A new empirical framework to quantify the hydraulic effects of soil and atmospheric drivers on plant water status

Metrics to quantify regulation of plant water status at the daily as opposed to the seasonal scale do not presently exist. This gap is significant since plants are hypothesised to regulate their water potential not only with respect to slowly changing soil drought but also with respect to faster changes in air vapour pressure deficit (VPD), a variable whose importance for plant physiology is expected to grow because of higher temperatures in the coming decades. We present a metric, the stringency of water potential regulation, that can be employed at the daily scale and quantifies the effects exerted on plants by the separate and combined effect of soil and atmospheric drought. We test our theory using datasets from two experiments where air temperature and VPD were experimentally manipulated. In contrast to existing metrics based on soil drought that can only be applied at the seasonal scale, our metric successfully detects the impact of atmospheric warming on the regulation of plant water status. We show that the thermodynamic effect of VPD on plant water status can be isolated and compared against that exerted by soil drought and the covariation between VPD and soil drought. Furthermore, in three of three cases, VPD accounted for more than 5 MPa of potential effect on leaf water potential. We explore the significance of our findings in the context of potential future applications of this metric from plant to ecosystem scale.

Impacts of climate timescale on the stability of trait-environment relationships

Predictive relationships between plant traits and environmental factors can be derived at global and regional scales, informing efforts to reorient ecological models around functional traits. However, in a changing climate, the environmental variables used as predictors in such relationships are far from stationary. This could yield errors in trait-environment model predictions if timescale is not accounted for. Here, the timescale dependence of trait-environment relationships is investigated by regressing in situ trait measurements of specific leaf area, leaf nitrogen content, and wood density on local climate characteristics summarized across several increasingly long timescales. We identify contrasting responses of leaf and wood traits to climate timescale. Leaf traits are best predicted by recent climate timescales, while wood density is a longer term memory trait. The use of sub-optimal climate timescales reduces the accuracy of the resulting trait-environment relationships. This study concludes that plant traits respond to climate conditions on the timescale of tissue lifespans rather than long-term climate normals, even at large spatial scales where multiple ecological and physiological mechanisms drive trait change. Thus, determining trait-environment relationships with temporally relevant climate variables may be critical for predicting trait change in a nonstationary climate system.

Diagnosing evapotranspiration responses to water deficit across biomes using deep learning

Accounting for water limitation is key to determining vegetation sensitivity to drought. Quantifying water limitation effects on evapotranspiration (ET) is challenged by the heterogeneity of vegetation types, climate zones and vertically along the rooting zone. Here, we train deep neural networks using flux measurements to study ET responses to progressing drought conditions. We determine a water stress factor (fET) that isolates ET reductions from effects of atmospheric aridity and other covarying drivers. We regress fET against the cumulative water deficit, which reveals the control of whole-column moisture availability. We find a variety of ET responses to water stress. Responses range from rapid declines of fET to 10% of its water-unlimited rate at several savannah and grassland sites, to mild fET reductions in most forests, despite substantial water deficits. Most sensitive responses are found at the most arid and warm sites. A combination of regulation of stomatal and hydraulic conductance and access to belowground water reservoirs, whether in groundwater or deep soil moisture, could explain the different behaviors observed across sites. This variety of responses is not captured by a standard land surface model, likely reflecting simplifications in its representation of belowground water storage.

Global net biome CO2 exchange predicted comparably well using parameter-environment relationships and plant functional types

Accurate estimation and forecasts of net biome CO₂ exchange (NBE) are vital for understanding the role of terrestrial ecosystems in a changing climate. Prior efforts to improve NBE predictions have predominantly focused on increasing models' structural realism (and thus complexity), but parametric error and uncertainty are also key determinants of model skill. Here, we investigate how different parameterization assumptions propagate into NBE prediction errors across the globe, pitting the traditional plant functional type (PFT)-based approach against a novel top-down, machine learning-based "environmental filtering" (EF) approach. To do so, we simulate these contrasting methods for parameter assignment within a flexible model-data fusion framework of the terrestrial carbon cycle (CARDAMOM) at a global scale. In the PFT-based approach, model parameters from a small number of select locations are applied uniformly within regions sharing similar land cover characteristics. In the EF-based approach, a pixel's parameters are predicted based on underlying relationships with climate, soil, and canopy properties. To isolate the role of parametric from structural uncertainty in our analysis, we benchmark the resulting PFT-based and EF-based NBE predictions with estimates from CARDAMOM's Bayesian optimization approach (whereby "true" parameters consistent with a suite of data constraints are retrieved on a pixel-by-pixel basis). When considering the mean absolute error of NBE predictions across time, we find that the EF-based approach matches or outperforms the PFT-based approach at 55% of pixels-a narrow majority. However, NBE estimates from the EF-based approach are susceptible to compensation between errors in component flux predictions and predicted parameters can align poorly with the assumed "true" values. Overall, though, the EF-based approach is comparable to conventional approaches and merits further investigation to better understand and resolve these limitations. This work provides insight into the relationship between terrestrial biosphere model performance and parametric uncertainty, informing efforts to improve model parameterization via PFT-free and trait-based approaches.

