Consider a horizontally homogeneous turbulent ABL of uniform density, with a vertical velocity distribution that is Gaussian with zero mean and standard deviation σw(z), and which has a Lagrangian decorrelation timescale τ(z). The canonical stochastic differential equation model e.g. for one-dimensional vertical dispersion in the ABL is dWt=-Wtτ+121+Wtσw2∂σw2∂zdt+2σw2τ1/2dBt,dZt=Wtdt. Here Wt and Zt are the vertical velocity and height of a given air parcel. Both are stochastic variables, with each individual realisation determined by that of the Brownian (or Wiener) process Bt. Further σw=σw(Zt) and τ=τ(Zt) are the values of σw(z) and τ(z) local to the parcel. In operational LPDMs, such as FLEXPART, appropriate vertical profiles for σw(z) and τ(z) are specified based on empirical fits to observations of different ABL conditions, as will be discussed below. The equation set () is typically augmented with reflecting boundary conditions at the Earth's surface and at the ABL top seefor a detailed discussion of the top boundary condition. For definiteness, for our test-case runs, the initial velocity for Eq. () at t=0 is sampled from a normal distribution W0∼N(0,σw2(z0)) and, for ease of comparison to the FPE results below, the initial position is sampled from a distribution Z0∼N(z0,σz2) centred on an initial height z0 with standard deviation σz.

For the purposes of numerical solution, it is more convenient e.g. Sect. 3.1 of to use Ito's lemma to express Eq. () in terms of the variables Ωt=Wt/σw(Zt) and Zt, leading to dΩt=-Ωtτ+∂σw∂zdt+2τ1/2dBt,Ω0∼N(0,1)dZt=Ωtσwdt,Z0∼N(z0,σz2). The simpler form () is exactly equivalent to Eq. (). Moreover, the FPE of Eq. () has a considerably simpler form than the corresponding FPE of Eq. (), a fact which will prove useful below.

It is simplest to view Eq. () as a non-dimensional equation, given that in particular Ωt is already a non-dimensional variable. The natural non-dimensionalisation has length, velocity, and timescales of ABL height h, surface friction velocity u*, and h/u* respectively. Under this non-dimensionalisation, the spatial domain for Eq. () is 0≤Zt≤1.

To specify our test-case problems it is necessary to choose suitable (non-dimensional) profiles for σw(z) and τ(z). Here we choose to focus on three such profiles, two of which are widely used empirical fits to observed statistics in stable and neutral conditions respectively. The third has τ(z) constant and a linear profile for σw(z), and is used to demonstrate a new LPDM scheme introduced below. The details of the profiles used are given in Table 1 and are plotted in Fig. 1. In practice, the exact profiles suggested by are modified slightly, as detailed in Table 1, to avoid singular behaviour at the ABL top and bottom. This is necessary because in Hanna's original profiles either σw→0 or τ→0 as z→0,1, with neither type of behaviour being physical.

In Sect. 4 large ensembles of numerical solutions of Eq. () will be calculated using different numerical integration schemes. The accuracy of each numerical scheme, as a function of time step Δt, will be assessed by comparison with the corresponding solution of the FPE, to be detailed next.

Following the standard procedure in stochastic calculus e.g. Sect. 3.4.1 of, the FPE which describes the time evolution of the probability density p(ω,z,t) of (Ωt,Zt) in Eq. () can be obtained as ∂p∂t=-∂ωσwp∂z-∂∂ω-ωτ+∂σw∂zp+1τ∂2p∂ω2. Explicitly, here ω=w/σw. The initial conditions consistent with those given in Eq. () are (for σz≪1 and z0 not near the boundaries) p(ω,z,0)=12πσzexp⁡-ω22-(z-z0)22σz2.

The FPE Eq. () also requires boundary conditions at z=0,1 which are consistent with the reflecting boundary conditions for the LPDM. The boundary conditions consistent with reflection are p(ω,0,t)=p(-ω,0,t),p(ω,1,t)=p(-ω,1,t), which in probabilistic terms is equivalent to the reflection condition Ωt→-Ωt being applied at the boundaries. found that this perfect reflection algorithm is exactly consistent with the “well-mixed constraint” in homogeneous Gaussian turbulence see also the appendix of Sect. 11 of.

