Studies on Lower Atmospheric Processes over South India using Numerical Atmospheric Models and Experiments

Contents

Preface

1. Introduction
1.1 General Introduction
1.2 Criticalities involved in the various Atmospheric Events
1.2.1 Scales ofAtmospheric Motion
1.2.2 Rapid changes in Atmospheric Parameterization
1.2.3 ComputingRequirements
1.2.4 Dissemination ofForecast Productto theUser’sCommunity
1.3 Examples of some Mesoscale Atmospheric Processes
1.3.1 Density or Gravity Current
1.3.2 Tropical Cyclones
1.3.3 Thunderstorms
1.3.4 SquallLines
1.3.5 Tornadoes
1.3.6 Sea/Land Breezes
1.3.7 Gravity or Buoyancy Waves
1.3.8 Mountain/Lee Waves
1.3.9 Slope andValleyWinds
1.4 History ofNumerical Weather Prediction (NWP)
1.5 Potentials of Atmospheric Model
1.6 Objective of the Thesis
1.7 Organization of the Thesis

2. Numerical Atmospheric Models used in the Study 26 -
2.1 GeneralIntroduction
2.1.1 What is an ‘Atmospheric Model’?
2.1.2 Limitations ofNWPModels
2.2 Basic Classification of Atmospheric Models
2.2.1 Hydrostatic Approximation
2.2.2 Atmospheric Model classification
2.3 High-resolution Regional Model (HRM)
2.3.1 Differentialformofthe ModelEquations
2.3.2 Physical Parameterization
2.3.2.1 Radiation and Clouds
2.3.2.2 GridScale Precipitation
2.3.2.3 Convection
2.3.2.4 Turbulent Fluxes in ABL and Free Atmosphere
2.3.2.5 SoilModel
2.3.3 Operational UseofHRMbasedonGMEData
2.4 Advanced Regional Prediction System (ARPS)
2.4.1 Dynamic Equations and Numerical Formulation
2.4.1.1 The Coordinate System
2.4.1.2 GoverningEquations
2.4.2 Parameterization
2.4.2.1 Sub-gridScale Turbulence
2A2.2 Microphysics
2.4.2.3 Planetary Boundary Layer Depth
2.4.2.4 Surface Flux
2.4.2.5 Land Surface Energy and Soil Vegetation
2.4.2.6 Cumulus Parameterization
2.4.3 Boundary Conditions
2.4.3.1 Lateral Boundary Conditions
2.4.3.2 Top and Bottom Boundary Conditions
2.5 Comparison between HRM and ARPS
2.6 Limitations ofHRM and ARPS
2.7 Rationale Behind the Choice of the Models
2.8 Discussion

3. Statistical Assessment of Atmospheric Models used in the Study 51-
3.1 Generallntroduction
3.2 Methodology ofEvaluation of Statistical Bias and its Significance
3.2.1 Model Bias and Simulation Errors: General Introduction
3.2.2 Statistical Significance of the Model Bias
3.2.3 Normalization ofModelBias
3.2.4 Incorporation ofModel Bias in Near Real Time Simulations
3.2.5 Statistical Skill Parameters: Correlation Coefficient and Root Mean Square Error (RMSE)
3.3 Statistical Evaluation ofHRM forecast fields
3.3.1 Error Analysis for the Entire Domain
3.3.2 Methodology
3.3.3 Model Performance for +24 hours and +48 hours simulations
3.3.4 Monthly Mean Normalized Bias in different Seasons
3.3.5 Evaluation of the Performance of the Model in the Entire Domain during different seasons
3.3.6 ModelBiasoverSHARandTrivandrum
3.4 Sensitivity of ARPS to the Initial Conditions and its Impact on the forecast fields
3.5 Discussion

4. Characterization of Sea/Land Breeze Circulation Along the West Coast of Indian Sub-continent 87-
4.1 Generallntroduction
4.2 Sea/Land Breeze Circulation
4.3 SLBC over the West Coast ofIndian Sub-continent: A Review
4.4 Domain of Study and Definition of Sea Breeze Component
4.5 Data
4.5.1 ModelData
4.5.2 Ship Data
4.5.3 IMD Data
4.6 Results
4.6.1 Thermal and wind structure of SLBC over Ocean and Land
4.6.2 Numerical Simulation of SLBC characteristics during pre-monsoon and winter
4.6.2.1 Diurnal Variation of Sea/Land Breeze Component
4.6.2.2 Spatial Variation of Sea Breeze component
4.6.2.3 Spatial variation ofBoundary Layer Height within the SLBC
4.7 Discussion

5. Mountain Wave Activity Over the Western Ghats of Indian Sub-continent
5.1 General Introduction
5.2 Mountain Wave Projects
5.3 Mountain Waves and their Significance
5.4 Domain ofStudyandDatabase
5.5 Theory
5.5.1 Mountain Waves
5.5.2 Orographic Lifting
5.6 Results
5.6.1 FroudeNumber
5.6.2 Scorer Parameter
5.6.3 Vertical Propagation and Wave Breaking Characteristics from the Model Simulations
5.6.4 Vertical Momentum Flux
5.7 Discussion

6. Thunderstorm Over the East and West Coast of Indian Sub-continent: A Case Study
6.1 GeneralIntroduction
6.2 Thunderstorm: Some Meteorological Aspects
6.2.1 WhatisaThunderstorm?
6.2.2 Related Natural Hazards andMeteorological Importance
6.2.3 Precursors to Thunderstorm Occurrence
6.2.3.1 Triggering Mechanism
6.2.3.2 Conditional Instability
6.2.3.3 High Dew-point Temperature
6.2.3.4 Low-levelWind Shear
6.2.4 Life Cycle ofThunderstorm
6.3 A Review of thunderstorm Studies
6.4 Thunderstorm Events over SHAR and Trivandrum: Typical case Study
6.5 Model Configuration
6.6 Results
6.6.1 Prediction ofThunderstormover SHAR:ACaseStudy
6.6.1.1 Prevailing Conditions
6.6.1.2 ModelSimulations
6.6.1.2.1 Surface Level Horizontal Divergence
6.6.1.2.2 PressurePerturbation
6.6.1.2.3 Vertical Velocity
6.6.1.3 Discussion: Prediction of Thunderstorm
6.6.2 Post Analysis of Thunderstorm event over Trivandrum
6.6.2.1 Meteorological Conditions Prevailing over the Location
6.6.2.2 Elevated Convection
6.6.2.3 Advection of Storms
6.6.2.4 Merger Process
6.6.2.5 Three Numerical Simulation Experiments over Trivandrum
6.6.2.5.1 Experiment
6.6.2.5.2 Experiment
6.6.2.5.3 Experiment
6.6.2.6 Discussion: PostAnalysis ofThunderstorm over Trivandrum
6.7 Conclusions

7. Summary and Conclusions

Scope for Future Work

References

Preface

Troposphere is the lowest layer of the atmosphere, where the temperature decreases with increasing altitude. The prime objective of this study is the characterization of various atmospheric processes ranging from a few kilometers to a few hundreds of kilometers taking place within this layer over the southern part of Indian sub-continent with the aid of numerical atmospheric models and observational data. With a view to broadening our understanding on some of the interesting atmospheric events such as Sea/Land Breeze Circulation (SLBC), Mountain Waves, and Thunderstorms, hitherto not attempted very much in an analytical and quantitative sense by many researchers the world over, the contents of this thesis titled “Studies on Lower Atmospheric Processes over South India using Numerical Atmospheric Models and Experiments” are organized into seven chapters.

