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

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.

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.
• Thanks for another answer. I have a few remarks: The geostrophic flow equation should have the Coriolis parameter $f$ in the demoninator, right? I do not see how you arrive at this formulation of the streamline equation... this reads to me as $u_1=\frac{\partial \psi}{\partial x_2}, u_2=-\frac{\partial \psi}{\partial x_1}$, and $u_3=0$, with $\epsilon_{ijk}$ the Levi-Civita symbol? – chriss Apr 15 '20 at 7:59