Appendix 1
The terms in (2.10) of Sect. 2 are as follows:
$$\begin{aligned} \begin{bmatrix} M \end{bmatrix}&= \begin{bmatrix} 1+\frac{a_h^2+\frac{1}{8}}{\mu r_\alpha ^2}&\frac{x_\alpha }{r_\alpha ^2}-\frac{a_h}{\mu r_\alpha ^2}&0&0\\ x_\alpha -\frac{a_h}{\mu }&1+\frac{1}{\mu }&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix}, \\ \left[ {\begin{array}{c} {D(\tau )} \\ \end{array} } \right] &= \left[ {\begin{array}{cccc} {\frac{{u^{\epsilon } \left( {\frac{1}{2} - a_{h} } \right)^{2} }}{{\mu r_{\alpha }^{2} }}} & { - \frac{{u^{\epsilon } \left( {\frac{1}{2} + a_{h} } \right)}}{{\mu r_{\alpha }^{2} }}} & { - \frac{{2u^{\epsilon } \left( {\frac{1}{2} + a_{h} } \right)\left( {A_{1}^{W} b_{1}^{W} + A_{2}^{W} b_{2}^{W} } \right)}}{{\mu r_{\alpha }^{2} }}} & { - \frac{{2\left( {\frac{1}{2} + a_{h} } \right)\left( {A_{1}^{K} b_{1}^{K} + A_{2}^{K} b_{2}^{K} } \right)}}{{\mu r_{\alpha }^{2} }}} \\ {\frac{{u^{\epsilon } \left( {\frac{3}{2} - a_{h} } \right)}}{\mu }} & {\frac{{u^{\epsilon } }}{\mu }} & {\frac{{2u^{\epsilon } \left( {A_{1}^{W} b_{1}^{W} + A_{2}^{W} b_{2}^{W} } \right)}}{\mu }} & {\frac{{2\left( {A_{1}^{K} b_{1}^{K} + A_{2}^{K} b_{2}^{K} } \right)}}{\mu }} \\ {a_{h} - \frac{1}{2}} & { - 1} & {b_{1}^{W} + b_{2}^{W} } & 0 \\ 0 & 0 & 0 & {b_{1}^{K} + b_{2}^{K} } \\ \end{array} } \right],\\ \begin{bmatrix} K(\tau )\end{bmatrix}&= \left[ \begin{array}{ccc} \frac{1}{U_m^2} - \frac{(u^{\epsilon })^2\left( \frac{1}{2}+a_h\right) }{\mu r_\alpha ^2} &{} 0 \\ \frac{(u^{\epsilon })^2}{\mu } &{} \frac{\left( \omega _h/\omega _\alpha \right) ^2}{U_m^2} \\ -u^{\epsilon } &{} 0 \\ 0 &{} 0 \end{array}\right. \left. \begin{array}{cc} -\frac{{(u^{\epsilon })^2\left( \frac{1}{2}+a_h\right) b_1^{W} b_2^W}}{\mu r_\alpha ^2} &{} -\frac{2\left( \frac{1}{2}+a_h\right) b_1^Kb_2^K}{\mu r_\alpha ^2} \\ \frac{u^{\epsilon } b_1^Wb_2^W}{\mu } &{} \frac{2b_1^Kb_2^K}{\mu } \\ b_1^Wb_2^W &{} 0 \\ 0 &{} b_1^Kb_2^K \end{array}\right] ,\\ \begin{bmatrix} K_3 \end{bmatrix}&= \begin{bmatrix} \frac{K_3}{U_m^2}&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix}, \end{aligned}$$
where \(u^{\epsilon }\mathop {=}\limits ^{\mathrm{def}}1+\epsilon u_T(\tau )\), \(w^{\epsilon }:=\epsilon \hbox {w}_T(\tau )\), and \(\omega _h\) and \(\omega _\alpha \) are the natural frequencies of heave and pitch [frequencies of the solutions to the decoupled, unforced Eq. (2.1)].
