Reports of the Academy of Sciences of the USSR
1960. Vol. 132, No. 2

GEOPHYSICS

Corresponding Member of the USSR Academy of Sciences I. A. KIBEL

A FINITE-DIFFERENCE SCHEME FOR SOLVING THE COMPLETE SYSTEM OF EQUATIONS OF SHORT-RANGE WEATHER FORECASTING AND QUASI-GEOSTROPHIC RELATIONS

The problem of short-range weather forecasting by means of the complete system of equations of hydrodynamics leads to the determination of four functions: the geopotential \(\Phi\), the horizontal wind components \(u, v\), and the quantity \(\bar{\omega}\), equivalent to the vertical component of velocity: \(\bar{\omega}=\frac{1}{P}\frac{dp}{dt}\) (\(p\) is pressure, \(P\) is the standard pressure at sea level, \(t\) is time). To determine these functions one may use the four equations

\[ \frac{\partial u}{\partial t}+\frac{\partial \Phi}{\partial x}-lv =-u\frac{\partial u}{\partial x}-v\frac{\partial u}{\partial y} -\bar{\omega}\frac{\partial u}{\partial \zeta}; \tag{1} \]

\[ \frac{\partial v}{\partial t}+\frac{\partial \Phi}{\partial y}+lu =-u\frac{\partial v}{\partial x}-v\frac{\partial v}{\partial y} -\bar{\omega}\frac{\partial v}{\partial \zeta}; \tag{2} \]

\[ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} +\frac{\partial \bar{\omega}}{\partial \zeta}=0; \tag{3} \]

\[ \zeta^{2}\frac{\partial^{2}\Phi}{\partial \zeta\,\partial t} +c^{2}\bar{\omega} =-\zeta^{2}\left( u\frac{\partial^{2}\Phi}{\partial x\,\partial \zeta} +v\frac{\partial^{2}\Phi}{\partial y\,\partial \zeta} \right). \tag{4} \]

Here the independent variables are: \(x, y\)—horizontal coordinates (the \(Y\)-axis is directed northward) and the reduced pressure \(\zeta=p/P\); \(l\) is the Coriolis parameter (depends on \(y\)); \(c^{2}=\alpha RT_{1}\); \(R\) is the gas constant; \(T_{1}\) is the mean temperature; \(\alpha=(\gamma_{a}-\gamma)R/g\) (\(g\) is the acceleration due to gravity, \(\gamma\) is the mean vertical temperature gradient, \(\gamma_{a}\) is the adiabatic (or pseudoadiabatic) gradient). In this case the vertical velocity \(w\) is determined from the relation

\[ -gw=\frac{RT_{1}}{\zeta}\bar{\omega} -\frac{\partial \Phi}{\partial t} -u\frac{\partial \Phi}{\partial x} -v\frac{\partial \Phi}{\partial y}, \tag{5} \]

and the temperature \(T\) and density \(\rho\) are found from the formulas

\[ T=-\frac{\zeta}{R}\frac{\partial \Phi}{\partial \zeta}, \qquad \frac{1}{\rho}=-\frac{1}{P}\frac{\partial \Phi}{\partial \zeta}. \tag{6} \]

The boundary conditions adopted are

\[ \zeta w=0 \quad \text{for } \zeta=0; \tag{7} \]

\[ w=0 \quad \text{for } \zeta=1 \tag{8} \]

(the case of absence of mountains is considered). The medium is assumed infinitely extended in the directions of the \(X\) and \(Y\) axes (the formulation of the short-range forecasting problem), and the curvature of the Earth is taken into account only by specifying \(l\).

as functions of \(y\). In the course of solving the problem, boundedness conditions are imposed.

The system (1)—(4) is of third order in time; it is sufficient to prescribe \(u\), \(v\), \(\Phi\) at the initial instant. In this case \(\bar{\omega}\) is determined from (3) and (7) in the form

\[ \bar{\omega}=-\int_0^\zeta\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)\,d\xi . \tag{9} \]

Eliminating \(\bar{\omega}\) from (4) by means of (3), we arrive at a system of three equations

\[ \frac{\partial u}{\partial t}+\frac{\partial \Phi}{\partial x}-lv=-B_u; \tag{10} \]

\[ \frac{\partial v}{\partial t}+\frac{\partial \Phi}{\partial y}+lu=-B_v; \tag{11} \]

\[ \frac{\partial}{\partial \zeta}\left(\zeta^2\frac{\partial^2\Phi}{\partial \zeta\,\partial t}\right) -c^2\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) =R\frac{\partial \zeta A_T}{\partial \zeta}, \tag{12} \]

where

\[ B_u=u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\bar{\omega}\frac{\partial u}{\partial \zeta},\qquad B_v=u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+\bar{\omega}\frac{\partial v}{\partial \zeta}; \tag{13} \]

\[ A_T=u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}. \]

\(\bar{\omega}\) and \(T\) are determined through \(u\), \(v\), \(\Phi\) by (6) and (9).

The system (10), (11), (12) is solved with the boundary condition

\[ \frac{\partial^2\Phi}{\partial \zeta\,\partial t}+\alpha\frac{\partial\Phi}{\partial t} =RA_T \quad \text{for } \zeta=1. \tag{14} \]

Let us approximately replace the time derivatives by finite-difference relations

\[ \frac{\partial u}{\partial t}\approx\frac{u-u^0}{\delta t},\qquad \frac{\partial v}{\partial t}\approx\frac{v-v^0}{\delta t},\qquad \frac{\partial\Phi}{\partial t}\approx\frac{\Phi-\Phi^0}{\delta t} \]

(\(u^0\), \(v^0\), \(\Phi^0\) are the values of the corresponding functions at the initial instant of time).

The system (10), (11), (12) is replaced by the finite-difference scheme

\[ u+\delta t\left(\frac{\partial\Phi}{\partial x}-lv\right) =-\delta t\cdot B_u+u^0; \tag{15} \]

\[ v+\delta t\left(\frac{\partial\Phi}{\partial y}+lu\right) =-\delta t\cdot B_v+v^0; \tag{16} \]

\[ \frac{\partial}{\partial \zeta}\left(\zeta^2\frac{\partial\Phi-\Phi^0}{\partial \zeta}\right) -c^2\delta t\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) =R\frac{\partial \zeta A_T}{\partial \zeta}, \tag{17} \]

where the boundary condition (14) takes the form

\[ \frac{\partial}{\partial \zeta}(\Phi-\Phi^0)+\alpha(\Phi-\Phi^0) =R\delta t\cdot A_T \quad \text{for } \zeta=1. \tag{18} \]

Eliminating the velocities \(u\), \(v\) from (17) by means of (15), (16) and introducing the dimensionless parameter \(l_0\delta t=\delta\) (\(l_0\) is the value of \(l\) at the origin of coordinates), we obtain…

we arrive at the equation

\[ \left[ \frac{1+\delta^2}{\delta^2}\frac{\partial}{\partial \zeta}\zeta^2\frac{\partial}{\partial \zeta} +\frac{c^2}{l_0^2}\left(\Delta+b\delta\frac{\partial}{\partial x}-b'\frac{\partial}{\partial y}\right) \right](\Phi-\Phi^0)= \]
\[ =\delta\left[ \frac{R(1+\delta^2)}{l_0\delta^2}\frac{\partial}{\partial \zeta}(\zeta A_T) -\frac{c^2}{l_0^2}\left(\frac{\partial B_v}{\partial x}-\frac{\partial B_u}{\partial y}\right) +b\frac{\partial \Phi^0}{\partial x} \right] - \]
\[ -\frac{c^2}{l_0^2}\left[ \frac{\partial B_u}{\partial x}+\frac{\partial B_v}{\partial y} +\Delta\Phi^0-l_0\left(\frac{\partial v^0}{\partial x}-\frac{\partial u^0}{\partial y}\right) -b'\frac{\partial\Phi^0}{\partial y} \right] +\frac{c^2}{l_0\delta}\left(\frac{\partial u^0}{\partial x}+\frac{\partial v^0}{\partial y}\right), \tag{19} \]

where \(\Delta\) is the plane Laplace operator,

\[ b=\left(\frac{1}{l}\frac{dl}{dy}\right)_0\frac{\delta^2-1}{\delta^2+1},\qquad b'=2\left(\frac{1}{l}\frac{dl}{dy}\right)_0\frac{\delta^2}{\delta^2+1}. \tag{20} \]

