2.1 The model in spherical coordinates
Let \((\varphi, \theta,r)\) be the spherical coordinates, where φ, θ, r represent the longitude, the latitude, and the radial coordinate, respectively. The unknown functions include the velocity field \(u=(u_{\varphi},u_{\theta},u_{r})\), the temperature function T, the pressure function p, the electromagnetic press function Φ and the magnetic field \(H=(H_{\varphi},H_{\theta},H_{r})\). Then the equations governing the atmospheric circulation with magnetic field [9, 21–23] in the spherical coordinates are given by
$$ \left\{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t}+(u\cdot\bar{ \nabla})u =\nu\bar {\triangle}u-\frac{1}{\rho_{0}}\nabla p-g\vec{k} [1- \alpha(T-T_{0}) ] + \frac{1}{\rho_{0}}(\nabla\times H)\times H, \\ \frac{\partial T}{\partial t}+(u \cdot\nabla)T=\kappa\triangle T, \\ \frac{\partial H}{\partial t}=\nabla\times(u\times H)+\eta\bar {\triangle} H+\nabla\Phi, \\ \operatorname{div} u=0, \\ \operatorname{div} H=0, \end{array}\displaystyle \right . $$
(2.1)
where \(0\leq\varphi\leq2\pi\), \(-\frac{\pi}{2}\leq\theta\leq\frac{\pi }{2}\), \(r_{0}< r< r_{1}\), \(r_{1}=r_{0}+d\), \(r_{0}\) is the radius of the sun, and d is the height of the troposphere, \(\vec{k}=(0,0,1)\), g is the gravitative constant, ν is the kinetic viscosity, κ is the thermal diffusivity, and η is the resistivity. The differential operators in the spherical coordinates are given as follows:
$$\begin{aligned}& \begin{aligned} (u\cdot\bar{\nabla})u={}& \biggl((u\cdot \nabla)u_{\varphi}+\frac {u_{r}u_{\varphi}}{r}+\frac{u_{\varphi}u_{\theta}}{r}\cot\theta, \\ & (u\cdot\nabla)u_{\theta}+ \frac{u_{r}u_{\theta}}{r}- \frac{u_{\varphi }^{2}}{r} \cot\theta,\\ &(u\cdot\nabla)u_{r}- \frac{u_{\theta }^{2}}{r}-\frac{u_{\varphi}^{2}}{r} \biggr)^{T},\end{aligned} \\& \nabla p= \biggl(\frac{1}{r\sin\theta}\frac{\partial p}{\partial\varphi },\frac{1}{r} \frac{\partial P}{\partial\theta}, \frac{\partial p}{\partial r} \biggr)^{T}, \\& \operatorname{div} u= \frac{1}{r\sin\theta} \frac{\partial u_{\varphi }}{\partial\varphi}+ \frac{1}{r\sin\theta} \frac{\partial (\sin\theta u_{\theta})}{\partial\theta}+\frac{1}{r^{2}}\frac{\partial }{\partial r} \bigl(r^{2}u_{r} \bigr), \\& \begin{aligned}\bar{\triangle}u={}& \biggl(\triangle u_{\varphi}+\frac{2}{r^{2}\sin \theta} \frac{\partial u_{r}}{\partial\varphi}+ \frac{2\cos\theta}{r^{2}\sin^{2}\theta}\frac{\partial u_{\theta }}{\partial\varphi}-\frac{u_{\varphi}}{r^{2}\sin^{2}\theta}, \\ &\triangle u_{\theta}-\frac{2\cos\theta}{r^{2}\sin \theta}\frac{\partial u_{\varphi}}{\partial\varphi}- \frac{u_{\theta}}{r^{2}\sin^{2}\theta}+\frac{2}{r^{2}}\frac{\partial u_{r}}{\partial\theta}, \\ &\triangle u_{r}-\frac{2u_{r}}{r^{2}} - \frac{2}{r^{2}\sin\theta}\frac{\partial(\sin\theta u_{\theta })}{\partial\theta}-\frac{2}{r^{2}\sin\theta} \frac{\partial