1. Internal Gravity Waves
MAST-602: Introduction to Physical Oceanography
Andreas Muenchow, Oct.-7, 2008
Internal Gravity Waves
Knauss (1997), chapter-2, p
Knauss (1997), chapter-10, p
Vertical Stratification
Descriptive view (wave characteristics)
Balance of forces, wave equation
Dispersion relation
Phase velocity
Same as
Surface waves

2. Ocean Stratification temperature salinity density surface depth, z
two random casts
from Baffin Bay
July/August 2003
500m

3. z Buoyant Force = Vertical pressure gradient =
Pressure of fluid at top - Pressure of fluid at bottom of object
acceleration = - pressure grad. + gravity
 ∂w/∂t = -∂p/∂z  g z

4. Buoyancy Frequency: acceleration = - pressure gradient + gravity
dw/dt = -1/ dp/dz g
but
p=grz so dp/dz= g z dr/dz + g r (chain rule)
and
d2z/dt = -g / dr/dz z
acceleration = restoring force
w = dz/dt:
thus
Solution is z(t) = z0 cos(N t)
and N2 = -g / dr/dz is stability or buoyancy frequency2

5. c2 = (/)2 = g*/ tanh[h]
Surface Gravity Wave Restoring  g (rwater-rair)/rwater ≈ g
because rwater >> rair
c2 = (/)2 = g/ tanh[h]
Internal Gravity Wave Restoring  g (r2-r1)/r2 ≈ g*
g* = g/r dr/dz Dz
= N Dz
because r1 ≈ r2
c2 = (/)2 = g*/ tanh[h]

6. c2 = (/T)2 = g (/2) tanh[2/ h]
Dispersion
Relation
c2 = (/T)2 = g (/2) tanh[2/ h]
c2 = g/ deep water waves
Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by shallow-water phase velocity √gh) as a function of relative depth h†/†λ.Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √gh) valid in shallow water.Drawn lines: dispersion relation valid in arbitrary depth.Dashed lines (blue and red): deep water limits.
Blue: Phase velocity (dash is deep water approximation)
Red: Group velocity (dash is deep water approximation)

7. c2 = (/T)2 = g (/2) tanh[2/ h]
Dispersion
Relation
c2 = (/T)2 = g (/2) tanh[2/ h]
c2 = g/ deep water waves
Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by deep-water phase velocity √Ω†gλ/†π as a function of relative depth h†/†λ.Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √gh) valid in shallow water.Drawn lines: dispersion relation valid in arbitrary depth.Dashed lines (blue and red): deep water limits.
Blue: Phase velocity (dash is deep water approximation)
Red: Group velocity (dash is deep water approximation)

8. Definitions: Wave number  = 2/wavelength = 2/
Wave frequency  = 2/waveperiod = 2/T
Phase velocity c = / = wavelength/waveperiod = /T

9. Superposition: Wave group = wave1 + wave2 + wave3
3 linear waves with different
amplitude, phase, period, and wavelength
Wave1
Wave2
Wave3
Frequency dispersion of surface gravity waves on deep water. The superposition (dark blue line) of three sinusoidal wave components (light blue lines) is shown. For the three components respectively 22 (bottom), 25 (middle) and 29 (top) wavelengths fit in a horizontal domain of 2,000 meter length. The component with the shortest wavelength (top) propagates slowest. The wave amplitudes of the components are respectively 1, 2 and 1 meter. The differences in wavelength and phase speed of the components results in a changing pattern of wave groups, due to amplification where the components are in phase, and reduction where they are in anti-phase.

10. Superposition: Wave group = wave1 + wave2 + wave3
Frequency dispersion of surface gravity waves on deep water. The superposition (dark blue line) of three sinusoidal wave components (light blue lines) is shown. For the three components respectively 22 (bottom), 25 (middle) and 29 (top) wavelengths fit in a horizontal domain of 2,000 meter length. The component with the shortest wavelength (top) propagates slowest. The wave amplitudes of the components are respectively 1, 2 and 1 meter. The differences in wavelength and phase speed of the components results in a changing pattern of wave groups, due to amplification where the components are in phase, and reduction where they are in anti-phase.
Phase (red dot) and group velocity (green dots) --> more later

11. Linear Waves (amplitude << wavelength)
X-mom.: acceleration = p-gradient
Z-mom: acceleration = p-gradient + gravity
Continuity: inflow = outflow
∂u/∂t = -1/ ∂p/∂x
∂w/∂t = -1/ ∂p/∂z + g
∂u/∂x + ∂w/∂z = 0
Boundary conditions:
Bottom z=-h is fixed
Surface z= (x,t) moves
@ bottom: w(z=-h) = 0
@surface: w(z= ) = ∂  /∂t

12. Wave Equation c2 ∂2/∂t2 = ∂2/∂x2
Combine dynamics and boundary conditions
to derive
Wave Equation
c2 ∂2/∂t2 = ∂2/∂x2
Try solutions of the form
(x,t) = a cos(x-t)

13. (x,t) = a cos(x-t) p(x,z,t) = … u(x,z,t) = … w(x,z,t) = …
Kinematics, pressure variations and drift under a 2 s Airy wave in 3m depth.
w(x,z,t) = …

14. (x,t) = a cos(x-t)
The wave moves with a “phase” speed c=wavelength/waveperiod
without changing its form. Pressure and velocity then vary as
p(x,z,t) = pa +  g  cosh[(h+z)]/cosh[h]
u(x,z,t) =   cosh[(h+z)]/sinh[h]

15. (x,t) = a cos(x-t) c2 = (/)2 = g/ tanh[h]
The wave moves with a “phase” speed c=wavelength/waveperiod
without changing its form. Pressure and velocity then vary as
p(x,z,t) = pa +  g  cosh[(h+z)]/cosh[h]
u(x,z,t) =   cosh[(h+z)]/sinh[h]
if, and only if
c2 = (/)2 = g/ tanh[h]

16. Dispersion: c2 = (/)2 = g/ tanh[h]
Dispersion refers to the sorting of waves with time. If wave phase speeds c depend on the wavenumber , the wave-field is dispersive. If the wave speed does not dependent on the wavenumber, the wave-field is non-dispersive.
One result of dispersion in deep-water waves is swell. Dispersion explains why swell can be so monochromatic (possessing a single wavelength) and so sinusoidal. Smaller wavelengths are dissipated out at sea and larger wavelengths remain and segregate with distance from their source.

17. c2 = (/T)2 = g (/2) tanh[2/ h]
c2 = (/)2 = g/ tanh[h]
h>>1
h<<1
c2 = (/T)2 = g (/2) tanh[2/ h]

18. c2 = gh shallow water waves
c2 = (/)2 = g/ tanh[h]
Dispersion means the wave phase speed varies
as a function of the wavenumber (=2/).
Limit-1: Assume h >> 1 (thus h >> ), then
tanh(h ) ~ 1 and
c2 = g/ deep water waves
Limit-2: Assume h << 1 (thus h << ), then
tanh(h) ~ h and
c2 = gh shallow water waves

19. Particle trajectories associated with linear waves
Deep water
Wave
Shallow water
wave

20. Particle trajectories associated with linear waves

21. c2 = g/ deep water waves phase velocity
red dot
cg = ∂/∂ = ∂(g )/∂ = 0.5g/ (g ) = 0.5 (g/) = c/2
requency dispersion in bichromatic groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity.
his deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots, when moving from the left to the right of the figure.New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.For gravity surface-waves, the water particle velocities are much smaller than the phase velocity, in most cases.
Deep water waves (depth >> wavelength)
Dispersive, long waves propagate faster than short waves
Group velocity half of the phase velocity