Linear and nonlinear responses to harmonic force in rotating flow

Wei, X. (2016). Linear and nonlinear responses to harmonic force in rotating flow. Journal of Fluid Mechanics, 796, 306–317. https://doi.org/10.1017/jfm.2016.267

pdf with notes

Equations

∂u∂t+u⋅∇u=−∇p+E∇2u+2u×z^+f\frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\cdot\nabla\boldsymbol{u}=-\nabla p+E\nabla^2\boldsymbol{u}+2\boldsymbol{u}\times\hat{\boldsymbol{z}}+\boldsymbol{f}

where the Ekman number E=ν/(Ωl2)E=\nu/(\Omega l^2) is the ratio of viscous time scale over rotational time scale.

f=Re{f^ei(k⋅x−ωt)},∇×f=kf\boldsymbol{f}=\text{Re}\{\hat{\boldsymbol{f}}e^{i(\boldsymbol{k}\cdot\boldsymbol{x}-\omega t)}\},\quad \nabla\times\boldsymbol{f}=k\boldsymbol{f}

Linear regime

∂u∂t=−∇p+E∇2u+2u×z^+f\frac{\partial \boldsymbol{u}}{\partial t}=-\nabla p+E\nabla^2\boldsymbol{u}+2\boldsymbol{u}\times\hat{\boldsymbol{z}}+\boldsymbol{f}

Assuming u=Re{u^ei(k⋅x−ωt)},p=Re{p^ei(k⋅x−ωt)}\boldsymbol{u}=\text{Re}\{\hat{\boldsymbol{u}}e^{i(\boldsymbol{k}\cdot\boldsymbol{x}-\omega t)}\},p=\text{Re}\{\hat{p}e^{i(\boldsymbol{k}\cdot\boldsymbol{x}-\omega t)}\}, and the solution is

u^=ikf^2kz+k(ω+iEk2)\hat{\boldsymbol{u}}=\frac{ik\hat{\boldsymbol{f}}}{2k_z+k(\omega+iEk^2)}

Nonlinear regime

The nonlinear effect due to the force amplitude aa and the frequency ω\omega:

Figure 4

Table 1

fluid
#rotating flow #linear solution
Linear and nonlinear responses to harmonic force in rotating flow
http://jingliangwei.github.io/blog-hexo/2026/03/24/Linear-and-nonlinear-responses-to-harmonic-force-in-rotating-flow/
Author
Arwell
Posted on
March 24, 2026
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