Rossby波是很重要的大尺度波动.在近几年中,在许多大规模尺度运动中,非线性Rossby波受到了大气海洋领域的高度重视. Long^[1]在20世纪60年代发现了切变纬向流中的Rossby孤立波.然后Benney^[2], Redekopp^[3]和其他学者推广了Long的结论.在20世纪70-80年代非线性Rossby孤立波的相关理论得到了飞速的发展.对于Rossby孤立波,许多研究者研究了一些弱非线性模型.其中, Korteweg-de Vries (KdV)方程^[4-6]常用来描述经典Rossby孤立波的.对于包络Rossby孤立波,非线性Rossby波的非线性Schrödinger(NLS)首先分别由Benney^[7]和Yamagata^[8]得到.之后,由于其广泛和重要的应用,许多研究人员已经致力于在理论和数值上研究该方程^[9-16].例如, Liao^[17]提出了变系数非线性薛定谔方程的变分积分方程. Chen^[18]利用时空伸长变化与摄动法描述了耦合非线性薛定谔方程的一种新的精确解. Yao^[19]描述了一些具有实际电位的线性和非线性薛定谔方程的均匀性. Li^[20]导出了$(2+1)$维耗散非线性薛定谔方程,用于描述在平面内传播耗散影响下的包络Rossby孤立波.以上都是非线性薛定谔方程的情况.然后, Luo^[21]推导了弱非线性Rossby波的高阶非线性薛定谔方程,之后, Li^[22]得到了关于高阶非线性薛定谔方程的Kuznetsov-Masoltion和Akhmediev的形式.

另一方面,地球自转对地球流体力学中的许多现象也有显著的影响,它的作用是通过流体动力学Navier-Stokes方程中出现额外加速度项$ 2 \overrightarrow{\Omega}\times\overrightarrow{V} $,其中$ \vec{\Omega}=\mid \Omega \mid (0, \cos\varphi, \sin \varphi) $为Coriolis力矢量, $ \overrightarrow{V}=(u, v, w ) $为三维速度矢量, $\mid \Omega \mid $是地球旋转角速度, $ \varphi $是纬度.忽略地球旋转作用的水平分量称为"传统近似",然而就动力学角度而言这也是一个受争议的问题^[23-25].近年来,在地球流体力学的许多研究中, Coriolis力水平分量的作用也越来越引起人们的重视^[26-28].通过尺度分析, White和Bromley^[29], Burger^[30]得到将行星尺度的大气运动保留到${f_{H}w}$是可取的. Draghide^[27]注意到在中尺度运动的范围内, ${f_{H}u}$的项远远大于${\frac{{\rm d}w}{{\rm d}t}}$的项. Zhao^[31]根据半地转近似给出了完整Coriolis力的非线性Rossby波的精确解. Song^[32]考虑了推广的平面作用下具有完整Coriolis力的Rossby孤立波.尽管他们考虑了完整Coriolis力对地球物理流体的影响,但都作了半地转近似.因此,本文的目的是获得一类高阶NLS方程,并讨论了Rossby波的调制不稳定性.

无量纲的正压位涡方程的形式^[33]如下

(2.1) $\begin{eqnarray}\Big (\frac{\partial}{\partial t} -\frac{\partial \psi}{\partial y}\frac{\partial}{\partial x} +\frac{\partial \psi}{\partial x}\frac{\partial}{\partial y}\Big)\Big (\frac{\partial^2 \psi}{\partial x^2} +\frac{\partial^2 \psi}{\partial y^2} -F\psi +\beta y +B -\lambda'\delta\frac{\partial B}{\partial y}\Big) =0, \end{eqnarray} $

其中引入了两个无量纲参数$\lambda'=\frac{f_{H}}{f_{0}}, \delta=\frac{H}{L}$.

边界条件为

(2.2) $ \begin{eqnarray} \frac{\partial \psi}{\partial x}=0, y=0, L_{y}. \end{eqnarray} $

平均流函数$\overline{\psi(y, t)}$必须满足

(2.3) $ \begin{eqnarray} \frac{\partial^2 \overline{\psi}}{\partial t\partial y}=0, y=0, L_{y}. \end{eqnarray} $

我们引进一个小参数$\varepsilon$,然后取如下形式

(2.4) $\begin{eqnarray} \psi=-\overline{u}y+\varepsilon\psi', B=\varepsilon^{3}\Omega(x, y), \end{eqnarray}$

其中$\overline{u}$为定常基本流.

