Nonlinear Equilibration Research Articles

Impact of sub-mesoscale physics on production and subduction of phytoplankton in an oligotrophic regime

Using a protocol of numerical experiments where horizontal resolution is progressively increased, we show that small-scale (or sub-mesoscale) physics has a strong impact on both mesoscale physics and phytoplankton production/subduction. Mesoscale and sub-mesoscale physics result from the nonlinear equilibration of an unstable baroclinic jet. The biogeochemical context is oligotrophy. The explicitly resolved sub-mesoscales, at least smaller than one e fth of the internal Rossby radius of deformation, reinforce the mesoscale eddy e eld and contribute to double the primary production and phytoplankton subduction budgets. This enhancement is due to the reinforced mesoscale physics and is also achieved by the small-scale frontal dynamics. This sub-mesoscale physics is associated with density and vorticity gradients around and between the eddies. It triggers a signie cant small-scale nutrient injection in the surface layers, leading to a phytoplankton e eld mostly dominated by e ne spatial structures. It is believed that, depending on wind forcings, this scenario should work alternately with that of Abraham (1998) which invokes horizontal stirring of nutrient injected at large scales. Results also reveal a strong relationship between new production and negative vorticity, in the absence of wind forcing and during the period of formation of the eddies.

Instabilities of magnetically induced shear layers and jets

We investigate numerically the flow of an electrically conducting fluid confined in a spherical shell, with the inner sphere rotating, the outer sphere stationary, and a strong magnetic field imposed parallel to the axis of rotation. It has previously been shown that the axisymmetric basic state depends strongly on the electromagnetic boundary conditions used, with insulating boundaries yielding a shear layer, but conducting boundaries yielding a counter–rotating jet, where in both cases these structures are located on the cylinder parallel to the imposed field and tangent to the inner sphere. Here we compute the non–axisymmetric instabilities of these basic states, and show that for sufficiently large rotation rates both the shear layer and the jet spawn a series of vortices encircling the tangent cylinder. Finally, we consider the fully three–dimensional nonlinear equilibration, and show that in the supercritical regime a secondary bifurcation occurs in which the number of vortices (for the shear layer) or vortex pairs (for the jet) is reduced by one.

Blocking as a local instability to zonally varying flows

AbstractAn idealized system consisting of a steady‐state, axisymmetric vortex patch with topography forcing variations in the basic‐state flow along the vortex edge is shown to possess local instabilities for a broad range of parameters. One branch of these instabilities corresponds in scale and structure to observed atmospheric blocks. The nonlinear equilibration of these blocking‐type instabilities reproduces many aspects of observed blocking onsets; the potential for rapid non‐modal growth and ready excitation of these local instabilities by transients suggests that this instability may lie at the root of observed blocking onsets.

Local Energetics of an Idealized Baroclinic Wave Using Extended Exergy

The concept of local extended exergy is here applied to an idealized, dry, and reversible-adiabatic cyclone development. The extended exergy as well as the kinetic energy are decomposed into a mean part, defined by a zonal average, and into a perturbation from the mean. The resulting local energy evolution equations provide an extension of the well-known Lorenz-type available energy equations. A term in the baroclinic conversion rate, connected with static stability anomalies, which is not usually considered, is of significance even in this idealized case study and contributes significantly to the nonlinear equilibration of the baroclinic wave.

Nonlinear Equilibration of Barotropic Waves in a Zonally Nonuniform Current

Abstract In this paper, nonlinear evolution and equilibration of the barotropic waves in a zonally inhomogeneous Bickley jet are investigated. The equilibrated state consists of a full spectrum of waves with a dominant frequency and its higher-frequency modes. The dominant wave scale in most equilibrated states is wavenumber 1 because of the changes in the local mean flow. Except for wave 1, the dominant wave scale varies with the governing parameters. The wavelength of the dominant wave increases for stronger and longer jets and for weaker dissipation. The equilibrations are highly dependent on the parametric settings. Steady wavy, limit cycle, and chaotic states are found as with other nonlinear equlibration studies. When the jet is weak but strong enough to produce waves, the equilibration behaves very smoothly. But when the jet becomes stronger, the equilibration process becomes much more complicated and the subregion in parameter space for each type of solution becomes very irregular. When the stream...

