CSU ATS601
5
5.1
Fall 2015
Nearly incompressible fluids and sound proofing
Boussinesq approximation: equations for a stratified ocean
Recall that when we discussed an approximate equation of state for sea-water, we realized that seawater
is nearly incompressible. That is, density variations ( ⇢) are small compared to a constant background
density (⇢0 ⇠ 1000 kg/m3 ):
⇢(x, t) = ⇢0 + ⇢(x, t),
with
⇢0 = const., and ⇢ << ⇢0
(5.1)
The approximation ⇢ << ⇢0 and its consequences for the structure of the equations of motion is frequently referred to as the Boussinesq approximation, named after French mathematician and physicist
Joseph Valentin Boussinesq (1842-1929).
In the ocean, density variations ⇢ ⇠ 10-2 ⇢0 , with the leading order contribution from vertical pressure
variations (as we discussed previously). It is therefore sometimes useful to further decompose ⇢ into two
components:
0
ˆ
⇢(x, t) = ⇢(z)
+ ⇢ (x, t)
(5.2)
where ⇢ˆ is the leading order density perturbation due to vertical pressure variations that are constant in time
0
and ⇢ represents the next order perturbations due to variations in other factors such as temperature and
salinity. Typically:
0
⇢ ⇠ 1kg/m3 ⇠ 10-1 ⇢ˆ ⇠ 10-3 ⇢0
(5.3)
Obviously the Boussinesq approximation is more appropriate for the ocean than the atmosphere! However, there is general conceptual value that can be obtained from this set of equations - and furthermore,
atmospheric phenomena smaller than typical density variations are actually well described by the Boussinesq system.
We first assume that the background density ⇢0 is in hydrostatic balance with the background pressure:
dp0
= -g⇢0
dz
i.e.
p(x, t) = p0 (z) + p(x, t)
with
p << p0
(5.4)
The horizontal and vertical momentum equations then become (without approximation):
Du
1
1
+f⇥u=rH (p0 + p) = rH p
Dt
⇢0 + ⇢
⇢0 + ⇢
E. A. Barnes
44
(5.5)
updated 10:09 on Wednesday 30th September, 2015
CSU ATS601
Fall 2015
✓
◆
Dw
1
@(p0 + p)
1
@p0 @ p
=-g = +
-g
Dt
(⇢0 + ⇢)
@z
⇢0 + ⇢ @z
@z
✓
◆
1
@ p
= -g⇢0 +
-g
⇢0 + ⇢
@z
✓
◆
1
@ p
= g ⇢+
⇢0 + ⇢
@z
(5.6)
(5.7)
(5.8)
We are now at the point of the derivation where we use the fact that ⇢ << ⇢0 and p << p0 . We
linearize the right-hand-side (i.e. neglect quadratic and higher-order terms) and obtain the Boussinesq momentum equations:
Du
+ f ⇥ u ⇡ -rH ,
Dt
Dw
@
⇡+b
Dt
@z
(5.9)
where
•
⌘ p/⇢0
• b ⌘ -g ⇢/⇢0 (the buoyancy)
Thus, the Boussinesq system neglects all density perturbations in the momentum equations except for those
coupled to gravity!
Note: For most large-scale motions in the ocean, the deviation pressure p and deviation density ⇢
are approximately in hydrostatic balance. However, for this to hold the
buoyancy (“reduced gravity”), not simply that
Dw
Dt
Dw
Dt
must be small compared to
be small compared to gravity (more on this later).
For the continuity equation, the unapproximated equation can be written as:
D⇢
D(⇢0 + ⇢)
D ⇢
=
=
Dt
Dt
Dt
where
D ⇢
+ (⇢0 + ⇢)r · v = 0
Dt
(5.10)
Expanding terms out:
D ⇢
D ⇢
+ (⇢0 + ⇢)r · v =
+ ⇢0 r · v + ⇢r · v = 0
Dt
Dt
(5.11)
Looking at these individual terms, we know that ⇢ << ⇢0 , and so the same is true about their associated
terms above. Thus, we can neglect ⇢r · v. If we next assume that
scales the same way as ⇢r · v, then
D ⇢
Dt
D ⇢
Dt
scales advectively as ⇢U/L i.e. it
<< ⇢0 r · v. Thus, we are led to conclude that the term ⇢0 r · v
is the dominant term in the Boussinesq continuity equation, and so:
⇢0 r · v ⇡ 0
)
r · v ⇡ 0.