将方程(2.4)代入方程(2.1),得

(2.5) $\begin{eqnarray} &&\Big (\frac{\partial}{\partial t} +\overline{u}\frac{\partial}{\partial x}\Big) (\nabla^{2}\psi'-F\psi') +\varepsilon J[\psi', \nabla^{2}\psi'] +\varepsilon^{2}\Big(\overline{u}\frac{\partial \Omega}{\partial x}-\overline{u}\frac{\partial^2 \Omega}{\partial x\partial y}\Big)\\ &&+\varepsilon^{3}\Big[J(\psi', \Omega)-\lambda'\delta J\Big(\psi', \frac{\partial \Omega}{\partial y}\Big)\Big] +(\beta+F\overline{u})\frac{\partial \psi'}{\partial x} =0. \end{eqnarray} $

引入缓变量

(2.6) $\begin{eqnarray} \xi=\varepsilon(x-C_{g}t), T=\varepsilon^{2}t. \end{eqnarray}$

则

(2.7) $\begin{eqnarray} \frac{\partial}{\partial x}=\frac{\partial}{\partial x}+\varepsilon\frac{\partial}{\partial \xi}, ~~~ \frac{\partial}{\partial t}=\frac{\partial}{\partial t}-\varepsilon C_{g}\frac{\partial}{\partial \xi}+\varepsilon^{2}\frac{\partial}{\partial T}. \end{eqnarray} $

将方程(2.7)带入方程(2.5),得到

(2.8) $\begin{eqnarray}&&L(\psi')+\varepsilon\bigg\{(\overline{u}-C_{g})\frac{\partial}{\partial \xi}(\nabla^{2}\psi'-F\psi')+2\Big(\frac{\partial}{\partial t}+\overline{u}\frac{\partial}{\partial x}\Big)\frac{\partial^2 \psi'}{\partial x\partial \xi}+(\beta+F\overline{u})\frac{\partial \psi'}{\partial \xi}\\&&+J(\psi', \nabla^{2}\psi')\bigg\}+\varepsilon^{2}\bigg\{\frac{\partial}{\partial T}(\nabla^{2}\psi'-F\psi')+2(\overline{u}-C_{g})\frac{\partial^3 \psi'}{\partial x\partial \xi^2}+\Big(\frac{\partial}{\partial t}+\overline{u}\frac{\partial}{\partial x}\Big)\frac{\partial^2 \psi'}{\partial \xi^2}\\&&+J\Big(\psi', 2\frac{\partial^2 \psi'}{\partial x\partial \xi}\Big)+\frac{\partial \psi'}{\partial \xi}\frac{\partial}{\partial y}\nabla^2\psi'-\frac{\partial \psi'}{\partial y}\frac{\partial}{\partial \xi}\nabla^2\psi'+\overline{u}\frac{\partial \Omega}{\partial x}-\overline{u}\frac{\partial^2 \Omega}{\partial x^2}\bigg\} \\&&+\varepsilon^3\bigg\{(\overline{u}-C_{g})\frac{\partial^3 \psi'}{\partial \xi^3}+2\frac{\partial^3 \psi'}{\partial x\partial \xi\partial T}+J\Big(\psi', \frac{\partial^2 \psi'}{\partial \xi^2}\Big)+2\frac{\partial \psi'}{\partial \xi}\frac{\partial^3 \psi'}{\partial x\partial y\partial \xi}-2\frac{\partial \psi'}{\partial y}\frac{\partial^3 \psi'}{\partial x\partial \xi^2}\\&&+\overline{u}\frac{\partial \Omega}{\partial \xi}-\overline{u}\frac{\partial^2 \Omega}{\partial x^2}+J(\psi', \Omega)-\lambda'\delta J\Big(\psi', \frac{\partial \Omega}{\partial y}\Big)\bigg\} +\varepsilon^4\bigg\{\frac{\partial^3 \psi'}{\partial T\partial \xi^2}+\frac{\partial \psi'}{\partial \xi}\frac{\partial^3 \psi'}{\partial y\partial \xi^2}\\&&-\frac{\partial \psi'}{\partial y}\frac{\partial^3 \psi'}{\partial \xi^3}+\frac{\partial \psi'}{\partial \xi}\frac{\partial \Omega}{\partial y} -\frac{\partial \psi'}{\partial y}\frac{\partial \Omega}{\partial \xi}+\frac{\partial \psi'}{\partial y}\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \psi'}{\partial \xi}\frac{\partial^2 \Omega}{\partial y^2}\bigg\}=0, \end{eqnarray} $

其中线性算子$L()=(\frac{\partial}{\partial t}+\overline{u}\frac{\partial}{\partial x})[\nabla^{2}()-F()]+(\beta+F \overline{u})\frac{\partial ()}{\partial x}$.

边界条件为

(2.9) $ \begin{eqnarray} \frac{\partial \xi}{\partial x}=0, \frac{\partial^2 \overline{\psi'}}{\partial y\partial t}=0, \frac{\partial^2 \overline{\psi'}}{\partial y\partial T}=0, y=0, L_{y}. \end{eqnarray}$

显然,对于偶极子结构,方程(2.8)的解可以表示为

(2.10) $ \begin{equation}\psi'=\psi_{0}+\varepsilon\psi_{1}(\xi, T, x, y, t)=A(T, \xi)\phi_{1}(y)\exp[{\rm i}(kx-\omega t)]+\varepsilon\psi_{1}(T, \xi, x, y, t)+c.c, \end{equation}$

其中, $A$是线性Rossby波列的复振幅, $k=n/6.371\cos(\phi_{0})$是Rossby波的第$n$个波的无量纲形式波数, $\phi_{1}(y)=(2/L_{y})^{\frac{1}{2}}\sin(my), m=-2\pi/L_{y}, \omega=\overline{u}k-\frac{(\beta+\overline{u}F)k}{k^2+m^2+F}$是线性Rossby列的频率, $c.c$表示前一项的复共轭.

