AOSS 401, Fall 2013 Lecture 3 Coriolis Force September 10, 2013 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

Class News Ctools site (AOSS 401 001 F13) New Postings to site Syllabus Lectures Homework (and solutions) New Postings to site Homework 1: Due September 12 Read Greene et al. paper: Discussion Tuesday

Weather National Weather Service Weather Underground Model forecasts: Weather Underground NCAR Research Applications Program

Outline Conservation of Momentum Forces Greene et al. Paper Discussion Coriolis Force Centrifugal Force Greene et al. Paper Discussion Should be review. So we are going fast. You have the power to slow me down.

Back to Basics: Newton’s Laws of Motion Law 1: Bodies in motion remain in motion with the same velocity, and bodies at rest remain at rest, unless acted upon by unbalanced forces. Law 2: The rate of change of momentum of a body with time is equal to the vector sum of all forces acting upon the body and is the same direction. Law 3: For every action (force) there is and equal and opposite reaction.

Newton’s Law of Motion Where i represents the different types of forces.

How do we express the forces? In general, we assume the existence of an idealized parcel or “particle” of fluid. We calculate the forces on this idealized parcel. We take the limit of this parcel being infinitesimally small. This yields a continuous, as opposed to discrete, expression of the force. Use the concept of the continuum to extend this notion to the entire fluid domain.

Back to basics: A couple of definitions Newton’s laws assume we have an “inertial” coordinate system; that is, and absolute frame of reference – fixed, absolutely, in space. Velocity is the change in position of a particle (or parcel). It is a vector and can vary either by a change in magnitude (speed) or direction.

Apparent forces: A mathematical approach Non-inertial, non-absolute coordinate system

One coordinate system related to another by:

Two coordinate systems z’ z’ axis is the same as z, and there is rotation of the x’ and y’ axis z y’ y x x’

One coordinate system related to another by: T is time needed to complete rotation.

Acceleration (force) in rotating coordinate system The apparent forces that are proportional to rotation and the velocities in the inertial system (x, y, z) are called the Coriolis forces. The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

The importance of rotation Non-rotating fluid http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_nonrot_MIT.mpg Rotating fluid http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_rotating_MIT.mpg

Apparent forces: A physical approach Coriolis Force http://climateknowledge.org/figures/AOSS401_coriolis.mov

Apparent forces With one coordinate system moving relative to the other, we have the velocity of a particle relative to the coordinate system and the velocity of one coordinate system relative to the other. This velocity of one coordinate system relative to the other leads to apparent forces. They are real, observable forces to the observer in the moving coordinate system. The apparent forces that are proportional to rotation and the velocities in the inertial system (x,y,z) are called the Coriolis forces. The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

Consider angular momentum We will consider the angular momentum of a “particle” of atmosphere to derive the Coriolis force.

Angular momentum? Like momentum, angular momentum is conserved in the absence of torques (forces) which change the angular momentum. This comes from considering the conservation of momentum of a body in constant body rotation in the polar coordinate system. If this seems obscure or is cloudy, need to review a introductory physics text.

Angular speed ω r (radius) v Δv Δθ v

What is the appropriate radius of rotation for a parcel on the surface of the Earth? Ω Direction away from axis of rotation R Earth

Magnitude of R the axis of rotation R=acos(f) Ω R a Φ = latitude Earth

Tangential coordinate system Place a coordinate system on the surface. x = east – west (longitude) y = north-south (latitude) z = local vertical Ω R a Φ Earth

Angle between R and axes Ω Φ R a Φ = latitude Earth

Assume magnitude of vector in direction R Ω Vector of magnitude B R a Φ = latitude Earth

Vertical component Ω z component = Bcos(f) R a Φ = latitude Earth

Meridional component Ω R y component = Bsin(f) a Φ = latitude Earth

Earth’s angular momentum (1) What is the speed of this point due only to the rotation of the Earth? Ω R a Φ = latitude Earth

Earth’s angular momentum (2) Angular momentum is Ω R a Φ = latitude Earth

Earth’s angular momentum (3) Angular momentum due only to rotation of Earth is Ω R a Φ = latitude Earth

Earth’s angular momentum (4) Angular momentum due only to rotation of Earth is Ω R a Φ = latitude Earth

Angular momentum of parcel (1) Assume there is some x velocity, u. Angular momentum associated with this velocity is Ω R a Φ = latitude Earth

Total angular momentum Angular momentum due both to rotation of Earth and relative velocity u is Ω R a Φ = latitude Earth

Displace parcel south (1) (Conservation of angular momentum) Let’s imagine we move our parcel of air south (or north). What happens? Δy Ω R a Φ Earth

Displace parcel south (2) (Conservation of angular momentum) We get some change ΔR Ω R a Φ Earth

Displace parcel south (3) (Conservation of angular momentum) But if angular momentum is conserved, then u must change. Ω R a Φ Earth

Displace parcel south (4) (Conservation of angular momentum) Expand right hand side, ignore squares and higher order difference terms. Expected mathematical knowledge

Displace parcel south (5) (Conservation of angular momentum) For our southward displacement

Displace parcel south (6) (Conservation of angular momentum) Divide by Δt and take the limit Coriolis term (check with previous mathematical derivation … what is the same? What is different?)

