Coriolis force in fluid mechanics

Gaspard-Gustave de Coriolis

Coriolis force in fluid mechanics is one of the forces of inertia acting on an ordered or fluctuating flow of a liquid or gas in a rotating non-inertial frame of reference .

The task of geophysical and astrophysical hydrodynamics consists in the physical description of the turbulent flow of a liquid (or gas) on rotating objects. In geophysics, it is natural to use a coordinate system rigidly connected with a rotating Earth. This coordinate system is non-inertial . To describe the relative motion in such a coordinate system, you can use the Navier – Stokes system of hydromechanical equations ^[1] , if you enter two additional inertia forces — the centrifugal force and the Coriolis force ^[2] . The Coriolis force in fluid mechanics, in contrast to solid mechanics, has its own characteristics and important applications.

Content

Definition

In a coordinate system rotating at angular velocity $\mathbf {\omega } ,$ ${\ displaystyle \ mathbf {\ omega},}$ material point moving at relative speed $\mathbf {v} ,$ ${\ displaystyle \ mathbf {v},}$ participates in complex motion and, according to the Coriolis theorem , acquires an additional pivotal or Coriolis acceleration equal to the vector product $2\mathbf {\omega } \times \mathbf {v}$ ${\ displaystyle 2 \ mathbf {\ omega} \ times \ mathbf {v}}$ . It is considered that the pseudovector $\mathbf {\omega }$ ${\ displaystyle \ mathbf {\ omega}}$ directed along the axis of rotation according to the rule of the right screw .

If a $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ - vector of relative flow rate of a liquid or gas having a density $\rho ,$ ${\ displaystyle \ rho,}$ then in the rotating coordinate system, the Coriolis force vector per unit volume is equal to

\mathbf {F_{c}} ={-}2\mathbf {\omega } \times \rho \mathbf {v} .\qquad (1)

{\ displaystyle \ mathbf {F_ {c}} = {-} 2 \ mathbf {\ omega} \ times \ rho \ mathbf {v}. \ qquad (1)}

In fluid mechanics, the flow velocity and the characteristics of the state of a substance, including density, are subject to fluctuations of a different nature - thermal movement of molecules, sound vibrations, turbulence . The influence of hydrodynamic fluctuations on the flow dynamics is investigated by the methods of statistical hydromechanics. In statistical hydromechanics, the equations of motion describing the behavior of average flow characteristics, in accordance with the method of O. Reynolds , are obtained by averaging the Navier – Stokes equations ^[3] . If, following the method of O. Reynolds, submit $\rho ={\overline {\rho }}+\rho ',\ \mathbf {v={\overline {v}}+v'} ,$ ${\ displaystyle \ rho = {\ overline {\ rho}} + \ rho ', \ \ mathbf {v = {\ overline {v}} + v'},}$ where the bar at the top is the averaging sign, and the bar is the deviation from the mean, then the vector of the averaged pulse density ^[3] takes the form:

{\overline {\rho \mathbf {v} }}={\overline {\rho }}~{\overline {\mathbf {v} }}+\mathbf {S} ,\qquad (2)

{\ displaystyle {\ overline {\ rho \ mathbf {v}}} = {\ overline {\ rho}} ~ {\ overline {\ mathbf {v}}} + \ mathbf {S}, \ qquad (2)}

Where $\mathbf {S} ={\overline {\rho '\mathbf {v} '}}$ ${\ displaystyle \ mathbf {S} = {\ overline {\ rho '\ mathbf {v}'}}}$ - the density vector of the fluctuating mass flux (or “ density of a turbulent pulse ” ^[3] ). Averaging (1) and taking into account (2), we find that the density of the averaged Coriolis force will consist of two parts:

{\overline {\mathbf {F_{c}} }}={-}2\mathbf {\omega } \times ({\overline {\rho }}~{\overline {\mathbf {v} }}+\mathbf {S} ).\qquad (3)

{\ displaystyle {\ overline {\ mathbf {F_ {c}}}} = {-} 2 \ mathbf {\ omega} \ times ({\ overline {\ rho}} ~ {\ overline {\ mathbf {v}} } + \ mathbf {S}). \ qquad (3)}

Thus, in the turbulent medium, a second part of the Coriolis force arose, called ^{[by whom?} ^] “ Coriolis Turbulent Force Density ” . It leads to the appearance of additional effects in hydrodynamic phenomena that are absent in solid mechanics.

