Download Ω The Coriolis force

Atmospheric Motion
Dynamics
Newton’s second law
F
m
x
Mass × Acceleration = Force
m
d2x
=F
dt 2
Thermodynamics
Concerned with changes in the internal
energy and state of moist air.
TMD Lecture 2
Newton’s second law
The Coriolis force
Newton’s second law
m
F
d2x
m 2 =F
dt
x
A line that rotates with
the roundabout
Ω
A line at rest in an
inertial system
Mass × Acceleration = Force applies in an inertial frame of
reference.
But we like to make measurements relative to the Earth,
which is rotating!
To do this we must add correction terms in the equation,
the centrifugal and Coriolis accelerations.
Apparent trajectory of
the ball in a rotating
coordinate system
Effective Gravity
The centripetal acceleration/centrifugal force
g is everywhere normal to the earth’s surface
outward
force
Ω
v2
= Ω2 r
r
Ω 2R
R
g*
v
g*
g
r
Inward
acceleration
−
Ω 2R
R
g
v2
= −Ω2r
r
g = g* + Ω2R
effective gravity g
on a spherical earth
Effective Gravity
If the earth were a perfect sphere and not rotating, the only
gravitational component g* would be radial.
Because the earth has a bulge and is rotating, the effective
gravitational force g is the vector sum of the normal gravity to
the mass distribution g*, together with a centrifugal force
Ω2R, and this has no tangential component at the earth’s
surface.
effective gravity on an earth
with a slight equatorial bulge
When frictional forces can be neglected, F is the pressure
gradient force
total pressure
F = −∇ p T
force per unit volume
⎛ du
⎞
ρ⎜
− 2Ω ∧ u ⎟ = F + ρg
⎝ dt
⎠
du
1
= − ∇pT + g − 2Ω ∧ u
dt
ρ
per
unit mass
g = g * +Ω 2R
This is Euler’s equation of motion in a rotating reference frame.
The Coriolis force does no work
Perturbation pressure, buoyancy force
Define
Ω
the Coriolis force acts normal to the
rotation vector and normal to the velocity.
u
is directly proportional to
the magnitude of u and Ω.
pT = p0 ( z) + p
where
dp 0
= − gρ0
dz
p0(z) and ρ0(z) are reference pressure and density fields
p is the perturbation pressure
Euler’s equation becomes
− 2Ω ∧ u
g = (0, 0, −g)
⎡ ρ − ρ0 ⎤
Du
1
+ 2Ω ∧ u = − ∇p + g ⎢
⎥
ρ
Dt
⎣ ρ ⎦
Note: the Coriolis force does no work because u ⋅ ( 2Ω ∧ u) ≡ 0
the buoyancy force
Important: the perturbation pressure gradient −
1
∇p
ρ
Mathematical formulation of the continuity
equation for an incompressible fluid
⎛ ρ − ρ0 ⎞
and buoyancy force g ⎜
⎟ are not uniquely defined.
⎝ ρ ⎠
⎛ ρ − ρ0 ⎞
1
But the total force − ∇ p + g ⎜
⎟
ρ
⎝ ρ ⎠
Indeed
v + δv
w + δw
δy
δx
is uniquely defined.
⎛ ρ − ρ0 ⎞
1
1
− ∇p + g ⎜
= − ∇p T + g
⎟
ρ
ρ
⎝ ρ ⎠
u
δz
v
w
u + δu
The mass continuity equation
Rigid body dynamics
∇⋅u = 0
Incompressible fluid
Mass m
Compressible fluid
Force F
∂ρ
+ ∇ ⋅ (ρu ) = 0
∂t
Anelastic approximation
∇ ⋅ (ρ (z)u ) = 0
o
x
Newton’s equation of motion is:
m
d2 x
dt 2
=F
Problem is to calculate x(t) given the force F
Fluid dynamics problems
Ø The force field is determined by the overall constraints
provided by
– the requirement of continuity
– the boundary conditions
Ø In particular, the pressure field at any instant is
determined by the flow configuration
– I will now illustrate this with an example!
– Let us forget about density differences and rotation
for this example
Fluid dynamics problems
Ø The aim of any fluid dynamics calculation is to calculate
the flow field U(x,y,z,t) in a given region subject to
appropriate boundary conditions and the constraint of
continuity.
