Are These Waves Dispersive or Non-Dispersive - Science Assignment Help

Assignment Task:

Task:

1 Consider an inviscid fluid in a uniformly rotating frame of reference. The effect of gravity is neglected. In the frame rotating with an angular velocity n, the momentum equation governing the fluid flow is given by: du Ou 1 dt = + (u • V) u Vpr — 211 x u,at where u is the velocity field, p is the density of the fluid and pr = p — (1111 lx11)2 /2 is the reduced pressure, where p is pressure and x_L is the perpendicular distance from the position vector x to the rotation axis. We consider the dynamics of inertial waves, i.e. small perturbations to the state of uniform rotation. 
(a) Write down an expression for the velocity field corresponding to unifo rotation. Find the vorticity corresponding to this flow. 
(b) 
Consider a small perturbation u' to the state of uniform rotation with angular velocity ft, which has the form of a plane harmonic wave 
_ A et (k..-wt) 
where k is the wavenumber vector, w is the frequency, A is the complex amplitude and A* is its complex conjugate. How small does the amplitude A need to be for the nonlinear (with respect to A) terms to be much smaller than the linear ones in the momentum equation? 
2 
(c) Consider the linearised momentum equation and derive the dispersion relation w = w (k). (Hint: Take the curl of the momentum equation and use the identity V x (SZ x u) = — (SI • V) u.) 
(d) What is the polarisation of the inertial waves? 
4 
(e) Are these waves dispersive or non-dispersive? Isotropic or anisotropic? Ex-plain why. Find the group velocity for the inertial waves and comment on its relative direction with respect to the wavenumber vector. 
Total: 2t 
 

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