Global patterns of water storage in the rooting zones of vegetation

The rooting-zone water-storage capacity-the amount of water accessible to plants-controls the sensitivity of land-atmosphere exchange of water and carbon during dry periods. How the rooting-zone water-storage capacity varies spatially is largely unknown and not directly observable. Here we estimate rooting-zone water-storage capacity globally from the relationship between remotely sensed vegetation activity, measured by combining evapotranspiration, sun-induced fluorescence and radiation estimates, and the cumulative water deficit calculated from daily time series of precipitation and evapotranspiration. Our findings indicate plant-available water stores that exceed the storage capacity of 2-m-deep soils across 37% of Earth's vegetated surface. We find that biome-level variations of rooting-zone water-storage capacities correlate with observed rooting-zone depth distributions and reflect the influence of hydroclimate, as measured by the magnitude of annual cumulative water-deficit extremes. Smaller-scale variations are linked to topography and land use. Our findings document large spatial variations in the effective root-zone water-storage capacity and illustrate a tight link among the climatology of water deficits, rooting depth of vegetation and its sensitivity to water stress.

Promoting Connectivity of Network-Like Structures by Enforcing Region Separation

We propose a novel, connectivity-oriented loss function for training deep convolutional networks to reconstruct network-like structures, like roads and irrigation canals, from aerial images. The main idea behind our loss is to express the connectivity of roads, or canals, in terms of disconnections that they create between background regions of the image. In simple terms, a gap in the predicted road causes two background regions, that lie on the opposite sides of a ground truth road, to touch in prediction. Our loss function is designed to prevent such unwanted connections between background regions, and therefore close the gaps in predicted roads. It also prevents predicting false positive roads and canals by penalizing unwarranted disconnections of background regions. In order to capture even short, dead-ending road segments, we evaluate the loss in small image crops. We show, in experiments on two standard road benchmarks and a new data set of irrigation canals, that convnets trained with our loss function recover road connectivity so well that it suffices to skeletonize their output to produce state of the art maps. A distinct advantage of our approach is that the loss can be plugged in to any existing training setup without further modifications.

Historically inconsistent productivity and respiration fluxes in the global terrestrial carbon cycle

The terrestrial carbon cycle is a major source of uncertainty in climate projections. Its dominant fluxes, gross primary productivity (GPP), and respiration (in particular soil respiration, R_S), are typically estimated from independent satellite-driven models and upscaled in situ measurements, respectively. We combine carbon-cycle flux estimates and partitioning coefficients to show that historical estimates of global GPP and R_S are irreconcilable. When we estimate GPP based on R_S measurements and some assumptions about R_S:GPP ratios, we found the resulted global GPP values (bootstrap mean [Formula: see text] Pg C yr^-1) are significantly higher than most GPP estimates reported in the literature ([Formula: see text] Pg C yr^-1). Similarly, historical GPP estimates imply a soil respiration flux (Rs_GPP, bootstrap mean of [Formula: see text] Pg C yr^-1) statistically inconsistent with most published R_S values ([Formula: see text] Pg C yr^-1), although recent, higher, GPP estimates are narrowing this gap. Furthermore, global R_S:GPP ratios are inconsistent with spatial averages of this ratio calculated from individual sites as well as CMIP6 model results. This discrepancy has implications for our understanding of carbon turnover times and the terrestrial sensitivity to climate change. Future efforts should reconcile the discrepancies associated with calculations for GPP and Rs to improve estimates of the global carbon budget.