The nonlinear evolution of unstable coupled equatorial ocean-atmosphere modes

Abstract. In this paper we investigate whether observed intraseasonal variability in the equatorial Pacific can be attributed to finite amplitude waves resulting from unstable air-sea interactions. Within a Zebiak - Cane type model of the coupled equatorial ocean - atmosphere, the nonlinear equilibration of instabilities of a simple basic state is considered with periodic conditions on the ocean boundaries. Three mechanisms exist which can induce a finite amplitude equilibration on a time scale ε2t. Here t is the characteristic time scale of growth of the disturbance and ε the relative distance from the instability threshold. For each equilibration mechanism, the finite amplitude and period of the equilibrium state are computed as a function of ε and substantial amplitude can be reached for a reasonable degree of supercriticality. Thereafter the analysis is extended to include time-dependent external forcing. It is shown that interannual variability may result through the interaction of the response of a weak annual external forcing and the finite amplitude development of the intraseasonal instabilities.

The Equilibration of Short Baroclinic Waves

The life cycles of short baroclinic waves are investigated with the intention of completing a simple classification of nonlinear equilibration scenarios. Short waves become important in moist environments as latent heating reduces the scale of maximum baroclinic instability. Long-wave life cycles (wavenumber 6) were previously found to depend on details of the low-level momentum fluxes established during the earliest stages of development. These fluxes also serve as a focal point for the present study. For a realistic, zonally symmetric jet on the sphere, the normal-mode life cycle of a short wave (wavenumber 8) under both dry and moist conditions is described. Latent heating intensifies the low pressure system and frontal zones but does not alter the broader details of the life cycle. The normal modes have predominantly equatorward momentum fluxes, in contrast to the mainly poleward momentum fluxes of long waves. The short waves are more meridionally confined. The equatorward momentum fluxes direct the waves toward cyclonic breaking. The feedback with the zonal-mean wind, the so-called barotropic governor, is less effective than in the standard long-wave life cycle, which ends in anticyclonic breaking. However, in contrast to long-wave life cycles that are “engineered” to produce equatorward momentum fluxes, relatively little potential vorticity and surface temperature anomaly roll up into isolated vortices. Therefore, the short wave undergoes protracted barotropic decay leading to complete zonalization. Short waves also have a brief period of baroclinic decay due to cold advection over the surface cyclones. Eliassen–Palm cross sections for the short-wave life cycles show the usual combination of upward and meridional propagation of wave activity. However, the meridional propagation is mainly toward the pole and there is a consequent zonal-mean deceleration at high latitudes. These details are included in the proposed classification of equilibration scenarios.

Nonlinear equilibration of a dynamo in a smooth helical flow

We investigate the nonlinear equilibration of magnetic fields in a smooth helical flow at large Reynolds number Re and magnetic Reynolds number Rm with Re[Gt ]Rm[Gt ]1. We start with a smooth spiral Couette flow driven by boundary conditions. Such flows act as dynamos, that is are unstable to growing magnetic fields; here we disregard purely hydrodynamic instabilities such as Taylor–Couette modes. The dominant feedback from a magnetic field mode is only on the mean flow and this yields a simplified ‘mean-flow system’ consisting of one magnetic mode and the mean flow, which we solve numerically. We also obtain the asymptotic structure of the equilibrated fields for weakly and strongly nonlinear regimes. In particular the field tends to concentrate in a cylindrical shell where all stretching and differential rotation is suppressed by the Lorentz force, and the fluid is in solid-body motion. This shell is bounded by thin diffusive layers where the stretching that maintains the field against diffusive decay is dominant.

