M1-Balanced Flow
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Transcript 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
Barbados (13N)
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:
A tornado:
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
For radical to be positive
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
R0
Anticyclonic low R 0
n outward
Increasing in n direction (low)
2
2
1/ 2
Radical >
fR
2
Positive root physical
Negative root unphysical
Called an “anomalous low” it is rarely
observed (technically since f is never 0 in
mid-latitudes, anticyclonic tornadoes are
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
Radical >
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
R0
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
or radical is imaginary
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,
R0
2
f R
2
R
4
n
therefore
direction (high)
or radical is imaginary
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
pressure gradient force in 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
Gradient Balance:
CEN + PGF + CO = 0