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We investigate the influence of steady periodic flows on the propagation of chemical fronts in an infinite channel domain. We focus on the sharp front arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large P\'eclet and Damk\"ohler numbers, $\mathrm{Pe}$ and $\mathrm{Da}$) and on its heuristic approximation by the G equation. We introduce a variational formulation that expresses the two front speeds in terms of periodic trajectories minimizing the time of travel across the period of the flow, under a constraint that differs between the FKPP and G equations. This formulation shows that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows used in experiments. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot. We verify our computations against closed-form expressions derived for $\mathrm{Da}\ll \mathrm{Pe}$ and for $\mathrm{Da}\gg \mathrm{Pe}$.

The tropical tropopause layer (TTL) is a key region controlling transport between the troposphere and the stratosphere. The efficiency of transport across the TTL depends on the continuous interaction between the large-scale advection and the small-scale intermittent convection that reaches the Level of Zero radiative Heating (LZH). The wide range of scales involved presents a significant challenge to determine the sources of convection and quantify transport across the TTL. Here, we use a simple Lagrangian model, termed TTL detrainment model, that combines a large ensemble of 200-day back trajectory calculations with high-resolution fields of brightness temperatures (provided by the CLAUS dataset) in order to determine the ensemble of trajectories that are detrained from convective sources. The trajectories are calculated using the ECMWF ERA-Interim winds and radiative heating rates, and in order to establish the radiative influence of clouds, the latter rates are derived both under all-sky and clear-sky conditions. We show that most trajectories are detrained near the mean LZH with the horizontal distributions of convective sources being highly-localized, even within the space defined by deep convection. As well as modifying the degree of source localization, the radiative heating from clouds facilitates the rapid upwelling of air across the TTL. However, large-scale motion near the fluctuating LZH can lead a significant proportion of trajectories to alternating clear-sky and cloudy regions, thus generating a large dispersion in the vertical transport times. The distributions of vertical transport times are wide and skewed and are largely insensitive to a bias of about ±1 km (∓5 K) in the altitude of cloud top heights (the main sensitivity appearing in the times to escape the immediate neighbourhood of the LZH) while some seasonal and regional transport characteristics are apparent for times up to 60 days. The strong horizontal mixing that characterizes the TTL ensures that most air of convective origin is well-mixed within the tropical and eventually within the extra-tropical lower-stratosphere.

The dispersion of a diffusive scalar in a fluid flowing through a network has many applications including to biological flows, porous media, water supply, and urban pollution. Motivated by this, we develop a large-deviation theory that predicts the evolution of the concentration of a scalar released in a rectangular network in the limit of large time t≫1. This theory provides an approximation for the concentration that remains valid for large distances from the center of mass, specifically for distances up to O(t) and thus much beyond the O(t^{1/2}) range where a standard Gaussian approximation holds. A byproduct of the approach is a closed-form expression for the effective diffusivity tensor that governs this Gaussian approximation. Monte Carlo simulations of Brownian particles confirm the large-deviation results and demonstrate their effectiveness in describing the scalar distribution when t is only moderately large.

We investigate the propagation of chemical fronts arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the presence of a steady cellular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large Peclet number, Pe≫1 ) and arbitrary reaction rate (arbitrary Damkohler number Da ). We identify three regimes corresponding to the distinguished limits Da=O(Pe −1 ) , Da=O((logPe) −1 ) and Da=O(Pe) and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of Pe and Da . Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on Da and Pe and valid in intermediate regimes, are deduced as limiting cases. Taken together, our asymptotic results provide a complete description of the complex dependence of the front speed on Da for Pe≫1 . They are confirmed by numerical solutions of the eigenvalue problem determining the front speed, and illustrated by a number of numerical simulations of the advection--diffusion--reaction equation.

Ozone air pollution is identified as one of the main threats bearing upon human health and ecosystems, with 25 000 deaths in 2005 attributed to surface ozone in Europe (IIASA 2013 TSAP Report #10). In addition, there is a concern that climate change could negate ozone pollution mitigation strategies, making them insufficient over the long run and jeopardising chances to meet the long term objective set by the European Union Directive of 2008 (Directive 2008/50/EC of the European Parliament and of the Council of 21 May 2008) (60 ppbv, daily maximum). This effect has been termed the ozone climate penalty. One way of assessing this climate penalty is by driving chemistry-transport models with future climate projections while holding the ozone precursor emissions constant (although the climate penalty may also be influenced by changes in emission of precursors). Here we present an analysis of the robustness of the climate penalty in Europe across time periods and scenarios by analysing the databases underlying 11 articles published on the topic since 2007, i.e. a total of 25 model projections. This substantial body of literature has never been explored to assess the uncertainty and robustness of the climate ozone penalty because of the use of different scenarios, time periods and ozone metrics. Despite the variability of model design and setup in this database of 25 model projection, the present meta-analysis demonstrates the significance and robustness of the impact of climate change on European surface ozone with a latitudinal gradient from a penalty bearing upon large parts of continental Europe and a benefit over the North Atlantic region of the domain. Future climate scenarios present a penalty for summertime (JJA) surface ozone by the end of the century (2071-2100) of at most 5 ppbv. Over European land surfaces, the 95% confidence interval of JJA ozone change is [0.44; 0.64] and [0.99; 1.50] ppbv for the 2041-2070 and 2071-2100 time windows, respectively. © 2015 IOP Publishing Ltd.

