Cloud Chamber

Overview

The Cloud Chamber problem is a single-level, Cartesian, fixed-density anelastic configuration for studying buoyancy-driven thermal and moist flow in a closed rectangular chamber. It supports prescribed-temperature walls, dry or SatAdj thermodynamics, and several wall-transfer closures for momentum, heat, and water vapor.

The Cloud Chamber implementation is intended for code-path verification, qualitative chamber-flow studies, wall-flux experiments, and conservation regression tests. It is not a quantitatively validated Pi-Chamber LES or a calibrated engineering wall-model package. It does not provide finite-rate droplet microphysics, droplet deposition, precipitation, natural-convection wall correlations, or experimentally calibrated transfer coefficients.

Most new inputs should use prob.thermodynamic_initialization = physical_temperature_rh. The older legacy_theta_qv mode is retained for existing numerical inputs but does not support the generalized per-channel wall-transfer models described below.

Choose the thermodynamic mode

9 Supported Cloud Chamber thermodynamic modes
Mode	Moisture model	Wall moisture	Typical use
Dry physical chamber	Omit `erf.moisture_model`	Every face is `dry`	Thermal convection and strict dry `rhoTheta` budget checks
SatAdj with dry walls	`erf.moisture_model = SatAdj`	Every face is `dry`	Moist thermodynamics in a water-impermeable chamber
SatAdj with wet walls	`erf.moisture_model = SatAdj`	One or more faces may be `wet`	Equilibrium vapor exchange with total-water budget checks
Legacy theta/qv	Omit moisture for dry legacy mode, or use `SatAdj` for cloudy legacy mode	Legacy numerical `theta`/`qv` wall values	Backward compatibility with existing Cloud Chamber inputs

A wall marked dry is impermeable to water but may still exchange heat. A wet wall is valid only when erf.moisture_model = SatAdj.

Terminology

rhoTheta: Density-weighted potential temperature, the conserved thermal scalar used by this Cloud Chamber configuration.
qv and qc: Water-vapor and cloud-water mixing ratios relative to dry air. In SatAdj cases ERF partitions total nonprecipitating water between these two phases.
SatAdj: ERF’s instantaneous saturation-adjustment thermodynamics, including the existing latent-heating adjustment associated with vapor–cloud-water equilibrium.
Resolved wall transfer: Transfer obtained from ERF’s resolved no-slip momentum treatment or from the adjacent-cell-to-wall half-cell scalar gradient. It is not an engineering wall-function correlation.
Bulk coefficient: A dimensionless transfer coefficient multiplying the actual tangential speed in the Cloud Chamber forced/shear-dependent wall-transfer formulas.

Global ERF requirements

The Cloud Chamber parser enforces a deliberately narrow numerical scope. Supported Cloud Chamber cases use one nonperiodic Cartesian level, fixed-density anelastic dynamics, gravity, and ConstantDz geometry. The generalized physical-temperature wall-transfer path additionally requires explicit vertical diffusion and ConstantAlpha scalar diffusion.

A typical common block is:

erf.prob_name = "Cloud Chamber"
erf.init_type = ConstantDensity
erf.anelastic = 1
erf.use_gravity = true
erf.mesh_type = ConstantDz
erf.vert_implicit = false
erf.molec_diff_type = ConstantAlpha

amr.max_level = 0
geometry.is_periodic = 0 0 0

The following constraints are part of the implemented Cloud Chamber contract:

10 Global Cloud Chamber requirements
Setting	Required value	Notes
`erf.anelastic`	`1`	Required in every Cloud Chamber mode.
`erf.use_gravity`	`true`	Required in every Cloud Chamber mode.
`amr.max_level`	`0`	AMR refinement is unsupported. The single level may still contain multiple AMReX boxes.
`geometry.is_periodic`	`0 0 0`	All three directions are closed and nonperiodic.
`erf.mesh_type`	omitted or `ConstantDz`	Stretched vertical grids are outside the supported Cloud Chamber path.
`erf.terrain_type`	omitted or `None`	Terrain and embedded-boundary configurations are unsupported.
`erf.buildings_type`	omitted or `None`	Immersed buildings are unsupported.
`erf.init_type`	`ConstantDensity`	The supported Cloud Chamber initializer requires a fixed-density anelastic base state in both physical and legacy thermodynamic modes.
`erf.molec_diff_type`	`ConstantAlpha`	Required by the physical wall-transfer path, including cases that also use bulk, neutral, or MOST wall closures.
`erf.vert_implicit`	`false`	Required by the physical wall-transfer path.

If prob.p_inf is supplied, the Cloud Chamber requires it to agree with ERF’s reference pressure p_0 = 100000 Pa within a relative tolerance of 1e-6. prob.p_inf does not redefine the reference pressure used by the Cloud Chamber potential-temperature conversion.

