psi =0$ at $r=0$, $\frac{\partial \psi}{\partial r}=0$ at $r=3$ degrees, and $\psi =0$ at $z=0$ and 25 km.

#### 3.3. Sensitivity Experiments of the SE Solution

## 4. Results

#### 4.1. Sensitivity Tests on Static Stability

^{−1}) at a height of less than 2 km, a relatively weak inflow in the mid-troposphere outside the eyewall at 3–9 km height, and a relatively deep outflow in the upper troposphere at 12–18 km height (Figure 2a). Note that inflow and outflow occurs at heights that are less than and over 19 km, respectively. The vertical velocity field shows an intense updraft at a radius of 0.5 degrees from the typhoon center which extends to the upper troposphere at a height of approximately 17 km, with the strongest upward motion evident at a height of about 9–12 km (Figure 2b). Moreover, there is a relatively strong updraft in the typhoon center, at a radius of 0.9 degrees, with the strongest updraft occurring at a height of nearly 12 km height and a relatively weak updraft inside the eyewall. Furthermore, there is a weak subsidence in the mid- to upper-troposphere inside the eyewall at a height of 9–18 km, and a relatively larger subsidence in the upper troposphere above heights of 17 km, at a radius of 0.4–0.7 degrees.

#### 4.2. Sensitivity Tests on the Residual Terms

#### 4.3. Sensitivity Tests on the Total Force Sources

^{−1}and 8 K h

^{−1}, respectively (Figure 10a). The two maxima diabatic heating rates are consistent with the two strong upward motions of CTL at 24 h (see Figure 4a), and their locations are also consistent with the positions of stronger force sources, as shown in Figure 7a. At 48 h, diabatic heating developed with three maximum zones (Figure 10b), which were collocated with the stronger total force, shown in Figure 7b. The maximum heating rate over 12 K h

^{−1}, at heights of approximately 6–8 km, is stronger than that at 24 h. The diabatic heating rate was largely confined to a vertical column in an annular region, with radii between 0.4–0.6 degrees for Tm1 (Figure 10c). However, the maximum diabatic heating rate occurred at heights less than 2 km for Tp1 (Figure 10d). The heating rate was mostly concentrated at heights less than 14 km. The diabatic heating distribution was similar to the vertical motion, as both followed the total force distribution shown in Figure 7. This implies that the transverse circulation of the intense typhoon was dominated by diabatic heating in the inner vortex.

#### 4.4. Tangential Velocity Tendency

#### 4.5. SE Diagnostics on the Original Vertical Heights

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Initial sea surface temperature at 0000 UTC, 22 October 2018. The dashed box indicates the ocean region where the associated temperature changed, according to the ocean temperature sensitivity experiments. (

**b**) Time evolution of simulated 10 m maximum wind speed (kt) for Yutu from 0000 UTC 22 (0 h) to 0600 UTC, 27 October 2018 (126 h). The best tracks in (

**a**,

**b**) are from JTWC (dashed black line), and the simulated typhoon tracks and intensities are shown for CTL (solid black line), Tp1 (red line), and Tm1 (blue line).

**Figure 2.**Azimuthal-mean (

**a**) radial velocity (shaded colors at units of m s

^{−1}) and (

**b**) vertical velocity (shaded colors at units of m s

^{−1}) of the nonlinear simulation at 48 h for CTL. (

**c**) and (

**d**), as in (

**a**) and (

**b**), respectively, but for N3. (

**e**) and (

**f**), as in (

**c**) and (

**d**), respectively, but for N4. (

**g**) and (

**h**), as in (

**c**) and (

**d**), respectively, but for N5. The wind vectors (m s

^{−1}) induced by the sources, overlapped at each panel, indicate the radial and vertical wind components, with their reference vectors given at the lower right corner.

**Figure 3.**Azimuthal-mean radial velocity (shaded colors at units of m s

^{−1}) at 24 h for (

**a**) the nonlinear model simulation of CTL, and the SE solutions for the different sensitivity experiments of (

**b**) U0V0W0, (

**c**) U1V0W0, (

**d**) U0V0W1, (

**e**) U1V0W1, and (

**f**) U1V1W1. The wind vectors (m s

^{−1}) induced by the sources, overlapped in each panel, indicate the radial and vertical wind components, with their reference vectors shown in the lower right corner.

