Eady Problem Research Articles

Edge-wave phase shifts versus normal-mode phase tilts in an Eady problem with a sloping boundary

One mechanistic interpretation of baroclinic instability is that of mutual constructive interference of Rossby edge waves. The suppression of baroclinic instability over slopes has been widely established, where previous research argues that a sloping boundary modifies the properties of these Rossby edge waves, but does not provide a mechanistic explanation for the suppression that is valid over all parameter space. In the context of an Eady problem modified by the presence of a sloping boundary, we provide a mechanistic rationalization for baroclinic instability in the presence of slopes that is valid over all parameter space, via an equivalent formulation explicitly in terms of Rossby edge waves. We also highlight the differences between edge-wave phase shifts and normal-mode phase tilts, showing that the edge-wave phase shifts should be the ones that are mechanistically relevant, and normal-mode phase tilt is a potentially misleading quantity to use. Further, we present evidence that the edge-wave phase shifts but not normal-mode phase tilts are well correlated with geometric quantities diagnosed from an analysis framework based on eddy variance ellipses. The result is noteworthy in that the geometric framework makes no explicit reference to the edge-wave structures in its construction, and the correlation suggests the geometric framework can be used in problems where edge-wave structures are not so well defined or readily available. Some implications for parametrization of baroclinic instability and relevant eddy-mean feedbacks are discussed. For completeness, we also provide an explicit demonstration that the linear instability problem of the present modified Eady problem is parity-time symmetric, and speculate about some suggestive links between parity-time symmetry, shear instability, and the edge-wave interaction mechanism. Published by the American Physical Society 2024

Transient energy growth in the ageostrophic Eady model

The problem of optimal initial disturbances in thermal wind shear is revisited and extended to include non-hydrostatic effects. This systematic study compares transient and modal growth rates of submesoscale instabilities over a large range of zonal and meridional wave numbers, aspect ratios, and different Richardson number regimes. Selection criteria were derived to remove spurious and unresolved instability modes that arise from the eigenvalue problem and we generalize the study of the hydrostatic Eady problem by Heifetz and Farrell (2003, 2007, 2008) to thin fronts, characterized by large aspect ratios. Such fronts are commonly found at the early stages of frontogenesis, for example, in the ocean mesoscale eddies and near the eye wall of hurricanes. In particular, we show that transient energy growth rates are up to two orders of magnitude larger than modal counterparts for a wide range of Richardson number and that the effects of transient energy gain become even greater when non-hydrostatic effects become important and/or for large Richardson numbers. This study also compares the dominant energy pathways contributing to the energy growth at short and long times. For symmetric modes, we recover the inertia-gravity instability described in Xu et al. (2007). These mechanisms are shown to be the most powerful mediator of vertical transport when compared with the fastest growing baroclinic and symmetric modes. These results highlight the importance of transient processes in the ocean and the atmosphere.

On energy exchanges between eddies and the mean flow in quasigeostrophic turbulence

We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.

Baroclinic Instability with a Simple Model for Vertical Mixing

AbstractHere, we examine baroclinic instability in the presence of vertical mixing in an idealized setting. Specifically, we use a simple model for vertical mixing of momentum and buoyancy and expand the buoyancy and vorticity in a series for small Rossby numbers. A flow in subinertial mixed layer (SML) balance (see the study by Young in 1994) exhibits a normal mode linear instability, which is studied here using linear stability analysis and numerical simulations. The most unstable modes grow by converting potential energy associated with the basic state into kinetic energy of the growing perturbations. However, unlike the inviscid Eady problem, the dominant energy balance is between the buoyancy flux and the energy dissipated by vertical mixing. Vertical mixing reduces the growth rate and changes the orientation of the most unstable modes with respect to the front. By comparing with numerical simulations, we find that the predicted scale of the most unstable mode matches the simulations for small Rossby numbers while the growth rate and orientation agree for a broader range of parameters. A stability analysis of a basic state in SML balance using the inviscid QG equations shows that the angle of the unstable modes is controlled by the orientation of the SML flow, while stratification associated with an advection/diffusion balance controls the size of growing perturbations for small Ekman numbers and/or large Rossby numbers. These results imply that baroclinic instability can be inhibited by small-scale turbulence when the Ekman number is sufficiently large and might explain the lack of submesoscale eddies in observations and numerical models of the ocean surface mixed layer during summer.

