Buoyancy

Buoyancy describes how a parcel’s density differs from that of its surroundings and how gravity acts on that difference. In ERF, the buoyancy term is the perturbational gravitational force per unit volume in the vertical momentum equation. A positive value acts upward; a negative value acts downward. It is not, by itself, the parcel’s vertical acceleration: pressure gradients and other momentum sources also contribute.

ERF offers several definitions of this force in compressible simulations. Some use the full density departure from a hydrostatic reference state; others approximate that departure using temperature or potential temperature. The anelastic equations use separate buoyancy expressions.

Choosing a formulation

In a compressible simulation, erf.buoyancy_type controls the expression used for the gravitational source. Gravity must also be enabled with erf.use_gravity = true; selecting a buoyancy type does not turn on gravity. With gravity disabled, the buoyancy force is zero.

17 Compressible buoyancy options

Value	Formulation	Scientific interpretation
1 (default)	Total-density perturbation	Retains the full density departure, including water loading and pressure-related density changes through the compressible equation of state. This is ERF’s default compressible option.
2 or 3	Temperature perturbation	The two values invoke the same implementation, in both dry and moist compressible runs. This is an approximate buoyancy expression, not the full compressible density force.
4	Potential-temperature perturbation	Uses dry potential-temperature anomalies, vapor anomalies, and condensate loading. It neglects an explicit pressure-density contribution and is therefore an approximation.

The choice is not merely a change in notation: these expressions can give different gravitational source terms for the same compressible state. Unless an approximation is part of the intended model formulation, the default type 1 retains the more complete density response.

For erf.anelastic = 1, ERF selects its dry or moist anelastic kernel irrespective of erf.buoyancy_type. Internally, the option is stored as 3 at anelastic levels, but this does not select the compressible type-3 formula. The input accepts values 1–4 only; there is no type 5. See Inputs for related configuration options.

Reference state, moisture, and sign convention

ERF uses dry-air density \(\rho_d\) and dry potential temperature \(\theta_d\). Water mixing ratios are measured per unit mass of dry air, not per unit mass of moist air. Write

\[q_t = q_v + q_{\mathrm{cond}},\qquad \rho_t = \rho_d(1+q_t),\]

where \(q_v\) is water-vapor mixing ratio, \(q_t\) is the total water-mass mixing ratio supplied to the buoyancy calculation, and \(q_{\mathrm{cond}}=q_t-q_v\) includes represented cloud and precipitation mass in both liquid and ice phases. This last quantity is not just liquid cloud water. ERF forms \(q_t\) from the moisture model’s conserved water-mass components; number concentrations and unrelated non-water species are not included in that sum.

Let \(\rho_0(z)\), \(\theta_0(z)\), \(p_0(z)\), and \(q_{v0}(z)\) denote the reference dry density, potential temperature, pressure, and vapor mixing ratio. There is no reference condensate, so the reference total moist density is

\[\rho_{t0}=\rho_0(1+q_{v0}).\]

For a hydrostatic reference state,

\[\frac{dp_0}{dz}=-\rho_{t0}g,\]

where \(g>0\) is the magnitude of gravity. We take height \(z\) positive upward and write \(g_z=-g\). Thus a negative density perturbation gives a positive (upward) gravitational force.

ERF evaluates the vertical buoyancy term at vertical-velocity faces. Throughout this page, an overbar denotes the arithmetic average of values from the two adjacent cell centers; the formulas below refer to interior, non-embedded-boundary faces unless stated otherwise.

The force and the momentum tendency

The buoyancy term \(B_z\) has units of \(\mathrm{N\,m^{-3}}=\mathrm{kg\,m^{-2}\,s^{-2}}\), rather than \(\mathrm{m\,s^{-2}}\). On a flat mesh, omitting other forces and any additional prescribed pressure forcing, the contribution of pressure and buoyancy to the dry vertical-momentum tendency is

\[\left.\frac{\partial(\rho_* w)}{\partial t}\right|_{p,B} =\frac{-G_{p,z}+B_z}{1+q_{t,f}},\qquad q_{t,f}=\frac{q_t(k-1)+q_t(k)}{2},\]

where \(\rho_*\) denotes \(\rho_d\) in compressible mode and the fixed reference dry density \(\rho_0\) in anelastic mode. The quantity \(G_{p,z}\) is ERF’s vertical pressure-gradient term. For compressible flow it is calculated from the EOS pressure perturbation \(p'=p-p_0\); for anelastic flow it is maintained through the pressure projection. Anelastic projection also applies a separate momentum correction, which is not represented by this isolated slow-RHS expression. Terrain metrics and other forcings enter elsewhere in the momentum update. In dry flow, \(q_{t,f}=0\).

