QG Dynamics – A Review
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Transcript of QG Dynamics – A Review
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!