diff --git a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst index 05a6f1e453..e47ad35ac6 100644 --- a/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst +++ b/documentation/source/science_guide/cloud_schemes/UMDP30_PC2CloudScheme.rst @@ -4,6 +4,25 @@ under which the code may be used. ----------------------------------------------------------------------------- +.. raw:: html + + + +.. role:: blue +.. role:: green +.. role:: purple + +.. role:: blue-lbl(strong) + :class: blue +.. role:: green-lbl(strong) + :class: green +.. role:: purple-lbl(strong) + :class: purple + .. attention:: This documentation has been transferred directly from the UM to LFRic; @@ -14,9 +33,13 @@ The PC2 Cloud Scheme ==================== -:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ :math:`^{1}`, +:Author: D. Wilson, A. Bushell, C. Morcrette, V. Varma\ [#affil1]_, M. Whitall + +.. [#affil1] National Institute of Water and Atmospheric Research, Wellington, + New Zealand + .. role:: raw-latex(raw) :format: latex .. @@ -89,7 +112,7 @@ valid structures to use in this respect.* - One may diagnose cloud fractions and condensate contents from knowledge of gridbox mean variables. This forms the basis of the - `Smith (1990)`_ scheme, which is described in . + `Smith (1990)`_ scheme, which is described in ``:umdp:029``. - A mixed scheme, such as `Sundqvist (1978)`_ uses a prediction of condensate contents, but a diagnostic cloud fraction. @@ -306,8 +329,8 @@ three-dimensional distribution in terms of three separate variables :math:`q_T`, :math:`T_L` and :math:`p`. This is the method used by `Smith (1990)`_, where a symmetric triangular distribution function is used. For further information on the -`Smith (1990)`_ scheme, please refer to . Physics and -dynamics schemes hence only need to provide increments to +`Smith (1990)`_ scheme, please refer to ``:umdp:029``. Physics +and dynamics schemes hence only need to provide increments to :math:`\overline{q_T}` and :math:`\overline{T_L}`, provided that a diagnostic scheme (such as `Smith (1990)`_) is called at some point in the timestep to partition :math:`\overline{q_T}` into @@ -332,46 +355,39 @@ distinct aspects of clouds, which may or may not overlap :numref:`Figure %s The equations for the five prognostic cloud variables can be written schematically: -.. math:: +.. math:: :label: eq:dqcldt_and_dcdt + \begin{aligned} \frac{\partial \overline{q_{cl}}}{\partial t} = \frac{\partial \overline{q_{cl}}}{\partial t} |_{advection} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{convection} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{boundary \, layer} + \frac{\partial \overline{q_{cl}}}{\partial t} |_{precipitation} + ... - -.. math:: - + \\ \frac{\partial \overline{q_{cf}}}{\partial t} = \frac{\partial \overline{q_{cf}}}{\partial t} |_{advection} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{convection} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{boundary \, layer} + \frac{\partial \overline{q_{cf}}}{\partial t} |_{precipitation} + ... - -.. math:: - + \\ \frac{\partial C_l}{\partial t} = \frac{\partial C_l}{\partial t} |_{advection} + \frac{\partial C_l}{\partial t} |_{convection} + \frac{\partial C_l}{\partial t} |_{boundary \, layer} + \frac{\partial C_l}{\partial t} |_{precipitation} + ... - -.. math:: - + \\ \frac{\partial C_i}{\partial t} = \frac{\partial C_i}{\partial t} |_{advection} + \frac{\partial C_i}{\partial t} |_{convection} + \frac{\partial C_i}{\partial t} |_{boundary \, layer} + \frac{\partial C_i}{\partial t} |_{precipitation} + ... - -.. math:: :label: eq:dqcldt_and_dcdt - + \\ \frac{\partial C_t}{\partial t} = \frac{\partial C_t}{\partial t} |_{advection} + \frac{\partial C_t}{\partial t} |_{convection} + \frac{\partial C_t}{\partial t} |_{boundary \, layer} + \frac{\partial C_t}{\partial t} |_{precipitation} + ... , - + \end{aligned} where :math:`\overline{q_{cf}}` is the ice water specific humidity, :math:`C_l` is the liquid cloud *volume* fraction, :math:`C_i` is the @@ -387,9 +403,9 @@ The idea is to parametrize each of the terms in the above equations. This approach removes the diagnostic method, hence it will be critical that we can write expressions for :math:`\frac{\partial \overline{q_{cl}}}{\partial t}` and -:math:`\frac{\partial C_l}{\partial t}` for *each process that alters -:math:`\overline{T}`, :math:`\overline{p}`, :math:`\overline{q}`, or -:math:`\overline{q_{cl}}` in the model* (and similarly for the ice +:math:`\frac{\partial C_l}{\partial t}` for *each process that alters* +:math:`\overline{T}` *,* :math:`\overline{p}` *,* :math:`\overline{q}` *, or* +:math:`\overline{q_{cl}}` *in the model* (and similarly for the ice terms). In doing so, we will not lose sight of underlying PDF approach given by :eq:`eq:int_gs_ds` and :eq:`eq:qclbar=int` since we will still use the @@ -549,7 +565,7 @@ power law near :math:`s=b_s`. G(s) ~ \propto ~ {(-s + b_s)}^n -provided :math:`s`. However, the checking routine (Q-Pos) involves a lot of communication between processors and can significantly increase -the run-time of the model. The option to “Ensure consistent sinks of qcl -and CFL” performs a check at the end of the homogeneous forcing routines +the run-time of the model. The option to "Ensure consistent sinks of qcl +and CFL" performs a check at the end of the homogeneous forcing routines to ensure that we are not trying to remove more condensate than was there to start with. @@ -853,7 +869,7 @@ is altered. This is, perhaps, the simplest method of representing a process that changes the shape of the PDF, and we will, in PC2, apply it to represent mixing of air within a gridbox, although this is a significant approximation of the process. Its application fulfils the -role of the “cloud erosion” term in the `Tiedtke (1993)`_ scheme. +role of the "cloud erosion" term in the `Tiedtke (1993)`_ scheme. By linking the term to the PDF shape we can place this term on a stronger mathematical footing than the simple reduction term parametrized by `Tiedtke (1993)`_. Equivalent arguments enabled @@ -953,7 +969,7 @@ of the process that is occuring. Note we don't need to calculate :math:`b_s` separately, just its *fractional* rate of change. Options for the parameterisation of :math:`\frac{1}{b_s}\frac{\partial b_s}{\partial t}` due to turbulent -“erosion” are described in section :ref:`PC2 erosion `, along with the +"erosion" are described in section :ref:`PC2 erosion `, along with the numerical methods used to integrate the equations. .. _sec_init: @@ -981,7 +997,7 @@ a choice of 2 different diagnostic cloud schemes that can be used for this; either a version of the Smith scheme (see UMDP 029), or the bimodal scheme (see UMDP 039). These two options are described below... -Initiation using a “Smith-like” method +Initiation using a "Smith-like" method ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -1008,8 +1024,8 @@ discussion of an equivalent width in the deposition / sublimation relationship for ice cloud). We still choose to define :math:`b_s` in terms of a critical relative humidity parameter, :math:`RH_{crit}`. Like the -`Smith (1990)`_ scheme (see ), we define the value of -:math:`b_s` as +`Smith (1990)`_ scheme (see ``:umdp:029``), we define the +value of :math:`b_s` as .. math:: :label: eq:bs @@ -1094,8 +1110,9 @@ value :math:`(2-RH_t)` (which is equivalent to the replacing of respectively. We then solve for the initiated cloud fraction :math:`C_l'`, using the -similar methods as described in , except that we allow the solution to -vary with the PDF shape :math:`n`. We first write :math:`Q_N` as +similar methods as described in ``:umdp:029``, except that we allow the +solution to vary with the PDF shape :math:`n`. We first write +:math:`Q_N` as .. math:: :label: eq:qn_def @@ -1204,7 +1221,7 @@ width of the moisture PDF at each point. The positions of the upper and lower truncated bounds of the moisture PDF relative to the saturation threshold are expressed in terms of a normalised :math:`Q_N` = :math:`Q_c` over PDF-width (see equation -:eq:`eq:qn_def`. In entrainment zones, the sum of the two +:eq:`eq:qn_def`). In entrainment zones, the sum of the two Gaussian modes can lead to a highly skewed distribution; hence :math:`Q_N` can have different values for the upper and lower bounds, each normalised by the different widths on either side of the PDF. The @@ -1702,7 +1719,7 @@ Smith scheme with a specified value of :math:`RH_{crit}` while the large :math:`RH` variability associated with these clouds implies much lower values than are typically used. -A profile of “forced cloud fraction”, :math:`C_{forced}`, is +A profile of "forced cloud fraction", :math:`C_{forced}`, is parametrized as linearly varying with height between a cloud-base value, at the lifting condensation level (LCL) from the convection diagnosis parcel ascent, and a cloud-top value of 0.1 at the top of the capping @@ -1713,7 +1730,7 @@ varies linearly between 0.1 and 0.3 for cloud depths between 100 m and layer depth, :math:`z_h` plus the inversion thickness, :math:`\Delta z_i` parametrized following `Beare (2008)`_ as: -.. math:: :label: dz_param +.. math:: :label: dz_param_inv \Delta z_i = 6.3 \, w_m^2 / \int_{z_h}^{z_h+\Delta z_i} b \, dz @@ -1721,7 +1738,7 @@ where :math:`w_m` is the boundary layer velocity scale (:math:`w_m^3 = u_*^3 + 0.25 w_*^3`) and :math:`b` is the parcel buoyancy that is integrated over the depth of the inversion assuming a piece-wise linear variation between grid-levels. Note that the constant -in :eq:`dz_param` is the same as in +in :eq:`dz_param_inv` is the same as in `Beare (2008)`_ because :math:`6.3 = 2.5 * 4^{2/3}` and :math:`w_m^3` differs by a factor of 4. @@ -1743,8 +1760,8 @@ the inversion thickness above. Also, there is an option to treat the calculated forced cumulus cloud fraction and water content as diagnostic quantities passed directly to -the radiation scheme as part of the “convective” cloud, instead of using -them to modify the prognostic “large-scale” cloud variables :math:`C` +the radiation scheme as part of the "convective" cloud, instead of using +them to modify the prognostic "large-scale" cloud variables :math:`C` and :math:`\overline{q_{cl}}`. If this option is used, the convective cloud fraction :math:`CCA` and water content :math:`CCW` output by the convection scheme are updated, by taking the forced cumulus profiles as @@ -1814,13 +1831,13 @@ ice supersaturation :math:`S_i=e_v/e_{sat\;ice}-1`: .. math:: :label: eqn:squires_eqn - \frac{D S_i}{D t} = -b_i B_0 {\cal M}_1 S_i + \frac{D S_i}{D t} = -b_i B_0 {\mathcal{M}}_1 S_i -\left(\frac{\varepsilon}{L^2}\right)^{1/3}(S_i-S_E) + a_i w, -where :math:`{\cal M}_1` is the first moment of ice particle size +where :math:`{\mathcal{M}}_1` is the first moment of ice particle size distribution (PSD), :math:`\varepsilon` is the turbulent dissipation rate, :math:`L` is a prescribed mixing length for the turbulence, -:math:`S_{\rm E}` is the ice supersaturation of the environment +:math:`S_{\mathrm{E}}` is the ice supersaturation of the environment surrounding the cloud and :math:`b_i,B_0` and :math:`a_i` are function of :math:`p` and :math:`T` given by @@ -1837,9 +1854,8 @@ of :math:`p` and :math:`T` given by a_i = \frac{g}{R T}\left( \frac{\epsilon L_s}{c_p T} - 1 \right), - The first term on the right hand side of -Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to +Eq. :eq:`eqn:squires_eqn` is the sink of vapor due to depositional growth of ice crystals, the second term models entrainment (mixing) of environmental air into the cloudy volume and the third term is a source term due to vertical air motions. @@ -1847,18 +1863,18 @@ is a source term due to vertical air motions. `Field et al. (2014)`_ modeled vertical velocity as a white-noise process with autocorrelation function: -.. math:: \overline{w(t)w(s)} = \sigma_w^2 \tau_{\rm d} \delta(t-s), +.. math:: \overline{w(t)w(s)} = \sigma_w^2 \tau_{\mathrm{d}} \delta(t-s), where :math:`\delta` is the Dirac distribution and the intensity of the noise, :math:`\sigma_w^2`, will be called the standard derivation of the vertical velocity fluctuations (due to the white nature of noise, a true expectation value :math:`\overline{w^2}` is not defined) and -:math:`\tau_{\rm d}` a Lagrangian decorrelation time define here by the +:math:`\tau_{\mathrm{d}}` a Lagrangian decorrelation time define here by the relation used by `Rodean (1997)`_: .. math:: :label: eqn:taud - \tau_{\rm d} = \frac{2\sigma_w^2}{\varepsilon C_0}, + \tau_{\mathrm{d}} = \frac{2\sigma_w^2}{\varepsilon C_0}, where :math:`C_0` is a known constant. @@ -1872,15 +1888,14 @@ solution PDF is Gaussian with mean and variance given by: .. math:: :label: eqn:si_avg \overline{S_i} = - S_{\rm E}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 {\cal M}_1 - + \left(\varepsilon/L^2\right)^{1/3} }. + S_{\mathrm{E}}\frac{ \left(\varepsilon/L^2\right)^{1/3} }{ b_i B_0 + {\mathcal{M}}_1 + \left(\varepsilon/L^2\right)^{1/3} }. .. math:: :label: eqn:si_var \overline{S_i^2} = - \frac{a^2_{\rm i} \sigma^2_w \tau_{\rm d}}{ 2\left(b_i B_0 {\cal M}_1 + - \left(\varepsilon/L^2\right)^{1/3}\right)}, - + \frac{a^2_{\mathrm{i}} \sigma^2_w \tau_{\mathrm{d}}}{ 2\left(b_i B_0 + {\mathcal{M}}_1 + \left(\varepsilon/L^2\right)^{1/3}\right)}, Equation :eq:`eqn:si_avg` and :eq:`eqn:si_var` completely specify the PDF, @@ -1895,12 +1910,11 @@ given by .. math:: :label: eqn:cloud_liquid q_{cl}^{sgt} = q_{sat\;ice}\int_{S_{i,wat}}^\infty d S_i (S_i -S_{i,wat}) - F(S_i) , - + F(S_i), where :math:`S_{i,wat} = e_{sat\;wat}/e_{sat\;ice}-1` is the value of ice supersaturation at water saturation. We use the superscription -‘:math:`sgt`’(=‘*s*\ ub\ *g*\ rid *t*\ urbulence’) to indicate that +':math:`sgt`'(='*s*\ ub\ *g*\ rid *t*\ urbulence') to indicate that :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` are values of cloud fraction and water content diagnosed from a parametrization of small-scale turbulent processes. @@ -1913,7 +1927,7 @@ Model implementation and closure relations To implement the model of Section :ref:`Model description ` in the Unified Model, closure relations are needed for the quantities -:math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\rm d}` +:math:`\sigma_w^2`, :math:`\varepsilon`, :math:`L`, :math:`\tau_{\mathrm{d}}` and :math:`S_E`, subject to the constraining relationship given by Eq. :eq:`eqn:taud`. In each model grid box, these parameters specify the subgrid PDF, :math:`F(S_i)`, and from this the liquid cloud @@ -1953,29 +1967,30 @@ can define (see Section :ref:`Other user options ` below), however it should be of order one. -To obtain :math:`\tau_{\rm d}` we impose an eddy size constraint: +To obtain :math:`\tau_{\mathrm{d}}` we impose an eddy size constraint: .. math:: :label: eqn:eddy_size - \tau_{\rm d} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta z}{\sigma_w} + \tau_{\mathrm{d}} = \frac{L}{\sigma_w} = \beta_{mix} \frac{\Delta + z}{\sigma_w} -Eq. :eq:`eqn:taud` then determines the dissipation rate, +Eq. :eq:`eqn:taud` then determines the dissipation rate, :math:`\varepsilon`, that is consistent with the other parameters. The constant :math:`C_0=10` by default, but can be adjusted by the user. The scheme is limited to act only in grid boxes where -:math:`\tau_{\rm d}` is less than a prescribed value, +:math:`\tau_{\mathrm{d}}` is less than a prescribed value, :math:`\tau_{d}^{max}`. The default is -:math:`\tau_d^{max}=1200\;{\rm sec}`, which typically coincides with a +:math:`\tau_d^{max}=1200\;{\mathrm{sec}}`, which typically coincides with a couple of model timesteps. The motivation for this is that a motion that takes longer than a few timestep to decorrelate will be partially -resolved by the dynamics and therefore cannot be considered as ‘subgrid’ +resolved by the dynamics and therefore cannot be considered as 'subgrid' turbulence. Finally, where :math:`T`, :math:`p` and :math:`q` appear in the expressions for :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, these are taken to be the grid box mean values. The first moment of the ice PSD, -:math:`{\cal M}_1`, is found from the parametrization, due to +:math:`{\mathcal{M}}_1`, is found from the parametrization, due to `Field et al. (2005)`_, described in Section 4.1 of UMDP26. .. _sec_sgt_increments: @@ -1998,7 +2013,7 @@ Given :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}`, two options are available for relating these to changes in the model prognostics: Option one: direct increments -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +""""""""""""""""""""""""""""" The values of :math:`C_l^{sgt}` and :math:`q_{cl}^{sgt}` can be added as increments to the model prognostic fields, :math:`C_l` and @@ -2025,7 +2040,6 @@ increments to the model prognostic fields, :math:`C_l` and \left( \Delta C \right)_{sgt} = C_l^{sgt} - where the left hand sides denote the increments to :math:`C_l`, :math:`q_{cl}`, :math:`T` and the total cloud fraction, :math:`C`, due to the subgrid scheme. Some bounds-checking is then applied to ensure @@ -2034,7 +2048,7 @@ scheme does not condense out more liquid than there is available moisture. Option two: PC2 Erosion method -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +"""""""""""""""""""""""""""""" Option one gives a simple method for incrementing the model prognostics, but it gives rise to a potential inconsistency with the PC2 cloud @@ -2058,7 +2072,6 @@ using PC2 Erosion. In this case: \left( \Delta T \right)_{sgt} = \frac{L_v}{c_p} \left( \Delta q_{cl} \right)_{sgt}, - where :math:`q_{cl}` is the liquid cloud amount prior to calling to the turbulent production scheme. The cloud fraction increments are calculated by calling PC2 Erosion with @@ -2080,7 +2093,7 @@ The following variables and logical switches are optional inputs: #. Setting the logical ``l_mixed_phase_t_limit`` to *TRUE* allows the user to use the variable ``mp_t_limit`` to define a temperature limit, :math:`T_{max}`, above which the scheme is not applied. The - default is :math:`T_{max}=0^\circ\;{\rm C}`, so the scheme is only + default is :math:`T_{max}=0^\circ\;{\mathrm{C}}`, so the scheme is only applied to cold clouds. #. The input variable ``mp_tau_d_lim`` defines the upper limit, @@ -2153,7 +2166,8 @@ Large-scale precipitation Precipitation processes have a large effect on cloud fractions. Here we present the simple physical models that are applied to the transfer terms included in the large-scale precipitation scheme. They are also -presented within the large-scale precipitation documentation (). +presented within the large-scale precipitation documentation +(``:umdp:026``). The basis of the physical model is that microphysical transfer processes can be calculated separately in different partitions of the model cloud @@ -2163,10 +2177,10 @@ consider here separately each process that is modelled in the large-scale precipitation scheme. The changes in :math:`\overline{q_{cl}}`, :math:`\overline{q_{cf}}` and :math:`\overline{q}` remain mathematically the same as in the non-PC2 -version of the code (), we only need to introduce calculations for the -changes in cloud fractions. We will see that many of these changes can -be well modelled by assuming no change to the cloud fractions, and the -others by using simple assumptions. +version of the code (``:umdp:026``), we only need to introduce +calculations for the changes in cloud fractions. We will see that many +of these changes can be well modelled by assuming no change to the cloud +fractions, and the others by using simple assumptions. Although the model may use two ice prognostic ice categories, only a single ice cloud fraction is stored, the assumption being that the two @@ -2237,9 +2251,9 @@ where :math:`A_{clear}` is the proportion of the gridbox that has neither ice nor liquid cloud present. **An inconsistency has been found in the way that the fall-of-ice term -is linked to the globally constant “wind-shear value” when calculting +is linked to the globally constant "wind-shear value" when calculting the ice cloud fraction overhang. Consequently, the option not to use the -“wind shear value” when calculating the overhang is available in the +"wind shear value" when calculating the overhang is available in the UMUI (from version 7.6 onwards).