In the present paper, we investigate the power-law behaviour of the magnetic field spectra in the Earth’s magnetosheath region using Cluster spacecraft data under solar minimum condition. The power spectral density of the magnetic field data and spectral slopes at various frequencies are analysed. Propagation angle, θkB, and compressibility, R∥, are used to test the nature of turbulent fluctuations. The magnetic field spectra have the spectral slopes, α, between −1.5 to 0 down to spatial scales of 20ρi (where ρi is ion gyroradius), and show clear evidence of a transition to steeper spectra for small scales with a second power-law, having α between −2.6 to −1.8. At low frequencies, fsc<0.3fci (where fci is ion gyro-frequency), θkB∼90∘ to the mean magnetic field, B0, and R∥ shows a broad distribution, 0.1≤R∥≤0.9. On the other hand at fsc>10fci, θkB exhibits a broad range, 30∘≤θkB≤90∘, while R∥ has a small variation: 0.2≤R∥≤0.5. We conjecture that at high frequencies, the perpendicularly propagating Alfvén waves could partly explain the statistical analysis of spectra. To support our prediction of kinetic Alfvén wave dominated spectral slope behaviour at high frequency, we also present a theoretical model and simulate the magnetic field turbulence spectra due to nonlinear evolution of kinetic Alfvén waves. The present study also shows the analogy between the observational and simulated spectra.