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\begin{document}

\thispagestyle{empty}

\begin{center}
{\large
{\bf 
Supplementary Material for Convectons in a Rotating Fluid Layer\\
\vspace{0.1cm}
By C. Beaume, A. Bergeon, H.-C. Kao and E. Knobloch\\
\vspace{0.1cm}
Published in the Journal of Fluid Mechanics\\
}
}
\end{center}

\section*{Appendix B. Derivation of the fifth order Ginzburg-Landau equation}

This file provides the pertinent details of the derivation of the nonlocal fifth order Ginzburg-Landau equation
\begin{equation}
\mu A+A_{XX}+i\left(\gamma A_X+a_1|A|^2A_X+a_2A^2A_X^*\right)+b|A|^2A-|A|^4A=0,\label{gle}
\end{equation}
and of the coefficients
\begin{eqnarray}
&&\mu=\mu_0+\mu_1\left<|A|^2\right>+\mu_2\left<|A|^4\right>+\mu_3\mbox{Im}\left[\left<AA_X^*\right>\right]+\mu_4\left<|A|^2\right>^2,\nonumber\\
&&\gamma=\gamma_0+\gamma_1\left<|A|^2\right>,\quad b=b_0+b_1\left<|A|^2\right>\label{nonlocal}
\end{eqnarray}
used in the accompanying paper Convectons in a Rotating Fluid Layer by C. Beaume, A. Bergeon, H.-C. Kao and E. Knobloch, published in the Journal of Fluid Mechanics.

We start with the equations of motion \citep{Veronis59}
\begin{eqnarray}
Ra\theta_x-Tv_z+\nabla^4\psi &=& \sigma^{-1}\left[\nabla^2\psi_t+J(\psi,\nabla^2\psi)\right],\label{psi}\\
\psi_x+\nabla^2\theta &=& \theta_t+J\left(\psi,\theta\right),\label{theta}\\
T\psi_z+\nabla^2 v &=& \sigma^{-1}\left[v_t+J(\psi,v)\right]\label{v}
\end{eqnarray}
subject to the boundary conditions 
\begin{eqnarray}
\psi=\psi_{zz}=\theta=v_z=0\mbox{ at }z\in\left\{0,1\right\}.
\label{bc}
\end{eqnarray}

Equation (\ref{gle}) describes the above problem near the critical Rayleigh number $Ra_c$ for the onset of convection and the critical Taylor number $T_c$ determined by the degeneracy condition $\xi^2=1/3$ \citep{Cox01}. Here $\xi\equiv\frac{T\pi^2}{\sqrt{3}pk^2\sigma}$, $p\equiv k^2+\pi^2$ and the wavenumber $k$ represents the critical wavenumber corresponding to $Ra_c$. Consequently, we write $Ra=Ra_c+\epsilon^2r_2+\epsilon^4r_4$ and $T=T_c+\epsilon^2\delta$, where $\epsilon\ll1$. We also introduce a slow spatial scale $X\equiv \epsilon^2x$ and write 
\begin{eqnarray}
\psi \sim \sum_{n=1}^{\infty}\epsilon^{n}\psi_{n}(x,X,z),\quad \theta \sim \sum_{n=1}^{\infty}\epsilon^{n}\theta_{n}(x,X,z),\quad v \sim v_0(X)+\sum_{n=1}^{\infty}\epsilon^{n}v_{n}(x,X,z).\label{exp}
\end{eqnarray}
The leading order term in the zonal velocity is now of order one instead of being of order $\epsilon$ \citep{Cox01}. To simplify the expressions that follow we use the notation
\begin{eqnarray*}
&&\nabla^2\equiv \partial_{xx}+\partial_{zz},\qquad J(u,w)\equiv u_xw_z-u_zw_x,\qquad\tilde{J}(u,w)\equiv u_Xw_z-u_zw_X,\\
&&p_{nm}\equiv -\left(n^2k^2+m^2\pi^2\right)^3+Ra_cn^2k^2-T_c^2m^2\pi^2.
\end{eqnarray*}
Expansion (\ref{exp}) leads, order by order, to linear inhomogeneous problems of the form
\begin{equation}
\mathbf{M}\mathbf{\Psi}_n \equiv \left(\begin{array}{ccc}\nabla^4 & Ra_c\partial_x & -T_c\partial_z \\ \partial_x & \nabla^2 & 0\\ T_c\partial_z & 0 & \nabla^2\end{array}\right)\mathbf{\Psi}_n= \mathbf{f}_n,\label{eq0}
\end{equation}
where $\mathbf{\Psi}_n$ represents $\left(\psi_n,\theta_n,v_n\right)^T$ and $\mathbf{f}_n$ is a vector with components
that are polynomials in $\psi_1,...,\psi_{n-1}$, $\theta_1,...,\theta_{n-1},v_0,...,v_{n-1}$, and their derivatives. 
Equation~(\ref{eq0}) can also be written as a single equation with respect to $\psi_n$,
\begin{equation}
M\psi_n\equiv\left(\nabla^6-Ra_c\partial_{xx}+T_c^2\partial_{zz}\right)\psi_n = \nabla^2f_{n1}-Ra_c\partial_xf_{n2}+T_c\partial_zf_{n3}.\label{eq01}
\end{equation}
We solve this equation for $\psi_n$ and determine the corresponding $\theta_n$ and $v_n$ from Eq.~(\ref{eq0}).

