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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^ v ) and a second axis is always normal to and to the left of the wind (
η^ v ).
We will assume that the vertical wind is negligible, and that the vertical unit vector is
v k.
Natural coordinate system notation:
Wind:
v V = V τv
Coordinate locations:
(^ s , n , z )
Axes (unit vectors):
τ^ v , ηv, k v ( )
The Navier-Stokes Equations in Natural Coordinates
Material Derivative
D
v V Dt
= τr DV Dt
+ V D^
τv Dt
Based on this geometry:
δψ =^ δ s R
and δψ =
δ τv τ
= δ τv
⇒ δ τv =^ δ s R
As
δ s → 0
δ τv is parallel to
η^ v , and:
d τv ds
η^ v R D τv Dt
= D^
τv Ds
Ds Dt = η^ v R
V
⇒ D^
v V Dt
= τv DV Dt
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.
Inertial Oscillations
For flow with no pressure gradient the governing equation reduces to:
V^2
R
V = − fR
What is the time period required for the flow to complete one revolution about its center of circulation?
T = 2 π R V
= 2 π f
= 2 π 2 Ωsin φ
= 1 day 2 sin φ
Cyclostrophic flow
The Rossby number in natural coordinates can be expressed as:
Ro = V^
2 R
fV = V fR
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
ρ
∂ pd ∂ n
What is the physical interpretation of this equation?
Geostrophic approximation
What value of Ro is required to for the geostrophic approximation to be valid?
In natural coordinates the geostrophic approximation can be written as:
fV = − 1 ρ
∂ 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
∂ pd ∂ n
⇒ V = − fR 2
± f^
2 R 2
− R
ρ
∂ pd ∂ n
(^12)
The gradient wind can also be expressed in terms of the geostrophic wind:
V = − fR 2
± f^
2 R 2
(^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?
Physical solutions for the Northern Hemisphere
Cyclonic (CCW) flow around L : R > 0 ; V = − fR 2
2 R 2
− R
ρ
∂ pd ∂ n
(^12)
Anticyclonic (CW) flow around H : R < 0 ; V = − fR 2
− f^
2 R 2
− R
ρ
∂ 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^2 R^2 4
> R
ρ
∂ 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
∂ pd ∂ n
V^2
R
⇒ Vg = V 1 + V fR
^ ^
^
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 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
f
∂σ ∂ y
∂ vg ∂ z
f
∂σ ∂ x
or
∂ ug ∂ z
= − g fT 00
∂ T
∂ y
∂ vg ∂ z
= g fT 00
∂ 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 Δ z.
For east/west oriented isotherms this gives:
u^ v
g ( z^^ +^ Δ z )^ =^
u v
g (^ z )^ +
∂ ug ∂ z
v i +
∂ vg ∂ z
v j
Δ z^ +^ O^ (Δ z^^2 )
≈ u v g ( z ) − g
fT 00
∂ T
∂ y
Δ z
v 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.
The terms in this equation scale as:
ug ~ U ua ~ RoU w ~ W x , 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
ρ
∂ pd ∂ x
∂ u ∂ t
u^ ∂ u ∂ x
v^ ∂ u ∂ y
w^ ∂ u ∂ z
= − fvg + fv
∂ ug ∂ t
+^ ∂ ua ∂ t
∂ (^) ( ug + ua ) ∂ x
∂ (^) ( u (^) g + ua ) ∂ y
∂ (^) ( u (^) g + ua ) ∂ z
= fva
Neglecting terms that scale to less the
U^2
L
gives:
∂ ug ∂ t
∂ ug ∂ x
∂ ug ∂ y
= fva
The full horizontal quasi-geostrophic momentum equation is:
∂ ug ∂ t
∂ ug ∂ x
∂ ug ∂ y
= fva
∂ vg ∂ t
∂ vg ∂ x
∂ vg ∂ y
= − fua
or in vector notation:
Dg u^ v g Dt
= − f
v k × u v a
Ageostrophic Flow
The quasi-geostrophic momentum equation can be rewritten in terms of the ageostrophic wind components:
ua = − 1 f
∂ vg ∂ t
f
ug
∂ vg ∂ x
∂ vg ∂ y
va = 1 f
∂ ug ∂ t
f
ug
∂ ug ∂ x
∂ 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 ρ 00 f^2
∂ 2 p ∂ x ∂ t
ρ 00 f^2
ug^ ∂^
(^2) p ∂ x^2
(^2) p ∂ x ∂ y
va = − 1 ρ 00 f^2
∂ 2 p ∂ y ∂ t
ρ 00 f^2
ug^ ∂^
(^2) p ∂ x ∂ y
(^2) p ∂ y^2
For a flow in which the time rate of change term is largest:
ua = − 1 ρ 00 f^2
∂ 2 p ∂ x ∂ t
va = − 1 ρ 00 f^2
∂ 2 p ∂ y ∂ t
and ua and va are referred to as the isallobaric wind.
Isallobar – line of constant
∂ pd ∂ t
When the time rate of change term is small the ageostrophic wind is given by the advective acceleration term:
ua = − 1 f
ug
∂ vg ∂ x
∂ vg ∂ y
=^ −^
ρ 00 f^2
ug^ ∂^
(^2) p ∂ x^2
(^2) p ∂ x ∂ y
va = 1 f
ug
∂ ug ∂ x
∂ ug ∂ y
=^ −^
ρ 00 f^2
ug^ ∂^
(^2) p ∂ x ∂ y
(^2) p ∂ y^2
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
