Terminology and notation for complicated roots I know that $a^*$ is the second largest root of the following equation:
$$32\left(\sqrt{50-10\sqrt{5}}\right)a^3-\frac{48}{5}(5+\sqrt{5}) \pi a + 8\sqrt{10(5+\sqrt{5})} = 0  $$
Note that $\pi$ is the constant, not a variable.

Q1: What would you call such an equation?

Perhaps a cubic equation with irrational coefficients?
I also want to express this as neatly as possible, maybe writing something like:
$$a^* = \text{Root}\left(32\left(\sqrt{50-10\sqrt{5}}\right)a^3-\frac{48}{5}(5+\sqrt{5}) \pi a + 8\sqrt{10(5+\sqrt{5})} = 0,2\right).$$

Q2: Does an operator (i.e. symbol denoting a mathematical operation, not this) like that exist?

I'm sure some will think it more elegant to just define $a^*$ as I did to start this question, and others will think it better to give the solution in radicals. The problem with the former is that, while more aesthetically appealing, it's a little more difficult to use in tables and perhaps a bit annoying if used repetitively. The problem with the latter is that it is very messy (relative to the operator approach I try above), even after simplifying as much as possible. That's why I'd prefer to define $a^*$ with an operator if an appropriate one exists.

Note: An ideal answer would answer both Q1 and Q2. It may be that such an answer would simply be "Yes and no, respectively." But, if that is the case, I would appreciate any commentary on how you would recommend representing the root in a journal article.

 A: Before answering the questions, it is desirable to convert the equation to the foreseeable mind. In this case we use the golden section Mauger constant
$$\varphi=\dfrac{\sqrt5+1}2\approx1.618,$$
then
$$\dfrac{\sqrt5-1}2=\dfrac1\varphi,$$
and original equation takes the form
$$\dfrac{64\cdot5^{3/4}}{\sqrt\varphi}a^3-\dfrac{96\pi\varphi}{\sqrt5}a+16\cdot5^{3/4}\sqrt\varphi = 0,$$
or
$$4a^3-\dfrac{6\pi\varphi^{3/2}}{5^{5/4}}a+\varphi=0.$$
Number $\pi$ is irrational (transcendental), multiplication and division on rational and irrational numbers does not change the situation. Consequently,
$$\boxed{\text{Q1: yes, it's a cubical equation with irrational (transcendental) coefficients.}}$$
Both the original and changed forms give the same result.
For an elegant answer to the second question, we introduce the coefficient
$$\mu=\dfrac{\sqrt{2\pi}\varphi^{3/4}}{5^{5/8}}\approx1.315,$$
then equation takes the form
$$4a^3-3\mu^2a+\varphi=0.\qquad(1)$$
The precise mathematical formula for recording the second-highest root is unknown to me. At the same time, the Mathcad package required value is given a convenient and understandable expression of the form
$$\boxed{\text{Q2a:}\qquad
a^* = \mathrm{polyroots}\left(
\begin{pmatrix}
\varphi\\
-3\mu^2\\
4
\end{pmatrix}
\right)_1}$$
The highest degree coefficient places below others and the result of polyroots() function is ordered vector of roots wherein the root with the greatest real part has index $0.$  
Also the equation $(1)$ can be presented in the form
$$4\left(\dfrac{a}\mu\right)^3-3\dfrac{a}\mu=-\dfrac\varphi{\mu^3},$$
or
$$\mathrm T_3\left(\dfrac{a}\mu\right) = -\dfrac\varphi{\mu^3},$$
where
$$\mathrm T_3(x)=4x^3-3x$$ is a Chebyshev polynomial of the first kind.
Note that
$$\left|-\dfrac\varphi{\mu^3}\right|\approx 0.711 <1,$$
so we can use the Chebyshev polynomial in form
$$\mathrm T_3(x) = \cos(3\arccos x)$$
and get the equation
$$\cos\left(3\arccos \left(\dfrac{a}\mu\right)\right) = -\dfrac\varphi{\mu^3}$$
with the ordered roots as in Mathcad
$$a_k=\mu\cos\left(\dfrac13\arccos\dfrac\varphi{\mu^3}+\dfrac{2k-1}3\pi\right),\quad k=0,1,2.$$
Thus, 
$$\boxed{\text{Q2b:}\quad a^*=\mu\cos\left(\dfrac13\arccos\dfrac\varphi{\mu^3}+\pi/3\right)}$$
A: Q1 : yes, it's a cubic equation with irrational coefficients. 
