Significance of Calculus in many areas of climate science
There is an excellent relationship between mathematics and physical phenomenon, climate change. Calculus entails analysis of continuous functions and small changes, while climatology is a geography branch that studies atmospheric patterns of climate and factors causing them. Climate is a result of many interlinks of several physical processes. Many concepts of climate change can be examined, expressed, and predicted mathematically. Climate scientists should be conversant with the many mathematical structures and how they can be integrated into their studies. Differentiation and integration, partial and ordinary differentials, and linear and non-linear equations are some of the ideas they should master. A mathematical model can be developed as a tool to establish issues of climate change. This project presents how calculus knowledge is useful in climate science. It includes the application of intermediate value and Borsuk-Ulam theorem in the argument that there exist antipodal points with the same temperature. It also shows the link between climate change models, prediction models with derivatives, and integrals of continuous functions.
Calculus on the temperature at antipodal points
Based on the continuous change of functions, Intermediate Value Theorem and the Borsuk-Ulam Theorem are the two main theorems that explain the temperature at antipodal points. Borsuk-Ulam theory postulates that given any continuous function f from a sphere mapping element to an n-dimensional Euclidean space, there are some two points on the opposite sides of the sphere mapped to the same temperature. It can be proved that there are pairs of points with the same temperature along the equator sing the two theorems. They are points which are exactly opposite to each other, i.e.; they are diametrically opposite.
Intuitively, in the context of climatologists, this theorem can be spanned-off to imply that at any given moment, there exist at least two points on the opposite sides of the sphere of the earth having the same temperature. To justify this argument, taking temperature measurements at any two antipodal points, say points a and b on the earth’s surface, the temperature readings may not be the same. Assume that the temperatures are different, taking the temperature at point to be higher than the temperature at the point b when thermometers are moved in such a way that they always remain antipodal at any given paths. In the process, swapping the positions is taking place, and it is obtained that there is at least one instance when the temperatures are equal at one pair of antipodal. To that extent, it has been established that there exist at least two antipodal locations on the surface of the earth whose temperature is the same. The repetition of the process uses many antipodal points, resulting in a band of these points on the earth’s surface. There will be no point without corresponding antipodal at the same temperature. It implies there are no gaps between the positions without finding an intersection where the temperature readings are the same. If there was a gap, then it means that it would be possible to change points ‘antipodally’ from a to b without getting the point where the temperature is the same, which is impossible. The same applies to atmospheric pressure.
The theoretical proof of this phenomenon follows from the Borsuk-Ulam theorem, which utilizes the intermediate value theorem in its evidence. They are both mathematical calculus theories. Given the continuous function defined on a circle, then there are two points and such that For climatologists, is a continuous function of temperature, and the sphere is the earth. In formal terms, the theorem states that given a continuous function, then there exists a point so that the function at , =. Where is the -sphere, and ℝ represents the set of real numbers. A unit circle centred in origin is used in the proof. The angle is used as the angular coordinate of the point. Another continuous function on a closed space [-1, 1] is defined. Then for any point in the closed interval, let be an intersection point of the upper half of the circle and the perpendicular position from with the horizontal axis and be the antipodal point to. The following equation is the result.
.
The values satisfying the function g(c) at the limits of the closed interval are . Then,
It implies that.
When g(1)=0, then the problem is solved. If it is not equal to zero, then the Intermediate Value Theorem is applied. It means there is a point c in the interval, so the g(c)=0. If x I the point corresponding to y, then the result and therefore and is obtained
These concepts cannot be proved practically since both atmospheric pressure and temperature and continuously changing. Therefore the only explanation and prove are based on mathematical knowledge of calculus.
Application of calculus on climate change
The change in climate can be explained in terms of rising or falling aspects of pressure, temperature, humidity, precipitation, or any other climate change component. The change is always in comparison with prevailing conditions. Moreover, it can be variations of dangerous occurrences like hurricane intensity and heavy winds or change in the total number of drought days or rainy days in a year. The various components of climate require different knowledge of mathematics, including statistics and analysis. This project demonstrates a significant relationship between calculus, a mathematics concept of real analysis, and climate change. Calculus knowledge is used to predict the future climatic changes, and to develop various climate model. A basic climate change model is generated in the form of partial differential equations or ordinary differential equations. For instance, comparisons help find the temperature of the earth or some parts of the planet by using the energy falling on the planet from the sun and the energy reflected by the earth’s surface. The radiations and re-radiation results in stabilizing the temperature of the earth. The earth’s temperature increases when the amount of energy re-radiated is less relative to the power falling from the sun to the surface. In this manner, global warming results from greenhouse gases. It absorbs the re-radiated thermal heat and radiates it back to heat the earth. The earth’s surface would be more relaxed when little greenhouse gas is emitted. The best temperature would be upheld when greenhouse gas is at optimum levels. There is global warming when greenhouse gas rises.