Physical-temperature initialization

Set:

prob.thermodynamic_initialization = physical_temperature_rh

The physical-temperature initializer constructs a linear vertical temperature profile and optionally adds a deterministic three-dimensional perturbation.

11 Physical initialization inputs
Key	Required	Default	Valid values / meaning
`prob.initial_temperature_bottom`	yes	none	finite and greater than zero, in K
`prob.initial_temperature_top`	yes	none	finite and greater than zero, in K
`prob.temperature_perturbation_amplitude`	no	`0`	finite amplitude in K
`prob.perturbation_mode`	no	`deterministic_sine`	the only currently supported perturbation mode
`prob.initial_relative_humidity`	SatAdj only	none	required with SatAdj; a fraction in `[0,1]`

For a domain with lower corner \((x_0,y_0,z_0)\) and lengths \((L_x,L_y,L_z)\), the temperature field is

\[T(x,y,z) = T_\mathrm{linear}(z) + A_T \sin\!\left(\frac{2\pi(x-x_0)}{L_x}\right) \sin\!\left(\frac{2\pi(y-y_0)}{L_y}\right) \sin\!\left(\frac{\pi(z-z_0)}{L_z}\right),\]

where \(A_T\) is prob.temperature_perturbation_amplitude and \(T_\mathrm{linear}\) interpolates between the prescribed bottom and top temperatures.

Potential temperature is derived from the local hydrostatic base-state pressure,

\[\theta = T \left(\frac{p_0}{p_\mathrm{hse}}\right)^{R_d/c_p},\]

using ERF’s configured rdOcp = R_d/c_p. Here \(p_\mathrm{hse}\) is the local hydrostatic pressure and \(p_0=100000\ \mathrm{Pa}\) is ERF’s reference pressure.

With SatAdj, the initial vapor mixing ratio is computed from the supplied relative humidity,

\[q_v = \frac{R_d}{R_v} \frac{RH\,e_s(T)}{p_\mathrm{hse}-RH\,e_s(T)},\]

where RH is a fraction rather than a percentage. The initializer sets qc = 0 and SatAdj subsequently establishes the equilibrium vapor–cloud water partition. In dry mode there are no qv or qc state fields, and prob.initial_relative_humidity must be omitted.

Do not combine physical initialization with the legacy profile keys theta_bottom, theta_top, theta_perturbation_amplitude, qv_bottom, or qv_top.

Legacy theta/qv initialization

Set:

prob.thermodynamic_initialization = legacy_theta_qv

Legacy mode is provided for existing numerical inputs. It uses potential temperature directly instead of physical temperature and does not support the generalized per-channel wall-transfer keys.

Dry legacy mode requires:

prob.theta_bottom = <positive value>
prob.theta_top = <positive value>
prob.theta_perturbation_amplitude = <optional finite value>
prob.perturbation_mode = deterministic_sine

For cloudy legacy mode also set erf.moisture_model = SatAdj and provide nonnegative prob.qv_bottom and prob.qv_top. Every face still requires <face>.type = NoSlipWall. Legacy walls use <face>.theta; cloudy legacy walls also require <face>.qv.

Do not combine legacy_theta_qv with physical profile keys such as initial_temperature_bottom, initial_temperature_top, initial_relative_humidity, or temperature_perturbation_amplitude. For new generalized wall-model studies, use physical_temperature_rh.

Optional deterministic initial velocity perturbation

prob.U_0 (m s^-1) controls an optional deterministic horizontal velocity perturbation. It is available with either thermodynamic initialization mode and is an initial-condition option only; it does not represent wall motion.

ERF initializes the perturbation on the staggered velocity grid so that its cell-centered finite-volume horizontal divergence is zero to roundoff and sets

\[w = 0.\]

The pattern is smooth, deterministic, and vanishes consistently with the stationary chamber-wall construction. On a square horizontal domain with equal horizontal resolution it reduces to

\[u = U_0 \sin^2\!\left(\frac{\pi(x-x_0)}{L_x}\right) \sin\!\left(\frac{2\pi(y-y_0)}{L_y}\right) \sin^2\!\left(\frac{\pi(z-z_0)}{L_z}\right),\]

\[v = -U_0 \sin\!\left(\frac{2\pi(x-x_0)}{L_x}\right) \sin^2\!\left(\frac{\pi(y-y_0)}{L_y}\right) \sin^2\!\left(\frac{\pi(z-z_0)}{L_z}\right).\]

For a general rectangular/discretized horizontal grid, ERF applies the corresponding grid-aspect scaling required for the staggered finite-volume divergence to cancel. Leave prob.U_0 unset, or set it to zero, for no initial velocity perturbation.

