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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

QG Dynamics A Review

- Secondary circulations induced by jet/streaks

QG Dynamics A Review

- Q-G perspective

QG Dynamics A Review

- Consider cyclonically and anticyclonically curved

jets Keyser and Bell, 1993, MWR.

QG Dynamics A Review

- A bit of lightness

QG Dynamics A Review

- We use Q-G equations all the time, either

explicitly or implicitly - Full omega equation

QG Dynamics A Review

- QG - omega equation
- Q-vector version

QG Dynamics A Review

- QG-Potential Vorticity

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)

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

QG Dynamics A Review

- More.

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.

QG Dynamics A Review

- Eqns become

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.

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.

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.

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.

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).

QG Dynamics A Review

- 7) Replace winds (u,v,z) by their geostrophic

values - 8) Assume a Frictionless AND adiabatic

atmosphere.

QG Dynamics A Review

- The Equations of motion and continutity
- So, there are the dynamic equations in QG-form,

or one approximation of them.

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.

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.

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.

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.

QG Dynamics A Review

- So Remember
- P-S-R

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!)

QG Dynamics A Review

- Here they are
- Then, lets reformulate the thermodynamic

equation

QG Dynamics A Review

- Thus, we can rework the first law of

thermodynamics, and after applying our Q-G theory

QG Dynamics A Review

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

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

QG Dynamics A Review

- Then, we are left with the vorticity equation in

an adiabatic, invicid, Q-G framework.

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!

QG Dynamics A Review Day 11

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

Day 11

- And get

QG Dynamics A Review

- Now add the Q-G vorticity and thermodynamic

equation (where ) and we dont have to

manipulate it

QG Dynamics A Review

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

QG Dynamics A Review

- This is the original height tendency equation!

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.

QG Dynamics A Review

- The full omega equation (The Beast!)

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.

QG Dynamics A Review

- Here we go

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

QG Dynamics A Review

- 3) Use hydrostatic balance in temp advection

term. - 4) divide through by sigma (oops equation too

big, next page)

QG Dynamics A Review

- Here we go

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!

QG Dynamics A Review

- Full version

QG Dynamics A Review

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

QG Dynamics A Review

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

QG Dynamics A Review

- Q-G Form 3!

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.

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!!

QG Dynamics A Review

- So,
- Also, you could start from our EPV expression

from earlier this year

QG Dynamics A Review

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

QG Dynamics A Review

- We assume that
- Thus (recall, this was an ln form, so we need

to multiply by 1/PV)

QG Dynamics A Review

- so,
- and

QG Dynamics A Review

- And QG
- Then

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)!

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

QG Dynamics A Review

- Then.

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)).

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?

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).

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

QG Dynamics A Review

- Generation

QG Dynamics A Review

- The Q - Vector approach (Hoskins et al., 1978,

QJRMS) Bluestein, pp. 350 - 370. - Start w/ Q-G Equations of motion

QG Dynamics A Review

- Here is the adiabatic form of the Q-G

thermodynamic equation

QG Dynamics A Review

- Then manipulation gives us Q1 and Q2

QG Dynamics A Review

- Then differentiate Q1 and Q2, w/r/t x and y,

respectively (in other words, take divergence). - Q1 Q2

QG Dynamics A Review

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

format!

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.

QG Dynamics A Review

- This form also gives a clear picture of omega on

a 2-D plot - Div. Q is sinking motion

QG Dynamics A Review

- Conv. Q is rising motion

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!

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