# A Simple Family of Tropical Cyclone Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Primitive Equation Model (PE),
- Hybrid Balanced Model 1 (HB1),
- Hybrid Balanced Model 2 (HB2),
- Gradient Balanced Model (GB).

## 2. Primitive Equation Model (PE)

**Table 1.**Summary of the PE model. The upper half of the table lists the diagnostic equations, while the lower half lists the nine prognostic equations. The right-hand sides of (1.11)–(1.16) are defined in (9) and (10). Because the radial components ${u}_{1}(r,t)$ and ${u}_{2}(r,t)$ are predicted, the PE model allows propagating inertia-gravity waves. Because the boundary layer radial and tangential components ${u}_{0}(r,t)$ and ${v}_{0}(r,t)$ are predicted, and because large inflow often occurs in the boundary layer, the PE model can develop boundary layer shocks. The diabatic mass flux ${Q}^{+}$ depends on boundary layer convergence ($w>0$) and the entrainment parameter $\eta $, which in turn depends on the boundary layer equivalent potential temperature ${\theta}_{e0}$, predicted by (1.19), and the upper tropospheric saturation equivalent potential temperature ${\theta}_{e2}^{\ast}$, diagnosed from (1.4).

Primitive Equation Model (PE) | |||||
---|---|---|---|---|---|

Diagnostic Equations: | |||||

Layer thicknesses: | Saturation equivalent potential temperatures: | ||||

${h}_{1}={\overline{h}}_{1}+\frac{1}{g\sigma}({\varphi}_{1}-\u03f5{\varphi}_{2})$ | (1.1) | $\theta}_{es}^{\ast}={\overline{\theta}}_{es}^{\ast}-\frac{2.0}{{c}_{p}}{\varphi}_{1$ | (1.3) | ||

${h}_{2}={\overline{h}}_{2}+\frac{1}{g\sigma}({\varphi}_{2}-{\varphi}_{1})$ | (1.2) | ${\theta}_{e2}^{\ast}={\overline{\theta}}_{e2}^{\ast}+\frac{10.3}{{c}_{p}}({\varphi}_{2}-{\varphi}_{1})$ | (1.4) | ||

Air-sea interaction: | Ekman pumping: | ||||

$U={\left({u}_{0}^{2}+{v}_{0}^{2}\right)}^{1/2}$ | (1.5) | $w=-{h}_{0}\frac{\partial \left(r{u}_{0}\right)}{r\partial r}$ | (1.7) | ||

$c}_{{}_{\mathrm{E}}}={c}_{{}_{\mathrm{D}}}=(0.5+0.06\phantom{\rule{0.166667em}{0ex}}U)\xb7{10}^{-3$ | (1.6) | ${w}^{\pm}={\textstyle \frac{1}{2}}\left(\left|w\right|\pm w\right)$ | (1.8) | ||

Diabatic mass flux: | |||||

$\eta =1+\frac{{\theta}_{e0}-{\theta}_{e2}^{\ast}}{{\theta}_{e2}^{\ast}-{\theta}_{e1}}$ | (1.9) | $Q={Q}^{+}-{Q}^{-}$ with $Q}^{+}=\eta {w}^{+$ | (1.10) | ||

Prognostic Equations: | |||||

Radial wind components: | |||||

$\frac{\partial {u}_{0}}{\partial t}+{u}_{0}\frac{\partial {u}_{0}}{\partial r}-\left(f+\frac{{v}_{0}}{r}\right){v}_{0}+\frac{\partial {\varphi}_{1}}{\partial r}={F}_{0}$ | (1.11) | ||||

$\frac{\partial {u}_{1}}{\partial t}+{u}_{1}\frac{\partial {u}_{1}}{\partial r}-\left(f+\frac{{v}_{1}}{r}\right){v}_{1}+\frac{\partial {\varphi}_{1}}{\partial r}={F}_{1}$ | (1.12) | ||||

$\frac{\partial {u}_{2}}{\partial t}+{u}_{2}\frac{\partial {u}_{2}}{\partial r}-\left(f+\frac{{v}_{2}}{r}\right){v}_{2}+\frac{\partial {\varphi}_{2}}{\partial r}={F}_{2}$ | (1.13) | ||||

Tangential wind components: | |||||

$\frac{\partial {v}_{0}}{\partial t}+{u}_{0}\frac{\partial {v}_{0}}{\partial r}+\left(f+\frac{{v}_{0}}{r}\right){u}_{0}={G}_{0}$ | (1.14) | ||||

$\frac{\partial {v}_{1}}{\partial t}+{u}_{1}\frac{\partial {v}_{1}}{\partial r}+\left(f+\frac{{v}_{1}}{r}\right){u}_{1}={G}_{1}$ | (1.15) | ||||

$\frac{\partial {v}_{2}}{\partial t}+{u}_{2}\frac{\partial {v}_{2}}{\partial r}+\left(f+\frac{{v}_{2}}{r}\right){u}_{2}={G}_{2}$ | (1.16) | ||||

Geopotential anomalies: | |||||

$\frac{\partial {\varphi}_{1}}{\partial t}+g\frac{\partial \left[r({u}_{1}{h}_{1}+{u}_{2}\u03f5{h}_{2})\right]}{r\partial r}=gw$ | (1.17) | ||||