**Figure 4.**As in Figure 3, but for vertical velocity (shaded colors in units of m s

^{−1}) for (

**a**) the nonlinear model simulation of CTL, and the SE solutions for the different sensitivity experiments of (

**b**) U0V0W0, (

**c**) U1V0W0, (

**d**) U0V0W1, (

**e**) U1V0W1, and (

**f**) U1V1W1.

**Figure 5.**(

**a**) Radial velocity (shaded colors at units of m s

^{−1}) and (

**b**) vertical velocity (shaded colors at units of m s

^{−1}) at 48 h for U0V0W0. (

**c**) and (

**d**), as in (

**a**) and (

**b**), respectively, but for U1V0W0; (

**e**) and (

**f**), as in (

**a**) and (

**b**), respectively, but for U0V0W1; (

**g**) and (

**h**), as in (

**a**) and (

**b**), respectively, but for U0V1W0. The wind vectors (m s

^{−1}) induced by the sources, overlapped at each panel, indicate the radial and vertical wind components, with their reference vectors shown in the lower right corner.

**Figure 6.**(

**a**) Radial velocity (shaded colors at units of m s

^{−1}) and (

**b**) vertical velocity (shaded colors at units of m s

^{−1}) at 48 h for U1. (

**c**) and (

**d**), as in (

**a**) and (

**b**), respectively, but for V1. (

**e**) and (

**f**), as in (

**a**) and (

**b**), respectively, but for W1. The wind vectors (m s

^{−1}) induced by the sources, overlapped at each panel, indicate the radial and vertical wind components, and their reference vectors are shown in the lower right corner.

**Figure 7.**The total force sources (shaded colors at units of 10

^{−12}K

^{−1}s

^{−2}h

^{−1}) at the radius–height cross-section at (

**a**) 24 h for CTL, (

**b**) 48 h for CTL, (

**c**) 48 h for Tm1, and (

**d**) 48 h for Tp1.

**Figure 8.**(

**a**) Radial velocity (shaded colors at units of m s

^{−1}) and (

**b**) vertical velocity (shaded colors at units of m s

^{−1}) at 24 h for S48Tm1. (

**c**) and (

**d**), as in (

**a**) and (

**b**), respectively, but for S48Tp1. (

**e**) and (

**f**), as in (

**a**) and (

**b**), respectively, but for 1.3xS24CTL. The wind vectors (m s

^{−1}) induced by the sources, overlapped at each panel, indicate the radial and vertical wind components, and their reference vectors are shown in the lower right corner.

**Figure 9.**As in Figure 8, but at 48 h for (

**a**) radial velocity (shaded colors at units of m s

^{−1}) and (

**b**) vertical velocity (shaded colors at units of m s

^{−1}) at 24 h for S48Tm1. (

**c**) and (

**d**), as in (

**a**) and (

**b**), respectively, but for S48Tp1. (

**e**) and (

**f**), as in (

**a**) and (

**b**), respectively, but for 1.3xS24CTL.

**Figure 10.**Diabatic heating rate (units of 3.6 K h

^{−1}) at the radius–height cross-section at (

**a**) 24 h for CTL, (

**b**) 48 h for CTL, (

**c**) 48 h for Tm1, and (

**d**) 48 h for Tp1.

**Figure 11.**Tangential velocity tendency (shaded colors at units of m s

^{−1}h

^{−1}) at the radius–height cross-section for CTL at 48 h for (

**a**) actual change in tangential wind, with the model output at a 20-s interval before 48 h; (

**b**) actual change in tangential wind, with the model output at a 20-s interval after 48 h; (

**c**) the azimuthal-mean budget of tangential wind velocity using the nonlinear simulation from the SE solutions for (

**d**) U1V0W1, (

**e**) U0V0W0, and (

**f**) U1V0W1_Vnew.

**Figure 12.**Azimuthal mean of tangential velocity tendency budgeting terms (shaded colors; m s

^{−1}h

^{−1}) at the radius–height cross-section, produced by U1V0W1, at 48 h, for (

**a**) net total budget, (

**b**) $-\overline{u}\overline{\eta}$, (

**c**) $-\overline{w}\frac{\partial \overline{v}}{\partial z}$, and (

**d**) $-\overline{{u}^{\prime}{\eta}^{\prime}}-\overline{{w}^{\prime}\frac{\partial {v}^{\prime}}{\partial z}}+\overline{\frac{{\rho}^{\prime}}{{\overline{\rho}}^{2}}\frac{1}{r}\frac{\partial {p}^{\prime}}{\partial \lambda}}+{\overline{F}}_{v}$ in Equation (22). (