** .. _sec_lsp_homo: @@ -2251,18 +2265,15 @@ This will freeze all supercooled liquid water when a temperature threshold is exceeded. Hence we turn all existing liquid and mixed phase cloud to ice cloud. The cloud fraction changes are: -.. math:: +.. math:: :label: eq:lsp_homo + \begin{aligned} C_l \leftarrow 0 - -.. math:: - + \\ C_i \leftarrow C_t - -.. math:: :label: eq:lsp_homo - + \\ \Delta C_t = 0. - + \end{aligned} Heterogeneous nucleation ^^^^^^^^^^^^^^^^^^^^^^^^ @@ -2272,18 +2283,15 @@ regardless of the previous presence of ice cloud. This will mean that previously existing 'liquid-only' cloud is converted to mixed phase cloud. These give the following changes: -.. math:: +.. math:: :label: eq:lsp_het + \begin{aligned} \Delta C_l = 0 - -.. math:: - + \\ C_i \leftarrow C_t - -.. math:: :label: eq:lsp_het - + \\ \Delta C_t = 0. - + \end{aligned} .. _sec_mp_depsub: @@ -2300,17 +2308,19 @@ framework (section :ref:`The 's' distribution `). However, since the same instantaneous condensation framework. It would be useful to investigate in the future whether the two descriptions of the moisture variability could be brought together. Because of its importance, we -describe the method below, although we note it is also described in . +describe the method below, although we note it is also described in +``:umdp:026``. We can calculate the local rate of change of :math:`q_{cf}`, given local :math:`T` and :math:`q` etc. using the standard microphysical growth -equations (see ). However, it is critical to know the way in which the -moisture is correlated with the ice in the gridbox. We will assume there -exists a distribution of vapour in the gridbox. We know that the regions -where liquid cloud exists must be saturated with respect to liquid -water, hence we need only consider the part of the gridbox that does not -have liquid water present. The average value, :math:`q_a`, of :math:`q` -within the liquid-free part of the gridbox is thus +equations (see ``:umdp:026``). However, it is critical to know the way in +which the moisture is correlated with the ice in the gridbox. We will +assume there exists a distribution of vapour in the gridbox. We know +that the regions where liquid cloud exists must be saturated with +respect to liquid water, hence we need only consider the part of the +gridbox that does not have liquid water present. The average value, +:math:`q_a`, of :math:`q` within the liquid-free part of the gridbox is +thus .. math:: :label: eq:qa @@ -2350,7 +2360,7 @@ cloud formulation, which considers an underlying PDF across the whole gridbox and does not have, in general, its width prescribed. Remember that we do not calculate on-line the whole of the liquid - vapour PDF, we only parametrize the single point :math:`G(-Qc)`, using equation -:eq:`eqn22`. +:eq:`eqn22`). The width is then limited further to be no greater than :math:`\overline{q}`, to make sure that there are no negative values of @@ -2373,7 +2383,6 @@ same assumption). Some algebra retrieves the expressions: q_{ice} = \frac {\overline{q} - C_l q_{sat~liq} - A_{clear} q_{clear} } {A_{ice}}, - where :math:`A_{ice}` is the proportion of the gridbox with ice cloud but not liquid cloud and :math:`A_{clear}` is the proportion of the gridbox without cloud. The numerical application will set @@ -2560,7 +2569,7 @@ use the values at the start of the microphysics (this includes the values of :math:`C_i` used in the calculation of 'in-cloud' water contents above. However, we do update the cloud fractions themselves sequentially. We also recalculate after each process the overlaps -between the rain fraction (see ) and the cloud fractions. +between the rain fraction (see ``:umdp:026``) and the cloud fractions. There is also a final set of checks that :math:`C_l` and :math:`C_i` lie between 0 and 1 and that :math:`C_t` is bounded between @@ -2624,7 +2633,7 @@ is close to :math:`1 \times 10^{-4} s^{-1}`. Note: the source-code for this erosion method (**pc2_hom_conv**, **pc2_homog_plus_turb**, **pc2_delta_hom_turb**) includes an additional -term “dbsdtbs1” which scales with the rate of homogeneous forcing +term "dbsdtbs1" which scales with the rate of homogeneous forcing :math:`\frac{\partial Q_c}{\partial t}`. However this term is always set to zero on input to these routines so is never used. @@ -2665,7 +2674,7 @@ discretize eq Similarly to :eq:`eq:c_l^n+1`, we then limit the cloud fraction to 0 and 1 and then apply a mid-point value of :math:`C_l` to calculate the change in :math:`\overline{q_{cl}}` (discretizing eq -:eq:`eq:dqcldt_width`: +:eq:`eq:dqcldt_width`): .. math:: :label: eq:dqcl_turb_final @@ -2678,7 +2687,7 @@ that no more :math:`\overline{q_{cl}}` is removed than the model has available. This was chosen to ensure that the erosion process itself contains this physical limit, not a numerical tidying-up process. -The option “l_fixbug_pc2_qcl_incr” ensures that qcl is set to zero if +The option "l_fixbug_pc2_qcl_incr" ensures that qcl is set to zero if the CFL has reached zero. Cloud-surface-area hybrid erosion method @@ -2688,7 +2697,7 @@ Cloud-surface-area hybrid erosion method `Morcrette and Petch (2010)`_ showed that changes to the erosion parameter (:math:`\Upsilon` in Eqn. -:eq:`eq:dbsbydtbs_turb` did not have as +:eq:`eq:dbsbydtbs_turb`) did not have as significant an impact on the global work done by the erosion process as might be expected. This was due to a feedback process whereby, reducing the erosion parameter leads to more cloud water, more autoconversion of @@ -2797,7 +2806,7 @@ Where the change in liquid water content This combination of a Tiedkte sink term for :math:`q_{cl}`, a PC2 term for :math:`C_l` and the introduction of some surface area dependence -leads to this formulation being referred to as a “hybrid” +leads to this formulation being referred to as a "hybrid" cloud-surface-area erosion method. .. _sec_erosion_numerics: @@ -2827,7 +2836,7 @@ simulations), this discretization can suffer severe numerical overshoot. i.e. the increment based on :math:`C_l^n` is large enough to reduce :math:`q_{cl}` (and hence also :math:`C_l`) to less than zero within a single timestep. If the continuous equation were solved analytically -this wouldn’t happen; as :math:`C_l` declines due to the erosion, so +this wouldn't happen; as :math:`C_l` declines due to the erosion, so will :math:`A(C_l)` and hence the erosion rate, so that :math:`q_{cl}` and :math:`C_l` smoothly decline towards zero. @@ -2838,11 +2847,12 @@ yield much less timestep sensitivity for detrained cloud in shallow cumulus regimes. #. **Retain the explicit discretization, but limit the resulting erosion - increments to ensure :math:`q_{cl}` and :math:`C_l` don’t go + increments to ensure** :math:`\boldsymbol{q_{cl}}` **and** + :math:`\boldsymbol{C_l}` **don't go negative. (i_pc2_erosion_numerics=1)** Also, to ensure that some cloud remains at end-of-timestep where shallow cumulus is detraining into dry environments, the erosion calculation is fed copies of the - fields with the current timestep’s convection increments subtracted + fields with the current timestep's convection increments subtracted off. This means any cloud detrained by convection during the current timestep cannot be eroded until the following timestep, and so is still present at end-of-timestep. As discussed in section @@ -2869,7 +2879,8 @@ cumulus regimes. erosion, to give some improvement in accuracy. #. **Use an approximate implicit discretisation, which intrinsically - yields a positive solution for :math:`q_{cl}` and :math:`C_l`. + yields a positive solution for** :math:`\boldsymbol{q_{cl}}` **and** + :math:`\boldsymbol{C_l}` **. (i_pc2_erosion_numerics=2)** The copies of the fields passed to the erosion calculation are fully updated with the convection increments. We then write equation :eq:`eq:dqcldt_hybrid` in @@ -2946,7 +2957,7 @@ cumulus regimes. :eq:`eq:dcdt_hybrid_discr`, except that the term :math:`\frac{1}{2} \Delta \overline{{q_{cl}}_{ero}}` is omitted (interpolating to the mid-point value of :math:`\overline{q_{cl}}` in - the denominator would be “double-counting” if we are already making + the denominator would be "double-counting" if we are already making an implicit correction to the full increment). In the case where the homogeneous forcing increments have already @@ -3005,7 +3016,7 @@ cumulus regimes. each case. In the code, we first test the value of :math:`Q_c` and compute the erosion increments as follows: - #. **Grid-mean subsaturation (:math:`Q_c < 0`):** + #. **Grid-mean subsaturation (** :math:`\boldsymbol{Q_c < 0}` **):** The relation between the erosion tendencies in liquid-cloud-fraction and liquid water content @@ -3025,9 +3036,9 @@ cumulus regimes. defined the PDF height at the saturation boundary when near the cloudy end of the PDF as :math:`G(-Q_c) = \frac{n+1}{n+2} \frac{C_l^2}{q_{cl}}` (eq - :eq:`eqn20`. In fact, :math:`G(-Q_c)` is set to some + :eq:`eqn20`). In fact, :math:`G(-Q_c)` is set to some blend between this and the value near the clear end of the PDF (eq - :eq:`eqn21`. But we will assume that when eroding cloud + :eq:`eqn21`). But we will assume that when eroding cloud under grid-mean subsaturated conditions (:math:`Q_c < 0`), :math:`G(-Q_c)` follows this scaling with :math:`\frac{C_l^2}{q_{cl}}` even if its value differs somewhat @@ -3082,8 +3093,8 @@ cumulus regimes. where :math:`SD` is the saturation defecit, and :math:`a_L` is the dimensionless factor defined in eq :eq:`eq:a_L`. Following the derivation in section - :ref:`“Smooth” initiation logic ` (eq - :eq:`eq:qc_plus_sd`, we can write this in terms + :ref:`"Smooth" initiation logic ` (eq + :eq:`eq:qc_plus_sd`), we can write this in terms of the liquid-water content: :math:`SD = q_{cl} - Q_c` (where :math:`Q_c` was defined in eq :eq:`eq:qc_eq_qt-qs`, and corresponds to the @@ -3146,11 +3157,11 @@ cumulus regimes. find the solution with the explicitly-treated terms (the exponent :math:`b_1 = \frac{ c_1 }{ 1 - \frac{q_{cl}}{C_l Q_c} }` and the terms :math:`(1 - C_l)` and :math:`(q_{cl}-Q_c)` in eq - :eq:`eq:qcl_int_hybrid` adjusted to values + :eq:`eq:qcl_int_hybrid`) adjusted to values linearly-interpolated to half-way between the start and end of the erosion timestep. - #. **Grid-mean supersaturation (:math:`Q_c > 0`):** + #. **Grid-mean supersaturation (** :math:`\boldsymbol{Q_c > 0}` **):** In this case, erosion does not act to reduce :math:`C_l` and :math:`q_{cl}` towards zero. Instead, the narrow PDF limit it @@ -3219,14 +3230,14 @@ cumulus regimes. iterations are then performed to find the solution with the explicitly-treated terms (the exponent :math:`b_2 = \frac{ c_2 }{ 1 + \frac{SD}{(1-C_l) Q_c} }` and the - term :math:`C_l` in eq :eq:`eq:sd_int_hybrid` + term :math:`C_l` in eq :eq:`eq:sd_int_hybrid`) adjusted to values linearly-interpolated to half-way between the start and end of the erosion timestep. Then the final values of :math:`SD` and :math:`1-C_l` are used to increment :math:`q_{cl} = Q_c + SD` and :math:`C_l`, as prognosed by the rest of the model. - #. **grid-mean saturation (:math:`Q_c` near-zero):** + #. **grid-mean saturation (** :math:`\boldsymbol{Q_c}` **near-zero):** In this case, the PDF is centred on the saturation boundary, so that narrowing it does not change the cloud-fraction. In the limit @@ -3322,7 +3333,7 @@ denominator is small, we will, to avoid numerical problems, set that we do not use the multiple phases injection source expressions (section :ref:`Multiple phases in the injection source ` and equation -:eq:`eq:cff_ts`. This is because the liquid water changes +:eq:`eq:cff_ts`). This is because the liquid water changes are not associated with the plume model. Equation :eq:`eq:qcf_ci` assumes that the change to the ice @@ -3436,48 +3447,51 @@ processes (e.g. total water content). In this case, .. math:: :label: eq:chibasic - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + {\chi}_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} To parametrize :eq:`eq:chibasic`, the current UM convection scheme takes a mass flux approximation .. math:: :label: eq:massflux - \left({\overline{\rho w^{'} {\chi}_{\rm{ }}^{\rm{E'}}}} \right)_{\rm{conv}} - = M^{\rm{P}} \, - \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) + \left({\overline{\rho w^{'} {\chi}_{\mathrm{ }}^{\mathrm{E'}}}} + \right)_{\mathrm{conv}} = M^{\mathrm{P}} \, + \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - {\chi}_{\mathrm{ }}^{\mathrm{E}} + } \right) which can be differentiated to give .. math:: :label: eq:eddyflux - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - {\chi}_{\rm{ }}^{\rm{E'}}}}{\partial \, z} = - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - - {\chi}_{\rm{ }}^{\rm{E}} \, \frac{\partial \, M^{\rm{P}}}{\partial \, p} - - - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} + {\chi}_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} = + \frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{P}} \, + M^{\mathrm{P}}}{\partial \, p} - + {\chi}_{\mathrm{ }}^{\mathrm{E}} \, \frac{\partial \, + M^{\mathrm{P}}}{\partial \, p} - + M^{\mathrm{P}} \, \frac{\partial \, {\chi}_{\mathrm{ + }}^{\mathrm{E}}}{\partial \, p} The bulk cloud model plume equations for mass and :math:`{\chi}` are: .. math:: :label: eq:dbydpmassflux - - \frac{\partial \, M^{\rm{P}}}{\partial \, p} = - \left({ \varepsilon \, M^{\rm{P}} - \mu \, M^{\rm{P}} - \delta \, M^{\rm{P}} - } \right) + - \frac{\partial \, M^{\mathrm{P}}}{\partial \, p} = + \left({ \varepsilon \, M^{\mathrm{P}} - \mu \, M^{\mathrm{P}} - \delta \, + M^{\mathrm{P}} } \right) .. math:: :label: eq:dbydpmfchi - - \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}} \, M^{\rm{P}}}{\partial \, p} - = \left({ - \varepsilon \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{E}} - - \mu \, M^{\rm{P}} \, {\chi}_{\rm{ }}^{\rm{R}} - \delta \, M^{\rm{P}} \, - {\chi}_{\rm{ }}^{\rm{P}} + - \frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{P}} \, + M^{\mathrm{P}}}{\partial \, p} = \left({ + \varepsilon \, M^{\mathrm{P}} \, {\chi}_{\mathrm{ }}^{\mathrm{E}} + - \mu \, M^{\mathrm{P}} \, {\chi}_{\mathrm{ }}^{\mathrm{R}} - \delta \, + M^{\mathrm{P}} \, {\chi}_{\mathrm{ }}^{\mathrm{P}} } \right) - Equations :eq:`eq:eddyflux`, :eq:`eq:dbydpmassflux` and :eq:`eq:dbydpmfchi` can then be substituted into @@ -3485,22 +3499,25 @@ Equations :eq:`eq:eddyflux`, .. math:: :label: eq:chimassflux - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = - - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, p} - + \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{R}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) - + \delta \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ }}^{\rm{E}} } \right) + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = + - M^{\mathrm{P}} \, \frac{\partial \, {\chi}_{\mathrm{ + }}^{\mathrm{E}}}{\partial \, p} + + \mu \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{R}} - {\chi}_{\mathrm{ }}^{\mathrm{E}} } \right) + + \delta \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - {\chi}_{\mathrm{ }}^{\mathrm{E}} } \right) -while :math:`{\chi}_{\rm{}}^{\rm{P}}` is obtained from the vertical +while :math:`{\chi}_{\mathrm{}}^{\mathrm{P}}` is obtained from the vertical gradient derived by combining :eq:`eq:dbydpmassflux` and :eq:`eq:dbydpmfchi` : .. math:: :label: eq:gradchipar - M^{\rm{P}} \, \frac{\partial \, {\chi}_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ - }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ {\chi}_{\rm{ }}^{\rm{P}} - {\chi}_{\rm{ - }}^{\rm{R}} } \right) + M^{\mathrm{P}} \, \frac{\partial \, {\chi}_{\mathrm{ + }}^{\mathrm{P}}}{\partial \, p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - + {\chi}_{\mathrm{ }}^{\mathrm{E}} } \right)- + \mu \, M^{\mathrm{P}} \, \left({ {\chi}_{\mathrm{ }}^{\mathrm{P}} - + {\chi}_{\mathrm{ }}^{\mathrm{R}} } \right) Within the model, eqn :eq:`eq:chimassflux` would take a discretized form which actually depends upon whether the model level, @@ -3512,28 +3529,29 @@ discretized form of :eq:`eq:chimassflux`, setting .. math:: :label: eq:chidisck - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, - k}} = m_{\rm{k+1/2}} \, - \frac{ \left({{\chi}_{\rm{k+1}}^{\rm{E}} - {\chi}_{\rm{k}}^{\rm{E}}} \right)} - {{\Delta z}_{\rm{k \, \rightarrow \, k+1}}} - + {\delta}_{\rm{k}} \, m_{\rm{k}} \, \left({ {\chi}_{\rm{k}}^{\rm{P}} - {\chi}_{\rm{k}}^{\rm{E}} } \right) - \qquad \ldots \; \mbox{for k $>$ cb} + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv, \, k}} = m_{\mathrm{k+1/2}} \, + \frac{ \left({{\chi}_{\mathrm{k+1}}^{\mathrm{E}} - + {\chi}_{\mathrm{k}}^{\mathrm{E}}} \right)} + {{\Delta z}_{\mathrm{k \, \rightarrow \, k+1}}} + + {\delta}_{\mathrm{k}} \, m_{\mathrm{k}} \, \left({ {\chi}_{\mathrm{k}}^{\mathrm{P}} - {\chi}_{\mathrm{k}}^{\mathrm{E}} } \right) + \qquad \ldots \; \mathrm{for k $>$ cb} .. math:: :label: eq:chidisccb - {\frac{\partial \, {\chi}_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv, \, - cb}} = m_{\rm{cb+1/2}} \, - \frac{ \left({{\chi}_{\rm{cb+1}}^{\rm{E}} - {\chi}_{\rm{cb}}^{\rm{E}}} - \right)} - {{\Delta z}_{\rm{cb \, \rightarrow \, cb+1}}} - - m_{\rm{cb}} \, - \left({ {\chi}_{\rm{i,cb}}^{\rm{P}} - {\chi}_{\rm{cb}}^{\rm{E}} } \right) - + {\frac{\partial \, {\chi}_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv, \, cb}} = m_{\mathrm{cb+1/2}} \, + \frac{ \left({{\chi}_{\mathrm{cb+1}}^{\mathrm{E}} - + {\chi}_{\mathrm{cb}}^{\mathrm{E}}} \right)} + {{\Delta z}_{\mathrm{cb \, \rightarrow \, cb+1}}} + - m_{\mathrm{cb}} \, + \left({ {\chi}_{\mathrm{i,cb}}^{\mathrm{P}} - + {\chi}_{\mathrm{cb}}^{\mathrm{E}} } \right) -where the initial parcel value :math:`{\chi}_{\rm{i,cb}}^{\rm{P}}` may +where the initial parcel value :math:`{\chi}_{\mathrm{i,cb}}^{\mathrm{P}}` may be chosen to produce a fixed increment or place a closure condition on -the cloud base flux. In fact, the convection equations (see ) differ -from :eq:`eq:chidisck` and +the cloud base flux. In fact, the convection equations (see ``:umdp:027``) +differ from :eq:`eq:chidisck` and :eq:`eq:chidisccb` because a different discretization is used, but the principle is unaltered. @@ -3541,27 +3559,26 @@ The model convection variables are NOT conserved under moist adiabatic processes because precipitation processes deplete the column moisture and condensation processes affect the temperature, specific humidity and cloud condensate variables. Surprisingly, however, the form of -eqn :eq:`eq:chimassflux` is retained even though the +eqn :eq:`eq:chimassflux` is retained even though the basic equation :eq:`eq:chibasic` acquires additional terms for temperature and specific humidity: .. math:: :label: eq:defineq1 - {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 - \equiv - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} + {\frac{\partial \, T_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q1 \equiv + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\mathrm{par}} - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - T_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + T_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} .. math:: :label: eq:defineq2 - {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 - \equiv - {\overline{Q}}_{\rm{par}} + {\frac{\partial \, q_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q2 \equiv - {\overline{Q}}_{\mathrm{par}} - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - q_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + q_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} - -where :math:`{\overline{Q}}_{\rm{par}}` is the rate of condensation +where :math:`{\overline{Q}}_{\mathrm{par}}` is the rate of condensation which occurs in the ascending plumes. The reason that :eq:`eq:defineq1` and @@ -3574,63 +3591,66 @@ gradient equations based upon :eq:`eq:gradchipar` .. math:: :label: eq:gradtpar - M^{\rm{P}} \, \frac{\partial \, T_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ - }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ T_{\rm{ }}^{\rm{P}} - T_{\rm{ - }}^{\rm{R}} } \right)- - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{par}} + M^{\mathrm{P}} \, \frac{\partial \, T_{\mathrm{ }}^{\mathrm{P}}}{\partial \, + p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ T_{\mathrm{ }}^{\mathrm{P}} - + T_{\mathrm{ }}^{\mathrm{E}} } \right)- + \mu \, M^{\mathrm{P}} \, \left({ T_{\mathrm{ }}^{\mathrm{P}} - + T_{\mathrm{ }}^{\mathrm{R}} } \right)- + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\mathrm{par}} .. math:: :label: eq:gradqpar - M^{\rm{P}} \, \frac{\partial \, q_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ - }}^{\rm{E}} } \right)- - \mu \, M^{\rm{P}} \, \left({ q_{\rm{ }}^{\rm{P}} - q_{\rm{ - }}^{\rm{R}} } \right)+ - {\overline{Q}}_{\rm{par}} + M^{\mathrm{P}} \, \frac{\partial \, q_{\mathrm{ }}^{\mathrm{P}}}{\partial \, + p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ q_{\mathrm{ }}^{\mathrm{P}} - + q_{\mathrm{ }}^{\mathrm{E}} } \right)- + \mu \, M^{\mathrm{P}} \, \left({ q_{\mathrm{ }}^{\mathrm{P}} - + q_{\mathrm{ }}^{\mathrm{R}} } \right)+ + {\overline{Q}}_{\mathrm{par}} .. math:: :label: eq:gradlpar - M^{\rm{P}} \, \frac{\partial \, l_{\rm{ }}^{\rm{P}}}{\partial \, p} = - \varepsilon \, M^{\rm{P}} \, \left({ l_{\rm{ }}^{\rm{P}} - l_{\rm{ - }}^{\rm{E}} } \right) - - {\overline{Q}}_{\rm{par}} + PPN - + M^{\mathrm{P}} \, \frac{\partial \, l_{\mathrm{ }}^{\mathrm{P}}}{\partial \, + p} = + \varepsilon \, M^{\mathrm{P}} \, \left({ l_{\mathrm{ }}^{\mathrm{P}} - + l_{\mathrm{ }}^{\mathrm{E}} } \right) + - {\overline{Q}}_{\mathrm{par}} + PPN -The final calculation of rates in the current condensation scheme (, -section 10) assumes a further condensation term, -:math:`{\overline{Q}}_{\rm{reset}}`, which acts to make the net rate of +The final calculation of rates in the current condensation scheme +(``:umdp:027``, section 10) assumes a further condensation term, +:math:`{\overline{Q}}_{\mathrm{reset}}`, which acts to make the net rate of change of condensate equal zero, and a final assumption is made that the environment values of condensate remain zero (and also that -:math:`l_{\rm{ }}^{\rm{R}}` = :math:`l_{\rm{ }}^{\rm{P}}`). The result +:math:`l_{\mathrm{ }}^{\mathrm{R}}` = :math:`l_{\mathrm{ }}^{\mathrm{P}}`). The +result is basic equations .. math:: :label: eq:basictold - {\frac{\partial \, T_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q1 - - \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\rm{reset}} + {\frac{\partial \, T_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q1 - + \left({ \frac{L}{c_{P}} } \right)\, {\overline{Q}}_{\mathrm{reset}} .. math:: :label: eq:basicqold - {\frac{\partial \, q_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} = Q2 + - {\overline{Q}}_{\rm{reset}} - -.. math:: - - 0 \equiv {\frac{\partial \, l_{\rm{ }}^{\rm{E}}}{\partial \, t}}_{\rm{conv}} - = {\overline{Q}}_{\rm{par}} - - {\overline{Q}}_{\rm{reset}} - PPN - - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{ }}^{\rm{E'}}}}{\partial \, z} + {\frac{\partial \, q_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} = Q2 + {\overline{Q}}_{\mathrm{reset}} .. math:: :label: eq:basiclold - = - \mu \, M^{\rm{P}} \, l_{\rm{ }}^{\rm{P}} + \delta \, M^{\rm{P}} \, l_{\rm{ - }}^{\rm{P}} - - {\overline{Q}}_{\rm{reset}} - + \begin{aligned} + 0 \equiv {\frac{\partial \, l_{\mathrm{ }}^{\mathrm{E}}}{\partial \, + t}}_{\mathrm{conv}} & = & {\overline{Q}}_{\mathrm{par}} - + {\overline{Q}}_{\mathrm{reset}} - PPN + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} + l_{\mathrm{ }}^{\mathrm{E'}}}}{\partial \, z} + \\ + & = & + \mu \, M^{\mathrm{P}} \, l_{\mathrm{ }}^{\mathrm{P}} + \delta \, + M^{\mathrm{P}} \, l_{\mathrm{ }}^{\mathrm{P}} - + {\overline{Q}}_{\mathrm{reset}} + \end{aligned} By analogy with equations :eq:`eq:defineq1` and :eq:`eq:defineq2`, we can define a :math:`Q4` from @@ -3652,30 +3672,31 @@ Define .. math:: :label: eq:defineq4l - \left({ \frac{\partial \, l_{\rm{l}}^{\rm{ }}}{\partial \, t} } - \right)_{\rm{conv}} = Q4_{\rm{l}} \equiv - {\overline{Q}}_{\rm{l, par}} - {\overline{Q}}_{\rm{l, reset}} - RAIN - + \left({ \frac{\partial \, l_{\mathrm{l}}^{\mathrm{ }}}{\partial \, t} } + \right)_{\mathrm{conv}} = Q4_{\mathrm{l}} \equiv + {\overline{Q}}_{\mathrm{l, par}} - {\overline{Q}}_{\mathrm{l, reset}} - RAIN + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{l}}^{\rm{'}}}}{\partial \, z} + l_{\mathrm{l}}^{\mathrm{'}}}}{\partial \, z} .. math:: :label: eq:defineq4f - \left({ \frac{\partial \, l_{\rm{f}}^{\rm{ }}}{\partial \, t} } - \right)_{\rm{conv}} = Q4_{\rm{f}} \equiv - {\overline{Q}}_{\rm{f, par}} - {\overline{Q}}_{\rm{f, reset}} - SNOW - + \left({ \frac{\partial \, l_{\mathrm{f}}^{\mathrm{ }}}{\partial \, t} } + \right)_{\mathrm{conv}} = Q4_{\mathrm{f}} \equiv + {\overline{Q}}_{\mathrm{f, par}} - {\overline{Q}}_{\mathrm{f, reset}} - SNOW + - \frac{1}{\overline{\rho}} \, \frac{\partial \, \overline{\rho w^{'} - l_{\rm{f}}^{\rm{'}}}}{\partial \, z} - + l_{\mathrm{f}}^{\mathrm{'}}}}{\partial \, z} where the PC2 assumption thus far has been that -:math:`{\overline{Q}}_{\rm{l, reset}} = 0 -= {\overline{Q}}_{\rm{f, reset}}`. +:math:`{\overline{Q}}_{\mathrm{l, reset}} = 0 += {\overline{Q}}_{\mathrm{f, reset}}`. - The current convection scheme assumes that parcel condensate is single phase (ie. either all liquid or all frozen) and this is seriously hard-wired into the code. Thus we can treat the precipitation and - parcel condensation processes in :math:`Q4_{\rm{l}}` and - :math:`Q4_{\rm{f}}` separately without worrying about cross-transfer + parcel condensation processes in :math:`Q4_{\mathrm{l}}` and + :math:`Q4_{\mathrm{f}}` separately without worrying about cross-transfer between the two because at most only one set will ever be active in a given grid box at one time. However, even for the inactive (zero parcel condensate) phase, convection mixes environmental air into the @@ -3689,70 +3710,73 @@ condensate is calculated as .. math:: :label: eq:vertparl - \frac{\partial \, l_{\rm{l}}^{\rm{P}}}{\partial \, p} = \varepsilon \, - \left({ l_{\rm{l}}^{\rm{P}} - l_{\rm{l}}^{\rm{E}} } \right)- - \frac{{\overline{Q}}_{\rm{l, par}}}{M^{\rm{P}}} - - \frac{RAIN}{M^{\rm{P}}} + \frac{\partial \, l_{\mathrm{l}}^{\mathrm{P}}}{\partial \, p} = + \varepsilon \, + \left({ l_{\mathrm{l}}^{\mathrm{P}} - l_{\mathrm{l}}^{\mathrm{E}} } \right)- + \frac{{\overline{Q}}_{\mathrm{l, par}}}{M^{\mathrm{P}}} - + \frac{RAIN}{M^{\mathrm{P}}} .. math:: :label: eq:vertparf - \frac{\partial \, l_{\rm{f}}^{\rm{P}}}{\partial \, p} = \varepsilon \, - \left({ l_{\rm{f}}^{\rm{P}} - l_{\rm{f}}^{\rm{E}} } \right)- - \frac{{\overline{Q}}_{\rm{f, par}}}{M^{\rm{P}}} - - \frac{SNOW}{M^{\rm{P}}} - + \frac{\partial \, l_{\mathrm{f}}^{\mathrm{P}}}{\partial \, p} = + \varepsilon \, + \left({ l_{\mathrm{f}}^{\mathrm{P}} - l_{\mathrm{f}}^{\mathrm{E}} } \right)- + \frac{{\overline{Q}}_{\mathrm{f, par}}}{M^{\mathrm{P}}} - + \frac{SNOW}{M^{\mathrm{P}}} -Following , equations :eq:`eq:dbydpmassflux`, +Following ``:umdp:027``, equations +:eq:`eq:dbydpmassflux`, :eq:`eq:vertparl` and :eq:`eq:vertparf` are discretized: .. math:: :label: eq:discdmfbydp - M_{\rm{k} + 1} = M_{\rm{k}} \, - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right)\, - EPSS_{\rm{k}} - -.. math:: - - l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ - l_{\rm{l \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{l \, k + 1}}^{\rm{E}} - } \right)\, / \, \left({EPSS_{\rm{k}}} \right) + M_{\mathrm{k} + 1} = M_{\mathrm{k}} \, + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right)\, + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right)\, + EPSS_{\mathrm{k}} .. math:: :label: eq:discvparl - { } { } + \left({ {\overline{Q}}_{\rm{l} \, \rm{k} + 1} \, / \, M_{\rm{k} - + 1}} \right) - - \left({ RAIN_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) - -.. math:: - - l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ - l_{\rm{f \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{f \, k + 1}}^{\rm{E}} - } \right)\, / \, \left({EPSS_{\rm{k}}} \right) + \begin{aligned} + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} & = & \left({ + l_{\mathrm{l \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{l \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{l \, k + 1}}^{\mathrm{E}} + } \right)\, / \, \left({EPSS_{\mathrm{k}}} \right) + \\ + { } & { } & + \left({ {\overline{Q}}_{\mathrm{l} \, \mathrm{k} + 1} \, / \, + M_{\mathrm{k} + 1}} \right) + - \left({ RAIN_{\mathrm{k} + 1} \, / \, M_{\mathrm{k} + 1} } \right) + \end{aligned} .. math:: :label: eq:discvparf - { } { } + \left({ {\overline{Q}}_{\rm{f} \, \rm{k} + 1} \, / \, M_{\rm{k} - + 1}} \right) - - \left({ SNOW_{\rm{k} + 1} \, / \, M_{\rm{k} + 1} } \right) - - -where :math:`EPSS_{\rm{k}} = -\left({1 + \varepsilon_{\rm{k} + 3 / 4} \, \Delta p_{\rm{k} + 3 / 4}} \right)\, -\left({1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} \right)`. + \begin{aligned} + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} & = & \left({ + l_{\mathrm{f \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{f \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{f \, k + 1}}^{\mathrm{E}} + } \right)\, / \, \left({EPSS_{\mathrm{k}}} \right) + \\ + { } & { } & + \left({ {\overline{Q}}_{\mathrm{f} \, \mathrm{k} + 1} \, / \, + M_{\mathrm{k} + 1}} \right) + - \left({ SNOW_{\mathrm{k} + 1} \, / \, M_{\mathrm{k} + 1} } \right) + \end{aligned} + +where :math:`EPSS_{\mathrm{k}} = +\left({1 + \varepsilon_{\mathrm{k} + 3 / 4} \, \Delta p_{\mathrm{k} + 3 / 4}} +\right)\, +\left({1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / 4}} +\right)`. The condensation and precipitation terms in equations :eq:`eq:discdmfbydp`, @@ -3763,50 +3787,51 @@ precipitation terms are suppressed: .. math:: :label: eq:discvparldry - l_{\rm{l \, k + 1}}^{\rm{P}} = \frac{\left({ - l_{\rm{l \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{l \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{l \, k + 1}}^{\rm{E}} - } \right)}{EPSS_{\rm{k}}} + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} = \frac{\left({ + l_{\mathrm{l \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{l \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{l \, k + 1}}^{\mathrm{E}} + } \right)}{EPSS_{\mathrm{k}}} .. math:: :label: eq:discvparfdry - l_{\rm{f \, k + 1}}^{\rm{P}} = \frac{\left({ - l_{\rm{f \, k}}^{\rm{P}} + - \varepsilon_{\rm{k} + 1/4} \, \Delta p_{\rm{k} + 1/4} \, l_{\rm{f \, - k}}^{\rm{E}} + - \varepsilon_{\rm{k} + 3/4} \, \Delta p_{\rm{k} + 3/4} \, - \left[{1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4}} - \right]\, - l_{\rm{f \, k + 1}}^{\rm{E}} - } \right)}{EPSS_{\rm{k}}} - + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} = \frac{\left({ + l_{\mathrm{f \, k}}^{\mathrm{P}} + + \varepsilon_{\mathrm{k} + 1/4} \, \Delta p_{\mathrm{k} + 1/4} \, + l_{\mathrm{f \, k}}^{\mathrm{E}} + + \varepsilon_{\mathrm{k} + 3/4} \, \Delta p_{\mathrm{k} + 3/4} \, + \left[{1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4}} \right]\, + l_{\mathrm{f \, k + 1}}^{\mathrm{E}} + } \right)}{EPSS_{\mathrm{k}}} At the base of the convective plume (ie. the level immediately above -cloud base), :math:`l_{\rm{l \, k}}^{\rm{P}}` is initialized to -:math:`l_{\rm{l \, i}}^{\rm{P}}` and :math:`l_{\rm{f \, k}}^{\rm{P}}` to -:math:`l_{\rm{f \, i}}^{\rm{P}}`, where the initial values are chosen +cloud base), :math:`l_{\mathrm{l \, k}}^{\mathrm{P}}` is initialized to +:math:`l_{\mathrm{l \, i}}^{\mathrm{P}}` and :math:`l_{\mathrm{f \, +k}}^{\mathrm{P}}` to +:math:`l_{\mathrm{f \, i}}^{\mathrm{P}}`, where the initial values are chosen such that the modified form of :eq:`eq:chidisccb` produces zero fluxes at cloud base: .. math:: :label: eq:q4lcbi - Q4_{\rm{l}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \frac{\partial \, l_{\rm{l}}^{\rm{E}}}{\partial \, p} - - M_{\rm{cb}}^{\rm{P}}\, - \left({ l_{\rm{l}}^{\rm{P \, i}} - l_{\rm{l}}^{\rm{E}}(\rm{cb}) } \right) + Q4_{\mathrm{l}}(cb) = 0 = M_{\mathrm{cb+1/2}}^{\mathrm{P}} \, + \frac{\partial \, l_{\mathrm{l}}^{\mathrm{E}}}{\partial \, p} - + M_{\mathrm{cb}}^{\mathrm{P}}\, + \left({ l_{\mathrm{l}}^{\mathrm{P \, i}} - + l_{\mathrm{l}}^{\mathrm{E}}(\mathrm{cb}) } \right) .. math:: :label: eq:q4fcbi - Q4_{\rm{f}}(cb) = 0 = M_{\rm{cb+1/2}}^{\rm{P}} \, - \frac{\partial \, l_{\rm{f}}^{\rm{E}}}{\partial \, p} - - M_{\rm{cb}}^{\rm{P}}\, - \left({ l_{\rm{f}}^{\rm{P \, i}} - l_{\rm{f}}^{\rm{E}}(\rm{cb}) } \right) - + Q4_{\mathrm{f}}(cb) = 0 = M_{\mathrm{cb+1/2}}^{\mathrm{P}} \, + \frac{\partial \, l_{\mathrm{f}}^{\mathrm{E}}}{\partial \, p} - + M_{\mathrm{cb}}^{\mathrm{P}}\, + \left({ l_{\mathrm{f}}^{\mathrm{P \, i}} - + l_{\mathrm{f}}^{\mathrm{E}}(\mathrm{cb}) } \right) As the convection scheme makes the single phase assumption for parcel condensate, it may be necessary to melt or freeze entrained condensate @@ -3814,26 +3839,26 @@ at this point and adjust the temperature accordingly. .. math:: :label: eqn:meltlf - \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} - - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{f \, k + - 