Boundary Layer Model for Moving Tropical Cyclones

Abstract

We propose a simple theoretical model for the boundary layer (BL) of moving tropical cyclones (TCs). The model estimates the horizontal and vertical wind velocity fields from a few TC characteristics: the maximum tangential wind speed V _max , the radius of maximum winds R _max , and Holland’s B parameter away from the surface boundary where gradient balance is approximately valid, in addition to the storm translation velocity V _t , the surface drag coefficient C _D , and the vertical diffusion coefficient of the horizontal momentum K.

The model is based on Smith’s (1968) formulation for stationary (axi-symmetric) tropical cyclones. Smith’s model is first extended to include storm motion and then solved using the momentum integral method. The scheme is computationally very efficient and is stable also for large B values and fast-moving storms.

Results are compared to those from other studies (Shapiro 1983; Kepert 2001) and validated using the Fifth-Generation Pennsylvania State University/NCAR Mesoscale Model (MM5). We find that Kepert’s (2001) BL model significantly underestimates the radial and vertical fluxes, whereas Shapiro’s (1983) slab-layer formulation produces radial and vertical winds that are a factor of about two higher than those produced by MM5. The velocity fields generated by the present model are consistent with MM5 and with tropical cyclone observations.

We use the model to study how the symmetric and asymmetric components of the wind field vary with the storm parameters mentioned above. In accordance with observations, we find that larger values of B and lower values of R _max produce horizontal and vertical wind profiles that are more picked near the radius of maximum winds. We also find that, when cyclones in the northern hemisphere move, the vertical and storm-relative radial winds intensify at the right-front quadrant of the vortex, whereas the storm-relative tangential winds are more intense in the left-front region. The asymmetry is higher for faster moving TCs and for higher surface drag coefficients C _D .

Appendix A

Analytical expressions for the parameters in Eq. (15)

Let $I_1 = \int\limits_0^\infty {f^2 d\eta }$, $I_2 = \int\limits_0^\infty {\left( {1 - g^2 } \right)d\eta }$, $I_3 = \int\limits_0^\infty {\left( {1 - g} \right)d\eta }$, $I_4 = \int\limits_0^\infty {fgd\eta }$, $I_5 = \int\limits_0^\infty {fd\eta }$, $I_6 = \int\limits_0^\infty {\left( {g^2 - g} \right)d\eta }$ and $I_7 = \int\limits_0^\infty {\left( {1 - g} \right)^2 d\eta }$, where f and g are the functions given in Eq. (11). Then

$$ \eqalign{& {{\partial f} \over {\partial \eta }}|_{\eta = 0} = a_2 - a_1 ,{{\partial g} \over {\partial \eta }}|_{\eta = 0} = a_1 - a_2 \cr & I_1 = {1 \over 8}\left( {a_1^2 + 2a_1 a_2 + 3a_2^2 } \right),I_2 = \left( {a_1 + a_2 } \right) - {1 \over 8}\left( {3a_1^2 + 2a_1 a_2 + a_2^2 } \right) \cr & I_3 = {{a_1 + a_2 } \over 2},I_4 = {1 \over 8}\left( {a_1^2 + 4a_1 a_2 + a_2^2 } \right) - {{a_1 + a_2 } \over 2} \cr & I_5 = - {{a_1 + a_2 } \over 2},I_6 = {1 \over 8}\left( {3a_1^2 + 2a_1 a_2 + a_2^2 } \right) - {{a_1 + a_2 } \over 2} \cr & I_7 = {1 \over 8}\left( {3a_1^2 + 2a_1 a_2 + a_2^2 } \right) \cr} $$

((A.1))

where a ₁ and a ₂ are calculated from Eq. (12). Under Eq.(A.1), the parameters A ₁−A ₂₀ in Eq. (15) are given by

$$A_1 = E^2 [v_1 ^2 \cos ^2 \theta I_7 + (v_1 \sin \theta - v_{gr} )^2 I_1 + (I_5 - I_4 )(v_1^2 \sin 2\theta - 2v_t v_{gr} \cos \theta )]$$

