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Hydromagnetic Reflection and Refraction at a Fluid Velocity Discontinuity
Abstract
The problem of hydromagnetic reflection and refraction at a fluid velocity discontinuity is solved. Expressions for the propagation vector of the refracted wave and for the reflection coefficient are derived. It is found that the reflected and refracted waves are pure characteristic waves if the incident wave is a pure characteristic wave.
... Miles (1957) [28] and Ribner (1957) [32] were the first who studied the overreflexion problem related to the transmission and reflexion of a sound waves at a vortex sheet separating by two regions of constant horizontal velocity U 1 and U 2 . This problem was extended by Fejer (1963) [14] to include the hydromagnetic effect. Further, McKenzie (1972) [26] included the effects due to buoyancy. ...
... Miles (1957) [28] and Ribner (1957) [32] were the first who studied the overreflexion problem related to the transmission and reflexion of a sound waves at a vortex sheet separating by two regions of constant horizontal velocity U 1 and U 2 . This problem was extended by Fejer (1963) [14] to include the hydromagnetic effect. Further, McKenzie (1972) [26] included the effects due to buoyancy. ...
In this research, we investigate the stability of a discontinuity interface in the shallow-water flow. We will focus on the effect of gravity waves on an interface between fluid regions which are moving parallel with different velocities. For the both problems of zero thickness layer (no shear layer) and non-zero thickness layer (shear layer) will be considered. The results of other researchers are also revisited to give an explanation for the effect of gravity waves clearly. The different gravity waves on the two sides of interface act on the stability, the critical value of Froude number M = U/c is a function of depth ratio r = H₁/ H₂. The critical value increases with r >1 while deceases with r < 1. We find that the minimum of the critical Froude number √8 occurs at r = 1 with the critical value √8 when both sides of interface have a same gravity wave as considered by Bezdenkov and Pogutse.
Without frictional bottom, the interface of tangential-velocity discontinuity in the shallow-water flow is stable if the Froude number is greater than the critical value. However, the bottom friction and the internal lateral friction both play significant roles in the linear stability of a two-dimensional shallow- water flow. Thereafter, we provide an example of the dissipation-induced instabilities that are ubiquitous in nature. The analysis is based on the Boussinesq shallow-water equations, this category of instability is usually considered for a small amount of dissipation. The instability persists in the regime of strong dissipation. We have obtained an unusual result that the instability mode is excited even for a large amount of dissipation; the discontinuity interface is linearly unstable over the entire range of drag coefficient as opposed to other models. In a closely related problem of a shear flow, only the effect of a small drag force was addressed.
Two above problems, the instability models are made by two infinite regions of fluid moving parallel with different uniform velocities. The interface is stable for large Froude number in the case of no frictional bottom and unstable for over the entire range of drag coefficient. Next, our analysis is made for the stability of the interface of non-zero thickness layer which the shear layer sandwiched between two infinite layers moving parallel with different velocities. The dispersion relation is found to involve the Whittaker functions and their first derivatives. The appropriate limits of these functions correspond to the various physical conditions of problem. We find that the linear-shear flow changes the stability property of the interface in zero thickness model and leads to the interface unstable for entire value of Froude number.
... The physical reason that Alfvén waves are not transmitted across a TD is that an Alfvén wave cannot propagate normally to the magnetic field; consequently, one cannot "adjust" the k x wavenumber to compensate for the different phase velocities on either side of a TD. Another way of expressing this is that the group velocity of Alfvén waves is along the ambient magnetic field, and so the wave always travels parallel to the HP (Fejer 1963). This difficulty is circumvented at a shock wave because the wave frequency is Doppler-shifted on crossing the shock. ...
... In the case of gas dynamics, this was described by Landau & Lifshitz (1987) for the example of r r = 1 2 on either side of the TD. The problem has been investigated in some detail in the context of the magnetopause, treating the magnetopause as a vortex sheet (Fejer 1963;McKenzie 1970;Verzariu 1973;Wolfe & Kaufmann 1975;Kwok & Lee 1984). The more specialized problem of wave transmission across a TD with no transverse flow on either side appears to have attracted less attention despite its reduced complexity. ...
Voyager 1 observed compressible magnetic turbulence in the very local interstellar medium (VLISM). We show that inner heliosheath (IHS) fast- and slow-mode waves incident on the heliopause (HP) generate VLISM fast-mode waves only that propagate into the VLISM. We suggest that this is the origin of compressible turbulence in the VLISM. We show that fast- and slow-mode waves transmitted across a tangential discontinuity such as the HP are strongly refracted on crossing the HP and subsequently propagate at highly oblique angles to the VLISM magnetic field. Thus, fast-mode waves in the VLISM contribute primarily to the compressible and not the transverse components of the VLISM fluctuating magnetic field variance since and . If the fast- and slow-mode waves in the IHS exhibit a Kolmogorov-like power spectral density, as appears to be observed by Voyager 1, then the corresponding transmitted spectral density in the VLISM forms an amplified power law. Consequently, the HP "radiates" fast-mode fluctuations into the VLISM, and the heliosphere therefore mediates the character of turbulence in the VLISM. In particular, we predict the form of the VLISM magnetic turbulence power spectral density to be a superposition of the background pristine interstellar turbulence spectrum and the fast-mode spectrum generated by the interaction of fast- and slow-mode IHS waves with the HP, i.e., a power law with an enhanced feature or "bump" corresponding to the contribution by fast-mode turbulence radiated by the HP. We note the applicability of these results to the generation of compressible turbulence in the local interstellar medium surrounding the asterospheres of stars.
... An incident wave either gives rise to only one type of propagating refracted wave or otherwise is totally reflected. 20 The absolute value of the reflection coefficient may, under certain circumstances, exceed unity. The energetic aspects of over-reflection show how the excess reflected energy is extracted from the mean motion, and the transmitted wave may be viewed as a carrier of so-called "negative energy." ...