1}}^{\rm{P}} - \; \ldots \; \mbox{ if l_{\rm{f \, k + 1}}^{\rm{P}} is melted } + \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} - + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} + \; \ldots \; \mbox{ if l_{\mathrm{f \, k + 1}}^{\mathrm{P}} is melted } .. math:: :label: eqn:freezell - \theta_{\rm{k + 1}}^{\rm{P}} = \theta_{\rm{k + 1}}^{\rm{P}} + - \left(\frac{L_{\rm{F}}}{C_{p} \, \Pi_{\rm{k + 1}}} \right)\, l_{\rm{l \, k + - 1}}^{\rm{P}} - \; \ldots \; \mbox{ if l_{\rm{l \, k + 1}}^{\rm{P}} is frozen } - + \theta_{\mathrm{k + 1}}^{\mathrm{P}} = \theta_{\mathrm{k + 1}}^{\mathrm{P}} + + \left(\frac{L_{\mathrm{F}}}{C_{p} \, \Pi_{\mathrm{k + 1}}} \right)\, + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} + \; \ldots \; \mbox{ if l_{\mathrm{l \, k + 1}}^{\mathrm{P}} is frozen } Once a final value for the condensation term -:math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1} \, / \, M_{\rm{k} + 1}` has +:math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1} \, / \, M_{\mathrm{k} + +1}` has been calculated from the parcel specific humidity equations, it can then be added to the parcel condensate to give a final pre-precipitation value. -- In practice, the rates :math:`{\overline{Q}}_{\rm{x} \, \rm{k} + 1}` +- In practice, the rates :math:`{\overline{Q}}_{\mathrm{x} \, \mathrm{k} + 1}` and :math:`PPN` are not calculated explicitly in the code. Instead, their effect is applied directly as increments to the temperature and moisture fields. @@ -3842,13 +3867,13 @@ The precipitation calculation is unaltered. .. math:: :label: eq:precip - P_{\rm{k} + 1} = \left({ l_{\rm{k + 1}}^{\rm{P}} - l_{\rm{MIN}}^{\rm{P}} } - \right)\, - M_{\rm{k} + 1} \, / \, g + P_{\mathrm{k} + 1} = \left({ l_{\mathrm{k + 1}}^{\mathrm{P}} - + l_{\mathrm{MIN}}^{\mathrm{P}} } \right)\, + M_{\mathrm{k} + 1} \, / \, g -where :math:`l_{\rm{k + 1}}^{\rm{P}}` = -:math:`l_{\rm{l \, k + 1}}^{\rm{P}}` + -:math:`l_{\rm{f \, k + 1}}^{\rm{P}}`. +where :math:`l_{\mathrm{k + 1}}^{\mathrm{P}}` = +:math:`l_{\mathrm{l \, k + 1}}^{\mathrm{P}}` + +:math:`l_{\mathrm{f \, k + 1}}^{\mathrm{P}}`. - Actually, given that the precipitation calculation appears to be based upon the hydrostatic equation, it is debatable whether it is even @@ -3859,159 +3884,157 @@ This reduces the parcel condensate to : .. math:: :label: eq:vparlfinal - l_{\rm{l \, k + 1}}^{\rm{P}} = \left({ - \frac{l_{\rm{l \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} - } \right)\, l_{\rm{MIN}}^{\rm{P}} + l_{\mathrm{l \, k + 1}}^{\mathrm{P}} = \left({ + \frac{l_{\mathrm{l \, k + 1}}^{\mathrm{P}}}{l_{\mathrm{k + 1}}^{\mathrm{P}}} + } \right)\, l_{\mathrm{MIN}}^{\mathrm{P}} .. math:: :label: eq:vparffinal - l_{\rm{f \, k + 1}}^{\rm{P}} = \left({ - \frac{l_{\rm{f \, k + 1}}^{\rm{P}}}{l_{\rm{k + 1}}^{\rm{P}}} - } \right)\, l_{\rm{MIN}}^{\rm{P}} - + l_{\mathrm{f \, k + 1}}^{\mathrm{P}} = \left({ + \frac{l_{\mathrm{f \, k + 1}}^{\mathrm{P}}}{l_{\mathrm{k + 1}}^{\mathrm{P}}} + } \right)\, l_{\mathrm{MIN}}^{\mathrm{P}} The final parcel condensate values are then used in the rate calculation -based upon eqn :eq:`eq:basiclold`: +based upon eqn :eq:`eq:basiclold`: .. math:: :label: eq:q4lmassf - Q4_{\rm{l}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, - l_{\rm{l}}^{\rm{E}}}{\partial \, p} + - \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + - {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ l_{\rm{l}}^{\rm{P}}(\rm{k}) - l_{\rm{l}}^{\rm{E}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{l, reset}} + Q4_{\mathrm{l}}(k) = M_{\mathrm{k+1/2}}^{\mathrm{P}} \, \frac{\partial \, + l_{\mathrm{l}}^{\mathrm{E}}}{\partial \, p} + + \left({ {\mu}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} + + {\delta}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} } \right)\, + \left({ l_{\mathrm{l}}^{\mathrm{P}}(\mathrm{k}) - + l_{\mathrm{l}}^{\mathrm{E}}(\mathrm{k}) } \right)- + {\overline{Q}}_{\mathrm{l, reset}} .. math:: :label: eq:q4fmassf - Q4_{\rm{f}}(k) = M_{\rm{k+1/2}}^{\rm{P}} \, \frac{\partial \, - l_{\rm{f}}^{\rm{E}}}{\partial \, p} + - \left({ {\mu}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} + - {\delta}_{\rm{k}} \, M_{\rm{k}}^{\rm{P}} } \right)\, - \left({ l_{\rm{f}}^{\rm{P}}(\rm{k}) - l_{\rm{f}}^{\rm{E}}(\rm{k}) } \right)- - {\overline{Q}}_{\rm{f, reset}} - + Q4_{\mathrm{f}}(k) = M_{\mathrm{k+1/2}}^{\mathrm{P}} \, \frac{\partial \, + l_{\mathrm{f}}^{\mathrm{E}}}{\partial \, p} + + \left({ {\mu}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} + + {\delta}_{\mathrm{k}} \, M_{\mathrm{k}}^{\mathrm{P}} } \right)\, + \left({ l_{\mathrm{f}}^{\mathrm{P}}(\mathrm{k}) - + l_{\mathrm{f}}^{\mathrm{E}}(\mathrm{k}) } \right)- + {\overline{Q}}_{\mathrm{f, reset}} -Note that, as a side-effect, the environment equations for potential -temperature and specific humidity are also altered because the +Note that, as a side-effect, the ``:umdp:027`` environment equations for +potential temperature and specific humidity are also altered because the condensate is no longer re-evaporated at the end -(:math:`{\overline{Q}}_{\rm{l, reset}} = 0 -= {\overline{Q}}_{\rm{f, reset}}`): - -.. math:: - - \frac{\Delta \, \theta_{\rm{k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) - \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } - \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \theta_{\rm{k + 1}}^{\rm{E}} - \theta_{\rm{k}}^{\rm{E}} } \right) - } \right . + - -.. math:: - - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \theta_{\rm{k}}^{\rm{R}} - \theta_{\rm{k}}^{\rm{E}} } \right) - + +(:math:`{\overline{Q}}_{\mathrm{l, reset}} = 0 += {\overline{Q}}_{\mathrm{f, reset}}`): .. math:: :label: eq:enviroth + \begin{aligned} + \frac{\Delta \, \theta_{\mathrm{k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ \theta_{\mathrm{k + 1}}^{\mathrm{E}} - + \theta_{\mathrm{k}}^{\mathrm{E}} } \right) + } \right . & + & + \\ + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ \theta_{\mathrm{k}}^{\mathrm{R}} - \theta_{\mathrm{k}}^{\mathrm{E}} + } \right) + & + & + \\ \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ \theta_{\rm{k}}^{\rm{P}} - \theta_{\rm{k}}^{\rm{E}} } \right) - } \right] { } - + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ \theta_{\mathrm{k}}^{\mathrm{P}} - \theta_{\mathrm{k}}^{\mathrm{E}} + } \right) + } \right]& { } & + \end{aligned} and -.. math:: +.. math:: :label: eq:enviroq - \frac{\Delta \, q_{\rm{k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) + \begin{aligned} + \frac{\Delta \, q_{\mathrm{k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ q_{\mathrm{k + 1}}^{\mathrm{E}} - q_{\mathrm{k}}^{\mathrm{E}} } \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ q_{\rm{k + 1}}^{\rm{E}} - q_{\rm{k}}^{\rm{E}} } \right) - } \right . + - -.. math:: - - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ q_{\rm{k}}^{\rm{R}} - q_{\rm{k}}^{\rm{E}} } \right) - + - -.. math:: :label: eq:enviroq - + } \right . & + & + \\ + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ q_{\mathrm{k}}^{\mathrm{R}} - q_{\mathrm{k}}^{\mathrm{E}} } \right) + & + & + \\ \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ q_{\rm{k}}^{\rm{P}} - q_{\rm{k}}^{\rm{E}} } \right) - } \right] { } - + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ q_{\mathrm{k}}^{\mathrm{P}} - q_{\mathrm{k}}^{\mathrm{E}} } \right) + } \right]& { } & + \end{aligned} Similarly, eqns :eq:`eq:q4lmassf` and :eq:`eq:q4fmassf` have a discretized form as follows: -.. math:: - - \frac{\Delta \, l_{\rm{l \, k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) - \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } - \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{l \, k + 1}}^{\rm{E}} - l_{\rm{l \, k}}^{\rm{E}} } \right) - } \right . + - -.. math:: - - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) - + - .. math:: :label: eq:enviroll + \begin{aligned} + \frac{\Delta \, l_{\mathrm{l \, k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{l \, k + 1}}^{\mathrm{E}} - l_{\mathrm{l \, + k}}^{\mathrm{E}} } \right) + } \right . & + & + \\ + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{l \, k}}^{\mathrm{P}} - l_{\mathrm{l \, k}}^{\mathrm{E}} + } \right) + & + & + \\ \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{l \, k}}^{\rm{P}} - l_{\rm{l \, k}}^{\rm{E}} } \right) - } \right] { } - + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{l \, k}}^{\mathrm{P}} - l_{\mathrm{l \, k}}^{\mathrm{E}} + } \right) + } \right]& { } & + \end{aligned} and -.. math:: - - \frac{\Delta \, l_{\rm{f \, k}}^{\rm{E}}}{\Delta \, t} = - \left(\frac{ M_{\rm{k}} }{ \Delta \, p_{\rm{k}} } \right) - \left[{ - \left({ 1 + \varepsilon_{\rm{k} + 1 / 4} \, \Delta p_{\rm{k} + 1 / 4} } - \right) - \left({ 1 - \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{f \, k + 1}}^{\rm{E}} - l_{\rm{f \, k}}^{\rm{E}} } \right) - } \right . + - -.. math:: - - \left({ \delta_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ 1 - \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{f \, k}}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) - + - .. math:: :label: eq:envirolf + \begin{aligned} + \frac{\Delta \, l_{\mathrm{f \, k}}^{\mathrm{E}}}{\Delta \, t} = + \left(\frac{ M_{\mathrm{k}} }{ \Delta \, p_{\mathrm{k}} } \right) + \left[{ + \left({ 1 + \varepsilon_{\mathrm{k} + 1 / 4} \, \Delta p_{\mathrm{k} + 1 / + 4} } \right) + \left({ 1 - \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{f \, k + 1}}^{\mathrm{E}} - l_{\mathrm{f \, + k}}^{\mathrm{E}} } \right) + } \right . & + & + \\ + \left({ \delta_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ 1 - \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{f \, k}}^{\mathrm{P}} - l_{\mathrm{f \, k}}^{\mathrm{E}} + } \right) + & + & + \\ \left . { - \left({ \mu_{\rm{k}} \, \Delta p_{\rm{k} + 1 / 2} } \right) - \left({ l_{\rm{f \, k }}^{\rm{P}} - l_{\rm{f \, k}}^{\rm{E}} } \right) - } \right] { } - + \left({ \mu_{\mathrm{k}} \, \Delta p_{\mathrm{k} + 1 / 2} } \right) + \left({ l_{\mathrm{f \, k }}^{\mathrm{P}} - l_{\mathrm{f \, k}}^{\mathrm{E}} + } \right) + } \right]& { } & + \end{aligned} .. _sec_conv_homog: @@ -4067,7 +4090,7 @@ Now we recognise that \Delta \overline{q} = Q2~ \Delta t where :math:`Q2` is the rate of moistening of the whole gridbox due to -convection. Remember that, at this stage, we haven’t done any +convection. Remember that, at this stage, we haven't done any condensation outside of the plume. Hence to calculate the condensation we should apply the background change in :math:`\overline{q}` as a uniform forcing for the background air. Hence @@ -4203,7 +4226,7 @@ Homogeneous forcing of the environment by convective-subsidence pressure change ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To this end, the code includes an option to perform the homogeneous -forcing of liquid cloud by convection using the “pressure forcing” from +forcing of liquid cloud by convection using the "pressure forcing" from the convective subsidence, consistent with the pressure forcing by large-scale advection (see sections :ref:`Advection ` and :ref:`Response to pressure changes `). This approach replaces the @@ -4382,18 +4405,19 @@ A prognostic dust approach is implemented in the micro-physics scheme under large-sale-precipitation where by the heterogeneous nucleation temperature can be defined to vary three dimensionally globally as an arc-tangent function of the mineral dust distribution in the model -(documented in ). By default, both liquid and ice are detrained -simultaneously at the same height, and the fraction of condensate that -is ice linearly ramps as a function of temperature. i.e. condensate is -assumed to be all-liquid when T is greater than one tuneable threshold; -all-ice when T is less than another tuneable threshold, and vary -linearly in-between (the threshold values are given by starticeTkelvin -and alliceTdegC in the UM cloud-scheme namelist. The new heterogeneous -nucleation temperatures calculated in the large-scale-precipitation are -passed to the convection scheme and are used as the above detrainment -temperature thresholds by maintaining a similar linear ramp. For e.g., -condensate is assumed to be all-liquid for T :math:`\geq` :math:`tnuc_n` -and all-ice for T :math:`\leq` :math:`tnuc_n` - 10.0 +(documented in ``:umdp:026``). By default, both liquid and ice are +detrained simultaneously at the same height, and the fraction of +condensate that is ice linearly ramps as a function of temperature. i.e. +condensate is assumed to be all-liquid when T is greater than one +tuneable threshold; all-ice when T is less than another tuneable +threshold, and vary linearly in-between (the threshold values are given +by starticeTkelvin and alliceTdegC in the UM cloud-scheme namelist. The +new heterogeneous nucleation temperatures calculated in the +large-scale-precipitation are passed to the convection scheme and are +used as the above detrainment temperature thresholds by maintaining a +similar linear ramp. For e.g., condensate is assumed to be all-liquid +for T :math:`\geq` :math:`tnuc_n` and all-ice for T :math:`\leq` +:math:`tnuc_n` - 10.0 .. _sec_conv_input_profs: @@ -4404,13 +4428,13 @@ The convection scheme itself is highly sensitive to the input environment temperature and moisture profiles *before* the convection increments (or PC2 response) are calculated. In particular, the parcel buoyancy (and hence the CAPE and mass-flux scaling) maybe radically -different depending on whether a “large-scale” condensation / +different depending on whether a "large-scale" condensation / evaporation adjustment is performed before the convection call. Where there is large-scale ascent, the profiles after Semi-Lagrangian advection may have become supersaturated and unrealistically unstable, until the expected condensation adjustment is performed. If the -convection scheme “sees” these unrealistic intermediate profiles, it is +convection scheme "sees" these unrealistic intermediate profiles, it is likely to predict an excessive, unrealistic mass-flux. To address this problem, there are two namelist switches that enable @@ -4572,11 +4596,11 @@ There are currently 3 options for the conditions under-which initiation may occur. For all of these options, if using the bimodal cloud scheme to do initiation within PC2, then the tests on :math:`RH_T` relative to :math:`RH_{crit}` are replaced by equivalent tests for whether the -saturation boundary lies within the bounds of the bimodal scheme’s +saturation boundary lies within the bounds of the bimodal scheme's assumed PDF, as described in section :ref:`Initiation using the bimodal scheme `. -“Original” initiation logic +"Original" initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -4615,7 +4639,7 @@ Equivalently, :math:`C_l` is initiated away from 1 if - :math:`RH_T^{[n+1]} < RH_T^{[n]}` . -“Simplified” initiation logic +"Simplified" initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -4646,7 +4670,7 @@ standard value is 0.005. Note this threshold is also used to remove small cloud-fractions after initiation; see section :ref:`Additional checks after PC2 initiation `. -This is very similar to the “Original” initiation logic described above, +This is very similar to the "Original" initiation logic described above, but with the following differences: - The condition that the boundary-layer hasn't diagnosed cumulus @@ -4662,7 +4686,7 @@ but with the following differences: .. _sec_smooth_initiation: -“Smooth” initiation logic +"Smooth" initiation logic ^^^^^^^^^^^^^^^^^^^^^^^^^ This option is selected by setting the UM namelist switch @@ -4754,7 +4778,7 @@ completely uninitiated state will have zero saturation deficit :math:`SD`, rather than zero :math:`q_{cl}`. Therefore, in this case the increment to :math:`C_l` is calculated based on the fractional increase in :math:`SD` from initiation (equation -:eq:`eq:dcl_init2`, instead of the fractional increase +:eq:`eq:dcl_init2`), instead of the fractional increase in :math:`q_{cl}`. Whether to increment :math:`C_l` based on the increase in :math:`q_{cl}` @@ -4839,22 +4863,17 @@ The thresholds :math:`C_{high}`, :math:`C_{high 2}`, :math:`C_{low}` and :math:`C_{low 2}` are set using the parameters :math:`C_{tol}` and :math:`C_{tol 2}`, according to: -.. math:: +.. math:: :label: eq:chigh-chigh2 + \begin{aligned} C_{high} = 1 - C_{tol}, - -.. math:: - + \\ C_{high 2} = 1 - C_{tol 2}, - -.. math:: - + \\ C_{low} = C_{tol}, - -.. math:: :label: eq:chigh-chigh2 - + \\ C_{low 2} = C_{tol 2}, - + \end{aligned} where the parameters :math:`C_{tol}` and :math:`C_{tol 2}` can be set via the UM namelist variables **cloud_pc2_tol** and **cloud_pc2_tol_2**. @@ -4932,18 +4951,15 @@ condition is violated, we condense water vapour to adjust to grid-mean saturation. :math:`-SD` corresponds to the amount of vapour that must be condensed to achieve this, so we have: -.. math:: +.. math:: :label: eq:qsdcheck1 + \begin{aligned} \overline{q} \leftarrow \overline{q} + SD - -.. math:: - + \\ \overline{q_{cl}} \leftarrow \overline{q_{cl}} - SD - -.. math:: :label: eq:qsdcheck1 - + \\ \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} SD - + \end{aligned} The original version of this check on :math:`SD` (which may increase :math:`\overline{q_{cl}}`), made no accompanying changes to liquid cloud @@ -5010,26 +5026,19 @@ behaviour is currently controlled by a temporary logical in the evaporate all the remaining liquid water, and reset :math:`C_l` to zero: - .. math:: + .. math:: :label: eq:qsdcheck2 + \begin{aligned} \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} - - .. math:: - + \\ \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} - - .. math:: - + \\ \overline{q_{cl}} \leftarrow 0 - - .. math:: - + \\ C_l \leftarrow 0 - - .. math:: :label: eq:qsdcheck2 - + \\ C_t \leftarrow C_i - + \end{aligned} .. _section-3: @@ -5040,18 +5049,15 @@ If :math:`\overline{q_{cl}} < q_{c0}` or :math:`C_l = 0` then we evaporate the small amount of :math:`\overline{q_{cl}}` that remains in the gridbox: -.. math:: +.. math:: :label: eq:qclcheck + \begin{aligned} \overline{q} \leftarrow \overline{q} + \overline{q_{cl}} - -.. math:: - + \\ \overline{q_{cl}} \leftarrow 0 - -.. math:: :label: eq:qclcheck - + \\ \overline{T} \leftarrow \overline{T} - \frac{L_c}{c_p} \overline{q_{cl}} - + \end{aligned} .. _section-4: @@ -5087,14 +5093,13 @@ ensure that it takes on a value that is physically possible, given the values of :math:`C_l` and :math:`C_i`. We have, firstly, the maximum overlap situation and then the minimum overlap situation. -.. math:: - - C_t \leftarrow \text{Max}( C_t, C_i, C_l ) - .. math:: :label: eq:ctchecks + \begin{aligned} + C_t \leftarrow \text{Max}( C_t, C_i, C_l ) + \\ C_t \leftarrow \text{Min}( C_t , C_l + C_i, 1) - + \end{aligned} .. _section-8: @@ -5106,26 +5111,19 @@ liquid cloud created by the initiation is allowed to persist in this phase if the temperature is cold enough. Hence, if :math:`\overline{T} < T_{homo}` then -.. math:: +.. math:: :label: eq:homochecks + \begin{aligned} \overline{q_{cf}} \leftarrow \overline{q_{cf}} + \overline{q_{cl}} - -.. math:: - + \\ \overline{q_{cl}} \leftarrow 0 - -.. math:: - + \\ \overline{T} \leftarrow \overline{T} + \frac{L_f}{c_p} \overline{q_{cl}} - -.. math:: - + \\ C_i \leftarrow C_t - -.. math:: :label: eq:homochecks - + \\ C_l \leftarrow 0. - + \end{aligned} .. _sec_qpos: @@ -5142,8 +5140,8 @@ During trialling prior to operational implementation, it was found that relying on Q-Pos to deal with negative condensate values was very expensive, as the Q-Pos routine does a lot of communications between different processors. It may be preferable to deal with the cause of -negative condensate amounts at their source. The option to “Ensure -consistent sinks of qcl and CFL” prevents the QCL increment from trying +negative condensate amounts at their source. The option to "Ensure +consistent sinks of qcl and CFL" prevents the QCL increment from trying to remove too much liquid condensate and hence reduces the models reliance on Q-Pos to deal with the inconsistencies. @@ -5248,10 +5246,10 @@ Area cloud fraction Two area cloud fraction parametrizations are available for use with PC2. -The area cloud fraction of Cusack (documented in ) has been adapted by -`Boutle and Morcrette (2010)`_ so it can be used with PC2 (and -is available from the UMUI as the “Cusack” option from version 7.6 -onwards). This method aims to reproduce some of the detail of the +The area cloud fraction of Cusack (documented in ``:umdp:029``) has been +adapted by `Boutle and Morcrette (2010)`_ so it can be used with +PC2 (and is available from the UMUI as the "Cusack" option from version +7.6 onwards). This method aims to reproduce some of the detail of the thermodynamic profile lost due to the coarseness of the grid. The interpolation/extrapolation technique is used prior to PC2 initiation (which is then called with three times as many levels) and it is used, @@ -5324,465 +5322,495 @@ called *l_pc2_reset*. Turning this on (not recommended!) does 2 things: The location of the various cloud scheme routine calls within the UM is summarised in the list below. -Subroutines only called for the Smith scheme are highlighted in blue, -those only called for PC2 are in green, and those only called for the -bimodal scheme are in purple. +Subroutines only called for the :blue:`Smith` scheme are highlighted in +:blue:`blue`, +those only called for :green:`PC2` are in :green:`green`, +and those only called for the :purple:`bimodal` scheme are in :purple:`purple`. Main Tree from atm_step_4a ^^^^^^^^^^^^^^^^^^^^^^^^^^ -.. container:: itemize | **atm_step_4a** - | \* (performs one timestep of the Unified Model...) + | (performs one timestep of the Unified Model...) + + | **atm_step_alloc_4a** + | (does miscellaneous initialisations in atm_step) + + | :green-lbl:`pc2_rhtl` + | (calculate start-of-timestep Relative Humidity, used by + PC2 initiation) + + | + + | **atmos_physics1** + | (calls explicit "slow" physics routines...) + + | **microphys_ctl** + | (interface to microphysics scheme) + + | :green-lbl:`pc2_turbulence_ctl` + | (Perform optional erosion of liquid-cloud; done + here if NOT doing erosion after convection, e.g. if + no convection scheme is used). - .. container:: itemize + | :green-lbl:`pc2_hom_conv` + | (called here just to do erosion) - .. container:: tcolorbox + | - | **atm_step_alloc_4a** - | \* (does miscellaneous initialisations in atm_step) + | :blue-lbl:`ls_cld` + | (Smith scheme without area cloud fraction + calculation, to set initial cloud fields passed into + microphysics) - - | pc2_rhtl - | \* (calculate start-of-timestep Relative Humidity, used by - PC2 initiation) + | - .. container:: tcolorbox + | **ls_ppn** + | (microphysics scheme) - | **atmos_physics1** - | \* (calls explicit “slow” physics routines...) + | - .. container:: itemize + | **mphys_turb_gen_mixed_phase** + | (turbulent production of liquid cloud) - .. container:: tcolorbox + | - | **microphys_ctl** - | \* (interface to microphysics scheme) + | :green-lbl:`pc2_turbulence_ctl` + | (optionally use the PC2 pdf-width-change code to + calculate the cloud-fraction change from the above + turbulent production of liquid cloud) - - | pc2_turbulence_ctl - | \* (Perform optional erosion of liquid-cloud; done - here if NOT doing erosion after convection, e.g. if - no convection scheme is used). + | :green-lbl:`pc2_hom_conv` + | (called here just to calculate the cloud + fraction increment consistent with the turbulent + qcl increment) - - | pc2_hom_conv - | \* (called here just to do erosion) + | - - | ls_cld - | \* (Smith scheme without area cloud fraction - calculation, to set initial cloud fields passed into - microphysics) + | **rad_ctl** + | (interface to radiation scheme) - - | **ls_ppn** - | \* (microphysics scheme) + | **sw_rad** + | (short-wave radiation scheme) - - | **mphys_turb_gen_mixed_phase** - | \* (turbulent production of liquid cloud) + | - - | pc2_turbulence_ctl - | \* (optionally use the PC2 pdf-width-change code to - calculate the cloud-fraction change from the above - turbulent production of liquid cloud) + | :green-lbl:`pc2_homog_plus_turb` + | (PC2 homogeneous forcing of liquid-cloud by SW + radiation heating) - - | pc2_hom_conv - | \* (called here just to calculate the cloud - fraction increment consistent with the turbulent - qcl increment) + | - .. container:: tcolorbox + | **lw_rad** + | (long-wave radiation scheme) - | **rad_ctl** - | \* (interface to radiation scheme) + | - - | **sw_rad** - | \* (short-wave radiation scheme) + | :green-lbl:`pc2_homog_plus_turb` + | (PC2 homogeneous forcing of liquid-cloud by LW + radiation tendency) - - | pc2_homog_plus_turb - | \* (PC2 homogeneous forcing of liquid-cloud by SW - radiation heating) + | - - | **lw_rad** - | \* (long-wave radiation scheme) + | **atmos_physics1_alloc_pc2** + | (wrapper for PC2 + self-consistency checks at end of atmos_physics1) - - | pc2_homog_plus_turb - | \* (PC2 homogeneous forcing of liquid-cloud by LW - radiation tendency) + | Add increments from microphysics + radiation onto + start-of-timestep fields to form updated fields. - .. container:: tcolorbox + | :green-lbl:`pc2_checks` + | (self-consistency checks on cloud fractions and + water contents) - **atmos_physics1_alloc_pc2** (wrapper for PC2 - self-consistency checks at end of atmos_physics1) + | Convert corrected updated fields back to increments. - - Add increments from microphysics + radiation onto - start-of-timestep fields to form updated fields. + | Begin loop over solver outer cycles - - | pc2_checks - | \* (self-consistency checks on cloud fractions and - water contents) - - Convert corrected updated fields back to increments. + | **atm_step_phys_reset** + | (for PC2, on subsequent solver outer cycles, reset + cloud-fractions to saved values after atmos_physics1) - Begin loop over solver outer cycles + | - .. container:: itemize + | **eg_sl_moisture** + | (large-scale advection of cloud water contents and fractions) - .. container:: tcolorbox + | - | **atm_step_phys_reset** - | \* (for PC2, on subsequent solver outer cycles, reset - cloud-fractions to saved values after atmos_physics1) + | :green-lbl:`pc2_pressure_forcing_only` + | (Optionally calculate homogeneous forcing of liquid + cloud by the pressure change along the trajectory from + departure point to arrival point). - .. container:: tcolorbox + | :green-lbl:`pc2_homog_plus_turb` + | (generic homogeneous forcing routine used here). - | **eg_sl_moisture** - | \* (large-scale advection of cloud water contents and - fractions) + | - .. container:: tcolorbox + | **atmos_physics2** + | (calls "fast" physics routines...) - | pc2_pressure_forcing_only - | \* (Optionally calculate homogeneous forcing of liquid - cloud by the pressure change along the trajectory from - departure point to arrival point). + | **ni_bl_ctl** + | (interface to explicit boundary-layer and surface + scheme calls, including calculation of TKE and + TKE-based :math:`RH_{crit}`) - - | pc2_homog_plus_turb - | \* (generic homogeneous forcing routine used here). + | - .. container:: tcolorbox + | :purple-lbl:`bm_calc_tau` + | (calculates turbulence properties used in the + bimodal cloud scheme, based on the boundary-layer + scheme TKE and mixing-length) - | **atmos_physics2** - | \* (calls “fast” physics routines...) + | - .. container:: itemize + | **cloud_call_b4_conv** + | (routine for optional cloud-scheme calls before convection) - .. container:: tcolorbox + | :blue-lbl:`ls_arcld` + | (Smith scheme with area cloud fraction; see + :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - | **ni_bl_ctl** - | \* (interface to explicit boundary-layer and surface - scheme calls, including calculation of TKE and - TKE-based :math:`RH_{crit}`) + | - .. container:: tcolorbox + | :purple-lbl:`bm_ctl` + | (bimodal scheme) - | bm_calc_tau - | \* (calculates turbulence properties used in the - bimodal cloud scheme, based on the boundary-layer - scheme TKE and mixing-length) + | Set area cloud fraction equal to bulk cloud fraction - .. container:: tcolorbox + | :green-lbl:`pc2_initiation_ctl` + | (interface to PC2 initiation and consistency-checks; see + :ref:`PC2 initiation ` for a + drill-down inside this routine) - | **cloud_call_b4_conv** - | \* (routine for optional cloud-scheme calls before - convection) + | - - | ls_arcld - | \* (Smith scheme with area cloud fraction; see - :ref:`Smith scheme with area cloud fraction ` for a drill-down - inside this routine) + | **ni_conv_ctl** or **other_conv_ctl** + | (interface routines to various convection schemes...) - - | bm_ctl - | \* (bimodal scheme) + | **glue_conv_5a/6a** + | (calls deep, shallow and mid-level convection schemes) - - Set area cloud fraction equal to bulk cloud fraction + | **deep/shallow/mid_conv** + | (convection scheme main routines) - - | pc2_initiation_ctl - | \* (interface to PC2 initiation and - consistency-checks; see - :ref:`PC2 initiation ` for a - drill-down inside this routine) + | **convec2** + | (completes lifting of the convective + parcel by one model-level) - .. container:: tcolorbox + | **parcel** + | (calculates new parcel properties at next level) - | **ni_conv_ctl** or **other_conv_ctl** - | \* (interface routines to various convection - schemes...) + | - - | **glue_conv_5a/6a** - | \* (calls deep, shallow and mid-level convection - schemes) + | **environ** + | (calculates grid-mean increments + to primary fields; includes PC2 partitioning + of detrained condensate mass between liquid + and ice phases) - - | **deep/shallow/mid_conv** - | \* (convection scheme main routines) + | - - **convec2** (completes lifting of the convective - parcel by one model-level) + | :green-lbl:`pc2_environ` + | (calculates increments to PC2 + cloud fractions due to convective detrainment + and subsidence) - - **parcel** (calculates new parcel properties - at next level) + | - - **environ** (calculates grid-mean increments - to primary fields; includes PC2 partitioning - of detrained condensate mass between liquid - and ice phases) + | :green-lbl:`pc2_from_conv_ctl` + | (PC2 calculations after convection) - - pc2_environ (calculates increments to PC2 - cloud fractions due to convective detrainment - and subsidence) + | :green-lbl:`pc2_hom_conv` + | (homogeneous forcing by convection, and + erosion of liquid-cloud) - - | pc2_from_conv_ctl - | \* (PC2 calculations after convection) + | - - | pc2_hom_conv - | \* (homogeneous forcing by convection, and - erosion of liquid-cloud) + | **ni_imp_ctl** + | (interface to boundary-layer implicit solver) - .. container:: tcolorbox + | **imp_solver** + | (implicitly solves vertical diffusion to find + :math:`T_l` and :math:`q_T` updated by turbulent fluxes). - | **ni_imp_ctl** - | \* (interface to boundary-layer implicit solver) + | - - | **imp_solver** - | \* (implicitly solves vertical diffusion to find - :math:`T_l` and :math:`q_T` updated by turbulent - fluxes). + | :green-lbl:`pc2_bl_inhom_ice` + | (inhomogeneous forcing of ice-cloud) - - | pc2_bl_inhom_ice - | \* (inhomogeneous forcing of ice-cloud) + | - - | pc2_delta_hom_turb - | \* (homogeneous forcing of liquid cloud by the - turbulent fluxes) + | :green-lbl:`pc2_delta_hom_turb` + | (homogeneous forcing of liquid cloud by the turbulent fluxes) - - | pc2_bl_forced_cu - | \* (adds diagnosed “forced cumulus” cloud fraction - and water content onto the PC2 prognostics) + | - - Calculate area cloud fraction: + | :green-lbl:`pc2_bl_forced_cu` + | (adds diagnosed "forced cumulus" cloud fraction + and water content onto the PC2 prognostics) - | ls_acf_brooks - | \* (for the Brooks epirical method) + | Calculate area cloud fraction: - | pc2_hom_arcld - | \* (for the Cusack vertical interpolation method) + | :green-lbl:`ls_acf_brooks` + | (for the Brooks epirical method) - - | pc2_homog_plus_turb - | \* (generic homogeneous forcing routine used to - interpolate) + | - - | ls_arcld - | \* (interface to diagnostic Smith scheme and area - cloud fraction; see - :ref:`Smith scheme with area cloud fraction ` for a drill-down - inside this routine) + | :green-lbl:`pc2_hom_arcld` + | (for the Cusack vertical interpolation method) - - | bm_ctl - | \* (bimodal cloud scheme) + | :green-lbl:`pc2_homog_plus_turb` + | (generic homogeneous forcing routine used to interpolate) - - Set area cloud fraction equal to bulk cloud fraction + | - - | **diagnostics_bl** - | \* (outputs boundary-layer diagnostics to STASH) + | :blue-lbl:`ls_arcld` + | (interface to diagnostic Smith scheme and area + cloud fraction; see + :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - - | **ls_cld** - | \* (Smith scheme used here to calculate various - diagnostics of near-surface temperature and - humidity, by extrapolating pressure, :math:`T_l` - and :math:`q_t` down to the desired height and - then re-diagnosing :math:`q_{cl}`. + | - .. container:: tcolorbox + | :purple-lbl:`bm_ctl` + | (bimodal cloud scheme) - | **atm_step_ac_assim** - | \* (interface to Data Assimilation analysis increments...) + | Set area cloud fraction equal to bulk cloud fraction - - **ac_ctl** (control routine for Data Assimilation analysis - increments...) + | **diagnostics_bl** + | (outputs boundary-layer diagnostics to STASH) - - **ac** (main analysis increment routine) + | **ls_cld** + | (Smith scheme used here to calculate various + diagnostics of near-surface temperature and + humidity, by extrapolating pressure, :math:`T_l` + and :math:`q_t` down to the desired height and + then re-diagnosing :math:`q_{cl}`. - - | pc2_assim - | \* (PC2 reponse to the analysis increments; see - :ref:`PC2 Data Assimilation ` for a drill-down - inside this routine) + | - - ls_acf_brooks (calculate area cloud fraction using - Brooks empirical method if active) + | **atm_step_ac_assim** + | (interface to Data Assimilation analysis increments...) - - ls_arcld (call diagnostic Smith scheme with area cloud - fraction again to account for the analysis increments; - see :ref:`Smith scheme with area cloud fraction - ` for a drill-down - inside this routine) + | **ac_ctl** + | (control routine for Data Assimilation analysis increments...) - .. container:: tcolorbox + | **ac** + | (main analysis increment routine) - | **eg_sl_helmholtz** - | \* (dynamics pressure solver; updates pressure, and the - winds used to perform advection on the next solver outer - cycle) + | - End loop over solver outer cycles + | :green-lbl:`pc2_assim` + | (PC2 reponse to the analysis increments; see + :ref:`PC2 Data Assimilation ` + for a drill-down inside this routine) - .. container:: tcolorbox + | - | pc2_pressure_forcing - | \* (interface to miscellaneous PC2 calculations at - end-of-timestep) + | :green-lbl:`ls_acf_brooks` + | (calculate area cloud fraction using + Brooks empirical method if active) - - | pc2_homog_plus_turb - | \* (homogeneous forcing of liquid-cloud by the dynamics - pressure change; optionally either uses total pressure - change including the Lagrangian component following the - winds, or only the Eulerian component from the dynamics - solver) + | - - | pc2_initiation_ctl - | \* (interface to PC2 initiation and consistency-checks; see - :ref:`PC2 initiation ` for a drill-down - inside this routine) + | :blue-lbl:`ls_arcld` + | (call diagnostic Smith scheme with area cloud + fraction again to account for the analysis increments; + see :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) - .. container:: tcolorbox + | - | **qt_bal_cld** - | \* (calculates end-of-timestep cloud state consistent with - final pressure...) + | **eg_sl_helmholtz** + | (dynamics pressure