Eric Simonnet

Spontaneous Generation of Low-Frequency Modes of Variability in the Wind-Driven Ocean Circulation

In idealized models that aim to understand the temporal variability of the wind-driven ocean circulation, lowfrequency instabilities associated with so-called oscillatory gyre modes have been found. For the double-gyre case, the spectral origin of these modes as well as the physical mechanism of the instability is explained. In a barotropic quasigeostrophic model, the low-frequency modes arise spontaneously from the merging between two nonoscillatory eigenmodes. Of the latter two, one is called here the P-mode and is responsible for the existence of multiple steady states. The other is called the L-mode and it controls the intensity of the gyres. This merging turns out to be robust over a hierarchy of models and can even be found in a low-order truncated quasigeostrophic model. The latter model is used to determine the physical mechanism of the instability. The low-frequency oscillation results from the conjugate effects of shear- and symmetry-breaking instabilities and is free of Rossby wave dynamics.

Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation

The wind-driven double-gyre circulation in a rectangular basin goes through several dynamical regimes as the amount of lateral friction is decreased. This paper studies the transition to irregular flow in the double-gyre circulation by applying dynamical systems methodology to a quasigeostrophic, equivalent-barotropic model with a 10-km resolution. The origin of the irregularities, in space and time, is the occurrence of homoclinic bifurcations that involve phase-space behavior far from stationary solutions. The connection between these homoclinic bifurcations and earlier transitions, which occur at larger lateral friction, is explained. The earlier transitions, such as pitchfork and asymmetric Hopf bifurcation, only involve the nonlinear saturation of linear instabilities, while the homoclinic bifurcations are associated with genuinely nonlinear behavior. The sequence of bifurcations—pitchfork, Hopf, and homoclinic—is independent of the lateral friction and may be described as the unfolding of a singularity that occurs in the frictionless, Hamiltonian limit of the governing equations. Two distinct chaotic regimes are identified: Lorenz chaos at relatively large lateral friction versus Shilnikov chaos at relatively small lateral friction. Both types of homoclinic bifurcations induce chaotic behavior of the recirculation gyres that is dominated by relaxation oscillations with a well-defined period. The relevance of these results to the mid-latitude oceans’ observed low-frequency variations is discussed. A previously documented 7-year peak in observed North-Atlantic variability is shown to exist across a hierarchy of models that share the gyre modes and homoclinic bifurcations discussed herein.

Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution

Successive bifurcations—from steady states through periodic to aperiodic solutions—are studied in a shallowwater, reduced-gravity, 2‰-layer model of the midlatitude ocean circulation subject to time-independent wind stress. The bifurcation sequence is studied in detail for a rectangular basin with an idealized spatial pattern of wind stress. The aperiodic behavior is studied also in a North Atlantic‐shaped basin with realistic continental contours. The bifurcation sequence in the rectangular basin is studied in Part I, the present article. It follows essentially the one reported for single-layer quasigeostrophic and 1‰-layer shallow-water models. As the intensity of the north‐ south-symmetric, zonal wind stress is increased, the nearly symmetric double-gyre circulation is destabilized through a perturbed pitchfork bifurcation. The low-stress steady solution, with its nearly equal subtropical and subpolar gyres, is replaced by an approximately mirror-symmetric pair of stable equilibria. The two solution branches so obtained are named after the inertial recirculation cell that is stronger, subtropical or subpolar, respectively. This perturbed pitchfork bifurcation and the associated Hopf bifurcations are robust to changes in the interface friction between the two active layers and the thickness H2 of the lower active layer. They persist in the presence of asymmetries in the wind stress and of changes in the model’s spatial resolution and finitedifference scheme. Time-dependent model behavior in the rectangular basin, as well as in the more realistic, North Atlantic‐shaped one, is studied in Part II.