Equations ()–() constitute a well-defined initial-value problem which is suitable for numerical solution. An important quantity obtained from the solution p(ω,z,t) is the physical concentration of parcels given by c(z,t)=∫-∞∞p(ω,z,t)dω. (In general, tracer concentrations and the marginal probability given in Eq.  can differ by a normalisation constant.) The concentration c(z,t) will be our main benchmark quantity in Sect. 4 below.

The non-dimensionalised FPE Eq. () is a hypo-elliptic differential equation defined on R×[0,1]. Our approach to its numerical solution is to seek a solution based on the following Hermite polynomial expansion: p(ω,z,t)=12π∑k=0∞Ck(z,t)Hek(ω)e-ω2/2. In statistics, this expansion is also known as the Gram-Charlier series of Type A see p. 23 of. Here the functions Ck(z,t) denote the projection, at the vertical level and time (z,t), of p(ω,z,t) onto the (probabilists') Hermite function Hek(ω)e-ω2/2/2π where Hek(ω) is the Hermite polynomial defined by Hek(ω)=-1keω2/2dkdωke-ω2/2. Notice that it follows that the particle concentration Eq. () satisfies c(z,t)=C0(z,t).

Before inserting the expansion Eq. () into the FPE Eq. () it is helpful to rewrite the FPE in the form ∂p∂t=1τ∂2p∂ω2+ω∂p∂ω+p-∂∂ω∂σw∂zp-∂ωσwp∂z. In this form the Hermite function identity Eq. () can be used to evaluate the first term on the right-hand side. Further, the second and third terms on the right-hand side can be simplified using the derivative and recursion formulae for Hermite polynomials (Eqs. , ). After some working the result is (using the convention C-1≡0) ∑k=0∞Hek(ω)e-ω2/2∂Ck∂t+kτCk+(k+1)∂∂zσwCk+1+σw∂Ck-1∂z=0. Using the orthogonality property of Hermite functions () it follows that ∂C0∂t=-∂∂zσwC1∂Ck∂t=-kτCk-(k+1)∂∂zσwCk+1-σw∂Ck-1∂z,fork≥1. The system () constitutes an infinite system of coupled 1 + 1 dimensional first-order partial differential equations for the coefficients Ck. For a numerical solution this series can be truncated as we describe below.

The initial conditions for Eq. () are easily obtained from () using the orthogonality property, C0(z,0)=12πσzexp⁡-(z-z0)22σz2,Ck(z,0)=0(fork≥1). The boundary conditions can be obtained using the symmetry Hek(ω)=-1kHek(-ω). Substituting the expansion () into the boundary condition (), it follows that ∑koddCk(z,t)Hek(ω)e-ω2/22π=0,atz=0,1, and consequently Ck(0,t)=Ck(1,t)=0,forkodd. It may seem surprising that the even equations have no boundary condition and the odd equations take two boundary conditions. However, as the system () consists of first-order PDEs it is clear that the total number of boundary conditions will be correct, provided that the series is truncated at k=K odd.

It is worth noting that the series () can also be truncated at K=0 by using an (approximate) quasi-stationary balance in the k=1 equation of the form C1=-σwτ∂C0∂z, which results in the diffusion equation ∂C0∂t=∂∂zσw2τ∂C0∂z,∂C0∂z(0,t)=∂C0∂z(1,t)=0. It is well known e.g. Sect. 3.5 of that the LPDM Eq. () can be approximated by a random walk (“random displacement” or RDM) model dZt=∂∂zσw2τdt+2σw2τ1/2dBt. Equation () is simply the Fokker–Planck equation of the RDM model (Eq. ), with the diffusivity κ of the RDM being κ=σw2τ. Note that the RDM model can be derived formally from the LPDM in the distinguished limit of a short decorrelation time, σw→∞, τ→0 with σw2τ=κ finite see e.g. Sect. 6.3 of. It is much easier to obtain accurate solutions of Eq. (), compared to Eq. () at relatively large time steps; hence, an interesting question concerns when exactly it is preferable to solve Eq. () rather than Eq. (). This question is best answered by quantifying the difference between the solution of Eqs. () and () and using this difference as a benchmark for assessing the errors in LPDM calculations, as will be done in Sect. 4 below.