As mentioned in Sect. 2, the damping and stiffness matrices [D] and [K] can be decomposed into their respective time invariant and time varying components:
$$\begin{aligned}{}[D(\tau )]&= [D_0] + \epsilon u_T (\tau ) [D_1], \\ [K(\tau )]&= [K_0] + \epsilon u_T (\tau ) [K_1], \end{aligned}$$
where
$$\begin{aligned} \begin{bmatrix} D_0 \end{bmatrix}&= \left[\begin{array}{cccc} \frac{\left( \frac{1}{2}-a_h\right) ^2}{\mu r_\alpha ^2} &-\frac{\left( \frac{1}{2}+a_h\right) }{\mu r_\alpha ^2} & -\frac{2\left( \frac{1}{2}+a_h\right) \left( A_1^Wb_1^W+A_2^Wb_2^W\right) }{\mu r_\alpha ^2}& -\frac{2\left( \frac{1}{2}+a_h\right) \left( A_1^Kb_1^K+A_2^Kb_2^K\right) }{\mu r_\alpha ^2}\\ \frac{\left( \frac{3}{2}-a_h\right) }{\mu } & \frac{1}{\mu } & \frac{2\left( A_1^Wb_1^W+A_2^Wb_2^W\right) }{\mu } & \frac{2\left( A_1^Kb_1^K+A_2^Kb_2^K\right) }{\mu }\\ a_h-\frac{1}{2} & -1 & b_1^W+b_2^W & 0\\ 0 & 0 & 0& b_1^K+b_2^K \end{array}\right] , \\ \begin{bmatrix} D_1 \end{bmatrix}&=\left[ \begin{array}{cc} \frac{\left( \frac{1}{2}-a_h\right) ^2}{\mu r_\alpha ^2} & -\frac{\left( \frac{1}{2}-a_h\right) }{\mu r_\alpha ^2} \\ \frac{1}{\mu }\left( \frac{3}{2}-a_h\right) & \frac{1}{\mu } \\ 0 & 0 \\ 0 & 0 \end{array}\right. \left. \begin{array}{cc} -\frac{2\left( \frac{1}{2}+a_h\right) \left( A_1^Wb_1^W+A_2^Wb_2^W\right) }{\mu r_\alpha ^2} & 0\\ \frac{2\left( A_1^Wb_1^W+A_2^Wb_2^W\right) }{\mu } & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right] , \\ \begin{bmatrix} K_0 \end{bmatrix}&=\left[ \begin{array}{cc} \frac{1}{U_m^2} - \frac{\left( \frac{1}{2}+a_h\right) }{\mu r_\alpha ^2}& 0\\ \frac{1}{\mu } & \frac{\left( \omega _h/\omega _\alpha \right) ^2}{U_m^2} \\ -1 & 0 \\ 0 & 0\end{array}\right. \left. \begin{array}{cc} -\frac{\left( \frac{1}{2}+a_h\right) b_1^Wb_2^W}{\mu r_\alpha ^2} & -\frac{2\left( \frac{1}{2}+a_h\right) b_1^Kb_2^K}{\mu r_\alpha ^2}\\ \frac{u^{\epsilon } b_1^Wb_2^W}{\mu } & \frac{2b_1^Kb_2^K}{\mu } \\ b_1^Wb_2^W & 0 \\ 0 & b_1^Kb_2^K \end{array}\right] , \\ \begin{bmatrix} K_1 \end{bmatrix}&= \left[ \begin{array}{cccc} - \frac{2\left( \frac{1}{2}+a_h\right) }{\mu r_\alpha ^2} & 0 & -\frac{2\left( \frac{1}{2}+a_h\right) b_1^Wb_2^W}{\mu r_\alpha ^2} & 0 \\ \frac{2}{\mu } & 0 & \frac{b_1^Wb_2^W}{\mu } & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] . \end{aligned}$$
The coefficient matrices in (2.11) of Sect. 2 are as follows:
$$\begin{aligned} \left[ A_0\right]= & {} \left[ Z\right] ^{-1}\left[ L\right] , \quad \left[ B_0\right] = \left[ Z\right] ^{-1}\left[ Q\right] , \\ \left[ C_0\right]= & {} \left[ Z\right] ^{-1}\left[ G\right] , \quad \left[ \hat{N}\right] = \left[ Z\right] ^{-1}\left[ N\right] , \end{aligned}$$
where
$$\begin{aligned} {[}Z]= & {} \begin{bmatrix} I&0 \\ 0&[M] \end{bmatrix}, \quad [L]= \begin{bmatrix} 0&I \\ -[K_0]&-[B_0] \end{bmatrix},\\ {[}Q]= & {} \begin{bmatrix} 0&0 \\ {-[K_1]}&{-[D_1]} \end{bmatrix}\quad [G]=\begin{bmatrix} 0&0 \\ -[K_3]&0 \end{bmatrix} \\ {[}N]= & {} \begin{bmatrix} 0 \\ 0 \\ 0 \\ \hbox {w}_T\end{bmatrix}. \end{aligned}$$
The terms in (2.13) of Sect. 2 are as follows:
$$\begin{aligned}&{\bar{b}}^0\left( {v^{\epsilon }_{\tau }}, U_m^c\right) = \left[ S A_0\left( U_m^c\right) T\right] \, {v^{\epsilon }_{\tau }}= \left[ A\left( U_m^c\right) \right] {v^{\epsilon }_{\tau }},\\&{\bar{b}}^1\left( {v^{\epsilon }_{\tau }}, u_T(\tau ), \hbox {w}_T (\tau )\right) = \wp [S] [\hat{N}] + [S B_0T] u_T (\tau ) {v^{\epsilon }_{\tau }}\\&\quad = \wp [\chi ] + [\bar{B}] u_T (\tau ) {v^{\epsilon }_{\tau }}, \\&{\bar{b}}^2\left( {v^{\epsilon }_{\tau }}, U_m^c\right) = \left[ F u_{\tau }^{\epsilon }+ G \left( {v^{\epsilon }_{\tau }}\right) ^3\right] ,\\&F =\beta \left[ S A_0 '\left( U_m^c\right) T\right] = \beta \left[ A '\left( U_m^c\right) \right] ,\\&G = [S] [C_0], \\&\left( {v^{\epsilon }_{\tau }}\right) ^3 = \left[ \left( \sum _{i=1}^{8}T_{1i} v_{\tau }^{\epsilon , (i)}\right) ^3, \dots ,\left( \sum _{i=1}^{8}T_{8i} v_{\tau }^{\epsilon ,(i)}\right) ^3\right] ^T,\\&[\chi ] = \hbox {w}_T(\tau ) [S_{18}, \dots , S_{88}]^T, \hbox {where} \ [S] = [T]^{-1}. \end{aligned}$$
