Reports of the Academy of Sciences of the USSR
1958. Volume 123, No. 3

GEOPHYSICS

Corresponding Member of the Academy of Sciences of the USSR E. N. BLINOVA

ON THE PROBLEM OF FORECASTING SMOOTHED VALUES OF METEOROLOGICAL ELEMENTS AT THE MIDDLE LEVEL OF THE ATMOSPHERE

For predicting the field of the stream function \(\psi\) at the middle level of the atmosphere, the equation of transport of absolute vorticity may serve:

\[ \frac{\partial \Delta \psi}{\partial t} +\frac{1}{a_0^2 \sin\theta} \left( \frac{\partial \psi}{\partial \theta}\frac{\partial \Delta \psi}{\partial \lambda} - \frac{\partial \psi}{\partial \lambda}\frac{\partial \Delta \psi}{\partial \theta} \right) +2\omega\frac{\partial \psi}{\partial \lambda}=0. \tag{1} \]

Here \(a_0\) is the radius of the Earth; \(\omega\) is the angular velocity of the Earth’s rotation; \(\theta\) is the complement of latitude (increasing southward); \(\lambda\) is the longitude of the place (increasing eastward); \(t\) is time;

\[ \Delta\psi= \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left( \sin\theta\frac{\partial \psi}{\partial\theta} \right) \frac{1}{\sin^2\theta} + \frac{\partial^2\psi}{\partial\lambda^2}. \tag{2} \]

We shall consider the motion at the middle level of the atmosphere to be turbulent and, starting from equation (1), shall try to find a cumulant-statistical description of this motion. In doing so we shall follow L. V. Keller \((^1)\), who showed how, for the complete system of equations of hydromechanics, one can construct a closed system of characteristics under the assumption that third-order correlation moments may be neglected.

Equation (1) contains only one function \(\psi(\theta,\lambda,t)\). We shall seek smoothed values of this function \(\bar{\psi}(\theta,\lambda,t)\). Deviations from the smoothed values will be denoted by \(\psi'(\theta,\lambda,t)\):

\[ \psi'(\theta,\lambda,t)=\psi(\theta,\lambda,t)-\bar{\psi}(\theta,\lambda,t). \tag{3} \]

Applying smoothing to equation (1), we obtain

\[ \frac{\partial \Delta \bar{\psi}}{\partial t} + \frac{1}{a_0^2\sin\theta} \left( \frac{\partial \bar{\psi}}{\partial\theta} \frac{\partial \Delta \bar{\psi}}{\partial\lambda} - \frac{\partial \bar{\psi}}{\partial\lambda} \frac{\partial \Delta \bar{\psi}}{\partial\theta} \right) + 2\omega\frac{\partial \bar{\psi}}{\partial\lambda} + \]

\[ + \frac{1}{a_0^2\sin\theta} \left[ \frac{\partial}{\partial\lambda} \left( \overline{\frac{\partial\psi'}{\partial\theta}\Delta\psi'} \right) - \frac{\partial}{\partial\theta} \left( \overline{\frac{\partial\psi'}{\partial\lambda}\Delta\psi'} \right) \right] =0. \tag{4} \]

Equation (4) contains, in addition to the sought function \(\bar{\psi}\), two more smoothed quantities

\[ \overline{\frac{\partial\psi'}{\partial\theta}\Delta\psi'}, \qquad \overline{\frac{\partial\psi'}{\partial\lambda}\Delta\psi'}. \tag{5} \]

These quantities can be expressed in terms of correlation moments. According to L. V. Keller, the correlation moment \(B_{\varphi}^{f}\) of two functions \(\varphi\) and \(f\) is defined as follows:

\[ B_{\varphi}^{f} = \overline{ \varphi'(\theta-\theta',\lambda-\lambda',t) f'(\theta+\theta',\lambda+\lambda',t) } \tag{6} \]

(\(\varphi'\) and \(f'\) are deviations of the functions \(\varphi\) and \(f\) from their mean values). This is a function of 5 variables: \(\theta,\theta',\lambda,\lambda',t\).

Then

\[ \overline{\frac{\partial \psi'}{\partial \theta}\Delta\psi'} = \left(B_{\partial\psi/\partial\theta}^{\Delta\psi}\right)_{\theta'=\lambda'=0}, \qquad \overline{\frac{\partial \psi'}{\partial \lambda}\Delta\psi'} = \left(B_{\partial\psi/\partial\lambda}^{\Delta\psi}\right)_{\theta'=\lambda'=0}. \]

