Diagnostics

Athelas supports optional diagnostics for derived quantities that are useful for interpreting a run but are not part of the evolved state. Diagnostics are opt-in and are configured under config.diagnostics in the input deck.

The current diagnostics are:

optical_depth

Computes the Rosseland optical depth from the outer boundary inward. The profile is written to the HDF5 .ath output in a field named diagnostics/optical_depth.

photosphere

Tracks the radius where the optical depth crosses a chosen threshold, defaulting to \(\tau = 2/3\). The tracker is written to the history output.

shock

Tracks the strongest velocity compression and records its radius, cell index, and compression strength in the history output.

Optical Depth

The optical depth diagnostic computes

(1)\[\tau(r) = \int_r^{R_\mathrm{out}} \rho \kappa_R \, dr,\]

where \(\kappa_R\) is the Rosseland mean opacity provided by the active opacity model. The integration is performed from the outer boundary inward, so the outermost boundary has \(\tau = 0\) and deeper points have larger optical depth.

When enabled, the optical-depth profile is stored in the HDF5 checkpoint output as a diagnostic field.

Note

Optical depth requires opacity data, so it is only valid when radiation physics is enabled. The input parser rejects diagnostics.optical_depth if physics.radiation = false.

Photosphere Tracker

The photosphere tracker uses the optical-depth profile to locate the radius where

(2)\[\tau(r_\mathrm{ph}) = \tau_\mathrm{ph}.\]

The threshold tau is configurable and defaults to 2.0 / 3.0. If the threshold is bracketed by the computed optical-depth profile, the tracker linearly interpolates between neighboring diagnostic samples and writes the photosphere radius to the history output.

The history quantities are:

• Photosphere Radius [cm]

• Photosphere Cell Index

• Photosphere Valid

Photosphere Valid is 1 when the requested optical depth was found and 0 otherwise.

Note

The photosphere tracker depends on the optical-depth diagnostic. The input parser rejects diagnostics.photosphere unless diagnostics.optical_depth.enabled = true.

Shock Tracker

The shock tracker locates the strongest velocity compression. It examines both nodal velocity gradients and velocity jumps across cell faces, then records the largest positive value of \(-\partial v / \partial r\).

Within each cell, the velocity is treated as the DG polynomial represented by the nodal values in that element. The tracker applies the basis differentiation matrix to those nodal values to estimate the reference-coordinate derivative,

(3)\[\left.{\partial v \over \partial \eta}\right|_{i,q} = \sum_p D_{q p} v_{i,p},\]

then converts to a physical gradient using the cell width,

(4)\[\left.{\partial v \over \partial r}\right|_{i,q} = {1 \over \Delta r_i} \left.{\partial v \over \partial \eta}\right|_{i,q}.\]

The nodal compression candidate is therefore \(-\partial v / \partial r\) evaluated at each interpolation node. This is different from a cell-average shock indicator based only on neighboring average velocities; it uses the local DG representation directly and can see compression inside a high-order element.

The tracker also checks jumps across cell faces. It evaluates the left cell polynomial at the outer face and the right cell polynomial at the inner face, then estimates

(5)\[- {v_R - v_L \over \Delta r_\mathrm{face}},\]

where \(\Delta r_\mathrm{face}\) is the average width of the two cells adjacent to the face. This catches compression concentrated at an element interface, where a purely interior nodal derivative can miss the sharpest part of the jump.

The reported shock position is the node or face with the largest positive compression. This is a lightweight tracker rather than a troubled-cell indicator or limiter; it does not modify the solution.

The history quantities are:

• Shock Radius [cm]

• Shock Cell Index

• Shock Compression [1 / s]

Shock Breakout

The shock and photosphere trackers are most useful together. Shock breakout – the point at which the shock reaches the stellar surface and its radiation is no longer trapped behind an optically thick layer – corresponds to the shock position crossing the photosphere. In terms of the tracked quantities, breakout is the first time

(6)\[r_\mathrm{shock} \gtrsim r_\mathrm{ph},\]

i.e. when Shock Radius [cm] first reaches or exceeds Photosphere Radius [cm]. Before breakout the shock sits below the photosphere (\(r_\mathrm{shock} < r_\mathrm{ph}\)); the ordering reverses at breakout.

Breakout is detected in post-processing rather than flagged by the solver: enable both shock and photosphere (and therefore optical_depth), and the Python athelas_tools interface locates the crossing from the history file. The breakout time is often a convenient analysis t = 0 reference.

from athelas_tools.athelas import Athelas

ds = Athelas("model_final.ath")
t_breakout = ds.breakout_time()    # None if breakout never occurs
t_rel = ds.time_since_breakout     # this checkpoint's time minus t_breakout

The crossing time is linearly interpolated between the bracketing history samples (pass interpolate=False for the first post-crossing sample time). Only rows with a valid photosphere and finite shock radius are considered.

Plotting Trackers

The Python athelas_tools plotting interface can overplot tracker positions from the history file on checkpoint profiles:

from athelas_tools.athelas import Athelas

ds = Athelas("sod_final.ath")
ax = ds.plot("density", overlay_trackers=True)

By default, this overlays all known trackers that are present in the history output. Tracker lines use black dashed vertical markers with labels such as Shock and Photosphere.

Input Deck

Diagnostics are configured in a dedicated input-deck table:

config.diagnostics = {
  optical_depth = {
    enabled = true,
  },
  photosphere = {
    enabled = true,
    tau = 2.0 / 3.0,
  },
  shock = {
    enabled = true,
  },
}

The controls are:

• optical_depth.enabled – compute and write the optical-depth profile. Requires physics.radiation = true.

• photosphere.enabled – write photosphere tracker quantities to the history output. Requires optical_depth.enabled = true.

• photosphere.tau – optical-depth threshold for the photosphere tracker. Must be positive. Defaults to 2.0 / 3.0.

• shock.enabled – write shock tracker quantities to the history output.