Taking the right-hand sides as known, we find \(\Phi\), integrating (19) under condition (18), in the form

\[ \Phi=\delta\Phi_{\mathrm I}+\Phi_{\mathrm{II}}+\frac{1}{\delta}\Phi_{\mathrm{III}}, \tag{21} \]

where

\[ \Phi_{\mathrm I}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int_0^1 \left[ G\left(\frac{\partial B_v}{\partial x'}-\frac{\partial B_u}{\partial y'}+b\frac{\partial\Phi^0}{\partial x'}\right) +\frac{Rl_0}{c^2}\frac{1+\delta^2}{\delta^2}\zeta'\frac{\partial G}{\partial \zeta'}A_T \right]\,d\zeta'\,dx'\,dy'; \tag{22} \]

\[ \Phi_{\mathrm{II}}=\Phi^0+\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int_0^1 G\left[ \frac{\partial B_u}{\partial x'}+\frac{\partial B_v}{\partial y'}+\Delta\Phi^0 -l_0\left(\frac{\partial v^0}{\partial x'}-\frac{\partial u^0}{\partial y'}\right) -b'\frac{\partial\Phi^0}{\partial y'} \right]\,d\zeta'\,dx'\,dy'; \tag{23} \]

\[ \Phi_{\mathrm{III}}=-\frac{l_0}{2\pi}\int_{-\infty}^{+\infty}\int_0^1 G\left(\frac{\partial u^0}{\partial x'}+\frac{\partial v^0}{\partial y'}\right)\,d\zeta'\,dx'\,dy'; \tag{24} \]

\[ G=\frac{1}{2\sqrt{\zeta\zeta'}} \left[ S\left(\ln\frac{\zeta'}{\zeta},r\right) +S\left(\ln\frac{1}{\zeta\zeta'},r\right) +\right. \]
\[ \left. +(1-2\alpha)(\zeta\zeta')^{1/2-\alpha} \int_{\infty}^{\ln\frac{1}{\zeta\zeta'}} e^{(1/2-\alpha)\xi}S(\xi,r)\,d\xi \right] \exp\left[-b\delta\frac{x-x'}{2}+b'\frac{y-y'}{2}\right]; \tag{25} \]

\[ S(\xi,r)= \left(\xi^2+\frac{1+\delta^2}{\delta^2}\frac{l_0^2}{c^2}r^2\right)^{-1/2} \exp\left[ -\sqrt{\frac{1+\alpha^2}{2}}\sqrt{\xi^2+\frac{1+\delta^2}{\delta^2}\frac{l_0^2}{c^2}r^2} \right]; \]

\[ r^2=(x-x')^2+(y-y')^2,\qquad \alpha^2=\frac{c^2}{l_0^2}\left(\frac{1}{l}\frac{dl}{dy}\right)_0^2 \frac{\delta^4}{1+\delta^2}. \]

In this case \(u\), \(v\), \(\bar{\omega}\) can be found from (15), (16), and (4) in the form

\[ u=\frac{\delta^2}{1+\delta^2} \left(\delta u_{\mathrm I}+u_{\mathrm{II}}+\frac{1}{\delta}u_{\mathrm{III}}+\frac{1}{\delta^2}u_{\mathrm{IV}}\right); \tag{26} \]

\[ v=\frac{\delta^2}{1+\delta^2} \left(\delta v_{\mathrm I}+v_{\mathrm{II}}+\frac{1}{\delta}v_{\mathrm{III}}+\frac{1}{\delta^2}v_{\mathrm{IV}}\right); \tag{27} \]

\[ \bar{\omega}=\bar{\omega}_{\mathrm{II}}+\frac{1}{\delta}\bar{\omega}_{\mathrm{III}}+\frac{1}{\delta^2}\bar{\omega}_{\mathrm{IV}}, \tag{28} \]

