The Solar Constant
It is remarkable that inspite of inconstant sun with large fluctuations in its radiation during solar flares and prominences, etc., the intensity of solar radiation reaching the outer boundary of the earth's atmosphere has remained more or less constant over the last few centuries. The reasons for this may be the following the large fluctuations that often occur in solar activity mostly affect the extreme ultraviolet and X-ray part of the solar spectrum which contains a very small amount of...
The Convective Layer
By the time, the outgoing radiation reaches the top of the radiative layer,the solar atmosphere becomes somewhat opaque to the outgoing radiation with the consequence that the heat energy piles up in a narrow transition zone at the top of the radiative layer causing the material below to be extremely hot as compared to that above. The pent-up energy of the transition zone then bursts into violent convection which rises to great heights, delivers the energy to the solar surface and then sinks....
Divergence Expansion D
This term signifies an expansion or contraction of the area or volume, in three dimensions of a closed physical curve. Let us consider here an infinitesimal area, A 5x 5y. Differentiating following the motion , we get d 5x 8y dt 5yd 5x dt 5xd 5y dt 13.10.6 Since 5x and 5y represent distances between particles along the co-ordinate axes, 5x5y represents an infinitesimal area, A, in the xy-plane and the differentiation represents simply an expansion or contraction of this area. Similarly,...
Adiabatic Changes in the Atmosphere
There are several thermodynamical processes in the atmosphere in which heat is prevented from entering or leaving a system during the process. Changes taking place during such a process are called adiabatic changes. This condition can be met in two ways either by thermally insulating the system, or by allowing the changes to occur so rapidly that there is no communication of the system with the outside, so far as heat transfer is concerned. The following are some examples of adiabatic changes...
Structure of the Sun its Interior
The sun has a layered structure which is suggested by the results of helioseismic soundings, measurement of neutrino flux, continuous monitoring of X-rays and gamma rays emanating from the sun's interior and inferred from the standard theoretical model. The layered structure is shown in Fig. 7.1. Some details about the layers in the interior of the sun are as follows Fig. 7.1 The layered structure of the sun Fig. 7.1 The layered structure of the sun The Core is the central region of the sun....
Geopotential Surfaces
If a body of unit mass is raised from the earth's surface to a height z in the atmosphere, the work that must be done against the earth's gravitational field is called its geopotential which is usually denoted by O and defined by the relation where the geopotential 0 0 at sea level is taken to be zero. Due to the spheroidal shape of the earth and variation of the acceleration due to gravity, g, over the earth's surface, the surfaces of constant geopotential are not quite parallel to the...
Scattering and Reflection
The intensity of the incoming solar radiation is depleted by the processes of scattering, reflection and absorption by air molecules or other particulate matter that may be suspended in the atmosphere and by clouds. In scattering, a straight parallel beam of radiation changes direction either sideways or backwards. However, the degree of scattering depends upon the size of the molecules or particles compared to the wavelength of the incident beam. If the size is equal to or comparable to the...
Carnot Engine
Sadi Carnot 1796-1832 , a French physicist, devised an engine which working in a four-stroke cycle under given conditions could be said to be the best possible heat engine to secure maximum useful mechanical energy out of the heat energy supplied to the engine. His engine worked between two reservoirs of heat, one at temperature T1 which we may call the source and the other at T3 which we may call the sink, with T1 gt T3. The working substance was 1 mol of a gas which was contained in a...
Berlageberlage 1956 And J. Bjerknes 1960
Agafonova EG, Monin AS 1972 On the origin of the thermohaline circulation in the ocean. Okeanologia Issue No.6. Aitken J 1923 Collected Scientific Papers 1880-1916 The University Press, Cambridge. Arakawa A 1972 Design of the UCLA General Circulation Model Tech Rpt No 7, Dept of Meteorology, Univ of California, Los Angeles, California. Arakawa A, Schubert WH 1974 Interaction of a cumulus cloud ensemble with the large-scale environment Part 1. J Atmos Sci 31 674-701. Arenberg D 1939 Turbulence...
