the simplest theoretical basis for understanding the location of significant vertical motions in an...
TRANSCRIPT
The simplest theoretical basis for understanding the location of significant vertical motions in an Eulerian framework is
QUASI-GEOSTROPHIC THEORY
QG Theory: Developed in late 1940s (before the age of computers) to help meteorologists diagnose the future state of the atmosphere.
Addressed the fundamental problem that divergence could not be calculated with the standard rawinsonde network
Is the theoretical framework for synoptic-scale dynamics taught to generations of meteorologists
We will look at standard development of the QG equations retaining diabatic and friction terms
We will look at Trenberth (1978) simplifications to QG equations
We will study in detail the Q vector form of Q-G Theory
xFfvxdt
du
We will first derive the vorticity equation
yFfuydt
dv
(1) (2)
Expand total derivative
xFfvxP
u
y
uv
x
uu
t
u
xFfu
yP
v
y
vv
x
vu
t
v
yxTake
)1()2(
y
F
x
F
xP
v
yP
u
y
v
x
uf
y
u
x
v
y
u
x
v
Pf
y
u
x
v
yvf
y
u
x
v
xu
y
u
x
v
t
yx
asy
u
x
vvorticityrelativewrite
y
F
x
F
xP
v
yP
u
y
v
x
uf
Pf
yvf
xu
tyx
y
F
x
F
xP
v
yP
u
y
v
x
uf
Pf
yvf
xu
tyx
Local rate ofchange of relativevorticity
Horizontal advectionof absolute vorticityon a pressure surface
Vertical advectionof relative vorticity
Divergence acting onAbsolute vorticity(twirling skater effect)
Tilting of verticallysheared flow
Gradients in forceOf friction
The vorticity equation
In English: Horizontal relative vorticity is increased at a point if 1) positive vorticity is advected to the point along the pressure surface, 2) or advected vertically to the point,3) if air rotating about the point undergoes convergence (like a skater twirling up),
4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion 5) if the force of friction varies in the horizontal.
Next we will derive the “Quasi-Geostrophic” Vorticity Equation
y
F
x
F
xP
v
yP
u
y
v
x
uf
Pf
yvf
xu
tyx
Start with vorticity equation
1 2 3 4 5 6
1. Based on scale analysis, we will ignore 3, 5, 6 and compared to f in 42. Assume relative vorticity in 1, 2 can be replaced by its geostrophic value g
3. Replace divergence in 4 using continuity equation4. Assume that the velocity (u,v) in 2 can be replaced by its geostrophic value5. Assume Coriolis force varies linearly across mid-latitudes (f = f0 + y)6. Ignore y where f is not differentiated.
P
ffVt ggg
0
1 2 4
Now derive the “Quasi-Geostrophic” Thermodynamic Equation
dt
dQ
cp
T
y
Tv
x
Tu
t
T
p
1
Start with conservation of energy equation
1 2 3 4 1. Ignore 4, diabatic heating2. Use Hydrostatic equation to replace T in 1 and 2.
3. Move outside derivatives (P is held constant in derivatives in P coordinate system)
4. Multiple equation by
pR
pT
R
p
p
R
pp
TR
pV
ptg
5. Write TR/P as the specific volume , and then write , the static stability.
p
pV
ptg
1 2 3
p
ffVt ggg
0
pV
ptg
Q-G vorticity equation
Q-G thermodynamic equation
Now we will use the geostrophic wind relationships and
to write g in terms of
2
02
2
2
2
000
1111
fyxfyfyxfxy
u
x
v ggg
2
0
1
fg
yfug
0
1
xfvg
0
1
p
ffVftf
g
0
2
0
2
0
11
PV
Ptg
Q-G vorticity equation
Q-G thermodynamic equation
We now have two equations in two unknowns, and
We will solve these to find an equation for , the vertical motion in pressure
coordinates, and for , the change of geopotential height with time.t
p
ffVftf
g
0
2
0
2
0
11
PV
ptg
1
2
Derive the Q-G omega equation (equation for vertical velocity in pressure coordinates)
1. Assume is constant
2. Take
3. Take of Eq. 1.3. Reverse order of differentiation on left side of equation4. Subtract: (1) – (2)5. Rearrange terms
2.1 2
0
Eqnf
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
p
The QG omega equation can be derived including the friction term and the diabatic heating term. We will not do this here, but I will simply show the result.
