# Land-Atmosphere Transfer Parameters in the Brazilian Pantanal during the Dry Season

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}and subjected to a displacement height d, and the atmospheric stability is a function of z/L that depends on at least one additional parameter (stability function parameter a). The surface length parameters z

_{0}and d are naturally site-dependent, and also the stability parameter a, in principle not depending on the surface characteristics, has shown some variability for measurements taken at different experimental sites [1].

_{0T}is different from z

_{0}, and the correction term in the temperature profile kB = ln(z

_{0}/z

_{0T}) is generally parameterized with different expressions for different surface roughness conditions, mainly validated in wind tunnel experiments [1]. The case is more complicated for the surface vapor flux, as the humidity profile has a surface value that is often not experimentally measurable at all.

## 2. Site, Experiment and Dataset

^{2}, periodically flooded by the Rio Paraguay and its affluents. Its surface shows an extremely large variability of conditions during the annual cycle, passing from very dry conditions (dry season: April–October), with actual fire risk (“queimadas”), to almost lagoon conditions (wet season: November–March), covered by a water layer that can reach several tens of centimeters in height. The climate is classified between sub-humid and semiarid with annual precipitations between 1000 and 1400 mm, mostly concentrated in the wet season. Temperature can reach 40 °C in summer, but can decrease down to 0 °C in winter, in association with cold fronts passing over the region. Due to the extension and the large seasonal variability, the vegetation cover is heterogeneous, including a large quantities of gramineous plants and bushes and different types of trees up to 20 meters in height.

## 3. Methodology

#### 3.1. Wind Speed and Temperature

_{0},d) = kU/u* + Ψ((z − d)/L) − Ψ(z

_{0}/L) over the measurement dataset, which is a two-parameter minimization problem for z

_{0}and d, and reduced it to a one-parameter conditioned minimization problem that is straightforward to solve numerically. (here, u* is the friction velocity, U the wind speed, k the Von Karman constant and Ψ the stability function of the measurement height z and the MO length L [1]). The M00 approach was originally developed for single-level sonic anemometer measurements; thus, some changes have been made to adapt it to the different condition of a multi-level slow response sensor mast. Indeed, the information available for five levels of slow response measurements has been used to look for an estimation of the stability parameter together with the roughness parameters. Then, the M00 approach has been modified to be used with couples of measurement levels together with the turbulent fluxes estimated by the fast response measurements on the mast top, in the following way.

_{S}

^{2}= ‹(S − ‹S›)

^{2}›, where S is now defined as a function of d and the stability parameter a:

_{2}− d)/L,a) − Ψ((z

_{1}− d)/L,a)

_{v}in the present case) between the two considered measurements levels z

_{1}and z

_{2}, q* is the turbulent scale of q obtained from the fast response measurements, Ψ((z

_{1}− d)/L,(z

_{2}− d)/L) is the stability function depending on the displacement height d and an unknown stability parameter a, also to be determined (see the end of this section), k is the Von Karman constant (k = 0.4) and ‹…› represents the average over the dataset [12].

_{S}

^{2}with respect to a only, with d calculated by the following constraint:

_{1}exp‹S(d,a)› −z

_{2})/(exp‹S(d,a)› −1)

_{S}

^{2}with respect to a, just numerically evaluating σ

_{S}

^{2}with a changing step by step between fixed maximum and minimum values and recalculating d by Equation (2) at each step. The estimates of a and d are obtained when the minimum of σ

_{S}

^{2}is found. Some examples of the procedure are shown in Figure 3.

_{0}(for the variable q) using a and d to numerically solve the equation:

_{0}) = ‹kΔq

_{0}/q* + Ψ((z − d)/L,a) − Ψ(z

_{0}/L,a)›

_{0}is the difference between q(z) and the surface value q

_{0}of q (that is, U or Θ

_{v}). Alternatively, the M00 approach can be used to determine both d and z

_{0}using the estimated value of a. Both methods have been used in the following sections for comparison.

