Structure and dynamical characteristics of mid-latitude fronts
Front: A boundary whose primary structural and dynamical characteristic is a larger then background density (temperature) contrast
A zero-order front: A front characterized by a discontinuity in temperature and density
This type of front does not occur in the atmosphere, but does exist where twofluids of different density approach one another as illustrated below
mixing associated with friction prevents atmospheric fronts from becoming zero-order
ATMOSPHERIC FRONTSGradients in temperature and density are discontinuous across fronts
Let’s for the moment consider a zero-order front
We will assume that: 1) pressure must be continuous across the front 2) front is parallel to x axis 3) front is steady-state
dzz
pdyy
pdp
Warm side of front dzz
pdy
y
pdp
ww
w
Cold side of front dzz
pdy
y
pdp
cc
c
Substitute hydrostatic equation and equate expressions:
gdzdyy
p
y
pwc
wc
0
Solve for the slope of the front
wc
wc
g
yp
yp
dy
dz
wc
wc
g
yp
yp
dy
dz
For cold air to underlie warm air, slope must be positive
Therefore:
1) Across front pressure gradient on the cold side must be larger that the pressure gradient on the warm side
y
p
fug
1
Substituting geostrophic wind relationship
wc
gcgw
g
uuf
dy
dzcw
cw gg uu
2) Front must be characterized by positive geostrophic relative vorticity
0dy
dug
Real (first order) fronts
1) Larger than background horizontal temperature (density) contrasts
2) Larger than background relative vorticity
3) Larger than background static stability
Working definition of a cold or warm front
The leading edge of a transitional zone that separates advancing cold (warm) air from warm (cold) air, the length of which is significantly greater than its width. The zone is characterized by high static stability as well as larger-than-background gradients in temperature and relative vorticity.
EXAMPLES OF FRONTS
EXAMPLES OF FRONTS
EXAMPLES OF FRONTS
FrontogenesisAgeostrophic Circulations associated with fronts and jetstreaks
dt
dF
The formation of a front is called frontogenesis
The decay of a front is called frontolysis
These processes are described quantitatively in terms of the Three-Dimensional Frontogenesis Function
Where is the magnitude of the 3-D potential temperature gradient
and the total derivativedt
d
implies that the change in the gradient is calculated following air-parcel motion
The processes by which a front forms or decays can be understood more directly by expanding the frontogenetical function
dt
dF
Algebraically, this involves expanding the total derivative
zw
yv
xu
tdt
d
expanding the term involving the magnitude of the gradient2/1
222
zyx
Reversing the order of differentiation, differentiating, and
then using the thermodynamic equation
to replace the term in the resulting equation.
dt
dQ
Cp
p
dt
d
p
k10
dt
d
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0
The Three-Dimensional Frontogenesis Function
)
dt
dF
The solution
becomes
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
The terms in the yellow box all contain the derivativewhich is the diabatic heating rate. These terms arecalled the diabatic terms.
dt
dQ
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
The terms in this yellow box represent the contributionto frontogenesis due to horizontal deformation flow.
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
The terms in this yellow box represent the contributionto frontogenesis due to vertical shear acting on a horizontal temperature gradient.
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
The terms in this yellow box represent the contributionto frontogenesis due to tilting.
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
The term in this yellow box represents the contributionto frontogenesis due to divergence.
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
dt
dQ
xp
p
C
xF
p
01/
Weightingfactor
Adjustmentfor specific
heat of air andair pressure
Horizontal gradient in diabatic heating or cooling rate
Magnitude of gradient in one directionMagnitude of total gradient
dt
dQ
xp
p
C
xF
p
01/
dt
dQ
yp
p
C
yF
p
01/
Gradient in diabatic heatingin x direction
Gradient in diabatic heatingin y direction
Can you think of other examples where this term might be important to frontogenesis?
