AOSS 401 Fall 2007 Lecture 5 September 17
AOSS 401, Fall 2007 Lecture 5 September 17, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich. edu 734 -647 -3530 Derek Posselt (Room 2517 D, SRB) dposselt@umich. edu 734 -936 -0502
Class News • Posselt office hours: Tues/Thurs AM and after class • Homework 2 due date postponed to this Wednesday • Homework 3 will be assigned this week
Weather • NCAR Research Applications Program – http: //www. rap. ucar. edu/weather/ • National Weather Service – http: //www. nws. noaa. gov/dtx/ • Weather Underground – http: //www. wunderground. com/modelmaps/m aps. asp? model=NAM&domain=US
Outline 1. Review: Material Derivative 2. Conservation of Mass and Scale analysis • • • Lagrangian and Eulerian derivations of continuity equation Scale analysis of the momentum equations Scale analysis of the continuity equation
From last time
Material Derivative Calculate the change in a property of the atmosphere (for example, temperature) over time Δt, following the parcel. Expand the change in temperature in a Taylor series around the temperature at the initial position, ignore second order and higher terms, and take the limit as Δt 0 Lagrangian Eulerian
Conservation of Mass • Conservation of mass leads to another equation; the continuity equation • Continuity Continuous • No holes in a fluid • Another fundamental property of the atmosphere • Need an equation that describes the time rate of change of mass (density)
Remember our particle of atmosphere, our parcel r ≡ density = mass per unit volume (DV) Dz DV = Dx. Dy. Dz m = r. Dx. Dy. Dz ------------------- Dy Dx p ≡ pressure = force per unit area acting on the particle of atmosphere
The Eulerian point of view our parcel is a fixed volume and the fluid flows through it. Dz Dy Dx
Introduce mass flux, ρu (x, y, z) Dz • ρu = mass flux at (x, y, z) in the x direction. . Dy Dx x • Flux is mass per unit time per area • Mass flux out =
Introduce mass flux, ρu (x, y, z) Dz • ρu = mass flux at (x, y, z) in the x direction. . Dy Dx x • Flux is mass per unit time per area • Mass flux in =
What is the change of mass inside the fixed volume? The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area) Mass out right (downstream) face Mass in left (upstream) face
Extend to 3 -Dimensions The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area) Note: this is change in mass per unit volume. Recognizing the definition of divergence
Eulerian Form of the Continuity Equation Dz In the Eulerian point of view, our parcel is a fixed volume and the fluid flows through it. Dy Dx
The Lagrangian point of view is that the parcel is moving. Dz Dy And it changes shape… Dx
In-Class Exercise: Derive the Lagrangian Form • Remember, we can write the continuity equation • Use the chain rule (e. g. , to go from the above equation to )
Lagrangian Form of the Continuity Equation The change in mass (density) following the motion is equal to the divergence Convergence = increase in density (compression) Divergence = decrease in density (expansion)
Our System of Equations We have u, v, w, ρ, p which depend on (x, y, z, t). We need one more equation for the time rate of change of pressure… (Wednesday)
Scale Analysis • Remember, we want to solve the system of equations that describes the atmosphere so that we can – understand how the atmosphere works – predict the motion and state of the atmosphere • Scale analysis simplify the equations • Identify which processes are most important
Our System of Equations (so far…)
Consider x and y components of the momentum equations Remember the units—each term must have units of acceleration m/sec 2 or L/t 2
Would like to define scales in terms of wind, pressure, and density distance = rate x time Estimate time as distance/(some average rate) L ≡ some characteristic distance U ≡ some characteristic speed Characteristic time ≡ L/U
So for our equations D ( )/Dt can be characterized by 1/(L/U)
Let us define: acceleration
What are the scales of the terms? Horizontal momentum equations: U*U/L U*W/a U*U/a Uf Wf
What are the scales of the terms? For “large-scale” mid-latitude U·U/L U·U/a U·W/ a ΔP/ρL Uf Wf νU/H 2 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12
What are the scales of the terms? For “large-scale” mid-latitude U·U/L U·U/a U·W/ a ΔP/ρL Uf Wf νU/H 2 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Largest Terms
Consider only the largest terms No D( )/Dt term = balance In this case: geostrophic balance Low Pressure High Pressure
What are the scales of the terms? For a tornado (In-class exercise) U·U/L U·U/a U·W/ a ΔP/ρL Uf Wf νU/H 2 10 -2 10 -3 102 10 -3 10 -9
What are the scales of the terms? For a tornado U·U/L U·U/a U·W/ a ΔP/ρL Uf Wf νU/H 2 10 -2 10 -3 102 10 -3 10 -9 Largest Terms
Cyclostrophic Balance: Pressure gradient vs. Centrifugal Acceleration here is the centrifugal acceleration
What are the scales of the terms? For the vertical motion g W*U/L U*U/a Uf
What are the scales of the terms? For the vertical motion W·U/L U·U/a Psfc/ρH g Uf νW/H 2 10 -7 10 -5 10 10 10 -3 10 -15 Hydrostatic relation The vertical acceleration Dw/Dt is 8 orders of magnitude smaller than this balance. The ability to use the vertical momentum equation to estimate w is essentially nonexistent.
