Gravity wavesatmos.ucla.edu/~fovell/AOSC115/gravwaves.pdfGravity waves AS 101 Fall, 2002 – Fovell...
Transcript of Gravity wavesatmos.ucla.edu/~fovell/AOSC115/gravwaves.pdfGravity waves AS 101 Fall, 2002 – Fovell...
Gravity waves
AS 101 Fall, 2002 – Fovell
Previously, we examined the oscillatory behavior of a displaced air parcel within a stable
environment. We obtained a pendulum-like equation in which the frequency was propor-
tional to the vertical gradient of potential temperature (the Brunt-Vaisalla frequency). Now,
we consider how this oscillating parcel provokes its surrounding environment, and how the
stable environment adjusts to being disturbed.
The adjustment is performed by the issuing of “gravity waves”. This rather poor name
comes from the fact that gravity – through density and the buoyancy force – provides the
restoring force that attempts to force vertically displaced air back to its original location
(in a stable environment).
Herein, we derive the “dispersion relationship” for gravity waves in a calm, dry, strictly
adiabatic and stable two-dimensional environment. The dispersion equation relates fre-
quency to wavelength. The derivation is simpler than that presented in Holton, because
additional restrictive assumptions have been made. I’m trying to step through the deriva-
tion providing sufficient detail so you can see what was done where, when and why.
The 2D Boussinesq equations
We are going to use the perturbation method applied to the equations describing 2D motions
in an incompressible (Boussinesq) fluid on a flat, non-rotating Earth. In the Boussinesq
approximation, the density of the environment is taken to be a constant. Since we know that
density is actually a very strong function of height, it behooves us to apply the Boussinesq
approximation only for very shallow phenomena. The approximation neglects the density
perturbation everywhere except when multiplied by the gravity acceleration g. That large
and important term can’t be neglected or we won’t have anything to study at all.
∂u
∂t+ u
∂u
∂x+ w
∂u
∂z+
1ρ
∂p
∂x= 0 (1)
∂w
∂t+ u
∂w
∂x+ w
∂w
∂z+
1ρ
∂p
∂z+ g = 0 (2)
∂u
∂x+
∂w
∂z= 0 (3)
∂θ
∂t+ u
∂θ
∂x+ w
∂θ
∂z= 0 (4)
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Now we apply the perturbation method presuming a calm, stable, dry and hydrostati-
cally balanced atmosphere. Mean pressure and θ are functions of height but mean density is
a constant value ρ0. Please note we are not neglecting the pressure perturbation here (like
we did straightaway for the air parcel oscillation derivation). Recall that perturbations are
assumed to be small.
u(x, z, t) = u′(x, z, t)
w(x, z, t) = w′(x, z, t)
ρ(x, z, t) = ρ0 + ρ′(x, z, t)
p(x, z, t) = p(z) + p′(x, z, t)
θ(x, z, t) = θ(z) + θ′(x, z, t)dp
dz= −ρ0g
Starting with the definition of θ, we can expand with logs to obtain:
ln θ =cv
cpln p− ln ρ + consts, (5)
where we are unconcerned with the constants (they’ll remove themselves anyway). The base
state satisfies
ln θ =cv
cpln p− ln ρ0 + consts. (6)
Apply the perturbation method to (5). We obtain:
ln(θ + θ′) =cv
cpln(p + p′)− ln(ρ0 + ρ′) + consts
ln[θ(1 +
θ′
θ)]
=cv
cpln
[p(1 +
p′
p)]− ln
[ρ0(1 +
ρ′
ρ0)]
+ consts.
Note that (6) may be used to cancel out the base state configuration (along with those
pesky constants). Note further that
ln(1 + x) ≈ x
if x << 1. This is true for the perturbations. Thus, we are left with the following:
θ′
θ=
cv
cp
p′
p− ρ′
ρ0
=p′
ρ0c2s
− ρ′
ρ0,
2
where c2s, the square of the adiabatic speed of sound, is given by
c2s =
cv
cpRdT .
Scale analysis shows that the term with p′ is considerably smaller than the density pertur-
bation term (cs is large; its square is huge) and thus, to a reasonable approximation, we
have:ρ′
ρ0≈ −θ′
θ.
This is the basis of the air parcel mechanical equilibrium assumption and permits us to
simply rewrite the buoyancy term in the w equation in terms of temperature instead of
density perturbation.
