## Two Level Primitive Equation Model

In the quasi-geostrophic model that we discussed in Sect. 17.3, the static stability o was assumed to be constant. But it is well-known that it varies in space and time depending upon the dynamical and thermodynamical processes in the atmosphere. We, therefore, need to take its variability into consideration in the primitive equation prediction models by computing d0 do at each time step. This requires temperature to be predicted in at least two levels, instead of one as it was in the case of...

## Ddt ddt u dudx v dvdy w ddz

Tzx( jd u dz) and Tzy( jd v dz) for the components of the shearing stresses tzx( p u' w') and xzy( p v' w') for the components of the shearing stresses due to eddy viscosity. In a fully turbulent flow, the shearing stresses due to eddy viscosity are much larger than those due to molecular viscosity. So, by neglecting the small effect of the molecular viscosity, the Eqs. (14.2.9) and (14.2.10) can be further simplified to the final form du dt (1 p) d p dx + fv +(1 p) dl'zx dz (14.2.11) dv dt (1...

## Primitive Equation Models

The filtered and the simplified models, discussed in the foregoing sections, looked promising for a while but obviously had their limitations so far as weather forecasts were concerned, simply because the real atmosphere and its behaviour are more complicated. So, the demand for better forecast naturally called for use of more complete and original hydrodynamical equations which at one time were dubbed as primitive equations. The primitive equation model of the atmosphere consisted of the three...

## Dynamical Models and Numerical Weather Prediction NWP

17.1 Introduction - Historical Background It was probably V. Bjerknes of Norway who first conceived of the idea of using the time-dependent hydrodynamical equations of motion, presented in Chap. 11, for atmospheric prediction. In 1904, in a German meteorological journal, he wrote If it is true, as every scientist believes, that subsequent atmospheric states develop from the preceding ones according to physical law, then it is apparent that the necessary and sufficient conditions for the...

## Waves and Oscillations in the Atmosphere and the Ocean

In Table 11.1, we referred to a few types of atmospheric waves which concern meteorologists most of the time because of their apparent relations with the formations of weather and climate. However, in the atmosphere as well as the ocean, depending upon the fluctuations in pressure, temperature and wind, several types of waves and oscillations may be excited. A fluid parcel when displaced from its position of equilibrium in a stable atmosphere by an external force will tend to return to its...

## 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....

## 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....

## 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...

## 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...

## 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...

## Ascent of Moist Air in the Atmosphere Pseudo Adiabatic 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...

## 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...

## The Steady State 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...

## 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...

## 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...

## 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 Earth Atmosphere 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...

## Supercooled Clouds and Ice Particles 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...

## Cloud Making 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...

## Density of Moist Air Virtual Temperature

If x be the humidity-mixing-ratio in a given volume V of moist air at temperature T and total pressure p and if e be the partial pressure of water vapour, the density of moist air is the sum of the density of water vapour and the density of dry air and is found as follows The density of water vapour as given by the equation of state for water vapour may be written in the form, where pv is the density and e the specific gravity of water vapour 0.622 and R the gas constant for dry air R Md and...