We investigate the influence of fluid flows on the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large Péclet (Pe) and Damköhler (Da) numbers. The front speed is expressed in terms of a periodic path--an instanton--that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for (logPe)(-1) ≪ Da ≪ Pe and for Da ≫ Pe. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection-diffusion-reaction equation.

A new bulk model of the convective boundary layer, the probabilistic bulk convection model (PBCM), is presented. Unlike prior bulk approaches that have modeled the mixed-layer-top buoyancy flux as a constant fraction of the surface buoyancy flux, PBCM implements a new mixed-layer-top entrainment closure based on the mass flux of updrafts overshooting the inversion. This mass flux is related to the variability of the surface state (potential temperature θ and specific humidity q) of an ensemble of updraft plumes. The authors evaluate the model against observed clear-sky weak and strong inversion cases and show that PBCM performs well. The height, state, and timing of the boundary layer growth are accurately reproduced. Sensitivity studies are performed highlighting the role of the main parameters (surface variances, lateral entrainment). The model is weakly sensitive to the exact specification of the variability at the surface and is most sensitive to the lateral entrainment of environmental air into the rising plumes. Apart from allowing time-dependent top-of-the-boundary-layer entrainment rates expressed in terms of surface properties, which can be observed in situ, PBCM naturally takes into account the transition to the shallow convection regime, as described in a companion paper. Thus, PBCM represents an important step toward a unified framework bridging parameterizations of mixed-layer entrainment velocity in both clear-sky and moist convective boundary layers.

We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source–sink distribution. We focus on the spatial variation of the scalar field, described by the dissipation wavenumber, ${k}_{d} $, that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Péclet number ($\mathit{Pe}\gg 1$) yield four distinct regimes for the scaling behaviour of ${k}_{d} $, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of $\mathit{Pe}$ and the ratio $\rho = {\ell }_{u} / {\ell }_{s} $, where ${\ell }_{u} $ and ${\ell }_{s} $ are, respectively, the characteristic length scales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a two-dimensional, chaotically mixing example flow and discuss their relation to previous bounds. Finally, we note some implications for three-dimensional turbulent flows.

This paper studies the spatial structure of decaying chemical fields generated by a chaotic-advection flow and maintained by a spatially smooth chemical source. Previous work showed that in a regime where diffusion can be neglected (large P\'eclet number), the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow. The scaling exponent, ${\ensuremath{\gamma}}_{q}$, of the $q\text{th}$-order structure function depends, at leading order, linearly on the ratio of the rate of decay of the chemical processes, $\ensuremath{\alpha}$, and the average rate of divergence of neighboring fluid parcel trajectories (Lyapunov exponent), $\overline{h}$. Under a homogeneous stretching approximation, ${\ensuremath{\gamma}}_{q}/q=\text{max}{\ensuremath{\alpha}/\overline{h},1}$ which implies that a well-defined filamental-smooth transition occurs at $\ensuremath{\alpha}=\overline{h}$. This approximation has been improved by using the distribution of finite-time Lyapunov exponents to characterize the inhomogeneous stretching of the flow. However, previous work focused more on the behavior of the exponents as $q$ varies and less on the effects of $\ensuremath{\alpha}$ and hence the implications for the filamental-smooth transition. Here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite-size (rather than finite-time) Lyapunov exponents. We clarify the relation between the two distributions. We show that the corrected exponents, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{q}$, depend nonlinearly on $\ensuremath{\alpha}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{q}l{\ensuremath{\gamma}}_{q}$ for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{q}lq$. The magnitude of the correction to the homogeneous stretching approximation, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{q}\ensuremath{-}{\ensuremath{\gamma}}_{q}$, grows as $\ensuremath{\alpha}$ increases, reaching a maximum when the leading-order transition is reached $(\ensuremath{\alpha}=\overline{h})$. The implication of these results is that there is no well-defined bulk filamental-smooth transition. Instead it is the case that the chemical field is unambiguously smooth for $\ensuremath{\alpha}g{h}_{\text{max}}$, where ${h}_{\text{max}}$ denotes the maximum finite-time Lyapunov exponent and unambiguously filamental for $\ensuremath{\alpha}l\overline{h}$, with an intermediate character for $\ensuremath{\alpha}$ between these two values. Theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the Cr\'amer function.

