地球旋转对地球流体中波的产生有及其重要的作用, 其中在大气海洋领域中它的作用也是显而易见. Rossby波同样与地球旋转是密不可分.国内外学者从不同角度出发对Rossby波的特性进行研究^[1-5].大气千变万化, 大气运动可以被一系列基本原始方程所描述, 如有连续方程、运动方程、能量方程.自从Long开创性用KdV方程来描述较为理想状态正压流体中的Rossby波的振幅演变规律后^[6].许多学者分析诱导与加强大气运动中Rossby波会受到beta效应、地形强迫、耗散和外源、基本流的切变效应、地形缓变效应以及行星波与天气波的相互作用等因素的影响^[7-9].其中在分析Rossby波振幅特性的过程中, 通常从准地砖位涡方程出发来进行研究. Dellar等^[10]利用变分原理推导含有完整Coriolis力作用的准地转位涡方程.在2010年, 他们^[11]扩展了这项工作, 并导出了具有完整Coriolis力的方程来描述无粘、不可压缩流体的多层流体背景下的Roosby波的流动.尹晓军从含有完整Coriolis力的准地转位涡方程出发^[12], 推导出了mKdV-Burgers方程, 进一步阐述了Rossby波的振幅演变规律会受到地球旋转水平分量、beta效应以及强耗散三个因素的影响.杨红卫从基本方程出发, 推到出分数阶BDO方程去描述Rossby波的振幅演变规律^[13].关于完整Coriolis力相关报道, 见文献[14-16].但是我们发现上述文献都没有讨论中高纬度Rossby波的波动形态, 实际上极端的天气现象(气旋、反气旋、寒潮等)主要发生在中高纬度地区, 极大的影响了人们的生活.另一方面, 由于描述波的波动形态是一系列微分方程, 因此寻找微分方程的解析解或者孤立波解近年来得到了迅速发展.如吕兴等应用双线性变换求解了(3+1)维非线性演变方程以及Boussinesq方程的各种精确解以及多孤立子解^[17-18].再比如Jacobi椭圆函数展开法^[19], 同伦摄动法^[20], Bäcklund变换法^[21]等.

本文主要对受到完整科里奥利力、地形效应、耗散和外源强迫共同作用的准地转位涡方程进行研究.首先对准地转位涡方程所表示的大尺度问题作了无量纲变换; 然后把流函数分为基本流函数和扰动流函数两部分, 在色散和非线性之间平衡的条件下, 通过作时空伸缩变换和摄动展开法推导出弱地形作用下的非齐次mKdV-Burgers方程, 阐述科里奥利力对方程齐次项系数产生影响, 所得的弱地形效应影响到方程中强迫项; 最后对得到的非齐次mKdV-Burgers方程应用简化的微分变化法和Maple数学软件进行了近似求解, 并且对一种特定Rossby波的振幅进行图形模拟后发现, Rossby波的振幅随时间在逐渐增大, Rossby波的波峰与波谷出现的经度位置随时间没有明显改变.

考虑含有完整科氏力以及带有耗散和外源的准地转位涡方程:

$ \begin{gather} (\frac{\partial}{\partial t}-\frac{\partial\phi}{\partial y}\frac{\partial}{\partial x}+\frac{\partial\phi}{\partial x}\frac{\partial}{\partial y})[\nabla^{2}\phi+\beta(y)y+\frac{f_{0}}{H}B(x, y)-f_{H}\frac{\partial B}{\partial y}] = -\mu_{0}\nabla^{2}\phi+Q, \end{gather} $

(2.1)

其中$ \phi(x, y, t) $表示总的流函数, $ x $、$ y $、$ t $分别表示经度和纬度变量以及时间变量, $ \nabla^{2}\phi = \frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}} $, $ f = f_{0}+\beta(y)y $和$ f_{H} $分别表示科里奥利力的垂直分量和水平分量, 且$ f_{0} $和$ f_{H} $为常数, $ H $表示垂直尺度, $ B(x, y) $表示底地形函数, $ \mu_{0}\nabla^{2}\phi $表示耗散, $ \mu_{0} $表示耗散强度, $ Q $表示外源.侧边界条件满足的刚壁条件

$ \begin{gather} \frac{\partial\phi}{\partial x} = 0, y = y_{1}, y = y_{2}. \end{gather} $

(2.2)

方程(2.2)中$ y = y_{1} $, $ y = y_{2} $分别表示地球南北方向的边界.

