4.1 Deep Convection

The process of deep convection is treated with a parameterization scheme developed by Zhang and McFarlane [199]. The scheme is based on a plume ensemble approach where it is assumed that an ensemble of convective scale updrafts (and the associated saturated downdrafts) may exist whenever the atmosphere is conditionally unstable in the lower troposphere. The updraft ensemble is comprised of plumes sufficiently buoyant so as to penetrate the unstable layer, where all plumes have the same upward mass flux at the bottom of the convective layer. Moist convection occurs only when there is convective available potential energy (CAPE) for which parcel ascent from the sub-cloud layer acts to destroy the CAPE at an exponential rate using a specified adjustment time scale. For the convenience of the reader we will review some aspects of the formulation, but refer the interested reader to Zhang and McFarlane [199] for additional detail, including behavioral characteristics of the parameterization scheme. Evaporation of convective precipitation is computed following the procedure described in section 4.3.

The large-scale budget equations distinguish between a cloud and sub-cloud layer where temperature and moisture response to convection in the cloud layer is written in terms of bulk convective fluxes as

$$c_p \left( \frac{\partial T}{\partial t} \right)_{cu} = - \frac{1}{\rho} \frac{\partial}{\partial z} \left( M_u S_u + M_d S_d - M_c S \right) + L(C - E)$$ (4.2)

$$\left( \frac{\partial q}{\partial t} \right)_{cu} = - \frac{1}{\rho} \frac{\partial}{\partial z} \left( M_u q_u + M_d q_d - M_c q \right) + E - C ,$$ (4.3)

for $z\ge z_b$, where $z_b$ is the height of the cloud base. For $z_s<z<z_b$, where $z_s$ is the surface height, the sub-cloud layer response is written as

$$c_p {\left( \rho \frac{\partial T}{\partial t} \right)}_{m} = - \frac{1}{z_b-z_s} \left( M_b [S(z_b) - S_u (z_b)] + M_d [S(z_b) - S_d (z_b)] \right)$$ (4.4)

$${\left( \rho \frac{\partial q}{\partial t} \right)}_{m} = - \frac{1}{z_b-z_s} \left( M_b [q(z_b) - q_u (z_b)] + M_d [q(z_b) - q_d (z_b)] \right) ,$$ (4.5)

where the net vertical mass flux in the convective region, $M_c$, is comprised of upward, $M_u$, and downward, $M_d$, components, $C$ and $E$ are the large-scale condensation and evaporation rates, $S$, $S_u$, $S_d$, $q$, $q_u$, $q_d$, are the corresponding values of the dry static energy and specific humidity, and $M_b$ is the cloud base mass flux.

4.1.1 Updraft Ensemble

The updraft ensemble is represented as a collection of entraining plumes, each with a characteristic fractional entrainment rate $\lambda$. The moist static energy in each plume $h_c$ is given by

$$\frac{\partial h_c}{\partial z} = \lambda (h - h_c), \quad z_b<z<z_D .$$ (4.6)

Mass carried upward by the plumes is detrained into the environment in a thin layer at the top of the plume, $z_D$, where the detrained air is assumed to have the same thermal properties as in the environment ($S_c=S$). Plumes with smaller $\lambda$ penetrate to larger $z_D$. The entrainment rate $\lambda_D$ for the plume which detrains at height $z$ is then determined by solving (4.6), with lower boundary condition $h_c(z_b)=h_b$:

$$\frac{\partial h_c}{\partial (z-z_b)} = \lambda_D (h - h_b) - \lambda_D (h_c - h_b)$$ (4.7)

$$\frac{\partial (h_c - h_b)}{\partial (z-z_b)} - \lambda_D (h_c - h_b) = \lambda_D (h - h_b)$$ (4.8)

$$\frac{\partial (h_c - h_b)e^{\lambda_D(z-z_b)}}{\partial (z-z_b)} = \lambda_D (h - h_b)e^{\lambda_D(z-z_b)}$$ (4.9)

$$(h_c - h_b)e^{\lambda_D(z-z_b)} = \int_{z_b}^z \lambda_D (h - h_b)e^{\lambda_D(z^\prime-z_b)} dz^\prime$$ (4.10)

$$(h_c - h_b) = \lambda_D \int_{z_b}^z (h - h_b)e^{\lambda_D(z^\prime-z)} dz^\prime  .$$ (4.11)

