Chapter 6: Simple Steady Motion Natural Coordinate System Natural coordinate system – a coordinate system in which one axis is always tangent to the horizontal wind (

€

τ ) and a second axis is always

normal to and to the left of the wind (

€

η ).

We will assume that the vertical wind is negligible, and that the vertical unit vector is

€

k .

Natural coordinate system notation: Wind:

€

V = V τ

Coordinate locations:

€

s,n,z( ) Axes (unit vectors):

€

τ , η , k ( )

The Navier-Stokes Equations in Natural Coordinates Material Derivative

€

D

V Dt

= τ

DVDt

+ V D τ Dt

Based on this geometry:

€

δψ =δsR

and δψ =δ τ

τ= δ τ

⇒ δ τ =

δsR

As

€

δs→ 0

€

δ τ is parallel to

€

η , and:

€

d τ ds

= η R

D τ Dt

=D τ Ds

DsDt

= η R

V

⇒D V

Dt= τ

DVDt

+ η

V 2

R

What is the physical interpretation of the two terms that make-up the parcel acceleration in this coordinate system? R – radius of curvature R is positive when center of curvature is to the left of the wind vector.

Coriolis force:

€

− fV η What is the direction of this term in the Northern and Southern hemispheres?

Pressure gradient force:

€

−1ρ∂pd∂s τ +

∂pd∂n η

'

( )

*

+ ,

Navier-Stokes equations in natural coordinates:

€

s component : DVDt

= −1ρ∂pd∂s

n component : V 2

R= − fV − 1

ρ∂pd∂n

What terms have been neglected in this set of equations? What do we know about vertical motion and forces for this system? Balanced Flow Balanced flow – a purely horizontal, frictionless flow that is also steady state For balanced flow the Navier-Stokes equations in natural coordinates reduces to:

€

s component : DVDt

= −1ρ∂pd∂s

= 0

n component : V 2

R= − fV − 1

ρ∂pd∂n

What does this imply about the orientation of the flow relative to isobars?

This system can be further simplified to:

€

V 2

R+ fV = −

1ρ∂pd∂n

What does each term in this equation represent?

Inertial Oscillations For flow with no pressure gradient the governing equation reduces to:

€

V 2

R+ fV = 0, and

€

V = − fR

What is the time period required for the flow to complete one revolution about its center of circulation?

€

T =2πRV

=2πf

=2π

2Ωsinφ=1 day2sinφ

Cyclostrophic flow The Rossby number in natural coordinates can be expressed as:

€

Ro =V 2

RfV =

VfR

What conditions result in a large value of Ro? What does this imply about the importance of the Coriolis term in the governing equations? For large Ro the governing equation reduces to:

€

V 2

R= −

1ρ∂pd∂n

What is the physical interpretation of this equation?

Geostrophic approximation What value of Ro is required for the geostrophic approximation to be valid? In natural coordinates the geostrophic approximation can be written as:

€

fV = −1ρ∂pd∂n

The Gradient Wind Approximation What force balance needs to be considered when Ro ~ 1? Gradient wind – the component of the flow that satisfies an exact balance between the centrifugal force, Coriolis force, and pressure gradient force

€

V 2

R+ fV +

1ρ∂pd∂n

= 0

⇒ V = −fR2

±f 2R2

4−Rρ∂pd∂n

&

' (

)

* +

12

Multiple solutions for V are possible with this equation (see table 6.1). What requirements exist for the sign of V in these equations? What other requirements are there that allow for a physical solution of this equation?

Table 6.1: Sign and magnitude of terms in the gradient wind equation for all possible flow regimes in the Northern Hemisphere.

Northern Hemisphere Cyclonic

(CCW) flow around L

Anticyclonic (CW) flow around H

Anticyclonic (CW) flow around L

Cyclonic (CCW) flow around H

€

f

+

+

+

+

€

R

+ -

-

+

€

∂p∂n

-

-

+

+

€

f 2R2

4−Rρ∂p∂n

%

& '

(

) *

12

always

€

>fR2

€

<fR2

or

imaginary for

€

f 2R2

4<Rρ∂p∂n

always

€

>fR2

€

<fR2

or

imaginary for

€

f 2R2

4<Rρ∂p∂n

€

−fR2

-

+

+

-

€

V positive for:

+ root only either root

but

€

f 2R2

4>Rρ∂p∂n

+ root only

never +

Physical solutions for the Northern Hemisphere

€

Cyclonic (CCW) flow around L : R > 0; V = −fR2

+f 2R2

4−Rρ∂pd∂n

%

& '

(

) *

12

Anticyclonic (CW) flow around H : R < 0; V = −fR2−

f 2R2

4−Rρ∂pd∂n

%

& '