Physical wall configuration

In physical-temperature mode, every face requires three basic declarations:

<face>.type = NoSlipWall
<face>.temperature = <temperature in K>
<face>.moisture = dry|wet

The face prefix is one of xlo, xhi, ylo, yhi, zlo, or zhi. Wall temperature must be finite and positive. A wet wall requires SatAdj. Cloud Chamber walls are stationary; <face>.velocity is rejected.

NoSlipWall is the required ERF boundary type for all supported Cloud Chamber wall models. Selecting an aerodynamic momentum model changes the Cloud Chamber tangential wall-stress closure; it does not change the required boundary type or create a moving wall.

Channel defaults

If no per-channel model is specified, the effective physical-wall defaults are:

12 Default wall-transfer models
Channel	Default model	Meaning
momentum	`resolved_noslip`	retain ERF’s resolved no-slip viscous momentum treatment
heat	`resolved_molecular`	use the adjacent-cell-to-wall half-cell resolved scalar gradient
vapor	`resolved_molecular`	use the resolved wet-wall gradient; dry walls remain exactly impermeable
cloud water	no selector	wall flux is always exactly zero

The retained aggregate compatibility key

<face>.wall_transfer_model = resolved_molecular

selects the fully resolved/default behavior for that face. Do not combine wall_transfer_model with per-channel model, coefficient, or roughness keys on the same face.

Choose a transfer model

The three configurable channels are selected independently:

<face>.momentum_transfer_model = resolved_noslip|bulk_aero|neutral_roughness_log|law_of_wall_momentum
<face>.heat_transfer_model = resolved_molecular|bulk_aero|neutral_roughness_log
<face>.vapor_transfer_model = resolved_molecular|bulk_aero|neutral_roughness_log

law_of_wall_momentum is a compatibility alias for neutral_roughness_log; it does not select a separate formulation.

13 Wall-model choices
Choice	Applicable channels	Additional inputs	Main behavior
resolved	momentum, heat, vapor	none beyond the basic wall settings and resolved diffusivities	retains ERF resolved no-slip momentum or half-cell scalar transfer
fixed `bulk_aero`	momentum, heat, vapor	`coefficient_source = fixed` plus the corresponding `C_D`, `C_H`, or `C_E`	user-supplied dimensionless transfer coefficient multiplied by actual tangential speed
`neutral_roughness_log`	momentum, heat, vapor	`z0_m` and the required scalar roughness lengths	neutral aerodynamic log-resistance closure
MOST-provided `bulk_aero`	momentum, heat, vapor	`coefficient_source = most` plus the required `z0_*` values	computes bulk coefficients from the current pointwise Monin–Obukhov stability state; `zlo`/`zhi` only

One coefficient provider per face

<face>.coefficient_source is a face-level provider shared by every ``bulk_aero`` channel on that face. It is required when at least one bulk channel is active and must be either fixed or most. Consequently, a single face cannot use fixed bulk momentum and MOST bulk heat at the same time. Non-bulk channels on that face may still be resolved or neutral.

With coefficient_source = fixed, provide a coefficient for every selected bulk channel:

C_D for bulk momentum;
C_H for bulk heat; and
C_E for bulk vapor.

The coefficients must be finite and nonnegative. They are dimensionless. With coefficient_source = most, do not provide C_D, C_H, or C_E; ERF computes them from the current wall-adjacent state.

Important

The parser validates selected vapor-model metadata even on a dry wall. Therefore a dry wall configured with fixed bulk_aero vapor still requires C_E; a dry wall configured with neutral or MOST vapor still requires the corresponding roughness metadata. The physical dry-wall vapor flux nevertheless remains exactly zero.

Roughness requirements

For neutral and MOST closures, the wall-adjacent reference distance is

\[z_\mathrm{ref}=\frac{\Delta n}{2},\]

where \(\Delta n\) is the cell width normal to the face. Every required roughness length must satisfy

\[0 < z_0 < z_\mathrm{ref}.\]

The parser does not clip invalid roughness values.

14 Roughness inputs
Key	Units	Required when
`<face>.z0_m`	m	any neutral channel, or any `bulk_aero` channel using MOST
`<face>.z0_h`	m	neutral heat, or MOST bulk heat
`<face>.z0_q`	m	neutral vapor, or MOST bulk vapor, including a configured dry vapor channel

When wet heat and wet vapor both use MOST on one face, z0_h and z0_q must be exactly equal because the implemented bulk-Richardson fixed point uses one scalar stability resistance. This equality is not imposed by a dry vapor channel because dry vapor is not an active scalar-transfer channel.

Per-face input reference

The following table summarizes the physical-temperature wall keys.

