# A Simple Parameterization to Enhance the Computational Time in the Three Layer Dry Deposition Model for Smooth Surfaces

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{+}

_{cbl}) and internal integral calculation intervals for each particle diameter D

_{p}(0.01–100 µm) and friction velocity u* (0.01–100 m/s). The particle concentration, as a solution to the particle flux equation, is obtained and modeled numerically by performing the left Riemann sum using MATLAB software. On the other hand, the number of subdivisions N of the Riemann sum was also parameterized for each D

_{p}and ${u}^{*}$ in order to lessen the numerical calculation time. From a numerical point of view, the new parameterizations were tested by several computers; about 78% on the average of the computation time was saved when compared with the original algorithm. In other words, on average, about 1.2 s/calculation was gained, which is valuable in climate models simulations when millions of dry deposition calculations are needed.

## 1. Introduction

_{d}), which is a measure of the effectiveness of the deposition mechanism, is derived from the particle flux equation. It is governed by certain boundary conditions [16,17,18,19]. For example, V

_{d}for particles being transported under gravitational settling and under the effect of turbophoresis is dominant for particles with large relaxation time τ

_{p}(i.e., particles with larger mass) [20]. Another example is the influence of the friction velocity (u*); increasing u* results in enhancement in V

_{d}for particles with small τ

_{p}(i.e., small particles). In the situations of inhomogeneous air turbulence, increasing u* can also enhance the deposition of particles with large τ

_{p}because of enhanced turbophoresis.

_{0}and y

_{cbl}, which are presented in Figure 1) is considered a major challenge. For example, Kallio and Reeks [22] developed a power law expression to achieve a mathematical representation based on a Direct Numerical Simulation (DNS) after Kim et al. [23]. After that, it was assumed that CBL height could be set to 30 (dimensionless units; y

^{+}

_{cbl}) above a smooth surface [18,19]. This height was set to 100 for large particles [15] and 200 for rough surfaces [21]. However, the chosen value for the CBL height (i.e., y

_{cbl}) and starting level (i.e., y

_{0}) above the surface area ought to be defined based on the particle concentration profile within the boundary layer. This depends on several factors, including u*, particle diameter (D

_{p}), and the deposition mechanism.

_{cbl}as a function of u* and D

_{p}. This parameterization was developed for a wide particle size range (D

_{p}0.01–100 µm) and u* within the range 0.01–100 m/s. This reflects typical conditions for dry deposition in most physical systems and applications. The parameterization was then implemented in the well-known three-layer dry deposition model in order to improve the computational time required to calculate the dry deposition velocity V

_{d}.

## 2. Materials and Methods

#### 2.1. Three-Layer Dry Deposition Model

_{Fickian}is the particle flux due to Fickian diffusion (Brownian and Eddy), J

_{Gravitational}is the particle flux due to gravitational settling, and the sum represents the particle fluxes due to other mechanisms not included in this part of research. For this equation to be valid, a number of assumptions must be confirmed: steady-state particle flux perpendicular to the surface, the particle concentration gradient exists only very close to the deposition surface (i.e., within the CBL), there are no sources or sinks of particles within the boundary layer, and the surface is a perfect sink for particles.

_{0}from the surface (i.e., y

_{0}= r

_{p}); thus, the particle concentration is zero in the fluid right above the surface. The CBL has an upper limit above which particle concentration becomes homogeneous (i.e., dC/dy = 0) [27]. This implies that the top of the CBL layer (i.e., y

_{cbl}) is set at the maximum concentration (i.e., C

_{∞}). In that sense, the key parameters in accurate estimation for the dry deposition velocity are to have the right value for the height of the concentration boundary layer (i.e., y

_{cbl}) and to have a well-behaved profile for the particle concentrations (i.e., C) within the boundary layer.

