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Atmospheric Science 4320 / 7320 Anthony R. Lupo

Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

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Page 1: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Atmospheric Science 4320 / 7320

Anthony R. Lupo

Page 2: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances:

Synoptic scale disturbance: 2000 - 6000 km (space) 1 - 7 days (time)

Page 3: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

Typical values:

3

4

1

110

10

mkg

sf

sm

Vh

1.0

101

10

10

23

23

23

fL

VRo

sm

np

smkg

np

sm

Vf

h

h

Page 4: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

Mesoscale disturbances: 10 - 2000 km, 1h - 1d

3

4

1

110

10

mkg

sf

sm

Vh

0.2

101

10

10

22

22

23

fL

VRo

sm

np

smkg

np

sm

Vf

h

h

Page 5: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

Hurricane

3

5

1

1105

50

mkg

sxf

sm

Vh

10

1051

105.2

22

23

fL

VRo

sm

xnp

sm

xVf

h

h

Page 6: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

Tornadoes: ][

3

4

1

110

50

mkg

sf

sm

Vh

5000

51

105

2

23

fL

VRo

sm

np

sm

xVf

h

h

Page 7: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

The horizontal Equation of motion for inviscid flow in terms of the geostrophic departure vector.

geohageo

hhh

VVV

Vkfpdt

dV

ˆ1

Page 8: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

The horizontal acceleration vector is always normal and to the right of the geostrophic departure vector

It’s magnitude is directly proportional to the magnitude of the departure vector.

Page 9: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

And here:

agh

agghh

depghh

Vkfdt

Vd

VkfVkfpdt

dV

VVkfpdt

dV

ˆ

ˆˆ1

ˆ1

Page 10: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

Then consider a case in which there are negligible spatial variations in Vgeo (spatially uniform PF) such that along the trajectory, Vg = constant

then, dt

Vd

dt

Vd

dt

Vd

dtVd agaggh

Page 11: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one becomes:

The time rate of change of the departure is always 90 degrees to the right of the Vag, and it’s magnitude is dependent on the magnitude of V departure!

agag Vkf

dt

Vd

ˆ

Page 12: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

The departure vector is of constant magnitude but rotates with time to the right - clockwise - with an inertial period of

f 2

Page 13: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one The Inertial Oscillation Is a

consequence of Unbalanced flow! Thus, it is an oscillation whose frequencey is tied to the Coriolis parameter.

This is a “gravity-type” wave that is emitted from an unbalanced flow situation. More appropriately, this is a “Kelvin” type wave or oscillation.

Page 14: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one

Interial oscillation influential in:

Intense convection (MCSs)

Unbalanced jet streaks associated with highly curved flow and rapidly developing cyclones

Page 15: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day one/two

Balanced flow in the friction layer.

Model I assume:

0

0

dt

Vd

FRPGFCF

VF

h

h

Page 16: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Thus the horizontal equations of motion become

The equation of motion in natural coords.

0ˆ1 FVfkp frh

nnp

ssp

PGF

nVfCO

sFF

fr

ˆ1

ˆ1

ˆ

ˆ

Page 17: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Thus for balance we must have for the s-component,

the frictional force balances the downstream component of the PGF.

ssp

sF ˆ1

ˆ0

Page 18: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Then for the n-component;

the CF balances the normal component of the PGF in Cartesian coordinates

nVfnnp

fr ˆˆ1

0

Page 19: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

The equation (BL)

Fkf

pkf

V he

ˆ1ˆ1

Page 20: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

The diagram

W/out Friction With Friction

Page 21: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Does our assumption for FR above hold up?

Balanced friction layer flow omitting the assumption of Fr acting directly opposite the horizontal velocity.

