气象学报  2020, Vol. 78 Issue (5): 805-815 PDF
http://dx.doi.org/10.11676/qxxb2020.048

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#### 文章信息

FENG Yerong, XUE Jishan, CHEN Dehui, WU Kaixin. 2020.
GRAPES区域扰动预报模式动力框架设计及检验
The dynamical core for GRAPES regional perturbation forecast model and verification

Acta Meteorologica Sinica, 78(5): 805-815.
http://dx.doi.org/10.11676/qxxb2020.048

### 文章历史

2019-05-15 收稿
2020-04-09 改回
GRAPES区域扰动预报模式动力框架设计及检验

1. 中国气象局广州热带海洋气象研究所/广东省区域数值天气预报重点实验室，广州，510641;
2. 中国气象科学研究院，北京，100081;
3. 国家气象中心，北京，100081

The dynamical core for GRAPES regional perturbation forecast model and verification
FENG Yerong1 , XUE Jishan2 , CHEN Dehui3 , WU Kaixin1
1. Guangzhou Institute of Tropical and Marine Meteorology/Key Laboratory of Regional Numerical Weather Prediction, Guangzhou 510641, China;
2. Chinese Academy of Meteorological Sciences, Beijing 100081, China;
3. National Meteorological Center, Beijing 100081, China
Abstract: In this study, the perturbation forecast model GRAPES_PF appropriate for the implementation of four dimensional variational data assimilation (4D-Var) system has been developed based on the regional numerical weather prediction model GRAPES. GRAPES_PF involves a set of linear perturbation forecast equations including momentum, thermodynamic, moisture and continuity, which are derived from the non-hydrostatic primitive equations used in GRAPES on a terrain-following vertical coordinate framework. A semi-implicit, semi-Lagrangian and two time-level integration scheme is applied to the linear equations. Spatial discretization is performed on the Arakawa staggered C-grid in the horizontal and the Charney-Phillips grid in the vertical. The Helmholtz equation that only contains perturbation Exner pressure at future time step of integration is obtained by eliminating other variables in the linear perturbation equations. Similar to the nonlinear model, the generalized conjugate residual (GCR) method is used to solve the perturbation Helmholtz equation. A numerical experiment has been designed to evaluate the GRAPES_PF model by applying an initial perturbation of mesoscale high pressure centered at model domain and predicting its evolution with time. The same initial perturbation of high pressure is also added to nonlinear model so that the evolution of the perturbation can be traced as truth for verification. We then verify the perturbations predicted by the linear GRAPES_PF model against those of the nonlinear GRAPES model. Results show that the initial pressure perturbation induces a fast-moving-outbound internal inertial gravity wave through the well-known geostrophic adaptation process. The linear GRAPES_PF model produces results similar to that of nonlinear GRAPES model with high accuracy: the initial pressure perturbation subsequently induces increments in the fields such as horizontal winds, vertical velocity, potential temperature and water vapor, which are almost identical to those of the nonlinear model. The main conclusion is that the perturbation forecast model GRAPES-PF, as a reasonable linear version of the nonlinear GRAPES model, can offer a good scientific base for the 4D-Var data assimilation system to be developed in the future.
Key words: Perturbation forecast model    GRAPES nonlinear model    Four dimensional variational data assimilation (4D-Var)    Semi-implicit semi-Lagrangian scheme    Helmholtz equation
1 引　言

2 扰动模式的设计 2.1 GRAPES非线性模式基本方程组

GRAPES非线性模式在地形追随垂直坐标下的球面基本方程组如下（薛纪善等，2008）：

 $\begin{split} \frac{{\rm d}{{{V}}}}{{\rm d}t}=&-{c}_{p}\theta {\left({\nabla }_{{\rm{h}}}\varPi \right)}_{\hat {\textit z}}-{c}_{p}\theta \frac{\partial \varPi }{\partial \hat{{\textit z}}}\left({Z}_{{{s}}{{x}}}{{\bf{i}}}+{Z}_{{{s}}{{y}}}{{\bf{j}}}+{Z}_{{{s}}\hat{{{\textit z}}}}{{\bf{k}}}\right)+\\&\left\{fv+{\delta }_{{\rm{M}}}\left(\frac{vu\tan\varphi }{r}-\frac{uw}{r}\right)-{\delta }_{{\rm{\varphi}}}{f}_{{\rm{\varphi}}}w\right\}{{\bf{i}}}- \\ &\left\{fu+{\delta }_{{\rm{M}}}\left(\frac{{u}^{2}\tan\varphi }{r}+\frac{vw}{r}\right)\right\}{{\bf{j}}}-\\&\left\{g-{\delta }_{\rm M}\frac{{u}^{2}+{v}^{2}}{r}-{\delta }_{{\rm{\varphi}}}{f}_{{\rm{\varphi}}}u\right\}{{\bf{k}}}+{{{F}}}\\[-12pt] \end{split}$ (1)