The stability of a flow in porous media relates to the velocity rate of injecting and withdrawing natural gases inside porous storage. We thus aim to analyze the stability of flows in porous media to accelerate the energy transition process. This research examines a flow model of a tangential-velocity discontinuity with porosity and viscosity changes in a three-dimensional (3D) compressible medium because of a co-existence of different gases in a storage. The fluids are assumed to move in a relative motion where the plane y=0 is a tangential-velocity discontinuity surface. We obtain that the critical value of the Mach number to stabilize a tangential discontinuity surface of flows via porous media is smaller than the one of flows in a plane. The critical value of the Mach number M to stabilize a discontinuity surface of the 3D flow is different by a factor |cos θ| compared to the two-dimensional (2D) flow. Here, θ is the angle between velocity and wavenumber vectors. Our results also show that the flow model with viscosity and porosity effects is stable faster than those without these terms. Our analysis is done for both infinite and finite flows. The effect of solid walls along the flow direction could suppress the instability, i.e., the tangential-discontinuity surface is stabilized faster.
... Strong shear can induce Kelvin-Helmholtz instability (KHI) (Chandrasekhar 1961;Zaqarashvili et al. 2015). In addition, the effect of overreflection may occur for MHD waves reflected from a plasma nonuniformity with a velocity shear (Fejer 1963;Sen 1963;McKenzie 1970;Nakaryakov & Stepanyants 1994;Gogichaishvili et al. 2014). In overreflection, the amplitude of the reflected wave is higher than the amplitude of the incident wave, i.e., the wave gains energy from the shear flow, or, more correctly, from the source that supports the shear flow. ...
Negative energy wave (NEW) phenomena may appear in shear flows in the presence of a wave decay mechanism and external energy supply. We study the appearance of negative energy surface waves in a plasma cylinder in the incompressible limit. The cylinder is surrounded by an axial magnetic field and by a plasma of different density. Considering flow inside and viscosity outside the flux tube, we derive dispersion relations and obtain analytical solutions for the phase speed and growth rate (increment) of the waves. It is found that the critical speed shear for the occurrence of the dissipative instability associated with NEWs and the threshold of Kelvin–Helmholtz instability (KHI) depend on the axial wavelength. The critical shear for the appearance of sausage NEW is lowest for the longest axial wavelengths, while for kink waves the minimum value of the critical shear is reached for the axial wavelength comparable to the diameter of the cylinder. The range between the critical speed of the dissipative instability and the KHI threshold is shown to depend on the difference of the Alfvén speeds inside and outside of the cylinder. For all axial wavenumbers, NEW appears for the shear flow speeds lower than the KHI threshold. It is easier to excite NEW in an underdense cylinder than in an overdense one. The negative energy surface waves can be effectively generated for an azimuthal number m = 0 with a large axial wavenumber and for higher modes ( m > 0) with a small axial wavenumber.
... Miles [6] and Ribner [7] were the first study of over-reflexion problem related to the transmission and reflexion of a sound waves at a vortex sheet separating by two regions of constant horizontal velocity U1 and U2. This problem was extended by Fejer [8] by including the effects of hydromagnetic, then McKenzie [9] included the effects due to buoyancy. The similar stability of tangential-velocity discontinuity in a shallow water in 2D was given by Bedenzkov and Pogutse [10] since a shallow water flow has an analogy with a compressible gas flow in 2D. ...
It is well known that for an incompressible flow in relative motion parallel to their interface, its interface is necessarily destabilized, regardless of the velocity difference's strength. This phenomenon is well so-called the Kelvin-Helmholtz instability (KHI). However, a large number of works demonstrated a surprising result that the instability is suppressed for shallow water flows; the interface is stabilized if the Froude number, defined by the velocity difference's ratio to the gravity wave's speed, is sufficiently large. In a limited way, these authors have been used the shallow-water equations without the higher-order effect of the dispersive terms. Thus, this investigation aims to examine these higher-order dispersive effects to analyze the interface stability problem of tangential-velocity discontinuity in shallow-water flows. In particular, we use the Green-Naghdi equations to introduce the dispersive terms related to the depth and the depth-averaged horizontal velocities of the fluid. We show that the interface stability depends on the Froude number (i.e., the velocity difference's strength) and the water depth. A critical value of the Froude number to stabilize the interface is smaller than the case of no dispersive terms, and the flow in a deeper region is more stable than in a shallower one. We also consider the distribution of kinetic and potential energy to clarify a feature characteristic of a large class of instabilities in shallow water flow. The instability of flows is caused by the decrease in the kinetic energy during the perturbation of waves. This phenomenon is known as negative energy modes and plays a vital role in applying the model to industrial equipment. A conclusion is that the equipartition of energies occurs if and only if the velocity difference is zero and the water depth is shallow enough to ignore the dispersive terms.
... Let hj be a unit vector, parallel to the field, pointing outwards from one end of the surface. The outward force on this end of the tube is ( 1. 68) This means that the region enclosed by the surface can be interpreted as being under a tension Z?2/ mo per unit cross-sectional area. If, however, we consider the outward force on an element of area lj 8 A perpendicular to the magnetic field, then the outward force per unit area is ...
... The method of analysis will be the same: the geometric optics approximation, assuming a high frequency plane wave moving towards the interface from one of the sides. The character of the reflected and refracted waves depends on the sound speed and the Alfvén velocity at each side of the interface; this has been studied, with different methods, for a number of configurations relevant in Astrophysics [15][16][17][18]. The possibilities are manifold: waves changing character between fast and slow, traveling waves transforming into evanescent ones, etc. ...
The decomposition of a magnetosonic wave when hitting a current-vortex sheet into a reflected and a transmitted wave is studied by geometric optics methods. The first order perturbations satisfy transport equation along rays that connect through the sheet by the Rankine–Hugoniot relations. Although in general the incoming wave determines uniquely the refracted and the reflected waves, in some simple situations the slow wave behaves differently. A comparison with a smooth sheet provides a useful analogy to explain the reason.
