Rossby wave resonance in the presence of a nonlinear critical layer

Abstract

The behavior of Rossby waves on a shear flow in the presence of a nonlinear critical layer is studied, with particular emphasis on the role played by the critical layer in a Rossby wave resonance mechanism. Previous steady analyses are extended to the resonant case and it is found that the forced wave dominates the solution, provided the flow configuration is not resonant for the higher harmonics induced by the critical layer. Numerical simulations for the forced initial value problem show that the solution evolves towards the analysed steady state when conditions are resonant for the forced wave, and demonstrate some of the complications that arise when they are resonant for higher harmonics. In relating the initial value and steady problems, it is argued that the time dependent solution does not require the large mean flow distortion that Haberman (1972) found to be necessary outside the critical layer in the steady case.     

The nonlinear stabilization of a zonal shear flow instability

Abstract

The development of initially small perturbations in a weakly supercritical zonal shear flow on a β-plane is studied. Two different scenarios of evolution are possible. If the supercriticality is sufficiently small, the growth of a perturbation is stopped in the viscous critical layer regime; for this case the evolution equation (corrected by the inclusion of a quintic nonlinearity) is derived. At greater supercriticality the nonlinearity cannot stop the growth of the perturbation in a linear (viscous or unsteady) critical layer regime, and the evolution is more complicated. Transition to a nonlinear critical layer regime leads to a reduction in the growth rate and to a slowing (but not a stopping) of the increase in amplitude, A. These are connected to the formation of a plateau (S=constant) of width L=O(A ^½) in the profile of absolute vorticity, S. Careful analysis reveals that the growth in amplitude ceases only when the whole instability domain (where the slope of unperturbed S-profile is positive) becomes covered again by the plateau.     

Nonlinear modal analysis of penetrative convection

Abstract

The modal expansion procedure has been used to analyze penetrative convection that arises when a thin unstable layer is embedded between two stable regions. The Boussinesq approximation is applied in which the effect of compressibility and stratification are neglected. Various calculations have been made, with one and two modes, for Rayleigh numbers ranging from the critical value to more than 10⁵ times critical. The effect of decreasing the Prandtl number has also been investigated.

It is found that in the nonlinear regime, the convective motions penetrate substantially into the stable regions. The flux of kinetic energy plays a crucial role in such penetration, and its existence puts some requirements on the motions: in the single-mode case, they need to be three-dimensional. The extent of penetration amounts to about half of the thickness of the unstable layer on each side of it when the degree of instability and that of stability are comparable in the two domains; it increases as the stability of the outer region is lowered. The penetration depth appears to be independent of all other parameters defining the problem.     

A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part II: Application to geophysical vortices

In a previous paper, Caillol [Geophys. Astrophys. Fluid Dyn., 2014, 108] investigated the steady nonlinear vortical structure of a singular vortex Rossby mode that has survived to a strong critical-layer-like interaction with a linearly stable, columnar, axisymmetric and dry vortex. We presented a general theory for this wave/mean flow interaction through the nonlinear critical layer theory and calculated the mean azimuthal and axial winds induced at the critical radius at the end of this interaction in the final stage. We here apply that theory to rapidly rotating geophysical vortices: tropical cyclones, cold-air mesocyclones and tornadoes. We find that the numerous assumptions invoked in that paper agree well with the reality of those intense vortices. We also find that in spite of a lack of moist-convection modelling, this dry vortex is fairly well accelerated at the critical radius by such a shear wave with a magnitude of order the square root of the damped-wave amplitude. The intensification level strongly depends on the aspect ratio, height of the system: rapid vortex and parent vortex, over core radius. The thinner the vortex is, the sharper the intensification is. This result is in sharp contrast to the numerous numerical simulations on VR wave/vortex interactions that yield a much smaller intensification of order the square of the wave amplitude. This weakly nonlinear approach nevertheless fails to model small vertical wavelength VR wave/vortex interactions for their related asymptotic expansions are divergent and for they yield strongly nonlinear VR waves coupled with evolving critical layers whose extent can no longer be considered as thin.     

