Finite-Volume Integration

Ripple treats spectral values as finite-volume cell averages. A value stored at N[i, j, m, n] represents the average over physical cell (i, j) and spectral control volume (m, n).

Spectral integrals are therefore sums of cell averages multiplied by exact cell measures:

integral = integrate_spectrum(cell_average_values, spectral_grid)

This is not a point-sample quadrature rule. For a scalar quantity q over a spectral cell C[m, n], the value supplied to integrate_spectrum should be the cell average

qbar[m, n] = (1 / measure(C[m, n])) * integral_C q dC

and the integral uses

sum(qbar[m, n] * measure(C[m, n]))

Spectral Measures

spectral_cell_measure and spectral_cell_measures provide the exact coordinate-space control-volume measures for Cartesian, polar, and frequency-direction spectral grids. For polar and frequency-direction grids, the measures include the radial Jacobian.

Power-law averages use analytic finite-volume formulas:

• spectral_radial_power_average
• spectral_frequency_power_average
• spectral_first_moment_measures
• spectral_second_moment_measures

These helpers are used by diagnostics and source terms so moments, energy diagnostics, source rates, and dissipation rates are consistent with the same cell geometry.

Directional wind-input and swell-dissipation spreading on polar and frequency-direction grids also uses angular finite-volume cell averages for nonnegative integer spreading powers. This avoids replacing a directional bin with its center angle when a wind direction cuts through the cell.

Q Transform Geometry

CWCM QTransform operators use exact vertical finite-volume cell integrals from CDF differences. Doppler velocity and pseudomomentum paths share the same cell geometry, which keeps the discrete wave-current coupling diagnostics consistent.

Practical Rule

When constructing analytic initial conditions or source rates, either provide a quantity that is constant over each spectral cell or use the finite-volume average helpers. Avoid replacing cell averages with midpoint samples on coarse spectral grids.