The introductory chapter provides a curtain raiser to the research theme of the thesis and describes the historical background in detail. A summary on some significant results obtained from previous studies and field campaigns over the study domain are detailed in this chapter. The study presented in thesis makes use of two numerical atmospheric models, namely: High-resolution Regional Model (HRM) and Advanced Regional Prediction System (ARPS). Various scientific and technical details on these models form the gist of Chapter 2. With a view to investigating the performance of these models over the study domain and to evaluating their bias in various meteorological fields, if any, chapter 3 of the thesis is aimed at addressing the general queries on the credibility of the model simulated forecast fields.

SLBC is one of the well-documented and thoroughly investigated lower atmospheric coastal processes. Yet some of its characteristics like typical horizontal extent, time-delay, if any, in onset timings of over sea and over land, the variation in atmospheric/marine boundary layer heights within the SLBC cell, any signature of return current which is the compensatory flow for the thermally driven sea breeze at higher altitudes are poorly documented through modeling studies especially in the western coast of Indian Sub-continent. Chapter 4 of this thesis is the outcome of the HRM model simulations for the western coastline of Indian sub-continent with a special emphasis on SLBC in association with the observational data obtained from a multi-disciplinary ship-borne field experiment, namely - Integrated Campaign for Aerosols gases and Radiation Budget (ICARB).

Mountain waves are often observed over some specific locations where topography induced alterations in wind pattern can take place. Analysis of meteorologically derived parameters like Froude Number and Scor]er parameter give a definite picture of genesis and characteristics of orographic waves over the mountainous regions. Chapter 5 of this thesis is aimed at studying the genesis and characteristics of mountain waves over the western ghats of the Indian sub-continent with the aid of HRM simulations. Analysis of Froude Number and Scorer Parameter over the western ghats for different seasons indicate enhancement of mountain wave activities in terms of orographic lifting and wave breaking during summer monsoon as against the wave blocking activities in the lower layers with suppressed orographic lifting observed in the transition periods of pre- and post-monsoon.

Both SLBC and Mountain waves can act as a triggering mechanism for locally induced thunderstorms. With a view to studying the thunderstorm processes over the east and west coast of Indian sub-continent, two stations like Sriharikota and Trivandrum respectively on each coasts are selected. Precursor analyses are carried out using the meteorological parameters existing over these locations and the thunderstorm characteristics are studied with the aid of ARPS model simulations. Chapter 6 of this thesis gives the gist of the prediction of thunderstorm over Sriharikota and the post analysis of a thunderstorm event over Trivandrum. The main findings from the study and the scope of future work are summarized in Chapter 7.

CHAPTER 1 INTRODUCTION

1.1 General Introduction

Atmosphere is a mixture of gases (fluid) and Meteorology is classical Newtonian Physics applied to the atmosphere. Consequently, equations and concepts in meteorology are similar to those in physics or engineering, although the jargon and conventions might look different (Stull, 2002). The basic idea of Numerical Atmospheric Modelling is to sample the state of the fluid at a given time and make use of the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future. An atmospheric model is a mathematical model constructed around the full set of dynamical equations, which govern atmospheric motions, and supplements these equations with optional parameterizations. Different parameterizations include turbulent diffusion, solar and terrestrial radiation, moist processes including the formation and interaction of clouds and precipitating liquid and ice hydrometeors, sensible and latent heat exchange, multiple soil layers, vegetation canopy, surface water, the kinematic effects of terrain, and cumulus convection. Most of the numerical atmospheric models can discretize equations of motion and to predict microscale phenomena such as tornadoes and boundary layer eddies, sub-microscale turbulent flow over buildings and in a wind tunnel, as well as synoptic, and global flows. The horizontal domain of a model is either global, covering the entire Earth, or regional, covering only part of the Earth.

These numerical models are initialized using observed data from radiosondes, weather satellites, and surface layer meteorological observations. The irregularly spaced observations are processed by Data Assimilation and Objective Analysis methods, which perform quality control and obtain values at locations usable by the model's mathematical algorithms (usually an evenly-spaced grid). The data are then used in the model as the starting point for a forecast. Commonly, the set of equations used is known as the primitive equations. These equations are initialized from the analysis data and rates of change are determined. The rates of change predict the state of the atmosphere a short time into the future. The equations are then applied to this new atmospheric state to find new rates of change, and these new rates of change predict the atmosphere at a yet further time into the future. This time stepping procedure is continually repeated until the solution reaches the desired forecast time. The length of the time step is related to the distance between the points on the computational grid. Time steps for global climate models may be on the order of tens of minutes, while time steps for regional models may be a few seconds to a few minutes. Some of the better known Global and Regional Numerical Models are tabulated in Table 1.1.

Table 1.1 Some of the better known Global and Regional Models

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In the present study two different mesoscale numerical models HRM and ARPS are used to broaden our understanding on the underlying physics of three different lower atmospheric processes like Sea/Land Breeze circulation, Mountain Wave Activity and Thunderstorm phenomena.

1.2 Criticalities involved in the various Atmospheric Events

1.2.1 Scalesof Atmospheric motion

Different atmospheric processes follow different scales of motion and observational data are sparse to study all the atmospheric processes in detail. Some of the atmospheric processes are initiated in some regions and its effects can be felt elsewhere at a distance quite apart, depending on the scale of atmospheric event. The fastest meteorologically significant disturbances are large gravity waves, or the air in jet streaks, and their speed rarely exceeds 100 m/s. In such situations, identification of the cause and problem are very difficult. Numerical Atmospheric Models are very useful in this kind of situations in expanding the knowledge of various atmospheric processes over different regions of the world, especially areas that are data sparse. This section of the chapter deals with a brief description of different scales of atmospheric processes in time and space.