Due to the block diagonal form of the linear operator, \(A(U_m^c)\), we can write
$$\begin{aligned} A\left( U_m^c\right)&= \left[ \begin{array}{lll}B &{} 0 &{} 0\\ 0 &{} R &{} 0\\ 0 &{} 0 &{} C\end{array}\right] , \,\hbox {where}\, B = \left[ \begin{array}{cc}0 &{} -\omega _0\\ \omega _0 &{} 0\end{array}\right] , \,\omega _0\in \mathbb {R}^{+}\\ R&= \begin{bmatrix}-\kappa&-\gamma \\ \ \gamma&-\kappa \end{bmatrix}, \,\hbox {and}\, C = \hbox {diag}(\lambda _i),\lambda _{i}<0, i=1,\dots ,4. \end{aligned}$$
The remaining terms are
$$\begin{aligned} A'\left( U_m^c\right) = \left[ \begin{array}{ll}D &{} E\\ H &{} J\end{array}\right] , [\bar{B}] = \left[ \begin{array}{ll}\bar{K} &{} \bar{M} \\ \bar{ N} &{} \bar{L}\end{array}\right] , G \left( u_{\tau }^{\epsilon }\right) ^3 = \left[ \begin{array}{l}\bar{g}_1 \left( u_{\tau }^{\epsilon }\right) \\ \vdots \\ \bar{g}_8 \left( u_{\tau }^{\epsilon }\right) \end{array}\right] , \end{aligned}$$
where D and \(\bar{K}\) are \(2\times 2\) matrices, E and \(\bar{M}\) are \(2\times 6\) matrices, H and \(\bar{N}\) are \(6\times 2\) matrices, and J, \(\bar{L}\) are \(6\times 6\) matrices, and
$$\begin{aligned} \bar{g}_m (x) \mathop {=}\limits ^{\mathrm{def}}{\hat{g}}_{m:ijk} x_i x_j x_k = {\hat{g}}_{m:111} x_1^3 + {\hat{g}}_{m:222} x_2^3+ 3{\hat{g}}_{m:122} x_1 x_2^2 + 3 {\hat{g}}_{m:112} x_1^2x_2 \end{aligned}$$
for \(m = 1,\dots ,8\), and \({\hat{g}}_{m:ijk}\) are constants.
Appendix 2
Here, we describe the calculations involved in Sect. 3.2. Substituting the expansion (3.6) into (3.5), we have a set of Poisson equations at increasing orders of \(\epsilon \):
$$\begin{aligned} \begin{aligned} \mathscr {L}_{F}^{(y,\theta ,z)} u_0(x,y,\theta ,z,\tau )&= b^{0}(y) \frac{\partial u_0}{\partial y}(x,y,\theta ,z,\tau ) +\omega _0 \frac{\partial u_0}{\partial \theta }(x,y,\theta ,z,\tau ) \\&\quad +\,\mathcal {G}u_0(x,y,\theta ,z,\tau ) = 0, \\ \mathscr {L}_{F}^{(y,\theta ,z)} u_1(x,y,\theta ,z,\tau )&= -a^0(x,y,\theta ,z) \frac{\partial u_0}{\partial x} (x,y,\theta ,z,\tau ) \\&\quad -\, b^{1}(x,y,\theta ,z) \frac{\partial u_0}{\partial y} (x,y,\theta ,z,\tau ), \\ \mathscr {L}_{F}^{(y,\theta ,z)} u_2(x,y,\theta ,z,\tau )&= -a^0(x,y,\theta ,z) \frac{\partial u_1}{\partial x} (x,y,\theta ,z,\tau ) \\&\quad - \,\,b^{1}(x,y,\theta ,z) \frac{\partial u_1}{\partial y} (x,y,\theta ,z,\tau ) \\&\quad - \,\left( a^{1}(x,y,\theta ) \frac{\partial u_0}{\partial x}(x,y,\theta ,z,\tau ) \right. \\&\quad \left. + \,b^{2}(x,y,\theta ) \frac{\partial u_0}{\partial y}(x,y,\theta ,z,\tau ) - \frac{\partial u_0}{\partial \tau }(x,y,\theta ,z,\tau )\right) , \\&\vdots \end{aligned} \end{aligned}$$
(7.1)
Since \(\mathcal {G}\) is an operator in z alone, it is clear from the first equation of (7.1) that \(u_0(x,y,\theta ,z,\tau )=u(x,\tau )\). Thus, the second and third equations of (7.1) simplifies to
$$\begin{aligned} \begin{aligned} \mathscr {L}_{F}^{(y,\theta ,z)} u_1(x,y,\theta ,z,\tau )&= -a^0(x,y,\theta ,z) \frac{\partial u}{\partial x} (x,\tau ), \\ \mathscr {L}_{F}^{(y,\theta ,z)} u_2(x,y,\theta ,z,\tau )&= -a^0(x,y,\theta ,z) \frac{\partial u_1}{\partial x} (x,y,\theta ,z,\tau ) \\&\quad - b^{1}(x,y,\theta ,z) \frac{\partial u_1}{\partial y} (x,y,\theta ,z,\tau ) \\&\quad - \left( a^{1}(x,y,\theta ) \frac{\partial u}{\partial x}(x,\tau ) - \frac{\partial u}{\partial \tau }(x,\tau )\right) . \\ \end{aligned}\end{aligned}$$
(7.2)
