This study develops a ridge-regularised Fourier autoregressive (Ridge-PFAR) model for periodic climate series in which harmonic lag regressors can become numerous and strongly correlated. A 500-replication Monte Carlo experiment compares unpenalised Fourier autoregression (FAR) and Ridge-PFAR at a fixed sample size and forecast horizon under low-, moderate-, and high-dimensional designs. In the high-dimensional design, FAR produces severe forecast outliers, with a median 24-step RMSE of 10.9645 and a mean RMSE of 2621.5584, whereas Ridge-PFAR reduces these values to 1.1576 and 1.1874, respectively. The median normal-equation condition number decreases from 3049.0586 to 7.2537 after regularisation. The empirical analysis uses 132 monthly Nigerian observations from January 2014 to December 2024, with the first 108 observations used for model selection and the final 24 reserved for testing. All combinations $p=1,\ldots,10$ and $K=1,\ldots,4$ are assessed using identical rolling-origin folds, and a seasonal-naive forecast provides a practical benchmark. For rainfall, FAR yields the lowest test RMSE (53.8078), followed by Ridge-PFAR (56.6733) and the seasonal-naive method (60.7139). For relative humidity, Ridge-PFAR lowers the RMSE to 5.8410, compared with 6.2069 for FAR and 8.2031 for the seasonal-naive method. The 95\% residual-bootstrap intervals achieve test-period coverage of 95.83\% for rainfall and 87.50\% for relative humidity. The results indicate that ridge regularisation is an effective safeguard against severe high-dimensional instability, although its forecasting gains in the empirical application remain target-dependent.