15 Cloud Chamber physical-wall input reference
Key	Default	Valid values	Required / rejected when
`<face>.type`	none	`NoSlipWall`	required on every face
`<face>.temperature`	none	finite, positive K	required on every physical wall
`<face>.moisture`	none	`dry` or `wet`	required on every physical wall; `wet` requires SatAdj
`<face>.wall_transfer_model`	omitted	`resolved_molecular` only	compatibility aggregate; rejected when combined with any per-channel transfer key
`<face>.momentum_transfer_model`	`resolved_noslip`	`resolved_noslip`, `bulk_aero`, `neutral_roughness_log`, `law_of_wall_momentum`	optional; alias maps to neutral roughness-log
`<face>.heat_transfer_model`	`resolved_molecular`	`resolved_molecular`, `bulk_aero`, `neutral_roughness_log`	optional
`<face>.vapor_transfer_model`	`resolved_molecular`	`resolved_molecular`, `bulk_aero`, `neutral_roughness_log`	optional; dry physical vapor flux remains exactly zero
`<face>.coefficient_source`	omitted	`fixed` or `most`	required if any channel is `bulk_aero`; rejected if no bulk channel is active; `most` only on `zlo`/`zhi`
`<face>.C_D`	none	finite and nonnegative	required only for fixed bulk momentum; rejected with MOST or non-bulk momentum
`<face>.C_H`	none	finite and nonnegative	required only for fixed bulk heat; rejected with MOST or non-bulk heat
`<face>.C_E`	none	finite and nonnegative	required only for fixed bulk vapor, including a configured dry vapor channel; rejected with MOST or non-bulk vapor
`<face>.z0_m`	none	`0 < z0_m < z_ref`	required by any neutral channel or any MOST bulk channel; otherwise rejected
`<face>.z0_h`	none	`0 < z0_h < z_ref`	required by neutral heat or MOST bulk heat; otherwise rejected
`<face>.z0_q`	none	`0 < z0_q < z_ref`	required by neutral vapor or MOST bulk vapor; otherwise rejected
`<face>.velocity`	stationary internally	unsupported	always rejected; moving walls are not implemented

Configuration examples

The snippets below illustrate syntax only. Numeric coefficients and roughness lengths are examples, not calibration recommendations.

Fully resolved physical wall:

zlo.type = NoSlipWall
zlo.temperature = 300.0
zlo.moisture = dry
zlo.wall_transfer_model = resolved_molecular

Fixed bulk momentum and heat with resolved wet vapor:

zlo.type = NoSlipWall
zlo.temperature = 300.0
zlo.moisture = wet
zlo.momentum_transfer_model = bulk_aero
zlo.heat_transfer_model = bulk_aero
zlo.vapor_transfer_model = resolved_molecular
zlo.coefficient_source = fixed
zlo.C_D = 0.01
zlo.C_H = 0.01

Neutral roughness-log momentum, heat, and vapor:

xlo.type = NoSlipWall
xlo.temperature = 292.0
xlo.moisture = wet
xlo.momentum_transfer_model = neutral_roughness_log
xlo.heat_transfer_model = neutral_roughness_log
xlo.vapor_transfer_model = neutral_roughness_log
xlo.z0_m = 1.0e-3
xlo.z0_h = 5.0e-4
xlo.z0_q = 5.0e-4

MOST-provided bulk transfer on a horizontal wall:

zlo.type = NoSlipWall
zlo.temperature = 300.0
zlo.moisture = wet
zlo.momentum_transfer_model = bulk_aero
zlo.heat_transfer_model = bulk_aero
zlo.vapor_transfer_model = bulk_aero
zlo.coefficient_source = most
zlo.z0_m = 2.0e-3
zlo.z0_h = 2.0e-3
zlo.z0_q = 2.0e-3

A face may mix non-bulk and bulk models. For example, neutral momentum may be combined with fixed bulk heat and resolved vapor:

ylo.momentum_transfer_model = neutral_roughness_log
ylo.heat_transfer_model = bulk_aero
ylo.vapor_transfer_model = resolved_molecular
ylo.z0_m = 1.0e-3
ylo.coefficient_source = fixed
ylo.C_H = 0.01

Wall-transfer formulations

Common definitions and sign convention

All Cloud Chamber wall closures use the adjacent wall cell and a reference distance \(z_\mathrm{ref}=\Delta n/2\). For outward unit normal \(\mathbf n\), stationary wall velocity \(\mathbf u_w=0\), and adjacent fluid velocity \(\mathbf u_c\), the tangential relative velocity is

\[\mathbf u_t = (\mathbf I-\mathbf n\mathbf n^T)(\mathbf u_c-\mathbf u_w), \qquad U_t=|\mathbf u_t|.\]

The normal velocity is removed before \(U_t\) is formed.