 $\frac{{{\rm{d}}\varPi }}{{{\rm{d}}t}} = - \hat \gamma \varPi {D_3} + \frac{{\hat \gamma }}{{{c_{{p}}}\theta }}\left({{Q_{\rm{R}}} + L{P_{\rm{w}}}} \right)$ (2)

 $\frac{{\rm{d}}\theta }{{\rm{d}}t}=\frac{{Q}_{{\rm{R}}}+L{P}_{{\rm{w}}}}{{c}_{{{p}}}\varPi }$ (3)

 $\frac{{\rm{d}}q}{{\rm{d}}t}=-{P}_{{\rm{w}}}$ (4)
2.2 扰动预报方程组推导

 $\begin{split} {\left(\frac{{\rm{d}}A}{{\rm{d}}t}\right)}'&={\left(\frac{\partial A}{\partial t}+{{{V}}}{\text •} \nabla A\right)}'\approx \frac{\partial {A}'}{\partial t}+{{{V}}}{\text •} \nabla {A}'+{{{{V}}}}'{\text •} \nabla A\\&=\frac{{\rm{D}}{A}'}{{\rm{D}}t}+{{{{V}}}}'{\text •} \nabla A \end{split}$ (5)

 $\frac{{{\rm{D}}u}'}{{\rm{D}}t}=-{{{{V}}}'}{\text •} \nabla u+{c}_{{{p}}}{{L}_{{\text{π}}{{x}}}\theta }'+{L}_{{\rm{\theta}}}({\varPi }_{x}'+{Z}_{sx}{\varPi }_{\hat{{\textit z}}}')+fv'$ (6)
 $\frac{{{\rm{D}}v}'}{{\rm{D}}t}={-{{{{V}}}'}{\text •} \nabla v+c}_{{{p}}}{L}_{{\text{π}}{y}}{\theta }'+{L}_{{\rm{\theta}}}({\varPi }_{y}'+{Z}_{{sy}}{\varPi }_{\hat{{\textit z}}}')-fu'$ (7)
 $\frac{{\rm{D}}{w}'}{{\rm{D}}t}=-{{{{V}}}'}{\text •} \nabla w+{c}_{{{p}}}{L}_{{\text{π}}\hat{{{\textit z}}}}\theta '+{L}_{{\rm{\theta}}}{\textit Z}_{{{s}}\hat{{{\textit z}}}}{\varPi }_{\hat{{\textit z}}}'$ (8)

 $\frac{{\rm{D}}{\varPi }'}{{\rm{D}}t}=-{{{{V}}}'}{\text •} \nabla \varPi -{\alpha }_{{\rm{D}}}{\varPi }'-{\alpha }_{\Pi}{D}_{3}'+{\gamma }_{{\rm{\theta}}}{P'_{{\rm{w}}}}$ (9)

 $\frac{{\rm{D}}{\theta }'}{{\rm{D}}t}=-{{{{V}}}'}{\text •} \nabla \theta +{\gamma }_{\Pi}{P'_{{{{\rm{w}}}}}}$ (10)

 $\frac{{{\rm{D}}q}'}{{\rm{D}}t}=-{{{{V}}}'}{\text •} \nabla q-{P'_{{\rm{w}}}}$ (11)

