A height-resolving tropical cyclone boundary layer model with vertical advection process

Abstract

The height-resolving model is thought to be an optimal scheme for modeling the tropical cyclone (TC) wind fields in the boundary layer because it explicitly depicts the wind structures in that layer as TC evolves over time. However, the vertical advection process which exists in TCs has not been well considered in previously proposed parametric TC models. Neglecting this process may cause deviations of the simulated wind field structure in the boundary layer. Herein, a height-resolving boundary layer wind field model incorporating both the vertical advection and vertical diffusion processes is proposed and a semi-analytical solution to the governing equations is developed. The adequacy of this model is evaluated by comparing with the Weather Research and Forecasting model simulations, the Hurricane Research Division’s H*Wind snapshots, GPS dropsonde datasets and ground measurements of several TC events. Results show that the proposed model with vertical advection can reasonably produce the wind fields of TCs, and its advantage lies in the production of a more realistic three-dimensional wind structure in the boundary layer.

Appendices

Appendix 1

In the cylindrical coordinate system, Eq. (3) can be expressed as

$$\begin{gathered} (u_{c} + u^{\prime})\frac{{\partial u^{\prime}}}{\partial r} + \frac{{v_{c} + v^{\prime}}}{r}\frac{{\partial u^{\prime}}}{\partial \lambda } + \frac{{v_{g} }}{r}\frac{{\partial u^{\prime}}}{\partial \lambda } + w^{\prime}\frac{{\partial u^{\prime}}}{\partial z} - \frac{{v_{c} v^{\prime} + v^{{\prime}{2}} + 2v^{\prime}v_{g} }}{r} = f \cdot v^{\prime} + \hfill \\ K_{v} \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{{\partial u^{\prime}}}{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} u^{\prime}}}{{\partial \lambda^{2} }} + \frac{{\partial^{2} u^{\prime}}}{{\partial z^{2} }} - \frac{1}{{r^{2} }}\left( {u^{\prime} + 2\frac{{\partial (v_{g} + v^{\prime})}}{\partial \lambda }} \right)} \right] \hfill \\ \end{gathered}$$

(33)

$$\begin{gathered} (u_{c} + u^{\prime})\frac{{\partial v^{\prime}}}{\partial r} + \frac{{v_{c} + v^{\prime}}}{r}\frac{{\partial v^{\prime}}}{\partial \lambda } + \frac{{v_{g} }}{r}\frac{{\partial v^{\prime}}}{\partial \lambda } + w^{\prime}\frac{{\partial v^{\prime}}}{\partial z} + \frac{{u_{c} v^{\prime} + u^{\prime}v^{\prime} + u^{\prime}u_{g} }}{r} + u^{\prime}\frac{{\partial v_{g} }}{\partial r} = - f \cdot u^{\prime} + \hfill \\ K_{v} \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{{\partial (v_{g} + v^{\prime})}}{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} (v_{g} + v^{\prime})}}{{\partial \lambda^{2} }} + \frac{{\partial^{2} (v_{g} + v^{\prime})}}{{\partial z^{2} }} - \frac{1}{{r^{2} }}\left( {(v_{g} + v^{\prime}) - 2\frac{{\partial u^{\prime}}}{\partial \lambda }} \right)} \right] \hfill \\ \end{gathered}$$

(34)

in which, u′, v′ and w′ are the radial, tangential and vertical friction caused wind components, respectively; r, λ and z are the radial, azimuthal and vertical variables, respectively; u_c and v_c are the radial and tangential components of V_c. It should be noted that the radial component of the gradient wind (u_g) is assumed to be zero. Scale analysis is used here to simplify Eqs. (33) and (34), and the variable magnitudes of a tropical cyclone are listed in Table

Table 3 Scales, magnitudes and units of variables in Eqs. (34) and (35)

3.

The scaling terms and magnitudes of Eqs. (33) and (34) are listed in Tables

Table 4 Scaling of the terms in Eq. (33)

4 and

Table 5 Scaling of the terms in Eq. (34)

5. For simplicity, we only consider the terms whose magnitudes larger than 10^–4. The Coriolis terms and vertical diffusion terms are retained for their importance in TC wind field simulations. Then, we have

$$\frac{{v_{g} }}{r}\frac{{\partial u^{\prime}}}{\partial \lambda } - \left( {\frac{{2v_{g} }}{r} + f} \right)v^{\prime} + w^{\prime}\frac{{\partial u^{\prime}}}{\partial z} = K_{v} \frac{{\partial^{2} u^{\prime}}}{{\partial z^{2} }}$$

(35)

$$\frac{{v_{g} }}{r}\frac{{\partial v^{\prime}}}{\partial \lambda }{ + }\left( {\frac{{\partial v_{g} }}{\partial r} + \frac{{v_{g} }}{r} + f} \right)u^{\prime} + w^{\prime}\frac{{\partial v^{\prime}}}{\partial z} = K_{v} \frac{{\partial^{2} v^{\prime}}}{{\partial z^{2} }}$$