... Such phenomenon which stems from the ability of a wave to extract energy and momentum from the mean flow was considered for the first time, apparently, by Miles [38] and Ribner [39] for acoustic waves as earlier as 1957, and then by many other authors for the different kinds of waves (see, for example, Refs. [40][41][42][43][44][45][46][47][48]; we cite here only the pioneering works and cannot present the full list of publications on this theme. Some other references can be found in the review [49]). ...
In the linear approximation, we study long-wave scattering on an axially symmetric flow in a shallow water basin with a drain in the center. Besides of academic interest, this problem is applicable to the interpretation of recent laboratory experiments with draining bathtub vortices and description of wave scattering in natural basins, and also can be considered as the hydrodynamic analog of scalar wave scattering on a rotating black hole in general relativity. The analytic solutions are derived in the low-frequency limit to describe both pure potential perturbations (surface gravity waves) and perturbations with nonzero potential vorticity. For the moderate frequencies, the solutions are obtained numerically and illustrated graphically. It is shown that there are two processes governing the dynamics of surface perturbations, the scattering of incident gravity water waves by a central vortex, and emission of gravity water waves stimulated by a potential vorticity. Some aspects of their synergetic actions are discussed.
... [7] When waves are incident on a nonrigid boundary such as the magnetopause, they can extract energy from the boundary and be reflected back with an enhanced amplitude and energy [e.g., McKenzie, 1970;Walker, 2000]. In this case, the transmitted wave can be considered to have negative energy in order to maintain energy conservation [Fejer, 1963;McKenzie, 1970;Mann et al., 1999]. However, for the more realistic scenario where wave-packets have finite length, causality suggests that the interactions can better be described by an active boundary where, instead of assigning negative energy to the transmitted waves, energy exchange between the waves and the flow is described in terms of the work done by Reynolds and Maxwell stresses at the boundary [Walker, 2000]. ...
1] Primary and secondary Kelvin-Helmholtz surface wave modes on the Earth's magnetopause are studied within the framework of warm plasma ideal magnetohydrodynamics (MHD) across an infinitely thin magnetopause tangential discontinuity (TD). With the increase of background flow velocity, a Kelvin-Helmholtz Instability (KHI) unstable boundary separating two uniform semi-infinite plasma regions is always ultimately stabilized to KHI growth at an upper cut-off while inclusion of an inner boundary in one plasma region removes this stabilization. Phase velocity Friedrichs diagrams are presented that allow us to identify unstable fast and slow modes that correspond to growing modes of the KHI under different magnetosphere and magnetosheath conditions. On the nightside magnetosphere, and the magnetotail, new KH unstable intermediate-fast modes are created, which cannot propagate exactly perpendicular to the magnetic field. In the plasma frame, primary unstable KH waves show fast/fast, while secondary KH waves show slow/fast mode behavior in the magnetosphere/magnetosheath. Secondary KHI occurs at slower flow speeds than the primary KHI and grows more slowly and at a narrow range of propagation angles. Our analysis is placed in the context of in situ satellite observations of the phase speed of KHI-related waves in the magnetosheath and magnetosphere in the long wavelength regime where our analysis applies. We conclude that KH unstable surface waves on the near-Earth magnetopause flanks are likely to be secondary KHI waves, while those further down the flanks and on the nightside magnetopause are likely to be primary KHI waves—the latter being the most important for energy transport at the magnetopause.
It is well known that an interface of tangential velocity discontinuity is necessarily unstable, regardless of the velocity difference's strength, so-called the Kelvin-Helmholtz instability (KHI). However, the instability is suppressed for shallow water flows if the Froude number, defined by the ratio of the velocity difference to the gravity wave's speed, is sufficiently large. In this investigation, we examine the effect of the depth difference of two fluid layers on the KHI. The depth difference enhances instability. The dispersion equation is obtained in a sextic polynomial, the interface is thus linearly stable if the dispersion equation has enough six real roots. Given the Froude number M1 in the instability range, the growth rate sensitively depends on the depth ratio r=H1/H2 and increases monotonically with the depth ratio difference from unity. The critical value of the Froude number for stabilization varies with the depth ratio and attains the minimum value 8 for equal depth. This behavior is verified by asymptotic and numerical analysis. Numerically, we illustrate that if the depth ratio r=0.5, the Froude number is equal or greater than 33≈5.745 to stabilize the interface and it is 4.08 for r=2.
We present measurements of ultra-low frequency (ULF) wave amplitudes measured by the THEMIS probes while crossing the plasmapause. During one crossing on 24 June 2007 which we study in detail, all three probes of which data were obtained show an increase in the ULF compressional wave amplitude between 10 and 50 mHz by about 30%. These results are confirmed by a statistical study that examines the ULF compressional wave amplitude in the same frequency range averaged over 132 sharp plasmapause crossings between June 2007 and December 2007 made by THEMIS C, D, and E. Sharp plasmapause crossings are defined as those where the density increases by more than a factor of 50 over a spatial scale of 500 km. We find that, assuming the plasmapause can be approximated by a tangential MHD discontinuity, a ULF wave amplitude enhancement of 30% is in good agreement with theoretical transmission coefficient calculations if the plasma density increases from about 4 cm-3 on the magnetospheric trough side to about 540 cm-3 on the plasmaspheric side while simultaneously the magnetic field magnitude increases from 250 nT to 350 nT. These results might have important implications for the detection of global fast modes by satellites as their amplitude is hence expected to be higher inside the plasmasphere than outside.