Some Practical Uses of Electromagnetic Radiation
Electromagnetic radiation has been applied using remote sensing techniques to several fields of human activity. These include ii Land surface mapping and analysis, and cartography iv Hydrology and water resources v Meteorology and oceanography vi Geology and mineral exploration viii Coastal resources management ix Monitoring biological activity in the ocean x Military surveillance, etc. However, owing to near-total absorption of ultra high frequency waves in the atmosphere, only a limited part...
Saturated Adiabatic Lapse Rate of Temperature
In a pseudo-adiabatic process, the lapse rate of temperature of a saturated parcel of air with height may be found from the entropy form of the First law of thermodynamics 3.2.1 as follows d ln T dz - k d lnp dz - L cp T dxs dz 4.6.1 where k R cp, and xs is the saturation mixing-ratio. With the aid of the hydrostatic approximation and the equation of state, 4.6.1 may be written as dT dz g cp - L cp d xs dz - L cp dxs dT dT dz 4.6.2 Since, by 3.4.8 , the dry adiabatic lapse rate, g cp -dT dz rd,...
Adiabatic Relationship Between Pressure Temperature and Volume
Under adiabatic condition, Q 0, and 3.2.2 may be written If U is expressed as a function of p and V, we can obtain a relationship between p and V by substituting for dU from 3.3.1 and making use of the equation of state 2.4.1 . Thus If we denote the ratio of the specific heats, cp cv, by y, then by integrating 3.4.2 we get Since y has a value of 1.4 for dry air, the adiabatic curves are steeper than the isothermals in a pV-diagram. The adiabatic relationships between T and V, or between p and...
Properties and Variables of the Atmosphere
In meteorology and thermodynamics, simplifying assumptions are made regarding the structure and behaviour of the gaseous molecules and the atmosphere is treated as an ideal gas. The main assumptions of an ideal gas concept are that the molecules do not occupy any finite space and hence have no volume and that there are no forces of attraction or repulsion between any two molecules. The properties of the air that find important applications in meteorology and thermodynamics under these...
D [dudx dvdy Z [dvdx dudyc F [dudx dvdy r [dvdx dudy
We now rotate the system of co-ordinates by a certain angle y0, such that r 0, i.e., dv dx -du dy. Equation 13.10.2 may then be written as u u0 1 2 Fx 1 2 Dx- 1 2 Zy v v0 - 1 2 F y 1 2 D y 1 2 Z x 13.10.4 If, instead of rotating the system of co-ordinates by y0 , we had rotated it by an angle y0 n 2 or y0 - n 2 , r would again be zero. So, we may choose between these rotations such that F is positive, while D and Z may have either sign. The derivations leading to 13.10.4 can easily be verified...
The Continuity Equation
The equation of continuity in isobaric co-ordinates may be derived as follows In Sect. 11.8, we showed that the rate of change of mass following motion in Cartesian co-ordinates is given by 11.8.1 . Since dw dz dm dp, we can write the continuity equation following motion in isobaric co-ordinates as dp dt -p du dx dv dy dm dp 12.2.5 However, if we transform the total derivative on the left-side of 12.2.5 to the isobaric co-ordinate system using 12.2.2 , we obtain, after re-arranging, the flux...
Dry Adiabatic Lapse Rate of Temperature with Height
The temperature of a parcel of dry or unsaturated air decreases when it ascends adiabatically in the atmosphere. If we take logarithm of 3.4.6 and differentiate it with respect to height and use the hydrostatic approximation and the ideal gas equation, we obtain the expression T 9 d9 dz dT dz g cp 3.4.7 For an atmosphere in which the potential temperature does not vary with height, where rd is the dry adiabatic lapse rate of temperature with height. Since g and cp vary little with height in the...
The Concept of Entropy
It is a unique property of heat that it always flows in one direction, viz, from a body at higher temperature to one at lower temperature when they are in contact with each other either directly or through some intermediate conductor. To show this, let us heat a piece of metal, say iron, of mass mi and specific heat ci to a temperature Ti and place it in water of mass m2, specific heat c2 and temperature T2, with T1 gt T2. In this case, heat will flow from the metal to the water and soon an...