Low-Frequency Variability in Shallow-Water Models of the Wind-Driven Ocean Circulation. Part II: Time-Dependent Solutions*

The time-dependent wind-driven ocean circulation is investigated for both a rectangular and a North Atlantic‐ shaped basin. Multiple steady states in a 2‰-layer shallow-water model and their dependence on various parameters and other model properties were studied in Part I for the rectangular basin. As the wind stress on the rectangular basin is increased, each steady-state branch is destabilized by a Hopf bifurcation. The periodic solutions that arise off the subpolar branch have a robust subannual periodicity of 4‐5 months. For the subtropical branch, the period varies between sub- and interannual, depending on the inverse Froude number F2 defined with respect to the lower active layer’s thickness H2 .A sF2 is lowered, the perturbed-symmetric branch is destabilized baroclinically, before the perturbed pitchfork bifurcation examined in detail in Part I occurs. T ransition to aperiodic behavior arises at first by a homoclinic explosion off the isolated branch that exists only for sufficiently high wind stress. Subsequent global and local bifurcations all involve the subpolar branch, which alone exists in the limit of vanishing wind stress. Purely subpolar solutions vary on an interannual scale, whereas combined subpolar and subtropical solutions exhibit complex transitions affected by a second, subpolar homoclinic orbit. In the latter case, the timescale of the variability is interdecadal. The role of the global bifurcations in the interdecadal variability is investigated. Numerical simulations were carried out for the North Atlantic with earth topography5 minute (ETOPO-5) coastline geometry in the presence of realistic, as well as idealized, wind stress forcing. The simulations exhibit a realistic Gulf Stream at 20-km resolution and with realistic wind stress. The variability at 12-km resolution exhibits spectral peaks at 6 months, 16 months, and 6‐7 years. The subannual mode is strongest in the subtropical gyre; the interannual modes are both strongest in the subpolar gyre.

Low-Frequency Variability in the Midlatitude Atmosphere Induced by an Oceanic Thermal Front

Abstract This study examines the flow induced in a highly idealized atmospheric model by an east–west-oriented oceanic thermal front. The model has a linear marine boundary layer coupled to a quasigeostrophic, equivalent- barotropic free atmosphere. The vertical velocity at the top of the boundary layer drives the flow in the free atmosphere and produces an eastward jet, parallel to the oceanic fronts isotherms. A large gyre develops on either side of this jet, cyclonic to the north and anticyclonic to the south of it. As the jet intensifies during spinup from rest, it becomes unstable. The most unstable wave has a length of about 500 km, it evolves into a meander, and eddies detach from the eastern edge of each gyre. The dependence of the atmospheric dynamics on the strength T∗ of the oceanic front is studied. The Gulf Stream and Kuroshio fronts correspond roughly, in the scaling used here, to T∗ ≅ 7°C. For weak fronts, T∗ ≤ 4°C, the circulation is steady and exhibits two large, antisymmetric gyres sepa...

Random changes of flow topology in two-dimensional and geophysical turbulence.

We study the two-dimensional (2D) stochastic Navier-Stokes (SNS) equations in the inertial limit of weak forcing and dissipation. The stationary measure is concentrated close to steady solutions of the 2D Euler equations. For such inertial flows, we prove that bifurcations in the flow topology occur either by changing the domain shape, the nonlinearity of the vorticity-stream-function relation, or the energy. Associated with this, we observe bistable behavior in SNS with random changes from dipoles to unidirectional flows. The theoretical explanation being very general, we infer the existence of similar phenomena in experiments and in some regimes of geophysical flows.

