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## AOSS 401, Fall 2007 Lecture 25 November 09 , 2007

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**AOSS 401, Fall 2007Lecture 25November 09, 2007**Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502**Class News November 09, 2007**• Computing assignment • Second component assigned next week • Due 5 December (or thereabouts) • Involves writing a very simple numerical model… • Important Dates: • November 12: Homework 6 due—questions? • November 16: Next Exam (Review on 14th) • November 21: No Class • December 10: Final Exam**Today**• Work through Homework 5 • Another perspective on planetary vorticity advection • Finish discussion of QG omega equation • Midlatitude cyclone energetics**5.1) Homework Problem**In pressure coordinates, the horizontal momentum equation is written as: Derive the equation for the conservation of vorticity. You should pursue the derivation until you have terms analogous to the divergence terms, tilting terms, and the baroclinic or solenoidal term (the term that included the pressure gradient). If there are additional or missing terms, then explain their presence or absence.**Solution to (5.1)**• Recognize that we are in pressure coordinates • Split the equation into u- and v- components**Solution to (5.1)**• Differentiate the v-equation with respect to x and the u-equation with respect to y**Solution to (5.1)**• Subtract the u-equation from the v-equation and expand the material derivatives**Solution to (5.1)**• Collect terms, use the definition of divergence and vorticity, and end up with No solenoidal term, as the equation is defined to be on a constant pressure surface**5.2) Homework Problem**This is a special homework problem. While we have not formally seen this equation, I think that we have the tools to do this problem. Given the equation for the conservation of vorticity, ζ, where the prime represents a small quantity and the overbar represents a larger, mean quantity: With the definition of the velocity field given below, show that the vorticity equation can be written as: What is the criterion for wave solutions to this equation?**Solution to (5.2)**• Plug in the definitions for • Then plug in • and scale out terms that are products of perturbations**Solution to (5.2)(Lecture 22, slide 39…)**Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.**Solution to (5.2)**• Rearrange the dispersion relation to find • Mean wind must be positive (from the west) for waves to form**Scaled equations in pressure coordinates(The**quasi-geostrophic (QG) equations) momentum equation geostrophic wind continuity equation thermodynamicequation**Application of QG:Prediction of Atmospheric Flow**• Want to know how distribution of geopotential will change in the atmosphere • changes in pressure gradient force (jet stream, convergence/divergence, cyclogenesis) • Derived geopotential height tendency equation**Geopotential Tendency Equation**f0 * Vorticity Advection Thickness Advection**Advection of vorticity**Advection of vorticity Advection of relative vorticity Advection of planetary vorticity**Advection of vorticity**ζ < 0; anticyclonic Advection of ζ tries to propagate the wave this way ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 Advection of f tries to propagate the wave this way ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic**Advection of planetary vorticity**Start with straight-line flow y, north x, east**Advection of planetary vorticity**Introduce perturbations and remember conservation of potential vorticity (assume no change in depth h) y, north x, east**Advection of planetary vorticity**North/south movement change in planetary vorticity Conservation of angular momentum change in ζ f = f0 ζ < 0; anticyclonic f > f0 ζ > 0; cyclonic f < f0 ζ > 0; cyclonic f < f0 y, north x, east**Advection of planetary vorticity**Flow associated with rotation advects adjacent parcels north/south ζ < 0; anticyclonic f > f0 ζ > 0; cyclonic f < f0 ζ > 0; cyclonic f < f0 y, north x, east**Advection of planetary vorticity**The wave propagates to the west (actual propagation direction depends on wind speed and wavelength…) ζ < 0; anticyclonic f > f0 ζ > 0; cyclonic f < f0 ζ > 0; cyclonic f < f0 y, north x, east**VERTICAL VELOCITY**• Important for • Clouds and precipitation • Cyclogenesis (spinup of vorticity through stretching) • Computed from governing equations**Scaled equations in pressure coordinates(The**quasi-geostrophic (QG) equations) momentum equation geostrophic wind continuity equation thermodynamicequation**Methods for Estimating Vertical Velocity**• Kinematic method (continuity equation) • Adiabatic method (thermodynamic eqn) • Diabatic method (thermodynamic eqn) • QG-omega equation (unified equation)**1. Kinematic Method:Link between and the ageostrophic**wind Continuity equation (pressure coordinates) and non-divergence of geostrophic wind lead to which can berewritten as: and solved for:**2. Adiabatic MethodLink between and temperature**advection Thermodynamic equation: Assume the diabatic heating term J is small (J=0), and there is no local time change in temperature Horizontal temperature advection term Stability parameter warm air advection: < 0, w ≈ -/g > 0 (ascending air) cold air advection: > 0, w ≈ -/g < 0 (descending air)**2. Adiabatic MethodLink between and temperature**advection Based on temperature advection, which is dominated by the geostrophic wind, which is large. Hence this is a reasonable way to estimate local vertical velocity when advection is strong. adiabatic = no change in θ WARM COLD θ θ + Δθ θ - Δθ**3. Diabatic MethodLink between and heating/cooling**Start from thermodynamic equation in p-coordinates: If you take an average over space and time, then the advection and time derivatives tend to cancel out. RadiationCondensationEvaporationMeltingFreezing Diabatic term**Vertical Velocity: Problems**1. Kinematic method: divergence vertical motion • Links the horizontal and vertical motions. Since geostrophy is such a good balance, the vertical motion is linked to the divergence of the ageostrophic wind (small). • Therefore: small errors in evaluating the winds <u> and <v> lead to large errors in . • The kinematic method is inaccurate.**Vertical Velocity: Problems**2. Adiabatic method: temperature advection vertical motion • Assumes steady state (no movement of weather systems) • Assumes no diabatic heating (no clouds or precipitation) • What about divergence/convergence? • The adiabatic method has severe limitations.**Vertical Velocity: Problems**3. Diabatic method: Heating/cooling vertical motion • Assumes definition of some average atmosphere • Assumes vertical motion only due to diabatic heating • What about divergence/convergence? • The diabatic method has severe limitations.**4. QG-omega equation Combine all QG equations**• None of the obvious methods work well for for midlatitude waves in general • Combine information from the full set of QG equations • Geopotential tendency equation(comes from vorticity equation--combines equation of motion, continuity equation, and geostrophic relationship) • Thermodynamic energy equation**4. QG-omega equation Combine all QG equations**1.) Apply the horizontal Laplacian operator ( 2 ) to the QG thermodynamic equation 2.) Differentiate the geopotential height tendency equation with respect to p 3.) Combine 1) and 2)**4. QG-omega equation Combine all QG equations**• Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion**4. QG-omega equation Combine all QG equations**• Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion • Link between temperature advection and vertical motion**4. QG-omega equation Combine all QG equations**• Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion • Link between temperature advection and vertical motion • Link between diabatic heating and vertical motion**4. QG-omega equation Combine all QG equations**• This is still a complicated equation to analyze directly—simplify it • Assume small diabatic heating (scale out last term) • Use the chain rule on the first and second terms on the right hand side and combine the remaining terms**4. QG-omega equation (simplified)**Simple, right? Advection of absolute vorticityby the thermal wind**“Advection” by thermal wind?**• How to analyze this on a map? is Perpendicular to ThermalWind Thickness Gradient of Look at contours of constant thickness**What about that Laplacian?**• QG omega equation relates vorticity advection by the thermal wind with the laplacian of omega • Assume omega has a wave-like form • This leads to • which means**Vertical Motion on Weather Maps**• Laplacian of omega is proportional to -ω • Omega can be analyzed as: • Remember, from definition of omega and scale analysis • Positive vorticity advection by the thermal wind indicates rising motion**Ascent**+ Descent Vertical Motion on Weather Maps • Positive vorticity advection by the thermal wind indicates rising motion Lines of constant thickness**Vertical Motion on Weather Maps**Surface 500 mb 700 mb**Vertical Motion on Weather Maps**Surface 500 mb 700 mb**Vertical Motion on Weather Maps**Surface 500 mb 700 mb**Vertical Motion on Weather Maps**Surface 500 mb 700 mb**Vertical Motion on Weather Maps**Surface 500 mb 700 mb