solver; updates pressure, and the + winds used to perform advection on the next solver outer cycle) - - | ls_arcld - | \* (interface to diagnostic Smith scheme and area cloud - fraction; see :ref:`Smith scheme with area cloud fraction ` for a - drill-down inside this routine) + | End loop over solver outer cycles - - | bm_ctl - | \* (bimodal cloud scheme) + | :green-lbl:`pc2_pressure_forcing` + | (interface to miscellaneous PC2 calculations at end-of-timestep) - - Set area cloud fraction equal to bulk cloud fraction + | :green-lbl:`pc2_homog_plus_turb` + | (homogeneous forcing of liquid-cloud by the dynamics + pressure change; optionally either uses total pressure + change including the Lagrangian component following the + winds, or only the Eulerian component from the dynamics + solver) - .. container:: tcolorbox + | - | **iau** - | \* (incremental analysis update; part of data assimilation) + | :green-lbl:`pc2_initiation_ctl` + | (interface to PC2 initiation and consistency-checks; see + :ref:`PC2 initiation ` for a drill-down + inside this routine) - - | pc2_assim - | \* (PC2 reponse to the analysis increments; see - :ref:`PC2 Data Assimilation ` for a drill-down inside - this routine) + | - - | initial_pc2_check - | \* (wrapper for optional self-consistency checks on + | **qt_bal_cld** + | (calculates end-of-timestep cloud state consistent with + final pressure...) + + | :blue-lbl:`ls_arcld` + | (interface to diagnostic Smith scheme and area cloud + fraction; see :ref:`Smith scheme with area cloud fraction + ` for a drill-down inside this routine) + + | + + | :purple-lbl:`bm_ctl` + | (bimodal cloud scheme) + + | Set area cloud fraction equal to bulk cloud fraction + + | + + | **iau** + | (incremental analysis update; part of data assimilation) + + | :green-lbl:`pc2_assim` + | (PC2 reponse to the analysis increments; see + :ref:`PC2 Data Assimilation ` + for a drill-down inside this routine) + + | + + | :green-lbl:`initial_pc2_check` + | (wrapper for optional self-consistency checks on prognostic cloud variables if not doing PC2 response to analysis increments) - - | pc2_checks - | \* (self-consistency checks on cloud fractions and water - contents) + | :green-lbl:`pc2_checks` + | (self-consistency checks on cloud fractions and water + contents) Drill-downs within some routines in the call tree are listed separately below, to avoid duplication (since these routines are called in multiple different places in the tree)... + .. _subsubsec_smith_acf: Smith scheme with area cloud fraction ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -.. container:: itemize + | :blue-lbl:`ls_arcld` + | (interface to diagnostic Smith scheme and area cloud fraction) - .. container:: tcolorbox + | If no area cloud fraction scheme: - | ls_arcld - | \* (interface to diagnostic Smith scheme and area cloud - fraction) + | :blue-lbl:`ls_cld` + | (just directly call Smith scheme) - - If no area cloud fraction scheme: + | Set area cloud fraction equal to bulk cloud fraction. - | ls_cld - | \* (just directly call Smith scheme) + | If using Cusack vertical interpolation method: - Set area cloud fraction equal to bulk cloud fraction. + | Interpolate fields onto finer vertical grid - - If using Cusack vertical interpolation method: + | :blue-lbl:`ls_cld` + | (call Smith scheme using higher vertical resolution fields) - Interpolate fields onto finer vertical grid + | Coarse-grain cloud fields back to model grid, but set area cloud + fraction to max of bulk cloud fraction over corresponding + fine-grid levels. - | ls_cld - | \* (call Smith scheme using higher vertical resolution fields) + | If using Brooks empirical area cloud fraction method: - Coarse-grain cloud fields back to model grid, but set area cloud - fraction to max of bulk cloud fraction over corresponding - fine-grid levels. + | :blue-lbl:`ls_cld` + | (just directly call Smith scheme) - - If using Brooks empirical area cloud fraction method: + | - | ls_cld - | \* (just directly call Smith scheme) + | :blue-lbl:`ls_acf_brooks` + | (estimate area cloud fraction) - | ls_acf_brooks - | \* (estimate area cloud fraction) .. _subsubsec_pc2_initiation: PC2 initiation ^^^^^^^^^^^^^^ -.. container:: itemize + | :green-lbl:`pc2_initiation_ctl` + | (interface to PC2 initiation and consistency-checks) - .. container:: tcolorbox - - | pc2_initiation_ctl - | \* (interface to PC2 initiation and consistency-checks) - - - | pc2_checks - | \* (self-consistency checks on cloud fractions and water + | :green-lbl:`pc2_checks` + | (self-consistency checks on cloud fractions and water contents) - - PC2 initiation of liquid-cloud: + | PC2 initiation of liquid-cloud: + + | :green-lbl:`pc2_bm_initiate` + | (using the bimodal cloud scheme) - | pc2_bm_initiate - | \* (using the bimodal cloud scheme) + | - | pc2_arcld - | \* (using the Smith scheme with the Cusack vertical + | :green-lbl:`pc2_arcld` + | (using the Smith scheme with the Cusack vertical interpolation method) - - | pc2_initiate - | \* (initiation using the Smith scheme, called here on a - finer vertical grid as per the Cusack method) + | :green-lbl:`pc2_initiate` + | (initiation using the Smith scheme, called here on a + finer vertical grid as per the Cusack method) - | pc2_initiate - | \* (using the Smith scheme with no area cloud representation) + | - - | pc2_checks2 - | \* (further self-consistency checks on cloud-fractions) + | :green-lbl:`pc2_initiate` + | (using the Smith scheme with no area cloud representation) - - | pc2_checks - | \* (repeat the first lot of self-consistency checks again, + | + + | :green-lbl:`pc2_checks2` + | (further self-consistency checks on cloud-fractions) + + | + + | :green-lbl:`pc2_checks` + | (repeat the first lot of self-consistency checks again, just in case we broke something in the mean-time!) - - | pc2_hom_arcld - | \* (finds area cloud fraction using a version of the Cusack + | + + | :green-lbl:`pc2_hom_arcld` + | (finds area cloud fraction using a version of the Cusack method, where the cloud fraction on the finer vertical grid is estimated by applying homogeneous forcing relative to the original grid fields) - - | pc2_homog_plus_turb - | \* (generic homogeneous forcing routine used to interpolate) + | :green-lbl:`pc2_homog_plus_turb` + | (generic homogeneous forcing routine used to interpolate) + .. _subsubsec_pc2_assim: PC2 Data Assimilation ^^^^^^^^^^^^^^^^^^^^^ -.. container:: itemize - - .. container:: tcolorbox - - | pc2_assim - | \* (PC2 reponse to the analysis increments) + | :green-lbl:`pc2_assim` + | (PC2 reponse to the analysis increments) - - | pc2_homog_plus_turb - | \* (generic PC2 homogeneous forcing routine used here for + | :green-lbl:`pc2_homog_plus_turb` + | (generic PC2 homogeneous forcing routine used here for liquid-cloud) - - Estimate change in ice-cloud fraction from the assimilation + | Estimate change in ice-cloud fraction from the assimilation increment to ice-cloud mass. - - | pc2_total_cf - | \* (update bulk cloud fraction due to change in ice cloud + | :green-lbl:`pc2_total_cf` + | (update bulk cloud fraction due to change in ice cloud fraction) - - | pc2_checks - | \* (self-consistency checks on prognostic cloud fractions and + | + + | :green-lbl:`pc2_checks` + | (self-consistency checks on prognostic cloud fractions and water contents) + .. _sec_diags: Diagnostics @@ -6378,13 +6406,12 @@ similar way to above gives .. math:: + \begin{aligned} SD = \int_{-b_s}^{-Q_c} G(s) (-Q_c + b_s) ds + \int_{-b_s}^{-Q_c} G(s) (-s - b_s) ds - -.. math:: - + \\ = (-Q_c + b_s) (1 - C_l) - I1 , - + \end{aligned} and hence :math:`I1` in terms of :math:`SD`. Using this value of :math:`I1` in :eq:`eqn:deltaqclmax` and cancelling @@ -6466,7 +6493,7 @@ We have (equivalent to B.6 from `Wilson and Gregory (2003)`_) .. math:: \frac{ (1-C_l)^2 }{SD} = G(-Q_c) \frac{n+2}{n+1}. -If n=0 (i.e. a ‘top-hat’ function) then +If n=0 (i.e. a 'top-hat' function) then :math:`G(-Q_c) = \frac{1}{2 b_s}` and we can write .. math:: C_l = 1 - \sqrt{ \frac{SD}{b_s} } . @@ -6599,7 +6626,7 @@ increment terms. If your scheme is currently using the homogeneous forcing then there is no need to update the cloud part of the scheme, *provided that you do -not alter values of :math:`T` and :math:`q` after the homogeneous +not alter values of* :math:`T` *and* :math:`q` *after the homogeneous forcing section is called* and that the physical interpretation of your :math:`q` and :math:`T` increments does not change. You need to be careful if you are moving code from one subroutine to another that you @@ -6611,7 +6638,7 @@ which necessitates that condensate increments are already calculated by the scheme, then there is also no need to update the cloud part of the scheme. This currently applies to the boundary layer, where :math:`q_{cf}` is altered by tracer mixing. Like for the homogeneous -schemes, this is provided that you *do not alter :math:`T`, :math:`q` or +schemes, this is provided that you *do not alter* :math:`T` *,* :math:`q` *or condensate values after the injection forcing subroutine is called* and that the physical interpretation of your :math:`q` and :math:`T` increments does not change. @@ -7109,7 +7136,7 @@ References .. _Mellor (1977): Mellor, G. (1977). - *The {Gaussian} cloud model relations*. + *The Gaussian cloud model relations*. J. Atmos. Sci., 34, 356-358. .. _Sommeria and Deardorff (1977): @@ -7169,14 +7196,14 @@ References .. _Field et al. (2014): Field, P.R. and Hill, A.A. and Furtado K. and Korolev, A. (2014). - *Mixed-phase clouds in a turbulent environment. {II: A}nalytic treatment*. + *Mixed-phase clouds in a turbulent environment. II: Analytic treatment*. Q. J. Roy. Meteor. Soc., 140, 870-880. .. _Jakob et al. (1999): Jakob, C. and Gregory, D. and Teixeria, J. (1999). *A package of cloud and convection changes for CY21R3*. - Research Department Memorandum, {ECMWF}, Shinfield Park, Reading {RG2 9AX}, + Research Department Memorandum, ECMWF, Shinfield Park, Reading RG2 9AX, United Kingdom. .. _Morcrette and Petch (2010): diff --git a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst index 300bbaf98c..ef494c000d 100644 --- a/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst +++ b/documentation/source/science_guide/turbulence_schemes/bl_scheme_doc.rst @@ -94,7 +94,7 @@ of specific quantities is also available and the details of the necessary changes are documented in appendix :ref:`Appendix: changing between specific humidities and mixing ratios `. Ultimately turbulent motions are dissipated as heat and so the source -term :math:`{\cal S}` in :eq:`cons_eqn_scal` can include an +term :math:`{\mathcal{S}}` in :eq:`cons_eqn_scal` can include an approximation for that frictional heating, as described in appendix :ref:`Appendix: including the heating from turbulence dissipation `. Finally, the ice cloud contributions in @@ -525,8 +525,8 @@ gradient between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` is compared with that for a parcel lifted adiabatically from grid-level :math:`k_{ct}-1`, in exactly the same way as for the SML parcel ascent (see section :ref:`Calculation of parcel buoyancy excess `). If -:math:`d\theta_v/dz|_{\rm env} > \Gamma_{\rm inv} d\theta_v/dz|_{\rm par}` -between grid-levels +:math:`d\theta_v/dz|_{\mathrm{env}} > \Gamma_{\mathrm{inv}} +d\theta_v/dz|_{\mathrm{par}}` between grid-levels :math:`k_{ct}` and :math:`k_{ct}-1` then NTDSC is set to :math:`k_{ct}-1`; if not then NTDSC is set to :math:`k_{ct}` (recall that grid-levels :math:`k_{ct}-1` and :math:`k_{ct}-2` have already been @@ -718,8 +718,8 @@ finite-difference form of :math:`\overline{w'b}`, see above, :math:`\overline{w'b}` is assumed to be linear between :math:`\overline{w'b}_S` at the surface and zero at a level which must be estimated. The surface layer integration is then from the surface up -to :math:`z_{{\rm K_{SURF}}}`, where :math:`\theta`-level K_SURF is the first -above +to :math:`z_{{\mathrm{K}_{SURF}}}`, where :math:`\theta`-level K_SURF is the +first above :math:`z_i/10`. The level where :math:`\overline{w'b}` is zero is found by linear interpolation across the grid-levels where the diagnosed cloud-free buoyancy flux would become negative. This is where @@ -986,8 +986,8 @@ namelist and :math:`z_{\mathrm{loc}}` is defined below. The orographic blending height, :math:`h_B` (only used within the boundary layer, as defined below), is given by -.. math:: h_B = {\rm max}\left[z_1+(z_{0m})_{\mbox{veg}}, 2^{1/2} \sigma_h - \right] +.. math:: h_B = {\mathrm{max}}\left[z_1+(z_{0m})_{\mathrm{veg}}, 2^{1/2} + \sigma_h \right] where :math:`\sigma_h` is the standard deviation of the height of the subgrid orography and :math:`(z_{0m})_{\mathrm{veg}}` is the vegetative @@ -1027,7 +1027,8 @@ As described in `Lock (2012)`_, the wind shear generated by drainage flows in complex terrain is thought to lead to additional vertical mixing. This wind shear can be approximated as -.. math:: S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, {\cal Z}_d +.. math:: S_d = \frac{\Delta B }{ \Delta z} \, \alpha_d \, t_d \, + {\mathcal{Z}}_d The representative slope of the local terrain, :math:`\alpha_d`, is given by @@ -1045,7 +1046,7 @@ such flows will be underresolved. The above formula is used so that tends to 0.2 for large values. To limit the vertical extent of :math:`S_d` to be below approximately :math:`z=\sigma_h`, a height-dependent factor is included, -:math:`{\cal Z}_d = 0.5( 1 - {\rm tanh}\left[ 4 ((z/\sigma_h)-1) +:math:`{\mathcal{Z}}_d = 0.5( 1 - {\mathrm{tanh}}\left[ 4 ((z/\sigma_h)-1) \right])`. The timescale, :math:`t_d`, takes a fixed value of 30 minutes, for simplicity. @@ -1100,12 +1101,12 @@ where :math:`Pr_N = 0.7`, and the constants :math:`b_{LEM}` and For stable conditions (:math:`Ri > 0`), several forms for the stability functions are available. The 'long-tailed' functions are -.. math:: f_{\rm stable} = \frac{1}{1+g_0 Ri} +.. math:: f_{\mathrm{stable}} = \frac{1}{1+g_0 Ri} Alternative functions, which decrease as :math:`1/Ri^2` with increasing stability are, from `Louis (1979)`_: -.. math:: f_{\rm stable} = \frac{1}{(1+ 5 Ri)^2} +.. math:: f_{\mathrm{stable}} = \frac{1}{(1+ 5 Ri)^2} and the family of "sharp" functions can be written in terms of a transitional Richardson number, :math:`Ri_{t}`, as: @@ -1138,7 +1139,7 @@ the surface to SHARPEST by 200m. A stability dependent Prandtl number (:math:`Pr=f_m/f_h`) is generally used following `Mailhot and Lock (2004)`_ with: -.. math:: Pr=\min \left( Pr_{\rm max}, \, Pr_N(1+2Ri) \, \right). +.. math:: Pr=\min \left( Pr_{\mathrm{max}}, \, Pr_N(1+2Ri) \, \right). The maximum permitted Prandtl number, :math:`Pr_{\mathrm{max}}`, is currently set to :math:`5` for model stability reasons. The stability functions @@ -1292,14 +1293,14 @@ therefore RI(K), are held on the 'half-level' below :math:`\rho`-level K, which is :math:`\theta`-level K-1. In addition to the above, the log profile correction applied to -:math:`{\cal L}_h` (to give :math:`\tilde{{\mathcal{L}}}_h`) must be applied -*after* +:math:`{\mathcal{L}}_h` (to give :math:`\tilde{{\mathcal{L}}}_h`) must be +applied *after* interpolation of :math:`K_h` to level :math:`k+\frac{1}{2}` in order that the correct cancellation with the finite difference scalar gradient in the flux calculation can occur. In the unstable stability functions :eq:`unstable_stab`, however, :math:`\tilde{{\mathcal{L}}}_h` must be calculated on :math:`\theta`-levels -(i.e., the same as :math:`\tilde{{\cal L}}_m` and :math:`Ri`) in order to +(i.e., the same as :math:`\tilde{{\mathcal{L}}}_m` and :math:`Ri`) in order to maintain the same stability dependence. @@ -1376,8 +1377,9 @@ adjustment by the non-local scheme, i.e., using level of turbulent instability (that incorporates the effects of shear) only needs extend some fractional distance into the cloud layer to disrupt the formation of cumulus elements. Thus, if -:math:`z_{\rm loc}> z_{\rm lcl}+ f_{\rm sh} -\left(z_{\rm par}-z_{\rm lcl}\right)`, where :math:`f_{\mathrm{sh}}` is a +:math:`z_{\mathrm{loc}}> z_{\mathrm{lcl}}+ f_{\mathrm{sh}} +\left(z_{\mathrm{par}}-z_{\mathrm{lcl}}\right)`, where :math:`f_{\mathrm{sh}}` +is a tunable parameter (:math:`0z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`, rather @@ -1528,8 +1531,8 @@ k (z/z_i) w_*^3 / u_*^3 )^{-(1/4)}`, would require a complex function of attempted. The formula for the Prandtl number used in the interior in HB93 is also -matched to that used in the surface exchange functions (:math:`Pr_{\rm surf}`, -say). For the UM, +matched to that used in the surface exchange functions +(:math:`Pr_{\mathrm{surf}}`, say). For the UM, .. math:: @@ -1540,8 +1543,8 @@ say). For the UM, giving :math:`Pr_{\mathrm{surf}} = 1` in the neutral limit (compared to 0.75 from :eq:`prandtl_nl`). In the convective limit, -:math:`Pr_{\rm surf}|_{0.1\, z_{\rm h}} \rightarrow 0.9 (w_*/u_*)^{-3/4} = 0.9 -\beta^{3/4} = 0.14` +:math:`Pr_{\mathrm{surf}}|_{0.1\, z_{\mathrm{h}}} \rightarrow 0.9 +(w_*/u_*)^{-3/4} = 0.9 \beta^{3/4} = 0.14` (compared to 0.375 from :eq:`prandtl_nl`). Thus, the Prandtl numbers do not match between the surface layer and interior formulations in the UM. @@ -1606,8 +1609,8 @@ that :math:`K_m^{\mathrm{Sc}}` will tend to :math:`K_h^{\mathrm{Sc}}` to :math:`K_h|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`), given by :eq:`khent`, as :math:`z` tends to :math:`z_{\mathrm{h}}` (and -here no restriction is made on the magnitude of either :math:`{\cal E}_m^{\rm -Sc}` or :math:`{\mathcal{E}}_h^{\mathrm{Sc}}`). +here no restriction is made on the magnitude of either +:math:`{\mathcal{E}}_m^{\mathrm{Sc}}` or :math:`{\mathcal{E}}_h^{\mathrm{Sc}}`). .. _sec_gradadj: @@ -1643,7 +1646,7 @@ for the different constants in :eq:`ws_defn`. Consistent with the mixed layer assumptions underlying the non-local scheme, the