Met 61: Dynamics
Conservation Laws:
There are three basic conservations laws that govern the way atmospheric
flows evolve. Understanding how these laws apply to the atmosphere and
their use is essential to a study of weather and climate.
I. Newton’s 2nd law of motion:


F  ma
or

DU
  Fi
Dt
i
(1)
(2)
The acceleration of a parcel of air of unit mass is equal to the sum of the
forces acting upon that mass. These equations are often referred to as the
momentum equations.
II. Conservation of energy (1st law of thermodynamics)
“The change in internal energy of a system is equal to the difference
between the heat added to the system and the work done by the system” or:
dU  dQ  dW
(3)
where, U – internal energy Q- heat and W work.
“The transformation of heat energy into various forms of mechanical
energy is the process which drives the atmospheric global circulation.”
III. Conservation of mass:
Mass conservation can be stated, ‘any change in mass associated with a
volume must be accomplished by a flux of mass into or out of that volume.
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Application of the conservation laws to the atmosphere:
I. Momentum equations
Newton’s 2nd law of motion is more generally written as,

DaU
 F  Dt
(4)
where the subscript, a, denotes an absolute reference frame.
Let’s look at the different forces:
1. Pressure gradient force:
(see handout)
PGF  
1

p
(5)
where  is the density, and p is the pressure gradient.
Note: This force is proportional to the gradient in pressure, not the actual
pressure itself.
2. Gravity
Gravity obviously plays an important role in the evolution of atmospheric
motions. Although the earth is not actually a sphere (more later), it is
typical to replace the gravitational force with a constant, g0.
3. Viscous Forces
Internal friction of the fluid or viscous forces are typically quite
complicated to calculate. For many atmospheric science problems, we can
2
approximate the viscous forces as proportional to a kinematic viscosity
coefficient multiplied by the shear of the wind. Mathematically, this is:
  2u  2u  2u 
Frx    2  2  2 
y
z 
 x
  2v  2v  2v 
Fry    2  2  2 
y
z 
 x
  2w  2w  2w 
Frz    2  2  2 
y
z 
 x
( 6)
(7)
(8)
where  is the viscosity coefficient = 1.46 x 10-5 m2 s-1.
4. Apparent (noninertial) forces
Newton’s first law of motion states that mass of uniform motion with
respect to a coordinate system fixed in space will remain in uniform
motion in the absence of any other forces. This is inertial motion, which
occurs on a fixed or inertial frame of reference. However, this object with
uniform motion on an inertial reference frame, is really viewed as
accelerating on a rotating reference frame. Newton’s laws can be applied
to a rotating reference frame if the acceleration of the rotating coordinates
is taken into account, via ‘apparent’ forces.
For our system, which rotates with uniform acceleration, two such
apparent forces result; the centrifugal force and the Coriolis force.
a. Centrifugal force:
The centripetal acceleration is the force that acts toward the axis of
rotation that keeps a ball on a string from flying away. If you are in a
rotating system, then the centrifugal force is the force opposite to the
centripetal force, or the force that balances the string tension.
3
A particle on earth is thus subject to both a centrifugal force, directed
normal to the axis of rotation, and a gravitational force. Because the
centrifugal force partly balances the gravitational force, it is convenient to
combine them as:

 
g  g *  2 R
(9)
where
 - angular speed of earth rotation (radians/sec) and R is the radius vector.
The new g, which is termed simply as ‘gravity’ is no longer directed
towards the center of the earth (except at the poles and equator), because
the centrifugal force is directed away from the axis of rotation. This
centrifugal force is responsible for the earth’s bulge! The equatorial radius
of the earth is around 21 km larger than at the pole.
b. Coriolis Force:
We have already developed some intuition for the Coriolis force and how
it works, but mathematically, we must pay closer attention. A parcel that
moves east-west, will experience an ‘apparent’ force both to the northsouth, and vertical planes. In particular,
 dv 
   2u sin 
 dt  Co
( 10 )
 dw 

