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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 temperature salinity density surface depth, z Ocean Stratification two random casts from Baffin Bay July/August 2003 500m 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 Buoyancy Frequency: acceleration = - pressure gradient + gravity dw/dt = -1/ dp/dz +g but p=gz so w = dz/dt: d2z/dt2 dp/dz= g z d/dz + g (chain rule) and thus = -g / d/dz z acceleration = restoring force Solution is z(t) = z0 cos(N t) and N2 = -g / d/dz is stability or buoyancy frequency2 Surface Gravity Wave Restoring g (water-air)/water ≈ g because water >> air c2 = (/)2 = g/ tanh[h] Internal Gravity Wave Restoring g (2-1)/2 ≈ g* g* = g/ d/dz Dz = N2 Dz c2 = (/)2 = g*/ tanh[h] because 1 ≈ 2 Dispersion Relation c2 = (/T)2 = g (/2) tanh[2/ h] Blue: Phase velocity (dash is deep water approximation) Red: Group velocity (dash is deep water approximation) Dispersion 2 c = (/T)2 = g (/2) tanh[2/ h] Relation 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 Superposition: Wave group = wave1 + wave2 + wave3 Wave1 Wave2 Wave3 Phase (red dot) and group velocity (green dots) --> more later Linear Waves (amplitude << wavelength) X-mom.: acceleration = p-gradient ∂u/∂t = -1/ ∂p/∂x Z-mom: acceleration = p-gradient + gravity ∂w/∂t = -1/ ∂p/∂z + g Continuity: inflow = outflow ∂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 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) = … (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) 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 wavefield is dispersive. If the wave speed does not dependent on the wavenumber, the wavefield 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] h>>1 h<<1 c2 = (/T)2 = g (/2) tanh[2/ h] 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 Deep water waves (depth >> wavelength) Dispersive, long waves propagate faster than short waves Group velocity half of the phase velocity