Atmospheric motion can be considered as a superimposition of various motions ranging from small turbulent eddies through thunderstorm and cyclones to large planetary-scale circulations. The spectrum of atmospheric motion extends from the mean free path of molecules at the lower end to the circumference of the earth at the upper end. Table 1.2 gives a detailed classification scheme for meteorological phenomena as a function of their time and space scales. Within such a wide range of motion, meteorologists have attempted to classify atmospheric flow according to the physical scale of the apparently coherent structures that appears generally or intermittently in an atmospheric flow (Tyagi, 2000). Mesoscale can be descriptively defined as having a horizontal scale of the order of few kilometers to several hundred kilometers or so, with a time scale of about 1 to 12 hours. The vertical scale extends from tens of meters to the depth of the troposphere. Mesoscale can also be applied to those atmospheric systems that have a horizontal extent large enough for the hydrostatic approximation to the vertical pressure distribution to be valid, yet small enough for the geostrophic and gradient winds to be inappropriate as approximation to the actual wind circulation above the Atmospheric Boundary Layer (ABL) (Pielke, 1984).

Forces cause winds on all scales. At synoptic scales, changing weather patterns cause fast winds on some days and slower winds on others. Each locale has unique landscape characteristics that modify the winds. These local winds affect our livelihood. Turbulent-scale motion disperses pollen necessary for agriculture, and disperses the pollutant by-products of industry. Mountain waves have significance in aviation safety and cause destructive down slope winds. Winds near the ground can be harnessed to provide electrical power, but extreme winds can damage bridges and tall buildings. Cold air can drain down gentle slopes at night to cause fog hazards to transportation and frost hazards to crops. At boundary-layer scales, surface roughness causes skin drag on the air, which causes the wind to be slower near the ground than aloft. At turbulent scales, gusts can be quantified statistically using standard deviation of wind. Figure 1.1 shows a schematic of three main scales of motion in the atmosphere.

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Figure 1.1 Schematic of three different scales of motion in the atmosphere

Small atmospheric phenomena of horizontal dimension less than about 2 km are frequently isotropic (their horizontal and vertical dimensions are roughly equal). The vertical dimensions of horizontally larger phenomena are generally limited by the depth of troposphere. Time scales τ of most phenomena are approximately proportional to horizontal scales λ as τ ~ aλ, where “a ~ 1s m-1”. For example, microscale turbulence about 1m in diameter might last about a second. Boundary layer thermals of diameter 1 km have circulation lifetimes of about 15min. Thunderstorms of size 10 km might last a couple of hours, while cyclones of size 100 km might last a week.

1.2.2 Rapid Changes in Atmospheric Parameterization

Numerical Atmospheric Models cannot resolve features and /or processes that occur within the confines of a single grid box. Thus, even mesoscale models cannot resolve local flows, swirls, or obstacles. Friction is larger near tall trees and buildings than it is over open areas. Figure 1.2 shows complex flow where turbulent eddies are created around obstacles. Weather models cannot resolve these features realistically at this scale, no matter how high the resolution. As a result, they must account for the total effect of these obstacles and surfaces on the flow with a single number that represents friction within the grid box. The method of accounting for such effects, without directly calculating them, is called parameterization. In other words parameterization is modelling the effects of a process (emulation) rather than modelling the process itself (simulation).

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Figure 1.2 Complex flow around the obstacles like buildings

Some of the many near surface physical processes that are typically parameterized are long wave radiation, soil moisture/temperature, vegetation, reflection, convection, evapo-transpiration, cloud processes etc. Figure 1.3 shows some of the many near-surface physical processes that are typically parameterized. The effects of these processes must be parameterized in a model for three main reasons:

1. Computers are not yet powerful enough to directly treat them because the phenomena are either too small or too complex to be resolved numerically,
2. The processes are often not understood well enough to be represented by an equation, and
3. The effects profoundly impact model fields and are crucial to create realistic forecasts.

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Figure 1.3 Image depicting some of the near-surface processes that are typically parameterized in mesoscale models

Even in very high-resolution mesoscale models, there are still other significant meteorological phenomena that can dramatically impact model forecasts and so they must be parameterized.

Many parameterization schemes exist for emulating the significant impacts of convective processes in mesoscale models. Even mesoscale models with high­resolution inner nests that can model convection without the use of parameterization schemes usually have to employ convective parameterization schemes in their outer nests. It is important to note that convective parameterization schemes are designed to reduce the atmospheric stability in the models. The prediction of precipitation is actuallyjust a by-product of the way in which these schemes do this. Consequently, these schemes may not predict the location and timing of the convective precipitation properly.

Even in the very high-resolution innermost nests of mesoscale models, there are still other significant meteorological phenomena that can dramatically impact model forecasts and so they must be parameterized. The use of different parameterization schemes typically has its greatest impact on predictions of sensible weather at the surface. This results from the fact that all sensible weather elements at the surface are inherently mesoscale, or even microscale, and are generally not accounted for by the analysis of synoptic situation. Consequently, simple synoptic forecast rules of thumb become less useful when using mesoscale models. This, in tum, requires forecasters to apply an understanding of physical processes on a case- by-case basis when forecasting mesoscale weather events.

Problems associated with using parameterizations can result from the increasing complexicity of parameterizations and from interactions between parameterization schemes. Unfortunately, forecast errors created by the interaction of parameterization schemes are more difficult to trace than errors resulting from a single scheme.

1.2.3 Computing Requirements

Using high-speed computer to tackle the computational demand, Numerical Weather Prediction (NWP) is the technique used to forecast weather by solving a set of equations within a numerical model that describes the evolution of meteorological variables representing the state of the atmosphere. These variables include temperature, wind, pressure and moisture content. In the model, the overall atmospheric state at any given instant is represented by the values of the variables at systematically arranged points set up within a three-dimensional grid. The larger the set of grid points, the higher the computational demand, the finer the model resolution and the more details in the future state of the atmosphere that can be described.

The past decade has seen significant advances in numerical prediction models, which has led to reliable forecasts of large-scale disturbances up to a week ahead. Several factors have contributed to these advances which include increases in model resolution, improvements in parameterization of physical processes, improvements in data assimilation, particularly with the implementation of 3D and 4D-VAR (Three dimensional and Four dimensional Variational method) and appropriate assimilation of satellite data and the implementation of numerics that are more accurate and stable. However, improved prediction of large-scale features does not necessarily lead to adequate weather forecasts. The need to provide detailed forecasts to the public has led many national weather forecasting centers to run regional models (in addition to global models, in many cases). The advances in computer technology have enabled rapid increases in horizontal and vertical resolutions of both global and regional models.