_{p}[m

^{2}s

^{−1}] is the turbulent (Eddy) diffusivity coefficient, C [m

^{−3}] is the particle concentration, D [m

^{2}s

^{−1}] is the Brownian diffusivity, y [m] is the height from the surface, i = 0, 1, −1 according to the surface orientations, vertical, horizontal facing up (floor) and horizontal facing down (ceiling), respectively, and V

_{s}[ms

^{−1}] is the particle velocity of gravitational settling, which is the constant velocity (terminal) towards the surface [28,29]:

^{−2}] is the gravitational acceleration, D

_{p}[m] is the particle diameter, ρ

_{p}[kg m

^{−3}] is the particle density, ρ [kg/m

^{−3}] is the gas density, C

_{D}[unitless] is the drag coefficient and C

_{c}[unitless] is the Cunningham slip correction coefficient. The deposition velocity ${V}_{d}$ is calculated as

^{+}as a function of the dimensionless height ${y}^{+}$ [32]

^{+}

_{o}and y

^{+}

_{cbl}leads to accurate calculation for V

_{d}. The determination of the appropriate value of y

^{+}

_{cbl}for each D

_{p}at a certain u* will be discussed hereafter.

#### 2.2. Parametrization for y^{+}_{cbl}

^{+}

_{cbl}as an unknown quantity to be determined for each particle size and friction velocity. In other words, the upper limit of the integral in Equation (7) was set to y

^{+}

_{max}, which satisfies the second boundary condition in the numerical investigation; when C

^{+}reaches 1. The method of left Reimann sum was adopted to evaluate the integral using MATLAB

^{TM}software (including Simulink toolboxes). It was evaluated by limiting the height from the surface with the parameter y

^{+}

_{max}to a certain value aiming for the determination of y

^{+}

_{cbl}. The distance from the surface to y

^{+}

_{max}was divided equally into a proper number of intervals (N) that lead to an accurate solution for V

_{d}.

^{+}

_{max}up to 1000, and N up to 1000 subdivisions for y

^{+}. The y

^{+}

_{cbl}and N for each parameter varied to obtain convergent solutions for V

_{d}. The variation of y

^{+}

_{cbl}(0–1000) was performed continuously and repeatedly using a MATLAB

^{TM}code that can repeat the solution of Equation (7) until the value of V

_{d}convergent to a certain value; at that end, the code gives us the values of V

_{d}and y

^{+}

_{cbl}as outputs at a certain friction velocity where N = 1000 at this stage. After we parameterize y

^{+}

_{cbl}and V

_{d}get known and convergent to non-parameterized value we proceeding the parameterization for N in order to diminish the calculation time further by using y

^{+}

_{cbl}as input and N as variable in this stage until the same V

_{d}achieved.

^{+}

_{cbl}according to certain assumptions that meet the conditions considered in the model needs. For example, it was assumed to have y

^{+}

_{cbl}= 30 for a smooth surface and fine particles [18,19,33]. For micron particles, it was suggested that y

^{+}

_{cbl}= 100 [15]. for a rough surface it was suggested that y

^{+}

_{cbl}= 200 [21]. Figure 2 illustrates the variation of y

^{+}

_{cbl}with D

^{+}

_{p}by using a certain u* (= 100 m/s) and N (= 1000 subdivisions)

^{+}

_{p}is about 50 and above, none of the previous assumptions for y

^{+}

_{cbl}is satisfactory. Furthermore, for D

^{+}

_{p}about 0.3, it is a waste of computational time for calculation to take the y

^{+}

_{cbl}larger than 5, so our new parameterizations, as will be shown in the next section, determine a proper y

^{+}

_{cbl}and N for each D

^{+}

_{p}at a certain u*.

^{+}

_{cbl}as a function of D

_{p}and u*. The second one for N as a function of D

_{p}and u*. These parameterizations were utilized in the three-layer deposition model so that the most suitable y

^{+}

_{cbl}and N are used as pre-set input parameters in the V

^{+}

_{d}calculation. The enhancement in the computational time was then compared between the original algorithm and the new one with these parameterizations.

## 3. Results and Discussion

#### 3.1. A Parameterization for Fickian Diffusion

^{+}

_{cbl}parameter, the integral in Equation (7), was evaluated using MATLAB code by taking the height from the surface y

^{+}

_{o}= r

^{+}

_{p}to y

^{+}

_{max}= 1000, aiming to determine y

^{+}

_{cbl}. The distance from the surface to y

^{+}

_{max}was divided equally into a number of subdivisions N = 1000, which leads to an accurate solution for V

^{+}

_{d}. That was calculated for each particle diameter and friction velocity determined in the range mentioned above. Figure 3a shows the y

^{+}

_{cbl}profile; from the figure, one can notice the dependence of y

^{+}

_{cbl}on the dimensionless particle relaxation time τ

^{+}

_{p}(or particle size D

_{p}) and u*. Figure 3b shows the 3-D matrix we obtained for y

^{+}

_{cbl}as a function D

_{p}and u*.