Energy-conserving Galerkin approximations for quasigeostrophic dynamics

A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) [1] to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recombined Legendre basis from Shen (1994) [2] for the streamfunction. The method is tested in the context of linear baroclinic instability calculations, where it is compared to the standard second-order finite-difference method and to a Chebyshev collocation method. The Galerkin scheme is quite accurate even for a small number of degrees of freedom N, and growth rates converge much more quickly with increasing N for the Galerkin scheme than for the finite-difference scheme. The Galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom. The energy-conserving Galerkin scheme is of comparable accuracy to the Chebyshev collocation scheme in most linear stability calculations, but not in the Eady problem where the Chebyshev scheme is significantly more accurate. Finally the three methods are compared in the context of a simplified version of the nonlinear equations: the two-surface model with zero potential vorticity. The Chebyshev scheme is the most accurate, followed by the Galerkin scheme and then the finite difference scheme. All three methods conserve energy with similar accuracy, despite not having any a priori guarantee of energy conservation for the Chebyshev scheme. Further nonlinear tests with non-zero potential vorticity to assess the merits of the methods will be performed in a future work.

Exact instantaneous optimals in the non-geostrophic Eady problem and the detrimental effects of discretization

We derive exact analytical expressions for flow configurations that optimize the instantaneous growth rate of energy in the linear Eady problem, along with the associated growth rates. These optimal perturbations are relevant linear stability analysis, but, more importantly, they are relevant for understanding the energetics of fully nonlinear baroclinic turbulence. The optimal perturbations and their growth rates are independent of the Richardson number. The growth rates of the optimal perturbations grow linearly as the horizontal wavelength of the perturbation decreases. Perturbation energy growth at large scales is driven by extraction of potential energy from the mean flow, while at small scales it is driven by extraction of kinetic energy from the mean shear. We also analyze the effect of spatial discretization on the optimal perturbations and their growth rates. A second order energy-conserving discretization on the Arakawa B grid generally has too-weak growth rates at small scales and is less accurate than two second order discretizations on the Arakawa C grid. The two C grid discretizations, one that conserves energy and another that conserves both energy and enstrophy, yield very similar optimal perturbation growth rates that are significantly more accurate than the B grid discretization at small scales.

Some effects of horizontal discretization on linear baroclinic and symmetric instabilities

The effects of horizontal discretization on linear baroclinic and symmetric instabilities are investigated by analyzing the behavior of the hydrostatic Eady problem in ocean models on the B and C grids. On the C grid a spurious baroclinic instability appears at small wavelengths. This instability does not disappear as the grid scale decreases; instead, it simply moves to smaller horizontal scales. The peak growth rate of the spurious instability is independent of the grid scale as the latter decreases. It is equal to cf/Ri where Ri is the balanced Richardson number, f is the Coriolis parameter, and c is a nondimensional constant that depends on the Richardson number. As the Richardson number increases c increases towards an upper bound of approximately 1/2; for large Richardson numbers the spurious instability is faster than the Eady instability. To suppress the spurious instability it is recommended to use fourth-order centered tracer advection along with biharmonic viscosity and diffusion with coefficients (Δx)4f/(32Ri) or larger where Δx is the grid scale.On the B grid, the growth rates of baroclinic and symmetric instabilities are too small, and converge upwards towards the correct values as the grid scale decreases; no spurious instabilities are observed. In B grid models at eddy-permitting resolution, the reduced growth rate of baroclinic instability may contribute to partially-resolved eddies being too weak. On the C grid the growth rate of symmetric instability is better (larger) than on the B grid, and converges upwards towards the correct value as the grid scale decreases.

Analytical approximations to spurious short-wave baroclinic instabilities in ocean models

Most community ocean models that use z- or s-coordinates stagger their variables in the vertical using a Lorenz grid. Spurious short-wave baroclinic instabilities have been shown to occur on that grid by Arakawa and Moorthi. As the vertical resolution of the grid is improved, the wavelength of the spurious modes decreases and they become more and more trapped near one of the boundaries but they continue to grow at almost the same rate as the deep Eady/Charney modes. The spurious instabilities in the case of the Eady problem are here shown to be accurately reproduced by an analytical calculation which reduces the stability problem to a quadratic equation for their complex phase speeds. The interpretation of these spurious instabilities as resulting from spurious sheets of potential vorticity is revisited. A new interpretation is presented using a finite difference analogue of the Charney–Stern–Pedlosky integral constraint. This indicates that the spurious instabilities result from a vertical averaging of the advection of the relative vorticity which leads to a spurious interior source term in the finite difference potential vorticity equation.

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.