On October 5 and 6, 2011, the three inner THEMIS spacecraft observed
magnetopause oscillations caused by the passage of a boundary surface
wave. The configuration of the spacecraft (in a plane basically
tangential to the magnetopause) allowed for determination of the
corresponding wave vector via cross-correlation analysis. We compared
measurements from both sides of the magnetopause and the wave
characteristics with ideal MHD theory of surface waves. Quantitative
agreement is achieved if a slightly higher than observed plasma velocity
is assumed in the plasma depletion layer. Based on this comparison, we
can infer that the surface wave was not generated or amplified by the
Kelvin-Helmholtz-instability. Instead, the propagation direction of the
surface wave and the direction of the interplanetary magnetic field
suggest that the wave's source is related to the foreshock or the
quasi-parallel bow shock. Additionally, slight spacecraft separations in
boundary normal direction allowed us to reconstruct the magnetopause
structure and wave form. Interestingly, the wave featured steeper
trailing edges, contrary to most other observations of magnetopause
surface waves as well as predictions from Kelvin-Helmholtz theory.
Before undertaking our examination and description of the magnetosphere, we shall discuss the solar wind. We do this because the solar wind is the dominant force on the outer surface of the magnetosphere and presumably is the source of energy for processes occurring within the magnetosphere as well as terrestrial observable events.
The interactions of transverse waves (electromagnetic or magnetohydrodynamic) with a moving ionizing shock in the presence of a magnetic field is investigated. The dispersion equation is derived, and the reflection coefficients and transmission coefficients are found including relativistic effects. The permeability and permittivity of the ionized gas are allowed to be arbitrary. The wave is considered for an arbitrary angle of incidence if it is initiated in the unionized gas region. There will be a wave reflected from the boundary and two modes of waves transmitted into the ionized gas. It is found that for interaction with a normal shock the propagation constants of the transmitted waves in the ionized gas are independent of the polarization and the direction of propagation of the incident electromagnetic wave. When the wave is initiated in the ionized gas region, two modes of waves will be reflected from the boundary and one transmitted into the un-ionized gas.
Reflection and transmission of small amplitude magnetohydrodynamic (MHD) waves of solar-wind origin through the earth's open magnetopause are studied. The open magnetopause with a nonzero normal component of magnetic field is assumed to be a rotational discontinuity. For any of the incident waves, there is one reflected wave (fast magnetosonic wave) and five transmitted waves (one fast magnetosonic wave, two slow magnetosonic waves, one Alfven wave and one entropy wave). In this paper, cases with different incident waves, i.e., Alfven wave, fast magnetosonic wave, forward slow magnetosonic wave and entropy wave are studied. It is found that depending on the type of incident wave, the incident angle and the orientation of the ambient magnetic field, the amplitudes of the emanating waves can be amplified. The research findings also suggest that the transmission of MHD waves across the rotational discontinuity at the earth's magnetopause can be an important mechanism for the energy transfer from the solar wind into the magnetosphere.
The coupling of magnetohydrodynamic waves to electromagnetic and sound waves at a plane interface between a plasma and a neutral gas is considered for arbitrary direction of the incident wave vector and for the constant external magnetic field lying either in the plane of incidence or perpendicular to it. The relations between the angle of incidence, reflection, and transmission, the ratio of the field amplitude transmitted or reflected to the corresponding amplitude incident, and the energy coupling coefficients are calculated. An incident fast magnetoacoustic mode generates a sound wave with energy coupling coefficient of the order of the ratio of the speed of sound to the Alfvén speed when this ratio is small. An incident slow magnetoacoustic mode generates a sound wave with energy coupling coefficient of the order of unity and an electromagnetic wave with energy coupling coefficient of the order of the ratio of the speed of sound to the speed of light.
The stability of the interface between a highly conducting liquid and a flowing, nonconducting gas is studied, supposing a magnetic field to be parallel to the interface. Arbitrary angles are allowed between magnetic field, flow direction and wave vector.
The reflection and refraction of a plane acoustic-gravity wave at an
interface separating two fluids in relative motion is calculated. It is
found, as for purely acoustic waves, that the reflection coefficient for
gravity waves can exceed unity (that is to say, the reflected wave can
be amplified) if the shear flow speed exceeds the horizontal phase speed
of the incident gravity wave. This result, which implies that gravity
waves extract energy and momentum from the mean flow, is discussed,
along with the idea of a critical layer at which the energy and momentum
of gravity waves are absorbed into the mean flow.
We find that slow magnetoacoustic waves produce the magnetic field
perturbations of the largest amplitude in the 0.01- to 0.1-Hz frequency
range within the ‘highly disturbed’ magnetosheath. The
frequencies quoted are frequencies seen by the satellite. Rotational
Alfvén waves with periods of several minutes and longer are also
detected on most orbits. The power spectrum of rotational waves usually
rises much more steeply below 0.01 Hz than the power spectrum of
magnetoacoustic waves. We conclude that the magnetoacoustic waves are
produced or strongly amplified at the earth's bow shock or in the outer
magnetosheath. The observed rotational waves may be produced beyond the
bow shock and carried into the magnetosheath by the solar wind.
In quasi-parallel geometry, the interaction of the solar wind with
planetary magnetospheres gives rise to upstream perturbations that are
carried back into the magnetosheath and (at Earth) have long been held
responsible for magnetospheric pulsations. The kinetic aspects of the
up-stream wave generation, transmission, and conversion are reviewed and
elucidated using results of hybrid (kinetic ions, fluid electrons)
simulations and linear kinetic theory. First, quasi-parallel shock
re-formation and its ensuing downstream, compressional perturbations are
examined. The temporal and spatial scales set significant constraints on
this process as a means of generating the magnetosheath fluctuations in
question. On the other hand, for θBn ˜ 30°
and medium Mach numbers (MA ˜ 2.3 to ˜ 3),
one-dimensional hybrid simulations have shown that upstream fast
magnetosonic (F/MS) waves are directly converted into downstream
Alfvén/ion-cyclotron (A/IC) waves in the Pc 3-4 band. Here,
θBn is the acute angle between the upstream magnetic
field and the shock normal. It is demonstrated that this process is
viable also at higher Mach numbers (MA = 4.5), at smaller
angles (θBn ˜ 10°), and in two-dimensional
simulations. Understanding the relevant wave processes requires a good
characterization of the low-frequency, warm plasma mode properties in a
collisionless plasma. The mode structure is reviewed, and some important
differences between kinetic and 2-fluid mode properties are shown. How
Alfvénic magnetosheath wave energy eventually transmits into the
magnetosphere is not currently understood. However, three examples are
given which demonstrate that kinetic aspects need to be taken into
account in the wave interaction with the (open) magnetopause. When
modelled as a rotational discontinuity (RD), the interaction with
downstream propagating A/IC waves leads to reflected and converted F/MS
waves that are not expected from MHD theory. The wave conversion also
depends on the handedness of the RD. Furthermore, the wave interaction
can trigger the break-up of RDs with rotation angle larger than
180°.
