Equation Of Motion PDF

4 CHAPTER 1 EQUATION OF MOTION Index PKC/DYN-P.1 Introduction 1. Dynamic Meteorology. Dynamic Meteorology is concerned with the a...

4 CHAPTER 1 EQUATION OF MOTION Index PKC/DYN-P.1 Introduction 1. Dynamic Meteorology. Dynamic Meteorology is concerned with the atmospheric motions and their time evolution. It starts from pure physical theory and attempts to give a systematic and quantitative description of the composition and physical behaviour of the atmosphere. The main objective of Dynamic Meteorology is to offer the complete explanation in terms of the atmospheric phenomena constituting the weather. This branch of Meteorology employs analytical approaches based upon basic Fluid Dynamics Hydrodynamics is four fundamental laws of conservation, which are:- (a) Law of conservation of Energy. (b) Law of conservation of Momentum. (c) Law of conservation of Mass. (d) Law of conservation of Water Vapour. 2. The air motion is fundamental feature for weather and climate. Understanding of these relies on application of basic concept of classical physics:- (a) Hydrodynamics. It deals with the motion of fluids in relation to the forces acting upon them, as the atmosphere can be regarded as continuous fluid medium. (b) Thermodynamics. It deals with the effect of influence of heat in its different forms on the vertical and horizontal structure of the atmosphere. 3. Fundamental Variables. The distribution of seven (07) basic characteristics / variables can specify the state of the atmosphere at any given time. These variables are as follows:- (a) Pressure (p) (b) Temperature (T) (c) Density (d) Water Vapour / Specific Humidity (q) (e) Horizontal Wind Speed & Direction (u) & (v) (f) Vertical Wind  5 4. Fundamental Equations. The behaviour of these seven variables is governed by a set of Seven equations; Four from the field of Hydrodynamics and Three from the field of Thermodynamics. These equations are:- (a) Hydrodynamics. (i) form of Law of Conservation of Momentum). (ii) Equation of Continuity (mathematical form of Law of Conservation of Mass). (b) Thermodynamics. (i) 1st Law of Thermodynamics (Law of Conservation of Energy). (ii) Ideal Gas Law (Equation of State). (iii) Moisture Continuity Equation (Law of Conservation of Water Vapour). 5. These seven equations and seven variables can explain any motion on any scale (from very small scale features, like Dust Devils, to planetary scale features, like Rossby Waves). However, importance of each term in each equation varies with scale. For example, Radiation in the heat equation is important on the largest scales, but not in the theory of Sea Breeze or short-range weather prediction; while on the other hand, vertical accelerations are important only for small scale problems (e.g., Cloud Dynamics etc.). These variables and their derivatives are assumed to be continuous functions of space and time. 6. These equations are expressed in terms of Partial Differential Equations (PDEs) involving the field variables as dependent variables and time & space as independent variables. This general set of non-linear PDEs governing the motion of the atmosphere is extremely complex and no analytical solutions are known to exist. Hence, various simplifying assumptions are required to solve these. It has been observed that these can be better solved using some of the numerical methods. 7. A particle would remain at its initial position in absence of any other force acting on it an absolute frame of reference would be equal to the vector sum of all the forces (per unit mass) acting on it, i.e.,. 8. However, this law is not valid for moving, or rotating, frames of reference (e.g., with respect to the rotating earth). If the motion (or rotation) of the frame of reference is  6 also taken into account, then some additional forces would also appear, which are Apparent Forces Fundamental and Apparent Forces 9. For a fluid or air mass, two types of forces act on it : Fundamental and Apparent Forces. 10. Fundamental Forces. Motion in a coordinate system fixed in space. These are the forces, which physically work on the body (or mass) or surface of the fluid / air mass. These are:- (a) Pressure Gradient Force. (b) Gravitational Force. (c) Frictional Force (or Friction). 11. Apparent forces. These forces are not due to any property, but appear due to the rotation of the earth only. These forces can be applied to a non-inertial coordinate system also. These forces are:- (a) Centrifugal Force. (b) Coriolis Force. Pressure Gradient Force 12. point P(xo, yo, zo), where the pressure is po (as shown in Fig. 1.1). Fig. 1.1 : X-component of Pressure Gradient Force acting on a fluid element Since Pressure is a (i.e., in YZ- plane) as:-  7 Neglecting the higher order terms in this expansion, the Pressure Force acting on the volume element at is:- where A -ve sign signifies the inward direction of force / pressure, i.e., against x-axis). Similarly, the force acting on the volume element at wall will be:- 13. Since the net Pressure Force acting on the volume in x-direction will be written as the algebraic sum of these forces, it can be written as:- m - Thus, the x-component of the Pressure Gradient Force per unit mass is:- 14. Similarly, it can easily be shown that the y- and z- components of the Pressure Gradient Force per unit mass are:- and Hence, the total (i.e., three dimensional) Pressure Gradient Force per unit mass will be:- (1.1) 15. It is important to note that this force is proportional to the Gradient of the Pressure field, not to the Pressure itself. Gravitational Force 16. mathematically as:- (1.2(a)) G M m r z the vertical coordinate, which is defined as:-  8 a z Fg But z 0 the Coriolis Force would deflect the particle towards Equator (i.e., in ve y- direction) and Upwards (i.e., +ve z-direction). (ii) u 0 Coriolis Force would deflect the particle towards East (i.e., in +ve x- direction). (c) Vertical Component. For pure vertical flow, u = 0; v = 0 and only vertical component of motion will be available. So, the Coriolis Force would become:- which means that the force would have only one component, which would deflect it in zonal direction, as can be seen for both the cases:- (i) Upward Motion. w>0 the Coriolis Force would deflect the particle towards West (i.e., in -ve x- direction). (ii) Downward Motion. w