_{U}(x,a) = 2ln[(1 + y)/2] + ln[(1 + y

^{2})/2] − 2tan

^{−1}x + π/2

^{1/4}in the unstable condition;

_{Θ}(x,a) = 2ln[(1 + y)/2]

^{1/2}in the unstable condition;

_{U,Θ}(x,a)= −ax

**Figure 3.**Example of σ

_{S}

^{2}(Equation (1)) for ΔU and for ΔΘ

_{v}calculated as a function of a (symbols: a for ΔU and A for ΔΘ

_{v}) and of d (symbols: d for ΔU and D for ΔΘ

_{v}), for the couple of Levels 2–3. The abscissa of the minima represents the estimated value of a and d for ΔU and for ΔΘ

_{v}in each case.

#### 3.2. Water Vapor

_{0}is not available for the water vapor. A “hybrid” formulation of the PM model was proposed by Martano [14] in which a surface resistance can be expressed by the parallel between a soil surface resistance R

_{s}and a canopy resistance R

_{v}. This is supposed to be the simplest way to capture the main characteristics of partially-vegetated surfaces, with bare soil and vegetated surfaces giving independent contributions to the total evaporation, and is justified by the lack of detailed surface information. Indeed, this model has the advantage of almost no necessity of additional surface information besides the regression parameters themselves, and actually, no additional external parameter is needed in the following procedure (e.g., no LAI value is used). For the sake of completeness, a short description follows.

_{em}is expressed in the PM model as [1]:

_{em}= [sE + ρC

_{p}q

_{sat}(1 − H

_{r})/R

_{a}][s + γ(1 + R

_{0}/R

_{a})]

^{−1}

_{r}is the air relative humidity, q

_{sat}the saturation-specific humidity, E the measured available energy flux (net radiation minus soil heat flux), s the slope of the saturation curve (from the Clapeyron formula), γ the psychrometric constant and C

_{p}and ρ the specific heat and the density of the air. All variables in the right-hand side are measured, with the exception of the surface resistance R

_{0}and the aerodynamic resistance R

_{a}. R

_{a}is expressed by a scalar drag coefficient C

_{h}that is considered as an unknown parameter to be determined in the form [1]:

_{a}= (C

_{h}U)

^{−1}

_{0}can be expressed as a parallel independent resistance contribution of the vegetated and the non-vegetated surface fractions, respectively R

_{v}and R

_{s}:

_{0}= 1/R

_{s}+ 1/R

_{v}

_{s}, is a strongly nonlinear function of the soil-specific water content w. It is modelled following the expression given by Kondo and Saigusa [20] for a “wetness parameter” that is mathematically equivalent to a surface resistance parameter. Using this equivalence, the surface resistance can be expressed as [14,20]:

_{s}= C

_{s}/D

_{a}(w

_{sat}– w)

^{p}

_{sat}is the saturation-specific water content, D

_{a}= 0.000023(Ts/273.2)

^{1.75}, T

_{s}is the soil surface temperature, C

_{s}is an unknown parameter depending on the soil texture and the actual non-vegetated fraction of the surface and p = 10 for loamy soil [20].

_{v}depends on the available energy and the air humidity through a similarity resistance form R

_{*}:

_{*}= ρL

_{e}q

_{sat}(1 − H

_{r})(1 + γ/s)/E

_{v}/R

_{a}= f(R

_{*}/R

_{a}), and used a linear approximation for the function f. Accordingly, the following expression is proposed for the vegetated fraction [14]:

_{v}/R

_{a}= C

_{v}R

_{*}/R

_{a}

_{v}is a parameter depending on the vegetation characteristics and the effective contribution of the vegetated surface fraction to the evapotranspiration.

_{h}, C

_{v}and C

_{s}. These determine the effective scalar drag coefficient and the evaporative response of the vegetated and non-vegetated fractions of the surface, respectively.

## 4. Parameter Estimation and Discussion

#### 4.1. Temperature and Wind Speed: Estimation of the Surface Length Scales and the Stability Parameter

_{S}

^{2}= ‹(S − ‹S›)

^{2}› incrementing step by step the value of the parameter a, with d calculated by numerically solving Equation (2).

_{S}

^{2}as a function of d and a, both for the wind speed and the temperature for the couple of Levels 2 and 3.

_{0}are shown in Table 1 for the couples of levels where a minimum of σ

_{S}

^{2}has been found. The surface value of the temperature T

_{0e}(T

_{0e}= q

_{0}in Equation (3)), has been estimated by the measured long wave upward emission Rl

_{e}considered as blackbody emission (Rl

_{e}= σT

_{0e}

^{4}), as no estimate was available for the soil surface emissivity. The radiometer was located at a four-meter height, which implies a field of view radius of about 40 meters (which captures about 99% of the total contribution to the surface flux [22]). Although much larger than the field of view of a typical infrared thermometer, this can be still significantly less than the flux footprint at the top tower level (see Section 4.3).