There is an increasing requirement for operational Centers to provide improved and timely forecasts for high impact (or severe) weather systems. Examples of such systems include heavy rainfall, thunderstorms, high winds, and tropical cyclone landfall. The prediction of such events will place increasing reliance on high­resolution mesoscale NWP systems. Such systems will require improvements across all components of a state-of-the-art system, namely model numerics, parameterization of physical processes and data assimilation.

Increasing resolution places increasing demands on computing resources because the model must calculate values for more grid points. When we divide the distance between model grid points by three, the number of grid points over the same increases 9-fold (Figure 1.4). Not only that, but as we decrease the grid-point spacing, we typically decrease the length of time between intermediate forecast steps. As a result, additional intermediate forecast steps are required to produce same length forecast. Nonetheless, higher resolution is worth the additional computational demands for many reasons, including improving a model’s ability to represent terrain. This, in turn, affects how accurately the model can predict terrain induced or terrain enhanced meteorological phenomena.

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Figure 1.4 Changing the grid distance from 27 km to 9 km, increases the grid points over the same area by 9-fold, and demands for high computations

The move towards higher horizontal and vertical resolutions will require implementation of non-hydrostatic formulation. The use of semi-Lagrangian formulation in many operational models has already resulted in the implementation of the non-hydrostatic formulation. The choice of vertical coordinates will also need to be revised, as the commonly used sigma coordinates will not be adequate at high resolutions. Further refinements in the numerics will be needed if the mesoscale models are applied to transport problems and air quality applications.

Even if a reasonable mesoscale analysis can be obtained, imbalances between the dynamic and thermodynamic fields can lead to model spin up which could degrade, for example, short-term precipitation forecasts, a key requirement for mesoscale systems. Finally, it is clear that high impact weather systems such as thunderstorms, heavy precipitation or tropical cyclones have a high level of uncertainty associated with them. It is therefore more appropriate to provide forecasts of these features in terms of probabilities rather than purely deterministic forecasts. This is already happening in medium range prediction with the implementation of ensemble prediction systems (EPS) at most operational Centers. The move towards high-resolution mesoscale prediction systems will also lead to the development of mesoscale EPS. This will require further research on development of appropriate means of generating initial perturbations and accounting for model uncertainties.

1.2.4 Dissemination of Forecast Products to the Users Community

Model forecast, if it is providing as such to the public, it is difficult for them to interpret different weather phenomena from the products. It is therefore important to provide the user-friendly inputs to the layman community. Most of the Operational Centers in the world provide general weather and weather forecast for major towns and tourist places, weather warning for strong winds and rough sea, severe weather warning like tropical cyclone and storm. In addition to this Aviation meteorology forecast and Marine meteorology forecasts are also provided by the Operational centers. Seismological information like earthquake, Tsunami, recent Earthquake charts etc are also provided to the public. Figure 1.5 shows how the final product is disseminated to the public.

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Information to the Public

Figure 1.5 Dissemination of the Model product to the public

Dissemination of model product to the public involves several steps including human and computational efforts. As the first step, observations (in situ observations, remote sensing, etc) are assimilated to the numerical models, and the model forecast fields are analyzed by a team of Analysts and the information is passed to the public through media.

1.3 Examples of Some Mesoscale Atmospheric Processes

Mesoscale systems may not be evident on synoptic charts. In mesoscale systems vertical motions may be as significant as the horizontal; and Coriolis force has little effect, due to the short time period, or to the over-riding magnitude of other forces. These mesoscale systems can be grouped in three broad classifications: Thermal Systems, Wave Systems and Orographic Systems. Some of the mesoscale systems are Density or gravity currents, Tropical Cyclones, Thunderstorms, Squall lines, Tornadoes, Sea/Land Breezes, Gravity or buoyancy waves, Mountain waves, Slope and Valley winds etc. (http://www.auf.asn.au/meteorology/section7.html).

1.3.1 Density or Gravity current

A density or gravity current is formed whenever denser air intrudes into, and displaces, less dense air, and is usually flowing along the surface. Density current motion is dependent on dynamic pressure, hydrostatic pressure and surface friction; which in turn are dependent on the height of the intrusion and the relative densities. The speed is also a function of the ambient wind flow.

1.3.2 Tropical Cyclones

Tropical cyclones form only in very moist air in ocean regions where surface water temperature exceeds 26°C, generally in latitudes 5° to 20° either side of equator. Coriolis effect within 5° of the equator is too weak to develop the initial vorticity and sea surface temperatures are too low at latitudes higher than 20°. To be named as a tropical cyclone, the storm must have sustained wind speeds exceeding 33 knots (16.5 m/s), if less than 34 knots (17 m/s) then it is a tropical depression. In the eastern Pacific and the Atlantic the tropical cyclone would be named as a tropical storm for wind speed between 34 and 62 knots then upgraded to hurricane status when the sustained wind speed exceeded 62 knots then downgraded back to tropical storm when it weakened.

1.3.3 Thunderstorms

A thunderstorm is merely a storm containing lightning and thunder. Sometimes a thunderstorm produces gusty surface winds with heavy rain and hail. The birth of a thunderstorm occurs when warm, humid air rises in a conditionally unstable environment. The trigger needed to start air moving upward may be the unequal heating of the surface, the effect of terrain, or the lifting of warm air along a frontal zone. Diverging upper-level winds, coupled with converging surface winds and rising air, also provide a favorable condition for thunderstorm development. Usually, several of these mechanisms work together to generate severe thunderstorms.

1.3.4 Squall Lines

The Squall line forms as a line of thunderstorms. Sometimes they are right along a cold front but they also form in the warm air 100 to 300 km out ahead of it. The line of storms may extend over 1000 km, with huge thunderstorms causing severe weather over much of its length.

1.3.5 Tornadoes

A tornado is a rapidly rotating narrow air column extending from the updraft base of a Cumulonimbus to the ground. Intense tornadoes usually develop from areas of rotation inside thunderstorms. One theory is that the horizontal vortices produced by the low level shear are tilted upward by the updraft inflow initiating the rotation within the cell, which develops into a mesocyclone. The vortex, deriving its energy from the latent heat of condensation released from the warm moist inflow, spins at perhaps 15 m/s, accelerating if the column contracts. Another theory is that the tornado forms when a smaller, more rapidly rotating updraft causes part of the storm base to lower and forming a rotating wall cloud from which a condensation funnel cloud appears which may or may not, reach the ground. The funnel is usually located on the edge of the storm's main updraft, close to the downdraft.

1.3.6 Sea/Land Breezes

Sea (and land) breezes are caused by unequal heating and cooling of adjacent land and sea surfaces. A sea breeze is one that blows from the sea to the land in consequence of this differential heating. During the day, solar radiation causes the land surface to become warmer than the sea surface. Therefore the contrast between land and sea surface temperatures becomes considerable during, the day, being greatest around mid-afternoon. Without going into the mechanics of the matter, suffice it to say that the warm air rises over the land surface and a local circulation commences, with cool air from the sea being drawn in over the land. At the same time the ascending air returns seaward, known as the upper return current (Figure 1.6).