Exact model of a nongeostrophic baroclinic instability

A nongeostrophic version of the classical problem of zonal flow instability with constant shear (the Eady problem) is considered. The linearized set of dynamic equations for two-dimensional disturbances is reduced to a single wave-type second-order equation relative to modified pressure (a linear combination of pressure and stream function). Dynamic features of disturbances with zero potential vorticity are studied in the framework of the equations formulated. Asymptotic solutions of the spectral problem of hydrodynamic stability theory are derived. The initial-value problem at large Richardson numbers is considered using multiple-time-scale expansions. The solution to the problem is represented as the sum of fast (wave) and slow (quasi-geostrophic) components. In the unstable regime, the slow component describes baroclinic waves (cyclones and anticyclones) generated by inhomogeneous initial buoyancy (potential temperature) distributions at the boundaries.

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.

Eddy parameterization challenge suite I: Eady spindown

The first set of results in a suite of eddy-resolving Boussinesq, hydrostatic simulations is presented. Each set member consists of an initially linear stratification and shear as in the Eady problem, but this profile occupies only a limited region of a channel and is allowed to spin-down via baroclinic instability. The diagnostic focus is on the spatial structure and scaling of the eddy transport tensor, which is the array of coefficients in a linear flux–gradient relationship. The advective (antisymmetric) and diffusive (symmetric) components of the tensor are diagnosed using passive tracers, and the resulting diagnosed tensor reproduces the horizontal transport of the active tracer (buoyancy) to within ±7% and the vertical transport to within ±12%. The derived scalings are shown to be close in form to the standard Gent–McWilliams (antisymmetric) and Redi diffusivity (symmetric) tensors with a magnitude that varies in space (concentrated in the horizontal and vertical near the center of the frontal shear) and time as the eddies energize. The Gent–McWilliams eddy coefficient is equal to the Redi isopycnal diffusivity to within ±6%, even as these coefficients vary with depth. The scaling for the magnitude of simulation parameters is determined empirically to within ±28%. To achieve this accuracy, the eddy velocities are diagnosed directly and used in the tensor scalings, rather than assuming a correlation between eddy velocity and the mean flow velocity where ±97% is the best accuracy achievable. Plans for the next set of models in the challenge suite are described.

Monge–Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem

We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge–Ampére method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis forces. This method can be used to resolve evolving small scale features in the flow. In this paper the method is then applied to the computation of the solution to the Eady problem which is observed to develop large gradients in a finite time. The moving mesh method is shown to work and be stable, and to give significantly better resolution of the evolving singularity than a fixed, uniform mesh.

A Framework for Parameterizing Eddy Potential Vorticity Fluxes

AbstractA framework for parameterizing eddy potential vorticity fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy flux. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and nondimensional parameters describing the mean shape and orientation of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are nondimensional and bounded in magnitude by unity. Moreover, these nondimensional geometric parameters have strong connections with classical stability theory. When applied to the Eady problem, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability. Moreover, in the limit in which Reynolds stresses are neglected, the framework reduces to a Gent and McWilliams type of eddy closure where the eddy diffusivity can be interpreted as the form proposed by Visbeck et al. Simulations of three-layer wind-driven gyres are used to diagnose the eddy shape and orientations in fully developed geostrophic turbulence. These fields are found to have large-scale structure that appears related to the structure of the mean flow. The eddy energy sets the magnitude of the eddy stress tensor and hence the eddy potential vorticity fluxes. Possible extensions of the framework to ensure potential vorticity is mixed on average are discussed.

Rapidly rotating plane layer convection with zonal flow

The onset of convection in a rapidly rotating layer in which a thermal wind is present is studied. Diffusive effects are included. The main motivation is from convection in planetary interiors, where thermal winds are expected due to temperature variations on the core–mantle boundary. The system admits both convective instability and baroclinic instability. We find a smooth transition between the two types of modes, and investigate where the transition region between the two types of instability occurs in parameter space. The thermal wind helps to destabilise the convective modes. Baroclinic instability can occur when the applied vertical temperature gradient is stable, and the critical Rayleigh number is then negative. Long wavelength modes are the first to become unstable. Asymptotic analysis is possible for the transition region and also for long wavelength instabilities, and the results agree well with our numerical solutions. We also investigate how the instabilities in this system relate to the classical baroclinic instability in the Eady problem. We conclude by noting that baroclinic instabilities in the Earth's core arising from heterogeneity in the lower mantle could possibly drive a dynamo even if the Earth's core were stably stratified and so not convecting.

A boundary-value problem for the parameterized mesoscale eddy transport

We present a physically and numerically motivated boundary-value problem for each vertical ocean column, whose solution yields a parameterized mesoscale eddy-induced transport streamfunction. The new streamfunction is a nonlocal function of the properties of the fluid column. It is constructed to have a low baroclinic mode vertical structure and to smoothly transition through regions of weak stratification such as boundary layers or mode waters. It requires no matching conditions or regularization in unstratified regions; it satisfies boundary conditions of zero transport at the ocean surface and bottom; and it provides a sink of available potential energy for each vertical seawater column, but not necessarily at each location within the column. Numerical implementation of the methodology requires the solution of a one-dimensional tridiagonal problem for each vertical column. To illustrate the approach, we present an analytical example based on the nonlinear Eady problem and two numerical simulations.

