Non-modal Analysis Research Articles

Non-modal analysis of the diocotron instability: Plane geometry

The comprehensive investigation of the temporal evolution of the diocotron instability of the plane electron strip on the linear stage of its development is performed. By using the method of Kelvin of the shearing modes, the role of the initial perturbations of the electron density is elucidated, which is connected with the problem of the continuous spectrum. The linear non-modal evolution process detected by the solution of the initial value problem, leads toward convergence to the phase-locking configuration of the mutually growing normal modes.

Non-Modal Stability and Optimal Perturbations in Unbounded Granular Shear Flow: Three-Dimensionality and Particle Spin

The effects of three-dimensional perturbations and particle rotations are analyzed on the non-modal stability characteristics of an unbounded granular shear flow for which the stability problem is solved as an initial value problem. A kinetic-theory constitutive model is used that incorporates the spin degrees of freedom along with certain micro-polar effects of granular materials. The singular values of the underlying non-normal linear operator play a central role in the non-modal analysis in contrast to the standard modal analysis where the eigenvalues determine the asymptotic (in)stability of the flow. For linearly stable flows, it is shown that the perturbation energy can be amplified by a few orders of magnitude at short times before decaying in the asymptotic time limit. Optimal perturbations, that correspond to maximum energy growth over all possible initial conditions, are found to be two-dimensional in a smooth granular fluid. The effect of particle rotation has been assessed by varying the tangential restitution coefficient (β) for smooth particles (β = −1) to perfectly rough particles (β = 1), with significant enhancement of maximum energy for rough particles. Since the non-modal mechanism can significantly amplify perturbation energy, this provides a viable alternate route for pattern formation in a sheared granular fluid.

Instabilities in superfluid plane Poiseuille flow

Modal and non-modal analysis of superfluid He-II is carried out in this paper. Landau's two fluid model for superfluid flow is assumed and stability of both normal fluid and superfluid are considered separately by taking the effect of the other through mutual friction. Stability of normal fluid, with a numerically calculated base flow, shows Tollmien-Schlichting mode of instability similar to classical fluids. Non-modal analysis shows the transient growth of perturbations at sub-critical Reynolds numbers. Mutual friction is found to have a stabilizing effect on normal fluid. Superfluid, with an inviscid uniform profile is found to be linearly stable for all wave numbers. Non-modal analysis rules out the possibility of transient growth of perturbations and sub-critical transition. The presence of turbulence in superfluid brings in a possibility that an initial uniform base flow is getting altered due to the action of mutual friction and this altered profile is likely to become unstable.

Self-localization of planetary wave structures in the ionosphere upon interaction with nonuniform geomagnetic field and zonal wind

The generation and further linear and nonlinear dynamics of planetary magnetized Rossby waves (MRWs) in the rotating dissipative ionosphere are studied in the presence of a zonal wind (shear flow). MRWs are caused by interaction with the spatially nonuniform geomagnetic field and are ionospheric manifestations of ordinary tropospheric Rossby waves. A simplified self-consistent set of model equations describing MRW-shear flow interaction is derived on the basis of complete equations of ionospheric magnetohydrodynamics. Based on an analysis of an exact analytical solution to the derived dynamic equations, an effective linear mechanism of MRW amplification in the interaction with nonuniform zonal wind is ascertained. It is shown that operators of linear problems are non-self-adjoint in the case of shear flows, and the corresponding eigenfunctions are nonorthogonal; therefore, the canonically modal approach is of little use when studying such flows; a so-called nonmodal mathematical analysis is required. It is ascertained that MRWs effectively get shear flow energy during the linear stage of evolution and significantly increase (by several orders of magnitude) their energy and amplitude. The necessary and sufficient condition of shear flow instability in an ionospheric medium is derived. Nonlinear self-localization begins with the development of shear instability and an increase in the amplitude, and the process ends with the self-organization of strongly localized isolated large-scale nonlinear vortex structures. Thus, a new degree of freedom and a way for perturbation evolution to occur appear in medium with shear flow. The nonlinear systems can be a pure monopole vortex, a vortex streets, or vortex chains depending of the shape of the sheared flow velocity profile. The accumulation of such vortices in the ionospheric medium can produce a strongly turbulent state.