u_{\varphi}}{\partial\varphi} \biggr)^{T},\end{aligned} \\& \begin{aligned}(\nabla\times H)\times H={}& \biggl(\frac{H_{r}}{r} \biggl[\frac{\partial (rH_{\varphi})}{\partial r}- \frac{1}{\sin\theta} \frac{\partial H_{r}}{\partial\theta} \biggr]-\frac{H_{\theta}}{r\sin\theta } \biggl[ \frac{\partial H_{\theta}}{\partial\varphi}- \frac{\partial(H_{\varphi}\sin\theta)}{\partial\theta} \biggr], \\ &\frac{H_{\varphi}}{r\sin\theta} \biggl[\frac {\partial H_{\theta}}{\partial\varphi}- \frac{\partial(H_{\varphi}\sin\theta)}{\partial\theta} \biggr]-\frac {H_{r}}{r} \biggl[\frac{\partial H_{r}}{\partial\theta}- \frac{\partial(rH_{\theta})}{\partial r} \biggr], \\ &\frac{H_{\theta}}{r} \biggl[\frac{\partial H_{r}}{\partial\theta}- \frac{\partial(rH_{\theta})}{\partial r} \biggr] -\frac{H_{\varphi}}{r} \biggl[\frac{\partial(rH_{\varphi})}{\partial r}- \frac {1}{\sin\theta}\frac{\partial H_{r}}{\partial\theta} \biggr] \biggr)^{T},\end{aligned} \\& \begin{aligned}\nabla\times(u\times H)= {}&\biggl(\frac{1}{r}\frac{\partial}{\partial r} \bigl[r(u_{\varphi }H_{r}-u_{r}H_{\varphi}) \bigr]- \frac{1}{r}\frac{\partial}{\partial\theta}(u_{\theta}H_{\varphi }-u_{\varphi}H_{\theta}), \\ & \frac{1}{r\sin\theta}\frac{\partial}{\partial\varphi}(u_{\theta }H_{\varphi}-u_{\varphi}H_{\theta})- \frac{1}{r\sin\theta}\frac{\partial}{\partial r} \bigl[r\sin\theta (u_{r}H_{\theta}-u_{\theta}H_{r}) \bigr], \\ &\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta} \bigl[r\sin\theta (u_{r}H_{\theta}-u_{\theta}H_{r}) \bigr]- \frac{1}{r^{2}\sin\theta} \bigl[r(u_{\varphi}H_{r}-u_{r}H_{\varphi}) \bigr] \biggr)^{T},\end{aligned} \end{aligned}$$
△̄H, ∇Φ, divH are similar to △̄u, ∇p, divu, and the operators Δ, \((u\cdot\nabla)\) are given by
$$ \begin{gathered} (u\cdot\nabla)=\frac{u_{\theta}}{r} \frac{\partial}{\partial\theta }+\frac{u_{\varphi}}{r\sin\theta}\frac{\partial}{\partial\varphi} +u_{r} \frac{\partial}{\partial r}, \\ \triangle=\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial \varphi^{2}}+\frac{1}{r^{2}\sin\theta} \frac{\partial}{\partial\theta} \biggl(\sin\theta\frac{\partial}{\partial\theta} \biggr)+\frac{1}{r^{2}} \frac {\partial}{\partial r} \biggl(r^{2}\frac{\partial}{\partial r} \biggr). \end{gathered} $$
In this paper, we mainly focus on the dynamic bifurcation for the granulation. For simplicity, we assume \(\theta=\frac{\pi}{2}\). Then the velocity component \(u_{\theta}\) and the magnetic field \(H_{\theta}\) are zero, and equations (2.1) become