Since the plume is saturated, the detraining air must have $h_c=h^*$, so that

$$(h_b - h^*) =\lambda_D \int_{z_b}^z (h_b - h)e^{\lambda_D(z^\prime-z)} dz^\prime .$$ (4.12)

Then, $\lambda_D$ is determined by solving (4.12) iteratively at each $z$.

The top of the shallowest of the convective plumes, $z_0$ is assumed to be no lower than the mid-tropospheric minimum in saturated moist static energy, $h^*$, ensuring that the cloud top detrainment is confined to the conditionally stable portion of the atmospheric column. All condensation is assumed to occur within the updraft plumes, so that $C = C_u$. Each plume is assumed to have the same value for the cloud base mass flux $M_b$, which is specified below. The vertical distribution of the cloud updraft mass flux is given by

$$M_u = M_b \int^{\lambda_D}_0 \frac{1}{\lambda_0} e^{\lambda (z - z_b)}d\lambda = M_b \frac{e^{\lambda_D (z - z_b)} - 1}{\lambda_0 (z - z_b)} ,$$ (4.13)

where $\lambda_0$ is the maximum detrainment rate, which occurs for the plume detraining at height $z_0$, and $\lambda_D$ is the entrainment rate for the updraft that detrains at height $z$. Detrainment is confined to regions where $\lambda_D$ decreases with height, so that the total detrainment $D_u = 0$ for $z < z_0$. Above $z_0$,

$$D_u = - \frac{M_b}{\lambda_0} \frac{\partial \lambda_D}{\partial z} .$$ (4.14)

The total entrainment rate is then just given by the change in mass flux and the total detrainment,

$$E_u = \frac{\partial M_u}{\partial z} - D_u .$$ (4.15)

The updraft budget equations for dry static energy, water vapor mixing ratio, moist static energy, and cloud liquid water, $\ell$, are:

$$\frac{\partial}{\partial z} \left ( M_u S_u \right ) = \left ( E_u - D_u \right ) S + \rho LC_u$$ (4.16)

$$\frac{\partial}{\partial z} \left ( M_u q_u \right ) = E_u q - D_u q^* + \rho C_u$$ (4.17)

$$\frac{\partial}{\partial z} \left ( M_u h_u \right ) = E_u h - D_u h^*$$ (4.18)

$$\frac{\partial}{\partial z} \left ( M_u \ell \right ) = - D_u \ell_d + \rho C_u - \rho R_u ,$$ (4.19)

where (4.18) is formed from (4.16) and (4.17) and detraining air has been assumed to be saturated ($q=q^*$ and $h=h^*$). It is also assumed that the liquid content of the detrained air is the same as the ensemble mean cloud water ( $\ell_d = \ell$). The conversion from cloud water to rain water is given by

$$\rho R_u = c_0 M_u \ell ,$$ (4.20)

following Lord et al. [116], with $c_0 = 2 \times 10^{-3} {\rm m}^{-1}$.

Since $M_u$, $E_u$ and $D_u$ are given by (4.13-4.15), and $h$ and $h^*$ are environmental profiles, (4.18) can be solved for $h_u$, given a lower boundary condition. The lower boundary condition is obtained by adding a $0.5$ K temperature perturbation to the dry (and moist) static energy at cloud base, or $h_u = h + c_p\times 0.5$ at $z=z_b$. Below the lifting condensation level (LCL), $S_u$ and $q_u$ are given by (4.16) and (4.17). Above the LCL, $q_u$ is reduced by condensation and $S_u$ is increased by the latent heat of vaporization. In order to obtain to obtain a saturated updraft at the temperature implied by $S_u$, we define $\Delta T$ as the temperature perturbation in the updraft, then:

$$h_u = S_u + L q_u$$ (4.21)

$$S_u = S + c_p \Delta T$$ (4.22)

$$q_u = q^* + \frac{d q^*}{dT}\Delta T .$$ (4.23)