Equivalent Potential Temperature
When condensation occurs in a sample of moist air which is lifted, the heat liberated in the process amounts to - L dxs, where L is the latent heat and xs the saturation-mixing-ratio at the temperature at which the air becomes saturated. This heat is added to the air. The entropy equation, 3.5.3 , may, therefore, be written as where T is the dry-bulb temperature at the level where air becomes saturated, cp the specific heat of dry air at constant pressure, and 9 the potential temperature of the...
v dfdy p[dpdxdpdy
where we have substituted Z for the vertical component of the relative vorticity, dv dx du dy . Since df dt vdf dy, by re-arranging, we can write 13.7.3 in the form d Z f dt Z f du dx dv dy dw dx dv dz dw dy du dz 1 p2 dp dxdp dy dp dy dp dx 13.7.4 The three terms on the right-hand side of 13.7.4 are called the divergence term, the tilting term and the solenoidal term respectively. In general, all the terms contribute to the rate of change of the vertical component of the absolute vorticity Z f...
Ascent of Moist Air in the Atmosphere PseudoAdiabatic Process
When a stream of moist but unsaturated air rises in the atmosphere, it first cools by dry-adiabatic expansion till it reaches the lifting condensation level where it becomes saturated. Further ascent leads to condensation of water vapour on nuclei that may be present in large numbers in the atmosphere. Usually, hygroscopic particles act as effective nuclei. Experiments in the laboratory have shown that the equilibrium vapour pressure required for condensation depends upon not only the...
Rotation
It is easy to show that the last terms on the right-hand side of 13.10.4 represent the horizontal components of a circulation as given by the Stokes's relation 13.8.1 , in which Z represents the vertical component of vorticity and, therefore, a rotation about a vertical axis. Thus, 13.11 Types of Wind Fields - Graphical Representation The component motions of a linear wind field which exhibit the above -mentioned differential properties of air flow individually are B, x-component of...
dp dpdx dx dpdy dy dpdz dz
From these expressions, it is evident that dp may be conceived as the scalar product of two vectors, dr and a vector dp dx i dp dy j dp dz k which is called the gradient of p. Therefore, we write When the isobaric surfaces are parallel to each other, then, according to rules of scalar products, the gradient of p vanishes along the isobaric surfaces, but becomes maximum at right angles to them. We now introduce a vector differential operator, called the Del or Nabla operator, usually denoted by...
Static Stability of Dry Air Buoyancy Oscillations
In view of 3.4.8 , 3.4.7 may be written as where we write re for -dT dz which is the actual lapse rate of temperature with height in the environment. From 3.4.9 , it follows that if the actual lapse rate re is less than the dry adiabatic lapse rate rd, d9 dz is positive, which means that the potential temperature increases with height. This makes the atmosphere statically stable. Thus, the atmosphere is vertically stable, neutral or unstable, according as re is less than, equal to or greater...
Special Cases of Baroclinic Instability
Before discussing the general properties of 18.3.13 , we consider two special cases In this case, we put UT 0 in 18.3.13 , and obtain two values of c given by c2 Um - P k2 2 2 18.3.15 The phase speeds c1 and c2 are real quantities which correspond to free stable normal mode oscillations of the two-level model with a vertically-averaged barotropic basic state zonal current,Um. It will be seen from 18.3.14 that c1 is simply the phase speed of the barotropic Rossby wave moving westward relative to...
Clouds in the Sky Types and Classification
The 'International cloud atlas' of 1932 divides clouds into various families, genera, and species, but here we give only a few examples of cloud types which are more commonly observed. Broadly, clouds are classified into four families depending upon the height and the layer of their formation. These are high clouds Fig. 5.4 a-c , medium clouds Fig. 5.4 d-e , low clouds Fig. 5.4 f-h and clouds with great vertical development Fig. 5.4 i . Table 5.4 gives a list of cloud types commonly observed in...