 $\begin{split} \frac{{{\rm{D}}u}'}{{\rm{D}}t}=&-\left({u}'{u}_{x}+{v}'{u}_{y}+{\hat{w}}'{u}_{\hat{{\textit z}}}\right)+{c_p}{{L}_{{\text{π}}x}\theta }'+\\&{L}_{{\rm{\theta}}}({\varPi }_{x}'+{Z}_{sx}{\varPi }_{\hat{{\textit z}}}')+fv' \end{split}$ (12)
 $\begin{split} \frac{{{\rm{D}}v}'}{{\rm{D}}t}=&{-\left({u}'{v}_{x}+{v}'{v}_{y}+{\hat{w}}'{v}_{\hat{{\textit z}}}\right)+c_p}{L}_{{\text{π}}{\rm{y}}}{\theta }'+\\&{L}_{{\rm{\theta}}}({\varPi }_{y}'+{Z}_{sy}{\varPi }_{\hat{{\textit z}}}')-fu' \end{split}$ (13)
 $\frac{{\rm{D}}{w}'}{{\rm{D}}t}=-\left({u}'{w}_{x}+{v}'{w}_{y}+{\hat{w}}'{w}_{\hat{{\textit z}}}\right)+{c_pL}_{{\text{π}}\hat{{{\textit z}}}}\theta '+{L}_{{{\theta}}}{Z}_{{{s}}\hat{{{\textit z}}}}{\varPi }_{\hat{{\textit z}}}'$ (14)
 $\begin{split} \frac{{\rm{D}}{\varPi }'}{{\rm{D}}t}=&-\left({u}'{\varPi }_{x}+{v}'{\varPi }_{y}+{\hat{w}}'{\varPi }_{\hat{{\textit z}}}\right)-{\alpha }_{{\rm{D}}}{\varPi }'-\\&{\alpha }_{\Pi}{D}_{3}'+{\gamma }_{{\rm{\theta}}}{P}_{{\rm{w}}}'\end{split}$ (15)
 $\frac{{\rm{D}}{\theta }'}{{\rm{D}}t}=-\left({u}'{\theta }_{x}+{v}'{\theta }_{y}+{\hat{w}}'{\theta }_{\hat{{\textit z}}}\right)+{\gamma }_{\Pi}{P}_{{\rm{w}}}'$ (16)
 $\frac{{{\rm{D}}q}'}{{\rm{D}}t}=-\left({u}'{q}_{x}+{v}'{q}_{y}+{\hat{w}}'{q}_{\hat{{\textit z}}}\right)-{P}_{{\rm{w}}}'$ (17)

 $\hat w' = {Z_{sx}}u' + {Z_{sy}}v' + {Z_{{{s\hat {\textit z}}}}}w'\qquad\qquad\qquad$
2.3 半隐式半拉格朗日时间积分

 $\frac{{\left({A}'\right)}_{{\rm{a}}}^{{{n+1}}}-{\left({A}'\right)}_{{\rm{d}}}^{n}}{\Delta t}=\alpha {\left({L}_{{\rm{A}}}\right)}_{{\rm{a}}}^{n+1}+(1-\alpha) {\left({L}_{{\rm{A}}}\right)}_{{\rm{d}}}^{n}$

 ${\left({A}'\right)}_{{\rm{a}}}^{n+1}-{\left({A}'\right)}_{{\rm{d}}}^{n}={\alpha }_{{\rm{\varepsilon}}}{\left({L}_{{\rm{A}}}\right)}_{{\rm{a}}}^{{n}+1}+ {\beta }_{{\rm{\varepsilon}}}{\left({L}_{{\rm{A}}}\right)}_{{\rm{d}}}^{{n}}$ (18)

 ${\left({u}'\right)}_{{\rm{a}}}^{{{n}}+1}-{\alpha }_{{\rm{\varepsilon}}}{\left[{L}_{{\rm{\theta}}}{(\varPi }_{x}'+{Z}_{{{sx}}}{\varPi }_{\hat{{\textit z}}}')+fv'\right]}_{{\rm{a}}}^{{{n}}+1}={X}_{u}'$ (19)

 ${u}'-{\alpha }_{{\rm{\varepsilon}}}f{v}'-{\alpha }_{{\rm{\varepsilon}}}{L}_{{\rm{\theta}}}{(\varPi }_{x}'+{Z}_{{{sx}}}{\varPi }_{\hat{{\textit z}}}')={X}_{{{u}}}'$ (20)

 $\begin{split} {X}_{u}'=&({u}')_{{\rm{d}}}^{{{n}}}+{\beta }_{{\rm{\varepsilon}}}f{\left({v}'\right)}_{{\rm{d}}}^{{{n}}}+{\beta }_{{\rm{\varepsilon}}}{L}_{{\rm{\theta}}}{{(\varPi }_{x}'+{Z}_{{{s}}{{x}}}{\varPi }_{\hat{{\textit z}}}')}_{{\rm{d}}}^{{{n}}}+\\&\Delta t{c}_{{{p}}}{L}_{{\text{π}}{{x}}}{\theta }'^{{{n}}}-\Delta t{\left({{{V}}'}{\text •} \nabla u\right)}^{{{n}}}\end{split}$