Now we multiply (2) by ρ and perform a perturbation analysis, setting products of
perturbations (such as u′ ∂u′
∂x ) to zero. We obtain:
ρ0∂w′
∂t+
dp
dz+
∂p′
∂z+ ρ0g + ρ′g = 0.
The hydrostatic mean state may be removed from the above. After dividing through by ρ0,
we get the ratio of the density perturbation to the mean coupled with gravity. Density may
be replaced with potential temperature as shown above, resulting in:
∂w′
∂t+
1ρ0
∂p′
∂z− g
θ′
θ= 0. (7)
The other linearized equations are similarly found to be:
∂θ′
∂t+ w
dθ
dz= 0 (8)
∂u′
∂t+
1ρ0
∂p′
∂x= 0 (9)
∂u′
∂x+
∂w′
∂z= 0. (10)
Note we’ve kept p′ in the pressure gradient acceleration term but neglected it in the buoyancy
term.
The horizontal vorticity form and further reduction
We define horizontal vorticity parallel to the y axis as:
η =∂u
∂z− ∂w
∂x.
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This represents spin in the x− z plane in which the spin axis is parallel to the y axis. By
convention, clockwise spin (viewed from the south, looking in the +y direction) is considered
positive vorticity. The perturbation vorticity η′ is simply
η′ =∂u′
∂z− ∂w′
∂x.
We can get the two velocity equations into a single equation in vorticity form in the following
manner. Take the horizontal derivative of (7) and subtract from it the vertical derivative
of (9). This yields:∂
∂t
[∂w′
∂x− ∂u′
∂z
]− g
θ
∂θ′
∂x= 0. (11)
Note the first term in the above is the time tendency of −η′. The equation tells us that
a horizontal temperature gradient in the x direction will cause the production of horizontal
vorticity in the x − z plane. This is easily seen as being manifested in the sea-breeze
circulation, depicted in Fig. 1.
cold warm
η < 0
Figure 1: Horizontal vorticity associated with horizontal temperature gradient.
From this point, we want to transform this equation into a form in which only one
perturbation variable appears. We choose w. We can take the horizontal derivative of (11)
and get (after interchanging the time and horizontal derivatives in the first term):
∂
∂t
[∂2w′
∂x2− ∂2u′
∂x∂z
]− g
θ
∂2θ′
∂x2= 0. (12)
Now for a few handy tools. The continuity equation (10), after rearrangement, is:
∂u′
∂x= −∂w′
∂z,
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and differentiating both sides with respect to z yields:
∂2u′
∂x∂z= −∂2w′
∂z2. (13)
The potential temperature equation (8) implies:
∂θ′
∂t= −w
dθ
dz,
which after differentiating twice with respect to x and rearranging yields:
∂
∂t
∂2θ′
∂x2= −dθ
dz
∂2w′
∂x2. (14)
Substitute (13) and (14) into (12), after having differentiated (12) with respect to time.
Note that the square of the Brunt-Vaisalla frequency of the mean state, defined by
N2 =g
θ
dθ
dz
appears in the equation. We end up with this expression:
∂2
∂t2
[∂2w′
∂x2+
∂2w′
∂z2
]+ N2 ∂2w′
∂x2= 0, (15)
which is another pendulum-like oscillation equation.
Solutions of the oscillation equation
The oscillation equation (15) has wave-like solutions describing gravity (buoyancy) waves.
We can reveal more about the wavy behavior by presuming a wave-like solution of a partic-
ular form. The waves will be characterized by an amplitude w and horizontal and vertical
wavelengths Lx and Lz. The particular form is:
w′ = w expi(kx+mz−ωt) = E, (16)
where k and m are wavenumbers related to wavelength by
k =2π
Lx
and
m =2π
Lz,
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and ω is the frequency, which is 2π divided by the oscillation period. (“E” is a convenient
shorthand.) From Euler’s relation, given by
expiq = cos q + i sin q
exp−iq = cos q − i sin q,
we see the solutions do indeed describe wavy structure in space as well as time.
Given (16), we can see that the horizontal derivatives are:
∂w′
∂x=
∂
∂x
[w expi(kx+mz−ωt)
]= w expi(kx+mz−ωt) ∂
∂x[i(kx + mz − ωt)]
= wik expi(kx+mz−ωt),
and so the second derivative of this with respect to x is
∂2w′
∂x2= −wk2 expi(kx+mz−ωt)
because i2 = −1. Continuing, we see that
∂
∂t
∂2w′
∂x2= −wk2iω expi(kx+mz−ωt)
and∂2
∂t2∂2w′
∂x2= k2ω2E.