On the other hand, by the very definition (6) we have

\[ B_{\varphi}^{\partial\chi/\partial s} = \frac{1}{2}\left(\frac{\partial B_{\varphi}^{\chi}}{\partial s} + \frac{\partial B_{\varphi}^{\chi}}{\partial \sigma}\right), \qquad B_{\partial\chi/\partial s}^{\varphi} = \frac{1}{2}\left(\frac{\partial B_{\chi}^{\varphi}}{\partial s} - \frac{\partial B_{\chi}^{\varphi}}{\partial \sigma}\right), \tag{7} \]

where \(\chi\) is an arbitrary function; \(s\) is taken in place of \(\theta\) or \(\lambda\); \(\sigma\) is taken in place of \(\theta'\) or \(\lambda'\), respectively; whence

\[ B_{\partial\psi/\partial\theta}^{\Delta\psi} = \frac{1}{2}\left( \frac{\partial B_{\psi}^{\Delta\psi}}{\partial \theta} - \frac{\partial B_{\psi}^{\Delta\psi}}{\partial \theta'} \right), \qquad B_{\partial\psi/\partial\lambda}^{\Delta\psi} = \frac{1}{2}\left( \frac{\partial B_{\psi}^{\Delta\psi}}{\partial \lambda} - \frac{\partial B_{\psi}^{\Delta\psi}}{\partial \lambda'} \right). \tag{8} \]

Using the representation for \(\Delta\psi\) (2), we can continue this transformation and express our correlation moments exclusively through the correlation moment \(B_{\psi}^{\psi}\). Along the way we introduce new variables:

\[ \theta_1=\theta-\theta',\qquad \theta_2=\theta+\theta',\qquad \lambda_1=\lambda-\lambda',\qquad \lambda_2=\lambda+\lambda'. \]

After simple calculations we obtain

\[ B_{\partial\psi/\partial\theta}^{\Delta\psi} = \frac{\partial}{\partial\theta_1}\left(\Delta_2 B_{\psi}^{\psi}\right), \qquad B_{\partial\psi/\partial\lambda}^{\Delta\psi} = \frac{\partial}{\partial\lambda_1}\left(\Delta_2 B_{\psi}^{\psi}\right), \tag{9} \]

where

\[ \Delta_2 = \frac{1}{\sin\theta_2}\frac{\partial}{\partial\theta_2} \left(\sin\theta_2\frac{\partial}{\partial\theta_2}\right) + \frac{1}{\sin^2\theta_2}\frac{\partial^2}{\partial\lambda_2^2}. \tag{10} \]

Finally, (4) takes the form

\[ \frac{\partial\Delta\overline{\psi}}{\partial t} + \frac{1}{a_0^2\sin\theta} \left( \frac{\partial\overline{\psi}}{\partial\theta} \frac{\partial\Delta\overline{\psi}}{\partial\lambda} - \frac{\partial\overline{\psi}}{\partial\lambda} \frac{\partial\Delta\overline{\psi}}{\partial\theta} \right) + 2\omega\frac{\partial\overline{\psi}}{\partial\lambda} + \]

\[ + \frac{1}{a_0^2\sin\theta} \left[ \left( \frac{\partial^2}{\partial\lambda_2\partial\theta_1} - \frac{\partial^2}{\partial\lambda_1\partial\theta_2} \right) \Delta_2 B_{\psi}^{\psi} \right] =0 \]

\[ \text{for }\ \theta_1=\theta_2=\theta,\quad \lambda_1=\lambda_2=\lambda. \tag{11} \]

To close the problem, we must construct a second equation relating \(\overline{\psi}\) and \(B_{\psi}^{\psi}\). For this purpose let us form the expression \(\partial B_{\Delta\psi}^{\Delta\psi}/\partial t\). Since

\[ \frac{\partial B_{\Delta\psi}^{\Delta\psi}}{\partial t} = B_{\Delta\psi}^{\partial\Delta\psi/\partial t} + B_{\Delta\psi}^{\partial\Delta\psi/\partial t}, \]

then, using (1) and (6), we find

\[ a_0^2\frac{\partial B_{\Delta\psi}^{\Delta\psi}}{\partial t} = \frac{1}{\sin\theta_1} \left[ B_{\partial\psi/\partial\lambda\cdot\partial\Delta\psi/\partial\theta}^{\Delta\psi} - B_{\partial\psi/\partial\theta\cdot\partial\Delta\psi/\partial\lambda}^{\Delta\psi} \right] + \]

\[ + \frac{1}{\sin\theta_2} \left( B_{\Delta\psi}^{\partial\psi/\partial\lambda\cdot\partial\Delta\psi/\partial\theta} - B_{\Delta\psi}^{\partial\psi/\partial\theta\cdot\partial\Delta\psi/\partial\lambda} \right) - 2\omega a_0^2 \left( B_{\partial\psi/\partial\lambda}^{\Delta\psi} + B_{\Delta\psi}^{\partial\psi/\partial\lambda} \right). \tag{12} \]

We now use the basic postulate of L. V. Keller on the possibility of neglecting third moments. In accordance with this postulate

\[ B_{\varphi}^{f\,\partial\chi/\partial s} = \frac{\partial\overline{\chi}_2}{\partial s_2}B_{\varphi}^{f} + \overline{f}_2\frac{\partial B_{\varphi}^{\chi}}{\partial s_2}, \]

\[ B_{f\,\partial\chi/\partial s}^{\varphi} = \frac{\partial\overline{\chi}_1}{\partial s_1}B_{f}^{\varphi} + \overline{f}_1\frac{\partial B_{\chi}^{\varphi}}{\partial s_1}. \tag{13} \]

Here \(\overline{\chi}_1=\overline{\chi}(\theta_1,\lambda_1,t)\), \(\overline{\chi}_2=\overline{\chi}(\theta_2,\lambda_2,t)\), \(\overline{f}_1=\overline{f}(\theta_1,\lambda_1,t)\), \(\overline{f}_2=\overline{f}(\theta_2,\lambda_2,t)\).

