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

<p>In Einstein notation, <span class="math-container">$$u_i=-\frac{1}{f \rho}\frac{\partial P}{\partial x_j}\epsilon_{ij3}$$</span></p>

<p>If we look at the equation for the streamline:<span class="math-container">$$u_i=-\frac{\partial \psi}{\partial x_j}\epsilon_{ij3}$$</span>, then we can see that </p>

<p><span class="math-container">$$-\frac{\partial \psi}{\partial x_j}\epsilon_{ij3}=-\frac{1}{f \rho}\frac{\partial P}{\partial x_j}\epsilon_{ij3}$$</span></p>

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

CC BY-SA 4.0
2020-04-15T07:59:58.153Z
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
CC BY-SA 4.0
2020-04-15T14:54:32.190Z
Yes, I will edit that. My bad. Streamlines are isopleths of the streamfunction (http://glossary.ametsoc.org/wiki/Streamfunction). I realize now that I have the order of the Levi-Civita symbol backwards (see link in comment). I'll edit my answer to reflect those corrections.
BarocliniCplusplus
CC BY-SA 4.0
2020-04-15T03:33:17.090Z
2020-04-15T14:57:57.030Z
2020-04-15T14:57:57.030Z
BarocliniCplusplus
<p>I am looking at Bernoulli's equation formulated in two dimensions, which reads
<span class="math-container">$$\frac{v^2}{2}+c_p\theta_0\pi+gz \equiv \mathrm{const.}$$</span>
along a given streamline <span class="math-container">$\gamma$</span>, where <span class="math-container">$v=|\mathrm{v}|$</span> for the flow field <span class="math-container">$\mathrm{v}(x,z)$</span> which is in steady state (i.e. <span class="math-container">$\frac{\partial\mathrm{v}}{\partial t}\equiv 0$</span>) and <span class="math-container">$\pi=\left(\frac{p}{p_0}\right)^\frac{R}{c_p}$</span> is the Exner function. The potential temperature <span class="math-container">$\theta$</span> is assumed to be constant equal to <span class="math-container">$\theta_0$</span> along <span class="math-container">$\gamma$</span>. Now if we assume <span class="math-container">$\pi$</span> to be constant along <span class="math-container">$\gamma$</span> as well, we end up with an upper bound to the maximum upward displacement an air parcel with initial velocity <span class="math-container">$v_0$</span> and initial height <span class="math-container">$z_0$</span> can attain, namely
<span class="math-container">$$z_{crit}= z_0+\frac{v_0^2}{2g},$$</span>
since 
<span class="math-container">$$v^2=0 \text{ if and only if } z-z_0=\frac{v_0^2}{2g}.$$</span>
But is the assumption that <span class="math-container">$\pi$</span> is constant along <span class="math-container">$\gamma$</span> 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.</p>

<meteorology><wind><fluid-dynamics>
Is it meaningful to assume static pressure to be constant along a streamline?
chriss
# Is it meaningful to assume static pressure to be constant along a streamline?

(Asked by: username_0 on 2020-04-14)

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.

**Tags:** meteorology, wind, fluid-dynamics


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## 1 answers

### Answer by username_1 on 2020-04-15. Score: 2

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: Yes, I will edit that. My bad. Streamlines are isopleths of the streamfunction (http://glossary.ametsoc.org/wiki/Streamfunction). I realize now that I have the order of the Levi-Civita symbol backwards (see link in comment). I'll edit my answer to reflect those corrections.  

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