4.2 Shallow/Middle Tropospheric Moist Convection

To characterize the convective forcing associated with shallow and middle-level convection (i.e., convective activity not treated by the primary convective parameterization scheme) we write the large-scale budget equations for dry static energy and total water as

$$\frac{\partial \overline s}{\partial t} = - \nabla \cdot \overline {\boldsymbol {V}} \overline s - \fr... ... \left( \overline{\omega^\prime s^\prime_\ell} \right) + L{\cal R} + c_p Q_R$$

$$= \frac{\partial \overline s}{\partial t} \bigg\vert _{R.S.} - \f... ... {\partial p} \left( \overline{\omega^\prime s^\prime_\ell} \right) + L{\cal R}$$ (4.74)
and

$$\frac{\partial \overline q}{\partial t} = - \nabla \cdot \overline {\boldsymbol {V}} \overline q - \fr... ...rline{\omega^\prime - \left( q^\prime + \ell^\prime \right)} \right) - {\cal R}$$

$$= \frac{\partial \overline q}{\partial t} \bigg\vert _{R.S.} - \f... ...rline{\omega^\prime \left( q^\prime + \ell^\prime \right)} \right) - {\cal R} ,$$ (4.75)

where $s \equiv c_p T + g z$ is the dry static energy; $\ell$ represents liquid water; $s_\ell \equiv s - L\ell$ is the static energy analogue of the liquid water potential temperature introduced by Betts [18]; ${\cal R}$ is the ``convective-scale'' liquid water sink (sometimes denoted by $C - E$); and $Q_R$ is the net radiative heating rate. The subscript $R.S.$ denotes the resolvable-scale contributions to the large-scale budget. Note that variations of the mean liquid water on the large scale have been neglected. The barred quantities represent horizontal averages over an area large enough to contain a collection of cloud elements, but small enough so as to cover only a fraction of a large-scale disturbance. By writing the mean thermodynamic variables in terms of their average cloud and environment properties, and assuming that the convection occupies only a small fraction of the averaging area, the vertical eddy transports $\overline{\omega^\prime s^\prime_\ell}$ and $\overline{\omega^\prime \left( q^\prime + \ell^\prime \right)}$ can be approximated by the difference between the upward flux inside a typical convective element and the downward flux (i.e. induced subsidence) in the environment (cf. Yanai et al. [193]). Mathematically, this approximation takes the form

$$F_{s_\ell}(p) = - \frac{1}{g} \left( \overline{\omega^\prime s^\prime_\ell} \ri... ...ine s \left( p \right) - s_c \left( p \right) + L \ell \left( p \right) \right)$$ (4.76)
and

$$F_{q+\ell} (p) = - \frac{1}{g} \left( \overline{\omega^\prime \left( q^\prime + ... ...ine q \left( p \right) - q_c \left( p \right) - \ell \left( p \right) \right) ,$$ (4.77)

where $M_c$ is a convective mass flux, and $s_c$, $q_c$, and $\ell$ represent cloud-scale properties. Thus, (4.74) and (4.75) can be written as

$$\frac{\partial \overline s}{\partial t} = \frac{\partial \overline s} {\partial t} \bigg\vert _{R.S.} + g \frac{\partial}{\partial p} F_{s_\ell} + L{\cal R} \; ,$$ (4.78)
and

$$\frac{\partial \overline q}{\partial t} \lq = \frac{\partial \overline q} {\partial t} \bigg\vert _{R.S.} + g \frac{\partial}{\partial p} F_{q+\ell} - {\cal R} \; .$$ (4.79)