将方程(2.10)代入方程(2.8)中,得到

(2.11) $ \begin{eqnarray} &&L(\psi_{1})+\varepsilon \bigg\{(\overline{u}-C_{g})\frac{\partial}{\partial \xi}(\nabla^2\psi_{1}-F\psi_{1})+(\beta+F \overline{u})\frac{\partial \psi_{1}}{\partial \xi}+J(\psi_{0}, \nabla^2\psi_{1})\\&&+J(\psi_{1}, \nabla^2\psi_{0})+\frac{\partial}{\partial T}(\nabla^2\psi_{0}-F\psi_{0})+2(\overline{u}-C_{g})\frac{\partial^3 \psi_{0}}{\partial x\partial \xi^2}+\Big(\frac{\partial}{\partial t}+\overline{u}\frac{\partial}{\partial x}\Big)\frac{\partial^2 \psi_{0}}{\partial \xi^2}\\&&+J\Big(\psi_{0}, 2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial}{\partial y}\nabla^{2}\psi_{0}-\frac{\partial \psi_{0}}{\partial y}\frac{\partial}{\partial \xi}\nabla^2\psi_{0}+\overline{u}\frac{\partial \Omega}{\partial x}-\overline{u}\frac{\partial^2 \Omega}{\partial x^2}\bigg\}\\&&+\varepsilon^2\bigg\{J(\psi_{1}, \nabla^2\psi_{1})+J\Big(\psi_{0}, 2\frac{\partial^2 \psi_{1}}{\partial x\partial \xi}\Big)+J\Big(\psi_{1}, 2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial}{\partial y}\nabla^{2}\psi_{1}+\frac{\partial \psi_{1}}{\partial \xi}\frac{\partial}{\partial y}\nabla^{2}\psi_{0}\\&&-\frac{\partial \psi_{0}}{\partial y}\frac{\partial}{\partial \xi}\nabla^{2}\psi_{1}-\frac{\partial \psi_{1}}{\partial y}\frac{\partial}{\partial \xi}\nabla^{2}\psi_{0}+(\overline{u}-C_{g})\frac{\partial^3 \psi_{0}}{\partial \xi^3}+2\frac{\partial^3 \psi_{0}}{\partial x\partial \xi\partial T}+J\Big(\psi_{0}, \frac{\partial^2 \psi_{0}}{\partial \xi^2}\Big)\\&&+2\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial^3 \psi_{0}}{\partial x\partial y\partial \xi}-2\frac{\partial \psi_{0}}{\partial y}\frac{\partial^3 \psi_{0}}{\partial x\partial \xi^2}+\overline{u}\frac{\partial \Omega}{\partial \xi}-\overline{u}\frac{\partial^2 \Omega}{\partial x\partial \xi}+J(\psi_{0}, \Omega)-\lambda'\delta J\Big(\psi_{0}, \frac{\partial \Omega}{\partial y}\Big)\bigg\}\\&&+\varepsilon^3\bigg\{(\overline{u}-C_{g})\frac{\partial^3 \psi_{1}}{\partial \xi^3}+J\Big(\psi_{0}, \frac{\partial^2 \psi_{1}}{\partial \xi^2}\Big)J\Big(\psi_{1}, \frac{\partial^2 \psi_{0}}{\partial \xi^2}\Big)+2\frac{\partial \psi_{1}}{\partial \xi}\frac{\partial^3 \psi_{0}}{\partial x\partial y\partial \xi}-2\frac{\partial \psi_{1}}{\partial y}\frac{\partial^3 \psi_{0}}{\partial x\partial \xi^2}\\&&+J(\psi_{1}, \Omega)-\lambda'\delta J\Big(\psi_{1}, \frac{\partial \Omega}{\partial y}\Big)+\frac{\partial^3 \psi_{0}}{\partial T\partial \xi^2}+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial^3 \psi_{0}}{\partial y\partial \xi^2}-\frac{\partial \psi_{0}}{\partial y}\frac{\partial^3 \psi_{0}}{\partial \xi^3}+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial \Omega}{\partial y}\\&&-\frac{\partial \psi_{0}}{\partial y}\frac{\partial \Omega}{\partial \xi}+\frac{\partial \psi_{0}}{\partial y}\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial^2 \Omega}{\partial y^2}\bigg\}=0. \end{eqnarray} $

方程(2.11)的解可以写成

(2.12) $\begin{equation}\psi_{1}=\psi_{11}(T, \xi, x, y, t)+\varepsilon\psi_{2}, \end{equation}$