$\frac{\partial {\varphi}_{2}}{\partial t}+g\frac{\partial \left[r({u}_{1}{h}_{1}+{u}_{2}{h}_{2})\right]}{r\partial r}=g\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (1.18) | ||||

Boundary layer equivalent potential temperature: | |||||

$\frac{\partial {\theta}_{e0}}{\partial t}+{u}_{0}\frac{\partial {\theta}_{e0}}{\partial r}+\frac{{w}^{-}}{{h}_{0}}({\theta}_{e0}-{\theta}_{e1})=\frac{{c}_{{}_{\mathrm{E}}}U}{{h}_{0}}({\theta}_{es}^{\ast}-{\theta}_{e0})$ | (1.19) |

- 1.
- Specify desired values for the various fixed parameters: g, ${c}_{p}$, f, $\u03f5$ or $\sigma =1-\u03f5$, ${h}_{0}$, ${\overline{h}}_{1}$, ${\overline{h}}_{2}$, ${\overline{\theta}}_{es}^{\ast}$, ${\overline{\theta}}_{e2}^{\ast}$, ${\theta}_{e1}$, ${Q}^{-}$, $\mu $, and K, where K is the constant kinematic coefficient of eddy viscosity appearing in the equations for the horizontal diffusive fluxes given in (A2) and (A4) in Appendix A.
- 2.
- Using the known geopotential anomalies ${\varphi}_{1}(r,t)$ and ${\varphi}_{2}(r,t)$, diagnose the layer depths ${h}_{1}(r,t)$ and ${h}_{2}(r,t)$ from (1.1) and (1.2), the sea-surface saturation equivalent potential temperature ${\theta}_{es}^{\ast}(r,t)$ from (1.3), and the upper tropospheric saturation equivalent potential temperature ${\theta}_{e2}^{\ast}(r,t)$ from (1.4).
- 3.
- Compute the boundary layer wind speed $U(r,t)$ from (1.5), and then the exchange coefficient ${c}_{{}_{\mathrm{E}}}(r,t)$ and the drag coefficient ${c}_{{}_{\mathrm{D}}}(r,t)$ from (1.6).
- 4.
- Using (1.7), diagnose the boundary layer vertical velocity $w(r,t)$ from the known boundary layer radial velocity ${u}_{0}(r,t)$, then compute ${w}^{+}(r,t)$ and ${w}^{-}(r,t)$ from (1.8).
- 5.
- Using the constant ${\theta}_{e1}$, the diagnosed ${\theta}_{e2}^{\ast}(r,t)$, and the known ${\theta}_{e0}(r,t)$, diagnose the entrainment parameter $\eta (r,t)$ from (1.9); then, compute the diabatic mass flux $Q(r,t)$ from (1.10).
- 6.
- 7.
- Using the known radial velocity components ${u}_{1}(r,t)$ and ${u}_{2}(r,t)$, predict the geopotential anomalies ${\varphi}_{1}(r,t)$ and ${\varphi}_{2}(r,t)$ via (1.17) and (1.18).
- 8.
- Using the known boundary layer radial velocity ${u}_{0}(r,t)$, the diagnosed boundary layer suction ${w}^{-}(r,t)$, and the diagnosed ${\theta}_{es}^{\ast}(r,t)$, predict the boundary layer equivalent potential temperature ${\theta}_{e0}(r,t)$ via (1.19), and then return to step 2 for the beginning of the next time step.

## 3. Hybrid Balanced Model 1 (HB1)

**Table 2.**Summary of the HB1 model. The upper half of the table lists the diagnostic equations, while the lower half lists the five prognostic equations. As indicated in (2.11) and (2.12), gradient balance is assumed in the upper two layers, but, as indicated in (2.15) and (2.16), the primitive forms of the momentum equations are retained in the boundary layer, thus allowing the formation of boundary layer shocks. Because the prognostic equations for ${u}_{1}(r,t)$ and ${u}_{2}(r,t)$ have been discarded and replaced by the coupled Eliassen Equations (2.13) and (2.14), propagating inertia-gravity waves are filtered in the HB1 model. A sufficient (but not necessary) condition for the uniqueness of the solutions ${\psi}_{1},{\psi}_{2}$ of the Eliassen equations is that ${S}_{1}\ge 0$ and ${S}_{2}\ge 0$. Numerical integrations of the Eliassen equations reveal that ${S}_{2}$ can become negative at some distance from the cyclone center, but no difficulty is encountered in the solution of the discretized Eliassen equations if a direct method of solution (such as Gaussian elimination) is used.

Hybrid Balanced Model 1 (HB1) | |||||
---|---|---|---|---|---|

Diagnostic Equations: | |||||

Layer thicknesses: | Saturation equivalent potential temperatures: | ||||

${h}_{1}={\overline{h}}_{1}+\frac{1}{g\sigma}({\varphi}_{1}-\u03f5{\varphi}_{2})$ | (2.1) | $\theta}_{es}^{\ast}={\overline{\theta}}_{es}^{\ast}-\frac{2.0}{{c}_{p}}{\varphi}_{1$ | (2.3) | ||

${h}_{2}={\overline{h}}_{2}+\frac{1}{g\sigma}({\varphi}_{2}-{\varphi}_{1})$ | (2.2) | ${\theta}_{e2}^{\ast}={\overline{\theta}}_{e2}^{\ast}+\frac{10.3}{{c}_{p}}({\varphi}_{2}-{\varphi}_{1})$ | (2.4) | ||