^{+}

_{cbl}decreases as the D

_{p}increases for small u* < 0.02 m/s, where for 0.02 m/s < u* < 0.3 m/s the y

^{+}

_{cbl}decreases for D

_{p}(<8 $\mu $m) and then increases smoothly as D

_{p}increases. For 0.3 m/s < u* < 10 m/s, the behavior is almost the same, but the y

^{+}

_{cbl}decreases for small D

_{p}(<0.3 $\mu $m) and then increases steeper than the previous range as D

_{p}increases. Finally, for u* > 10 m/s, the y

^{+}

_{cbl}decreases for small D

_{p}(<0.3 $\mu $m) and then increases steeply as D

_{p}increases.

^{+}

_{d}) for each particle diameter at a certain friction velocity, we varied N in the code until we obtained the accurate value of V

^{+}

_{d}for each D

_{p}and u*. The 3-D matrix we obtained for N as a function D

_{p}and u* is illustrated in Figure 4. The largest N (= 100) is for the smallest D

_{p}and u* and then decreases (the smallest value is 50) in general as D

_{p}or u* increases (Figure 4). This behavior has varied for many particle diameters, where N fluctuated for large values of u* and D

_{p}.

#### 3.2. The Inclusion of Gravitational Settling

^{+}

_{cbl}parameter was investigated by solving the integral in equation (6) was evaluated using MATLAB code and taking the height from the surface with the parameter y

^{+}

_{max}= 1000 again. The area under the curve was calculated by partitioning the area to N = 1000 subdivisions, and then the integral was calculated for each D

_{p}and u* and in the range determined. The profile of the y

^{+}

_{cbl}and the 3-D matrices we obtained for y

^{+}

_{cbl}as a function D

_{p}and u* is shown in Figure 5a,b; respectively.

^{+}

_{d}at the same N for the same D

_{p}and u* in this case is identical with that of the Fickian diffusion effect alone.

^{+}

_{cbl}and N as a function D

_{p}and u*. At this stage, we transformed to another phase, which is the goal of our study, which was the optimization of calculation time by updating the code to select the appropriate y

^{+}

_{cbl}and N based on the input u* and D

_{p}.

#### 3.3. Computation Advantage by the Parameterization

^{+}

_{d}. The calculations were performed by four computers for the two codes for selected particle diameters for three different friction velocities (u* = 0.01, 0.1, and 1 m/s). Specifications of computers used for testing running time are summarized in Table 1.

^{+}

_{d}. To add a flavor to our calculations, we found the time gained as percent time gained (% time) by the percent error method and as the time difference. The results are summarized in Table 2, Table 3 and Table 4. Notice that Table 2 is for friction velocity u* = 0.01 m/s, Table 3 is for friction velocity u* = 0.1 m/s, and Table 4 is for friction velocity u* = 1 m/s.

^{2}) [34] into a grid whose spatial resolution is 50 km × 50 km or at most 100 km × 100 km, which is relatively high resolution, the computer time on the fastest computers to simulate an experiment along one century may spend several weeks typically due to the large number of calculations required [14]. It seems affordable to use parameterizations that can save about 78% of the computational time (i.e., increasing the computing power), which stimulates us to include more factors that affect the dry deposition and/or use a grid with higher resolution.

^{+}

_{cbl}and N parameters are the factors that affect the running time; the time gained in the case of gravitational settling and Fickian diffusion together is identical to that we obtained for Fickian diffusion alone because N does not change in the two cases, and we use in the optimized code the y

^{+}

_{cbl}matrix that shown in Figure 5b since it includes y

^{+}

_{cbl}matrix for Fickian diffusion (i.e., we have used just one matrix) in order to shorten the time elapsed for codes to obtain the accurate value of V

^{+}

_{d}.