_{0T}is given. The average values of a are in reasonable agreement with the literature (a ≈ 16 unstable; a ≈ 4.7 stable [1]), taking into account the statistical errors, in the case of the wind speed. For the potential temperature, a is underestimated, but the uncertainty in the measurements is greater in this case; and there is no result obtained for Θ

_{v}in the unstable case, in which the measured vertical temperature differences can be hardly significant due to the strong turbulent mixing.

**Table 1.**a, d and z

_{0}calculated from the shown couples of levels for U and Θ

_{v}(Equations (1)–(3)).

U Stable | |||

Levels | a | d (m) | z_{0} (m) |

1–3 | 4.5 | 1.9 | 0.13 |

1–5 | 4.1 | 3.7 | 0.07 |

2–3 | 4.9 | 4.2 | 0.09 |

2–5 | 4.2 | 5.3 | 0.06 |

3–5 | 3.8 | 6.9 | 0.04 |

Averages | 4.3 ± 0.4 | 4.4 ± 1.8 | 0.08 ± 0.03 |

U Unstable | |||

Levels | a | d (m) | z_{0} (m) |

1–3 | 10.2 | 2.7 | 0.05 |

Θ _{v} Stable | |||

Levels | a | d (m) | z_{0} (m) |

1–2 | 3.0 | 3.6 | 0.17 |

1–3 | 1.4 | 6.9 | 0.03 |

2–3 | 3.3 | 8.5 | 0.03 |

Averages | 2.6 ± 1.0 | 6.3 ± 2.5 | 0.08 ± 0.08 |

_{0}and d by the M00 method are reported for comparison in Table 2 for each measurement level, using the average values of a from Table 1.

**Table 2.**d and z

_{0}calculated as in M00 (Martano approach), with average values of a from Table 1.

U Stable | |||||

Levels | 1 | 2 | 3 | 5 | Averages |

z_{0} (m) | 0.07 | 0.08 | 0.09 | 0.07 | 0.08 ± 0.01 |

d (m) | 3.8 | 4.1 | 3.5 | 4.4 | 4.0 ± 0.4 |

U Unstable | |||||

Levels | 1 | 2 | 3 | 5 | Averages |

z_{0} (m) | 0.017 | 0.024 | 0.017 | - | 0.019 ± 0.004 |

d (m) | 6.8 | 7.7 | 11.0 | - | 8.5 ± 2.2 |

Θ_{v} Stable | |||||

Levels | 1 | 2 | 3 | 5 | Averages |

z_{0} (m) | 0.12 | 0.13 | 0.08 | - | 0.11 ± 0.03 |

d (m) | 4.8 | 4.8 | 4.9 | - | 4.8 ± 0.1 |

**Figure 4.**Test for ΔU (stable: circles, unstable: squares) and Δθ

_{v}(+) calculated between the surface and all measurement levels by the similarity profiles with the averaged parameters in Table 1, versus the measured values.

_{0}should be equal for different physical quantities, the surface transfer for wind speed, temperature and scalars being governed by different physical processes. In general, z

_{0}could depend also on the flow condition [1], and a theoretical analysis gives different expressions for d in the case of wind speed and temperature [11]. Nevertheless, within the relevant uncertainty in the temperature parameters (Table 1 and the Appendix), the obtained results do not point out relevant differences in z

_{0}and d between wind speed and temperature, at least in order of magnitude and for stable conditions.

_{0}) to test the ability of the average values of a, d and z

_{0}in reproducing the observed profiles for the wind speed and the potential temperature. The results are shown in Figure 4, where the values of U and ΔΘ

_{v}have been calculated using the averaged values of a, d and z

_{0}from Table 1 and compared with the measured data for all measurement levels. The agreement is fairly good, mainly for the wind speed in the stable case, while a larger dispersion is present for the temperature and the wind speed in the unstable case, as expected.

