Baroclinic Instability Problem Research Articles

Planetary geostrophic Boussinesq dynamics: Barotropic flow, baroclinic instability and forced stationary waves

AbstractMotions on planetary spatial scales in the atmosphere are governed by the planetary geostrophic equations. However, little attention has been paid to the interaction between the baroclinic and barotropic flows within the planetary geostrophic scaling. This is the focus of the present study, which utilizes planetary geostrophic equations for a Boussinesq fluid supplemented by a novel evolution equation for the barotropic flow. The latter is affected by meridional momentum flux due to baroclinic flow and drag by the surface wind. The barotropic wind, on the other hand, affects the baroclinic flow through buoyancy advection. Via a relaxation towards a prescribed buoyancy profile the model produces realistic major features of the zonally symmetric wind and temperature fields. We show that there is considerable cancellation between the barotropic and the baroclinic surface zonal mean zonal winds. Linear and nonlinear model responses to steady diabatic zonally asymmetric forcing are investigated, and the arising stationary waves are interpreted in terms of analytical solutions. We also study the problem of baroclinic instability on the sphere within the present model.

BAROCLINIC INSTABILITY IN THE SOLAR TACHOCLINE. II. THE EADY PROBLEM

ABSTRACT We solve the nongeostrophic baroclinic instability problem for the tachocline for a continuous model with a constant vertical rotation gradient (the Eady problem), using power series generated by the Frobenius method. The results confirm and greatly extend those from a previous two-layer model. For effective gravity G independent of height, growth rates and ranges of unstable longitudinal wavenumbers m and latitudes increase with decreasing G. As with the two-layer model, the overshoot tachocline is much more unstable than the radiative tachocline. The e-folding growth times range from as short as 10 days to as long as several years, depending on latitude, G, and wavenumber. For a more realistic effective gravity that decreases linearly from the radiative interior to near zero at the top of the tachocline, we find that only m = 1, 2 modes are unstable, with growth rates somewhat larger than for constant G, with the same value as at the bottom of the tachocline. All results are the same whether we assume that the vertical velocity or the perturbation pressure is zero at the top of the layer; this is a direct consquence of not employing the geostrophic assumption for perturbations. We explain most of the properties of the instability in terms of the Rossby deformation radius. We discuss further improvements in the realism of the model, particularly adding toroidal fields that vary in height, and including latitudinal gradients of both rotation and toroidal fields.

Submesoscale baroclinic instability in the balance equations

AbstractOcean submesoscale baroclinic instability is studied in the framework of the balance equations. These equations are an intermediate model that includes balanced ageostrophic effects with higher accuracy than the quasigeostrophic approximation, but rules out unbalanced wave motions. As such, the balance equations are particularly suited to the study of baroclinic instability in submesoscale ocean dynamics. The linear baroclinic instability problem is developed in generality and then specialized to the case of constant vertical shear. It is found that non-quasigeostrophic effects appear only for perturbations with cross-front variation, and that perturbation energy can be generated through both baroclinic production and shear production. The Eady problem is solved analytically in the balance equation framework. Ageostrophic effects are shown to increase the range of unstable modes and the growth rate of the instability for perturbations with cross-front variation. The increased level of instability is attributed to both ageostrophic baroclinic production and shear production of perturbation energy; these results are verified in the primitive equations. Finally, submesoscale baroclinic instability is examined in a case where the buoyancy frequency increases rapidly near the bottom boundary, mimicking the increase of stratification at the base of the oceanic mixed layer. The qualitative results of the Eady problem are repeated in this case, with increased growth rates attributed to the production of perturbation energy by the ageostrophic velocity. The results show that submesoscale baroclinic instability acts to reduce lateral buoyancy gradients and their associated geostrophic shear simultaneously through lateral buoyancy fluxes and vertical momentum fluxes.