Applying (13) to (12), we obtain

\[ a_0^2 \frac{\partial B_{\Delta\psi}^{\Delta\psi}}{dt} = \frac{1}{\sin\theta_1} \left[ \frac{\partial \bar{\psi}_1}{\partial \lambda_1} \frac{\partial B_{\Delta\psi}^{\Delta\psi}}{\partial \theta_1} - \frac{\partial \bar{\psi}_1}{\partial \theta_1} \frac{\partial B_{\Delta\psi}^{\Delta\psi}}{\partial \lambda_1} - \frac{\partial \Delta_1 \bar{\psi}_1}{\partial \lambda_1} B_{\Delta\psi}^{\Delta\psi\mid\theta} - \left( 2\omega a_0^2 \sin\theta_1 - \frac{\partial \Delta_1 \bar{\psi}_1}{\partial \theta_1} \right) B_{\Delta\psi}^{\Delta\psi\mid\partial\lambda} \right] + \]

\[ + \frac{1}{\sin\theta_2} \left[ \frac{\partial \bar{\psi}_2}{\partial \lambda_2} \frac{\partial B_{\Delta\psi}^{\Delta\psi}}{\partial \theta_2} - \frac{\partial \bar{\psi}_2}{\partial \theta_2} \frac{\partial B_{\Delta\psi}^{\Delta\psi}}{\partial \lambda_2} - \frac{\partial \Delta_2 \bar{\psi}_2}{\partial \lambda_2} B_{\Delta\psi}^{\Delta\psi\mid\theta} - \left( 2\omega a_0^2 \sin\theta_2 - \frac{\partial \Delta_2 \bar{\psi}_2}{\partial \theta_2} \right) B_{\Delta\psi}^{\Delta\psi\mid\partial\lambda} \right]. \]

Then applying a transformation of type (9), we finally obtain:

\[ a_0^2 \Delta_1 \Delta_2 \frac{\partial B_\psi^\psi}{\partial t} = \frac{1}{\sin\theta_1} \left[ \frac{\partial \bar{\psi}_1}{\partial \lambda_1} \frac{\partial}{\partial \theta_1} (\Delta_1 \Delta_2 B_\psi^\psi) - \frac{\partial \bar{\psi}_1}{\partial \theta_1} \Delta_1 \Delta_2 \frac{\partial B_\psi^\psi}{\partial \lambda_1} - \right. \]

\[ \left. - \frac{\partial \Delta_1 \bar{\psi}_1}{\partial \lambda_1} \Delta_2 \frac{\partial B_\psi^\psi}{\partial \theta_1} - \left( 2\omega a_0^2 \sin\theta_1 - \frac{\partial \Delta_1 \bar{\psi}_1}{\partial \theta_1} \right) \Delta_2 \frac{\partial B_\psi^\psi}{\partial \lambda_1} \right] + \]

\[ + \frac{1}{\sin\theta_2} \left[ \frac{\partial \bar{\psi}_2}{\partial \lambda_2} \frac{\partial}{\partial \theta_2} (\Delta_1 \Delta_2 B_\psi^\psi) - \frac{\partial \bar{\psi}_2}{\partial \theta_2} \Delta_1 \Delta_2 \frac{\partial B_\psi^\psi}{\partial \lambda_2} - \right. \]

\[ \left. - \frac{\partial \Delta_2 \bar{\psi}_2}{\partial \lambda_2} \Delta_1 \frac{\partial B_\psi^\psi}{\partial \theta_2} - \left( 2\omega a_0^2 \sin\theta_2 - \frac{\partial \Delta_2 \bar{\psi}_2}{\partial \theta_2} \right) \Delta_1 \frac{\partial B_\psi^\psi}{\partial \lambda_2} \right]. \tag{14} \]

The two functional-differential equations (11) and (14) make it possible to determine the two functions \(\bar{\psi}\) and \(B_\psi^\psi\) from their prescribed initial values. The solution may be carried out in time steps, first determining from equation (11) (Poisson’s equation) \(\partial \bar{\psi}/\partial t\), and from equation (14) \(\partial B_\psi^\psi/\partial t\) (a double solution of Poisson’s equation).

Smoothing may be carried out in various ways. In particular, if smoothing is understood as averaging over a circle of latitude, then we obtain a system of equations by solving which one can give a direct forecast of the circulation index (bypassing the forecast of the zonal circulation).

Institute of Applied Geophysics
Academy of Sciences of the USSR

Received
6 VIII 1958

CITED LITERATURE

1. L. V. Keller, Zhurn. geofiz. i meteorol., 2, No. 3–4 (1925).