#### 3.4. The Effect of Parameterization on V^{+}_{d} Calculations

^{+}

_{d}itself before and after optimization. The results we obtained after optimization, which appear as dashed curves, are almost identical to the results we have before optimization, which appear as solid curves, as shown in Figure 6a for Fickian diffusion. This figure implicitly tells us that we gain time by using the optimized code without accuracy loss. We verify that by quantitative comparison between parameterized versus non-parameterized dry deposition velocities shown in Figure 6b, notice that the residuals of the order of magnitude of 10

^{−5}and the residual points are distributed around zero, which indicates that the parameterization is valuable from the point of calculation of view since the parameterization will not affect the accuracy of the calculation in any application including climate modeling.

^{+}

_{p}> 0.07) for small friction velocity (u* = 0.01 m/s) when we take all the residuals into account, and the other residuals point almost lie on residual = 0. In fact, the behavior of residuals, in this case, is selective due to the gravitational settling mechanism.

^{+}

_{p}> 6.46 × 10

^{−6}(i.e., D

_{p}> 0.4 μm), where the residuals diverge when τ

^{+}

_{p}> 8.45 × 10

^{−5}(i.e., D

_{p}> 1 μm) for friction velocity (u* = 0.1 m/s), and finally the divergence accrue after τ

^{+}

_{p}= 7.24 × 10

^{−4}(i.e., D

_{p}> 5 μm) for friction velocity (u* = 1 m/s), as shown in Figure 7c. These details led us to conclude that our parameterizations, in the case of gravitational settling mechanism is included, are excellent in the case of sub-micron (i.e., fine and ultrafine) particles, whereas in the case of super-micron particles (1 μm < D

_{p}< 10 μm) the parameterizations are excellent for u* ≥ 1 m/s since the residuals of an order of magnitude 10

^{−5}and the residual points surrounding the residual is zero.

## 4. Conclusions

^{+}

_{cbl}) that gives accurate dry deposition velocity (V

^{+}

_{d}) above a smooth surface in a shorter calculation time according to the particle’s diameter (D

_{p}) and the friction velocity (u*).

_{p}range (10 nm–100 μm) and u* range (0.01–100 m/s) in addition to the y

^{+}

_{cbl}.

^{+}

_{cbl}and N depending on D

_{p}and u* as the first step of the calculation. This procedure saved up to 78% (average) for each calculation when compared to the time taken for the same code without parameterization, where the comparison was by four computers that have different specifications. In other words, on average, 1.2 s/calculation can be saved, which means that our parameterization can lessen the accumulated time in the case of big data calculations in large-scale climate models.

^{+}

_{d}we obtained without parameterization with that we obtained with parameterization. The results confirm that the accuracy did not affect the results for the dry deposition calculation value, but the calculation time gain is valuable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Unit | Description |

C | m^{−3} | Particle concentration within the boundary layer. In dimensionless form C ^{+} = C/C_{∞} C _{∞} is the particle concentration above the boundary layer or far away from the surface |

C_{c} | -- | Cunningham slip correction coefficient |

D | m^{2} s^{−1} | Brownian diffusivity of the particle, D = k_{B} T Cc/3πμ D_{p}in dimensionless form D ^{+} = (ε_{p} + D)/ν |

D_{p} | m | Particle diameter, in dimensionless form D^{+}_{p} = D_{p} u*/ν |

J | m^{−2} s^{−1} | Total particle flux across the concentration boundary layer towards the surface. ${J}_{Fickian}$ is particle flux due to Brownian and Eddy diffusions. ${J}_{n}$ is the particle flux across the concentration boundary layer due to other mechanisms to be included in the model in the future |

k_{B} | Joule/K | Boltzmann constant |

m_{p} | kg | Particle mass |

r_{p} | m | Particle radius, in dimensionless form r^{+}_{p} = r_{p} u*/ν |

T | K | Absolute temperature |

u* | m s^{−1} | Friction velocity |

V_{d} | m s^{−1} | Deposition velocity onto a surface, in dimensionless form V^{+}_{d} = V_{d}/u* |

$\u2329V{\prime}_{y}^{2}\u232a$ | m^{2} s^{−2} | Air wall normal fluctuating velocity intensity, in dimensionless [16,22]: ${\u2329V{\prime}_{y}^{2}\u232a}^{+}=\frac{\u2329V{\prime}_{y}^{2}\u232a}{{\left(u*\right)}^{2}}={\left[\frac{0.005{\left({y}^{+}\right)}^{2}}{1+0.002923{\left({y}^{+}\right)}^{2.128}}\right]}^{2}$ |