Physical scalar fluxes in this section are defined as positive into the chamber fluid and are denoted \(J_\mathrm{in}\). ERF stores a face flux in the positive coordinate direction, so

\[\begin{split}F_\mathrm{coord}= \begin{cases} +J_\mathrm{in}, & \text{low face},\\ -J_\mathrm{in}, & \text{high face}. \end{cases}\end{split}\]

The wall potential temperature is

\[\Pi_w = \left(\frac{p_\mathrm{hse}}{p_0}\right)^{R_d/c_p}, \qquad \theta_w = \frac{T_w}{\Pi_w} = T_w\left(\frac{p_0}{p_\mathrm{hse}}\right)^{R_d/c_p}.\]

Dry vapor and cloud water have exact zero physical wall flux for every supported transfer model:

\[J_{\rho q_v,\mathrm{in}}=0 \quad \text{for a dry wall}, \qquad J_{\rho q_c,\mathrm{in}}=0 \quad \text{for every wall}.\]

A dry wall is therefore not a qv_wall = 0 Dirichlet condition.

Resolved transfer

momentum_transfer_model = resolved_noslip leaves momentum to ERF’s existing resolved no-slip viscous treatment.

For heat_transfer_model = resolved_molecular, the physical-temperature wall replaces the scalar face flux with the half-cell resolved transfer

\[J_{\rho\theta,\mathrm{in}} = \rho\,\alpha_T\,(\theta_w-\theta_a)\frac{2}{\Delta n}.\]

For a wet wall with vapor_transfer_model = resolved_molecular,

\[J_{\rho q_v,\mathrm{in}} = \rho\,\alpha_C \left[q_\mathrm{sat}(T_w,p_\mathrm{hse})-q_{v,a}\right] \frac{2}{\Delta n}.\]

The resolved scalar coefficients are configured through erf.alpha_T for heat and erf.alpha_C for moisture. Both have units of m² s^-1. In this Cloud Chamber path they set the half-cell resolved wall-normal scalar transfer; they are not calibrated aerodynamic exchange coefficients.

momentum_transfer_model = resolved_noslip retains ERF’s resolved viscous no-slip momentum-stress treatment and does not use a Cloud Chamber C_D or roughness length.

Fixed bulk-aerodynamic transfer

With coefficient_source = fixed, the user supplies the dimensionless bulk coefficients. Momentum traction on the fluid is

\[\mathbf t_\mathrm{wall\to fluid} = -\rho C_D U_t\mathbf u_t.\]

The traction is tangent to the wall, opposes tangential relative motion, and is exactly zero when \(U_t=0\).

Bulk heat transfer is

\[J_{\rho\theta,\mathrm{in}} = \rho C_H U_t(\theta_w-\theta_a).\]

For a wet wall, bulk vapor transfer is

\[J_{\rho q_v,\mathrm{in}} = \rho C_E U_t \left[q_\mathrm{sat}(T_w,p_\mathrm{hse})-q_{v,a}\right].\]

The physical transfer uses the actual \(U_t\); there is no hidden minimum speed, gustiness velocity, or free-convection augmentation. Fixed bulk transfer is therefore a forced/shear-dependent closure, not a complete natural- or mixed-convection wall correlation.

Neutral roughness-log transfer

For neutral_roughness_log, let

\[D_m = \ln\left(\frac{z_\mathrm{ref}}{z_{0m}}\right), \qquad \kappa=0.41.\]

The neutral momentum state is

\[u_* = \frac{\kappa U_t}{D_m}, \qquad C_D=\left(\frac{\kappa}{D_m}\right)^2,\]

and momentum uses the same physical drag form

\[\mathbf t_\mathrm{wall\to fluid} = -\rho C_D U_t\mathbf u_t.\]

Neutral heat and wet-vapor coefficients are

\[C_H = \frac{\kappa^2} {D_m\ln(z_\mathrm{ref}/z_{0h})}, \qquad C_E = \frac{\kappa^2} {D_m\ln(z_\mathrm{ref}/z_{0q})},\]

and the bulk scalar-flux formulas above are then used. A scalar-only neutral heat or vapor channel still requires z0_m because its resistance contains \(D_m\). Neutral channels do not use coefficient_source or user-supplied C_D, C_H, or C_E.

MOST-provided bulk coefficients

coefficient_source = most keeps the bulk_aero physical flux formulas but obtains their coefficients from one pointwise Monin–Obukhov stability state on the wall face. It is supported only on zlo and zhi.