其中

(2.13) $ \begin{equation} \psi_{11}=|A|^2\sum\limits_{n=1}^\infty q_{n}g_{n}\cos(n+\frac{1}{2})my. \end{equation}$

$\psi_{2}$需要满足方程

(2.14) $\begin{eqnarray} &&L(\psi_{2})+\frac{\partial}{\partial T}(\nabla^2\psi_{0}-F\psi_{0})+2(\overline{u}-C_{g})\frac{\partial^3 \psi_{0}}{\partial x\partial \xi^2}+\Big(\frac{\partial}{\partial t}+\overline{u}\frac{\partial}{\partial x}\Big)\frac{\partial^2 \psi_{0}}{\partial \xi^2}+J(\psi_{11}, \nabla^2\psi_{0})\\&&+J(\psi_{0}, \nabla^2\psi_{11})+J\Big(\psi_{0}, 2\frac{\partial^2 \psi_{0}}{\partial x\partial\xi}\Big)+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial \nabla^2\psi_{0}}{\partial y}-\frac{\partial \psi_{0}}{\partial y}\frac{\partial \nabla^2\psi_{0}}{\partial \xi}+\overline{u}\frac{\partial \Omega}{\partial x}-\overline{u}\frac{\partial^2 \Omega}{\partial x^2}\\&&+\varepsilon\bigg\{(\overline{u}-C_{g})\frac{\partial^3 \psi_{0}}{\partial \xi^3}+2\frac{\partial^3 \psi_{0}}{\partial x\partial \xi\partial T}+J\Big(\psi_{0}, \frac{\partial^2 \psi_{0}}{\partial \xi^2}\Big)+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial}{\partial y}\Big(2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)\\&&-\frac{\partial \psi_{0}}{\partial y}\frac{\partial}{\partial \xi}\Big(2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)+J\Big(\psi_{0}, 2\frac{\partial^2 \psi_{11}}{\partial x\partial \xi}\Big)+J\Big(\psi_{11}, 2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial}{\partial y}\nabla^2\psi_{11}\\&&+\frac{\partial \psi_{11}}{\partial \xi}\frac{\partial}{\partial y}\nabla^2\psi_{0}-\frac{\partial \psi_{0}}{\partial y}\frac{\partial}{\partial \xi}\nabla^2\psi_{11}-\frac{\partial \psi_{11}}{\partial y}\frac{\partial}{\partial \xi}\nabla^2\psi_{0}+\overline{u}\frac{\partial \Omega}{\partial \xi}-\overline{u}\frac{\partial^2 \Omega}{\partial x\partial \xi}\\&&+J(\psi_{0}, \Omega)-\lambda'\delta J\Big(\psi_{0}, \frac{\partial \Omega}{\partial y}\Big)\bigg\}+\varepsilon^2\bigg\{\frac{\partial^3 \psi_{0}}{\partial T\partial \xi^2}+(\overline{u}-C_{g})\frac{\partial^3 \psi_{11}}{\partial \xi^3}+J\Big(\psi_{11}, \frac{\partial^2 \psi_{0}}{\partial \xi^2}\Big)\\&&+J\Big(\psi_{0}, \frac{\partial^2 \psi_{11}}{\partial \xi^2}\Big)+\frac{\partial \psi_{11}}{\partial \xi}\frac{\partial}{\partial y}\Big(2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)-\frac{\partial \psi_{11}}{\partial y}\frac{\partial}{\partial \xi}\Big(2\frac{\partial^2 \psi_{0}}{\partial x\partial \xi}\Big)+J(\psi_{11}, \Omega)\\&&-\lambda'\delta J\Big(\psi_{11}, \frac{\partial \Omega}{\partial y}\Big)+\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial \Omega}{\partial y}-\frac{\partial \psi_{0}}{\partial y}\frac{\partial \Omega}{\partial \xi}+\frac{\partial \psi_{0}}{\partial y}\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \psi_{0}}{\partial \xi}\frac{\partial^2 \Omega}{\partial y^2}\bigg\}=0, \end{eqnarray} $

其中方程(2.13)中的系数请参照附录A.