$$ \left\{ \textstyle\begin{array}{l} \frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r} =\nu (\triangle u_{\varphi}+ \frac{2}{r^{2}}\frac{\partial u_{r}}{\partial\varphi} -\frac{u_{\varphi}}{r^{2}} ) - \frac{1}{\rho_{0}r}\frac{\partial p}{\partial\varphi}+\frac {H_{r}}{\rho_{0}r} [\frac{\partial(rH_{\varphi})}{\partial r} - \frac{\partial H_{r}}{\partial\varphi} ], \\ \frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac {u_{\varphi}^{2}}{r} =\nu ( \triangle u_{r}-\frac{2u_{r}}{r^{2}} -\frac{2}{r^{2}} \frac{\partial u_{\varphi}}{\partial\varphi} ) -\frac{1}{\rho_{0}}\frac{\partial p}{\partial r} -g [1- \alpha(T-T_{0}) ] \\ \hphantom{\frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac {u_{\varphi}^{2}}{r} =}+ \frac{H_{r}}{\rho_{0}r} [\frac{\partial H_{r}}{\partial\varphi}-\frac {\partial(rH_{\varphi})}{\partial r} ], \\ \frac{\partial T}{\partial t}+(u \cdot\nabla)T=\kappa\triangle T, \\ \frac{\partial H_{\varphi}}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r} [r(u_{\varphi}H_{r}-u_{r}H_{\varphi}) ] +\eta (\triangle H_{\varphi}+\frac{2}{r^{2}}\frac{\partial H_{r}}{\partial\varphi} -\frac{H_{\varphi}}{r^{2}} )+\frac{1}{r}\frac{\partial\Phi}{\partial \varphi}, \\ \frac{\partial H_{r}}{\partial t}=\frac{1}{r^{2}}\frac{\partial }{\partial\varphi} [r(u_{r}H_{\varphi}-u_{\varphi}H_{r}) ] +\eta (\triangle H_{r}-\frac{2H_{r}}{r^{2}} - \frac{2}{r^{2}}\frac{\partial H_{\varphi}}{\partial\varphi} )+\frac {\partial\Phi}{\partial r}, \\ \operatorname{div}u=0, \\ \operatorname{div}H=0, \end{array}\displaystyle \right . $$
(2.2)
where
$$ \begin{aligned} &(u\cdot\nabla)= \frac{u_{\varphi}}{r} \frac{\partial}{\partial\varphi }+u_{r}\frac{\partial}{\partial r}, \\ &\triangle=\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}+ \frac{1}{r^{2}} \frac{\partial}{\partial r} \biggl(r^{2}\frac{\partial }{\partial r} \biggr), \\ &\operatorname{div}u=\frac{1}{r}\frac{\partial u_{\varphi}}{\partial\varphi }+\frac{1}{r^{2}} \frac{\partial}{\partial r} \bigl(r^{2}u_{r} \bigr). \end{aligned} $$
Furthermore, equations (2.2) are supplemented with the following boundary condition:
$$ \left\{ \textstyle\begin{array}{l} (u,T,H) (\varphi,r)=(u,T,H) ( \varphi+2k\pi, r), \\ u_{r}=0,\qquad H_{r}=H_{0},\qquad T=T_{0},\qquad \frac{\partial u_{\varphi}}{\partial r}=\frac{\partial H_{\varphi}}{\partial r}=0,\qquad r=r_{0}, \\ u_{r}=0,\qquad H_{r}=H_{1},\qquad T=T_{1},\qquad \frac{\partial u_{\varphi }}{\partial r}=\frac{\partial H_{\varphi}}{\partial r}=0,\qquad r=r_{0}+d. \end{array}\displaystyle \right . $$
(2.3)
2.2 Perturbed dimensionless equations
We determine the basic flow by following assumptions.
-
1.
\(U=(u_{\varphi},u_{r},T,H_{\varphi} ,H_{r})=(0,0,\tilde {T}(r),0,\tilde{H}_{r})\), \(p=\tilde{p}(r)\), \(\Phi=0\); that is, the pressure function, the temperature function and the magnetic field function in r-direction are not zero, and which are only depending on r.
-
2.
The functions \(\tilde{T}(r)\), \(\tilde{H}(r)\) and \(\tilde{p}(r)\) satisfy
$$ \left\{ \textstyle\begin{array}{l} -\frac{1}{\rho_{0}}\frac{\partial\tilde{p}}{\partial r}-g [1-\alpha (\tilde{T}-T_{0}) ]=0, \\ \triangle\tilde{T}=0, \\ \triangle\tilde{H_{r}}-\frac{2}{r^{2}}\tilde{H_{r}}=0. \end{array}\displaystyle \right . $$
-
3.
