Internal Gravity Waves MAST-602: Introduction to Physical Oceanography Andreas Muenchow, Oct.-7, 2008 Internal Gravity Waves Knauss (1997), chapter-2, p. 24-34 Knauss (1997), chapter-10, p. 229-234 Vertical Stratification Descriptive view (wave characteristics) Balance of forces, wave equation Dispersion relation Phase velocity Same as Surface waves

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

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 http://www.youtube.com/watch?v=hkT3ulsGWyA

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/dt2 = -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

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 = N2 Dz because r1 ≈ r2 c2 = (/)2 = g*/ tanh[h]

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)

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)

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

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.

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

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

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)

(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. http://commons.wikimedia.org/wiki/Image:2sec_3mdepth_puv_drift.png w(x,z,t) = …

(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]

(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]

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.

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]

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

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

Particle trajectories associated with linear waves

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