The SteadyState Solution Geostrophic Adjustment
We now assume that the gravitational adjustment process leads ultimately to a steady state which can be given by the time-independent solution of 15.8.10 . Further, we assume that in the steady state, a geostrophic balance is reached in which the pressure gradient is balanced by the Coriolis acceleration. Since the initial condition is independent of y, we assume that the solution at all subsequent times will be independent of y. Thus, from 15.8.1 and 15.8.2 , we obtain the geostrophic balance...
Stability of Moist Air
In Chap. 3, Eq. 3.4.14 , we derived the condition for the vertical stability of an atmosphere in which the presence of water vapour was ignored and the atmosphere was treated as totally dry. But the fact is that the terrestial atmosphere always holds some moisture which, when lifted, would produce condensation at some level which is called the lifting condensation level, whatever may be the mixing-ratio. A parcel of air rising in such a moist atmosphere would follow the dry adiabatic lapse rate...
Thermodynamic Diagrams
Several thermodynamic diagrams have been devised to study the static stability conditions of the atmosphere. Stability parameters in these diagrams vary but they all seem to have the same common objective to find out by comparing the environment temperatures at different heights with the dry and moist adiabats at those heights whether stable and unstable conditions exist in any layer, and then, in some diagrams, if the atmosphere is conditionally unstable, to assess the amount of net...
Seasonal and Latitudinal Variations of Surface Temperature
Variations of the incoming solar radiation with season and latitude cause corresponding changes in surface temperatures which are observed all over the globe. An example is presented Fig. 8.6 for two stations in Asia an equatorial station in Borneo and Beijing near 40 N latitude. It is evident from Fig. 8.6 that the amplitude of the seasonal oscillation is close to 1 C in Borneo, whereas it is about 15 C at Beijing. DEC FEB MARCH MAY JUNE AUG SEPT NOV DEC DEC FEB MARCH MAY JUNE AUG SEPT NOV DEC...
Nondivergent Models
The basis for this type of models is a simplified version of the divergence equation 17.2.5 . Elimination of gravity waves from the solution of the equations of motion by putting dD dt 0 in 17.2.5 leaves us with a complicated diagnostic relation involving 0, V, and m in a balance. On scaling considerations, further simplification of this balance equation may be made. According to a theorem of Helmholtz, any velocity field V can be partitioned into two independent velocity components, a...
Evaporative Cooling
Evaporation of water from the surface of a body whether a gas, liquid or solid leads to the cooling of the body. Numerous examples can be cited from everyday experience where this property of evaporation has been observed and utilized by humankind. The following are a few of them These are mechanical devices used in several tropical countries to cool air in living rooms by drawing outside dry hot air which may be at a temperature of about 45 C through a rectangular wooden frame which is stuffed...
The Reversing Layer
Immediately above the photosphere lies a thin layer of the solar atmosphere, approximately 600 km thick, which is comparatively cooler than the photosphere and contains gases which can selectively absorb some of the radiation coming out of the photosphere. This is called the reversing layer, because the absorption gives rise to appearance of dark lines in the emission spectrum of the sun as received by the earth, at wavelengths characteristic of the gas present in the reversing layer. These...
The Thermodynamic Energy Equation
In the isobaric co-ordinate system, the thermodynamic energy equation 11.9.1 may be written in the form dT dt am 8Q dt cp 12.2.7 Expanding the left side of 12.2.7 with the aid of 12.2.2 and re-arranging, we obtain dT dt -V V T am 1 cp 5Q dt 12.2.8 where a KT p - dT dp is the static stability parameter, m dp dt is the vertical p-velocity, and k R cp. Since T -p R dO dp, the Eq. 12.2.8 may also be expressed in terms of dO dp. As already mentioned in Chap. 10, diabatic heating or cooling occurs...