Let us now turn our attention to a vertically discrete model atmosphere and consider the case where layers $k$ and $k+1$ are moist adiabatically unstable, i.e. a non-entraining parcel of air at level $k+1$ (with moist static energy ${h}_{c }$) would be unstable if raised to level $k$. We assume the existence of a non-entraining convective element with roots in level $k+1$, condensation and rainout processes in level $k$, and limited detrainment in level $k-1$ (see Figure 4.1). In accordance with (4.78) and (4.79), the discrete dry static energy and specific humidity budget equations for these three layers can be written as

$$\hat{\overline{s}}_{k-1} = \overline{s}_{k-1} + \frac{2\Delta t g}{\Delta p_{k-1}} \left\{... ..._{c} \left( s_{c} - \overline{s}_{k-\frac{1}{2}} - L\ell_k \right) \right\} ,$$ (4.80)

$$\hat{\overline{s}}_{k} = \overline{s}_{k} + \frac{2\Delta t g}{\Delta p_{k}} \left\{ {m}... ...t( {s}_{c} - L\ell_k - \overline{s}_{k-\frac{1}{2}} \right) + LR_k \right\} ,$$ (4.81)

$$\hat{\overline{s}}_{k+1} = \overline{s}_{k+1} + \frac{2\Delta t g}{\Delta p_{k+1}} \left\{ {m}_{c} \left(\overline{s}_{k+\frac{1}{2}} - {s}_{c} \right) \right\} ,$$ (4.82)

$$\hat{\overline{q}}_{k-1} = \overline{q}_{k-1} + \frac{2\Delta t g}{\Delta p_{k-1}} \left\{ \beta{}{m}_{c} \left( {q}_{c} - \overline{q}_{k-\frac{1}{2}} \right)\right\} ,$$ (4.83)

$$\hat{\overline{q}}_{k} = \overline{q}_{k} + \frac{2\Delta t g}{\Delta p_{k}} \left\{ {m}... ...m}_{c} \left( {q}_{c} - \overline{q}_{k-\frac{1}{2}} - \right) R_k \right\} ,$$ (4.84)

$$\hat{\overline{q}}_{k+1} = \overline{q}_{k+1} + \frac{2\Delta t g}{\Delta p_{k+1}} \left\{ m_{c} \left( \overline{q}_{k+\frac{1}{2}} - {q}_{c} \right) \right\} ,$$ (4.85)

where the subscript $c$ denotes cloud properties in the ascent region, $m_c$ is a convective mass flux at the bottom of the condensation layer (level $k+\frac{1}{2}$, ``cloud base''), and $\beta{}$ is a yet to be determined ``detrainment parameter'' at level $k-\frac{1}{2}$ that will take a value between zero and one. Note that the convective-scale liquid water sink ${\cal R}$ has been redefined in terms of mass per unit area per unit time (denoted by $R$), and the resolvable-scale components have been dropped for the convenience of the following discussion. In the general case, the thermodynamic properties of the updraft region can be assumed to be equal to their large-scale values in the sub-cloud layer, level $k+1$, plus some arbitrary thermodynamic perturbation; i.e.

$${s}_{c} = \overline{s}_{k+1} + s^{\prime} ,$$ (4.86)

$${q}_{c} = \overline{q}_{k+1} + q^{\prime} ,$$ (4.87)
and

$${h}_{c} = {s}_{c} + L {q}_{c} .$$ (4.88)

In the CAM 3.0 implementation of this scheme, when a sub-cloud layer lies within the diagnosed atmospheric boundary layer, the perturbation quantities $q^{\prime}$ and $s^\prime$ are assumed to be equal to $b\ \frac{(\overline{w^\prime q^\prime})_s}{w_m}$ (e.g. see 4.470 and the atmospheric boundary layer discussion) and zero.

The liquid water generation rate at level $k$ is given by

$$m_{c} \ell_k = m_{c} \left[ q_{c} - (q_c)_{_k} \right] .$$ (4.89)

Using the saturation relation

$$(q_c)_{_k} = \overline{q}^{*}_k + \frac{\gamma_{_k}}{1 + \gamma_{_k}}\; \frac{1}{L} \left({h}_{c} - \overline{h}^{*}_k \right) ,$$ (4.90)

where $\overline{q}^*$ denotes the saturated specific humidity

$${\overline q}^{*} = \epsilon \frac{e_{s}}{p - (1-\epsilon)e_{s}} ,$$ (4.91)