将方程(2.10), (2.12)和(2.13)代入到方程(2.14)中,要想不含$\exp[{\rm i}(kx-\omega t)]$项,必须满足条件

(2.15) $\begin{eqnarray} &&{\rm i}\frac{\partial A}{\partial T}+\gamma\frac{\partial^2 A}{\partial \xi^2}+RA|A|^2\\&=&{\rm i}\bigg\{\varepsilon\bigg[\frac{\overline{u}-C_{g}}{m^2+k^2+F}\frac{\partial^3 A}{\partial \xi^3}+\Big(\frac{R}{k}+R_{1}\Big)\frac{\partial (|A|^2A)}{\partial \xi}+(-R_{1})A\frac{\partial |A|^2}{\partial \xi}\\&&+\frac{2{\rm i} k}{k^2+m^2+F}\frac{\partial^2 A}{\partial T\partial \xi}+{\rm i}\frac{(\frac{\partial \Omega}{\partial y}-\lambda'\delta\frac{\partial^2 \Omega}{\partial y^2})k}{m^2+k^2+F}A+\frac{(\lambda'\delta\frac{\partial^2 \Omega}{\partial x\partial y}-\frac{\partial \Omega}{\partial x})m\cot my}{m^2+k^2+F}A\bigg]\\&&+\varepsilon^2\bigg[\frac{1}{m^2+k^2+F}\frac{\partial^3 A}{\partial T\partial \xi^2}+{\rm i}\frac{R_{1}}{2k}\frac{\partial}{\partial \xi}\Big(A\frac{\partial |A|^2}{\partial \xi}\Big)-{\rm i}\frac{3R_{1}}{2k}|A|^2\frac{\partial^2 A}{\partial \xi^2}-{\rm i}\frac{3R_{1}}{2k}\frac{\partial A}{\partial \xi}\frac{\partial |A|^2}{\partial \xi}\\&&+\frac{\frac{\partial \Omega}{\partial y}-\frac{\partial^2 \Omega}{\partial y^2}}{m^2+k^2+F}\frac{\partial A}{\partial \xi}+\frac{(\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \Omega}{\partial \xi})m \cotmy}{m^2+k^2+F}A\bigg]\bigg\}+O(\varepsilon^3)=0, \end{eqnarray} $

其中的系数$\gamma, R, R_{1}$在附录B中给出.

在这一节中,我们来研究Rossby波列的调制不稳定性.方程(2.15)中的振幅取如下形式

(3.1) $\begin{equation}A=[A_{0}+\varphi(\xi, T)]\exp[{\rm i}\delta A_{0}^2T], \end{equation}$

$\varphi(\xi, T)$代表调制性.

将方程(3.1)代入方程(2.15)中,得到$\varphi(\xi, T)$的线性化方程

(3.2) $\begin{eqnarray} &&{\rm i}\frac{\partial \varphi}{\partial T}+\gamma\frac{\partial^2 \varphi}{\partial \xi^2}+RA_{0}^{2}(\varphi+\varphi^*)\\&=&{\rm i}\bigg\{\varepsilon\bigg[\gamma_{1}\frac{\partial^3 \varphi}{\partial \xi^3}+\gamma_{2}A_{0}^2(2\varphi_{\xi}+\psi_{\xi}^*)+\gamma_{3}A_{0}^2(\varphi_{\xi}+\psi_{\xi}^*)+\gamma_{4}RA_{0}^2\frac{\partial \varphi}{\partial \xi}+{\rm i}\gamma_{5}\varphi_{\xi T}\\&&+{\rm i}\frac{k(\frac{\partial \Omega}{\partial y}-\lambda'\delta\frac{\partial^2 \Omega}{\partial y^2})}{m^2+k^2+F}(A_{0}+\varphi(\xi, T))+\frac{(\lambda'\delta\frac{\partial^2 \Omega}{\partial x\partial y}-\frac{\partial \Omega}{\partial x})m \cot my}{m^2+k^2+F}(A_{0}+\varphi(\xi, T))\bigg]\\&&+\varepsilon^2\bigg[{\rm i}\gamma_{6}RA_{0}^2\frac{\partial^2 \varphi}{\partial \xi^2}+{\rm i}\gamma_{7}A_{0}^2(\varphi_{\xi\xi}+\varphi_{\xi\xi}^*)-{\rm i}\gamma_{8}A_{0}^2\frac{\partial^2 \varphi}{\partial \xi^2}+\frac{\frac{\partial \Omega}{\partial y}-\frac{\partial^2 \Omega}{\partial y^2}}{m^2+k^2+F}\varphi_{\xi}\\&&+\frac{(\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \Omega}{\partial \xi})m \cot my}{m^2+k^2+F}(A_{0}+\varphi(\xi, T))\bigg]\bigg\}, \end{eqnarray} $

其中$\gamma_{1}=(\overline{u}-C_{g})/(k^2+m^2+F), $$\gamma_{2}=R/k+R_{1}, \gamma_{3}=-R_{1}, $$\gamma_{4}=-(2k/(k^2+m^2+F)), $$\gamma_{5}=2k/(k^2+m^2+F), $$\gamma_{6}=1/(k^2+m^2+F), \gamma_{7}=R_{1}/2k, \gamma_{8}=3R/2k$.

