### M1-Balanced Flow

```Balanced Flow
The momentum equation in natural coordinates:
r
r2
  )   ) 
dV ) V )
)
i 
n   
i 
n   fV n
 s
dt
R
n 
Let’s break this up into component equations:

dV


s
dt
V
2
 fV 

R
n
0
If that the flow is parallel to the height contours, then
dV
dt


s
0
Under these conditions, the flow is uniquely described by the equation in
the yellow box. If PGF normal to the flow direction is a constant, then the
radius of curve is also a constant.
Rossby Number
R0 
U
fL
where U and L are, respectively, characteristic velocity and length scales of the
phenomenon and f = 2 Ω sin φ is the Coriolis frequency.
V  V
fk V
V t
fk V

U


2
L
fU

U (L U )
fU
U
fL

 R0
U
fL
 R0
Ratio of advection to the CF
Ratio of local acceleration to the CF
A small Rossby number signifies a system which is strongly affected by the
Coriolis force, and a large Rossby number signifies a system in which inertial
and centrifugal forces dominate.
V
2
R
 fV 

n
0
Geostrophic flow
Geostrophic flow occurs when the PGF = CO, implying that R
Strictly speaking, for geostrophic flow to occur the flow must be straight
and parallel to the latitude circles.
Vg  
1 
f n
Pure geostrophic flow is
uncommon in the atmosphere,
but the geostrophic flow is a
good approximation when R0 is
small:
V 2
R 0 ~ 
 R

/ fV

 V 
   
 fR 
V
2
 fV 
R

n
0
Inertial flow
Inertial flow occurs in the absence of a PGF
R 
V
V   fR
or
f
This type of flow follows circular, anticyclonic paths since fR is negative

Time to complete a circle:
t
2 R

V
 is one half rotation
 is one full rotation/day

 sin 
2 R

fR

0 . 5 day
sin 
2
f

2
2  sin 


 sin 
called a half-pendulum day
Power Spectrum of
kinetic energy at 30 m
in the ocean near
0 . 5 day
sin 13 
 2 . 23 days
Pure inertial oscillation is rare in the atmosphere but common in the
oceans where transient wind stress drives currents
V
2
 fV 
R

n
0
Cyclostrophic flow
When the horizontal scale of the motion is small (e.g., tornados, dust
devils, water spouts), the Coriolis force can be neglected:
Flow is approximately cyclostrophic when the Centrifugal force is
much larger than the Coriolis force or R0 is much larger than 1.
V 2 
R 0 ~  / fV
 R 
A synoptic scale wave:

 V 
   
 fR 
V

fR
V
fR
10 m s
10

4
s
1
10
s
1
6
 0 .1
NO
10 m
100 m s
4
1
1
3
10 m
 1000
YES
1
V
2
R


n
0
  2

V   R

n 

In cyclostrophic flow, circulation can rotate counterclockwise or
clockwise (anticyclonic and cyclonic tornadoes and smaller vortices

are observed),
but it is always associated with a low.
The centrifugal force points away from the center of curvature
so the PGF must point toward the center of curvature.
V
2
R
 fV 

n
0
Gradient flow: a three-way balance among CO, PGF and CEN
2
2
fR  f R
  
V 
 
R

2
4
n


1/2
This expression has a number of mathematically possible
roots, not all of which conform to reality

Is V a
nonnegative
real number?

the unit vector nis everywhere normal to the flow and positive to the left of
the flow, and  is the geopotential height

n
is the height gradient in the direction of

n
R is the radius of curvature following parcel motion
R  0
R 0

n

n
directed toward center of curve (counterclockwise flow)
directed toward outside of curve (clockwise flow)
Let’s consider the
Northern
Hemisphere (f>0):
R>0: cyclonic
R<0: anticyclonic
V is always positive in the natural coordinate system
 f R
 


V 

R
2
 n 
 4
fR

Solutions for
n
2
1/ 2
2
 0,
Therefore:
Cyclonic high
2
R 0
 f 2R2
 


 
R
2
 n 
 4
fR
f R
4
2
R
1/ 2
is always negative.
V = negative = UNPHYSICAL


n
V 
Solutions for

n
 0,
 f R
 

 
R

2
4

n


fR
R0
Anticyclonic low R  0

n outward

 Increasing in n direction (low)
2
2
1/ 2
fR
2
Positive root physical
Negative root unphysical
Called an “anomalous low” it is rarely
observed (technically since f is never 0 in
actually anomalous lows
)
n

V 
Solutions for
Cyclonic

n inward


n
 0,
 f R
 

 
R

2
4

n


fR
R 0
2
2
1/ 2
fR
2
Positive root physical
Negative root unphysical
R  0
decreasing in

n direction (low)
Called an “regular low” it is commonly
observed (synoptic scale lows to cycloni-cally
rotating dust devils all fit this category
)
n

V 
Solutions for

n
 0,
 f R
 

 
R

2
4

n


fR
2
2
R0
2
f R
decreasing in
2
 R
4
Antiyclonic R  0

n outward

1/ 2
Then

n direction (high)

V 
Positive Root

n
fR
2
or
V
2
R

fV
2
CEN>CO/2, so CEN>PGF
Called an “anomalous high” (CEN>PGF)

)
n

V 
Solutions for

n
Antiyclonic R  0

n outward

 decreasing inn
 f R
 

 
R

2
4

n


fR
2
2
1/ 2
Negative Root
 0,
R0
2
f R
2
 R
4

n
therefore
direction (high)
V 
fR
2
Called a “regular high” : PGF exceeds
the centrifugal force

)
n
PGF
CEN

Condition for both regular and anomalous highs
 f 2 R 2
  
V 
 
R

2
n 
 4
1/2
fR
2
f R

4
2
R

n
For a regular high, we have
2
f R
0
4
R


n
0
2
R

n

n

f
2
R
4
This is a strong constraint on the magnitude of the
vicinity of high


pressure systems
Close to the high, the pressure gradient must be
weak, and must disappear at the high center
Force Balance in a Regular Low and a
Regular High
What if V is non-zero at small radii?
Note pressure gradients in vicinity of highs and lows
Force Balance in a Regular Low and a
Regular High
If PGF has the same magnitude, which one, the high or the low, has
stronger wind speed?
The ageostrophic wind in natural coordinates
V
2
 fV 
R
Note that
Vg  
We have
V
2
R

0
n
1 
f n
 f V  V g   0
V  Vg  
V
2
fR
For cyclonic flow (fR > 0) gradient wind is less than geostrophic wind (V<Vg)

For anticyclonic flow (fR < 0) gradient wind is greater than geostrophic
wind V>Vg.
Summary
V
2
 fV 
R
Centrifugal Force

n
0
PGF
Coriolis Force
Geostrophic Balance: PGF = CO
Inertial Balance:
CEN = CO
Cyclostrophic Balance: CEN = PGF