We first investigate the \(\mathcal O(\frac{1}{\epsilon ^2})\) dynamics in (3.3) to obtain the transient and invariant measures of the fast generator \(\mathscr {L}_F^{y,\theta ,z}\). Let us denote by \(\psi _{\tau }\) the flow map induced by the deterministic vector field \(b^{0} (y)\) of (3.3), i.e. \(\psi _{\tau }(y)\) is the flow of \(Y^{\epsilon }\) starting from y if it was driven only by the \(\mathcal O(\frac{1}{\epsilon ^2})\) dynamics. The transient measure is a delta measure centered at \(\psi _{\tau }(y)\): \(\delta _{\psi _{\tau }(y)}(\eta )\). Since the vector field is asymptotically stable, we can associate a measure defined by the following limit
$$\begin{aligned} \delta _{\infty }(\eta ) = \delta _{0}(\eta ) \mathop {=}\limits ^{\mathrm{def}}\lim _{\tau \rightarrow \infty } \delta _{_{\psi _s(y)}}(\eta ). \end{aligned}$$
For the \(\theta ^{\epsilon }\) process, all orbits live on a circle whose radius is dictated by the values of the initial conditions \(x_1\) and \(x_2\) of the process \(X_{\tau }^{\epsilon }\). More precisely, contained in a closed orbit, we can associate a measure defined by the following limit
$$\begin{aligned} \delta _{\infty }(\xi )=\frac{1}{2\pi }= \lim _{\tau \rightarrow \infty } \frac{1}{\tau } \int _0^\tau \delta _{{\theta + \omega _0 s}}(\xi ) \; ds. \end{aligned}$$
For the noise process, it is natural to obtain the final results in terms of the spectral densities of the input noise \(Z^{\epsilon }\) whose generator is \(\mathcal {G}\). To this end, we express the solution in terms of the Green’s function \(g(\zeta ,\tau ;z,0)\) for \(\mathcal {G}^{*}\). The Green’s function for \(\mathcal {G}^{*}\) is the solution of
$$\begin{aligned} \frac{\partial g}{\partial \tau } = \mathcal {G}^{*}g, \quad g(\zeta ,0;z,0) = \delta _0(z-\zeta ). \end{aligned}$$
Since they are independent, the transient density of the \(\mathcal O(\frac{1}{\epsilon ^2})\) components is
$$\begin{aligned} p_s(y, \theta ,z, \eta ,\xi ,\zeta ) = \delta _{_{\psi _s(y)}}(\eta ) \cdot \delta _{_{\omega _0 s + \theta }}(\xi ) \cdot g(\zeta ,s;z,0). \end{aligned}$$
Now we are in a position to evaluate the PDEs (7.2). The coefficients \(a^0(x,y,\theta ,z)\) and \(b^{1}(x,y,\theta ,z)\) in (3.3) are both linearly dependent on the process \(Z^{\epsilon }\), which invariant distribution has zero mean, so
$$\begin{aligned} \mathbb {E}\left[ a^0(x,Y^{\epsilon },\theta ^{\epsilon },Z^{\epsilon })\right] =\mathbb {E}\left[ b^{1}(x,Y^{\epsilon },\theta ^{\epsilon },Z^{\epsilon })\right] =0, \end{aligned}$$
where \(\mathbb {E}[\cdot ]\) is expectation with respect to the invariant density \(p_\infty \). Thus, the Fredholm alternative implies that the first equation of (7.2) has a bounded solution. Now employing the Feynman–Kac formula (see, for example, Chapter 5 of Karatzas and Shreve [27]) and defining
$$\begin{aligned} u_{x}^{\prime }(x,\tau )\mathop {=}\limits ^{\mathrm{def}}\frac{\partial u}{\partial x} (x,\tau ), \quad h(x,y,\theta ,z;\tau ) \mathop {=}\limits ^{\mathrm{def}}a^0(x,y,\theta ,z) u_{x}^{\prime }, \end{aligned}$$
the bounded solution of the first equation of (7.2) is given by
$$\begin{aligned} u_1(x,y,\theta ,z,\tau )= \mathbb{E}_{y,\theta ,z}\left[ \int _0^\infty h\left( Y_s^x,\theta ^x_s,Z_s^x; x,\tau \right) \, ds\right] = \int _0^\infty ds \; \int _{\mathbb {R}^6} \int _{\mathbb{S}} \int _{\mathbb{R}^3} a^0(x,\eta ,\xi ,\zeta ) u_{x}^{\prime } (x,\tau ) p_s(\eta ,\xi ,\zeta ; y,\theta ,z) \; d\eta d\xi d\zeta , \end{aligned} $$
(7.3)
where x and \(\tau \) are parameters and the transient density does not depend on x due to the fact that the fast components of \(Y^{\epsilon }, \theta ^{\epsilon }, Z^{\epsilon }\) are independent of
the slow process \(X^{\epsilon }\). Furthermore, we note once again that y, \(\theta \) and z represent the starting points of the processes \(Y^{\epsilon }_{\tau }, \theta ^x_{\tau }, Z^{\epsilon }_{\tau }\) respectively. Since \(\mathbb {E}\left[ a^0(x,Y^{\epsilon },\theta ^{\epsilon },Z^{\epsilon })\right] =0\) (see above), the centering condition is automatically satisfied:
