Steady Zonal Flow Research Articles

On the linear and non-linear fluid response to the circular forcing in a rotating spherical shell

Fluid flow excited by a core oscillating in a rotating spherical cavity is experimentally investigated. Oscillations are set by an external inertial field so that in the reference frame of the cavity, the core moves along a circular trajectory around the rotation axis. Two situations are considered: when the core oscillations are co-directed or counter-directed with respect to the rotation of the cavity. The oscillating core is a source of non-axisymmetric inertial waves that form a system of cone-shaped shear layers in fluid bulk. Depending on the oscillation frequency, various inertial flow regimes arise, the spatial structure of which strongly depends on the sign of the oscillations. It is found that a strong non-linear response in the form of a steady zonal flow corresponds to each flow regime. The flow structure is a system of nested liquid geostrophic cylinders, one of which is associated with the critical latitude at the core boundary, where inertial waves are generated. The next one is associated with the wave reflection from the cavity boundary and is clearly manifested when they are focused on the wave attractor. The most intense zonal flow occurs when inertial waves are superposed and global vortex structures are resonantly excited.

Inertial Waves and Steady Flows in a Liquid Filled Librating Cylinder

The fluid flow in a non-uniformly rotating (librating) cylinder about a horizontal axis is experimentally studied. In the absence of librations the fluid performs a solid-body rotation together with the cavity. Librations lead to the appearance of steady zonal flow in the whole cylinder and the intensive steady toroidal flows near the cavity corners. If the frequency of librations is twice lower than the mean rotation rate the inertial waves are excited. The oscillating motion associated with the propagation of inertial wave in the fluid bulk leads to the appearance of an additional steady flow in the Stokes boundary layers on the cavity side wall. In this case the heavy particles of the visualizer are assembled on the side wall into ring structures. The patterns are determined by the structure of steady flow, which in turn depends on the number of reflections of inertial wave beams from the cavity side wall. For some frequencies, inertial waves experience spatial resonance, resulting in inertial modes, which are eigenmodes of the cavity geometry. The resonance of the inertial modes modifies the steady flow structure close to the boundary layer that is manifested in the direct rebuilding of patterns. It is shown that the intensity of zonal flow, as well as the intensity of steady flows excited by inertial waves, is proportional to the square of the amplitude of librations.

Shallow‐water equations on a spherical multiple‐cell grid

The shallow‐water equations (SWEs) are discretized on a spherical multiple‐cell (SMC) grid and validated with classical tests, including steady zonal flow over flat or hill floor, the Rossby–Haurwitz wave and the unstable zonal jet. The numerical schemes follow the conventional latitude–longitude (lat‐lon) grid ones at low latitudes but switch to a fixed‐reference direction system to define vector variables at high latitudes for reduction of polar curvature errors. Semi‐implicit schemes are used for both the Coriolis terms and the potential energy gradients. A C‐grid mass‐conserving advection–diffusion scheme is used for the thickness variable and mixed A‐ and D‐grid schemes are applied on the momentum equation. Tests demonstrate that the two reference directions work fine on the SMC grid and the SWEs model is stable as long as numerical noises are suppressed with enough smoothing. This implies that other reduced grids could be used for dynamical models if the vector polar problem is properly solved. The unstructured SMC grid also supports multi‐resolutions in mesh refinement style and a refined area is included to show grid refinement effects. For steady smooth flows, the refinement is almost unnoticeable while in unstable jet case, it may stir up instability ripples in the jet flow unless strong average is applied.

Isotope Effects on Trapped-Electron-Mode Driven Turbulence and Zonal Flows in Helical and Tokamak Plasmas.

Impacts of isotope ion mass on trapped-electron-mode (TEM)-driven turbulence and zonal flows in magnetically confined fusion plasmas are investigated. Gyrokinetic simulations of TEM-driven turbulence in three-dimensional magnetic configuration of helical plasmas with hydrogen isotope ions and real-mass kinetic electrons are realized for the first time, and the linear and the nonlinear nature of the isotope and collisional effects on the turbulent transport and zonal-flow generation are clarified. It is newly found that combined effects of the collisional TEM stabilization by the isotope ions and the associated increase in the impacts of the steady zonal flows at the near-marginal linear stability lead to the significant transport reduction with the opposite ion mass dependence in comparison to the conventional gyro-Bohm scaling. The universal nature of the isotope effects on the TEM-driven turbulence and zonal flows is verified for a wide variety of toroidal plasmas, e.g., axisymmetric tokamak and non-axisymmetric helical or stellarator systems.

