Axel Deloncle

Nonlinear evolution of the zigzag instability in stratified fluids : a shortcut on the route to dissipation

We present high-resolution direct numerical simulations of the nonlinear evolution of a pair of counter-rotating vertical vortices in a stratified fluid for various high Reynolds numbers Re and low Froude numbers F h . The vortices are bent by the zigzag instability producing high vertical shear. There is no nonlinear saturation so that the exponential growth is stopped only when the viscous dissipation by vertical shear is of the same order as the horizontal transport, i.e. when Z h max /Re = O(1) where Z h max is the maximum horizontal enstrophy non-dimensionalized by the vortex turnover frequency. The zigzag instability therefore directly transfers the energy from large scales to the small dissipative vertical scales. However, for high Reynolds number, the vertical shear created by the zigzag instability is so intense that the minimum local Richardson number Ri decreases below a threshold of around 1/4 and small-scale Kelvin-Helmholtz instabilities develop. We show that this can only occur when ReF 2 h is above a threshold estimated as 340. Movies are available with the online version of the paper.

Weather regime prediction using statistical learning

Abstract Two novel statistical methods are applied to the prediction of transitions between weather regimes. The methods are tested using a long, 6000-day simulation of a three-layer, quasigeostrophic (QG3) model on the sphere at T21 resolution. The two methods are the k nearest neighbor classifier and the random forest method. Both methods are widely used in statistical classification and machine learning; they are applied here to forecast the break of a regime and subsequent onset of another one. The QG3 model has been previously shown to possess realistic weather regimes in its northern hemisphere and preferred transitions between these have been determined. The two methods are applied to the three more robust transitions; they both demonstrate a skill of 35%–40% better than random and are thus encouraging for use on real data. Moreover, the random forest method allows one, while keeping the overall skill unchanged, to efficiently adjust the ratio of correctly predicted transitions to false alarms. A l...

Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.

The three-dimensional stability of vertical vortex pairs in stratified and rotating fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-rotating and counter-rotating vortex pairs are most unstable to the zigzag instability when the Froude number F h = Γ/(2π R 2 N ) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Vaisala frequency) is lower than unity independently of the Rossby number Ro = Γ/(4π R 2 Ω b ) (Ω b is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2π b 2 ) ( b is the separation distance between the vortices) and is almost independent of F h and Ro . The most amplified wavelength scales like F h b when the Rossby number is large and like F h b /| Ro | when | Ro | ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects. When F h is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Vaisala frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous fluids. However, its growth rate is lower than in homogeneous fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution. Physically, the different stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.

Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid

This paper investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude number, F h , is varied from ∞ to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude number for F h ≪ 1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally.

Towards a theory for vortex filaments in stratified-rotating fluids

In inviscid fluids with uniform density, it is common to idealize three-dimensional vortex tubes by filaments (i.e., single lines of an infinitesimal cross section). Thanks to the Kelvin and Helmholtz theorems, it is known that these vortex filaments are transported with the fluid and their circulation is conserved. The induced motions can be computed by the Biot–Savart law, with an appropriate cut off in the integral to avoid singularity. Hence, this approach allows one to model the linear or nonlinear dynamics of vortex flows. A priori, vortex filaments cannot be used in density-stratified and rotating fluids since the circulation is not conserved and the vortex lines are not material lines. However, in this paper we review a theory that is equivalent to vortex filaments. It is based on matched asymptotic expansions for small vortex-core size, weak curvature, and small vortex displacements. The resulting stability equations are formally identical to those of vortex filaments in homogeneous fluids. However, striking differences between homogeneous and stratified-rotating fluids exist, such as the reversal of the self-induced motion for strong stratification or complex self-induction for moderate stratification due to the presence of critical points. The three-dimensional linear stability of vertical vortex pairs and vortex arrays (Karman street, double symmetric row) in stratified and rotating fluids has been investigated using this analytical approach. The results are in very good agreement with the results of direct numerical stability analyses of smooth vortex configurations. Possible extensions to include nonlinear and baroclinic effects are briefly discussed.

Zigzag instability of the Kármán vortex street in stratified and rotating fluids

We investigate the three-dimensional stability of the Karman vortex street in a stratified and rotating fluid by means of an asymptotic theory for long-vertical wavelength and well-separated vortices. It is found that the Karman street with close rows is unstable to the zigzag instability when the fluid is strongly stratified independently of the background rotation. The zigzag instability bends the vortices with almost no internal deformation. The results are in excellent agreement with direct numerical stability analyses and may explain the formation of layers commonly observed in stratified flows.

Three-Dimensional Stability of Vortices in a Stratified Fluid

The three-dimensional linear stabilities of vertically uniform shear flows and vortex configurations (dipole, couple, von Karman street and double symmetric row) are investigated through experiments, theoretical and numerical analysis when the fluid is stratified. For strong stratification, all the vortex configurations are unstable to the zigzag instability associated to vertically sheared horizontal translations that develop spontaneously. The most unstable wavelength decreases with the strength of the stratification, whereas the maximum growthrate is independent of the stratification and solely proportional to the strain felt by the vortex core. Experiments and direct numerical simulation show that the zigzag instability eventually decorrelates the flow on the vertical. The zigzag instability is therefore a generic instability that constrains turbulent energy cascade in stratified fluid and contributes to structure oceanic and atmospheric flows.

Nonlinear evolution of the zigzag instability in a stratified fluid

In a strongly stratified fluid, a columnar counter-rotating vortex pair is subject to the zigzag instability which bends the vortices and ultimately produces layers. We have investigated the nonlinear evolution of this linear instability by means of DNS. We show that the instability grows exponentially without nonlinear saturation and therefore produces rapidly intense vertical shear. We show that this growth is only stopped when vertical viscous effects become important and that it occurs when F 2 h Re = O(1) where Fh is the horizontal Froude number and Re the Reynolds number. Energy is then rapidly dissipated through viscous effects. We also show that for sufficiently high initial values of F 2 h Re, the intense vertical shear created by the initial zigzag instability is not directly dissipated through viscous effects but first leads to Kelvin-Helmholtz instabilities. This makes the flow turbulent and again rapidly dissipates energy. In both cases, this means that the zigzag instability is a mechanism capable of directly transferring the energy from large scales to small vertical scales where it is dissipated. Key-words : stratified flow ; instability ; energy transfer