Effects of an Asymmetric Friction on the Nonlinear Equilibration of a Baroclinic System

Following some recent linear and nonlinear studies the authors examine, using numerical simulations of a classical two-layer model, the effect of an asymmetric friction on the nonlinear equilibrium of moderately unstable baroclinic systems. The results show that the presence of an asymmetric friction leads to a significant wave scale selection: ‘‘long’’ waves (in terms of their zonal wavelengths) emerge with a traditional asymmetric friction (with the upper layer less viscous than the lower layer), while only ‘‘short’’ waves dominate with a nontraditional asymmetric friction (with the lower layer less viscous than the upper layer). The role of the nonlinear interactions and, more precisely, the effects of an asymmetric friction on the wave‐mean flow and wave‐wave interactions and their consequences on the wave scale selection are examined.

Inertial nonlinear equilibration of equatorial flows

We explore the nature of inertial equilibration of equatorial flows in the presence of mean meridional and vertical shears of the basic state, with oceanic applications in mind. The study is motivated by the observational evidence that the subthermocline equatorial mean circulation displays nearly zero Ertel potential vorticity away from the equator, when taking into account the non-traditional horizontal component of the Earth rotation. This observed state precisely verifies the marginal condition for inertial instability: a linear analysis for the equatorial β-plane confirms that the usual condition of instability, namely that Ertel potential vorticity should be of opposite sign to the vertical Coriolis parameter, remains valid even when the traditional approximation is relaxed. Analytical linear normal modes reveal that a meridional shear of the basic state leads to a vertical stacking of equatorially-trapped zonal flows of alternate signs, with a new centre of symmetry located at the dynamical equator. A vertical shear of the basic state causes a meridional stacking of extra-equatorial zonal flows.In an inviscid framework, a two-dimensional formulation is ill-posed and we resort to non-hydrostatic viscous simulations to determine the nonlinear normal forms of the system. The influence of a small-scale eddy diffusivity and a large-scale Rayleigh damping on the equilibrated vertical scale is determined numerically. The nonlinear equilibration occurs through a steady-state bifurcation from a basic state without jets to another steady state with secondary jets of alternate signs. The final state corresponds to eastward jets located on the geographic equator, while westward jets are located near the dynamical equator. These results are consistent with in situ observations of equatorial deep jets.The analogy between the equatorial meridional shear flow and the cylindrical Couette–Taylor flow with an axial density stratification is detailed. There is a strong similarity in the general symmetries and nonlinear normal forms of the two problems. Similarly to the homogeneous Couette–Taylor flow, the gap width between the two cylinders is important for determining the axial scale of the secondary flow through the Reynolds number. For the equatorial problem, an upper bound for the height scale of inertial jets is such that the corresponding equatorial radius of deformation times √2 fits between the geographic and dynamic equators.One of our main conclusions is that the raisond’être of the observed region of zero Ertel potential vorticity is to facilitate angular momentum exchanges between the two hemispheres and inertial deep jets are the byproducts of this angular momentum mixing.

Baroclinic Adjustment in a Two-Level Model with Barotropic Shear

Baroclinic instability in two-level models is characterized by a critical vertical shear, for values above which the flow is unstable. Existing studies of nonlinear baroclinic equilibration in two-level models suggest that, while equilibration does occur, it does so for values of vertical shear that are supercritical. The criterion used for the critical shear, however, ignores nonlinear changes in barotropic (meridional) shear. The present note estimates, using the two-scale formalism, the effect of both jet scale and damping on the critical value of vertical shear. The results suggest the barotropic shear in the equilibrated states may be sufficient, in the presence of damping, to render the equilibrated states neutral. More generally, it appears important to take account of the nature of the evolved flow when assessing the stability properties of the equilibrated state.