Coriolis Force in Atmospheric and Oceanic Physics

Cyclone over Iceland September 4, 2003

The most important role is played by the Coriolis force in global geophysical processes. The equilibrium of the horizontal component of the force of the baric gradient and the Coriolis force leads to the establishment of a flow whose velocity is directed along the isobars ( geostrophic wind ). With the exception of the equatorial zone outside the planetary boundary layer, the motion of the atmosphere is close to geostrophic. Additional consideration of centrifugal force and frictional force gives a more accurate result. The combined action of these forces leads to the formation of cyclones in the atmosphere in which the wind rotates counterclockwise in the Northern Hemisphere, leaving a low pressure area to the left of itself. In the anticyclone , in the center of which there is an area of high pressure, rotation occurs in the opposite direction ^[4] . In the southern hemisphere, the direction of rotation is reversed.

Cyclones and anticyclones are large-scale eddies involved in the general circulation of the atmosphere . In the troposphere as a whole, the general circulation of the atmosphere is formed under the action of the force of the pressure gradient and the force of Coriolis. Three circulation cells are formed in each hemisphere: from the equator to the latitude of 30 °, the Hadley cell , approximately between 30 ° and 65 °, the Ferrell cell , and in the polar region, the Polar cell . The atmospheric heat engine drives these six "wheels" of circulation into rotation. The force of Coriolis, deflecting the wind circulating in a vertical plane, leads to the appearance of trade winds - eastern winds in the lower part of the atmosphere in the tropical belt . The deflecting action of the Coriolis force in the Ferrell cell leads to the predominance of westerly winds of the temperate belt . At the top of the troposphere, the direction of the winds is opposite.

Coriolis force in the same way participates in the formation of the general circulation of the ocean .

Ekman's Spiral

In the boundary layers of the atmosphere and the ocean, including the transition layer between the atmosphere and the ocean, along with the Coriolis force and the pressure gradient force, the internal friction force also plays a significant role. The action of friction in the boundary layer ( Ekman layer ) leads to the deviation of the wind from the geostrophic to the area of reduced pressure. As a result, in the lower part of the cyclone, air is directed to its center. There is a "suction" of air rising up in the center of the cyclone, which, due to condensation of water vapor leads to the release of heat of vaporization , the formation of precipitation and the maintenance of its rotational energy. In anticyclones, the movement of the wind is the opposite, which leads to the lowering of air in its center and the dispersion of clouds. As the distance from the underlying surface increases, the role of the friction force decreases, which leads to the rotation of the flow velocity vector in the direction of the geostrophic wind. The rotation of the wind with a height in the boundary layer of the atmosphere at an angle of ~ 20-40 ° is called the Ekman spiral . This effect is clearly manifested in the deviation of the direction of ice drift from the velocity vector of the geostrophic wind, which was first discovered by F. Nansen during the polar expedition of 1893-1896. on the ship "Fram". The theory of the phenomenon was presented by V. Ekman in 1905.

Inertia Circle

In the inertial frame of reference, a uniform and rectilinear motion is inertial. And on a rotating planet, on each material point (and also on a stream), moving freely along a curved trajectory, there are two inertia forces - the centrifugal force and the Coriolis force. These forces can balance each other. Let be $\ v$ ${\ displaystyle \ v}$ - the relative linear velocity of a point, directed horizontally clockwise in the North, and counterclockwise in the Southern Hemisphere (as in the anticyclone ). Then, the balance of inertia forces comes, if

{\frac {v^{2}}{R}}=fv

{\ displaystyle {\ frac {v ^ {2}} {R}} = fv}

,

Where $\ R=v/f$ ${\ displaystyle \ R = v / f}$ - the radius of curvature of the particle trajectory, $\ f=2\omega \sin \varphi$ ${\ displaystyle \ f = 2 \ omega \ sin \ varphi}$ - Coriolis parameter , $\varphi$ ${\ displaystyle \ varphi}$ - geographical latitude. In the absence of other forces, an equilibrium Coriolis force and centrifugal force will cause the particle (flow) to rotate along an arc, called the "circle of inertia" , having a radius $\ R$ ${\ displaystyle \ R}$ . The material point makes a complete revolution in a circle of inertia for a period equal to $\ 2\pi /f$ ${\ displaystyle \ 2 \ pi / f}$ - half of the pendulum day .