  2u cos 
dt

Co
( 11 )
For a particle that moves in the north/south plane, the conservation of
angular momentum explains that for a particle that is given an equatorward
push, a westward velocity must develop. Mathematically, this is:
4
 du 
   2v sin   2w cos 
 dt  Co
( 12 )
Without deriving the explicit relationship for the Coriolis force (this you
will encounter in Metr 121A, the shorthand notation for the Coriolis force
as a vector is expressed as:

 
Fc  2  U
5
( 13
)
5. Total Differentiation:
The conservation laws contain expressions following the motion
(Lagrangian reference frame), and yet it is often more convenient to
express these relationship in an Eulerian perspective. In atmospheric
science, we often refer to the total derivative (material or substantial) as
the Lagrangian and the local derivative as the Eulerian. The relationship
between the two is as:
D
 
 U 
Dt t
( 14 )
So that the total derivative of temperature, T is:
DT T 

 U  T
Dt
t
T
T
T
T

u
v
w
t
x
y
z
( 15 )
( 16 )
Thus, the local derivative for temperature is:
T DT 

 U  T
t
Dt
( 17 )
where the term,(-U·T) is called the temperature advection (note the
negative sign!)
If we want to determine the temperature tendency at a particular point,
then by assuming that the total derivative is zero (e.g. DT/Dt=0), then

T
 U  T   U T cos
t
where  is the angle between the two vectors.
If ,-U·T > 0 then warm advection
If ,-U·T < 0 then cold advection
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( 18 )


B
A


C
D
At points A, B C and D, indicate whether there is cold or warm
advection.
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6.
Component form of the momentum equations:
If we first consider the momentum equations, given as:

  
DU
1

  P  g  2  U  F
Dt

where,
(18B)

U  u, v, w,

F  Fx , Fy , Fz 

   x ,  y ,  z 

g  (0,0, g )




i
j k
x y
z
In component form, the momentum equations are:
Du
1 p

 2v sin   2w cos   Fx
Dt
 x
( 19 )
Dv
1 p

 2u sin   Fy
Dt
 y
( 20 )
Dw
1 p

 g  2u cos   Fz
Dt
 z
( 21 )
where,
8
D 

  U  
Dt  t

If we wanted to study synoptic scale motions, then based on observations,
we could define the following characteristics scales:
U (horizontal wind)
W (vertical wind)
L (length scale)
H~(depth scale)
P/ (horizontal pressure fluctuation scale)
L/U (time scale)
Midlatitude (latitude)
Coriolis parameter(f0=2sin0)
7.
~ 10 m/s
~ 1 cm/s
~ 1000km
~ 10,000m
~103m2s-2
~ 105s
45º
~ 10-4
Scale analysis example:
The acceleration scales as follows:
(Du/Dt) when scaled produces: U / (L/U) or U2/L or 10-4
One can do this for all the terms in the momentum equations to determine
the predominate force balance for each equation.
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Important balances
a) Geostrophic Balance:
 fv  
1 p
 x
1 p
 y
( 22 )

1
Vg  kˆ 
P
f
( 23 )
fu  
or

where Vg  (u g , v g )
Question: Write out the geostrophic wind in component form
Prognostic Equations
If we wanted to develop a prognostic equation for u and v, then we could
retain the acceleration terms in (19) and (20) as:
Du
1 p
 fv 
 f (v  v g )
Dt
 x
( 24 )
Dv
1 p
 fu 
 f (u  u g )
Dt
 y
( 25 )
In theory, these equations would predict the evolution of the horizontal
wind with time. However, note that the acceleration terms are really the
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difference between two much larger terms, and thus, small errors in the
measurements of the wind fields would produce very large errors in
estimating the velocity. This type is situation is a persistent difficulty in
fluid dynamics.
Rossby Number:
One way to estimate the importance of the earth’s rotations (i.e. Coriolis
force) on fluid motions is to calculate the Rossby Number, which is simply
the ratio of the wind acceleration to the Coriolis acceleration,
U2 L U