The Second Law of Thermodynamics
The Second law of thermodynamics makes an important statement about the heat balance in a thermodynamic system, such as the atmosphere. Like the First law, it also emphasizes that the total heat energy supplied to a system remains constant and goes partly to increase the internal energy of the system and partly to do work against external pressure. But there is a difference. The First law does not tell us how much of the given heat energy goes to increase the internal energy and how much to do...
The General Circulation of the Atmosphere
19.1 Introduction - Historical Background Historically, there must been a time when people had little idea of a circulation in the earth's atmosphere. Few were aware that the wind at their locality was related to the wind at another location on the face of the earth. It appears that the first to visualize a circulation in the atmosphere was the British scientist, Halley 1686 , who made a detailed study of the wind systems over the tropical belt with the data then available and hypothesized that...
Scale Analysis of the Vertical Momentum Equation
The scale analysis of the vertical component of the momentum equation 11.7.10 carried out on the same lines as in Table 11.2 shows that the order of magnitude of the various terms are as given in Table 11.3. It follows from Table 11.3 that the two terms which are in close balance with each other are the 3rd and the 5th terms. They yield the well-known hydrostatic relation 2.3.4 Table 11.3 Scale analysis of the vertical momentum equation Table 11.3 Scale analysis of the vertical momentum...
Free Enthalpy or Gibbs Potential or Gibbs Free Energy
If the pressure in an isothermal system is kept constant, 3.7.2 may be written in the form d U - TS pV dG 0 3.7.6 where G U - TS pV is called the 'Free enthalpy' or Gibbs potential, or Gibbs free energy. As with free energy in an isothermal-isometric system, the Gibbs free energy in an isothermal-isobaric system tends to be at its minimum value at equilibrium, i.e., where the sign of equality refers to a reversible system. The Gibbs equilibrium condition 3.7.7 has been widely applied to study...
ddx Xrn fooAp[dUmddx dvdxdx dUTdvdx dvdxdx dddxdx Udddxdxj
We may interpret the terms on the right-hand side of 18.4.1 as follows. The first term represents the Laplacian of the advection of the perturbation thickness by the basic state vertically-averaged mean wind. The second term is proportional to the Laplacian of the advection of the basic state thickness by the vertically-averaged perturbation meridional wind. The third term represents the differential advection of the perturbation vorticity by the basic state wind. Thus, it appears that three...
Heat Balance of the EarthAtmosphere System Heat Sources and Sinks
10.1 Introduction - definition of heat sources and sinks We showed in Chap. 8 that the intensity of the incoming solar radiation at the earth's surface varies widely with latitude and the transmissivity of the overlying atmosphere, being, in the annual mean, maximum at the equator and minimum at the poles. As against this and as shown by measurements and computations, the intensity of the outgoing longwave radiation varies only slightly with latitude. Thus, the difference between the incoming...
Adiabatic Propagation of Sound Waves
Another example of adiabatic change in the atmosphere is found in the propagation of sound waves. Sound waves are longitudinal waves which travel by compression and rarefaction of air parcels. Newton was the first to compute the velocity of sound in air by using the relation where E is isothermal volume elasticity which in the case of an ideal gas is equal to pressure, p. However, the velocity of sound computed by 3.4.15 differed from observed values. The cause for the discrepancy was found by...
Spectral Distribution of Radiant Energy
Planck's law enables us to determine the complete spectral distribution of energy emitted by a black body at different temperatures at various wave-lengths. Prevost's finding that in the physical world, every body, regardless of its surroundings and temperature, emits its own radiation makes it possible for us to examine the characteristic spectrum of radiation from any body of interest to us. In fact, the spectral analysis has been used widely as a powerful tool to examine the physical...
Plancks Law of Black Body Radiation
Finally, in 1900, Planck, using the results of careful experiments and on the basis of quantum theory came out with his celebrated law of black body radiation which states as follows EXdX 2hc2 X5 1 exp hc KXT - 1 dX 6.4.4 where Ex is the intensity of radiation from a black body at temperature T at wavelength X, and h is Planck' constant, c the velocity of light and k the Boltzmann's constant. Planck's law 6.4.4 satisfactorily explains the distribution of energy amongst the whole range of...
