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)

1

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

]

7

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.

13

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.

16

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.

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