The instability of precessing flow

Kerswell, R. R. (1993). The instability of precessing flow. Geophysical & Astrophysical Fluid Dynamics, 72(1–4), 107–144. https://doi.org/10.1080/03091929308203609

pdf with notes

Parameters

Parameters	Expression
Oblateness	$\eta,c$: $r^2+(1+\eta)z^2=r^2+\dfrac{z^2}{c^2}=1$
Precessional vector	$\boldsymbol{\Omega}=[\Omega_1,0,\Omega_3]^T$
Shearing of the streamlines	$\varepsilon=\dfrac{\eta\mu}{2+(2+\eta)\mu^2}$
Elliptical distortion of the streamlines	$\beta=\dfrac{\eta\mu^2}{2+(2+\eta)\mu^2}$
Relative magnitude of elliptical over shearing	$\mu=\dfrac{2\Omega_1}{\eta+2(1+\eta)\Omega_3}$

1 Formulation

Poincaré’s solution

$$\boldsymbol{u}=\boldsymbol{\omega}\times\boldsymbol{r}+\nabla A$$

$$\boldsymbol{\omega}=\hat{\boldsymbol{z}}-\frac{2+\eta}{\eta+2(1+\eta)\boldsymbol{\Omega}\cdot\hat{\boldsymbol{z}}}\hat{\boldsymbol{z}}\times(\hat{\boldsymbol{z}}\times\boldsymbol{\Omega})$$

$$A=\frac{\eta}{\eta+2(1+\eta)\boldsymbol{\Omega}\cdot\hat{\boldsymbol{z}}}(\boldsymbol{\Omega}\times\hat{\boldsymbol{z}}\cdot\boldsymbol{r})(\hat{\boldsymbol{z}\cdot\boldsymbol{r}})$$

Derivation of equation

From $\boldsymbol{u}=u_x\hat{\boldsymbol{x}}+u_y\hat{\boldsymbol{y}}+u_z\hat{\boldsymbol{z}}$ to $\boldsymbol{u}=u\tilde{\boldsymbol{s}}+v\tilde{\boldsymbol{\phi}}+w\tilde{\bar{\boldsymbol{z}}}$

$$\boldsymbol{u}=\frac{2+(2+\eta)\mu^2}{2\sqrt{1+\mu^2}}\sqrt{1-\beta^2}s\tilde{\boldsymbol{\phi}}=\bar{\Omega}s\tilde{\boldsymbol{\phi}} \tag{1.6}$$

The momentum equation projecting onto $\tilde{\boldsymbol{l}},\tilde{\boldsymbol{m}},\tilde{\boldsymbol{n}}$ reads

$$\begin{align} \frac{\partial u}{\partial t}+&(\boldsymbol{u}\cdot\nabla)u-\frac{v^2}{s}-2\left(\frac{\Omega_3^*+\Omega_1^*\varepsilon}{\sqrt{1-\beta^2}}\right)v+\frac{1}{(1-\beta^2)}\left[1+\frac{2\varepsilon^2}{1+\beta}\right]\frac{\partial p}{\partial s} \\ =&\frac{2\varepsilon\cos\phi}{\sqrt{1+\beta}\sqrt{1-\beta^2}}\frac{\partial p}{\partial\bar{z}}+\frac{2\sin\phi}{\sqrt{1+\beta}}\left(\Omega_1^*-\frac{2\Omega_3^*\varepsilon}{\sqrt{1+\beta}}\right)w \\ &+\cos 2\phi\left\{\frac{2(\Omega_1^*\varepsilon+\Omega_3^*\beta)}{\sqrt{1-\beta^2}}v-\frac{1}{(1-\beta^2)}\left[\beta+\frac{2\varepsilon^2}{1+\beta}\right]\frac{\partial p}{\partial s}\right\} \\ &+\sin 2\phi\left\{\frac{2(\Omega_1^*\varepsilon+\Omega_3^*\beta)}{\sqrt{1-\beta^2}}u+\frac{1}{(1-\beta^2)}\left[\beta+\frac{2\varepsilon^2}{1+\beta}\right]\frac{1}{s}\frac{\partial p}{\partial \phi}\right\} \end{align} \tag{1.7}$$

$$\begin{align} \frac{\partial v}{\partial t}+&(\boldsymbol{u}\cdot\nabla)v-\frac{uv}{s}+2\left(\frac{\Omega_3^*+\Omega_1^*\varepsilon}{\sqrt{1-\beta^2}}\right)u+\frac{1}{(1-\beta^2)}\left[1+\frac{2\varepsilon^2}{1+\beta}\right]\frac{1}{s}\frac{\partial p}{\partial \phi} \\ =&-\frac{2\varepsilon\sin\phi}{\sqrt{1+\beta}\sqrt{1-\beta^2}}\frac{\partial p}{\partial\bar{z}}+\frac{2\cos\phi}{\sqrt{1+\beta}}\left(\Omega_1^*-\frac{2\Omega_3^*\varepsilon}{1+\beta}\right)w \\ &+\cos 2\phi\left\{\frac{2(\Omega_1^*\varepsilon+\Omega_3^*\beta)}{\sqrt{1-\beta^2}}u+\frac{1}{(1-\beta^2)}\left[\beta+\frac{2\varepsilon^2}{1+\beta}\right]\frac{1}{s}\frac{\partial p}{\partial \phi}\right\} \\ &+\sin 2\phi\left\{-\frac{2(\Omega_1^*\varepsilon+\Omega_3^*\beta)}{\sqrt{1-\beta^2}}v+\frac{1}{(1-\beta^2)}\left[\beta+\frac{2\varepsilon^2}{1+\beta}\right]\frac{\partial p}{\partial s}\right\} \end{align} \tag{1.8}$$