Ø The calculation of the force field (i.e. the pressure field)
may not be necessary, depending on the solution method.
LO
A mathematical demonstration
HI
U
isobars
Du
1
= − ∇p '
Dt
ρ
HI
Momentum equation
∇⋅u = 0
Continuity equation
The divergence of the momentum equation gives:
U
HI
LO
∇2p' = −∇⋅ (ρu ⋅∇u)
pump
streamlines
This is a diagnostic equation!
But what about the effects of rotation?
Assumptions: inviscid, irrotational, incompressible flow
buoyancy form
Newton’s 2nd law vertical component
mass × acceleration = force
ρ
Dw
∂p
= − T − gρ
Dt
∂z
Put
p
T
= p (z) + p′
o
ρ = ρ (z) + ρ′
o
Then
Dw
1 ∂p′
=−
+b
ρ ∂z
Dt
where
where
dpo
= −gρ
o
dz
⎛ ρ−ρ o ⎞
b = −g ⎜
⎟
⎝ ρ ⎠
Buoyancy force in a hurricane
buoyancy force is NOT unique
ρ (z)
⎛ ρ − ρo ⎞
b = −g ⎜
⎟
⎝ ρ ⎠
o
it depends on choice of reference density ρo(z)
but
−
1 ∂pT
1 ∂p '
−g = −
+b
ρ ∂z
ρ ∂z
is unique
ρ (z)
o
Initiation of a thunderstorm
z
tropopause
negative buoyancy
outflow
θ = constant
original heated air
θ = constant
positive
buoyancy
LFC
LCL
Τ + ΔΤ
U(z)
T
inflow
negative buoyancy
positive buoyancy
Some questions
HI
outflow
p'
Ø How does the flow evolve after the original thermal has
reached the upper troposphere?
Ø What drives the updraught at low levels?
original heated air
– Observation in severe thunderstorms: the updraught at
cloud base is negatively buoyant!
LO
LFC
– Answer: - the perturbation pressure gradient
LCL
HI
inflow
HI
HI
negative buoyancy
The geostrophic approximation
For frictionless motion (D = 0) the momentum equation is
Choose rectangular
coordinates:
k = (0,0,1)
z
Ω = Ωk
Du
1
+ 2 Ω ∧ u = − ∇p
Dt
ρ
Let Ro → 0
k
perturbation pressure
uh
1
2Ω ∧ u = − ∇ p
ρ
y
x
This is called the geostrophic equation
We expect this equation to hold approximately in synoptic
scale motions in the atmosphere and oceans, except possibly
near the equator.
velocity components u = (u,v,w),
u = uh + wk
uh = (u,v,0) is the horizontal flow velocity
Take k ∧
1
2Ω ∧ u = − ∇ p
ρ
The geostrophic wind
(k ⋅ u)k = (0, 0 w)
1
2 Ω k ∧ ( k ∧ u ) = 2Ω [( k ⋅ u ) k − u ] = − k ∧ ∇ p
ρ
− uh
uh =
and
s h p = (∂p/∂x, ∂p/∂y, 0)
1
k ∧ ∇h p
2Ωρ
0=
∂p
∂z
uh =
1
k ∧ ∇h p
2Ωρ
Ø The geostrophic wind blows parallel to the lines (or more
strictly surfaces) of constant pressure - the isobars, with
low pressure to the left.
Ø Well known to the layman who tries to interpret the
newspaper "weather map", which is a chart showing
isobaric lines at mean sea level.
Ø In the southern hemisphere, low pressure is to the right.
This is the solution for geostrophic flow.
Choice of coordinates
Ø For simplicity, let us orientate the coordinates so that x
points in the direction of the geostrophic wind.
Ø Then v = 0, implying that ∂p/∂x = 0 .
u=−
Geostrophic flow
pressure gradient force
isobar
low p
1 ∂p
2Ωρ ∂y
Ø Note that for fixed Ω , the winds are stronger when the
isobars are closer together and, for a given isobar
separation, they are stronger for smaller |Ω|.
u
high p
isobar
Coriolis force
(Northern hemisphere case: > 0)
A mean sea level isobaric chart over Australia
Note also that the solution
and
uh =
1
k ∧ ∇hp
2Ωρ
0=
∂p
∂z
tells us nothing about the vertical velocity w.
