Section 8 Vertical Circulation at Fronts

1. Structural and dynamical characteristics of mid-latitude fronts

2. Frontogenesis

3. Semi-geostrophic equations

4. Symmetric instability

1. Structure and dynamical characteristics of mid-latitude fronts

EXAMPLES OF FRONTS

A front is a transition zone between different air masses. It is characterized by:

1. Larger than background horizontal temperature (density) contrasts ( strong vertical shear)

2. Larger than background relative vorticity

3. Larger than background static stability

4. a quasi linear structure (length >> width)

Let’s for the moment consider a zero-order front

We will assume that: 1) front is parallel to x axis 2) front is steady-state 3) pressure is continuous across the front 4) density and T are discontinuous across the front

dzz

pdy

y

pdp

Warm side of front dzz

pdy

y

pdp

ww

w

Cold side of front

€

dpc =∂p

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟c

dy +∂p

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟c

dz

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

€

dpc = dpwWe have

€

∂p

∂z= −ρgand

wc

wc

g

yp

yp

dy

dz

For cold air to underlie warm air, slope must be positive

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

The stronger the density (T) contrast becomes, the stronger is the vorticity at the front.

First-order fronts

1) Larger than background horizontal temperature (density) gradient

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 temperature gradient and relative vorticity.

2. Frontogenetic Function

the Lagrangian rate of change of the magnitude of potential temperature gradient

Move to the whiteboard and talk about 1D frontogenesis

dt

dF

Expanding the total derivative

€

d

dt=

∂

∂t+ u

∂

∂x+ v

∂

∂y+ w

∂

∂z

expanding the term involving the magnitude of the gradient

2/1222

zyx

3D Frontogenesis

(

€

F =1

∇θ

∂θ

∂x

1

Cp

p0

p

⎛

⎝ ⎜

⎞

⎠ ⎟

κ∂

∂x

dQ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥−

∂u

∂x

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂x

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂w

∂x

∂θ

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

€

+∂∂y

1

Cp

p0

p

⎛

⎝ ⎜

⎞

⎠ ⎟

κ∂

∂y

dQ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥−

∂u

∂y

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂y

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂w

∂y

∂θ

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

zz

w

yz

v

xz

u

dt

dQp

zC

p

z p

0

The Three-Dimensional Frontogenesis Function

)

dt

dF

The solution

becomes

€

F1D =d

dt(∂θ

∂x) =

∂

∂x(dθ

dt) −

∂u

∂x

∂θ

∂x−

∂v

∂x

∂θ

∂y−

∂ω

∂x

∂θ

∂pCompared to

• Confluence terms (or stretching deformation): with

• Shearing terms (or shearing deformation):

involved with

• Tilting terms: with derivative of omega

€

∂u

∂x,

∂v

∂y

€

∂v

∂x,

∂u

∂y

€

F =1

∇θ

∂θ

∂x

1

Cp

p0

p

⎛

⎝ ⎜

⎞

⎠ ⎟

κ∂

∂x

dQ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥−

∂u

∂x

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂x

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂w

∂x

∂θ

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪(

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 )

€

F =1

∇θ

∂θ

∂x

1

Cp

p0

p

⎛

⎝ ⎜

⎞

⎠ ⎟

κ∂

∂x

dQ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Temperature gradient Horizontal gradient in diabatic heating or cooling rate

If and have the same sign, it means the diabatic heating will increase the temperature gradient.

€

∂∂x

dQ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

€

∂∂x

dt

dQp

zC

pzF

p

0/

Vertical cross section of potential temperature

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 ) €

−∂u

∂x

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂x

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

−∂u

∂y

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂y

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

Stretching deformation

Shearing deformation

yy

vy

xx

uxF

//

Stretching Deformation

Deformation acting on

temperature gradient

Deformation acting on

temperature gradient

x

y

x

Time = t

T

T-

T- 2T

T- 3T

T- 4T

T- 5T

T- 6T

T- 7T

T- 8T

y

Time = t + t

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 ) €

−∂u

∂x

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂x

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

−∂u

∂y

∂θ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟−

∂v

∂y

∂θ

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

Stretching deformation

Shearing deformation

xy

uy

yx

vxF

//

Shearing Deformation

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 )

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 )

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

x or y

z

After

x or y

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 )

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 )

yz

v

xz

uzF

/

Vertical shear acting on a horizontal temperature gradient(also called vertical deformation term)

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 )

The term in this yellow box represents the contributionto frontogenesis due to divergence.

zz

wzF

/

Compressionof vertical Gradient

by differential vertical motion

Differential vertical motion

zz

wzF

/

Before

x or y

z

After

x or y

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 )

2D Frontogenetic Function

x

y

T

T-

T- 2T

T- 3T

T- 4T

T- 5T

T- 6T

T- 7T

T- 8T

x

y

TT-

T- 2TT- 3T

T- 4TT- 5T

T- 6TT- 7T

T- 8T

The stretching and shearing deformations “look like” one another:

yy

v

xy

u

yyx

v

xx

u

xF D

12

Another view of the 2D frontogenesis function

y

v

x

uD

€

ζ ∂v

∂x−

∂u

∂y

€

F1 =∂u

∂x−

∂v

∂y

21FD

x

u

21FD

y

v

22F

x

v

ζ

Recall the kinematic quantities: divergence (D)vorticity (ζ)

stretching deformation (F1)shearing deformation (F1).

