Jan O Pralits

Alessandro Bottaro

Stefania Cherubini

Weakly nonlinear optimal perturbations

A simple approach is described for computing spatially extended, weakly nonlinear optimal disturbances, apt at maintaining a disturbance-regeneration cycle in a simple shear flow. Weakly nonlinear optimals, computed over a short time interval for the expansion used to remain tenable, are oblique waves which display a shorter streamwise and a longer spanwise wavelength than their linear counterparts. Threshold values of the initial excitation energy, separating the region of damped waves from that where disturbances grow without bounds, are found. Weakly nonlinear optimal solutions of varying initial amplitudes are then fed as initial conditions to direct numerical simulations of the Navier-Stokes equations and it is shown that the weakly nonlinear model permits to identify flow states which cause rapid breakdown to turbulence.

Introduction

Optimal perturbations represent a fundamental concept of the linear stability theory of shear flows and one highly cited textbook [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF] centers much of the discussion on this concept and on the associated idea of transient growth of disturbances for systems ruled by non-normal linear operators. It has been thought for quite some time that, under nominally stable conditions, the most dangerous initial disturbances computed in a linear setting might undergo a substantial amplification, eventually capable of bringing the fluid system outside of the domain of attraction of the laminar solution (and also outside of the domain of validity of the linearized equations), so that transition can occur because of nonlinear interactions. This has provided the rationale for studying linear optimal disturbances, via a number of approaches including eigenfunction expansion [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF], expansion in periodic functions [START_REF] Criminale | Perturbation dynamics in viscous channel flows[END_REF]) and adjoint equations [START_REF] Andersson | Optimal disturbances and bypass transition in boundary layers[END_REF][START_REF] Luchini | Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations[END_REF][START_REF] Corbett | Optimal linear growth in swept boundary layers[END_REF].

Linear, optimal perturbations in simple parallel shear flows, such as Couette, Poiseuille or Hagen-Poiseuille flows, take the form of streamwise or quasi-streamwise vortices (the streamwise wavenumber α either vanishes or is very small, in the latter case decreasing like Re -1 , with Re the Reynolds number), and their downstream evolution turns them into spanwise-periodic streaks, elongated along the main flow direction, x. While the presence of vortices and streaks seems appropriate in view of experimental observations of transitional flow features, their very early presence in the flow field is unwarranted since x-invariant flow structures trigger nonlinear interactions very slowly and only when their amplitude is large, through the onset and growth of a traveling wave instability [START_REF] Andersson | On the breakdown of boundary layer streaks[END_REF][START_REF] Gustavsson | Non-linear wave interactions from transient growth in plane-parallel shear flows[END_REF].

The traditional concept of optimal disturbances (those which give rise to the largest disturbance energy growth over a given time window, or over a given spatial development length, should one consider flows developing in space) is thus misleading if one searches for the initial waveform most suited to provoke transition, at least for the simplest shear flows. This has been understood for a few years now, and has produced a rising number of contributions on non-linear optimals [START_REF] Zuccher | Algebraic growth in a Blasius boundary layer: Nonlinear optimal disturbances[END_REF][START_REF] Cherubini | The minimal seed of turbulent transition in the boundary layer[END_REF][START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF][START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF][START_REF] Monokrousos | Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows[END_REF][START_REF] Pringle | Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos[END_REF][START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF][START_REF] Duguet | Minimal transition thresholds in plane Couette flow[END_REF] which have highlighted the profound differences between linear and nonlinear transient growth. In the abstract of their contribution [START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF] state that all of their results "highlight the irrelevance of the linear energy gain optimal perturbation for predicting or describing the lowest-energy flow structure which triggers turbulence." Nonlinear optimals are typically found via adjoint looping [START_REF] Luchini | Adjoint equations in stability analysis[END_REF]) and are very demanding in terms of computer memory and CPU time, because of the need to carry out several direct simulations of the Navier-Stokes equations and to store (or recompute, via a checkpointing procedure) the whole flow field at all times for use in the adjoint equations. Thus, a thorough parametric search (in terms of wavenumbers, disturbance amplitude and target time of the optimization) is still out of reach.

A weakly nonlinear alternative, much cheaper computationally than the full nonlinear optimization approach, is explored in the present paper. The approach pursued aims at identifying initial perturbations which distort efficiently -via primary nonlinear interactions -the main flow. It will be shown that such optimal solutions take the form of oblique structures. It has been known since [START_REF] Schmid | A new mechanism for rapid transition involving a pair of oblique waves[END_REF] that the early appearance of oblique waves elicits faster transition than either optimal streamwise vortices or Tollmien-Schlichting waves (see also [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]. The mechanism described by [START_REF] Schmid | A new mechanism for rapid transition involving a pair of oblique waves[END_REF] has been called oblique transition; they have expressed the generic disturbance field, φ, in Fouries series, e.g. mαx+nβz) , where x, y and z indicate, respectively, the streamwise, wall-normal and spanwise direction; α and β are the streamwise and spanwise wavenumbers; M and N are positive integers that define the truncation of their direct numerical simulation. By imposing at t = 0 a disturbance field in the form of a pair of oblique waves with (m, n) = (1, ±1), [START_REF] Schmid | A new mechanism for rapid transition involving a pair of oblique waves[END_REF] have observed a rapid amplification of the mean flow distortion, mode (0, 0), accompanied by a growth (equally rapid but less vigorous) of the streamwise-independent component, (0, ±2); conversely, the (±2, 0) mode remained an order of magnitude smaller. The (0, ±2) component is ruled by a nonnormal evolution operator, very sensitive to external forcing, and can thus be enhanced linearly by a transient growth mechanism, yielding large amplitude streaks via the lift-up effect [START_REF] Landahl | Wave breakdown and turbulence[END_REF]. Much attention has been paid to the instability of such streaks, with lower emphasis placed on the fact that the streaks develop on top of a base flow which is being strongly distorted. The predominance of the (0, 0) non-linearly generated component, even above the (1, ±1) mode, is, however, evident in figs. 1 and 2 of the paper by [START_REF] Schmid | A new mechanism for rapid transition involving a pair of oblique waves[END_REF]. It is thus clear the interest of studying the optimal linear (modal or nonmodal) growth of an oblique disturbance developing over a base flow which is being deformed nonlinearly by the primary wave itself. This calls for an autonomous cycle, schematized in fig. 1, with the first generation of non-linear interactions (the Reynolds stresses), which feeds onto the mean flow. The present paper describes the simplest possible model of such a cycle: the creation of harmonics, past the (0, 0) mode which concentrates in itself the majority of growth, is neglected. We thus study the rapid transient growth of an oblique wave and search for that optimal initial disturbance capable of creating and maintaining the self-sustained, neutral cycle sketched in fig. 1. The appropriateness of the present minimal model will be demonstrated by comparisons with direct numerical simulations.

φ(x, y, z, t) = M m=-M N n=-N φm,n (y, t)e i(

Oblique

Formulation of the weakly non-linear problem

A self-sustained cycle can be created decomposing velocity and pressure into a steady, laminar parallel state, a traveling wave and a slowly varying time-dependent base flow distortion. The idea for this decomposition stems from [START_REF] Biau | An optimal path to transition in a duct[END_REF]; however, other papers in the recent literature [START_REF] Hall | Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures[END_REF][START_REF] Blackburn | Lower branch equilibria in coeutte flow: the emergence of canonical states for arbitrary shear flows[END_REF][START_REF] Beaume | Reduced escription of exact coherent states in parallel shear flows[END_REF] have also split the disturbance field in a streamwise-invariant (i.e. mean) and a streamwise-varying (i.e. fluctuations) component, reaching a set of equations in which the Reynolds stresses force the growth of the mean flow. The decomposition employed here assumes no a priori scaling on the Reynolds number and reads:

    U (y) 0 0 P (x)     +     ũ(x, y, z, t) ṽ(x, y, z, t) w(x, y, z, t) p(x, y, z, t)     + 2     u 00 (y, t) v 00 (y, t) w 00 (y, t) p 00 (y, t)     , (2.1)
where ∈ R denotes the wave amplitude. In the following, the simplest wall-bounded flow is considered but the procedure is general and can easily be applied to semi-infinite or infinite domains. The O( ) terms correspond to the traveling wave which we wish to optimize for; those of O( 2 ) are the mean flow distortion terms which dominate the nonlinear interactions at early times. Clearly the decomposition proposed is insufficient at large times when higher modes are created and interact; it is however adequate, as we will show, to approximate the lower bound of the amplitude beyond which the flow can undergo transition to turbulence. The flow decomposition (2.1) is introduced into the Navier-Stokes equations and like-order terms are collected. At order O(1) we obtain

P x = 1 Re U yy , (2.2)
with subscripts denoting partial differentiation, plus no-slip boundary conditions for U . The Reynolds number, Re, is based on the half-channel thickness and the wall velocity.