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
dt
dQp
zC
pzF
p
0/
Weightingfactor
Adjustmentfor specificheat of air
Vertical gradient in diabatic heating or cooling rateadjusted for pressure altitude
Magnitude of gradient in one directionMagnitude of total gradient
dt
dQp
zC
pzF
p
0/
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
yx
v
xx
u
yy
v
xy
u
Stretching deformation
Shearing deformation
yy
vy
xx
uxF
//
Stretching Deformation
Weighting factors
Deformation acting on
temperature gradient
Deformation acting on
temperature gradient
Magnitude of gradient in one directionMagnitude of total gradient
x
y
x
y
Time = t Time = t + t
T
T-
T- 2T
T- 3T
T- 4T
T- 5T
T- 6T
T- 7T
T- 8T
TT- T- 2TT- 3T
T- 4TT- 5T
T- 6TT- 7TT- 8T
yy
vy
xx
uxF
//
Stretching Deformation
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
yx
v
xx
u
yy
v
xy
u
Stretching deformation
Shearing deformation
xy
uy
yx
vxF
//
Shearing Deformation
Weighting factors
Magnitude of gradient in one directionMagnitude of total gradient
Deformation acting on
temperature gradient
Deformation acting on
temperature gradient
x
y
TT-
T- 2TT- 3T
T- 4TT- 5T
T- 6TT- 7T
T- 8T
x
y
TT-
T- 2TT- 3T
T- 4TT- 5T
T- 6TT- 7T
T- 8T
xy
uy
yx
vxF
//
Shearing Deformation
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
yz
v
xz
uzF
/
Vertical shear acting on a horizontal temperature gradient(also called vertical deformation term)
Weighting factor
Magnitude of gradient in one directionMagnitude of total gradient
Vertical shear of E-W windComponent acting on
a horizontal temp gradient in xdirection
Vertical shear of N-S windcomponent acting on
a horizontal temp gradient in ydirection
yz
v
xz
uzF
/
Vertical shear acting on a horizontal temperature gradient
Before
x
z z
x
After
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
Tilting terms
Weighting factor
Magnitude of gradient in one directionMagnitude of total gradient
TiltingOf vertical Gradient
(E-W direction)
zy
wy
zx
wxF
//
TiltingOf vertical Gradient
(N-S direction)
Tilting terms
zy
wy
zx
wxF
//
Before
After
x or y x or y
z z
zx
w
yx
v
xx
u
dt
dQ
xp
p
CxF
p
011 (
zy
w
yy
v
xy
u
dt
dQ
yp
p
Cy p
01
zz
w
yz
v
xz
u
dt
dQp
zC
p
z p
0 )
Differential vertical motion (also called divergence term because
w/ z is related to divergence through continuity equation)
Weighting factor
Magnitude of gradient in one directionMagnitude of total gradient
zz
wzF
/
Compressionof vertical Gradient
by differential vertical motion
Differential vertical motion
zz
wzF
/
Before
x or y
z
After
x or y
z
yy
v
xy
u
yyx
v
xx
u
xF D
12
Another view of the 2D frontogenesis function
y
v
x
uD
y
u
x
v
y
v
x
uF
1
21FD
x
u
21FD
y
v
22F
x
v
Recall the kinematic quantities: divergence (D)vorticity ()
stretching deformation (F1)shearing deformation (F1).
y
u
x
vF
2
and note that:
22
F
y
u
Substituting:
y
FD
x
F
yy
F
x
FD
xF D
2222
1 12212
y
FD
x
F
yy
F
x
FD
xF D
2222
1 12212
This expression can be reduced to:
yxF
yxF
yxDF D
2
22
1
22
2 22
1
x
y
x
y Shearing and stretching
deformation“look alike” with
axes rotated
We can simplify the 2D frontogenesis equation by rotating our coordinate axes to align with the axis of dilitation of the flow (x´)
22
1
2
2 2
1
yxFDF D
This equation illustrates that horizontal frontogenesis is only associated with divergence and deformation, but not vorticity
22
1
2
2 2
1
yxFDF D
Yet another view of the 2D frontogenesis function
yy
v
xy
u
yyx
v
xx
u
xF ggggDg
12
Let’s replace u and v with their geostrophic components and examine geostrophic frontogenesis:
jy
Vi
x
VfQ gg ˆ,ˆ
Recalling the Q vector
Therefore:
1
1Q
f 2
1Q
f
Q
fF Dg
1
2
Magnitude of geostrophic frontogenesis is a scalar multiple
of the cross isentropic component of the Q vector
Convergence of Q vectors associated with rising motion
Implication: Direct circulation (warm air rising and cold air
sinking) associated with frontogenesis
Divergence of Q vectors associated with descending
motion
Is geostrophic frontogenesis, as represented by the Q vector, sufficient to describe the circulation about a front?
xk
xx
u
dx
d
dt
d g
Consider a simple north-south front undergoing frontogenesis by the geostrophic wind
Assume that the confluence occurs at a constant rate k
dtx
kdx
dd
integrate to get:
kt
t
exx 0
Using typical values of it takes 105 seconds or about 1 day for geostrophic
confluence to increase the temperature gradient by a factor of e (2.5)
x
ug
example from real atmosphere
In 6 hours, temperature gradient doubles, a factor of 8 larger than that expected from scale analysis of geostrophic
confluence
Implication: ageostrophic non-QG forcing is important to the
circulations on cross frontal scale
QUASI-GEOSTROPHIC THEORY IS INSUFFICIENT TO ACCOUNT FOR THE VERTICAL MOTIONS IN THE
VICINITY OF FRONTS