Substituting (4.22) and (4.23) into (4.21),

$$h_u = S + L q^* + c_p \left(1 + \frac{L}{c_p}\frac{d q^*}{dT} \right)\Delta T$$ (4.24)

$$= h^* + c_p\left(1+\gamma \right)\Delta T$$ (4.25)

$$\gamma \equiv \frac{L}{c_p}\frac{d q^*}{dT}$$ (4.26)

$$\Delta T = \frac{1}{c_p}\frac{h_u - h^*}{1+\gamma}  .$$ (4.27)

The required updraft quantities are then

$$S_u = S + \frac{h_u - h^*}{1+\gamma}$$ (4.28)

$$q_u = q^* + \frac{\gamma}{L} \frac{h_u - h^*}{1+\gamma} .$$ (4.29)

With $S_u$ given by (4.28), (4.16) can be solved for $C_u$, then (4.19) and (4.20) can be solved for $\ell$ and $R_u$.

The expressions above require both the saturation specific humidity to be

$$q^* = \frac{\epsilon e^*}{p-e^*}, \qquad e^* < p ,$$ (4.30)

where $e^*$ is the saturation vapor pressure, and its dependence on temperature (in order to maintain saturation as the temperature varies) to be

$$\frac{d q^*}{d T} = \frac{\epsilon}{p-e^*} \frac{d e^*}{d T} - \frac{\epsilon e^*}{(p-e^*)^2}\frac{d (p-e^*)}{d T}$$ (4.31)

$$= \frac{\epsilon}{p-e^*}\left(1 + \frac{1}{p-e^*}\right) \frac{d e^*}{d T}$$ (4.32)

$$= \frac{\epsilon}{p-e^*}\left(1 + \frac{q^*}{\epsilon e^*}\right) \frac{d e^*}{d T}  .$$ (4.33)

The deep convection scheme does not use the same approximation for the saturation vapor pressure $e^*$ as is used in the rest of the model. Instead,

$$e^* = c_1 \exp\left[\frac{c_2(T - T_f)}{(T-T_f+c_3)} \right] ,$$ (4.34)

where $c_1=6.112$, $c_2=17.67$, $c_3=243.5$ K and $T_f=273.16$ K is the freezing point. For this approximation,

$$\frac{d e^*}{d T} = e^* \frac{d}{dT} \left[\frac{c_2(T - T_f)}{(T-T_f+c_3)} \right]$$ (4.35)

$$= e^* \left[\frac{c_2}{(T-T_f+c_3)} - \frac{c_2(T - T_f)}{(T-T_f+c_3)^2} \right]$$ (4.36)

$$= e^* \frac{c_2 c_3}{(T-T_f+c_3)^2}$$ (4.37)

$$\frac{d q^*}{d T} = q^*\left(1+ \frac{q^*}{\epsilon e^*}\right) \frac{c_2 c_3}{(T-T_f+c_3)^2} .$$ (4.38)

We note that the expression for $\gamma$ in the code gives

$$\frac{d q^*}{d T} = \frac{c_p}{L}\gamma = q^*\left(1+ \frac{q^*}{\epsilon}\right) \frac{\epsilon L}{RT^2} .$$ (4.39)

The expressions for ${d q^*}/{d T}$ in (4.38) and (4.39) are not identical. Also, $T-T_f+c_3 \neq T$ and $c_2 c_3 \neq \epsilon L/R$.