$\u2329V{\prime}_{py}^{2}\u232a$ | m^{2} s^{−2} | Particle wall normal fluctuating velocity intensity [31]: $\u2329V{\prime}_{py}^{2}\u232a=\u2329V{\prime}_{y}^{2}\u232a{\left[1+\frac{{\tau}_{p}}{{\tau}_{L}}\right]}^{-1}$ ${\u2329V{\prime}_{py}^{2}\u232a}^{+}=\u2329V{\prime}_{py}^{2}\u232a/{\left(u*\right)}^{2}$ |

y | m | Vertical distance from the surface, in dimensionless form y^{+} = y u*/ν |

y_{0} | m | Distance from the surface at which the particle with a radius r_{p} is deposited, in dimensionless form y^{+}_{o} = y_{0} u*/ν |

y_{cbl} | m | Depth of the concentration boundary layer above which dC/dy = 0 in dimensionless form y ^{+}_{cbl} = y_{cbl} u*/ν |

μ | kg m^{−1} s^{−1} | Dynamic viscosity of the fluid |

ρ | kg m^{−3} | Fluid density |

τ_{L} | s | Lagrangian time-scale of the fluid [31]: ${\tau}_{L}={\nu}_{t}/\u2329V{\prime}_{y}^{2}\u232a$ ${\tau}_{L}^{+}={\tau}_{L}{\left(u*\right)}^{2}/\nu $ |

τ_{p} | s | Particle relaxation time ${\tau}_{p}={m}_{p}{C}_{c}/3\pi \mu {D}_{p}$ ${\tau}_{p}^{+}={\tau}_{p}{\left(u*\right)}^{2}/\nu $ |

ε_{p} | m^{2} s^{−1} | Eddy diffusivity of the particle. For relatively small particles and homogeneous isotropic turbulence [18] ${\epsilon}_{p}={\nu}_{t}$ For any particle size [21,29] ${\epsilon}_{p}={\left[1+\frac{{\tau}_{p}}{{\tau}_{L}}\right]}^{-1}{\nu}_{t}$ |

ν | m^{2} s^{−1} | Kinematic viscosity of the fluid, ν = μ/ρ |

ντ | m^{2} s^{−1} | Air turbulent viscosity. For smooth surfaces it is [17] $\frac{{\nu}_{t}}{\nu}=\left\{\begin{array}{ll}7.67\times {10}^{-4}{\left({y}^{+}\right)}^{3},& 0\le {y}^{+}\le 4.3\\ {10}^{-3}{\left({y}^{+}\right)}^{2.8214},& 4.3\le {y}^{+}\le 12.5\\ 1.07\times {10}^{-2}{\left({y}^{+}\right)}^{1.8895},& 12.5\le {y}^{+}\le 30\end{array}\right.$ and for rough surfaces it is [16] $\frac{{\nu}_{t}}{\nu}=\left\{\begin{array}{ll}{\left(\frac{{y}^{+}}{11.15}\right)}^{3},& 0\le {y}^{+}\le 3\\ {\left(\frac{{y}^{+}}{11.4}\right)}^{3}-0.049774,& 3\le {y}^{+}\le 52.108\\ 0.4{y}^{+},& 52.108\le {y}^{+}\end{array}\right.$ |

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**Figure 1.**Schematic diagram for dry deposition on a smooth surface of airborne particle with radius r

_{p}across the concentration boundary layer. The concentration of particle number across the boundary layer grows from lowest value (C = 0) at height y

_{0}(= r

_{p}) to its highest value (C = C

_{∞}) at height y

_{cbl}. V

_{d}is the dry deposition velocity through the concentration boundary layer.

**Figure 2.**Illustration of the Variation of the concentration boundary layer height above the surface in dimensionless unit (y

^{+}) with the dimensionless particle diameter (D

^{+}

_{p}) for the current study. The horizontal dashed lines resemble the limit value for y

^{+}

_{cbl}used in the calculation by Zhao and Wu [19], Hussain et al. [15], and Lai and Nazaroff [18]. Friction velocity u* = 100 m/s, and number of subdivisions N = 1000 subdivisions.