The hydromagnetic stability of a vortex sheet with respect to all modulations is examined. A domain of absolute stability not previously noted is found for high compressibility. Stability criteria are given. Onset of instability is related to radiation of hydromagnetic waves.
The stability of a combined current and vortex sheet in compressible and perfectly conducting fluids is studied. In contrast with the incompressible fluid case, a current sheet is found to have, for a given average magnetic energy, a destabilizing influence.
Relationships are examined between interplanetary parameters and local daytime geomagnetic energy (at Pittsburg, New Hampshire; L approximately 3.5) in three period bands. Ground-based magnetometer data, for 336 hours during July and August 1975, are studied when good interplanetary data coverage was available from instruments on the Imp J spacecraft. Multiple linear regression analyses show that the most important interplanetary parameter controlling ground hourly magnetic energy is the solar wind speed and that hourly magnetic energy in the 30- to 60-s period range is controlled by the solar wind speed and the interplanetary magnetic field direction. The velocity dependence supports the idea of wave production at the magnetopause via the Kelvin-Helmholtz instability which are later observed as magnetic pulsations (30-240 s) on the ground.
We investigate the relationship of local daytime geomagnetic energy (measured in the period range approx.30--300 s at a mid-latitude ground-based station) to interplanetary parameters during two time intervals when the solar wind/interplanetary field conditions were quite different. We find that the geomagnetic energy levels in the entire period band are consistent with production by a Kelvin-Helmholtz instability source at the earth's magnetopause. We further find that at the shorter periods (approx.30--60 s) the energy may also have a source contribution from the mechanism proposed by Greenstadt, which involves wave energy transfer from the earth's bow shock to the magnetopause. This result represents the first evidence of frequency dependencies in the interplanetary source mechanisms for hydromagnetic energy production in the magnetosphere.
The magnetopause is assumed to be a stable, plane, tangential discontinuity so that we can calculate the amplitudes of waves that are transmitted through, reflected from, and produced at this interface. The calculations show that the ratio of magnetosphere to magnetosheath wave power spectral densities should depend most sensitively upon the ratios of field strengths and plasma densities and upon the magnetic field direction change at the magnetopause if all waves seen in the magnetosphere are transmitted through the interface. These calculations are compared with Explorer 12 measurements of waves with periods between 30 s and 2 min. The observations show that the source of most wave power seen in the magnetosheath near the earth-sun line lies in the magnetosheath or beyond. Transmission could account for a substantial fraction of the wave power seen in the magnetosphere in this subsolar region. Beyond about 35 deg from the earth-sun line, magnetosphere wave power levels are much higher than is predicted by transmission through a stable closed magnetopause.
Evidence is reviewed indicating that the main features of geomagnetic storms of both types are caused by frictional interaction with the solar wind rather than by mere pressure on the geomagnetic cavity. If this is so then recurrent or M-region storms are caused by a property of the (steady) solar wind which controls the frictional interaction. This property is the component of the interplanetary magnetic field in the geomagnetic equatorial plane. Hence M-regionsm are regions above which (projected along the streamlines) the magnetic field is unusually weak or lies at an angle to this plane.РефератPaccMatPивaюtcя дaнныe, Укaзывaющиe нa to, чto глaвныe чaPaкtePиctики Maгниtныч бУPь oбoeгo tипa пPoиcчoдяt ot фPикциoннoгo взaиMoдeйctвин c coлнeчныM вetPoM, a нe лишь ot дaвлeния нa гeoMaгниtнУю пoлoctь. ecли эto taк, to пPичинoй пoвtoPяющичcя бУгь в oблactи M являetcя cвoйctвo (Уctoйчивoгo) coлнeчнoгo вetPa, PeгУлиPУющeгo фPикциoннoe взaиMoдeйctвиe. Эto cвoйctвo—coctaвнoй элeMeнt Meжплaнetнoгo Maгниtнoгo пoля и гeoMaгниtнoй эквatoPиaльнoй плocкoctи. taкиM oбPaзoM, oблactи M—эto oблactи, пoвePч кotoPыч (пPoeкtиPoвaннoe вдoль линии вoздУшнoгo нotoкa) Maгниtнoe пoлe нeoбчaйнo cлaбoe, или жe нaчoдиtcя пoд ≈π/2 УглoM к эtoй плocкoctи.
The theory presented deals with propagation of magnetohydrodynamic waves in a plasma whose propagation characteristics vary along the path of propagation. Plane waves with wave normals in the direction of the gradient of the medium are assumed. Modified Alfvén waves and acoustic waves obey a set of coupled wave equations. Alfvén waves are decoupled by a restricting assumption on the variation of the static magnetic field. The main aim of the paper is to demonstrate the generation of fairly intense acoustic waves in propagation of modified Alfvén waves in a steep gradient of the propagation conditions. This effect is investigated at a boundary which is equivalent to a very steep gradient.