\subsubsection*{First \& second order}

At $O(\epsilon)$ $\mathbf{f}_1=0$. The resulting homogeneous problem has a solution of the form
\begin{eqnarray*}
\psi_1 &=& \frac{a(X)}{2}e^{ikx}\sin(\pi z)+c.c.,\\
\theta_1 &=& \frac{ika(X)}{2p}e^{ikx}\sin(\pi z)+c.c.,\\
v_1 &=& \frac{\pi T_ca(X)}{2p}e^{ikx}\cos(\pi z)+c.c.,
\end{eqnarray*}
where $k$ and $\pi$ are, respectively, the wavenumbers in the $x$ and $z$ directions. The critical value $Ra_c$ and the critical wavenumber $k$ satisfy 
\begin{equation}
Ra_c=3p^2,\qquad p^2\left(2k^2-\pi^2\right)=T^2\pi^2.
\end{equation}
Note that the linear operator $M$ in Eq.~(\ref{eq01}) is self-adjoint with kernel $e^{\pm ikx}\sin(\pi z)$. This fact
simplifies the solvability condition applied at each subsequent order.

At $O(\epsilon^2)$ $\mathbf{f}_2$ is given by
\begin{eqnarray*}
f_{21}&=&\sigma^{-1}J(\psi_1,\nabla^2\psi_1),\\
f_{22}&=&-J(\theta_1,\psi_1),\\
f_{23}&=&-J(v_1,\psi_1).
\end{eqnarray*}
The solvability condition for $\psi_2$ is always satisfied. Thus $\psi_2$ may be set to zero and $\mathbf{\Psi}_2$ is given by
\begin{eqnarray*}
\psi_{2}&=&0,\\
\theta_{2}&=& -\frac{k^2|a|^2}{8\pi p}\,\sin(2\pi z),\\
v_{2}&=&v_{20}(X)+\frac{iT_c\pi^2 a^2}{16kp\sigma}\,e^{2ikx}+c.c.
\end{eqnarray*}
The homogeneous term $v_{20}(X)$ plays an important role in what follows.
%{\bf Except $v_{20}(X)$,} no solutions of the homogeneous problem are included here and elsewhere since these can be removed by redefining the expansion parameter $\epsilon$.