Modal and non-modal stability of particle-laden channel flow

Modal and non-modal linear stability analysis of channel flow with a dilute particle suspension is presented where particles are assumed to be solid, spherical, and heavy. The two-way coupling between particle and fluid flow is therefore modeled by the Stokes drag only. The results are presented as function of the particle relaxation time and mass fraction. First, we consider exponentially growing perturbations and extend previous findings showing the potential for a significant increase of the critical Reynolds number. The largest stabilization is observed when the ratio between the particle relaxation time and the oscillation period of the wave is of order one. By examining the energy budget, we show that this stabilization is due to the increase of the dissipation caused by the Stokes drag. The observed stabilization has led to the hypothesis that dusty flows can be more stable. However, transition to turbulence is most often subcritical in canonical shear flows where non-modal growth mechanisms are responsible for the initial growth of external disturbances. The non-modal analysis of the particle-laden flow, presented here for the first time, reveals that the transient energy growth is, surprisingly, increased by the presence of particles, in proportion to the particle mass fraction. The generation of streamwise streaks via the lift-up mechanism is still the dominant disturbance-growth mechanism in the particle laden flow; the length scales of the most dangerous disturbances are unaffected, while the initial disturbance growth can be delayed. These results are explained in terms of a dimensionless parameter relating the particle relaxation time to the time scale of the instability. The presence of a dilute solid phase therefore may not always work as a flow-control strategy for maintaining the flow as laminar. Despite the stabilizing effect on modal instabilities, non-modal mechanisms are still strong in internal flows seeded with heavy particles. Our results indicate that the initial stages of transition in dilute suspensions of small particles are similar to the stages in a single phase flow.

Self-organization of large-scale ULF electromagnetic wave structures in their interaction with nonuniform zonal winds in the ionospheric E region

A study is made of the generation and subsequent linear and nonlinear evolution of ultralow-frequency planetary electromagnetic waves in the E region of a dissipative ionosphere in the presence of a nonuniform zonal wind (a sheared flow). Hall currents flowing in the E region and such permanent global factors as the spatial nonuniformity of the geomagnetic field and of the normal component of the Earth’s angular velocity give rise to fast and slow planetary-scale electromagnetic waves. The efficiency of the linear amplification of planetary electromagnetic waves in their interaction with a nonuniform zonal wind is analyzed. When there are sheared flows, the operators of linear problems are non-self-conjugate and the corresponding eigenfunctions are nonorthogonal, so the canonical modal approach is poorly suited for studying such motions and it is necessary to utilize the so-called nonmodal mathematical analysis. It is shown that, in the linear evolutionary stage, planetary electromagnetic waves efficiently extract energy from the sheared flow, thereby substantially increasing their amplitude and, accordingly, energy. The criterion for instability of a sheared flow in an ionospheric medium is derived. As the shear instability develops and the perturbation amplitude grows, a nonlinear self-localization mechanism comes into play and the process ends with the self-organization of nonlinear, highly localized, solitary vortex structures. The system thus acquires a new degree of freedom, thereby providing a new way for the perturbation to evolve in a medium with a sheared flow. Depending on the shape of the sheared flow velocity profile, nonlinear structures can be either purely monopole vortices or vortex streets against the background of the zonal wind. The accumulation of such vortices can lead to a strongly turbulent state in an ionospheric medium.

Global Linear Instability

This article reviews linear instability analysis of flows over or through complex two-dimensional (2D) and 3D geometries. In the three decades since it first appeared in the literature, global instability analysis, based on the solution of the multidimensional eigenvalue and/or initial value problem, is continuously broadening both in scope and in depth. To date it has dealt successfully with a wide range of applications arising in aerospace engineering, physiological flows, food processing, and nuclear-reactor safety. In recent years, nonmodal analysis has complemented the more traditional modal approach and increased knowledge of flow instability physics. Recent highlights delivered by the application of either modal or nonmodal global analysis are briefly discussed. A conscious effort is made to demystify both the tools currently utilized and the jargon employed to describe them, demonstrating the simplicity of the analysis. Hopefully this will provide new impulses for the creation of next-generation algorithms capable of coping with the main open research areas in which step-change progress can be expected by the application of the theory: instability analysis of fully inhomogeneous, 3D flows and control thereof.