 ${v}'+{\alpha }_{{\rm{\varepsilon}}}f{u}'-{\alpha }_{{\rm{\varepsilon}}}{L}_{{\rm{\theta}}}{(\varPi }_{y}'+{Z}_{{\rm{s}}{\rm{y}}}{\varPi }_{\hat{{\textit z}}}')={X}_{v}'$ (21)

 ${w}'+{\alpha }_{{\rm{\varepsilon}}}{c}_{{{p}}}\theta {'Z}_{{{s}}\hat{{{\textit z}}}}{\varPi }_{\hat{{\textit z}}}+{\alpha }_{{\rm{\varepsilon}}}{c}_{{{p}}}\theta {Z}_{{s}\hat{{\textit z}}}{\varPi }_{\hat{{\textit z}}}'={X}_{{{w}}}'$ (22)

 $\begin{split}&\hat w' - {Z_{{{sx}}}}u' - {Z_{{{sy}}}}v' + {\alpha _\varepsilon }{c_{{p}}}\theta {Z_{{{s\hat z}}}^2}\varPi {'_{\hat {\textit z}}} + {\alpha _\varepsilon }{c_{{p}}}{Z_{{{s\hat z}}}^2}{\varPi _{\hat {\textit z}}}\theta ' = X_{{{\hat w}}}' \end{split}$ (23)

 $\begin{split}& {\varPi }'+{\alpha }_{{\rm{\varepsilon}}}\left({u}'{\varPi }_{x}+{v}'{\varPi }_{y}+{\hat{w}}'{\varPi }_{\hat{\textit z}}\right)+{\alpha }_{{\rm{\varepsilon}}}{\alpha }_{{\rm{D}}}{\varPi }'+{\alpha }_{{\rm{\varepsilon}}}{\alpha }_{\Pi}{D}_{3}'={X}_{\varPi }' \end{split}$ (24)

${D'_{3}}={u'_{x}}+{v'_{y}}+{{\hat w}'}_{\hat{\textit z}}-\dfrac{{\partial Z}_{{sx}}}{\partial \hat{\textit z}}u'-\dfrac{{\partial Z}_{{sy}}}{\partial \hat{\textit z}}v'$ 代入式（24），整理得到

 $\begin{split}& {\varPi }'={A}_{\varPi1}u'+{A}_{\varPi2}v'+{A}_{\varPi3}{\hat{w}}'+\\&{A}_{\varPi4}\left({u}'_{x}+{v}'_{y}+\hat{w}'_{\hat{{\textit z}}}\right)+{A}_{\varPi0} \end{split}$ (25)

 $\begin{split} &{A_{{{\varPi }}1}} = {\alpha _{\rm{\varepsilon }}}\dfrac{{{\alpha _{{\varPi }}}\dfrac{{\partial {Z_{sx}}}}{{\partial \hat {\textit z}}} - {\varPi _x}}}{{1 + {\alpha _{\rm{\varepsilon }}}{\alpha _{\rm{D}}}}},\quad{A_{{{\varPi }}2}} = {\alpha _\varepsilon }\dfrac{{{\alpha _{{\varPi }}}\dfrac{{\partial {Z_{sy}}}}{{\partial \hat {\textit z}}} - {\varPi _y}}}{{1 + {\alpha _{\rm{\varepsilon }}}{\alpha _{\rm{D}}}}},\\&{A_{{{\varPi }}3}} = - \frac{{{\alpha _\varepsilon }{\varPi _{\hat {\textit z}}}}}{{1 + {\alpha _{\rm{\varepsilon }}}{\alpha _{\rm{D}}}}},\quad{A_{{{\varPi }}4}} = - \frac{{{\alpha _{\rm{\varepsilon }}}{\alpha _{{\varPi }}}}}{{1 + {\alpha _{\rm{\varepsilon }}}{\alpha _{\rm{D}}}}},\quad{A_{{{\varPi }}0}} = \frac{{X_{{\varPi }}'}}{{1 + {\alpha _{\rm{\varepsilon }}}{\alpha _{\rm{D}}}}} \end{split}$

 ${q}'=-{\alpha }_{{\rm{\varepsilon}}}{q}_{\hat{{\textit z}}}{\hat{w}}'+{X}_{{\rm{q}}}'$ (26)

