1. The simplest theoretical basis for understanding the location of significant vertical motions in an Eulerian framework is QUASI-GEOSTROPHIC THEORY QG Theory: Developed in late 1940s (before the age of computers) to help meteorologists diagnose the future state of the atmosphere. Addressed the fundamental problem that divergence could not be calculated with the standard rawinsonde network Is the theoretical framework for synoptic-scale dynamics taught to generations of meteorologists We will look at standard development of the QG equations retaining diabatic and friction terms We will look at Trenberth (1978) simplifications to QG equations We will study in detail the Q vector form of Q-G Theory

2. We will first derive the vorticity equation (1)(2) Expand total derivative

3. Local rate of change of relative vorticity Horizontal advection of absolute vorticity on a pressure surface Vertical advection of relative vorticity Divergence acting on Absolute vorticity (twirling skater effect) Tilting of vertically sheared flow Gradients in force Of friction The vorticity equation In English: Horizontal relative vorticity is increased at a point if 1) positive vorticity is advected to the point along the pressure surface, 2) or advected vertically to the point, 3) if air rotating about the point undergoes convergence (like a skater twirling up), 4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion 5) if the force of friction varies in the horizontal.

4. Next we will derive the “Quasi-Geostrophic” Vorticity Equation Start with vorticity equation 1 2 3 4 5 6 1.Based on scale analysis, we will ignore 3, 5, 6 and  compared to f in 4 2.Assume relative vorticity in 1, 2 can be replaced by its geostrophic value  g 3.Replace divergence in 4 using continuity equation 4.Assume that the velocity (u,v) in 2 can be replaced by its geostrophic value 5.Assume Coriolis force varies linearly across mid-latitudes (f = f 0 +  y) 6.Ignore  y where f is not differentiated. 1 2 4

5. Now derive the “Quasi-Geostrophic” Thermodynamic Equation Start with conservation of energy equation 1 2 3 4 1.Ignore 4, diabatic heating 2.Use Hydrostatic equation to replace T in 1 and 2. 3.Move outside derivatives (P is held constant in derivatives in P coordinate system) 4.Multiple equation by 5. Write TR/P as the specific volume , and then write, the static stability. 1 2 3

6. Q-G vorticity equation Q-G thermodynamic equation Now we will use the geostrophic wind relationships and to write  g in terms of 

7. Q-G vorticity equation Q-G thermodynamic equation We now have two equations in two unknowns,  and  We will solve these to find an equation for , the vertical motion in pressure coordinates, and for, the change of geopotential height with time.

8. 1 2 Derive the Q-G omega equation (equation for vertical velocity in pressure coordinates) 1.Assume  is constant 2.Take 3.Take of Eq. 1. 3.Reverse order of differentiation on left side of equation 4.Subtract: (1) – (2) 5.Rearrange terms

9. The QG omega equation can be derived including the friction term and the diabatic heating term. We will not do this here, but I will simply show the result. QG OMEGA EQUATION

10. BEFORE WE PROCEED TO TRY TO UNDERSTAND THIS EQUATIONS ANSWER THIS QUESTION: WHAT PHENOMENA OF IMPORTANCE ARE WE IMPROPERLY REPRESENTING IN THE DERIVATION OF THESE EQUATIONS?

11. By assuming that the static stability,is constant WE HAVE ELIMINATED A TERM RELATED TO FRONTS! We took x, y, and P derivatives assuming  is constant Real Atmosphere (  ) QG Atmosphere (  )

12. Where does all the precipitation occur?

13. ALONG FRONTS!

14. By assuming that the static stability,is constant WE HAVE ELIMINATED A TERM RELATED TO JETSTREAKS! We took x, y, and P derivatives assuming  is constant – this implies that the vertical wind shear is constant, which can’t happen in a jetstreak environment

15. To illustrate quasi-geostrophic vertical motion in a real system consider the cyclone below

16. 700 mb height and rainwater field simulated by MM5 model

17. Vertical motion calculated at 700 mb by MM5 Model

18. Vertical motion attributed to terms in QG Omega equation

19. Residual vertical motion (not attributable to QG forcing)

20. Quasi-Geostrophic Equations describe the broad scale ascent and descent in troughs and ridges and the propagation of these troughs and ridges. Quasi-Geostrophic interpretation of the atmosphere: Quasi-Geostrophic Equations do not describe the mesoscale ascent and descent associated with ageostrophic circulation near fronts and within jetstreaks, nor the propagation of low and high pressure systems due to jetstreaks. What the quasi-geostrophic solutions do: What the quasi-geostrophic solutions do not do: (These are the motions are most responsible for significant weather)

21. So what does the QG system  equation tell us about the atmosphere and how should we use it to diagnose vertical motions?