(

) *

12

Physical solutions for the Southern hemisphere

€

Cyclonic (CW) flow around L : R < 0; V = −fR2

+f 2R2

4−Rρ∂pd∂n

%

& '

(

) *

12

Anticyclonic (CCW) flow around H : R > 0; V = −fR2−

f 2R2

4−Rρ∂pd∂n

%

& '

(

) *

12

Force balance for Northern Hemisphere gradient wind

In order for the solution of this equation to be real for the anticyclonic case:

€

f 2R2

4>Rρ∂pd∂n

This requires a light pressure gradient near the center of a high, and thus also light winds. No such limit exists for flow around low pressure. What is the physical explanation for this limit?

The geostrophic wind can be written as a function of the gradient wind:

€

V 2

R+ fV +

1ρ∂pd∂n

= 0

V 2

R+ fV − fVg = 0

⇒ Vg =V 1+VfR

&

' (

)

* +

What does this imply about the magnitude of the geostrophic wind relative to the gradient wind for flow around low and high pressure centers? We can also express the geostrophic wind as a function of the gradient wind and the Rossby number:

€

Vg =V 1+ Ro( ) What does this imply about the magnitude of the geostrophic wind relative to the gradient wind as Ro increases? Example: Comparison of observed, gradient, and geostrophic winds from a surface weather map

What role does friction play in altering the gradient wind balance?

The Boussinesq Approximation Boussinesq approximation – allows variations in density to give rise to buoyancy forces in the vertical momentum equation, but have no impact on the horizontal force balance To apply the Boussinesq approximation we must define: ρ00 – constant reference density ρ0(z) – vertically varying density (consistent with hydrostatic pressure profile) ρ(x,y) – horizontally varying density (consistent with horizontally varying dynamic pressure, pd) Using these definitions the geostrophic equation becomes:

€

fVg = −1ρ00

∂pd∂n

in natural coordinates, and

€

fug = −1ρ00

∂pd∂y

fvg =1ρ00

∂pd∂x

in an Earth-based coordinate system.

The vertical momentum equation is given by:

€

−ρ − ρ0( )ρ00

g − 1ρ00

∂pd∂z

= 0

The buoyancy force can be rewritten as:

€

σ = −ρ − ρ0 z( )( )ρ00

g =T −T0 z( )( )T00

g

and can be combined with the vertical momentum equation to give:

€

σ =1ρ00

∂pd∂z

The Thermal Wind Taking the horizontal derivatives of the vertical momentum equation and substituting these into the vertical derivative of the geostrophic equation gives:

€

∂ug∂z

= −1f∂σ∂y

∂vg∂z

=1f∂σ∂x

or

€

∂ug∂z

= −gfT00

∂T∂y

∂vg∂z

=gfT00

∂T∂x

What is the physical interpretation of this equation? How can we use this equation to explain the increase of westerly winds with height in the mid-latitude troposphere?

Thermal advection Warm air advection (WAA) – the wind blows from a region of warmer temperatures to a region of cooler temperatures Cold air advection (CAA) – the wind blows from a region of cooler temperatures to a region of warmer temperatures

We can use the thermal wind relationship to evaluate the change in geostrophic wind over a layer of depth Δz. For east/west oriented isotherms this gives:

€

u g z + Δz( ) = u g z( ) +

∂ug

∂z i +

∂vg

∂z j

$

% &

'

( ) Δz + O Δz 2( )

≈ u g z( ) − g

fT00∂T∂y

Δz i

Warm and cold advection cases in the Northern hemisphere:

In what direction does the geostrophic wind turn for the warm advection (cold advection) case? Veering – wind turns clockwise with height Backing – wind turns counterclockwise with height What would these cases look like in the Southern hemisphere? For both hemispheres: Cold air advection leads to cyclonic turning of the geostrophic wind with height. Warm air advection leads to anticyclonic turning of the geostrophic wind with height.

Departures from Balance Quasi-geostrophic flow – a flow in which small departures from geostrophic flow occur For this type of flow Ro is small, but finite. Derivation of the quasi-geostrophic equations: Start with horizontal momentum equation scaled for mid-latitude systems

€

Dh u h

Dt= −

1ρ∂pd

∂x i + ∂pd

∂y j

%

& '

(

) * − f k × u h

and use the hydrostatic approximation for the vertical momentum equation. Ageostrophic wind – component of the wind that is not in geostrophic balance

€

u h = u g + u a

We will define:

€

u a u g~ Ro <<1

Assume that variations in density are negligible so:

∂u∂x

+∂v∂y

+∂w∂z

= 0 and∂ug∂x

+∂vg∂y

= 0

Note that the geostrophic wind is non-divergent.

