282 Journal of the Meteorological Society of Japan Vol. 60, No. 1. A Theory and Method of Long-Range Numerical

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1 282 Journal of the Meteorological Society of Japan Vol. 60, No. 1 A Theory and Method of Long-Range Numerical Weather Forecasts By Chao Jih-Ping, Guo Yu-Fu and Xin Ru-Nan Institute of Atmospheric Physics, Academia Sinica, Beijing (Manuscript received 24 September 1981) Abstract By parameterizing various types of heat sources (sinks) a linear model of coupled oceanatmospheric system has been established. By means of frequency analysis it is shown that in this model there are two different types of waves, one corresponding essentially to the travelling Rossby wave in non-adiabatic atmosphere which has a period of several days, and the other is a slowly varying wave which are driven by anomaly heating in ocean. Evidently the existence of this short time-scale weather process presents difficulties for the long-range numerical weather forecasting, because the evolution of the long-range weather process of smaller amplitude will be distorted by the short-range weather process of large amplitude. In order to overcome these difficulties the travelling Rossby wave are filtered as a "noise" from the model of the long-range weather forecast. A simple method of filtering is given. Then a practical model for predicting the monthly mean anomalous fields of 500mb geopotential height and earth's surface temperature is given and the experimental results are reported. This tentative study shows that the filtering method mentioned in this paper may be a promising way for making long-range weather forecast. 1. Introduction The success of short-range numerical weather prediction has aroused the interest of meteorologists in the possibility of long-range numerical weather prediction in past two decades. One of goals of recent rapid development of numerical simulation of global circulation model (GCM) is to find out the successful way of long-range weather forecasting. There are several ways for making long-range numerical weather forecast. Usually we may design a non-adiabatic GCM and integrate it numerically as the short-range numerical forecast. On the other hand, we may take another entirely different way. As well known, Namias (1948) used 30-day mean map and predicted the change of the two successive monthly mean maps. It may also be possible to do it along this line by means of numerical method instead of Namias's synoptic method. Naturally, some difficulties have to be overcome. In this aspect, Adem (1964) suggests a model for predicting monthly mean temperature field, but his model did not forecast the monthly mean air-flow. We follow the second line mentioned above, a theory and method have been developed (GLRNWF* 1977, 1979). In this paper, this model will be further discussed and the predicted examples of one month mean anomalous state of atmospheric circulation and the temperature of earth's surface will be given. 2. Basic equations In x, y, p, t coordinates, the vorticity equation and the first law of thermodynamics become respectively f is Coriolis parameter; * is the density of air and Cp is the specific heat under constant pressure; *T is the static stability; the operator in which u, *, * are the velocity components respectively. In Eq. (2), the turbulent heat exchange *T is

2 February 1982 Jih-Ping Chao, Yu-Fu Guo and Ru-Nan Xin 283 Kp=*2g2KT, KT is the coefficient heat conductivity, g is gravity. The radiational heat exchange SR, according to the scheme of Kuo (1968) may be given by s is the solar radiation, k' is the absorption coefficient of the short wave radiation, Te is the temperature of the environment, and here we have assumed the reference temperature Te to be equal to the climatological monthly mean air temperature T. By introducing the geostrophic approximation ks and k* are the mean absorption coefficients of the absorptive matter in the region of strong absorption and weak absorption respectively r is the proportion (in %) of the black body radiation in the region of strong absorption to the total black body radiation. For the heat exchange of condensation EQ, it may be simply parameterized as follows: and considering the hydrostatic relation and Eqs. (8) and (9), the equations of anomalous state may be obtained from Eqs. (2) and (7) as follows qs is saturated specific humidity, L is the latent heat and Zb is the level of condensation. According to the theory of the boundary layer we may obtain R is the gas constant of the air, and lb is the thickness of the boundary layer, is the geostrophic vorticity on the earth's *0g surface, substituting (6) into (5), we get and the operator d/dt now becomes Eliminating *' from Eqs. (12) and (13), we obtain the non-adiabatic vorticity equation as follows Substituting Eqs. (3), (4) and (5') into Eq. (2), then the first law of the thermodynamics becomes K=Kp+KR. It is assumed that the climatological monthly mean process satisfies the following equation Group of Long-Range. Numerical Weather Forecasting. On the other hand, we suppose that the temperature variation in the soil obeys the heat conductivity equation, and in the ocean it is also

