QG Dynamics – A Review

1. QG Dynamics – A Review Anthony R. Lupo Department of Soil, Environmental, and Atmospheric Science 302 E ABNR Building University of Missouri Columbia, MO 65211

2. QG Dynamics – A Review • Secondary circulations induced by jet/streaks:

3. QG Dynamics – A Review • Q-G perspective

4. QG Dynamics – A Review • Consider cyclonically and anticyclonically curved jets: Keyser and Bell, 1993, MWR.

5. QG Dynamics – A Review • A bit of lightness…

6. QG Dynamics – A Review • We use Q-G equations all the time, either explicitly or implicitly • Full omega equation

7. QG Dynamics – A Review • QG - omega equation • Q-vector version

8. QG Dynamics – A Review • QG-Potential Vorticity

9. QG Dynamics – A Review • Introduction to Q-G Theory: • Recall what we mean by a geostrophic system: • 2-D system, no divergence or vertical motion • no variation in f • incompressible flow • steady state • barotropic (constant wind profile)

10. QG Dynamics – A Review • We once again start with our fundamental equations of geophysical hydrodynamics: • (4 ind. variables, seven dependent variables, 7 equations) • x,y,z,t u,v,w or w,q,p,T or q, r

11. QG Dynamics – A Review • More…….

12. QG Dynamics – A Review • Our observation network is in (x,y,p,t). We’ll ignore curvature of earth: • Our first basic assumption: We are working in a dry adiabatic atmosphere, thus no Eq. of water mass cont. Also, we assume that g, Rd, Cp are constants. We assume Po = a reference level (1000 hPa), and atmosphere is hydrostatically balanced.

13. QG Dynamics – A Review • Eqns become:

14. QG Dynamics – A Review • Now to solve these equations, we need to specify the initial state and boundary conditions to solve. This represents a closed set of equations, ie the set of equations is solvable, and given the above we can solve for all future states of the system. • Thus, as V. Bjerknes (1903) realizes, weather forecasting becomes an initial value problem.

15. QG Dynamics – A Review • These (non-linear partial differential equations) equations should yield all future states of the system provided the proper initial and boundary conditions. • However, as we know, the solutions of these equations are sensitive to the initial cond. (solutions are chaotic). • Thus, there are no obvious analytical solutions, unless we make some gross simplifications.

16. QG Dynamics – A Review • So we solve these using numerical techniques. • One of the largest problems: inherent uncertainty in specifying (measuring) the true state of the atmosphere, given the observation network. This is especially true of the wind data. • So our goal is to come up with a system that is somewhere between the full equations and pure geostrophic flow.

17. QG Dynamics – A Review • We can start by scaling the terms: • 1) f = fo = 10-4 s-1 (except where it appears in a differential) • 2) We will allow for small divergences, and small vertical, and ageostrophic motions. Roughly 1 mb/s • 3) We will assume that are small in the du/dt and dv/dt terms of the equations of motion.

18. QG Dynamics – A Review • 4) Thus, assume the flow is still 2 - D. • 5) We assume synoptic motions are fairly weak (u = v = 10 m/s). • Also, flow heavily influenced by CO thus (z <<< f).

19. QG Dynamics – A Review • 7) Replace winds (u,v,z) by their geostrophic values • 8) Assume a Frictionless AND adiabatic atmosphere.

20. QG Dynamics – A Review • The Equations of motion and continutity • So, there are the dynamic equations in QG-form, or one approximation of them.

21. QG Dynamics – A Review • TIME OUT! • Still have the problem that we need to use height data (measured to 2% uncertainty), and wind data (5-10% uncertainty). Thus we still have a problem! • Much of the development of modern meteorology was built on Q-G theory. (In some places it’s still used heavily). Q-G theory was developed to simplify and get around the problems of the Equations of motion.

22. QG Dynamics – A Review • Why is QG theory important? • 1) It’s a practical approach  we eliminate the use of wind data, and use more “accurate’ height data. Thus we need to calculate geopotential for ug and vg. Use these simpler equations in place of Primitive equations.

