Knowlpedia

Definition

Viscous damping of a Fourier mode

The exponential decay produced by viscosity at a specified spatial wavevector.

For a fixed kk, the mode w(t,x)=a(t)eik⋅xw(t,x)=a(t)e^{ik\cdot x} satisfies ∂tw=νΔw\partial_tw=\nu\Delta w exactly when

a′=−ν∣k∣2a,a(t)=e−ν∣k∣2(t−s)a(s).a'=-\nu|k|^2a, \qquad a(t)=e^{-\nu|k|^2(t-s)}a(s).

This is viscous damping at rate ν∣k∣2\nu|k|^2, since Δeik⋅x=−∣k∣2eik⋅x\Delta e^{ik\cdot x}=-|k|^2e^{ik\cdot x}.

A varying wavevector

If an amplitude equation has scalar damping −ν∣k(t)∣2a-\nu|k(t)|^2a, its damping factor is exp⁡(−ν∫st∣k(r)∣2 dr)\exp(-\nu\int_s^t|k(r)|^2\,dr). Other matrix terms can simultaneously amplify or rotate the amplitude. For eiκmΦe^{i\kappa m\Phi}, the leading viscous rate is νκ2m2∣∇Φ∣2\nu\kappa^2m^2|\nabla\Phi|^2; lower-order amplitude and phase derivatives remain in the full Laplacian.

References