L
Ø For an incompressible fluid, ∇ ⋅ u = 0 .
Ø Also, for geostrophic flow, ∇h ⋅ uh = 0 .
Ø then ∂w/∂z = 0 implying that w is independent of z.
H
H
H
If w = 0 at some particular z, say z = 0, which might be the
ground, then w ≡ 0.
The geostrophic equation is degenerate!
Ø The geostrophic equation is degenerate, i.e. time
derivatives have been eliminated in the approximation.
Ø We cannot use the equation to predict how the flow will
evolve.
Ø Such equations are called diagnostic equations.
Ø In the case of the geostrophic equation, for example, a
knowledge of the isobar spacing at a given time allows
us to calculate, or 'diagnose', the geostrophic wind.
Ø We cannot use the equation to forecast how the wind
velocity will change with time.
Vortex flows: the gradient wind equation
Ø Strict geostrophic motion requires that the isobars be
straight, or, equivalently, that the flow be uni-directional.
Ø To investigate balanced flows with curved isobars, including
vortical flows, it is convenient to express Euler's equation in
cylindrical coordinates.
Ø To do this we need an expression for the total horizontal
acceleration Duh/Dt in cylindrical coordinates.
The case of pure circular motion with u = 0 and ∂/∂θ ≡ 0.
The radial and tangential components of Euler's equation
may be written
v2
1 ∂p
+ fv =
r
ρ ∂r
∂u
∂u v ∂u
∂u v
1 ∂p
+u
+
+w
−
− fv = −
∂t
∂r r ∂θ
∂z r
ρ ∂r
2
∂v uv
1 ∂p
∂v
∂v v ∂v
+u
+
+w
+
+ fu = −
∂r r ∂θ
∂z
r
ρr ∂θ
∂t
The axial component is
∂w
∂w
∂w v ∂ w
1 ∂p
+u
+
+w
=−
∂t
∂r
∂z
ρ ∂z
r ∂θ
Ø This is called the gradient wind equation.
Ø It is a generalization of the geostrophic equation which takes
into account centrifugal as well as Coriolis forces.
Ø This is necessary when the curvature of the isobars is large,
as in an extra-tropical depression or in a tropical cyclone.
The gradient wind equation
Write
Force balances in low and high pressure systems
1 ∂p v 2
0=−
+
+ fv
ρ ∂r
r
terms interpreted as forces
Cyclone
V
V
Ø The equation expresses a balance of the centrifugal force
(v2/r) and Coriolis force (fv) with the radial pressure
gradient.
Ø This interpretation is appropriate in the coordinate system
defined by r and θ , which rotates with angular velocity v/r.
Anticyclone
LO
HI
PG
HI
CO
CO
CE
LO
CE
PG
The equation
0=−
1
1
r ∂p
v = − fr + f 2 r 2 +
ρ ∂r
2
4
1 ∂p v 2
+
+ fv
ρ ∂r
r
is a diagnostic equation for the tangential velocity v in terms
of the pressure gradient:
r ∂p
1
1
v = − fr + f 2 r 2 +
ρ ∂r
2
4
1
2
Ø In a low pressure system, ∂p/∂r > 0 and there is no
theoretical limit to the tangential velocity v.
Ø In a high pressure system, ∂p/∂r < 0 and the local value of
the pressure gradient cannot be less than −ρrf2/4 in a
balanced state.
Choose the positive sign so that geostrophic balance is
recovered as r → ∞ (for finite v, the centrifugal force
tends to zero as r → ∞ ).
Ø Therefore the tangential wind speed cannot locally exceed
rf/2 in magnitude.
Ø This accords with observations in that wind speeds in
anticyclones are generally light, whereas wind speeds in
cyclones may be quite high.
Limited wind speed in anticyclones
In the anticyclone, the Coriolis force increases only in
proportion to v: => this explains the upper limit on v
predicted by the gradient wind equation.
V
CO = fv
HI
CE =
v2
r
CO
CE
PG
1
2
End of L2