Β-plane Channel Research Articles

Low-frequency finite-amplitude oscillations in a near resonant topographically forced barotropic flow

Abstract In this paper, an inviscid, barotropic, nondivergent potential vorticity equation with topography in a β-plane channel is used to investigate nonlinear interaction between a travelling Rossby wave in a uniform basic flow and a single-wave topography with the same zonal wavenumber. It is found that for the CDV-type monopole near resonance, when the uniform zonal basic westerly wind is near the resonant basic westerly wind, 30–60 day low-frequency oscillations (LFO) can arise from the wave-topography interaction. It has both travelling and standing components. In mid-latitudes (30°–45°N), westward-travelling 30–60 day LFO are dominated by Zonal Wavenumbers 1–3, while eastward-travelling 30–60 day LFO are dominated by Zonal Wavenumbers 1 and 3. In high latitudes (near 60°N), 30–60 day LEO are observed to travel westward, and are dominated by Zonal Wavenumber 1. However, for the LG-type (Legras and Ghil, 1985) dipole near resonance it requires a weaker, more realistic basic flow. When the uniform zonal basic westerly wind is near the resonant basic westerly wind, 30–60 day LFO can be excited by the wave-topography interaction. In the mid-bigh latitudes(30°–60°N), 30–60 day LFO are observed to slowly travel either eastward or westward, depending on the intensity of the near resonant westerly wind, and are dominated by Zonal Wavenumbers 1–3. This demonstrates that nonlinear wave-topography interaction appears to be able to induce low-frequency finite-amplitude oscillations with periods of 30–60 days in the Northern Hemisphere (NH) extratropics.

Maintenance of Multiple Jets in a Baroclinic Flow

This article describes the generation and maintenance of persistent zonal jets in a two-layer quasigeostrophic, β-plane channel model, focusing on the transition from a one to a two jet state. For weak and moderate values of surface friction and supercriticality, the transition occurs abruptly as the width of the baroclinically unstable region of the initial flow is gradually increased. Across the transition point, the persistent two jet state is characterized by a smaller value of eddy energy than that for the one jet state. This reduction in eddy energy is due to increased barotropic energy conversion from the eddies to the zonal mean flow. Consistent with this result, the abrupt emergence of two persistent jets is accompanied by sharply defined eddy momentum flux divergence maxima at the critical latitudes between the two jet maxima. Essentially the same behavior is found in the transition from two to three, and three to four jets. When two or more jets are present, baroclinically growing waves are found to exist along inter-jet minima, which are referred to as “inter-jet disturbances.” More importantly, the momentum fluxes of the interjet disturbances diverge at the interjet minimum, further decelerating the jets. Unstable normal modes similar to the inter-jet disturbances are also found. It is argued that the systematic wave absorption at the critical latitudes and the momentum flux divergence by the interjet disturbances may play a central role in the persistence of the multiple jets.

The effect of barotropic shear on baroclinic instability Part II: The initial value problem

Abstract The effect of barotropic shear on baroclinic instability has been investigated using both a linear quasi-geostrophic β-plane channel model and a multilevel primitive equation model on the sphere when a nonmodal disturbance is used as the initial perturbation condition. The analysis of the initial value problem has demonstrated the existence of a rapid transient growth phase of the most unstable mode. The inclusion of a linear barotropic shear reduces initial rapid transient growth, although at intermediate times the transient growth rates of the sheared cases can be larger than in the unsheared case owing to downgradient eddy momentum fluxes. Certain disturbances can amplify by factors of 4.5–60 times (for the L2 norm), or 3–30 times (for the perturbation amplitude maximum), as large as disturbances based on the linear normal modes. However, linear horizontal shear always reduces the amplification factors. The mechanism is that the shear confines the disturbance meriodionally and therefore limits the energy conversion from the zonal available potential energy to eddy energy. The effect of barotropic shear on the transient growth is not changed much in the presence of either thermal damping or Ekman pumping. Nonmodal integrations of baroclinic wave lifecycles show that the energy level reached by eddies is not very sensitive to the structure of the initial disturbance if the amplitude of the initial disturbance is small. Although in some cases the eddy kinetic energy level reached by the wave integrated from nonmodal disturbance can be 25–150% larger than the normal mode integrations, barotropic shear, characterized by large shear vorticity with small horizontal curvature, always reduces the eddy kinetic energy level reached by the wave, confirming the results of normal mode studies.