In mid-latitudes, the Coriolis parameter is of the order of 10 ⁻⁴ s ⁻¹ . The geostrophic speed in the troposphere is about 10 m / s , which corresponds to the inertia circle with a radius of about 100 km . The average speed of the current in the ocean of 10 cm / s corresponds to the inertia circle, which has a radius of about 1 km . Circulation of the flow in a circle of inertia forms an anticyclonic vortex for the occurrence of which does not require any other reason than inertia ^[5] .

Inertial oscillations and waves

If, for a liquid (or gas), the Coriolis force is the main force that returns a particle to a state of equilibrium, then its action leads to the appearance of planetary inertial waves (also called “ inertial oscillations ”). The period of such oscillations is $\ 2\pi /f$ ${\ displaystyle \ 2 \ pi / f}$ , and the oscillatory process develops in the direction transverse to the vector of the velocity of wave propagation. A mathematical description of inertial waves can, in particular, be obtained within the framework of the shallow water theory ^[6] . In middle latitudes, the period of inertial oscillations is about 17 hours .

Changing the Coriolis parameter with latitude creates the conditions for the occurrence of Rossby waves in the atmosphere or in the ocean. These waves lead to the meandering of jet streams , as a result of which the main synoptic processes are formed.

The work of Coriolis Turbulent Force

In hydromechanics, the amount of mechanical work produced by a force per unit volume per unit of time (i.e. power) is the scalar product of the force vector and the flow velocity vector. (It is believed that the concept of work was introduced into the mechanics of Coriolis ). Since in the mechanics of a material point the Coriolis force is always directed at a right angle to its velocity, the work of this force is identically zero . Therefore, the Coriolis force cannot change the kinetic energy as a whole, however, it may be responsible for the redistribution of this energy between its components. In statistical hydromechanics, there are two equations of kinetic energy — the equation of the kinetic energy of an ordered motion and the balance equation for the energy of turbulence ^[3] . In this case, the concept of the work of the turbulent Coriolis force arises, which determines the exchange of kinetic energy between the ordered and turbulent motion that occurs under the action of this force ^[7] . For a unit of time in a unit of volume, the Coriolis turbulent force produces work equal to

\ N_{c}=(\mathbf {{\overline {F_{c}}}{\overline {v}}} )=-2\mathbf {(\omega \times S{\overline {v}})}

{\ displaystyle \ N_ {c} = (\ mathbf {{\ overline {F_ {c}}} {\ overline {v}}) = - 2 \ mathbf {(\ omega \ times S {\ overline {v} })}}

.

A positive value $\ N_{c}$ ${\ displaystyle \ N_ {c}}$ corresponds to the transition of the kinetic energy of an ordered motion into the energy of turbulence ^[3] .

The Coriolis force plays a key role in geophysical hydrodynamics, however, only the work of the relatively small, but important, turbulent Coriolis force contributes to the energy of hydrodynamic processes. Analysis of the upper-air data ^[8] indicates that this effect makes the main contribution to the energy of the ordered movement, leading to the superrotation of the atmosphere.

Similar physical mechanisms, based on the Coriolis force, form atmospheric circulation on other planets, (possibly) circulation in the liquid core of planets, as well as in stars, in accretion disks , in the gas components of rotating galaxies. ^[9] , ^[10] , ^[11]

Gyroturbulent instability

If the liquid (or gas) is non-uniform (in particular, if it is unevenly heated), then a fluctuating flow of matter occurs in it $\ \mathbf {S\neq 0}$ ${\ displaystyle \ \ mathbf {S \ neq 0}}$ . This flow depends on both the density gradient and the energy of turbulent fluctuations. In a rotating fluid, this flow generates a turbulent Coriolis force, whose work leads to a reversible exchange of kinetic energy between the ordered and turbulent components. But since the turbulent flow of matter depends on the energy of turbulence, a feedback occurs. Under favorable conditions, such feedback leads to the emergence of the so-called gyroturbulent instability ^[12] . In the process of gyroturbulent oscillations, a periodic transfer of energy occurs between the ordered and disordered forms of motion. Since these oscillations arise as a result of the action of the turbulent Coriolis force, they should be considered as a special type of inertial oscillations.

