Inertial wave

The inertial waves or inertial oscillations are a type of Onde S mechanics which one finds in a Fluide in Rotation. Contrary to the Vague S which animate surface, the inertial waves affect the mass of the fluid. They result from the tendency of return to the state intial of the movements induced by a Inertia. The force generally at the origin of these waves is that of Coriolis, which gives him a Wavelength and a frequency having certain particular values. The inertial waves most known are the waves of Rossby, the geostrophic winds and currents in the atmosphere. One also finds them in the Océan S and the magma.

The inertial waves are responsible for the variations with large scales of the movement of the fluids, which leads to the mixtures between parts of various densities or compositions, for example to the mixture of the masses of air of the atmosphere or the zones different Salinité S from the ocean.

Force reaction

To include/understand the idea of a force of reaction, let us think of a cord of tended guitar and which one moves downwards. The elasticity of the cord makes it return while giving him a Accélération proportional to the distance to the initial point. It thus acquires a speed which makes it exceed this last, then it undergoes a force of deceleration which slows down it gradually to reach a maximum distance from the other side of balance; then the process is repeated in the opposite direction. Without Friction, this effect of Pendule would continue indefinitely, but possibly the Vibration will stop because of this one.

In a fluid, balance occurs when it is perfectly at rest and thus that its surface is uniform. When that a disturbance increases or lowers the level in a point, the Gravité will be exerted to turn over surface to balance and to induce a wave known as Onde of gravity .

In the case of an inertial wave, the fluid being in rotation, the state of balance is that it always turns to the same distance from the axis of rotation. For example, in the case of the atmosphere, the state of balance is carried out when the air with a Latitude given always accompanies rotation by the Earth along this one. The piece of fluid in rotation thus has a turning moment which depends on this latitude by the factor of Coriolis. If it is changed latitude, the size of this force changes and the force of Coriolis, acting with 90 degrees of the direction of movement, brings back it towards its point of balance. Its force is also proportional to the rate of rotation according to the latitude. These two characteristics give a formulation of the balance of power in this movement which is not very intuitive.

Mathematical description

A fluid in rotation obeys the Navier-Stokes equations:

$\ frac {\ partial \ vec {U}} {\ partial T}
+ (\ vec {U} \ cdot \ vec {\ nabla}) \ vec {U}$

- \ frac {1} {\ rho} \ vec {\ nabla} P

+ \ naked \ nabla^2 \ vec {U} - 2 \ vec {\ Omega} \ times \ vec {U}

On the left, one finds the variation speed $\, \ vec {U}$ of a piece of fluid according to time $\, t$ and spaces it.
On the right, one finds three terms in the usual order:

that of the variation of pressure $\, P$ which contains in its center a term of correction for the centripetal Force induced by the rotation movement
that of diffusion and convection by the Viscosité $\, \ nu$
that of the force of Coriolis with the number of revolutions $\, \ Omega$

If the rate of rotation becomes important, one can neglect the second term of right-hand side of the equation because the forces of Coriolis and centripetal become dominant. One can then multiply by rotational two sides of the equation and solve the resulting vectorial equation as follows:

$\ frac {\ partial} {\ partial T} \ times \ vec {U}$

2 (\ vec {\ Omega} \ cdot \ vec {\ nabla}) \ vec {U}.

The solutions with this equation give the representation of the inertial waves which must satisfy the following conditions:

1) $\ vec {U} \ cdot \ vec {K} = 0$ , Where $\, \ vec {K}$ thus represents a transverse wave (perpendicular to the movement).

and

2) $\ Omega = 2 \ hat {K} \ cdot \ vec {\ Omega} = 2 \ Omega \ cos {\ theta},$ , Where $\, \ theta$ is the angle between the axis of rotation and the direction of the wave.

Characteristics

The first condition of the solution of these equations defines that the force of restoration is perpendicular to the movement. Moreover, the solution of these equations gives waves whose Speed of phase, which gives its movement, is perpendicular to that of group which gives the direction of the transport of energy. That gives undulations which similarly lead to the variation of the electric fields and magnetic in a electromagnetic Onde compared to its propagation.

The second condition implies that the solutions with these equations have values between 0 and 2 times the rate of rotation of the reference mark only in one open system. Actually, in a real system, like the ocean or the atmosphere, certain restrictions decrease the choice and give discrete values.

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