(5.12)
That is, the Boussinesq approximation results in non-divergent flow, which is physically related to the assumption of near-incompressibility. Note that this does not allow one to go back and say that
E. A. Barnes
45
D ⇢
Dt
= 0.
updated 10:09 on Wednesday 30th September, 2015
CSU ATS601
Fall 2015
For the thermodynamic equation, we will consider (for simplicity) adiabatic flows without salinity
changes. From the equation of state for seawater under these assumptions (see Chapter 2.4.2):
⇢ = -⇢0
T
T + p/c2s
(5.13)
Taking the total derivative of each side leads to
D⇢
D ⇢
=
= -⇢0
Dt
Dt
In seawater,
DT
Dt
T
DT
1 Dp
+ 2
Dt
cs Dt
(5.14)
⇡ 0 (since ✓ ⇡ T ; see Vallis 1.6.2), and so, we obtain the thermodynamic equation
D⇢
1 Dp
= 2
Dt
cs Dt
(5.15)
(This turns out to be the exact thermodynamic equation for adiabatic flow without salinity changes).
For nearly incompressible fluids the pressure term will always be small since sound waves cannot exist
in purely incompressible fluids (corresponding to c2s ! 1). In this case we obtain the thermodynamic
equation for a (simple) Boussinesq fluid (still adiabatic):
D⇢
D ⇢
=
=0
Dt
Dt
or
Db
=0
Dt
(5.16)
0
ˆ
recalling that the buoyancy b ⌘ -g ⇢/⇢0 . Using the decomposition ⇢ = ⇢0 + ⇢(z)
+ ⇢ (x, t) we can also
write this as:
0
D⇢
@p̂
+w
=0
Dt
@z
or
0
Db
+ Ñ2 w = 0
Dt
(5.17)
where
0
0
• b ⌘ -g⇢ /⇢0 = 0
ˆ 0 (square of the buoyancy frequency - a measure of stratification)
• Ñ2 ⌘ -g@z ⇢/⇢
For stable stratification (light fluid above a heavy fluid):
@⇢ˆ
<0
@z
i.e.
Ñ2 > 0
(5.18)
where the buoyancy frequency (Ñ2 ) measures the frequency with which a fluid parcel, if displaced vertically,
oscillates about its equilibrium position.
E. A. Barnes
46
updated 10:09 on Wednesday 30th September, 2015
CSU ATS601
5.2
Fall 2015
Anelastic approximation: equations for a stratified atmosphere
5.2.1
Sound waves
Sound waves represent the high-frequency end of the scales of motion. They do not interact with the
lower-frequency scales and are not resolved in numerical models since the time-step required would be very
short! Hence, it is desirable to filter this wave type from the equations of motion. Filtered equations that do
not allow for sound waves are sometimes referred to as sound-proof equations.
Sound waves are compression waves and belong to the general kind of wave called a longitudinal
wave. These waves propagate in the same direction (or opposite) as the particle perturbations (compression/rarefaction). Pretty much any perturbation in a compressible fluid will propagate away, in part, as a
sound wave. The larger the compressibility of the fluid the slower the speed of sound, e.g.:
• seawater: cs ⇠ 1500 m/s
• air: cs ⇠ 300 m/s
Sound waves tend to operate so fast (on such short timescales) that they are also largely adiabatic since they
are too fast for heat exchange with the environment!
Sound wave examples, movies, transverse v.s. longitudinal
The pressure perturbations associated with sound waves are tiny compared to other dynamic pressure
perturbations:
• sound pressure perturbation: ⇠ 10-3 hPa.