Dividing \(B_z\) by a representative density gives a buoyancy scale with units of acceleration, but the source alone is not a complete prediction of vertical acceleration.

Type 1: total-density perturbation

The type-1 expression is the gravitational force associated with the full total-density departure from the reference state:

\[B_{z,f}^{(1)} =g_z\,\overline{\left[\rho_d(1+q_t)-\rho_0(1+q_{v0})\right]}.\]

The entire bracketed density perturbation is averaged to the face. A reference-state column with \(\rho_d=\rho_0\) and \(q_t=q_{v0}\) has zero buoyancy. At otherwise fixed dry density, adding condensate makes the air heavier and gives a downward contribution. The compressible EOS also allows pressure perturbations to modify the density, and type 1 retains that contribution. Here “full” describes the density departure in this gravitational source; it does not imply that every other term in ERF’s momentum equation is an exact moist-mixture treatment.

Types 2 and 3: temperature perturbation

These two input values use the same approximate expression. Define

\[\epsilon_v=\frac{R_v}{R_d}-1,\qquad \beta_T=\frac{T-T_0}{T_0} +\epsilon_v(q_v-q_{v0})-q_{\mathrm{cond}},\]

where \(R_d\) and \(R_v\) are the gas constants of dry air and water vapor, \(T\) is diagnosed from the compressible thermodynamic state, and \(T_0\) is diagnosed from \(p_0\) and \(\theta_0\). In the active buoyancy helper, that reference conversion uses the fixed exponent \(R_d/C_{p,d}\), with \(C_{p,d}=1004.5\,\mathrm{J\,kg^{-1}\,K^{-1}}\).

The interior-face force is

\[B_{z,f}^{(2)}=B_{z,f}^{(3)} =-\overline{\rho_0}\,g_z\,\overline{\beta_T}.\]

The three contributions to \(\beta_T\) have familiar interpretations: warmer air favors upward buoyancy, additional water vapor favors upward buoyancy through its gas constant, and condensate or precipitation mass favors downward buoyancy. In a dry simulation, the water terms are zero. This temperature-based expression approximates the full density force when water mixing ratios are dilute and pressure-induced density differences are negligible compared with the thermal and moisture contributions retained by the approximation. A small value of \(|p'|/p_0\) alone does not guarantee agreement with type 1: for a pressure-only perturbation, the two gravitational-source expressions differ already at first order.

Type 4: potential-temperature perturbation

Type 4 uses the dry potential-temperature anomaly instead of the actual temperature anomaly. Define

\[\beta_\theta=\frac{\theta_d-\theta_0}{\theta_0} +\epsilon_v(q_v-q_{v0})-q_{\mathrm{cond}},\]

and compute

\[B_{z,f}^{(4)} =-\overline{\rho_0}\,g_z\,\overline{\beta_\theta}.\]

In dry flow this reduces to the potential-temperature perturbation term. A warm potential-temperature perturbation gives an upward contribution; a negative perturbation gives a downward contribution. As in the temperature formulation, moisture enhances buoyancy through vapor anomalies and reduces it through condensate loading.

This is an approximation to the full compressible density force, appropriate when the omitted pressure-induced density contribution is negligible compared with the retained potential-temperature and moisture perturbations, and the dilute-moisture approximation is acceptable. In particular, a pressure perturbation at fixed potential temperature has no explicit contribution to \(\beta_\theta\). Merely reducing \(|p'|/p_0\) does not make this formula identical to type 1.

Anelastic buoyancy

In anelastic mode, ERF fixes the prognostic dry density to its base-state value and imposes the anelastic velocity constraint. The active buoyancy calculation does not use the compressible numeric selector.

