The simulation of DYNAMO active phase (within 15 days; from 2011/10/15 to 2011/10/29) in this study is simulated by the Vector Vorticity Model (VVM), using 256 km x 256 km horizontal domain with 1km grid size, and 34 vertical stretching grids from 100m at lower boundary to 1000m at about 19km height. To realize the statistics of sub-grid scale convection fluxes in different GCM-like grid sizes, we divide the original 256km x 256km domain into different sizes of sub-domain (1, 2, 4, 8, 16, 32, 64, 128, 256km), regarding each sub-domain as a grid cell in GCM, and evaluate the sub-grid scale convection strength. Fig. 2 is the example of whole domain divided by 32 km size of sub-domains.
Figure 2 Snapshot of whole horizontal domain at 500th time steps, 3km height. Shaded color
represents the vertical velocity, and red dot line grids are 32 km sub-domain grids. Only the
sub-domains with any convective updraft grid (w ≥ 0.5m/s) are chosen as samples to calculate
< 𝑤ℎ̅̅̅̅ > and < 𝑤̅̅̅̅̅̅ >. ′ℎ′
We use the definition following AW13: 𝑤̅, ℎ̅ are the averaged vertical velocity and moist static energy of CRM grids in a single sub-domain grid, which can be
regarded as GCM-like grids, and 𝑤′, ℎ′ are the deviation of CRM grid values from 𝑤̅, ℎ̅, respectively. For the sub-domain size grid cells, 𝑤̅, ℎ̅ are resolvable while 𝑤′, ℎ′
are the unresolvable variables. The sub-grid scale vertical eddy fluxes of moist static
energy (MSE) can be written as 𝑤̅̅̅̅̅̅ where the overbar represents the sub-domain ′ℎ′ average values. Since the parameterized convection is only triggered in the grid points that reach a particular threshold, 𝑤̅̅̅̅̅̅ of sub-domains with any grid point that have ′ℎ′ vertical velocity larger than or equal to 0.5m/s, are chosen as the convective ensemble members. The ensemble-averaged 𝑤̅̅̅̅̅̅ is denoted as < 𝑤′ℎ′ ̅̅̅̅̅̅ >, which represents ′ℎ′ the convection fluxes that need to be parameterized in the ensemble members.
The degree of parameterization that is required for sub-grid scale convection in
sub-domains can be evaluated by the ratio between vertical eddy fluxes and total fluxes of MSE (< 𝑤̅̅̅̅̅̅ >/< 𝑤ℎ′ℎ′ ̅̅̅̅ >) as shown in fig. 3 for a selected level at 3km height,
which is close to the layer of largest < 𝑤̅̅̅̅̅̅ >. The results show that when the ′ℎ′ sub-domain sizes are much larger than the scale of cumulus convection, the sub-grid scale convections dominate the total convection strength. The degree of required parameterization dramatically decreases as the sub-domain sizes become closer to 1km since sub-grid scale convection is more resolvable in finer resolutions. If there is an ideal unified convection parameterization, it should pick up the main sources of convection at conventional GCM resolutions and ease its task as the resolution gradually reaches to cumulus convection scales, as the results of AW13. The relations between total and eddy convection fluxes at other heights also show the similar results
as 3km height (fig. 4 and fig. 5).
Figure 3 < 𝑤̅̅̅̅̅̅ > and < 𝑤ℎ′ℎ′ ̅̅̅̅ > divided by 𝐶𝑝 (m/s K) for different sub-domain sizes (km) at 3 km height.
Figure 4 The vertical profile of < 𝑤ℎ̅̅̅̅ > divided by 𝐶𝑝 (m/s K) for different sub-domain sizes (km).
Figure 5 The vertical profile of < 𝑤̅̅̅̅̅̅ > divided by 𝐶′ℎ′ 𝑝 (m/s K) for different sub-domain sizes (km).
AW13 pointed out that the standard deviations of < 𝑤̅̅̅̅̅̅ > and < 𝑤ℎ′ℎ′ ̅̅̅̅ > in the same sub-domain size are quite large, about the scaling of variables themselves, showing that there exist significant uncertainty in the resolution dependence. The statistics of resolution dependence in DYNAMO case also show the similar results (fig. 6). Using resolution as the index of sub-grid scale convection is not an ideal method since different phases of convection are all categorized in the same groups.