Confronting the water potential information gap

Water potential directly controls the function of leaves, roots, and microbes, and gradients in water potential drive water flows throughout the soil-plant-atmosphere continuum. Notwithstanding its clear relevance for many ecosystem processes, soil water potential is rarely measured in-situ, and plant water potential observations are generally discrete, sparse, and not yet aggregated into accessible databases. These gaps limit our conceptual understanding of biophysical responses to moisture stress and inject large uncertainty into hydrologic and land surface models. Here, we outline the conceptual and predictive gains that could be made with more continuous and discoverable observations of water potential in soils and plants. We discuss improvements to sensor technologies that facilitate in situ characterization of water potential, as well as strategies for building new networks that aggregate water potential data across sites. We end by highlighting novel opportunities for linking more representative site-level observations of water potential to remotely-sensed proxies. Together, these considerations offer a roadmap for clearer links between ecohydrological processes and the water potential gradients that have the 'potential' to substantially reduce conceptual and modeling uncertainties.

Plant-water sensitivity regulates wildfire vulnerability

Extreme wildfires extensively impact human health and the environment. Increasing vapour pressure deficit (VPD) has led to a chronic increase in wildfire area in the western United States, yet some regions have been more affected than others. Here we show that for the same increase in VPD, burned area increases more in regions where vegetation moisture shows greater sensitivity to water limitation (plant-water sensitivity; R² = 0.71). This has led to rapid increases in human exposure to wildfire risk, both because the population living in areas with high plant-water sensitivity grew 50% faster during 1990–2010 than in other wildland–urban interfaces and because VPD has risen most rapidly in these vulnerable areas. As plant-water sensitivity is strongly linked to wildfire vulnerability, accounting for ecophysiological controls should improve wildfire forecasts. If recent trends in VPD and demographic shifts continue, human wildfire risk will probably continue to increase.

The authors show that an ecosystem’s sensitivity to drought, measured as the amount of change in vegetation moisture content for a given change in background moisture, predicts the fire hazard in that location.

Detecting forest response to droughts with global observations of vegetation water content

Droughts in a warming climate have become more common and more extreme, making understanding forest responses to water stress increasingly pressing. Analysis of water stress in trees has long focused on water potential in xylem and leaves, which influences stomatal closure and water flow through the soil-plant-atmosphere continuum. At the same time, changes of vegetation water content (VWC) are linked to a range of tree responses, including fluxes of water and carbon, mortality, flammability, and more. Unlike water potential, which requires demanding in situ measurements, VWC can be retrieved from remote sensing measurements, particularly at microwave frequencies using radar and radiometry. Here, we highlight key frontiers through which VWC has the potential to significantly increase our understanding of forest responses to water stress. To validate remote sensing observations of VWC at landscape scale and to better relate them to data assimilation model parameters, we introduce an ecosystem-scale analog of the pressure-volume curve, the non-linear relationship between average leaf or branch water potential and water content commonly used in plant hydraulics. The sources of variability in these ecosystem-scale pressure-volume curves and their relationship to forest response to water stress are discussed. We further show to what extent diel, seasonal, and decadal dynamics of VWC reflect variations in different processes relating the tree response to water stress. VWC can also be used for inferring belowground conditions-which are difficult to impossible to observe directly. Lastly, we discuss how a dedicated geostationary spaceborne observational system for VWC, when combined with existing datasets, can capture diel and seasonal water dynamics to advance the science and applications of global forest vulnerability to future droughts.

Leaf surface water, not plant water stress, drives diurnal variation in tropical forest canopy water content

Variation in canopy water content (CWC) that can be detected from microwave remote sensing of vegetation optical depth (VOD) has been proposed as an important measure of vegetation water stress. However, the contribution of leaf surface water (LW_s ), arising from dew formation and rainfall interception, to CWC is largely unknown, particularly in tropical forests and other high-humidity ecosystems. We compared VOD data from the Advanced Microwave Scanning Radiometer for the Earth Observing System (AMSR-E) and CWC predicted by a plant hydrodynamics model at four tropical sites in Brazil spanning a rainfall gradient. We assessed how LW_s influenced the relationship between VOD and CWC. The analysis indicates that while CWC is strongly correlated with VOD (R²  = 0.62 across all sites), LW_s accounts for 61-76% of the diurnal variation in CWC despite being < 10% of CWC. Ignoring LW_s weakens the near-linear relationship between CWC and VOD and reduces the consistency in diurnal variation. The contribution of LW_s to CWC variation, however, decreases at longer, seasonal to inter-annual, time scales. Our results demonstrate that diurnal patterns of dew formation and rainfall interception can be an important driver of diurnal variation in CWC and VOD over tropical ecosystems and therefore should be accounted for when inferring plant diurnal water stress from VOD measurements.