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
Cp
RK
p
f
Pg
120
QG OMEGA EQUATION
BEFORE WE PROCEED TO TRY TO UNDERSTAND THIS EQUATIONS
ANSWER THIS QUESTION:
WHAT PHENOMENA OF IMPORTANCE ARE WE IMPROPERLY REPRESENTING IN THE DERIVATION
OF THESE EQUATIONS?
By assuming that the static stability,
p
is constant
WE HAVE ELIMINATED A TERM RELATED TO FRONTS!
We took x, y, and P derivatives assuming is constant
Real Atmosphere () QG Atmosphere ()
Where does all the precipitation occur?
Where does all the precipitation occur?
ALONG FRONTS!
By assuming that the static stability,
p
is constant
WE HAVE ELIMINATED A TERM RELATED TO JETSTREAKS!
We took x, y, and P derivatives assuming is constant – this implies that thevertical wind shear is constant, which can’t happen in a jetstreak environment
To illustrate quasi-geostrophic vertical motion in a real system consider the cyclone below
700 mb height and rainwater field simulated by MM5 model
Vertical motion calculated at 700 mb by MM5 Model
Vertical motion attributed to terms in QG Omega equation
Residual vertical motion (not attributable to QG forcing)
Quasi-Geostrophic Equations describe the broad scale ascent and descentin troughs and ridges and the propagation of these troughs and ridges.
Quasi-Geostrophic interpretation of the atmosphere:
Quasi-Geostrophic Equations do not describe the mesoscale ascent and descentassociated with ageostrophic circulation near fronts and within jetstreaks,
nor the propagation of low and high pressure systems due to jetstreaks.
What the quasi-geostrophic solutions do:
What the quasi-geostrophic solutions do not do:
(These are the motions are most responsible for significant weather)
So what does the QG system equation tell us about the atmosphereand how should we use it to diagnose vertical motions?
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
cp
RK
p
f
Pg
120
QG OMEGA EQUATION
We are taking two derivatives of the vertical motion
Let’s write as a Fourier series in x: nxbnxa nnn
sincos1
When we take two x derivatives of this series we get
nxnbnxna nnn
sincos 22
1
2
Therefore: First term is proportional to - Read the first term as “Rising motion”
P
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
cp
RK
p
f
Pg
120
QG OMEGA EQUATION
The second term is the vertical derivative of the absolute vorticity advection:
Rising motion is proportional to“Differential Positive Vorticity Advection”
The geostrophic wind is strong at jetstream level and the height gradients are greatest.
ff
Vp
fg
2
0
0 1
The geostrophic wind is weak at low levels and the height gradients are weak.
Vertical gradient in vorticityadvection is strongest, onaverage, in middle troposphere
(so absolute vorticity is typicallyPlotted on the 500 mb surface)
Relationship between
Shear and curvature in the Jetstream
Absolute vorticity and Abs. Vorticity Advection
andVertical Motion
ff
Vp
fg
2
0
0 1
P
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
cp
RK
p
f
Pg
120
QG OMEGA EQUATION
We are taking two derivatives of the thickness advection,
nxbnxaA nnn
sincos1
When we take two derivatives of this series we get
Ap
V g
Let’s write A as a Fourier series in x:
p
is the depth, or thickness, of the layer between two pressure surfaces. This depth
is proportional to the mean temperature of the layer.
nxnbnxnaA nnn
sincos 22
1
2
Read the term inside the parentheses as temperature advection
However, it is the Laplacian of temperature advection we are interested in…
Let’s rewrite this term:
p
Vp
V gg
22
Multiply term by -1 twice to get in form consistent with temperature advection
p
Vp
V gg
2 If the gradient of temperature advection
is constant, then the term is zero!