 ${\theta }'=-{\alpha }_{{\rm{\varepsilon}}}{\hat{w}}'{\theta }_{\hat{{\textit z}}}+{X}_{{\rm{\theta}}}'$ (27)

2.4 扰动变量亥姆霍兹方程推导

 $u'={A}_{{u}1}{\varPi }_{x}'+{A}_{{u}2}{\varPi }_{y}'+{A}_{{u}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{{u}0}$ (28)
 $v'={A}_{{v}1}{\varPi }_{x}'+{A}_{{v}2}{\varPi }_{y}'+{A}_{{v}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{{v}0}$ (29)
 $\begin{split} \text{其中，} &{A_{{{u}}1}} = \frac{{{\alpha _{\rm{\varepsilon }}}{L_{\rm{\theta }}}}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}},\quad{A_{{{u}}2}} = \frac{{{\alpha _{\rm{\varepsilon }}}^2f{L_{\rm{\theta }}}}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}},\\&{A_{{{u}}3}} = \frac{{{\alpha _{\rm{\varepsilon }}}{L_{\rm{\theta }}}\left({{Z_{{sx}}} + {\alpha _{\rm{\varepsilon }}}f{Z_{sy}}} \right)}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}},\quad{A_{{{u}}0}} = \frac{{{\alpha _\varepsilon }fX_{{v}}' + X_{{u}}'}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}}\end{split}$
 $\begin{split} &{A_{{{v}}1}} = - \frac{{{\alpha _{\rm{\varepsilon }}}^2f{L_{\rm{\theta }}}}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}},\quad{A_{{{v}}2}} = \frac{{{\alpha _{\rm{\varepsilon }}}{L_{\rm{\theta }}}}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}},\\ &{A_{{{v}}3}} = \frac{{{\alpha _{\rm{\varepsilon }}}{L_{\rm{\theta }}}\left({{Z_{{{sy}}}} - {Z_{{sx}}}{\alpha _{\rm{\varepsilon }}}f} \right)}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}},\quad{A_{{{v}}0}} = \frac{{X_v' - {\alpha _\epsilon }fX_u'}}{{1 + {{({\alpha _{\rm{\varepsilon }}}f)}^2}}}\end{split}$

 $\begin{split} {\hat{w}}'=&\frac{{Z}_{{{sx}}}}{1-{\alpha }_{\varepsilon }^{2}{c}_{{{p}}}{Z}_{{{s}}\hat{{{\textit z}}}}^{2}{\varPi }_{\hat{{\textit z}}}{\theta }_{\hat{{\textit z}}}}{u}'+\frac{{Z}_{{{sy}}}}{1-{\alpha }_{\varepsilon }^{2}{c}_{{{p}}}{Z}_{{s}\hat{{{\textit z}}}}^{2}{\varPi }_{\hat{{\textit z}}}{\theta }_{\hat{{\textit z}}}}{v}'-\\ &\frac{{\alpha }_{\varepsilon }{c}_{{{p}}}\theta {Z}_{{s}\hat{{\textit z}}}^{2}}{1-{\alpha }_{\varepsilon }^{2}{c}_{{{p}}}{Z}_{{s}\hat{{\textit z}}}^{2}{\varPi }_{\hat{{\textit z}}}{\theta }_{\hat{{\textit z}}}}{{\varPi }}'_{\hat{{\textit z}}}+\frac{{X}_{\hat{{{w}}}}'-{\alpha }_{{\rm{\varepsilon}}}{c}_{{{p}}}{Z}_{{{s}}\hat{{{\textit z}}}}^{2}{\varPi }_{\hat{{\textit z}}}{X}_{\theta }'}{1-{\alpha }_{\varepsilon }^{2}{c}_{{{p}}}{Z}_{{{s}}\hat{{{\textit z}}}}^{2}{\varPi }_{\hat{{\textit z}}}{\theta }_{\hat{{\textit z}}}} \end{split}$

 ${\hat{w}}'={A}_{\hat{{{w}}}1}{\varPi }_{x}'+{A}_{\hat{{{w}}}2}{\varPi }_{y}'+{A}_{\hat{{{w}}}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{\hat{{{w}}}0}$ (30)