Interannual variability of ecosystem iso/anisohydry is regulated by environmental dryness

●Plants are characterized by the iso/anisohydry continuum depending on how they regulate leaf water potential (Ψ_L ). However, how iso/anisohydry changes over time in response to year-to-year variations in environmental dryness and how such responses vary across different regions remains poorly characterized. ●We investigated how dryness, represented by aridity index, affects the interannual variability of ecosystem iso/anisohydry at the regional scale, estimated using satellite microwave vegetation optical depth (VOD) observations. This ecosystem-level analysis was further complemented with published field observations of species-level Ψ_L . ●We found different behaviors in the directionality and sensitivity of isohydricity (σ) with respect to the interannual variation of dryness in different ecosystems. These behaviors can largely be differentiated by the average dryness of the ecosystem itself: in mesic ecosystems, σ decreases in drier years with a higher sensitivity to dryness; in xeric ecosystems, σ increases in drier years with a lower sensitivity to dryness. These results were supported by the species-level synthesis. ●Our study suggests that how plants adjust their water use across years - as revealed by their interannual variability in isohydricity - depends on the dryness of plants' living environment. This finding advances our understanding of plant responses to drought at regional scales.

Empirical estimates of regional carbon budgets imply reduced global soil heterotrophic respiration

Resolving regional carbon budgets is critical for informing land-based mitigation policy. For nine regions covering nearly the whole globe, we collected inventory estimates of carbon-stock changes complemented by satellite estimates of biomass changes where inventory data are missing. The net land–atmospheric carbon exchange (NEE) was calculated by taking the sum of the carbon-stock change and lateral carbon fluxes from crop and wood trade, and riverine-carbon export to the ocean. Summing up NEE from all regions, we obtained a global ‘bottom-up’ NEE for net land anthropogenic CO₂ uptake of –2.2 ± 0.6 PgC yr⁻¹ consistent with the independent top-down NEE from the global atmospheric carbon budget during 2000–2009. This estimate is so far the most comprehensive global bottom-up carbon budget accounting, which set up an important milestone for global carbon-cycle studies. By decomposing NEE into component fluxes, we found that global soil heterotrophic respiration amounts to a source of CO₂ of 39 PgC yr⁻¹ with an interquartile of 33–46 PgC yr⁻¹—a much smaller portion of net primary productivity than previously reported.

Macro to micro: microwave remote sensing of plant water content for physiology and ecology

Although primarily valued for their suitability for oceanographic applications and soil moisture estimation, microwave remote sensing observations are also sensitive to plant water content (M_w ). Since M_w depends on both plant water status and biomass, these observations have the potential to be useful for a range of plant drought response studies. In this paper, we introduce the principles behind microwave remote sensing observations to illustrate how they are sensitive to plant water content and discuss the relationship between landscape-scale M_w and common stand-scale metrics, including plant-scale relative water content, live fuel moisture content and leaf water potential. Lastly, we discuss how various sensor types can be leveraged for specific applications depending on the spatio-temporal resolution needed.

Beyond soil water potential: An expanded view on isohydricity including land-atmosphere interactions and phenology

Over the past decade, the concept of isohydry or anisohydry, which describes the link between soil water potential (Ψ_S ), leaf water potential (Ψ_L ), and stomatal conductance (g_s ), has soared in popularity. However, its utility has recently been questioned, and a surprising lack of coordination between the dynamics of Ψ_L and g_s across biomes has been reported. Here, we offer a more expanded view of the isohydricity concept that considers effects of vapour pressure deficit (VPD) and leaf area index (A_L ) on the apparent sensitivities of Ψ_L and g_s to drought. After validating the model with tree- and ecosystem-scale data, we find that within a site, isohydricity is a strong predictor of limitations to stomatal function, though variation in VPD and leaf area, among other factors, can challenge its diagnosis. Across sites, the theory predicts that the degree of isohydricity is a good predictor of the sensitivity of g_s to declining soil water in the absence of confounding effects from other drivers. However, if VPD effects are significant, they alone are sufficient to decouple the dynamics of Ψ_L and g_s entirely. We conclude with a set of practical recommendations for future applications of the isohydricity framework within and across sites.

