1. TropicalM. D. Eastin Cylindrical Coordinate System

2. TropicalM. D. Eastin Three Coordinates: Radius (r): “Spokes on a bicycle wheel” r = 0 at the cylinder center (circulation center) r > 0 as one moves away from the center Azimuth (λ) : “Angle along rings around the center” λ = 0 at due North λ > 0 as one moves clockwise around the cylinder center Altitude (z): “Different vertical layers” z = 0 at the lowest level (land/ocean surface) z > 0 as moves upward Cylindrical Coordinates

3. TropicalM. D. Eastin Three Components: Radial Velocity: Flow along the spokes on a bicycle wheel Flow away from (+) or toward (-) the center Azimuthal Velocity: Flow along rings around the center Flow counterclockwise (+) or clockwise (-) around the rings Also called “tangential velocity” Vertical Velocity: Flow between different vertical layers Flow up (+) or down (-) Velocity Components

4. TropicalM. D. Eastin Total Derivative (of any variable): Equations of Motion Total Derivative of a variable Local change with time of a variable Change / Acceleration induced by the radial flow Change / Acceleration induced by the tangential flow Note the “1/r” that is always associated with azimuthal (λ) derivatives Change / Acceleration induced by the vertical flow

5. TropicalM. D. Eastin Radial Equation of Motion:(or Radial Momentum Equation) Equations of Motion Total Derivative of radial velocity Radial Acceleration due to the Coriolis force turning of the tangential flow into the radial direction Radial Acceleration due to the Centrifugal Force Like releasing a ball while riding on a rotating merry-go round Radial Acceleration induced by the Radial Pressure Gradient Force The most dominant horizontal pressure gradient in a TC Increasing pressure with radial will accelerate parcels inward Radial Accelerations induced by friction and/or turbulence

6. TropicalM. D. Eastin Gradient Wind Balance: (from Radial Momentum Equation) Equations of Motion Assumptions:Radial velocity does not change with time (neglect total derivative) Frictional forces are negligible (neglect friction) A balanced state exists between:Coriolis force Centrifugal force Radial pressure gradient force A dominant balance in a tropical cyclone: At large radii, the Coriolis force plays a greater role At small radii, the Centrifugal force plays a greater role

7. TropicalM. D. Eastin Tangential Equation of Motion:(or Tangential Momentum Equation) Equations of Motion Total Derivative of tangential velocity Tangential Acceleration due to the Coriolis Force turning of radial flow into the azimuthal direction Tangential Acceleration due to Centrifugal force Tangential Acceleration induced by the Tangential pressure gradient force The weaker of the two horizontal pressure gradients in a TC Tangential Accelerations induced by friction and/or turbulence

8. TropicalM. D. Eastin Vertical Equation of Motion:(or Vertical Momentum Equation) Equations of Motion Total Derivative of vertical velocity Vertical Acceleration induced by the Vertical pressure gradient force A decrease in pressure with altitude will accelerate parcels upward Vertical Acceleration due to Gravitational forces Gravity accelerates parcels downward (toward the earth’s center of mass) Vertical Accelerations induced by Friction and/or Turbulence

9. TropicalM. D. Eastin Hydrostatic Balance:(from the Vertical Momentum Equation) Equations of Motion Assumptions:Vertical velocity does not change with time (neglect total derivative) Frictional forces are negligible (neglect friction) A balanced state exists between:Gravitational force Vertical pressure gradient force A dominant balance in a tropical cyclone for the large scale

10. TropicalM. D. Eastin Thermal Wind Balance:(from Equation of State and Hydrostatic Balance) Equations of Motion A balanced state exists between:Radial Temperature gradient Vertical (pressure) gradient in tangential wind A dominant balance in a tropical cyclone Similar to the Mid-latitude Jet Stream

11. TropicalM. D. Eastin Vertical Vorticity: (from Radial and Tangential Momentum Equations) Equations of Motion Like in Cartesian Coordinates: Vertical component of the curl of the velocity vector Related to:Shear in the horizontal winds Azimuthal flow Coriolis force Since hurricanes are spinning, vorticity values can be very large when compared to typical values observed mid-latitude synoptic-scale systems

12. TropicalM. D. Eastin Angular Momentum: Equations of Motion The rate at which an air parcel changes azimuth angle Related to:Radius and tangential velocity Coriolis force In the absence of friction, air parcels conserve their angular momentum As air parcels move toward the circulation center, their tangential velocity increases

13. TropicalM. D. Eastin Summary: Three coordinates (r, λ, z) Three velocity components (v r, v t, w) Three equations of motion Gradient balance Hydrostatic Balance Thermal Wind Balance Vertical Vorticity Angular Momentum Cylindrical Coordinate System