How to compute vertical motions? • The vertical acceleration Dw/Dt is 8 orders of magnitude smaller than hydrostatic balance. • The ability to use the vertical momentum equation to estimate w is essentially nonexistent. • Vertical motion is important: rising motion leads to clouds and precipitation… • Hence: – w must be “diagnosed” from some balance – exception: small scales: thunderstorms, tornadoes • We will return to this in a moment…
Scale Analysis of the Continuity Equation
Define a background pressure field • “Average” pressure and density at each level in the atmosphere • No variation in x, y, or time • Hydrostatic balance applies to the background pressure and density • Total pressure and density = sum of background + perturbations (perturbations vary in x, y, z, t)
Use the continuity equation Eulerian form Perturbation Base State Advection Divergence
Divide by ρ0 and assume ρ’/ρ0 ~ 10 -2 <<1 (ρ’U)/(Lρ0) W/H 10 -7 s-1 10 -6 s-1
Look at the velocity divergence (ρ’U)/(Lρ0) 10 -7 s-1 W/H 10 -6 s-1 These are U/L, but they balance ~ 10 -6 s-1
Cancel relatively small terms Remember, horizontal variations of ρ0 are zero and we can add them back in Multiply through by ρ0 and we can write
What can we do with scale analysis? • Remember our goals – Analysis of how the atmosphere works – Prediction of how the atmosphere will evolve in time
What are the scales of the terms? For “large-scale” mid-latitude U·U/L U·U/a U·W/ a ΔP/ρL Uf Wf νU/H 2 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Largest Terms
Geostrophic Balance There is no D( )/Dt term. Hence, no acceleration, no change with time. This is a DIAGNOSTIC equation that can be used to analyze how the atmosphere works
Geostrophic & observed wind 300 mb
What are the scales of the terms? For “large-scale” mid-latitude U·U/L U·U/a U·W/ a ΔP/ρL Uf Wf νU/H 2 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Prediction (Prognosis) Ageostrophic Analysis (Diagnosis) Geostrophic
Our prediction equation for large scale midlatitudes
Our prediction equation for large scale midlatitudes We used the definition of the geostrophic component of the wind. Within 10 -15% of real wind in middle latitudes, large-scale.
Our prediction equation for large scale midlatitudes For middle latitudes and large scales, the acceleration can be computed directly as the difference from geostrophic balance. Remember: pressure and density are buried inside the definition of the geostrophic wind. The mass field and velocity field are linked.
Ageostrophic Wind • Acceleration can be computed from the difference between the real wind and the geostrophic wind • Acceleration defined as: – Change in direction (curvature/rotation) – Along-flow change in speed (convergence/divergence)
Geostrophic & Full wind 500 mb
Next time • Conservation of Energy (Newton’s first law of thermodynamics) (Holton, 2. 6, 2. 7) • Thermodynamic energy equation • Thermodynamics of a dry atmosphere – Potential temperature – Lapse rates – Scale analysis
- Slides: 51