For the case of the Couette flow treated here P is constant and U = y, for y ∈ [-1, 1]. The disturbance at O( ) is taken to be a simple monochromatic wave which travels along a direction inclined by tan -1 (β/α) with respect to the x-axis:

(ũ, p)(x, y, z, t) = (u 11 , p 11 )(y, t)e i(αx+βz) + (u 11 * , p * 11 )(y, t)e -i(αx+βz) , (2.3)

where superscript * denotes complex conjugation, ũ = (ũ, ṽ, w), and α, β ∈ R are the streamwise and spanwise wave numbers, respectively. Using expression (2.3), the governing equations, linearized around the perturbed and time-varying mean flow [U (y) + 2 u 00 (y, t), 0, 2 w 00 (y, t)], are given by iαu 11 + v 11y + iβw 11 = 0, (2.4)

u 11t + iα(U + 2 u 00 )u 11 + v 11 (U + 2 u 00 ) y + iβ( 2 w 00 )u 11 + iαp 11 = 1 Re ∆ k u 11 , (2.5) v 11t + iα(U + 2 u 00 )v 11 + iβ( 2 w 00 )v 11 + p 11y = 1 Re ∆ k v 11 , (2.6) w 11t + iα(U + 2 u 00 )w 11 + v 11 ( 2 w 00y ) + iβ( 2 w 00 )w 11 + iβp 11 = 1 Re ∆ k w 11 ,(2.7)
where ∆ k = ∂ 2 /∂y 2 -k 2 and k 2 = α 2 + β 2 . Equations (2.4)-(2.7) are accompanied by no-slip conditions on both the upper and lower walls and an initial condition u 11 (y, t = 0) which we will optimize for. At order O( 2) the equations read

v 00 = 0, (2.8) u 00t - 1 Re u 00yy = -[v 11 u * 11y + iβw * 11 u 11 + c.c.],
(2.9)

p 00y = -[iαu * 11 v 11 + v 11 v * 11y + iβw * 11 v 11 + c.c.],
(2.10)

w 00t - 1 Re w 00yy = -[iαu * 11 w 11 + v 11 w * 11y + c.c.],
(2.11) plus the usual no-slip conditions on both walls. The key point of the analysis is the fact of retaining the x-and z-component of the velocity at O( 2 ) in the expression of the base state above which the mode (1, 1) is superposed (note also that we have anticipated the fact that v 00 = 0 in writing equations 2.5-2.7). The fact of having lifted a higher order term to equations at lower order is motivated by the need to allow for a distorted base state. From the mathematical point of view, this corresponds to replacing the original series (2.1) in powers of into a sequence of unknown (and decreasing) functions of [START_REF] Van Dyke | Perturbation Methods in Fluid Mechanics[END_REF][START_REF] Cousteix | Asymptotic Analysis and Boundary Layers[END_REF]. The Orr-Sommerfeld stability equation for the Blasius boundary layer flow is the best-known example of one such instance. System (2.4)-(2.7) plus equations (2.9) and (2.11) are solved by maximising the energy of the oblique mode at the target time t = T , defined as

e(T ) = 2 2 1 -1 (u 11 u * 11 + v 11 v * 11 + w 11 w * 11 )dy t=T .
(2.12)

Alternative measures of growth could have been considered, for example based on the r.h.s. terms of equations (2.9-2.11). In light of previous experience (e.g., [START_REF] Bottaro | Formation of secondary vortices in turbulent square-duct flow[END_REF][START_REF] Monokrousos | Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows[END_REF][START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF], the conclusions reached here would probably be but mildly affected; this remains however to be verified. For assigned the nonlinear problem is solved iteratively in the following manner: (a) Maximise the energy e(T ), constrained by equations (2.4)-(2.7), over a given time span T , to find u 11 , v 11 , w 11 . The optimization procedure is performed via adjoint looping; in the first iteration set u 00 = w 00 = 0 ∀y, t.

(b) Compute the right hand side of equations (2.9) and (2.11), and solve for u 00 (y, t) and w 00 (y, t) under the initial condition u 00 (y, 0) = w 00 (y, 0) = 0; equations (2.9) and (2.11) are heat equations with Reynolds stress terms acting as source. Then, go back to (a).

(c) Convergence is declared when the final wave energy e(T ) converges to within a defined precision. The normalization employed is e(0) = 2 . In order to quantify the results, suitable quantities are defined which can also be used in the limit → 0 to compare with linear optimal results from the literature. The gain G is

G( , Re, α, β, T ) = e(T ) e(0) , (2.13)
and is obtained from the converged solution of the procedure outlined above, for given values of α, β, Re and in a defined temporal interval [0, T ]. Finally, the mean-flow distortion is quantified by evaluating its "energy":

E U (t) = 4 4 1 -1 u 2 00 (y, t)dy and E W (t) = 4 4 1 -1
w 2 00 (y, t)dy.

(2.14)

Numerical procedure

The problem outlined at step (a) in the solution procedure is a constrained maximization problem which can be solved using Lagrange multipliers. The governing equations are written in primitive variable form and the resulting adjoint equations are derived using a discrete approach (see [START_REF] Luchini | Adjoint equations in stability analysis[END_REF]. The spatial derivatives are discretized using second order finite differences and a semi-implicit second-order scheme is used to advance in time. A uniform grid is used in the y-direction and 300 discrete points are sufficient to obtain a converged solution. The code has been tested on several cases found in the literature for → 0 (i.e. the mean flow is not distorted); in particular, the value of the optimal gain G = 0.00118 Re 2 and the corresponding time at which it is achieved, T = 0.117 Re, with α = 35/Re and β = 1.60, are recovered with less than 0.1% error [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]. Results are obtained imposing that convergence is reached when the relative difference in gain between two consecutive iterations is less than 10 -8 .

Weakly non-linear results

To illustrate results, the case Re = 400 is chosen. We take four values of and fix the wavenumbers α and β as well as the final time T ; all parameters selected are given in the caption of figure 2, which displays the curves of gain (equation 2.13) together with the energies of the mean flow modification for the cases in which > 0. We choose a rather small value of the final time in order to avoid situations in which the nonlinear effects become dominant for which an expansion of the waves including higher harmonics would be necessary.

For < 0.0153 all solutions are damped for t > 15. For ≈ 0.0153 a limit condition is reached, with the different energy measures neither increasing nor decreasing. This corresponds to a threshold between initial disturbance states which relaminarize and states which might lead the flow to turbulence. When > 0.0153 the measure of energy in figure 2 shows a monotonic increase. We can note that the two components of the energy of the mean flow distortion differ by almost one order of magnitude. This fact might lead to the idea that including the 2 w 00 terms in the wave equations, (2.5)-(2.7), yields a negligible effect. This will be further discussed in section 3.1. Figure 3 shows, for = 0.0153, the mean flow U and its distortion at t = 0, 10, 15, 20, highlighting the presence of an inflection point, in both the streamwise and the spanwise velocity components, at y = 0.

The optimal disturbance wave at t = 0 is displayed in modulus and phase in figure 4, and through isocontours of u 11 and (v 11 , w 11 ) vectors in figure 5. The modulus of the optimal disturbance peaks, both at t = 0 and at the target time, in correspondence of the inflection point of the mean flow. The initial perturbation is tilted in the (y, z) plane at t = 0, and grows in time developing a slant in the opposite sense, under the forcing of the mean flow distortion shown in figure 3. values of α and β, where the peak of G is found, remain at 0.93 and 1.44, respectively. We denote these weakly nonlinear optimal values of the wave numbers as α opt and β opt .

In figure 7(a) we report α opt and β opt for different values of the final time T used in the optimization procedure. For T < 15 the streamwise wavenumber is larger than the spanwise wavenumber, in particular at T = 10 α ≈ 2 β. For T > 15 the opposite is true and both α opt and β opt have asymptotic behavior for large values of T . In particular the asymptotic value of the spanwise wavenumber approaches its linear counterpart, i.e. 1.60. The value of which indicates a limit condition is here denoted crit and its value as a function of the target time T of the optimization is shown in figure 7(b). As one could expect, crit decreases with T , reaching the asymptotic value crit ≈ 0.015 already for T as low as 30.

Neglecting the correction to the transverse velocity

In figure 2 is was shown that the measure E W was about one order of magnitude smaller than E U . This seems to imply that the mean-flow component w 00 can be safely omitted from the equations for the optimal oblique disturbance; such an approximation has been made, for example, by [START_REF] Farrell | Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow[END_REF]. This is, however, not true -at least in the present model -as the example here will show. Let us consider the case in which Re = 400 and the terms 2 w 00 are not included in equations (2.5)-(2.7). The corresponding optimal Gain G as a function of time t for = 0.0078, 0.0084, 0.0086, and αopt = 0.27, βopt = 1.083 (obtained in the case excluding the 2 w00 terms) and Re = 400. Lines (-) denote the case including 2 w00-terms, (--) is the case without the 2 w00 terms; the arrow displays increasing values of . (b) Energy of the mean flow modification when = 0.0084 for both EU and EW in the case (--) without 2 w00 terms and (-) including 2 w00 terms.

wavenumbers, optimized also with respect to T , are α opt = 0.27 and β opt = 1.083. In figure 8 we report the values of the gain G and energies E U and E W for three different values of . A limiting condition is found for ≈ 0.0084, similar to what shown in figure 2.

Here, the value of w 00 is not included in the governing equations and the value of E W is only shown for comparison with the value of E U . Clearly E W is two orders of magnitude smaller than E U . If we now include the 2 w 00 terms in (2.5)-(2.7) and compute the gain and energies due to the mean-flow distortion using the values of α opt and β opt previously obtained, no limiting condition is found, and all solutions are damped for t > 40 (cf. figure 8 (a), thick solid lines). This holds for all values of used and also for larger values of (not shown here).

The result is somewhat unexpected and it confirms that acting on the spanwise mean flow is an efficient strategy to control fluid motion [START_REF] Gatti | Performance losses of drag-reducing spanwise forcing at moderate values of the Reynolds number[END_REF].

Non-linear results and comparison

The weakly non-linear optimal solutions obtained for T = 20, α opt = 0.93 and β opt = 1.44, and different values of ranging from 0 to 0.016 have been used to initialize several Direct Numerical Simulations (DNSs). The computational domain has length 2π/α opt in the streamwise direction and 2π/β opt in the spanwise domain. The chosen domain is discretized using a staggered grid and the Navier-Stokes equations are solved by a fractional-step method with second-order accuracy in space and time [START_REF] Verzicco | A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates[END_REF]). In order to provide a measure of the perturbation energy in the whole computational domain we define two integral energies, E and E wavy , the former based on the whole perturbation, u, and the latter only on its wavy part, u wavy = u -2 u 00 . The perturbation and wavy integral energies are thus defined as:

E = 1 2V V u 2 + v 2 + w 2 dV E wavy = 1 2V V u 2 wavy + v 2 wavy + w 2 wavy dV, (4.1)
with V = 8π 2 αoptβopt the computational domain. As in the previous section, gains are defined with respect to the energy at the initial time. In particular, the gain G predicted by the weakly non-linear model will be compared with the wavy energy gain, E wavy (t)/E wavy (0) provided by the DNS, which, for brevity, will be labeled G as well. Figure 9 (a) shows the wavy energy gain obtained by DNS using the initial optimal solution for = 0.014, compared with that predicted by the weakly non-linear model for the same value of . The two curves show a remarkable agreement, indicating that the weakly non-linear expansion succeeds in predicting the evolution of the perturbations, provided that non-linear effects are not too strong. On the other hand, for considerably larger values of (in particular, for > crit ), the prediction provided by the weakly non-linear model for = 0.016, after a finite time (equal to about 10 in figure 9 (b)), begins to differ from the DNS. This can be observed in figure 9 (b) by comparing the red dotted and the solid black curves. This discrepancy is clearly due to the fact that, for large enough amplitudes, non-linear effects become too strong to be correctly approximated by a weakly non-linear expansion. However, despite the differences in energy amplification, the perturbation resulting from the weakly non-linear model is able to induce a self-sustaining cycle leading to turbulence. The dashed line in figure 9 (b) shows that, when the field obtained at t = 12 by the model is injected as initial perturbation into the DNS, sudden transition to turbulence is observed. A similar behaviour is observed when using optimal solutions extracted at larger times, provided that the initial value of is sufficiently large.