Air-sea interaction: | Ekman pumping: | ||||

$U={\left({u}_{0}^{2}+{v}_{0}^{2}\right)}^{1/2}$ | (2.5) | $w=-{h}_{0}\frac{\partial \left(r{u}_{0}\right)}{r\partial r}$ | (2.7) | ||

$c}_{{}_{\mathrm{E}}}={c}_{{}_{\mathrm{D}}}=(0.5+0.06\phantom{\rule{0.166667em}{0ex}}U)\xb7{10}^{-3$ | (2.6) | ${w}^{\pm}={\textstyle \frac{1}{2}}\left(\left|w\right|\pm w\right)$ | (2.8) | ||

Diabatic mass flux: | Gradient balance equations: | ||||

$\eta =1+\frac{{\theta}_{e0}-{\theta}_{e2}^{\ast}}{{\theta}_{e2}^{\ast}-{\theta}_{e1}}$ | (2.9) | $\left(f+\frac{{v}_{1}}{r}\right){v}_{1}=\frac{\partial {\varphi}_{1}}{\partial r}$ | (2.11) | ||

$Q={Q}^{+}-{Q}^{-}$ with $Q}^{+}=\eta {w}^{+$ | (2.10) | $\left(f+\frac{{v}_{2}}{r}\right){v}_{2}=\frac{\partial {\varphi}_{2}}{\partial r}$ | (2.12) | ||

Transverse circulation: (where ${\psi}_{1}=-r{u}_{1}{h}_{1}$ and ${\psi}_{2}=-r{u}_{2}\u03f5{h}_{2}$) | |||||

$\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}r\frac{\partial}{\partial r}\left(\frac{\partial ({\psi}_{1}+{\psi}_{2})}{r\partial r}\right)-{S}_{1}{\psi}_{1}=\frac{1}{g}\left(f+\frac{2{v}_{1}}{r}\right)r{G}_{1}-r\frac{\partial w}{\partial r}$ | (2.13) | ||||

$r\frac{\partial}{\partial r}\left(\frac{\partial ({\psi}_{1}+{\u03f5}^{-1}{\psi}_{2})}{r\partial r}\right)-{S}_{2}{\u03f5}^{-1}{\psi}_{2}=\frac{1}{g}\left(f+\frac{2{v}_{2}}{r}\right)r{G}_{2}-r\frac{\partial}{\partial r}\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (2.14) | ||||

Prognostic Equations: | |||||

Boundary layer radial and tangential wind components: | |||||

$\frac{\partial {u}_{0}}{\partial t}+{u}_{0}\frac{\partial {u}_{0}}{\partial r}-\left(f+\frac{{v}_{0}}{r}\right){v}_{0}+\frac{\partial {\varphi}_{1}}{\partial r}={F}_{0}$ | (2.15) | ||||

$\frac{\partial {v}_{0}}{\partial t}+{u}_{0}\frac{\partial {v}_{0}}{\partial r}+\left(f+\frac{{v}_{0}}{r}\right){u}_{0}={G}_{0}$ | (2.16) | ||||

Geopotential anomalies: | |||||

$\frac{\partial {\varphi}_{1}}{\partial t}=g\frac{\partial ({\psi}_{1}+{\psi}_{2})}{r\partial r}+gw$ | (2.17) | ||||

$\frac{\partial {\varphi}_{2}}{\partial t}=g\frac{\partial ({\psi}_{1}+{\u03f5}^{-1}{\psi}_{2})}{r\partial r}+g\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (2.18) | ||||

Boundary layer equivalent potential temperature: | |||||

$\frac{\partial {\theta}_{e0}}{\partial t}+{u}_{0}\frac{\partial {\theta}_{e0}}{\partial r}+\frac{{w}^{-}}{{h}_{0}}({\theta}_{e0}-{\theta}_{e1})=\frac{{c}_{{}_{\mathrm{E}}}U}{{h}_{0}}({\theta}_{es}^{\ast}-{\theta}_{e0})$ | (2.19) |

## 4. Hybrid Balanced Model 2 (HB2)

## 5. Gradient Balanced Model (GB)

**Table 3.**Summary of the HB2 model. The upper half of the table lists the diagnostic equations, while the lower half lists the three prognostic equations. Some results from this model were given by Ooyama [1], who discussed how the results differ from the GB model shown in Table 4. These differences between HB2 and GB are discussed in Section 8, along with the observation that the numerical solution of the nonlocal diagnostic Equations (3.15) and (3.16) can lead to mathematical difficulties.

Hybrid Balanced Model 2 (HB2) | |||||
---|---|---|---|---|---|

Diagnostic Equations: | |||||

Layer thicknesses: | Saturation equivalent potential temperatures: | ||||

${h}_{1}={\overline{h}}_{1}+\frac{1}{g\sigma}({\varphi}_{1}-\u03f5{\varphi}_{2})$ | (3.1) | $\theta}_{es}^{\ast}={\overline{\theta}}_{es}^{\ast}-\frac{2.0}{{c}_{p}}{\varphi}_{1$ | (3.3) | ||

${h}_{2}={\overline{h}}_{2}+\frac{1}{g\sigma}({\varphi}_{2}-{\varphi}_{1})$ | (3.2) | ${\theta}_{e2}^{\ast}={\overline{\theta}}_{e2}^{\ast}+\frac{10.3}{{c}_{p}}({\varphi}_{2}-{\varphi}_{1})$ | (3.4) | ||