Based on the boundary condition (2.3), the value of basic flow on the boundary is given by
$$ \begin{gathered} \tilde{H_{r}}=H_{0},\qquad \tilde{T }=T_{0},\qquad r=r_{0}, \\ \tilde{H_{r}}=H_{1},\qquad \tilde{T }=T_{1},\qquad r=r_{0}+d. \end{gathered} $$
From the above assumptions, we derive the basic flow as follows:
$$ \left \{ \textstyle\begin{array}{l} \tilde{T}=\frac{C_{1}}{r}+C_{0}, \\ \tilde{H_{r}}=K_{1}r+\frac{K_{0}}{r^{2}}, \\ \tilde{p}= \int^{r_{0+d}}_{r_{0}}-\rho_{0}g [1-\alpha( \tilde{T}-T_{0}) ]\,dr, \end{array}\displaystyle \right . $$
(2.4)
where
$$ \begin{gathered} C_{0}=\frac{T_{1}r_{1}-T_{0}r_{0}}{r_{1}-r_{0}},\qquad C_{1}=\frac {(T_{0}-T_{1})r_{0}r_{1}}{r_{1}-r_{0}}, \\ H_{0}=K_{1}r_{0}+ \frac{K_{0}}{r_{0}^{2}},\qquad H_{1}=K_{1}r_{1}+ \frac {K_{0}}{r_{1}^{2}}. \end{gathered} $$
(2.5)
It is noticed that \(K_{0}\) and \(K_{1}\) are related to the boundary value \(H_{0}\) and \(H_{1}\). Furthermore, in order to get the perturbation equations related to the variables r and φ, we make the following translations:
$$p=p'+\tilde{p},\qquad T=T'+\tilde{T},\qquad H_{r}=H_{r}'+\tilde{H_{r}}. $$
Omitting the primes, equations (2.2) can be rewritten as
$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r} =\nu (\triangle u_{\varphi}+ \frac{2}{r^{2}}\frac{\partial u_{r}}{\partial\varphi} -\frac{u_{\varphi}}{r^{2}} ) - \frac{1}{\rho_{0}r}\frac{\partial p}{\partial\varphi}+\frac {H_{r}}{\rho_{0}r} [\frac{\partial(rH_{\varphi})}{\partial r} - \frac{\partial H_{r}}{\partial\varphi} ] \\ \hphantom{\frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r} =} +\frac{\tilde{H_{r}}}{\rho_{0}r} [\frac{\partial(rH_{\varphi})}{\partial r}-\frac{\partial H_{r}}{\partial\varphi} ], \\ \frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac {u_{\varphi}^{2}}{r} =\nu ( \triangle u_{r}-\frac{2u_{r}}{r^{2}} -\frac{2}{r^{2}} \frac{\partial u_{\varphi}}{\partial\varphi} ) -\frac{1}{\rho_{0}}\frac{\partial p}{\partial r} -g\alpha T, \\ \frac{\partial T}{\partial t}+(u \cdot\nabla)T=\kappa\triangle T+\frac{C_{1}}{r^{2}}u_{r}, \\ \frac{\partial H_{\varphi}}{\partial t}=\frac{1}{r}\frac{\partial }{\partial r} [r(u_{\varphi}H_{r}-u_{r}H_{\varphi}) ] +\eta (\triangle H_{\varphi}+\frac{2}{r^{2}}\frac{\partial H_{r}}{\partial\varphi} -\frac{H_{\varphi}}{r^{2}} )+\frac{1}{r}\frac{\partial\Phi}{\partial \varphi}+ \frac{1}{r}\frac{\partial}{\partial r}(ru_{\varphi}\tilde{H_{r}}), \\ \frac{\partial H_{r}}{\partial t}=\frac{1}{r^{2}}\frac{\partial }{\partial\varphi} [r(u_{r}H_{\varphi}-u_{\varphi}H_{r}) ] +\eta (\triangle H_{r}-\frac{2H_{r}}{r^{2}} - \frac{2}{r^{2}}\frac{\partial H_{\varphi}}{\partial\varphi} )+\frac {\partial\Phi}{\partial r} -\frac{1}{r} \tilde{H_{r}}\frac{\partial u_{\varphi}}{\partial\varphi}, \\ \operatorname{div}u=0, \\ \operatorname{div}H=0. \end{array}\displaystyle \right . $$
(2.6)