 $\begin{split} &\ {A_{{{\hat w}}2}} = \frac{{{Z_{{{sx}}}}{A_{{{u}}2}} + {Z_{{{sy}}}}{A_{{{v}}2}}}}{{1 - \alpha _\varepsilon ^2{c_{{p}}}Z_{{{s\hat {\textit z}}}}^2{\varPi _{\hat {\textit z}}}{\theta _{\hat {\textit z}}}}},\\ &\ {A_{{{\hat w}}3}} = \frac{{{Z_{{{sx}}}}{A_{{{u}}3}} + {Z_{{sy}}}{A_{{{v}}3}} - {\alpha _\varepsilon }{c_{{p}}}\theta {Z_{{{s\hat {\textit z}}}}^2}}}{{1 - \alpha _\varepsilon ^2{c_{{p}}}{Z_{{s\hat {\textit z}}}^2}{\varPi _{\hat {\textit z}}}{\theta _{\hat {\textit z}}}}},\\ &\ {A_{{{\hat w}}0}} = \frac{{{Z_{{{sx}}}}{A_{{{u}}0}} + {Z_{{{sy}}}}{A_{{{v}}0}} + X_{{{\hat w}}}' - {\alpha _{\rm{\varepsilon }}}{c_{{p}}}{Z_{{s\hat {\textit z}}}^2}{\varPi _{\hat {\textit z}}}X_\theta '}}{{1 - \alpha _\varepsilon ^2{c_{{p}}}{Z_{{{s\hat {\textit z}}}}^2}{\varPi _{\hat {\textit z}}}{\theta _{\hat {\textit z}}}}} \end{split}$

 ${\theta }'={A}_{{{\theta}}1}{\varPi }_{x}'+{A}_{{{\theta}}2}{\varPi }_{y}'+{A}_{{{\theta}}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{{{\theta}}0}$ (31)
 $\begin{split} \text{其中，}\qquad &{A_{{{\theta }}1}} = - {\alpha _{{\varepsilon }}}{\theta _{{{\hat {\textit z}}}}}{A_{{{\hat w}}1}},\quad{A_{{{\theta }}2}} = - {\alpha _{{\varepsilon }}}{\theta _{{{\hat {\textit z}}}}}{A_{{{\hat w}}2}},\qquad\qquad\\ &{A_{{{\theta }}3}} = - {\alpha _{{\varepsilon }}}{\theta _{{{\hat {\textit z}}}}}{A_{{{\hat w}}3}},\quad{A_{{\rm{\theta }}0}} = - {\alpha _{\rm{\varepsilon }}}{\theta _{{{\hat {\textit z}}}}}{A_{{{\hat w}}0}} + X_{{\theta }}'\\ &q' = {A_{{{q}}1}}\varPi _x' + {A_{{{q}}2}}\varPi _y' + {A_{{{q}}3}}\varPi _{\hat {\textit z}}' + {A_{{{q}}0}}\\[-10pt] \end{split}$ (32)
 $\begin{split} \text{其中，}\qquad &{A_{{{q}}1}} = - {\alpha _{{\varepsilon }}}{q_{{{\hat {\textit z}}}}}{A_{{{\hat w}}1}},\quad{A_{{{q}}2}} = - {\alpha _{{\varepsilon }}}{q_{{{\hat {\textit z}}}}}{A_{{{\hat w}}2}},\\ &{A_{{{q}}3}} = - {\alpha _{{\varepsilon }}}{q_{{{\hat {\textit z}}}}}{A_{{{\hat w}}3}},\quad{A_{q0}} = - {\alpha _\varepsilon }{q_{{{\hat {\textit z}}}}}{A_{{{\hat w}}0}} + X_{{q}}'\end{split} \hspace{27pt}$

 ${u}'={A}_{{{u}}1}{\varPi }_{x}'+{A}_{{{u}}2}{\left({\overline{{\varPi }'}}^{yx}\right)}_{y}+{A}_{{{u}}3}{\left({\overline{{\varPi }'}}^{\hat{{\textit z}}x}\right)}_{\hat{{\textit z}}}+{A}_{{{u}}0}$
 ${v}'={A}_{{{v}}1}{\left({\overline{{\varPi }'}}^{xy}\right)}_{x}+{A}_{{{v}}2}{\varPi }_{y}'+{A}_{{{v}}3}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{\hat{{\textit z}}}+{A}_{{{v}}0}$
 ${\hat{w}}'={A}_{\hat{{{w}}}1}{\left({\overline{{\varPi }'}}^{x\hat{{\textit z}}}\right)}_{x}+{A}_{\hat{{{w}}}2}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{y}+{A}_{\hat{{{w}}}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{\hat{{{w}}}0}$

