1	Lecture 2: Basic Conservation Laws Conservation of Momentum Conservation of Mass Conservation of Energy Scaling Analysis
2	Conservation Law of Momentum The conservation law for momentum (Newton’s second law of motion) relates the rate of change of the absolute momentum following the motion in an inertial reference frame to the sum of the forces acting on the fluid.
3	Apply the Law to a Rotating Coordinate For most applications in meteorology it is desirable to refer the motion to a reference frame rotating with the earth. Transformation of the momentum equation to a rotating coordinate system requires a relationship between the total derivative of a vector in an inertial reference frame and the corresponding total derivative in a rotating system.
4	Total Derivative in a Rotating Coordinate
5	Newton’s 2^nd Law in a Rotating Frame
6	Momentum Conservation in a Rotating Frame
7	Momentum Equation on a Spherical Coordinate
8	Rate of Change of U
9	Coriolis Force (for n-s motion)
10	Momentum Eq. on Spherical Coordinate
11	Momentum Eq. on Spherical Coordinate
12	Scaling Analysis Scale analysis, or scaling, is a convenient technique for estimating the magnitudes of various terms in the governing equations for a particular type of motion. In scaling, typical expected values of the following quantities are specified: (1) magnitudes of the field variables; (2) amplitudes of fluctuations in the field variables; (3) the characteristic length, depth, and time scales on which these fluctuations occur. These typical values are then used to compare the magnitudes of various terms in the governing equations.
13	Scales of Atmospheric Motions
14	Scaling for Synoptic-Scale Motion The complete set of the momentum equations describe all scales of atmospheric motions. We need to simplify the equation for synoptic-scale motions. We need to use the following characteristic scales of the field variables for mid-latitude synoptic systems:
15	Pressure Gradients Pressure Gradients The pressure gradient force initiates movement of atmospheric mass, wind, from areas of higher to areas of lower pressure Horizontal Pressure Gradients Typically only small gradients exist across large spatial scales (1mb/100km) Smaller scale weather features, such as hurricanes and tornadoes, display larger pressure gradients across small areas (1mb/6km) Vertical Pressure Gradients Average vertical pressure gradients are usually greater than extreme examples of horizontal pressure gradients as pressure always decreases with altitude (1mb/10m)
16	Scaling Results for the Horizontal Momentum Equations
17	Geostrophic Approximation, Balance, Wind
18	Weather Prediction In order to obtain prediction equations, it is necessary to retain the acceleration term in the momentum equations. The geostrophic balance make the weather prognosis (prediction) difficult because acceleration is given by the small difference between two large terms. A small error in measurement of either velocity or pressure gradient will lead to very large errors in estimating the acceleration.
19	Rossby Number Rossby number is a non-dimensional measure of the magnitude of the acceleration compared to the Coriolis force: The smaller the Rossby number, the better the geostrophic balance can be used. Rossby number measure the relative importnace of the inertial term and the Coriolis term. This number is about O(0.1) for Synoptic weather and about O(1) for ocean.
20	Scaling Analysis for Vertical Momentum Eq.
21	Hydrostatic Balance The acceleration term is several orders smaller than the hydrostatic balance terms. è Therefore, for synoptic scale motions, vertical accelerations are negligible and the vertical velocity cannot be determined from the vertical momentum equation.
22	Vertical Motions For synoptic-scale motions, the vertical velocity component is typically of the order of a few centimeters per second. Routine meteorological soundings, however, only give the wind speed to an accuracy of about a meter per second. Thus, in general the vertical velocity is not measured directly but must be inferred from the fields that are measured directly. Two commonly used methods for inferring the vertical motion field are (1) the kinematic method, based on the equation of continuity, and (2) the adiabatic method, based on the thermodynamic energy equation.
23	Primitive Equations
24	The Kinematic Method We can integrate the continuity equation in the vertical to get the vertical velocity.
25	The Adiabatic Method The adiabatic method is not so sensitive to errors in the measured horizontal velocities, is based on the thermodynamic energy equation.
26	Conservation of Mass The mathematical relationship that expresses conservation of mass for a fluid is called the continuity equation.
27	Mass Divergence Form (Eulerian View)
28	Velocity Divergence Form (Lagragian View)
29	Scaling Analysis of Continuity Eq.
30	Meaning of the Scaled Continuity Eq.
31	The First Law of Thermodynamics This law states that (1) heat is a form of energy that (2) its conversion into other forms of energy is such that total energy is conserved. The change in the internal energy of a system is equal to the heat added to the system minus the work down by the system:
32	"Therefore" Therefore, when heat is added to a gas, there will be some combination of an expansion of the gas (i.e. the work) and an increase in its temperature (i.e. the increase in internal energy):
33	Heat and Temperature Heat and temperature are both related to the internal kinetic energy of air molecules, and therefore can be related to each other in the following way:
34	Specific Heat
35	Apply the Energy Conservation to a Control Volume The first law of thermodynamics is usually derived by considering a system in thermodynamic equilibrium, that is, a system that is initially at rest and after exchanging heat with its surroundings and doing work on the surroundings is again at rest. A Lagrangian control volume consisting of a specified mass of fluid may be regarded as a thermodynamic system. However, unless the fluid is at rest, it will not be in thermodynamic equilibrium. Nevertheless, the first law of thermodynamics still applies. The thermodynamic energy of the control volume is considered to consist of the sum of the internal energy (due to the kinetic energy of the individual molecules) and the kinetic energy due to the macroscopic motion of the fluid. The rate of change of this total thermodynamic energy is equal to the rate of diabatic heating plus the rate at which work is done on the fluid parcel by external forces.
36	Total Thermodynamic Energy If we let e designate the internal energy per unit mass, then the total thermodynamic energy contained in a Lagrangian fluid element of density ρ and volume δV is
37	External Forces The external forces that act on a fluid element may be divided into surface forces, such as pressure and viscosity, and body forces, such as gravity or the Coriolis force. However, because the Coriolis force is perpendicular to the velocity vector, it can do no work.
38	Work done by Pressure
39	Thermodynamic Eq. for a Control Volume
40	Final Form of the Thermodynamic Eq. After many derivations, this is the usual form of the thermodynamic energy equation. The second term on the left, representing the rate of working by the fluid system (per unit mass), represents a conversion between thermal and mechanical energy. This conversion process enables the solar heat energy to drive the motions of the atmosphere.
41	Entropy Form of Energy Eq.
42	Potential Temperature (θ) For an ideal gas undergoing an adiabatic process (i.e., a reversible process in which no heat is exchanged with the surroundings; J=0), the first law of thermodynamics can be written in differential form as:
43	Static Stability
44	Scaling of the Thermodynamic Eq.
45	Pressure Gradients Pressure Gradients The pressure gradient force initiates movement of atmospheric mass, wind, from areas of higher to areas of lower pressure Horizontal Pressure Gradients Typically only small gradients exist across large spatial scales (1mb/100km) Smaller scale weather features, such as hurricanes and tornadoes, display larger pressure gradients across small areas (1mb/6km) Vertical Pressure Gradients Average vertical pressure gradients are usually greater than extreme examples of horizontal pressure gradients as pressure always decreases with altitude (1mb/10m)
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47	Temperature Tendency Equation Term A: Diabatic Heating Term B: Horizontal Advection Term C: Adiabatic Effects
48	Primitive Equations The scaling analyses results in a set of approximate equations that describe the conservation of momentum, mass, and energy for the atmosphere. These sets of equations are called the primitive equations, which are very close to the original equations are used for numerical weather prediction. The primitive equations does not describe the moist process and are for a dry atmosphere.
49	Primitive Equations