We now look at the significance of H from another point of view, using a simple cyclostrophic model of a tropical cyclone applied in a region where H < 0. For H locally constant, (6) can be integrated to yield, where u = u0 at T = T0. Hence if H < 0, u increases as T decreases. This is precisely the situation in a hurricane, which is characterised by a warm core. Conversely in the tropics in regions in which H > 0, there is a tendency for hurricane development to be suppressed.

In order to gain a greater insight into the significance of H, we may incorporate (10) into a standard model of hurricane dynamics. A suitable expression is the cyclostrophic 'thermal' wind relation (18.17) in Gordon et al. (1998), dKE/dt =KE/t

where r is the radial co-ordinate, and oz is vertically upward, which may be integrated with respect to height assuming the radial temperature gradient is a constant, to yield, u2 =-gh/T @T/@lnr (11b)

in which h is the height at which the (azimuthal) wind speed becomes zero. On substituting for u in (11b) from (10) and integrating with respect to lnr, assuming that (gh/T) is a constant, we obtain, lnr/r0 = B(1 - exp(-2H(T - T0))) (12)

in which T = T0 and u = u0 at r = r0, and B = -(gh /T)/(2H u02).

Equations (10) and (12) are a model for a developing hurricane in which r0, u0 and T0 are the conditions at the outer radius of the hurricane. On evaluating (10) and (12) at the outer radius of the warm core (r1), assuming that (gh/T) is constant, we can determine u0 as a function of H for a series of values of warm core temperature increment, T = T1 - T0 , where T1 = T(r1) is the temperature of the warm core. For the hurricane parameters h = 10 km, T = 300 K and g = 9.8 ms-2, and a warm core of radius 1/10 that of the radius of the hurricane (r1/r0 = 0.1), we obtain the results shown in Table 1.

It is apparent that at small values of H, intense tropical storms (u0) cannot be supported for any realistic warm core temperature increment (AT). As the magnitude of H increases, however, intense systems can occur for realistic warm core increments. Table 1 indicates that H 1 K-1 marks an approximate upper limit for their development. The existence of regions of H of large magnitude, therefore, is a necessary condition for hurricane development, however it is not a sufficient condition since systems with small AT can also be supported, in which u0 only attains a modest value.

H (K-1) |
AT (K) |
u0 (ms-1) |

-0.3 |
0.5 |
9 |

-0.3 |
1 |
14 |

-0.3 |
2 |
23 |

-0.3 |
3 |
35 |

-0.3 |
4 |
49 |

-1 |
0.5 |
11 |

-1 |
1 |
21 |

-1 |
2 |
62 |

-3 |
0.5 |
21 |

3 |
1 |
98 |

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