The stationary-state spatial structure of reacting scalar fields, chaotically advected by a two-dimensional large-scale flow, is examined for the case for which the reaction equations contain delay terms. Previous theoretical investigations have shown that, in the absence of delay terms and in a regime where diffusion can be neglected (large P\'eclet number), the emergent spatial structures are filamental and characterized by a single scaling regime with a H\"older exponent that depends on the rate of convergence of the reactive processes and the strength of the stirring measured by the average stretching rate. In the presence of delay terms, we show that for sufficiently small scales all interacting fields should share the same spatial structure, as found in the absence of delay terms. Depending on the strength of the stirring and the magnitude of the delay time, two further scaling regimes that are unique to the delay system may appear at intermediate length scales. An expression for the transition length scale dividing small-scale and intermediate-scale regimes is obtained and the scaling behavior of the scalar field is explained. The theoretical results are illustrated by numerical calculations for two types of reaction models, both based on delay differential equations, coupled to a two-dimensional chaotic advection flow. The first corresponds to a single reactive scalar and the second to a nonlinear biological model that includes nutrients, phytoplankton and zooplankton. As in the no-delay case, the presence of asymmetrical couplings among the biological species results in a non-generic scaling behavior.

Ozone air pollution is identified as one of the main threats bearing upon human health and ecosystems, with 25 000 deaths in 2005 attributed to surface ozone in Europe (IIASA 2013 TSAP Report #10). In addition, there is a concern that climate change could negate ozone pollution mitigation strategies, making them insufficient over the long run and jeopardising chances to meet the long term objective set by the European Union Directive of 2008 (Directive 2008/50/EC of the European Parliament and of the Council of 21 May 2008) (60 ppbv, daily maximum). This effect has been termed the ozone climate penalty. One way of assessing this climate penalty is by driving chemistry-transport models with future climate projections while holding the ozone precursor emissions constant (although the climate penalty may also be influenced by changes in emission of precursors). Here we present an analysis of the robustness of the climate penalty in Europe across time periods and scenarios by analysing the databases underlying 11 articles published on the topic since 2007, i.e. a total of 25 model projections. This substantial body of literature has never been explored to assess the uncertainty and robustness of the climate ozone penalty because of the use of different scenarios, time periods and ozone metrics. Despite the variability of model design and setup in this database of 25 model projection, the present meta-analysis demonstrates the significance and robustness of the impact of climate change on European surface ozone with a latitudinal gradient from a penalty bearing upon large parts of continental Europe and a benefit over the North Atlantic region of the domain. Future climate scenarios present a penalty for summertime (JJA) surface ozone by the end of the century (2071-2100) of at most 5 ppbv. Over European land surfaces, the 95% confidence interval of JJA ozone change is [0.44; 0.64] and [0.99; 1.50] ppbv for the 2041-2070 and 2071-2100 time windows, respectively.

The observed filamental nature of plankton populations suggests that stirring plays an important role in determining their spatial structure. If diffusive mixing is neglected, the various interacting biological species within a fluid parcel are determined by the parcel time history. The induced spatial structure has been shown to be a result of competition between the time evolution of the biological processes involved and the stirring induced by the flow as measured, for example, by the rate of divergence of the distance of neighbouring fluid parcels. In the work presented here we examine a simple biological model based on delay-differential equations, previously seen in Abraham (1998) including nutrients, phytoplankton and zooplankton, coupled to a strain flow. Previous theoretical investigations made on a differential equation model (Hernández-Garcia et al., 2002) imply that the latter two should share the same small-scale structure. The generalization from differential equations to delay-differential equations, associated with the addition of a maturation time to the zooplankton growth, should not make a difference, provided sufficiently small spatial scales are considered. However, this theoretical prediction is in contradiction with the results of Abraham (1998) where the phytoplankton and zooplankton structures remain uncorrelated at all length scales. A new set of numerical experiments is performed here which show that these two regimes coexist. On larger scales , there is a decoupling of the spatial structure of the zooplankton distribution on the one hand, and the phytoplankton and nutrient on the other. On the other hand, at small enough length scales, the phytoplankton and zooplankton share the same spatial structure as expected by the theory involving no maturation time.