At night, the land cools off quicker than the ocean due to differences in their specific heat values, which forces the dying of the daytime sea breeze. If the land cools below that of the adjacent, sea surface temperature the pressure over the water will be lower than that of the land, setting up a land breeze as long as the environmental surface wind pattern is not strong enough to oppose it. If there is sufficient moisture and instability available, the land breeze can cause showers or even thunderstorms, over the water.

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Figure 1.6 Schematic of Sea Breeze circulation

1.3.7 Gravity or Buoyancy waves

Wave motion is the basic mechanism by which local disturbances are transferred from one part of the atmosphere to another, without net mass transport. Gravity, or buoyancy waves, is pressure waves generated by disturbances within the atmosphere, where the restoring forces (potential energy) for the wave motion is provided by buoyancy and gravity, rather than compression/expansion as in higher frequency acoustic waves. The inertia (kinetic energy) is provided by mass, i.e. an air parcel, vertically displaced by a disturbance, will be acted on by gravity because its density differs from its environment. The potential energy of displacement is converted to kinetic energy when buoyancy returns the parcel to its original level. However kinetic energy reaches a maximum at its original position, thus the parcel overshoots that position and once again is returned by the restoring force of buoyancy and the air parcel tends to oscillate around its undisturbed position, at a typical frequency of 5 to 10 minutes. If successive parcels of air are subject to displacement then a gravity wave is generated in the direction of propagation.

Gravity waves can be external waves or internal waves. External waves are those propagating on a discontinuous surface such as an inversion; or, in regions where the gradient is strong enough to guide the propagation in a direction perpendicular to the gradient. Ocean waves are external gravity waves. Internal waves propagate horizontally or obliquely to the density strata and transport energy to the upper atmosphere, producing clear air turbulence.

1.3.8 Mountain/Lee Waves

In meteorology, lee waves, are atmospheric standing waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots. They are periodic changes of atmospheric pressure, temperature and stability in a current of air caused by vertical displacement, for example orographic lift when the wind blows over a mountain or mountain range. They can also be caused by the surface wind blowing over a plateau or even by upper winds deflected over a thermal updraft. Figure 1.7 shows the schematic of mountain waves and associated characteristics.

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Figure 1.7 Schematic of mountain waves and associated characteristics

The vertical motion forces periodic changes in speed and direction of the air within this air current. They always occur in groups on the lee side of the terrain that triggers them. Usually a turbulent horizontal vortex is generated around the first trough, the so-called rotor. The strongest lee waves are produced when the lapse rate shows a stable layer above the obstruction, with an unstable layer above and below.

1.3.9 Slope and Valley winds

Valleys tend to develop their own air circulation, somewhat independent of the ambient wind overflow, and having a tendency to flow up or down the valley regardless of the prevailing wind direction. Solar heating of the valley slopes modifies this circulation.

Anabatic winds form during the day when hillside slopes are heated more than the valley floor. The differential heating of contact air causes air to flow upslope. Wind speeds of 5m/s plus may be achieved. Katabatic winds normally form in the evening, the result of re-radiative cooling of upper slopes lowering the temperature of air in contact with it and the colder, denser air sinking rapidly down-slope. In some circumstances katabatic winds can grow to strong breeze force during the night but cease with morning warming.

1.4 History ofNumerical Weather Prediction (NWP)

The primitive equations for fluid mechanics were first formulated in Germany by H. von Helmholtz in 1888. About a decade later Vilhelm Bjerknes in Norway suggested that these same equations could be used for the atmosphere, and he proposed weather forecasting as a deterministic initial value problem based on the laws of physics (Bjerknes, 1904).

In 1922, L.F. Richardson in England published the first experimental numerical weather forecast - made by solving the primitive equations with mechanical desk calculators. Richardson (1922) first attempted a numerical solution of atmospheric flow solutions by hand!! The resulting prediction was highly unrealistic (due to Courant-Friedrich-Levy (CFL) problems!). John von Neumann, a physicist at Princeston University’s Institute for Advanced Studies, and Vladimir Zworykin, an electronics scientist at Princeston Laboratories and key inventor of television, proposed in 1945 to initiate NWP using recently invented electronic computers. Von Neumann brought together a group of theoretical meteorologist, including Carl Rossby, Arnt Eliassen, Jule Charney and George Platzman. Eventually, they realized the necessity to simplify the full primitive equations in order to focus their limited computer power on the long waves of the general circulation.

The first successful numerical prediction was performed in 1950 by a team composed of the American meteorologists Jule Charney, Philip Thompson, Larry Gates, and Norwegian meteorologist Ragnar Fjortoft and applied mathematician John von Neumann, using the ENIAC (Electronic Numerical Integrator and Computer) digital computer. They used a simplified form of atmospheric dynamics based on the barotropic vorticity equation. This simplification greatly reduced demands on computer time and memory, so that the computations could be performed on the relatively primitive computers available at the time. Later models used more complete equations for atmospheric dynamics and thermodynamics. In 2002, 640- processor node Earth Simulator is introduced. Each node consists of eight vector processors connected via a high-speed network; having a peak performance of 8 Gig flops per processor. A brief history ofNWP is shown in Figure 1.8.

Abbildung in dieser Leseprobe nicht enthalten

Figure 1.8 Brief History of NWP (http://www.met.rdg.ac.uk/cag/courses)

1.5 PotentialsofAtmospheric Model

While the atmospheric motion can be classified into various scales, their own characteristics and underlying physics also form an interesting domain to researchers. Since it is virtually impossible to get observational data that can cover all the scales of atmospheric motion we have to resort to NWP to broaden our understanding of the various atmospheric phenomena. A mesoscale model is a NWP model with sufficiently high horizontal and vertical resolution to forecast mesoscale weather phenomena. These phenomena are often forced by topography or coastlines, or are related to convection. One frequently depends on guidance from mesoscale models, particularly in tactical situations where real-time weather observations are sparse or nonexistent. The complex mathematical representations of physical laws of motion and conservation of energy that govern the atmosphere make up the core of NWP. These equations are known as forecast or prognostic equations, because they predict what will happen in the future. The variables in these equations represent different aspects of weather (for example, wind, temperature, pressure, etc.). Since these equations govern how the variables change with time, these can be solved for a later time and obtain new values of those variables, if the initial conditions of the atmosphere are known.