Page 22: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

The balance equation (Eq 1)

dtVd

FVfkp hfrh

0ˆ1

Page 23: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Thus we must determine by observation and calculation the direction of the friction vector:

PGF can be observed the coriolis force can be calculated

ph1

frVkfCF

ˆ

Page 24: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two and Fr can be determined from

equation 1 as a residual:

Diagrammatically, you’ll get a result similar to our other one, at least it will be “CEFGW”

FVfkp frh

ˆ1

Page 25: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Friction in terms of the ageostrophic velocity vector

but recall that by geostrophy,

0ˆ1 FVfkp frh

geoh Vfkp

ˆ1

Page 26: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

thus,

or,

0ˆˆ FVfkVfk frgeo

geofr VVkfF

ˆ

Page 27: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

so in this context,

Ffk

V

VkfF

VVV

ageo

ageo

ageogeofr

ˆ

ˆ

Page 28: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two Note of caution: some folks adhere to this

position, that Vageo is equal to the difference between the Vfric and Vgeo.

We have shown this mathematically to be the case. We also know that Vgeo is non-divergent in it’s most basic form.

Thus what we have stated above is that Vfric is COMPLETELY the divergent part of the wind in a balanced flow.

Page 29: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

However, you can also show mathematically, by Helmholtz partitioning that Vgeo contains a rotational and divergent part, and Vageo contains a divergent and rotational part (Keyser et al., 1989, MWR; Loughe et al., 1995 MWR, May, or Lupo, 2002, MWR )

dt

dV ag

ag

,

Page 30: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two Example:

Where “chi” is the velocity potential.

h

d

r

dr

Vp

and

fgz

where

V

kV

VVV

N

22

ˆ

Page 31: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Diagramatic Example of Vfr development

Start with a system in geostrophic balance.

Page 32: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Wind adjusts such that PGF is balanced by CO, and is steady state.

Introduce a Frictional decelerating force (Fric.) (opposite the wind)

Page 33: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Friction reduces Vh to a value less that Vgeo which then reduces the CF, to a value less than that of the PGF resulting in a net force in the normal direction across the isobars towards the lower pressure causing air to turn to it’s left.

Page 34: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Thus final balance can be achieved if Vfr adjusts so that:

Page 35: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two Properties of balanced flow in the friction

layer

A) Horizontal non accelerating flow

B) Vfr < Vgeo or subgeostrophic wind speeds

C) Vfr has a component across the isobars from high to low

D) cross isobaric component is proportional to the Frictional force

Page 36: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two Friction layer convergence and

divergence results in vertical motions.

This will have important implications for when we examine the Vorticity, Omega Equations, etc....

Page 37: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

Low pressure (surface convergence) gives upward motion. However, with regard to a developing system where vertical motion is forced aloft, the friction acts against cyclone development.

Page 38: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day two

High pressure (surface divergence) gives downward motion. However, with regard to a developing system where vertical motion is forced aloft, the friction acts against anticyclone development.

Page 39: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three Gradient Flow

(Holton 65 - 68 and Hess p 180 ff)

Gradient flow is unbalanced frictionless flow along a curved path at a constant speed and parallel to the isobars (isoheights)

Page 40: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

Recall we observe that Vh tends to be parallel to the isobars in straight and curved flow regions.

Assumptions:1)

2) 3)

hVV

0F

nR

V

dt

Vd

c

hh ˆ2

Page 41: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three Normal component of centripetal acceleration

only. Rc (Radius of curvature) is positive for cyclonic flow and negative for anticyclonic flow.

4) We’ll also assume that Rc = Ri, or the curvature of the flow is the same as the curvature of the isobars.

5) acceleration = PGF - CF, normal acceleration due to unbalanced flow between PGF and CO.

Page 42: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

The gradient wind equation

The horizontal equation of motion reduces to only the normal (n) component equation. (No (s) component, since there is no coriolis component in that direction)

Page 43: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three The expression:

so, In scalar form

Ri = radius of curvature of the isobars

nVfnnp

nR

Vgr

i

gr ˆˆ1

ˆ2

gri

gr Vfnp

R

V

1

2

Page 44: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

Equation above in form of (see it?):

where Vgr:

npRRffR

V iiigr

42

22

cbxax 2

Page 45: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

Recall: We must solve for the gradient wind by either “completing the square”, i.e., add 0.5 the “b” term and squaring, then adding to both sides. Alternatively, we can use the quadratic formula:

aacbb

r2

42

2

2,1

Page 46: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

Examine the possible solutions of the Gradient Wind (Vgr) equation.

In agreement with polar coordinate convention, counterclockwise (clockwise) flow is a positive (negative) value of Vgr.

