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

Problem
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.
Background
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.
Solve

Solution Stats

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

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