Early operational weather models could resolve only synoptic scale features. As computational resources have increased, it has become feasible to model and predict mesoscale weather phenomena in a timely manner. Predicting mesoscale weather features requires much finer model resolution as well as some other significant changes in the traditional synoptic and global modelling approaches. Mesoscale models can provide extra information that helps forecasters to make the best forecast possible, in regions where the environmental conditions change rapidly in time and space. Mesoscale models often produce superior forecasts in coastal and mountainous regions when compared to traditional larger-scale models. They do this by taking advantage of high-resolution topography datasets and detailed sea surface temperature information.

In the last few decades, while the field of atmospheric modelling has seen several new developments in simulation skills and data assimilation techniques; systematic bias and error analysis on the model simulated products have also pulled the attention of several researchers (Richardson 1922; Thompson 1961; Monin 1972; Potter 1973; Marchuk 1974; Oran and Boris 1987; Murray 1989; Gershenfeld 1999, Haltiner and Williams 1980; Daley 1991; Kalnay 2003). To mention a few remarkable developments with respect to the error analysis in model simulations, Keyser and Anthes (1977) developed a method for estimating forecast skill of a weather prediction model and Wilmott et al (1985) expanded upon this technique and developed an index of agreement between observation and simulation. In a systematic error analysis study of the European Centre for Medium Weather Forecast (ECMWF) model, some of the key features like the evolution of the systematic model errors during the last two decades, characteristics of the growth of systematic errors throughout the forecast including medium range, monthly and seasonal time scales and the sensitivity of systematic model errors of horizontal resolution are addressed (Jung and Tompkins, 2003).

Forecasting requirements for launch vehicle program can be satisfactorily met only with a high-resolution Numerical Weather Prediction (NWP) models. Use of mesoscale models to provide forecasts for local and regional applications is well accepted widely. McQueen et al. (1995) performed mesoscale simulations for the northeastern US and Manobianco et al (1996) using a mesoscale model to provide forecasts for the Kennedy Space Center in Cape Canaveral. Snook et al. (1977) demonstrated the use of mesoscale models during the 1996 summer Olympics at Atlanta. Radhika et.al. (2006) described the skill of mesoscale models for local and regional applications with a special emphasis towards the Indian Space Research Organization's (ISRO) satellite launch station - Sriharikota. In their study, Radhika et.al. (2006) compared the performance of two mesoscale models HRM and MM5 with the observations and showed that with the improvements in the initial conditions to the model produce considerable improvements in model simulations as well. The performance of HRM model in the tropical Indian region is evaluated and well documented during different rocket launches of ISRO (Radhika et. al. (2007, 2006, 2004, 2003a,b)). In order to use HRM to capture different lower atmospheric processes over south India, a systematic statistical error analysis is done for different model simulated parameters with respect to the reanalysis field in different seasons (Rani et al., 2007, andSubrahamanyam et al., 2008).

In the Space Physics Laboratory (SPL) of Vikram Sarabhai Space Centre (VSSC), ISRO, ARPS model has been used in operational mode for predicting winds, wind shears and thunderstorm activity over Sriharikota (13.7°N, 80.3°E), and in parallel as a specific research activity with similar goals to understand and improve the capabilities of the model in a tropical environment (Dolas et al., 2001; Radhika, 2001; Radhika and Dolas, 2001a; 2001b; 2000; Radhika et al., 2002). The sensitivity studies conducted using ARPS model shows that a 15% error in the input wind value causes a change in the 12-hour forecast that is well within the model prediction error margins. For 2-D and 3-D configuration runs using ARPS, topography, soil and vegetation data with fine resolution and near real time observations are essential for forecast studies. Accurate lateral boundary conditions from global regional model are most essential. A suitable choice of moist processes, soil model and surface layer parameterization for the tropics is vital (Radhika, 2001; 1999; Radhika and Raman, 1997).

Continuous observations of different atmospheric events over the globe are not available presently. Even though not continuous, now a days, satellite products are available all over the globe. In such situations, meteorologists can rely only on Numerical Atmospheric Models, which can simulate the future state of the atmosphere provided the initial and lateral boundary conditions are known. Numerical modelling of different atmospheric events help to understand the underlying physics of the events.

Research on a marine fog event that occurred near the Pearl River Estuary region on 26 March 2002 was investigated by FAN Qi et al. (2004) using MM5. The results of the numerical experiment are very consistent with the surface observations, especially in the processes of marine fog formation and evolution. Besides, a series of sensitivity numerical experiments were performed by varying the distribution of land use type and the turbulence exchange process. Dangerous events (floods, spouts, storms etc) connected with atmospheric fronts and cyclones and their cloud systems were selected by Pirnach et al. (2008) for their investigations. They used 3-D numerical diagnostic and prognostic models included dynamic and thermodynamic full equations; kinetic equations for cloud particles. They paid special attention to rotor structure of convective cells and its environments. Rotor features in frontal systems were investigated by calculation of vertical component of vorticity and several components of rotor equations. Numerical modelling of convective cells and widespread cloudiness developed in central Ukraine on August of 2006 when aircrash took place and the passage of a storm on March of 2004 were also considered in detail in their study by Pirnach et al (2008).

Eruption clouds in explosive volcanic eruptions are a kind of free boundary shear flow with very high Reynolds numbers (Re > 108), and their dynamics are governed by the entrainment of ambient air into eruption clouds by turbulent mixing and the density change of eruption clouds accompanied by turbulent mixing. Suzuki and Koyaguchi (2007) developed a numerical pseudo-gas model, which correctly simulates turbulent mixing in and around eruption clouds by employing three­dimensional coordinates, a high-order accuracy scheme, and a fine grid size. Their model has successfully reproduced the quantitative features of turbulent mixing at high Reynolds numbers observed in laboratory experiments as well as fundamental features of the dynamics of eruption clouds. Goncalves et al (2002) investigated the atmospheric scavenging processes taking into consideration a numerical simulation through the model RAMS, the below-cloud scavenging model, local atmospheric conditions and local emissions in the Serra do Mar region in southeastern Brazil. Szeto and Chan (2006) examined the possibility of forecasting severe squalls using a numerical weather prediction model for a squall line case in Hong Kong on 9 May 2005. Gevaerd et al (2006) studied the atmospheric transport of carbon monoxide from the biomass burning, focusing on the role of deep convective systems on the 3D redistribution using the Eulerian approach for CO mixing ratio determination. They carried out the simulation using CATT-BRAMS (Coupled Aerosol and Tracer Transport model to the Brazilian developments on the Regional Atmospheric Modelling System).