To determine the critical value of for transition to turbulence, we perform several DNSs initializing the computations with fields extracted at t = 15; the initial conditions providing the t = 15 solutions are the weakly nonlinear optimal disturbances found at different values of for the same target time, T = 20. This choice is motivated by the fact that the energy gain predicted by the model reaches a maximum at approximately t ≈ 15, as shown, e.g., in figure 9.

Figure 10 shows the integral energy curves obtained by DNS for 4 values of ranging from 0.0151 to 0.0154. For the lowest value of the perturbation energy decreases in time, leading the flow to relaminarization. Whereas, for 0.0152, the energy increases reaching a statistically steady value, while the flow is characterized by a chaotic motion. This critical value of for transition to turbulence, determined by DNS, is very close to the critical one predicted by the model, crit = 0.0153. This result might have been anticipated, considering that the strong increase of the energy gain observed by the model for > crit is associated with the development of strong non-linear effects. In order to investigate the mechanisms suddenly leading the optimal perturbation towards a self-sustained state, we have performed a spatial Fourier transform of the streamwise disturbance velocity on the y = -0.4 plane, at each 2.5 time units (by selecting a different wall-normal position or variable the results change but marginally). We have tracked the development in time of modes having different wavenumbers, using their amplitudes

A m,n = (ũ m,n ) 2 + (ũ m,n ) 2 .
The Fourier amplitudes of the most relevant modes, normalized with respect to the initial amplitude of the mode A 0,0 , are provided in Figure 11 for = 0.0151 and 0.0152. The mode (1, 1), corresponding to the oblique wave, is represented by a dashed red line, the mean flow deformation (0, 0) is given by the solid line, and streaky structures, (0, 2) and (0, 1) by the blue and green lines. For the lower value of the scenario appears similar to the early phases of oblique transition. At the initial time of the DNS (t = 15), the flow is composed by the wavy disturbance and its corresponding mean flow deformation. While the energy of the former decreases in time, the latter increases and saturates towards an almost steady state; at the same time, streaky modes begin to be generated from the initial numerical noise, and slowly saturate in time. However, the amplitude of the induced mean flow distortion is not sufficiently high to sustain a regeneration cycle cycle, and after a long saturation phase, the streaks decay towards the laminar state (frame (a) of figure 11). On the other hand, when is slightly larger, the self-sustained cycle is triggered very suddenly, roughly after 10 time units. As one can observe in frame (b) of figure 11, the streaky modes generated by the initial noise do not experience the slow phase of saturation and secondary instability observed in the classical bypass transition scenario, but they rapidly reach a chaotic dynamics. This behaviour is not surprising, since the perturbation has been optimized by taking into account non-linear effects; thus, it should be able to skip some of the linear phases of growth of the perturbations, similarly to fully non-linear optimals (cf., for instance, [START_REF] Cherubini | The minimal seed of turbulent transition in the boundary layer[END_REF]).

The different phases of the weakly non-linear optimal transition are displayed in figure (2013)). Finally, after 55 time units, these chaotic structures lose memory of the initial inclination provided by the vortices, and a self-sustained chaotic motion is observed. On the other hand, for a lower value of , although similar structures are created at early times, they are not able to self-sustain. Figure 13 shows that, after 30 time units, rolls of vorticity normal to the direction of the oblique wave begin to be generated. However, their amplitude being too weak, they cannot be sustained in time by non-linear effects; thus, they begin to decay. During this slow decay phase, the streamwise vorticity component of those rolls slowly begin to generate streamwise streaks, which can be observed in the fourth frame of figure 13. However, since the regeneration cycle is not sustained due to the weak effect of non-linearities, the streaks saturate and then slowly decay.

The trajectory of the flow in these two cases can be observed on a projection of the phase space given by the energy input rate, I, and the integral dissipation rate, D, normalized by their respective laminar values and defined as: where ω represents the vorticity vector. This projection of the phase space (see, e.g. [START_REF] Kawahara | Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst[END_REF]) allows to visualize the balance between the energy injected in the system through the moving walls and that dissipated by viscosity in the whole domain. As discussed by [START_REF] Kawahara | Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst[END_REF], the laminar attractor sits in the point I = D = 1; the turbulent statistically stationary state is close to I = D ≈ 3, whereas all other equilibrium solutions are situated on the I = D diagonal, indicated in the two frames of figure 14 by dashed lines. As shown in figure 14 (a), the trajectories start from the upper left part of the plane (the empty triangles indicating the starting point of the trajectory and its direction), corresponding to an energy input very close to the laminar one and a rather high dissipation value. This is not surprising since the initial flow structures are not placed close to the wall, where energy enters the system, but rather around the center of the domain in y = 0. The light blue trajectory shows that, for = 0.0151, after a rapid decrease of the dissipation, the trajectory bends towards the I = D bisector, locus of equilibrium solutions. However, the perturbation amplitudes reached are not sufficiently high to escape from the laminar attractor. On the other hand, for = 0.0153, the trajectory spends much time close to the bisector, and then leaves towards higher values of the energy input and dissipation rate, which are characteristic of the turbulent dynamics. Finally, in order to evaluate the efficiency of the weakly non-linear optimal perturbations in inducing turbulence, we have compared these trajectories with those followed by a linear optimal perturbation, computed for the same target time. The linear optimal perturbation has been rescaled in amplitude in order to have at t = 15 the same integral energy of the weakly non-linear solutions for = 0.0151 and 0.0157. The light blue and green curves in figure 14 (b) show that, for these amplitudes, the linear optimal perturbation is not able to induce transition, but rather leads to very fast relaminarization. Increasing the initial energy of these perturbations by 5 times still does not lead the flow to transition (as shown by the red line in figure 14 (b)), but rather induces a strong bursting event which slowly decays in time. Thus, we can conclude that weakly non-linear optimal perturbations are indeed more effective in inducing transition than linear ones.

I = α opt β opt 4π 2 2π/

Conclusions

A formulation of the optimal-perturbation problem in shear flows is proposed, which takes into account the distortion of the base flow by the action of the Reynolds stresses, modeled by averaging the products of single-mode disturbance waves. The technique proposed permits to consider the effect of the amplitude of the initial disturbance field and to isolate a strongly receptive region of wavenumber space, for T sufficiently small for the theory to remain tenable, which differs significantly from the most selective range of linear optimal disturbances.

The simplicity of the weakly nonlinear model permits to compute thresholds separating initial states which relaminarize from those which grow without bound, in a fraction of the time compared to full DNS and thereby the model represents a valuable tool for parametric studies of the early stages of the laminar-turbulent transition. The importance of including the spanwise mean-flow distortion terms in the equations describing the transient amplification of oblique flow structures has been demonstrated: although the 2 w 00 terms are small in magnitude their influence on the cross-stream terms in the wave equations is not negligible.

It would, however, be illusory to expect for the present highly truncated model to yield accurate transition thresholds, since transition relies on the growth and interaction of several modes. We have thus performed Direct Numerical Simulations to unravel the transition scenarios induced by the weakly nonlinear optimal disturbances. Concerning the energy amplification, a remarkable agreement is found between the DNS and the weakly nonlinear model results, for initial amplitudes below threshold. For larger amplitudes, non-linear effects become too strong to be correctly captured by a weakly non-linear expansion. However, when crit , the optimal perturbation resulting at finite time from the weakly non-linear model is able to induce a self-sustaining cycle leading to turbulence, whereas a linear optimal disturbance computed for the same target time leads to relaminarization, even when the initial energy is rescaled to very large values. The self-sustained cycle identified is triggered by an inflectional instability at the center of the domain, resulting from the mean flow distortion induced by Reynolds stresses. This inflectional instability generates an array of vorticity rolls in the direction normal to the oblique wave, that, while increasing in amplitude, transport the base flow momentum, creating inclined zones of high and low streamwise velocity. Finally, these chaotic structures lose memory of the initial inclination provided by the vortices, and sustained chaotic motion is observed.

The financial support to the PRIN 2012 project, no. D38C1300061000, from the Italian Ministry of Education is gratefully acknowledged.

Figure 1 .

 1 Figure 1. Self-sustaining cycle centered on the optimal amplification of oblique waves.

Figure 2 .

 2 Figure 2. (a) Gain G as a function of the final time t for → 0, = 0.015, 0.0153, 0.016, for αopt = 0.93, βopt = 1.44 (obtained by choosing T = 20 as target time for the optimization) and Re = 400. (b) Energy of the mean flow modification, EU (-) and EW (--), for = 0.015, 0.0153, 0.016. The arrows display increasing values of .

Figure 3 .

 3 Figure 3. Components of the mean flow U + 2 u00 (a) and 2 w00 (b) at t = 0, 10, 15, 20 for the case Re = 400, = 0.0153, αopt = 0.93 and βopt = 1.44. The arrows display increasing values of t.

Figure 6 Figure 4 .

 64 Figure3shows, for = 0.0153, the mean flow U and its distortion at t = 0, 10, 15, 20, highlighting the presence of an inflection point, in both the streamwise and the spanwise velocity components, at y = 0.The optimal disturbance wave at t = 0 is displayed in modulus and phase in figure4, and through isocontours of u 11 and (v 11 , w 11 ) vectors in figure5. The modulus of the optimal disturbance peaks, both at t = 0 and at the target time, in correspondence of the inflection point of the mean flow. The initial perturbation is tilted in the (y, z) plane at t = 0, and grows in time developing a slant in the opposite sense, under the forcing of the mean flow distortion shown in figure3.Figure6displays the gain G at T = 20 in the Fourier plane, for four different values of . For = 0 the results are those from the classical optimal perturbation theory and the maximum value of G is found for (α, β) = (0.34, 1.72). As the value of exceeds ≈ 0.015 a second region emerges for larger values of α and smaller values of β. This is more evident in figure6(c) with = 0.0153, while in figure 6(d) this new peak, with (α, β) = (0.93, 1.44), is clearly dominant. If is further increased (not shown here) the

Figure 5 .

