Transcript Lesson 06 - United States Naval Academy

Balanced Flows
SO 254 – Spring 2017
LCDR Matt Burich
Balanced Flows
Before discussing states of balance between the various
forces in the equation of motion, we’ll first define a slightly
different coordinate system called “natural coordinates”
which will maintain a consistent orientation relative to a
moving air parcel
Natural coordinates consist of unit vector components:
𝐧
𝐧
𝐧: Normal to horizontal velocity and
𝐭
oriented positive leftward
𝐭
𝐭: Tangential to horizontal velocity
𝐤: Vertical
unimportant for this discussion since we’ll only be
considering horizontal flows
Balanced Flows
In this system, if a distance traveled along the
path of the parcel is 𝛿𝑆, then the horizontal
velocity may be expressed as:
𝐷𝑆
= 𝑉𝐭
𝐷𝑡
𝐷𝑉 1 𝜕𝑝
and
=
𝐷𝑡 𝜌 𝜕𝑠
𝐧
𝐭
𝐭
𝛿𝑠
Pressure gradient force along the motion
path (Coriolis and centrifugal force act in
the 𝐧-direction so they do not exist here
For our purposes, we’ll assume the flow is unaccelerated
thus there is no pressure gradient in the 𝐭-direction
In the 𝐧-direction we find the forces:
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝑅
𝜌 𝜕𝑛
centrifugal
force
pressure
gradient force
Coriolis
𝐧
𝜕𝑝
𝜕𝑠
𝐷𝑉
𝐷𝑡
= 0 and
=0
Coriolis
𝑅
PGF
𝑅: radius of curvature
𝑓 ≡ 2Ω sin 𝜃
centrf.
Balanced Flows
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝑅
𝜌 𝜕𝑛
centrifugal
force
pressure
gradient force
Coriolis
There are four possible balances that may exist between the forces…
three of which we’ll consider directly
1)
For flows with no curvature (𝑅 → ∞), centrifugal force is zero and we
have:
1 𝜕𝑝
+ 𝑓𝑉 = 0
𝜌 𝜕𝑛
⇒
1 𝜕𝑝
= −𝑓𝑉
𝜌 𝜕𝑛
L
PGF
Coriolis
This balance of pressure gradient
force and Coriolis is referred to as
geostrophic balance
height contours
H
1 𝜕𝑝
= −𝑓𝑉
𝜌 𝜕𝑛
PGF
Geostrophic Balance
(500 mb surface)
Coriolis
Low pressure
Why does an air parcel
begin to move?
528 dm
An external force (PGF) is
applied to it
Once the parcel is moving,
what additional force is
acting?
PGF
velocity
Coriolis
Forces in
balance
North
540 dm
546 dm
552 dm
558 dm
Coriolis…to the right of
direction of motion (Northern
Hemisphere)
PGF never changes in this
example (equally spaced height
contours) but the magnitude of
Coriolis is dependent upon
parcel velocity
534 dm
High pressure
PGF
500 mb
North
(side view)
558 dm
552 dm
546 dm
540 dm
534 dm
528 dm
1 𝜕𝑝
= −𝑓𝑉
𝜌 𝜕𝑛
PGF
Geostrophic Balance
Coriolis
In reality, height contours are rarely
perfectly straight, so the state of
geostrophic balance is an
approximation
The validity of the “geostrophic approximation” for a particular
flow situation is checked via computation of the Rossby
𝑈
number:
𝑅0 =
𝑓𝐿
𝑈 = characteristic velocity scale
𝐿 = characteristic length scale
𝑓 = 2Ω sin 𝜃
The smaller the value of 𝑅0 , the
more valid the geostrophic
approximation
For large scale motions: 𝑈~10 m s −1 ,
𝐿~106 m, and 𝑓~10−4 𝑠 −1 (in midlatitudes)
Thus, 𝑅0 ~0.1
(a reasonable approximation)
Balanced Flows
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝑅
𝜌 𝜕𝑛
centrifugal
force
2)
pressure
gradient force
Coriolis
For flows where Coriolis force is insignificant, we have:
𝑉2
1 𝜕𝑝
+
= 0
𝑅
𝜌 𝜕𝑛
⇒
1 𝜕𝑝
𝑉2
= −
𝜌 𝜕𝑛
𝑅
These are generally small scale flows
Compute the Rossby number for a tornado:
𝑈~100 m s −1 , 𝐿~102 m, and 𝑓~10−4 𝑠 −1
𝑅0
~104
L
PGF
Centrf.
Not geostrophic!
height contours
H
Balanced Flows
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝑅
𝜌 𝜕𝑛
centrifugal
force
2)
pressure
gradient force
Coriolis
For flows where Coriolis force is insignificant, we have:
𝑉2
1 𝜕𝑝
+
= 0
𝑅
𝜌 𝜕𝑛
⇒
1 𝜕𝑝
𝑉2
= −
𝜌 𝜕𝑛
𝑅
These are generally small scale flows
This balance of pressure gradient
force and centrifugal force is
referred to as cyclostrophic
balance
L
PGF
Centrf.
height contours
H
Balanced Flows
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝑅
𝜌 𝜕𝑛
centrifugal
force
3)
pressure
gradient force
Coriolis
For flows where all three forces are significant, we have:
1 𝜕𝑝
𝑉2
= −
− 𝑓𝑉
𝜌 𝜕𝑛
𝑅
These are often medium (meso) scale flows
Compute the Rossby number for a
hurricane: 𝑈~50 m s −1 , 𝐿~105 m, and
𝑓~10−4 𝑠 −1
L
Centrf.
PGF
Coriolis
𝑅0 ~5
Neither very large nor
very small
height contours
H
Balanced Flows
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝑅
𝜌 𝜕𝑛
centrifugal
force
3)
pressure
gradient force
Coriolis
For flows where all three forces are significant, we have:
1 𝜕𝑝
𝑉2
= −
− 𝑓𝑉
𝜌 𝜕𝑛
𝑅
These are often medium (meso) scale flows
This balance of pressure gradient
force, Coriolis, and centrifugal force
is referred to as gradient (wind)
balance
L
Centrf.
PGF
Coriolis
height contours
H
Summary
𝑉2
1 𝜕𝑝
+
+ 𝑓𝑉 = 0
𝜌 𝜕𝑛
𝑅
centrifugal
force
pressure
gradient force
Coriolis
Small scale
rotation (e.g.,
tornado / dust
devil)
Geostrophic balance
Gradient (wind) balance
Cyclostrophic balance
1 𝜕𝑝
= −𝑓𝑉
𝜌 𝜕𝑛
1 𝜕𝑝
𝑉2
= −
− 𝑓𝑉
For
𝜌 𝜕𝑛example,
𝑅 jet
stream flow that is
relatively straight
𝑅0 ≪ 1
𝑅0 ≈ 1
𝑅0 ≫ 1
Large scale flows
(synoptic scale)
Medium scale flows
(mesoscale)
Small scale flows
1 𝜕𝑝
𝑉2
= −jet
For example,
𝜌 𝜕𝑛
𝑅
stream flow that is
strongly curved (at
the base of a
trough)
What happens when friction gets involved?
H
L