The Meridional Flow of Source-Driven Abyssal Currents in a Stratified Basin with Topography. Part I: Model Development and Dynamical Properties

Abstract The equatorward flow of source-driven grounded deep western boundary currents within a stratified basin with variable topography is examined. The model is the two-layer quasigeostrophic (QG) equations, describing the overlying ocean, coupled to the finite-amplitude planetary geostrophic (PG) equations, describing the abyssal layer, on a midlatitude β plane. The model retains subapproximations such as classical Stommel–Arons theory, the Nof abyssal dynamical balance, the so-called planetary shock wave balance (describing the finite-amplitude β-induced westward propagation of abyssal anomalies), and baroclinic instability. The abyssal height field can possess groundings. In the reduced gravity limit, a new nonlinear steady-state balance is identified that connects source-driven equatorward abyssal flow (as predicted by Stommel–Arons theory) and the inertial topographically steered deep flow described by Nof dynamics. This model is solved explicitly, and the meridional structure of the predicted grounded abyssal flow is described. In the fully baroclinic limit, a variational principle is established and is exploited to obtain general stability conditions for meridional abyssal flow over variable topography on a β plane. The baroclinic coupling of the PG abyssal layer with the QG overlying ocean eliminates the ultraviolet catastrophe known to occur in inviscid PG reduced gravity models. The baroclinic instability problem for a constant-velocity meridional abyssal current flowing over sloping topography with β present is solved and the stability characteristics are described.

A further study on the stability of a baroclinic flow in cyclostrophic balance

[1] Stability of a baroclinic basic flow in gradient wind balance is examined for a very wide range of parameters. When the Rossby number is sufficiently small, it is found that the baroclinic instability occurs not only in the basic flow in geostrophic regime, but also in that in cyclostrophic one. As the Rossby number increases, the baroclinic instability disappears in the nearly geostrophic regime. However, the unstable mode appears again when the dynamic balance changes into the cyclostrophic one; its growth rate is fairly larger than that of the baroclinic instability in the geostrophic regime. It is shown that the baroclinic instability problem in the Venus cloud layer (45–70 km) is not so deviated from that in the terrestrial troposphere, while the cyclostrophic balance is essential in a situation of the Venus lower atmosphere, where the unstable baroclinic wave is predicted to appear.

Baroclinic Instability of Frontal Geostrophic Currents over a Slope

Abstract The Phillips problem of baroclinic instability is generalized in a frontal geostrophic model. The configuration used here is a two-layer flow (with quasigeostrophic upper-layer current) over a sloping bottom. Baroclinic instability in the frontal model has a single unstable mode, corresponding to isobaths and isopycnals sloping in the same direction, contrary to the quasigeostrophic model, which has two unstable modes. In physical terms, this is explained by the absence of relative vorticity in the lower (frontal) layer. Indeed, the frontal geostrophic model can be related to the quasigeostrophic model in the limit of very small thickness of the lower layer, implying that potential vorticity reduces to vortex stretching in this layer. This stability study is then extended to unsteady flows. In the frontal geostrophic model, a mean flow oscillation can stabilize an unstable steady flow; it can destabilize a stable steady flow only for a discrete spectrum of low frequencies. In this case, the model equations reduce to the Mathieu equation, the properties of which are well known.

Baroclinic instability in a two‐layer model with a free boundary and β effect

The classical Phillips problem of baroclinic instability is generalized, allowing for free deformations of the bottom boundary. The simplicity of the model is exploited to analyze the effects of the variation of the Coriolis parameter with latitude (the so‐called β effect) on the stability/instability problem. Conservation laws of energy, momentum, and vorticity‐related Casimirs are used to establish nonlinear stability conditions. A spectral analysis reveals that unlike the case of Phillips problem, the β effect can either strengthen or weaken the stability of the basic current, depending on the perturbation scale and the slope of the bottom relative to that of the interface. In particular, the maximal instability occurs in the limit of weak stratification when the planetary and the topographic β effects compensate each other. The maximal unstable wave has an intermediate scale between the internal and the external deformation radii. Nonlinear saturation bounds on unstable basics states are also determined using Shepherd's method. It is found that the enstrophy of the most unstable wave can only be bounded by the total enstrophy of the system.

