# Joint PDF Model¶

Note

See Kim et al. (2020b) for details.

## Cool outflow model¶

The cool outflow ($$T<2\times10^4{\rm K}$$) in the TIGRESS suite is well described by a model combining log-normal and generalized gamma distribution:

$\tilde{f}_{M}^{\rm cool}(u,w) = A_c \left(\frac{v_{\rm out}}{v_{\rm out,0}}\right)^2 \exp\left[-\left(\frac{v_{\rm out}}{v_{\rm out,0}}\right)\right] \exp\left[-\frac{1}{2}\left(\frac{\ln(c_{\rm s}/c_{\rm s,0})}{\sigma}\right)^2\right]$

where $$A_c=(\ln 10)^2/(2\pi\sigma^2)^{1/2}=2.12/\sigma$$.

$\frac{v_{\rm out,0}}{{\rm km/s}} = v0 \left(\frac{\Sigma_{\rm SFR}}{M_{\odot}{\rm kpc^{-2}yr^{-1}}}\right)^{0.23}+3$

At $$|z|=H$$, we adopt $$v0=25$$, $$c_{s,0}=6.7{\rm km/s}$$, and $$\sigma=0.1$$. We found the same function form with $$(v0,c_{s,0})=(45,7.5)$$, (45,8.5), and (60,10) works reasonably well at $$|z|=2H$$, 500pc, and 1kpc.

## Hot outflow model¶

The hot outflow ($$T>5\times10^5{\rm K}$$) in the TIGRESS suite is well described by a model combining two generalized gamma distributions:

$\tilde{f}_{M}^{\rm hot}(u,w) = A_h \left(\frac{v_{\mathcal{B},z}}{v_{\mathcal{B},0}}\right)^2 \exp\left[-\left(\frac{v_{\mathcal{B},z}}{v_{\mathcal{B},0}}\right)^4\right] \left(\frac{\mathcal{M}}{\mathcal{M}_0}\right)^3 \exp\left[-\left(\frac{\mathcal{M}}{\mathcal{M}_0}\right)\right]$

where $$A_h\equiv 2(\ln 10)^2/\Gamma(1/2)=5.98$$, $$v_{\mathcal{B},z}\equiv(v_{\rm out}^2+c_s^2)^{1/2}$$, and $$\mathcal{M}=v_{\rm out}/c_s$$.

$\frac{v_{\mathcal{B},0}}{10^3{\rm km/s}} = 2.4 \left(\frac{\Sigma_{\rm SFR,0}^{1/2}}{2+\Sigma_{\rm SFR,0}^{1/2}}\right)+0.8$

where $$\Sigma_{\rm SFR,0}\equiv \Sigma_{\rm SFR}/(M_{\odot}{\rm kpc^{-2}yr^{-1}})$$

We adopt $$\mathcal{M}_0=0.5$$ irrespective of $$z$$.