The wall virtual potential-temperature state is

\[\theta_v=\theta(1+\epsilon_v q_v),\]

where

\[\epsilon_v=\frac{R_v}{R_d}-1,\]

using ERF’s thermodynamic constants. With

\[\begin{split}q_{v,w}=\begin{cases} q_\mathrm{sat}(T_w,p_\mathrm{hse}), & \text{wet wall},\\ q_{v,a}, & \text{dry wall}. \end{cases}\end{split}\]

The stability calculation uses

\[U_\mathrm{MOST}=\max(U_t,0.01),\]

and the bulk Richardson number

\[Ri_b = s_g\, \frac{g z_\mathrm{ref}}{\theta_a} \frac{\theta_{v,a}-\theta_{v,w}}{U_\mathrm{MOST}^2},\]

where \(s_g=+1\) on zlo and \(s_g=-1\) on zhi. ERF clamps

\[Ri=\mathrm{clamp}(Ri_b,-4,4).\]

Cloud-water loading is not included in the virtual-potential-temperature bulk Richardson number used by this Cloud Chamber MOST closure.

Define

\[D_m=\ln(z_\mathrm{ref}/z_{0m}),\qquad D_h=\ln(z_\mathrm{ref}/z_{0h}),\qquad D_q=\ln(z_\mathrm{ref}/z_{0q}).\]

The scalar resistance used by the stability fixed point is represented by

\[\begin{split}D_s = \begin{cases} D_h, & \text{if MOST heat is active},\\ D_q, & \text{if heat is inactive and active wet MOST vapor is present},\\ D_m, & \text{for momentum-only MOST}. \end{cases}\end{split}\]

Thus z0_h supplies the scalar stability resistance when MOST heat is active, z0_q supplies it otherwise for active wet MOST vapor, and z0_m is used for momentum-only MOST. When wet heat and wet vapor are both active, the parser requires z0_h == z0_q.

Starting from \(\zeta=0\), ERF evaluates its existing calc_psi_m2 and calc_psi_h2 similarity functions and iterates

\[A_m=\max(D_m-\psi_m,1),\qquad A_s=\max(D_s-\psi_h,1),\]

\[\zeta_\mathrm{new} =0.5\,\zeta_\mathrm{old} +0.5\,Ri\frac{A_m^2}{A_s}.\]

The iteration stops when \(|\zeta_\mathrm{new}-\zeta_\mathrm{old}|\le 10^{-3}\) and allows at most 100 iterations. Nonconvergence or a non-finite result is an error; there is no silent neutral fallback.

After convergence,

\[A_m=\max(D_m-\psi_m,1),\qquad A_h=\max(D_h-\psi_h,1),\qquad A_q=\max(D_q-\psi_h,1),\]

and

\[C_D=\left(\frac{\kappa}{A_m}\right)^2,\qquad C_H=\frac{\kappa^2}{A_mA_h},\qquad C_E=\frac{\kappa^2}{A_mA_q}.\]

The internal MOST closure also forms the friction velocity

\[u_* = \frac{\kappa U_\mathrm{MOST}}{A_m}.\]

The 0.01 speed floor is used only to condition the stability calculation. Physical momentum, heat, and vapor transfer still multiply the actual \(U_t\), so all shear-dependent physical transfer is exactly zero at rest.

A configured dry MOST vapor channel is metadata only: its z0_q is still required and validated, but dry vapor does not participate in the active scalar stability resistance and its physical wall flux remains exactly zero. When a moisture state is present, ambient qv still enters the virtual potential temperature used by any active MOST momentum or heat calculation.

Momentum stress application

Bulk and neutral momentum closures return physical wall-on-fluid tangential traction. The Cloud Chamber stress adapter maps that traction onto ERF’s retained cross-stress components and the existing DiffusionSrcForMom operator applies the momentum divergence. The wall closure does not replace normal diagonal stress. This distinction matters when comparing analytic traction with stored stress-array signs.

Wall-transfer timestep guard

Bulk and neutral transfer can impose a stronger explicit timestep restriction than the ordinary advective/diffusive CFL limits. ERF therefore scans the wall-adjacent state and constructs a wall rate.

For heat or active wet vapor,

\[\lambda_s = C_s U_t\,\Delta n^{-1}.\]

Neutral scalars use their derived coefficient. Dry vapor does not contribute because its wall flux is exactly zero.

For a bulk or neutral tangential momentum face, the local conservative infinity-row-sum factor is

\[K_\infty=\frac{3+\sqrt{2}}{2},\]

so the per-face momentum contribution is

\[\lambda_{m,f} = K_\infty C_{D,f}U_{t,f}\Delta n_f^{-1}.\]

At an edge or corner, contributions from perpendicular active walls are added for each free velocity component before taking the maximum. MOST coefficients are evaluated once for the current wall/cell sample and then treated as frozen for this local rate estimate; this is not an exact Jacobian of the nonlinear MOST iteration.