首先通过无量纲化方程(2.1)和(2.2)变为

$ \begin{gather} (\frac{\partial}{\partial t}-\frac{\partial\phi}{\partial y}\frac{\partial}{\partial x}+\frac{\partial\phi}{\partial x}\frac{\partial}{\partial y})[\nabla^{2}\phi+\beta(y)y+B(x, y)-\lambda\delta\frac{\partial B}{\partial y}] = -\mu\nabla^{2}\phi+Q, \end{gather} $

(2.3)

$ \begin{gather} \frac{\partial\phi}{\partial x} = 0, y = 0, y = 1. \end{gather} $

(2.4)

引入两个无量纲参数$ \delta = \frac{H}{L_{0}} $, $ \delta $表示形态比. $ \lambda = \frac{f_{H}}{f_{0}} $.其中$ L_{0} $表示水平尺度.假设总的流函数$ \phi(x, y, t) $是由基本流函数和扰动流函数两部分构成, 即

$ \begin{gather} \phi(x, y, t) = -\int_{0}^{y}[u(s)-c_{0}]ds+\varepsilon\phi'(x, y, t), \end{gather} $

(2.5)

把(2.5)式代入到方程(2.3)中得

$ \begin{eqnarray} &&\varepsilon[\frac{\partial}{\partial t}+(U-c_{0})\frac{\partial}{\partial x}]\nabla^{2}\phi'+\varepsilon p(y)\frac{\partial\phi'}{\partial x}+\varepsilon J[\phi', B]+\varepsilon^{2}J[\phi', \nabla^{2}\phi']-\lambda\delta\varepsilon J [\phi', \frac{\partial B}{\partial y}]\\ &&-\lambda\delta(U-c_{0})\frac{\partial^{2}B}{\partial y\partial x}+(U-c_{0})\frac{\partial B}{\partial x} = -\mu\varepsilon\nabla^{2}\phi'+\mu U'+Q, \end{eqnarray} $

(2.6)

其中$ p(y) = [\beta(y)y]'-U'' $, $ J[E, F] = \frac{\partial E}{\partial x}\frac{\partial F}{\partial y}-\frac{\partial E}{\partial y}\frac{\partial F}{\partial x} $.为了色散和非线性之间达到平衡, 令

$ \begin{gather} \mu = \varepsilon^{3}\hat{\mu}, Q = -\mu U', \end{gather} $

(2.7)

把变换(2.7)式代入到(2.6)式得

$ \begin{align} &\varepsilon[\frac{\partial}{\partial t}+(U-c_{0})\frac{\partial}{\partial x}]\nabla^{2}\phi'+\varepsilon p(y)\frac{\partial\phi'}{\partial x}+\varepsilon J[\phi', B]+\varepsilon^{2}J[\phi', \nabla^{2}\phi']-\lambda\delta\varepsilon J [\phi', \frac{\partial B}{\partial y}]\\ &-\lambda\delta(U-c_{0})\frac{\partial^{2}B}{\partial y\partial x}+(U-c_{0})\frac{\partial B}{\partial x} = -\hat{\mu}\varepsilon^{4}\nabla^{2}\phi', \end{align} $

(2.8)

为了讨论非线性长波, 可作时空伸缩变换, 即Gardner-Morikawa变换

$ \begin{gather} X = \varepsilon x, T = \varepsilon^{3}t, \end{gather} $

(2.9)

其中$ X $, $ T $分别为经度和时间的缓变量, 同时由变换(2.9)得

$ \begin{gather} \frac{\partial}{\partial x} = \varepsilon\frac{\partial}{\partial X}, \frac{\partial}{\partial t} = \varepsilon^{3}\frac{\partial}{\partial T}. \end{gather} $

(2.10)