THEREFORE: will be non-zero only in areas where the thermal advection pattern in non uniform
Heterogeneity in warm advection on a pressure surface implies upward motion
When air is ascending on an isentropic surface…
…the projection of the wind on to a constant pressure surface appears as warm air flowing toward cold air (warm advection)
But what if isentropic surface is moving northward at the same time?
Warm advection involves both flow on the isentropic surface and movement of the surface
When air is descending on an isentropic surface…
…the projection of the wind on to a constant pressure surface appears as cold air flowing toward warm air (cold advection)
Heterogeneity in cold advection on a pressure surface implies downward motion
But what if isentropic surface is moving southward at the same time?
Cold advection involves both flow on the isentropic surface and movement of the surface
The vertical derivative of friction acting on the geostrophic vorticity
Or more simply: The rate that friction decreases with height in thepresence of cyclonic vorticity in the boundary layer
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
cp
RK
p
f
Pg
120
QG OMEGA EQUATION
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
cp
RK
p
f
Pg
120
QG OMEGA EQUATION
Again: Two derivatives in x,y ……….proportional to negative
dt
dQ
cP
1
With additional minus sign:
Rising motion is proportional to the Laplacian of the diabatic heating rate (modulated by static stability)
Sinking motion is proportional to the Laplacian of the diabatic cooling rate (modulated by static stability)
Heterogeneity in the diabatic heating leads to rising motion
low static stability
p
small
335
650
250
135
mbKmb
K
p/06.0
315
20
mbKmb
K
p/65.0
115
75
high static stability
p
large
From our earlier discussion of isentropic coordinates:
p
dtd
For a given amount of diabatic heating, a parcel in a layer with high static stabilityWill have a smaller vertical displacement than a parcel in a layer with low static stability
From our earlier discussion of isentropic coordinates:
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
cp
RK
p
f
Pg
120
QG OMEGA EQUATION
Broad scale (synoptic scale) rising motion in the atmosphere is proportional to:Differential positive vorticity advectionHeterogeneity in the warm advection fieldThe rate of decrease with height of friction in the presence of vortexHeterogeneity in the diabatic heating rate
SUMMARY
Broad scale (synoptic scale) desending motion in the atmosphere is proportional to:Differential negative vorticity advectionHeterogeneity in the cold advection fieldThe rate of decrease with height of friction in the presence of vortexHeterogeneity in the diabatic cooling rate
Recall for isentropic coordinates:
dt
dppV
t
p
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
CP
R
P
12
Diabatic term
dt
dp
TemperatureAdvection
VerticalMotion
Let’s compare the QG Omega Equation* with the Omega equation we derived a long time ago when we considered isentropic coordinates
*Note… I dropped friction term since we did not consider it in our discussion of isentropic coordinates
Let’s compare the QG Omega Equation* with the Omega equation we derived a long time ago when we considered isentropic coordinates
Recall for isentropic coordinates:
dt
dppV
t
p
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
CP
R
P
12
Diabatic term
dt
dp
TemperatureAdvection
VerticalMotion Must be
Related!
*Note… I dropped friction term since we did not consider it in our discussion of isentropic coordinates
Cold advection
850 mb heights, temperature, winds
Warm advection
TEMPERATURE ADVECTION TERM
Descent on isentropic surface
Pressure, winds and RH on 290 K isentropic surface
Ascent onisentropic surface
TEMPERATURE ADVECTION TERM
dt
dppV
t
p
p
Vff
Vp
f
p
fgg
22
0
02
2202 11
dt
dQ
CP
R
P
12
So how are orange terms related???
Differential absolute vorticity advection measures the rate at which potential temperature surfaces rise or fall as ridges and troughs propagate along!
Vertical displacement of isentrope =
t
P
(Air must rise for isentrope to be displaced upward)
500 mb w 700 mb w