The wall limit is

\[dt_\mathrm{wall}=\frac{0.5}{\lambda_\mathrm{max}}.\]

Adaptive stepping takes the minimum of this limit and the ordinary ERF constraints. If a positive user-specified erf.fixed_dt exceeds the wall limit, ERF aborts and reports the requested fixed step, wall limit, and maximum wall rate; it does not silently clip the requested fixed step.

Conserved-scalar budget diagnostics

Set

erf.cloud_chamber_budget_interval = <positive number of steps>

to write cloud_chamber_budget.dat. The diagnostic uses the same retained physical wall face flux that is applied to the scalar RHS; it does not re-evaluate a separate wall closure.

Each output row contains the interval start/end step and time, scalar name and units, six face contributions, net boundary contribution, volume change, internal source, residual, tolerance, and status. Face signs follow the coordinate-flux convention above, so the net inward boundary contribution is formed from low-minus-high stored coordinate fluxes.

For total nonprecipitating water,

\[\Delta\int_\Omega \rho(q_v+q_c)\,dV = \int_{t_a}^{t_b}\sum_f J_{v,\mathrm{in}}\,dA\,dt.\]

SatAdj exchange between qv and qc cancels in this total-water row. With six dry walls the total-water boundary contribution is exactly zero.

Budget statuses are:

PASS: A supported conservation row closes within its printed tolerance.
FAIL: A supported conservation row is outside its printed tolerance.
UNSUPPORTED_SOURCE: The row is not a supported conservation verdict because the diagnostic does not account for every required internal source. In SatAdj Cloud Chamber runs, rhoTheta is always labeled UNSUPPORTED_SOURCE because the current diagnostic omits part of the moist latent-heating source, regardless of the numerical residual. Use dry rhoTheta for strict thermal closure and total nonprecipitating water for moist conservation checks.

Enabling the budget diagnostic must not change the simulated state.

Validated regression examples

The repository contains tracked regression inputs for each major supported path. They are useful syntax references, but their grid, timestep, LES settings, coefficients, and roughness lengths are software-test parameters, not calibrated physical recommendations.

16 Tracked Cloud Chamber examples
Case	Input file
Dry physical, resolved walls	`Tests/test_files/CloudChamber_Dry/CloudChamber_Dry.i`
SatAdj wet, resolved walls and budgets	`Tests/test_files/CloudChamber_SatAdj_WetBudget/CloudChamber_SatAdj_WetBudget.i`
SatAdj mixed fixed-bulk wall transfer	`Tests/test_files/CloudChamber_SatAdj_BulkMixedWet/CloudChamber_SatAdj_BulkMixedWet.i`
SatAdj neutral roughness-log transfer	`Tests/test_files/CloudChamber_SatAdj_NeutralWet/CloudChamber_SatAdj_NeutralWet.i`
SatAdj MOST wet-wall transfer and budgets	`Tests/test_files/CloudChamber_SatAdj_MOSTWetBudget/CloudChamber_SatAdj_MOSTWetBudget.i`
Mixed neutral sidewalls and MOST horizontal walls	`Tests/test_files/CloudChamber_SatAdj_MOSTMixedWalls/CloudChamber_SatAdj_MOSTMixedWalls.i`
Legacy theta/qv configuration	`Tests/test_files/CloudChamber_Legacy_Config/CloudChamber_Legacy_Config.i`

A complete dry physical example is included below.

1 Dry physical Cloud Chamber with resolved wall transfer

# Cloud Chamber dry physical-temperature regression.
erf.prob_name = "Cloud Chamber"
erf.init_type = ConstantDensity
erf.anelastic = 1
erf.use_gravity = true
erf.vert_implicit = false
erf.molec_diff_type = ConstantAlpha
erf.dynamic_viscosity = 0.0
erf.alpha_T = 0.001
erf.alpha_C = 0.001
erf.les_type = Smagorinsky
erf.Cs = 0.1
erf.fixed_dt = 0.001
erf.sum_interval = -1
erf.check_int = -1
erf.cloud_chamber_budget_interval = 2
erf.plot_file_1 = plt
erf.plot_int_1 = 2
erf.plot_vars_1 = density theta temp pressure x_velocity y_velocity z_velocity

max_step = 6
geometry.prob_lo = 0.0 0.0 0.0
geometry.prob_hi = 2.0 2.0 1.0
geometry.is_periodic = 0 0 0
amr.n_cell = 16 16 8
amr.max_level = 0

prob.p_inf = 100000.0
prob.T_0 = 292.0
prob.thermodynamic_initialization = physical_temperature_rh
prob.initial_temperature_bottom = 300.0
prob.initial_temperature_top = 284.0
prob.temperature_perturbation_amplitude = 0.02
prob.perturbation_mode = deterministic_sine