$$\begin{aligned} \int _{\mathbb {R}^6} \int _{\mathbb {S}} \int _{\mathbb {R}^3} h(\eta ,\xi ,\zeta ; x,\tau ) \ p_{\infty }(\eta ,\xi ,\zeta ) \; d\eta d\xi d\zeta = 0. \end{aligned}$$
Making use of the above expression for the transient density, (7.3) can be written as
$$\begin{aligned} u_1(x,y,\theta ,z,\tau ) = \int _0^\infty \int _{\mathbb {R}^6} \int _{\mathbb {S}} \int _{\mathbb {R}^3} a^0(x,\eta ,\xi ,\zeta ) u_{x}^{\prime } (x,\tau ) \delta _{_{\psi _s(y)}}(\eta ) \cdot \delta _{_{\omega _0 s + \theta }}(\xi ) \cdot g(\zeta ,s;z,0) \; d\eta d\xi d\zeta . \end{aligned}$$
Now, let us consider the last of the Poisson equations in (7.2) more carefully:
$$\begin{aligned} \mathscr {L}_{F}^{(y,\theta ,z)} u_2(x,y,\theta ,z,\tau )= -a^0_{j}(x,y,\theta ,z) \frac{\partial u_1}{\partial x_{j}} (x,y,\theta ,z,\tau ) - b^{1}_{j}(x,y,\theta ,z) \frac{\partial u_1}{\partial y_{j}} (x,y,\theta ,z,\tau ) - \left( a^{1}_{j}(x,y,\theta ) \frac{\partial u}{\partial x_{j}}(x,\tau ) - \frac{\partial u}{\partial \tau }(x,\tau )\right) . \end{aligned}$$
(7.4)
Note that the transient and invariant measures are not functions of the slow variable x, but only of the initial point \((y,\theta ,z)\). Keeping this in mind and integrating with respect to \(\eta (=y)\) and \(\xi (=\theta )\) we can rewrite (7.4) as
$$\begin{aligned} \mathscr {L}_{F}^{(y,\theta ,z)} u_2(x,y,\theta ,z,\tau ) = -a^0_{j}(x,y,\theta ,z) \int _0^\infty ds \left( \int _{\zeta \in \mathbb {R}^3} \left( \frac{\partial a^0_i}{\partial x_{j}} \left( x,{\psi _s(y)},{\omega _0 s + \theta },\zeta \right) u_{x_{i}}^{\prime } (x,\tau ) + a^0_{i} \left( x,{\psi _s(y)},{\omega _0 s + \theta },\zeta \right) u_{x_{i}x_{j}}^{\prime \prime } (x,\tau )\right) g(\zeta ,s;z,0) \; d\zeta \right) - b^{1}_{j}(x,y,\theta ,z) \int _0^\infty ds \; \frac{\partial }{\partial y_{j}} \left( \int _{\zeta \in \mathbb {R}^3} a^0_{i}(x,{\psi _s(y)},{\omega _0 s + \theta },\zeta ) u_{x_{i}}^{\prime } (x,\tau ) g(\zeta ,s;z,0) \; d\zeta \right) - \left( a^{1}_{j}(x,y,\theta ) \frac{\partial u}{\partial x_{j}}(x,\tau ) - \frac{\partial u}{\partial \tau }(x,\tau )\right) =: \varphi (x,y,\theta ,z,\tau ). \end{aligned}$$
Applying the solvability condition \(\left\langle \varphi (x,y,\theta ,z,\tau ), p_{\infty }(y,\theta ,z) \right\rangle = 0\), where \(p_{\infty }(y,\theta ,z)\) is in the kernel of \(\mathscr {L}_{F}^{(y,\theta ,z)}\):
$$\begin{aligned} &-\int _{\mathbb {R}^6} \int _{\mathbb {S}}\int _{\mathbb {R}^3} a^0_{j}(x,y,\theta ,z) p_{\infty }(y,\theta ,z) \int _0^\infty ds \left\{ \int _{\zeta \in \mathbb {R}^3} \left( \frac{\partial a^0_{i}}{\partial x_{j}} (x,{\psi _s (y)},{\omega _0 s + \theta },\zeta ) u_{x_{i}}^{\prime } (x,\tau )\right. \right. \left. \left. + a^0_i (x,{\psi _s (y)},{\omega _0 s + \theta },\zeta ) u_{x_{i}x_{j}}^{\prime \prime } (x,\tau )\right) \right. \left. g(\zeta ,s;z,0) \; d\zeta \right) \; dy d\theta dz \\&- \int _{\mathbb {R}^6} \int _{\mathbb {S}} \int _{\mathbb {R}^3} b^{1}_{j}(x,y,\theta ,z) p_{\infty }(y,\theta ,z) \int _0^\infty ds \left( \int _{\zeta \in \mathbb {R}^3} \frac{\partial a^0_{i}}{\partial \zeta _{l}} (x,\zeta ,{\omega _0 s + \theta },\zeta )|_{\zeta ={\psi _s(y)}} \frac{\partial {\psi _s^{l}(y)}}{\partial y_j} u_{x_{i}}^{\prime } (x,\tau ) g(\zeta ,s;z,0) \; d\zeta \right) dy d\theta dz \\&- \int _{\mathbb {R}^6} \int _{\mathbb {S}}\int _{\mathbb {R}^3} p_{\infty }(y,\theta ,z) \left( a^{1}_{j}(x,y,\theta ) \frac{\partial u}{\partial x_{j}}(x,\tau ) - \frac{\partial u}{\partial \tau }(x,\tau )\right\} dy d\theta dz = 0. \end{aligned}$$
(7.5)
Equation (7.5) yields the homogenized equation for \(u(x,\tau )\).