 5 Figure 5. Vectors and contours, in the (y, z) plane, of the optimal oblique wave at t = 0 (top) and at the target time T = 20 (bottom), shown over two spanwise periods. The shaded contours represent the positive and negative streamwise velocity component, whereas the vectors represent wall-normal and spanwise components. Same parameters as in figure 3.

Figure 6 .Figure 7 .

 67 Figure 6. Gain G in the α -β plane for Re = 400 and T = 20; (a) → 0 with contour values from 20 to 120, (b) = 0.0145 with contour values from 20 to 240, (c) = 0.0153 with contour values from 20 to 300, (d) = 0.01603 with contour values from 20 to 380. The interval among adjacent isolines is ∆G=20 in frames. The maximum value of G for each is denoted by a filled circle.

  Figure8. (a) Gain G as a function of time t for = 0.0078, 0.0084, 0.0086, and αopt = 0.27, βopt = 1.083 (obtained in the case excluding the 2 w00 terms) and Re = 400. Lines (-) denote the case including 2 w00-terms, (--) is the case without the 2 w00 terms; the arrow displays increasing values of . (b) Energy of the mean flow modification when = 0.0084 for both EU and EW in the case (--) without 2 w00 terms and (-) including 2 w00 terms.

Figure 9 .

 9 Figure 9. (a) Energy gain computed by the weakly non-linear model (dashed line) and by the DNS (solid line) both initialized with the optimal solution obtained at t = 0 for T = 20, = 0.014. (b) Energy gain versus time obtained by DNS (black solid line) and by the weakly non-linear model (red dotted line) initialized with the initial optimal solutions for = 0.016. The black dashed line represent the wavy energy gain for a DNS initialized by the optimal solution extracted at t = 12 for the same value of .

Figure 10 .

 10 Figure 10. Disturbance energy E versus time measured by a DNS initialized with the weakly nonlinear optimal states at t = 15, = 0.0151, 0.0152, 0.0153, 0.0154 (increasing from bottom to top).

Figure 11 .

 11 Figure 11. Variation in time of the amplitude of the different modes extracted by Fourier transform of the velocity snapshots obtained by DNS initialized with the weakly nonlinear solutions at t = 15, = 0.0151 (a) and = 0.0152 (b). The legend indicates the streamwise and spanwise wavenumbers, as multiples of the α and β values chosen for the optimisation.

Figure 12 .

 12 Figure 12. Isosurfaces of the streamwise perturbation to the base flow (yellow and blue for u = ±0.3, respectively), extracted from a DNS initialized with the optimal solution at t = 15 for = 0.0152. The snapshots are extracted at t = 25, 30, 35, 45.

Figure 13 .

 13 Figure 13. Isosurfaces of the streamwise perturbation to the base flow (yellow and blue for u = ±0.2, respectively), extracted from a DNS initialized with the optimal solution at t = 15 for = 0.0151. The snapshots are extracted at t = 25, 30, 45, 75.

  The slow-down of the trajectory in correspondence of the bisector, in a region characterized by I and D values intermediate between the laminar and the turbulent ones, might be the signature of the passage of the perturbation close to a relative attractor on the laminar-turbulent boundary (the edge state, see[START_REF] Schneider | Laminarturbulent boundary in plane Couette flow[END_REF];[START_REF] Schneider | Edge of chaos in pipe flow[END_REF];[START_REF] Skufca | Edge of chaos in a parallel shear flow[END_REF]). Indeed, for an initial amplitude far from the laminar-turbulent boundary, = 0.0157 (green line), the trajectory does not pass close to the bisector before reaching turbulence, but displays strong values of D, typical of a bursting phenomenon (see also[START_REF] Van Veen | Homoclinic tangle on the edge of shear turbulence[END_REF];[START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF]).

Figure 14 .

 14 Figure 14. Trajectories on a I -D projection of the phase space of the (a) weakly non-linear optimal solutions obtained for = 0.0151, 0.0153, 0.0157 and (b) of the corresponding linear optimal solutions rescaled with increasing amplitudes. Note that the arrows indicate increasing values of . The starting points of the trajectories, as well as their direction, are indicated by the white triangles, whereas the full triangles indicate the ending points.

† Email address for correspondence: jan.pralits@unige.it


Eugênia Dória
email: eugeniadoria@gmail.com

Viana Cerqueira

Benjamin Motte-Baumvol
email: benjamin.motte@u-bourgogne.fr

Leslie Belton Chevallier
email: leslie.belton-chevallier@univ-eiffel.fr

Olivier Bonin
email: olivier.bonin@univ-eiffel.fr

Does working from home reduce CO2 emissions? An analysis of travel patterns as dictated by workplaces

Keywords:

This research provides new evidence about the relationship between travel behavior, workplace diversification, and environmental impact in the United Kingdom using data from the National Travel Survey for the period between 2002 and 2017. The path analysis approach based on SEM handles both direct and indirect effects and allows for a comprehensive study of travel behavior, trade-off effects, and work and non-work trips. The results suggest that workplace diversification is often reflected by longer average distances for work trips, which are often associated with more remote residential locations. Findings also show that for some categories, such as teleworkers and home-based workers, trade-off effects are observed between work and non-work trips, which increase CO2 emission levels.

Introduction

A number of key developments have affected the labor market over recent decades. These include the increased participation of women in the labor force, the changing nature of the employer-employee relationship, and the growing role of information and communication technologies (ICT). These developments have tended to modify daily work patterns in terms of both time-schedules and workplace locations. The changes in the working day and, in particular, the diversification of workplaces (as with teleworking) may potentially contribute to a strategy for managing travel demand so as to reduce congestion and the negative environmental impacts of transportation such as greenhouse gas emissions (GHG). Nonetheless, studies of teleworking tend to identify a series of rebound effects, such as the trade-off between commuting and non-work trips, or residential relocation [START_REF] Ory | Which came first, the telecommuting or the residential relocation? An empirical analysis of causality[END_REF][START_REF] Ravalet | Télétravail et mobilité : pour le meilleur et pour le pire? Étude de cas en Suisse[END_REF]. Although previous scholars have explored the impacts of telework on travel patterns, few studies have highlighted how other changes in the workplace, such as multiple work locations or home-based work [START_REF] Mokhtarian | Analyzing the Travel Behavior of Home-Based Workers in the 1991 CALTRANS Statewide Travel Survey[END_REF] might affect travel behavior.

Commuting and business-related trips aside, the changes affecting workplaces also have significant effects on non-work trips. Work location and schedules play a structuring role in the daily activity program and are likely to affect other trip purposes as well, such as shopping, recreational, and health-related trips [START_REF] Aguilera | Exploring the relationship between work and travel behavior on weekdays: analysis of Paris region travel survey over 20 years[END_REF]. On the one hand, a large proportion of individuals' non-work trips are undertaken in conjunction with work-related travel. On the other hand, the development of ICT increases the https://doi.org/10. 1016/j.trd.2020.102338 porosity between work and non-work activities, creating more complex travel patterns [START_REF] Schieman | Is there a downside to schedule control for the work-family interface?[END_REF]. Therefore, this paper aims to evaluate the impacts of the changing nature of workplaces on travel behavior and, consequently, its environmental effects. The underlying research question is: Does the evolution of workplace locations (teleworking, working at home, multiple workplaces) contribute to restructuring travel patterns and reducing CO2 emissions? The "teleworkers" group accounts for individuals who do not identify themselves as "homeworkers" and who declared they worked from home "three or more times a week" or "once or twice a week", while home-based workers are defined as individuals whose main place of work is at home or in the same building as their home.

Thus, we investigate the relationship between workplace, travel behavior, and CO2 emissions, based on travel survey data for the United Kingdom (NTS 2002(NTS -2017 dataset) dataset), which collects data from trip diaries for each surveyed household during a week. The seven-day observation period provides a much more representative picture than traditional one-day travel surveys do.

The main findings are based on Structural Equation Modeling (SEM) implementing path analysis and notably highlight the effects of homeworking on travel behavior and related CO2 emissions. Results show that individuals who work at home travel more kilometers in total for non-work trips on weekdays. Thus, the CO2 emission levels corresponding to the total weekly trips of home-based workers are higher than for those working in a fixed non-home-based place. Moreover, teleworkers are responsible for the highest CO2 emission levels, as they tend to run up a higher number of kilometers traveled for both work and non-work trips. Trade-off effects are observed, as teleworkers tend to offset commuting trips against non-work travel on telework days.

The article is structured as follows. In Section 2, previous studies of the change of workplace and its impacts on travel behavior are reviewed. The case for focusing on the influence of work on non-work trips is also made, and previous contributions in this area are summarized. Section 3 presents the approach and the data and it briefly describes the modeling method used. Results are presented in Section 4 and Section 5 provides a discussion of the main conclusions and some lessons that can be drawn for future studies.

Literature review

Changes in the workplace and travel behavior

In recent years, scholars have drawn attention to the growing diversification of workplaces and the notion that the world of work is spreading through time and space. Working hours have become less standardized and work can be done in a myriad of places, at the office, at home, on the train, at a coffee shop [START_REF] Felstead | Rapid change or slow evolution? Changing places of work and their consequences in the UK[END_REF]. Over the past two decades, the proportion of individuals who work in a fixed workplace has declined continually, and now concerns certain jobs to a much greater extent than others (e.g., manufacturing jobs). Although this category remains the majority among all groups of workers, alternative work arrangements, such as multi-location work and home-based work (in different forms), now make up a growing component of the labor market. [START_REF] Lesnard | Investigating scheduling of work: a two-stage optimal matching analysis of workdays and workweeks: Investigating Scheduling of Work[END_REF] provide important insights into the de-standardization of working patterns. Those authors demonstrate that standard workdays account for just over a half of total workdays, and standard workweeks constitute about a fourth of workweeks in the United Kingdom between 2000 and 2001.

