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Concept
Beltrami vector field
Summary
In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that
Thus and are parallel vectors in other words, .
If is solenoidal - that is, if such as for an incompressible fluid or a magnetic field, the identity becomes and this leads to
and if we further assume that is a constant, we arrive at the simple form
Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.
The vector field
is a multiple of the standard contact structure −z i + j, and furnishes an example of a Beltrami vector field.
Beltrami fields with a constant proportionality factor are a distinct category of vector fields that act as eigenfunctions of the curl operator. In essence, they are functions that map points in a three-dimensional space, either in (Euclidean space) or on a flat torus , to other points in the same space. Mathematically, this can be represented as:
(for Euclidean space) or (for the flat torus).
These vector fields are unique due to the special relationship between the curl of the vector field and the field itself. This relationship can be expressed using the following equation:
In this equation, is a non-zero constant, which indicates that the curl of the vector field is proportional to the field itself.
Beltrami fields are relevant in fluid dynamics, as they offer a classical family of stationary solutions to the Euler equation in three dimensions. The Euler equations describe the motion of an ideal, incompressible fluid and can be written as a system of two equations:
For stationary flows, where the velocity field does not change with time, i.e. , we can introduce the Bernoulli function, , and the vorticity, . These new variables simplify the Euler equations into the following system:
The simplification is possible due to a vector identity, which relates the convective term to the gradient of the kinetic energy and the cross product of the velocity field and its curl:
When the Bernoulli function is constant, Beltrami fields become valid solutions to the simplified Euler equations.
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