#### 4.2. Water Vapor: Estimation of the Scalar Drag Coefficient

_{0}for the air humidity [1] and the large uncertainty in the humidity gradients estimated between the measurement levels. Instead, to obtain an estimate of the surface transfer coefficients, the approach outlined in Section 3.2 has been used together with the regression method described in [14], which will be shortly outlined here. To estimate the unknown parameters C

_{h}, C

_{v}and C

_{s}for the simple model outlined in Section 3.2, a cost function F depending on both the latent heat flux and the surface temperature is written in the form [14]:

_{h},C

_{v},C

_{s}) = (2N)

^{−1}∑

_{1}

^{N}([Q

_{e}– Q

_{em}(C

_{h},C

_{v},C

_{s})]

^{2}/σ

_{Q}

^{2}+ [ΔΘ

_{v0}– ΔΘ

_{vm}(C

_{h},C

_{v},C

_{s})]

^{2}/σ

_{Θ}

^{2})

_{e}represents the experimental latent heat flux data, estimated as Q

_{e}= E – Q

_{s}(the difference between the measured net radiation, soil heat flux and measured sensible heat flux Q

_{s}), thus assuming the surface energy budget closure. ΔΘ

_{v0}is the measured surface-atmosphere potential virtual temperature difference (depending on the considered level of measurement) from a dataset of N measurements. The modelled temperature difference ΔΘ

_{vm}(C

_{h},C

_{v},C

_{s}) is expressed using the energy budget closure, the modelled latent flux Q

_{em}(Equation (1)) and the drag laws [1,14] as:

_{vm}(C

_{h},C

_{v},C

_{s}) = [E − Q

_{em}(C

_{h},C

_{v},C

_{s})]/(C

_{p}ρU C

_{h})

_{Q}, σ

_{T}for Q

_{e}and Θ

_{v}have been chosen as 30 W∙m

^{−2}and 3 K, respectively. The first value comes from an evaluation of the instrumental errors and is in general agreement with an expected uncertainty coming from the flux estimation by the energy budget [23]. The second comes from a calculation of the variance between the radiometric surface temperature estimate and the soil temperature at a 1-cm depth. It is also comparable with other estimates obtained elsewhere [14,24], and this choice for the statistical uncertainties is in agreement with the obtained convergence values for F that are expected to be close to one [24].

_{h}for heat and water vapor.

_{s}in Equation (10) is adjusted by the regression procedure (the values used were w = 0.3 and w

_{s}= 0.49 with p = 10 [20]). Eventually, the function F has been minimized by the Levenberg–Marquardt regression method [24,26] for the measurement Levels 3 and 5.

_{h}for the same Levels 3 and 5 are reported. In Column 2, they have been calculated from the experimental data of temperature and heat flux (flux-gradient approach), as averaged values of the scalar drag for the sensible heat transfer in the unstable case. They can also be compared with the values obtained from the averages of z

_{0}and d from Table 1 and calculated as C

_{h}= k

^{2}[(ln(z − d)/z

_{0})(ln(z − d)/z

_{0T})]

^{−1}[1], which are 7.5 × 10

^{−3}and 5.6 × 10

^{−3}for Levels 3 and 5, respectively (stable case). Figure 5 and Figure 6 test the reliability of the obtained parameters by plotting the parameterized versus the measured latent heat fluxes and surface-air temperature differences from Levels 3 and 5 during the daytime. Both have a fairly good correlation, but with a larger scatter in the case of the temperature differences. This is somehow expected, as the PM model itself is derived by an approximation that eliminates the surface temperature from the expression of the latent flux [1], thus not being directly sensitive to this parameter.

**Table 3.**C

_{h}from both the Penman–Monteith (PM) model and measurements and the evaporative vegetation fractions (evf).

Level | C_{h} PM Model (10^{−3}) | C_{h} Data Average (10^{−3}) | evf |
---|---|---|---|

3 | 7.6 ± 1.5 | 6.8 ± 0.4 | 0.72 ± 0.11 |

5 | 6.5 ± 1.2 | 5.4 ± 0.3 | 0.83 ± 0.08 |

_{es}and Q

_{ev}, respectively, it is straightforward to show that Q

_{es}/Q

_{e}= (R

_{s}//R

_{v})/R

_{s}and Q

_{ev}/Q

_{e}= (R

_{s}//R

_{v})/R

_{v}(with the obvious condition Q

_{es}/Q

_{e}+ Q

_{ev}/Q

_{e}= 1). The results show that the vegetated areas should be responsible for 70%–80% of the latent heat flux in the dry season. Thus, the vegetation has a dominant contribution, even in the presence of the quite large soil moisture content that is expected from the generally small values of the Bowen ratio in the whole period of measurements.