Simulation of moist mountain waves with an anelastic model

Abstract

A two-dimensional, nonlinear, time-dependent, non-hydrostatic, anelastic, numerical model is used to assess the effect of condensation on the evolution and structure of gravity waves generated by the passage of a stable, moist stream over topography. Precipation is ignored but water phase changes are taken into account explicitly.

The main effect of condensation is to damp the wave intensity and to reduce the wave drag, which can be diminished by as much as 50% compared to its value in dry simulations. This result agrees with some earlier analytical models and some more recent fully compressible numerical models.

This model also confirms that the presence of condensation delays the overturning of isentropes, and the formation of the critical layer that accompanies wave-breaking.     

A model of thermal convection in the major planets with a strongly density-stratified atmospheric layer

Abstract

As an extension of a model by Busse (1983a), a two-layer model of thermal convection in the self-gravitating rotating spherical fluid is considered. The upper layer with arbitrary vertical distributions of density and potential temperature representing the atmospheric layer of major planets is imposed on the spherical Boussinesq fluid. The Prandtl number P and the ratio of the mass of the upper layer to that of the lower layer are used as small expansion parameters. The modification of the critical Rayleigh number by imposing the upper layer are clearly separated into two parts, proportional to (1) the mass of the upper layer and to (2) an integral representing a measure of convective instability of the upper layer. Some implications for atmospheric dynamics of the major planets are also presented.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

The instability in relativistic,plane parallel,compressible shear flows

Abstract

The growth rates of instabilities in these shear flows are bounded by the rate of shear and the relativistic y factor. It is also shown that (i) the Howard Semi-Circle theorem is still valid, and (ii) the critical layer lies in the flow. Beaming effects are taken into account.     

Finite-amplitude mountain waves in a compressible atmosphere including the effects of dissipation

Abstract

Adiabatic, two-dimensional, steady-state finite-amplitude, hydrostatic gravity waves produced by flow over a ridge are considered. Nonlinear self advection steepens the wave until the streamlines attain a vertical slope at a critical height z_c. The height z_c , where this occurs, depends on the ridge crest height and adiabatic expansion of the atmosphere. Dissipation is introduced in order to balance nonlinear self advection, and to maintain a marginal state above z_c. The approach is to assume that the wave is inviscid except in a thin layer, small compared to a vertical wavelength, where dissipation cannot be neglected. The solutions in each region are matched to obtain a continuous solution for the streamline displacement δ. Solutions are presented for different values of the nondimensional dissipation parameter β. Eddy viscosity coefficients and the thickness of the dissipative layer are expressed as functions of β, and their magnitudes are compared to other theoretical evaluations and to values inferred from radar measurements of the stratosphere.

The Fourier spectrum of the solution for z ≫ z_c is shown to decay exponentially at large vertical wave numbers n. In comparison, a spectral decay law n ^?-8/3 characterizes the marginal state of the wave at z = z_c .     

An analytical approach to capture zone delineation for a well near a stream with a leaky layer

Abstract

An analytical solution is developed to delineate the capture zone of a pumping well in an aquifer with a regional flow perpendicular to a stream, assuming a leaky layer between the stream and the aquifer. Three different scenarios are considered for different pumping rates. At low pumping rates, the capture zone boundary will be completely contained in the aquifer. At medium pumping rates, the tip of the capture zone boundary will intrude into the leaky layer. Under these two scenarios, all the pumped water is supplied from the regional groundwater flow in the aquifer. At high pumping rates, however, the capture zone boundary intersects the stream and pumped water is supplied from both the aquifer and the stream. The two critical pumping rates which separate these three scenarios, as well as the proportion of pumped water from the stream and the aquifer, are determined for different hydraulic settings.

Editor D. Koutsoyiannis; Associate editor A. Koussis

Citation Asadi-Aghbolaghi, M., Rakhshandehroo, G.R., and Kompani-Zare, M., 2013. An analytical approach to capture zone delineation for a well near a stream with a leaky layer. Hydrological Sciences Journal, 58 (8), 1813–1823.     