Substitute these expressions into (15) and note that “E” appears in every term and thus
may be divided out. The entire equation simplifies dramatically and may be solved for the
frequency ω:
ω2(k2 + m2)−N2k2 = 0,
or
ω = ± Nk√(k2 + m2)
. (17)
It is seen that the gravity wave frequency (and thus the period) depends on the stability
of the environment (N) and the horizontal and vertical wavelengths of the waves. These
wavelengths may be determined by the size and shape of the phenomenon that is provoking
the environmental response (i.e., the air parcel characteristics). Note that the more stable
the environment is (larger N), the larger the frequency is and thus the period of the wave
shortens. In an adiabatically neutral environment, there is no gravity wave activity.
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Phase speed and phase line tilt
If you toss a rock in a pond, this disturbance instigates an adjustment in the pond surface
that radiates outward from the disturbance point at some speed. Actually, individual fluid
elements do not experience any net motion as the disturbance waves pass by; instead, they
merely bob up and down. This is an example of external gravity waves, which propagate
along a boundary or the interface between fluids of differing densities (such as air and water).
We’ve been deriving the frequency equation for internal gravity waves, which propagate
within the fluid. However, there are similarities in terms of concept.
The phase speed represents the speed in which a wave’s trough or ridge is moving
within the fluid. This speed is given by the frequency divided by the wavenumber. Thus,
the horizontal phase speed (cx) for internal gravity waves in this instance is:
cx =ω
k= ± N√
(k2 + m2).
Note there are two signs. This is because a disturbance at some point will provoke two,
oppositely propagating waves.
The phase speed written above is an intrinsic speed, relative to the fluid in which the
wave is moving. If the fluid itself is moving relative to the ground, the ground-relative
phase speed is the sum of the intrinsic flow-relative speed plus the ground-relative speed of
the flow. It is quite possible that a wave can move relative to the fluid but be stationary
relative to the ground. In that case, say the wave is moving west at a speed of X but the
flow is coming from the west at that same speed. The ground-relative motion of the wave
is zero. This occurs for mountain-induced gravity waves which play an important role in
the development of downslope windstorms.
If the vertical wavenumber (wavelength) is nonzero (not infinite), the waves will be seen
to tilt with height from the vertical. This tilting angle (α) is a function of the wave’s
frequency relative to the Brunt-Vaisalla frequency of the environment.
To see this, examine Fig. 2. The angle α is related to the wavelengths by:
α = arctan[Lx
Lz
]= arccos
[Lz√
L2x + L2
z
]
= arccos[
k√k2 + m2
]
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Lx
Lz
α
Figure 2: Gravity wave phase line tilt geometry.
Using the definition of ω (17), we obtain:
α = arccos[ ω
N
].
Types of forcings
The forcing for gravity waves may be of finite duration, a single pulse, such as a single rock
that disturbs a pond. It can also be periodic (oscillatory and ongoing) or steady. Sources
of such forcings include flow over mountains (or other obstacles) and by the development of
short-lived convective cells. In flow over a mountain, the disturbance comes in the form of
a succession of air parcels impelled to rise over a fixed obstacle. Individual convective cells
grow upward within the troposphere and successively impinge on the stable stratosphere
from below, making for a temporally unsteady forcing with frequency ω and period related
to the separation time between the cells. Gravity waves are produced, with phase tilts
depending on the ratio of ω and N .
Gravity waves can also be excited by heat sources, including condensation warming
and evaporation cooling. Again, the gravity waves represent the environment’s coming
to terms with the introduction of a disturbance. If the sources are maintained – i.e, the
forcing frequency is zero – then the gravity waves spread outward, permanently altering
the environment through which they have passed. Permanent, that is, until the source is
removed or deactivated; then the environment responds to that provocation – by producing
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another gravity wave, of course.
Synthesis and examples
Here is a summary of the principal points made thusfar, with some additional discussion,
presented in outline form:
• An air parcel oscillating within a stable environment disturbs its surroundings;
• The environment adjusts to the disturbance through the issuance of gravity waves
that propagate away from the disturbed area;
• The horizontal and vertical wavelengths of the gravity waves reflect the size of the dis-
turbance source, while the waves’ frequency is determined by the oscillation (forcing)
frequency of the disturbance, ω.
• The phase speed of the waves relative to the flow in which they are embedded is a
function of the wave’s forcing frequency and wavelength.