Air-sea interaction: | Ekman pumping: | ||||

$U={\left({u}_{0}^{2}+{v}_{0}^{2}\right)}^{1/2}$ | (3.5) | $w=-{h}_{0}\frac{\partial \left(r{u}_{0}\right)}{r\partial r}$ | (3.7) | ||

$c}_{{}_{\mathrm{E}}}={c}_{{}_{\mathrm{D}}}=(0.5+0.06\phantom{\rule{0.166667em}{0ex}}U)\xb7{10}^{-3$ | (3.6) | ${w}^{\pm}={\textstyle \frac{1}{2}}\left(\left|w\right|\pm w\right)$ | (3.8) | ||

Diabatic mass flux: | Gradient balance equations: | ||||

$\eta =1+\frac{{\theta}_{e0}-{\theta}_{e2}^{\ast}}{{\theta}_{e2}^{\ast}-{\theta}_{e1}}$ | (3.9) | $\left(f+\frac{{v}_{1}}{r}\right){v}_{1}=\frac{\partial {\varphi}_{1}}{\partial r}$ | (3.11) | ||

$Q={Q}^{+}-{Q}^{-}$ with $Q}^{+}=\eta {w}^{+$ | (3.10) | $\left(f+\frac{{v}_{2}}{r}\right){v}_{2}=\frac{\partial {\varphi}_{2}}{\partial r}$ | (3.12) | ||

Transverse circulation: (where ${\psi}_{1}=-r{u}_{1}{h}_{1}$ and ${\psi}_{2}=-r{u}_{2}\u03f5{h}_{2}$) | |||||

$\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}r\frac{\partial}{\partial r}\left(\frac{\partial ({\psi}_{1}+{\psi}_{2})}{r\partial r}\right)-{S}_{1}{\psi}_{1}=\frac{1}{g}\left(f+\frac{2{v}_{1}}{r}\right)r{G}_{1}-r\frac{\partial w}{\partial r}$ | (3.13) | ||||

$r\frac{\partial}{\partial r}\left(\frac{\partial ({\psi}_{1}+{\u03f5}^{-1}{\psi}_{2})}{r\partial r}\right)-{S}_{2}{\u03f5}^{-1}{\psi}_{2}=\frac{1}{g}\left(f+\frac{2{v}_{2}}{r}\right)r{G}_{2}-r\frac{\partial}{\partial r}\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (3.14) | ||||

Boundary layer flow: | |||||

$u}_{0}\frac{\partial {u}_{0}}{\partial r}-\left(f+\frac{{v}_{0}}{r}\right){v}_{0}+\frac{\partial {\varphi}_{1}}{\partial r}={F}_{0$ | (3.15) | $\left(f+{\zeta}_{0}\right){u}_{0}+\frac{{w}^{-}({v}_{0}-{v}_{1})}{{h}_{0}}+\frac{{c}_{{}_{\mathrm{D}}}U}{{h}_{0}}{v}_{0}=0$ | (3.16) | ||

Prognostic Equations: | |||||

Geopotential anomalies: | |||||

$\frac{\partial {\varphi}_{1}}{\partial t}=g\frac{\partial ({\psi}_{1}+{\psi}_{2})}{r\partial r}+gw$ | (3.17) | ||||

$\frac{\partial {\varphi}_{2}}{\partial t}=g\frac{\partial ({\psi}_{1}+{\u03f5}^{-1}{\psi}_{2})}{r\partial r}+g\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (3.18) | ||||

Boundary layer equivalent potential temperature: | |||||

$\frac{\partial {\theta}_{e0}}{\partial t}+{u}_{0}\frac{\partial {\theta}_{e0}}{\partial r}+\frac{{w}^{-}}{{h}_{0}}({\theta}_{e0}-{\theta}_{e1})=\frac{{c}_{{}_{\mathrm{E}}}U}{{h}_{0}}({\theta}_{es}^{\ast}-{\theta}_{e0})$ | (3.19) |

**Table 4.**Summary of the GB model. The upper half of the table lists the diagnostic equations, while the lower half lists the three prognostic equations. Note that the boundary layer radial inflow ${u}_{0}(r,t)$ and the boundary layer pumping $w(r,t)$ are computed from (4.15) and (4.7), which are local diagnostic relations. This differs from the other three models in which the boundary layer dynamics is nonlocal. Although the boundary layer radial flow ${u}_{0}(r,t)$, determined by (4.15), can vary rapidly in r because ${\zeta}_{1}(r,t)$ varies rapidly in r, true boundary layer shocks are not produced in the GB model because of the local nature of the simplified boundary layer dynamics. An extensive set of numerical integrations using this model is given by Ooyama [2].