Barotropic Model Simulations of the North Pacific Retrograde Disturbances

Branstator–Kushnir-type large-scale westward propagating waves are investigated using linear and nonlinear global barotropic models with an idealized zonally asymmetric basic state. Retrograde waves are found in the most unstable normal mode of the zonally asymmetric basic state with a jet in the Northern Hemisphere. West-ward propagating waves also exist in nonlinear equilibrium states under a wide range of supercriticality and in both periodic and chaotic regimes. The frequency of the most unstable mode remains as a peak in the frequency spectrum through the nonlinear equilibration process. That frequency matches the frequency of the westward propagating waves in the nonlinear equilibrium states. Local energetics analyses of the linear and nonlinear cases show that the barotropic energy conversion concentrated in the jet exit supplies the perturbation energy of the disturbances all over the globe. Under a traditional spherical-harmonic decomposition, the westward propagating waves consist of several spherical-harmonic components. In the weakly chaotic nonlinear equilibrium states, these components show higher regularity in time than the others and may possess higher predictability.

Weather Regimes: Recurrence and Quasi Stationarity

Abstract Two different definitions of midlatitude weather regimes are compared. The first seeks recurrent atmospheric patterns. The second seeks quasi-stationary patterns, whose average tendency vanishes. Recurrent patterns are identified by cluster analysis, and quasi-stationary patterns are identified by solving a nonlinear equilibration equation. Both methods are applied on the same dataset: the NMC final analyses of 700-hPa geopotential heights covering 44 winters. The analysis is performed separately over the Atlantic and Pacific sectors. The two methods give the same number of weather regimes—four over the Atlantic sector and three over the Pacific sector. However, the patterns differ significantly. The investigation of the tendency, or drift, of the clusters shows that recurrent flows have a systematic slow evolution, explaining this difference. The patterns are in agreement with the ones obtained from previous studies, but their number differs. The cluster analysis algorithm used here is a partiti...

Nonlinear Equilibration of Localized Instabilities on a Baroclinic Jet

Dynamical mechanisms underlying the equilibration of absolute instability are examined in a nonlinear, quasigeostrophic, two-layer model. The key to understanding the nonlinear equilibration is in recognizing that linear absolute instabilities can be stabilized both by a reduction of the vertical shear and by enhancement of the mean barotropic velocity. In a localized domain, the equilibration process proceeds with the creation of locally convectively unstable regions downstream, which encroach onto the locally absolutely unstable region until the local instability is suppressed. That local instabilities exist only if absolutely unstable regions span a minimum size is verified by eigenvalue calculations of three-dimensional flows. Numerical examples suggest that this critical size is at least 9000 km for a wide range of parameter values chosen to investigate the midlatitude storm tracks. Fluctuations arising from local absolute instability obtain maximum amplitude in the downstream convectively unstable regions rather than in the absolutely unstable regions themselves. Together, these results suggest that if an equilibrated absolute instability were to occur in midlatitudes, a zonal band of surface easterlies exceeding 9000 km would be required and the associated enhanced variances would not be found coincident with the regions of absolute instability.

Nonlinear equilibration of two-dimensional optimal perturbations in viscous shear flow

Two-dimensional perturbations configured for maximum energy growth in laminar viscous shear flow are shown to develop into quasisteady finite amplitude structures, provided that the initial perturbation has sufficient energy and a nearby nonlinear mode exists. For Poiseuille flow, which supports finite amplitude equilibria for Reynolds numbers above ∼2900, an optimal perturbation with initial energy density equal to or greater than 0.1% of the mean flow energy density closely approaches the quasiequilibrium state within 10 advective time units. For Couette flow, which has no finite amplitude solution, the optimal perturbations decay rapidly after reaching maximum amplitude unless the configuration is sufficiently close to a linear mode with slow exponential decay rate. While the quasiequilibrium structure for Poiseuille flow is locally inflectional, it supports only weak instabilities with scales larger than the local region.