Displace parcel south (7) (Conservation of angular momentum) What’s this? “Curvature or metric term.” It takes into account that y curves, it is defined on the surface of the Earth. More later. Remember this is ONLY FOR a NORTH-SOUTH displacement.

Coriolis Force in Three Dimensions Do a similar analysis displacing a parcel upwards and displacing a parcel east and west. This approach of making a small displacement of a parcel, using conversation, and exploring the behavior of the parcel is a common method of analysis. This usually relies on some sort of series approximation; hence, is implicitly linear. Works when we are looking at continuous limits.

Coriolis Force in 3-D

Definition of Coriolis parameter (f) Consider only the horizontal equations (assume w small) For synoptic-scale systems in middle latitudes (weather) first terms are much larger than the second terms and we have

Our momentum equation

Highs and Lows Motion initiated by pressure gradient In Northern Hemisphere velocity is deflected to the right by the Coriolis force Motion initiated by pressure gradient Opposed by viscosity

Where’s the low pressure?

Geostrophic and observed wind 1000 mb (ocean)

Do you notice two “types” of motion? Hurricane Charley Do you notice two “types” of motion?

Our momentum equation Surface Body Apparent This equation is a statement of conservation of momentum. We are more than half-way to forming a set of equations that can be used to describe and predict the motion of the atmosphere! Once we add conservation of mass and energy, we will spend the rest of the course studying what we can learn from these equations.

Our momentum equation Surface Body Apparent Acceleration (change in momentum) Pressure Gradient Force: Initiates Motion Friction/Viscosity: Opposes Motion Gravity: Stratification and buoyancy Coriolis: Modifies Motion

The importance of rotation Non-rotating fluid http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_nonrot_MIT.mpg Rotating fluid http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_rotating_MIT.mpg

Discussion of Greene et al. Paper

End

Apparent forces: A mathematical approach Non-inertial, non-absolute coordinate system

Two coordinate systems Can describe the velocity and forces (acceleration) in either coordinate system. x y z x’ y’ z’

One coordinate system related to another by:

Velocity (x’ direction) So we have the velocity relative to the coordinate system and the velocity of one coordinate system relative to the other. This velocity of one coordinate system relative to the other leads to apparent forces. They are real, observable forces to the observer in the moving coordinate system.

Two coordinate systems z’ z’ axis is the same as z, and there is rotation of the x’ and y’ axis z y’ y x x’

One coordinate system related to another by: T is time needed to complete rotation.

Acceleration (force) in rotating coordinate system The apparent forces that are proportional to rotation and the velocities in the inertial system (x, y, z) are called the Coriolis forces. The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

Derivation of Centrifugal Force

Apparent forces: A physical approach

Circle Basics Arc length ≡ s = rθ ω r (radius) θ Magnitude s = rθ

Centrifugal force: Treatment from Holton ω r (radius) Δv Δθ Magnitude

Angular speed ω r (radius) v Δv Δθ v

Centrifugal force: for our purposes

Now we are going to think about the Earth The preceding was a schematic to think about the centrifugal acceleration problem. Note that the r vector above and below are not the same!

What direction does the Earth’s centrifugal force point? Ω Ω2R R Earth

What direction does gravity point? Ω R a Earth

What direction does the Earth’s centrifugal force point? So there is a component that is in the same coordinate direction as gravity (and local vertical). And there is a component pointing towards the equator We are now explicitly considering a coordinate system tangent to the Earth’s surface. Ω Ω2R R Earth

What direction does the Earth’s centrifugal force point? So there is a component that is in the same coordinate direction as gravity: ~ aΩ2cos2(f) And there is a component pointing towards the equator ~ - aΩ2cos(f)sin(f) Ω Ω2R R Φ = latitude Earth

So we re-define gravity as

What direction does the Earth’s centrifugal force point? And there is a component pointing towards the equator. The Earth has bulged to compensate for the equatorward component. Hence we don’t have to consider the horizontal component explicitly. Ω Ω2R R Earth

Centrifugal force of Earth Vertical component incorporated into re-definition of gravity. Horizontal component does not need to be considered when we consider a coordinate system tangent to the Earth’s surface, because the Earth has bulged to compensate for this force. Hence, centrifugal force does not appear EXPLICITLY in the equations.

That’s all