quasi-geostrophic theory chapter 5climate.ucdavis.edu/atm121/...chapter05-part02... · scale...

Quasi-Geostrophic Theory Chapter 5

Paul A. Ullrich

[email protected]

Part 2: The Equations of Quasi-Geostrophic Theory

This approach allows us to simplify the 3D equations of motion while retaining the time derivatives (prognostic equations) Suitable model to forecast the dynamics of weather systems in midlatitudes (on isobaric surfaces)

This approach is the foundation for many other methods of analyzing atmospheric motions.

Quasi-Geostrophic Theory

Observe: For extratropical synoptic-scale motions: •  Horizontal velocities approximately geostrophic (quasi-geostrophic) •  Atmosphere is approximately hydrostatic

Paul Ullrich Quasi-Geostrophic Theory March 2014

Scale Analysis

⇣ =@v

@x

� @u

@y

⇡ U

L

⇡ 10�5 s�1Relative Vorticity:

Planetary Vorticity: f0 ⇡ 10�4 s�1

⇣

f0⇡ 10�1

Definition: The Rossby number of a flow is a dimensionless quantity which represents the ratio of inertia to Coriolis force.

⇣

f0⇡ U

f0L⌘ Ro

In the mid-latitudes planetary vorticity is generally larger than relative vorticity.


Momentum Equation

Hydrostatic Relation

Continuity Equation

Thermodynamic Equation

Duh

Dt+ fk⇥ uh = �rp�

@�

@p= �RdT

p✓@u

@x

+@v

@y

◆

p

+@!

@p

= 0

@T

@t

+ u

@T

@x

+ v

@T

@y

� Sp! =J

cp

p = ⇢RdTIdeal Gas Law

D

Dt

=@

@t

+ u

@

@x

+ v

@

@y

+ !

@

@p

Material Derivative


Dynamical Equations Pressure Coordinates


u = ug + ua ug =1

f0k⇥r�

We assume that advection is dominated by the geostrophic winds:

Recall the separation of the real wind into geostrophic and ageostrophic components:


Approximation (A) Separation of geostrophic and ageostrophic wind

D

Dt

⇡ Dg

Dt

⌘ @

@t

+ ug@

@x

+ vg@

@y


f = f0 + �y

Expand the Coriolis parameter as a Taylor series about a fixed latitude:

f0 = 2⌦ sin�0 � =

2⌦ cos�0

a

y = a(�� 0)

with

Using the Cartesian coordinate

with a defined as the radius of the Earth

Note: A constant f0 is still used for computing the geostrophic wind.


Approximation (B) The mid-latitude beta plane approximation

Quasi-Geostrophic Theory Approximation (B) The mid-latitude beta plane approximation

f = f0 + �y

�L

f0⇡ L cos�0

a sin�0⇡ Ro ⇡ 10

�1

The second term in the beta-plane approximation is approximately an order of magnitude smaller than the first term.


From the Coriolis parameter expansion:

and scale analysis for large-scale midlatitudinal systems:

with

Quasi-Geostrophic Theory Approximation (C) Small vertical temperature perturbations

��@T0

@p

�� @T 0

@p

��

Expand the total temperature in terms of a background temperature and a perturbation:

T (x, y, p, t) = T0(p) + T

0(x, y, p, t)


QG Momentum Equation

Duh

Dt+ fk⇥ uh = �rp�

Starting from here:

Definition of geostrophic wind

Duh

Dt= �fk⇥ uh � f0k⇥ ug

Duh

Dt= �(f0 + �y)k⇥ (ug + ua)� f0k⇥ ug


f = f0 + �y u = ug + uaUsing


Expand and Cancel

Duh

Dt= �(f0 + �y)k⇥ (ug + ua)� f0k⇥ ug

10-4 m/s2 Scales: 10-4 m/s2 10-5 m/s2

Observe: this term is much

smaller than the others


Duh

Dt= �f0k⇥ ua � �yk⇥ ug � �yk⇥ ua

Duh

Dt= �f0k⇥ ua � �yk⇥ ug � �yk⇥ ua


Du

Dt⇡ Dgug

Dt



Dgug

Dt= �f0k⇥ ua � �yk⇥ ug

Duh


This equation states that the time rate of change of the geostrophic wind is related to:

1.  The Coriolis force due to the ageostrophic wind.

2.  The variation of the Coriolis force with latitude (β) multiplied by the geostrophic wind.

Both of these terms are smaller than the geostrophic wind itself.