Based on the analysis above, Eq. () can be solved numerically by integrating the system () with boundary conditions (Eq. ), truncated at k=K odd. Our approach is to use a standard finite-difference discretisation with Nz grid points, equally spaced with Δz=1/Nz, on a staggered cell-centred grid (i.e. zi=i-1/2Δz, for i=1,…,Nz) in order to apply the boundary conditions at z=0,1 systematically. The details of the implementation of the boundary conditions are described in Appendix B.

The set () is stiff and a naive solution method would have the time step Δt bounded above by Δt≲Minzτ(z)/K, i.e. the timescale of exponential decay of the highest Hermite function mode. However, considerably longer time steps can be used if an exponential time-stepping scheme is chosen. Our choice is the Exponential Time-Differencing fourth-order Runge–Kutta (ETDRK4) scheme of , with the “linear” operator in that scheme taken to be the first term on the right-hand side of Eq. () only, because it is this first term that is responsible for the stiffness of Eq. ().

To obtain our benchmark solutions of Eq. () and therefore Eq. (), tests of the convergence of the solutions as both Δt and Δz are decreased and K is increased have been performed. For all three case studies, it was found to be adequate to take K=19 to obtain fully converged solutions, because the Hermite series was found to converge rapidly, i.e. |C19|≲10-16 everywhere in the domain. Comparison of a sequence of solutions with Δz=1/Nz with Nz=27,28,…,212 revealed quadratic convergence with Δz as expected for our scheme. Figure 2 shows the relative error Ej(t), with reference to the next-highest resolution solution, in the L2 norm for the mean concentration c(z,t) at fixed times, for the two test cases. That is, Ej(t)=∫01C0j(z,t)-C0j+1(z,t)2dz1/2, where C0j(z,t) denotes the solution with Nz=2j. Quadratic convergence is evident from the slope of the graphs in Fig. 2. For example, typical numerical errors at Nz=212 (highest resolution) are E12(t1)=9.7×10-5 (stable boundary layer at t1=h/u*) and 1.3×10-4 (neutral boundary layer at t1=3h/u*) respectively. The numerical accuracy above is sufficient for benchmarking our LPDM solutions, because the statistical error associated with reasonable-sized ensembles (N=106) of the LPDM is of order E(t1)≈10-2, as will be discussed below.

Figure 3 shows snapshots of the particle concentration c(z,t) for each of the three FPE benchmark solutions described above. The left panel shows the constant τ case, the middle panel shows the stable ABL case, and the right panel the neutral ABL. In all three cases particles are initialised close to z=z0=1/2 and disperse to become well mixed throughout the ABL at late times. The neutral and stable cases differ in that mixing is rather more rapid (in terms of the dimensional timescale h/u*) for the stable case compared to the neutral case. Also, in the neutral case, mixing is relatively slow towards the top of the ABL where the amplitude of turbulent fluctuations decays exponentially.

Tables 2–3 summarise the SDE numerical schemes to be investigated. The first, most obvious scheme to test is the Euler–Maruyama (E-M) scheme , i.e. the simplest and lowest order time-stepping scheme for SDEs. Next, as with ordinary differential equations (ODEs), it is possible to construct schemes with higher orders of formal accuracy in the spirit of Runge–Kutta schemes for ODEs. Here we test the performance of Platen's “explicit order 2.0 weak scheme” (EXPLICIT 2.0) and “explicit order 3.0 weak scheme” (EXPLICIT 3.0) see Chap. 15 of. In common with schemes for ODEs, higher order schemes are somewhat more complicated to implement, and are more computationally expensive per time step Δt. The advantage, however, is that the schemes have weak order Δt2 (EXPLICIT 2.0) and Δt3 (EXPLICIT 3.0) compared to Δt for E-M.