On the Linear Approximation of Velocity and Density Profiles in the Problem of Baroclinic Instability

The linear approximation of density and velocity profiles is compared to more realistic models with vertically inhomogeneous density gradients and nonzero anomalous vorticity (i.e., the nonplanetary part of potential vorticity). Calculations based on the parameters of ‘‘real life’’ currents in the Northern Pacific demonstrate that these effects, acting together, can make baroclinic instability 2.5‐6 times stronger and dramatically expand the spectral range of unstable disturbances toward the short-wave region (by a factor of more than 20‐30).

On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution

Abstract With the purpose of studying the upper part of the ocean, the shallow water equations (in a `reduced gravity' setting) have been extended in the last decades by allowing for horizontal and temporal variations of the buoyancy field ϑ , while keeping it as well as the velocity field u as depth-independent. In spite of the widespread use of this `slab' model, there has been neither a discussion on the range of validity of the system nor an explanation of points such as the existence of peculiar zero-frequency normal modes, the nature of the instability of a uniform u flow, and the lack of explicit vertical shear associated with horizontal density gradients. These questions are addressed here through the development of a subinertial model with more vertical resolution, i.e., one where the buoyancy ϑ varies linearly with depth. This model describes satisfactorily the problem of baroclinic instability with a free boundary, even for short perturbations and large interface slopes. An enhancement of the instability is found when the planetary β effect is compensated with the topographic one, due to the slope of the free boundary, allowing for a `resonance' of the equivalent barotropic and first baroclinic modes. Other low-frequency models, for which buoyancy stratification does not play a dynamical role, are invalid for short perturbations and have spurious terms in their energy-like integral of motion.

Bounds on the phase speed and growth rate of barotropic–baroclinic instability problem

We consider the barotropic–baroclinic instability problem for zonal flows on the β-plane satisfying the boundary conditions ∂U/ ∂z=0 at z=0, z T . First, we obtain a new estimate for the growth rate of any unstable normal mode. Then, we prove the boundedness of the wave velocity of non-singular neutral modes. Finally, we obtain parabolic instability regions, which improve Pedlosky’s instability region for two classes of flows.

Secondary instability, EOF reduction, and the transition to baroclinic chaos

High-resolution numerical simulations illustrate how an initial baroclinic instability may become chaotic as the degree of supercritically is increased. The route to chaos is via bifurcations to periodic and quasi-periodic states, but the spatial structures involved are much more complex than those assumed in previous low-order models. The fundamental participants in the transition can be educed by studying the symmetry-breaking secondary instabilities of fixed points of the primary baroclinic instability problem. These unstable disturbances to the equilibrium way baroclinic flow serve to eradicate the fundamental zonal currents. Reduced low-dimensional models that replicate the full 10 4-mode computer simulations can be formulated by projecting the governing equations onto the EOFs (empirical orothogonal functions) of the large-scale calculations. The successful constructions of such models using relatively few EOFs appears to be related to the fact that the EOFs are almost identical to the secondary instabilities of the primary baroclinic wave state, and these secondary instabilities are, in turn, the primary participants in the supercritical periodic and quasiperiodic states.

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 ...

Four-dimensional assimilation in the presence of baroclinic instability

Current operational assimilation methods have revealed deficiencies in cases of strong baroclinic development. Baroclinic conditions are therefore appropriate for evaluating the potential for improvement which could be achieved through the implementation of a fully four-dimensional data assimilation. In this paper the behaviour of a variational scheme is investigated for a typical baroclinic instability problem, where a wave develops and retroacts on to the basic zonal flow. the tangent-linear model is shown to lead to a good approximation of the time evolution of the wave over a range of 48 hours, even at the most intense cyclogenesis period. For the assimilation experiments the twin experiment approach is applied. Only part of the flow evolution is observed, either the zonal component or the eddy component. In either case the method proves successful in reconstructing the unobserved part of the flow, taking advantage of the nonlinear coupling that exists between those components. However, nonlinearities can lead to difficulties when the range of validity of the tangent-linear model is exceeded or when the cost function exhibits multiple minima.