4.1.2 Downdraft Ensemble

Downdrafts are assumed to exist whenever there is precipitation production in the updraft ensemble where the downdrafts start at or below the bottom of the updraft detrainment layer. Detrainment from the downdrafts is confined to the sub-cloud layer, where all downdrafts have the same mass flux at the top of the downdraft region. Accordingly, the ensemble downdraft mass flux takes a similar form to (4.13) but includes a ``proportionality factor'' to ensure that the downdraft strength is physically consistent with precipitation availability. This coefficient takes the form

$$\alpha = \mu \left [ \frac{P}{P + E_d} \right ] ,$$ (4.40)

where $P$ is the total precipitation in the convective layer and $E_d$ is the rain water evaporation required to maintain the downdraft in a saturated state. This formalism ensures that the downdraft mass flux vanishes in the absence of precipitation, and that evaporation cannot exceed some fraction, $\mu$, of the precipitation, where $\mu$ = 0.2.

4.1.3 Closure

The parameterization is closed, i.e., the cloud base mass fluxes are determined, as a function of the rate at which the cumulus consume convective available potential energy (CAPE). Since the large-scale temperature and moisture changes in both the cloud and sub-cloud layer are linearly proportional to the cloud base updraft mass flux (e.g. see eq. 4.2 - 4.5), the CAPE change due to convective activity can be written as

$$\left( \frac{\partial A}{\partial t} \right)_{cu} = -M_b F ,$$ (4.41)

where $F$ is the CAPE consumption rate per unit cloud base mass flux. The closure condition is that the CAPE is consumed at an exponential rate by cumulus convection with characteristic adjustment time scale $\tau = 7200$ s:

$$M_b = \frac{A}{\tau F} .$$ (4.42)

4.1.4 Numerical Approximations

The quantities $M_{u,d}$, $\ell$, $S_{u,d}$, $q_{u,d}$, $h_{u,d}$ are defined on layer interfaces, while $D_u$, $C_u$, $R_u$ are defined on layer midpoints. $S$, $q$, $h$, $\gamma$ are required on both midpoints and interfaces and the interface values $\psi^{k\pm}$ are determined from the midpoint values $\psi^k$ as

$$\psi^{k-} = \log\left(\frac{\psi^{k-1}}{\psi^k}\right) \frac{\psi^{k-1} \psi^k}{\psi^{k-1} - \psi^k} .$$ (4.43)

All of the differencing within the deep convection is in height coordinates. The differences are naturally taken as

$$\frac{\partial \psi}{\partial z} = \frac{\psi^{k-} - \psi^{k+}}{z^{k-} - z^{k+}} ,$$ (4.44)

where $\psi^{k-}$ and $\psi^{k+}$ represent values on the upper and lower interfaces, respectively for layer $k$. The convention elsewhere in this note (and elsewhere in the code) is $\delta^k\psi = \psi^{k+} - \psi^{k-}$. Therefore, we avoid using the compact $\delta^k$ notation, except for height, and define

$$d^kz \equiv z^{k-} - z^{k+} = -\delta^k z ,$$ (4.45)

so that $d^kz$ corresponds to the variable dz(k) in the deep convection code.

Although differences are in height coordinates, the equations are cast in flux form and the tendencies are computed in units $\rm kg m^{-3}\ s^{-1}$. The expected units are recovered at the end by multiplying by $g\delta z/\delta p$.

The environmental profiles at midpoints are

$$S^k = c_p T^k + g z^k$$ (4.46)

$$h^k = S^k + L q^k$$ (4.47)

$$h^{*k} = S^k + L q^{*k}$$ (4.48)

$$q^{*k} = \epsilon e^{*k} / (p^k - e^{*k})$$ (4.49)

$$e^{*k} = c_1 \exp\left[\frac{c_2(T^k - T_f)}{(T^k-T_f+c_3)} \right]$$ (4.50)

$$\gamma^k = q^{*k}\left(1+ \frac{q^{*k}}{\epsilon}\right) \frac{\epsilon L^2}{c_pR{T^k}^2}  .$$ (4.51)