Low-Frequency Variability in the Midlatitude Baroclinic Atmosphere Induced by an Oceanic Thermal Front

This study examines the flow induced by an east–west-oriented oceanic thermal front in a highly idealized baroclinic model. Previous work showed that thermal fronts could produce energetic midlatitude jets in an equivalent-barotropic atmosphere and that barotropic instabilities of this jet had dominant periods of 25–30 and 65–75 days. The present study extends this work to a two-mode baroclinic free atmosphere. The baroclinic jet produced in this case is subject to both barotropic and baroclinic instabilities. A barotropic symmetric instability propagates westward with periods of roughly 30 days and is similar to those found in the equivalent-barotropic model. A baroclinic instability results in standing-dipole anomalies and oscillates with a period of 6–8 months. A mixed barotropic–baroclinic instability results in anomalies that propagate northward, perpendicular to the jet, with a period of 2–3 months. The later anomalies are reminiscent of the 70-day oscillation found over the North Atlantic in observed fields. The atmospheric flow has two distinct states: the flow in the high-energy state exhibits two large gyres and a strong eastward jet; its antisymmetric component is dominant. The low-energy flow is characterized by small gyres and a weak jet. The model’s dynamics depends on the layer-depth ratio. When the model is nearly equivalent-barotropic, symmetric oscillatory modes dominate. As the two layers become nearly equal, antisymmetric oscillatory modes become significant and the mean energy of the flow increases. When the oceanic thermal front’s strength T * is weak (T * 1.5°C), the flow is steady. For intermediate values of the strength (1.5°C T * 3°C), several oscillatory instabilities set in. As the frontal strength increases further (T * 3°C), the flow becomes more turbulent. These results all depend on the atmospheric model’s horizontal resolution being sufficiently high.

Atmospheric Circulations Induced by a Midlatitude SST Front: A GCM Study

The atmospheric effects of sea surface temperature (SST) anomalies over and near western boundary currents are a matter of renewed interest. The general circulation model (GCM) of the Laboratoire de Meteorologie Dynamique (LMD-Z) has a zooming capability that allows a regionally increased resolution. This GCM is used to analyze the impact of a sharp SST front in the North Atlantic Ocean: two simulations are compared, one with climatological SSTs and the other with an enhanced Gulf Stream front. The results corroborate the theory developed previously by the present team to explain the impact of oceanic fronts. In this theory, the vertical velocity at the top of the atmospheric boundary layer has two components: mechanical and thermal. It is the latter that is dominant in the tropics, while in midlatitudes both play a role in determining the wind convergence above the boundary layer. The strengthened SST front does generate the previously predicted stronger ascent above the warmer water south of the front and stronger descent above the colder waters to the north. In the GCM simulations, the ascent over the warm anomalies is deeper and more intense than the descent.

Quantization of the Low-Frequency Variability of the Double-Gyre Circulation

Abstract The low-frequency dynamics of the double-gyre wind-driven circulation in large midlatitude oceanic basins is investigated. It is shown that for quasigeostrophic models linear (Rayleigh) friction is necessary to obtain realistic recirculation gyres and elongated jet streams with small meridional-to-zonal aspect ratio. It is also found that the use of either no-slip or free-slip boundary conditions does not change the drastic effects of bottom drag on the large scales. These long oceanic jets are alternatively destabilized and restabilized through successive (subcritical) supercritical symmetry-breaking bifurcations that are linked to the (non) existence of stationary Rossby waves. These waves are strongly localized along the oceanic front and are thus hardly affected by the basin geometry. Numerical and analytical results show that these waves are “quantized” with respect to the length of the jet, and an explicit dispersion relation is given. Numerical computations of branches of steady states, to...

Hopf bifurcation in quasi-geostrophic channel flow

In this article, we conduct a rigorous stability and bifurcation analysis for a highly idealized model of planetary-scale atmospheric and oceanic flows. The model is governed by the two-dimensional, quasi-geostrophic equation for the conservation of vorticity in an east-west oriented, periodic channel. The main result is the existence of Hopf bifurcation of the flow as the Reynolds number crosses a critical value.The key idea in proving this result is translating the eigenvalue problem into a difference equation and treating the latter by continued-fraction methods. Numerical results are obtained by using a finite-difference scheme with high spatial resolution and these results agree closely with the theoretical predictions. The spatio-temporal structure of the limit cycle corresponds to a wave that propagates slowly westward and is symmetric about the midaxis of the channel. For plausible paramater values that correspond to midlatitude atmospheric flows, the period of this wave is 20--25 days.