Dynamics of Vortex Rossby Waves in Tropical Cyclones, Part 1: Linear Time‐Dependent Evolution on an <i>f</i>‐Plane

Nikitina, Lidia; Campbell, Lucy

doi:10.1111/sapm.12094

Cited by 9 publications

(32 citation statements)

References 38 publications

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“…In [, ], we presented a model for barotropic vortex Rossby waves in a cyclonic vortex on a β‐plane, a horizontal plane on which the Coriolis parameter is approximated by a linear function of latitude. In this two‐dimensional configuration, the flow can be described in terms of a streamfunction Ψ and represented by the nondimensional barotropic vorticity equation, which is given in polar coordinates r and λ by

\nabla^{2} Ψ_{t} - \frac{1}{r} Ψ_{λ} \nabla^{2} Ψ_{r} + \frac{1}{r} Ψ_{r} \nabla^{2} Ψ_{λ} - \frac{β}{r} Ψ_{λ} sin λ + β Ψ_{r} cos λ = 0,

where the subscripts denote partial differentiation with respect r and λ.…”

Section: The Nonlinear Equation For Vortex Rossby Waves On a β‐Planementioning

confidence: 99%

“…In , we made the f ‐plane approximation and linearized this equation by ignoring all the terms involving β and ε and then derived analytical solutions for the simplified equation in an annular region given by

r_{1} \leq r < \infty, 0 \leq λ < 2 π

. We considered the case where the vortex waves are forced at

r = r_{1}

by a sinusoidal boundary condition of the form

ψ (r_{1}, λ, t) = e^{i (k λ - ω t)} + e^{- i (k λ - ω t)},

where the azimuthal wavenumber k is a positive integer and ω is the circular frequency of the wave.…”

Section: The Nonlinear Equation For Vortex Rossby Waves On a β‐Planementioning

confidence: 99%

“…If

β ≪ ɛ ≪ 1

and

β ≲ ɛ^{2}

in , then either case (i) or (ii) holds and the solution can be expressed as a perturbation series of the form

ψ (r, λ, t) \sim ψ^{(0)} (r, λ, t) + ɛ ψ^{(1)} (r, λ, t) + O (ɛ^{2}) + O (β ɛ^{- 1}) + O (β),

where

ψ^{(1)} (r, λ, t) \sim O (1)

. The leading‐order term is the solution obtained in for the linear problem on an f ‐plane. It satisfies the boundary condition

ψ^{(0)} (r_{1}, λ, t) = e^{i k λ} + c.c.

and can thus be written in the form

ψ^{(0)} (r, λ, t) = φ^{(0)} (r, t) e^{i k λ} + c.c.,

where

φ^{(0)} (r, t)

is given by the expressions (84), (85), and (86) in and c.c.…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

“…The leading‐order term is the solution obtained in for the linear problem on an f ‐plane. It satisfies the boundary condition

ψ^{(0)} (r_{1}, λ, t) = e^{i k λ} + c.c.

and can thus be written in the form

ψ^{(0)} (r, λ, t) = φ^{(0)} (r, t) e^{i k λ} + c.c.,

where

φ^{(0)} (r, t)

is given by the expressions (84), (85), and (86) in and c.c. denotes its complex conjugate.…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

“…The second term on the right‐hand side of and its complex conjugate are the leading‐order contributions to the second harmonic. The function

φ^{(1)} (r, t)

satisfies

\begin{matrix} true \\ right (() \frac{\partial}{\partial t} + 2 i k \bar{Ω}) (() φ_{r r}^{(1)} + \frac{1}{r} φ_{r}^{(1)} - \frac{4 k^{2}}{r^{2}} φ^{(1)}) + 16 i k Ω_{0} α φ^{(1)} \\ right = - \frac{i k}{r^{2}} ({(φ_{r}^{(0)})}^{2} - φ^{(0)} φ_{r r}^{(0)} + r φ_{r}^{(0)} φ_{r r}^{(0)} - r φ^{(0)} φ_{r r r}^{(0)}) . \end{matrix}

In the outer region where

| r - r_{c} | t ≫ 1

, for

t ≫ 1

, the linear solution

φ^{(0)} (r, t)

, according to equation (84) in , is

\begin{matrix} true \\ right φ^{(0)} (r, t) & \sim & left φ_{\infty} (r) e^{- i ω t} \\ left + e^{- i k \overset{Ω}{true} (r) t} {(\frac{r}{r_{1}})}^{- k} ([, \frac{2 h_{1} (r)}{(i k Ω_{0}}) \end{matrix}

…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

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Dynamics of Vortex Rossby Waves in Tropical Cyclones, Part 2: Nonlinear Time‐Dependent Asymptotic Analysis on a β‐Plane

Nikitina

Campbell

2015

Stud Appl Math

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Vortex Rossby waves in cyclones in the tropical atmosphere are believed to play a role in the observed eyewall replacement cycle, a phenomenon in which concentric rings of intense rainbands develop outside the wall of the cyclone eye, strengthen and then contract inward to replace the original eyewall. In this paper, we present a two‐dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location. The equations studied include the nonlinear terms that describe wave‐mean‐flow interactions, as well as the terms arising from the latitudinal gradient of the Coriolis parameter. Asymptotic methods based on perturbation theory and weakly nonlinear analysis are used to obtain the solution as an expansion in powers of two small parameters that represent nonlinearity and the Coriolis effects. The asymptotic solutions obtained give us insight into the temporal evolution of the forced waves and their effects on the mean vortex. In particular, there is an inward displacement of the location of the critical radius with time which can be interpreted as part of the secondary eyewall cycle.

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