Note: This tutorial was generated from an IPython notebook that can be downloaded here.


Step 1: initialize Twind class

We are initilizing the TigressWindModel class with default parameters set to match the TIGRESS simulation suite results at \(|z|=H\) (see Simulation PDFs). Possible options are z0=['H', '2H', '500', '1000'].

import twind

number of wind phase = 2
galactic parameter = sfr
reference height = H
    A_v = 2.302585092994046
    p_v = 1
    d_v = 2
    A_cs = 9.185985478173896
    cs0 = 6.7
    sigma = 0.1
    vout0 = 25.0
    A_vB = 5.196378098798331
    p_vB = 4
    d_vB = 2
    A_M = 1.151292546497023
    Mach0 = 0.5
    p_M = 1
    d_M = 3
    Esn = 1e+51 erg
    mstar = 95.5 solMass
    vcool = 200.0 km / s
    Mej = 10.0 solMass
    ZSN = 0.2
    ZISM0 = 0.02
    vej = 3171.4804794827423 km / s
    Mref = 95.5 solMass
    pref = 2.5e+48 erg s / km
    Eref = 1e+51 erg
    Zref = 2.0 solMass
    a = [-0.067 -1.216 -0.857  0.013 -1.426 -2.151 -1.013 -0.879 -2.228 -2.627
 -0.698 -0.695  0.171 -0.942 -0.375  0.281]
    b = [-0.441 -0.227 -0.069 -0.418 -0.287 -0.149  0.016 -0.157 -0.117 -0.076
  0.136  0.108 -0.364 -0.146  0.04  -0.335]
    A = [0.86 0.06 0.14 1.03 0.04 0.01 0.1  0.13 0.01 0.   0.2  0.2  1.48 0.11
 0.42 1.91]
    p = [0.559 0.773 0.931 0.582 0.713 0.851 1.016 0.843 0.883 0.924 1.136 1.108
 0.636 0.854 1.04  0.665]

With the verbose=True option, key attributes are printed.

  • cool_params: parameters for cool mass loading PDF. See Joint PDF Model
  • hot_params: parameters for hot mass loading PDF. See Joint PDF Model
  • params: other physical parameters related to the particular choices of the TIGRESS simulation suite (see Kim et al. (2020a))
  • ref_params: outflow rates are normalized by \(\Sigma_{\rm SFR}q_{\rm ref}/m_*\) to obtain loading factors
  • scaling_params: fitting results for velocity-integrated loading factors as a function of SFR surface density (in log-log space) presented in Kim et al. (2020a). Each array contains the results for four loading factors (mass, momentum, energy, metal) of four phases (cool, intermediate, hot, whole). E.g., first four values are the results for the mass loading factor of cool, intermediate, hot, and whole gas.
  • a is the intercept
  • b is the slope
  • A is \(10^a\)
  • p is \(b+1\) for flux scalings.
\[\eta_q = A\Sigma_{\rm SFR}^b\]


\(\Sigma_{\rm SFR}\) is in \(M_\odot{\rm kpc^{-2} yr^{-1}}\) everywhere in this document.


\(u \equiv \log v_{\rm out}\) and \(w \equiv \log c_s\) as defined in Kim et al. (2020b).

Step 2: setup axes

We use xarray extensibly for easier manipulation with broadcasting, indexing, slicing, and interpolation.

The TigressWindModel.set_axes() method accept either the simulated PDF (in the form of xarray.Dataset) or list of ranges (in log) and number of bins for vout and cs axes (sfr can either be a scalar or an array). Default is

  • vout = (0,4,500)
  • cs = (0,4,500)
  • sfr = (-6,2,100)

This function will set attributes u=logvout and w=logcs as 1D DataArray as well as vBz and Mach as 2D DataArray for future use. If a range of sfr is passed, it will also set a member logsfr as 1D DataArray with different coordinates so that the final PDFs would be 3D DataArray.