3 284 Journal of the Meteorological Society of Japan Vol. 60, No. 1 affected by the advection of ocean current, then we get the following equation tion; a is the earth's albedo. In general, these radiations may be calculated as follows: *=1 for ocean and *=0 for land, *s is the stream function of ocean current. Similarly, if the variation of the climatological monthly mean earth's surface temperature satisfies the equation then the equation for the anomalous part from the climatological monthly mean may be written in the form *s can be calculated from the climatologlcal monthly average of oceanic current. Assuming that the anomaly part *s' be yielded by the air flow, then, the use of Ekman winddriven current theory leads to the symbol "*" denotes the value in cloudless condition, and symbol "0" denotes the value on the earth's surface; cs, ci are the coefficients showing respectively the effects of cloud on the total radiation and on the effective radiation; n is the amount of cloudiness. Thus, the expression of the radiational flux of heat becomes What we are interested in is the anomaly rather than the climatological mean itself. Therefore we express n by n+n', the symbols "-" and "*" denote the climatological monthly mean and the anomalous value respectively. Then the anomalous flux of radiation becomes approximately Now we suppose that the cloud is mainly produced by the large-scale vertical motion induced by frictional convergence in the boundary layer, and further that the cloudiness n is proportional to Wb which is the vertical velocity on the top of boundary layer. We get *0 is the latitude and W* is an empirical parameter. From Eq. (6) we get 3. Heat balance on the earth's surface The heat balance on earth's surface (land or ocean) may be expressed by (Budyko 1956) R is the radiational flux of heat, or the radiational balance; P is the turbulent heat flux between the underlying surface and the atmosphere; A is the heat flux between the underlying surface and the lower layer; LE is the expenditure of heat by evaporation (L, the latent heat of evaporation and E, the rate of evaporation). The radiation balance is the difference of total solar radiation received by the earth's surface and the effective outgoing radiation. This may be expressed as follows: By assuming *0g=*0g+*0g', we may give the anomaly cloudiness Substituting Eq. (26) into Eq. (23), we get The turbulent heat flux is Its anomaly value may be given by So and so are total direct and diffused radiations respectively; I is the effective radia-

4 February 1982 Jih-Ping Chao, Yu-Fu Guo and Ru-Nan Xin 285 The heat flux between the underlying surface and the lower layer is with its anomaly value and its states that the anomalous flux of evaporation is proportional to the anomalous SST. According to the results mentioned above, the heat balance of the anomalous state of the earth's surface may be written as *s, Cps, KS are the density, the specific heat and the coefficient of heat conductivity of the soil or the water respectively. The heat flux of evaporation is 4. Waves in the ocean-atmospheric system Its anomaly value may be written in the form q is the specific humidity, Kq is the coefficient of turbulent exchange, and Kq*KT approximately. In general, the evaporation is important only on the sea surface. In this case the process of evaporation may be parameterized by another convenient form. Assuming the specific humidity of the air above the sea surface reaches its saturated value, we get Let us next discuss the characters of this model. One of the boundary conditions for Eq. (14) is that at p=p0 (sea level) By applying the condition (37) to Eq. (13) at sea level, replacing the reference temperature Te in Eq. (7) by the earth's surface temperature Te=Ts+T's, and taking T=Ts at sea level, and the condensation is disappeared, then we get at p=p0, Rw is the gas constant of the water vapour, es is the saturation vapour pressure. Differentiating the logarithm of the above equation, we may obtain the following approximate expression: This is one of the boundary conditions for solving Eq. (14). The upper boundary condition that we take motionless,, at n=0_ i, e., Its anomaly value becomes Supposing that the water vapour in the air immediately above the sea surface be just saturated at the sea surface and this water vapour evaporated acquires the temperature of the sea surface, then we may assume reasonably For simplicity, only one layer geopotential surface i.e., 500mb is considered. Using the boundary conditions (38) and (39), we integrate Eq. (14) from p= p0 to p=0, and taking the mean value over height on the right hand, then we get from Eq. (14) The anomalous part of the heat flux of evaporation becomes G1 is the average value of G1 over whole layers of condensation, and