Gradient Balanced Model (GB) | |||||
---|---|---|---|---|---|

Diagnostic Equations: | |||||

Layer thicknesses: | Saturation equivalent potential temperatures: | ||||

${h}_{1}={\overline{h}}_{1}+\frac{1}{g\sigma}({\varphi}_{1}-\u03f5{\varphi}_{2})$ | (4.1) | $\theta}_{es}^{\ast}={\overline{\theta}}_{es}^{\ast}-\frac{2.0}{{c}_{p}}{\varphi}_{1$ | (4.3) | ||

${h}_{2}={\overline{h}}_{2}+\frac{1}{g\sigma}({\varphi}_{2}-{\varphi}_{1})$ | (4.2) | (4.4) | |||

Air-sea interaction: | Ekman pumping: | ||||

$U=|{v}_{0}|$ | (4.5) | $w=-{h}_{0}\frac{\partial \left(r{u}_{0}\right)}{r\partial r}$ | (4.7) | ||

$c}_{{}_{\mathrm{E}}}={c}_{{}_{\mathrm{D}}}=(0.5+0.06\phantom{\rule{0.166667em}{0ex}}U)\xb7{10}^{-3$ | (4.6) | ${w}^{\pm}={\textstyle \frac{1}{2}}\left(\left|w\right|\pm w\right)$ | (4.8) | ||

Diabatic mass flux: | Gradient balance equations: | ||||

$\eta =1+\frac{{\theta}_{e0}-{\theta}_{e2}^{\ast}}{{\theta}_{e2}^{\ast}-{\theta}_{e1}}$ | (4.9) | $\left(f+\frac{{v}_{1}}{r}\right){v}_{1}=\frac{\partial {\varphi}_{1}}{\partial r}$ | (4.11) | ||

$Q={Q}^{+}-{Q}^{-}$ with $Q}^{+}=\eta {w}^{+$ | (4.10) | $\left(f+\frac{{v}_{2}}{r}\right){v}_{2}=\frac{\partial {\varphi}_{2}}{\partial r}$ | (4.12) | ||

Transverse circulation: (where ${\psi}_{1}=-r{u}_{1}{h}_{1}$ and ${\psi}_{2}=-r{u}_{2}\u03f5{h}_{2}$) | |||||

$\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}r\frac{\partial}{\partial r}\left(\frac{\partial ({\psi}_{1}+{\psi}_{2})}{r\partial r}\right)-{S}_{1}{\psi}_{1}=\frac{1}{g}\left(f+\frac{2{v}_{1}}{r}\right)r{G}_{1}-r\frac{\partial w}{\partial r}$ | (4.13) | ||||

$r\frac{\partial}{\partial r}\left(\frac{\partial ({\psi}_{1}+{\u03f5}^{-1}{\psi}_{2})}{r\partial r}\right)-{S}_{2}{\u03f5}^{-1}{\psi}_{2}=\frac{1}{g}\left(f+\frac{2{v}_{2}}{r}\right)r{G}_{2}-r\frac{\partial}{\partial r}\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (4.14) | ||||

Boundary layer flow: | |||||

$u}_{0}=-\frac{{c}_{{}_{\mathrm{D}}}U{v}_{0}}{{h}_{0}(f+{\zeta}_{1})$ | (4.15) | $v}_{0}={v}_{1$ | (4.16) | ||

Prognostic Equations: | |||||

Geopotential anomalies: | |||||

$\frac{\partial {\varphi}_{1}}{\partial t}=g\frac{\partial ({\psi}_{1}+{\psi}_{2})}{r\partial r}+gw$ | (4.17) | ||||

$\frac{\partial {\varphi}_{2}}{\partial t}=g\frac{\partial ({\psi}_{1}+{\u03f5}^{-1}{\psi}_{2})}{r\partial r}+g\left(w+\frac{\sigma}{\u03f5}Q\right)$ | (4.18) | ||||

Boundary layer equivalent potential temperature: | |||||

(4.19) |

## 6. Isolating the Slab Boundary Layer Model from the PE and HB1 Models

## 7. Potential Vorticity Aspects of the Models

## 8. Concluding Remarks

Parameter | Value | Parameter | Value |
---|---|---|---|

$\u03f5$ | 0.9 | ${h}_{0}$ | 1000 m |

$\sigma =1-\u03f5$ | 0.1 | ${\overline{h}}_{1}$ | 5000 m |

g | 9.81 m s${}^{-2}$ | ${\overline{h}}_{2}$ | 5000 m |

f | $5\times {10}^{-5}$ s${}^{-1}$ | ${\overline{\theta}}_{es}^{\ast}$ | 372 K |

${c}_{p}$ | 1004.5 J kg${}^{-1}$ K${}^{-1}$ | ${\overline{\theta}}_{e2}^{\ast}$ | 342 K |