On thermonuclear convection: I shellular instability

Abstract

As the sun evolves, a sharp compositional peak of He ³ builds up in the core. Nuclear reactions involving He ³ are very temperature sensitive, as a result, this He ³ layer is susceptible to thermal instability. The small horizontal wavenumber g-modes have large time scales, comparable to the thermal time scale. Using a two-layer model, we find that such “shellular modes” are the most unstable. As a result of nuclear heating, these modes may be excited in the solar core in a shallow layer confined to the He ³ zone. A possible effect of such shellular convection on the solar neutrino problem is discussed. In this paper we discuss the linear theory; the nonlinear effects will be treated in a subsequent paper.     

Internal gravity waves in a slowly varying,dissipative medium

Nonlinear internal gravity waves in a slightly dissipative, slightly compressible fluid are discussed for the case when the properties of the medium vary slowly on a scale determined by the local wave structure. A two‐timing technique is used to obtain transport equations which describe the changes in amplitude, phase and mean flow of a wave packet. Various solutions of these transport equations are discussed, with relevance to critical layer absorption.     

The effect of stratification and compressibility on anelastic convection in a rotating plane layer

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation (τ^?1/12???|θ|???1, where τ is the Taylor number and θ is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999, 11, 1469–1483] to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio γ is larger than two.     

On the frontal condition for finite amplitude α2Ω-dynamo wave trains in stellar shells

Abstract

In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443–1481, 1999) (BKS) investigated finite amplitude αΩ-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from α-quenching. There asymptotic solutions were developed based upon the small aspect ratio ε of the shell. Specifically, as a consequence of a prescribed latitudinally dependent α-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude θ_F. Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point θ_f; this is essentially the Dee–Langer criterion, which determines both the wave frequency and front location.

Recently, Griffiths et al. (Geophys. Astrophys. Fluid Dynam. 94, 85–133, 2001) (GBSK) obtained solutions to the α²Ω-extension of the model by application of the Dee—Langer criterion. Its justification depends on the linear solution in a narrow layer ahead of the front on the short O(θ_f—θ_F) length scale; here conventional WKB-theory, used to describe the solution elsewhere, is inadequate because of mode coalescence. This becomes a highly sensitive issue, when considering the transition from the linear solution, which occurs when the dynamo number D takes its critical value D _c corresponding to the onset of kinematic dynamo action, to the fully nonlinear solutions, for which the Dee—Langer criterion pertains.

In this paper we investigate the nature of the narrow layer for α²Ω-dynamos in the limit of relatively small but finite α-effect Reynolds numbers R _α, explicitly ε^½ ? R ² _α ? 1. Though there is a multiplicity of solutions, our results show that the space occupied by the corresponding wave train is generally maximised by a solution with θ_f—θ_F small; such solutions are preferred as evinced by numerical simulations. This feature justifies the application by GBSK of the Dee—Langer criterion for all D down to the minimum D _min that the condition admits. Significantly, the frontal solutions are subcritical in the sense that |D _min| ≤ |D _c|; equality occurs as the α-effect Reynolds number tends to zero. We demonstrate that the critical linear solution is not connected by any parameter track to the preferred nonlinear solution associated with D _min. By implication, a complicated bifurcation sequence is required to make the connection between the linear and nonlinear states. This feature is in stark contrast to the corresponding results for αΩ-dynamos obtained by BKS valid in the limit R _{2 α} ? ε^½, which, though exhibiting a weak subcriticality, showed that the connection follows a clearly identifiable nonbifurcating track.     

Linear Stability of Thermal Convection in Rotating Systems with Fixed Heat Flux Boundaries

Linear stability of rotating thermal convection in a horizontal layer of Boussinesq fluid under the fixed heat flux boundary condition is examined by the use of a vertically truncated system up to wavenumber one. When the rotation axis is in the vertical direction, the asymptotic behavior of the critical convection for large rotation rates is almost the same as that under the fixed temperature boundary condition. However, when the rotation axis is horizontal and the lateral boundaries are inclined, the mode with zero horizontal wavenumber remains as the critical mode regardless of the rotation rate. The neutral curve has another local minimum at a nonzero horizontal wavenumber, whose asymptotic behavior coincides with the critical mode under the fixed temperature condition. The difference of the critical horizontal wavenumber between those two geometries is qualitatively understood by the difference of wave characteristics; inertial waves and Rossby waves, respectively.     