Transient growth in stable linearized Vlasov–Maxwell plasmas

Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, Ωit⪢1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov–Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov–Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov–Maxwell equations, namely, the one dimensional Vlasov–Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2π/ωpe, thus demonstrating the existence of transient growth of solutions for the linearized Vlasov–Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov–Maxwell system as well as for Landau fluid models.

Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability

A systematic approach for non-modal stability analysis of thermoacoustic systems with a localized heat source is proposed. The response of the heat source to flow perturbations is obtained from unsteady computational fluid dynamics combined with correlation-based linear system identification. A model for the complete thermoacoustic system is formulated with a Galerkin expansion technique, where the heat source is included as an acoustically compact element. The eigenvalues of the resulting system are obtained from discretization of the solution operator, the maximum growth factor is estimated from the pseudospectra using Kreiss’ theorem.The approach is illustrated with a simple Rijke tube configuration. Results obtained with a simple “baseline” model for the heat source dynamics based on King’s law – widely used in hot wire anemometry – are compared against the more advanced treatment developed here. Analysis of pseudospectra diagrams shows that the choice of the heat source model does influence the sensitivity of eigenvalues to perturbations and hence the non-normal behavior. The maximum growth factor for the system with the heat source model based on King’s law is more sensitive to changes in the heat source location than the CFD-based heat source model.

Shear flow driven magnetized planetary wave structures in the ionosphere

The generation and further linear and nonlinear dynamics of planetary ultra-low-frequency (ULF) waves are investigated in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary ULF magnetized Rossby type waves appear as a result of interaction of the medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the intensification and mutual transformation of large scale magnetized Rossby type and small scale inertial waves is found. For shear flows, the operators of the linear problem are not self-conjugate, and therefore the eigenfunctions of the problem may not be orthogonal and can hardly be studied by the canonical modal approach. Hence, it becomes necessary to use the so-called nonmodal mathematical analysis. The nonmodal analysis shows that the transformation of wave disturbances in shear flows is due to the non-orthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Using numerical modeling, the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances are illustrated in the ionosphere. It has been shown that the shear flow driven wave perturbations effectively extract an energy of the shear flow increasing the own energy and amplitude. These perturbations undergo self-organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices or vortex streets and vortex chains.

Onset of convection over a transient base-state in anisotropic and layered porous media

The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initial-value problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapakaet al. (J. Fluid Mech., vol. 609, 2008, pp. 285–303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-Kinget al. (Phys. Fluids, vol. 17, 2005, paper no. 084107). For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.

Anomalous transport of particles in plasma flow with strong inhomogeneous velocity shear

The temporal evolution of the drift modes and resulting anomalous transport are considered under the conditions of strong inhomogeneous flow shear [flow shear parameter dv0(r)∕dr is greater or comparable to the drift frequency] on the ground of the nonmodal approach with application to boundary regions of tokamaks. The nonmodal linear analysis of the effect of flow shear nonuniformity on the temporal evolution of the drift modes, performed on the base of the Hasegava–Wakatani model, has shown, that terms reflecting velocity profile curvature decay more rapidly with time, as compared with those containing only velocity shearing rate. Therefore, the linear effect of the flow shear nonuniformity appears to be subdominant and the long-time evolution of the drift modes is determined by more slowly damped shear rate contained terms. The anomalous transport of particles in shear flow due to nonmodal drift perturbations exhibits a subdiffusive behavior with the diffusion coefficient reducing in time as t−3.