The effect of barotropic shear on baroclinic instability Part I: Normal mode problem

Abstract The effect of barotropic shear in the basic flow on baroclinic instability is investigated using a linear multilevel quasi-geostrophic β-plane channel model and a nonlinear spherical primitive equation model. Barotropic shear has a profound effect on baroclinic instability. It reduces the growth rates of normal modes by severely restricting their structure, confirming earlier results with a two-layer model. Dissipation, in the form of Ekman pumping and Newtonian cooling, does not change the main characteristics of the effect of the shear on normal mode instability. Barotropic shear in the basic state, characterized by large shear vorticity with small horizontal curvature, also effects the nonlinear development of baroclinic waves. The shear limits the energy conversion from the zonal available potential energy to eddy energy, reducing the maximum eddy kinetic energy level reached by baroclinic waves. Barotropic shear, which controls the level of eddy activity, is a major factor which should be considered when parameterizing the eddy temperature and momentum fluxes induced by baroclinic waves in a climate model.

Forced Baroclinic Wave Dynamics at Minimum Critical Shear: Potential Vorticity Homogenization, Vacillation, and Equilibration

Abstract Finite-amplitude dynamics of a slightly dissipative baroclinic wave in a two-layer, β-plane channel model at the point of minimum critical shear are examined. At this point, both the potential vorticity gradient and the Doppler-shifted frequency vanish within the lower layer. Previous studies have shown that for this parameter setting both the magnitude of the dissipation and the harmonics of the fundamental wave play important roles in the nonlinear dynamics of the system. In the present study, the response of the nonlinear dynamical system to zonally varying potential vorticity forcing is examined. When the forcing and dissipation are asymptotically small and of equal magnitude, an analytical analysis indicates that the fundamental wave equilibrates to a steady amplitude regardless of the mode being forced. For sufficiently strong forcing, the system must be solved numerically, in which case it is shown that when a harmonic of the fundamental is forced, the system can exhibit one of two dynamic...

A Linear Homogeneous Model of Wind-Driven Circulation in a β-Plane Channel

Abstract An analytical solution is sought for a wind-driven circulation in the inviscid limit in a linear barotropic channel model of the Antarctic Circumpolar Ocean in the presence of a bottom ridge. There is a critical height of the ridge, above which all geostrophic contours in the channel are blocked. In the subcritical case, the Sverdrup balance does not apply and there is no solution in the inviscid limit. In the supercritical case, however, the Sverdrup balance applies and an explicit form for the zonal transport in the channel is obtained. In the case with a uniform wind stress, the transport in the β-plane channel is independent of the width of the ridge, linearly proportional to the wind stress and the length of the channel, while inversely linearly proportional to the ridge height. In the f plane with β = 0, the transport is even independent of the width of the channel. In the case with a nonuniform wind stress τx = τ0(1-−cosπy/D), the Sverdrup flow driven by the vorticity input always induces ...

A note on the generalized Fofonoff modes in a β-plane channel and their stabilities

A class of generalized Fofonoff modes, both barotropic and baroclinic, are found in a β-plane channel presumably to model the circumpolar ocean in the presence of a bottom ridge. These generalized Fofonoff modes share many dynamic features with those in closed basins. Conserved quantities are employed to show that these generalized Fofonoff modes, like those in closed basins, are stable with respect to finite amplitude perturbations.

Nonlinear evolution of spatially growing baroclinic waves

Abstract The linear and weakly nonlinear stability characteristics of spatially growing, long, low frequency baroclinic waves are examined in a continuous atmosphere on a midlatitude β-plane channel in the presence of Ekman friction and Newtonian cooling (NC). Under the assumption that the damping mechanisms are sufficiently weak, the linear spatial instability problem is solved analytically and explicit expressions for the zonal wavenumber and spatial growth rate are obtained. The expressions show that Ekman damping is stabilizing whereas NC is destabilizing. The tendency of NC to dominate over Ekman damping increases linearly with frequency and β, and quadratically with reduction in zonal wind at the surface. The zonal wavelength of the disturbance is affected only by the flow supercriticality and NC, which always oppose each other; increasing the NC (supercriticality) increases (decreases) the zonal wavelength.

