УДК 519.6


Прахова Софья Александровна1, Volker Rehbock2, Сулейманов Игорь Нугуманович3
1Университет Кёртин, Перт, Западная Австралия, аспирант факультета математики и статистики
2Университет Кёртин, Перт, Западная Австралия, доцент факультета математики и статистики
3Уфимский государственный нефтяной технический университет, Россия, кандидат технических наук, доцент кафедры математики

В данной работе мы модифицировали инсоляционную компоненту одной из широко известных глобальных климатических моделей C-GOLDSTEIN. Поступающая радиация была смоделирована как поток через поверхность (широтный пояс), а изменения освещенности в течение года были представлены эллипсом с изменяющимися параметрами. Данный подход позволил нам получить результаты для каждого широтного пояса в любой момент времени. В результате средняя точность моделирования поступающей радиации была увеличена с 96 до 97%. Также данная модификация позволит пользователям C-GOLDSTEIN осуществлять более детализированные эксперименты (такие как изучение сезонных изменений и т.д.)

Ключевые слова: глобальные климатические модели, изменение климата, моделирование инсоляции, поступающая радиация, сlimate change, сезонные изменения, широтный пояс


Prakhova Sofya Alexandrovna1, Volker Rehbock2, Suleimanov Igor Nugumanovich3
1Curtin University, Perth, Western Australia, PhD student department of mathematics and statistics
2Curtin University, Perth, Western Australia, associate Professor department of mathematics and statistics
3Ufa State Petroleum Technological University, Russia, associate Professor department of mathematics

In the current paper we have modified the insolation component of the well-established global climate model C-GOLDSTEIN. The incoming radiation was modelled as a flux passing through a cross-section (latitudinal belt), and the changes of the light throughout the year were represented by an ellipse with changing parameters. This approach has allowed us to get the results for any latitude at any particular time. As the result, the average insolation accuracy has been increased from 96 percent to 97. Also, this modification allows the users of C-GOLDSTEIN to conduct more detailed studies (such as examining seasonality etc.).

Keywords: global climate models, incoming radiation, insolation modeling, latitudinal belt, seasonal changes


Библиографическая ссылка на статью:
Прахова С.А., Volker R., Сулейманов И.Н. Modified Global Climate Model // Современные научные исследования и инновации. 2013. № 11 [Электронный ресурс]. URL: http://web.snauka.ru/issues/2013/11/28602 (дата обращения: 05.02.2020).


Nowadays the problem of climate change is especially important. Many scientists all over the world are concerned about it as negative climate changes will likely affect the life of every creature. There is significant interest in developing climate models, incorporating as many effects as possible to try to accurately predict future climate changes. Most of the existing climate models are atmosphere-ocean general circulation models (AOGCMs) or Earth System Models of intermediate complexity (EMICs).

The first category of models [1, 2] consists of comprehensive and high-resolution models which provide the most accurate forecasts. However, those models are computationally expensive and the forecasting period cannot be extended further than a few centuries [3, 4].

On the other hand, the Earth system Models of Intermediate Complexity (EMICs) [5, 6, 7] are simpler and can be used to run forecasts for a few thousand years [8, 9], as well as to perform some extensive sensitivity studies. Also, EMICs are good for determining the main trends of the future changes, which can be then examining in greater detail by the AOGCMs. As the result, these two types of model complement each other very well and one’s particular choice of model depends on one’s specific goals.

In this paper we aim to modify the insolation component of one of the mentioned well-established EMIC C-GOLDSTEIN [10] in order to increase the accuracy of forecasting and to adjust this model for performing seasonal experiments. The derivation uses the methods of vector field theory and the surface integrals. This approach has allowed us to model insolation for any latitude at any particular time.

Numerical results show that running the global climate model with a new insolation component has increased the average accuracy of the incoming radiation by one percent.

1. Model’s description
In the C-GOLDSTEIN model the atmosphere, the ocean, land surface and sea ice are coupled together. These components interact with each other through momentum, heat and fresh water fluxes. The ocean component was based on thermocline equations with an additional linear drag term in the horizontal momentum equations. The land component has no dynamical land-surface scheme and only determinates the runoff of fresh water. The surface temperature was assumed to be equal to the atmospheric temperature and the evaporation is set to zero. The sea ice component contains dynamic equations which were solved for the fraction of the ocean surface covered by sea ice and the average height of sea ice.

The atmosphere component balances heat and moisture within the atmosphere. Here incoming and outgoing fluxes, sensible heat exchange with the underlying surface, latent heat release due to precipitation and horizontal transport processes were modelled. The incoming radiation was modelled based on tabulated data [11] and produces latitudinal-dependent annual average values.

The standard time step is 0.73 days for the atmosphere and 1.46 days for the ocean. Longitudinal resolution is 10°, while latitudinal resolution varies from 3° near the equator to 20° for polar regions.