把变换(2.9)和(2.10)代入到方程(2.8)中得

$ \begin{align} &\varepsilon^{4}\frac{\partial G}{\partial T}+\varepsilon^{3}J[\phi', G]+\varepsilon^{2}\{(U-c_{0})\frac{\partial G}{\partial X}+\frac{\partial\phi'}{\partial X}p(y)+J[\phi', B]+\lambda\delta J[\frac{\partial B}{\partial y}, \phi']\}\\ &+\varepsilon (U-c_{0})(\frac{\partial B}{\partial X}-\lambda\delta\frac{\partial^{2}B}{\partial y\partial X}) = -\hat{\mu}\varepsilon^{4}G, \end{align} $

(2.11)

其中$ G = (\varepsilon^{2}\frac{\partial^{2}\phi'}{\partial X^{2}}+\frac{\partial^{2}\phi'}{\partial y^{2}}) $, $ J[E, F] = \frac{\partial E}{\partial X}\frac{\partial F}{\partial y}-\frac{\partial E}{\partial y}\frac{\partial F}{\partial X} $.假设底地形函数

$ \begin{gather} B(x, y) = \varepsilon^{i}h(X, y), \end{gather} $

(2.12)

把(2.12)式代入(2.11)式中得

$ \begin{align} &\varepsilon^{2}\frac{\partial G}{\partial T}+\varepsilon J[\phi', G]+(U-c_{0})\frac{\partial G}{\partial X} +\frac{\partial\phi'}{\partial X}p(y)+\varepsilon^{i}J[\phi', h] +\lambda\delta \varepsilon^{i}J[\frac{\partial h}{\partial y}, \phi']\\ &+\varepsilon^{i-1}(U-c_{0})(\frac{\partial h}{\partial X}-\lambda\delta\frac{\partial^{2}h}{\partial y\partial X}) = -\hat{\mu}\varepsilon^{2}G, \end{align} $

(2.13)

通过后续(2.17), (2.21), (2.24)式分析得

$ \begin{gather} i = 3. \end{gather} $

(2.14)

从而底地形函数$ B(x, y) = \varepsilon^{3}h(X, y) $, 因为科氏参数$ \varepsilon $$ \ll $1, 所以底地形函数$ B(x, y) $表示相对非常小的量, 即为弱地形作用.进一步方程(2.13)可变为

$ \begin{align} &\varepsilon^{2}\frac{\partial G}{\partial T}+\varepsilon J[\phi', G] +(U-c_{0})\frac{\partial G}{\partial X}+\frac{\partial\phi'}{\partial X}p(y) +\varepsilon^{3}J[\phi', h]+\lambda\delta \varepsilon^{3}J[\frac{\partial h}{\partial y}, \phi']\\ &+\varepsilon^{2}(U-c_{0})(\frac{\partial h}{\partial X}-\lambda\delta\frac{\partial^{2}h}{\partial y\partial X}) = -\hat{\mu}\varepsilon^{2}G. \end{align} $

(2.15)

下面采用摄动展开法.首先设扰动流函数有如下的小参数展开式

$ \begin{gather} \phi' = \phi_{0}(X, y, T)+\varepsilon \phi_{1}(X, y, T)+\varepsilon^{2} \phi_{2}(X, y, T)+\cdots. \end{gather} $

(2.16)

把方程(2.16)代入到方程(2.15)中, 通过比较$ \varepsilon^{0} $的系数得

$ \begin{gather} o(\varepsilon^{0}):(U-c_{0})\frac{\partial}{\partial X}\frac{\partial^{2}\phi_{0}}{\partial y^{2}}+\frac{\partial\phi_{0}}{\partial X}p(y) = 0. \end{gather} $

(2.17)

假设$ \phi_{0} $具有下列形式的分离变量解

$ \begin{gather} \phi_{0} = A(X, T)\Phi_{0}(y), \end{gather} $

(2.18)

其中$ A(X, T) $表示Rossby波的振幅, 把方程(2.18)代入方程(2.17)中得$ \Phi_{0} $满足

$ \begin{gather} \Phi''_{0}+\frac{p(y)}{U-c_{0}}\Phi_{0} = 0, \end{gather} $

(2.19)

$ \begin{gather} \Phi_{0}(0) = \Phi_{0}(1) = 0. \end{gather} $

(2.20)