$\mu $ | $5\times {10}^{-4}$ m s${}^{-1}$ | ${\theta}_{e1}$ | 332 K |

K | 1000 m${}^{2}$ s${}^{-1}$ | $\widehat{v}$ | 10 m s${}^{-1}$ |

${Q}^{-}$ | 0 m s${}^{-1}$ | $\widehat{r}$ | 50 km |

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Momentum Equations

## References

- Ooyama, K. Numerical simulation of tropical cyclones with an axisymmetric model. In Proceedings of the WMO/IUGG Symposium on Numerical Weather Prediction, Tokyo, Japan, 26 November–4 December 1968; Japan Meteorological Agency: Minato, Tokyo, 1969. Session III. pp. 81–88. [Google Scholar]
- Ooyama, K. Numerical simulation of the life cycle of tropical cyclones. J. Atmos. Sci.
**1969**, 26, 3–40. [Google Scholar] [CrossRef] - DeMaria, M.; Pickle, J.D. A simplified system of equations for simulation of tropical cyclones. J. Atmos. Sci.
**1988**, 45, 1542–1554. [Google Scholar] [CrossRef] - Eliassen, A. Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norv.
**1951**, 5, 19–60. [Google Scholar] - Greenspan, H.P.; Howard, L.N. On a time-dependent motion of a rotating fluid. J. Fluid Mech.
**1963**, 17, 385–404. [Google Scholar] [CrossRef] - Greenspan, H.P. The Theory of Rotating Fluids; Cambridge University Press: Cambridge, UK, 1968; 325p. [Google Scholar]
- Eliassen, A.; Lystad, M. The Ekman layer of a circular vortex: A numerical and theoretical study. Geophys. Norv.
**1977**, 31, 1–16. [Google Scholar] - Frisius, T.; Lee, M. The impact of gradient wind imbalance on tropical cyclone intensification within Ooyama’s three-layer model. J. Atmos. Sci.
**2016**, 73, 3659–3679. [Google Scholar] [CrossRef] - Kepert, J.D. Slab- and height-resolving models of the tropical cyclone boundary layer. Part II: Why the simulations differ. Q. J. R. Meteorol. Soc.
**2010**, 136, 1700–1711. [Google Scholar] [CrossRef] - Smith, R.K.; Vogl, S. A simple model of the hurricane boundary layer revisited. Q. J. R. Meteorol. Soc.
**2008**, 134, 337–351. [Google Scholar] [CrossRef] - Williams, G.J.; Taft, R.K.; McNoldy, B.D.; Schubert, W.H. Shock-like structures in the tropical cyclone boundary layer. J. Adv. Model. Earth Syst.
**2013**, 5, 338–353. [Google Scholar] [CrossRef] - Slocum, C.J.; Williams, G.J.; Taft, R.K.; Schubert, W.H. Tropical cyclone boundary layer shocks. arXiv
**2014**, arXiv:1405.7939. [Google Scholar] - Tsujino, S.; Kuo, H.C. Potential vorticity mixing and rapid intensification in the numerically simulated Supertyphoon Haiyan (2013). J. Atmos. Sci.
**2020**, 77, 2067–2090. [Google Scholar] [CrossRef][Green Version] - Schubert, W.H.; Alworth, B.T. Evolution of potential vorticity in tropical cyclones. Q. J. R. Meteorol. Soc.
**1987**, 113, 147–162. [Google Scholar] [CrossRef] - Möller, J.D.; Smith, R.K. The development of potential vorticity in a hurricane-like vortex. Q. J. R. Meteorol. Soc.
**1994**, 120, 1255–1265. [Google Scholar] [CrossRef] - Hausman, S.A.; Ooyama, K.V.; Schubert, W.H. Potential vorticity structure of simulated hurricanes. J. Atmos. Sci.
**2006**, 63, 87–108. [Google Scholar] [CrossRef][Green Version] - Martinez, J.; Bell, M.M.; Rogers, R.F.; Doyle, J.D. Axisymmetric potential vorticity evolution of Hurricane Patricia (2015). J. Atmos. Sci.
**2019**, 76, 2043–2063. [Google Scholar] [CrossRef][Green Version] - Hendricks, E.A.; Vigh, J.L.; Rozoff, C.M. Forced, balanced, axisymmetric shallow water model for understanding short-term tropical cyclone intensity and wind structure changes. Atmosphere
**2021**, 12, 1308. [Google Scholar] [CrossRef] - Kepert, J.D. Slab- and height-resolving models of the tropical cyclone boundary layer. Part I: Comparing the simulations. Q. J. R. Meteorol. Soc.
**2010**, 136, 1686–1699. [Google Scholar] [CrossRef] - Marks, F.D.; Black, P.G.; Montgomery, M.T.; Burpee, R.W. Structure of the eye and eyewall of Hurricane Hugo (1989). Mon. Weather Rev.
**2008**, 136, 1237–1259. [Google Scholar] [CrossRef][Green Version] - von Neumann, J.; Richtmyer, R.D. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys.
**1950**, 21, 232–237. [Google Scholar] [CrossRef] - Charney, J.G.; Eliassen, A. On the growth of the hurricane depression. J. Atmos. Sci.
**1964**, 21, 68–75. [Google Scholar] [CrossRef] - Yanai, M.; Esbensen, S.; Chu, J.H. Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos. Sci.
**1973**, 30, 611–627. [Google Scholar] [CrossRef] - Gray, W.M. Cumulus convection and larger scale circulations I. Broadscale and mesoscale considerations. Mon. Weather Rev.
**1973**, 101, 839–855. [Google Scholar] [CrossRef] - Lopez, R.E. Cumulus convection and larger scale circulations II. Cumulus and mesoscale interactions. Mon. Weather Rev.
**1973**, 101, 856–870. [Google Scholar] [CrossRef] - Ogura, Y.; Cho, H.R. Diagnostic determination of cumulus cloud populations from observed large-scale variables. J. Atmos. Sci.
**1973**, 30, 1276–1286. [Google Scholar] [CrossRef] - Smith, R.K.; Kilroy, G.; Montgomery, M.T. Tropical cyclone life cycle in a three-dimensional numerical simulation. Q. J. R. Meteorol. Soc.
**2021**, 147, 3373–3393. [Google Scholar] [CrossRef] - Bryan, G.H.; Fritsch, J.M. A benchmark simulation for moist nonhydrostatic numerical models. Mon. Weather Rev.
**2002**, 130, 2917–2928. [Google Scholar] [CrossRef] - Zhu, H.; Smith, R.K.; Ulrich, W. A minimal three-dimensional tropical cyclone model. J. Atmos. Sci.
**2001**, 58, 1924–1944. [Google Scholar] [CrossRef] - Arakawa, A. Parameterization of cumulus convection. In Proceedings of the WMO/IUGG Symposium on Numerical Weather Prediction, Tokyo, Japan, 26 November–4 December 1968; Japan Meteorological Agency: Minato, Tokyo, 1969. Session IV. pp. 1–6. [Google Scholar]
- Nguyen, C.M.; Smith, R.K.; Zhu, H.; Ulrich, W. A minimal axisymmetric hurricane model. Q. J. R. Meteorol. Soc.