The environmental profiles at interfaces of $S$, $q$, $q^*$, and $\gamma$ are determined using (4.43) if $\vert\psi^{k-1}-\psi^{k}\vert$ is large enough. However, there are inconsistencies in what happens if $\vert\psi^{k-1}-\psi^{k}\vert$ is not large enough. For $S$ and $q$ the condition is

$$\psi^{k-} = (\psi^{k-1}+\psi^k)/2, \quad \frac{\vert\psi^{k-1}-\psi^{k}\vert}{\max(\psi^{k-1}-\psi^{k})} \leq 10^{-6} .$$ (4.52)

For $q^*$ and $\gamma$ the condition is

$$\psi^{k-} = \psi^{k}, \quad \vert\psi^{k-1}-\psi^{k}\vert \leq 10^{-6} .$$ (4.53)

Interface values of $h$ are not needed and interface values of $h^*$ are given by

$$h^{*k-} = S^{k-} + L q^{*k-}  .$$ (4.54)

The unitless updraft mass flux (scaled by the inverse of the cloud base mass flux) is given by differencing (4.13) as

$$M_u^{k-} = \frac{1}{\lambda_0(z^{k-}-z_b)} \left( e^{\lambda_D^k (z^{k-}-z_b)} -1 \right) ,$$ (4.55)

with the boundary condition that $M_u^{M+} =1$. The entrainment and detrainment are calculated using

$$m_u^{k-} = \frac{1}{\lambda_0(z^{k-}-z_b)} \left( e^{\lambda_D^{k+1} (z^{k-}-z_b)} -1 \right)$$ (4.56)

$$E_u^k = \frac{m_u^{k-} - M_u^{k+}}{d^kz}$$ (4.57)

$$D_u^k = \frac{m_u^{k-} - M_u^{k-}}{d^kz}  .$$ (4.58)

Note that $M_u^{k-}$ and $m_u^{k-}$ differ only by the value of $\lambda_D$.

The updraft moist static energy is determined by differencing (4.18)

$$\frac{M_u^{k-}h_u^{k-} - M_u^{k+}h_u^{k+}}{d^kz} = E_u^k h^k - D_u^k h^{*k}$$ (4.59)

$$h_u^{k-} = \frac{1}{M_u^{k-}}\left[M_u^{k+} h_u^{k+} + d^kz\left( E_u^k h^k - D_u^k h^{*k} \right)\right] ,$$ (4.60)

with $h_u^{M-} = h^M + c_p/2$, where $M$ is the layer of maximum $h$.

Once $h_u$ is determined, the lifting condensation level is found by differencing (4.16) and (4.17) similarly to (4.18):

$$S_u^{k-} = \frac{1}{M_u^{k-}}\left[M_u^{k+} S_u^{k+} + d^kz\left( E_u^k S^k - D_u^k S^{k} \right)\right]$$ (4.61)

$$q_u^{k-} = \frac{1}{M_u^{k-}}\left[M_u^{k+} q_u^{k+} + d^kz\left( E_u^k q^k - D_u^k q^{*k} \right)\right] .$$ (4.62)

The detrainment of $S_u$ is given by $D_u^kS^k$ not by $D_u^kS_u^k$, since detrainment occurs at the environmental value of $S$. The detrainment of $q_u$ is given by $D_u^k q^{*k}$, even though the updraft is not yet saturated. The LCL will usually occur below $z_0$, the level at which detrainment begins, but this is not guaranteed.

The lower boundary conditions, $S_u^{M-} = S^M + c_p/2$ and $q_u^{M-} = q^M$, are determined from the first midpoint values in the plume, rather than from the interface values of $S$ and $q$. The solution of (4.61) and (4.62) continues upward until the updraft is saturated according to the condition

$$q_u^{k-} > q^{*}(T_u^{k-}),$$ (4.63)

$$T_u^{k-} = \frac{1}{c_p}\left( S_u^{k-} - gz^{k-}\right)  .$$ (4.64)

