<p>In meteorological terminology within dynamics, there are usually one of two unique assumptions made:</p>

<ol>
<li>Baroclinic- density is a function of both temperature and pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT(x,y,z,t)}$</li>
<li>Barotropic- density is a function of only pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT_0}$</li>
</ol>

<p>However, these simple, yet often used and powerful definitions miss something important- the effect of water vapor. To get around this, often meteorologists make virtual temperature, that is $$T_v(x,y,z,t)=T(x,y,z,t)[1+\epsilon q(x,y,z,t) ]$$ where $q$ is the mixing ratio and $\epsilon$ is the ratio of the molar mass of water vapor to the molar mass of dry air.</p>

<p>Incorporating this into the definition of baroclinic is easy: $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R_dT_v(x,y,z,t)}$</p>

<p>Incorporating this into a barotropic assumption however, is difficult. What would the approach be? I would consider this important conceptually for the tropics, where the barotropic assumption is most often made, but is mostly covered in water.</p>

CC BY-SA 3.0
2018-02-16T18:06:27.640Z
Wouldn't that remove the mathematical benefit of the barotropic assumption (that temperature doesn't vary over time/space)?  What's more, systems without significant temperature gradients are physical realities... but not sure you can say the same about moisture.  It's an interesting idea, though, look forward to see what more you/others have towards the thought.
JeopardyTempest
CC BY-SA 3.0
2018-02-16T18:08:42.723Z
And feel like really you're in the realm of of $θ$ and [$θ_e$](https://en.wikipedia.org/wiki/Equivalent_potential_temperature) rather than $T$ and $T_V$ since T is always a function of height?
JeopardyTempest
<meteorology><thermodynamics>
Is there such a thing as virtual barotropic?
BarocliniCplusplus
# Is there such a thing as virtual barotropic?

(Asked by: username_0 on 2018-02-16)

In meteorological terminology within dynamics, there are usually one of two unique assumptions made:

1. Baroclinic- density is a function of both temperature and pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R\_dT(x,y,z,t)}$
2. Barotropic- density is a function of only pressure : $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R\_dT\_0}$

However, these simple, yet often used and powerful definitions miss something important- the effect of water vapor. To get around this, often meteorologists make virtual temperature, that is $$T\_v(x,y,z,t)=T(x,y,z,t)[1+\epsilon q(x,y,z,t) ]$$ where $q$ is the mixing ratio and $\epsilon$ is the ratio of the molar mass of water vapor to the molar mass of dry air.

Incorporating this into the definition of baroclinic is easy: $\rho(x,y,z,t)=\frac{P(x,y,z,t)}{R\_dT\_v(x,y,z,t)}$

Incorporating this into a barotropic assumption however, is difficult. What would the approach be? I would consider this important conceptually for the tropics, where the barotropic assumption is most often made, but is mostly covered in water.

**Tags:** meteorology, thermodynamics

> **Comments:**
> **username_1**: Wouldn't that remove the mathematical benefit of the barotropic assumption (that temperature doesn't vary over time/space)?  What's more, systems without significant temperature gradients are physical realities... but not sure you can say the same about moisture.  It's an interesting idea, though, look forward to see what more you/others have towards the thought.  
> **username_1**: And feel like really you're in the realm of of $θ$ and [$θ_e$](https://en.wikipedia.org/wiki/Equivalent_potential_temperature) rather than $T$ and $T_V$ since T is always a function of height?  

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## 0 answers