Nonlinear Equilibration of Two-Dimensional Eady Waves: Simulations with Viscous Geostrophic Momentum Equations

Abstract Equilibration of two-dimensional Eady waves is numerically investigated using the geostrophic momentum equations incorporating heat and momentum diffusion. Extended solutions are obtained beyond what would be the collapse of surface fronts in the inviscid theory, and are found to accurately reproduce the equilibration of baroclinic waves simulated with the primitive equations. Potential vorticity anomalies produced at the surface fronts are essential in the initial amplitude saturation and in the asymptotic behavior of the equilibrated flow. A supergeostrophic shear spun up nonlinearly in the zonal flow also plays an important role in causing the reversal of the tilt. The zonal mean potential temperature profile in the equilibrium state is similar to the prediction by the adjustment hypothesis of Gutowski when only horizontal diffusion is present. However, it is closer to an Eady's basic state with enhanced static stability when vertical diffusion is also present. The difference in the ageostroph...

A Geometric Model of the Nonlinear Equilibration of Two-Dimensional Eady Waves

Abstract The initial-value problem for Eady's model is reexamined using a two-dimensional (x, z) Lagrangian semigeostrophic model. A basic-flow state in a periodic domain with a perturbation field imposed is represented by piecewise constant data. The Lagrangian conservation laws governing the motion allow the construction of a solution that describes the evolution of the unstable wave as it passes through the point of frontal collapse, reaches a maximum amplitude, and then decays. The integration is continued and a second cycle is observed. A similar experiment was performed by Nakamura and Held, using a primitive equation model with diffusion. Two integrations are carried out to investigate the sensitivity of the solution to the representation of the data by piecewise constant values and present observations and comparisons between these results and those of Nakamura and Held.

Nonlinear Equilibration of Two-Dimensional Eady Waves: A New Perspective

Abstract The equilibration of two-dimensional baroclinic waves differs fundamentally from equilibration in three dimensions because two-dimensional eddies cannot develop meridional temperature or velocity structure. It was shown in an earlier paper that frontogenesis together with diffusive mixing in a two-dimensional Eady wave brings positive potential vorticity (PV) anomalies deep into the atmosphere from both boundaries and allows the disturbance to settle into a steady state without meridional gradients. Here we depart from the earlier explanation of this equilibration and associate the PV intrusions with essentially the same kind of vortex “roll-up” that characterizes the evolution of barotropic shear layers. To avoid subgrid turbulence parameterizations and computational diffusion, the analogy is developed using Eady's generalized baroclinic instability problem. Eady's generalized model has two semi-infinite regions of large PV surrounding a layer of relatively small PV. Without boundaries, frontal ...

Nonlinear spatial equilibration of an externally excited instability wave in a free shear layer

A two-dimensional disturbance evolving from a strictly linear, finite-growth-rate instability wave with nonlinear effects first becoming important in the critical layer is considered. The analysis is carried out for a general weakly non-parallel mean flow using matched asymptotic expansions. The flow in the critical layer is governed by a nonlinear vorticity equation which includes a spatial-evolution term. As in Goldstein & Hultgren (1988), the critical layer ages into a quasi-equilibrium one and the initial exponential growth of the instability wave is converted into a weak algebraic growth during the roll-up process. This leads to a next stage of evolution where the instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects and is eventually converted to decay. Expansions for the various streamwise regions of the flow are combined into a single composite formula accounting for both shear-layer spreading and nonlinear critical-layer effects and good agreement with the experimental results of Thomas & Chu (1989), Freymuth (1966), and C.-M. Ho & Y. Zohar (1989, private communication) is demonstrated.

Nonlinear Baroclinic Adjustment

Abstract The concept of baroclinic adjustment is reexamined in the context of a fully nonlinear two-layer model on a β-plane. Based on our results we propose a single, conceptually very simple mechanism of the nonlinear equilibration of waves and the mean flow, which we term “nonlinear baroclinic adjustment.” The new concept appears applicable to cases that currently require different explanations, varying from case to case, to account for the equilibration of the mean temperature gradient in the presence of external driving.