(36)

Appendix 2

Here, we assume the friction wind components u′ and v′ to be zero as z →  + ∞. The p_k in Eq. (26) can be solved as

$$p_{k} = \delta - \sqrt {2i\sqrt {\alpha \beta } + 2ik\gamma + \delta^{2} }$$

(37)

Let

$$t_{k} = \sqrt {2i\sqrt {\alpha \beta } + 2ik\gamma + \delta^{2} } = a + bi$$

(38)

where a = Re(t_k), b = Im(t_k). For $\sqrt {\alpha \beta } + k\gamma \ge 0$, $p_{k} = \delta - \left[ {\left| a \right| + \left| b \right|i} \right]$; for $\sqrt {\alpha \beta } + k\gamma < 0$, $p^{\prime}_{k} = \delta - \left[ {\left| a \right| - \left| b \right|i} \right]$.

After some manipulations of terms in Eqs. (27) and (28), for $\sqrt {\alpha \beta } + k\gamma \ge 0$ and $\left| k \right| \le 1$, we have the following equations.

For k = 0,

$$- \sqrt {\frac{\alpha }{\beta }} A_{0} p_{0} - \sqrt {\frac{\alpha }{\beta }} A_{0}^{*} p_{0}^{*} + \sqrt {\frac{\alpha }{\beta }} \frac{{A_{0} C_{d} v_{g} }}{{K_{v} }} + \sqrt {\frac{\alpha }{\beta }} \frac{{A_{0}^{*} C_{d} v_{g} }}{{K_{v} }} = 0$$

(39)

$$A_{0} p_{0} - A_{0}^{*} p_{0}^{*} - \frac{{2A_{0} C_{d} v_{g} }}{{K_{v} }} + \frac{{2A_{0}^{*} C_{d} v_{g} }}{{K_{v} }} - \frac{{2iC_{d} v_{g}^{2} }}{{K_{v} }} = 0$$

(40)

For k = 1,

$$- \sqrt {\frac{\alpha }{\beta }} A_{1} p_{1} - \sqrt {\frac{\alpha }{\beta }} A_{ - 1}^{*} p_{ - 1}^{*} + \sqrt {\frac{\alpha }{\beta }} \frac{{A_{1} C_{d} v_{g} }}{{K_{v} }} + \sqrt {\frac{\alpha }{\beta }} \frac{{A_{ - 1}^{*} C_{d} v_{g} }}{{K_{v} }} + \frac{{{\mathbf{V}}_{c} C_{d} v_{g} }}{{K_{v} }} = 0$$

(41)

$$A_{1} p_{1} - A_{ - 1}^{*} p_{ - 1}^{*} - \frac{{2A_{1} C_{d} v_{g} }}{{K_{v} }} + \frac{{2A_{ - 1}^{*} C_{d} v_{g} }}{{K_{v} }} + \frac{{2V_{c} C_{d} v_{g} }}{{K_{v} }} = 0$$

(42)

For k = −1,

$$- \sqrt {\frac{\alpha }{\beta }} A_{ - 1} p_{ - 1} - \sqrt {\frac{\alpha }{\beta }} A_{1}^{*} p_{1}^{*} + \sqrt {\frac{\alpha }{\beta }} \frac{{A_{ - 1} C_{d} v_{g} }}{{K_{v} }} + \sqrt {\frac{\alpha }{\beta }} \frac{{A_{1}^{*} C_{d} v_{g} }}{{K_{v} }} + \frac{{{\mathbf{V}}_{c} C_{d} v_{g} }}{{K_{v} }} = 0$$

(43)

$$A_{ - 1} p_{ - 1} - A_{1}^{*} p_{1}^{*} - \frac{{2A_{ - 1} C_{d} v_{g} }}{{K_{v} }} + \frac{{2A_{1}^{*} C_{d} v_{g} }}{{K_{v} }} - \frac{{2V_{c} C_{d} v_{g} }}{{K_{v} }} = 0$$

(44)

In the above equations, $*$ indicates a complex conjugate. A₀, A₁ and A_-1 can be derived from Eqs. (39) and (40), (41) and (42), (43) and (44), respectively, and we have

$$A_{0} = - \frac{{2iC_{d} v_{g}^{2} ( - K_{v} p_{0}^{*} + C_{d} v_{g} )}}{{2K_{v}^{2} p_{0} p_{0}^{*} - 3K_{v} C_{d} p_{0} v_{g} - 3K_{v} C_{d} p_{0}^{*} v_{g} + 4C_{d}^{2} v_{g}^{2} }}$$