Generation of zonal flows through symmetry breaking of statistical homogeneity

In geophysical and plasma contexts, zonal flows (ZFs) are well known to arise out of turbulence. We elucidate the transition from homogeneous turbulence without ZFs to inhomogeneous turbulence with steady ZFs. Starting from the equation for barotropic flow on a β plane, we employ both the quasilinear approximation and a statistical average, which retains a great deal of the qualitative behavior of the full system. Within the resulting framework known as CE2, we extend recent understanding of the symmetry-breaking zonostrophic instability and show that it is an example of a Type instability within the pattern formation literature. The broken symmetry is statistical homogeneity. Near the bifurcation point, the slow dynamics of CE2 are governed by a well-known amplitude equation. The important features of this amplitude equation, and therefore of the CE2 system, are multiple. First, the ZF wavelength is not unique. In an idealized, infinite system, there is a continuous band of ZF wavelengths that allow a nonlinear equilibrium. Second, of these wavelengths, only those within a smaller subband are stable. Unstable wavelengths must evolve to reach a stable wavelength; this process manifests as merging jets. These behaviors are shown numerically to hold in the CE2 system. We also conclude that the stability of the equilibria near the bifurcation point, which is governed by the Eckhaus instability, is independent of the Rayleigh–Kuo criterion.

Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity

We study both experimentally and numerically the steady zonal flow generated by longitudinal librations of a spherical rotating container. This study follows the recent weakly nonlinear analysis of Busse (J. Fluid Mech., vol. 650, 2010, pp. 505–512), developed in the limit of small libration frequency–rotation rate ratio and large libration frequency–spin-up time product. Using particle image velocimetry measurements as well as results from axisymmetric numerical simulations, we confirm quantitatively the main features of Busse's analytical solution: the zonal flow takes the form of a retrograde solid-body rotation in the fluid interior, which does not depend on the libration frequency nor on the Ekman number, and which varies as the square of the amplitude of excitation. We also report the presence of an unpredicted prograde flow at the equator near the outer wall.

Mean zonal flows generated by librations of a rotating spherical cavity

Longitudinal librations represent oscillations about the axis of a rotating axisymmetric fluid-filled cavity. An analytical theory is developed for the case of a spherical cavity in the limit when the libration frequency is small in comparison with the rotation rate, but large in comparison with the inverse of the spin-up time. It is shown that longitudinal librations create a steady zonal flow through the nonlinear advection in the Ekman layers. The theory can be applied to laboratory experiments as well as to solid planets and satellites with a liquid core.

The Effect of Tilted Rotation on Shear Instabilities at Low Stratifications

Abstract A linear stability analysis of the inviscid stratified Boussinesq equations is presented given a steady zonal flow with constant vertical shear in a tilted f plane. Full nonhydrostatic terms are included: 1) acceleration of vertical velocity and 2) Coriolis force terms arising from the meridional component of earth’s rotation vector. Calculations of growth rates, critical wavenumbers, and dominance regimes for baroclinic and symmetric instabilities are compared with results from the traditional nonhydrostatic equations, which include a strictly vertical rotation vector, as well as results from the hydrostatic equations. The authors find that for positive zonal z shear, tilted rotation enhances the dominance regime of symmetric instabilities at the expense of baroclinic instabilities and maintains symmetric instabilities at larger scales than previously indicated. Furthermore, in contrast to former studies, it is determined that hydrostatic growth rates for both instabilities are not maximal. Rather, growth rates peak in the fully nonhydrostatic equations for parameter regimes physically relevant and consistent with abyssal ocean stratifications and weak zonal z shears and oceanic measurements of the Labrador Sea and Southern Ocean. In addition, the authors find that zonal shear modifies the frequency range of subinertial inertio–gravity waves. Tilted rotation effects break the base flow shear reflection symmetry present in the traditional and hydrostatic models. Thus, only in the fully nonhydrostatic model does weak negative zonal z shear stabilize the flow and decrease the subinertial frequency range.

A model of GAM intermittency near critical gradient in toroidal plasmas

We have constructed a four-field minimal model that describes the growing intermittency of turbulence associated with the geodesic acoustic mode (GAM) observed in our toroidal Landau-fluid simulations [K Miki et al. 2007 Phys. Rev. Lett. 99, 145003]. The intermittent dynamics are well reproduced by the model for the reference parameters used in the simulation. The model can also reproduce characteristics of turbulent transport associated with the GAM, such as a single burst leading to a full quench of turbulence and also a steady state turbulence mixed with steady zonal flows and GAMs. Investigating the behaviour of the solution trajectories around the fixed points in four-dimensional phase space, we study the comprehensive properties of the model and identify the bifurcation property between Dimits shift and steady state turbulence regimes, which correspond to different eigen-states.

