In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
, the Taylor–Proudman theorem (after
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as " ...
and
Joseph Proudman
Joseph Proudman (30 December 1888 – 26 June 1975), CBE, FRS was a distinguished British mathematician and oceanographer of international repute. His theoretical studies into the oceanic tides not only "solved practically all the remaining t ...
) states that when a solid body is moved slowly within a fluid that is steadily rotated with a high
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
, the fluid
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
will be uniform along any line parallel to the axis of rotation.
must be large compared to the movement of the solid body in order to make the
Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
large compared to the acceleration terms.
Derivation
The
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
for steady flow, with zero
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
and a body force corresponding to the Coriolis force, are
:
where
is the fluid velocity,
is the fluid density, and
the pressure. If we assume that
is a
scalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
and the
advective term on the left may be neglected (reasonable if the
Rossby number
The Rossby number (Ro), named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial force to Coriolis force, terms , \mathbf \cdot \nabla \mathbf, \sim U^2 / L and \Omega ...
is much less than unity) and that the
flow is incompressible (density is constant), the equations become:
:
where
is the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector. If the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of this equation is taken, the result is the Taylor–Proudman theorem:
:
To derive this, one needs the
vector identities
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
\o ...
:
and
:
and
:
(because the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the gradient is always equal to zero).
Note that
is also needed (angular velocity is divergence-free).
The vector form of the Taylor–Proudman theorem is perhaps better understood by expanding the dot product:
:
In coordinates for which
, the equations reduce to
:
if
. Thus, ''all three'' components of the velocity vector are uniform along any line parallel to the z-axis.
Taylor column
The
Taylor column
A Taylor column is a fluid dynamics phenomenon that occurs as a result of the Coriolis effect. It was named after Geoffrey Ingram Taylor. Rotating fluids that are perturbed by a solid body tend to form columns parallel to the axis of rotation ca ...
is an imaginary cylinder projected above and below a real cylinder that has been placed parallel to the rotation axis (anywhere in the flow, not necessarily in the center). The flow will curve around the imaginary cylinders just like the real due to the Taylor–Proudman theorem, which states that the flow in a rotating, homogeneous, inviscid fluid are 2-dimensional in the plane orthogonal to the rotation axis and thus there is no variation in the flow along the
axis, often taken to be the
axis.
The Taylor column is a simplified, experimentally observed effect of what transpires in the Earth's atmospheres and oceans.
History
The result known as the Taylor-Proudman theorem was first derived by Sydney Samuel Hough (1870-1923), a mathematician at Cambridge University, in 1897. Proudman published another derivation in 1916 and Taylor in 1917, then the effect was demonstrated experimentally by Taylor in 1923.
References
{{DEFAULTSORT:Taylor-Proudman theorem
Fluid dynamics
Physics theorems