where

\[ u_{\mathrm I}=-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm I}}{\partial y}; \qquad v_{\mathrm I}=\frac{1}{l_0}\frac{\partial\Phi_{\mathrm I}}{\partial x}; \]

\[ u_{\mathrm{II}}=-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{II}}}{\partial y}-\frac{B_v}{l_0}-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{I}}}{\partial x}; \qquad v_{\mathrm{II}}=\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{II}}}{\partial x}+\frac{B_u}{l_0}-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{I}}}{\partial y}; \]

\[ u_{\mathrm{III}}=-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{III}}}{\partial y}-\frac{1}{l_0}B_u+v^0-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{III}}}{\partial x}; \tag{29} \]

\[ v_{\mathrm{III}}=\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{III}}}{\partial x}-\frac{B_v}{l_0}-u^0-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{II}}}{\partial y}; \]

\[ u_{\mathrm{IV}}=u^0-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{III}}}{\partial x}; \qquad v_{\mathrm{IV}}=v^0-\frac{1}{l_0}\frac{\partial\Phi_{\mathrm{III}}}{\partial y}; \]

\[ \overline{\omega}_{\mathrm{II}}=\frac{l_0\zeta}{c^2}\left(RA_T-\zeta\frac{\partial\Phi_{\mathrm{I}}}{\partial\zeta}\right); \qquad \overline{\omega}_{\mathrm{III}}=\frac{l_0\zeta^2}{c^2}\frac{\partial\Phi^0-\Phi_{\mathrm{II}}}{\partial\zeta}; \qquad \overline{\omega}_{\mathrm{IV}}=-\frac{l_0\zeta^2}{c^2}\frac{\partial\Phi_{\mathrm{III}}}{\partial\zeta}. \]

The first of the three terms on the right-hand side of (21) (containing the factor \(\delta\)) may conventionally be called the evolutionary term, the second the stationary term, and the third the damping term (we assume \(\delta\gg 1\), which means that \(\delta t\gg 2\) h 30 min). Analogous terms are contained in the right-hand sides of \(u, v\); \(\omega\) does not contain an evolutionary term. Let us note that, completely discarding the nonlinear terms \((B_u, B_v, A_T)\) and neglecting \(\dfrac{dl}{dy}\) \((b=b'=0)\), we obtain \(\Phi_{\mathrm{I}}=u_{\mathrm{I}}=v_{\mathrm{I}}=\overline{\omega}_{\mathrm{II}}=0\). \(\Phi, u, v\) will contain only stationary terms (which will be connected by the condition of geostrophy) and damping terms. This result is in agreement with \((^1,^2)\). This solution has no prognostic meaning.

Discarding \(B_u, B_v, A_T\), but retaining the terms with \(b\) and \(b'\), we obtain evolution \((\Phi_{\mathrm{I}}\ne 0)\). A more complete solution, which can already be applied to forecasting, corresponds to a linearization of \(B_u, B_v, A_T\) with respect to some basic motion. Finally, the general case of forecasting is obtained by retaining the nonlinear terms in full.

It is remarkable that, in the most general case, the principal (evolutionary) terms for \(\Phi, u, v\) will always be connected, according to (21), (29), by the conditions of geostrophy.

The time steps used up to now in quasigeostrophic short-range forecasting \((^3)\) are obtained from our formula (21) if in it only the evolutionary term is retained, while \(B, B_v, A_T\) are replaced by \(B_u^0, B_v^0, A_T^0\), substituting for \(u, v\) their evolutionary parts (and zero instead of \(\overline{\omega}\)), setting in (25) \(b=b'=a=0\), and replacing \(\dfrac{\delta^2}{1+\delta^2}\) by 1.

Institute of Applied Geophysics
Academy of Sciences of the USSR

Received
19 II 1960

REFERENCES CITED

\(^1\) I. A. Kibel’, DAN, 118, No. 4 (1958).
\(^2\) A. M. Obukhov, Izv. AN SSSR, geophys. ser., No. 4 (1949).
\(^3\) N. I. Buleev, G. I. Marchuk, Tr. Inst. fiz. atmosfery AN SSSR, No. 2 (1958).