由于函数$ p(y) $的未知性, 所以从本征值问题(2.19)和(2.20)来确定本征函数$ \Phi_{0} $和本征值$ c_{0} $的精确解是比较困难.为了确定Rossby波振幅$ A(X, T) $的数学演化模型, 继续比较$ \varepsilon $的系数得

$ \begin{gather} o(\varepsilon):(U-c_{0})\frac{\partial ^{3}\phi_{1}}{\partial y^{2}\partial X}+p(y)\frac{\partial\phi_{1}}{\partial X} = -\frac{\partial\phi_{0}}{\partial X}\frac{\partial^{3}\phi_{0}}{\partial y^{3}}+\frac{\partial\phi_{0}}{\partial y}\frac{\partial^{3}\phi_{0}}{\partial y^{2}\partial X}. \end{gather} $

(2.21)

假设$ \phi_{1} $具有下列形式的分离变量解

$ \begin{gather} \phi_{1} = \tilde{A}(X, T)\Phi_{1}(y). \end{gather} $

(2.22)

把(2.22)式代入(2.21)式, 通过分析可以得到$ \tilde{A} = \frac{1}{2}A^{2} $, 得$ \Phi_{1} $满足

$ \begin{gather} \Phi''_{1}+\frac{p(y)}{U-c_{0}}\Phi_{1} = \frac{1}{U-c_{0}}[\frac{p(y)}{U-c_{0}}]'\Phi^{2}_{0}. \end{gather} $

(2.23)

通过分析, 还不能从方程(2.23)中确定Rossby波振幅的演化规律所满足的数学模型, 需要提高精度, 继续比较$ \varepsilon^{2} $的系数得

$ \begin{gather} o(\varepsilon^{2}):(U-c_{0})\frac{\partial ^{3}\phi_{2}}{\partial y^{2}\partial X}+p(y)\frac{\partial\phi_{2}}{\partial X} = -F, \end{gather} $

(2.24)

其中

$ \begin{align} F = &\frac{\partial^{3}\phi_{0}}{\partial y^{2}\partial T} +(U-c_{0})\frac{\partial^{3}\phi_{0}}{\partial X^{3}}+J[\phi_{1}, \frac{\partial^{2}\phi_{0}}{\partial y^{2}}] +J[\phi_{0}, \frac{\partial^{2}\phi_{1}}{\partial y^{2}}]\\ &+(U-c_{0})(\frac{\partial h}{\partial X}-\lambda\delta\frac{\partial^{2}h}{\partial y\partial X})+\hat{\mu}\frac{\partial^{2}\phi_{0}}{\partial y^{2}}, \end{align} $

(2.25)

把(2.18)和(2.22)式代入(2.25)式中, 并利用(2.19)和(2.23)式得

$ \begin{align} F = &[-\frac{p(y)}{U-c_{0}}\frac{\partial A}{\partial T}+(U-c_{0})\frac{\partial^{3} A}{\partial X^{3}}]\Phi_{0}+(U-c_{0})(\frac{\partial h}{\partial X}-\lambda\delta\frac{\partial^{2}h}{\partial h\partial X}) +\hat{\mu}A\Phi''_{0}\\ &+(\Phi'''_{0}\Phi_{1}+\frac{1}{2}\Phi_{0}\Phi'''_{1}-\frac{1}{2}\Phi''_{0}\Phi'_{1}-\Phi'_{0}\Phi''_{1})A^{2}\frac{\partial A}{\partial X}. \end{align} $

(2.26)

利用本征函数的正交性和消奇异条件

$ \begin{gather} \int^{1}_{0}\Phi_{0}\frac{F}{U-c_{0}}dy = 0, \end{gather} $

(2.27)

可以得到Rossby波的振幅满足下列非齐次mKdV-Burgers方程

$ \begin{gather} \frac{\partial A}{\partial T}+\alpha A^{2}\frac{\partial A}{\partial X}+\beta\frac{\partial^{3}A}{\partial X^{3}}+\gamma A = \frac{dH_{1}}{dX}+\lambda\delta\frac{dH_{2}}{dX}, \end{gather} $

(2.28)