$h^{*}$ denotes the saturated moist state energy, $e_s$ is the saturation vapor pressure (determined from a precomputed table), and $\gamma \equiv ({L}/{c}_{p}) (\partial \overline{q}^{ *} / \partial \overline{T})_p,$ and assuming that the large-scale liquid water divergence in layer $k$ is zero, (4.89) can be manipulated to give the rainout term in layer $k$ as

$$LR_k \equiv L(1 - \beta{}) m_{c}\ell_k = (1 - \beta{}) m_{c} \l... ...1}{1 + \gamma_{_k}} \left( {h}_{c} - \overline{h}^{ *}_k \right) \right\} ,$$ (4.92)

and the liquid water flux into layer $k-1$ as

$$\beta{}m_{c} L\ell_k = \beta{}m_{c} \left\{ \overline{s}_k - {s}_... ...1}{1 + \gamma_{_k}} \left( {h}_{c} - \overline{h}^{ *}_k \right) \right\} .$$ (4.93)

Figure 4.1: Conceptual three-level non-entraining cloud model

Equations (4.82) and (4.85) can be combined to give an equation for moist static energy in layer $k+1$

$$\frac{\partial \overline{h}_{k+1}}{\partial t} = \frac{g}{\Delta ... ...\frac{1}{2}} - {h}_{c} \right) \approx \frac{\partial {h}_{c}}{\partial t} ,$$ (4.94)

where the approximation follows from the assumption that ${\partial h^{\prime} / \partial t}$ can be neglected. Using the relation $(1 + \gamma_{_k}) \frac{\partial \overline{s}_k}{\partial t} = {\partial \overline{h}^{ *}_k}{\partial t}$, (4.81) can be manipulated to give an expression for the time rate of change of saturated moist static energy in layer $k$

$$\frac{\partial \overline{h}^{ *}_k}{\partial t} = \frac{gm_{c}}... ...) - \beta{} \; \left( {s}_{c} - \overline{s}_{k-\frac{1}{2}} \right) \Big\} .$$ (4.95)

Subtracting (4.95) from (4.94) results in

$$\frac{\partial \left({h}_{c} - \overline{h}^{ *}_k \right)}{\partial t} = m_{c} \Big\{ \frac{g}{\Delta p_{k+1}} \left( \overline{h}_{k+\frac{1}{2}} - {h}_{c} \right)$$

$$\phantom{=} - \frac{g}{\Delta p_k} \left(1 + \gamma_{_k} \right) \left[ \left... ...\beta{} \left({s}_{c} - \overline{s}_{k-\frac{1}{2}} \right) \right] \Big\} ,$$ (4.96)

from which the convective mass flux $m_c$ can be written as

$$m_{c} = \frac{h_c - \overline{h}^{ *}_k} {g\tau \left\{ \frac{(... ...{1}{\Delta p_{k+1}} \left[\overline{h}_{k+\frac{1}{2}}- h_{c} \right] \right\}}$$ (4.97)

where $\tau$ is a characteristic convective adjustment time scale.

Physically realistic solutions require that the convective mass flux $m_c$ be positive, implying the following constraint on the detrainment parameter $\beta{}$

$$\beta{} \left( 1 + \gamma_{k} \right) \left(s_{c} - \overline{s}_... ... p_k}{\Delta p_{k+1}}\; \left( \overline{h}_{k+\frac{1}{2}} - h_{c} \right) .$$ (4.98)

A second physical constraint is imposed to ensure that the adjustment process does not supersaturate the ``detrainment layer'', $k-1$, which leads to the following constraint on the detrainment parameter, $\beta{}$:

$$\frac{1}{\Delta p_k} \Big[ \left( 1 + \gamma_{k} \right) \left(s_... ... \frac{1}{\Delta p_{k+1}} \left[ \overline{h}_{k+\frac{1}{2}} - h_{c} \right] >$$

$$\beta{} \Bigg\{ \left( \frac{2\Delta t}{\tau} \right) \frac{h_{c}... ...gg[ \gamma_{k-1} \bigg\{ \overline{s}_{k-\frac{1}{2}} - s_{c} + L\ell_k \bigg\}$$

$$+ h_{c} - \overline{h}_{k-\frac{1}{2}} -s_{c} + \overline{s}_{k-\... ...mma_{k} \right) \left( s_{c} - \overline{s}_{k-\frac{1}{2}} \right) \Bigg\} .$$ (4.99)