23. QG Dynamics – A Review • 2) Use QG theory to balance and replace initial wind data (PGF = CO) using geostrophic values. Thus, understanding and using QG theory (a simpler problem) will lead to an understanding of fundamental physical process, and in the case of forecasts identifying mechanisms that aren’t well understood.

24. QG Dynamics – A Review • 3) QG theory provides us with a reasonable conceptual framework for understanding the behavior of synoptic scale, mid-latitude features. PE equations may me too complex, and pure geostrophy too simple. QG dynamics retains the presence of convergence divergence patterns and vertical motions (secondary circulations), which are all important for the understanding of mid-latitude dynamics.

25. QG Dynamics – A Review • So Remember…… • “P-S-R”

26. QG Dynamics – A Review • Informal Scale analysis derivation of the Quasi - Geostrophic Equations (QG’s) • We’ll work with geopotential (gz): • Rewrite (back to) equations of motion: (We’ll reduce these for now!)

27. QG Dynamics – A Review • Here they are; • Then, let’s reformulate the thermodynamic equation:

28. QG Dynamics – A Review • Thus, we can rework the first law of thermodynamics, and after applying our Q-G theory:

29. QG Dynamics – A Review • Next, let’s rework the vorticity equation: • In isobaric coordinates:

30. QG Dynamics – A Review • Let’s start applying some of the approximations: • 1) Vh = Vg • 2) Vorticity is it’s geostrophic value • 3) assume zeta is much smaller than f = fo except where differentiable. • 4) Neglect vertical advection • 5) neglect tilting term • 6) Invicid flow

31. QG Dynamics – A Review • Then, we are left with the vorticity equation in an adiabatic, invicid, Q-G framework.

32. QG Dynamics – A Review • Now let’s derive the height tendency equation from this set • We will get another “Sutcliffe-type” equation, like the Z-O equation, the omega equation, the vorticity equation. • Like the others before them, they seek to describe height tendency, as a function of dynamic and thermodynamic forcing!

33. QG Dynamics – A Review Day 11 • Take the thermodynamic equation and: • 1) Introduce: • 2) switch: • 3) apply:

34. Day 11 • And get:

35. QG Dynamics – A Review • Now add the Q-G vorticity and thermodynamic equation (where ) and we don’t have to manipulate it:

36. QG Dynamics – A Review • The result • becomes after addition: • (Dynamics – Vorticity eqn, vort adv) (Thermodynamics – 1st Law, temp adv)

37. QG Dynamics – A Review • This is the original height tendency equation!

38. QG Dynamics – A Review • The Omega Equation (Q-G Form) • We could derive this equation by taking of the thermodynamic equation, and of the vorticity equation (similar to the original derivation). However, let’s just apply our assumptions to the full omega equation.

39. QG Dynamics – A Review • The full omega equation (The Beast!):

40. QG Dynamics – A Review • Apply our Q-G assumptions (round 1): • Assume: • Vh = Vgeo, z = zg, and zr <<< fo • f= fo, except where differentiable • frictionless, adiabatic • s = s(p) = const.

41. QG Dynamics – A Review • Here we go:

42. QG Dynamics – A Review • Then let’s assume: • 1) vertical derivatives times omega are small, or vertical derivatives of omega, or horizontal gradients of omega are small. • 2) substitute:

43. QG Dynamics – A Review • 3) Use hydrostatic balance in temp advection term. • 4) divide through by sigma (oops equation too big, next page)

44. QG Dynamics – A Review • Here we go;

45. QG Dynamics – A Review • Of course there are dynamics and thermodynamics there, can you pick them out? • Q-G form of the Z-O equation (Zwack and Okossi, 1986, Vasilj and Smith, 1997, Lupo and Bosart, 1999) • We will not derive this, we’ll just start with full version and give final version. Good test question on you getting there!

46. QG Dynamics – A Review • Full version:

47. QG Dynamics – A Review • Q-G version #1 (From Lupo and Bosart, 1999):

48. QG Dynamics – A Review • Q-G Form #2 (Zwack and Okossi, 1986; and others)

49. QG Dynamics – A Review • Q-G Form #3!

50. QG Dynamics – A Review • Quasi - Geostropic potential Vorticity • We can start with the Q-G height tendency, with no assumption that static stability is not constant.