Arakawa and Wu (2013) regarded the convective updraft fraction (σ) dependence of vertical eddy MSE flux in CRM simulation as a solution of parameterizing sub-grid scale convection in variant-resolved GCMs, and also applied σ dependence in the derivation of unified parameterization closure. Instead of using GATE, whose forcing terms are invariant with time, as the analyzed case in AW13, we use DYNAMO case
that applies time-variant forcing terms in the simulation to represent different phases of convection development cycles. The analysis of σ dependence of convection in DYNAMO is similar to the convection in GATE, so this parameter is also appropriate for unified expression of sub-grid scale convection in variant forcing cases. We also compare the σ dependence of different convection strength in the simulation of DYNAMO case. Variables of all time steps in CRM simulation are categorized into four groups according to the precipitation rates. It shows that within all groups of convection, σ dependence are less variant than resolution dependence, thus σ dependence is also an appropriate parameter for various strength of convection.
A UP closure, using the parameter σ and some variables output from the conventional parameterization, is derived in AW13. This closure uses σ to relax the full adjustment convection fluxes in conventional parameterization scheme because the σ in conventional parameterization is usually assumed to be much smaller than 1 and thus can be neglected. Theoretically, when the σ become closer to 1, the grid-scale variables should perceive significant components of sub-grid scale convection, so the sub-grid scale convection shouldn’t be as large as the condition when σ is ignorable, or it will double-count the convection fluxes both in grid-scale and sub-grid scale. Numerically, the UP closure multiplies the convection fluxes by
(1 − σ)2, which means that if the convective updraft is close to grid-scale, the role of
sub-grid scale convection can almost be neglected. For the GCM with the resolution
of CRM, the function of parameterization can be smoothly “closed” when multiplying (1 − σ)2 because all convection fluxes in the simulation are grid scale. In our
research, we combine a conventional cumulus convection parameterization:
Zhang-McFarlane parameterization scheme, with the UP closure in AW13, to diagnose the unified convection fluxes in the simulation of DYNAMO. The average variables of sub-domain in DYNAMO are input to ZM scheme to parameterize sub-grid convection fluxes and use σ derived from UP closure to adjust the convection fluxes. We use the ratio of ZM parameterized convection fluxes, and the multiplication of in-cloud moist static energy and the in-cloud vertical velocity derived from the boundary convection parameterization (VKE budget) to derive σ.
This convection fluxes can also be recognized as the consumption rate of environment instability by the parameterized in-cloud convection.
The result shows that the distribution of σ derived from UP closure performs better at cloud-base (here defined as LCL) than higher layers, but still have some inconsistence in both scale and position with the σ derived from CRM simulation.
This situation could be result from the triggering mechanics in conventional
parameterization, since it decides the CAPE and parameterized convection fluxes, and the coefficients and closure of in-cloud vertical velocity. The ensemble averaged sub-grid scale convection fluxes derived from UP scheme are similar to the ones from
ZM scheme since 𝜎𝑈𝑃 are concentrate at lower values, making the adjustment parameter (1 − σ)2 close to 1, and also far from the fluxes derived from CRM. The
main purpose of UP scheme is to put the awareness of σ into the parameterized convection fluxes, so the relaxation adjustment for parameterized fluxes at higher σ has shown the effects of UP scheme. The gap between fluxes in CRM and UP is mainly attributed to the closure in ZM scheme, which can relax or strengthen the adjustment rate by its closure. The possible progression in the future might be the methods to deal with tilting convection, the coefficients of in-cloud vertical velocities and the closure to decide the strength of cloud base mass fluxes.
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Appendix
The following contents are the more details about deriving the sub-grid scale convection fluxes of DYNAMO case from Zhang-McFarlane cumulus convection parameterization scheme (ZM Scheme) by offline method in this study. The variables that will be input into ZM scheme are first simulated in VVM, which is a CRM with discretized places of variables (see the figure (2) in Jung and Arakawa (2008)), so the horizontal interpolation of u, v and vertical interpolation of w to the point of temperature is necessary for place consistency. The original horizontal size of variables, with 256 x 256 grids in horizontal and 34 layers in vertical, is expanded to 258 x 258 grids, in order to include the periodic boundary of domain variables u and v into the numerical interpolation method. The variables of first layer are excluded, except the layer of w, since the first layer of u, v and θ is under the ground.
To interpret the variables in GCM-like grid cells, the profiles of temperature, mixing ratio and etc. are averaged in the specific sub-domains of VVM grids. Before inputting the variables into ZM scheme, we make two profiles that including one current profile, which is assumed to be equilibrium state here (see the theorem in section 3.3), and one imaginary profile that the cumulus convection has not yet adjust the profile toward current profile. The tendencies due to grid-scale advection, apparent
heat and humidity are removed from, and upward heat flux and moisture flux at surface (within planetary boundary) layer from the mean profiles at current time are added to the imaginary profile. In ZM scheme, the program deals with one vertical column in each iterative call, so it’s necessary to mapping variables from 3-dimension (i, k, j) to 1-dimension (k). The vertical layers of variables are also flipped so that the
largest number of k represents the lowest layer.