flux profile produced by the scheme is assumed to be essentially determined by the specified surface and entrainment values. -Thus, the effect of including this non-local term (:math:`K_h^{\rm surf} +Thus, the effect of including this non-local term (:math:`K_h^{\mathrm{surf}} \gamma_{\theta_{\ell}}`) is to allow the model to maintain more well-mixed :math:`\theta_{\ell}` profiles (i.e., with :math:`\partial \theta_{\ell}/ \partial z` less negative or even @@ -1770,7 +1773,7 @@ The components of :eq:`fg_new` are: - :math:`K_{h,m}^{\mathrm{surf}}= k z_h w_{h,m} \frac{z}{z_h}\left(1-\frac{z}{z_h}\right)^2` -- :math:`K_h^{\rm Sc}= 3.6 k V_{\rm Sc}z_{ml} +- :math:`K_h^{\mathrm{Sc}}= 3.6 k V_{\mathrm{Sc}}z_{ml} \left(\frac{z'}{z_{ml}}\right)^{3}\left(1-\frac{z'}{z_{ml}} \right)^{2}` - @@ -1778,9 +1781,9 @@ The components of :eq:`fg_new` are: with :math:`\gamma_{\chi}=A_{ga}\frac{\overline{w'\chi'}_S}{w_h z_h}` and :math:`A_{ga}=10` -- :math:`\overline{w'\chi'}_{ng}^{\rm Sc}= f^{Sc} \left(F_{\chi}|_{z_h}- - F_{\chi}^{NT}|_{z_{\rm b}} \right)` with - :math:`f^{Sc}=3.5 \, k \, \frac{V_{\rm Sc}}{V_{\rm sum}} +- :math:`\overline{w'\chi'}_{ng}^{\mathrm{Sc}}= f^{Sc} \left(F_{\chi}|_{z_h}- + F_{\chi}^{NT}|_{z_{\mathrm{b}}} \right)` with + :math:`f^{Sc}=3.5 \, k \, \frac{V_{\mathrm{Sc}}}{V_{\mathrm{sum}}} \left(\frac{z}{z_h}\right)^{3}\left(1-\frac{z}{z_h}\right)` - :math:`f_2 = 0.5 \, \frac{z}{z_h}\, 2^{(z/z_h)^4}` @@ -1815,7 +1818,7 @@ except that :math:`w_m` is replaced by its neutral value: Pr_{\mathrm{conv}} )} and the range is now :math:`Pr_{\mathrm{neut}} = 0.75` to -:math:`Pr_{\rm conv} = +:math:`Pr_{\mathrm{conv}} = 0.6`. As with the standard scheme, a constant Prandtl number of 0.75 is used to calculate :math:`K_m^{\mathrm{Sc}}`. @@ -1826,7 +1829,6 @@ Discussion of some of the revisions :name: tab:vscales :header-rows: 2 - * - Formulation - Convective limit - @@ -1903,8 +1905,8 @@ on which the UM was based. This seems an appealing feature (HB's and probably should be considered for the revised scheme (the dash-dotted line in :numref:`Fig. %s ` sets :math:`d^{std} = 10 w_*^2/w_h^2`). Similarly, :math:`f_2` might benefit -from an additional factor of the form :math:`(V_{\rm surf}^3+ -V_{\rm Sc}^3)/ V_{\rm sum}^3` so that it too tends to zero in the +from an additional factor of the form :math:`(V_{\mathrm{surf}}^3+ +V_{\mathrm{Sc}}^3)/ V_{\mathrm{sum}}^3` so that it too tends to zero in the neutral limit. Further analysis of LES and SCM tests will be required to verify this. @@ -1993,7 +1995,7 @@ only difference is in the mixing length, which is calculated as l_{\mathrm{blend}} = W_{1D}l_{\mathrm{bl}}+(1-W_{1D})l_{\mathrm{smag}}, where :math:`l_{\mathrm{bl}}^{-1} = (\kappa z)^{-1} + \lambda_0^{-1}` and -:math:`l_{\rm smag}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, +:math:`l_{\mathrm{smag}}^{-2} = (\kappa z)^{-2} + (c_s \Delta x)^{-2}`, :math:`\kappa` is the von Karman constant and :math:`c_s` is the Smagorinsky constant. Near the surface :math:`l_{\mathrm{bl}}` and :math:`l_{\mathrm{smag}}` are @@ -2010,15 +2012,15 @@ component to the turbulent flux, and this is simply down-weighted by turbulence becomes better resolved. Therefore the full eddy diffusivity is given by -.. math:: K_\chi = \max\left[W_{1D}K_\chi^{\rm NL}, K_\chi(Ri)\right], +.. math:: K_\chi = \max\left[W_{1D}K_\chi^{\mathrm{NL}}, K_\chi(Ri)\right], where :math:`K_\chi^{\mathrm{NL}}` is the non-local diffusivity and :math:`l` in Eq. :eq:`eq-kri` is given by :math:`l_{\mathrm{blend}}` in Eq. :eq:`eq-lblend`. The turbulent flux is then calculated as -.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + W_{1D}F_\chi^{\rm - NL}, +.. math:: F_\chi=-K_\chi\frac{\partial \chi}{\partial z} + + W_{1D}F_\chi^{\mathrm{NL}}, where :math:`F_\chi^{\mathrm{NL}}` is the non-local flux. Therefore when :math:`W_{1D}=1`, the scheme of `Lock et al. (2000)`_ is recovered, @@ -2058,8 +2060,8 @@ The simplest case is for a well-mixed boundary layer, where the appropriate lengthscale is the boundary-layer depth (inversion height). Therefore we set :math:`z_{\mathrm{turb}}=z_h`, which is broadly consistent with `Malavelle et al. (2014)`_, and choose -:math:`\beta=\beta_{\rm bl}=0.15` to give the best match of our function to -that of +:math:`\beta=\beta_{\mathrm{bl}}=0.15` to give the best match of our function +to that of `Honnert et al. (2011)`_. These functions are shown in :numref:`Figure %s `\ (a) and are only dissimilar for small :math:`\Delta @@ -2253,10 +2255,10 @@ be written where the thickness of the inversion is parametrized as :math:`\Delta z_i = -\mbox{min}[V_{\rm sum}^2/\Delta b, 100]` and :math:`L_{rad}` is a +\mathrm{min}[V_{\mathrm{sum}}^2/\Delta b, 100]` and :math:`L_{rad}` is a depth-scale for the radiatively-cooled layer (taken to be 15 :math:`\times -\,\mbox{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To +\,\mathrm{max}[200/z_c, 1]`, where :math:`z_c` is the cloud depth). To allow for a feedback with forcing of entrainment by buoyancy reversal (see appendix :ref:`Appendix: Definitions of the velocity scales `), @@ -2331,8 +2333,8 @@ F|_{z_i}`, so that where the total heat flux :math:`{\mathcal{H}} = \overline{w'\theta_{\ell}'}+ F_{\mathrm{net}}` and -:math:`F_{\rm net} -= F- F|_{z_{\rm b}}`. The net radiative flux relative to the base of the +:math:`F_{\mathrm{net}} += F- F|_{z_{\mathrm{b}}}`. The net radiative flux relative to the base of the mixed layers is simply calculated as .. math:: @@ -2344,7 +2346,7 @@ mixed layers is simply calculated as where NBDSC\ :math:`=1` in SMLs, :math:`{\mathcal{S}}_F` are the temperature increments (in Ks\ :math:`^{-1}`) from the radiation scheme and :math:`F_{\mathrm{net}}|_h` is estimated by extrapolating down from -:math:`F|_{z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} +:math:`F|_{z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} }` using the flux-divergence in grid-level NTML\ :math:`+2` (and similarly for DSC layers). @@ -2440,8 +2442,8 @@ entrainment arising from the model's resolved vertical advection (as discussed in `Lock (2001)`_). This is performed at whichever grid-level the entrainment fluxes are specified, to allow for any entrainment implied by a :math:`\theta_{\ell}` subsidence increment, -:math:`\Theta^{\rm S}` (Ks\ :math:`^{-1}`), at the model grid-level below. The -subsidence +:math:`\Theta^{\mathrm{S}}` (Ks\ :math:`^{-1}`), at the model grid-level below. +The subsidence increments could be obtained directly in the SCM but in the full 3D UM advection increments are dominated by the horizontal component. The subsidence increments are calculated, therefore, from the vertical @@ -2453,8 +2455,8 @@ not be very significant). The interpolated entrainment fluxes given by :math:`w_e` but using an entrainment velocity, :math:`\tilde{w_e}`, that is reduced to allow for any subsidence increments applied to the grid-level below the entrainment flux. To take the case of -:math:`z_{\mbox{\tiny \rm NTDSC}+\frac{1}{2}} < -z_i^{n+1} < z_{\mbox{\tiny \rm NTDSC}+\frac{3}{2}}` as an example, this +:math:`z_{\mathrm{ \mathrm{NTDSC}}+\frac{1}{2}} < +z_i^{n+1} < z_{\mathrm{ \mathrm{NTDSC}}+\frac{3}{2}}` as an example, this reduced entrainment velocity is given by .. math:: \tilde{w_e} = w_e + \tilde{w_S} @@ -2518,7 +2520,7 @@ extended up to :math:`z_i`, while the stable lapse in the free atmosphere, between grid-levels NTML\ :math:`+2` and NTML\ :math:`+3`, :math:`\gamma^{\scriptsize \mathrm{FA}}`, is extrapolated down. Equating these areas gives a quadratic equation in :math:`\Delta z_{disc} = -z_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - z_i` which can be written +z_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} - z_i` which can be written .. math:: :label: zi_interp @@ -2609,8 +2611,8 @@ inaccurate as :math:`z_i` tends to :math:`{\chi}_{\mathrm{ \mathrm{NTML}}}`). Consequently, if the fraction on the right hand side of :eq:`dqt_disc` is greater than 10, double grid-level jumps are used (i.e., -:math:`\Delta \chi = {\chi}_{\mbox{\tiny \rm NTML}+2} - -{\chi}_{\mbox{\tiny \rm NTML}}`). Finally, note that +:math:`\Delta \chi = {\chi}_{\mathrm{ \mathrm{NTML}}+2} - +{\chi}_{\mathrm{ \mathrm{NTML}}}`). Finally, note that :eq:`dqt_disc` implicitly assumes the structure of the :math:`\theta_{\ell}` and :math:`q_t` profiles across the inversion grid-level are consistent with the diagnosed :math:`z_i`. This is very @@ -2721,11 +2723,11 @@ radiative flux is extrapolated down from divergence in the grid-level above the inversion as representative of the free-atmospheric divergence. Second, since the microphysical flux is generated within the cloud, :math:`F_{\chi}^{ppn}|_{z_t} = -{F_{\chi}}^{ppn}_{\mbox{\tiny \rm NTML}+\frac{3}{2}}`. Finally, the +{F_{\chi}}^{ppn}_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}}`. Finally, the subsidence flux-divergence across level :math:`\mathrm{ \mathrm{NTML}}` and :math:`\mathrm{ \mathrm{NTML}}+1` is assumed to be associated with the inversion so :math:`{F_{\chi}}^{Subs}|_{z_h} = -{F_{\chi}}^{Subs}_{\mbox{\tiny \rm NTML}-\frac{1}{2}}`. Thus, the +{F_{\chi}}^{Subs}_{\mathrm{ \mathrm{NTML}}-\frac{1}{2}}`. Thus, the finite-difference form of :eq:`fxtot_zi` becomes .. math:: :label: fxtot_zi_fd @@ -2769,8 +2771,8 @@ the total grid-level flux, :eq:`fxtot_interp`, is used to calculate the entrainment fluxes, it is straightforward to ensure that the net budget of the inversion grid-level, namely :math:`- ( -F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{3}{2}} - -F_{\chi}^{Tot}|_{\mbox{\tiny \rm NTML}+\frac{1}{2}})/\Delta z`, +F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{3}{2}} - +F_{\chi}^{Tot}|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}})/\Delta z`, is consistent with the entrainment/subsidence balance. In other words, to use :math:`\theta_{\ell}` as an example, if the inversion is rising (falling) then @@ -2966,7 +2968,7 @@ imposed at the height of the temperature inversion :math:`z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}`) and :math:`K_m|_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}}` is calculated from :eq:`kmsurf` and :eq:`kmtop`, noting the use of -the :math:`{\cal E}` factors. +the :math:`{\mathcal{E}}` factors. .. _sec_entr_prof: @@ -2977,8 +2979,8 @@ An inversion is defined as being resolved when it extends above the flux-level above the usual entrainment interface level (see section :ref:`Diagnosis of inversion thickness `), i.e. when -.. math:: z_{\mbox{\tiny \rm NTML}+\frac{1}{2}} + \Delta z_i > z_{\mbox{\tiny - \rm NTML}+\frac{3}{2}} +.. math:: z_{\mathrm{ \mathrm{NTML}}+\frac{1}{2}} + \Delta z_i > z_{\mathrm{ + \mathrm{NTML}}+\frac{3}{2}} When this happens, there is no subgrid inversion diagnosis and the entrainment parametrization follows the methodology given in @@ -3029,8 +3031,8 @@ turbulence forcing and the inversion jump change slowly compared to the timestep. Whilst this is true for atmospheric :math:`\theta_{\ell}` and :math:`q_t`, the latter is not true for tracers with a small boundary layer concentration. Consequently, for a tracer field :math:`\chi`, the -parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mbox{\tiny \rm -NTML}+\frac{1}{2}} }` are calculated from +parametrized entrainment fluxes :math:`\overline{w'\chi'}_{ z_{\mathrm{ +\mathrm{NTML}}+\frac{1}{2}} }` are calculated from :eq:`fluxinterp` but are implemented through an equivalent entrainment eddy-diffusivity given by: @@ -3751,8 +3753,8 @@ representing, respectively, the large-scale, gust and small-scale turbulent contributions to the velocity. Locally, Monin-Obukhov theory then gives -.. math:: |\bar{\bf u} + {\bf u}_g({\bf x}) | = \frac{u_*({\bf x})}{k} - \Phi_m({\bf x}). +.. math:: |\bar{\mathbf{u}} + {\mathbf{u}}_g({\mathbf{x}}) | = + \frac{u_*({\mathbf{x}})}{k} \Phi_m({\mathbf{x}}). We ignore the spatial variation of :math:`\Phi_m`, expecting that the principal effect of locally stronger winds is to increase the local @@ -4287,10 +4289,10 @@ Two approaches are available in uncoupled configurations of the model. #. Explicit Treatment of Ice Form Drag - `L{\ (2012)`_ have suggested a simple + `Lupkes et al. (2012)`_ have suggested a simple parametrization of the form drag coefficient of marginal ice that has been found to perform well in comparison to aircraft measurements - (`Elvidge et al. (2016)`_). `L{\ (2015)`_ + (`Elvidge et al. (2016)`_). `Lupkes and Gryanik (2015)`_ have extended the parametrization to include the effects of stability. When coupled to CICE, it is intended that a more elaborate scheme will be used, but this scheme is useful for application in @@ -4303,12 +4305,12 @@ Two approaches are available in uncoupled configurations of the model. :math:`u(z)` will in general exhibit a mixed character, but it may be taken as the developed flow over open sea, as in - `L{\ (2012)`_, or may be interpolated between the + `Lupkes et al. (2012)`_, or may be interpolated between the developed flows over open sea or pack ice, depending on the ice - fraction, as in `L{\ (2015)`_. In principle, it will + fraction, as in `Lupkes and Gryanik (2015)`_. In principle, it will be subject to the effects of stability, but since the free-board does not much exceed 0.5m, these effects are small - (`L{\ (2015)`_) and the flow may be taken as neutral + (`Lupkes and Gryanik (2015)`_) and the flow may be taken as neutral up to :math:`h_f`. Hence, .. math:: :label: eq:int_u2 @@ -4321,18 +4323,18 @@ Two approaches are available in uncoupled configurations of the model. the wind on the model's lowest atmospheric level. Because this will be significantly above :math:`h_f`, the stability dependence of :math:`C_d` should be considered here (again see - `L{\ (2015)`_). :math:`U_1` may be interpreted as + `Lupkes and Gryanik (2015)`_). :math:`U_1` may be interpreted as the wind at a specific height, or, consistenly with the flux-difference form of the momentum equation, as the layer-averaged velocity. This distinction affects the numerical value of :math:`C_d`, but does not otherwise affect the foregoing equation. If using the original version of the scheme - (`L{\ (2012)`_), :math:`C_d` must be taken as the + (`Lupkes et al. (2012)`_), :math:`C_d` must be taken as the neutral drag coefficient. Note also that various approximations may be made in Equation :eq:`eq:int_u2`. - `L{\ (2012)`_ approximate + `Lupkes et al. (2012)`_ approximate :math:`(\log(h_f/z_0) -1)^2 +1` as :math:`(\log(h_f/z_0) )^2`; while - `L{\ (2015)`_ approximate it as + `Lupkes and Gryanik (2015)`_ approximate it as :math:`(\log(h_f/z_0) -1)^2`. Here we retain the full expression. If, in a unit area, there are :math:`N` floes, each of crosswind @@ -4369,19 +4371,19 @@ Two approaches are available in uncoupled configurations of the model. (1-A) C_{ds} L_s + A C_{di} L_i \right ], where the sheltering factor is taken to be the same over ice and - water. `L{\ (2012)`_ provides parametrizations for + water. `Lupkes et al. (2012)`_ provides parametrizations for quantities such as :math:`h_f`, while `Elvidge et al. (2016)`_ provide suggested values for the constants in the scheme, based on observations. In using these values in the Unified Model, :math:`c_e` should be increased by about 30% to represent the effect of differing approximations of the logarithmic wind profile. - For scalar transfer `L{\ (2015)`_ suggest adding a + For scalar transfer `Lupkes and Gryanik (2015)`_ suggest adding a contribution to the sensible heat flux to represent the impact of form drag; however, the mechanistic physical basis of the scheme they propose is unclear. Moreover, when combined with the interfacial drag, this suggests scalar transfer much larger than observed by - `Schr{\ (2003)`_. Consequently, no enhancement of the + `Schroder et al. (2003)`_. Consequently, no enhancement of the scalar transfer coefficient by form drag is included. The overall drag coefficients are now set by interpolation in the ice @@ -4541,10 +4543,10 @@ where \right| The effective roughness for momentum is derived by setting the total -effective surface stress, **:math:`\tau`**\ :math:`_{0(eff)}`, to the +effective surface stress, :math:`\boldsymbol{\tau}`\ :math:`_{0(eff)}`, to the sum of the surface stress over a flat surface with the same vegetative -roughness, **:math:`\tau`**\ :math:`_{0(f)}`, and the orographic -pressure drag force at the surface, **:math:`\tau`**\ :math:`_{0(p)}`. +roughness, :math:`\boldsymbol{\tau}`\ :math:`_{0(f)}`, and the orographic +pressure drag force at the surface, :math:`\boldsymbol{\tau}`\ :math:`_{0(p)}`. The stresses are evaluated in terms of the velocity at height z\ :math:`_{c}` above the surface. z\ :math:`_{c}` is currently set to 2\ :math:`^{1/2}\sigma _{h}` where :math:`\sigma _{h}` is the standard @@ -4994,7 +4996,7 @@ included in the Unified Model as an additional explicit (in terms of time discretisation) stress. Following `Wood et al. (2001)`_ we define :math:`{\bf\tau}_{\mathrm{orog}}` to be -.. math:: {\bf\tau}_{\rm orog}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, +.. math:: {\bf\tau}_{\mathrm{orog}}(z)=\left({F_p}_x,{F_p}_y\right)e^{-z/\ell}, where :math:`{\mathbf{F}_p}=({F_p}_x,{F_p}_y)`, :math:`{F_p}_x` and :math:`{F_p}_y` are the grid-box average :math:`x` and :math:`y` @@ -5013,7 +5015,7 @@ constrained to be at least 100 m. If the steep-hill expression is to be used, the surface stress applied is almost identical to that used in the effective roughness -parametrization (Eq. :eq:`2.1.10`, namely: +parametrization (Eq. :eq:`2.1.10`), namely: .. math:: :label: eq:dragsteep @@ -5023,7 +5025,7 @@ parametrization (Eq. :eq:`2.1.10`, namely: the main difference being the dependence on the height scale :math:`\ell` rather than :math:`z_c`. Similarly, if the `Wood and Mason (1993)`_ low-hill expression