 Ro
fU
fL
( 26 )
For typical synoptic scale motions, it turns out that the acceleration is
small (~10% or less) compared to the Coriolis, and as we have already
seen, the wind is nearly geostrophic. Thus, for geostrophic balance to be
valid, the Rossby number must be small (Ro ~ 0.1), or the accelerations of
the winds are about an order of magnitude smaller than the Coriolis
acceleration. In the tropics, this is not the case, and the Rossby number is
the of the order 1. This means the flow acceleration and the Coriolis
acceleration are about the same.
Q: Provide a physical explanation for why the Rossby number is O(1) in
the tropics.
The important point is that if the Rossby number becomes large, then
geostrophic balance does not strictly hold. Note that geostrophic balance
assumes frictionless flow, and flow that is not highly curved.
b) Hydrostatic balance
When you do a scale analysis on the w-momentum equation, you find that
the predominate balance in the vertical is:
11
p
  g
z
( 27 )
This balance only breaks down for small-scale phenomena, such as in the
strong updrafts of thunderstorms. For scales larger than 10km, hydrostatic
balance is typically valid.
II. Thermodynamic Energy Equation
The conservation of energy (1st law of thermodynamics), as applied to the
atmosphere, in the end relates how solar energy can produce movements in
the atmosphere,
cv
DT
D
p
J
Dt
Dt
( 28 )
where, T is the temperature,  is the specific volume (1/ ), and J is the
rate of heating (per unit mass) due to radiation, conduction and latent heat
release.
Thus, the diabatic heating rate, J, can produce changes in the local
temperature (1st term), while the 2nd term (pD/Dt) represents a conversion
between thermal and mechanical energy. Note that thermal energy is
related to temperature, while mechanical energy related to wind.
Recall: The first law of thermo says:
The change in internal energy of the system (DT) is equal to the difference
between the heat added to the system (J) minus the work done by the
system (pD/Dt).
For atmospheric flows, the work done is by pressure and gravity (viscosity
has been neglected and Coriolis does no work).
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For a dry atmosphere, in the absence of strong diabatic heating, scaling
considerations can recast the thermodynamic energy equation in terms of
potential temperature as:
d ( z )
D
w 0
0
Dt
dz
( 29 )
where the potential temperature has been expressed as
   0 ( z )   ( x, y, z, t )
Equation 29 can also be expressed as:
T
T
T
u
v
 wd     0
t
x
y
III. The Continuity Equation
The third conservation principle, mass conservation, for atmospheric flows
is expressed as the continuity equation, namely,
1 D
U  0
 Dt
( 30 )
This equation states that the fractional increase in the density following the
motion is equal to minus the velocity divergence, where divergence is
defined as the flow out of a unit volume.
U 
u v w


x y z
( 31 )
After an appropriate scale analysis for synoptic scales, (30) can be stated
as
13
   0 U   0
( 32 )
where 0(z) only.
Thus, the mass flux for synoptic-scale motions using the basic state density
is nondivergent; the divergence is zero. This is similar to the idea of
incompressibility which is used in fluid mechanics.
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Gradient wind balance
Geostrophic balance is only one of the types of balances that may exist in
the atmosphere, and best describes large scale flow without friction and
where the flow is not highly curved. The more general case, where the
flow does curve can also be explored.
Again, consider the momentum equations only now, we include the
centrifugal force as:
VV
1 p
0
 f cV  s
 x
R
UU
1 p
0
 f cU  s
 y
R
( 33 )
( 34 )
where, fc is the Coriolis parameter (2sin) and s is the sign of the
centrifugal force (page 183 Stull) and R is the radius of curvature for the
flow. If we define Ur=U and Vr=V , the component to the gradient wind,
then the result of this force balance is the gradient wind, where the
magnitude of the gradient wind is Mr, as in,
M r  U r  Vr
2
2
( 35 )
The gradient wind balance again like geostrophic wind gives us clues
about the real atmosphere, even in cases where ‘pure’ gradient wind
balance does not exist. For example, from the solution to the gradient
wind, we find that around a low,
fc R 
4G 
Mr 
 1  1 

