Multistability and rare spontaneous transitions in barotropic β-plane turbulence

We demonstrate that turbulent zonal jets, analogous to Jovian ones, which are quasi-stationary, are actually metastable. After extremely long times, they randomly switch to new configurations with a different number of jets. The genericity of this phenomenon suggests that most quasi-stationary turbulent planetary atmospheres might have many climates and attractors for fixed values of the external forcing parameters. A key message is that this situation will usually not be detected by simply running the numerical models, because of the extremely long mean transition time to change from one climate to another. In order to study such phenomena, we need to use specific tools: rare event algorithms and large deviation theory. With these tools, we make a full statistical mechanics study of a classical barotropic beta-plane quasigeostrophic model. It exhibits robust bimodality with abrupt transitions. We show that new jets spontaneously nucleate from westward jets. The numerically computed mean transition time is consistent with an Arrhenius law showing an exponential decrease of the probability as the Ekman dissipation decreases. This phenomenology is controlled by rare noise-driven paths called instantons. Moreover, we compute the saddles of the corresponding effective dynamics. For the dynamics of states with three alternating jets, we uncover an unexpectedly rich dynamics governed by the symmetric group of permutations, with two distinct families of instantons, which is a surprise for a system where everything seemed stationary in the hundreds of previous simulations of this model. We discuss the future generalization of our approach to more realistic models.

Piecewise potential vorticity inversion without far-field response?

Given a flow domain D with subdomains D1 and D2, piecewise potential vorticity inversion (PPVI) inverts a potential vorticity (PV) anomaly in D2 and assumes vanishing PV in D1 where boundary conditions must be taken into account. It is a widely held view that the PV anomaly exerts a far-field influence on D1 which is revealed by PPVI. Tests of this assertion are conducted using a simple quasigeostrophic model where an upper layer D2 contains a PV anomaly and D1 is the layer underneath. This anomaly is inverted. Any downward physical impact of PV in D2 must also be represented in the results of a downward piecewise density inversion (PDI) based on the hydrostatic relation and the density in D2 as following from PPVI. There is no doubt about the impact of the mass in D2 on the flow in the lower layer D1. Thus results of PPVI and PDI have to agree closely. First, PPVI is applied to a locally confined PV-anomaly in D2. There is no far-field ’response’ in D1 if stationarity is imposed. Modifications of boundary conditions lead to “induced” flows in D1 but the results of PPVI and PDI differ widely. This leads to a simple proof that there is no physical far-field influence of PV-anomalies in D2. Wave patterns of the streamfunction restricted to D2 are prescribed in a second series of tests. The related PV-anomalies are obtained by differentiation and are also confined to D2 in this case. This approach illustrates the basic procedure to derive PV-fields from observations which excludes a far-field response.

Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 2: Finite-Core-Vortex Approach and Oceanographic Application

The three-layer version of the contour dynamics/surgery method is used to study the interaction mechanisms of a large-scale surface vortex with a smaller vortex/vortices of the middle layer (prototypes of intrathermocline vortices in the ocean) belonging to the middle layer of a three-layer rotating fluid. The lower layer is assumed to be dynamically passive. The piecewise constant vertical density distribution approximates the average long-term profile for the North Atlantic, where intrathermocline eddies are observed most often at depths of 300–1600 m. Numerical experiments were carried out with different initial configurations of vortices, to evaluate several effects. Firstly, the stability of the vortex compound was evaluated. Most often, it remains compact, but when unstable, it can break as vertically coupled vortex dipoles (called hetons). Secondly, we studied the interaction between a vertically tilted cyclone and lenses. Then, the lenses first undergo anticlockwise rotation determined by the surface cyclone. The lenses can induce alignment or coupling with cyclonic vorticity above them. Only very weak lenses are destroyed by the shear stress exerted by the surface cyclone. Thirdly, under the influence of lens dipoles, the surface cyclone can be torn apart. In particular, the shedding of rapidly moving vortex pairs at the surface reflects the presence of lens dipoles below. More slowly moving small eddies can also be torn away from the main surface cyclone. In this case, they do not appear to be coupled with middle layer vortices. They are the result of large shear-induced deformation. Common and differing features of the vortex interaction, modeled in the framework of the theory of point and finite-core vortices, are noted.

Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach

The theory of point vortices is used to explain the interaction of a surface vortex with subsurface vortices in the framework of a three-layer quasigeostrophic model. Theory and numerical experiments are used to calculate the interaction between one surface and one subsurface vortex. Then, the configuration with one surface vortex and two subsurface vortices of equal and opposite vorticities (a subsurface vortex dipole) is considered. Numerical experiments show that the self-propelling dipole can either be captured by the surface vortex, move in its vicinity, or finally be completely ejected on an unbounded trajectory. Asymmetric dipoles make loop-like motions and remain in the vicinity of the surface vortex. This model can help interpret the motions of Lagrangian floats at various depths in the ocean.

The Effect of the Madden–Julian Oscillation on the North Atlantic Oscillation Using Idealized Numerical Experiments

The aim of the paper is to investigate the influence of the Madden–Julian oscillation (MJO) on the North Atlantic Oscillation (NAO) using a quasigeostrophic model on the sphere. A simplified forcing based on potential vorticity anomalies in the tropics is used to mimic the MJO. The idealized nature of our setup allows us to determine the distinct roles played by stationary and synoptic waves. This is done by means of several series of almost 10 000 short runs of 30 days. Ensemble averages and a streamfunction budget analysis are used to study the modifications of the flow induced by the MJO. We find that a stationary Rossby wave is excited in the tropics during MJO phase 3. The western part of the Pacific jet is displaced poleward, which modifies the transient eddy activity in that basin. These changes create a ridge south of Alaska, which favors equatorward propagation of synoptic waves and larger poleward eddy momentum fluxes from the eastern Pacific toward the Atlantic, increasing the frequency of occurrence of the positive NAO events. The situation is essentially reversed following phase 6 of the MJO and conducive to the negative phase of the NAO. For a realistic MJO forcing amplitude, we find increases in both NAO phases to be around 30%, in reasonable agreement with the observations given the model simplicity. Finally, we present a series of experiments to assess the relative importance of linear versus nonlinear effects.

Single-Precision in the Tangent-Linear and Adjoint Models of Incremental 4D-Var

The use of single-precision arithmetic in ECMWF’s forecasting model gave a 40% reduction in wall-clock time over double-precision, with no decrease in forecast quality. However, using reduced-precision in 4D-Var data assimilation is relatively unexplored and there are potential issues with using single-precision in the tangent-linear and adjoint models. Here, we present the results of reducing numerical precision in an incremental 4D-Var data assimilation scheme, with an underlying two-layer quasigeostrophic model. The minimizer used is the conjugate gradient method. We show how reducing precision increases the asymmetry between the tangent-linear and adjoint models. For ill-conditioned problems, this leads to a loss of orthogonality among the residuals of the conjugate gradient algorithm, which slows the convergence of the minimization procedure. However, we also show that a standard technique, reorthogonalization, eliminates these issues and therefore could allow the use of single-precision arithmetic. This work is carried out within ECMWF’s data assimilation framework, the Object Oriented Prediction System.

A Nonmodal Instability Perspective of the Declining Northern Midlatitude Synoptic Variability in Boreal Summer

The Pacific–North America–North Atlantic sector in general experienced a dryer and warmer climate in summer during the past 40 years. These changes are partly associated with declining midlatitude synoptic variability in boreal summer, especially over the two ocean basins. A nonmodal instability analysis of the boreal summer background flow is conducted for two periods, 1979–94 and 2000–15, to understand dynamical processes potentially responsible for the observed decline of synoptic variability. The synoptic variability associated with fast, nonmodal growth of atmospheric disturbances shows a decline over northern midlatitudes in the later period, in both a barotropic model and a two-level quasigeostrophic model. These results highlight the importance of the changing summer background flow in contributing to the observed changes in synoptic variability. Also discussed are factors likely associated with background flow changes including sea surface temperature and sea ice change.

Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear

The transport by materially coherent surface-layer eddies was studied in a two-layer quasigeostrophic model driven by eastward mean shear. The coherent eddies were identified by closed contours of the Lagrangian-averaged vorticity deviation obtained from Lagrangian particles advected by the flow. Attention was restricted to eastward mean flows, but a wide range of flow regimes with different bottom friction strengths, layer thickness ratios, and background potential vorticity (PV) gradients were otherwise considered. It was found that coherent eddies become more prevalent and longer-lasting as the strength of bottom drag increases and the stratification becomes more surface-intensified. The number of coherent eddies is minimal when the shear-induced PV gradient is 10–20 times the planetary PV gradient and increases for both larger and smaller values of the planetary PV gradient. These coherent eddies, with an average core radius close to the deformation radius, propagate meridionally with a preference for cyclones to propagate poleward and anticyclones to propagate equatorward. The meridional propagation preference of the coherent eddies gives rise to a systematic upgradient PV transport, which is in the opposite direction as the background PV transport and not captured by standard Lagrangian diffusivity estimates. The upgradient PV transport by coherent eddy cores is less than 15% of the total PV transport, but the PV transport by the periphery flow induced by the PV inside coherent eddies is significant and downgradient. These results clarify the distinct roles of the trapping and stirring effect of coherent eddies in PV transport in geophysical turbulence.

A Multiscale Alignment Method for Ensemble Filtering with Displacement Errors

High-resolution models nowadays simulate phenomena across many scales and pose challenges to the design of efficient data assimilation methods that reduce errors at all scales. Smaller-scale features experience rapid nonlinear error growth that gives rise to displacement errors, which cause suboptimal ensemble filter performance. Previous studies have started exploring methods that can reduce displacement errors. In this study, a multiscale alignment (MSA) method is proposed for ensemble filtering. The MSA method iteratively processes the model state from the largest to the smallest scales. At each scale, an ensemble filter is applied to update the state with observations, and the analysis increments are utilized to derive displacement vectors for each member that align the ensemble at smaller scales before the next iteration. The nonlinearity in smaller-scale priors is reduced by removing larger-scale displacement errors. Because the displacement vectors are derived from analysis increments in the state space rather than the nonlinear observation-space cost function formulated in previous studies, this method provides a less costly and more robust way to solve for the displacement vectors. Observing system simulation experiments using a two-layer quasigeostrophic model were conducted to provide a proof of concept of the MSA method. Results show that the MSA method significantly improves the accuracy of posteriors compared to the existing ensemble filter methods with or without multiscale localization. Advantage of the MSA method are more evident when the ensemble size is relatively small and the cycling period is comparable to the average eddy turnover time of the dynamical system.

The Ocean Mesoscale Regime of the Reduced-Gravity Quasigeostrophic Model

A statistical-equilibrium, geostrophic-turbulence regime of the stochastically forced, one-layer, reduced-gravity, quasigeostrophic model is identified in which the numerical solutions are representative of global mean, midlatitude, open-ocean mesoscale variability. Solutions are forced near the internal deformation wavenumber and damped linearly and by high-wavenumber enstrophy dissipation. The results partially rationalize a recent semiempirical stochastic field model of mesoscale variability motivated by a global eddy identification and tracking analysis of two decades of satellite altimeter sea surface height (SSH) observations. Comparisons of model results with observed SSH variance, autocorrelation, eddy, and spectral statistics place constraints on the model parameters. A nominal best fit is obtained for a dimensional SSH stochastic-forcing variance production rate of 1/4 cm2 day−1, an SSH damping rate of 1/62 week−1, and a stochastic forcing autocorrelation time scale near or greater than 1 week. This ocean mesoscale regime is nonlinear and appears to fall near the stochastic limit, at which wave-mean interaction is just strong enough to begin to reduce the local mesoscale variance production, but is still weak relative to the overall nonlinearity. Comparison of linearly inverted wavenumber–frequency spectra shows that a strong effect of nonlinearity, the removal of energy from the resonant linear wave field, is resolved by the gridded altimeter SSH data. These inversions further suggest a possible signature in the merged altimeter SSH dataset of signal propagation characteristics from the objective analysis procedure.