Effects of dissipation on internal waves in a contained rotating stratified fluid

Abstract

Boundary layer techniques are used to examine the modifications due to dissipation in the normal modes of a uniformly rotating, density stratified, Boussinesq fluid in a rigid container. Arbitrary relative influence of rotation and stratification is considered. The existence of critical regions of the container boundary is discussed. In cylindrical geometry a formula is derived for the decay factor on the homogeneous “spin-up” time scale which reveals how the dominant dissipation varies as a function of several parameters. For the situation where the buoyancy and inertial frequency are exactly equal, all boundaries are everywhere critical. In this case the method of multiple time-scales is employed to investigate the confluence inertial-gravity mode which is shown to persist until the diffusive time-scale is achieved.     

Instability of a geostrophic front and its energetics

Abstract

The instability of a current with a geostrophic surface density front is investigated by means of a reduced gravity model having a velocity profile with nearly uniform potential vorticity. It is shown that currents are unstable when the mean potential vorticity decreases toward the surface front at the critical point of the frontal trapped waves investigated by Paldor (1983). This instability is identical with that demonstrated by Killworth (1983) in the longwave limit.

The cross-stream component of mass flux and the rates of energy conversions among the five energy forms defined by Orlanski (1968) are also calculated. The main results are as follows, (a) The mass flux toward the surface front is positive near the front and negative around the critical point. The positive mass flux near the front does not vanish at the position of the undisturbed surface front, so that the mean position of the front moves outward and the region of the strong current spreads. (b) The potential energy of the mean flow integrated over the fluid is released through the work done by the force of the pressure gradient of the mean flow on the fluid, and is converted into the kinetic energy of the mean flow. (c) In the critical layer, the mean flow is rapidly accelerated with the growth of the unstable wave. This acceleration is caused by the rapid phase shift of the unstable wave in the critical layer.     

Dynamo in Asymmetric Square Convection

Linear and nonlinear dynamo action is investigated for square patterns in nonrotating and weakly rotating Boussinesq Rayleigh-Bénard convection in a plane horizontal layer. The square-pattern solutions may or may not be symmetric to up-down reflections. Vertically symmetric solutions correspond to checkerboard patterns. They do not possess a net kinetic helicity and are found to be incapable of kinematic dynamo action at least up to magnetic Reynolds numbers of , 12 000. There also exist vertically asymmetric squares, characterized by rising (descending) motion in the centers and descending (rising) motion near the boundaries, among them such that possess full horizontal square symmetry and others lacking also this symmetry. The flows lacking both the vertical and horizontal symmetries possess kinetic helicity and show kinematic dynamo action even without rotation. The generated magnetic fields are concentrated in vertically oriented filamentary structures. Without rotation these dynamos are, however, always only kinematic, not nonlinear dynamos since the back-reaction of the magnetic field then forces the solution into the basin of attraction of a roll pattern incapable of dynamo action. But with rotation added parameter regions are found where stationary asymmetric squares are also nonlinear dynamos. These nonlinear dynamos are characterized by a subtle balance between the Coriolis and Lorentz forces. In some parameter regions also nonlinear dynamos with flows in the form of oscillating squares or stationary modulated rolls are found.     

Inertial effects in an oceanic circulation model

Abstract

The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime.

An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive.

To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is found by examining the dynamics controlling this additional transport from the western boundary layer as the transport recirculates through the rest of the ocean basin. The strong constraint of total recirculation within the western boundary layer (zero net inertial transport) is derived.

The calculated primary inertial effects are in agreement with the observations of the laboratory model of Baker and Robinson (1969).

The analysis indicates the extent to which three-dimensional non-linear circulation can be reduced to a two dimensional problem.