Page 47: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

Consider positive root:

As the pressure gradient goes to zero, Vgr goes to zero

npRRffR

V iiigr

42

22

Page 48: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

Oh yeah? Show me!

0np

42

22ii

gr

RffRV

Page 49: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three

But, if the pressure gradient force > 0, then Vgr is positive. (Observed Low, counterclockwise flow)

0,0

iRnp

!,42

22

cyclonicV

PosRffR

V

gr

iigr

Page 50: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three/four

Diagram!

Page 51: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day three/four

If the pressure gradient force < 0, then Vgr is negative (Observed High, clockwise flow)

0np

!,

...42

22

icanticyclonV

RffRV

gr

iigr

Page 52: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Diagram!

Page 53: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Consider negative root of Vgr:

As the pressure gradients go to zero, Vgr approaches -f Ri (anticyclonic motion with no pressure gradient limit). This is the motion of the Inertia Circle.

Page 54: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

The Math!

igr

iigr

iiigr

fRV

fRfRV

npRRffR

V

22

42

22

Page 55: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

If the pressure gradient force > 0 (Low pressure), Vgr is negative (clockwise flow around a Low)

gr

iigr

iiigr

V

PosRffR

V

npRRffR

V

.42

4222

22

Page 56: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four Thus, this is mathematically possible,

but not observed on synoptic and larger-scales. Clockwise tornado?

][

Now, if the pressure gradient force < 0 (High pressure), get two solutions:

Page 57: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

1) anomalously strong anticyclonic (clockwise) flow

(but “radical term” stays +)

npRRffR

V iiigr

42

22

Page 58: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

2) impossible cyclonic (counterclockwise flow) around high pressure.

npRRf

npRRffR

V

ii

iiigr

4

4222

22

Page 59: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Consider the Common Case (positive roots):

If pressure gradient is (+) or Low Pressure:

The radical: 04

22

npRRf ii

Page 60: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four never becomes negative, so there’s no

theoretical pressure gradient limit to the strength of low pressure.

Hurricanes and tornadoes can have incredible wind speeds and pressure gradients!

Also, their radius of curvature can be quite small. E.g., hurricane eye can be as small as 2 km!

Page 61: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

There is a limit on how strong a “high” can get. The radical:

cannot be less than zero thus,

04

22

npRRf ii

npRf

npRRf ii

1

4

42

22

Page 62: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Or, you can look at this equation and determine that there are no limits on the Radius of curvature for low pressures, while, for a high pressure there is a limit which the Radius of curvature cannot fall below.

Page 63: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Compare Geostrophic and Gradient Wind Values

First, Vg as a special case of Vgr.

Vgr:gr

i

gr Vfnp

R

V

1

2

Cent. PGF CO

Page 64: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

In the case of geostrophic flow, the radius of curvature goes to infinity, and:

geogr

gr

Vnp

fV

COPGF

orR

VR

1

0,2

Page 65: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Gradient Flow vs. Geostrophic Flow

Gradient wind equation:

gri

gr Vfnp

R

V

1

2

Page 66: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Substitute the geostrophic component into the gradient wind equation.

We get:

np

fVgeo

1

grgeoi

gr VVfR

V

2

Page 67: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four

Now Cyclonic Gradient Flow:

Note PGF > Co

and Ri = Rc > 0

so,geo

i

grgr V

fR

VV

2

Page 68: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four So,

(1)

(1) is now a “correction term”

Thus Vgr < Vgeo (subgeostrophic)

geoi

grgr V

fR

VV

2

Page 69: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four Anticyclonic Gradient Flow

CO > PGF

Ri = Rc < 0

so,

Thus, Vgr > Vgeo

geoi

grgr V

fR

VV

2

Page 70: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day four/five

Gradient Flow imbalance and Adjustments in a ridge trough system, containing a region in the ridge where |Ri| < |Ri min|

(see diagram and handout)

Page 71: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day five Divergence / Convergence patterns associated with

gradient flow through a ridge and trough system with uniform isoheight contour spacing.

Uniform contour spacing : Geostrophic flow everywhere is the same, thus in “the straightaways” Vgeo = Vgr.

This depends on the assumption that f = fo though, because when f varies, some of the divergence/convergence is the result of the variance of f. (Recall, I said we could show this via Helmholtz partitions also). Though we can assume based on our drawings that this divergence is small.

Page 72: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day five

Convergence: Since winds in ridge are super-geostrophic, we get speed convergence between ridge and trough line. Convergence aloft implies divergence at the surface and high pressure. Can maintain or develop high pressure.

Page 73: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day five/six

A map:

Page 74: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day five/six

Divergence: Since winds in trough are sub-geostrophic, we get speed divergence between ridge and trough line. Divergence aloft implies convergence at the surface and low pressure. Can maintain or develop low pressure.

Page 75: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Eulerian flow

1. assumptions

!.

0.

0.

.

onlyPGFd

COc

Frb

VVa hs

Page 76: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Equation of Motion becomes

Eulerian flow types (there are 3!):

Type 1 (straight isobars):

dtVd

dtVd sh

Page 77: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Type II (tangential and centripetal acceleration) (curved isobars):

nRV

dt

Vd

dt

Vd

c

sh ˆ2

Page 78: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Type III (Cyclostrophic – closed isobars):

nRV

dt

Vd

c

h ˆ2

Page 79: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six Cyclostrophic flow (Eulerian flow

type 3)

Assumptions:

1)Flow is horizontal (2- dimensional) 2) Friction = 0 3) PGF >> CO of Ro >> 10

Page 80: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

4) flow is parallel to the isobars 5) acceleration is due to

centripetal acceleration only (PGF = Centripetal force)

nRV

dtVd

ac

h ˆ2

Page 81: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Thus the equation of motion in natural coordinates reduces to:

nnp

nR

V

i

cyc ˆ1

ˆ2

Page 82: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

in scalar form,

or,np

R

V

i

cyc

1

2

npR

V icyc

Page 83: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six Thus, we must now examine the

behavior of the radical term, which is negative, but only possesses two real solutions:

Cyclonic

Anticyclonic 0,0

iRnp

0,0

iRnp

Page 84: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

The diagrams:Cyclostrophic - cyclonic Cyclostrophic anticyclonic

Page 85: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six We can only interpret such flows

as having a low pressure center, no high pressure centers, else we get imaginary roots.

This type of flow reasonably approximates the eyewall of intense hurricanes, tornadoes, and dustdevils.

Page 86: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Hurricanes are all cyclonic disturbaces

95 - 99% of tornadoes are cyclonic

but dustdevils tend to be distributed 50 - 50.

Page 87: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six Inertial Flow

A horizontal-frictionless flow in which CO is the dominant force:

Assume: V3 = Vh (2 dimensional flow)

Coriolis force dominant.

Page 88: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

Horizontal equation of motion in natural coordinates reduces to:

nVfnR

V

or

nVfnR

V

hc

cyc

hc

cyc

ˆˆ

ˆˆ

2

2

Page 89: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six Since the Co is always normal and to the right

of Vh, the acceleration is normal and to the right of Vh.

Diagrammatically:

We have a constant Vh being turned anticyclonically (toward the right), thus motion similar to the inertial oscillation, even though each was brought about in a slightly different manner.

Page 90: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six

The Inertial period with latitude ():

anticyclonic!

0ˆˆ2

cchc

cyc RfV

RnVfnR

V

sin

12

22

ffR

RVR

c

c

h

ci

Page 91: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day six (These are

observed in oceanic circulations – Inertial Gyres)

Latitude (degrees)

Time (hours)

15 46.2

30 23.9

45 16.9

60 13.8

75 12.4

90 12.0

Page 92: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven Acceleration of the horizontal

winds and the development of geostrophic departures.

Recall from earlier:

The Navier - Stokes equations for frictionless flow

Page 93: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

The equation:

From the definition of geostrophic flow

hhh Vkfp

dtVd

ˆ1

geoh Vkfp

ˆ1

Page 94: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

and we get:

where Vageo = V - Vgeo

hgeoh VkfVkf

dtVd

ˆˆ

ageoh Vkf

dt

Vd ˆ

Page 95: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

so, if we solve for the ageostrophic wind,

(1)

this is the most common formulation so we’ll use it.

dt

Vd

fk

V hageo

ˆ

Page 96: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

Here’s an alternative (Trenberth and Chen, 1988–JAS, Lupo, 2001).

then recall that the acceleration is perpendicular and to the right of Vageo.