Linear stability, transient energy growth, and the role of viscosity stratification in compressible plane Couette flow

Linear stability and the nonmodal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the uniform shear flow with constant viscosity, and (b) the nonuniform shear flow with stratified viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers (M). For a given M , the critical Reynolds number (Re) is significantly smaller for the uniform shear flow than its nonuniform shear counterpart; for a given Re, the dominant instability (over all streamwise wave numbers, alpha ) of each mean flow belongs to different modes for a range of supersonic M . An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean flow to perturbations. It is shown that the energy transfer from mean flow occurs close to the moving top wall for "mode I" instability, whereas it occurs in the bulk of the flow domain for "mode II." For the nonmodal transient growth analysis, it is shown that the maximum temporal amplification of perturbation energy, G(max), and the corresponding time scale are significantly larger for the uniform shear case compared to those for its nonuniform counterpart. For alpha=0 , the linear stability operator can be partitioned into L ~ L+Re(2) L(p), and the Re-dependent operator L(p) is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: G(t/Re) ~ Re(2). In contrast, the dominance of L(p) is responsible for the invalidity of this scaling law in nonuniform shear flow. An inviscid reduced model, based on Ellingsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and nonmodal instability, it is shown that the viscosity stratification of the underlying mean flow would lead to a delayed transition in compressible Couette flow.

Hydrodynamic stability and mode coupling in Keplerian flows: local strato-rotational analysis

Aims. We present a qualitative analysis of key (but yet unappreciated) linear phenomena in stratified hydrodynamic Keplerian flows: (i) the occurrence of a vortex mode, as a consequence of strato-rotational balance, with its transient dynamics; (ii) the generation of spiral-density waves (also called inertia-gravity or gΩ waves) by the vortex mode through linear mode coupling in shear flows. Methods. Non-modal analysis of linearized Boussinesq equations were written in the shearing sheet approximation of accretion disk flows. Results. It is shown that the combined action of rotation and stratification introduces a new degree of freedom, vortex mode perturbation, which is in turn linearly coupled with the spiral-density waves. These two modes are jointly able to extract energy from the background flow, and they govern the disk dynamics in the small-scale range. The transient behavior of these modes is determined by the non-normality of the Keplerian shear flow. Tightly leading vortex mode perturbations undergo substantial transient growth, then, becoming trailing, inevitably generate trailing spiral-density waves by linear mode coupling. This course of events – transient growth plus coupling – is particularly pronounced for perturbation harmonics with comparable azimuthal and vertical scales, and it renders the energy dynamics similar to the 3D unbounded plane Couette flow case. Conclusions. Our investigation strongly suggests that the so-called bypass concept of turbulence, which has been recently developed by the hydrodynamic community for spectrally stable shear flows, can also be applied to Keplerian disks. This conjecture may be confirmed by appropriate numerical simulations that take the vertical stratification and consequent mode coupling into account in the high Reynolds number regime.

Global two-dimensional stability measures of the flat plate boundary-layer flow

The stability of the two-dimensional flat plate boundary-layer is studied by means of global eigenmodes. These eigenmodes depend both on the streamwise and wall-normal coordinate, hence there are no assumptions on the streamwise length scales of the disturbances. Expanding the perturbation velocity field in the basis of eigenmodes yields a reduced order model from which the stability characteristics of the flow, i.e. the initial condition and forcing function leading to the largest energy growth, are extracted by means of non-modal analysis. In this paper we show that, even when performing stability analysis using global eigenmodes, it is not sufficient to consider only a few of the least damped seemingly relevant eigenmodes. Instead it is the task of the optimization procedure, inherent in the non-modal analysis, to decide which eigenmodes are relevant. We show that both the optimal initial condition and the optimal forcing structure have the form of upstream tilted structures. Time integration reveals that these structures gain energy through the so called Orr mechanism, where the instabilities extract energy from the mean shear. This provides the optimal way of initiating Tollmien–Schlichting waves in the boundary layer. The optimal initial condition results in a localized Tollmien–Schlichting wavepacket that propagates downstream, whereas the optimal forcing results in a persistent Tollmien–Schlichting wave train.