We now provide a brief overview of the computation involved at this point. The complete details can be found in Singh [11]. In order to simplify the calculations further, we look at \({\bar{b}}^1 (v^{\epsilon }, u_T,\hbox {w}_T)\), \({\bar{b}}^2 (v^{\epsilon },U_m^c)\) in (2.13) and note the change of coordinates in (3.2) that motivates an appropriate structure for the variables \(a^0(x,y,\theta ,z)\), \(b^{1}(x,y,\theta ,z)\), \(a^{1}(x,y,\theta )\). These variables are defined in terms of the matrices \(\bar{K}\), \(\bar{M}\), S, \(\bar{N}\), \(\bar{L}\), D as given in (2.13) as well as in terms of functions of the respective noise processes: \(\varphi _1 (u_T)\), \(\varphi _2 (c,\hbox {w}_T)\). Furthermore, the invariant measure is given as
$$\begin{aligned} p_{\infty }(y,\theta ,z)=\frac{\delta _{0}(y) \cdot \nu _1(z_1) \cdot \nu _2(z_2,z_3)}{2 \pi }, \end{aligned}$$
where \(\nu _1\cdot \nu _2\) is the invariant measure of the real noise process, which are independent. The transition density of the noise process can be decomposed as
$$\begin{aligned} g(\zeta ,s;z,0) = g_1 (\zeta _1,s;z_1,0) \cdot g_2 (\zeta _2,\zeta _3,s;z_2,z_3,0). \end{aligned}$$
We treat \(\left( \zeta _2,\zeta _3\right) \) in \(g_2(\zeta _2,\zeta _3,s;z,0)\) as one quantity in the analysis that follows.
We then integrate out both y and \(\theta \) variables, respectively, in the solvability condition that follows from the previous remarks. We further note that the cross terms between \(\varphi _1 (u_T)\), \(\varphi _2 (c,\hbox {w}_T)\) have been dropped because of the independence between the horizontal and vertical noise. Additionally, we define the covariances (time correlation) in terms of functions of the respective noise processes \(\varphi _1 (u_T)\), \(\varphi _2 (c,\hbox {w}_T)\) as
$$\begin{aligned}\mathscr {R}_{\varphi _1}(s) \mathop {=}\limits ^{\mathrm{def}}&\mathbb {E}[\varphi _1 (u_T(\tau ))\varphi _1 (u_T({\tau +s}))] = \int _{\mathbb {R}} \varphi _1 (\zeta ) \nu _1 (\zeta ) \left( \int _{\zeta '\in \mathbb {R}} \varphi _1 (\zeta ') g_1 (\zeta ',s;z,0) \; d\zeta ' \right) d \zeta ,\\ \mathscr {R}_{\varphi _2}(s) \mathop {=}\limits ^{\mathrm{def}}&\mathbb {E}[\varphi _2 (c(\tau ),\hbox {w}_T(\tau ))\varphi _2 (c({\tau +s}),\hbox {w}_T({\tau +s}))] \int _{\mathbb {R}^2} \varphi _2 (\zeta ) \nu _2 (\zeta ) \left( \int _{\zeta '\in \mathbb {R}^2} \varphi _2 (\zeta ') g_2 (\zeta ',s;z,0) \; d\zeta ' \right) d \zeta . \end{aligned}$$
Their time integrals are the power spectral densities:
$$\begin{aligned} \begin{aligned} \mathbb {S}_{u_T} (0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty \mathscr {R}_{\varphi _1}(s) ds, \\ \mathbb {S}_{\mathrm{w}_T} (0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty \mathscr {R}_{\varphi _2}(s) ds,\\ \mathbb {S}_{u_T}^{\cos } (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty \mathscr {R}_{\varphi _1}(s) C^0 (\omega _0 s) ds,\\ \mathbb {S}_{u_T}^{\sin } (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty \mathscr {R}_{\varphi _1}(s) S^0 (\omega _0 s) ds,\\ \mathbb {S}_{\mathrm{w}_T}^{\cos } (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty \mathscr {R}_{\varphi _2}(s) C^0 (\omega _0 s) ds,\\ \mathbb {S}_{\mathrm{w}_T}^{\sin } (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty \mathscr {R}_{\varphi _2}(s) S^0 (\omega _0 s) ds, \end{aligned} \end{aligned}$$
which can be readily found from the Dryden Model. We also define the (l, j)th element:
$$\begin{aligned} {\mathcal {S}}^{\cos ,l}_{u_T,j} (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}\int _0^\infty \mathscr {R}_{\varphi _1}(s) C^0 (\omega _0 s) \frac{\partial {\psi _s^{l}(y)}}{\partial y_j}|_{y=0} \ ds, \\ {\mathcal {S}}^{\sin ,l}_{u_T,j} (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}\int _0^\infty \mathscr {R}_{\varphi _1}(s) S^0 (\omega _0 s) \frac{\partial {\psi _s^{l}(y)}}{\partial y_j}|_{y=0} \ ds. \end{aligned}$$
We define the various damped spectra as follows:
$$\begin{aligned} \mathbb {S}_{u_T}^{\cos , \lambda _i} (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty e^{\lambda _i s} \mathscr {R}_{\varphi _1}(s) C^0 (\omega _0 s) ds, \\ \mathbb {S}_{u_T}^{\sin , \lambda _i} (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty e^{\lambda _i s} \mathscr {R}_{\varphi _1}(s) S^0 (\omega _0 s) ds \end{aligned}$$
for \(i= 1,\ldots ,4\), and