For this example, we use a single value of SFR surface density and reduced number of bins for velocity axes.

cs: min=0, max=4, N=200
vout: min=0, max=4, N=200

We make sure that vBz and Mach are 2D while u=logvout and w=logcs are 1D.

print('u shpae:',tw.u.shape)
print('w shape:',tw.w.shape)
print('vBz shpae:',tw.vBz.shape)
print('Mach shape:',tw.Mach.shape)
u shpae: (200,)
w shape: (200,)
vBz shpae: (200, 200)
Mach shape: (200, 200)

Step 3: build mass loading PDFs

We have a method TigressWindModel.build_Mpdf() that automatically builds model PDFs for mass loading factor and return a xarray.Dataset. Note that if the range of (u,w) is not large enough, the mass PDF may not integrate to 1 (use verbose=True to check this).

Depending on the choice of the sfr axis, the resulting PDF can either be 2D or 3D. The returned Dataset have variables for PDFs (Mpdf, Mpdf-cool, Mpdf-hot) for total, cool, and hot outflow components. This also contains vBz and Mach as 2D arrays for convenience. In addition, the integrated loading factor (etaM and their phase-separated values, i.e., etaM-cool and etaM-hot) as a function of sfr are saved. If sfr is a scalar, these are also scalars.

pdf = tw.build_Mpdf(verbose=True)
Mass PDFs are integrated to: cool=0.997 hot=1
<xarray.plot.facetgrid.FacetGrid at 0x7f88b069a400>

Step 4: build all PDFs

We have a method TigressWindModel.build_model() that automatically builds model PDFs for mass, momentum, energy, and metal loading factors and return a xarray.Dataset containing all. The last three PDFs are reconstructed from the mass PDF as outlined in Kim et al. (2020b). By default, they are renormalized to ensure the integration over the entire (u,w) gives 1. Note that the metal PDF is not normalized for the input ZISM but for ZISM0.

Again, depending on the choice of the sfr axis, the resulting PDFs can either be 2D or 3D. The returned Dataset have variables for PDFs (Mpdf, ppdf, Epdf, Zpdf) and their phase-separated counterparts (e.g., Mpdf-cool, Mpdf-hot). The velocity-integrated loading factors (etaM, etap, etaE, etaZ) and their phase-separated counterparts (e.g., etaM-cool and etaM-hot) as a function of sfr are also stored. Finally, if renormalize=True (default), it also stores the renormalization factors (p_renorm, E_renorm, Z_renorm), which are also a function of sfr.

The Dataset has attributes for the choice of ZISM for the metal loading PDF as well as the bin sizes dlogcs and dlogvout for convenience.


As it builds a model PDF, it automatically checks whether the mass PDFs are integrated to 1. I.e., both cool and hot PDFs should satisfy

\[\int\int \tilde{f}_M^{\rm ph} dudw =1\]

individually. Again, this may depend on the (u,w) range. We then apply loading factor ratios to combine the mass loading PDF as

\[\tilde{f}_M = \frac{\eta_M^{\rm cool}}{\eta_M}\tilde{f}_M^{\rm cool}+\frac{\eta_M^{\rm hot}}{\eta_M}\tilde{f}_M^{\rm hot}\]

Note that Mpdf-cool and Mpdf-hot (and corresponding other PDFs) in the returned Dataset are not \(\tilde{f}_M^{\rm ph}\) but \(\frac{\eta_M^{\rm ph}}{\eta_M}\tilde{f}_M^{\rm ph}\).

print('contribution to')
print('mass outflow rate from cool is {:.3f} and hot is {:.3f}'.format(
print('energy outflow rate from cool is {:.3f} and hot is {:.3f}'.format(
contribution to
mass outflow rate from cool is 0.968 and hot is 0.029
energy outflow rate from cool is 0.081 and hot is 0.919

Finally, 2D PDFs for mass, momentum, energy, and metal loadings at \(\Sigma_{\rm SFR}=10^{-2}\) look as follows.

<xarray.plot.facetgrid.FacetGrid at 0x7f8870725898>