Recent developments in ICT have eroded the spatial fixity of the workplace, allowing more individuals to work from home, either permanently or several times a week. Telework has benefitted from improvements in information and communication technologies, and it was expected to be a potential way of reducing travel demand and, consequently, congestion and greenhouse gas (GHG) emissions. Findings show that teleworking is a strategy often used by individuals to mitigate the cost and time spent commuting (De Abreu e Silva and [START_REF] Abreu E Silva | Home telework, travel behavior, and land-use patterns: A path analysis of British single-worker households[END_REF]. Thus, workers living in low-density areas and with longer commutes tend to telework more frequently. However, teleworkers tend to have longer commutes than their non-teleworker counterparts, which means that they tend to live further from the workplace [START_REF] Ory | Which came first, the telecommuting or the residential relocation? An empirical analysis of causality[END_REF][START_REF] Zhu | Telecommuting, household commute and location choice[END_REF]. Reducing the frequency of commuting seems to allow teleworkers to accept longer commute distances [START_REF] Ettema | The impact of telecommuting on residential relocation and residential preferences: A latent class modelling approach[END_REF][START_REF] Zhu | Telecommuting, household commute and location choice[END_REF].

The positive effects of telework, such as the reduction in the number of trips and kilometers traveled, have been actively supported by policymakers and demonstrated by some scholars [START_REF] Choo | Does telecommuting reduce vehicle-miles traveled? An aggregate time series analysis for the U[END_REF][START_REF] Helminen | Relationships between commuting distance, frequency and telework in Finland[END_REF]. More recently, several studies have indicated the limited impact of teleworking on reducing kilometers traveled and GHG emissions, and they have also identified a variety of rebound effects [START_REF] He | Telecommuting, income, and out-of-home activities[END_REF]. Findings suggest that while telework may reduce travel costs and time, the time saved on commuting may be used to engage in other activities and be converted into additional personal non-work trips or longer trips [START_REF] Mokhtarian | If telecommunication is such a good substitute for travel, why does congestion continue to get worse?[END_REF][START_REF] Zhu | Are telecommuting and personal travel complements or substitutes?[END_REF][START_REF] Zhu | The impact of telecommuting on personal vehicle usage and environmental sustainability[END_REF]. Thus, telecommuters may drive more for both daily work and non-work trips than non-telecommuters [START_REF] Volosin | Exploring the dynamics in travel time frontiers[END_REF]. Workers who are able to work from home report longer travel time, often because they are in higher-paying managerial and professional positions or live farther from work. Moreover, recent studies have demonstrated that households with at least one teleworker tend to travel more and have larger travel budgets than other households do [START_REF] Melo | Home telework and household commuting patterns in Great Britain[END_REF].

Though telework has been a trending topic in the scholarly literature, few studies have focused on travel patterns of home-based workers (non-teleworkers). Growth in home-based work is related to advances in ICT and also to the increased proportion of contingent and self-employed workers. [START_REF] Mokhtarian | Analyzing the Travel Behavior of Home-Based Workers in the 1991 CALTRANS Statewide Travel Survey[END_REF] demonstrate that home-based business workers make a higher average number of daily trips than non-home-based workers, both for work-related trips and for other purposes such as shopping, social life, and recreation. On the other hand, home-based workers have the lowest average daily travel time and the shortest travel distances. While home-based workers typically might not have traditional commute trips, their work-related travel might still be significant.

Thus, scholars have concluded that individuals who work from home tend to display reduced activity spaces [START_REF] Pendyala | Impact of telecommuting on spatial and temporal patterns of household travel[END_REF] because their activities tend to be centered around their residence, contrary to what happens for other categories of workers whose activities tend to be oriented toward their employment area. Moreover, there is evidence that teleworkers and home-based workers tend to live in more suburban areas and that there is a higher likelihood of their local trips being made by car [START_REF] Yen | Interpreting employee telecommuting adoption: An economics perspective[END_REF]. Although there is evidence that homeworking can reduce emissions related to transport use, [START_REF] Baker | Improving the prediction of UK domestic energy-demand using annual consumption-data[END_REF] demonstrate that working from home increases domestic energy consumption during the day. Homeworkers need adequate warmth and lighting and usually some electrical equipment for their work, leading to increased energy consumption and CO2 emissions.

Changes in work arrangements also mean that some workers have multiple places of work [START_REF] Koroma | Looking for people, places and connections: hindrances when working in multiple locations: a review[END_REF]. This often relates to a diversity of occupations: construction workers, artists, executives. The rapid development of mobility over the past decades has allowed workers to access multiple places faster. Most studies analyze commuting trips only, largely ignoring business trips, such as delivering goods and visiting customers. [START_REF] Aguilera | Exploring the relationship between work and travel behavior on weekdays: analysis of Paris region travel survey over 20 years[END_REF] emphasize that, for Greater Paris, workers with multiple workplaces have considerably different travel-behavior than regular commuters with a single workplace. These individuals tend to make more work trips than regular workers. In addition, the authors show that the fact that the average distance traveled for business purposes is short implies that non-regular workplaces are mainly situated close to the main workplace.

The influence of work on non-work trips

Work has been considered to play a key role on how travel patterns are structured: individuals' trips are largely determined by workplace location, work schedules, and trips between home and the workplace. A large proportion of individuals' non-work trips is undertaken in conjunction with work-related travel, which creates a network of multi-purpose trips [START_REF] Hanson | The importance of the multi-purpose journey to work in urban travel behavior[END_REF]. However, most studies still contemplate work trips as single straight commuting trips, ignoring the fact that individuals may make stops to shop or use other service activities in conjunction with work-related trips. Thus, it is noteworthy that work-related travel tends to play a major structural role in individuals' travel behavior, including non-work trips: other activities tend to be planned around work, that remains the primary trip purpose [START_REF] Westelius | The individual's way of choosing between alternative outlets[END_REF]. A substantial number of non-work trips are centered around the workplace, around home, or on the journey between the two [START_REF] Aguilera | Exploring the relationship between work and travel behavior on weekdays: analysis of Paris region travel survey over 20 years[END_REF].

The significant variability in daily travel suggests there is a complex logic behind allocating time to work and non-work activities [START_REF] Astroza | Understanding activity engagement across weekdays and weekend days: A multivariate multiple discrete-continuous modeling approach[END_REF]. Some studies have demonstrated that on working days, non-work trips are limited in number and distance by space-time constraints related to work: on working days, non-work trips are usually shorter and more frequently for mandatory activities, such as grocery shopping and escort trips, while on non-working days, non-work travel involves discretionary activities such as leisure, recreation, visiting family, and planned constrained activities such as medical consultations [START_REF] Soo | Analysis of Travel Time in Multiple-Purpose Trips[END_REF].

It has been previously noted that long working hours result in less time for travel associated with discretionary activities and greater variability in the number of non-work trips during the week for workers [START_REF] Tarigan | Week-to-Week Leisure Trip Frequency and Its Variability[END_REF]. Thus, for individuals working long hours, non-work activities, are often limited by the location of discretionary activities and mainly by the location of the workplace. Hence, some non-work trips are made within a short radius around home or the workplace especially during break-time [START_REF] Aguilera | Exploring the relationship between work and travel behavior on weekdays: analysis of Paris region travel survey over 20 years[END_REF]). It appears that long working hours limit and strongly shape non-work trips, in terms of number, distance, and duration.

Furthermore, socio-economic characteristics of workers, at an individual and household level, might influence their travel patterns for non-work activities. Studies have concluded that the presence of children contributes to an increase in the average number of non-work trips, mostly related to escort trips [START_REF] Aguilera | Exploring the relationship between work and travel behavior on weekdays: analysis of Paris region travel survey over 20 years[END_REF]. Likewise, the influence of gender has been confirmed in explaining the average number of daily trips and the average number of non-work trips with female workers tending to make more non-work trips than male workers [START_REF] Gordon | Gender differences in metropolitan travel behaviour[END_REF][START_REF] Fan | Household structure and gender differences in travel time: spouse/partner presence, parenthood, and breadwinner status[END_REF].

Data and methods

The NTS survey and GHG emissions

The data used were obtained from the United Kingdom National Travel Survey (NTS) for the period between 2002 and 2017, in order to ensure a healthy sample size for the non-traditional working patterns, which are relatively rare occurrences. Data used refer to the travel of English residents within Great Britain. The NTS is a continuous household survey of personal travel of UK residents that has been carried out yearly since 1988. Data is collected via interviews from a seven-day trip diary for each individual from a household. Several scholars have demonstrated that one-day data is inadequate to fully explain travel patterns, in particular nonwork trips, which are less frequent and regular than work-related travel. [START_REF] Pas | An examination of the determinants of day-to-day variability in individuals' urban travel behavior[END_REF] show that individual variability explains up to 50% of the number of non-work activities, challenging the ability of many models to predict demand for travel. [START_REF] Axhausen | Observing the rhythms of daily life: A six-week travel diary[END_REF] demonstrate that the most relevant period for studying the frequency of all motives of movement of individuals is two weeks, but that a whole week of travel makes for an acceptable compromise.

Among the individuals surveyed in the NTS, we selected workers over the age of 18 who traveled on at least two days during the survey week. This last condition is designed to remove individuals with extreme mobility from the sample as they could potentially disrupt the statistical models employed. In the end, the sample size is 113,153 observations. In this study, we use five different groups of workers based on their workplace: 1 -workers who work in a fixed place all week (73%); 2 -workers who work in a fixed place at least two days a week (6%); 3 -workers who have multiple places of work (11.1%); 4 -home-based workers (3.8%); 5 -teleworkers (6%).