A final constraint on the adjustment process attempts to minimize the introduction of $2\Delta \eta$ computational structures in the thermodynamic field by not allowing the procedure to increase the vertical gradient of $h$ when $\frac{\partial h} {\partial p} < 0$ in the upper pair of layers. Mathematically this constraint is formulated by discretizing in time the moist static energy equations in layers $k$ and $k-1$, leading to the following constraint on $\beta$

$$\frac{\overline{h}_k - \overline{h}_{k-1} - G}{(h_c - \overline{h... ...p_{k + \frac{1}{2}}} \left[ \overline{h}_{k+ \frac{1}{2}} - h_c \right] \right)$$

$$+ \frac{1}{\Delta p_k} \left[h_c - \overline{h}_{k + \frac{1}{2}}... ...mma_{k}}{\Delta p_k} \right) \left(s_c - \overline{s}_{k - \frac{1}{2}} \right)$$

$$+ \left( h_c - \overline{h}_{k - \frac{1}{2}} - L\ell_k \right) \left( \frac{1}{\Delta p_k} + \frac{1}{\Delta p_{k + 1}} \right) \bigg\} .$$ (4.100)

where $G$ is an arbitrary vertical difference in the adjusted moist static energy profile (cf. Hack et al. [66]).

The first guess for the detrainment parameter, $\beta{}$, comes from a crude buoyancy argument where

$$\beta = {\rm max} \begin{cases}\beta_{\rm min} \ [1ex] {\rm min}... ...a p_{k-1}} {(h_{c} - \overline{h}^{ *}_k) \Delta p_k} \end{cases} \end{cases}$$ (4.101)

and $\beta_{\rm min}$ is assumed to be 0.10 (i.e., 10% detrainment). Since $\beta{}$ effectively determines the actual autoconversion from cloud water to rainwater, $\beta_{\rm max}$ is determined from a minimum autoconversion requirement which is mathematically written as

$$\beta_{\rm max} = {\rm max} \begin{cases}\beta_{\rm min} \ [1ex] 1 - c_0 ( \delta z - \delta z_{\rm min}) \end{cases}$$ (4.102)

where $c_0$ is a constant autoconversion coefficient assumed to be equal to 1.0$\times$10$^{-4}$ m$^{-1}$, $\delta z$ is the depth of contiguous convective activity (i.e. layers in which condensation and rainout takes place) including and below layer $k$, and $\delta z_{\rm min}$ is a minimum depth for precipitating convection. The physical constraints on the adjustment process are then applied to determine the actual value of $\beta{}$ appropriate to the stabilization of levels $k$ and $k+1$.

In summary, the adjustment procedure is applied as follows. A first guess at $\beta{}$ is determined from (4.101) and (4.102), and further refined using (4.98), (4.99), and (4.100). The convective mass flux, $m_c$, is then determined from (4.97), followed by application of budget equations (4.80)-(4.85) to complete the thermodynamic adjustment in layers $k-1$ through $k+1$. By repeated application of this procedure from the bottom of the model to the top, the thermodynamic structure is locally stabilized, and a vertical profile of the total cloud mass flux associated with shallow and mid-level convection, $M_c$ (where $M_{c_{k+\frac{1}{2}}} = {m}_{{c}_{k + \frac{1}{2}}} + \beta{} {m}_{{c}_{k+\frac{3}{2}}}$) can be constructed. This mass flux profile can also be used to estimate the convective-scale transport of arbitrary passive scalars. The production rate of convective precipitation $R_k$ is supplied to the parameterization of evaporation of convective precipitation described in section 4.3. The free parameters for the convection scheme consist of a minimum convective detrainment, $\beta_{\rm min}$, a characteristic adjustment time scale for the convection, $\tau$, a cloud-water to rain-water autoconversion coefficient $c_0$, and a minimum depth for precipitating convection $\delta z_{\rm min}$.