Moisture pulse-reserve in the soil-plant continuum observed across biomes

The degree to which individual pulses of available water drive plant activity across diverse biomes and climates is not well understood. It has previously only been investigated in a few dryland locations. Here, plant water uptake following pulses of surface soil moisture, an indicator for the pulse-reserve hypothesis, is investigated across South America, Africa and Australia with satellite-based estimates of surface soil and canopy water content. Our findings show that this behaviour is widespread: occurring over half of the vegetated landscapes. We estimate spatially varying soil moisture thresholds at which plant water uptake ceases, noting dependence on soil texture and proximity to the wilting point. The soil type and biome-dependent soil moisture threshold and the plant soil water uptake patterns at the scale of Earth system models allow a unique opportunity to test and improve model parameterization of vegetation function under water limitation.

Hydraulic diversity of forests regulates ecosystem resilience during drought

Plants influence the atmosphere through fluxes of carbon, water and energy¹, and can intensify drought through land-atmosphere feedback effects^2-4. The diversity of plant functional traits in forests, especially physiological traits related to water (hydraulic) transport, may have a critical role in land-atmosphere feedback, particularly during drought. Here we combine 352 site-years of eddy covariance measurements from 40 forest sites, remote-sensing observations of plant water content and plant functional-trait data to test whether the diversity in plant traits affects the response of the ecosystem to drought. We find evidence that higher hydraulic diversity buffers variation in ecosystem flux during dry periods across temperate and boreal forests. Hydraulic traits were the predominant significant predictors of cross-site patterns in drought response. By contrast, standard leaf and wood traits, such as specific leaf area and wood density, had little explanatory power. Our results demonstrate that diversity in the hydraulic traits of trees mediates ecosystem resilience to drought and is likely to have an important role in future ecosystem-atmosphere feedback effects in a changing climate.

Water, Energy, and Carbon with Artificial Neural Networks (WECANN): A statistically-based estimate of global surface turbulent fluxes and gross primary productivity using solar-induced fluorescence

A new global estimate of surface turbulent fluxes, latent heat flux (LE) and sensible heat flux (H), and gross primary production (GPP) is developed using a machine learning approach informed by novel remotely sensed Solar-Induced Fluorescence (SIF) and other radiative and meteorological variables. This is the first study to jointly retrieve LE, H and GPP using SIF observations. The approach uses an artificial neural network (ANN) with a target dataset generated from three independent data sources, weighted based on triple collocation (TC) algorithm. The new retrieval, named Water, Energy, and Carbon with Artificial Neural Networks (WECANN), provides estimates of LE, H and GPP from 2007 to 2015 at 1° × 1° spatial resolution and on monthly time resolution. The quality of ANN training is assessed using the target data, and the WECANN retrievals are evaluated using eddy covariance tower estimates from FLUXNET network across various climates and conditions. When compared to eddy covariance estimates, WECANN typically outperforms other products, particularly for sensible and latent heat fluxes. Analysing WECANN retrievals across three extreme drought and heatwave events demonstrates the capability of the retrievals in capturing the extent of these events. Uncertainty estimates of the retrievals are analysed and the inter-annual variability in average global and regional fluxes show the impact of distinct climatic events - such as the 2015 El Niño - on surface turbulent fluxes and GPP.

Regionally strong feedbacks between the atmosphere and terrestrial biosphere

The terrestrial biosphere and atmosphere interact through a series of feedback loops. Variability in terrestrial vegetation growth and phenology can modulate fluxes of water and energy to the atmosphere, and thus affect the climatic conditions that in turn regulate vegetation dynamics. Here we analyze satellite observations of solar-induced fluorescence, precipitation, and radiation using a multivariate statistical technique. We find that biosphere-atmosphere feedbacks are globally widespread and regionally strong: they explain up to 30% of precipitation and surface radiation variance. Substantial biosphere-precipitation feedbacks are often found in regions that are transitional between energy and water limitation, such as semi-arid or monsoonal regions. Substantial biosphere-radiation feedbacks are often present in several moderately wet regions and in the Mediterranean, where precipitation and radiation increase vegetation growth. Enhancement of latent and sensible heat transfer from vegetation accompanies this growth, which increases boundary layer height and convection, affecting cloudiness, and consequently incident surface radiation. Enhanced evapotranspiration can increase moist convection, leading to increased precipitation. Earth system models underestimate these precipitation and radiation feedbacks mainly because they underestimate the biosphere response to radiation and water availability. We conclude that biosphere-atmosphere feedbacks cluster in specific climatic regions that help determine the net CO₂ balance of the biosphere.