The different categories of workplaces are defined in the survey on the basis of a main criterion which is the "usual" place of work. This workplace must be "a place they visit on at least two consecutive days per week for at least four consecutive weeks". It is this criterion which makes it possible to differentiate those who work away from home, at home, and those who work in different places (do not have a usual place of work). Those who have a usual place of work outside the home, and also a secondary place of work (which may be the home) have been included under the category "go to the same place on at least two days". This traditional categorization of workplaces, which is found in many travel surveys in Europe [START_REF] Armoogum | Survey Harmonisation with New Technologies Improvement[END_REF] presents numerous limitations with regards to changes in the job market. In particular ICT enables individuals to exercise many professional activities outside of a "usual" place of work. Teleworking is one of the notable forms of these changes and its effects on travel-behavior have been already well documented over the last decade [START_REF] Ravalet | Télétravail et mobilité : pour le meilleur et pour le pire? Étude de cas en Suisse[END_REF][START_REF] Abreu E Silva | Home telework, travel behavior, and land-use patterns: A path analysis of British single-worker households[END_REF]. Thus, we opted to include the category of "teleworkers" in our analysis. The "teleworkers" group was created from the variable "How often do you work from home?" We selected individuals who do not identify themselves as "homeworkers" and who declared they worked from home "three or more times a week" or "once or twice a week". This means that the teleworker category in this work is relatively restrictive compared to the definitions that can be found in the literature and includes only individuals with a usual place of work outside of home, i.e., whose teleworking does not exceed three days a week. The teleworkers group is derived mainly from the categories "Same place" and "Different places" (Table 1).

In order to calculate GHG emissions, we applied the DECC & DEFRA CO2 conversion factors for each trip to the data set using a method similar to that developed by [START_REF] Mattioli | Gross polluters for food shopping travel: An activity-based typology[END_REF]. For private vehicles, we assigned different factors according to vehicle type (car or motorbike), vehicle size (small, medium, large, 4x4) and type of fuel (petrol or diesel). 1 For trips with more than one passenger we allocated emissions equally among passengers. [START_REF] Heinen | Multimodality and CO2 emissions: A relationship moderated by distance[END_REF] argue that attributing all vehicle emissions to the driver might lead to an underestimation of the emissions of individuals who often travel as passengers, while equally assigning emissions among passengers entails a "leakage" of emissions, as some emissions are attributed to minors. As the sample in this study includes workers only, the choice was to allocate emissions equally among passengers.

We also assigned CO2 coefficients to trips made by public transport according to the type of mode of transport used (bus, light rail, London Bus). No carbon emissions factor was assigned to cycling and walking trips. Finally, the total CO2 trip emissions were obtained by multiplying the trip distance 2 by the CO2 emission factor for each mode of transport. It is important to note that the NTS Survey provides CO2 emission factors derived from the Driver & Vehicle Licensing Agency (DVLA) dataset, as adopted by [START_REF] Heinen | Multimodality and CO2 emissions: A relationship moderated by distance[END_REF]. Nonetheless, the emission factors are not available for the majority of vehicles in the older waves of the survey, leading us to adopt the DECC & DEFRA CO2 conversion factors.

Conceptual framework and SEM model

The main research questions outlined in this paper are related to the effects of the variety of workplaces on travel behavior and CO2 emissions. In order to answer these questions, the modeling framework was developed to understand the relationship between socio-economic characteristics and workplaces of the active population, travel patterns, and CO2 emissions. The model seeks to understand the different effects and travel patterns by taking into account the number of weekly trips and the distance traveled. The model is built in two stages. The first stage aims at a better understanding of the link between workplaces and travel patterns, while the second stage measures the effects of travel patterns on CO2 emissions. It is important to note that the analysis applied the NTS weights.

Structural Equation Modeling (SEM) is used here and has an advantage over classical path analysis via multiple regressions. We can precisely analyze the covariances between the variables, particularly between determined and predictor variables. The estimation is performed in one step via a maximization of the likelihood of the model, allowing for non-spherical error terms. For the first stage of the model, four regressions are used to explain the number of trips and the distances traveled for work (work and in course of work) and non-work trips (other purposes except returning home). The first level is illustrated in the upper part of Fig. 1. The regressions are represented by the four arrows that link the four trip-related variables.

At this stage, travel to return home was ignored. Indeed, the weight of the return home is highly dependent on other trip purposes. The low variability of return-to-home trips relative to all other motives justifies not including these trips in the model. Indeed, they are a function of other trips. This can be verified with the share of home returns in the total number of trips (i.e., an indicator of trip chaining), which varies very little with the type of workplace (Table 2). Thus, their formal inclusion in the model would expose it to problems of multicollinearity, while doing very little to further our understanding of the mechanisms at work.

The explanatory variables are the same for the four regressions and take into account the socio-economic characteristics of workers and their households, including the types of workplace. All of the variables observed come from the UK NTS and are presented in Table 4.

1 Some vehicles did not have an assigned size. An average CO2 coefficient was allocated *vehicles* for which vehicle size was unspecified and, consequently, to all the trip stages that individuals made *using those vehicles. 2 Trips longer than 80km were 'top-coded' as 80km, which corresponds to the threshold for long-distance journeys in the NTS. The presence of extreme distances could potentially lead to misinterpreted results.

Results:

The determinants of trips made by the working population

Descriptive analysis and co-variance

Table 2 presents the variables used in the structural equation modeling and their mean values. Variables account for socioeconomic characteristics of the individuals and their households (occupational category, income, gender, age, car access, presence of children, number of adults) and travel patterns (number of trips, kilometers traveled).

The descriptive analysis shows that workers with a fixed workplace report a higher share of walking/biking, as well as higher than average use of public transport. These travel patterns are directly associated with land-use indicators, as workers with a fixed workplace tend to live in high-density areas. Home-based workers report a high share of car use, but they account for a higher share of walking/biking than the average. Moreover, the average distance traveled shows that their trips tend to be local or more clustered at a local scale. Thus, these two groups exhibit lower average CO2 emissions, with home-based workers displaying a slightly higher level of emissions due to their higher share of car use. In contrast, individuals with a fixed workplace for two days, individuals with multiple workplaces, and teleworkers are shown to be particularly car-dependent. Teleworkers report the highest average CO2 emission levels, directly associated with the average number of work and non-work trips, as well as weekly kilometers traveled. For teleworkers, the high share of car use is directly related to their residential location, which is more often in low density areas. In addition, they report more than average kilometers traveled for both work and non-work trips, entailing high CO2 emission levels. To a lesser extent, workers with multiple workplaces also account for long distances traveled, especially for work-related trips.

Workers with a fixed place of work account for a majority of clerical, full-time employees, earning less than £25,000. Likewise, homeworkers also account for a higher share of clerical occupations, but they differ from individuals with a single workplace as they are most often self-employed and with a higher average salary. The category of teleworkers has a higher proportion of professional/ managerial occupations, as well as an average higher income, much like workers with a fixed workplace for two consecutive days. Nonetheless, teleworkers are more often self-employed than the latter. In contrast, workers with multiple places of work include a higher proportion of skilled manual workers than average, and self-employed workers. They tend to have slightly higher salaries than people with a fixed workplace.

At the first stage, it is also possible to observe the covariance between the different variables analyzed (Table 3). The covariances are represented by two-way arrows in Fig. 1. The results thus show a trade-off between the number of work trips and the number of non-work trips. This means that workers who make more work trips tend to travel less often for non-work purposes. This result mainly reflects the fact that telecommuters, part-time workers, or even those whose main place of work is home make more out-of- *The missing coefficients correspond to non-significant values, p < 0.001 work trips than those working full-time away from home. In addition, it can be observed that the more commuting trips that are made, the longer they tend to be in terms of kilometers traveled. This may be because part-time workers make fewer work-related trips and have shorter commute distances. However, this particular result counters the findings described in the literature review section regarding telework. It will therefore be necessary to verify the detailed results within the model. Moreover, covariance analysis shows an inverse effect for non-work trips, which tend to be shorter when the number of trips made is higher. This could indicate that individuals who travel more frequently tend to cluster their trips around home, as do individuals who live in highdensity areas, who tend to travel more for non-work purposes. Finally, we observe that the kilometers traveled for work and nonwork trips are positively correlated.

The second stage of the model is based on a single regression, where the dependent variable is CO2 emission levels, and the independent variables are the number and average distances of work and non-work trips. In addition, all the socio-economic variables included in the previous regressions also act as independent variables. By observing the direct effects of these variables, we can measure all the effects on CO2 emissions that are not taken into account among the number of trips and kilometers traveled.

Finally, our approach allows for a better understanding of the multiple factors explaining CO2 emission levels, related to the workplace. The model can be considered as comprehensive, as it introduces indirect effects (through the number of trips and kilometers traveled). A model based on direct effects alone might produce misleading conclusions (De Abreu e Silva and [START_REF] Abreu E Silva | Home telework, travel behavior, and land-use patterns: A path analysis of British single-worker households[END_REF] since omitting any indirect effects may nullify or lead to adverse effects for some determinants. Or conversely, the direct effects may be amplified by the indirect effects. The results of this work illustrate these different possibilities as well as highlighting the relevance of path analysis within the framework of a more comprehensive approach.

The model obtained is considered a good model with respect to the three indicators generally used for the purpose [START_REF] Golob | Structural equation modeling for travel behavior research[END_REF]. The root mean square error of approximation (RMSEA) and the standardized root mean square residual (SRMR) are both less than 0.05, which is judged highly satisfactory. Moreover, the comparative fit index (CFI), which compares the proposed model with an unrestricted base model exhibits a value of 0.99 (a good model should have a value of more than 0.95). The model is therefore well fitted to the data and has good explanatory power.

SEM results

In this section, SEM results are presented in two steps, according to the two levels of the model, as presented in the conceptual framework. The first part emphasizes the determinant factors of work and non-work trips (Table 4). The second part focuses on the direct, indirect, and total effects that explain the CO2 emission levels according to the type of workplace (Table 5).

Determinant factors of the number of work trips

The first regression examines the determinants of the number of weekly work trips (1st column of Table 4). First, it should be noted that workers with different workplaces travel more for work, especially workers who work in the same place for two days. On the other hand, individuals who work from home, fully or partially (home-based workers and teleworkers), travel less for work purposes weekly. Findings also show that workers with professional and managerial functions make more work-related trips than clerical workers. In contrast, skilled manual workers travel less for work purposes. In addition, the number of weekly work trips depends on the working status of individuals: part-time workers make fewer work-related trips than other categories. Finally, the regression points out that the profile of workers with the highest probability of having a large number of weekly trips for work are men, on low incomes, aged under 60, who are the main drivers of a car, and part of a household with at least one child.

