Helmholtz-decomposition-code.md

@@ -2,12 +2,12 @@
 
 # Introduction
 
-Analysis of turbulence and vortices requires knowledge of the kinetic energy, is conservation and its modes. This tool performs the Helmholtz Decomposition, in order to extract the compressive and rotational components of kinetic energy of the flow as shown in **Tsubota, Fujimoto, Yui** (2017) [**Numerical Studies of Quantum Turbulence**](https://arxiv.org/abs/1704.02566).
+Analysis of turbulence and vortices requires knowledge of the kinetic energy, its conservation and its modes. This tool performs the Helmholtz Decomposition, in order to extract the compressive and rotational components of kinetic energy density of the flow as shown in **Tsubota, Fujimoto, Yui** (2017) [**Numerical Studies of Quantum Turbulence**](https://arxiv.org/abs/1704.02566).
 
 We define the effective velocity field as $`\omega(\textbf{r},t)=\sqrt{\rho(\textbf{r},t)}v(\textbf{r},t)`$. From this, we can calculate the kinetic energy as: 
 
 ```math
-E_k=\frac{m}{2V}\int\omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2,
+\mathcal{E}_k=\frac{m}{2V}\int\omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2,
 ```
 
 where $`\widetilde{\omega}(\textbf{k},t)=\mathfrak{F}[\omega(\textbf{r},t)]=\int\omega(\textbf{r},t)\frac{e^{-i\textbf{k}\cdot\textbf{r}}}{V}d\textbf{r}`$ is the Fourier Transform of the effective velocity.
@@ -63,7 +63,7 @@ The code outputs four files:
   5. rotational energy calculated via spectral decomposition
 * One `hhspec*.txt` file, containing data blocks for every timestep and showing the compressive and rotational spectra as a function of k. In current version, only rotational spectra is reliable. **[TODO: Fix compressive.]**
 
-[ADD UNITS]
+Units for every quantity are expressed in terms of **density** $\rho$, since we take $m=\hbar=k_B=1$. As a result, energies have measure units of $\rho^{5/6}$. 
 
 # Limitations
 Presently the code supports only 3D data. However, you can apply it to 2D as well. In order to do that, you need to increase data dimensionality of the datasets you wish to analyze. You can do it using wdata tool `wdata-datadim-up`. For example: