Lighthill Waves In Fluids

[Jacqualine Henington]

Wavepropagation in fluids is the study of how waves travel through different types of fluids, such as water, air, and other liquids or gases. This includes understanding the properties of waves, such as their speed, frequency, and wavelength, as well as how they interact with the medium they are traveling through.

Some common examples of wave propagation in fluids include ocean waves, sound waves in air or water, seismic waves, and electromagnetic waves in the Earth's ionosphere. These waves can have different properties and behaviors depending on the type of fluid they are traveling through.

Wave propagation in fluids differs from solids in several ways. For example, in fluids, the particles are free to move and can be displaced by the wave, while in solids, the particles are tightly packed and can only vibrate in place. Additionally, fluids have varying densities and viscosities, which can affect the speed and behavior of waves.

Several factors can affect wave propagation in fluids, including the density and viscosity of the fluid, the depth or thickness of the fluid, and the properties of the wave itself, such as its amplitude and frequency. Other factors, such as temperature, pressure, and the presence of obstacles, can also influence wave propagation in fluids.

Understanding wave propagation in fluids is essential in many fields, including oceanography, meteorology, acoustics, and geophysics. It can help us predict and understand natural phenomena, such as weather patterns and earthquakes, and also has practical applications in areas such as communication, transportation, and energy production.

Waves in an ionized fluid, for which the only restoring force is magnetic. This definition allows for different background media (homogeneous or not) under an external magnetic field (which may be uniform, or vary in strength and/or direction), for the presence of dissipation and other effects (displacement, Ohmic or Hall currents, fluid viscosity, mean flow, multiple ion species); it excludes other types of waves (sound, gravity and inertial) associated with restoring forces of non-magnetic origin.

In a compressible fluid, the pressure acts as a restoring force, and one obtains sound waves. The combination is magneto-acoustic waves [a3], [a4], which have three modes: i) unchanged Alfvn mode, because it is incompressible; and sound waves modified into two coupled slow ii) and fast iii) modes. Considering a stratified fluid (e.g., an atmosphere) and adding gravity as a restoring force, one has magneto-acoustic-gravity waves [a2], [a5], [a6], [a7] and Alfvn-gravity waves decouple only if the horizontal wave-vector (which exists only in the direction transverse to stratification) lies in the plane of gravity and the external magnetic field. Adding rotation and the Coriolis force as the fourth restoring force leads to magneto-acoustic-gravity-inertial waves [a2], [a8], for which decoupling of Alfvn-gravity modes is generally not possible. Below, the Alfvn waves are uncoupled to other types of waves in fluids.

Alfvn waves in a stratified medium, e.g., with density a function of altitude, but under a uniform external magnetic field [a9], and isothermal conditions, satisfy different equations [a10] for the velocity and magnetic field perturbations:

where the Alfvn speed is, in general, non-uniform. The relation (a3) no longer holds, and thus equi-partition of energies breaks down (a4), (a5), and the simplification of the energy flux (a6) fails. Examples of a non-constant magnetic field are: i) a radial magnetic field [a10], for which Alfvn waves remain incompressible; and ii) a spiral magnetic field [a12], for which they are not divergence free (compressive Alfvn waves). Alfvn waves have also been considered in non-isothermal atmospheres [a13], [a14], and in magnetic flux tubes [a15].

The Alfvn wave equations are usually deduced from the equations of magneto-hydrodynamics [a16], neglecting the displacement current in comparison with the electric current (cf. also Magneto-hydrodynamics, mathematical problems in). The effect of the displacement current on Alfvn waves has been studied [a17]. Another effect is Hall currents, resulting from the spiralling of electrons around the magnetic field [a18], [a19]. If the external magnetic field is non-uniform, the ion-gyro (or spiralling) frequency varies with altitude, and where it equals the wave frequency a critical layer occurs [a20], [a21]. A critical layer is a singularity of the wave equation, where wave absorption, reflection or transformation can occur. A critical layer also occurs for Alfvn-gravity waves in the presence of Ohmic currents, i.e., electrical resistance [a22]. Since Alfvn waves are transversal, i.e., incompressible, there are no thermal effects (e.g., conduction or radiation), and the other dissipation mechanism is shear viscosity [a23], [a24]. In a homogeneous medium the dissipative Alfvn wave equation reads [a25]:

$$ \taga10 \left \ \frac\partial ^ 2 \partial t ^ 2 - A ^ 2 \frac\partial ^ 2 \partial l - ( \chi + \eta ) \frac \partial \partial t \nabla ^ 2 + \chi \eta \nabla ^ 4 \right \ \vecv , \vech ( \vecx ,t ) = 0,$$

Alfvn waves have been observed in the laboratory [a29], and occur in plasma machines and fusion reactors. They may be present in the Earth's molten core, where inertial effects could be important. Alfvn waves have been observed in the solar atmosphere [a30], and could be a mechanism for: i) heating the atmosphere by dissipation [a31], [a32]; or ii) accelerating the solar wind [a33]. Alfvn waves [a34], [a35] propagate with the solar wind to the Earth's ionosphere, and exist in the interplanetary [a15] and probably in the interstellar medium. The case of the solar wind [a33] combines several of the possible influences on Alfvn waves: i) non-uniform background density, decaying towards the Earth; ii) external magnetic field varying in strength and direction along Parker's spiral; iii) background mean flow with velocity exceeding the Alfvn speed, beyond the critical point; iv) presence of multiple ion species. In the distant solar wind, as particle density decreases, Alfvn waves should be considered in the context of plasmas [a36], rather than of magneto-hydrodynamics [a37].

The generation of Alfvn waves results from hydromagnetic turbulence and ionized inhomogeneities [a4], [a38], [a39], e.g., in the photosphere of the sun, where the Alfvn waves propagating in the solar atmosphere and solar wind originate. The solution of the Alfvn wave equation uses:

iii) hypergeometric functions (cf. also Hypergeometric function) when critical layers are present [a21], [a22], [a23], [a24], [a27], [a28], [a32]. In this case the three singularities represent the initial and asymptotic wave fields, and the wave fields near the critical layer.

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