y

u

x

vF

2

and note that:

22 ζ

F

y

u

Substituting:

y

FD

x

F

yy

F

x

FD

xF D

ζζ 2222

1 12212

y

FD

x

F

yy

F

x

FD

xF D

ζζ 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 dilatation of the flow (x´)

€

F2D =1

2∇θD ∇θ

2( ) + F

∂θ

∂ ′ x

⎛

⎝ ⎜

⎞

⎠ ⎟2

−∂θ

∂ ′ y

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

€

F = (F12 + F2

2)1/ 2where F is the total deformation

This equation illustrates that horizontal frontogenesis is only associated with divergence and deformation, but not vorticity€

F2D =1

2∇θD ∇θ

2( ) + F

∂θ

∂ ′ x

⎛

⎝ ⎜

⎞

⎠ ⎟2

−∂θ

∂ ′ y

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

€

F2D =∇θ

2(F cos2β − D)

Where F is the total deformation of the flow, β is the angle between the isentropes and the dilatation axis of the total deformation field, and D is divergence (D <0 for convergence)

€

F

∇θ2

∂θ

∂ ′ x

⎛

⎝ ⎜

⎞

⎠ ⎟2

−∂θ

∂ ′ y

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥= F cos2α = −F cos2βNote that

€

F2D =∇θ

2D +

F

∇θ2

∂θ

∂ ′ x

⎛

⎝ ⎜

⎞

⎠ ⎟2

−∂θ

∂ ′ y

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

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

€

F2D =∇θ

2(F cos2β − D)

Frontogenesis occurs 1)if non-zero is coincident with convergence (D<0)2)if the total deformation field (F) acts on isentropes that are between 0 and 45° of the dilatation axis of the total deformation. deformation. €

∇

3. S.G. vs. Q.G. Approximations

• Q.G.: S.G.

€

u = ug, v = vg + va

dug

dt= fva

€

u = ug + ua, v = vg + va , f = const

du

dt= fv −

∂φ

∂xdv

dt= − fu −

∂φ

∂y

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

€

ug >> ua , vg >> va

dgug

dt= fva

dgvg

dt= − fua

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

€

dg

dt=

∂

∂t+ ug

∂

∂x+ vg

∂

∂ywhere

Right side of equation represent the forcing (known from measurements or in model solution)

Geostrophic deformation Diabatic heating

, the streamfunction, is the response. V and ω can be derived from

€

−γ∂∂p

⎛

⎝ ⎜

⎞

⎠ ⎟∂2ψ

∂y 2 + 2∂M

∂p

⎛

⎝ ⎜

⎞

⎠ ⎟∂2ψ

∂p∂y+ −

∂M

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟∂2ψ

∂p2 = Qg −γ∂

∂y

dθ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

Sawyer-Eliassen Equation

Questions: 1) How is the thermal wind balance maintained by the transverse circ.? 2) Where should we expect upward motion (precipitation)?

€

dug

dt= fva

coldwarm

Cold

warm

Nature of the solution of the Sawyer-Eliassen Equation:

A direct circulation (warm air rising and cold air sinking) will result with positive forcing.

An indirect circulation (warm air sinking and cold air rising) will result with negative forcing.

Cold air Warm air

Isentrope

Cold air Warm air

Isentrope

€

−∂∂y

dθ

dt

⎛

⎝ ⎜

⎞

⎠ ⎟> 0

(heating in the warm sideand cooling in the cold side) will produceA thermally direct circulation and promoteFrontogenesis.

yy

V

xy

UQ gg

g

γ2

Geostrophic shearing

deformation

Geostrophic stretching

deformation

yy

VQ g

STg

γ2 Geostrophic stretching deformation

Note in this figure that both and are negative, implying

frontogenesis and a direct circulation in which warm air is rising and

cold air sinking.

y

Vg

y

Entrance region of jet

xy

UQ g

sHg

γ2 Geostrophic shearing deformation

Note in this figure that both and are positive, implying

frontogenesis and a direct circulation in which warm air is rising and

cold air sinking.

y

U g

x

confluent flow along front

Why does a spinning top stay upright?

• Buoyancy tends to stabilize air parcels against vertical displacements, and rotation tends to stabilize parcels with respect to horizontal displacements.

• If ordinary static and inertial stabilities are satisfied, is the flow always stable?

4. Symmetric Instability

• hydrostatic instability

• Inertial instability

0dz

d v stable 0dz

d v neutral 0dz

d v unstable

€

f f −∂ug

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟> 0 stable

€

f f −∂ug

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟= 0 neutral

€

f f −∂ug

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟< 0 unstable

MSI: an intuitive explanation

M = fy-ug

70

60

4030

dM/dy>0

M = absolute zonal momentum

see also: Jim Moore’s meted module on frontogenetic circulations & stability)

-

-

-

-

-

-

-

-

--

Dash: e

Solid: Mg

Potential Symmetric Stability PotentialPotential Symmetric INstability

e ee 2y

z

gM

z z

y y

gg MM 2

e ee 2

gM

gg MM 2

Stable Neutral Unstable

e ee 2

gM

g

g

M

M

2

€

red :θ e or θ v gMblue :

Symmetric instability evaluation

The flow satisfies and0dz

d v

€

f f −∂ug

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟z

> 0