The laws of reflection and refraction of incompressible magnetohydrodynamic (MHD) waves at a fluid-solid interface are derived by matching inhomogeneous electromagnetic plane wave solutions in a finitely conducting solid with homogeneous MHD plane wave solutions in a semi-infinite, perfectly conducting, inviscid fluid. In the undisturbed state both media contain a constant and uniform magnetic field H0 of arbitrary orientation with respect to the interface. The solid solution represents a refracted wave which attenuates due to ohmic dissipation as it penetrates the solid, while the fluid solution consists of unattenuated incident and reflected waves plus a disturbance travelling parallel to the interface whose amplitude decreases exponentially (without dissipation) with distance from the interface. Modification of the Ferraro-Roberts law of reflection is required to insure that incident and reflected waves propagate in opposite directions along H0. Since the law is independent of the fluid and solid parameters, it is also valid in the fluid-fluid and fluid-free-space cases. Owing to the anisotropy imparted to the fluid by H0, the law of refraction is antisymmetric about normal incidence and, as a result, most waves from an isotropic source are refracted into the quadrant of the plane of incidence containing a component of H0.
Some basic properties of the Kelvin-Helmholtz instability are reviewed, and some papers dealing with various refinements of this phenomenon are discussed.
An outgoing group velocity criterion is used to determine the regions of existence of radiation modes for a tangential velocity discontinuity in a collisionless hydromagnetic re´gime. It is shown how previously published results must be corrected.
Formal conditions are derived for the stability of a velocity
discontinuity at a plane interface between two perfectly conducting, inviscid,
compressible fluids, in the presence of uniform magnetic fields. Certain special
cases, in which the stability conditions can be simplified, are discussed in
detail. Previous results for incompressible fluids are rederived and it is shown
that the introduction of a slight compressibility always reduces the stabilizing
effect of magnetic fields. It is further shown that in two special cases the
magnetic field required to stabilize an interface continues to increase with the
introduction of more compressibility until it reaches a limiting value that is
about twice as large as the value appropriate to incompressible fluids. (auth)
The Kelvin-Helmholtz (KH) instability of a tangential discontinuity between two compressible plasmas in relative motion is investigated, by solving the dispersion equation for two cases. In the first, neutrals are excluded; in the second, collisions between neutrals and ions are introduced in the form of a drag force in the momentum equation. The velocity of neutrals is assumed to be perpendicular to the interface. In both cases the growth rate of the KH instability is obtained as a function of the density jump between the plasmas. Although it has often been remarked that compressibility should, in general, stabilize a plasma, it is found that this ceases to be true when allowance is made for a significant density jump at the interface. Thus, for a large density jump and a large velocity shear, the instability growth rate in a compressible plasma may considerably exceed the growth rate obtained when incompressibility is assumed. Collisions, it is shown, may either stabilize or destabilize a tangential discontinuity, depending on the change in the product of density and collision frequency (pv), as one moves with the neutrals across the interface; when pv decreases, the instability is enhanced (and vice versa).
Reflexion coefficients greater than unity have now been predicted for a variety of different systems involving waves propagating towards a shear layer, but almost invariably only in regions of parameter space for which the layer exhibits Kelvin-Helmholtz instability. This paper contains a study of two examples in which, for appropriate parameter values, there are no such instabilities to obscure (or even prevent) the ‘over-reflexion’ of an incident wave, namely (a) hydro-magnetic internal gravity waves meeting a vortex-current sheet in a stratified fluid and (b) magneto-acoustic waves meeting a vortex sheet in a compressible fluid. In the former case the energetic aspects of over-reflexion are examined in detail, thus displaying the way in which the excess reflected energy is extracted from the mean motion and the sense in which the transmitted wave may be viewed, by analogy with certain concepts employed in plasma physics, as a carrier of so-called ‘negative energy’.
A magnetohydrodynamic (MHD) wave, incident on the boundary between
two MHD media in relative motion, may be amplified or attenuated by exchanging
energy with the kinetic energy of the background flow. The conventional treatment
of this problem uses a definition of the wave energy such that energy is conserved
at the boundary. An amplified reflected wave then leads to the requirement of a
transmitted wave carrying negative energy. Such an approach, while producing
correct results, obscures the nature and location of the energy interchange. In this
paper, the proper definitions of energy density and flux in a moving plasma are
discussed, and the relationship of the group velocity and the energy flow is clarified.
The mechanism by which energy is exchanged between streaming plasma and wave
is through the work done by the Maxwell and Reynolds stresses on the gradient of
velocity at the boundary. The location of the energy exchange is identified as the
active boundary, with no need to invoke ideas of negative energy. The relationship
between the two approaches is critically discussed.
The spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic ray particles in interplanetary space is calculated in the WKB approximation. The model considers only Alfven waves of solar origin to be responsible for scattering of moderate energy particles. After developing the general theory results are presented for the asymptotic case corresponding to radial distancesr greater than about 1 to 2 AU. The radial diffusion coefficient
r
increases with energyE like
r
E
, wherev2/3. The radial mean free path turns out to increase proportional tor
3 at medium and low heliographic latitudes. This behaviour is consistent with a very small radial cosmic ray gradient and the existence of a free boundary for particle diffusion. At equal radial distances the high latitude mean free path is not only much smaller than the one calculated at the lower latitudes but in addition increases only weakly with distance. Some conceivable dynamical implications for the outer solar system are indicated.
The plasma model for the magnetosphere and ionosphere is first discussed. A review of some parts of the theory for a warm collisionless plasma of interest in the magnetosphere in connection with waves of periods between 0.1 and 1000 seconds is given. The theory for magnetohydro-dynamic waves in a slightly ionized gas is then summarized. The available observational data about magnetospheric and ionospheric phenomena, which may be interpreted in terms of waves with periods between 0.1 and 1000 seconds, are briefly surveyed and some theoretical applications to the ionosphere and magnetosphere are finally discussed. The theory of shock phenomena and transients in the magnetosphere is not included in the report.