A single candidate from a second class of schemes, the so-called “small-noise” schemes, to be investigated is the HON-SRKII scheme of . Small-noise schemes typically have the same weak order (Δt) as E-M see e.g. the discussion in Chap. 3 of, but the schemes are designed so that the leading-order error depends on the “noise amplitude” in the equation, which in many practical situations is sufficiently small that higher-order convergence is observed in practice (at least for intermediate length time steps; see discussion below). The HON-SRKII scheme will be shown below to converge with global error ∼Δt2 in this intermediate time-step regime.

A third class of schemes to be investigated is designed to work well with long time steps. Such schemes are of interest operationally, because the practical advantages of calculating large ensembles efficiently are thought to outweigh the disadvantage of loss of accuracy due to time-stepping errors. The FLEXPART model , for example, switches between using E-M and a long time-stepping scheme due to LEGGRAUP. It is of some interest to verify that long time-stepping schemes such as LEGGRAUP do indeed outperform E-M at operationally relevant values of Δt. In fact, in Appendix C we review the derivation of the LEGGRAUP scheme, and show that additional care is needed to couple the velocity and position equations. A corrected scheme (LONGSTEP) is derived in Appendix C and is then compared with the schemes listed above in Sect. 4.2.

The method used to compare the results from a particular scheme, at fixed time step Δt, to the particle concentration c(z,t) obtained from the numerical solution of the FPE, is as follows. First, a large ensemble (typically N=106) of trajectories is calculated using the scheme under investigation. Next, the density of particles c^ is reconstructed from the resulting ensemble {Zt(i),i=1,…,N} using kernel density estimation c^(z,t;hb)=1Nhb∑i=1NKz-Zt(i)hb+“image terms”. Here hb>0 is a (small) smoothing parameter known as the bandwidth, and “image terms” refer to contributions from the images of trajectories, introduced to satisfy the boundary conditions. The function K(⋅) is the kernel function, and is non-negative with zero mean and has unit integral. Here we use a Gaussian kernel. Details, including how the optimal bandwidth hb=h* is chosen in practice, are given in Appendix D.

The error associated with a given scheme, at time step Δt, is measured by the L2 norm ‖c-c^‖2=∫01(c(z,t)-c^(z,t;h*))2dz1/2. In practice the error (Eq. ) is effectively bounded below by the so-called statistical error, which is defined as the expected value of ‖c-c^‖2 in the event that the ensemble {Zt(i),i=1,…,N} were sampled from the exact distribution c(z,t) itself. It is important to emphasise that it is not possible, using our method, to investigate schemes with errors below the statistical error. The statistical error can of course be reduced by using a larger ensemble N, but convergence is slow as the dependency is N-1/5, as discussed in Appendix D where details are given.

In the results below, in the interests of reproducibility, the error is presented as a function of the fixed time step Δt for each scheme. However, the schemes have different computational costs per time step, which will depend on both the method of implementation of each algorithm and on the machine used for the simulations. To give a rough idea of representative computational costs, in Table 4 the relative cost measured with reference to the E-M scheme is shown for our calculations. Following best practice in large operational calculations see e.g., the random numbers used to simulate the Wiener processes are pre-calculated so the costs of their generation are not included in the comparison. Another possible computational saving comes from the use of variable time steps. To test whether or not a significant computational saving is easily attainable, we have made some calculations in which Δt∝τ (the local Lagrangian decorrelation time). For each scheme tested, the use of variable time steps was found to lead to a computational saving of a factor of around 2 to 3 compared to fixed time steps, with the schemes otherwise performing as detailed below. More details on variable time-stepping schemes will be given elsewhere.