Global variations in ecosystem-scale isohydricity

Droughts are expected to become more frequent and more intense under climate change. Plant mortality rates and biomass declines in response to drought depend on stomatal and xylem flow regulation. Plants operate on a continuum of xylem and stomatal regulation strategies from very isohydric (strict regulation) to very anisohydric. Coexisting species may display a variety of isohydricity behaviors. As such, it can be difficult to predict how to model the degree of isohydricity at the ecosystem scale by aggregating studies of individual species. This is nonetheless essential for accurate prediction of ecosystem drought resilience. In this study, we define a metric for the degree of isohydricity at the ecosystem scale in analogy with a recent metric introduced at the species level. Using data from the AMSR-E satellite, this metric is evaluated globally based on diurnal variations in microwave vegetation optical depth (VOD), which is directly related to leaf water potential. Areas with low annual mean radiation are found to be more anisohydric. Except for evergreen broadleaf forests in the tropics, which are very isohydric, and croplands, which are very anisohydric, land cover type is a poor predictor of ecosystem isohydricity, in accordance with previous species-scale observations. It is therefore also a poor basis for parameterizing water stress response in land-surface models. For taller ecosystems, canopy height is correlated with higher isohydricity (so that rainforests are mostly isohydric). Highly anisohydric areas show either high or low underlying water use efficiency. In seasonally dry locations, most ecosystems display a more isohydric response (increased stomatal regulation) during the dry season. In several seasonally dry tropical forests, this trend is reversed, as dry-season leaf-out appears to coincide with a shift toward more anisohydric strategies. The metric developed in this study allows for detailed investigations of spatial and temporal variations in plant water behavior.

Water, Energy, and Carbon with Artificial Neural Networks (WECANN): A statistically-based estimate of global surface turbulent fluxes and gross primary productivity using solar-induced fluorescence

A new global estimate of surface turbulent fluxes, latent heat flux (LE) and sensible heat flux (H), and gross primary production (GPP) is developed using a machine learning approach informed by novel remotely sensed Solar-Induced Fluorescence (SIF) and other radiative and meteorological variables. This is the first study to jointly retrieve LE, H and GPP using SIF observations. The approach uses an artificial neural network (ANN) with a target dataset generated from three independent data sources, weighted based on triple collocation (TC) algorithm. The new retrieval, named Water, Energy, and Carbon with Artificial Neural Networks (WECANN), provides estimates of LE, H and GPP from 2007 to 2015 at 1° × 1° spatial resolution and on monthly time resolution. The quality of ANN training is assessed using the target data, and the WECANN retrievals are evaluated using eddy covariance tower estimates from FLUXNET network across various climates and conditions. When compared to eddy covariance estimates, WECANN typically outperforms other products, particularly for sensible and latent heat fluxes. Analysing WECANN retrievals across three extreme drought and heatwave events demonstrates the capability of the retrievals in capturing the extent of these events. Uncertainty estimates of the retrievals are analysed and the inter-annual variability in average global and regional fluxes show the impact of distinct climatic events - such as the 2015 El Niño - on surface turbulent fluxes and GPP.

Mean velocity profile in a sheared and thermally stratified atmospheric boundary layer

A stability correction function φ(m)(ζ) that accounts for distortions to the logarithmic mean velocity profile (MVP) in the lower atmosphere caused by thermal stratification was proposed by Monin and Obukhov in the 1950s using dimensional analysis. Its universal character was established from many field experiments. However, theories that describe the canonical shape of φ(m)(ζ) are still lacking. A previous link between the spectrum of turbulence and the MVP is expanded here to include the effects of thermal stratification on the turbulent kinetic energy dissipation rate and eddy-size anisotropy. The resulting theory provides a novel explanation for the power-law exponents and coefficients already reported for φ(m)(ζ) from numerous field experiments.