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