Semi-deep UMJS

Compatibility conditions: $$\begin{align*} \frac{u^2}{r} \frac{\sin(\phi)}{\cos(\phi)} + 2 \Omega u \sin \phi + \frac{RT}{r} \partial_\phi \log(p) &= 0\\ -\frac{u^2}{r} - 2\Omega u \cos\phi + g \hat{r}^2 + RT \partial_r \log(p) &= 0\\ \end{align*}$$

$$\begin{align*} \partial_r\left[ \frac{r}{T}\left(\frac{u^2}{r} \frac{\sin(\phi)}{\cos(\phi)} + 2 \Omega u \sin \phi\right) \right] + R \partial_{r\phi} \log(p) &= 0\\ \partial_\phi \left[\frac{1}{T}\left(-\frac{u^2}{r} - 2\Omega u \cos\phi + g \hat{r}^2 \right)\right]+ R \partial_{\phi r} \log(p) &= 0\\ \end{align*}$$

$$\begin{align*} \partial_r\left[ \frac{ur}{T}\left(\frac{u}{r} \frac{\sin(\phi)}{\cos(\phi)} + 2 \Omega \sin \phi\right) \right] &= \partial_\phi \left[\frac{u}{T}\left(-\frac{u}{r} - 2\Omega \cos\phi + g \hat{r}^2 \right)\right] \end{align*}$$

$$\begin{align*} \partial_r\left[ \sin(\phi)\frac{r}{T}\left(\frac{u^2}{r\cos(\phi)} + 2 \Omega u\right) \right] &= \partial_\phi \left[\frac{\cos\phi}{T}\left(-\frac{u^2}{r\cos\phi } - 2\Omega u + g \hat{r}^2 \right)\right] \end{align*}$$

$$\begin{align*} \partial_r\left[ r\sin(\phi)\frac{\frac{u^2}{r\cos(\phi)} + 2 \Omega u}{T} \right] &= -\partial_\phi \left[\cos\phi\frac{\frac{u^2}{r\cos\phi } + 2\Omega u}{T}\right] + \partial_\phi \left[\frac{\cos\phi}{T}g \hat{r}^2 \right] \end{align*}$$

$$\begin{align*} \partial_r\left[ r\sin(\phi)\frac{U}{T} \right] &= -\partial_\phi \left[\cos\phi\frac{U}{T}\right] + \partial_\phi \left[\frac{\cos\phi}{T}g \hat{r}^2 \right] \end{align*}$$

$$\begin{align*} \sin(\phi)\frac{U}{T} + r\sin(\phi)\partial_r\left[\frac{U}{T} \right] &= \sin\phi \frac{U}{T} - \cos\phi\partial_\phi \left[\frac{U}{T}\right] + \partial_\phi \left[\frac{1}{T}g \hat{r}^2 \right] \end{align*}$$

$$\begin{align*} \left[r\sin(\phi)\partial_r - \cos\phi \partial_\phi \right]\left(\frac{U}{T} \right) &= \partial_\phi \left[\frac{1}{T}g \hat{r}^2 \right] \end{align*}$$

$$\begin{align*} \left[\sin(\phi)\partial_r - \frac{\cos\phi}{r} \partial_\phi \right]\left(\frac{U}{T} \right) &= \partial_\phi \left[\frac{1}{rT}g \hat{r}^2 \right] \end{align*}$$

SW form of eqn: $$\begin{align*} \left[\sin(\phi)\partial_r - \frac{\cos\phi}{r} \partial_\phi \right]\left(\frac{U}{T} \right) &= \frac{g}{a}\partial_\phi \left[\hat{r}^2\frac{a}{rT} \right] \end{align*}$$

Choose functional form $$\begin{align*} T(r, \phi) = \frac{a}{r} \hat{r}^2 \left[\tau_1(r) - \tau_2(r) \left( \left(\frac{r}{a} \cos\phi\right)^k - \left(\frac{k}{k+2} \left(\frac{r}{a} \cos\phi \right)^{k+2} \right) \right) \right] \end{align*}$$

$$\begin{align*} \left[\sin(\phi)\partial_r - \frac{\cos\phi}{r} \partial_\phi \right]\left(\frac{U}{T} \right) &= \frac{g}{a}\partial_\phi \left[\hat{r}^2\frac{a}{rT} \right] \end{align*}$$