**e**), (

**f**), (

**g**), and (

**h**), as in (

**a**), (

**b**), (

**c**), and (

**d**), respectively, but for U1V0W1_Vnew.

**Figure 13.**Azimuthal mean of tangential velocity tendency budgeting terms (shaded colors) at the radius–height cross-section, produced by U1V0W1, at 48 h for (

**a**) $-\overline{u}\overline{\eta}$ (unit: m s

^{−1}h

^{−1}), (

**b**) $-\overline{w}\frac{\partial \overline{v}}{\partial z}$ (unit: m s

^{−1}h

^{−1}), (

**c**) $-\overline{{u}^{\prime}{\eta}^{\prime}}$ (unit: m s

^{−1}h

^{−1}), (

**d**) $-\overline{{w}^{\prime}\frac{\partial {v}^{\prime}}{\partial z}}$ (unit: m s

^{−1}h

^{−1}), (

**e**) $\overline{\frac{{\rho}^{\prime}}{{\overline{\rho}}^{2}}\frac{1}{r}\frac{\partial {p}^{\prime}}{\partial \lambda}}$ (unit: 10

^{−1}m s

^{−1}h

^{−1}), and (

**f**) ${\overline{F}}_{v}$ (unit: m s

^{−1}h

^{−1}) in Equation (22).

**Figure 14.**Azimuthal-mean radial velocity (shaded colors at units of m s

^{−1}) calculated at 48 h for (

**a**) the nonlinear model simulation of CTL, and the SE solutions for different sensitivity experiments of (

**b**) U1V0W1, (

**c**) U1V0W0, (

**d**) U0V01W0, (

**e**) U0V0W1, and (

**f**) U1V1W1. Note that these results were calculated using the original vertical coordinate of the HWRF model. The wind vectors (m s

^{−1}) induced by the sources, overlapped at each panel, indicate the radial and vertical wind components, with their reference vectors shown in the lower right corner.

Experiments | Including Residual Terms | Changing Total Forcing Sources | ||
---|---|---|---|---|

$\dot{\mathit{U}}$ | $\dot{\mathit{V}}$ | $\dot{\mathit{W}}$ | ||

U0V0W0 | No | No | No | No |

U1V0W0 | Yes | No | No | No |

U0V1W0 | No | Yes | No | No |

U0V0W1 | No | No | Yes | No |

U1V0W1 | Yes | No | Yes | No |

U1V1W1 | Yes | Yes | Yes | No |

S48Tm1 | Yes | No | Yes | S at 48 h of Tm1 |

S48Tp1 | Yes | No | Yes | S at 48 h of Tp1 |

1.3xS24CTL | Yes | No | Yes | 1.3 times S at 24 h of CTL |

1.3xS48CTL | Yes | No | Yes | 1.3 times S at 48 h of CTL |

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**MDPI and ACS Style**

Nguyen, T.-C.; Huang, C.-Y.
Investigation on the Intensification of Supertyphoon Yutu (2018) Based on Symmetric Vortex Dynamics Using the Sawyer–Eliassen Equation. *Atmosphere* **2023**, *14*, 1683.
https://doi.org/10.3390/atmos14111683

**AMA Style**

Nguyen T-C, Huang C-Y.
Investigation on the Intensification of Supertyphoon Yutu (2018) Based on Symmetric Vortex Dynamics Using the Sawyer–Eliassen Equation. *Atmosphere*. 2023; 14(11):1683.
https://doi.org/10.3390/atmos14111683

**Chicago/Turabian Style**

Nguyen, Thi-Chinh, and Ching-Yuang Huang.
2023. "Investigation on the Intensification of Supertyphoon Yutu (2018) Based on Symmetric Vortex Dynamics Using the Sawyer–Eliassen Equation" *Atmosphere* 14, no. 11: 1683.
https://doi.org/10.3390/atmos14111683