We analytically solve a forced linear problem of vortex Rossby waves (VRWs) associated with the vortex resiliency of tropical cyclones. We consider VRWs on a basic barotropic axisymmetric vortex. VRWs, which are initially absent, are successively forced by a vertically sheared unidirectional environmental flow. The problem is formulated in the quasigeostrophic equations, linearized about the basic vortex. The basic potential vorticity (PV) is assumed to be piecewise constant in the radial

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... ion so that the problem can be analytically solved. The obtained solutions show the following. When the vertical interaction (VI) between the VRWs is weak, a stationary mode (called the pseudo mode) is selectively forced and grows linearly in time, and the vortex is eventually destroyed by the environmental vertical shear. When the VI is moderate, an almost form-preserving quasi-mode (simply called the quasi mode) of the VRWs appears and precesses about a downshear-left tilt equilibrium (DSLTE). The precession does not grow and the vortex maintains vertical coherence. In particular, in the presence of the inward radial gradient of the basic PV at the critical radius, the precession damps and the quasi mode eventually approaches the DSLTE. When the VI is strong, the VRWs are simply advected by the basic angular velocity at each radius to be axisymmetrized to some extent about the DSLTE, and the vortex maintains vertical coherence. To examine the diabatic effect near the eyewall, the solution with the basic buoyancy frequency being small in the central region and large in the outer region is also obtained. The small and large buoyancy frequencies imply strong and weak VIs, respectively. The central VRWs are simply advected by the basic vortex flow. While, the outer VRWs precess about the DSLTE just like a quasi mode, and the vortex maintains vertical coherence.