Odd function base state

The idea is to look and see if the Staniforth & White generalized thermal wind solutions can be generalized further if we place less of a premium on closed-form solutions. Can we derive an interesting stratosphere using this method?

The compatibility condition is, letting U≡2Ωu+u2rcos⁡ϕU \equiv 2\Omega u + \frac{u^2}{r\cos\phi} and T(r,ϕ)=(ar)3[τ1(r)−τ2(r)τ3(racos⁡ϕ)]−1 T(r, \phi) = \left(\frac{a}{r} \right)^3 \left[ \tau_1(r) - \tau_2(r) \tau_3\left(\frac{r}{a}\cos\phi \right) \right]^{-1}, (sin⁡(ϕ)∂∂r+cos⁡ϕr∂∂ϕ)(UT)=ga∂∂ϕ(a3r3T)=−gaτ2(r)∂∂ϕ[τ3(racos⁡ϕ)] \left(\sin(\phi) \pder{}{r} + \frac{\cos\phi}{r} \pder{}{\phi}\right)\left(\frac{U}{T}\right) = \frac{g}{a} \pder{}{\phi} \left(\frac{a^3}{r^3 T} \right) = -\frac{g}{a} \tau_2(r) \pder{}{\phi} \left[ \tau_3\left( \frac{r}{a} \cos\phi \right) \right]

The 2011 paper indicates that one specifies the latitudinal dependence and then leave τ1,τ2\tau_1, \tau_2 underdetermined.

τ2∂∂ϕ[τ3(racos⁡ϕ)]=(sin⁡ϕ∂∂r+cos⁡ϕr∂∂ϕ)[(a2r2∫arr′aτ2(r′) dr)] \tau_2\pder{}{\phi} \left[ \tau_3 \left(\frac{r}{a} \cos \phi \right) \right] = \left(\sin \phi \pder{}{r} + \frac{\cos \phi}{r} \pder{}{\phi} \right) \left[ \left( \frac{a^2}{r^2} \int_a^r \frac{r'}{a}\tau_2(r') \intd{r} \right) \right]

(sin⁡ϕ∂∂r+cos⁡ϕr∂∂ϕ)[a2r2∫ar′r′aτ2(r′) dr′((ra)k+1cos⁡(ϕ)k−1−(ra)k+3cos⁡(ϕ)k+1)]=(sin⁡ϕ∂∂r)[a2r2∫ar′r′aτ2(r′) dr′((ra)k+1cos⁡(ϕ)k−1−(ra)k+3cos⁡(ϕ)k+1)]+(cos⁡ϕr∂∂ϕ)[a2r2∫ar′r′aτ2(r′) dr′((ra)k+1cos⁡(ϕ)k−1−(ra)k+3cos⁡(ϕ)k+1)]=(sin⁡ϕ∂∂r)[a2r2∫ar′r′aτ2(r′) dr′((ra)k+1cos⁡(ϕ)k−1−(ra)k+3cos⁡(ϕ)k+1)]+(cos⁡ϕr∂∂ϕ)[a2r2∫ar′r′aτ2(r′) dr′((ra)k+1cos⁡(ϕ)k−1−(ra)k+3cos⁡(ϕ)k+1)] \begin{align*} &\left(\sin\phi \pder{}{r} + \frac{\cos\phi}{r} \pder{}{\phi} \right) \left[ \frac{a^2}{r^2} \int_a^{r'} \frac{r'}{a} \tau_2(r') \intd{r'}\left( \left(\frac{r}{a}\right)^{k+1} \cos(\phi)^{k-1} - \left(\frac{r}{a} \right)^{k+3} \cos(\phi)^{k+1} \right)\right]\\ &= \left(\sin\phi \pder{}{r} \right) \left[ \frac{a^2}{r^2} \int_a^{r'} \frac{r'}{a} \tau_2(r') \intd{r'}\left( \left(\frac{r}{a}\right)^{k+1} \cos(\phi)^{k-1} - \left(\frac{r}{a} \right)^{k+3} \cos(\phi)^{k+1} \right)\right] + \\ &\qquad \left(\frac{\cos\phi}{r} \pder{}{\phi}\right)\left[ \frac{a^2}{r^2} \int_a^{r'} \frac{r'}{a} \tau_2(r') \intd{r'}\left( \left(\frac{r}{a}\right)^{k+1} \cos(\phi)^{k-1} - \left(\frac{r}{a} \right)^{k+3} \cos(\phi)^{k+1} \right)\right] \\ &= \left(\sin\phi \pder{}{r} \right) \left[ \frac{a^2}{r^2} \int_a^{r'} \frac{r'}{a} \tau_2(r') \intd{r'}\left( \left(\frac{r}{a}\right)^{k+1} \cos(\phi)^{k-1} - \left(\frac{r}{a} \right)^{k+3} \cos(\phi)^{k+1} \right)\right] + \\ &\qquad \left(\frac{\cos\phi}{r} \pder{}{\phi}\right)\left[ \frac{a^2}{r^2} \int_a^{r'} \frac{r'}{a} \tau_2(r') \intd{r'}\left( \left(\frac{r}{a}\right)^{k+1} \cos(\phi)^{k-1} - \left(\frac{r}{a} \right)^{k+3} \cos(\phi)^{k+1} \right)\right] \\ \end{align*}

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