我们将$\varphi(\xi, T)$写成实部与虚部的形式

(3.3) $ \begin{equation}\varphi=u+{\rm i}\rho. \end{equation}$

将方程(3.3)代入到方程(3.2),得到关于$u, \rho$的两个微分方程

(3.4) $\begin{eqnarray}u_{T}+\gamma\rho_{\xi\xi}&=&\varepsilon\bigg[\gamma_{1}u_{\xi\xi\xi}+(3\gamma_{2}+2\gamma_{3}+\gamma_{4}R)A_{0}^2u_{\xi}-\gamma_{5} \rho_{T\xi}\\&&-\frac{\frac{\partial \Omega}{\partial y}-\lambda'\delta\frac{\partial^2 \Omega}{\partial y^2}}{m^2+k^2+F}\rho+\frac{(\lambda'\delta\frac{\partial^2 \Omega}{\partial x\partial y}-\frac{\partial \Omega}{\partial y})m\cotmy}{m^2+k^2+F}(A_{0}+u)\bigg]\\&&+\varepsilon^2\bigg[(\gamma_{8}-\gamma_{6}\delta)A_{0}^2\rho_{\xi\xi} +\gamma_{6}u_{\xi\xi T}+\frac{\frac{\partial \Omega}{\partial y}-\frac{\partial^2 \Omega}{\partial y^2}}{m^2+k^2+F}u_{\xi}\\&&+\frac{(\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \Omega}{\partial \xi})m\cot my}{m^2+k^2+F}(A_{0}+u)\bigg], \end{eqnarray} $

(3.5) $\begin{eqnarray}\rho_{T}-\gamma u_{\xi\xi}-2RA_{0}^2u&=&\varepsilon\bigg[\gamma_{1}\rho_{\xi\xi\xi}+(\gamma_{2}+\gamma_{4}R)A_{0}^2\rho_{\xi}+\gamma_{5}u_{\xi}T \\&& -\frac{(\frac{\partial \Omega}{\partial y}-\lambda'\delta\frac{\partial^2 \Omega}{\partial y^2})k}{m^2+k^2+F}(A_{0}+u)-\frac{(\lambda'\delta\frac{\partial^2 \Omega}{\partial x\partial y}-\frac{\partial \Omega}{\partial x})m\cot\ my}{m^2+k^2+F}\rho\bigg]\\&& +\varepsilon^2\bigg[(\gamma_{6}\delta+2\gamma_{7}-\gamma_{8})A_{0}^2u_{\xi\xi}+\gamma_{6}\rho_{\xi\xi T} -\frac{\frac{\partial \Omega}{\partial y}-\frac{\partial^2 \Omega}{\partial y^2}}{m^2+k^2+F}\rho_{\xi}\\&&-\frac{(\frac{\partial^2 \Omega}{\partial y\partial \xi}-\frac{\partial \Omega}{\partial \xi})m \cotmy}{m^2+k^2+F}\rho\bigg]. \end{eqnarray} $

由于调制的周期性,我们取如下形式

(3.6) $ u=u_{0}\cos(K\xi-\Omega T), $

(3.7) $\rho=\rho_{0}\sin(K\xi-\Omega T), $

其中$u_{0}, \rho_{0}$是常数.

1)若$\frac{\partial \Omega}{\partial y}=Ak^2, \frac{\partial^2 \Omega}{\partial y^2}=0, $将方程(3.6), (3.7)代入到方程(3.4), (3.5)中,得到

(3.8) $\begin{eqnarray} &&\Omega^2[(1+\varepsilon^2k^2\gamma_{6})^2-\varepsilon^2k^2\gamma_{5}^2]-\Omega\bigg[\varepsilon(P_{1}+P_{2}) (1+\varepsilon^2k^2\gamma_{6})-(P_{3}+P_{4})\varepsilon K\gamma_{5}-\frac{A\varepsilon^2k^4\gamma_{5}}{m^2+k^2+F}\bigg]\\&& +\varepsilon^2P_{1}P_{2}-P_{3}P_{4}-\frac{A\varepsilon\gamma k^5}{m^2+k^2+F}+\frac{A(\gamma_{8}-\gamma_{6}R)\varepsilon^3A_{0}^2k^5}{m^2+k^2+F}=0, \end{eqnarray} $

其中

$P_{1}=\gamma_{1}K^3-(3\gamma_{2}+2\gamma_{3}+\gamma_{4}R)A_{0}^2K, P_{2}=\gamma_{1}K^3-(\gamma_{2}+\gamma_{4}R)A_{0}^2K, $

(3.9) $\begin{equation} P_{3}=\gamma K^2+\varepsilon^2[-(\gamma_{8}-\gamma_{6}R)A_{0}^2K^2], P_{4}=\gamma K^2-2RA_{0}^2+\varepsilon^2(\gamma_{6}R+2\gamma_{7}-\gamma {8})A_{0}^2K^2. \end{equation}$

我们定义$M_{0}=\varepsilon A_{0}, \varepsilon K=pk, \Omega=W/\varepsilon^2, $则方程(3.8)可以重新表示为

(3.10) $W^2-WP+Q=0, $

其中$P=I_{1}/I, Q=I_{2}/I, I=(1+\gamma_{6}(pk)^2)^2-\gamma_{5}^2(pk)^2, $

(3.11) $\begin{eqnarray} I_{1}&=&[2\gamma_{1}(pk)^3-2(2\gamma_{2}+\gamma_{3}+\gamma_{4}R)M_{0}^2pk][1+\gamma_{6}(pk)^2]-\gamma_{5}(pk) [2\gamma(pk)^2-2RM_{0}^2\\&&+2(\gamma_{6}R-\gamma_{8}+\gamma_{7})M_{0}^2(pk)^2]-\frac{A\gamma_{5}}{m^2+k^2+F}(pk)^4, \end{eqnarray} $