The turbulent force of Coriolis is relatively small. But despite this, gyroturbulent instability is responsible for the relatively slow, but very powerful geophysical and astrophysical natural processes such as the index cycle .

Literature

↑ Landau LD , Lifshits E.M. Hydrodynamics. - M .: Science, 1988. - C. 736
↑ Khaikin S. E. Physical foundations of mechanics. - M .: Science, 1971. - p. 752
↑ ¹ ² ³ ⁴ ⁵ Monin A. S. , Yaglom A. M. Statistical hydromechanics. Part 1. - M .: Science, 1965. - 639 p.
↑ Matveev, L. T. The course of general meteorology. Physics of the atmosphere. - L .: Gidrometeoizdat, 1984. - p. 751
↑ Haltiner J. Martin F. Dynamic and physical meteorology. M .: Foreign literature .— 1960.— 436 p.
↑ Gill A. Dynamics of the atmosphere and the ocean. In 2 parts. - M .: Mir, 1986.
Oph Geogys . Astrophys. Fluid Dynamics. — 1980. —16 .— p. 1-18.
↑ Krigel A.M. Analysis of the mechanisms of transformation of turbulent energy into an ordered circulation of the atmosphere // Leningrad State University Herald. Ser. 7. - 1989. - Vol. 2 (No. 14). - p. 91-94.
↑ Kriegel A. M. Theory of stationary disk accretion onto stars and galactic nuclei // Astrophysics. - 1989. - 31 . - Issue 1. - p.137-143.
↑ Krigel, A.M., Influence of Turbulence on Radial Motion in the Gas Disks of Galaxies, Kinematics and Physics of Celestial Bodies. - 1990. - 6 . - №1. - p.73-78.
↑ Kriegel, A.M., Numerical Simulation of Gyroturbulent Oscillations of X-ray Stars, Astronomical Journal. - 1990. - 67 . - Issue 6. - p. 1174-1180.
↑ Kriegel A. M. The instability of a jet stream in a turbulent rotating inhomogeneous fluid // Journal of Technical Physics. - 1985. - 55 . - Vol. 2. - p. 442-444.

[1] Landau LD , Lifshits E.M. Hydrodynamics. - M .: Science, 1988. - C. 736

[2] Khaikin S. E. Physical foundations of mechanics. - M .: Science, 1971. - p. 752

[Monin-3] ¹ ² ³ ⁴ ⁵ Monin A. S. , Yaglom A. M. Statistical hydromechanics. Part 1. - M .: Science, 1965. - 639 p.

[4] Matveev, L. T. The course of general meteorology. Physics of the atmosphere. - L .: Gidrometeoizdat, 1984. - p. 751

[5] Haltiner J. Martin F. Dynamic and physical meteorology. M .: Foreign literature .— 1960.— 436 p.

[6] Gill A. Dynamics of the atmosphere and the ocean. In 2 parts. - M .: Mir, 1986.

[7] Oph Geogys . Astrophys. Fluid Dynamics. — 1980. —16 .— p. 1-18.

[8] Krigel A.M. Analysis of the mechanisms of transformation of turbulent energy into an ordered circulation of the atmosphere // Leningrad State University Herald. Ser. 7. - 1989. - Vol. 2 (No. 14). - p. 91-94.

[9] Kriegel A. M. Theory of stationary disk accretion onto stars and galactic nuclei // Astrophysics. - 1989. - 31 . - Issue 1. - p.137-143.

[10] Krigel, A.M., Influence of Turbulence on Radial Motion in the Gas Disks of Galaxies, Kinematics and Physics of Celestial Bodies. - 1990. - 6 . - №1. - p.73-78.

[11] Kriegel, A.M., Numerical Simulation of Gyroturbulent Oscillations of X-ray Stars, Astronomical Journal. - 1990. - 67 . - Issue 6. - p. 1174-1180.

[12] Kriegel A. M. The instability of a jet stream in a turbulent rotating inhomogeneous fluid // Journal of Technical Physics. - 1985. - 55 . - Vol. 2. - p. 442-444.