QG Dynamics

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QG Dynamics A Review

• Anthony R. Lupo
• Department of Soil, Environmental, and
  Atmospheric Science
• 302 E ABNR Building
• University of Missouri
• Columbia, MO 65211

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QG Dynamics A Review

• Secondary circulations induced by jet/streaks

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QG Dynamics A Review

• Q-G perspective

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QG Dynamics A Review

• Consider cyclonically and anticyclonically curved
  jets Keyser and Bell, 1993, MWR.

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QG Dynamics A Review

• A bit of lightness

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QG Dynamics A Review

• We use Q-G equations all the time, either
  explicitly or implicitly
• Full omega equation

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QG Dynamics A Review

• QG - omega equation
• Q-vector version

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QG Dynamics A Review

• QG-Potential Vorticity

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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)

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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

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QG Dynamics A Review

• More.

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QG Dynamics A Review

• Our observation network is in (x,y,p,t). Well
  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.

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QG Dynamics A Review

• Eqns become

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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.

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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.

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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.

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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.

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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 ltltlt
  f).

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QG Dynamics A Review

• 7) Replace winds (u,v,z) by their geostrophic
  values
• 8) Assume a Frictionless AND adiabatic
  atmosphere.

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QG Dynamics A Review

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

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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 its still
  used heavily). Q-G theory was developed to
  simplify and get around the problems of the
  Equations of motion.

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QG Dynamics A Review

• Why is QG theory important?
• 1) Its 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.

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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 arent well
  understood.

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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.

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QG Dynamics A Review

• So Remember
• P-S-R

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QG Dynamics A Review

• Informal Scale analysis derivation of the Quasi -
  Geostrophic Equations (QGs)
• Well work with geopotential (gz)
• Rewrite (back to) equations of motion
• (Well reduce these for now!)

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QG Dynamics A Review

• Here they are
• Then, lets reformulate the thermodynamic
  equation

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QG Dynamics A Review

• Thus, we can rework the first law of
  thermodynamics, and after applying our Q-G theory

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QG Dynamics A Review

• Next, lets rework the vorticity equation
• In isobaric coordinates

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QG Dynamics A Review

• Lets start applying some of the approximations
• 1) Vh Vg
• 2) Vorticity is its 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

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QG Dynamics A Review

• Then, we are left with the vorticity equation in
  an adiabatic, invicid, Q-G framework.

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QG Dynamics A Review

• Now lets 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!

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QG Dynamics A Review Day 11

• Take the thermodynamic equation and
• 1) Introduce
• 2) switch
• 3) apply

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Day 11

• And get

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QG Dynamics A Review

• Now add the Q-G vorticity and thermodynamic
  equation (where ) and we dont have to
  manipulate it

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QG Dynamics A Review

• The result
• becomes after addition
• (Dynamics Vorticity eqn, vort adv)
• (Thermodynamics 1st Law, temp adv)

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QG Dynamics A Review

• This is the original height tendency equation!

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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, lets just apply our assumptions to the
  full omega equation.

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QG Dynamics A Review

• The full omega equation (The Beast!)

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QG Dynamics A Review

• Apply our Q-G assumptions (round 1)
• Assume
• Vh Vgeo, z zg, and zr ltltlt fo
• f fo, except where differentiable
• frictionless, adiabatic
• s s(p) const.

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QG Dynamics A Review

• Here we go

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QG Dynamics A Review

• Then lets assume
• 1) vertical derivatives times omega are small, or
  vertical derivatives of omega, or horizontal
  gradients of omega are small.
• 2) substitute

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QG Dynamics A Review

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

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QG Dynamics A Review

• Here we go

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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, well just start with
  full version and give final version. Good test
  question on you getting there!

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QG Dynamics A Review

• Full version

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QG Dynamics A Review

• Q-G version 1 (From Lupo and Bosart, 1999)

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QG Dynamics A Review

• Q-G Form 2 (Zwack and Okossi, 1986 and others)

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QG Dynamics A Review

• Q-G Form 3!

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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.

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QG Dynamics A Review

• Vorticity Stability
• This is quasi-geostropic potential vorticity!
  (See Hakim, 1995, 1996, MWR Henderson, 1999,
  MWR, March)
• Note after manipulation that we combined dynamic
  and thermodynamic forcing!!

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QG Dynamics A Review

• So,
• Also, you could start from our EPV expression
  from earlier this year

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QG Dynamics A Review

• Or in (x,y,p,t) coordinates
• In two dimensions

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QG Dynamics A Review

• We assume that
• Thus (recall, this was an ln form, so we need
  to multiply by 1/PV)

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QG Dynamics A Review

• so,
• and

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QG Dynamics A Review

• And QG
• Then

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QG Dynamics A Review

• we get QGPV!
• Again, we have both thermodynamic and dynamic
  forcing tied up in one variable QGPV (as was the
  case for EPV)!

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QG Dynamics A Review

• Thus, QGPV can also be tied to one variable, the
  height field, thus we can invert QGPV field and
  recover the height field.
• We can also linearize this equation, dividing
  the height field into a mean and perturbation
  height fields, then

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QG Dynamics A Review

• Then.

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QG Dynamics A Review

• Thus, when we invert the PV fields we get the
  perturbation potential vorticity fields.
  Ostensibly, we can recover all fields
  (Temperature, heights, winds, etc. from one
  variable, Potential Vorticity, subject to the
  prescribed balance condition (QG)).

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QG Dynamics A Review

• Weve boiled down all the physics into one
  equation! Impressive development! Thus, we dont
  have to worry about non-linear interactions
  between forcing mechanisms, its all there,
  simple and elegant!
• Disadvantage we cannot isolate individual
  forcing mechanisms. We must also calculate PV to
  begin with! Also, does this really give us
  anything new?

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QG Dynamics A Review

• Forecasting using QGPV or EPV
• Local tendency just equal to the advection (see
  Lupo and Bosart, 1999 Atallah and Bosart, 2003).

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QG Dynamics A Review

• EPV and QGPV NOT conserved in a diabatically
  driven event. Diabatic heating is a source or
  sink of vorticity or Potential Vorticity.
• Potential Vorticity Generation

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QG Dynamics A Review

• Generation

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QG Dynamics A Review

• The Q - Vector approach (Hoskins et al., 1978,
  QJRMS) Bluestein, pp. 350 - 370.
• Start w/ Q-G Equations of motion

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QG Dynamics A Review

• Here is the adiabatic form of the Q-G
  thermodynamic equation

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QG Dynamics A Review

• Then manipulation gives us Q1 and Q2

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QG Dynamics A Review

• Then differentiate Q1 and Q2, w/r/t x and y,
  respectively (in other words, take divergence).
• Q1 Q2

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QG Dynamics A Review

• Then use continuity
• This give us the omega equation in Q-vector
  format!

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QG Dynamics A Review

• Note that on the RHS, we have the dynamic and
  thermodynamic forcing combined into one term.
• Also, note that we can calculate these on p
  -surfaces (no vertical derivatives). The forcing
  function is exact differential (ie, not path
  dependent), and dynamics or thermodynamics not
  neglected.

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QG Dynamics A Review

• This form also gives a clear picture of omega on
  a 2-D plot
• Div. Q is sinking motion

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QG Dynamics A Review

• Conv. Q is rising motion

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QG Dynamics A Review

• Forcing function is Galilean Invariant which
  simply means that the forcing function is the
  same in a fixed coordinate system as it is in a
  moving one (i.e., no explicit advection terms!)
• And this is the end of Dynamics!