(3.12) $\begin{eqnarray} I_{2}&=&[\gamma_{1}(pk)^3-(3\gamma_{2}+2\gamma_{3}+\gamma_{4}R)M_{0}^2(pk)][\gamma_{1}(pk)^3-(\gamma_{2}+\gamma_{4}R)M_{0}^2(pk)] \\&&-[\gamma(pk)^2-(\gamma_{8} -\gamma_{6}R))M_{0}^2(pk)^2][\gamma(pk)^2-2RM_{0}^2+(\gamma_{6}R+2\gamma_{7}-\gamma_{8})M_{0}^2(pk)^2]\\&& -\frac{A\gamma}{m^2+k^2+F}(pk)^5+\frac{A(\gamma_{8}-\gamma_{6}R)}{m^2+k^2+F}M_{0}^2(pk)^5. \end{eqnarray} $

2)若$\frac{\partial \Omega}{\partial y}=0, \frac{\partial^2 \Omega}{\partial y^2}=Ak^2, $将方程(3.6), (3.7)代入到方程(3.4), (3.5)中,得到

(3.13) $\begin{eqnarray} &&\Omega^2[(1+\varepsilon^2k^2\gamma_{6})^2-\varepsilon^2k^2\gamma_{5}^2]-\Omega\bigg[\varepsilon(P_{1}+P_{2}) (1+\varepsilon^2k^2\gamma_{6})-(P_{3}+P_{4})\varepsilon K\gamma_{5}+\frac{A\lambda'\delta k^4\gamma_{5}}{m^2+k^2+F}\bigg]\\&& +\varepsilon^2P_{1}P_{2}-P_{3}P_{4}+\frac{A\lambda'\delta\varepsilon k^5}{m^2+k^2+F}-\frac{A\lambda'\delta(\gamma_{8}-\gamma_{6}R)\varepsilon^3A_{0}^2k^5}{m^2+k^2+F}=0, \end{eqnarray} $

其中

$P_{1}=\gamma_{1}K^3-(3\gamma_{2}+2\gamma_{3}+\gamma_{4}R)A_{0}^2K, P_{2}=\gamma_{1}K^3-(\gamma_{2}+\gamma_{4}R)A_{0}^2K, $

(3.14) $\begin{equation} P_{3}=\gamma K^2+\varepsilon^2[-(\gamma_{8}-\gamma_{6}R)A_{0}^2K^2], P_{4}=\gamma K^2-2RA_{0}^2+\varepsilon^2(\gamma_{6}R+2\gamma_{7}-\gamma_{8})A_{0}^2K^2. \end{equation}$

我们定义$M_{0}=\varepsilon A_{0}, \varepsilon K=pk, \Omega=W/\varepsilon^2, $则方程(3.13)可以重新表示为

(3.15) $W^2-WP+Q=0, $

这里$P=I_{1}/I, Q=I_{2}/I, I=(1+\gamma_{6}(pk)^2)^2-\gamma_{5}^2(pk)^2, $

(3.16) $\begin{eqnarray} I_{1}&=&[2\gamma_{1}(pk)^3-2(2\gamma_{2}+\gamma_{3}+\gamma_{4}R)M_{0}^2pk][1+\gamma_{6}(pk)^2]-\gamma_{5}pk[2\gamma(pk)^2 \\&& -2RM_{0}^2+2(\gamma_{6}R-\gamma_{8}+\gamma_{7})M_{0}^2(pk)^2]+\frac{A\lambda'\delta\gamma_{5}}{m^2+k^2+F}(pk)^4, \end{eqnarray}$

(3.17) $\begin{eqnarray} I_{2}&=&[\gamma_{1}(pk)^3-(3\gamma_{2}+2\gamma_{3}+\gamma_{4}R)M_{0}^2(pk)][\gamma_{1}(pk)^3- (\gamma_{2}+\gamma_{4}R)M_{0}^2(pk)]\\&&-[\gamma(pk)^2-(\gamma_{8} -\gamma_{6}R)M_{0}^2(pk)^2][\gamma(pk)^2-2RM_{0}+(\gamma_{6}R+2\gamma_{7}-\gamma_{8})M_{0}^2(pk)^2]\\&& +\frac{A\lambda'\delta\gamma}{m^2+k^2+F}(pk)^5-\frac{A\lambda'\delta(\gamma_{8}-\gamma_{6}R)}{m^2+k^2+F}M_{0}^2(pk)^5. \end{eqnarray} $