As we know, the radius of the sun is about \(7\times10^{5}\mbox{ km}\) and the thickness of the photosphere is about 500 km. Then the ratio of the thickness of the photosphere to the radius of the sun is small. Hence, we adopt the approximations that \(1/r \simeq1/r_{0}\), \((r_{0}+d)/r_{0}\simeq1 \). For simplicity, we assume \(\nu/\kappa=\eta/\kappa=1\). Also, we introduce the following dimensionless variables:
$$\begin{gathered} u=\frac{\kappa}{d}u',\qquad r=dr', \qquad T= \frac{T_{0}-T_{1}}{\sqrt {R}}T',\qquad p=\frac{\rho_{0}\kappa^{2}}{d^{2}}p', \\ H=H_{0}H',\qquad\Phi=H_{0} \frac{\kappa}{d} \Phi',\qquad t=\frac {d^{2}}{\kappa}t', \end{gathered} $$
where R, called the Rayleigh number, is a dimensionless parameter and
$$R=\frac{g\alpha(T_{0}-T_{1})}{\kappa\nu}d^{3}. $$
Let \((\varphi'', r'')=(r_{0}\varphi', r')\). With the above assumptions, we omit the primes and get the approximate equations, which are given as follows:
$$ \left\{ \textstyle\begin{array}{l} \frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r_{0}} =\triangle u_{\varphi}+ \frac{2}{r_{0}}\frac{\partial u_{r}}{\partial \varphi} -\frac{u_{\varphi}}{r^{2}_{0}} -\frac{\partial p}{\partial\varphi}+ \frac{1}{\rho_{0}}\frac {d^{2}}{\kappa^{2}}H_{0}^{2}H_{r} (\frac{\partial H_{\varphi}}{\partial r} -\frac{\partial H_{r}}{\partial\varphi} ) \\ \hphantom{\frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r_{0}} =} + \frac{1}{\rho_{0}}\frac{d^{2}}{\kappa^{2}}H_{0} (K_{1}dr_{0}+ \frac {K_{0}}{d^{2}r_{0}^{2}} ) (\frac{\partial H_{\varphi}}{\partial r}-\frac{\partial H_{r}}{\partial \varphi} ), \\ \frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac {u_{\varphi}^{2}}{r} =\triangle u_{r}-\frac{2u_{r}}{r_{0}^{2}} -\frac{2}{r_{0}}\frac{\partial u_{\varphi}}{\partial\varphi} - \frac{\partial p}{\partial r}+\sqrt{R}T, \\ \frac{\partial T}{\partial t}+(u \cdot\nabla)T=\triangle T+\sqrt{R}u_{r}, \\ \frac{\partial H_{\varphi}}{\partial t}=\frac{\partial}{\partial r}(u_{\varphi}H_{r}-u_{r}H_{\varphi}) +\triangle H_{\varphi}+\frac{2}{r_{0}}\frac{\partial H_{r}}{\partial \varphi} - \frac{H_{\varphi}}{r_{0}^{2}}+\frac{1}{r}\frac{\partial\Phi}{\partial \varphi}+\frac{K_{1}d}{H_{0}}u_{\varphi} \\ \hphantom{\frac{\partial H_{\varphi}}{\partial t}=} + \frac{1}{H_{0}} (K_{1}dr_{0}+\frac{K_{0}}{d^{2}r_{0}^{2}} )\frac {\partial u_{\varphi}}{\partial r}, \\ \frac{\partial H_{r}}{\partial t}=\frac{\partial}{\partial\varphi }(u_{r}H_{\varphi}-u_{\varphi}H_{r}) +\triangle H_{r}-\frac{2H_{r}}{r_{0}^{2}} -\frac{2}{r_{0}} \frac{\partial H_{\varphi}}{\partial\varphi}+\frac {\partial\Phi}{\partial r} -\frac{1}{H_{0}} (K_{1}dr_{0}+ \frac{K_{0}}{d^{2}r_{0}^{2}} )\frac {\partial u_{\varphi}}{\partial\varphi}, \\ \operatorname{div}u=0, \\ \operatorname{div}H=0, \end{array}\displaystyle \right . $$
(2.7)
where \((\varphi,r)\in M=(0,L)\times(r_{0},r_{0}+1)\), \(r_{0}\) is the radius of the sun with the unit of d, \(L=2\pi r_{0}\), and \(K_{0}\), \(K_{1}\) are given as (2.5). \((u\cdot\nabla)\), div and △ are general differential operators.