ZM scheme use CAPE of every vertical column to evaluate the strength of convection updraft in the vertical column. To integrate CAPE, the base and the top of convection should be determined first. The launching level, which is assumed to be the base of convection, of parcel in ZM is defined as the layer of maximum moist static energy under PBL top (about 470m in VVM). The original parcel properties are set to be the same as launching level, and additionally be added the temperature perturbation which we arbitrarily set equal to 0.1K to trigger convection easier. For level higher than launching level, we use the arithmetic average value of temperature, mixing ratio at level k and k+1 to represent environment variables (just as the entrainment of mid-level between k and k+1). ZM scheme mix up the environment and parcel entropy, total mixing ratio, and mass flux relative to cloud base, with specified entrainment rate (∂(normalized mass of parcel))/ ∂z = −1.0x10−3 /m (parcel fractional mass
entrainment rate), from launching level to current level, and the entropy of parcels is then inverted to determine temperature and saturated mixing ratio. If total mixing ratio is larger than saturated mixing ratio, drop part of the liquid water out (ZM assumes a certain amount of liquid water is hold in cloud, about 1.0x10−3) and add latent heat released from the water back to the profile. LCL (lifting condensation level) is also determined in this process, and if LCL is higher than 600 mb, no deep convection is permitted. Note that the added latent heat cause the increase of temperature, thus the saturated mixing ratio have to be re-calculated until the error is small enough.
After adjusting vertical profile of parcel to saturated or unsaturated condition, ZM scheme calculate the virtual temperature of parcel and environment above launching level. The difference of virtual temperature between parcel and environment is used to evaluate the obtained buoyancy of parcel at every layer, and 0.5K is also added to the parcel and in-cloud properties to trigger convection. The CAPE of vertical column is derived from the integration of buoyancy from the launching level to the level that buoyancy reversal takes part. Note that buoyancy may reverse several times in a column, so the scheme chooses the largest CAPE to be the determined value.
ZM scheme specify the properties of updraft and downdraft, including mass flux, entrainment, detrainment, and dry static energy of plumes by the equations in
Description of the NCAR Community Atmosphere Model (CAM 3.0). The values
of mass flux and entrainment/detrainment are normalized by the updraft mass flux at cloud base. These in-cloud properties can be related to the adjustment rate of CAPE toward equilibrium state and thus determine the cloud base mass flux. To evaluate the adjustment rate of CAPE, ZM scheme regards the vertical column as several boxes, and use the advection of dry static energy, mixing ratio of updraft, downdraft and environment by updraft and downdraft mass flux to derive the change rate of environment temperature and mixing ratio by per unit mass flux. The change of CAPE equals to the integration of vertical buoyancy change due to the effect of cumulus convection (see equation (24)). The parcel temperature change due to the change of sub-cloud layer properties during convection is also included.
∂A ⁄ ∂t = ∑lel(convection top) ∂(buoyancy) ⁄ ∂t ∗ dz
z=mx(launching level) (24) Cloud base mass flux =
max (– (CAPE − CAPEequlibrium) (time scale ∗ (∂A ∂t⁄ ⁄ )), 0) (25)
For cloud base mass flux, we use the modified closure instead of the conventional one in ZM scheme. The cloud base mass flux derived in conventional ZM scheme
(equation (25)) is replaced by an arbitrarily value for convenience, and re-calculated in our functions by the closure out of ZM scheme (will be shown later). After the deriving of cloud mass flux, dry static energy and mixing ratio of updraft, downdraft and environment of each layer are determined, thus we can also derive the vertical eddy flux of grid by the combination of dry static energy and mixing ratio with the updraft/downdraft mass flux, and in-cloud moist static energy by the updraft properties.
The parameterized vertical eddy flux (dimension of density is included), in-cloud MSE and in-cloud w are used to derive the convective updraft coverage (σ) of UP scheme closure (see section 3.2). We also revise the cloud base mass flux as
cloud base mass fluxrevised = (CAPEadv− CAPEnoadv) (time scale ∗ (∂A ∂t⁄ )
adv)
⁄ ,
(26)
which is from the definition of Xiao et al. (2015), so the revised vertical eddy flux is
𝑤′ℎ′
̅̅̅̅̅
𝑍𝑀 =( 𝑤̅̅̅̅̅′ℎ′
𝑍𝑀) /ρ ∗ cloud base mass fluxrevised/arbitrary defined mass flux . (27)