$$\begin{aligned} \mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty e^{-\kappa s} \mathscr {R}_{\varphi _1}(s) C^0 (\omega _0 s) ds, \\ \mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0)&\mathop {=}\limits ^{\mathrm{def}}2 \int _0^\infty e^{-\kappa s} \mathscr {R}_{\varphi _1}(s) S^0 (\omega _0 s) ds. \end{aligned}$$
Thus,
$$\begin{aligned} {\mathcal {S}}^{\cos ,l}_{u_T,j} (\omega _0)&= \begin{bmatrix} \frac{1}{8} [\mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0-\gamma ) + \mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0+\gamma ) ]&\frac{1}{8} [\mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0-\gamma )-\mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0+\gamma )]&0&0&0&0 \\ \frac{1}{8} [\mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0+\gamma )-\mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0-\gamma )]&\frac{1}{8} [\mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0-\gamma ) + \mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0+\gamma )]&0&0&0&0 \\ 0&0&\frac{1}{4}\mathbb {S}_{u_T}^{\cos , \lambda _1} (\omega _0)&0&0&0 \\ 0&0&0&\frac{1}{4}\mathbb {S}_{u_T}^{\cos , \lambda _2} (\omega _0)&0&0 \\ 0&0&0&0&\frac{1}{4}\mathbb {S}_{u_T}^{\cos , \lambda _3} (\omega _0)&0 \\ 0&0&0&0&0&\frac{1}{4}\mathbb {S}_{u_T}^{\cos , \lambda _4} (\omega _0) \end{bmatrix} , \\ {\mathcal {S}}^{\sin ,l}_{u_T,j} (\omega _0)&= \begin{bmatrix} \frac{1}{8} [\mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0-\gamma ) + \mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0+\gamma ) ]&\frac{1}{8} [\mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0+\gamma )-\mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0-\gamma )]&0&0&0&0 \\ \frac{1}{8} [\mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0-\gamma )-\mathbb {S}_{u_T}^{\cos , \kappa } (\omega _0+\gamma ) ]&\frac{1}{8} [\mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0-\gamma ) + \mathbb {S}_{u_T}^{\sin , \kappa } (\omega _0+\gamma )]&0&0&0&0 \\ 0&0&\frac{1}{4} \mathbb {S}_{u_T}^{\sin , \lambda _1} (\omega _0)&0&0&0 \\ 0&0&0&\frac{1}{4} \mathbb {S}_{u_T}^{\sin , \lambda _2} (\omega _0)&0&0 \\ 0&0&0&0&\frac{1}{4} \mathbb {S}_{u_T}^{\sin , \lambda _3} (\omega _0)&0 \\ 0&0&0&0&0&\frac{1}{4} \mathbb {S}_{u_T}^{\sin , \lambda _4} (\omega _0) \end{bmatrix}. \end{aligned}$$
Now, the partial differential equation (7.5) is written in terms of these power spectral densities. The homogenized generator \(\mathscr {L}^{\dagger \dagger }\) will be an operator that depends only on the two dimensional slow variables \(x_{\tau }\), hence the test functions associated with \(\mathscr {L}^{\dagger \dagger }\) will be of the form \(f(x_{\tau })\). Therefore, let f be a smooth function of x only, and the generator that produces the slow process is given by
$$\begin{aligned} \mathscr {L}^{\dagger \dagger } \mathop {=}\limits ^{\mathrm{def}}\sum _{i=1}^2 {\bar{b}}_i(x) \frac{\partial }{\partial x_i} + \frac{1}{2} \sum _{i,j=1}^2 {\hat{a}}_{ij}(x) \frac{\partial ^2 }{\partial x_i \partial x_j } \end{aligned}$$
(7.6)
with the homogenized coefficients given by
$$\begin{aligned} {\bar{b}}_i(x)= x_k\left( \beta \pi ^{\beta }_{ik} + (x_1^2 +x_2^2 ) \pi ^{\bar{g}}_{ik}\right. + \frac{1}{8}\left[ \pi ^1_{ik} \mathbb {S}_{u_T}^{\cos } (2 \omega _0) +\pi ^2_{ik} \mathbb {S}_{u_T}^{\sin } (2 \omega _0) + \pi ^3_{ik} \mathbb {S}_{u_T} (0) \right] +\frac{1}{8}\left[ \pi ^4_{ik} \mathbb {S}_{u_T}^{\cos ,\kappa } (\omega _0+\gamma ) + \pi ^5_{ik} \mathbb {S}_{u_T}^{\cos ,\kappa } (\omega _0-\gamma ) \right. \left. + \pi ^6_{ik} \mathbb {S}_{u_T}^{\sin ,\kappa } (\omega _0+\gamma ) + \pi ^7_{ik} \mathbb {S}_{u_T}^{\sin ,\kappa } (\omega _0-\gamma )\right] \left. + \frac{1}{4}\left[ \sum _{r=1}^4 \pi _{ik}^{\sin , \lambda _r} \mathbb {S}_{u_T}^{\sin ,\lambda _r} (\omega _0) + \sum _{r=1}^4 \pi _{ik}^{\cos , \lambda _r} \mathbb {S}_{u_T}^{\cos ,\lambda _r} (\omega _0)\right] \right) \end{aligned}$$
and
$$\begin{aligned} {\hat{a}}_{ij} (x)= & {} \frac{x_1^2 +x_2^2 }{8} \left[ \pi ^1_{ij} \mathbb {S}_{u_T}^{\cos } (2 \omega _0) + \pi ^2_{ij} \mathbb {S}_{u_T}^{\sin } (2 \omega _0) \right] \\&+ \frac{\wp }{2} \left[ \pi ^{8}_{ij} \mathbb {S}^{\cos }_{\mathrm{w}_T} (\omega _0) + \pi ^{9}_{ij} \mathbb {S}^{\sin }_{\mathrm{w}_T} (\omega _0) \right] \\&+ \frac{\mathbb {S}_{u_T} (0) }{4} \left( \pi ^{10}_{ij} x_1^2 + \pi ^{11}_{ij} x_2^2 + \pi ^{12}_{ij} x_1 x_2 \right) , \end{aligned}$$
where the definitions of the various \(2 \times 2\) matrices \(\pi ^{r}\) are given in “Appendix 3”.
