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Register a new userStability of Rossby waves in the beta-plane approximation
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Floquet theory is used to describe the unstable spectrum at large scales of the beta-plane equation linearized about Rossby waves. Base flows consisting of one to three Rossby wave are considered analytically using continued fractions and the method of multiple scales, while base flow with more than three Rossby waves are studied numerically. It is demonstrated that the mechanism for instability changes from inflectional to triad resonance at an O(1) transition Rhines number Rh, independent of the Reynolds number. For a single Rossby wave base flow, the critical Reynolds number Re^c for instability is found in various limits. In the limits Rh --> infinity and k --> 0, the classical value Re^c = sqrt(2) is recovered. For Rh --> 0 and all orientations of the Rossby wave except zonal and meridional, the base flow is unstable for all Reynolds numbers; a zonal Rossby wave is stable, while a meridional Rossby wave has critical Reynolds number Re^c = sqrt(2). For more isotropic base flows consisting of many Rossby waves (up to forty), the most unstable mode is purely zonal for 2 <= Rh < infinity and is nearly zonal for Rh = 1/2, where the transition Rhines number is again O(1), independent of the Reynolds number and consistent with a change in the mechanism for instability from inflectional to triad resonance.
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The connectivity pattern of networks, which are based on a correlation between ground level temperature time series, shows a dominant dense stripe of links in the southern ocean. We show that statistical categorization of these links yields a clear association with the pattern of an atmospheric Rossby wave, one of the major mechanisms associated with the weather system and with planetary scale energy transport. It is shown that alternating densities of negative and positive links (correlations) are arranged in half Rossby wave distances around 3,500 km, 7,000 km and 10,000 km and are aligned with the expected direction of energy flow, distribution of time delays and the seasonality of these waves. It is also shown that long distance links (i.e., of distances larger than 2,000 km) that are associated with Rossby waves are the most dominant in the climate network. Climate networks may thus be used as an efficient new way to detect and analyze Rossby waves, based on reliable and available ground level measurements, in addition to the frequently used 300 hPa reanalysis meridional wind data.
Using a one-layer QG model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow $U(t)$ and smaller-scale macroturbulence (both standing and transient eddies). The flow $U(t)$ is retarded by Ekman drag and by the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated solutions delineates the main flow regimes and the dependence of the time-mean $U$ on the problem parameters and the statistical properties of the topography. If $beta$ is smaller than the topographic PV gradient then the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. If $beta$ is comparable to, or larger than, the topographic PV gradient then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this case there is an eddy saturation regime in which $U$ is insensitive to changes in the wind stress. We show that eddy saturation requires strong transient eddies that act as PV diffusion. This PV diffusion does not alter the energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between topographic slope and the standing-eddy pressure field. Last, using bounds based on the energy and enstrophy we show that as the wind stress increases the flow transitions from a regime in which form stress balances the wind stress to a regime in which the form stress is very small and large transport ensues.
Retrograde-propagating waves of vertical vorticity with longitudinal wavenumbers between 3 and 15 have been observed on the Sun with a dispersion relation close to that of classical sectoral Rossby waves. The observed vorticity eigenfunctions are symmetric in latitude, peak at the equator, switch sign near $20^circ$-$30^circ$, and decrease at higher latitudes. We search for an explanation that takes into account solar latitudinal differential rotation. In the equatorial $beta$ plane, we study the propagation of linear Rossby waves (phase speed $c <0$) in a parabolic zonal shear flow, $U = - overline{U} xi^2<0$, where $overline{U} = 244$ m/s and $xi$ is the sine of latitude. In the inviscid case, the eigenvalue spectrum is real and continuous and the velocity stream functions are singular at the critical latitudes where $U = c$. We add eddy viscosity in the problem to account for wave attenuation. In the viscous case, the stream functions are solution of a fourth-order modified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. For reasonable values of the eddy viscosity corresponding to supergranular scales and above (Reynolds number $100 le Re le 700$), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode with a real frequency close to that of the classical Rossby mode, which we call the R mode. For $Re approx 300$, the attenuation and the real part of the eigenfunction is in qualitative agreement with the observations (unlike the imaginary part of the eigenfunction, which has a larger amplitude in the model. Conclusion: Each longitudinal wavenumber is associated with a latitudinally symmetric R mode trapped at low latitudes by solar differential rotation. In the viscous model, R modes transport significant angular momentum from the dissipation layers towards the equator.
Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (`zonostrophic instability) which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg-Landau (G-L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G-L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide ranges of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G-L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G-L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G-L dynamics that is able to capture the asymmetric evolution for weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability.
Jets coexist with planetary scale waves in the turbulence of planetary atmospheres. The coherent component of these structures arises from cooperative interaction between the coherent structures and the incoherent small-scale turbulence in which they are embedded. It follows that theoretical understanding of the dynamics of jets and planetary scale waves requires adopting the perspective of statistical state dynamics (SSD) which comprises the dynamics of the interaction between coherent and incoherent components in the turbulent state. In this work the S3T implementation of SSD for barotropic beta-plane turbulence is used to develop a theory for the jet/wave coexistence regime by separating the coherent motions consisting of the zonal jets together with a selection of large-scale waves from the smaller scale motions which constitute the incoherent component. It is found that mean flow/turbulence interaction gives rise to jets that coexist with large-scale coherent waves in a synergistic manner. Large-scale waves that would only exist as damped modes in the laminar jet are found to be transformed into exponentially growing waves by interaction with the incoherent small scale turbulence which results in a change in the mode structure allowing the mode to tap the energy of the mean jet. This mechanism of destabilization differs fundamentally and serves to augment the more familiar S3T instabilities in which jets and waves arise from homogeneous turbulence with energy source exclusively from the incoherent eddy field and provides further insight into the cooperative dynamics of the jet/waves coexistence regime in planetary turbulence.
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