The Lagrangian stochastic model for estimating footprint and water vapor fluxes over inhomogeneous surfaces

Abstract

This study investigated a two-dimensional Lagrangian stochastic dispersion model for estimating water vapor fluxes and footprint over homogeneous and inhomogeneous surfaces. Over the homogeneous surface, particle trajectories were computed from a 2-D Lagrangian model forced by Eulerian velocity statistics determined by Monin–Obukhov similarity theory (MOST). For an inhomogeneous surface, the velocity and atmospheric stability profiles were computed using a second-order Eulerian closure model, and these local profiles were then used to drive the Lagrangian model. The model simulations were compared with water vapor flux measurements carried out above an irrigated bare soil site and an irrigated potato site. The inhomogeneity involved a step change in surface roughness, humidity, and temperature. Good agreement between eddy-correlation-measured and Lagrangian-model-predicted water vapor fluxes was found for both sites. Hence, this analysis demonstrates the practical utility of second-order closure models in conjunction with Lagrangian analysis to estimate the scalar footprint in planar inhomogeneous flows.

Appendices

Appendix A: stability correction functions and Lagrangian time scale profiles

• 1. For unstable conditions (z/L < 0):

  $$\psi _m = - 2\ln \left( {\frac{{1 + \phi }}{2}} \right) - \ln \left( {\frac{{1 + \phi ^2 }}{2}} \right) + 2\tan ^{ - 1} \left( \phi \right) - \frac{\pi }{2}$$

  (A.1)
  $$t_L = \frac{{k\left( {z - d} \right)u_* }}{{\sigma _w^2 \phi _h }}$$

  (A.2)
  $$\phi = \left( {{{1 - 16\left( {z - d} \right)} \mathord{\left/ {\vphantom {{1 - 16\left( {z - d} \right)} L}} \right. \kern-\nulldelimiterspace} L}} \right)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} $$

  (A.3)
  $$\phi _h = 0.37\left( {{{0.03 - 3\left( {z - d} \right)} \mathord{\left/ {\vphantom {{0.03 - 3\left( {z - d} \right)} L}} \right. \kern-\nulldelimiterspace} L}} \right)^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-\nulldelimiterspace} 3}} $$

  (A.4)

• 2. For neutral and stable conditions (z/L ≥ 0):

  $$\psi _m = {{1 + 5\left( {z - d} \right)} \mathord{\left/ {\vphantom {{1 + 5\left( {z - d} \right)} L}} \right. \kern-\nulldelimiterspace} L}$$

  (A.5)
  $$t_L = \frac{{k\left( {z - d} \right)u_* }}{{\sigma _w^2 \phi _h }}$$

  (A.6)
  $$\phi _h = {{1 + 5\left( {z - d} \right)} \mathord{\left/ {\vphantom {{1 + 5\left( {z - d} \right)} L}} \right. \kern-\nulldelimiterspace} L}$$

  (A.7)

Appendix B: boundary conditions for the second-order closure model

The boundary conditions are based on Monin-Obukhov similarity theory (MOST) under neutral atmospheric stratification. Near the ground, the downwind boundary values (i.e., at z ≈z  _o2; z _o2 is the downstream surface roughness) for velocities are: U(z _o2) = 0 and W(z _o2) = 0. The boundary values for temperature are solved by assuming that the available energy is conserved through the downwind surface. In other words, the following equation is valid.

$${\text{H}}_{{\text{o2}}} + {\text{LE}}_{{\text{o2}}} = {\text{Rn}}_{{\text{o2}}} - {\text{G}}_{{\text{o2}}} = {\text{constant}}$$

(A.8)

where the subscript o2 denotes the downwind boundary parameter, Rn is the net radiation and G is the soil heat flux. With the equilibrium flux-profile relations, Eq. A.8 can be expressed as

$$\left. { - \rho {\text{C}}_p kzu_{*o2} \frac{{\partial \Theta }}{{\partial z}}} \right|_{zo2} = Rn_{o2} - G_{o2} - LE_{o2} ,$$

(A.9)

where ρ is the air density, C _p is the specific heat of air at constant pressure, and u _*o2 is specified as

$$u_{*o2} = kz\left. {\frac{{\partial U}}{{\partial z}}} \right|_{zo2} $$

(A.10)

Equations A.9 and A.10 are solved together to determine the downwind boundary values for temperature.

With u _* calculated by Eq. A.10 and θ _* calculated by

$$\theta _{{\text{*o2}}} = 0.74kz\left. {\frac{{\partial \Theta }}{{\partial z}}} \right|_{zo2} ,$$

(A.11)

the boundary conditions for the second moments of velocity and temperature and the dissipate rate of turbulence kinetic energy are specified as follows:

$$\overline {u\prime u\prime } = a_{uu} u_*^2 ;\;\overline {v\prime v\prime } = a_{vv} u_*^2 ;\;\overline {w\prime w\prime } = a_{ww} u_*^2 ;\;\overline {u\prime w\prime } = - u_*^2 $$

(A.12)

$$\overline {\theta u\prime } = a_{\theta {\kern 1pt} u} \theta _* u_* ;\;\overline {\theta w\prime } = - \theta _* u_* $$

(A.13)

$$\overline \varepsilon = {{k_\varepsilon (\overline {u\prime u\prime } + \overline {v\prime v\prime } + \overline {w\prime w\prime } )^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} } \mathord{\left/ {\vphantom {{k_\varepsilon (\overline {u\prime u\prime } + \overline {v\prime v\prime } + \overline {w\prime w\prime } )^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} } {kz_{o2} }}} \right. \kern-\nulldelimiterspace} {kz_{o2} }}$$

(A.14)

In Eqs. A.12 and A.13, a _{uu vv ww} , and a _θu , are similarity constants.

As to the upstream boundary conditions, Θ _o1, u _*o1, and θ _*o1 are known/given (the subscript o1 denotes the upstream boundary parameter), and the second moments of velocity and temperature and the dissipate rate of turbulence kinetic energy are calculated by Eqs. A.12–A.14; U(z _o1) and W(z _o1) are set to zero (z _o1 is the upstream surface roughness.