Wave-Mean Flow Interaction during the Life Cycles of Baroclinic Waves

A two-layer quasigeostrophic β-plane channel model is used to examine the role of the wave-mean flow interaction during the life cycles of baroclinic waves. Two cases are examined: a wide and a narrow jet limit. These two limits are required to satisfy the property that their instability lead to a realistic baroclinic life cycle consisting of baroclinic growth and barotropic decay. In order to characterize the properties of the zonal-wind tendency in the two cases, scaling arguments based on a study by Andrews and McIntyre are used. This scaling procedure is then used to explain the nonlocal (local) zonal-wind tendency during the realistic baroclinic life cycle for the wide (narrow) jet limit.Several differences between the properties of the two jet limits are found. For the wide jet limit, the acceleration at the center of the jet is confined to the growth stage. This contrasts the narrow jet limit where the jet is accelerated throughout the entire life cycle. These differences depend upon the lower-layer potential vorticity fluxes, which exhibit the same timing properties as the zonal-wind tendency. In addition, for both the wide and narrow jet limits, irreversible potential vorticity mixing is shown to force nonlocal and local permanent changes to the zonal wind, respectively. A comparison is also made between the vorticity flux and potential vorticity flux to determine which is a better predictor of the zonal-wind tendency. It is shown that in the wide (narrow) jet limit, the vorticity (potential vorticity) flux does better at predicting the zonal-wind tendency. It is also argued that one can use a barotropic model to study the temporal evolution of the upper-layer flow for both the narrow and wide jet limits.Last, it is shown that the properties of the inviscid calculations are retained when thermal forcing and surface Ekman friction are included. Calculations are performed with different values for the surface Ekman friction coefficient and with the thermal forcing coefficient fixed. For the wide (narrow) jet limit, it is found that the disturbance grows to a larger (smaller) total energy as the Ekman friction coefficient is increased (decreased). This behavior for the wide jet limit is explained in terms of an enhancement of the baroclinic energy conversions that overcome the barotropic governor mechanism of James and Gray.

A Comparison of the Weakly Nonlinear Instability of Westerly and Easterly Jets in a Two-Layer Beta-Plane Model

Abstract The nonlinear evolution of disturbances that emanate from unstable westerly and easterly jet profiles is examined with a two-layer quasi-geostrophic β-plane channel model. Significant differences arise between the solutions for westerly and easterly jets. For unstable narrow (width less than the deformation radius) westerly jets, the disturbance undergoes a sequence of life cycles characterized by barotropic growth and barotropic decay. Unstable easterly jets also give rise to a series of life cycles in which the disturbance grows and decays in a combined baroclinic/barotropic manner. In each of these cases, the disturbance structure remains close to its unstable normal mode form throughout the life cycle. This contrasts with earlier results of Feldstein and Held who find that disturbances emanating from unstable wide (width greater than the deformation radius) westerly jets undergo a change in meridional structure, enhanced lateral radiation, and a single life cycle consisting of baroclinic grow...

Wind-Driven Flow over Topography in a Zonal β-Plane Channel: A Quasi-geostrophic Model of the Antarctic Circumpolar Current

The paper gives a detailed account of the dynamical balance of a wind-driven zonally unbounded flow over topography. The problem is investigated with a quasi-geostrophic β-plane channel with two layers and eddy resolution. The channel has a width of 1500 km and a zonal periodicity of 4000 km. Apart from the dimensions, the model structure is similar to the one used by McWilliams et al. The experiments with this model address the problem of the relative role of transient and standing eddies as well as bottom friction and topographic form stress in the balance of a current driven by a steady surface windstress. The response of the system is investigated for different values of the friction parameter and various locations of topographic obstacles in the bottom layer of the channel. The principal momentum balance emerging from these experiments supports the concept of Munk and Palmén for the dynamics of the Antarctic Circumpolar Current, which proposes that the momentum input by windstress is transferred to the deep ocean—where it leaves the system activity—where it leaves the system by topographic form stress. Frictional effects in the balance of the circumpolar flow may thus be of minor importance. This concept of the momentum balance is confirmed in simulations over more complex topographies. Here we have taken two differently scaled versions of the highly resolved bottom relief in the Macquarie Ridge area. The flow in these simulations is virtually frictionless in the momentum balance. The flow pattern reflects some features of the Circumpolar Current in this areas.