其中系数如下

$ \begin{equation} \left\{\begin{aligned} I& = \int^{1}_{0}\frac{p(y)}{(U-c_{0})^{2}}\Phi^{2}_{0}dy, \\ \alpha& = -\frac{1}{2I}\int^{1}_{0}\frac{1}{U-c_{0}}\{[\frac{1}{U-c_{0}}(\frac{p(y)}{U-c_{0}})_{y}]_{y}\Phi^{4}_{0}-3(\frac{p(y)}{U-c_{0}})_{y} \Phi^{2}_{0}\Phi_{1}\}dy, \\ \beta& = -\frac{1}{I}\int^{1}_{0}\Phi^{2}_{0}dy, \\ \gamma& = \frac{\hat{\mu}}{I}\int^{1}_{0}\frac{\Phi^{2}_{0}p(y)}{(U-c_{0})^{2}}dy, H_{1} = \frac{1}{I}\int^{1}_{0}h\Phi_{0}dy, H_{2} = -\frac{1}{I}\int^{1}_{0}\Phi_{0}\frac{\partial h}{\partial y}dy.\\ \end{aligned} \right. \end{equation} $

(2.29)

从方程组(2.28)和(2.29)中的$ \frac{dH_{1}}{dX} $和$ \lambda\delta\frac{dH_{2}}{dX} $可以看出底地形效应对方程起到了强迫作用, 其中$ \gamma A $表示耗散的作用, 与标准Burgers方程中的$ \frac{\partial^{2}A}{\partial X^{2}} $具有相同的物理意义, 体现科里奥利力的水平分量参数$ \lambda $加强了地形的强迫作用.因此方程(2.28)表明Rossby波的振幅满足带有地形强迫的非齐次mKdV-Burgers方程, 其中系数依赖于基本剪切流以及$ \beta $效应.

下面采用简化的微分变化法求解方程(2.28).假设方程(2.28)具有如下形式的解

$ \begin{gather} A(X, T) = \sum\limits_{j = 0}^\infty A_{j}(X)T^{j}. \end{gather} $

(3.1)

为书写简便, 同时方程(2.28)的非齐次项记为$ g(X) $, 即

$ \begin{gather} g(X)\triangleq\frac{dH_{1}}{dX}+\lambda\delta\frac{dH_{2}}{dX}. \end{gather} $

(3.2)

把(3.1)和(3.2)式代入到(2.28)式得

$ \begin{gather} \sum\limits_{j = 1}^\infty jA_{j}T^{j-1}+\alpha(\sum\limits_{j = 0}^\infty A_{j}T^{j})^{2}\sum\limits_{j = 0}^\infty A'_{j}T^{j}+\beta\sum\limits_{j = 0}^\infty A'''_{j}T^{j}+\gamma\sum\limits_{j = 0}^\infty A_{j}T^{j} = g. \end{gather} $

(3.3)

分别比较(3.3)式中$ T^{0} $, $ T^{1} $, $ \cdots $, $ T^{j-1} $, $ \cdots $前的系数得$ T^{0}:A_{1}+\alpha A^{2}_{0}A'_{0}+\beta A'''_{0}+\gamma A_{0} = g $, 由此可得

$ \begin{gather} A_{1} = g-\alpha A^{2}_{0}A'_{0}-\beta A'''_{0}-\gamma A_{0}; \end{gather} $

(3.4)

$ T^{1}:2A_{2}+\alpha A_{0}(A_{0}A'_{1}+2A_{1}A'_{0})+\beta A'''_{1}+\gamma A_{1} = 0 $, 由此可得

$ \begin{gather} A_{2} = -\frac{\alpha A_{0}(A_{0}A'_{1}+2A_{1}A'_{0})+\beta A'''_{1}+\gamma A_{1}}{2}; \end{gather} $

(3.5)

$ T^{j-1}:jA_{j}+\alpha (\sum\limits_{k = 0}^{l}A_{k}A_{l-k})A'_{j-l-1}+\beta A'''_{j-1}+\gamma A_{j-1} = 0(j\geq2) $, 由此可得

$ \begin{gather} A_{j} = \frac{-\alpha (\sum\limits_{k = 0}^{l}A_{k}A_{l-k})A'_{j-l-1}-\beta A'''_{j-1}-\gamma A_{j-1}}{j}. \end{gather} $

