Cht' IV The Equation of Motion

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Cht. IV
The Equation of Motion

• Prof. Alison Bridger
• 10/2004

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Overview

• In this chapter we will
• Develop - using Newtons 2nd Law of Motion - an
  equation that in principal - will enable us to
  predict flows.
• Identify the forces which cause air motions.
• Adapt the resulting equation to Earth.
• See how the various forces work.
• See what the resulting equation(s) look like in
  component form.

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Introduction

• In the previous chapter, a flow was assumed.
• The object of this chapter is to develop an
  equation that will allow us to predict (forecast)
  the flow of air - and thus show us where the
  previously-assumed flow came from!
• We will call this the Momentum Equation, or the
  Equation of Motion.

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Introduction

• To solve this in general, we will need to supply
  various external parameters, such as friction and
  thermal forcing.
• In the next chapter, well begin the process of
  solving this equation.
• NOTE that the flow evolution depends upon
  thermodynamic variables too as well as flow
  variables.
• For example,we know (MET 61) that flows are
  forced by pressure gradients.

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Introduction

• These can be generated by differential heating
  patterns in the atmosphere, and will have
  temperature and density gradients associated with
  them.
• By the end of the semester, we will have
  developed additional equations that forecast the
  entire evolution of the flow - dynamic and
  thermodynamic (via the variables V, p, T, ?
  etc.).
• The entire set of equations are called the
  Equations of Motion.

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Newtons 2nd Law of Motion

• Read the full definition on p. 142.
• We have

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Newtons 2nd Law of Motion
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Newtons 2nd Law of Motion

• We sum over all possible forces (Fi) per unit
  mass.
• By solving the resulting equation, we gain
  knowledge of the acceleration of the air parcel.

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Newtons 2nd Law of Motion

• Here are some steps we need to follow
• identify the forces that affect air parcel
  motions
• formulate how these work (mathematically)
• substitute these forms back into the Eqn. above
  (Fma), and then...

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Newtons 2nd Law of Motion

• There is a problem (of course!)
• Earth is rotating - this makes it a
  non-inertial frame of reference.
• However, Newtons 2nd Law is formulated for an
  inertial frame of reference.

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Newtons 2nd Law of Motion

• We therefore have to adapt the equation that we
  have developed to a non-inertial frame of
  reference.
• Well do this after we identify forces.

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Forces...

• We can start by identifying
  2 basic types of forces
• body forces
• affect the entire body of the fluid (not just the
  surface)
• act at a distance

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Forces...

• examples are
• gravitational forces (not the same as gravity!)
• magnetic forces
• electrical forces
• we will ignore the latter two

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Forces...

• surface forces
• affect the surface of a fluid parcel
• caused by contact between fluid parcel
• examples are
• pressure forces
• viscous forces (friction)

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Gravitation

• Newtons Law of Gravitationp.137
• the magnitude of the force is given by
• Ga GmM / r2
• G is the Universal Gravitation Constant
• m and M are the two masses
• r is the distance between the two centers of
  masses

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Gravitation

• We assume M Earths mass (Me) and m mass of
  air parcel (we will eventually set m1).
• For the direction, we assume
• Earth is a sphere
• Earth is not rotating
• Earth is homogeneous (so the center of mass is at
  the Earths geometric center)
• As a result, we can write

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Gravitation
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Gravitation

• For the case m1 (unit mass), Ga ? ga and we have

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Gravitation
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Gravitation

• Here, Ge is Earths Gravitation Constant (GMe).
• And note that the lower case (ga) means per unit
  mass.
• So - the expression above is just one that goes
  into the RHS of our expression of Newtons 2nd
  Law (above).

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Friction

• This topic will be treated in detail in MET 130.
• In dynamics we typically take one of two
  approaches
• ignore friction - its usually a second order
  correction to the important first-order
  dynamics
• assume something very simple for friction

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Friction

• examplewe may set friction to depend linearly on
  the strength of the existing wind, as in

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Friction
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Friction

• Here, ? is a constant that determines the rate of
  decay of V with time due to friction.
• When we wish to allow for friction in our work,
  we will often simply add F to our Eqn. Of
  Motion - you should remember that this stands for
  friction.

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Pressure Gradient Force

• The last force that is important in driving
  motions is the pressure gradient force.
• In many respects, it is the most important force.
• It is important to remember that it is pressure
  gradients that matter - not absolute pressures
  themselves.

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Pressure Gradient Force

• Fluid parcels experience pressure forces due to
  contact with surrounding parcels.
• When these forces are spatially variable, the
  parcel will experience a net motion.
• We need to quantify this...