In vector calculus, a topic in pure and applied mathematics, a **poloidal–toroidal decomposition** is a restricted form of the Helmholtz decomposition. It is often used in the spherical coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids.

For a three-dimensional vector field **F** with zero divergence

∇
⋅
F
=
0
,
this **F** can be expressed as the sum of a toroidal field **T** and poloidal vector field **P**

F
=
T
+
P
where **r** is a radial vector in spherical coordinates (*r*, *θ*, *φ*). The toroidal field is obtained from a scalar field, Ψ(*r*, *θ*, *φ*), as the following curl,

T
=
∇
×
(
r
Ψ
(
r
)
)
and the poloidal field is derived from another scalar field Φ(*r*, *θ*, *φ*), as a twice-iterated curl,

P
=
∇
×
(
∇
×
(
r
Φ
(
r
)
)
)
.
This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal.

A toroidal vector field is tangential to spheres around the origin,

r
⋅
T
=
0
while the curl of a poloidal field is tangential to those spheres

r
⋅
(
∇
×
P
)
=
0
.

The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius *r*.

A poloidal–toroidal decomposition also exists in Cartesian coordinates, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as

F
(
x
,
y
,
z
)
=
∇
×
g
(
x
,
y
,
z
)
z
^
+
∇
×
(
∇
×
h
(
x
,
y
,
z
)
z
^
)
+
b
x
(
z
)
x
^
+
b
y
(
z
)
y
^
,
where
x
^
,
y
^
,
z
^
denote the unit vectors in the coordinate directions.