2 
f c R 
where,
15
( 36 )
If G is defined as:
G  U g  Vg
2
2
( 37 )
The result of such an analysis gives us:
1. Flow subgeostrophic around a low pressure.
2. Flow supergeostrphic around a high pressure.
In addition, gradient balance requires certain constraints on the fluid
motion (to allow mathematical solutions), which provides more guidance
for how flows develop. For example,
Around high pressure systems, gradient flow is violated when the isobars
are closely spaced AND sharply curved. Thus, pressure gradient around
high pressure systems are not typically tightly spaced.
There is no such restriction around low pressures.
Cyclostrophic Balance
As in the preceding analysis, it is always important to consider the
dominant force balance for a particular type of flow. Consider the
situation of intense vortices such as tornadoes, water spouts and dust
devils, where the wind rotates around a very tight circle.
Question:
1. Write down the horizontal momentum equations that would best
represent this type of circulation.
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Thermal Wind:
Not really a wind, but rather the vertical shear of the geostrophic wind.
Consider that the geostrophic wind is proportional to a pressure gradient.
This pressure gradient is proportional to the gradient in geopotential, so the
geostrophic wind is proportional to the geopotential. Recall that
geopotential is proportional to the thickness of the layer, or the
temperature. Therefore, the geostrophic wind is proportional to the
temperature in some manner.
It turns out than an important relationship between the vertical wind shear
and horizontal temperature variations, is very useful in analyses of
atmospheric motions. This relationship is called the ‘thermal wind’
relationship. Formally, the thermal wind is:

Vg
 ln p

R ˆ
k   pT
f
( 38 )
which says that the vertical shear of the geostrophic wind is proportional to
the horizontal gradient in temperature. So, if there is a horizontal
temperature gradient, then the geostrophic wind will change direction with
altitude.
One of the results of this relationship is that the temperature advection can
be determined by comparing the geostrophic wind at two levels.
If the geostrophic wind turns counterclockwise (backs) with height, then
cold advection results
If the geostrophic wind turns clockwise (veers) with height, then warm
advection results
Example:
V0 (850mb) is a north wind
V1 (250mb) is a northwest wind: Advection is ____________
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Barotropic and baroclinic atmospheres:
Barotropic:
   ( p)
or on a pressure surface, density is constant.
In addition, Temperature is constant on pressure surfaces, and from the
thermal wind expression, the vertical wind shear is also constant, or the
wind doesn’t change with altitude. Thus in a barotropic atmosphere, the
wind (and thus the temperature gradient) is constant as you move up in
altitude. Although this doesn’t seem a likely state of the atmosphere in
general, this assumptions greatly simplifies solving the governing
equations, and was used in the first weather prediction models.
Baroclinic:
   ( p, T )
For this case, the density depends on both pressure and temperature. Thus,
for this case, the geostrophic wind (and temperature) has a vertical shear,
related to a temperature gradient. Thus a baroclinic atmosphere, where the
wind changes with altitude, is a more realistic condition in the dynamical
atmosphere.
Vorticity:
The conservation of angular momentum (which explains how an ice skater
spins faster with their arms in tight compared to outstretched) is a powerful
constraint on the rotational aspect of atmospheric circulations.
The vorticity is a property that describes the internal rotation of a fluid.
In meteorology we consider both absolute and relative vorticity, with the
distinction having to do with the Earth’s own rotation.
The absolute vorticity is:

  k̂    Ua 
while the relative vorticity:

  k̂    U
18

( 39 )
( 40 )
where, the absolute vorticity is equal to the relative vorticity plus the
planetary vorticity,
f
( 41 )
We also note that the vertical component of the relative vorticity is equal
to:

v u

x y
( 42 )
In atmospheric studies, the relative vorticity (rotation relative to the
rotating Earth) is of primary importance.
Note: for a low pressure system (N. Hem), the relative vorticity is positive,
for a high pressure system, the relative vorticity is negative.
Relative vorticity is affected by a few different processes in the
atmosphere including horizontal divergence and convergence, which
decreases and increases the vorticity respectively.
One can derive a diagnostic or prognostic equation for vorticity, both
extremely useful for understanding atmospheric circulations. A diagnostic
equation for vorticity uses current weather information, such as
geopotential height, to determine both the horizontal and vertical motion
fields. A prognostic equation, which predicts a future state of the
atmosphere, can be formed for vorticity as well, and represents a common
approach in numerical models of the atmosphere.
Examples: http://www.met.sjsu.edu/
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Further Vorticity Notes:
A. Vorticity in general is defined as:

