I'm trying to follow the derivation of a 9D Lorenz type system in the following paper by Reiterer et al (1998), and am having difficulties:
https://www.academia.edu/17449586/A_nine-dimensional_Lorenz_system_to_study_high-dimensional_chaos
Background
Considering a fluid between $z\in [0,\pi]$ and with $x,y$ being unbounded, I am happy getting up to the point where the equations: $$\nabla \cdot \vec u = 0, \\ \partial_t\vec \omega - \nabla\times(\vec u \times \vec \omega)=\sigma \nabla \times (\theta\vec e_z) + \sigma \nabla^2\vec \omega, \\ \partial_t \theta + (\vec u \cdot \nabla)\theta = \nabla^2 \theta + Rw, $$ have been derived, with $\vec u = (u, v, w), \,\,\vec \omega = \nabla \times \vec u$ and $R, \sigma$ being the Rayleigh and Prandtl numbers respectively.
I am also happy with the derivation of the boundary conditions on $z= 0,\pi$: $$w = \partial^2_z w = \partial_z u = \partial_z v = \theta = 0.$$
Then since the flow is divergence free, we can write: $$ \vec u = \nabla \times \vec A $$ for some vector field $\vec A$, and since we're interested in square convection cells, we choose $1/\alpha$ to be the period in the $x, y$ directions.
Difficulties
Where I run into difficulties is getting the form of the Fourier series ansatz the authors use.
My Fourier series for the $i$th component of $\vec A$ is as follows:
$$A_i(\vec x, t) = \sum_{\vec n \in \mathbb Z^d} \hat {A_i}(\vec n, t) \exp\left(2\pi i \left({ n_1 }{\alpha} x + { n_2 }{\alpha} y +\frac{ n_3 }{\pi} z \right) \right)$$
This is quite different from the form the authors have - I have a stray factor of $2\pi$ floating a round. Also, theirs is in a much simpler form using only a cosine/sine for each component.
I'm unsure of where to go from here to get my ansatz into the form that theirs takes.
I wondered if perhaps this was some sort of physics convention.
Any help or hints would be very much appreciated. This is for a master's thesis and I'm not terribly experienced with Fourier series.