A semi-implicit, two time-level, three-step iterative time-difference scheme is proposed for the two-dimensional nonlinear shallow-water equations in a conservative flux form. After a semi-implicit linearization of the governing equations, the linear gravity wave terms are time discretized implicitly using a second-order trapezoidal scheme applied over each iterative step, whereas the nonlinear terms including horizontal advection and other terms left over from the semi-implicit linearization are time discretized explicitly using a third-order Runge–Kutta scheme. The effectiveness of the scheme in terms of numerical accuracy, stability, and efficiency is established through a forced initial-boundary value problem studied using a two-dimensional shallow-water model.

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SUMMARYEvidence is presented of upper-tropospheric ow transitions during rapid tropical cyclone (TC) intensication. Transitions occur when a mid-latitude upper trough, with a wind maximum on its eastern ank, located well to the west of a storm, relaxes as anticyclogenesis occurs near to, and east of, its equatorward extremity. During these episodes, the ow is characterized by weak inertial stability. It is proposed that this allows extremely rapid and large-scale changes to occur in the upper-level environment of the storm. The new environment provides seemingly favourable conditions of reduced wind shear, development of a downstream trough very near the storm, and access for the storm out ow to the tropical easterlies and mid-latitude westerlies.A global shallow-water model, initialized with objective analyses at the 200 hPa level, is used to study the phenomenon, and the interaction between the environmental ow and local sources of mass and momentum. The sources are used to represent the effects of inner-core deep convection in the out ow layer. Using the technique it seems possible, as a rst approximation, to isolate the environment from the vortex development. It is shown that ow transitions are mostly independent of the presence of the TC. Short term, local enhancement of divergence over the TC occurs during superposition of environmentally-induced divergence with the mass source. However, formation of the upper-level vortex is the distinguishing feature of the intensi cation, and this occurs during an upper-tropospheric ow transition. It is shown: (i) that the reduction in ventilation is associated with ow transitions, (ii) that these transitions directly in uence the development of the upper vortex of the TC via downstream development of a weak environmental trough, and (iii) that the transitions indirectly in uence the vortex development by allowing the momentum and mass sources to operate more ef ciently to assist in both the development of the vortex and the out ow channels at small radii.Several examples of ow transitions during intensi cation are presented to support the proposed hypothesis.

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