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DYNAMICAL OCEANOGRAPHY
We can write the resulting equations as
C W Φ
n+1
i−1 + C C Φ
n+1
i
+ C E Φ
n+1
i+1 = g(Φ
n
i−1 , Φ
n
i , Φ
n
i+1 )
During each time step, we have to solve a system of equations Ax = b with a
tridiagional matrix A.
a. Determine the coefficients C W ,C C ,C E and the function g in the expression
above.
b. Write a program in your favorable language to solve the system of
equations.
c. Take r =0 .1 and determine how long we have to integrate for different
values of Λ.T a k eΛ=1and Λ = 100 and determine a suitable integration
time. What are optimal time steps in both cases?
d. Consider now the convergence of solutions with m en Δt for r =0 .1.
Choose the same two values of Λ as under b). Justify the use of specific m
and Δt in connection with the accuracy of the solutions.
e. Choose an optimal value of m and Δt for Λ=0 . Perform a set of simulations with different r ∈ [0.001, 0.1] and describe the different phenomena
(wave propagation, boundary layers) you see in the solutions.
f. Do the same for Λ = 100 and provide an overview of the different adjustment phenomena in the parameter plane (r, Λ). Give a physical explanation
for these phenomena.
DYNAMICAL OCEANOGRAPHY
We can write the resulting equations as
C W Φ
n+1
i−1 + C C Φ
n+1
i
+ C E Φ
n+1
i+1 = g(Φ
n
i−1 , Φ
n
i , Φ
n
i+1 )
During each time step, we have to solve a system of equations Ax = b with a
tridiagional matrix A.
a. Determine the coefficients C W ,C C ,C E and the function g in the expression
above.
b. Write a program in your favorable language to solve the system of
equations.
c. Take r =0 .1 and determine how long we have to integrate for different
values of Λ.T a k eΛ=1and Λ = 100 and determine a suitable integration
time. What are optimal time steps in both cases?
d. Consider now the convergence of solutions with m en Δt for r =0 .1.
Choose the same two values of Λ as under b). Justify the use of specific m
and Δt in connection with the accuracy of the solutions.
e. Choose an optimal value of m and Δt for Λ=0 . Perform a set of simulations with different r ∈ [0.001, 0.1] and describe the different phenomena
(wave propagation, boundary layers) you see in the solutions.
f. Do the same for Λ = 100 and provide an overview of the different adjustment phenomena in the parameter plane (r, Λ). Give a physical explanation
for these phenomena.