$$(u\cdot\nabla)=u_{\varphi}\frac{\partial}{\partial\varphi}+u_{r} \frac {\partial}{\partial r},\qquad \triangle=\frac{\partial^{2}}{\partial\varphi^{2}}+\frac{\partial ^{2}}{\partial r ^{2}}, \qquad \operatorname{div}u=\frac{\partial u_{\varphi}}{\partial\varphi}+\frac {\partial u_{r}}{\partial r}. $$
The boundary conditions (2.3) are rewritten as
$$ \left \{ \textstyle\begin{array}{l} (u,T,H) (\varphi,r)=(u,T,H) ( \varphi+L, r), \\ u_{r}=0,\qquad H_{r}=0,\qquad T=0,\qquad \frac{\partial u_{\varphi}}{\partial r}= \frac {\partial H_{\varphi}}{\partial r}=0,\qquad r=r_{0}, \\ u_{r}=0,\qquad H_{r}=0,\qquad T=0,\qquad \frac{\partial u_{\varphi}}{\partial r}= \frac{\partial H_{\varphi}}{\partial r}=0,\qquad r=r_{0}+1. \end{array}\displaystyle \right . $$
(2.8)
2.3 Abstract operator equation
Now we will show that equations (2.7) can be written in the operator form. Since the velocity field u and the magnetic field H on M are divergence-free, there exist the following stream functions \(f_{1}\) and \(f_{2}\) satisfying the given boundary condition:
$$\begin{gathered} u_{\varphi}=\frac{\partial f_{1}}{\partial r}, \qquad u_{r}=- \frac {\partial f_{1}}{\partial\varphi}, \\ H_{\varphi}=\frac{\partial f_{2}}{\partial r}, \quad\quad H_{r}=- \frac {\partial f_{2}}{\partial\varphi}. \end{gathered} $$
Moreover, the following two vector fields:
$$\begin{gathered} \biggl(\frac{2}{r_{0}}\frac{\partial u_{r}}{\partial\varphi}, -\frac {2}{r_{0}} \frac{\partial u_{\varphi}}{\partial\varphi} \biggr) =-\frac{2}{r_{0}}\nabla\frac{\partial f_{1}}{\partial\varphi}, \\ \biggl(\frac{2}{r_{0}}\frac{\partial H_{r}}{\partial\varphi}, - \frac {2}{r_{0}} \frac{\partial H_{\varphi}}{\partial\varphi} \biggr) =-\frac{2}{r_{0}}\nabla \frac{\partial f_{2}}{\partial\varphi}, \end{gathered} $$
are gradient fields, which can be balanced by ∇p and ∇Φ in (2.7). Hence, (2.7) are equivalent to the following equations:
$$ \left\{ \textstyle\begin{array}{l} \frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r_{0}} =\triangle u_{\varphi} - \frac{u_{\varphi}}{r^{2}_{0}} -\frac{\partial p}{\partial\varphi}+\frac{1}{\rho_{0}}\frac {d^{2}}{\kappa^{2}}H_{0}^{2}H_{r} (\frac{\partial H_{\varphi}}{\partial r} -\frac{\partial H_{r}}{\partial\varphi} ) \\ \hphantom{\frac{\partial u_{\varphi}}{\partial t}+(u\cdot \nabla)u_{\varphi }+\frac{u_{r}u_{\varphi}}{r_{0}} =} + \frac{1}{\rho_{0}}\frac{d^{2}}{\kappa^{2}}H_{0} (K_{1}dr_{0}+ \frac {K_{0}}{d^{2}r_{0}^{2}} ) (\frac{\partial H_{\varphi}}{\partial r}-\frac{\partial H_{r}}{\partial \varphi} ), \\ \frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac {u_{\varphi}^{2}}{r} =\triangle u_{r}-\frac{2u_{r}}{r_{0}^{2}} -\frac{\partial p}{\partial r}+\sqrt{R}T, \\ \frac{\partial T}{\partial t}+(u \cdot\nabla)T=\triangle T+\sqrt{R}u_{r}, \\ \frac{\partial H_{\varphi}}{\partial t}=\frac{\partial}{\partial