Beta-plane Channel Research Articles

Meridional Distribution of Stratospheric Trace Constituents

Vertically stratified stratospheric tracers such as methane and nitrous oxide tend to have constant mixing ratio surfaces that slope downward toward the poles in the meridional plane. The equilibrium tracer slope results from the competition between the slope steepening effects of advection by the diabatic circulation and the slope flattening effects of quasi-isentropic eddy transport and photochemical loss. The diabatic circulation itself is, however, driven primarily by eddy transports, which maintain the departure of stratosphere temperatures from radiative equilibrium. If the eddy transports are weak, the diabatic circulation is also weak and the slope is small. Using a simple beta-plane channel model and an eddy diffusion parameterization for the eddy potential vorticity and tracer transports, we show that the slope is a maximum for a value of eddy diffusion such that the dynamical time scale is between the radiative and chemical time scales.

Excitation of Nearly Steady Finite-Amplitude Barotropic Waves

An exact nonlinear stationary solution is obtained for barotropic waves in a beta-plane channel and it is shown that it can be excited under a range of initial conditions. Results show that a finite-amplitude wave in a constant shear flow, given an initial phase tilt against the shear and a sufficient initial amplitude, interacts with the mean flow to produce a nearly steady state close to the exact stationary solution. This equilibration process involves nonlinear transients; in particular, as the flow equilibrates, the emergence of critical levels is accompanied by the neutralization of local mean vorticity gradients at these levels, thus allowing the solution to attain a nonsingular modal structure.

Resonance of Topographically Forced Waves in a Quasi-Geostrophic Model

Abstract The resonance of stationary waves forced by topography is examined using a quasi-geostrophic model on a beta-plane channel. It is shown analytically that among the factors favoring the resonance of large, rather than synoptic or small, scale waves is the fact that the sensitivity of the large resonant responses to a change of zonal wind decreases as the scale of the resonant wave increases. A numerical model is used to examine resonance in the presence of topography having zonal wavenumber 2 with zonal flows having horizontal and vertical shear and including the effects of damping and nonlinear interactions. Although the effects of resonance are found to be important even in the presence of damping mechanisms, linear experiments with topographical forcing of reasonable amplitude indicate that a period or several weeks is required for a resonant internal mode to achieve large amplitude in the troposphere. However, as the structure of the resonant mode is such that it has much larger amplitudes in ...

A Further Study of Gravity Wave Induced Drag and Diffusion in the Mesosphere

Lindzen's (1967) parameterization for the drag and eddy diffusion produced by breaking internal gravity waves in the mesosphere and lower thermosphere is applied to a modified version of the beta-plane channel model of Holton (1982) in which an isotropic source spectrum of waves is specified similar to that given in 1982 by Matsuno (1982). The transmission for each wave component is influenced by Newtonian cooling and by eddy diffusion induced by the breaking of other wave components. In general the waves with smallest Doppler-shifted phase speeds break first and produce sufficient eddy diffusion to significantly raise the breaking heights for the higher speed components. Thus, the wave drag and diffusion is spread through a deep layer and the resulting mean wind profiles for both summer and winter solstice conditions are more realistic than those computed previously by Holton.

Resonant interactions between unstable and neutral baroclinic waves in a continuous model of the atmosphere

Resonant interactions between a finite-amplitude marginally unstable wave and two neutral baroclinic waves are investigated in a quasi-geostrophic, continuous, infinite depth model on the beta-plane channel. Since the neutral waves are characterized by total horizontal wavenumbers less than that of the unstable wave, the kinematic resonant conditions can only be satisfied when the meridional scale of the unstable wave is sufficiently small. Employing asymptotic methods, equations are obtained governing the temporal evolution of the amplitudes and phases of the triad. The neutral waves are scaled smaller in magnitude than the unstable wave as a result of an asymptotic imbalance between the advective non-linearity and instability. Numerical solutions of the amplitude equations reveal a separation between the instability and interaction time scales. The neutral wave amplitudes vacillate on the longer interaction time scale. The unstable wave amplitude vacillates locally on the instability time scale while its envelope vacillates on the interaction time scale in direct proportion to the initial neutral wave amplitudes. DOI: 10.1111/j.1600-0870.1984.tb00251.x

The Charney-Stern theorem on barotropic-baroclinic instability

The Rayleigh stability criterion for linear, incompressible and inviscid flow, of which the Charney—Stern theorem is a generalization, is reviewed together with Fjortoft’s theorem. The linear equations governing small—amplitude adiabatic disturbances superimposed on a baroclinic zonal current in a beta—plane channel are expressed in isentropic coordinates. Using an integral constraint technique introduced by G. I. Taylor, and introducing geostrophy by assuming smallness of the particle frequency, it is shown that the Charney—Stern theorem follows without making any further assumptions.

The Role of Gravity Wave Induced Drag and Diffusion in the Momentum Budget of the Mesosphere

A slight modification of the parameterization suggested by Lindzen (1981) for the zonal drag and eddy diffusion effects generated by breaking internal gravity waves in the mesosphere is tested using a severely truncated midlatitude beta-plane channel model. It is found that realistic mean zonal flow profiles with zonal wind reversals above the mesopause can be simulated for both winter and summer radiative heating conditions provided that a gravity-wave spectrum is assumed which includes both stationary waves and waves of relatively large phase speeds. These results contrast greatly with the unrealistic mean wind profiles produced when Rayleigh friction is used to parameterize the effects of small scale motions on the mean flow.

Eigenvalues of a Baroclinic Stability Problem with Ekman Damping

An analytical solution is presented for the baroclinic stability problem of a Boussinesq fluid in a beta-plane channel with Ekman suction boundary conditions. All of the modes, stable and unstable, belonging to this problem are identified. It is found that an unstable mode exists for only a certain range of values of the Burger number. The value of the Burger number at the upper limit of this range increases as the Ekman number decreases. Beyond this upper limit only a damped mode exists. It is also found that this transition in parameter space from the unstable to the stable mode occurs in a discontinuous manner.

Theoretical Regime Diagrams for Thermally Driven Flows in a Beta-Plane Channel

It is noted that thermally driven flows in rotating laboratory containers with cylindrical geometry can be axially symmetric or wavelike depending on the experimental parameters. In anticipation that rotating fluid experiments might soon be done in spherical shell geometry, Barcilon's model has been extended to a beta-plane channel in order to gain a rough understanding of the effects of rotating spherical geometry. An incompressible fluid version of the Charney (1947) model of baroclinic instability, modified to include Ekman pumping at rigid horizontal boundaries is used. With this model, stability boundaries are mapped out for individual zonal wavenumbers in the parameter space used by Barcilon.

PROPAGATION OF SYSTEMATIC ERRORS IN A ONE LAYER PRIMITIVE-EQUATION MODEL FOR SYNOPTIC SCALE MOTION

Abstract A one-layer, mid-latitude, beta-plane channel model of an incompressible homogeneous fluid is constructed to study the propagation of systematic errors on a nearly stationary synoptic scale wave. A time- and space-centered difference scheme is used to evaluate the governing primitive equations. Data fields resulting from height field perturbations injected at various locations in the synoptic wave are compared to the unperturbed synoptic wave at 3-hr intervals for 5 model days. Results show that the low-frequency or quasi-geostrophic component of the error tends to move toward the core of maximum velocity in the basic state and that, after 5 days, these maximum height errors are in the core regardless of the location of the initial perturbation.