$$\begin{align} \frac{\partial w}{\partial t}+(\boldsymbol{u}\cdot\nabla)w+\frac{\partial p}{\partial \bar{z}}=&-\sin\phi\left[\frac{2\varepsilon}{\sqrt{1+\beta}\sqrt{1-\beta^2}}\frac{1}{s}\frac{\partial p}{\partial \phi}+2\Omega_1^*\sqrt{1+\beta}u\right] \\ &+\cos\phi\left[\frac{2\varepsilon}{\sqrt{1+\beta}\sqrt{1-\beta^2}}\frac{\partial p}{\partial s}-2\Omega_1^*\sqrt{1+\beta}v\right] \end{align} \tag{1.9}$$

Consider small disturbances $\boldsymbol{u}$ upon the basic state $\boldsymbol{U}=\bar{\Omega}s\tilde{\phi}$ in a frame rotating with this flow,

$$\frac{\partial \boldsymbol{u}^*}{\partial t^*}+2\hat{\boldsymbol{z}}\times\boldsymbol{u}^*+\bar{\nabla}p=\cdots\mathscr{L}(\cdots) \tag{1.11}$$

Scale of parameters

For the earth,

$$\Omega_1=4\times 10^{-8}\ll\eta\approx\frac{1}{200}\ll 1$$

The hierachy,

$$\varepsilon^2,\beta\ll\varepsilon\ll 1 \tag{1.18}$$

Shearing and elliptical parts

$$\begin{align} \frac{\partial \boldsymbol{u}}{\partial t}+2\left[\begin{array}{c}-v \\ u \\ 0 \end{array}\right]+\bar{\nabla}p=&\varepsilon\left[e^{i(\phi+t)}\mathscr{L}_S(\boldsymbol{u},p)+e^{-i(\phi+t)}\mathscr{L}_S^*(\boldsymbol{u},p)\right] \\ &-\frac{1}{2}\beta\left[e^{2i(\phi+t)}\mathscr{L}_E(\boldsymbol{u},p)+e^{-2i(\phi+t)}\mathscr{L}_E^*(\boldsymbol{u},p)\right] \end{align}\tag{1.22}$$

2 The shearing instability

Poincaré mode

The Poincaré mode $[\boldsymbol{u}(\boldsymbol{x},t),p(\boldsymbol{u},t)]=[\boldsymbol{Q}_{n,m,k}(\boldsymbol{x}),\Phi_{n,m,k}(\boldsymbol{x})]e^{i\lambda t}$ when $\varepsilon=\beta=0$

$$\boldsymbol{Q}_{n,m,k}=\left[\begin{array}{c} \dfrac{-i}{4-\lambda^2}\left(\lambda\Phi_r+\dfrac{2m}{r}\Phi\right) \\ \dfrac{1}{4-\lambda^2}\left(2\Phi_r+\dfrac{m\lambda}{r}\Phi\right) \\ \dfrac{i}{\lambda}\Phi_z \end{array}\right]e^{i(m\phi+\lambda t)} \tag{2.1}$$

the eigenfrequency $\lambda$

$$\nu+2\sum_1^N \frac{\lambda^2c^2}{\lambda^2c^2-\bar{x}_j^2\{4-\lambda^2(1-c^2)\}}=\frac{mc^2\lambda}{2-\lambda} \tag{2.4}$$

Shearing resonance condition

Assuming $\varepsilon\neq 0$ and ignoring $\beta$ for this section, consider perturbation to the basic flow (1.6)

$$\boldsymbol{u}=A(t)\boldsymbol{Q}_a+B(t)\boldsymbol{Q}_b+\varepsilon \boldsymbol{u}_1+\mathrm{O}(\varepsilon^2)$$

Projecting equation (1.22) onto $\boldsymbol{Q}_a$ and $\boldsymbol{Q}_b$, the resonance conditions read,

$$m_b=m_a+1$$

$$n_b=n_a$$

$$-2\varepsilon\sqrt{(1+\lambda_a)(1-\lambda_b)}|I|<\lambda_b-\lambda_a-1<2\varepsilon\sqrt{(1+\lambda_a)(1-\lambda_b)}|I|$$

so the interaction integral read,

$$\begin{align} \langle\boldsymbol{Q}_a,e^{-i\phi}\mathscr{L}_S^*\rangle&=(1-\lambda_b)I,\\ \langle\boldsymbol{Q}_b,e^{i\phi}\mathscr{L}_S\rangle&=(1+\lambda_a)I^* \end{align} \tag{2.7}$$