QG Momentum Equation Dug



QG Continuity Equation

rp · u+@!

@p= 0

Geostrophic wind is non-divergent on pressure

surfaces rp · u = rp · ua

Starting from here:

QG Continuity Equation

@ua

@x

+@va

@y

+@!

@p

= 0


with

QG Thermodynamic Eq’n

@T

@t

+ u

@T

@x

+ v

@T

@y

� Sp! =J

cp

Starting from here:

T (x, y, p, t) = T0(p) + T

0(x, y, p, t)


✓@

@t+ ug ·r

◆T �

✓�p

Rd

◆! =

J

cp

� ⌘ �RdT0

p

@ ln ✓0@p

QG Thermodynamic Equation

QG Thermodynamic Eq’n

Hydrostatic Relation @�

@p= �RdT

p

✓@

@t+ ug ·r

◆✓�@�

@p

◆� �! =

J

p

⌘ Rd

cpwith


✓@

@t+ ug ·r

◆T �

✓�p

Rd

◆! =

J

cp

� ⌘ �RdT0

p

@ ln ✓0@p

are independent variables, form a complete set if heating rate J is known


QG Equations

Dgug


ug =1

f0k⇥r�

@ua

@x

+@va

@y

+@!

@p

= 0

✓@

@t+ ug ·r

◆✓�@�

@p

◆� �! =

J

p

Momentum Equation

Geostrophic Wind

Continuity Equation

Thermodynamic Equation

Auxiliary Equations ⌘ Rd

cp

�,ug,ua,!

� ⌘ �RdT0

p

@ ln ✓0@p

QG Equations

Quasi-geostrophic system is good for: •  Synoptic scales •  Middle latitudes •  Situations in which ua is important •  Flows in approximate geostrophic and

hydrostatic balance •  Mid-latitude cyclones

In deriving the QG equations we used scale analysis. That is, we made assumptions on the scales of the phenomena we are studying.


QG Equations In deriving the QG equations we used scale analysis. That is, we made assumptions on the scales of the phenomena we are studying.

Quasi-geostrophic system is not good for: •  Very small or very large scales •  Flows with large vertical velocities •  Situations in which ua ≈ ug •  Flows not in approximate geostrophic and

hydrostatic balance •  Thunderstorms/convection, boundary layer,

tropics, etc…


QG Equations

Question: What’s the point?

A set of equations that describes synoptic-scale motions and includes the effects of ageostrophic wind (vertical motion). This moves us towards a set of simple predictive equations for atmospheric motions in the mid-latitudes.


QG Equations

Question: What’s the point?

We want to describe the evolution of two key features of the atmosphere: •  Large-scale waves (in particular, the connection between large-

scale waves and geopotential) •  Midlatitude cyclones (that is, the development of low pressure

systems in the lower troposphere)


The next goal is to derive a vorticity equation for these scaled equations. •  This equation actually provides a “suitable”

prognostic equation because will include the divergence of the ageostrophic wind (and hence account for vertical motion).

•  Recall that divergence is a dominant mechanism for the generation of vorticity in the vorticity equation…

QG Vorticity Equation



Dgug



Dgug

Dt� f0va � �yvg = 0

DgvgDt

+ f0ua + �yug = 0

@

@x

✓Dgvg

Dt

◆� @

@y

✓Dgug

Dt

◆

Component Form

Then compute


QG Vorticity Equation Dg⇣g

Dt

= �f0

✓@ua

@x

+@va

@y

◆� �vg

@⇣g

@t

+ ug ·r⇣g = �f0

✓@ua

@x

+@va

@y

◆� �vg

@⇣g@t

= f0@!

@p� ug ·r⇣g � �vg

@⇣g@t

= f0@!

@p� ug ·r(⇣g + f)

Expand Dg/Dt


QG Vorticity Equation These forms of the

QG vorticity equation are all

equivalent

Dg⇣g

Dt

= �f0

✓@ua

@x

+@va

@y

◆� �vg


Dh⇣

Dt

= �(⇣ + f)

✓@u

@x

+@v

@y

◆� v

@f

@y

Scaled Vorticity Equation

QG Vorticity Equation The QG vorticity equation is very similar to the scaled vorticity equation we developed earlier, with a few additional assumptions (highlighted).


@⇣g@t

= f0@!

@p� ug ·r(⇣g + f)

@⇣g@t

= f0@!

@p� ug ·r⇣g � �vg

Stretching Term

Advection of relative vorticity

Advection of planetary vorticity Competing



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