Is it meaningful to assume static pressure to be constant along a streamline?

Question

I am looking at Bernoulli's equation formulated in two dimensions, which reads $$\frac{v^2}{2}+c_p\theta_0\pi+gz \equiv \mathrm{const.}$$ along a given streamline $\gamma$, where $v=|\mathrm{v}|$ for the flow field $\mathrm{v}(x,z)$ which is in steady state (i.e. $\frac{\partial\mathrm{v}}{\partial t}\equiv 0$) and $\pi=\left(\frac{p}{p_0}\right)^\frac{R}{c_p}$ is the Exner function. The potential temperature $\theta$ is assumed to be constant equal to $\theta_0$ along $\gamma$. Now if we assume $\pi$ to be constant along $\gamma$ as well, we end up with an upper bound to the maximum upward displacement an air parcel with initial velocity $v_0$ and initial height $z_0$ can attain, namely $$z_{crit}= z_0+\frac{v_0^2}{2g},$$ since $$v^2=0 \text{ if and only if } z-z_0=\frac{v_0^2}{2g}.$$ But is the assumption that $\pi$ is constant along $\gamma$ a physically useful one? I cannot find a discussion concerning this in any textbook on meteorology I currently posses, neither in internet resources. Any advice/resource would be appreciated.

Answer 1

If you assume both hydrostatic and pure geostrophic balance, that is a valid assumption.

In Einstein notation, $$u_i=-\frac{1}{f \rho}\frac{\partial P}{\partial x_j}\epsilon_{ij3}$$

If we look at the equation for the streamline:$$u_i=-\frac{\partial \psi}{\partial x_j}\epsilon_{ij3}$$, then we can see that

$$-\frac{\partial \psi}{\partial x_j}\epsilon_{ij3}=-\frac{1}{f \rho}\frac{\partial P}{\partial x_j}\epsilon_{ij3}$$

which simplifies down to $$\frac{\partial \psi}{\partial x_j}=\frac{1}{f \rho}\frac{\partial P}{\partial x_j}$$. Thus, for isochoric (incompressible) geostrophic flow on an f-plane, the streamfunction only depends solely on pressure.