Amplification and energy transformation of the magnetized Rossby waves in the ionosphere with an inhomogeneous zonal wind. I. A theory

The generation and further dynamics of the planetary magnetized Rossby waves and inertial waves in a dissipative ionosphere in the presence of a smooth inhomogeneous zonal wind (shear flow) have been studied. The magnetized Rossby waves are caused by the interaction with the spatially inhomogeneous geomagnetic field and represent the ionospheric manifestations of usual tropospheric Rossby waves. The effective linear mechanism of amplification and mutual transformation of the Rossby and inertial waves has been revealed. For shear flows, the operators of linear problems are not self-adjoint, and the corresponding eigenfunctions are non-orthogonal; therefore, a canonical modal approach is of little use in studying such motions. It becomes necessary to apply the so-called nonmodal mathematical analysis, which has actively been developed for the last years. The nonmodal approach makes it possible to reveal that the transformation of wave-like disturbances in shear flows is caused by the nonorthogonality of eigenfunctions in the problem of linear dynamics. Thus, there appear a new degree of the system freedom and a new way of disturbance evolution in the medium.

Mechanism of the planetary Rossby wave energy amplification and transformation in the ionosphere with an inhomogeneous zonal smooth shear wind

The generation and further dynamics of planetary magnetized Rossby waves and inertia waves are investigated in the rotating dissipative ionosphere in the presence of a smooth inhomogeneous zonal wind (shear flow). Magnetized Rossby waves appear as a result of the interaction of the medium with the spatially inhomogeneous geomagnetic field and are an ionospheric manifestation of usual tropospheric Rossby waves. An effective linear mechanism responsible for the intensification and mutual transformation of Rossby and inertia waves is found. For shear flows, the operators of the linear problem are not self‐adjoint, and therefore the eigenfunctions of the problem maybe nonorthogonal and can hardly be studied by the canonical modal approach. Hence it becomes necessary to use the so‐called nonmodal mathematical analysis which has been actively developed in recent years. The nonmodal approach shows that the transformation of wave disturbances in shear flows is due to the nonorthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Thus there arises a new degree of freedom and a new way for the evolution of disturbances in the medium. Using numerical modeling, we illustrate the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances in the D, E, and F regions of the ionosphere. It is established that the presence of a geomagnetic field, Hall and Pedersen currents in the ionospheric medium, improves the interaction and mutual energy exchange between waves and a shear flow.

Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Nonmodal transient growth studies and an estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a nonuniform shear rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, Gmax, is found to increase with increasing Reynolds number Re, but decreases with increasing Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of Gmax over both the streamwise and spanwise wave numbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wave number, αopt, is nonzero at M=0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G(t∕Re)∼Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with an increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (Ė1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of Ė1 for the present mean flow.

Linear Dynamics of Perturbations in Flows with Constant Horizontal Shear

The dynamics of perturbations in shallow water and incompressible stratified fluid flows with constant horizontal shear are described using the nonmodal analysis. It is shown that the shear flow perturbations can be divided into two classes on the basis of the potential vorticity: rapidly oscillating wave perturbations with zero potential vorticity and slow vortex perturbations with nonzero potential vorticity. In the cases of weak and strong shear the main features of the dynamics of wave and vortex perturbations are studied analytically (using the WKBG method) and numerically. It is shown that for large times the wave perturbation energy increases linearly, i.e., the shear flow is algebraically unstable due to the growth of rapid wave perturbations. This instability can be of importance in processes of turbulence development and surface and internal wave generation.

On the effect of the inhomogeneous subsurface flows on the high degree solarp-modes

The observed power spectrum of high-degree solar p-modes (l>200) shows discrepancies with the power spectrum predicted by the stochastic excitement and damping theory. In an attempt to explain these discrepancies, the present paper is concerned with the influence of the observed subsurface flows on the trapped acoustic modes (p-modes). The effect of these inhomogeneous background flows is investigated by means of a non-modal analysis and a multi-layer model. It is shown that the rotational and meridional components of the velocity field change the wavelengths of the oscillation modes which, in turn, results in modifications of the corresponding modal frequencies. The magnitudes of the frequency residuals depend on the spatial scales of the modes and on the gradients of the different components of the flow velocity. Together with other mechanisms (e.g. the scattering of modes by the large scale convection (Goldreich & Murray 1994), the non-modal effect of the variation of the frequencies in time may contribute: 1) to the observed widening of the corresponding peaks in the observed power spectrum with increasing angular degree; 2) to the partial dissipation of spectral power, and, as a result, 3) to the discrepancies between the predicted and the observed power spectrum of solar p-modes.