(45)

$$A_{1} = \frac{{{\mathbf{V}}_{c} C_{d} v_{g} \left( {K_{v} p_{ - 1}^{*} - 2K_{v} \sqrt {\frac{\alpha }{\beta }} p_{ - 1}^{*} - 2C_{d} v_{g} + 2\sqrt {\frac{\alpha }{\beta }} C_{d} v_{g} } \right)}}{{\sqrt {\frac{\alpha }{\beta }} \left( {2K_{v}^{2} p_{1} p_{ - 1}^{*} - 3K_{v} C_{d} p_{1} v_{g} - 3K_{v} C_{d} p_{ - 1}^{*} v_{g} + 4C_{d}^{2} v_{g}^{2} } \right)}}$$

(46)

$$A_{ - 1} = \frac{{{\mathbf{V}}_{c} C_{d} v_{g} \left( {K_{v} p_{1}^{*} + 2K_{v} \sqrt {\frac{\alpha }{\beta }} p_{1}^{*} - 2C_{d} v_{g} - 2\sqrt {\frac{\alpha }{\beta }} C_{d} v_{g} } \right)}}{{\sqrt {\frac{\alpha }{\beta }} \left( {2K_{v}^{2} p_{ - 1} p_{1}^{*} - 3K_{v} C_{d} p_{ - 1} v_{g} - 3K_{v} C_{d} p_{1}^{*} v_{g} + 4C_{d}^{2} v_{g}^{2} } \right)}}$$

(47)

For $\sqrt {\alpha \beta } + k\gamma < 0$, one can easily have

$$A^{\prime}_{1} = \frac{{V_{c} C_{d} v_{g} \left( {K_{v} p_{ - 1}^{*\prime } - 2K_{v} \sqrt {\frac{\alpha }{\beta }} p_{ - 1}^{*\prime } - 2C_{d} v_{g} + 2\sqrt {\frac{\alpha }{\beta }} C_{d} v_{g} } \right)}}{{\sqrt {\frac{\alpha }{\beta }} \left( {2K_{v}^{2} p_{1} p_{ - 1}^{*\prime } - 3K_{v} C_{d} p_{1} v_{g} - 3K_{v} C_{d} p_{ - 1}^{*\prime } v_{g} + 4C_{d}^{2} v_{g}^{2} } \right)}}$$

(48)

$$A_{ - 1}^{\prime } = \frac{{{\mathbf{V}}_{c} C_{d} v_{g} \left( {K_{v} p_{1}^{*} + 2K_{v} \sqrt {\frac{\alpha }{\beta }} p_{1}^{*} - 2C_{d} v_{g} - 2\sqrt {\frac{\alpha }{\beta }} C_{d} v_{g} } \right)}}{{\sqrt {\frac{\alpha }{\beta }} \left( {2K_{v}^{2} p_{ - 1}^{\prime } p_{1}^{*} - 3K_{v} C_{d} p_{ - 1}^{\prime } v_{g} - 3K_{v} C_{d} p_{1}^{*} v_{g} + 4C_{d}^{2} v_{g}^{2} } \right)}}$$

(49)

Finally, the friction caused wind components (uʹ, vʹ) in the TC boundary layer are,

for $\sqrt {\alpha \beta } + k\gamma \ge 0$,

$$u^{\prime}{\kern 1pt} {\kern 1pt} (r,\lambda ,z) = \sqrt {\frac{\alpha }{\beta }} \cdot {\text{Re}} \left[ {A_{ - 1} \exp (p_{ - 1} \cdot z - i\lambda ) + A_{0} \exp (p_{0} \cdot z) + A_{1} \exp (p_{1} \cdot z + i\lambda )} \right]$$

(50)

$$v^{\prime}{\kern 1pt} {\kern 1pt} (r,\lambda ,z) = {\text{Im}} \left[ {A_{ - 1} \exp (p_{ - 1} \cdot z - i\lambda ) + A_{0} \exp (p_{0} \cdot z) + A_{1} \exp (p_{1} \cdot z + i\lambda )} \right]$$

(51)

for $\sqrt {\alpha \beta } + k\gamma < 0$,

$$u^{\prime}{\kern 1pt} {\kern 1pt} (r,\lambda ,z) = \sqrt {\frac{\alpha }{\beta }} \cdot {\text{Re}} \left[ {A^{\prime}_{ - 1} \exp (p^{\prime}_{ - 1} \cdot z - i\lambda ) + A^{\prime}_{0} \exp (p^{\prime}_{0} \cdot z) + A^{\prime}_{1} \exp (p^{\prime}_{1} \cdot z + i\lambda )} \right]$$

(52)

$$v^{\prime}{\kern 1pt} {\kern 1pt} (r,\lambda ,z) = {\text{Im}} \left[ {A^{\prime}_{ - 1} \exp (p^{\prime}_{ - 1} \cdot z - i\lambda ) + A^{\prime}_{0} \exp (p^{\prime}_{0} \cdot z) + A^{\prime}_{1} \exp (p^{\prime}_{1} \cdot z + i\lambda )} \right]$$

(53)