€

⇒∂ua∂x

+∂va∂y

+∂w∂z

= 0

€

∂w∂z

= −∂ua∂x

+∂va∂y

$

% &

'

( )

Only the ageostrophic component of the wind can cause divergence!

The terms in this equation scale as:

€

ug ~U ua ~ RoU w ~Wx,y ~ L z ~ H

How do the vertical and horizontal velocity scales compare? Use the definition of the ageostrophic wind to rewrite the horizontal momentum equation:

€

∂u∂t

+ u∂u∂x

+ v∂u∂y

+ w∂u∂z

= −1ρ∂pd∂x

+ fv

∂u∂t

+ u∂u∂x

+ v∂u∂y

+ w∂u∂z

= − fvg + fv

∂ug∂t

+∂ua∂t

+ ug + ua( )∂ ug + ua( )

∂x+ vg + va( )

∂ ug + ua( )∂y

+ w∂ ug + ua( )

∂z= fva

Neglecting terms that scale to less the

€

U 2

L gives:

€

∂ug∂t

+ ug∂ug∂x

+ vg∂ug∂y

= fva

The full horizontal quasi-geostrophic momentum equation is:

€

∂ug∂t

+ ug∂ug∂x

+ vg∂ug∂y

= fva

∂vg∂t

+ ug∂vg∂x

+ vg∂vg∂y

= − fua

or in vector notation:

€

Dg u g

Dt= − f k × u a

Ageostrophic Flow The quasi-geostrophic momentum equation can be rewritten in terms of the ageostrophic wind components:

€

ua = −1f∂vg∂t

−1fug∂vg∂x

+ vg∂vg∂y

$

% &

'

( )

va =1f∂ug∂t

+1fug∂ug∂x

+ vg∂ug∂y

$

% &

'

( )

What is the direction of the ageostrophic wind relative to the acceleration vector? The equations for the ageostrophic wind components can be expressed in terms of pressure gradients using the definition of the geostrophic wind:

€

ua = −1

ρ00 f2∂ 2p∂x∂t

−1

ρ00 f2 ug

∂ 2p∂x2

+ vg∂ 2p∂x∂y

%

& '

(

) *

va = −1

ρ00 f2∂ 2p∂y∂t

−1

ρ00 f2 ug

∂ 2p∂x∂y

+ vg∂ 2p∂y2

%

& '

(

) *

For a flow in which the time rate of change term is largest:

€

ua = −1

ρ00 f2∂ 2p∂x∂t

va = −1

ρ00 f2∂ 2p∂y∂t

and ua and va are referred to as the isallobaric wind. Isallobar – line of constant

€

∂pd ∂t

What is the isallobaric wind for the example below?

Based on this map

€

∂ 2p∂x∂t

=∂∂x∂p∂t

< 0 and

€

∂ 2p∂y∂t

=∂∂y∂p∂t

= 0,

Then for a Northern hemisphere location:

€

f > 0, so

€

ua > 0 and

€

va = 0 How will the change in pressure with time in this example alter the geostrophic wind? What is the direction of the Coriolis force associated with this ageostrophic component of the wind? The Coriolis force associated with ua in this example provides an acceleration of the northerly wind, allowing the flow to accelerate towards a balanced geostrophic state.

When the time rate of change term is small the ageostrophic wind is given by the advective acceleration term:

ua = −1fug∂vg∂x

+ vg∂vg∂y

⎛

⎝⎜

⎞

⎠⎟

va =1fug∂ug∂x

+ vg∂ug∂y

⎛

⎝⎜

⎞

⎠⎟

What is the ageostrophic wind for the jetstreak example below?

On the left side:

€

∂ug ∂x > 0, so

€

va > 0

On the right side:

€

∂ug ∂x < 0, so

€

va < 0

What is the direction of the Coriolis force associated with the ageostrophic flow on each side of this jet? On either side of the jet the Coriolis force associated with the ageostrophic flow accelerates the flow towards a geostrophic balance.

In general, an ageostrophic wind directed towards low pressure will accelerate the flow in the direction of the geostrophic wind, while an ageostrophic wind directed towards high pressure will decelerate the flow in the direction of the geostrophic wind. Geostrophic Adjustment – a process of restoring the flow to geostrophic balance What role does the ageostrophic flow play at the surface and 500mb in the example below?

- accelerate the geostrophic wind to bring the flow back towards geostrophic balance - alter the vertical shear of the geostrophic wind to bring the flow back to thermal wind balance - alter the horizontal temperature gradient, through rising and sinking motion, to bring the flow back to thermal wind balance