 $\begin{split} {\varPi }'=&{A}_{{\varPi}1}{\overline{\left[{A}_{{{u}}1}{\varPi }_{x}'+{A}_{{{u}}2}{\left({\overline{{\varPi }'}}^{yx}\right)}_{y}+{A}_{{{u}}3}{\left({\overline{{\varPi }'}}^{\hat{{\textit z}}x}\right)}_{\hat{{\textit z}}}+{A}_{u0}\right]}}^{x}+\\ &{A}_{\varPi2}{\overline{\left[{A}_{{{v}}1}{\left({\overline{{\varPi }'}}^{xy}\right)}_{x}+{A}_{{{v}}2}{\varPi }_{y}'+{A}_{{{v}}3}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{\hat{{\textit z}}}+{A}_{{{v}}0}\right]}}^{y}+\\ &{A}_{\varPi3}{\overline{\left[{A}_{\hat{{{w}}}1}{\left({\overline{{\varPi }'}}^{x\hat{{\textit z}}}\right)}_{x}+{A}_{\hat{{{w}}}2}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{y}+{A}_{\hat{{{w}}}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{\hat{w}0}\right]}}^{\hat{{\textit z}}}+ \\&{A}_{\varPi4}\left\{{\left[{A}_{{{u}}1}{\varPi }_{x}'+{A}_{{{u}}2}{\left({\overline{{\varPi }'}}^{yx}\right)}_{y}+{A}_{{{u}}3}{\left({\overline{{\varPi }'}}^{\hat{{\textit z}}x}\right)}_{\hat{{\textit z}}}+{A}_{u0}\right]}_{x}\right.+ \\&{\left[{A}_{{{v}}1}{\left({\overline{{\varPi }'}}^{xy}\right)}_{x}+{A}_{{{v}}2}{\varPi }_{y}'+{A}_{{{v}}3}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{\hat{{\textit z}}}+{A}_{{{v}}0}\right]}_{y}+ \\&\left.{\left[{A}_{\hat{w}1}{\left({\overline{{\varPi }'}}^{x\hat{{\textit z}}}\right)}_{x}+{A}_{\hat{{\rm{w}}}2}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{y}+ {A}_{\hat{w}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{\hat{{{w}}}0}\right]}_{\hat{{\textit z}}}\right\}+{A}_{\varPi 0} \end{split}$ (33)

 $\begin{split} & {B}_{1}{\varPi }_{{{i}},{{j}},{{k}}}+{B}_{2}{\varPi }_{{{i}}-1,{{j}},{{k}}}+{B}_{3}{\varPi }_{{{i}}+1,{{j}},{{k}}}+{B}_{4}{\varPi }_{{{i}},{{j}}-1,{{k}}}+{B}_{5}{\varPi }_{{{i}},{{j}}+1,{{k}}}+\\ &\quad{B}_{6}{\varPi }_{{{i}}+1,{{j}}+1,{{k}}}+{B}_{7}{\varPi }_{{{i}}+1,{{j}}-1,{{k}}}+{B}_{8}{\varPi }_{{{i}}-1,{{j}}-1,{{k}}}+{B}_{9}{\varPi }_{{{i}}-1,{{j}}+1,{{k}}}+\\ &\quad{B}_{10}{\varPi }_{{{i}},{{j}},{{k}}-1}+{B}_{11}{\varPi }_{{{i}}-1,{{j}},{{k}}-1}+{B}_{12}{\varPi }_{{{i}}+1,{{j}},{{k}}-1}+{B}_{13}{\varPi }_{{{i}},{{j}}-1,{{k}}-1}+\\ &\quad{B}_{14}{\varPi }_{{{i}},{{j}}+1,{{k}}-1} +{B}_{15}{\varPi }_{{{i}},{{j}},{{k}}+1}+{B}_{16}{\varPi }_{{{i}}-1,{{j}},{{k}}+1}+{B}_{17}{\varPi }_{{{i}}+1,{{j}},{{k}}+1}+\\&\quad{B}_{18}{\varPi }_{{{i}},{{j}}-1,{{k}}+1}+{B}_{19}{\varPi }_{{{i}},{{j}}+1,{{k}}+1}={B}_{0}\\[-10pt] \end{split}$ (34)