((A.2))

$$A_2 = E[(v_t^2 \cos 2\theta + v_t v_{gr} \sin \theta )I_5 + ({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}v_t^2 + ({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}v_t^2 \sin 2\theta - v_t v_{gr} \cos \theta - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}I_7 v_t^2 \sin 2\theta ]$$

((A.3))

$$A_3 = - v_t^2 I_1 \cos ^2 \theta + v_t^2 I_5 \sin 2\theta + (2v_t v_{gr} \cos \theta - v_t^2 \sin 2\theta )I_4 - v_t^2 I_7 \sin ^2 \theta + v_{gr} I_2 + 2v_t v_{gr} I_{_6 } \sin \theta $$

((A.4))

$$A_4 = v_t^2 I_1 \cos ^2 \theta - v_t^2 I_5 \sin 2\theta + (v_t^2 \sin 2\theta - 2v_t v_{gr} \cos \theta )I_4 + v_t^2 I_7 \sin ^2 \theta - 2v_t v_{gr} I_6 \sin \theta $$

((A.5))

$$A_5 = v_t I_5 \cos \theta + (v_{gr} - v_t \sin \theta )I_3 $$

((A.6))

$$ A_6 = E\left[ {v_t \cos \theta {{\partial g} \over {\partial \eta }}|_{\eta = 0} + \left( {v_{gr} - v_t \sin \theta } \right){{\partial f} \over {\partial \eta }}|_{\eta = 0} } \right] $$

((A.7))

$$A_7 = E[(v_{gr} - v_t \sin \theta )I_5 - v_t I_3 \cos \theta ],\,\,A_8 = - v_t \cos \theta I_5 + v_t I_3 \sin \theta $$

((A.8))

$$ A_9 = - v_t \cos \theta {{\partial f} \over {\partial \eta }}|_{\eta = 0} + \left( {v_{gr} - v_t \sin \theta } \right){{\partial g} \over {\partial \eta }}|_{\eta = 0} $$

((A.9))

$$A_{10} = 2E[v_t^2 I_7 \cos ^2 \theta + I_1 (v_{gr} - v_t \sin \theta )^2 + (v_t^2 \sin 2\theta - 2v_1 v_{gr} \cos \theta )(I_5 - I_4 )]$$

((A.10))

$$ A_{11} = E^2 \left[ {v_t^2 \cos ^2 \theta {{\partial I_7 } \over {\partial r}} - 2I_1 \left( {v_t \sin \theta - v_{gr} } \right){{\partial v_{gr} } \over {\partial r}} + \left( {v_{gr} - v_t \sin \theta } \right)^2 {{\partial I_1 } \over {\partial r}} - 2v_t \left( {I_5 - I_4 } \right){{\partial v_{gr} } \over {\partial r}}\cos \theta + \left( {v_t^2 \sin 2\theta - 2v_t v_{gr} \cos \theta } \right){{\partial \left( {I_5 - I_4 } \right)} \over {\partial r}}} \right] $$

((A.11))

$$A_{12} = (v_t^2 \cos 2\theta + v_t v_{gr} \sin \theta )I_5 + ({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}v_t^2 \sin 2\theta - v_t v_{gr} \cos \theta )I_1 + (v_{gr} ^2 - 2v_t v_{gr} \sin \theta - v_t^2 \cos 2\theta )I_4 + v_t v_{gr} I_6 \cos \theta - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}v_t^2 I_7 \sin 2\theta $$

((A.12))

$$ A_{13} = E\left[ {\left( { - 2v_t^2 \sin 2\theta + v_t v_{gr} \cos \theta } \right)I_5 + \left( {v_t^2 \cos 2\theta + v_t v_{gr} \sin \theta } \right)\left( {{{\partial I_5 } \over {\partial \theta }} + I_1 } \right) + \left( {{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}v_t^2 \sin 2\theta - v_t v_{gr} \cos \theta } \right){{\partial I_1 } \over {\partial \theta }} + \left( {2v_t^2 \sin 2\theta - 2v_t v_{gr} \cos \theta } \right)I_4 + \left( {v_{gr}^2 - 2v_t v_{gr} \sin \theta - v_t^2 \cos 2\theta } \right){{\partial I_4 } \over {\partial \theta }} - v_t v_{gr} I_6 \sin \theta + v_t v_{gr} \cos \theta {{\partial I_6 } \over {\partial \theta }} - v_t^2 I_7 \cos 2\theta - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}v_t^2 \sin 2\theta {{\partial I_7 } \over {\partial \theta }}} \right] $$