0
We can therefore linearize the equations about a basic state of rest: e.g. p = p0 + p , and similarly for other
variables. This results in the same horizontal momentum equations as we obtained in the Boussinesq system
(but not the vertical).
Since sound waves are longitudinal waves we can think about them in one dimension without loss of
generality:
0
0
0
@u
@x p
=
,
@t
⇢0
0
@⇢
+ ⇢0 @x u = 0
@t
(5.19)
0
where we assume the background zonal flow u0 = 0 so that u = u . For adiabatic processes we also have
dp =
E. A. Barnes
@p
|✓ d⇢
@⇢
i.e.
0
p = c2s ⇢
47
0
since
c2s ⌘
@p
|✓
@⇢
(5.20)
updated 10:09 on Wednesday 30th September, 2015
CSU ATS601
Fall 2015
0
If we combine the above equations we obtain an equation for the perturbation pressure (substituting p for
0
⇢ ):
0
0
@tt p - c2s @xx p = 0
(5.21)
This is a generic wave equation for waves propagating with the speed of sound cs .
A wave equation is an equation of the form:
@2 x
= c2 r2 x.
@t2
For example, for the atmosphere p = ⇢RT , c2s =
(5.22)
RT where
= cp /cv = 1.4 and so at room
temperature where T = 293 K, cs = 340 m/s.
Typical flow speeds in the atmosphere (ocean) are at least one (four) order(s) of magnitude smaller than
cs . This can also be seen by a simple scaling argument. Consider the scaling for the x-momentum equation
with an advective timescale (i.e. T ⇠ L/U):
Du U2
⇠
,
Dt
L
0
0
0
@x p
p
⇢
⇠
⇠ c2s
⇢0
L⇢0
L⇢0
2
)
U ⇠
0
⇢
c2s
⇢0
(5.23)
Thus, for flow speeds much smaller than the speed of sound we obtain:
U2
<< 1
c2s
0
)
⇢
<< 1
⇢0
(5.24)
That is, for flow speeds much smaller than the speed of sound the fluid can be treated as nearly incompressible! It is clear that this holds much better in the ocean than the atmosphere given the ocean’s much faster
speed of sound. The ratio U/cs is called the Mach number, named after Austrian physicist and philosopher Ernst Mach (1838-1916). Thus, we now see why letting cs ! 1 corresponds to treating the fluid as
incompressible!
5.2.2
Anelastic approximation equations
The Boussinesq approximation assumed a constant background density, which was appropriate for the
ocean. The atmosphere’s density, however, is certainly not constant since it varies strongly with height.
However, in the case of the atmosphere we can make the assumption that the background density and
pressure are in hydrostatic balance:
⇢(x, t) = ⇢0 (z) + ⇢(x, t),
E. A. Barnes
with
p(x, t) = p0 (z) + p(x, t)
48
and
@p0
= -g⇢0
@z
(5.25)
updated 10:09 on Wednesday 30th September, 2015
CSU ATS601
Fall 2015
Since only the horizontal pressure gradients enter the horizontal momentum equations, the anelastic horizontal momentum equations are identical to those of the Boussinesq. The vertical momentum equations are
not identical, because now, ⇢0 = ⇢0 (z) unlike in the Boussinesq equations where ⇢0 = constant. Thus, for
the anelastic vertical momentum equation we have:
Dw
Dt
1
@(p0 (z) + p)
-g
(⇢0 (z) + ⇢)
@z
✓
◆
1
@p0 (z) @( p)
+
-g
(⇢0 (z) + ⇢)
@z
@z
✓
◆
1
@( p)
-g⇢0 +
-g
(⇢0 (z) + ⇢)
@z
g⇢0
@z ( p)
-g
⇢0 + ⇢ ⇢0 + ⇢
-g ⇢
@z ( p)
⇢0 + ⇢ ⇢0 + ⇢
and now using the approximation that ⇢ << ⇢
g ⇢ @z ( p)
⇢0
⇢0
p
⇢
-@z - 2 @z ⇢0 - g
⇢0
⇢0
= -
(5.26)
=
(5.27)
=
=
=
⇡
=
(5.28)
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
Note that up until the final line we followed the exact same steps as we did for the Boussinesq approximation,
where
⌘ p/⇢0 .