is used, the surface -stress is given by the equivalent of (Eq. :eq:`2.1.16`, +stress is given by the equivalent of (Eq. :eq:`2.1.16`), namely: .. math:: :label: eq:draglow @@ -5314,7 +5316,7 @@ the bottom row discretization is obtained: where -.. math:: A_{0}=-{\cal I}_{1}\frac{\Delta +.. math:: A_{0}=-{\mathcal{I}}_{1}\frac{\Delta tK_{u}\Big|_{1}}{z_{1}(z_{3/2}-z_{1/2})},\quad B_{0}=1-A_{0}. Equations :eq:`eq:tridiag`, @@ -5330,9 +5332,8 @@ equations. When the elimination procedure takes place where :math:`\delta u_{1/2}^{'}`, :math:`\beta` are available quantities. Furthermore, -.. math:: \bar{\tau}_{x}^{*}\Big|_{0}=\left({\cal I}_{1}-{\cal - E}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\cal I}_{1}\left(K_{u}\frac{\partial\delta - u^{*}}{\partial z}\right)\Big|_{0} +.. math:: + \bar{\tau}_{x}^{*}\Big|_{0}=\left({\mathcal{I}}_{1}-{\mathcal{E}}_{1}\right)\tau_{x}^{n}\Big|_{0}+{\mathcal{I}}_{1}\left(K_{u}\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0} Approximating :math:`\left(\frac{\partial\delta u^{*}}{\partial z}\right)\Big|_{0}\approx\frac{\delta u_{1/2}^{*}-\delta u_{0}^{*}}{z_{1/2}}`, @@ -5396,10 +5397,10 @@ Considering that, :eq:`eq:dX_star` would re-produce :eq:`eq:sppf_inc1`, which is re-written below, -.. math:: \frac{\delta X^{*}}{\Delta t}=({\cal I}_{1}-{\cal - E}_{1})\left(\frac{\partial F^{n}}{\partial z}+S\right)+{\cal - I}_{1}\frac{\partial}{\partial z}\left(K_{X}\frac{\partial\delta - X^{*}}{\partial z}\right) +.. math:: \frac{\delta X^{*}}{\Delta + t}=({\mathcal{I}}_{1}-{\mathcal{E}}_{1})\left(\frac{\partial F^{n}}{\partial + z}+S\right)+{\mathcal{I}}_{1}\frac{\partial}{\partial + z}\left(K_{X}\frac{\partial\delta X^{*}}{\partial z}\right) and thus the following discretization is obtained, on :math:`\theta`-levels: @@ -5422,9 +5423,9 @@ or, where, -.. math:: A_{k}=-{\cal I}_{1}\frac{\Delta - tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}=-{\cal - I}_{1}\frac{\Delta +.. math:: A_{k}=-{\mathcal{I}}_{1}\frac{\Delta + tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; + C_{k}=-{\mathcal{I}}_{1}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}=1-A_{k}-C_{k}. @@ -5519,15 +5520,15 @@ Similarly the corresponding discrete equations for where, -.. math:: A_{k}^{'}=-{\cal I}_{2}\frac{\Delta - tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; C_{k}^{'}=-{\cal - I}_{2}\frac{\Delta +.. math:: A_{k}^{'}=-{\mathcal{I}}_{2}\frac{\Delta + tK_{X}\Big|_{k+1/2}}{(z_{k+1}-z_{k})(z_{k+1/2}-z_{k-1/2})},\; + C_{k}^{'}=-{\mathcal{I}}_{2}\frac{\Delta tK_{X}\Big|_{k-1/2}}{(z_{k+1/2}-z_{k-1/2})(z_{k}-z_{k-1})},\quad B_{k}^{'}=1-A_{k}^{'}-C_{k}^{'}, for :math:`k=L,\ldots,2,\quad A_{L}=0`. -.. math:: A_{1}^{'}=-{\cal I}_{2}\Delta +.. math:: A_{1}^{'}=-{\mathcal{I}}_{2}\Delta t\frac{K_{X}\Big|_{3/2}}{z_{3/2}(z_{2}-z_{1})},\quad B_{1}^{'}=1-A_{1}^{'} and the approximation @@ -5544,7 +5545,6 @@ calculations take place for the scalar variables: .. list-table:: - * - CALL bdy_impl3(): - set up coefficients for :eq:`eq:dX_disc_top`, :eq:`eq:dX_disc` and do a downward sweep; @@ -5761,8 +5761,8 @@ hand-side of :eq:`eq:FTLstar`, relationship, which is simply the definition of the time-weighted averaging consistent with the new scheme: -.. math:: \overline{H^{*}}\equiv{\cal I}_{1}H^{*}-{\cal - E}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\cal I}_{1}E^{*}-{\cal E}_{1}E^{n}. +.. math:: + \overline{H^{*}}\equiv{\mathcal{I}}_{1}H^{*}-{\mathcal{E}}_{1}H^{n},\qquad\overline{E^{*}}\equiv{\mathcal{I}}_{1}E^{*}-{\mathcal{E}}_{1}E^{n}. Therefore, once :math:`\overline{H^{*}},\overline{E^{*}}` have been computed, :math:`H^{*}`, :math:`E^{*}` can be computed as follows: @@ -5797,7 +5797,7 @@ has to be adjusted if evaporation exhausts any of the moisture stores during the timestep or if the tile has a melting snowcover. Limited evaporation -^^^^^^^^^^^^^^^^^^^ +""""""""""""""""""" Downward surface moisture fluxes are added to canopy moisture or, if the surface temperature is below freezing, snowcover. @@ -5841,7 +5841,7 @@ surface) is not limited and does not draw on the conserved moisture stores. Snowmelt -^^^^^^^^ +"""""""" Classical surface energy balance neglects snowmelt heat fluxes. If :math:`T_*>T_m` for a snow-covered tile and sufficient snow is @@ -5879,8 +5879,8 @@ to the tile heat and moisture fluxes. The model with the above changes coded seems to work stably but the surface fluxes (and therefore the boundary layer increments) are only qualitatively correct. They seem to be overestimated by the above -scheme. [\ *Could it be that coefficients :math:`D_{j}`, -:math:`\psi_{j}` need to be modified in the second sweep?*] +scheme. [\ *Could it be that coefficients* :math:`D_{j}` *,* +:math:`\psi_{j}` *need to be modified in the second sweep?*] Blending height coupling ^^^^^^^^^^^^^^^^^^^^^^^^ @@ -5942,7 +5942,8 @@ parametrization of non-local momentum fluxes (see section stability dependence in :eq:`tau_nl` that can be written as -.. math:: f_{stab} = - \frac{a_{stab} z_{\rm h}/L }{1 - a_{stab} z_{\rm h}/L} +.. math:: f_{stab} = - \frac{a_{stab} z_{\mathrm{h}}/L }{1 - a_{stab} + z_{\mathrm{h}}/L} for the Obhukov length, :eq:`1.1.4`, :math:`<0` (i.e., unstable boundary layers) and the empirical constant @@ -5951,7 +5952,7 @@ decreases in magnitude (i.e., surface heating increases and wind stress decreases). The final thermal speed, in units of ms\ :math:`^{-1}`, is given simply by -.. math:: {\rm Thermal} \, {\rm Speed} = f_{stab} \, w_* +.. math:: {\mathrm{Thermal}} \, {\mathrm{Speed}} = f_{stab} \, w_* Wind gust: stash 3,463 and 3,515 (scale-dependent) -------------------------------------------------- @@ -6085,15 +6086,16 @@ Combining :eq:`bl_scaling` with and subsuming the Prandtl number into the other constants, for surface-driven boundary layer mixing we can write: -.. math:: K_m^{\rm surf}= \frac{\tau_{\rm surf}}{2} \, \overline{w'^2} = - \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, f(z') = k z_{\rm - h}w_m f(z') +.. math:: K_m^{\mathrm{surf}}= \frac{\tau_{\mathrm{surf}}}{2} \, + \overline{w'^2} = \frac{\tau_{turb}}{2} \, \frac{2.66 }{C_{ws}^{2/3}} w_m^2 \, + f(z') = k z_{\mathrm{h}}w_m f(z') which then gives :math:`\tau_{\mathrm{surf}} = C_{ws}^{2/3} k z_{\mathrm{h}}/ (1.33 w_m)`. An analogous timescale can be derived for top-driven mixing in decoupled -stratocumulus layers, :math:`\tau_{\rm Sc} = g_1 k z_{\rm ml}/ (1.33 \, -V_{\rm Sc})`. +stratocumulus layers, :math:`\tau_{\mathrm{Sc}} = g_1 k z_{\mathrm{ml}}/ (1.33 +\, +V_{\mathrm{Sc}})`. There are two options to derive a TKE diagnosis from the Ri-based scheme and then combine with the non-local TKE (selected via var_diags_opt). @@ -6156,7 +6158,7 @@ done via the convection scheme. Therefore an option *l_conv_tke* is provided to include an estimate of the TKE due to parametrized convection within the diagnostic. This is given by: -.. math:: e_{\rm conv} = \left(\frac{M}{g\rho \times CCA}\right)^2 +.. math:: e_{\mathrm{conv}} = \left(\frac{M}{g\rho \times CCA}\right)^2 where :math:`M` is the convective updraft mass flux (Pa s\ :math:`^{-1}`) and CCA is the convective cloud area. The final @@ -6236,7 +6238,7 @@ where the subscript :math:`_S` indicates the surface flux; calculation is described in section :ref:`Calculation of \Delta_F `. Various depth parameters are given by :math:`\zeta_s = -(z_{\rm ml}-\tilde{z_c})/z_{\rm ml}`, +(z_{\mathrm{ml}}-\tilde{z_c})/z_{\mathrm{ml}}`, :math:`\zeta = (z_{\mathrm{ml}}-z_c)/z_{\mathrm{ml}}` and :math:`\zeta_r = \zeta + Br (1-\zeta)`. :math:`z_{\mathrm{ml}}` is the mixed-layer depth, :math:`z_c` is the cloud depth and :math:`\tilde{z_c}` is the @@ -6350,7 +6352,7 @@ zero should then give a reasonably accurate measure of cloud-base. The formula for :math:`V_{\mathrm{br}}` was derived using dimensional arguments and comparison with LES data: -:math:`\chi_s = -{q_{\ell}}_{\rm ct}(1+(L/c_p)\alpha_L) / ( +:math:`\chi_s = -{q_{\ell}}_{\mathrm{ct}}(1+(L/c_p)\alpha_L) / ( \Delta q_t - \alpha_L \Delta \theta_{\ell})`, where :math:`{q_{\ell}}_{\mathrm{ct}}` is the cloud-top liquid water mixing ratio, :math:`L` is the latent heat of vaporisation of water, :math:`c_p` the @@ -6407,13 +6409,13 @@ and similarly for :math:`q_f` (noting that currently The only other explicit account of variable cloud fraction is in :eq:`vbr` for which it is assumed that buoyancy reversal can only occur for cloudy air underlying cloud-free air (assuming maximum -overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mbox{max}[ +overlap). Thus, the cloud fraction factor, :math:`C_{fac} = \mathrm{max}[ 0.0, -\Delta C_F ]`, where -:math:`\Delta C_F = {C_F}_{\mbox{\tiny \rm NTML}+2} - -{C_F}_{\mbox{\tiny \rm NTML}}` if a subgrid inversion is diagnosed +:math:`\Delta C_F = {C_F}_{\mathrm{ \mathrm{NTML}}+2} - +{C_F}_{\mathrm{ \mathrm{NTML}}}` if a subgrid inversion is diagnosed (because :math:`{C_F}_{\mathrm{ \mathrm{NTML}}+1}` is currently meaningless) and :math:`\Delta C_F = -{C_F}_{\mbox{\tiny \rm NTML}+1} - {C_F}_{\mbox{\tiny \rm NTML}}` if not. +{C_F}_{\mathrm{ \mathrm{NTML}}+1} - {C_F}_{\mathrm{ \mathrm{NTML}}}` if not. A more complete decomposition is not possible given a cloud scheme in the model :raw-latex:`\cite[]{smith90}` which does not allow discrete identification of in-cloud and out-of-cloud profiles. The cloud-fraction @@ -6856,7 +6858,8 @@ its local :math:`Ri`-based scheme are used. In :eq:`asymp_ml`, the definition of :math:`\lambda_m` only is altered to -.. math:: \lambda_m = \mbox{max}\left[40,\, 0.3 z_{\rm loc}, 2 h_B \right] +.. math:: \lambda_m = \mathrm{max}\left[40,\, 0.3 z_{\mathrm{loc}}, 2 h_B + \right] and both :math:`\lambda_m` and :math:`\lambda_h` are not reduced (to 40m) above the boundary layer top. It is possible these modifications @@ -6876,7 +6879,6 @@ the UKCA code owner before lodging the change. .. list-table:: - * - Boundary layer inputs to UKCA - - @@ -7034,7 +7036,6 @@ Appendix: Notation .. list-table:: - * - Finite difference notation - @@ -7061,7 +7062,6 @@ Appendix: Notation .. list-table:: - * - Model variables - @@ -7092,7 +7092,6 @@ Appendix: Notation .. list-table:: - * - Thresholds - @@ -7123,7 +7122,6 @@ Appendix: Notation .. list-table:: - * - Layer definitions and parameters - @@ -7189,7 +7187,6 @@ Appendix: Notation .. list-table:: - * - Other parameters - @@ -7275,8 +7272,8 @@ References A. P. Lock and A. R. Brown and M. R. Bush and G. M. Martin and R. N. B. Smith (2000). - *{A New Boundary Layer Mixing Scheme. Part I: Scheme Description and - Single-Column Model Tests}*. + *A New Boundary Layer Mixing Scheme. Part I: Scheme Description and + Single-Column Model Tests*. Mon. Wea. Rev., 128, 3187-3199. .. _Lock et al. (2001): @@ -7290,15 +7287,15 @@ References G. M. Martin and M. R. Bush and A. R. Brown and A. P. Lock and R. N. B. Smith (2000). - *{A New Boundary Layer Mixing Scheme. Part II: Tests in Climate and - Mesoscale Models}*. + *A New Boundary Layer Mixing Scheme. Part II: Tests in Climate and + Mesoscale Models*. Mon. Wea. Rev., 128, 3200-3217. .. _Lock (2001): A. P. Lock (2001). - *{The Numerical Representation of Entrainment in Parametrizations of - Boundary Layer Turbulent Mixing}*. + *The Numerical Representation of Entrainment in Parametrizations of + Boundary Layer Turbulent Mixing*. Mon. Wea. Rev., 129, 1148-1163. .. _Brown and Grant (1997): @@ -7310,16 +7307,16 @@ References .. _Wood et al. (2001): N. Wood and A. R. Brown and F. E. Hewer (2001). - *{Parametrizing the effects of orography on the boundary layer: An - alternative to effective roughness lengths}*. + *Parametrizing the effects of orography on the boundary layer: An + alternative to effective roughness lengths*. Quart. J. Roy. Meteor. Soc., 127, 759-777. .. _Brown et al. (2008): A. R. Brown and R. J. Beare and J. M. Edwards and A. P. Lock and S. J. Keogh and S. F. Milton and D. N. Walters (2008). - *{Upgrades to the Boundary-Layer Scheme in the Met Office Numerical Weather - Prediction Model}*. + *Upgrades to the Boundary-Layer Scheme in the Met Office Numerical Weather + Prediction Model*. Bound.-Layer Meteor., 128, 117-132. .. _Troen and Mahrt (1986): @@ -7332,8 +7329,8 @@ References .. _Holtslag and Moeng (1991): A. A. M. Holtslag and C.-H. Moeng (1991). - *{Eddy Diffusivity and Countergradient Transport in the Convective - Atmospheric Boundary Layer}*. + *Eddy Diffusivity and Countergradient Transport in the Convective + Atmospheric Boundary Layer*. J. Atmos. Sci., 48, 1690-1698. .. _Lock (1998): @@ -7365,7 +7362,7 @@ References .. _Beljaars and Holtslag (1991): A.C.M. Beljaars and A.A.M. Holtslag (1991). - *{Flux Parametrization over Land Surfaces for Atmospheric Models}*. + *Flux Parametrization over Land Surfaces for Atmospheric Models*. J. Appl. Meteor., 30, 327-341. .. _Wood and Mason (1993): @@ -7384,21 +7381,20 @@ References .. _Redelsperger et al. (2000): J.-L. Redelsperger and F. Guichard and S. Mondon (2000). - *{A Parametrization of Mesoscale Enhancement of Surface Fluxes for - Large-Scale Models}*. + *A Parametrization of Mesoscale Enhancement of Surface Fluxes for + Large-Scale Models*. J. Climate, 13, 402-421. .. _Holtslag and Boville (1993): A. A. M. Holtslag and B. A. Boville (1993). - *{Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate - Model}*. + *Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate Model*. J. Climate, 6, 1825-1842. .. _Bolton (1980): Bolton, D. (1980). - *{The Computation of Equivalent Potential Temperature}*. + *The Computation of Equivalent Potential Temperature*. Mon. Wea. Rev., 108, 1046-1053. .. _Brown (1996): @@ -7425,7 +7421,7 @@ References .. _Bush et al. (1999): Bush, M. R. and A. P. Lock and R. N. B. Smith (1999). - *{Testing of the new boundary layer scheme in the Mesoscale Model}*. + *Testing of the new boundary layer scheme in the Mesoscale Model*. NWP Tech Report, 260. .. _Cullen and James (1994): @@ -7437,7 +7433,7 @@ References .. _Derbyshire (1997): Derbyshire, S. H. (1997). - *{Recommendations for UM parametrization of stable boundary layers}*. + *Recommendations for UM parametrization of stable boundary layers*. Cardington Tech Note, 38. .. _Driedonks (1982): @@ -7449,8 +7445,8 @@ References .. _Edwards (2007): Edwards, J. M. (2007). - *{Oceanic Latent Heat Fluxes: Consistency with the atmospheric hydrological - and energy cycles and general circulation modeling}*. + *Oceanic Latent Heat Fluxes: Consistency with the atmospheric hydrological + and energy cycles and general circulation modeling*. J. Geophys. Res., 112, D06115. .. _Essery et al. (2001): @@ -7485,7 +7481,7 @@ References .. _Lock (2012): Lock, A. P. (2012). - *{Stable boundary layer modelling at the Met Office}*. + *Stable boundary layer modelling at the Met Office*. ECMWF Workshop on Diurnal Cycles and the Stable Boundary Layer, 137-148. .. _Louis (1979): @@ -7498,21 +7494,21 @@ References Mason, P. J. (1986). *On the parametrization of orographic drag*. - {ECMWF Seminar on Physical Parametrization for Numerical Models of the - Atmosphere}, 2, 139-165. + ECMWF Seminar on Physical Parametrization for Numerical Models of the + Atmosphere, 2, 139-165. .. _Nicholls and Turton (1986): Nicholls, S. and J. D. Turton (1986). - *{An observational study of the structure of stratiform cloud sheets. Part - II: Entrainment}*. + *An observational study of the structure of stratiform cloud sheets. Part + II: Entrainment*. Quart. J. Roy. Meteor. Soc., 112, 461-480. .. _Stage and Businger (1981): Stage, S. A. and J. A. Businger (1981). - *{A model for entrainment into a cloud-topped marine boundary layer. Part - I: Model description and application to a cold-air outbreak episode}*. + *A model for entrainment into a cloud-topped marine boundary layer. Part I: + Model description and application to a cold-air outbreak episode*. J. Atmos. Sci., 38, 2213-2229. .. _Turton and Nicholls (1987): @@ -7545,8 +7541,8 @@ References .. _Zilitinkevich (1975): Zilitinkevich, S. S. (1975). - *{Comments on ''A model for the dynamics of the inversion above a - convective boundary layer''}*. + *Comments on ''A model for the dynamics of the inversion above a convective + boundary layer''*. J. Atmos. Sci., 32, 991-992. .. _Siems et al. (1990): @@ -7560,7 +7556,7 @@ References Mailhot, J. and Lock, A. P. (2004). *An examination of several parametrizations of mixing lengths in a stable - boundary layer: the {GABLS} case*. + boundary layer: the GABLS case*. .. _Honnert et al. (2011): @@ -7607,17 +7603,17 @@ References zone*. Q. J. R. Meteorol. Soc., 136, 927-943. -.. _L{\ (2012): +.. _Lupkes et al. (2012): - C. L{\ (2012). + C. Lupkes and V. M. Gryanik and J. Hartmann and E. L. Andreas (2012). *A parametrization, based on sea ice morphology of the neutral atmospheric drag coefficients for weather prediction and climate models*. J. Geophys. Res., 117, D13112. https://doi.org/10.1029/2012JD017630 -.. _L{\ (2015): +.. _Lupkes and Gryanik (2015): - C. L{\ (2015). + C. Lupkes and V. M. Gryanik (2015). *A stability-dependent parametrization of transfer coefficients for momentum and heat over polar sea to be used in climate models*. J. Geophys. Res. Atmos., 120, 552-581. @@ -7632,9 +7628,9 @@ References Atmos. Chem. Phys., 16, 1545-1563. https://doi.org/10.5194/acp-16-1545-2016 -.. _Schr{\ (2003): +.. _Schroder et al. (2003): - D. Schr{\ (2003). + D. Schroder and T. Vihma and A. Kerber and B. Brummer (2003). *On the parameterization of turbulent surface fluxes over heterogeneous sea ice surfaces*. J. Geophys. Res., 108(C6), 3195. @@ -7658,7 +7654,6 @@ References .. _Hsu et al. (2017): Hsu, J. and Lien, R. and D'Asaro, E. A. and Sanford, T. B. (2017). - *Estimates of Surface Wind Stress and Drag Coefficients in {T}yphoon - {M}egi*. + *Estimates of Surface Wind Stress and Drag Coefficients in Typhoon Megi*. J. Phys. Oceanogr., 47, 545-565. https://doi.org/10.1175/JPO-D-16-0069.1