The condensation (in units of m$^{-1}$) is determined by a centered differencing of (4.16):

$$\frac{M_u^{k-}S_u^{k-} - M_u^{k+}S_u^{k+}}{d^kz} = (E_u^k - D_u^k) S^k + L C_u^k$$ (4.65)

$$C_u^k = \frac{1}{L} \left[ \frac{M_u^{k-}S_u^{k-} - M_u^{k+}S_u^{k+}}{d^kz} - (E_u^k - D_u^k) S^k \right]  .$$ (4.66)

The rain production (in units of m$^{-1}$) and condensed liquid are then determined by differencing (4.19) as

$$\frac{M_u^{k-}\ell^{k-} - M_u^{k+}\ell^{k+}}{d^kz} = -D_u^k \ell^{k+} + C_u^k - R_u^k ,$$ (4.67)

and (4.20) as

$$R_u^k = c_0 M_u^{k-} \ell^{k-} .$$ (4.68)

Then

$$M_u^{k-}\ell^{k-} = M_u^{k+}\ell^{k+} - d^kz \left( D_u^k \ell^{k+} - C_u^k + c_0 M_u^{k-} \ell^{k-} \right)$$ (4.69)

$$M_u^{k-}\ell^{k-} \left(1 + c_0 d^kz \right) = M_u^{k+}\ell^{k+} + d^kz \left( D_u^k \ell^{k+} - C_u^k \right)$$ (4.70)

$$\ell^{k-} = \frac{1}{M_u^{k-}\left(1 + c_0 d^kz \right)} \left[ M_u^{k+}\ell^{k+} - d^kz \left(D_u^k \ell^{k+} - C_u^k \right) \right]  .$$ (4.71)

4.1.5 Deep Convective Tracer Transport

The CAM 3.0 provides the ability to transport constituents via convection. The method used for constituent transport by deep convection is a modification of the formulation described in Zhang and McFarlane [199].

We assume the updrafts and downdrafts are described by a steady state mass continuity equation for a ``bulk'' updraft or downdraft

$${\partial (M_x q_x) \over \partial p} = E_x q_e - D_x q_x .$$ (4.72)

The subscript $x$ is used to denote the updraft ($u$) or downdraft ($d$) quantity. $M_x$ here is the mass flux in units of Pa/s defined at the layer interfaces, $q_x$ is the mixing ratio of the updraft or downdraft. $q_e$ is the mixing ratio of the quantity in the environment (that part of the grid volume not occupied by the up and downdrafts). $E_x$ and $D_x$ are the entrainment and detrainment rates (units of s$^{-1}$) for the up- and down-drafts. Updrafts are allowed to entrain or detrain in any layer. Downdrafts are assumed to entrain only, and all of the mass is assumed to be deposited into the surface layer.

Equation 4.72 is first solved for up and downdraft mixing ratios $q_u$ and $q_d$, assuming the environmental mixing ratio $q_e$ is the same as the gridbox averaged mixing ratio $\bar q$.

Given the up- and down-draft mixing ratios, the mass continuity equation used to solve for the gridbox averaged mixing ratio $\bar q$ is

$${\partial \bar q \over \partial t} = {\partial \over \partial p} (M_u (q_u-\bar q) + M_d (q_d-\bar q)) .$$ (4.73)

These equations are solved for in subroutine CONVTRAN. There are a few numerical details employed in CONVTRAN that are worth mentioning here as well.

• mixing quantities needed at interfaces are calculated using the geometric mean of the layer mean values.
• simple first order upstream biased finite differences are used to solve 4.72 and 4.73.
• fluxes calculated at the interfaces are constrained so that the resulting mixing ratios are positive definite. This means that this parameterization is not suitable for moving mixing ratios of quantities meant to represent perturbations of a trace constituent about a mean value (in which case the quantity can meaningfully take on positive and negative mixing ratios). The algorithm can be modified in a straightforward fashion to remove this constraint, and provide meaningful transport of perturbation quantities if necessary. the reader is warned however that there are other places in the model code where similar modifications are required because the model assumes that all mixing ratios should be positive definite quantities.