def residual(x,U,dU): ux_r = U[0:ct.nz] ux_i = U[ct.nz:2*ct.nz] b_par_r = U[2*ct.nz:3*ct.nz] b_par_i = U[3*ct.nz:4*ct.nz] u_perp_r = U[4*ct.nz:5*ct.nz] u_perp_i = U[5*ct.nz:] dux_r = dU[0:ct.nz] dux_i = dU[ct.nz:2*ct.nz] db_par_r = dU[2*ct.nz:3*ct.nz] db_par_i = dU[3*ct.nz:4*ct.nz] du_perp_r = dU[4*ct.nz:5*ct.nz] du_perp_i = dU[5*ct.nz:] ux = ux_r + 1j * ux_i b_par = b_par_r + 1j * b_par_i u_perp = u_perp_r + 1j * u_perp_i dux = dux_r + 1j * dux_i db_par = db_par_r + 1j * db_par_i du_perp = du_perp_r + 1j * du_perp_i nabla_perp_b_par = 1j * ct.k_perp * b_par - np.sin(ct.alpha) * psdiff(b_par, period = 2 * ct.Lz) nabla_perp_u_perp = 1j * ct.k_perp * u_perp - np.sin(ct.alpha) * psdiff(u_perp, period = 2 * ct.Lz) L_ux = np.cos(ct.alpha) ** 2 * psdiff(ux, order = 2, period = 2 * ct.Lz) \ + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * ux L_u_perp = np.cos(ct.alpha) ** 2 * psdiff(u_perp, order = 2, period = 2 * ct.Lz) \ + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * u_perp # nabla_perp_b_par = 1j * ct.k_perp * b_par - 1j * np.sin(ct.alpha) * b_par # nabla_perp_u_perp = 1j * ct.k_perp * u_perp - 1j * np.sin(ct.alpha) * u_perp # L_ux = -ct.kz ** 2 * np.cos(ct.alpha) ** 2 * ux \ # + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * ux # L_u_perp = -ct.kz ** 2 * np.cos(ct.alpha) ** 2 * u_perp \ # + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * u_perp res = np.zeros(6 * ct.nz) res[0:ct.nz] = np.real(dux + 1j * ct.omega * b_par + nabla_perp_u_perp) res[ct.nz:2*ct.nz] = np.imag(dux + 1j * ct.omega * b_par + nabla_perp_u_perp) res[2*ct.nz:3*ct.nz] = np.real(db_par + 1j / ct.omega * L_ux) res[3*ct.nz:4*ct.nz] = np.imag(db_par + 1j / ct.omega * L_ux) res[4*ct.nz:5*ct.nz] = np.real(L_u_perp - 1j * ct.omega * nabla_perp_b_par) res[5*ct.nz:] = np.imag(L_u_perp - 1j * ct.omega * nabla_perp_b_par) return res
def dUdx(x, U):
ux = U[0:ct.nz]
b_par = U[ct.nz:]
u_perp = td.calc_u_perp(x, b_par)
dux_dx = -(1j * ct.omega * b_par + 1j * ct.k_perp * u_perp - \
np.sin(ct.alpha) / (2 * ct.dz) * (np.roll(u_perp,-1) - np.roll(u_perp,1)))
db_par_dx = -1j / ct.omega * ( \
np.cos(ct.alpha) ** 2 / ct.dz ** 2 * (np.roll(ux,-1) - 2 * ux + np.roll(ux,1)) + \
ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * ux)
return np.concatenate((dux_dx, db_par_dx))
def plot_vA(ix0, iz0):
fig = plt.figure()
fig_size = fig.get_size_inches()
fig_size[0] = 3 * fig_size[0]
fig.set_size_inches(fig_size)
ax = fig.add_subplot(131)
ax.plot(ct.z, ct.vA(ct.x[ix0], ct.z))
ax.set_xlabel('z')
ax.set_title(r'$v_A(z)$' + r' at $x = $' + "{:.2f}".format(ct.x[ix0]))
ax = fig.add_subplot(132)
ax.plot(ct.x / abs(ct.xi), ct.vA(ct.x, ct.z[iz0]))
ax.set_xlabel(r'$x / |x_i|$')
ax.set_title(r'$v_A(x)$' + r' at $z = $' + "{:.2f}".format(ct.z[iz0]))
ax = fig.add_subplot(133, projection='3d')
surf = ax.plot_surface(ct.X, ct.Z, ct.vA(ct.X, ct.Z), cmap=cm.cool)
ax.set_xlabel(r'$x / |x_i|$')
ax.set_ylabel(r'$z$')
ax.set_title(r'$v_A(x,z)$')
fig.savefig('Figures/vA.png', bbox_inches='tight')
def calc_A_u_v(x): a = np.full(ct.nz, np.cos(ct.alpha) ** 2 / ct.dz ** 2, dtype=complex) # Off diagonal elements b = -2 * a + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 # Diagonal elements b0 = b[0] b[0] += b0 b[-1] += a[0] ** 2 / b0 ab = np.zeros((3, ct.nz), dtype=complex) ab[0,:] = a ab[1,:] = b ab[2,:] = a u = np.zeros(ct.nz, dtype=complex) v = np.zeros(ct.nz, dtype=complex) u[0] = -b0 u[-1] = a[0] v[0] = 1 v[-1] = -a[0] / b0 return [ab, u, v]