Problem 59696. Solve an ODE: Ekman spiral on a solid surface

Write a function to solve for u and v as a function of z in this system of ordinary differential equations:
-fv = -fV + eta d^2u/dz^2
fu = fU + eta d^2v/dz^2
where f, U, V, and η are constants. The boundary conditions are that u = v = 0 at z = 0 and u = U and v = V as z --> infinity.
This set of equations results from simplifying the Navier-Stokes equations (i.e., conservation of momentum for a fluid with a linear stress-rate of strain relation) for large-scale flow subjected to rotation. The horizontal velocity components are u and v. The Coriolis parameter f is related to Earth’s rotation rate and the latitude, and η is the kinematic viscosity, which can be interpreted as an eddy viscosity to account for the effects of turbulence.
The velocities far above the surface, U and V, result from the pressure gradient: U = -(1/rho f) dp/dy and V = (1/rho f) dp/dx, where p is pressure and ρ is density. Notice that far above the surface, the flow is along the isobars, or lines of constant pressure. As the surface is approached, the velocity vector rotates—or spirals.

Solution Stats

66.67% Correct | 33.33% Incorrect
Last Solution submitted on Jun 07, 2024

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