2

ˆ

ffk

V dageo

Page 97: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

Let’s expand the total derivative on the RHS into its natural components (s,n,z,t)

z

Vw

s

VV

t

V

dt

Vd hhh

hh

Page 98: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven the approximate Vh as Vgeo (nothing

magic here, just an approximation.

(2)

then substitute 2 into 1 and get,

(A) (B) (C)

z

Vw

s

VV

t

V

dt

Vd geogeoh

geoh

z

Vw

fk

s

VV

fk

t

V

fk

V geogeoh

geoageo

ˆˆˆ

Page 99: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven Term (A) The ageostrophic component due

to local changes in the wind with time.

Term (B) The Ageostrophic component due to horizontal advection

Term (C) The ageostrophic component due to vertical advection or vertical motion

Page 100: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

Term A The isallobaric wind.

If,

and by definition (recall our substitution earlier)

t

V

fk

V geoageo

ˆ

pfk

V hgeo ˆ

Page 101: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

then taking the derivative with time,

and,

pfk

tt

Vh

geo

ˆ

tp

fk

t

Vh

geo

ˆ

Page 102: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven Where ( ) is the local pressure

tendency

lines of constant pressure tendency are called isallobars

so,

is the isallobaric gradient

tp

tp

h

Page 103: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven

and,

is the isallobaric component of the wind.

tp

fV hageo 2

1

Page 104: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven The ageostrophic component of the

wind has a component due to the pressure field change, which is proportional to the magnitude of the isallobaric pressure gradient.

This ageostophic component is normal to the isallobars and directed toward pressure fall regions.

Page 105: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven Using typical values for pressure

changes in the atmsphere, and gradients of pressure changes over say, 1000 km.

The Isallobaric component of the ageostrophic wind is on order of 5 - 10 m/s. Very significant!

Page 106: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day seven/ eight

Term B: the ageostrophic component due to horizontal advection

Vageo;

s

Vk

f

VV geoh

Bageo

ˆ)(

s

VV geo

ageo

Page 107: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Local geostrophic imbalance, “geostrophic adjustment”, as we go from one geostrophic state to another.

Consider first straight isobars (diffluent) :

0s

Vgeo

Page 108: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day Eight

Last page courtesy of Uccellini and Kocin (1987)

Vgeo slows down (since it is proportional to PGF), thus locally CO exceeds PGF and Vgeo turns toward the right (anticyclonic curvature). (Jet Exit!)

Page 109: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day Eight

3-D Jet streak (J.T. Moore)

Page 110: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day Eight

Rising or sinking air (Univ. Illinois)

Page 111: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day Eight

Page 112: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Consider then confluent isobars

Vgeo speeds up (since it is proportional to PGF), thus locally PGF exceeds CO and Vgeo turns toward the left (cyclonic curvature).

0s

Vgeo

Page 113: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Consider this component for typical synoptic scales: (it’s on order of 1 m/s)

If the Isobars were straight, then Vageo would be 0, as would the partial V / partial s term.

Page 114: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Consider Curved isobars

curved isobars, but uniformly spaced, we covered with the gradient wind. (where the Vageo is forced by centripetal acceleration)

Page 115: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

You could consider the two together, curved isobars and diffluent or confluent flow.

ageogeogr VVV

s

VV h

dep

Page 116: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Term C: The Ageostrophic component due to vertical motions.

With vertical motions, we must have shear as dictated by:

z

V

fw

kV geoCageo

ˆ

)(z

VV geo

ageo

Page 117: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

The vertical shear of the geostrophic wind is of course, the thermal wind!!

Consider Vh or Vgeo of constant direction, but increasing with height:

Page 118: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Like This

Page 119: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Or consider Vgeo of constant magnitude but veering with height (thus w > 0)

Page 120: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eightOr consider Vgeo of constant magnitude but veering with height (thus w > 0)

Page 121: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Or

Page 122: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

Vh = super geostrophic for veering (+) with upward motion (+) or backing (-) with downward (-) motion

Vh = subgeostrophic for backing (-) with upward (+) motion or for veering (+) with downward (-) motion.