   U

( 43 )
However, it is the vertical component of relative vorticity which is the
most important component for synoptic scale motions (as determined
from a scale analysis). Thus, in meteorology when we say vorticity, we
typically mean the vertical component of the vorticity:

  k̂    U

( 44 )
B. The absolute vorticity, which is defined as,
f
can be a useful diagnostic for atmospheric motions. For example, the
advection of absolute vorticity at times is directly related
 to the vertical
motion field. Recall that advection is defined as U   , so that the
advection of absolute vorticity is:

U  
( 45 )
If you consider a situation where the geostrophic wind is advecting a
cyclonic disturbance,
U
then,
You



 U    u
v
x
y
or
20
( 46
)



 U    u
v
x
y
( 47 )
where the planetary vorticity is assumed constant since there is no change
in latitude. So, if you are downstream from the vorticity disturbance, you
will first encounter:
u

 ()(  )( )  ( ) or positive vorticity advection
x
Thus, from a weather map, one can determine regions of positive vorticity
advection (PVA) and negative vorticity advection (NVA).
It turns out that for synoptic scale motions, areas of PVA are often (but
not always!) associated with rising motions, and areas of NVA are
associated with sinking motion. Recall, that determining the sign and
strength of vertical motions fields are important components to
understanding the evolution of weather systems.
C. Potential Vorticity:
Potential vorticity is a term that relates both the absolute vorticity
(rotation) and the static stability (temperature) together. Thus, it’s not
really vorticity, but rather a new quantity that has both rotation and
temperature in one term. The extremely useful nature of potential vorticity
is that for certain conditions, potential vorticity is conserved. Remember,
that conserved quantities are really useful for understanding fluid flow,
and are great tools for atmospheric scientists!
If we consider a cylindrical column of air, conserving potential vorticity,
with the depth of the column represented as, z, then potential vorticity is:
  f   const
z
21
( 48 )
Rossby Waves
The largest scale waves in the atmosphere are called planetary waves, or
Rossby waves, and are of primary importance for large-scale
meteorological processes. If we consider a barotropic atmosphere, then
the Rossby wave is an absolute vorticity conserving motion, which owes
its existence to the variation of the Coriolis force with latitude, or the socalled beta effect.
The Rossby wave: Basic mechanism
We first consider the conservation of absolute vorticity
d
  f   0
dt
( 49 )
Suppose at t0, that the relative vorticity is zero, and at t1, that the parcel is
displaced a distance dy (north-south). Because of the conservation
statement above, we can then say,
f 0    f 1
( 50 )
or
 1  f 0  f1  2 sin   2 sin   y 
( 51 )
If dy <<1 then we can expand the last term in a Taylor series which is
truncated, giving

df 
1  2 sin   2 sin   y 
dy 

( 52 )
or
1   y
( 53 )
where,

df
dy
22
( 54 )
So, if a chain of parcels is subject to a southward displacement, so that dy
is negative, then the vorticity that results will be positive (cyclonic); for a
northward displacement, a negative (anticyclonic) vorticity will result.
Thus, the mechanism that causes large scale waves, which we call Rossby
waves, is the change in Coriolis force with latitude, or .
Characteristics of Rossby waves are:
Wavelength () = 6000km
Amplitude: 1500 km (meridionally)
The wave speed (c) is
  
c0    

2



( 55 )
which means Rossby waves propagate to the west!
However, the total wave speed is C=U0 + C0, and for typical disturbances,
the overall disturbance wave speed is eastward.
However, this relationship states that longer waves move faster to the
west, while shorter waves have slower westward tendency. Physically,
this is related to the fact that for longer waves, f (Coriolis) dominates,
while for shorter waves, the relative vorticity dominates.
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