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

$${\displaystyle \mathbf {F} \times (\nabla \times \mathbf {F} )=0.}$$

Thus ${\displaystyle \mathbf {F} }$ and ${\displaystyle \nabla \times \mathbf {F} }$ are parallel vectors in other words, ${\displaystyle \nabla \times \mathbf {F} =\lambda \mathbf {F} }$.

If ${\displaystyle \mathbf {F} }$ is solenoidal - that is, if ${\displaystyle \nabla \cdot \mathbf {F} =0}$ such as for an incompressible fluid or a magnetic field, the identity ${\displaystyle \nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F} +\nabla (\nabla \cdot \mathbf {F} )}$ becomes ${\displaystyle \nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F} }$ and this leads to

$${\displaystyle -\nabla ^{2}\mathbf {F} =\nabla \times (\lambda \mathbf {F} )}$$

and if we further assume that ${\displaystyle \lambda }$ is a constant, we arrive at the simple form

$${\displaystyle \nabla ^{2}\mathbf {F} =-\lambda ^{2}\mathbf {F} .}$$

Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.

The vector field

$${\displaystyle \mathbf {F} =-{\frac {z}{\sqrt {1+z^{2}}}}\mathbf {i} +{\frac {1}{\sqrt {1+z^{2}}}}\mathbf {j} }$$

is a multiple of the standard contact structure −z i + j, and furnishes an example of a Beltrami vector field.

Beltrami fields and fluid mechanics

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 ${\displaystyle \mathbb {R} ^{3}}$ (Euclidean space) or on a flat torus ${\displaystyle \mathbb {T} ^{3}}$, to other points in the same space. Mathematically, this can be represented as:

${\displaystyle u:\mathbb {R} ^{3}\to \mathbb {R} ^{3}}$ (for Euclidean space) or ${\displaystyle u:\mathbb {T} ^{3}\to \mathbb {T} ^{3}}$ (for the flat torus).

These vector fields are unique due to the special relationship between the curl of the vector field ${\displaystyle u}$ and the field itself. This relationship can be expressed using the following equation:

${\displaystyle \nabla \times u=\lambda u}$

In this equation, ${\displaystyle \lambda }$ is a non-zero constant, which indicates that the curl of the vector field ${\displaystyle u}$ 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.^[1] The Euler equations describe the motion of an ideal, incompressible fluid and can be written as a system of two equations:

${\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}+(u\cdot \nabla )u=-\nabla p,\\\nabla \cdot u=0.\end{cases}}}$ For stationary flows, where the velocity field ${\displaystyle u}$ does not change with time, i.e. ${\displaystyle {\frac {\partial u}{\partial t}}=0}$, we can introduce the Bernoulli function, ${\displaystyle B:=p+{\frac {1}{2}}\lVert u\rVert ^{2}}$, and the vorticity, ${\displaystyle \omega$ :=\nabla \times u} . These new variables simplify the Euler equations into the following system:

${\displaystyle {\begin{cases}u\times \omega =\nabla B,\\\nabla \cdot u=0.\end{cases}}}$ The simplification is possible due to a vector identity, which relates the convective term ${\displaystyle (u\cdot \nabla )u}$ to the gradient of the kinetic energy and the cross product of the velocity field and its curl:

${\displaystyle (u\cdot \nabla )u={\frac {1}{2}}\nabla \lVert u\rVert ^{2}-u\times (\nabla \times u)}$

When the Bernoulli function ${\displaystyle B}$ is constant, Beltrami fields become valid solutions to the simplified Euler equations. Note that we do not need the proportionality factor to be constant for the proof to work.

See also

• Beltrami flow
• Complex lamellar vector field
• Conservative vector field

Bibliography

• Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0-486-66110-5
• Lakhtakia, Akhlesh (1994), Beltrami fields in chiral media, World Scientific, ISBN 981-02-1403-0
• Etnyre, J.; Ghrist, R. (2000), "Contact topology and hydrodynamics. I. Beltrami fields and the Seifert conjecture", Nonlinearity, 13 (2): 441–448, Bibcode:2000Nonli..13..441E, doi:10.1088/0951-7715/13/2/306.

References