Yes there is.
To me, the best way to understand this is through the idea of multipole expansion. You can even do this for an electrostatic potential --- and then, via the symmetry apparent with Maxwell equations, figure out a similar expansion for a magnetic (vector) potential.
To see the derivation of a multipole expansion, you can consult Griffith's Electrodynamics. This pdf seems to offer a similar, if not identical, derivation.
In short, the multipole expansion for an electrostatic potential, achieved with little more than a Taylor series expansion, reads, in SI:
$$V(r) = \frac{1}{4 \pi \varepsilon_0}\frac{Q}{r}\sum_{i=0}^{\infty}\left(\frac{r'}{r}\right)^n P_n(\cos \theta)$$
Where
- $Q$ is the total charge of the system,
- $r$ is a distance from a
point of interest to the origin (you choose the origin arbitrarily;
generally, your expansion depends on the choice of origin),
- $r'$ is a
distance from any given charge to the origin, and
- $P_n(\cos \theta)$
are Legendre
polynomials.
Note that if the total charge of the system is zero, our monopole (zeroth) moment term, $\frac{Q}{r}$, is also zero. The term that follows the zeroth term is a familiar dipole term.
By symmetry, you can easily verify that the corresponding multipole expansion for a magnetic (vector) potential is as follows:
$$\vec{A}(r) = \frac{\mu_0}{4 \pi} \sum_{i=0}^{\infty} \left( \frac{1}{r} \right)^{n+1} \oint (r')^n P_n (\cos \theta) \vec{I}(r') dl' $$
Mathematically speaking, I just transcribed $\frac{1}{\varepsilon_{0}} \Rightarrow \mu_{0}$, and replaced a small piece of charge $dq$ with $I(r')dl'$. Note that here the monopole term is always zero. The rest gives an expansion of a magnetic (vector) potential. You will have a dipole term, a quadrupole term, an octopole term, etc.
For a detailed derivation, you can, again, read Griffiths or consult this source.
Now that you have a magnetic potential, you can, of course, calculate the magnetic field. It is usually cumbersome now that we don't have a scalar quantity, but is seldom useful in certain applications.