Ling and Kimura (2007) numerically investigated the characteristics of stratified flow over an isolated mountain ridge. They solved the two-dimensional model equations; based on the time-dependent Reynolds averaged Navier- Stokes equations using implicit time integration in a fitted body grid arrangement to simulate stratified flow over an isolated ideally bell-shaped mountain. Their simulation results are in good agreement with the existing corresponding analytical and approximate solutions. Their study showed that for atmospheric conditions where non-hydrostatic effects become dominant, the model is able to reproduce typical flow features. A detailed modelling study of a mountain wave PSC event on 25-27 January 2000 over Scandinavia were presented by Fueglistaler et al (2003). Afanasyev and Peltier (2001) performed a series of new analyses of the problem of the evolution of the internal gravity wave field that is excited when a uniformly stratified fluid flows over monochromatic topography and their results demonstrate that upward-propagating waves overturn and break when they reach sufficient amplitude.

To examine the detailed features of urban heat island phenomena, Ashie et al (2007) developed a thermal environmental analysis system, which can simulate the atmospheric environment considering urban effects such as artificial pavement, building height and anthropogenic heat. Ferreira and Miranda (2005) used coupled ocean/atmospheric model to study the interactions between the sea breeze and the coastal up welling over a limited area of the Atlantic Ocean, between SW Europe and Africa near the entrance to the Mediterranean Sea. Jamima and Lakshinarasimhan (2004) studied Sea breeze characteristics around Kalpakkam tropical coastal site using ARPS mesoscale model. Ramis and Romero (1995) conducted a numerical study of the development and structure of the sea breeze in Mallorca using a meso-beta numerical model and the results showed that the model reproduces the main known features of the circulation and new ones appear, which seem to have an appreciable effect on the circulation during the decay of the sea breeze. Wang (2007) examined the thermodynamic structure on top of a numerically simulated severe storm to explain the satellite observed plume formation above thunderstorm anvils.

1.6 Objective of the Thesis

Different atmospheric processes confined within the lowest portion of the earth’s atmosphere where the temperature decreases with altitude, which is known as troposphere, are poorly understood and documented due to the lack of high-resolution observation and stable observational platforms. Observational techniques and field experiments have its own demerits. For example it could not cover large spatial area for long duration. Numerical atmospheric models can supplement these observational data wherever the observations are sparse, provided the models are well validated. In this thesis the underlying physics of three such lower atmospheric processes like Sea/Land Breeze Circulation (SLBC), Mountain wave Activity, and Thunderstorm phenomena are dealt in detail with the aid of two different numerical atmospheric models, High-resolution Regional Model (HRM) and Advanced Regional Prediction System (ARPS) and the model simulations are compared with available observations.

Sea/land breeze circulation and mountain waves are two different lower atmospheric processes, which are produced due to the topographic effects. Sea/land Breeze circulation, which is the direct outcome due to the different heat capacities of land and adjacent sea, whereas the mountain waves are produced due to the heterogeneity over the land surface. Both phenomena can generate circulations having considerable horizontal and vertical extent. The horizontal extent of Sea/land breeze circulation is around 100 km, while its vertical extent is limited to around 1km. The mountain waves which are induced by the changes in the topography can travel hundreds of km in the downwind region, at the same time orography effects can propagate the waves in the vertical which can reach up to an altitude of upper troposphere or lower stratosphere before breaking. Both the sea/land breeze circulation and mountain waves can act as a triggering mechanism for localized thunderstorms, which are short lived and have a maximum horizontal extent of 16 km in the mature stage, and the convection can reach up to the altitude of upper troposphere provided the ambient conditions are favorable for storm development.

1.7 Organization of the thesis

With the objectives and goal of the thesis - confined to the usage of numerical atmospheric models for investigating the lower atmospheric processes, the contents of the thesis are organized into seven chapters. Detailed description of the two different numerical models High-resolution Regional Model (HRM) and Advanced regional Prediction System (ARPS) used for the study is provided in the second chapter of the thesis, entitled “Numerical Atmospheric Models used in the Study”.

Before using the Numerical Models for the simulation study it is customary to carry out the error analysis of the model, and such a bias estimation of the two models used in the present study are described in Chapter 3 of the thesis, entitled “Statistical Assessment of Atmospheric Models used in the Study”.

After validating the models, simulations are used to broaden our current understanding on the underlying physics of different lower atmospheric processes like Sea/Land Breeze Circulation, Mountain wave Activity and Thunderstorms over the peninsular Indian Region.

Chapter 4, entitled “Characterization of Sea/Land Breeze Circulation along the West Coast of Indian subcontinent” is the outcome of the HRM model simulations for the western coastline of Indian sub-continent with a special emphasis on Sea/Land Breeze Circulation (SLBC) in association with the observational data obtained from a multi-disciplinary ship-borne field experiment, namely - Integrated Campaign for Aerosols gases and Radiation Budget (ICARB). Since the winter months and pre­monsoon periods are quite favorable for SLBC, the study is confined to these months. In this study special emphasis is paid towards some of the inherent characteristics of the SLBC e.g., the typical horizontal extent of SLBC over the study domain, time­delay, if any, in onset timings of SLBC over sea and over land, the variation in atmospheric/marine boundary layer heights within the SLBC cell, any signature of return current which is the compensatory flow for the thermally driven sea breeze at higher altitudes, etc. Upper air atmospheric sounding data obtained from Trivandrum and ICARB field experiment are used to support the model simulations.

Chapter 5, entitled “Mountain Wave Activity over the Western Ghats of Indian sub-continent” is aimed at extending the domain of HRM simulations to capture one of the interesting atmospheric processes, namely - Mountain Waves which are often observed over some specific locations where topography induced alterations in wind pattern can take place. This particular study is confined to the Western Ghats of Indian sub-continent with a special emphasis on the genesis and characteristics of mountain waves such as Froude Number and Scorer Parameter in different seasons. Analysis of Froude Number and Scorer Parameter over the Western Ghats for different seasons indicate enhancement of mountain wave activities in terms of orographic lifting and wave breaking during summer monsoon as against the wave blocking activities in the lower layers in the transition periods of pre and post­monsoon.

Chapter 6, entitled “Thunderstorm over East and West coast of Indian sub­continent- A Case Study” provide the gist of ARPS model simulations carried out to study the atmospheric dynamics involved in formation and dissipation of a thunderstorm event. The study is confined to two typical cases of thunderstorms that took place over Sriharikota (SHAR) (East coast) and Trivandrum (West coast). Out of the two locations mentioned above, ARPS is used for the prediction of thunderstorm over SHAR whereas ARPS simulations are used for the post analysis of the thunderstorm features over Trivandrum. The results from this study revealed the fact that ARPS could indeed resolve the thunderstorm event even from a single localized sounding with a potential temperature perturbation of approximately 3K.

