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AOSS 401, Fall 2013 Lecture 5 Pressure September 17 , 2013

AOSS 401, Fall 2013 Lecture 5 Pressure September 17 , 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 ) Disruption of schedule There will be no class on Thursday, September 19 th

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AOSS 401, Fall 2013 Lecture 5 Pressure September 17 , 2013

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  1. AOSS 401, Fall 2013Lecture 5Pressure September 17, 2013 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

  2. Class News • Ctools site (AOSS 401 001 F13) • Disruption of schedule • There will be no class on Thursday, September 19 th • Class will be remote on Tuesday, September 24th • gotomeeting / • https://www3.gotomeeting.com/join/286347606 • Dial in 1 866 692 4541 // 3066034 • Homework will be posted today • Stay tuned for details

  3. Weather • National Weather Service • Model forecasts: • Weather Underground • Model forecasts: • NCAR Research Applications Program

  4. Dynamic Atmosphere:Extratropical storm systems • Satellite image • Storm system in the Gulf of Alaska • Scale of the motion:3000-5000 km • What are the differences/similarities of these weather systems?

  5. Outline • Any Questions • Material Derivative / Redux • Pressure Coordinates Should be review. So we are going fast. You have the power to slow me down.

  6. Material Derivative Written in terms of the local change leads to advection We will use this again later…

  7. Class Exercise: Gradients and Advection • The temperature at a point 50 km north of a station is three degrees C cooler than at the station. • If the wind is blowing from the north at 50 km h-1 and the air is being heated by radiation at the rate of 1 degree C h-1, what is the local temperature change at the station? • Hints: • You should not need a calculator • Use the definition of the material derivative and of advection

  8. Class Exercise: Gradients and Advection Material Derivative of T Write in terms of localderivative No east-west or vertical velocity Plug in values

  9. Class Exercise: Gradients and Advection Material Derivative of T Write in terms of localderivative No east-west or vertical velocity Plug in values

  10. Class Exercise: Gradients and Advection Material Derivative of T Write in terms of localderivative No east-west or vertical velocity Plug in values

  11. Class Exercise: Gradients and Advection Material Derivative of T Write in terms of localderivative No east-west or vertical velocity Plug in values

  12. Pressure as Vertical Coordinate

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

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

  15. Vertical Momentum Equation • How did I get to this equation? • What are the terms in this equation called?

  16. Vertical Momentum Equation • What if I tell you … for now … • Acceleration is small • Curvature term is small • Coriolis term is small • Viscosity term is small

  17. Vertical Momentum Equation • Scaled momentum equation for large-scale motion • Hydrostatic equation

  18. Vertical StructurePressure as a vertical coordinate

  19. Pressure Coordinates: Why? • From Holton, p2: “The general set of … equations governing the motion of the atmosphere is extremely complex; no general solutions are known to exist. …it is necessary to develop models based on systematic simplification of the fundamental governing equations.” • Two goals of dynamic meteorology: • Understand atmospheric motions (diagnosis) • Predict future atmospheric motions (prognosis) • Use of pressure coordinates simplifies the equations of motion

  20. Some basics of Earth’s atmosphere • Atmosphere is composed of air, which is a mixture of gases, which is treated as an ideal gas, and which below ~ altitude of 1.0 x 105 m (100 km) behaves like a fluid – a continuum. • Hint: Know and use the ideal gas law. • What is a continuum?

  21. Some basics of Earth’s atmosphere atmosphere: depth ~ 1.0 x 105 m Mountain: height ~ 5.0 x 103 m Ocean Land Biosphere Earth: radius ≡ a = 6.37 x 106 m

  22. Some basics of Earth’s atmosphere Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10-3 Earth radius Troposphere: depth ~ 1.0 x 104 m Scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.

  23. Some basics of Earth’s atmosphere Do you know these units? Pressure: mb = millibars hPa=hecto Pascals Troposphere: depth ~ 900 mb Scale analysis tells us that most of the mass of the atmosphere is in the troposphere.

  24. Pressure Units • 1000 mb • 1000 hPa • Pa = Pascal = N/m2 = Newton/m2 • Newton = kg m/s2

  25. Temperature2 Pressure1 Pressure2 Temperature1 (P2-P1)/(x2-x1) height Longitude, our x

  26. Pressure altitude Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart.

  27. Horizontal Momentum Equations • How did I get to this equation? • What are the terms in this equation called? • Where are the advection terms?

  28. Use pressure as a vertical coordinate? • What do we need. • Pressure gradient force in pressure coordinates. • Way to express derivatives in pressure coordinates. • Way to express vertical velocity in pressure coordinates.

  29. Expressing pressure gradient force • We are going to use the hydrostatic relation. • You remember the hydrostatic relation? • Where does the hydrostatic relationship come from?

  30. Integrate in altitude Pressure at height z is force (weight) of air above height z.

  31. Concept of geopotential Define a variable F such that the gradient of F is equal to g. This is called a potential function. We have assumed here that F is a function of only z.

  32. Integrating with height

  33. What is geopotential? • Potential energy that a parcel would have if it was lifted from surface to the height z.

  34. Vertical Structure and Pressure as a Vertical Coordinate • Remember, the geopotential is defined as • Remember your first-semester Physics? • Gravitational Potential Energy:

  35. What is geopotential? • Potential energy that a parcel would have if it was lifted from surface to the height z. • It is analogous to the height of a pressure surface. • We seek to have an analogue for pressure on a height surface, which will be height on a pressure surface.

  36. Linking geopotential to pressure Definition of specific volume Ideal gas law

  37. Remembering some calculus

  38. Define geopotential height(assumption of constant g = g0) Z2-Z1 = ZT ≡ Thickness - is proportional to temperature is often used in weather forecasting to determine, for instance, the rain-snow transition. (We will return to this.)

  39. Vertical Structure and Pressure as a Vertical Coordinate • Integrate from pressure p1 to p2 at heights z1 and z2 • From the definition of geopotential we get thickness and the fact that thickness is proportional to temperature • So, hydrostatic balance and the ideal gas law form the basis for the relationship between and

  40. Calculate the “pressure” gradient

  41. Pressure Gradient Force Change in height on a constant pressure surface Change in pressure on a constant height surface A Physical Perspective What is the force that initiates motion?

  42. Getting pressure gradient in pressure coordinates z Constant pressure p0 Constant pressure p0+Δp x

  43. Getting pressure gradient in pressure coordinates z Constant pressure p0 Constant pressure p0+Δp We have, for instance, ∂p/∂x on a constant z surface in our derivation of the momentum equation. x Δx ((p0+Δp)-p0)/Δx

  44. Getting pressure gradient in pressure coordinates z Constant pressure p0 Constant pressure p0+Δp We can also calculate how pressure changes on on a z surface as we hold x constant. Δz x (p0-(p0 + Δp))/Δz

  45. Getting pressure gradient in pressure coordinates z Constant pressure p0 Constant pressure p0+Δp Which we project onto the x direction by how much z changes with x on the pressure surface. Δz/Δx Δz x (p0-(p0 +Δp))/Δz

  46. Getting pressure gradient in pressure coordinates z Constant pressure p0 Constant pressure p0+Δp Δz x Δx

  47. Implicit that this is on a constant z surface Implicit that this is on a constant p surface

  48. Horizontal pressure gradient force in pressure coordinate is the gradient of geopotential

  49. Our momentum equation(height (z) coordinates) Horizontal momentum equations (u, v), no viscosity

  50. Our horizontal momentum equation(pressure coordinate) Assume no viscosity

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