22. QG OMEGA EQUATION We are taking two derivatives of the vertical motion  Let’s write  as a Fourier series in x: When we take two x derivatives of this series we get Therefore: First term is proportional to -  Read the first term as “Rising motion”

23. QG OMEGA EQUATION The second term is the vertical derivative of the absolute vorticity advection: Rising motion is proportional to “Differential Positive Vorticity Advection”

24. The geostrophic wind is strong at jetstream level and the height gradients are greatest. The geostrophic wind is weak at low levels and the height gradients are weak. Vertical gradient in vorticity advection is strongest, on average, in middle troposphere (so absolute vorticity is typically Plotted on the 500 mb surface)

25. Relationship between Shear and curvature in the Jetstream Absolute vorticity and Abs. Vorticity Advection and Vertical Motion

26. QG OMEGA EQUATION We are taking two derivatives of the thickness advection, When we take two derivatives of this series we get Let’s write A as a Fourier series in x: is the depth, or thickness, of the layer between two pressure surfaces. This depth is proportional to the mean temperature of the layer.

27. Read the term inside the parentheses as temperature advection However, it is the Laplacian of temperature advection we are interested in… Let’s rewrite this term: Multiply term by -1 twice to get in form consistent with temperature advection If the gradient of temperature advection is constant, then the term is zero! THEREFORE:  will be non-zero only in areas where the thermal advection pattern in non uniform

28. Heterogeneity in warm advection on a pressure surface implies upward motion When air is ascending on an isentropic surface… …the projection of the wind on to a constant pressure surface appears as warm air flowing toward cold air (warm advection) But what if isentropic surface is moving northward at the same time? Warm advection involves both flow on the isentropic surface and movement of the surface

29. When air is descending on an isentropic surface… …the projection of the wind on to a constant pressure surface appears as cold air flowing toward warm air (cold advection) Heterogeneity in cold advection on a pressure surface implies downward motion But what if isentropic surface is moving southward at the same time? Cold advection involves both flow on the isentropic surface and movement of the surface

30. The vertical derivative of friction acting on the geostrophic vorticity Or more simply: The rate that friction decreases with height in the presence of cyclonic vorticity in the boundary layer QG OMEGA EQUATION

31. Again: Two derivatives in x,y ……….proportional to negative With additional minus sign: Rising motion is proportional to the Laplacian of the diabatic heating rate (modulated by static stability) Sinking motion is proportional to the Laplacian of the diabatic cooling rate (modulated by static stability) Heterogeneity in the diabatic heating leads to rising motion

32. low static stabilitysmall 335 650 250 135 high static stabilitylarge From our earlier discussion of isentropic coordinates:

33. For a given amount of diabatic heating, a parcel in a layer with high static stability Will have a smaller vertical displacement than a parcel in a layer with low static stability From our earlier discussion of isentropic coordinates:

34. QG OMEGA EQUATION Broad scale (synoptic scale) rising motion in the atmosphere is proportional to: Differential positive vorticity advection Heterogeneity in the warm advection field The rate of decrease with height of friction in the presence of vortex Heterogeneity in the diabatic heating rate SUMMARY Broad scale (synoptic scale) desending motion in the atmosphere is proportional to: Differential negative vorticity advection Heterogeneity in the cold advection field The rate of decrease with height of friction in the presence of vortex Heterogeneity in the diabatic cooling rate

35. Recall for isentropic coordinates: Diabatic termTemperature Advection Vertical Motion Let’s compare the QG Omega Equation* with the Omega equation we derived a long time ago when we considered isentropic coordinates *Note… I dropped friction term since we did not consider it in our discussion of isentropic coordinates

36. Let’s compare the QG Omega Equation* with the Omega equation we derived a long time ago when we considered isentropic coordinates Recall for isentropic coordinates: Diabatic termTemperature Advection Vertical Motion Must be Related! *Note… I dropped friction term since we did not consider it in our discussion of isentropic coordinates

37. Cold advection 850 mb heights, temperature, winds Warm advection TEMPERATURE ADVECTION TERM

38. Descent on isentropic surface Pressure, winds and RH on 290 K isentropic surface Ascent on isentropic surface TEMPERATURE ADVECTION TERM

39. So how are orange terms related???

40. Differential absolute vorticity advection measures the rate at which potential temperature surfaces rise or fall as ridges and troughs propagate along! Vertical displacement of isentrope = (Air must rise for isentrope to be displaced upward)

41. 500 mb w700 mb w