Page 123: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight We can consider the two combined

for our typical warm air advection situation (veering w/height)

Vh has a component across isobars toward lower pressure

Page 124: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight

backing w/ height is cold air advection

Cross isobaric flow toward higher pressure.

Term C using typical synoptic scale values is on order of 2 m/s

Page 125: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day eight/nine

The Isobaric Coordinate system:

Until now, we have worked our basic equations in the x,y,z,t coordinate system, let’s work some of our basic concepts in the x,y,p coordinate system.

Page 126: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

Now:

dtdp

vdtdy

udtdx

,,

Page 127: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine Consider a small portion of an

isobaric sfc in the x -z plane

Consider the change in variable Q along two paths at the same instant in time:

322131 QQQ

322131 QQQ

Page 128: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

The picture:

Page 129: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

Or

x varies z varies

(1) (2)

tyxtzytpy QQQ ,,,,,,

Page 130: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

but,

(1)

(2)

xxQ

Q tzy ,,

zzQ

Q tyx ,,

Page 131: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

so,

xz

zQ

xQ

xQ

or

zzQ

xxQ

Q

tzytpy

,,,,

Page 132: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

then since z/x is the slope of the isobaric surface in x,

tpytyxtzytpy xz

zQ

xQ

xQ

,,,,,,,,

Page 133: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

In the y - z plane

tpytyxtzytpy yz

zQ

yQ

yQ

,,,,,,,,

Page 134: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

Then let’s put in vector form, thus,

tpytyx

tzytpy

jyz

ixz

zQ

jyQ

ixQ

jyQ

ixQ

,,,,

,,,,

ˆˆ

ˆˆˆˆ

Page 135: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine or,

where,

= The gradient of Q on a p-surface

= The gradient of Q on a z-surface

= The gradient of z on a p-surface

zzQ

QQ pzp

QpQzzp

Page 136: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

The horizontal pressure gradient force:

In (x,y,z,t) coords:

pPGF h1

Page 137: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

substitute Q = Pressure

zzp

pp

then

zzQ

QQ

pph

pph

Page 138: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine

and get:

Since there is no pressure gradient on pressure surface!

zzp

p ph

Page 139: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day nine/ten

but from hydrostatic balance:

then,

gzp

zgpPGF ph 1

Page 140: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Why do we talk about PGF on a constant pressure surface? We can represent mass field in terms of pressure surfaces or height surfaces. Since these fields are “sloped” there is a component acting on them due to gravity. Recall definition of geopotential surface!

Page 141: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

The Inviscid 2 -D Navier-Stokes equation in x,y,p

We hammered on the geostrophic wind in lab!

hph Vkf

dtVd

ˆ

Page 142: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

The 3-D Del operator in the x,y,p,t coordinate system

The total derivative in x,y,z,t and x,y,p,t

kp

jy

ix

ˆˆˆ3

dtdp

pQ

yQ

vxQ

utQ

dtdQ

dtdz

wzQ

wyQ

vxQ

utQ

dtdQ

p

z

,

,

Page 143: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

The vertical coordinate of Newton’s 2nd Law: Hydrostatic Balance

zp

pQ

zQ

Page 144: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

now Q equals z;

In other words, NO Change!

gzp

gpz

zz

1

Page 145: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten Forms of the Hydrostatic equation

We can rewrite in other forms, to make it more consistent with coordinate system!

pTR

p

pTR

pzg

pzg

vd

vd

)3

)2

1)1

Page 146: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

The equation of continuity in (x,y,p,t)(Our cube)

Page 147: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Total mass in the left face:

substitute: to get:

dzdyu11

gdp

dz

dpdygu1

Page 148: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Total mass out of the right face:

But,

dzdyu22 dpdygu2

dxxu

uup

12

Page 149: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Net Flux of mass in/out of cube

dxdydpxu

thus

dxdydpxu

dpdygu

dpdygu

p

p

11

Page 150: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Thus in y direction

and the “p” dimension.

dxdydpyv

p

dxdydpp p

Page 151: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten Add ‘em up!