5 286 Journal of the Meteorological Society of Japan Vol. 60, No. 1 here the assumption (*)0=b(*) has been used in the term of condensation, *' is the geopotential height at 500mb level. Furthermore, it is assumed that the earth is covered totally by an ocean and the climatological states of atmosphere is taken as u. Linearizing Eq. (40) and one-dimensional problem is considered, we have sional form as Omitting the terms of advection, Eq. (17) becomes Assuming the solutions of Eqs. (49) and (50) to be of the form The sensible heat flux in heat balance equation at the earth's surface is also ignored, and then we get from (36) we obtain the frequencies of the system as follows At the bottom of mixed layer in ocean, we may use the condition Integrating Eq. (42) with Z from -D to 0, we obtain Considering *<0(1), we get the approximate formula an assumption was made: There are two time scales in this oceanatmosphere coupled system that are t1 is the characteristic time of Rossby wave, as t2 is the characteristic time of mixing process of heat in the ocean. The ratio of these two time scales is smaller than the order of one, i.e., Therefore, there are two roots as follows because in the common cases we may take u= 103cm/sec, *=10-13/cm*sec, KS=10cm2/sec and D=3*103cm. Introducing the non-dimensional variables we transform the equations to the non-dimen-

6 February 1982 Jih-Ping Ghao, Yu-Fu Guo and Ru-Nan Xin 287 Obviously, the order of the ratio of the two frequencies is that It shows that the frequency *1* is fast one as *2* is slow one. For illustrating the character of the fast one, let us omit the heating effect of ocean, i.e., putting *=0, then we obtain This is clearly the frequency of Ross,by wave with the growth rate which comes from the condensation. However, under this situation another solution disappears, i.e., *2*=0. From this that we may understand that the slow one is produced by the heating of ocean. On the other hand, if we omit the term of local time variation in equation (41), we get rapidly if they are not controlled in some way in the model. One way to overcome this difficulty is to filter out these noises in the model. The other is to control their growth in the model. In making the long-range numerical weather forecast and overcoming the same difficulty, we may apply the similar treatments as done in short-range numerical weather forecast. We now employ the first method and treat the travelling Rossby wave as a "noise-wave" and filter it from the model of long-range numerical weather forecasting. As mentioned above, a simple method of filtering is that, we may omit the term of local time variation in vorticity equation for atmosphere. It is equivalent to replace the equation of motion Eq. (41) by the relation between the field of atmospheric flow and that of heating anomaly, for two dimensional case that is The latter formula agree well with (55), except it loses a small term with the order of * in denominator, and also the fast Rossby wave is filtered out. 5. Adaptation equation The calculation of the above section indicates that in an ocean-atmospheric system there are two basic types of dynamical processes corresponding to two different time-scales. Obviously, the evolution of the long-range process of smaller amplitude will be distorted by the short-range process with large amplitude, because the growth rate of the shorter time-scale waves is about one order of magnitude larger than that of the long time-scale waves. This is a difficulty for making the long-range numerical forecast. As well known, a similar problem also occurs in short-range numerical forecast, the solution of the model of primitive equation contains both meteorologically significant motions and meteorologically noises (gravity waves and sound waves). The existence of these noise waves gives on one hand rapid oscillations in the hydrodynamical field and thus cause large fictitious tendency which will completely distort the slower change connected with the meteorologically significant motion. On the other hand, these noise-waves may grow very Although this equation is stationary in form, the geopotential field still varies with time because the equation describing the field of the underlying temperature anomaly is non-stationary. In the non-linear case we have a similar relation This equilibrium relation may be called adaptation equation. Physically it means that after the travelling Rossby wave are dispersed, an adjustment relationship between the height anomalous field and the earth's surface temperature field would be established. In order to test this adjustment relationship, we may use the observed TS' field to compute the ' field from Eq. (61), and then compared it with * the observed *' field of the same month to examine the accuracy of this adaptation equation. The over-relaxation method may be used for getting the solution of this equation. The corresponding 12 monthly mean anomaly field of 500mb height are calculated from the observed 12 monthly anomaly field on the earth's surface temperature in The calculation covers only the northern hemisphere, the grid distance being 540km and the polar stereographic projection being used. Since data of soil surface temperature on land are indequate now, we use the air temperature at sea level instead. Table 1

7 288 Journal of the Meteorological Society of Japan Vol. 60, No. 1 Table 1 Correlation Coefficients gives the correlation coefficients between the relation coefficients between the anomaly fields calculated anomaly field of 500mb height and of two successive months. the observed ones of the corresponding months. Figs. 1a and 1b are respectively the calculated The mean correlation coefficient for whole year and observed anomaly fields of 500mb height is The last line in Table 1 gives the cor- for October 1965, Comparing these two figures, Fig. la Anomalous field of 500mb height for Oc- Fig. 2a Anomalous field of earth's surface temperatober 1965 (calculated). ture for September 1965 (predicted). Fig. lb Anomalous field of 500mb height for Oc- Fig. 2b Anomalous field of 500mb height for Septober 1965 (observed), tember 1965 (predicted).