#### 4.3. Discussion

_{r}over the roughness elements. A discussion can be found in [8], where a suggested expression for z

_{r}is proposed as z

_{r}= d + 20z

_{0}, allowing a test of the self-consistency of the obtained length scales. Thus, using the results for z

_{0}and d, it is found z

_{r}≈ 7 m, still below the tower lower measurement level of 8 m, which means that the obtained results are consistent with the measurement level heights.

**Figure 5.**Test for the latent heat flux calculated by the PM model with the calibrated parameters, versus the latent heat flux estimated by the measured surface energy budget (squares: Level 3, circles: Level 5).

**Figure 6.**Test for Δθ

_{v}obtained by the PM model with the calibrated parameters versus the measured values (squares: Level 3, circles: Level 5).

_{0}and L [27]. Using the estimated values for the surface length scales and typical average values of L for daytime and nighttime from the dataset (nighttime L ≈ 10 m and daytime L ≈ −10 m), it results that the 90% flux footprint varies from more than 10 km upwind (nighttime) to about 500 m (daytime) for the upper measurement level and from 2.5 km to about 200 m for the lower level. The strong reduction of the footprint in the daytime may be related to the difficulties in obtaining the transfer parameters for the unstable case, because the heterogeneities of the terrain roughness can be more relevant on this smaller source area and, also, they can show stronger variability with the measurement level. It also appears that the obtained transfer parameters, estimated in the stable case, are representative of an area of some square kilometers around the tower.

## 5. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

_{e}measured by a radiometer above a radiant surface of emissivity ε and (real) surface temperature T

_{0}is due to the surface thermal emission Rl

_{0}= εσT

_{0}

^{4}plus the reflected environment radiation Rl

_{a}and is given by (σ is the Stefan–Boltzmann constant):

_{e}= εσT

_{0}

^{4}+ (1 − ε)Rl

_{a}

_{e}= σT

_{0e}

^{4}and α = (1 − ε)/ε where T

_{0e}is the estimated “blackbody” surface temperature, it is straightforward to show that, for α[1 − R]<<1 and R = Rl

_{a}/Rl

_{e}:

_{0}= T

_{0e}(1 + α − αR)

^{1/4}≈ T

_{0e}(1 + α[1 − R]/4)

_{0}− T

_{0e}|/T

_{0e}≤ 0.002

_{0}= 300 K) is comparable with the typical uncertainties of brightness surface temperature from satellite data and comparable or less than the inherent uncertainties associated with the heterogeneities of the surface in the source area containing the footprint of the turbulent fluxes, which can be of some degrees [14].

_{T}, which is directly related to the uncertainty factor β on the estimation of the scalar roughness parameter z

_{0T}. An approximate flux gradient relation for these uncertainties can be written as:

_{T}) ≈ (T*/k)ln(z/βz

_{0T})

_{0T}), it results:

_{T}/T* ≈ ln(1/β)

_{T}≈ 0.15 (calculated as the ratio of the previous estimate of the uncertainty of T

_{0}of 0.6 K and the average value |ΔT| ≈ 4 K), it appears that β can affect the uncertainty of z

_{0T}by a factor three.

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**MDPI and ACS Style**

Martano, P.; Filho, E.P.M.; De Abreu Sá, L.D.
Land-Atmosphere Transfer Parameters in the Brazilian Pantanal during the Dry Season. *Atmosphere* **2015**, *6*, 805-821.
https://doi.org/10.3390/atmos6060805

**AMA Style**

Martano P, Filho EPM, De Abreu Sá LD.
Land-Atmosphere Transfer Parameters in the Brazilian Pantanal during the Dry Season. *Atmosphere*. 2015; 6(6):805-821.
https://doi.org/10.3390/atmos6060805

**Chicago/Turabian Style**

Martano, Paolo, Edson Pereira Marques Filho, and Leonardo Deane De Abreu Sá.
2015. "Land-Atmosphere Transfer Parameters in the Brazilian Pantanal during the Dry Season" *Atmosphere* 6, no. 6: 805-821.
https://doi.org/10.3390/atmos6060805