The energy balance model is an essential climate model that uses the calculus concept of differentiation. Its validity is on predicted temperature. If the future temperature can be predicted correctly, the equation holds.
CE
Such that A is obtained from Stefan – Boltzmann theorem given as A =.
The is the emissivity of a surface
– Infrared transmissivity of the atmospheres
Is the surface temperature
Is the planetary albedo
CE is the effective heat capacity
Represents the total irradiance of solar energy
The equation shows the temperature of the future can be established if the transmitivity of the infrared radiation in the future atmosphere can be obtained from the anticipated greenhouse gas emissions. Nonetheless, there is uncertainty in the expected amount of greenhouse gas. It depends on the measures and policies formulated to curb climate change by various countries. Scientists work with the statisticians and economists to avail resources and the data required to establish limits of emissions of the greenhouse gas.
The energy balance model establishes the average earth’s temperature, spread over the earth’s surface. The temperatures are high at the tropics and very low at the poles are redistributed by the ocean circulation and the atmospheric circulation.
The oceanic and wind flows are simulated to stand for regional climate at levels. It involves the clubbing of Navier Stokes’ equations, which are pivoted on conservation of energy, momentum, and the mass. They are the governing expressions, and there are other physical courses like natural and biological reactions in the air and ice melting. They are all exemplified by mathematical equations. Most physical processes are well expressed and understood mathematically, but for some, there is no well-scripted knowledge relating them with mathematics. In such situations, they are shown by semi-empirical or empirical mathematical equations established on observation.
The equations are evaluated in steps using numerical methods.
In the energy balance model, the irradiations changes with time. At night it is zero; hence the adoption of the step of time denoted which is always less than 24 hours.
To understand the process well, a time step of between three to six hours is the most appropriate because it results in better results. Nonetheless, it can also be solved by taking the average of solar irradiance, a total of the night and day combined, and keeping the time change as one day. The anticipated accuracy determines the choice. To that extent, the heat equation is solved in time.
The variables of climate such as humidity, pressure, temperature, and factors influencing them varies with the time of the location on the earth’s surface. In this regard, the earth’s surface is divided into a grid of three-dimension with a slight change and in directions and respectively. The spacing of the grids is based on the rate of climate change. It can be from east to west, north to south, up and down on the earth’s surface. Therefore the climate variables are evaluated along with these points in a given time. The already observed climatic conditions are simulated once such a general model-general circulation model is complete. It is then used to predict future conditions.
In applying the two models, a rational approach is used because they use approximations, and therefore, predictions cannot be taken blindly. Some climate aspects are simulated well-using climate models, but some are yet to be analyzed better. For such variables that are not captured well, the results are not reliable. The only better option is to use the well-analyzed variables to relate the poorly captured ones as a proxy to predict the poorly captured variables. Moreover, the east-west and north-south spacing use longitude and latitudes, which is over 100 kilometres. The quantity of a variable between the points on the grid can be obtained by interpolation. However, the variables of climate can change non-linearly and faster over the given distance of 100 kilometres. To settle this problem, climatologists can use the statistical connection between the different variables and those which do not vary fast over the given space. Therefore, it can be noticed that climatology can be solved using calculus and incorporating statistics and numerical analysis.
Calculus on pressure gradient force
Higher-order differential equations can also be of importance to the science of climate. For instance, the slope, which is the gradient, indicates the changing path of a mountain compared to the horizontal change in distance, i.e., “the rise overrun.” The slope can be changing and is obtained by differentiating the height twice with respect to the distance measured horizontally. It shows how the fast gradient of the mountain is changing with the horizontal distance. The knowledge of calculus is also used by climate scientists to establish pressure gradient force with Coriolis force. The air in the atmosphere is acted upon by various forces, including apparent force resulting from the rotation of the earth, pressure gradient force, gravitational force, Coriolis force, and frictional force. Pressure difference on the surface of the earth causes the pressure gradient force. It results in a force from the region of high pressure and directed to the places where the pressure is low. The force results in acceleration as per Newton’s law of motion expressed by. Let to be the field of pressure at any given time; the pressure gradient force is represented by
=
The symbol is the density of the mass of air. The equation illustrates the dimension of [], which is equivalent to Force/mass. It implies that the pressure gradient force is acceleration whose units are Force per unit Mass, i.e., newton/kilogram.