r}(u_{\varphi}H_{r}-u_{r}H_{\varphi}) +\triangle H_{\varphi} -\frac{H_{\varphi}}{r_{0}^{2}}+\frac{\partial\Phi}{\partial\varphi }+ \frac{K_{1}d}{H_{0}}u_{\varphi} \\ \hphantom{\frac{\partial H_{\varphi}}{\partial t}=} + \frac{1}{H_{0}} (K_{1}dr_{0}+\frac{K_{0}}{d^{2}r_{0}^{2}} )\frac {\partial u_{\varphi}}{\partial r}, \\ \frac{\partial H_{r}}{\partial t}=\frac{\partial}{\partial\varphi }(u_{r}H_{\varphi}-u_{\varphi}H_{r}) +\triangle H_{r}-\frac{2H_{r}}{r_{0}^{2}} +\frac{\partial\Phi}{\partial r} - \frac{1}{H_{0}} (K_{1}dr_{0}+\frac{K_{0}}{d^{2}r_{0}^{2}} )\frac {\partial u_{\varphi}}{\partial\varphi}, \\ \operatorname{div}u=0, \\ \operatorname{div}H=0. \end{array}\displaystyle \right . $$
(2.9)
To get the abstract form of (2.9), we define the following spaces:
$$\begin{gathered} H= \bigl\{ (u,T,H)\in L^{2} \bigl(M, R^{5} \bigr)\mid\operatorname{div} u= \operatorname{div} H=0,(u,T,H) \text{ are periodic in } \varphi \text{-direction} \bigr\} , \\ H_{1}= \bigl\{ (u,T,H)\in H^{2} \bigl(M,R^{5} \bigr) \cap H\mid(u,T,H) \text{ satisfy (2.8)} \bigr\} . \end{gathered} $$
Now, we define the operators \(L=-A+B: H_{1}\rightarrow H\) and \(G: H_{1}\rightarrow H\) by
$$\begin{aligned}& \begin{aligned} AU={}&{-}P \biggl(\triangle u_{\varphi} - \frac{u_{\varphi}}{r^{2}_{0}}+\widetilde{A}F \biggl(\frac{\partial H_{\varphi }}{\partial r}-\frac{\partial H_{r}}{\partial\varphi} \biggr), \triangle u_{r} -\frac{u_{r}}{r^{2}_{0}}, \\ & \triangle T, \triangle H_{\varphi} - \frac{1}{r^{2}_{0}}H_{\varphi}+\frac {1}{H_{0}}F\frac{\partial u_{\varphi}}{\partial r}, \triangle H_{r} -\frac{1}{r^{2}_{0}}H_{r}- \frac{1}{H_{0}}F\frac {\partial u_{\varphi}}{\partial\varphi} \biggr)^{T},\end{aligned} \\& BU=P \biggl(0, \sqrt{R}T, \sqrt{R}u_{r}, \frac {K_{1}d}{H_{0}}u_{\varphi}, 0 \biggr)^{T}, \\& \begin{aligned}GU={}&P \biggl(-(u\cdot\nabla)u_{\varphi}-\frac{u_{r}u_{\varphi }}{r_{0}}+ \frac{1}{\rho_{0}} \frac{d^{2}}{\kappa^{2}}H^{2}_{0}H_{r} \biggl( \frac{\partial H_{\varphi}}{\partial r}-\frac{\partial H_{r}}{\partial \varphi} \biggr), \\ & {-}(u\cdot\nabla)u_{r}- \frac{u_{\varphi }^{2}}{r_{0}}, -(u\cdot \nabla)T, \frac{\partial}{\partial r}(u_{\varphi}H_{r}-u_{r}H_{\varphi }), \frac{\partial}{\partial\varphi}(u_{r}H_{\varphi}-u_{\varphi}H_{r}) \biggr)^{T}, \end{aligned} \end{aligned}$$
where \(P: L^{2}(M, R^{5})\rightarrow H\) is the Leray projection, \(U=(u,T,H)\in H_{1}\) and
$$ F=K_{1}r_{0}d+\frac{K_{0}}{r_{0}^{2}d^{2}},\qquad \widetilde{A}=\frac {H_{0}d^{2}}{\rho_{0}\kappa^{2}}. $$
(2.10)
Therefore, the problem (2.9) with boundary conditions (2.8) are equivalent to the following abstract equation.
$$ \left \{ \textstyle\begin{array}{l} \frac{\text{d}U}{\text{d}t}=L_{\lambda}U+GU, \\ U(0)=U_{0}, \end{array}\displaystyle \right . $$
(2.11)
where \(\lambda=(R,F)\in R^{2}\) is the parameter and \(U_{0}\) is the initial value of (2.9).