xlo.type = NoSlipWall
xhi.type = NoSlipWall
ylo.type = NoSlipWall
yhi.type = NoSlipWall
zlo.type = NoSlipWall
zhi.type = NoSlipWall
xlo.temperature = 292.0
xlo.moisture = dry
xlo.wall_transfer_model = resolved_molecular
xhi.temperature = 292.0
xhi.moisture = dry
xhi.wall_transfer_model = resolved_molecular
ylo.temperature = 292.0
ylo.moisture = dry
ylo.wall_transfer_model = resolved_molecular
yhi.temperature = 292.0
yhi.moisture = dry
yhi.wall_transfer_model = resolved_molecular
zlo.temperature = 300.0
zlo.moisture = dry
zlo.wall_transfer_model = resolved_molecular
zhi.temperature = 284.0
zhi.moisture = dry
zhi.wall_transfer_model = resolved_molecular

Run checklist

Choose physical_temperature_rh for new work, or legacy_theta_qv only for compatibility with an existing legacy input.
Use one ConstantDz, nonperiodic Cartesian level with fixed-density anelastic dynamics and gravity.
In physical_temperature_rh, define bottom/top initial temperature and add RH only when SatAdj is enabled.
For physical_temperature_rh, define all six stationary NoSlipWall faces with physical wall temperature and dry or wet moisture state. Choose the momentum, heat, and vapor model on each face; remember that one coefficient_source is shared by every bulk channel on that face. Supply every required coefficient or roughness length, check 0 < z0_* < Delta_n/2 on the actual grid, and use equal z0_h and z0_q for wet heat+vapor MOST on one face.
For legacy_theta_qv, use the legacy <face>.theta wall values and, for cloudy legacy cases, provide <face>.qv. Do not use generalized per-channel wall-transfer keys in legacy mode.
Choose a timestep that satisfies the wall-rate guard, or use adaptive stepping.
Run a short case and inspect velocity, temperature/potential temperature, and, for SatAdj, qv and qc.
Enable erf.cloud_chamber_budget_interval and check dry thermal or total nonprecipitating-water conservation as appropriate.

Supported scope and limitations

The implemented Cloud Chamber wall-model scope is intentionally limited to:

one Cartesian ConstantDz level with no AMR refinement;
six stationary NoSlipWall faces;
no periodic direction;
no terrain, embedded boundaries, or immersed buildings;
prescribed physical wall temperature in the generalized wall-model path;
resolved, fixed bulk, neutral roughness-log, and horizontal MOST transfer as documented above; and
forced/shear-dependent bulk transfer with no gustiness or natural/mixed convection augmentation.

The implementation does not currently provide moving walls, smooth-wall y+/viscous-sublayer laws, stretched-dz wall sampling, terrain-aware wall models, EB/AMR wall models, natural-convection wall correlations, ocean/COARE closures, cloud-water deposition, precipitation, droplet activation, or facility-specific coefficient calibration.

A numerically stable run is not by itself evidence of quantitative Pi-Chamber agreement or grid-independent LES behavior.

Troubleshooting

initial_relative_humidity is rejected in dry mode: Omit prob.initial_relative_humidity unless erf.moisture_model = SatAdj. Enter RH as a fraction such as 0.95, not 95.
A wet wall is rejected: Wet walls require erf.moisture_model = SatAdj.
coefficient_source is rejected: Use it only when at least one channel on that face is bulk_aero. The same provider applies to every bulk channel on the face.
C_D, C_H, or C_E is missing: Fixed bulk transfer requires the coefficient corresponding to every bulk channel. A configured dry bulk-vapor channel still requires C_E even though its physical vapor flux is exactly zero.
A C_* coefficient is rejected with MOST: MOST computes the bulk coefficients. Remove user-supplied C_D, C_H, and C_E for the channels using the face-level MOST provider.
MOST is rejected on a side wall: coefficient_source = most is implemented only on zlo and zhi. Use resolved, fixed bulk, or neutral roughness-log transfer on side walls.
A roughness length is rejected: Check that the selected model actually consumes the key and that 0 < z0_* < Delta_n/2 for the face-normal cell spacing.
Wet heat+vapor MOST rejects unequal z0_h and z0_q: The implemented prescribed-temperature bulk-Richardson fixed point uses one scalar stability resistance, so coupled wet MOST heat and vapor require equal scalar roughness lengths.
Dry vapor appears to have no transfer: This is intentional. moisture = dry enforces exact zero vapor wall flux for resolved, fixed bulk, neutral, and MOST selections. Configured model metadata is still validated.
A positive fixed timestep aborts: Compare the reported fixed_dt with wall_dt. Bulk, neutral, and MOST wall transfer participate in an explicit wall-rate stability guard.
The cloudy rhoTheta budget reports UNSUPPORTED_SOURCE: The current diagnostic does not include the complete moist latent-heating source. Use total nonprecipitating water for the supported cloudy conservation check.