A major advantage of C-GOLDSTEIN is that both short-term and multi-millennium forecasts can be performed within a relatively short computational time. In order to obtain near present-day climate, a 2000 year experiment is performed (known as SPINUP) which starts from some unrealistic conditions (such as zero mean global air temperature) and then progresses until the system comes close to equilibrium. All subsequent experiments are performed by starting from these equilibrium initial conditions.

In this paper we are going to modify the insolation component within C-GOLDSTEIN which is currently based on tabulated data and uses the annual average values by incorporating the seasonality into it. For this we are going to model the amount of radiation for each latitudinal belt continuously throughout the year using the vector field theory. This approach is expected to increase the accuracy of the insolation computation. It will also allow users of C-GOLDSTEIN investigate the impact of seasonal and random variations of insolation on climate.

2. A modified insolation component

Consider Figure 1.

Figure 1- Coordinate systems and radiation vector (annual cycle)

Here the radiation vector is F = 1368 Wt/mis the Solar constant, and α is the angle of the Earth’s rotation around the Sun, where α=0 corresponds to the winter solstice.

The (x, y, z) coordinate axes are aligned with the equatorial plane. Here OZ axis represents the Earth’s axis of rotation and OY is in the Earth’s equatorial plane. The (x1, y1, z1) axes are aligned with the ecliptic plane and this is inclined to the equatorial plane by an obliquity angle ε = 23°26′. This coordinate system remains fixed as the Earth rotates around the Sun.

All the calculations are performed in the (x1, y1, z1) coordinates. The reason for this choice is that the formulation of the radiation vector  remains the same as the Earth rotates around the Sun. The direction of radiation vector in the Earth plane is shown on Figure 2.

Figure 2 – Coordinate systems and radiation vector (Earth plane)

The radiation flux through the surface can be calculated as the product of the radiation vector () and the unit normal vector () to the surface (S) integrated over the chosen side of the surface (see Figure 3):


Figure 3 – The choice of surface orientation

This also can be written in coordinate form. In order to calculate the flux we separate the integral into an integral over the positively-oriented surface and an integral over the negatively-oriented surface.

Now we need to find the projection of a latitudinal belt in the (x1, y1, z1)
plane (see Figure 4). The Y1OZ1 plane was chosen as it gives the easiest projections (Figure 4). In the two other projection planes rotating ellipses arise.

Figure 4 – Projections plane choice

Here r and h are the distances from the equator to the upper and lower latitude limits of the belt, respectively.

The equation of the surface will be the equation of the sphere:

where R is the Earth’s radius.

In the chosen projection plane the positively-oriented surface will be the part of the sphere closer to us. The part pointing away from us will be the negatively-oriented surface.

For each of these, a surface integral was calculated as a double integral over the projection of the area in the chosen coordinate plane:


Here z1α, y1α and z1β, y1β are the lower and upper limits of integration, respectively.

The amount of light received by a particular area changes as Earth rotates around the Sun. In a chosen projection plane it will be represented by an ellipse with changing parameters (see Figure 5). The slashed area of the latitudinal belt will be referred as the illumination area from this point onwards.

Figure 5 – Time dependence of light boundaries

Here we have derived the equations of the ellipse, circle, equator line and upper and lower latitudes, which will be used later to determine the limits of integration.

The forward-slashed area is on the negatively-oriented surface, so it will be a strip coming from the side located further from us and a small part on the side located closer to us. The back-slashed area is on the positively-oriented surface (a small piece on the side located closer to us).

For different latitudes the light boundaries will vary with time. This is illustrated in Figure 6.

Here we consider an angle β which determines the latitude and express it in terms of ε and α. Now β ranges from 0° to 23°26′ and there can be four possible types of light boundaries which are indicated in the figure.

Figure 6 – The changing boundaries of light for different latitudes

As the ellipse’s parameters change with time the point Rsin(α) moves and the type of light boundary may be changed. For each type of light boundary the illumination area will be different.

Clearly, the middle latitudinal belts are not affected by time and the illumination area for them will be ε – (π/2 – ε) for any α as β for them will always be less than ε or equal to it. An example for the 30°-40° belt is illustrated in Figure 7.


Figure 7 – Illumination areas for 30°-40° latitudinal belt for the period of time from the winter solstice to the vernal equinox (a-c)

However, for equatorial latitudes the area does change with time. For instance, the 10°-20° belt starts at βε and then as time progresses and the ellipse is growing, it becomes 0 – β (see Figure 8).

Figure 8 – Illumination areas for 10°-20° latitudinal belt for the period of time from the winter solstice to the vernal equinox (a-c)

For polar latitudes the area is between (π/2 – ε) – π/2. However, here we observe the phenomena of polar night and polar day. An example of a polar night and its disappearance from the polar region is presented in Figure 9. 

Figure 9 – Illumination areas for the latitudes above the polar circle for the period of time from the winter solstice to the vernal equinox (a-c)

For the Southern Hemisphere the illumination areas below equator need to be considered. However, for computational simplicity we used a symmetrical area in the Northern Hemisphere. An example for the area 0 – β for the Southern Hemisphere is shown in Figure 10.