Since we intend to apply our set of equations to the atmosphere, we can now use the ideal gas law.
Namely, we are going to do a little fancy footwork to get potential temperature into our equations. First,
recall the definition of potential temperature:
✓=T
✓
p0
p
◆R/cp
)
R
ln ✓ = ln(T ) +
ln
cp
✓
p0
p
◆
= ln T +
R
R
ln p0 ln p
cp
cp
(5.34)
Now, the ideal gas law can be written in differential form:
⇢=
E. A. Barnes
p
RT
)
(5.35)
d(ln ⇢) = d(ln p) - d(ln RT )
= d(ln p) - [d(ln R) + d(ln T )] = d(ln p) - d(ln T )
✓
◆
✓
◆
R
R
= d(ln p) - d(ln ✓) + d
ln p0 - d
ln p
cp
cp
✓
◆
R
= d(ln p) - d(ln ✓) - d
ln p
cp
= (1 - R/cp )d(ln p) - d(ln ✓)
(5.36)
= (1 - )d(ln p) - d(ln ✓)
(5.40)
49
(5.37)
(5.38)
(5.39)
updated 10:09 on Wednesday 30th September, 2015
CSU ATS601
Fall 2015
Plugging-in for (1/⇢0 )@z ⇢0 in (5.33):
p
p
p
p
p
p
@z ⇢0 =
@z ln(⇢0 ) = (1 - ) @z (ln p0 ) @z (ln ✓0 ) = -g(1 - )
@z (ln ✓0 ) (5.41)
2
⇢0
⇢0
⇢0
p0
⇢0
⇢0
where the first term on the right-hand-side comes from using hydrostatic balance (and yes, the denominator
is supposed to be a p0 , not a ⇢0 ). In the above equation, ✓0 is the potential temperature associated with the
background state given by ⇢0 and p0 .
In a similar manner, the linearized perturbations ⇢, p, ✓ are related through the ideal gas law and
definition of potential temperature:
⇢
p
✓
⇡ (1 - )
⇢0
p0
✓0
(5.42)
from which we can finally re-write (5.33) as:
Dw
✓
p
⇡ -@z + g +
@z (ln ✓0 )
Dt
✓0
⇢0
(5.43)
To estimate the last term on the r.h.s. we note the potential temperature scale height:
H✓ ⌘ (@z ln ✓0 )-1 =
g
⇠ 100 km for tropospheric N2
N2
(5.44)
For vertical scales much smaller than H✓ we can neglect this last term and obtain:
Dw
✓
⇡ -@z + g
Dt
✓0
(5.45)
This is the vertical momentum equation in the anelastic approximation.
You may wonder where the term anelastic comes from. This originates from the approximation that is
used in the system’s continuity equation:
0 = @t ⇢ + r[(⇢0 + ⇢)v] ⇡ r(⇢0 v),
or
rH · u +
1
@z (⇢0 w) ⇡ 0
⇢0
(5.46)
This approximation corresponds to neglecting the local rate of change of density. This is consistent with
scaling time advectively as we did for the Boussinesq system. That is, we neglect the elastic compressibility
of the fluid, and this serves to eliminate sound waves. However, unlike the Boussinesq system, the anelastic
system allows ⇢0 to vary with altitude, and so the continuity equation is not simply a statement of divergencefree flow (i.e. the density cannot be simply taken out of the divergence term).
In closing, we note that there are other variants of the anelastic approximation, e.g. Lipps and Hemler
(1982), see also Durran (1989). Here, we have derived the system a la Ogura and Phillips (1962).
E. A. Barnes
50
updated 10:09 on Wednesday 30th September, 2015