Chapter 7 summarizes the important findings of the study and provides the probable scope for future work. In order to bring the model simulations closer to the observational field, HRM and ARPS need to be fine-tuned with a special emphasis on various data assimilation techniques, thereby providing a wide scope for continuation of the simulation studies presented in this thesis.

CHAPTER 2 NUMERICAL ATMOSPHERIC MODELS USED IN THE STUDY

2.1 General Introduction

In the modem world, there is an ever-increasing demand for more accurate weather forecasts. From factories to farms, from satellite launching stations to commerce and industries and even from general public there is a persistent demand for more reliable weather forecasts. In this regard, it is undoubtedly true that all human activities are directly or indirectly affected by the vagaries of weather elements, thereby stressing the importance of accurate weather forecasts. That is why we are not satisfied with only short-range weather forecasts, but there is, more than even before, an ever-growing demand for quite accurate and reliable medium-range and even long-range weather predictions. In most cases, accurate weather forecasting happens to be the ultimate target of atmospheric research. It is also one of the most sophisticated areas in meteorology. It involves a sound knowledge of higher mathematics, physics and other branches of pure science. Stated simply, the objective of Numerical Weather Prediction (NWP) is to predict the future state of the atmospheric circulation from knowledge of its present state by use of the dynamics equations. To fulfill this objective the following information is required: (1) the initial state of the field variables; (2) a closed set of prediction equations relating the field variables; and (3) a method of integrating the equations in time to obtain the future distribution of the field variables. Modern weather forecasting relies heavily on NWP. The word “numerical” is somewhat misleading, for all types of weather forecasting are based on some quantitative data and therefore could fit under this heading. NWP is based on the fact that the gases of the atmosphere obey a number ofknown physical principles. Ideally, these physical laws can be used to predict the future state of the atmosphere, given the current conditions. Still, the large number of variables that must be included when considering the dynamic atmosphere makes this task extremely difficult. To simplify the problem, numerical models are developed that omit some variables by assuming that certain aspects of the atmosphere do not change significantly with time. Though these models do not represent all the real atmospheric processes completely, their usefulness in prediction is well established. The domain of this thesis comprises two atmospheric models - namely, High-resolution Regional Model (HRM) and Advanced Regional Prediction System (ARPS) looking at different prospects of atmospheric events in different scales of motion. This chapter of the thesis is aimed to introduce these models and provide a detailed insight on these models.

2.1.1 What is an 'Atmospheric Model'?

In general, the term 'Model' can be used for simplifications of complexities in nature. These models should be carefully matched with the problems they attempt to solve. An 'Atmospheric Model' can be defined as a hypothesis (frequently in the form of mathematical statements) that describes some process or processes that we think are physically important for the workings of the atmosphere that has physical consistency in the model formulation and the agreement with observation serving to test the hypothesis. This means that any atmospheric model is simply a “box” that represents how we think the atmosphere works. An atmospheric model itself is composed of seven basic mathematical equations representing seven basic variables, which describe the instantaneous state of the atmosphere. Weather forecasts are made by solving the equations of motion for the atmosphere. To be more precise, these are the non-linear partial differential equations of dynamics, thermodynamics, mass continuity equations and moisture conservation. While it is quite difficult to obtain an analytical solution to these governing equations, there are alternate ways to find an appropriate set of solutions, for example:

1. First to find an exact analytical solution to a simplified version of the governing equations: The geostrophic wind in atmosphere is one of the examples of this kind, which represent the case of steady state winds in which coriolis force and the horizontal pressure gradient force balances above the boundary layer where friction can be neglected.
2. Second to conceive a simplified physical model for -which exact equations can be solved: For example, Rossby derived a simplified set of equations for the early computers by modelling the atmosphere as it were as a layer of air surrounding the earth. The initial state of NWP was more or less confined to this method.
3. Third to find an approximate numerical solution to the full governing equations: Modern NWP techniques make use of this method. Here the lull, partial differential equations are solved using finite difference approximations, but only at discrete locations called Grid Points.

2.1.2 Limitations of NWP Models

The physical assumptions adopted and the degree of sophistication of each physical process incorporated differs from model to model. Numerical forecasting is beset with a large number of problems. These problems are related with the need for collecting requisite amount and type of data so that initial condition of the atmosphere may be known. Further, there is the problem of determining and allowing for changes in the conditions at the boundary of the model. The eddies of various sizes and forms produced in the atmosphere also pose problems. If there is any error in the initial stage, the errors go on increasing each time. Another problem inherent in this method of weather prediction is that there are numerous violent weather phenomena, which adversely affect human activity, but they are so small in extent that they cannot be resolved by current operational numerical models. In addition to the inherent problems defined above, the field of atmospheric modelling often comes across with four potential causes of errors in numerical weather prediction:

(1) Round off error: Round off error exists because computers represent numerics by a limited number of binary bits (e.g. 32, 64 or 80 bits). As a result, some real decimal numbers can be only approximately represented in the computer, thereby loosing its exact precision.
(2) Truncation error: While handling some analytical solutions, we often come across few equations, where one parameter is represented as a function of the other and the dependable variable can be of very high order. Incorporation of higher-order terms will surely improve the accuracy but for practical reasons, the numerical forecast can consider only the first few terms in such equations (like Taylor series). Such a series is said to be truncated, meaning that higher order terms are neglected in the real calculations.
(3) Numerical instability: Numerical instabilities can come into picture when the numerical solution rapidly diverges from the true solution or blow up. Truncation error is one of the causes of numerical instabilities. Numerical instability can also occur if the wind speeds are large, the grid size is small, and the time step is large. Taking a small time step can minimize such errors. The time step of any numerical simulation of the atmosphere or ocean is constrained by the Courant-Friedrich-Levy (CFL) criterion. According to this criterion, the time step must be too small for the fastest traveling disturbance to have time to traverse the distance of the grid spacing. Higher-resolution models require shorter time-steps, so that more calculations are needed to simulate climate over the same period.
(4) Dynamical instability: NWP is an initial value problem, where the initial values are based on observed weather conditions. Unfortunately, the observations include instrumentation and sampling errors. The dynamical equations of motion, which are non-linear and very sensitive to initial conditions. Such sensitivity means that substantially different weather forecasts can result from slightly different initial conditions and will lead to dynamical errors.

2.2 Basic Classification of Atmospheric Models

2.2.1 Hydrostatic Approximation

In hydrostatic approximation, it is assumed that the downward weight of the atmosphere is balanced with the upward directed pressure gradient force and the hydrostatic equation is the vertical equation of motion in the absence of vertical acceleration. The term hydrostatic is used because it describes a stationary (static) balance in a fluid (hydro) between pressure pushing up and gravity pulling down.

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