8 February 1982 Jih-Ping Chao, Yu-Fu Guo and Ru-Nan Xin 289 it can be seen that their general patterns are quite similar. The calculations and comparisons made above naturally lead to the conclusion that the atmospheric motion tends to adapt itself to the earth's surface temperature to a certain degree for the process of the time-scale of one month. 6. Forecasting experiments It may be seen from Table 1 that the persistence between successive monthly mean anomaly fields of 500mb height is rather poor, so the accuracy of the extrapolation of the persistent prediction for forecasting the anomaly field of 500mb height of next month would generally be very low, as if the prediction of the anomaly field of the earth's surface temperature could be made accurately, it would be possible to predict the height field of the following month numerically by adjusting the height field to the earth's surface temperature field. From this idea mentioned above, a predicted method of monthly mean state will be developed. At first, we may predicted the earth's surface temperature field by Eq. (17) with boundary conditions (36) and Fig. 3a Anomalous field of earth's surface temperature for September 1965 (observed). ture for February 1978 Fig. 4a Anomalous field of eath's surface tempera- (predicted). Fig. 3b Anomalous field of 500mb height for September 1965 (observed). Fig. 4b Anomalous field of 500mb height for February 1978 (predicted).

9 290 Journal of the Meteorological Society of Japan Vol. 60, No. 1 and monthly mean anomaly state of initial month. The predicted method see other paper (GLRNWF 1979). Once the temperature of earth's surface has been obtained, the 500mb anomalous geopotential height field is calculated by the adaptation equation (61). In the following, we will give two forecasting examples. This is forecast for one month. a. Example of September Figs. 2a and 2b are the predicted anomaly fields of the earth's surface temperature and 500mb height in September Figs. 3a and 3b are the corresponding observed maps respectively in same Fig. 6 Anomalous field of 500mb height for February 1978 by three layers model (predicted). Fig. 5a Anomalous field of 500mb height for February 1978 (observed). Fig: 5b Anomalous field of 500mb height for Feb (observed). month. It may be seen from these figures that the location and intensity of the main centers of anomaly field in the prediction chart of the earth's surface temperature are close with the observed except that the intensity of the minus anomaly center near Berin sea is too weak and the result of the prediction near artic region is not satisfactory. The location of the main centers of height anomaly are all close with the observed except that a small positive anomaly center in Europe has not been predicted and the predicted positive anomaly center near the pole has not been observed. The correlation coefficient of the persistence of height anomaly from August to September in 1965 is 0.24 (see Table 1), but the correlation coefficient between the predicted and the observed is 0.36, which is higher than the levell of the inertial forecast. b. Example of February Figs. 4a and 4b indicate the predicted anomaly fields of earth's surface temperature and 500mb height in February 1978 respectively. Figs. 5a and 5b denote the corresponding observed maps of the same month. The main discrepancy between prediction and observations is the location of positive anomaly center of temperature and also 500mb height over America. The calculated center is further northwest than the observed. However, this discrepancy can be improved by three layers model (for atmosphere). The new result of 500mb anomaly height ' is given in Fig. 6. From these predicted examples it may be seen that although the predicted skill is not so high, but this tentative investigation on long-range numerical weather forecasts is a promising way worthy of further study.

10 February 1982 Jih-Ping Chao, Yu-Fu Guo and Ru-Nan Xin 291 References Adem, J., 1964: On the physical basis for the numerical prediction of monthly and seasonal temperatures in the troposphere-ocean-continent system. Mon. Wea. Rev., 92, Budyko, M. I., 1956: The heat balance of earth's surface. PB Group of Long-Range Numerical Weather Forecasting, 1977: On the physical basis of a model of long-range numerical weather forecasting. Scientia Sinica, 20, : A -, filtering method for long-range numerical weather forecasting. Scientia Sinica, 22, Kuo, H-L., 1968: On a simpilfied radiative-conductive heat transfer equation, Scientific Report, No. 14. The planetary circulation, project, the University of Chicago. Namias, J., 1948: Evolution of monthly mean circulation and weather patterns. Trans. Amer. Geophys. Union, 29,