For dry anelastic flow, ERF first averages the potential temperatures to the vertical face and then forms the fractional perturbation:

\[B_{z,f}^{A,\mathrm{dry}} =-\overline{\rho_0}\,g_z\, \frac{\overline{\theta_d}-\overline{\theta_0}} {\overline{\theta_0}}.\]

For moist anelastic flow, ERF first forms \(\beta_\theta\) at each cell center and then averages it:

\[B_{z,f}^{A,\mathrm{moist}} =-\overline{\rho_0}\,g_z\,\overline{\beta_\theta}.\]

The order of averaging is different. Consequently, the two formulas do not have to coincide exactly on a vertically varying reference profile even when the water-mixing-ratio terms are zero. The pressure-gradient term in anelastic momentum is supplied by the projection, not by recomputing the compressible EOS pressure perturbation.

Why the compressible formulas differ

The compressible equation of state implemented in ERF is

\[p=P_{00} \left[\frac{R_d\rho_d\theta_d(1+\alpha q_v)}{P_{00}}\right]^\Gamma, \qquad \alpha=\frac{R_v}{R_d},\]

where \(P_{00}=10^5\,\mathrm{Pa}\) is the fixed pressure used to define potential temperature, not the hydrostatic profile \(p_0(z)\). The EOS exponent is fixed at \(\Gamma=1.4\). For states consistent with this EOS, the exact total-density ratio is

\[\frac{\rho_t}{\rho_{t0}} =\left(\frac{p}{p_0}\right)^{1/\Gamma} \frac{\theta_0}{\theta_d} \frac{1+q_t}{1+q_{v0}} \frac{1+\alpha q_{v0}}{1+\alpha q_v}.\]

Writing \(p'=p-p_0\), \(\theta_d'=\theta_d-\theta_0\), \(\delta q_v=q_v-q_{v0}\), and \(\delta\rho_t=\rho_t-\rho_{t0}\), a first-order expansion gives

\[\frac{\delta\rho_t}{\rho_{t0}} =\frac{1}{\Gamma}\frac{p'}{p_0} -\frac{\theta_d'}{\theta_0} +\left[\frac{1}{1+q_{v0}} -\frac{\alpha}{1+\alpha q_{v0}}\right]\delta q_v +\frac{q_{\mathrm{cond}}}{1+q_{v0}}+O(\delta^2),\]

where \(O(\delta^2)\) collects products and higher orders of small state departures. Ignoring the pressure-density term and then taking the dilute-water limit yields the type-4 expression. The currently implemented approximation is not the exact finite-humidity linearization, whose vapor and condensate coefficients depend on \(q_{v0}\).

For small perturbations, with \(T'=T-T_0\), \(T'/T_0\simeq\theta_d'/\theta_0+ (R_d/C_{p,d})p'/p_0\). Therefore types 2 and 3 retain a different pressure-related response than type 4; neither is identical to the full-density type 1.

Note

A dry pressure-only perturbation illustrates the distinction. If \(p>p_0\) while \(\theta_d=\theta_0\), type 1 gives a downward density-perturbation force, types 2/3 an upward temperature-perturbation force, and type 4 zero. These are separate gravitational-source contributions. They must not be interpreted as predictions of the complete vertical momentum tendency without also including the pressure-gradient term.

The runtime input erf.c_p defaults to the fixed reference heat capacity \(C_{p,d}\) but is a separate configurable parameter. Changing erf.c_p affects selected temperature and forcing calculations but does not change the fixed EOS \(\Gamma\). Consequently, results that mix the runtime \(R_d/c_p\) with the fixed EOS need not satisfy all inverse thermodynamic identities away from the reference value. See Governing Equations and Constants and Units.

Supported configurations and further reading

Embedded-boundary buoyancy has a separate implementation. With gravity enabled, its established choices are dry compressible type 1 and dry anelastic; the other embedded-boundary combinations should not be treated as supported buoyancy configurations. Compressible Kessler_NoRain, SAM, and SAM_NoPrecip_NoIce require type 1 in the current source. These restrictions do not change the meaning of the selector for other supported simulations.

For background on cloud-resolving models and the sensitivity of moist models to their governing-equation choices, see Khairoutdinov and Randall (2003), Journal of the Atmospheric Sciences, 60, 607–625, and Bryan and Fritsch (2002), Monthly Weather Review, 130, 2917–2928. These are background references; the expressions above describe ERF’s implemented options.