Following the analyses in AW13, the convective updraft coverage ratio of each
sub-domain is used as an alternative index of sub-grid scale convection in our experiment. The convective updraft coverage ratio (denoted as σ) is defined as the number of CRM grid points with vertical velocity larger than or equal to 0.5 m/s
divided by the number of total grid points in the sub-domain (GCM-like grid). Fig. 7 is the σ dependence of < 𝑤̅̅̅̅̅̅ > and < 𝑤ℎ′ℎ′ ̅̅̅̅ > in the case of 4km sub-domain size at 3km height. The distribution of < 𝑤̅̅̅̅̅̅ > is likely a bimodal distribution, which ′ℎ′ shows that the sub-grid scale convection decline for both higher and lower σ.
Furthermore, < 𝑤̅̅̅̅̅̅ > dominates < 𝑤ℎ′ℎ′ ̅̅̅̅ > not only in coarser resolutions but also for lower σ in the relatively high resolutions (shown in fig. 7). For higher σ,
< 𝑤̅̅̅̅̅̅ >/< 𝑤ℎ′ℎ′ ̅̅̅̅ > decrease since the sub-domains themselves are more dominated
by convective updrafts, making the sub-grid scale convection more precisely resolved by grid scale processes.
Figure 6 < 𝑤̅̅̅̅̅̅ > (green line) and < 𝑤ℎ′ℎ′ ̅̅̅̅ > (red line) divided by 𝐶𝑝 (m/s K) and the corresponding standard deviation for different sub-domain sizes (km).
Figure 7 < 𝑤̅̅̅̅̅̅ > (green line) and < 𝑤ℎ′ℎ′ ̅̅̅̅ > (red line) divided by 𝐶𝑝 (m/s K) for different σ at 3 km height and 4km sub-domain size. Blue line is for < 𝑤̅̅̅̅̅̅ > that use the single ′ℎ′ top-hat assumption.
To compare the resolution dependence and σ dependence at the same chart,
< 𝑤̅̅̅̅̅̅ >/< 𝑤ℎ′ℎ′ ̅̅̅̅ >) of different sub-domain sizes and different σ at 3 km height are
shown in fig. 8. Similar to the results of AW13, the ratio of eddy and total vertical fluxes of MSE is more likely to be dependent on σ, rather than sub-domain sizes. If we choose < 𝑤̅̅̅̅̅̅ >/< 𝑤ℎ′ℎ′ ̅̅̅̅ > of sub-domain size = 4km as examples, the distribution of
ratio range from 10% for the largest σ to 88% for about σ = 0.1. If σ = 0.5 is considered, the < 𝑤̅̅̅̅̅̅ >/< 𝑤ℎ′ℎ′ ̅̅̅̅ > range from 41% to 57%, which is much narrower than the distribution range of sub-domain size = 4km. The ratio for other σ and heights also show the similar results, indicating that the σ dependence of < 𝑤̅̅̅̅̅̅ >/< 𝑤ℎ′ℎ′ ̅̅̅̅ > is
more consistent than resolution dependence. The results also show that < 𝑤̅̅̅̅̅̅ >/<′ℎ′ 𝑤ℎ̅̅̅̅ > of sub-domains with larger σ is smaller than those with lower σ because that if
the convective updrafts develop to sub-domain grid size, the variables of grid cell will resolve more of its sub-grid scale processes, and its degree of parameterization should be reduced to prevent double counting.
Figure 8 The ratio of < 𝑤̅̅̅̅̅̅ >/ < 𝑤ℎ′ℎ′ ̅̅̅̅ > (%) for various combination of sub-domain size
(horizontal axis) and σ (vertical axis) at 3km height
In the conventional cumulus parameterization, the updrafts of in-cloud and environment in the same grid cell of GCM are assumed to be homogeneous. This assumption is called single top-hat profile assumption, which means that there exists only one kind of vertical MSE flux for in-cloud and another for the environment in each GCM grid cell. In the following contents, we are going to testify this assumption in DYNAMO case by applying the analysis methods in AW13. The vertical velocity and MSE of CRM grids in each sub-domain are classified into two categories: in-cloud and environment, according to whether the vertical velocity of CRM grid is convective (w ≥ 0.5m/s) or not. For the in-cloud grids, the vertical velocities and MSE are replaced by the in-cloud average variables. These variables are used to derive the in-cloud vertical flux of MSE, and the processes are also
conducted for those environment grids to derive the environment vertical flux of MSE.