Determinant factors of the number of non-work trips

The effects of the socio-economic characteristics of workers on the number of non-work trips (2nd column of Table 4) are often the contrary of the effects observed for the number of work-related trips. For part-time workers, a trade-off is observed between the number of work trips and the number of non-work trips. Thus, part-time workers tend to report a particularly high number of nonwork trips. This is also the case for home-based workers. The latter make very few work-related trips (if any), but they tend to compensate by traveling more for non-work purposes. Also, teleworkers report a higher number of non-work trips than the average. In contrast, for workers with the same workplace for two days, the higher number of non-work trips does not entail fewer work trips.

Moreover, the model shows that all occupational categories travel less for non-work purpose than the reference category (clerical workers) and these effects are even more accentuated for skilled manual or other skilled workers. Thus, there are no trade-off effects between work and non-work trips. Overall, women, main drivers of a vehicle, individuals living in high-density areas, and people without children tend to make more non-work trips than the average.

Determinant factors of average kilometers traveled for work

Regarding average kilometers traveled for work (3rd column of Table 4), there is a contrast between workers with a fixed workplace, as well as home-based workers, and workers with multiple workplaces including teleworkers, workers with a fixed workplace for two days, and workers with different workplaces. For the former the average distance traveled is shorter, while for the latter the distances are greater, especially for teleworkers. In addition, results show that professional/managerial groups are those that report the longest commuting distances, while skilled manual workers have shorter commutes. Also, compared to full-time workers, all other categories (part-time workers and self-employed) travel shorter distances for work purposes. Finally, the probability of traveling longer distances for work is higher for men, high-income workers, main drivers of a vehicle, individuals who are part of households with children, and people living in rural areas.

Determinant factors of average kilometers traveled for non-work purposes

The effects of the main determinants on average kilometers traveled for non-work purposes (4th column of Table 4) are similar to those observed for average work trip distances. There is also a contrast between workers with a fixed workplace or home-based workers, who report fewer kilometers traveled for non-work purposes than do workers with multiple workplaces. Professional/ managerial workers report the most kilometers traveled for non-work purposes, in contrast to skilled manual workers. Also, regarding working status, part-time workers report shorter average distances for non-work trips than do other categories. Overall, men on highincomes who are the main drivers of a vehicle and have children and individuals living in low-density/rural areas tend to travel the longest distances for non-work purposes. This final results table shows the results of the last regression, explaining the CO2 emissions of workers. The determinants are the same as for the previous regressions to which were added the four variables previously determined, namely the number and average distances of travel for work and for non-work trips. As the SEM is able to model several endogenous variables and handles both direct and indirect relationships between them. Table 5 includes direct effects and indirect effects which are the sum of the effects mediated by the variable above. The last column shows the total effect, which is the sum of the direct and indirect effects.

Unsurprisingly the factor that affects CO2 emissions most is being the main driver of a car, the effect being greater than for the other determinants. Regarding car access, the direct effect seems to be the most significant as it relates directly to the mode of transport actually used (which is one of the travel characteristics not specifically taken into account in the model). Moreover, car access effects show that the increased number of non-work trips weighs heavily on CO2 emission levels of main drivers, followed by the average distance and the number of work trips.

Secondly, the model emphasizes that the type of workplace has significant effects on CO2 emissions. It is interesting to note that the greatest effect is observed for workers with multiple workplaces, as well as workers with a fixed workplace for at least two days. Regarding the type of workplace, the average kilometers traveled for work purposes, which is directly related to the mode of transport used, has the most significant effect on the carbon footprint of workers. The effects related to the number of work trips and the average kilometers traveled for non-work trips are less significant. In addition, teleworkers are associated with high CO2 emission levels, due mainly to average kilometers traveled for work and, to a lesser extent, the number of non-work trips and the average kilometers traveled for those trips. In addition, for teleworkers the effect of the number of work trips on CO2 emissions is negative, as expected. Nonetheless, this negative effect does not prevent teleworkers from having one of the highest CO2 emission levels. On the other hand, home-based workers report one of the lowest carbon footprints, although it is still higher than for workers with a single workplace. In fact, home-based workers report few work-related trips, but we observe a trade-off effect with the number of non-work trips.

Moreover, the results of the model show that professional/managerial workers report higher CO2 emissions levels than skilled manual workers. Their carbon footprint is directly related to average kilometers traveled for work, modal share, and the number of weekly work-related trips. Conversely, for skilled manual workers the effects are significantly lower than for the reference category. Part-time workers report lower emission levels due to fewer commutes and shorter average commuting distances for work. However, part-time workers report a trade-off between work and non-work trips, much as for home-based workers.

The model also demonstrates that high-income workers have higher CO2 emissions associated with higher average kilometers traveled for both work and non-work purposes. In contrast, the lower carbon footprint of low-income workers is explained by average kilometers traveled and, to a lesser extent, the modes of transport used and the number of non-work trips.

Second, the model shows that men's carbon footprints are higher than women's due to kilometers traveled for work, mode of transport, and number of work-related trips. Finally, workers who are part of households with children are responsible for lower CO2 emission levels for travel for non-work purposes, although they display longer average distances for work and non-work trips combined.

Discussion

This study developed a Structural Equation Modeling approach using individual and household level data from the National Travel Survey (NTS) to shed light on the diversification of workplaces and its impacts on travel behavior and CO2 emissions in the United Kingdom. We advocate that, with the continuous decrease over the past two decades in the proportion of individuals with a fixed workplace, it is important to capture both the recent shifts in travel patterns and their environmental impact. We also hypothesize that the development of ICT and the consequent increase in the proportion of home-based workers and teleworkers does not necessarily generate fewer CO2 emissions due to potential rebound effects. The main findings of this study are noteworthy for policymakers, as current urban travel demand forecasting practice and environmental policies rarely consider work location (in-home or out-of-home) or employment type as explanatory variables. It is also important to note that there are some complexities and limitations in the analysis. While we find that non-traditional working arrangements are associated with higher CO2 emissions, this is based on a cross-sectional analysis. In order to definitely prove the existence of a "rebound effect", one would probably need longitudinal datae.g., to demonstrate the effect on distance traveled by car/ CO2 of transitioning from a traditional arrangement to teleworking.

The model developed in this study aims at analyzing the complex relationship between workplace and CO2 emissions beyond direct effects. The methodological approach introduces a comprehensive study of travel behavior, trade-off effects, and work and non-work trips by breaking down and distinguishing travel characteristics that influence CO2 emissions. Thus, the model reveals the many interactions between workers, workplace, and travel patterns. For instance, although CO2 emissions for teleworkers and workers with multiple workplaces are mainly related to kilometers traveled for work, the findings show non-work trips also have a significant environmental impact for the former. Similarly, home-based workers account for slightly higher levels of CO2 emissions than workers with a single workplace because their non-work-related travel offsets the absence of regular commuting trips. Therefore, the comprehensive approach and findings introduced by the SEM calls for an in-depth discussion of GHG reduction policies.

The model demonstrates the key effects of socio-economic characteristics and travel behavior over CO2 emissions. First, we observe that CO2 emission levels are considerably higher than other determinants for individuals with multiple work places. These levels are mainly associated with total work kilometers traveled per week, but also direct effects, such as modal share and car access.

Results show that, overall, the diversification of the workplace is reflected by longer average distances for work trips, which are often associated with more distant residential locations. For teleworkers, CO2 emissions are more closely related to distances traveled to work: they tend to live further away and have longer commuting distances. To a lesser extent, the trade-off observed between work and non-work trips also affects the total CO2 emissions for individuals who telework. For home-based workers, the effects are more directly related to trade-off effects between work and non-work travel. Secondly, car access also appears to be a significant factor, not only because the automobile is a highly polluting mode of transportation, but also because its use reflects longer distances traveled for work. Part-time workers are associated with lower CO2 emissions because they travel shorter distances for work trips, indicating that they live closer to their workplace. Moreover, the model confirms some expected findings: men, on high-incomes and who are skilled are responsible for higher CO2 emission levels as they tend to travel longer distances, particularly for work trips. Men in particular display more significant direct effects, possibly because they use cars more than women do.

Moreover, an original contribution from this paper is that it highlights the effects of non-work trips on total weekly travel. Current urban travel policies are constructed mostly on the basis of studies focusing on commuting trips and neglect the effects of non-work activities on travel behavior. Thus, the use of a seven-day trip diary overcomes some of the limitations with one-day trip diaries, which tend to overlook the majority of non-work trips or trade-off effects. Contrary to the generally accepted idea that commuting trips have a greater environmental impact, the findings of this study show that non-work trips account for a significant share of CO2 emissions.

Policy implications

The recent development of telework arrangements is directly related to residential location: as highlighted in the literature review section, the NTS database confirms that teleworkers tend to live further from central business districts, resulting in significant car dependency and longer commute distances. Some recent studies, such as Shabanpour et al.'s, (2018) have made a case for teleworking policies, showing through an activity-based model that they have the potential to reduce network congestion and vehicular emissions specifically during rush hours. Our results show that although planners and policymakers have hypothesized that the development of telework could potentially reduce travel demand, those forecasts remain largely unmet. These opposite results could be due to several factors: the first study considers all frequencies of telework, while our approach takes into account individuals who telework at least once a week. Moreover, the 7-day trip diary allows to better understanding the full-array of non-work trips. Finally, the differences between the American and British contexts should be considered: we could hypothesize that American teleworkers are more likely to commute by car, thus telecommuting would have a more significant impact on GHG emissions.

The model shows that, in the United Kingdom, individuals who telework seem to have a greater impact on CO2 emissions than individuals with a fixed work location. The environmental impact is directly related to total weekly kilometers traveled for both work and non-work trips. As previously noted, rebound effects cancel out the positive effects of teleworking or further increase GHG emissions because more kilometers are traveled. With the increasing development of technological progress, we may expect that more workers will opt for telecommuting and that travel-behavior related to this practice will increase: "It appears that without holistic consideration of the options and potential outcomes for GHG reduction policies, anticipated decreases in some arenas will permit increases in other types of travel behavior in the form of substitution effects." (Zhu and Mason, 2014, p.11). [START_REF] Moos | The sustainability of telework: an ecological-footprinting approach[END_REF] suggest that policies aimed at decreasing automobile commuting so as to limit harmful emissions might not be effective if they are thought about individually. Thus, it is important to understand that many environmental effects of the development of ICT arise through a compound set of individual adjustments.