• For finite pulse or periodic forcings, the fluid is bumped up and down as a gravity wave
passes a particular point, but no net change in position is effected. For steady forcings,
the waves cause fluid displacements that persist until the forcing is deactivated.
• The tilt of the propagating gravity waves’ phase lines is a function of the ratio of the
forcing and environmental Brunt-Vaisalla frequencies.
Excitation of stratospheric waves by convection
The successive development of individual convective cells in an organized thunderstorm
complex can provide a means of mechanically exciting gravity waves in the overlying strato-
sphere. Each individual cell, or cloud, develops upward and impinges on the stable strato-
sphere from below as it reaches (and overshoots slightly) its equilibrium level. The waves so
excited are dependent upon the size and shape of the individual cells, the separation period
between cell developments and the motion of the cells within the thunderstorm complex.
This is illustrated by the results of a rather simple, 2D compressible model simulation
in which a mechanical oscillator of specified size and period was inserted into the model’s
middle troposphere (Fovell, Durran and Holton 1992). The oscillator was an elliptically
shaped object with a half-width of 10 km and a half-height of 3 km, centered at 8 km above
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the ground. It was forced at a period of 20 min, corresponding to the separation interval
between individual cloud developments.
In panel (a) of the figure below, the oscillator was configured to be fixed in space and
vertically erect. As the object rises and impinges on the tropopause (located at about 12
km), gravity waves are generated that propagate upward and horizontally away from the
disturbance point. In this case, waves propagate both westward and eastward from the
source, and the results are symmetric. Both the phase speed and phase line tilt appear as
expected from our simplified analysis of the equations.
Figure 3: Gravity waves induced by a simple mechanical oscillator.
It will be shown later that in actual storms the stratospheric gravity wave activity
tends to be concentrated on the rear side of the storm. The simple oscillator model was
employed to elucidate the reason(s) for this behavior. First, it was noted that convective
cells are not perfectly erect, that they tend to tilt towards the west for an eastward-moving
storm complex. In panel (b), we made the source tilt to the left slightly. It is seen that
this orientation favors the westward-propagating stratospheric waves, but that eastward
traveling waves are still excited.
More important is the fact that the cells, once established, tend to travel towards the
storm’s rear with time. This is westward motion for an eastward propagating storm. In
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panel (c), the source not only tilts westward but also translates westward as it oscillates.
Note the absence of eastward traveling waves in the stratosphere.
Finally, the primacy of translation over tilt is illustrated in panel (d). In this case,
the source still translates westward but is vertically erect. So, westward translation is a
sufficient condition for suppressing the eastward propagating stratospheric waves.
Figure 4 shows the results of a simulation of an organized squall-line in a 2D cloud
model, after it has reached its mature stage. Both panels depict potential temperature
(note the contours are much closer together vertically in the stable stratosphere) and the
cloud outline (bold curve). New convective updrafts are established in the lower troposphere
on the storm’s east side (around x = 340 km) and subsequently propagate upward and
westward with time.
The upper and lower panels superpose horizontal and vertical velocity, respectively.
A series of gravity waves spread like a fan outward from a single point, representing the
location in the storm where each cell tends to strike the tropopause as it develops and
translates. Note the absence of stratospheric activity on the east side.
Ground stationary gravity waves
Consider a mountain sitting within a mean flow, coming from the west for sake of example.
Parcels are being forced to rise over the obstacle. This disturbance will initially provoke both
westward and eastward propagating gravity waves which will translate relative to the flow.
Because the flow is from the west, the eastward moving waves will move very quickly away
from the obstacle. In contrast, the westward propagating waves can be rendered stationary
relative to the ground, even though they are (and remain) propagating phenomena relative
to the flow. These stationary gravity waves hang around above the mountain, and represent
mountain waves. Figure 5 depicts a situation in which these stationary waves have resulted
in the establishment of a strong, downslope wind event.
Figure 6 shows stationary stratospheric gravity waves situated above another organized
storm. The principal difference now is that the storm experiences strong relative flow in
the stratosphere. (In contrast, the case in Fig. 4 had no mean storm-relaitve flow in the
stratosphere.) For this strong flow, the principal gravity wave activity results from the flow
being gently but definitely lifted over the storm. The storm has impinged into the lower
stratosphere and has presented an effective obstacle to the flow.