The Geostrophic balance is a condition whereby the Coriolis force and the pressure gradient force balance with others. Most large-scale oceanic flows and atmospheric flows are always nearly in geostrophic balance.
When Newton’s laws of motion are manipulated to fit the rotation state of the earth, the Coriolis force emerges. However, the earth requires 24 hours to complete a rotation. Therefore the consequences of Coriolis force are more significant when the movement takes a long time and considerable distance. The Coriolis force, FC acting on a given mass in a rotating system whose angular velocity is is expressed mathematically as,
.
Where are the cross product of angular velocity and the linear velocity of the mass, the implication of the equation relating to Coriolis force, and linear velocity is that the force is significant when the linear velocity of the object is high. The cross product of the two speeds is another force whose direction is guided by Fleming’s right-hand rule. The magnitude of the Coriolis force can be computed as
. Where is the angle between the two vectors
Calculus and geopotential
Climate scientists apply the knowledge of calculus-integration, to determine geopotential and studies around it. Geopotential is a point in the atmosphere where the amount of work done against the gravitational field of the earth to raise 1-kilogram mass from the lowest altitude to that point. It is the gravitational potential for a unit mass. Its units are joules per kilogram. The 1kg mass at a height is acted upon by a force of which is arithmetically equal to. The equivalent amount of energy or work is done in lifting the unit mass from point to a point is. Therefore,
Thus the geopotential Φ(x) at a point is obtained by performing the integral below.
The geopotential at any given position in the air is affected only by the height from the zero height above the sea level and not the path it takes to that point. The potential at altitude, is trivially zero. The energy consumed in lifting a mass of 1kg from position a to b is obtained by getting the difference in the geopotential, i.e., Φb- Φa.
Geopotential height x is defined as,
X = = dx
The value g0 is the average acceleration as a result of the gravity of the earth. It is accepted worldwide as 9.81m/s2. Geopotential height is always utilized as a coordinate in the air where most applications where energy is of significant consideration, mostly large-scale motions of the atmosphere. In climatology practices, it is not easy to use the density of a gas because it cannot be generally established. To obtain the thickness or pressure difference between two points, climate scientists integrate geopotential Φ1 and Φ2 between the two pressure levels P1 and P2 respectively,
= –
= –
On dividing all through by g0 and interchanging the integral limits results in
= –
Where is the thickness of the gas layer enclosed between pressure levels P1 and P2
Climatologists express force as the slope of the geopotential over a surface of the same pressure because it has properties force that conforms. For instance, the use of hydrostatic pressure is made as a vertical coordinate and used in models to forecast weather. It articulates the gradients of the pressure surfaces with potential heights.
More generally, the pressure due to air at a given height is due to the force of gravity acting on mass air above the height and expressed as
Therefore, the change in surface pressure at a height due to all the air above it
Or
The hydrostatic equation shows that the pressure at various levels of the atmosphere changes by falling or increasing exponentially with the vertical height. However, it is not appropriate to use it in approximating fast heavy and complex thunderstorms convective processes, including tornadoes. In many worst climatic conditions, the downward and upward acceleration of air packages is smaller than gravity.
From the above information, the rapidness of change of logarithm of pressure with the height varies indirectly with the absolute temperature and is independent of pressure. Basing on we obtain the following equations.
Integrating both sides and making the vertical height the subject of the equation,
X= and for any small change in X, ,
And therefore, the pressure reduces exponentially with height.
The hydrostatic equation shows that the pressure in various levels of the atmosphere changes by falling or rising exponentially with the vertical height. In the same way, the molecule concentration varies exponentially. Climatologists can, therefore, use the differentials to obtain the exact value of pressure and molecule concentration. They can also relate this knowledge to scattering, which is directly related to molecule concentration.
Conclusion
Calculus plays a significant role in many areas of climate science. Intermediate Mean Value Theorem and the Borsuk-Ulam theorem are used to show that there exist antipodal points on the sphere of the earth having the same temperature and pressure. The energy balance model is a climate model that uses the calculus concept of differentiation. It is done on the assumption that changes in temperature and pressure are continuous. It is also shown that integration is used in the establishment of a geopotential point. Calculus is used in the calculation of various atmospheric thicknesses by integration between any two geopotential since it is not easy to use the density of a gas that cannot be generally established. To obtain the thickness or pressure difference between two points, climate scientists integrate geopotential Φ1 and Φ2 between the two pressure levels P1 and P2, respectively. Therefore, climate scientists must be well equipped with the various calculus concepts of continuous functions and know how they are applied in climatology.