\subsubsection*{Third order}

At $O(\epsilon^3)$ $\mathbf{f}_3$ is given by
\begin{eqnarray*}
f_{31} &=& -4\nabla^2\psi_{1,Xx}-Ra_c\theta_{1,X}+\delta v_{1,z}+\sigma^{-1}\left[J(\psi_2,\nabla^2\psi_1)+J(\psi_1,\nabla^2\psi_2)\right]-r_2\theta_{1,x},\\
f_{32}&=&J\left(\psi_1,\theta_2\right)+J(\psi_2,\theta_1)-\psi_{1,X}-2\theta_{1,Xx},\\
f_{33}&=&-\delta\psi_{1,z}-2v_{1,Xx}+\sigma^{-1}\left[J\left(\psi_1,v_2\right)+J(\psi_2,v_1)-\psi_{1,z}v_{0,X}\right].
\end{eqnarray*}
The solvability condition at this order gives
\begin{equation}
\left(\frac{k^2 r_2}{2}-T_c\pi^2\delta\right)a-\frac{T_c\pi^2}{2\sigma}av_{0,X}+\left(\frac{T_c^2\pi^4}{16p\sigma^2}-\frac{3pk^4}{16}\right)|a|^2a=0.\label{cod3}
\end{equation}

In traditional approaches which do not include spatial modulation the result (\ref{cod3}) with $r_2=\delta=0$ reduces to $\xi^2=1$, i.e., the relation that determines the location of the codimension two point at which a subcritical periodic wavetrain becomes supercritical \citep{Veronis59}.

The solution $\mathbf{\Psi}_3$ is
\begin{eqnarray*}
\psi_3&=&-\frac{T_c^2\pi^4a^3}{16p\sigma^2p_{31}}\,e^{3ikx}\sin(\pi z)-\frac{3pk^4|a|^2a}{16p_{13}}\,e^{ikx}\sin(3\pi z)+c.c.,\\
\theta_3&=&\frac{i k^3(p_{13}-3p^2k^2)|a|^2a}{16pp_{13}(k^2+9\pi^2)}\,e^{ikx}\sin(3 \pi z)-\frac{3iT_c^2k \pi^4a^3}{16pp_{31}\sigma^2(9k^2+\pi^2) }\,e^{3 ikx}\sin(\pi z)\nonumber\\
&&+\frac{(\pi^2-k^2)a_{X}}{2 p^2}\,e^{ikx}\sin(\pi z)-\frac{ik^3 |a|^2a}{16 p^2}\,e^{ikx}\sin(\pi z)+c.c.,\\
v_3&=&\biggl[\frac{\pi\left(v_{0,X}+\delta\sigma\right)a}{2\sigma p}+\frac{iT_ck\pi a_{X}}{p^2}-\frac{T_c\pi^3|a|^2a}{16 p^2\sigma^2}\biggr]e^{ikx}\cos(\pi z)\nonumber\\
&&-\frac{9pT_ck^4 \pi|a|^2a}{16p_{13}(k^2+9\pi^2)}\,e^{ikx}\cos(3\pi z)-\frac{T_c\pi^3a^3\left(T_c^2\pi^2+p_{31}\right)}{16 \sigma^2pp_{31} (9 k^2+\pi^2)}\,e^{3 ikx}\cos(\pi z)+c.c.
\end{eqnarray*}

\subsubsection*{Fourth order}

At $O(\epsilon^4)$ $\mathbf{f}_4$ is given by
\begin{eqnarray*}
f_{41}&=&\sigma^{-1}\left[2J\left(\psi_1,\psi_{1,Xx}\right)+\tilde{J}\left(\psi_1,\nabla^2\psi_1\right)+J\left(\psi_1,\nabla^2\psi_3\right)+J\left(\psi_3,\nabla^2\psi_1\right)\right.\\
&&\left.+J(\psi_2,\nabla^2\psi_2)\right]-r_2\theta_{2,x}-Ra_c\theta_{2,X}+\delta v_{2,z}-4\nabla^2\psi_{2,Xx},\\
f_{42}&=&-\psi_{2,X}-2\theta_{2,Xx}-\tilde{J}\left(\theta_1,\psi_1\right)-J\left(\theta_3,\psi_1\right)-J\left(\theta_2,\psi_2\right)-J\left(\theta_1,\psi_3\right),\\
f_{43}&=&-\sigma^{-1}\left[v_{0,X}\psi_{2,z}+\tilde{J}\left(v_1,\psi_1\right)+J\left(v_1,\psi_3\right)+J\left(v_2,\psi_2\right)+J\left(v_3,\psi_1\right)\right]\\
&&-2v_{2,Xx}-v_{0,XX}-\delta\psi_{2,z}.
\end{eqnarray*}