The paper considers wave coupling for an arbitrary direction of propagation on the basis of single fluid hydromagnetic equations appropriate for a rarefied plasma. The analysis is used to study the transfer of solar wind momenta into the magnetosphere. It is found that wave refraction is significant only during disturbed conditions for waves travelling with the wind. Enhanced reflection of waves might be important even under quiet conditions in the flanks of the magnetosphere.
The Kelvin-Helmholtz instability on the magnetopause has frequently been invoked as a mechanism for driving geomagnetic pulsations in the Pc3–Pc5 range, as well as to explain the occurrence of surface waves on the magnetopause observed by satellites. Most theories of the instability represent the magnetopause by a sharp boundary with velocity shear. In this paper a linear theory is developed which takes into account the finite thickness of the low-latitude boundary layer on the magnetopause. The theory is in a form suitable for numerical computation and can take into account the effect of gradients in the plasma pressure, magnetic field magnitude and direction, and density. Computations show that the instability is suppressed at wavelengths short compared with the scale width of the boundary. There is thus a wavelength for which the growth rate is maximum. Extensive computations have been carried out and they show that growth can take place for a very wide range of conditions. The computations confirm earlier results snowing that maximum growth occurs for a wave vector which is perpendicular to the magnetic field. For typical solar wind conditions the theory predicts wavelengths on the magnetopause of the order of 10 times the thickness of the low-latitude boundary layer and periods in the Pc3–Pc5 range. The possible non-linear development of the instability is discussed qualitatively. The predicted results are consistent with satellite observations of pulsations.
The reflection and refraction of MHD waves through an “open” magnetopause (rotational discontinuity) is studied. It is found that most of the incident wave energy can be transmitted through the open magnetopause. A transverse Alfvén wave (or a compressional magnetosonic wave) from the solar wind incident upon the open magnetopause would generally lead to the generation of both the transverse Alfvén and compressional magnetosonic waves in the magnetosphere. Transmission of Alfvén waves in the coplanar rotational discontinuity is studied in detail. The integral power of the Alfvén-wave transfer is found to be proportional to the open magnetic flux of the magnetosphere and is typically ∼ 1% of the power of the total electromagnetic energy transfer through the open magnetopause. The transmitted wave power may contribute significantly to the geomagnetic pulsations observed on the ground, especially in the open-field-line region.
Low frequency hydromagnetic waves are studied in the magnetosphere by using the Explorer XII magnetometer observations. A method is developed to separate transverse Alfvén and magnetosonie waves from the satellite results. Several waves of 10 gamma amplitude and 100-200 second period are found. The transverse waves are circularly and elliptically polarized. Evidence of transverse waves originating in the magnetosphere and propagated to the surface of the Earth is found. A model is presented to explain the generation of such waves in the magnetosphere. This study of the hydromagnetic waves in the magnetosphere covers 7 to 10 Earth radii in radial distance and 75 degrees from the Sun-Earth line toward morning side in longitudinal extent, close to the geomagnetic equator. Polarizations of transverse waves have dependence on local time. It is found that the polarization directions are opposite in sense about 1045 and not 1200 hours local time. Direct evidence for such an asymmetry, suspected from several ground observations, is found for the first time in the magnetosphere.
The results of analyses of hydromagnetic reflection and refraction at a shear layer and at a shock are applied to situations representative of the magnetopause and the Earth's bow shock.The almost complete absence of magnetic fluctuations in the outer fringes of the magnetopause is explicable in terms of the magnetopause behaving like a near perfect reflector to the turbulent hydromagnetic waves in the magnetosheath. Therefore it seems unlikely that the turbulent sound wave refraction mechanism is very effective in producing a viscous-like interaction between the solar wind and the magnetosphere. For conditions typical of the solar wind and the magnetotail during quiet times the stability analysis given here indicates that the tail of the magnetopause is unlikely to undergo Kelvin-Helmholtz instability. However, during times when the solar wind blows harder than it usually does the distant tail of the magnetopause will exhibit not only instability but also greatly enhanced hydromagnetic wave transmission into the tail from the magnetosheath. It is noted that both longitudinal and transverse waves can be amplified on passage through a strong shock. Thus the amplification of the turbulent spectrum of hydromagnetic waves in the solar wind on passage through the Earth's bow shock may account for the (at least) order of magnitude increase of the noise spectrum in the magnetosheath over that in the unshocked solar wind.
The effect of finite ion Larmor radius on the Kelvin-Hehnholtz instability of the Earth's magnetopause is theoretically investigated when a wave vector is perpendicular to a magnetic field. It is found that a dawn-dusk asymmetry in excited waves is caused by this effect. This result is discussed in comparison with satellite observations.
Reflection and transmission coefficients of MHD waves are obtained at a stable, plane interface which separates two compressible, perfectly conducting media in relative motion to each other. The coefficients are evaluated for representative conditions of the quiettime, near-Earth magnetopause. The transmission coefficient averaged over a hemispherical distribution of incident waves is found to be 1–2 per cent. Yet the magnitude of the energy flux deposited into the magnetosphere in a day averaged over a hemispherical distribution of waves having amplitudes of say 2–3 gamma, is estimated to be of the order 1022 erg. Therefore the energy input of MHD waves must contribute significantly to the energy budget of the magnetosphere. The assumption that the boundary surface is a tangential discontinuity with no curvature limits the present theory to hydromagnetic frequencies higher than about 10−1 Hz. The ion gyrofrequencies for the models assumed here lie above ∼2 × 10−1 Hz. Therefore the present treatment applies to MHD waves near 10−1 Hz.