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Environmental response to heat sources
Figure 7 shows a time-space or “Hovmuller” diagram for a numerically simulated squall-line
storm from the beginning of the simulation through maturity, taken from Fovell (2002). The
plotted field is the difference between the lower and middle tropospheric horizontal velocity,
dubbed ∆u. The storm was initiated with a warm, saturated bubble which quickly grew into
a deep cumulus cloud that extended throughout the troposphere. The environment’s im-
mediate response to the commencement of deep tropospheric heating and uplift was equally
deep subsidence which propagated quickly away from the cloud. This response, identified
as “initial subsidence wave” on the figure, moved downstream and upstream (relative to
storm motion) at speeds of 30 and 45 m/s, respectively. The storm is moving towards the
right (east) with time, and thus the east side represents its upstream environment. The
speeds cited are relative to the ground.
The initial subsidence wave can be understood by realizing that the cloud represents a
maintained heat source of depth H of about 12 km. This source can be considered as half
(the warm part) of a sinusoidal structure with vertical wavelength Lz = 2H. The horizontal
wavelength is effectively infinite for a maintained source, and so k → 0. This results in an
expected phase speed of
c =NH
π,
yielding about 38 m/s when N = 0.01 s−1, a common tropospheric value. Recall this is
an intrinsic speed, a speed relative to the flow. These waves are embedded in an airflow
which is directed towards the east at 7.5 m/s in the middle troposphere. This speeds up the
eastbound wave relative to the ground, and slows the westward propagating wave. That’s
why the downstream wave moves more slowly than its upstream counterpart.
This initial wave spreads subsidence a substantial distance upstream of the storm. Sub-
sidence makes the environment both warmer (owing to adiabatic compression) and drier
(since vapor mixing ratio decreases with height). Figure 8a presents a schematic model
illustrating the updraft maintained by the deep heating and the environmental response to
that heating. The heating profile is shown at right, is a half-sine wave of depth H. Con-
densation heating released in the middle troposphere makes the atmosphere below more
stable as it decreases the lapse rate beneath the heating. Adiabatic compression warming
in the subsidence wave accomplishes the very same thing. Thus, convection stabilizes the
atmosphere not only through its own motions but also through the environment’s reaction
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to it.
Inspection of Figure 7 shows that additional, more slowly moving waves propagate
through the original subsidence region’s wake. Some of these waves are excited in part
by the fact that latent heat release in the cloud is not temporally steady. However, the
principal slower mode has an intrinsic wave speed consistent with a heat source having a
half-wavelength of about 5 km, superposed on the original deep heating profile. One can
create such a mode in two ways: by deforming the simple half-sine heating profile, making it
more “top-heavy” (see Fig. 8b, panel at right) or by introducing some adiabatic or diabatic
cooling into the midtroposphere, on top of the already established deep heating. Either
way, the environment responds by generating a shallower gravity wave, this one consisting
of lower tropospheric uplift, that spreads in the wake of the original subsidence wave. The
second mode travels more slowly since it its excited by a shallower source. Fovell (2002)
discussed the effect of this second mode on storm structure, maintenance and the initiation
of subsequent convection.
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Figure 4: Gravity waves excited above an organized but oscillatory convective storm complex.
14
Figure 5: Downslope flow provoked on the lee side of a mountain.
15
Figure 6: Stationary stratospheric waves induced by the obstacle effect.
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100 200 300 400 500
x(km)
1
2
3
4
5
time
(hr)
∆u
domain speed 12.0 m s-1
1-sm54
1-sm
82
1- sm
71
1- sm
51
1-sm
22
30 ms -1
initial s
ubsidence
wave
initial subsidence wave
∆u
< -
10 m
s-1
Figure 7: Hovmuller diagram of ∆u for a squall-line storm; see text. Speeds cited on the figure areground-relative but the reference frame has been translated eastward at 12 m/s.
17
warm
source-induced
updraught
Q1
Q1+Q2
Q1
(a) idealized single-mode response
(b) idealized two-mode response
(c) less idealized numerical solution
z (k
m)
x (km)
zz
���yyywarmer
cool
warm
source-induced
updraught
z
<<< -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 >>>
260 280 300 320 3400
4
8
θ' (shaded), u' (black contours), w (white contours)
Q (K/s)
updraught
downdraught
0 .005
0
H
0
H
Q_
�
Figure 8: Environmental response to maintained heat sources. Top two panels qualitatively depictresponse to symmetric heating functions with (a) one and (b) two vertical modes, drawn fromNicholls et al. (1991) and Mapes (1993); only upstream side is drawn. Panel (c) presents numericalresult for a less idealized situation. Heating functions shown at right; for (c) the function is spatiallyaveraged over the vertically tilted source region.
18