QG Dynamics – A Review

Anthony R. LupoDepartment of Soil, Environmental, and Atmospheric

Science302 E ABNR BuildingUniversity of MissouriColumbia, 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

Fkp

Vk

tpV

pf

ppf

c

QTV

p

R

pf

h

agaah

a

pha

ˆˆ

22

22

QG Dynamics – A Review QG - omega equation

Q-vector version

pV

ff

Vp

f

p

f

gho

ogh

o

o

o

o

2

22

222

1

1

Ty

V

p

RQ

Tx

V

p

RQ

where

x

T

p

RQ

p

f

g

g

o

2

1

22

222

QG Dynamics – A Review QG-Potential Vorticity

then

ppf

ffQGPVq

where

ppff

fQGPV

and

QGPVdtd

oo

oo

11

11

,0

2*

2

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 ,q,p,T or ,

kssourcesdtdm

Vdtd

dtdp

dtdT

cQRTP p

sin,

,

QG Dynamics – A Review More…….

z

y

x

Fgzp

zw

wyw

vxw

utw

dtdw

Ffuxp

zv

wyv

vxv

utv

dtdv

Ffvxp

zu

wyu

vxu

utu

dtdu

1

1

1

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.

QG Dynamics – A Review Eqns become:

p

TR

p

Ffuyp

v

y

vv

x

vu

t

v

dt

dv

Ffvxp

u

y

uv

x

uu

t

u

dt

du

d

y

x

p

c

R

o

c

Q

pV

t

p

pT

V

p

d

lnlnln

,033

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 b/s

3) We will assume that are small in the du/dt and dv/dt terms of the equations of motion.

pv

pu

,

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 ( <<< f).

QG Dynamics – A Review 7) Replace winds (u,v,) 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.

hh

ohh

ohh

Vp

ufy

vVt

v

dt

dv

vfx

uVt

u

dt

du

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.

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.

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.

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 (QG’s)

We’ll work with geopotential (gz):

Rewrite (back to) equations of motion:(We’ll reduce these for now!)

QG Dynamics – A Review Here they are;

Then, let’s reformulate the thermodynamic equation:

ufy

vVtv

dtdv

vfx

uVtu

dtdu

ohh

ohh

TcQ

Vtdt

d

p

lnlnln

QG Dynamics – A Review Thus, we can rework the first law of

thermodynamics, and after applying our Q-G theory:

0

ogh p

Vpt

QG Dynamics – A Review Next, let’s rework the vorticity

equation:

In isobaric coordinates:

Fk

pV

kVp

Vt

hhha

aah

a

ˆ

ˆ

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

QG Dynamics – A Review Then, we are left with the vorticity

equation in an adiabatic, invicid, Q-G framework.

pff

fVf

t oo

go

222 1

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!

QG Dynamics – A Review Day 11 Take the thermodynamic equation and:

1) Introduce:

2) switch: 3) apply:

t

pt

,

p

f

o

o

2

Day 11 And get:

0

0

22

2

22

pf

pV

pf

pf

and

pV

p

o

ogh

o

o

o

o

ogh

QG Dynamics – A Review Now add the Q-G vorticity and

thermodynamic equation (where ) and we don’t have to manipulate it:

t

pff

fVf o

ogo

222 1

QG Dynamics – A Review The result

becomes after addition:

(Dynamics – Vorticity eqn, vort adv)

(Thermodynamics – 1st Law, temp adv)

pV

p

f

ff

Vfp

f

gho

o

ogo

o

o

2

22

222 1

QG Dynamics – A Review This is the original height tendency

equation!

pV

pf

ff

Vfp

f

gho

o

ogo

o

o

2

22

222 1

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.

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

Fkp

Vk

tpV

pf

ppf

c

QTV

p

R

pf

h

agaah

a

pha

ˆˆ

22

22

QG Dynamics – A Review Apply our Q-G assumptions (round 1):

Assume:

Vh = Vgeo, = g, and r <<< fo f= fo, except where differentiable frictionless, adiabatic = (p) = const.

