Structure and dynamical characteristics of mid-latitude fronts

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


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




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






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


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


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


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

Documents

Structure and dynamical characteristics of mid-latitude fronts