1	Lecture 1: Introduction and Review Review of fundamental mathematical tools Fundamental and apparent forces
2	Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics is the study of the causes of motion.
3	Basic Conservation Laws Atmospheric motions are governed by three fundamental physical principles: conservation of mass (continuity equation) conservation of momentum (Newton’s 2^nd law of motion) conservation of energy (1^st law of thermodynamics)
4	Control Volume The mathematical relations that express these laws may be derived by considering the budgets of mass, momentum, and energy for an infinitesimal control volume in the fluid. Two types of control volume are commonly used in fluid dynamics: Eulerian and Lagrangian.
5	Eulerian View of Control Volume In the Eulerian frame of reference the control volume consists of a parallelepiped of sides δx, δy, δz, whose position is fixed relative to the coordinate axes. Mass, momentum, and energy budgets will depend on fluxes caused by the flow of fluid through the boundaries of the control volume.
6	Lagrangian View of Control Volume In the Lagrangian frame, the control volume consists of an infinitesimal mass of “tagged” fluid particles. The control volume moves about following the motion of the fluid, always containing the same fluid particles.
7	Eulerian and Lagrangian Views Eulerian view of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows. Lagrangian view of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time.
8	Linking Lagrangian and Eulerian Views The conservation laws to be derived contain expressions for the rates of change of density, momentum, and thermodynamic energy following the motion of particular fluid parcels. è The Lagrangian frame is particularly useful for deriving conservation laws. However, observations are usually taken at fixed locations. è The conservation laws are often applied in the Eulerian frame.
9	Linking Total Derivative to Local Derivative
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11	Example Q: The surface pressure decreases by 3 hPa per 180 km in the eastward direction. A ship steaming eastward at 10 km/h measures a pressure fall of 1 hPa per 3 h. What is the pressure change on an island that the ship is passing? A: The pressure change on the island ( ) can be linked to the pressure change on the ship ( ) in the following way: è
12	Coordinate System A coordinate system is needed to describe the location in space.
13	State Variable Fundamental state variables (A) in the atmosphere (such as temperature, pressure, moisture, geopotential height, and 3-D wind) are function of the independent variables of space (x, y, z) and time (t):
14	Scalar and Vector Many physical quantities in the atmosphere are described entirely in terms of magnitude, known as scalars (such as pressure and temperature). There are other physical quantities (such as 3D-wind or gradient of scalar) are characterized by both magnitude and direction, such quantities are known as vectors. Any description of the fluid atmosphere contains reference to both scalars and vectors. The mathematical descriptions of these quantities are known as vector analysis.
15	Representation of Vector
16	Vector Multiplication Multiplication by a scalar Dot product (scalar product) è scalar Cross product (vector product) è vector
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18	Four Most Important Vector Operations
19	Gradient (Derivative) Operator We will often need to describe both the magnitude and direction of the derivative of a scalar field, by employing a mathematical operator known as the del operator.
20	Curl (Rotor) Operator The curl (or rotor) is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector. The length and direction of the vector characterize the rotation at that point. The curl is a form of differentiation for vector fields. A vector field whose curl is zero is called irrotational.
21	Example
22	Divergence Operator divergence is an operator that measures the magnitude of a vector field’s source or sink at a given point, in terms of a signed scalar. Negative values of divergence is also known as “convergence”.
23	Laplacian Operator The Laplace operator is used in the modeling of wave propagation, heat flow, and fluid mechanics.
24	Fundamental and Apparent Forces Newton’s second law of motion states that the rate of change of momentum (i.e., the acceleration) of an object, as measured relative to coordinates fixed in space, equals the sum of all the forces acting. For atmospheric motions of meteorological interest, the forces that are of primary concern are the pressure gradient force, the gravitational force, and friction. These are the fundamental forces. For a coordinate system rotating with the earth, Newton’s second law may still be applied provided that certain apparent forces, the centrifugal force and the Coriolis force, are included among the forces acting.
25	Inertial and Noninertial Reference Frames In formulating the laws of atmospheric dynamics it is natural to use a geocentric reference frame, that is, a frame of reference at rest with respect to the rotating earth. Newton’s first law of motion states that a mass in uniform motion relative to a coordinate system fixed in space will remain in uniform motion in the absence of any forces. Such motion is referred to as inertial motion; and the fixed reference frame is an inertial, or absolute, frame of reference. It is clear, however, that an object at rest or in uniform motion with respect to the rotating earth is not at rest or in uniform motion relative to a coordinate system fixed in space. Therefore, motion that appears to be inertial motion to an observer in a geocentric reference frame is really accelerated motion. Hence, a geocentric reference frame is a noninertial reference frame. Newton’s laws of motion can only be applied in such a frame if the acceleration of the coordinates is taken into account. The most satisfactory way of including the effects of coordinate acceleration is to introduce “apparent” forces in the statement of Newton’s second law. These apparent forces are the inertial reaction terms that arise because of the coordinate acceleration. For a coordinate system in uniform rotation, two such apparent forces are required: the centrifugal force and the Coriolis force.
26	Convention of Using Cartesian Coordinate X increases toward the east. Y increases toward the north. Z is zero at surface of earth and increases upward.
27	Taylor Series Expansion Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is common practice to use a finite number of terms of the series to approximate a function.
28	Partial Differential
29	Pressure Gradient Force
30	Gravitational Force
31	Frictional (Viscous) Force
32	Centrifugal Force
33	Coriolis Force By adding the “apparent” centrifugal force, we can use Newton’s 2nd law of motion to describe the force balance for an object at rest on the surface of the earth. We need to add an additional “apparent” Coriolis force in the 2^nd law if the object is in motion with respect to the surface of the earth.
34	Coriolis Force (for n-s motion)
35	Coriolis Force (for e-w motion)
36	Summary of Coriolis Force
37	Apparent Gravity (g) and Geopotential (Φ)