((A.13))

$$ A_{14} = E\left[ {v_t \sin \theta I_5 {{\partial v_{gr} } \over {\partial r}} + \left( {v_t^2 \cos 2\theta + v_t v_{gr} \sin \theta } \right){{\partial I_5 } \over {\partial r}} - v_t {{\partial v_{gr} } \over {\partial r}}I_1 \cos \theta + \left( {{1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}v_t^2 \sin 2\theta - v_t v_{gr} \cos \theta } \right){{\partial I_1 } \over {\partial r}} + 2I_4 \left( {v_{gr} - v_t \sin \theta } \right){{\partial v_{gr} } \over {\partial r}} + I_6 v_t \cos \theta {{\partial v_{gr} } \over {\partial _r }} + \left( {v_{gr}^2 - 2v_t v_{gr} \sin \theta - v_t^2 \cos 2\theta } \right){{\partial I_4 } \over {\partial r}} + v_t v_{gr} \cos \theta {{\partial I_6 } \over {\partial r}} - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}v_t^2 \sin 2\theta {{\partial I_7 } \over {\partial r}}} \right] $$

((A.14))

$$ A_{15} = - v_t^2 \sin 2\theta + v_t^2 \cos ^2 \theta {{\partial I_1 } \over {\partial \theta }} - 2v_t^2 I_5 \cos 2\theta - v_t^2 \sin 2\theta {{\partial I_5 } \over {\partial \theta }} + \left( {2v_t^2 \cos 2\theta + 2v_t v_{gr} \sin \theta } \right)I_4 + \left( {v_t^2 \sin 2\theta - 2v_t v_{gr} \cos \theta } \right){{\partial I_4 } \over {\partial \theta }} + v_t^2 I_7 \sin 2\theta + v_t^2 \sin ^2 \theta {{\partial I_7 } \over {\partial \theta }} - 2v_t v_{gr} \sin \theta {{\partial I_6 } \over {\partial \theta }} $$

((A.15))

$$A_{16} = (v_{gr} - v_t \,\sin \theta )I_5 - v_t \,I_3 \cos \theta $$

((A.16))

$$ A_{17} = E\left[ {I_5 {{\partial v_{gr} } \over {\partial r}} + \left( {v_{gr} - v_t \sin \theta } \right){{\partial I_5 } \over {\partial r}} - v_t \cos \theta {{\partial I_3 } \over {\partial r}}} \right] $$

((A.17))

$$ A_{180} = v_t I_5 \sin \theta - v_t \cos \theta {{\partial I_5 } \over {\partial \theta }} + v_t I_3 \cos \theta + v_t \sin \theta {{\partial I_3 } \over {\partial \theta }} $$

((A.18))

$$ A_{19} = v_{gr}^2 I_6 ,A_{20} = v_{gr} {{\partial I_6 } \over {\partial \theta }} $$

((A.19))

In Eq. (A.2)-(A.19), the derivatives ${{\partial I_j } \over {\partial r}}$ and ${{\partial I_j } \over {\partial \theta }}$ for j=1,..,7 can be calculated analytically using the chain rule

$$ {{\partial I_j } \over {\partial s}} = {{\partial I_j } \over {\partial a_1 }}{{\partial a_1 } \over {\partial s}} + {{\partial I_j } \over {\partial a_2 }}{{\partial a_2 } \over {\partial s}},j = 1,..,7 $$

((A.20))

where s is either s or θ and ${{\partial a_1 } \over {\partial s}}$ and ${\partial a_2 } \over {\partial s}$ can be analytically derived from Eq. (12) and (13).