**Figure 3.**Dependence of y

^{+}

_{cbl}on particle size (τ

^{+}

_{p}or D

_{p}) and friction velocity (u*) for Fickian diffusion: (

**a**) profile of y

^{+}

_{cbl}as a function of τ

^{+}

_{p}for three friction velocities. (

**b**) y

^{+}

_{cbl}as a function of D

_{p}(10 nm–100 µm) and u* (0.01–100 m/s).

**Figure 4.**Appropriate number of subdivisions N for each pair of D

_{p}(10 nm–100 µm) and u* (0.01–100 m/s).

**Figure 5.**Dependence of y

^{+}

_{cbl}on particle size (τ

^{+}

_{p}or D

_{p}) and friction velocity (u*) when gravitational settling is added to Fickian diffusion: (

**a**) profile of y

^{+}

_{cbl}as a function of the dimensionless relaxation time τ

^{+}

_{p}for three friction velocities. (

**b**) y

^{+}

_{cbl}as a function of D

_{p}(10 nm–100 µm) and u* (0.01–100 m/s).

**Figure 6.**(

**a**) A comparison, for Fickian diffusion only, between the optimized code (dashed curves) and the unoptimized code (solid curves) for dry deposition velocity V

^{+}

_{d}. (

**b**) the residuals between both codes.

**Figure 7.**(

**a**) A comparison, for gravitational settling and Fickian diffusion together, between the optimized code (dashed curves) and the unoptimized code (solid curves) for dry deposition velocity V

^{+}

_{d}. (

**b**–

**c**) the residuals between both codes.

Computer | Processor (CPU) | Memory (RAM) | Storage |
---|---|---|---|

PC-1 | Core i7 10th, generation | 8 GB | 256 SSD |

PC-2 | Core i5 2nd, generation | 4 GB | 256 SSD |

PC-3 | AMD RYZON 3, 3rd generation | 4 GB | 256 SSD |

PC-4 | Core i7 3rd, generation | 8 GB | 250 SSD |

**Table 2.**A comparison between computation time for unoptimized code (without parameterization) and optimized one (with parameterization). for four computers for a selected particle diameters (D

_{p}) for friction velocity u* = 0.01 m/s.

PC | D_{p} (µm) | Calculation Time (s) | % Time | Time Difference (s) | |
---|---|---|---|---|---|

Without Parameterization | With Parameterization | ||||

PC-1 | 0.01 | 0.60 | 0.15 | 75% | 0.45 |

0.1 | 0.67 | 0.17 | 75% | 0.50 | |

1 | 0.63 | 0.15 | 76% | 0.48 | |

10 | 0.61 | 0.13 | 78% | 0.48 | |

100 | 0.56 | 0.14 | 74% | 0.42 | |

PC-2 | 0.01 | 1.6 | 0.38 | 76% | 1.2 |

0.1 | 1.5 | 0.40 | 73% | 1.1 | |

1 | 1.5 | 0.38 | 74% | 1.1 | |

10 | 1.4 | 0.37 | 74% | 1.0 | |

100 | 1.3 | 0.36 | 72% | 0.94 | |

PC-3 | 0.01 | 2.5 | 0.52 | 79% | 2.0 |

0.1 | 2.8 | 0.52 | 81% | 2.3 | |

1 | 2.3 | 0.51 | 78% | 1.8 | |

10 | 2.4 | 0.48 | 80% | 1.9 | |

100 | 2.7 | 0.46 | 83% | 2.2 | |

PC-4 | 0.01 | 1.4 | 0.31 | 78% | 1.1 |

0.1 | 1.4 | 0.28 | 80% | 1.1 | |

1 | 1.3 | 0.30 | 77% | 1.0 | |

10 | 1.3 | 0.26 | 80% | 1.0 | |

100 | 1.3 | 0.25 | 80% | 1.0 |

**Table 3.**A comparison between calculation time for unoptimized code (without parameterization) and optimized one (with parameterization) for four computers for a selected particle diameters (D

_{p}) for friction velocity u* = 0.1 m/s.