Four‐Dimensional Assimilation In the Presence of Baroclinic Instability

AbstractCurrent operational assimilation methods have revealed deficiencies in cases of strong baroclinic development. Baroclinic conditions are therefore appropriate for evaluating the potential for improvement which could be achieved through the implementation of a fully four‐dimensional data assimilation. In this paper the behaviour of a variational scheme is investigated for a typical baroclinic instability problem, where a wave develops and retroacts on to the basic zonal flow. the tangent‐linear model is shown to lead to a good approximation of the time evolution of the wave over a range of 48 hours, even at the most intense cyclogenesis period. For the assimilation experiments the twin experiment approach is applied. Only part of the flow evolution is observed, either the zonal component or the eddy component. In either case the method proves successful in reconstructing the unobserved part of the flow, taking advantage of the nonlinear coupling that exists between those components. However, nonlinearities can lead to difficulties when the range of validity of the tangent‐linear model is exceeded or when the cost function exhibits multiple minima.

On the Accuracy of the WKBJ Approximation to the Nonseparable Quasi-geostrophic Baroclinic Instability Problem

Abstract The accuracy Of the WKBJ approximation to the nonseparable quasi-geostrophic baroclinic instability problem is investigated. By direct comparison with an effectively exact solution, we demonstrate for a particular class of frontal zones that the range of parameters for which agreement is attained is rather limited. Most importantly, the WKBJ solution is shown to be significantly in error for frontal zones representative of typical atmospheric conditions.

Nongeostrophic Corrections to the Eigensolutions of a Moist Baroclinic Instability Problem

The analysis of Emanuel et al. is repeated for the linearized primitive equations. The consideration of the Rossby number is shown to reduce growth rates and increase the width of updrafts. The singularity exhibited by the semigeostrophic equations in the limit qe → 0 is moved to N2 → 0. The result of a vanishing width of the updraft persists in the latter limit and it is shown to be a consequence of the hydrostatic approximation. The value qe = 0 in the updraft is still used as representative of slantwise convective adjustment, but the relevant stability parameter for baroclinic waves is now the Brunt–Väisälä frequency and not the potential vorticity. Their difference is o(Ro2), so that the PE contributions become dominant when (qe/qd) ≲ o(Ro2). The eddy fluxes of heat in the limit of a symmetrically neutral environment are compared with the results for dry baro-clinic waves.

Bounds on the eigenvalues of the planetary-scale baroclinic instability problem

In the framework of the linear baroclinic instability problem for planetary geostrophic flows, bounds on the related complex eigenvalues are deduced. The main feature of this result is the independence of these bounds from the latitude and the density stratification, so that it generalizes the results of previous work on the subject.

The effects of friction and heating of convective condensation in the baroclinic instability problem

By using theΒ-plane, two-layer quasi-geostrophic baroclinic model, this paper discusses the baroclinic instability problem concerning the effects of friction and heating of convective condensation. By linear analysis it is shown that the combination ofβ effect, friction and convective heating brings about the asymmetric phenomenon of margin curves. The convective heating plays a role in the increased baroclinic instability. As the heating increases (m*→1), the short wave cutoff can increase infinitely. Besides, the numerical integration of the finite-amplitude equations shows that the trajectory on the phase plane oscillates periodically in the case of non-dissipation. When the friction dissipation is considered, the trajectory of phase decays and oscillates to the equilibrium. The stronger convective heating not only makes the unstable wave length shorter and the amplitude of the equilibrium decrease, but also makes multiple equilibrium into single equilibrium.

Eigenvalue confinement in the problem of linear baroclinic instability for oceanic zonal flows

The classical problem of baroclinic instability is examined from the point of view of eigenvalue confinement in the complex plane. The analysis is performed for a baroclinic zonal flow perturbed by a monochromatic plane wave, harmonic in the latitude. The main results, obtained from thea priori properties of the eigenvalue equation, yield estimates of the phase velocity and the growth rate of the perturbation. As a consequence, the spectrum results to be confined within a bounded region of the complex plane, which depends upon the zonal flux only through its extreme values and the maximum of the shear. The larger is the wave number of the meridional structure of the perturbation, the narrower is the obtained confinement: in the case of higher harmonics the present estimates sharpen the results known from the literature.

Eigenvalue confinement in the problem of linear baroclinic instability for oceanic zonal flows

Eigenvalue confinement in the problem of linear baroclinic instability for oceanic zonal flows