Mason, R. M., & Kerswell, R. R. (2002). Chaotic dynamics in a strained rotating flow: A precessing plane fluid layer. Journal of Fluid Mechanics , 471 , 71–106. https://doi.org/10.1017/S0022112002001994
The inertial waves
The inertial waves which statisfy
∂ u ⃗ ∂ t + 2 z ^ × u ⃗ + ∇ p = 0 , \frac{\partial\vec{u}}{\partial t}+2\hat{z}\times\vec{u}+\nabla p=0,
∂ t ∂ u + 2 z ^ × u + ∇ p = 0 ,
∇ ⋅ u ⃗ = 0 , w ( x , y , ± 1 2 ) = 0 , \nabla\cdot\vec{u}=0,\quad w(x,y,\pm\frac{1}{2})=0,
∇ ⋅ u = 0 , w ( x , y , ± 2 1 ) = 0 ,
are
[ u v w p ] = ( k 2 ( k x λ − 2 i k y ) cos ( n π [ z + 1 2 ] ) / 4 k ⊥ 2 k 2 ( k y λ + 2 i k x ) cos ( n π [ z + 1 2 ] ) / 4 k ⊥ 2 − i k z sin ( n π [ z + 1 2 ] ) / λ cos ( n π [ z + 1 2 ] ) ) exp ( i ( k x x + k y y + λ t ) ) , (4.4) \left[\begin{array}{c} u \\ v \\ w \\ p\end{array}\right]=
\left(
\begin{array}{c}
k^2(k_x\lambda-2ik_y)\cos(n\pi[z+\frac{1}{2}])/4k_\perp^2 \\
k^2(k_y\lambda+2ik_x)\cos(n\pi[z+\frac{1}{2}])/4k_\perp^2 \\
-ik_z\sin(n\pi[z+\frac{1}{2}])/\lambda \\
\cos(n\pi[z+\frac{1}{2}])
\end{array}
\right)
\exp(i(k_xx+k_yy+\lambda t)),\tag{4.4}
u v w p = k 2 ( k x λ − 2 i k y ) cos ( nπ [ z + 2 1 ]) /4 k ⊥ 2 k 2 ( k y λ + 2 i k x ) cos ( nπ [ z + 2 1 ]) /4 k ⊥ 2 − i k z sin ( nπ [ z + 2 1 ]) / λ cos ( nπ [ z + 2 1 ]) exp ( i ( k x x + k y y + λ t )) , ( 4.4 )
with the dispersion relation
λ = ± 2 k z k . (4.5) \lambda=\frac{\pm 2k_z}{k}.\tag{4.5}
λ = k ± 2 k z . ( 4.5 )
The derivation:
{ ∂ u ⃗ ∂ t + 2 z ^ × u ⃗ + ∇ p = 0 [ u ⃗ p ] = [ u v w p ] exp ( i ( k x x + k y y + λ t ) ) ⇒ { u = u ( p ) v = v ( p ) w = w ( p ) \left\{ \begin{array}{l} \dfrac{\partial\vec{u}}{\partial t}+2\hat{z}\times\vec{u}+\nabla p=0 \\ \left[ \begin{array}{c} \vec{u} \\ p \end{array} \right]= \left[ \begin{array}{c} u \\ v \\ w \\ p \end{array} \right]\exp(i(k_xx+k_yy+\lambda t)) \end{array} \right.\Rightarrow \left\{ \begin{array}{l} u=u(p) \\ v=v(p) \\ w=w(p) \end{array} \right. ⎩ ⎨ ⎧ ∂ t ∂ u + 2 z ^ × u + ∇ p = 0 [ u p ] = u v w p exp ( i ( k x x + k y y + λ t )) ⇒ ⎩ ⎨ ⎧ u = u ( p ) v = v ( p ) w = w ( p )
{ ∇ ⋅ u ⃗ = 0 w ( x , y , ± 1 2 ) = 0 ⇒ p \left\{ \begin{array}{l} \nabla\cdot\vec{u}=0 \\ w(x,y,\pm\frac{1}{2})=0 \end{array} \right. \Rightarrow p { ∇ ⋅ u = 0 w ( x , y , ± 2 1 ) = 0 ⇒ p