1. Slide 1The Wave Model: 1.3. Energy Balance Eqn (Physics) 1.3. Energy Balance Eqn (Physics): Discuss wind input and nonlinear transfer in some detail. Dissipation is just given. Common feature: Resonant Interaction Wind: Critical layer: c(k) = U o (z c ). Resonant interaction between air at z c and wave Nonlinear: i = 0 3 and 4 wave interaction zczc U o =c

2. Slide 2The Wave Model: 1.3. Energy Balance Eqn (Physics) Transfer from Wind: Instability of plane parallel shear flow (2D) Perturb equilibrium: Displacement of streamlines W = U o - c (c = /k)

3. Slide 3The Wave Model: 1.3. Energy Balance Eqn (Physics) give Im(c) possible growth of the wave Simplify by taking no current and constant density in water and air. Result: Here, = a / w ~ 10 -3 << 1, hence for 0, ! Perturbation expansion growth rate = Im(k c 1 ) c 2 = g/k

4. Slide 4The Wave Model: 1.3. Energy Balance Eqn (Physics) Further simplification gives for = w/w(0): Growth rate: Wronskian:

5. Slide 5The Wave Model: 1.3. Energy Balance Eqn (Physics) 1.Wronskian W is related to wave-induced stress Indeed, with and the normal mode formulation for u 1, w 1 : (e.g. ) 2.Wronskian is a simple function, namely constant except at critical height z c To see this, calculate dW/dz using Rayleigh equation with proper treatment of the singularity at z=z c where subscript c refers to evaluation at critical height z c (W o = 0) zczc z w

6. Slide 6The Wave Model: 1.3. Energy Balance Eqn (Physics) This finally gives for the growth rate (by integrating dW/dz to get W (z=0) ) Miles (1957): waves grow for which the curvature of wind profile at z c is negative (e.g. log profile). Consequence:waves grow slowing down wind: Force = d w / dz ~ (z) (step function) For a single wave, this is singular! Nonlinear theory.

7. Slide 7The Wave Model: 1.3. Energy Balance Eqn (Physics) Linear stability calculation Choose a logarithmic wind profile (neutral stability) = 0.41 (von Karman), u * = friction velocity, =u * Roughness lengrth, z o : Charnock (1955) z o = u * 2 / g, 0.015 Note: Growth rate,, of the waves ~ = a / w and depends on so, short waves have the largest growth. Action balance equation

8. Slide 8The Wave Model: 1.3. Energy Balance Eqn (Physics) Wave growth versus phase speed (comparison of Miles' theory and observations)

9. Slide 9The Wave Model: 1.3. Energy Balance Eqn (Physics) Nonlinear effect: slowing down of wind Continuum: w is nice function, because of continuum of critical layers w is wave induced stress ( u, w): wave induced velocity in air (from Rayleigh Eqn).... with (sea-state dep. through N!). D w > 0 slowing down the wind.

10. Slide 10The Wave Model: 1.3. Energy Balance Eqn (Physics) Example: Young wind sea steep waves. Old wind sea gentle waves. Charnock parameter depends on sea-state! (variation of a factor of 5 or so). Wind speed as function of height for young & old wind sea old windsea young windsea

11. Slide 11The Wave Model: 1.3. Energy Balance Eqn (Physics) Non-linear Transfer (finite steepness effects) Briefly describe procedure how to obtain: 1. 2.Express in terms of canonical variables and = (z = ) by solving iteratively using Fourier transformation. 3.Introduce complex action-variable

12. Slide 12The Wave Model: 1.3. Energy Balance Eqn (Physics) gives energy of wave system: with, … etc. Hamilton equations: become Result: Here, V and W are known functions of

13. Slide 13The Wave Model: 1.3. Energy Balance Eqn (Physics) Three-wave interactions: Four-wave interactions: Gravity waves: No three-wave interactions possible. Sum of two waves does not end up on dispersion curve. dispersion curve 2, k 2 1, k 1 k

14. Slide 14The Wave Model: 1.3. Energy Balance Eqn (Physics) Phillips (1960) has shown that 4-wave interactions do exist! Phillips figure of 8 Next step is to derive the statistical evolution equation for with N 1 is the action density. Nonlinear Evolution Equation

15. Slide 15The Wave Model: 1.3. Energy Balance Eqn (Physics) Closure is achieved by consistently utilising the assumption of Gaussian probability Near-Gaussian Here, R is zero for a Gaussian. Eventual result: obtained by Hasselmann (1962).

16. Slide 16The Wave Model: 1.3. Energy Balance Eqn (Physics) Properties: 1.N never becomes negative. 2.Conservation laws: action: momentum: energy: Wave field cannot gain or loose energy through four- wave interactions.

17. Slide 17The Wave Model: 1.3. Energy Balance Eqn (Physics) Properties: (Contd) 3.Energy transfer Conservation of two scalar quantities has implications for energy transfer Two lobe structure is impossible because if action is conserved, energy ~ N cannot be conserved!

18. Slide 18The Wave Model: 1.3. Energy Balance Eqn (Physics) Dissipation due to Wave Breaking Define: with Quasi-linear source term: dissipation increases with increasing integral wave steepness