On the stratorotational instability in the quasi-hydrostatic semigeostrophic limit

The linear normal mode stratorotational instability (SRI) is analytically re-examined in the inviscid limit where the length-scales of horizontal disturbances are large compared their vertical and radial counterparts. Boundary conditions different from channel walls are also considered. This quasi-hydrostatic, semigeostrophic (QHSG) approximation allows one to examine the effect of a vertically varying Brunt–Vaisaila frequency, N2. It is found that the normal mode instability persists when N2 increases quadratically with respect to the disc vertical coordinate. However, we also find that the SRI seems to exist in this inviscid QHSG extreme only for channel wall conditions: when one or both of the reflecting walls are removed, there is no instability in the asymptotic limit explored here. It is also found that only exponential-type SRI modes (as defined by Dubrulle et al.) exist under these conditions. These equations also admit non-normal mode behaviour. Fixed Lagrangian pressure conditions on both radial boundaries predict there to be no normal mode behaviour in the QHSG limit. The mathematical relationship between the results obtained here and that of the classic Eady problem for baroclinic instability is drawn. We conjecture as to the mathematical/physical nature of the SRI. The general linear problem, analysed without approximation in the context of the Boussinesq equations, admits a potential vorticity-like quantity that is advectively conserved by the shear. Its existence means that a continuous spectrum is a generic feature of this system. It also implies that in places where the Brunt–Vaisaila frequency becomes dominant the linearized flow may two-dimensionalize by advectively conserving its vertical vorticity.

Parameterizing ocean eddy transports from surface to bottom

We consider new parameterization schemes for the extra transport velocity by eddies in a stratified fluid, with particular emphasis on baroclinic instability. These come in the form of elliptic equations, previously unmentioned, which we derive for the eddy‐induced overturning stream function. They guarantee decrease of the mean field potential energy. Our principal example gives a relationship between the vertical shear of the overturning velocity and the buoyancy torque of the main geostrophic current. The parameterized velocity is nonsingular at the bottom and the sea surface, contrasting with previous constant‐coefficient theories based on depth diffusion. Idealized two‐dimensional numerical experiments successfully reproduce meridional overturning circulation even when the background density gradient is uniform everywhere (the Eady problem) or when the bottom is steeply sloped. We further demonstrate that adding an eddy form drag (wave stress) term in the TRM momentum equations yields overturning of the velocity field.

Wave Over-reflection and Baroclinic Instability of the Eady Problem

Abstract The concept of wave over-reflection is applied to the unstable normal mode of the Eady problem. In order to consider propagation properties of the Rossby waves trapped at the boundaries, two thin boundary layers are introduced at the top and bottom of the fluid layer of the original model. It is shown that the Rossby waves in the boundary layers are always over-reflected as long as their critical levels exist in the constant-shear evanescent region and the waves are transmitted to the opposite boundary layer. The dispersion relation obtained by using laser formula and quantization qualitatively coincides with that of the normal mode. Although the growth rate is systematically overestimated, the short-wave cutoff is well described by the over-reflection solution. The mechanism of over-reflection obtained in this study is understood by conservation of momentum. Since the pseudomomentum of the transmitted and the incident waves have opposite signs to each other, the amplitude of the reflected wave s...

Influence of the β‐effect on non‐modal baroclinic instability

AbstractWe focus on the influence of the β‐effect on the singular modes of baroclinic instability in the Phillips model. An analytical intercomparison between normal and singular modes for the Eady problem on an f‐plane has already been performed, showing that the amplification rate of the singular mode for the kinetic‐energy norm after a dimensional time of three days is about 1.2 times larger than for the normal mode. We show here that, in the presence of the β‐effect, the maximum amplification rate of the singular mode can be 1.5 times larger than for the normal‐mode case. This difference is due to the lesser stabilizing effect of β on the singular modes compared with the normal modes. This result is obtained for the case of equal layer depths which is relevant for the atmospheric situation. For the oceanic situation with unequal layer depths the amplification due to β is further increased leading to a factor of 1.8 in amplification rates between the two types of modes after a dimensional time of about 20 days. This behaviour is explained by the fact that β only intervenes in the anti‐Hermitian part of the linear dynamical operator, the main effect of which is to alter the vertical phase tilt of the perturbations. As a consequence, the structure of the bi‐orthogonal mode becomes more inclined to the vertical in the presence of β while, on the contrary, the structure of the unstable normal mode becomes less inclined.