Figure 10 – An example of actual illumination area for latitudinal belt in Southern Hemisphere (a) and a symmetrical one used for calculation (b)

The example of polar day and its disappearance is shown in Figure 11. 

Figure 11 – Polar day and its disappearance (a-c)

After the areas of illumination have been determined the amount of radiation for each latitude for any particular time can be calculated. Here we illustrate one example for the illumination area 0 – β (see Figure 12).

Figure 12 – Illumination area 0 – β

The amount of radiation is the sum of integrals over the positively-oriented surface (5) and integrals over the negatively-oriented surface (4). The surface has been spread out into the areas convenient for integration. As such, the total amount of radiation passing through the latitudinal belt for a fixed α will be:

Here the limits of integration with respect to z1 and y1 were determined from Fig.5. The limits with respect to y1 will be the following:

In a similar way the limits of integration can be obtained for the βε, ε – (π/2 – ε) and (π/2 – ε) – π/2 illumination areas.

In order to find the amount of radiation received per m2 in the current latitudinal belt per second (I), the total amount of radiation needs to be divided by the surface area of latitudinal belt:

The projection of latitudinal belt in YOZ plane will be the following (figure 13):

Figure 13 – The projection of latitudinal belt in YOZ plane

Here rr and rh are the radius of upper and lower limits of the belt, respectively.

For computational simplicity the projection in the XOY plane is shown in Figure 14:

Figure 14 – The projection of latitudinal belt in XOY plane

Here, the latitudinal belt area is calculated as follows:

The amount of radiation received per m2 in the current latitudinal belt per second (daily average) is calculated for each latitudinal belt of a width of 10°. Calculations were performed for the Northern and Southern Hemisphere for the period of time from the winter solstice to the spring equinox (see Figure 15). It is also easy to modify the procedure to thinner belts for more accuracy.

Figure 15 – Period of calculations

In the first quadrant α ranges from 0° to 90°. The step size for α was chosen to be 10°, which approximately equals 10 days (as 360°~365 days). Note that it is easy to choose smaller α at the expensive of more computation.

The amount of radiation for the rest of the year was obtained as shown in Table 1.

Table 1 – Calculations of the insolation for the whole year period for Northern Hemisphere and Southern Hemisphere


















(α )









3. Results

The results of insolation for any particular time of the year for any latitudinal belt were calculated from the previously derived formulae using the Maple software (see Figure 16 and Figure 17).

Figure 16 – The amount of radiation received for different latitudinal belts in the Northern Hemisphere. On vertical axis – the amount of radiation (Wt/m2 per second)

Figure 17 – The amount of radiation received for different latitudinal belts in the Southern Hemisphere. On vertical axis – the amount of radiation (Wt/m2 per second)

Note that smoother curves would result if we used a smaller step size for α.

We then calculated the annual average value for each latitudinal belt and compared it with satellite data [12] and with those obtained by the old insolation component within C-GOLDSTIN. For the results of comparison see Table 2.

Table 2 – Comparison of the obtained results

Latitudinal belt

Satellite data


Seasonal insolation component


Tabulated-data insolation component



Seasonal insolation component


Tabulated-data insolation component

0° -10°






10° -20°






20° -30°






30° -40°






40° -50°






50° -60°






60° -70°






70° -80°






80° -90°









The results show very good agreement for equatorial and middle latitudes regions, and slightly less agreement in the polar regions which are known to be difficult to model. Also, an average accuracy has been improved from 96% to 97% compared with one obtained by previous insolation model within C-GOLDSTEIN.

We have then replaced the initial insolation component by a new one and reinitialised the atmosphere starting from initial conditions and leaving all other parameters to be set to zero level (carbon dioxide increase rate etc.). Note that due to the grid resolution in C-GOLDSTEIN, it was necessary to average the insolation components for the 70-80 degree and 80-90 degree latitudinal belts (this does not affect the average accuracy reported in Table 2).

After running the global climate model with new atmospheric component realistic latitudinal temperature distribution has been obtained (Figure 18).

Figure 18 – Latitudinal temperature distribution obtain by using C-GOLDSTEIN climate model

4. Conclusion and Discussion

After modification the insolation component in C-GOLDSTEIN, some new experiments can be performed within the tool, such as investigating seasonal and random variations of insolation responses to climate change.

In addition, a new approach for modelling the process of receiving energy by Earth has been introduced. In the current paper this process was modelled from the space perspective, while all the existing models use the Earth’s point of view. This approach can be useful in the future for modelling external insolation forces.

We should note that the aspherical shape of the Earth has not been considered in our proposed insolation component. However, this effect is very small (only 1/300) and has a negligible impact on an EMIC like C-GOLDSTEIN.

Another approach for modelling seasonal variations of insolation is available within the GENIE GOLDSTEIN model (Marsh and Edwards 2005), which uses the Earth point of view. However, it was not used for investigating seasonal variations there, since only the annual average values of insolation were ultimately used.

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