On the Lagrangian Dynamics of Atmospheric Zonal Jets and the Permeability of the Stratospheric Polar Vortex

Abstract The Lagrangian dynamics of zonal jets in the atmosphere are considered, with particular attention paid to explaining why, under commonly encountered conditions, zonal jets serve as barriers to meridional transport. The velocity field is assumed to be two-dimensional and incompressible, and composed of a steady zonal flow with an isolated maximum (a zonal jet) on which two or more traveling Rossby waves are superimposed. The associated Lagrangian motion is studied with the aid of the Kolmogorov–Arnold–Moser (KAM) theory, including nontrivial extensions of well-known results. These extensions include applicability of the theory when the usual statements of nondegeneracy are violated, and applicability of the theory to multiply periodic systems, including the absence of Arnold diffusion in such systems. These results, together with numerical simulations based on a model system, provide an explanation of the mechanism by which zonal jets serve as barriers to the meridional transport of passive tracers under commonly encountered conditions. Causes for the breakdown of such a barrier are discussed. It is argued that a barrier of this type accounts for the sharp boundary of the Antarctic ozone hole at the perimeter of the stratospheric polar vortex in the austral spring.

Steady zonal flows in spherical shell dynamos

Convective dynamos in a rotating spherical shell feature steady zonal flows. This process is studied numerically for Prandtl numbers of 0.1 and 1, Ekman numbers in the range the rotation rate. As a consequence of the corotation law, this scaling relationship is remarkably independent of the magnetic field amplitude. It does not depend on thermal, kinematic and magnetic diffusivities, owing to the large-scale and steady nature of forcing and dissipative processes. The scaling law is in agreement with the zonal-flow amplitude at the external boundary of the Earth's liquid core.

Normal-Mode Global Rossby Waves: Theory and Observations

Abstract This study addresses first the question of what normal-mode global Rossby waves might exist in the Earth's atmosphere. Then it identifies fourteen of these theoretically predicted waves in the NMC global tropospheric analyses. Normal modes of linearized global primitive shallow water equations were found given a basic state of latitudinally dependent steady zonal flow. The solutions are free Rossby and gravity waves. Many of the waves' north-south structures are similar to Hough functions, which are the solutions of the simpler problem of free waves in an atmosphere at rest. By projecting 1200 consecutive days of twice-daily NMC global tropospheric analyses of velocity and geopotential onto idealized three-dimensional, normal-mode Rossby wave structures, time series of wave amplitudes and phases were formed. Spectral analyses of these time series for zonal wavenumbers 1–4 revealed statistically significant peaks at eight out of 25 theoretical Rossby wave frequencies. Six additional woes may exist...

Baroclinically Unstable Modes of a Two-Layer Model on the Sphere

Abstract A model consisting of two layers of inviscid, constant-density fluid in stable stratification on a rotating gravitating sphere is linearized about a basic state consisting of a steady zonal flow with arbitrary latitudinal variation in the upper layer, balanced geostrophically by mean free surface and interfacial slopes. Normal modes solutions are assumed and a Galerkin procedure employed to obtain approximate solutions to the resulting eigenproblem for various basic state wind profiles. Baroclinically unstable modes for the basic states of constant angular rotation in the upper layer and of jet-type wind profiles are compared with the results of mid-latitude β-plane studies. In general, it is found that the spherical model produces modes with growth rates in reasonable agreement to those predicted by β-plane theory, but for the parameter ranges consistent with the model their growth rates are somewhat smaller than those suitable for typical mid-latitude storms.

PLANETARY WAVES IN THE ATMOSPHERE

Abstract A numerical method is presented for solving the non-linear barotropic vorticity equation in spherical coordinates. It is shown that, when the stream function is expressed as a sum of surface spherical harmonics, the barotropic vorticity equation gives rise to harmonic tendency equations which express the time rate of change of the harmonic coefficients as functions of all the harmonic coefficients. The harmonic tendency equations may be used in an iterative process to find the flow pattern at a future time from a knowledge of the harmonic coefficients at an initial time. The amount of computation is reduced greatly if the planetary waves are regarded as composed of perturbations superimposed on a steady zonal flow in which the angular velocity varies with colatitude. An example based on an actual synoptic situation is given. The effect of large scale lateral mixing is to curb the development of harmonics of large degree.