Now since hydrostatic balance holds, there is NO mass change, since the total mass between two isobaric surfaces remains the same! Thus, mass is conserved in x,y,p system. (Sutcliffe and Godart (1942)).

0

gdxdydp

pyv

xu

Page 152: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

In vector form:

But commonly expressed as:

033 V

pVhh

Page 153: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Compare to x,y,z form:

dtd

V

1

33

zw

Vhh

Page 154: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

As we saw earlier in the semester, we can use continuity to calculate vertical motions (velocity).

What is the relationship between w and ?

p

po

hho dpVp

Page 155: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Well,

where,

zp

wpVtp

dtdp

hh

pVpVpV ageogeoh

Page 156: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Recall:

and substitute to get:

A B C

pVBABA g

,cos

zp

wpVtp

dtdp

hageo

Page 157: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten

Typical orders of magnitude

Term A 10 hPa day-1

Term B 1 hPa day-1

Term C 100 hPa day-1

Page 158: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day ten/11

Thus,

Where,

gw

dtdz

wdtdp ,

Page 159: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

Note the sign convention via hydrostatic balance!

Upward motion +w -

Downward motion -w +

Page 160: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11 Let’s look at typical values for synoptic and mesoscales:

Synoptic Scale: w

Typical 1 cm s-1 10-3 hPa s-1

Strong Low 10 cm s-1 10 mb s-1

Mesoscale:

Cu 1 m s-1 100 mb s-1

Cb 10 m s-1 1 mb s-1

Page 161: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11 Advantages of the Isobaric Coordinate system (A

summary)

On an isobaric surface, isolines of temp (T) are also isolines of density, since P is const. unlike on a z-surface. Isotherms are also isentropes!

The geostrophic wind does not depend on gradients of pressure and density and “f” variations, it now depends on height gradients and ‘f’ variations.

The thickness is directly proportional to temperatures (Thermal wind and hysometric equations).

Page 162: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11 The vertical shear of the Vg is also

related to temperature gradients (thermal wind)

The equation of continuity is much simpler.

density no longer appears explicitly in N-S or Continuity equations.

Page 163: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11 The isentropic and sigma coordinate

systems

We define a variable Q(x,y,,t)

Recall to go from x,y,z,t to x,y,p,t, we examined some geometry. Let’s cut to the chase, since the derivations the same:

Page 164: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

Here’s the equation:

Note we see a pattern developing, for a general conversion to x,y,z,t to x,y,c,t

where c = any vertical coordinate:

zzQ

QQh

Page 165: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

The expression:

zzQ

QQ cch

Page 166: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

So, now we must transform the N-S and Continuity equations to isentropic and sigma coordinates:

isentropic: (let c = and Q = Pressure)

pPGF 1

Page 167: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

we get;

But,

zzp

ppz

gzp

Page 168: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

so,

(1)

then,

zgppz 11

pc

R

o

pp

T

Page 169: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

Now hold on,

and thus (after product rule),

pc

R

o

p

pT

Tp

pT

p

p

p

p

c

R pp c

R

ooc

R

o

p

1

Page 170: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

and using some math/algebra (first break up exponent):

and, what is the gradient of theta on a theta surface?

TT

T

p

pTp

p

p

p

p

p

p

c

R pp c

R

oo

o

c

R

o

p

)(

2

Page 171: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

See?

Let’s cancel some things and use the Eqn of state:

TT

ppc

R

p

1

TT

pTcp

111

Page 172: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

One more workout:

Tcp

and

Tcp

p

p

1

Page 173: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

now substitute into Eqn (1) and we get,

gzTczgTcp ppz 1

Page 174: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11

Wait! Isn’t M = CpT + gz, dry static energy (a potential surface in a dry atmosphere), or the Montgomery streamfunction.

Mpz 1

Page 175: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 11/12

Now, in sigma coordinates, where we assume sigma is a terrain following coordinate, and we specify a top (modeler’s coordinate!).