**2002**, 128, 2641–2661. [Google Scholar] [CrossRef] - Zhu, H.; Smith, R.K. The importance of three physical processes in a three-dimensional tropical cyclone model. J. Atmos. Sci.
**2002**, 59, 1825–1840. [Google Scholar] [CrossRef] - Zhu, H.; Smith, R.K. Effects of vertical differencing in a minimal hurricane model. Q. J. R. Meteorol. Soc.
**2003**, 129, 1051–1069. [Google Scholar] [CrossRef] - Zhu, H.; Ulrich, W.; Smith, R.K. Ocean effects on tropical cyclone intensification and inner-core asymmetries. J. Atmos. Sci.
**2004**, 61, 1245–1258. [Google Scholar] [CrossRef] - Emanuel, K.A. An air–sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci.
**1986**, 43, 585–604. [Google Scholar] [CrossRef] - Emanuel, K.A. The finite-amplitude nature of tropical cyclogenesis. J. Atmos. Sci.
**1989**, 46, 3431–3456. [Google Scholar] [CrossRef] - Emanuel, K.A. The behavior of a simple hurricane model using a convective scheme based on subcloud-layer entropy equilibrium. J. Atmos. Sci.
**1995**, 52, 3960–3968. [Google Scholar] [CrossRef] - Montgomery, M.T.; Smith, R.K. Mininal conceptual models for tropical cyclone intensification. Trop. Cyclone Res. Rev.
**2022**, 11, 61–75. [Google Scholar] [CrossRef] - Smith, R.K.; Montgomery, M.T. Tropical Cyclones: Observations and Basic Processes; Elsevier: Amsterdam, The Netherlands, 2023. [Google Scholar]
- Bliss, V.L. Numerical Simulation of Tropical Cyclone Genesis. Ph.D. Thesis, Department of Atmospheric Science, University of Washington, Seattle, WA, USA, 1980; 268p. [Google Scholar]
- DeMaria, M.; Schubert, W.H. Experiments with a spectral tropical cyclone model. J. Atmos. Sci.
**1984**, 41, 901–924. [Google Scholar] [CrossRef] - Schubert, W.H.; DeMaria, M. Axisymmetric, primitive equation, spectral tropical cyclone model. Part I: Formulation. J. Atmos. Sci.
**1985**, 42, 1213–1224. [Google Scholar] [CrossRef] - DeMaria, M.; Schubert, W.H. Axisymmetric, primitive equation, spectral tropical cyclone model. Part II: Normal mode initialization. J. Atmos. Sci.
**1985**, 42, 1225–1236. [Google Scholar] [CrossRef] - Arakawa, A. Non-geostrophic effects in the baroclinic prognostic equations. In Proceedings of the International Symposium on Numerical Weather Prediction, Tokyo, Japan, 7–13 November 1960; The Meteorological Society of Japan: Tokyo, Japan, 1962; pp. 161–175. [Google Scholar]
- Thompson, P.D. A theory of large-scale disturbances in non-geostrophic flow. J. Meteorol.
**1956**, 13, 251–261. [Google Scholar] [CrossRef] - Thompson, P.D. A short-range prediction scheme based on conservation principles and the generalized balance condition. Contrib. Atmos. Phys.
**1980**, 53, 256–263. [Google Scholar] - Shapiro, L.J.; Montgomery, M.T. A three-dimensional balance theory for rapidly rotating vortices. J. Atmos. Sci.
**1993**, 50, 3322–3335. [Google Scholar] [CrossRef] - Emanuel, K.A. Tropical cyclones. Annu. Rev. Earth Planet. Sci.
**2003**, 31, 75–104. [Google Scholar] [CrossRef] - Emanuel, K.A. 100 years of progress in tropical cyclone research. Meteorol. Monogr.
**2018**, 59, 15.1–15.68. [Google Scholar] [CrossRef] - Shapiro, L.J. Hurricane vortex motion and evolution in a three-layer model. J. Atmos. Sci.
**1992**, 49, 140–153. [Google Scholar] [CrossRef] - Camp, J.P.; Montgomery, M.T. Hurricane maximum intensity: Past and present. Mon. Weather Rev.
**2001**, 129, 1704–1717. [Google Scholar] [CrossRef] - Smith, R.K.; Montgomery, M.T. Balanced boundary layers used in hurricane models. Q. J. R. Meteorol. Soc.
**2008**, 134, 1385–1395. [Google Scholar] [CrossRef][Green Version] - Schecter, D.A.; Dunkerton, T.J. Hurricane formation in diabatic Ekman turbulence. Q. J. R. Meteorol. Soc.
**2009**, 135, 823–838. [Google Scholar] [CrossRef] - Schecter, D.A. Hurricane intensity in the Ooyama (1969) paradigm. Q. J. R. Meteorol. Soc.
**2010**, 136, 1920–1926. [Google Scholar] [CrossRef] - Schecter, D.A. Evaluation of a reduced model for investigating hurricane formation from turbulence. Q. J. R. Meteorol. Soc.
**2011**, 137, 155–178. [Google Scholar] [CrossRef] - Frisius, T.; Schönemann, D.; Vigh, J. The impact of gradient wind imbalance on potential intensity of tropical cyclones in an unbalanced slab boundary layer model. J. Atmos. Sci.
**2013**, 70, 1874–1890. [Google Scholar] [CrossRef] - Rogers, R.F.; Aberson, S.; Bell, M.M.; Cecil, D.J.; Doyle, J.D.; Kimberlain, T.B.; Morgerman, J.; Shay, L.K.; Velden, C. Rewriting the tropical record books: The extraordinary intensification of Hurricane Patricia (2015). Bull. Am. Meteorol. Soc.
**2017**, 98, 2091–2112. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of the axisymmetric, three-layer, f-plane family of models. The constant density in the lower two layers is ${\rho}_{0}={\rho}_{1}=\rho $, while the constant density in the upper layer is ${\rho}_{2}=\u03f5\rho $, with $\u03f5<1$. The thicknesses of the layers are ${h}_{0}$, ${h}_{1}$, and ${h}_{2}$, with ${h}_{0}$ a specified constant, and with ${h}_{1}$ and ${h}_{2}$ functions of $(r,t)$. The radial velocity components are ${u}_{0}$, ${u}_{1}$, and ${u}_{2}$, and the radial mass fluxes are ${\psi}_{0}=-r{u}_{0}{h}_{0}$, ${\psi}_{1}=-r{u}_{1}{h}_{1}$, and ${\psi}_{2}=-r{u}_{2}\u03f5{h}_{2}$. The tangential velocity components are ${v}_{0}$, ${v}_{1}$, and ${v}_{2}$. The surface stress ${\tau}_{s}$ drives the boundary layer radial inflow, which results in boundary layer pumping ($w>0$) in the inner region and boundary layer suction ($w<0$) in the outer region. Diabatic processes are parameterized through the mass transport terms ${Q}^{+}$ and ${Q}^{-}$. Adapted from ([2], Figure 1).