Topographic Influences on Wind-Driven, Stratified Flow in a β-Plane Channel: An Idealized Model for the Antarctic Circumpolar Current

Abstract Topographic influences are examined in an eddy-resolving model of oceanic channel flow forced by steady zonal winds. With small explicit lateral friction, transient eddies generated by the baroclinic instability of the mean flow transfer momentum downward to the bottom layer. In the flat-bottom case, bottom friction is the only efficient sink of eastward momentum. When bottom topography is present, the topographic form stress can replace the bottom friction sink in the momentum budget, and a large decrease of the zonal transport results. Large wale topography (of the scale of the forcing) provides the largest form stress. Topographic effects decay with height as suggested by the Prandit scaling, and therefore only topographic scales larger than the Rossby radius can affect the whole water column. In that case, the interfaces are deformed by standing eddies on topographic length scales, and standing eddies replace transient eddies in transferring momentum downward. The bottom-layer mean streamfunc...

Baroclinic instability and the summer southern hemisphere wavenumber 5 circulation

Abstract We examine the linear instability of the observed two-dimensional (latitude/height) January 1979 zonal wind of the mid-latitude Southern Hemisphere. The model used is a 10-level, linear, quasi-geostrophic 45°S β-plane channel model, with 30 Fourier harmonics in the meridional direction and a single harmonic in the zonal direction. The fastest growing mode has a zonal wavelength corresponding to wavenumber 12 at 45°S. The mode corresponding to zonal wavenumber 5 at 45°S is also baroclinically unstable. Its latitude-height structure bears qualitative resemblance to the observed wavenumber 5 circulation, which frequently dominates the summer Southern Hemisphere mid-latitude circulation. The latitude of the maximum eddy amplitude at 50°s is simulated, but the maximum is located near the surface instead of aloft. Effects of surface dissipation and a normalization of the zonal wind by the cosine of latitude are also considered.

Inviscid nonlinear stability of barotropic steady flow

Abstract The nonlinear stability of inviscid forced flow in a β-plane channel is investigated with a barotropic model. The equilibrium-forced flow whose stability is considered results from the interaction of a constant westerly mean zonal flow over a finite-amplitude orography. Fully nonlinear integrations of the quasi-geostrophic vorticity equation, initialized with equilibrium flows which are slightly disturbed by their fastest-growing linear disturbances, reveals the possibility of explosive nonlinear instability. Along with this period of explosive disturbance growth there exists an interesting mean flow vacillation arising from the periodic interference between a forced. quasi-stationary wave on the scale of the topography and a free travelling wave (with the same horizontal wavenumber). Although most of this vacillation can be explained using simple linear interference theory some nonlinear effects are also noted. The breakdown of linear theory, as well as, the subsequent finite-amplitude behaviour...

Simulation and models of the role of topographic instability in the formation of atmospheric teleconnection patterns

Simulation and models of the role of topographic instability are used to investigate the evolution of atmospheric teleconnection patterns. Both simple theoretical and complex realistic models are considered in β-plane channel and spherical atmospheric models based on the quasi-geostrophic vorticity equation.

Optimal Excitation of Neutral Rossby Waves

Abstract Properly configured disturbances are known to be effective in transferring the kinetic energy of a mean shear flow to neutrally stable modal and nonmodal waves. Consideration of perturbation energetics requires that such disturbances produce down-gradient momentum fluxes which are associated with perturbation phase lines oriented against the mean shear. Initial conditions chosen arbitrarily, except that they satisfy this requirement, have been shown to result in robust excitation of neutral waves. A question naturally arising from such studies is whether there exists, in some well-defined sense, a best or most effective choice of initial conditions which optimally excites the waves. This question is addressed as a variational problem, and examples of optimal initial conditions are identified for the barotropic β-plane channel. These examples include the most effective excitation of a given neutral Rossby mode and the most rapidly growing perturbation for a given time period without restriction on...