< 𝑤̅̅̅̅̅̅ > and < 𝑤ℎ′ℎ′ ̅̅̅̅ > that are modified by the single top-hat assumption are calculated and plotted with σ index (shown as the blue line in fig. 7). The difference
between green and blue lines is mainly attributed to the multi-structure of in-cloud grid cells for large σ. Although the single top-hat assumption used in conventional
parameterization underestimates the sub-grid scale convection strength, this simplifying assumption still do well for sub-domains with small σ, which are the main groups of all convection samples.
The results mentioned above are the statistics of DYNAMO case including convective grids (𝑤̅ + 𝑤′≥ 0.5m/s) for all time steps, but the difference of active
and suppress phase of convection might be covered. Xiao et al. (2015) pointed out that the resolution dependence of ZM-parameterized sub-grid scale convection is sensitive to the convection strength, implying that weaker and stronger convection might have inconsistency on resolution dependence and σ dependence. We categorize the 15-day CRM simulation data (1080 time steps) into four groups according to the domain-averaged precipitation rates (see fig. 9), and use the analysis methods of resolution and σ dependence just as we conducted to 15-day simulation before. The analysis results of these four groups of convection are shown in fig. 10 and 11. The resolution-dependent < 𝑤̅̅̅̅̅̅ > and < 𝑤ℎ′ℎ′ ̅̅̅̅ > rank according to their precipitation rates, indicating that the sub-domain size dependence of convection is dependent on precipitation rates. The σ dependence of four precipitation rate quartiles shown in fig.
12, 13 and 14 illustrate that < 𝑤ℎ̅̅̅̅ > (red lines), < 𝑤̅̅̅̅̅̅ > (green lines) and single ′ℎ′
top-hat sub-grid scale convection (blue lines) of different precipitation rates are similar to each other for σ < 0.4. Overall, for variant thermodynamic forcing applied in cloud-resolved simulation, σ dependence is a better choice to evaluate the sub-grid scale convection than the sub-domain sizes since most of the samples are under σ <
0.4. The results of sub-grid scale convection analysis and four precipitation rate quartiles analysis, using the CRM simulation of DYNAMO case, put the application of the σ dependence of sub-grid scale convection that proposed in AW13 to more extensive condition.
Figure 9 Space-averaged precipitation (mm/hr) of simulation using DYNAMO time-variant
forcing. The percentile rank of precipitation rate is showed at the right side of the chart.
Figure 10 < 𝑤ℎ̅̅̅̅ > divided by 𝐶𝑝 (m/s K) for different sub-domain sizes (km) at 3 km height.
The data of four quartiles are shown as following marks: plus, cross, square, and circle
according to its precipitation rank.
Figure 11 < 𝑤̅̅̅̅̅̅ > divided by 𝐶′ℎ′ 𝑝 (m/s K) for different sub-domain sizes (km) at 3 km height. The data of four quartiles are shown as following marks: plus, cross, square, and circle
according to its precipitation rank.
Figure 12 < 𝑤ℎ̅̅̅̅ > divided by 𝐶𝑝 (m/s K) for different σ at 3 km height. The data of four quartiles are shown as following marks: circle, square, cross and plus, according to its
precipitation rank.
Figure 13 < 𝑤̅̅̅̅̅̅ > divided by 𝐶′ℎ′ 𝑝 (m/s K) for different σ at 3 km height. The data of four quartiles are shown as following marks: circle, square, cross and plus, according to its
precipitation rank.
Figure 14 < 𝑤̅̅̅̅̅̅ >, that use single top-hat assumption, divided by 𝐶′ℎ′ 𝑝 (m/s K) for different σ at 3 km height. The data of four quartiles are shown as following marks: circle, square, cross and plus, according to its precipitation rank.