The solvability condition for $\psi_4$ is always satisfied at this order. However, the Laplace operator appearing 
in the equation for $v_4$ has a nonempty kernel spanned by multiples of $v_4=1$, a fact that is responsible for 
the presence of a second solvability condition, viz.,
\begin{equation}
v_{0,XX}+\frac{T_c\pi^2}{4p\sigma}(|a|^2)_X=0.\label{cod4}
\end{equation}
From Eqs.~(\ref{cod3}) and (\ref{cod4}) we obtain the condition $\xi^2=1/3$ that defines the critical Taylor number $T_c$:
\begin{equation}
T_c=T_{\pm}^{mod}\equiv\frac{\sigma\pi^2(2\pm\sqrt{1-\sigma^2})}{\left(1\pm\sqrt{1-\sigma^2}\right)^2}.
\end{equation}
Note that two such critical values of $T$ are present.

Now that all solvability conditions have been imposed we can proceed to solve the $O(\epsilon^4)$ problem. Here we only list the terms which enter into the computation of the solvability conditions arising at $O(\epsilon^5)$ and $O(\epsilon^6)$. These are
\begin{eqnarray*}
\psi_4 &=&-\frac{k^2 \pi(2 T_c^2 \pi^2+4 p^2 \pi^2+3 \sigma p^3)(|a|^2)_X}{2 p^2 p_{02} \sigma}\,\sin(2 \pi z)+...,\\
\theta_4&=&\frac{k^4(p_{13}-3p^2k^2)(k^2+5\pi^2)|a|^4}{32\pi p^2p_{13}(k^2+9 \pi^2)}\,\sin(2\pi z)+\left\{\frac{ik\pi a^*a_X}{8 p^2}\,\sin(2\pi z)+c.c.\right\}+...,\\
v_4&=&-\frac{k^4(4T_c^2\pi^2+8p^2\pi^2+6\sigma p^3+p_{02})(|a|^2)_X}{8pp_{02}\pi^2}\,\cos(2\pi z)\nonumber\\
&&+\left\{-\frac{ik\pi^2(T_c^2\pi^2+p_{31})(5 k^2+\pi^2)|a|^2a^2}{64pp_{31}\sigma^2(9 k^2+\pi^2)}\,e^{2ikx}+\frac{i\pi^2(v_{0,X}+\delta\sigma)a^2}{16kp\sigma^2}\,e^{2ikx}\right.\nonumber\\
&&-\left.\frac{(3k^2+\pi^2)aa_X}{16p}\,e^{2ikx}+c.c.\right\}+....
\end{eqnarray*}