QG Dynamics – A Review Here we go:

p

Vk

pV

pf

ppfTV

p

R

pf

ghaagho

aoghoo

ˆ

22

222

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: 21

or f

QG Dynamics – A Review 3) Use hydrostatic balance in temp

advection term.

4) divide through by sigma (oops equation too big, next page)

pRT

p

QG Dynamics – A Review Here we go;

pV

ff

Vp

f

p

f

and

pV

ff

Vp

fp

f

gho

ogh

o

o

o

o

gh

oghooo

2

22

222

2

22

222

1

1

1

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!

QG Dynamics – A Review Full version:

dpp

dpcQ

STVfR

Pd

dp

Fkp

Vk

t

ppV

Pdt

po

pt

p

po ph

po

pt hag

aa

ah

p

go

2

ˆˆ

tppPd

1

QG Dynamics – A Review Q-G version #1 (From Lupo and Bosart,

1999):

dpp

dpSTV

fR

Pd

dpff

VPdft

po

pt

p

po

og

po

pt og

po

2

22 11

QG Dynamics – A Review Q-G Form #2 (Zwack and Okossi, 1986;

and others)

dpp

dpTV

fR

Pd

dpff

VPdft

po

pt

p

po

g

po

pt og

po

2

22 11

QG Dynamics – A Review Q-G Form #3!

dpdpp

Vf

Pd

dpff

VPdft

po

pt

p

po

g

po

pt og

po

2

22

1

11

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.

pV

pf

ff

Vftpp

f

gho

ogoo

1

11

2

222

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!!

011 2

ppff

fV

t oo

g

QG Dynamics – A Review So,

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

ppff

fQGPV o

o

11 2

aEPV

QG Dynamics – A Review Or in (x,y,p,t) coordinates:

In two dimensions:

agEPV

constp

gPV a

QG Dynamics – A Review We assume that:

Thus (recall, this was an “ln” form, so we need to multiply by 1/PV):

pPV

wheredtPVd

a

,0

0

1 dtPVd

PV

QG Dynamics – A Review so,

and

0lnln

pdt

ddtd

a

01

pdtdp

dtd

aa

QG Dynamics – A Review And “QG”

Then

01 2

pdtdp

ffdt

da

o

pp

ppRT

foa

1

,

011 2

pdt

d

pff

fdt

do

o

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)!

011 2

ppff

fdtd

oo

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….

then

ppf

ffQGPVq

where

ppff

fQGPV

and

QGPVdtd

oo

oo

11

11

,0

2*

2

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)).

pp

ff

q oo

11 2*

QG Dynamics – A Review We’ve boiled down all the physics into one

equation! Impressive development! Thus, we don’t have to worry about non-linear interactions between forcing mechanisms, it’s 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).

EPVVEPVt

QGPVVQGPVt

EPVdtd

QGPVdtd

0,0

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:

.sin FricDiabaticskssourcesEPVdtd

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:

gygog

gyagog

uufdt

dv

vvfdt

du

QG Dynamics – A Review Here is the adiabatic form of the Q-G

thermodynamic equation:

0

RP

yT

vxT

utT

gg

QG Dynamics – A Review Then manipulation gives us Q1 and Q2:

jQiQQ

where

xT

pR

Ty

V

pR

p

vf

yQ

yT

pR

Tx

V

pR

p

uf

xQ

ygao

ygao

ˆ2ˆ1

22

21

2

2

QG Dynamics – A Review Then differentiate Q1 and Q2, w/r/t x and

y, respectively (in other words, take divergence).

Q1 Q2

xT

pR

Ty

V

yT

x

V

xpR

yv

xu

pf

gg

aao

2

22

QG Dynamics – A Review Then use continuity:

This give us the omega equation in Q-vector format!

py

v

x

uV aa

Ty

V

p

RQ

Tx

V

p

RQ

where

x

T

p

RQ

p

f

g

g

o

2

1

22

222

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!