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Conservation of mass
t
M w
If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove fluid – it would be…
inflow rate – outflow rate
Now consider a small cube, fixed in space, through which air is flowing
The x direction mass flux is given by 12 smkgu
Using a Taylor series expansion about the center point, the rate of inflow through side A would be:
zyx
ux
u
2
and through side B would be:
zyx
ux
u
2
As with the basin, the rate of accumulation of mass in the cube must be:
zyx
ux
uzyx
ux
ut
M x
22
zyxuxt
M x
Similarly:
zyxuxt
M x
zyxvyt
M y
zyxwzt
M z
The net mass accumulation in the cube would be:
zyxwz
vy
uxt
M
Dividing by the volume zyx
wz
vy
uxt
wz
vy
uxt
This equation can be written as:
Vt
The flux form of the mass
continuity equation
Using the vector identity:
VVV
this equation can be written as:
0
VVt
or:0
1 V
dt
d
The velocity divergence form of the mass
continuity equation
Vt
The flux form of the mass
continuity equation
01
Vdt
d
The velocity divergence form of the mass
continuity equation
For the special case of an incompressible fluid, density doesn’t change following parcel motion so the continuity equation reduces to:
0 V The incompressible form
of the mass continuity equation
Finally, let’s derive the mass continuity equation in pressure coordinates…
zyxV Consider a volume of air:
Using the hydrostatic equation gz
p
zg
p
or
We can write our volume as:g
pyxV
g
pyxVM
and the mass of the parcel as:
If we follow parcel motion, the mass of parcel should be conserved:
g
pyx
dt
dM
dt
d 0
g
pyx
dt
dM
dt
d 0
Divide both sides by M
g
pyx
dt
d
pyx
gM
dt
d
M
01
Apply chain rule to RHS
0
1
yx
dt
pdpx
dt
ydpy
dt
xd
pyx
Recalling that
dt
pdv
dt
ydu
dt
xd,,
The equation can be reduced, in the limit that the volume approaches 0, to be
0
py
v
x
u The pressure coordinate form of the mass continuity equation
0
py
v
x
u The pressure coordinate form of the mass continuity equation
Vt
The flux form of the mass
continuity equation
01
Vdt
d
The velocity divergence form of the mass
continuity equation
0 V The incompressible form
of the mass continuity equation
The various forms of the mass continuity equation so far
Finally, substituting the geostrophic and ageostrophic wind:
yfug
1
xfvg
1
0
py
v
x
u
0
py
vv
x
uu aggagg
011 22
py
v
x
u
yxfyxfagag
0
py
v
x
u agag
0
p
Vag
The ageostrophic wind form of the mass continuity equation
0
py
v
x
u The pressure coordinate form of the mass continuity equation
Vt
The flux form of the mass
continuity equation
01
Vdt
d
The velocity divergence form of the mass
continuity equation
0 V The incompressible form
of the mass continuity equation
The various forms of the mass continuity equation
0
p
Vag
The ageostrophic wind form of the mass continuity equation
The pressure coordinate form is particularly enlightening
0
py
v
x
u
We can integrate this equation between the top of a column and the surface
t
s
th
sh
pVV
V
h
stPhPh stVV
stPhPh stVV
If there is no vertical air motion at the surface and there is convergence, air will rise, a direct outcome of mass continuity. Vertical motion and column divergence patterns are directly related.
Conservation of energy
We will consider mechanical energy, thermal energy, and total energy
cos2sin2
1tan2
2
wvz
uK
x
p
a
uw
a
uv
dt
du
sin2
1tan2
22
uz
vK
y
p
a
vw
a
u
dt
dv
The complete momentum equations
Multiply (1) by u, (2) by v, and (3) by w to get energy equations
cos2122
ugz
p
a
vu
dt
dw
cos2sin2
tan
2
12
2222
uwuvz
uuK
x
pu
a
wu
a
vu
dt
du
sin2
tan
2
12
2222
uvz
vvK
y
pv
a
wv
a
vu
dt
dv
cos22
1 222
uwwgz
pw
a
wvwu
dt
dw
cos2sin2
tan
2
12
2222
uwuvz
uuK
x
pu
a
wu
a
vu
dt
du
sin2
tan
2
12
2222
uvz
vvK
y
pv
a
wv
a
vu
dt
dv
cos22
1 222
uwwgz
pw
a
wvwu
dt
dw
Add equations together: note that earth curvature and Coriolis force terms all cancel!
hh Vz
KVgwpVwvu
dt
d 2
2222 1
2
Note that:dt
d
dt
dzggw
Move this to left side
hh Vz
KVpVwvu
dt
d 2
2222 1
2
hh Vz
KVpVwvu
dt
d 2
2222 1
2
Kinetic Energy
Potential Energy
Work done by PFG Work done by Friction
A change in total mechanical energy of a parcel of air must come about by work done by the pressure gradient and frictional forces
Note that in geostrophic flow so the first term on the LHS is zero in geostrophic flow. Only the Ageostrophic wind component does work.
ptoparallelisV
The thermodynamic energy equation
First law of thermodynamics can be expressed as:
dt
dp
dt
dTcQ v
dt
dp
dt
dTcQ p or
where is the diabatic heating rate and is the specific volumeQ
FVpVwvu
dt
dh
1
20
222simplifyingfriction notation
Mechanical energy equation
Total energy equation (add TEE and MEE)
FVpVwvu
dt
d
dt
dp
dt
dTcQ hv
1
2
222
FVpVwvu
dt
d
dt
dp
dt
dTcQ hv
1
2
222
pVpV
1
pVt
p
dt
dp
Note that: and
So:
t
p
dt
dppV
1
Substituting: FVt
p
dt
dpwvu
dt
d
dt
dp
dt
dTcQ hv
2
222
FVt
pwvupTc
dt
dQ hv
2
222
The Energy Equation
If flow is adiabatic , frictionless and
steady state , then:
0Q 0F
FVt
pwvupTc
dt
dQ hv
2
222
0t
p
.2
222
constwvu
pTcv
-For an atmosphere at rest, an increase in elevation results in a decrease in hydrostatic pressure
Special case of Bernoulli’s equation for incompressible flow
.2
222
constwvu
pTcv
Other implications
In accelerating flow over a hill the pressure difference between p2 and p1 must be > hydrostatic
Potential Temperature
Temperature a parcel of air would have if it were brought dryadiabatically to a pressure of 1000 mb. “Dry adiabatically” impliesNo exchange of mass or energy with the environment, and noCondensation or evaporation occurring within the air parcel.
Potential temperature equation derived from 1st law of thermodynamics:
pdRTdcdS dp lnln S = Entropycp = Specific Heat at constant pressureRd = dry air gas constant
For an adiabatic process, dS = 0
0lnln pdRTdc dp
0lnln pdRTdc dp
Integrate equation from an arbitrary temperature and pressure tothe potential temperature and a pressure of 1000 mb.
1000
lnlnP
Tp
d pdc
RTd
pc
R
T p
d 1000lnln
p
d
c
R
pT
1000
