Download - Internal Gravity Waves
Internal Gravity Waves
Knauss (1997), chapter-2, p. 24-34Knauss (1997), chapter-10, p. 229-234
Vertical StratificationDescriptive view (wave characteristics)Balance of forces, wave equationDispersion relationPhase velocity
MAST-602: Introduction to Physical OceanographyAndreas Muenchow, Oct.-7, 2008
Same asSurface waves
OceanStratification
500m
dept
h, z
surface
temperature salinity density
two random castsfrom Baffin BayJuly/August 2003
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
acceleration = - pressure gradient + gravitydw/dt = -1/ dp/dz + g
p=gz so dp/dz= g z d/dz + g (chain rule)
but
w = dz/dt:
Solution is z(t) = z0 cos(N t)
and N2 = -g / d/dz is stability or buoyancy frequency2
thus
and
Buoyancy Frequency:
d2z/dt2 = -g / d/dz z
acceleration = restoring force
Surface Gravity Wave Restoring g water-air)/water ≈ g
because water >> air
Internal Gravity Wave Restoring g 2-1)/2 ≈ g*
because 1 ≈ 2
c2 = (/)2 = g/ tanh[h]
c2 = (/)2 = g*/ tanh[h]
g* = g/ d/dz z = N2 z
Blue: Phase velocity (dash is deep water approximation)Red: Group velocity (dash is deep water approximation)
DispersionRelation
c2 = (/T)2 = g (/2) tanh[2/ h]c2 =
g/
dee
p w
ater
wav
es
Blue: Phase velocity (dash is deep water approximation)Red: Group velocity (dash is deep water approximation)
DispersionRelation
c2 = (/T)2 = g (/2) tanh[2/ h]c2 =
g/
dee
p w
ater
wav
es
Definitions:
Wave number = 2/wavelength = 2/
Wave frequency = 2/waveperiod = 2/T
Phase velocity c = / = wavelength/waveperiod = /T
Wave1Wave2Wave3
Superposition: Wave group = wave1 + wave2 + wave3
3 linear waves with differentamplitude, phase, period, and wavelength
Wave1Wave2Wave3
Superposition: Wave group = wave1 + wave2 + wave3
Phase (red dot) and group velocity (green dots) --> more later
Linear Waves (amplitude << wavelength)
∂u/∂t = -1/ ∂p/∂x
∂w/∂t = -1/ ∂p/∂z + g
∂u/∂x + ∂w/∂z = 0
X-mom.: acceleration = p-gradient
Z-mom: acceleration = p-gradient + gravity
Continuity: inflow = outflow
Boundary conditions:
@ bottom: w(z=-h) = 0
@surface: w(z= ) = ∂ /∂t
Bottom z=-h is fixed
Surface z= (x,t) moves
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)
p(x,z,t) = …
(x,t) = a cos(x-t)
u(x,z,t) = …
w(x,z,t) = …
(x,t) = a cos(x-t)
The wave moves with a “phase” speed c=wavelength/waveperiodwithout 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]
(x,t) = a cos(x-t)
The wave moves with a “phase” speed c=wavelength/waveperiodwithout 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]
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.
c2 = (/)2 = g/ tanh[h]Dispersion:
c2 = (/)2 = g/ tanh[h]
c2 = (/T)2 = g (/2) tanh[2/ h]
h>>1
h<<1
c2 = (/)2 = g/ tanh[h]
Dispersion means the wave phase speed variesas 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
Deep waterWave
Shallow waterwave
Particle trajectories associated with linear waves
Particle trajectories associated with linear waves
Deep water waves (depth >> wavelength)Dispersive, long waves propagate faster than short wavesGroup velocity half of the phase velocity
c2 = g/ deep water waves phase velocityred dot
cg = ∂/∂ = ∂(g )/∂ = 0.5g/ (g ) = 0.5 (g/) = c/2