It is argued that the traditional interpretation of the Kp index is no longer tenable. The Kp index, generally taken to be a measure of the strength of the solar-wind flux, may be more acceptably interpreted as a measure of the time rate of change of the sum of plasma plus magnetic pressure acting on the magnetosphere. The stability of the magnetospheric surface in the solar wind is demonstrated theoretically when reasonable assumptions are made for the plasma density just inside the magnetosphere. The magnetic irregularities that have been observed outside the magnetosphere do not appear to be hydromagnetic waves, but most likely are quasi-static irregularities that are swept past the detectors by the solar-wind flow. As a corollary to this new interpretation of Kp it is proposed that M-region geomagnetic storms are due to sheets of turbulence or irregularities that are generated by the collision of a region of high solar-wind velocity with a low velocity region.РефератДoкaзыbaeтcя, чтo пpeжняя интepпpeтaция индeкcaKp бoльшe нe пpи-гoднa. ИндeкcKp, кoтopый oбычнo пpинят кaкmepa cилы пoтoкa coляpнoгo beтpa, moжeт быть бoлee пpиemлemo интepпpeтиpobaн, кaк mepabpemeннoй cкopocти изmeнэнияcymmы плaзmы и maгнитнoгoдabлeния, дeйcтbyющeгo нa maгнитocфepнoи пobepчнocти bcoляpнom beтpe ; этo пoкaзaнo тeopeтичecки, кoгдa дeлaютcя пpиemлemыe дoпyщeния oтнocитeльнo плoтнocти плaтmы bнyтpи camoй maгни тo-cфepы. maгнитныe нepabнomepнocти, кoтopыe нaблюдaлиcь bнe maгнитocфepы, oкaзыbaютcя нe гидpomaгнитныmи boлнamи, нo, пo bceй bepoятнocти, кba зиcтaтичecкиmи нepabнomepнocтяmи, кoтopыe пpoнocятcя mиmo пpиemникob пoтoкom coляpнoгo beтpa. bcлeдcтbиe этoй нoboй интepпpeтaции Kp, пpeдлaгaeтcя, чтo гeomaгнитныe бypи b m-oблacти oбycлobлиbaютcя нobepчнocтяmи тypбyлeнтнocти или нepabнomepнocтяmи, oбpaзobaнныmи cтoлкнobeниem paйoнa coляpнoгo beтpa bыcoкoй cкopocти c paйoнom maлoй cкopocти.
Order of magnitude calculations are used to show that viscous interaction between the solar wind and the Earth's magnetosphere can satisfy the energy requirements of a typical magnetic storm. The viscous interaction is considered to be due mainly to turbulence of a compressible nature in the solar wind, and it is shown that this can provide the necessary drag forces, although other mechanisms are not excluded.РефератПopядoк вычиcлeния эвeзднoй вeличины пpимeняeтcя c тeм, чтoбы пoкaзaть, чтo вязкoe взaимoдeйcтвиe мeждy coлнeчным вeтpoм и мaгнитocфepoй зeмли мoжeт yдoвлeтвopить пoтpeбнocти в энepгии типичнoй мaгнитнoй бypи. Пoлaгaют, чтo вязкoe взaимoдeйcтвиe oбycлoвлeнo, глaвным. oбpaзoм, cжaтoгo poдa тypбyлeнтнocтью в coлнeчнoм вeтpe и этo, пoвидимoиy, мoжeт cнaбдить нyжиыe cилы coпpoтивлeния, чoтя нe иcключaютcя и дpyгиe мeчaнизмы.
Explorer 18 magnetometer and plasma probe data are used to calculate the magnitude of the Kelvin-Helmholtz instability criterion for 42 crossings of the magnetopause. The 30 crossings through the dipole field region show that for low levels of geomagnetic disturbance ( Kp less than or equal to 2) large regions of the magnetopause near the nose are stable, whereas for higher levels of geomagnetic disturbance (Kp greater than 2) only a small region near the nose is stable. The unstable regions are found further downstream. Similarly, the 12 crossings through the tail show that the unstable regions are downstream of the stable regions. The Kelvin-Helmholtz mechanism might be the 'viscous interaction' discussed by Axford and Hines (1961) and the relevant parameter for worldwide geomagnetic disturbance, just as the southward interplanetary field has been firmly established as the major relevant parameter in the case of substorm activity.
Kelvin-Helmholtz instability at magnetopause, initiating semiannual variation of geomagnetic disturbances
The reflection of a plane wave of sound at an interface between two perfect fluids is considered. Previous analyses of Rudnick, Keller, and Franken and Ingard are found in error as a result of improper boundary conditions. It is found that, in addition to the possibility of total reflection in some range of angles of incidence at all finite, relative speeds, there exists the possibility of a reflection coefficient exceeding unity for sufficiently high, supersonic speeds; in particular, resonance may occur at one or more angles of incidence. The question of stability of the vortex sheet separating the two fluids also is discussed.
The reflection and transmission process is analyzed for plane sound waves originating in air at rest and impinging obliquely on a plane interface with a moving stream. Use of a moving reference frame provides transformation to an equivalent aerodynamic problem of flows past a wavy wall—the rippled interface. The angles of incidence, reflection, and refraction are identified with the Mach angles. The angular relations and the amplitude relations (coefficients of reflection and transmission) are evaluated in closed form. In a graph three zones can be distinguished the plane of angle of incidence v.Mach number of the moving medium: ordinary reflection and transmission, total reflection, and amplified reflection and transmission. Included are three loci of infinite reflection: i.e., serf‐excited waves. The energy balance is examined, and the source of amplification is concluded to be the energy of the moving stream. In appendices the results are generalized (1) for the case of two moving media and (2) for differing density and speed of sound in the two media.
The well-known result for the stability of a plane vortex sheet in an incom-pressible fluid is generalized to include magnetic fields and three-dimensional effects. The resulting current-vortex sheet is shown to be stabilized by sufficiently strong magnetic fields, and weaker fields are shown to give a partial stability. The effect of walls is examined, and in general, they are found to add to the stability of the flow. The dispersion relationships are found for several configurations of walls and current-vortex sheets, and conditions for stability are given.