The model also reveals that, although home-based workers report fewer work-related trips weekly, they still exert a greater effect on CO2 emissions than workers with a fixed workplace. As with the results observed for the category of teleworkers, part of the commuting trips avoided by home-based workers are converted into personal or non-work trips. Therefore, although the average journey of home-based workers tends to be shorter and clustered around home, a significant environmental impact is still perceived, suggesting a higher proportion of car use. Likewise, the results emphasize that individuals with multiple work locations tend to report significant effects on CO2 emissions. Unlike the previous two categories of workers discussed, their environmental impact is mostly associated with work-related trips. It is interesting to note that the number of their work trips per week is only slightly higher than for workers with a fixed workplace, but they account for the longest total work kilometers traveled weekly. Most policies fail to consider that a proportion of workers have multiple workplaces, and that this often entails travel patterns that are less regular, more variable, and may require significant flexibility in terms of travel behavior, as well as car dependency.

Thus, GHG emission reduction policies should be implemented at different levels and scales. It is important to note that, in a context of strong overall growth in rail use in the UK, public transport services need to respond adequately to changing demands arising in the context of the growing diversification of workplaces. First, we note that the travel patterns observed for home-based workers, mainly clustered around a local scale, call for policies that encourage the use of environmentally-friendly modes of transport, such as electrically assisted cycles. As this category of workers is already more likely to cycle than their counterparts, such policies could reduce the automobile dependency of workers living in low-density areas. As far as teleworkers are concerned, a comprehensive approach to both work and non-work trips should be considered while devising GHG emission reduction policies. First, it is important to tackle long commuting distances by encouraging the increase in the modal share of public transport (e.g., long-distance rail). Thus, the implementation of telework in companies could be accompanied by incentives to increase public transit use and multimodality. Multimodality should be studied carefully, though, since [START_REF] Heinen | Multimodality and CO2 emissions: A relationship moderated by distance[END_REF] demonstrate its limited impacts in terms of CO2 emissions and environmental policies in the transport sector. The authors also demonstrate that multimodal trips tend to have higher CO2 emissions than unimodal trips. It is emphasized that high emissions levels are more related to high levels of travel activity, which should be the main focus of transport policies. In addition, policies designed to reduce emissions from non-work trips could be introduced in a similar fashion as the recommendations made for home-based workers. Nonetheless, GHG emissions of individuals with multiple places of work are more challenging to address through transport policies: their weekly travel patterns are more flexible and varied than those of other categories of workers. Often, they do not have a regular commuting pattern, which entails significant car dependency.

In conclusion, this paper presents a novel exploration of the relationship between workplace and travel behavior by comparing different categories of workers by workplace. Thus, more comprehensive transportation and environmental policies should be considered in order to tackle travel planning in the light of the increasingly diverse and complex travel patterns of the labor force. The discussion emphasizes the importance of capturing the full array of trade-off effects, as well as demonstrating the potential range of travel patterns that vary with individual characteristics and preferences. It seems that specific climate-mitigation transport policies fall victim to unexpected changes in travel behavior because new possibilities arising from the development of ICT allow individuals to drive longer distances and emit more GHG. In view of the results found here and the growing diversification of the workplace, advances in policy making with respect to sustainability should be extending the scope of travel-behavior impacts.

Fig. 1 .

 1 Fig. 1. Conceptual framework.

Table 1

 1 Frequency of working from home according to workplace.

	Work place location	How often do you work at home? (%)		All
		Do not apply	Less than once a week or never	Once or more times a week
	Same place	30.7	42.1	2.5	75.3
	Same place on at least 2 consecutive days	2.1	3.8	1.1	7.0
	Different places	4.4	6.8	2.6	13.8
	Home / same building as home	3.9	0.0	0.0	3.9
	All	41.0	52.8	6.2	100.0

Table 2

 2 Characteristics of trips and individuals according to type of workplace.

		Same place Same place on at least 2 consecutive	Different places Home / same building as	Teleworking
			days			home
	# work trips	4.3	5.4		4.5	1.6	3.7
	# non work trips	6.5	7.1		6.2	9.5	7.6
	# home trips	8.0	8.1		7.6	8.0	7.8
	# total trips	18.8	20.6		18.4	19.1	19.2
	% home trips*	42.4	39.4		41.4	42.0	40.8
	Km. work (mean per trip)	13.8	22.3		22.2	21.5	27.0
	Km. non work (mean per trip)	11.1	12.8		12.2	11.3	12.9
	Km. home (mean per trip)	12.1	17.0		16.2	12.5	17.9
	Km. totals (mean per trip)	12.1	16.9		16.3	12.6	17.7
	Kg. CO2 work	9.4	20.3		17.4	5.9	16.7
	Kg. CO2 non work	12.0	15.7		13.1	18.6	16.9
	Kg. CO2 home	15.6	23.3		21.2	17.3	23.3
	Kg. CO2 total	36.9	59.2		51.7	41.8	56.9
	Kg. CO2 / km.	0.16	0.17		0.17	0.17	0.17
	% Walk, Bicycle	8.8	6.9		6.5	8.6	8.0
	% Car Driver	64.5	73.0		73.7	71.6	72.2
	% Car Passenger	14.7	10.3		11.2	13.9	9.9
	% Public Transport	9.6	7.6		6.6	4.3	7.9
	% Other	2.5	2.3		2.0	1.7	2.0
	% Professional / managerial	18.6	40.4		17.2	24.8	44.6
	% Clerical	45.0	38.0		23.7	39.4	39.0
	% Skilled manual	14.0	12.4		43.4	29.1	11.8
	% Other manual and other	22.4	9.2		15.7	6.7	4.6
	% Employees: full-time	73.6	77.9		50.6	23.6	57.3
	% Employees: part-time	22.2	12.2		12.2	13.0	11.8
	% Self-employed: full-time	3.5	8.2		28.8	39.0	21.9
	% Self-employed: part-time	0.8	1.7		8.4	24.3	8.9
	% Less than £25,000	68.8	41.0		61.0	62.3	37.8
	% £25,000 to £49,999	25.4	40.9		30.8	25.5	38.2
	% £50,000 and over	5.9	18.1		8.2	12.2	24.0
	% Men	49.0	60.8		72.1	48.2	59.7
	% 60 and more	6.8	5.4		9.8	19.8	10.5
	% Main driver	67.7	82.0		77.9	78.0	82.7
	% Only one adult	11.0	13.9		10.7	11.5	13.7
	% One or more child	35.7	37.7		38.3	38.1	39.7
	% Less than 5 persons/hectare	24.7	28.2		28.0	38.6	32.7
	% 5 to 75 persons/hectare	69.2	66.4		66.1	54.8	59.4
	% More than 75 persons/hectare	6.1	5.4		6.0	6.6	7.9
	% London Boroughs	13.1	11.5		14.4	13.3	16.2
	% Metropolitan built-up areas	15.1	14.1		13.1	10.0	9.9
	% Large urban (over 250,000)	16.2	17.8		15.1	13.5	15.6
	% Medium urban (25,000-250,000)	27.8	25.7		25.3	21.6	22.4
	% Small/medium urban (10,000-25,000) 8.1	7.9		8.3	6.6	7.7
	% Small urban (3000-10,000)	6.7	7.7		7.4	7.0	7.2
	% Rural	13.0	15.3		16.4	27.9	21.0
	# individuals (weighted)	87 988	7 218		13 377	4 618	7 255
	# individuals (unweighted)	81 897	6 837		12 663	4 669	7 087
	% individuals (weighted)	73.0	6.0		11.1	3.8	6.0
	Table 3
	Covariances between determined variables.
		# work trips	# non work trips	Km. Work (mean)
	# non work trips	-1.57
	Km. Work (mean)	4.15		-10.06
	Km. non work (mean)	-2.73	-5.05	18.89

Table 4

 4 Direct effects on weekly travel indicators*

	# work trips # non work	Km. work	Km. non work
	trips	(mean)	(mean)

* The missing coefficients correspond to non-significant values.

Table 5

 5 Indirect, Direct, and Total effects on CO2 emissions.

	Variable	Category	Indirect effects			Direct effect Total effects
			# work # non work Km. work Km. non
						work
	Type of workplace (ref: Same place)	Same place on at least 2	1.91	0.83	5.61	0.74	4.02	13.11
		consecutive days
		Different places	0.35		6.88	0.88	4.38	12.50
		Home / same building as home	-6.29	5.16			3.19	2.06
		Teleworking	-2.02	1.63	9.85	1.04		10.50
	Socio Economic Group (ref: Clerical)	Professional / managerial	1.09	-0.49	2.32	0.61	1.52	5.05
		Skilled manual	-0.86	-1.38	-3.08	-1.11	0.97	-5.45
		Other manual and other		-1.38	-3.51	-1.30		-6.19
	Working status (ref: Employees: full-	Employees: part-time	-3.00	4.34	-3.83	-0.78		-3.28
	time)	Self-employed: full-time	0.39	-0.40	-5.06			-5.07
		Self-employed: part-time	-3.32	5.20	-7.10			-5.22
	Individual income (ref: £25,000 to	Less than £25,000		-0.33	-4.56	-1.25	-1.59	-7.73
	£49,999)	£50,000 and over		-0.74	5.12	0.96	-2.23	3.11
	Gender	Men	0.92	-2.74	4.85	0.59	3.81	7.43
	Age	60 and over	-0.78		-2.26			-3.05
	Density	Person/ha.		-0.03		-0.01	-0.05	-0.09
	Area Type	Urban -> Rural			1.08	0.78	1.44	3.30
	Car access	Main driver	1.21	6.17	3.27	0.85	9.59	21.10
	Type of household	Only one adult		0.79	-0.81		-2.06	-2.08
	Type of household	One or more child	0.80	-3.70	0.44	1.77		-0.69

Acknowledgements

This research was funded by the Mobile Lives Forum, as part of its research program on mobility transition. The Mobile Lives Forum is a research and prospective institute created by SNCF (France).

The NTS special license dataset of the UK Department of Transport was kindly provided by the UK Data Service.