1. 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

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

3. 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/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

5. c2 = (/)2 = g/ tanh[h] c2 = (/)2 = g*/ tanh[h] Surface Gravity Wave Restoring  g (rwater-rair)/rwater ≈ g because rwater >> rair Internal Gravity Wave Restoring  g (r2-r1)/r2 ≈ g* g* = g/r dr/dz Dz = N2Dz because r1 ≈ r2

6. c2 = (/T)2 = g (/2) tanh[2/ h] Dispersion Relation c2 = g/ deep water waves 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 = g/ deep water waves 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

10. Superposition: Wave group = wave1 + wave2 + wave3 Wave1 Wave2 Wave3 Phase (red dot) and group velocity (green dots) --> more later

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

12. 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) = …

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. c2 = (/)2 = g/ tanh[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

16. c2 = (/)2 = g/ tanh[h] Dispersion: 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

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

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 Deep water waves (depth >> wavelength) Dispersive, long waves propagate faster than short waves Group velocity half of the phase velocity