Appendix 3
$$\begin{aligned} &\pi ^\beta= \begin{bmatrix} \delta '&\gamma ' \\ -\gamma '&\delta ' \end{bmatrix}, \ \pi ^{\bar{g}} = \begin{bmatrix} -\bar{R}&-\tilde{R} \\ \tilde{R}&-\bar{R} \end{bmatrix}, \\ &\pi ^1= \begin{bmatrix} \kappa _2&0 \\ 0&\kappa _2 \end{bmatrix},\ \pi ^2 = \begin{bmatrix} 0&\kappa _2 \\ -\kappa _2&0 \end{bmatrix}, \\ &\pi ^3= \begin{bmatrix} \kappa _1 - \kappa _{7}&2\sqrt{\kappa _1 \kappa _{7}} \\ -2\sqrt{\kappa _1 \kappa _{7}}&\kappa _1 - \kappa _{7} \end{bmatrix}, \ \pi ^4 = \begin{bmatrix} \kappa _3&\kappa _5 \\ -\kappa _5&\kappa _3 \end{bmatrix}, \\ &\pi ^5= \begin{bmatrix} \kappa _4&\kappa _6 \\ -\kappa _6&\kappa _4 \end{bmatrix}, \ \pi ^6 = \begin{bmatrix} -\kappa _5&\kappa _3 \\ -\kappa _3&-\kappa _5 \end{bmatrix}, \\ &\pi ^7= \begin{bmatrix} -\kappa _6&\kappa _4 \\ -\kappa _4&-\kappa _6 \end{bmatrix}, \ \pi ^{\sin , \lambda _r} = \begin{bmatrix} \kappa ^{\sin , \lambda _r}&\kappa ^{\cos , \lambda _r} \\ -\kappa ^{\cos , \lambda _r}&\kappa ^{\sin , \lambda _r} \end{bmatrix}, \\ &\pi ^{\cos , \lambda _r}= \begin{bmatrix} \kappa ^{\cos , \lambda _r}&-\kappa ^{\sin , \lambda _r} \\ \kappa ^{\sin , \lambda _r}&\kappa ^{\cos , \lambda _r} \end{bmatrix}, \ \pi ^{8} = \begin{bmatrix} \kappa _{8}&0 \\ 0&\kappa _{8} \end{bmatrix}, \\ &\pi ^{9}= \begin{bmatrix} 0&\kappa _{8} \\ -\kappa _{8}&0 \end{bmatrix}, \ \pi ^{10} = \begin{bmatrix} \kappa _1&-\sqrt{\kappa _1\kappa _{7}} \\ -\sqrt{\kappa _1\kappa _{7}}&\kappa _7 \end{bmatrix}, \\ &\pi ^{11}= \begin{bmatrix} \kappa _7&\sqrt{\kappa _1\kappa _{7}} \\ \sqrt{\kappa _1\kappa _{7}}&\kappa _1 \end{bmatrix}, \ \pi ^{12} = \begin{bmatrix} 2\sqrt{\kappa _1\kappa _{7}}&\kappa _1-\kappa _7 \\ \kappa _1-\kappa _7&- 2\sqrt{\kappa _1\kappa _{7}} \end{bmatrix},\\ &\delta' = \frac{d_{11}+d_{22}}{2},\ \gamma ' = \frac{d_{12}-d_{21}}{2},\ d_{ij} \in D,\\ &\kappa _1= ( k_{11} + k_{22})^2, \ \kappa _2 = \left\{ (k_{11} - k_{22})^2 + ( k_{12} + k_{21})^2 \right\} , \\ & \kappa _3= (m_{22}-m_{11})(n_{22}-n_{11}) + (m_{12}+m_{21})(n_{12}+n_{21}), \\ & \kappa _4= (m_{22}+m_{11})(n_{22}+n_{11}) - (m_{12}-m_{21})(n_{12}-n_{21}), \\ & \kappa _5= (m_{12}+m_{21})(n_{22}-n_{11}) - (m_{22}-m_{11})(n_{12}+n_{21}), \\ & \kappa _6= (m_{22}+m_{11})(n_{12}-n_{21}) + (m_{12}-m_{21})(n_{22}+n_{11}), \end{aligned}$$
where \(m_{ij} \in \bar{M}\) and \(n_{ij} \in \bar{N}\),
$$\begin{aligned} \kappa ^{\sin , \lambda _r}= & {} \left\{ -m_{1(r+2)}n_{(r+2)2} + m_{2(r+2)}n_{(r+2)1} \right\} , \\ \kappa ^{\cos , \lambda _r}= & {} \left\{ m_{1(r+2)}n_{(r+2)1} + m_{2(r+2)}n_{(r+2)2} \right\} , \end{aligned}$$
for \(r = 1,\ldots ,4\),
$$\begin{aligned} \kappa _{7}= & {} (k_{12} - k_{21} )^2,\ k_{ij} \in \bar{K}, \\ \kappa _{8}= & {} \left\{ f_1^2 + f_2^2 \right\} , \end{aligned}$$
and
$$\begin{aligned} \bar{R}= & {} - \frac{3}{8} \left\{ {\hat{g}}_{1:111} +{\hat{g}}_{1:122} + {\hat{g}}_{2:112} + {\hat{g}}_{2:222} \right\} , \\ \tilde{R}= & {} - \frac{3}{8} \left\{ {\hat{g}}_{1:112} +{\hat{g}}_{1:222} - {\hat{g}}_{2:111} - {\hat{g}}_{2:122} \right\} . \end{aligned}$$