\subsubsection*{Fifth \& sixth order}

At $O(\epsilon^5)$ and $O(\epsilon^6)$, $\mathbf{f}_5$ and $f_{63}$ are given by
\begin{eqnarray*}
f_{51}&=&\sigma^{-1}\left[\sum_{n=1}^2\left(2J(\psi_n,\psi_{3-n,Xx})+\tilde{J}(\psi_n,\nabla^2\psi_{3-n})\right)+\sum_{n=1}^{4}J(\psi_n,\nabla^2\psi_{5-n})\right]-4\psi_{1,XXxx}\nonumber\\
&&-2\nabla^2\psi_{1,XX}-4\nabla^2\psi_{3,Xx}-Ra_c\theta_{3,X}-r_4\theta_{1,x}+\delta v_{3,z}-r_2\theta_{1,X}-r_2\theta_{3,x},\\
f_{52}&=&-\psi_{3,X}-\theta_{1,XX}-2\theta_{3,Xx}-\sum_{n=1}^4J(\theta_n,\psi_{5-n})-\sum_{n=1}^2\tilde{J}(\theta_n,\psi_{3-n}),\\
f_{53}&=&-\sigma^{-1}\left[\sum_{n=1}^4J(v_n,\psi_{5-n})+\sum_{n=1}^2\tilde{J}(v_n,\psi_{3-n})+v_{0,X}\psi_{3,z}\right]-v_{1,XX}-2v_{3,Xx}-\delta\psi_{3,z},\\
f_{63}&=&-\sigma^{-1}\left[\sum_{n=1}^3\tilde{J}(v_n,\psi_{4-n})+\sum_{n=1}^5J(v_n,\psi_{6-n})+v_{0,X}\psi_{4,z}\right]-v_{2,XX}-2v_{4,Xx}-\delta\psi_{4,z}.
\end{eqnarray*}
The solvability condition for $\psi_5$ yields
\begin{eqnarray}
&&\left(\tilde{\mu}_0+\tilde{\mu}_1v_{0,X}+i\tilde{\mu}_2v_{0,XX}\right)a+\tilde{d}a_{XX}+i\left[\left(\tilde{\gamma}_0 +\tilde{\gamma}_1v_{0,X}\right)a_X+\tilde{a}_{10}|a|^2a_{X}+\tilde{a}_{20}a^2\bar{a}_{X}\right]\nonumber\\
&&+\left(\tilde{b}_0+\tilde{b}_1v_{0,X}\right)|a|^2a-\tilde{c}_0|a|^4a-\frac{pk^2}{2}av_{20,X}=0\label{cod5}
\end{eqnarray}
with coefficients given by
\begin{eqnarray*}
\tilde{\mu}_0&=& \frac{k^2r_4-\delta^2\pi^2}{2},\quad \tilde{\mu}_1=-\frac{\delta\pi^2}{2\sigma},\quad\tilde{d}=6k^2p,\quad\tilde{\gamma}_0=-\frac{k\pi^2(r_2+2T_c\delta)}{p},\\
\tilde{\mu}_2&=&\tilde{\gamma}_1=-k^3,\quad\tilde{b}_0=\frac{2p\delta\pi^2-k^2r_2\sigma}{16p\sigma}k^2,\quad\tilde{b}_1=\frac{k^2\pi^2}{16\sigma^2},\\
\tilde{a}_{10}&=&\frac{k^3p(4k^2+2\pi^2+3p\sigma)(2k^4-9\pi^4+7k^2\pi^2+6k^2\pi^2\sigma)}{8p_{02}\sigma^2}+\frac{9k^3(3\pi^2+k^2)}{32}+\frac{k^7}{16\pi^2},\\
\tilde{a}_{20}&=&\frac{k^3p(4k^2+2\pi^2+3p\sigma)(2k^4-9\pi^4+7k^2\pi^2+6k^2\pi^2\sigma)}{8p_{02}\sigma^2}+\frac{k^3(3\pi^2+k^2)}{32}+\frac{k^7}{16\pi^2},\\
\tilde{c}_0&=&\frac{p^2k^8(19k^2+3\pi^2)}{128p_{31}(9k^2+\pi^2)}+\frac{k^4\pi^2(5k^2+\pi^2)}{64\sigma^2(9k^2+\pi^2)}+\frac{9k^8p^2}{128p_{13}}-\frac{3k^6(k^2+5\pi^2)(p_{13}-3k^2p^2)}{64p_{13}(k^2+9\pi^2)}.
\end{eqnarray*}
The solvability condition for $v_6$ yields
\begin{equation}
v_{20,XX}+\left(\frac{k^3}{2p}\mbox{Im}[aa_X^*]+\frac{\pi^2|a|^2v_{0,X}}{4p\sigma^2}+\frac{\delta\pi^2|a|^2}{4p\sigma}-\frac{k^2\pi^2|a|^4}{32p\sigma^2}\right)_X=0.\label{cod6}
\end{equation}
Equation (\ref{gle}) follows from (\ref{cod5}) on integrating conditions (\ref{cod4}) and (\ref{cod6}) with respect to $X$.

\bibliographystyle{jfm}
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\end{document}