The main results, showing the performance of the six schemes described in Tables 2–3 over a wide range of time steps Δt, are shown in Figs. 4–6. Figures 4–6 detail the results for the constant τ test case, the stable ABL test case and the neutral ABL test case respectively (see Table 1). In each figure, the L2 error (Eq. ) is plotted as a function of non-dimensional time step Δtu*/h. Logarithmic scales are used so that lines of constant slope correspond to the observed order of the schemes. Blue lines with slopes 1, 2, and 3 are plotted for reference. The statistical error, which is the lowest possible error that can be measured for a given scheme, is plotted as a solid black line in each panel.

Also plotted in Figs. 4–6, as a dotted black line, is the L2-norm difference ‖c-C0‖2 between the concentration field c(z,t) obtained from the solution of the FPE Eq. () and C0(z,t) obtained from the diffusion Eq. (). The dotted black line marks an important boundary on each panel. If the time step Δt is such that the error of a given scheme lies above this line, then it is preferable to solve the RDM Eq. () in place of Eq. (), because (at fixed Δt) the numerical error for the former is more easily controlled.

Figure 4 shows results for the constant τ test case at time t=h/u* (see Fig. 3 and Table 1 for details). The lowest order schemes, LEGGRAUP (blue circles) and E-M (black squares), are seen to realise their formal weak order Δt. EXPLICIT 2.0 (red hexagons) and HON-SRKII (green solid triangles) have weak order Δt2, whereas EXPLICIT 3.0 (blue triangles) has weak order Δt3, as expected. The best performing scheme for this particular case is the new scheme LONGSTEP (purple diamonds) derived in Appendix C. The rationale for LONGSTEP is that there is a conceptual error in the derivation of LEGGRAUP, which results in its performance at large Δt being no better than E-M. When this error is corrected in LONGSTEP, the performance is better than even EXPLICIT 3.0. LONGSTEP in effect uses exact solutions of the LPDM equations for constant τ and linear σw, meaning that if the same calculations had been performed in an infinite domain, the numerical error would be zero. In the constant τ test case, errors in LONGSTEP arise only from the reflection boundary conditions at z=0,1. However, LONGSTEP does not fare well in the remaining two test cases to be described next.

Figure 5 shows results for the stable ABL test case at intermediate time t=h/u* (upper panel) and at late time t=4h/u* (lower panel), when the concentration is almost well mixed across the ABL (see Fig. 3). The results are similar to those of the constant τ case, except LONGSTEP (purple diamonds) now performs as poorly as E-M. Both E-M and LONGSTEP outperform LEGGRAUP. It was not found to be possible to obtain solutions for EXPLICIT 3.0 using time steps longer than Δt=0.02h/u* because of problems with the reflective boundary conditions.

Figure 6 shows the results for the neutral ABL case at intermediate time t=3h/u* (upper panel) and at late time t=12h/u* (lower panel). In this case the performance of LONGSTEP and LEGGRAUP are comparable, but with the E-M scheme performing better than both, except at very long time steps where LEGGRAUP has slightly better accuracy at long time steps. As for the previous test cases, the EXPLICIT 3.0 (blue triangles) scheme gives the lowest errors (weak order Δt3), and EXPLICIT 2.0 (red hexagons) along with HON-SRKII (green solid triangles) perform consistently well with weak order Δt2.

To give an impression of where the particle concentration errors are accumulating, Fig. 7 shows snapshots of particle density c^(z,t) for the stable ABL case, at t=h/u*. Results are shown for each scheme when a long time step Δt=0.05h/u* is used (left panel) and a moderate time step Δt=0.007h/u* (right panel). The errors in the long time-step case are large and are largely due to issues with the reflection of trajectories at the surface (z=0). Numerical accuracy requires that Δt≪τ, which is evidently violated close to the boundary where τ(z) is small (see Fig. 1). Errors due to reflection are particularly acute for the higher order schemes (such as EXPLICIT 2.0 and HON-SRKII) that require the treatment of an intermediate step(s). See the discussion in Appendix B for how this step is implemented. The stable boundary layer case, where τ decays most rapidly near the z=0 boundary, is the case which appears to be the most sensitive to the treatment of reflection there.