Selective Decay of Enstrophy and the Excitation of Barotropic Waves in a Channel

Abstract If the momentum, energy and circulation of a fluid in a periodic, quasi-geostrophic, β-plane channel are specified, then there is a minimum enstrophy implied. This minimum enstrophy flow is obtained using the calculus of variations and is found to be also a solution of the quasi-geostrophic equations. It is either a parallel flow or a finite-amplitude Rossby wave, depending on the aspect ratio of the channel and the amount of energy and momentum within it. The most geophysically relevant case is a channel whose zonal length is substantially greater than its meridional breadth. In this instance the form of the minimum enstrophy solution is decided by the ratio of the energy to the squared momentum. When this parameter is below a Critical value one has a parallel flow, while if this value is exceeded, the minimum enstrophy, solution is a Rossby wave. Heuristic arguments based on the enstrophy cascade in two-dimensional turbulence suggest a “selective decay hypothesis”. This is that scale-selective ...

Large-Scale Dynamical Response to Differential Heating: Statistical Equilibrium States and Amplitude Vacillation

Abstract This is a study of the statistical behavior of a very low order “general circulation model,” consisting of a single finite-amplitude baroclinic wave interacting with a mean zonal shear flow, maintained against dissipation by differential heating. Starting with the equations for two-level quasi-geostrophic flow in a β-plane channel, we derive a closed system of evolution equations for five zonally-averaged quantities at 45° latitude—including net poleward heat transport, meridional kinetic energy and mean vertical shear (or mean horizontal temperature gradient). This system of equations enables us to relate the equilibrium values of mean vertical shear between 250 and 750 mb and rms northward component of velocity at 500 mb to the rate of differential heating. The former is estimated to be 10.7 m s−1 at 45°S: Oort's observed statistics show that it is actually about 12 m s−1. The theoretical estimate of the rms northward component of velocity at 45°S is 11.0 m s−1. Oort's and Trenberth's statistic...

On the multiplicity of equilibria of baroclinic waves

It is shown that, including the null solution (the Hadley state), there can be six different equilibria in a quasi-geostrophic forced dissipative β-plane channel model of a triadic baroclinic wave system. The dependence of the structural, energetics, and stability properties of each equilibrium solution upon the forcing and dissipation parameters have been delineated. Furthermore, the fairly subtle relations among these equilibria have been uniquely identified on the basis of the local bifurcation theory. Analytic solutions for the viscous single-wave equilibria are obtained. They are unique in contrast to those under an inviscid condition. The latter additionally depend upon the initial state. The diagnosis of the physical character of the single-wave equilibria reveals that such a wave is dynamically neutral with respect to the modified zonal flow. It has identical structural properties as those of a neutral wave according to the linear instability theory. Two distinctly different classes of multiple-wave equilibria are found numerically. One class is essentially a modified form of the single-wave equilibria. The other class uniquely stems from the presence of wave-wave interaction. A stable single-wave equilibrium and a stable multiple-wave equilibrium are found to coexist in a small part of the parameter domain. The instability of the multiple wave equilibria would result in triad-limit-cycles in this model as previously reported by Mak. DOI: 10.1111/j.1600-0870.1987.tb00294.x

順圧大気における強制ロスビー波の非線型的な振舞い第1部強制ロスビー波の安定性

A barotropic quasi-geostrophic inviscid β-plane channel model with simple surface topography is constructed to investigate the nonlinear temporal behavior of forced Rossby waves. In particular, an emphasis is placed on the consequent evolution of forced Rossby waves due to growing unstable perturbations, in connection with the wave amplification observed during the blocking and the sudden warming.This paper consists of two parts: In Part I, we investigate linear stability properties of forced Rossby waves. The presence of surface topography is found to have a significant effect to destabilize the basic flow. When the forced Rossby wave has the largest horizontal scale, the flow becomes unstable and produces two kinds of instabilities: One is the topographic instability for the superresonant flow and the other is for the subresonant flow. The former, which can be represented in a loworder model of Charney and DeVore (1979), produces a standing type perturbation characterized by the form-drag through topography, while the latter, which cannot be represented in the low-order model, is characterized by the interaction between wave components of the perturbation and the basic forced Rossby wave. With the decrease of the zonal flow speed, the horizontal scale of the perturbation becomes smaller, as is expected from the necessary condition for instability.It is also found that the topographic instability does not appear when the meridional wavenumber of the forced Rossby wave is even.