Finite-amplitude Rossby Waves Research Articles

Why Are Stratospheric Sudden Warmings Sudden (and Intermittent)?

Abstract This paper examines the role of wave–mean flow interaction in the onset and suddenness of stratospheric sudden warmings (SSWs). Evidence is presented that SSWs are, on average, a threshold behavior of finite-amplitude Rossby waves arising from the competition between an increasing wave activity A and a decreasing zonal-mean zonal wind u¯. The competition puts a limit to the wave activity flux that a stationary Rossby wave can transmit upward. A rapid, spontaneous vortex breakdown occurs once the upwelling wave activity flux reaches the limit, or equivalently, once u¯ drops below a certain fraction of uREF, a wave-free, reference-state wind inverted from the zonalized quasigeostrophic potential vorticity. This fraction is 0.5 in theory and about 0.3 in reanalyses. We propose r≡u¯/uREF as a local, instantaneous measure of the proximity to vortex breakdown (i.e., preconditioning). The ratio r generally stays above the threshold during strong-vortex winters until a pronounced final warming, whereas during weak-vortex winters it approaches the threshold early in the season, culminating in a precipitous drop in midwinter as SSWs form. The essence of the threshold behavior is captured by a semiempirical 1D model of SSWs, similar to the “traffic jam” model of Nakamura and Huang for atmospheric blocking. This model predicts salient features of SSWs including rapid vortex breakdown and downward migration of the wave activity/zonal wind anomalies, with analytical expressions for the respective time scales. The model’s response to a variety of transient wave forcing and damping is discussed.

Role of Finite-Amplitude Rossby Waves and Nonconservative Processes in Downward Migration of Extratropical Flow Anomalies

There is growing evidence that stratospheric variability exerts a noticeable imprint on tropospheric weather and climate. Despite clear evidence of these impacts, the principal mechanism whereby stratospheric variability influences tropospheric circulation has remained elusive. Here, the authors introduce a novel approach, based on the theory of finite-amplitude wave activity, for quantifying the role of adiabatic and nonconservative effects on the mean flow that shape the downward coupling from the stratosphere to the troposphere during stratospheric vortex weakening (SVW) events. The advantage of using this theory is that eddy effects (at finite amplitude) on the mean flow can be more readily distinguished from nonconservative effects. The results show (in confirmation of previous work) that the downward migration of extratropical wind anomalies is largely attributable to dynamical adjustments induced by fluctuating finite-amplitude wave forcing. The nonconservative effects, on the other hand, contribute to maintaining the downward signals in the recovery stage within the stratosphere, hinting at the importance of mixing and diabatic heating. The analysis further indicates that variations in stratospheric finite-amplitude wave forcing are too weak to account for the attendant changes and shapes in the tropospheric flow. It is suggested that the indirect effect of tropospheric finite-amplitude wave activity through the residual displacements is needed to amplify and prolong the tropospheric wind responses over several weeks. The results also reveal that the local tropospheric wave activity over the North Pacific and North Atlantic sectors plays a significant role in shaping the high-latitude tropospheric wind response to SVW events.

Dispersive effects in Rossby-wave hydraulics

This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson & Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these ‘supercritical leaps’ can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.

Finite-amplitude topographic Rossby waves in a channel

This paper describes Rossby waves propagating along a channel with a discontinuity in potential vorticity. A finite-amplitude, long-wave equation giving the displacement of the material interface separating two constant vorticity regions is derived and solved for both steady and unsteady finite-amplitude waves. Finite-amplitude Rossby waves that propagate in the opposite direction to infinitesimal waves are identified, and the mechanism for these waves is elucidated. The full parameter space of possible flows is described quantitatively. The weakly nonlinear limit of this finite-amplitude equation is also presented, including the novel case of kink solitons in an equation with only quadratic nonlinearity.

A numerical method for solving the evolution equation of solitary Rossby waves on a weak shear

In this paper an evoluion equation in integral-differential form for finite amplitude Rossby waves on a weak shear is presented and an efficient method for its numerical solution is set up. It is shown that a propagation of solitary wave is possible whenever a proper weak shear in basic flows acts with the nonlinear effects and dispersion of the media, both in the atmosphere and in the ocean. To test the numerical method for solving the evolution equation, a series of experiments are carried out. The results indicate that the solitary solutions do exist and interact with each other in quite a succinct, manner. Therefore the method is successful and efficient for solving initial value problems of the above equation. The time decoupling problem arising in the numerical scheme and the related filtering technique are discussed. A variety of interesting phenomena such as the interaction of solitary Rossby waves, damping, dispersion and the development of nonlinear wave train are numerically studied.

Finite-amplitude Rossby waves in the presence of weak shear and topography

The evolution of finite-amplitude Rossby waves in an average zonal flow with weak shear and topography has been developed. The evolution equation developed is a general one and can be treated as an extension of Benney's equation to the time-varying case with topography. As an application to geophysical flow, the evolution equation has been applied to the problem of the formation of a monsoon onset vortex over the Indian region.

Nonlinear waves in barotropic model

In this paper, from the system of equation describing a barotropic atmosphere using the method of Taylor expansion for the nonlinear terms, the periodic solutions of the nonlinear inertio-surface gravity waves and Rossby waves have been obtained. The finite-amplitude nonlinear inertio-surface gravity waves and Rossby waves with horizontal divergence satisfy all the KdV equation. The solutions are all the cnoidal function, i. e., the cnoidal waves which include the linear waves and form the solitary waves under certain conditions. For the finite-amplitude Rossby waves with horizontal divergence, we find the new dispersive relation including both the wave number and the amplitude parameter. In case of small amplitude it is reduced to the Yeh formula. It is shown that the larger the amplitude and width, the faster the finite-amplitude inertio-surface gravity waves and the slower the finite-amplitude Rossby waves with horizontal divergence propagate. The blocking or cut-off system in which the amplitude and width are large may be considered as Rossby solitary waves.

On the theory of solitary Rossby waves

The evolution of long, finite amplitude Rossby waves in a horizontally sheared zonal current is studied. The wave evolution is described by the Korteweg–de Vries equation or the modified Korteweg-de Vries equation depending on the atmospheric stratification. In either case, the cross-stream modal structure of these waves is given by the long-wave limit of the neutral eigensolutions of the barotropic stability equation. Both non-singular and singular eigensolutions are considered and the appropriate analysis is developed to yield a uniformly valid description of the motion in the critical-layer region where the wave speed matches the flow velocity. The analysis demonstrates that coherent, propagating, eddy structures can exist in stable shear flows and that these eddies have peculiar interaction properties quite distinct from the traditional views of turbulent motion.