方程(3.15)的解为

(3.18) $ \begin{equation}W=\frac{1}{2}(P\pm\sqrt{P^2-4Q}). \end{equation}$

在方程(3.18)中,当$P^2-4Q<0$时,此时均匀Rossby波列是不稳定的,对于Rossby波列, Plumb^[34]首次提出调制不稳定性,之后Yamagata^[8]在NLS方程阐明了这种调制不稳定性.显然,我们可以看出如果忽略带有水平分量项以及地形项,那么方程(3.18)可退化为罗德海的结果^[21].与此同时,我们可以发现,由于水平分量项和地形项的存在会影响调制不稳定性的不稳定区域.因此,为了讨论这些项对其不稳定区域的影响,我们作以下定义:Im$W=100\sqrt{4Q-P^2}$,并且$P^2-4Q<0$.这里$\pm k(\varepsilon k=pk)$是一对作用在水平波数为$k$的边界扰动, $p$为变量.不失一般性,我们取参数$F=1.0, L_{y}=5.0, M_{0}=0.3$.若取$\overline{u}=0.0, 0.75, 1.0, 1.25$,水平波数$k=2/[6.371\cos(\phi_{0})]$,且在北纬60度,我们可以通过图 1和图 2来描述$W与P$的关系.

第一种情形:从图 1中我们可以看到,在北纬60度下,当$\overline{u}=0.0$时,在地形项以及水平分量项下的均匀Rossby波列的不稳定区域要小于仅含高阶项的区域,另一方面,在此情形下,当$0<p<0.80$,此时的区域要窄与仅含高阶项的区域,当$\overline{u}=0.75$时,这时仅含高阶项的不稳定区域相对变小,然而,若加上地形项以及水平分量项,此时的不稳定区域为$0<p<0.81$.与此同时,加上地形项以及水平分量项,此时的不稳定区域为$0<p<0.6$,它要稍稍高于仅含高阶项的区域.当$\overline{u}=1.0$,只含高阶项的不稳定区域为$0<p<1.0$,而加上地形项以及水平分量项,那么不稳定区域为$0<p<0.79$.值得注意的是,在$0<p<0.57$,这两条曲线相差的区域在减少.当$\overline{u}=1.25$我们可以看到仅含高阶项的不稳定区域位于$0<p<0.9$,而含有地形项与水平分量项的区域是$0<p<0.74$.由此看来,随着基本气流的增大,对不稳定区域的影响在逐渐减弱.因此,我们得到的结论是,基本气流可以影响Rossby列的调制不稳定性.

第二种情形:从图 2中我们可以看出,在北纬60度下,当$\overline{u}=0.0$时,在地形项和水平分量下的均匀Rossby波列的不稳定性区域仍然小于仅含高阶项的区域,另一方面,在此情形下,当$0<p<0.80$时,此时的区域要窄于仅含高阶项的区域.当$\overline{u}=0.75$这时仅含高阶项的不稳定区域相对变小,然而,若加上地形项与水平分量项,此时的不稳定区域要超过1,与此同时,我们可以从图中看到实线逐渐逼近虚线.当$\overline{u}=1.0在0<p<0.55$实线要低于虚线,但是当$p>0.55$实线要高于虚线.当$\overline{u}=1.25$我们可以看到在区域$0<p<0.47$,含有地形项与水平分量项实线逐渐逼近虚线,而$p>0.47$含有地形项与高阶项的不稳定区域要远远大于仅含高阶项的区域.从图 1与图 2中可以得到,均匀基本气流可以对Rossby列产生一定的影响.

在本文中,应用了摄动展开的方法推导了在含有地形项作用和完整Coriolis力作用下的一类高阶Schrödinger方程(HNLS方程).基于此方程,讨论了均匀Rossby波列的调制不稳定性.我们可以得到完整Coriolis力下的高阶项$O(\varepsilon)$和$O(\varepsilon^2)$对均匀Rossby波列的调制不稳定性会产生一定的影响.倘若不考虑地球旋转水平分量,即$\lambda'=0$,那么方程(2.15)则退回罗德海的结果^[21].

方程(2.13)的系数可以定义为

(A.1) $g_{m}=\frac{8}{m[4-(n+1/2)^2]L_{y}}, q_{n}=\frac{4k^2m}{L_{y}\{\beta+F\overline{u}-(\overline{u}-C_{g})[F+(n+1/2)^2m^2]\}}.$

方程(2.15)的系数可以定义为

$\gamma=\frac{\beta k(3m^2-k^2)}{(m^2+k^2)^3}, \gamma_{1}=\frac{\overline{u}-C_{g}}{m^2+k^2}, \gamma_{2}=\frac{R}{k}+R_{1}, \gamma_{3}=-R_{1}, \gamma_{4}=-\frac{2k}{m^2+k^2}, $

$\gamma_{5}=-\gamma_{4}, \gamma_{6}=\frac{1}{m^2+k^2}, \gamma_{7}=\frac{R1}{2k}, \gamma_{8}=\frac{3R}{2k}, R_{1}=\frac{2mk^2}{m^2+k^2}\sum\limits_{n=1}^{\infty}q_{n}g_{n}^2, $

(B.1) $R=\frac{km\sum\limits_{n=1}^{\infty}[m^2+k^2-(n+\frac{1}{2}m^2)]q_{n}g_{n}^2}{m^2+k^2}.$