 ${\left({\hat{w}}'\right)}_{{\rm{T}}}={\left[{A}_{\hat{{{w}}}1}{\left({\overline{{\varPi }'}}^{x\hat{{\textit z}}}\right)}_{x}+{A}_{\hat{{{w}}}2}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{y}+{A}_{\hat{{{w}}}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{\hat{{{w}}}0}\right]}_{{\rm{T}}}=0$ (35)

 ${\left({\hat{w}}'\right)}_{{\rm{B}}}={\left[{A}_{\hat{{{w}}}1}{\left({\overline{{\varPi }'}}^{x\hat{{\textit z}}}\right)}_{x}+{A}_{\hat{{{w}}}2}{\left({\overline{{\varPi }'}}^{y\hat{{\textit z}}}\right)}_{y}+{A}_{\hat{{{w}}}3}{\varPi }_{\hat{{\textit z}}}'+{A}_{\hat{w}0}\right]}_{{\rm{B}}}=0$ (36)
3 数值求解试验和分析

 图 1  扰动模式试验范围和初始高压扰动分布 （模式第25层） 情况 Fig. 1  Model domain for the perturbation forecast experiment and the initial pressure perturbation （on the 25th model level）

 图 2  积分11步后第20层的水平扰动风场 （a. 非线性模式预报的扰动风场，b. 扰动模式预报的扰动风场，c. 扰动模式预报与非线性模式预报的风场之差） Fig. 2  Perturbation winds on the 20th model level after the 11th time step of integration （a. from NLM，b. from PFM，c. difference between NLM and PFM）

 图 3  非线性模式和扰动模式分别在t=2（a），5（b），10（c），15（d），20（e）和25（f）积分步第25层的预报扰动风场（箭头）和气压场（等值线） （a1—f1. 非线性模式的结果，a2—f2. 扰动模式的结果） Fig. 3  Perturbation wind （arrows） and pressure （contours） fields predicted by nonlinear and perturbation models at the 2nd（a）, 5th （b）, 10th （c）, 15th （d）, 20th （e） and 25th （f） time steps on the 25th level （a1—f1. nonlinear model，a2—f2. perturbation model）

 图 4  积分11步后扰动量的垂直剖面 （a1—f1. 非线性模式的预报，a2—f2. 扰动模式的预报） Fig. 4  Latitude-height cross-sections of perturbation variables at the 11th time step of integration （a1—f1. nonlinear forecast model，a2—f2. perturbation forecast model）

 ${\rm{NLM}}\left({x}_{0}+\gamma {{\rm{\delta}}x}_{0}\right)={\rm{NLM}}\left({x}_{0}\right)+{R}_{0\to i}\gamma {{\rm{\delta}}x}_{0}+O\left({\gamma }^{2}\right)$ (37)

${\rm{PFM}}\left(\gamma {{\rm{\delta}}x}_{0}\right)$ 为从初值（ $\gamma {{\rm{\delta}}x}_{0}$ ）出发的一次扰动模式积分。为清楚表示，扰动模式表示为 ${\rm{PFM}}\left(\gamma {{\rm{\delta}}x}_{0}\right)= {M}_{0\to i}\gamma {{\rm{\delta}}x}_{0}$ 。令测试指数为：

 $\begin{split} F\left(\gamma \right)&=\frac{\left\|{\rm{NLM}}\left({x}_{0}+\gamma {{\rm{\delta}}x}_{0}\right)-{\rm{NLM}}\left({x}_{0}\right)\right\|}{\left\|\rm{PFM}\left(\gamma {{\rm{\delta}}x}_{0}\right)\right\|} \\& = \frac{\left\|{R}_{0\to i}{{\rm{\delta}}x}_{0}\right\|}{\left\|{M}_{0\to i}{{\rm{\delta}}x}_{0}\right\|}+O\left(\gamma \right) \end{split}$ (38)

 ${\left({\varPi }_{\hat{{\textit z}}}'\right)}_{{\rm{bottom}}\;{\rm{or}}\;{\rm{top}}}=\frac{g{\theta }'}{{c}_{p}{Z}_{{{s}}\hat{{{ {\textit z}}}}}{\theta }^{2}}$ (39)

 参数γ 虚层方案 静力平衡方案 1.0 0.8990750313 0.9092131257 0.1 1.038652658 0.9952505231 0.01 1.054983139 1.004807591 0.001 1.058226705 1.005833268 0.0001 1.192282557 1.037582755
4 结论和讨论

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