PC | D_{p} (µm) | Calculation Time (s) | % Time | Time Difference (s) | |
---|---|---|---|---|---|

Without Parameterization | With Parameterization | ||||

PC-1 | 0.01 | 0.70 | 0.10 | 86% | 0.60 |

0.1 | 0.66 | 0.15 | 77% | 0.51 | |

1 | 0.63 | 0.13 | 79% | 0.50 | |

10 | 0.60 | 0.13 | 78% | 0.47 | |

100 | 0.58 | 0.13 | 77% | 0.45 | |

PC-2 | 0.01 | 1.6 | 0.40 | 76% | 1.2 |

0.1 | 1.5 | 0.37 | 75% | 1.1 | |

1 | 1.4 | 0.36 | 75% | 1.0 | |

10 | 1.4 | 0.40 | 70% | 1.0 | |

100 | 1.2 | 0.37 | 68% | 0.8 | |

PC-3 | 0.01 | 2.7 | 0.49 | 82% | 2.2 |

0.1 | 2.2 | 0.48 | 78% | 1.7 | |

1 | 2.4 | 0.46 | 81% | 1.9 | |

10 | 2.2 | 0.50 | 78% | 1.7 | |

100 | 2.7 | 0.55 | 79% | 2.2 | |

PC-4 | 0.01 | 1.5 | 0.27 | 81% | 1.2 |

0.1 | 1.3 | 0.25 | 81% | 1.0 | |

1 | 1.8 | 0.24 | 87% | 1.6 | |

10 | 1.2 | 0.30 | 74% | 0.9 | |

100 | 1.4 | 0.26 | 82% | 1.1 |

**Table 4.**A comparison between calculation time for unoptimized code (without parameterization) and optimized one (with parameterization) for four computers for a selected particle diameters (D

_{p}) for friction velocity u* = 1 m/s.

PC | D_{p} (µm) | Calculation Time (s) | % Time | Time Difference (s) | |
---|---|---|---|---|---|

Without Parameterization | With Parameterization | ||||

PC-1 | 0.01 | 0.68 | 0.13 | 81% | 0.55 |

0.1 | 0.65 | 0.13 | 80% | 0.52 | |

1 | 0.75 | 0.13 | 83% | 0.62 | |

10 | 0.62 | 0.13 | 79% | 0.49 | |

100 | 0.59 | 0.13 | 78% | 0.46 | |

PC-2 | 0.01 | 1.6 | 0.41 | 74% | 1.2 |

0.1 | 1.6 | 0.38 | 76% | 1.2 | |

1 | 1.5 | 0.39 | 74% | 1.1 | |

10 | 1.2 | 0.38 | 69% | 0.82 | |

100 | 1.1 | 0.26 | 77% | 0.84 | |

PC-3 | 0.01 | 2.7 | 0.46 | 83% | 2.2 |

0.1 | 2.4 | 0.50 | 79% | 1.9 | |

1 | 2.6 | 0.53 | 80% | 2.1 | |

10 | 2.4 | 0.45 | 81% | 2.0 | |

100 | 2.3 | 0.44 | 81% | 1.9 | |

PC-4 | 0.01 | 1.2 | 0.26 | 78% | 0.94 |

0.1 | 1.2 | 0.24 | 80% | 1.0 | |

1 | 1.5 | 0.26 | 83% | 1.2 | |

10 | 1.3 | 0.30 | 77% | 1.0 | |

100 | 1.7 | 0.26 | 85% | 1.4 |

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## Share and Cite

**MDPI and ACS Style**

Nofal, O.M.M.; Al-Jaghbeer, O.; Bakri, Z.; Hussein, T.
A Simple Parameterization to Enhance the Computational Time in the Three Layer Dry Deposition Model for Smooth Surfaces. *Atmosphere* **2022**, *13*, 1190.
https://doi.org/10.3390/atmos13081190

**AMA Style**

Nofal OMM, Al-Jaghbeer O, Bakri Z, Hussein T.
A Simple Parameterization to Enhance the Computational Time in the Three Layer Dry Deposition Model for Smooth Surfaces. *Atmosphere*. 2022; 13(8):1190.
https://doi.org/10.3390/atmos13081190

**Chicago/Turabian Style**

Nofal, Omar M. M., Omar Al-Jaghbeer, Zaid Bakri, and Tareq Hussein.
2022. "A Simple Parameterization to Enhance the Computational Time in the Three Layer Dry Deposition Model for Smooth Surfaces" *Atmosphere* 13, no. 8: 1190.
https://doi.org/10.3390/atmos13081190