(3.6)

把(3.4)、(3.5)和(3.6)式代入(3.1)式中, 可得方程(2.28)的解

$ \begin{align} A(X, T) = A_{0}+(g-\alpha A^{2}_{0}A'_{0}-\beta A'''_{0}-\gamma A_{0})T\\ &+\sum\limits_{j = 2}^\infty\frac{-\alpha (\sum\limits_{k = 0}^{l}A_{k}A_{l-k})A'_{j-l-1}-\beta A'''_{j-1}-\gamma A_{j-1}}{j}T^{j}. \end{align} $

(3.7)

由(3.4)、(3.5)、(3.6)和(3.7)式得, 当系数$ \alpha $、$ \beta $、$ \gamma $、$ A_{0} $和$ g(X) $确定后, 方程(2.28)的解也就确定了, 即确定了Rossby波振幅的演化规律.由(3.1)和(3.7)式得$ A(X, T) $的近似解^[7]为

$ \begin{align} A = A_{0}(X)+A_{1}(X)T+A_{2}(X)T^{2}. \end{align} $

(3.8)

假设初始项$ A_{0}(X) = m \sec h X $和非齐次项$ g(X) = \sin nX $, 由递推关系(3.4)式和(3.5)式应用Maple数学软件得

$ \begin{align} A_{1}(X) = &\sin nX+m\sec h X[\alpha m^{2}\sec h^{2}X\tan h X-\beta\tan h X(5-6\tan h^{2}X)-\gamma], \end{align}$

(3.9)

$ \begin{align}A_{2}(X) = &\alpha^{2}m^{5}\sec h^{5} X(3\tan h^{2} X-0.5)\\ &+\alpha m^{3}\sec h^{3} X(78\beta\tan h^{4}X-71\beta\tan h^{2}X-2\gamma\tan h X+8\beta)\\ &+\alpha m^{2}\sec h^{2} X(\tan h X\sin nX-0.5n\cos nX)\\ &+\beta m\sec h X\tan h X(360\beta\tan h^{5}X-660\beta\tan h^{3}X- 6\gamma\tan h^{2}X+331\beta\tan h X+5\gamma)\\ &+0.5\beta n^{3}\cos nX-30.5\beta^{2}m\sec h X+0.5\gamma^{2}m\sec h X-0.5\gamma\sin nX. \end{align} $

(3.10)

把(3.9)和(3.10)式代入(3.8)式中, 方程(2.28)的近似解就即可确定.从(3.4)、(3.5)式和(3.6)式体现出弱地形效应会对方程解的系数$ A_{j}(X), j\in N $均会产生影响.进一步得出, 在正压模式下底地形效应对Rossby波的振幅起主要作用.

依据初始项$ A_{0}(X) = m\sec h X $和非齐次项$ g(X) = \sin nX $, 通过(3.9)和(3.10)式方程(2.28)的近似解就可以完全确定.当$ \alpha = 0.5, \beta = 1, \gamma = 1, m = 1, n = 0.5 $时, 方程(2.28)的近似解的图形如上.从图 1和图 2可以看出:这种模拟式Rossby波的振幅在随时间的改变而振幅逐渐在增大.波峰和波谷出现的经度位置随时间发生略微改变.

本文从中高纬度含有完整Coriolis力的准地转位涡方程出发, 利用时空伸缩变换, 得到了描述Rossby波的振幅形态满足的非齐次mKdV-Burgers方程.在推导模型过程中, 发现底地形函数$ B(x, y) = \varepsilon^{3}h(X, y) $, 因为这里考虑的是大尺度问题, 即$\varepsilon \ll 1$时, 所以底地形函数就表示弱地形效应.当底地形效应彻底消失的时候就是文献[12]的情形.最后对非齐次mKdV-Burgers方程利用简化的微分变换法做近似求解, 对影响解的因素做出分析.

通过所得的分析结论, 可以看出在理想状态正压模式下底地形效应对Rossby波振幅影响较大.在大气海洋学中, 该结论为研究Rossby波在接近地球低层的振幅形态提供了理论依据.