Supercooled Clouds and IceParticles Sublimation
Not all clouds are, however, made up of water droplets. Observations show that high-level cirrus clouds, appearing at heights of 6 to 12 km, where the temperature is well below the freezing point see chap. 2 , are composed mostly of ice-particles. Between clouds consisting entirely of water droplets and those consisting entirely of ice particles, there are, however, many other types, composed partly of water droplets and partly of ice. The surprising fact is that clouds consisting even entirely...
The Vorticity Equation in Isobaric Coordinates
A somewhat simpler form of the vorticity equation may be derived by using the equations of motion in isobaric co-ordinates. Combining 12.2.3 and 12.2.4 and using vector notations, we can write the equations of motion in isobaric co-ordinates as dV dt V-V pV mdV dp -VO - k x f V Or, dV dt -V O V-V 2 - kx V Z f - mdV dp 13.7.5 where we have used the vector identity, V-V V V2 V V k x ZV, and put Z k-V x V. We now operate on the vector equation 13.7.5 with the Del operator Vx, and obtain, after...
dadxc dadxz dzdxc dadz
A similar expression can be written for the y-component of the gradient. The transformation of the gradient of any arbitrary function from z to c coordinate is then given by the vector operator Vc Vz Vc z d dz i.e., Vz Vc - Vcz d dz 11.6.3 Equation 11.6.3 is quite general and may be used for transformation to any vertical co-ordinate 'c' which is a function of z or p, such as an isobaric surface, an isentropic surface or a sigma a surface where a p p0 . As examples of the use of 11.6.3 , let us...
Simplified Equations of Motion QuasiBalanced Winds
The scale analysis in Chap. 11 has shown that in the equations of motion 11.7.811.7.10 , all terms are not equally important and that it is possible to eliminate some of them in order to retain the more important ones which may be in approximate balance. The sorting of this kind permits us to obtain useful relationships between the distributions of pressure, temperature and wind in the atmosphere. It will be shown in the present chapter that the simplified equations offer relationships of great...
CloudMaking in the Laboratory Condensation Nuclei
Aitken 1923 's method was to expand suddenly a closed volume of air standing over a water surface containing saturated water vapour into a larger volume. This is not exactly what happens in Nature, for when air saturated with water vapour goes up, say to a height of 2 km, the pressure falls and the mass of air expands gradually in volume and the expansion is not large. In the laboratory experiment, the expansion is sudden, and generally large. However, in the atmosphere as well as in the...
The Gradient Wind
In 12.3.3 , dV dt denotes the tangential acceleration, while V2 R gives the radial acceleration of the moving parcel towards the center of the curve. The latter is also called the centripetal acceleration, since for a unit mass of the parcel it signifies the force with which it is continuously attracted towards the center while it moves along the curve. The curvature effect produces a centrifugal acceleration along the outward normal. Thus, the centripetal and the centrifugal accelerations are...
Coexistence of the Three Phases of Water the Triple Point
The Clausius-Clapeyron equation may be applied to study the variation of saturated vapour pressure with temperature between any two phases of water, for example, between vapour V and liquid water W , or between liquid and solid ice I , or even directly between vapour and solid. For this purpose, we integrate Eq. 4.8.2 from the initial temperature 273.16A where the saturation vapour pressure is experimentally known to be 6.11 mb, to temperature T, by noting that the latent heat of a water...
Melting Point of Ice Variation with Pressure
The Clausius-Clapeyron equation 4.8.1 is exact and may be applied to the case of the variation of the melting-point of ice with pressure. For this, we invert the Eq. 4.8.1 and obtain dT des T v2 - v3 Le T 4.8.7 where we write Le T for the latent heat of melting of ice or fusion of water and v3 for the volume of 1 mol of ice. Since the specific volume of ice is greater than that of water, i.e., v3 gt v2, the right-hand side of 4.8.7 is negative. This means that the melting-point of ice is...