3 The elliptical instability

In the case that $\varepsilon^2,\beta\ll\varepsilon\ll 1$

$$\boldsymbol{u}=A\boldsymbol{Q}_a+B\boldsymbol{Q}_b+\varepsilon[\boldsymbol{v}_{11}e^{i(\lambda_a-1)t}+\boldsymbol{v}_{12}e^{i(\lambda_a+1)t}+\boldsymbol{v}_{13}e^{i(\lambda_b+1)t}]+\beta \boldsymbol{v}_2+\mathrm{O}(\varepsilon\beta)$$

4 Exact linear solutions

Taking precessional vector $\boldsymbol{\Omega}=\Omega\hat{\boldsymbol{x}}$ (that is $\Omega_1=\Omega, \Omega_3=0$ ), the Poincaré’s basic state can be written as

$$\boldsymbol{U}=\left[\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & -(1+\eta)\mu \\ 0 & \mu & 0 \end{array}\right]\boldsymbol{x}=\boldsymbol{A}\cdot\boldsymbol{x}\tag{4.1}$$

The linearised disturbance equations in the precessing frame are

$$\frac{\partial \boldsymbol{u}}{\partial t}+2\boldsymbol{\Omega}\times\boldsymbol{u}+\boldsymbol{A}\cdot\boldsymbol{x}\cdot\nabla\boldsymbol{u}+\boldsymbol{A}\cdot\boldsymbol{u}+\nabla p=\boldsymbol{0} \tag{4.2}$$

The orthogonal vector spaces $\mathscr{V}_n$

$$\mathscr{V}_n=\langle\boldsymbol{Q}_{nmk};-n\le m\le n,k=1,...,k_{max}(n,m)\rangle$$

4.1 Linear velocities

$$\mathscr{V}_2=\{\boldsymbol{Q}_{2,-1,1},\boldsymbol{Q}_{2,0,1},\boldsymbol{Q}_{2,1,1}\}$$

$$\boldsymbol{u}(\boldsymbol{x},t)=\sum_{i=1}^3 \alpha_i(t)\boldsymbol{u}_i(\boldsymbol{x})$$

$$\alpha_i(t)=\alpha_i e^{\sigma t}$$

Three eigenfrequencies are

$$\sigma_1=0,\quad \sigma_2=i\kappa,\quad \sigma_3=-i\kappa,\quad \kappa=\frac{1-c^2}{1+c^2}$$

No instability in $\mathscr{V}_2$, need to consider quadratic disturbances.

4.2 Quadratic velocities

$$\mathscr{V}_3=\{\boldsymbol{Q_{3,-2,1}},\boldsymbol{Q_{3,-1,1}},\boldsymbol{Q_{3,-1,2}},\boldsymbol{Q_{3,0,1}},\boldsymbol{Q_{3,0,2}},\boldsymbol{Q_{3,1,1}},\boldsymbol{Q_{3,1,2}},\boldsymbol{Q_{3,2,1}}\}$$

5 Unbounded streamlines

The unbounded basic flow is

$$\boldsymbol{u}=\left[\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & -2\varepsilon \\ 0 & 0 & 0 \end{array}\right]\boldsymbol{x}=\boldsymbol{D}\cdot\boldsymbol{x}\tag{5.1}$$

The perturbation equation is

$$\frac{\mathrm{d}\hat{\boldsymbol{u}}}{\mathrm{d}t}+\boldsymbol{D}\cdot\hat{\boldsymbol{u}}+2\boldsymbol{\Omega}\times\hat{\boldsymbol{u}}+i\boldsymbol{k}\hat{p}=0 \tag{5.3}$$

Floquet method

$$\frac{\mathrm{d}\hat{u}_i}{\mathrm{d}t}=A_{il}\hat{u}_l$$

this is a Floquet problem, $\hat{\boldsymbol{u}}$ has a growth rate

$$\bar{\sigma}(\alpha,\varepsilon)=\frac{1}{2\pi}\ln|\mu|$$

where $\mu$ is the eigenvalue of matrix $\boldsymbol{M}(2\pi)$.

Perturbation method

$$\hat{\boldsymbol{u}}=[\hat{\boldsymbol{u}}_0(t)+\varepsilon\hat{\boldsymbol{u}}_1(t)+\mathrm{O}(\varepsilon^2)]e^{\sigma\varepsilon t}$$

$$\left\langle\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2}|\hat{\boldsymbol{u}}|^2\right)\right\rangle=\varepsilon\tan\alpha\left[\frac{\gamma+1}{2}\right]\left\langle\cos\left(\frac{4}{\gamma}-1\right)t\right\rangle+\mathrm{O}(\varepsilon^2)$$

For $\varepsilon\rightarrow 0$, growth occurs only at $\gamma=4,2,\dfrac{4}{3},1$ or $\cos\alpha=\dfrac{1}{4},\dfrac{1}{2},\dfrac{3}{4},1$.

6 Discussion

Considering viscous, a pair of inertial waves can only grow if their joint rate of excitation exceeds the viscous decay rates.