Q = Q(x,y,,t) topsfc

top

pp

pp

Page 176: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

Thus, the horizontal PGF is:

this will not simplify any more!

ppz

11

Page 177: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

The Navier Stokes equations in Theta and sigma coordinates:

hh

hh

Vkfpdt

Vd

VkfMdt

Vd

ˆ1

ˆ

Page 178: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

The geostrophic wind! (PGF = CO)

Theta coords

M

fk

V

VkfM

g

h

ˆ

ˆ

Page 179: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

Sigma Coords

fk

pfk

V

Vkfp

g

h

ˆˆ

ˆ1

xfxp

fv

yfyp

fu

g

g

11

11

Page 180: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

The 3-d del operator in the general (c) theta () and sigma () coordinates:

kjy

ix

kjy

ix

kc

jy

ix

ˆˆˆ

ˆˆˆ

ˆˆˆ

3

3

3

Page 181: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

The total derivative in the general (c), theta () and sigma () coordinates:

QQdtd

QVtQ

dtdQ

QQdtd

QVtQ

dtdQ

cQ

ccQ

dtdc

QVtQ

dtdQ

hh

hh

hh

Page 182: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

Hydrostatic Balance: Recall general form or chain rule:

Then: general coordinate:

gzc

cp

pc

gzc

Page 183: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

Now hydrostatic balance in theta coords:

Eq (1)

and;

p

pg

z

pc

R

o

pp

T

Page 184: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

becomes (How?);

apply chain rule;

tp

pcR

tT

Tt p

11

p

pcRT

T p

11

Page 185: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

substitute Eq. (1) into RHS.

Now use the Ideal Gas Law on RHS:

pcRT

T p

11

T

cTTcT

T pp

11111

Page 186: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

And then;

MTc

and

gzTcTcTc

p

ppp

Page 187: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

In sigma coordinates:

Equation 1

zg

p

and

gzc

cp

Page 188: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12

Now:

take derivative w/r/t p:

topsfc

top

pp

pp

topsfc ppp

1

Page 189: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 12/13

Substitute this into Equation (1):

ts

topsfc

pp

then

pp

Page 190: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13 Continuity equation:

In (x,y,,t), Let’s take a different approach this time, instead of looking at our “cube”, let’s look as some math instead:

(1)

Continuity in (x,y,z,t)dtd

V

1

33

Page 191: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

We can examine a “generalized” form of the divergence relationship.

)2(

33

cz

dtd

zc

zzc

c

VV

zw

VV ccchh

Page 192: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

Now we know that:

w = dz / dt

(3)cz

czVtz

dtdz

w hhc

Page 193: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

take the partial of equation (3) w/r/t “c” (See prod. Rule on RHS?)

)4(2

2

cz

ccz

cc

zcV

cz

Vcz

tdtdz

ccw

chhc

Page 194: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

Then, by chain rule:

(5)zc

cw

zw

Page 195: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

And substituting (4) into (2) and using (5), (3) and eventually into (1):

cc

cz

dtd

cz

Vdtd

cc

cz

dtd

cz

Vzw

V

cc

cz

ccV

tzc

Vzw

V

cc

cchh

cccchh

11

1

Page 196: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

Now A has the form of ??? (term A is the RHS of the final form above)

dtdA

AdtAd 1ln

Page 197: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

Then:

cc

Vcz

dtd

and

cc

Vcz

dtd

dtd

cc

cc

ln

ln1

Page 198: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

A generalized continuity equation:

cc

Vcz

dtd

cc

ln

Page 199: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

In theta coordinates:

or

V

zdtd

ln

V

pdtd

ln

Page 200: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

In sigma coordinates:

use the hydrostatic relationship:

Vz

dtd

ln

ts pp

Page 201: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

and get:

Vppdtd

tsln

Page 202: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

Which becomes:

V

ppVppppt

or

Vppdtd

tshtsts

tsln

Page 203: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

Let’s talk about fundamental Kinematic Concepts

In lab, we talked about divergence, which is a scalar quantity:

hh V

Page 204: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

We can prove that divergence is the fractional change with time of some horizontal area A.

dtdA

AVhh

1

Page 205: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13

and, then

dtyd

xdtxd

yyxdt

AdA

and

dtyxd

yxdtAd

A

11

11

Page 206: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Day one The orders of magnitude of PGF, CO, and Ro for a synoptic scale disturbances: Synoptic scale

Day 13