**Figure 2.**Comparison of the diagnostic and prognostic variables from the four models. The left column lists the model variables, all of which are functions of r and t. For the PE model, the bottom nine variables are prognostic, and all the remaining variables (above them in the table) are diagnostic. For the HB1 model, only the bottom five variables are prognostic, and four variables ($\{{u}_{1},{u}_{2}\}$ or $\{{\psi}_{1},{\psi}_{2}\}$, ${v}_{1}$, and ${v}_{2}$) that are prognostic for the PE model are now diagnostic. For the HB2 and GB models, just the bottom three variables are prognostic with two variables (${u}_{0}$ and ${v}_{0}$) that are prognostic for the PE and HB1 models now being diagnostic. The PE model allows propagating inertia-gravity waves in the upper two layers, while the other three models filter these waves because the radial components ${u}_{1}$ and ${u}_{2}$ are determined diagnostically, as indicated by the outer arrow on the right. Because of the form of their ${u}_{0}$ and ${v}_{0}$ equations, the PE and HB1 models allow the formation of well-defined boundary layer shocks, as indicated by the inner arrow on the right.

**Figure 3.**Comparison of the tangential winds ${v}_{0}$ and ${v}_{1}$ (

**a**) and the Ekman pumping w (

**b**) for the gradient balanced model (GB, dashed lines) and the second hybrid model (HB2, solid lines) at $t=146$ h for a typical life cycle experiment. The vortex in HB2 is tighter and has gone through a more rapid intensification than the vortex in GB. This difference in structure and intensification rate is due to the striking differences in Ekman pumping, with the GB model having its 2 m s${}^{-1}$ maximum w at $r\approx 60$ km and the HB2 model having a similar maximum but at $r\approx 15$ km. This illustrates the fundamental importance of the ${u}_{0}(\partial {u}_{0}/\partial r)$ term in the radial equation of boundary layer motion in the HB2 model. Note that the boundary layer tangential flow ${v}_{0}$ in the HB2 model is slightly subgradient for $r>30$ km and slightly supergradient for $r<30$ km. This subgradient/supergradient effect can be more apparent in strong storms such as Hurricane Hugo (see [11,20]). Adapted from ([1], Figure 2).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schubert, W.H.; Taft, R.K.; Slocum, C.J. A Simple Family of Tropical Cyclone Models. *Meteorology* **2023**, *2*, 149-170.
https://doi.org/10.3390/meteorology2020011

**AMA Style**

Schubert WH, Taft RK, Slocum CJ. A Simple Family of Tropical Cyclone Models. *Meteorology*. 2023; 2(2):149-170.
https://doi.org/10.3390/meteorology2020011

**Chicago/Turabian Style**

Schubert, Wayne H., Richard K. Taft, and Christopher J. Slocum. 2023. "A Simple Family of Tropical Cyclone Models" *Meteorology* 2, no. 2: 149-170.
https://doi.org/10.3390/meteorology2020011