Loughborough University
Leicestershire, UK
LE11 3TU
+44 (0)1509 263171
Loughborough University

Loughborough University Research Publications

Publications for Christopher Keylock

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Journal Articles

Keylock, C, Singh, A, Passalacqua, P, Foufoula‐Georgiou, E (2021) Evaluating landscape complexity and the contribution of non‐locality to geomorphometry, Journal of Geophysical Research: Earth Surface, 126(4), e2020JF005765, ISSN: 2169-9003. DOI: 10.1029/2020jf005765.

Keylock, C, Singh, A, Passalacqua, P, Foufoula-Georgiou, E (2020) Hölder-Conditioned Hypsometry: A Refinement to a Classical Approach for the Characterization of Topography, Water Resources Research, 56(5), ARTN e2019WR025412, ISSN: 0043-1397. DOI: 10.1029/2019WR025412.

Keylock, CJ (Accepted for publication) Turbulence at the Lee bound: maximally non-normal vortex filaments and the decay of a local dissipation rate, Journal of Fluid Mechanics, 881, pp.283-312, ISSN: 0022-1120. DOI: 10.1017/jfm.2019.779.

Beaumard, P, Buxton, ORH, Keylock, C (2019) The importance of non-normal contributions to velocity gradient tensor dynamics for spatially developing, inhomogeneous, turbulent flows, Journal of Turbulence, 20(9), pp.577-598, DOI: 10.1080/14685248.2019.1685095.

Keylock, C, Ghisalberti, M, Katul, GG, Nepf, HM (2019) A joint velocity-intermittency analysis reveals similarity in the vertical structure of atmospheric and hydrospheric canopy turbulence, Environmental Fluid Mechanics, 20(1), pp.77-101, ISSN: 1567-7419. DOI: 10.1007/s10652-019-09694-w.

Kesserwani, G, Shaw, J, Sharifian, MK, Bau, D, Keylock, C, Bates, PD, Ryan, JK (2019) (Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models, Advances in Water Resources, 129, pp.31-55, ISSN: 0309-1708. DOI: 10.1016/j.advwatres.2019.04.019.

Keylock, CJ (2018) The Schur decomposition of the velocity gradient tensor for turbulent flows, Journal of Fluid Mechanics, 848, pp.876-905, ISSN: 0022-1120. DOI: 10.1017/jfm.2018.344.

Truong, HK, Keylock, CJ, Eckert, N, Bellot, H, Naaïm, M (2018) Refining the processing of paired time series data to improve velocity estimation in snow flows, Cold Regions Science and Technology, 151, pp.75-88, ISSN: 0165-232X. DOI: 10.1016/j.coldregions.2018.03.004.

Higham, JE, Brevis, W, Keylock, CJ (2018) Implications of the selection of a particular modal decomposition technique for the analysis of shallow flows, Journal of Hydraulic Research, 56(6), pp.796-805, ISSN: 0022-1686. DOI: 10.1080/00221686.2017.1419990.

Keylock, C (2018) Hypothesis testing for nonlinear phenomena in the geosciences using synthetic, surrogate data, Earth and Space Science, 6(1), pp.41-58, ISSN: 2333-5084. DOI: 10.1029/2018EA000435.

Keylock, C (2017) Synthetic velocity gradient tensors and the identification of statistically significant aspects of the structure of turbulence, Physical Review Fluids, 2(8), 084607, DOI: 10.1103/PhysRevFluids.2.084607.

Keylock, CJ (2017) Multifractal surrogate-data generation algorithm that preserves pointwise Hölder regularity structure, with initial applications to turbulence, Physical Review E, 95(3), 032123, ISSN: 2470-0045. DOI: 10.1103/physreve.95.032123.

Higham, JE, Brevis, W, Keylock, C, Safarzadeh, A (2017) Using modal decompositions to explain the sudden expansion of the mixing layer in the wake of a groyne in a shallow flow, Advances in Water Resources, 107, pp.451-459, ISSN: 0309-1708. DOI: 10.1016/j.advwatres.2017.05.010.

Keylock, C (2017) Gradual multifractal reconstruction of time-series: Formulation of the method and an application to the coupling between stock market indices and their Hoelder exponents, Physica D: Nonlinear Phenomena, 368, pp.1-9, ISSN: 0167-2789. DOI: 10.1016/j.physd.2017.11.011.

Keylock, C, Chang, KS, Constantinescu, GS (2016) Large eddy simulation of the velocity-intermittency structure for flow over a field of symmetric dunes, Journal of Fluid Mechanics, 805, pp.656-685, ISSN: 0022-1120. DOI: 10.1017/jfm.2016.519.

Keylock, CJ, Ganapathasubramani, B, Monty, J, Hutchins, N, Marusic, I (2016) The coupling between inner and outer scales in a zero pressure boundary layer evaluated using a Hölder exponent framework, Fluid Dynamics Research, 48(2), pp.021405-021405, ISSN: 0169-5983. DOI: 10.1088/0169-5983/48/2/021405.

Keylock, CJ and Nishimura, K (2016) Wavelet phase analysis of two velocity components to infer the structure of interscale transfers in a turbulent boundary-layer, Fluid Dynamics Research, 48(2), pp.021406-021406, ISSN: 0169-5983. DOI: 10.1088/0169-5983/48/2/021406.

Keylock, C, Kida, S, Peters, N (2016) JSPS Supported Symposium on Interscale Transfers and Flow Topology in Equilibrium and Non-equilibrium Turbulence (Sheffield, UK, September 2014), Fluid Dynamics Research, 48(2), pp.020001-020001, ISSN: 0169-5983. DOI: 10.1088/0169-5983/48/2/020001.

Pitsch, H and Keylock, C (2016) Obituary for Norbert Peters, Fluid Dynamics Research, 48(2), pp.021001-021001, ISSN: 0169-5983. DOI: 10.1088/0169-5983/48/2/021001.

Higham, JE, Brevis, W, Keylock, C (2016) A rapid non-iterative proper orthogonal decomposition based outlier detection and correction for PIV data, Measurement Science and Technology, 27(12), ISSN: 0957-0233. DOI: 10.1088/0957-0233/27/12/125303.

Keylock, CJ (2015) Flow resistance in natural, turbulent channel flows: The need for a fluvial fluid mechanics, Water Resources Research, 51(6), pp.4374-4390, ISSN: 0043-1397. DOI: 10.1002/2015wr016989.

Keylock, CJ, Stresing, R, Peinke, J (2015) Gradual wavelet reconstruction of the velocity increments for turbulent wakes, Physics of Fluids, 27(2), pp.025104-025104, ISSN: 1070-6631. DOI: 10.1063/1.4907740.

Keylock, CJ, Singh, A, Venditti, JG, Foufoula‐Georgiou, E (2014) Robust classification for the joint velocity‐intermittency structure of turbulent flow over fixed and mobile bedforms, Earth Surface Processes and Landforms, 39(13), pp.1717-1728, ISSN: 0197-9337. DOI: 10.1002/esp.3550.

Ash, M, Brennan, PV, Keylock, CJ, Vriend, NM, McElwaine, JN, Sovilla, B (2014) Two-dimensional radar imaging of flowing avalanches, Cold Regions Science and Technology, 102, pp.41-51, ISSN: 0165-232X. DOI: 10.1016/j.coldregions.2014.02.004.

Keylock, CJ (2014) Discussion of “Testing Stationarity with Wavelet-Based Surrogates” by Megan McCullough and Ahsan Kareem, Journal of Engineering Mechanics, 140(4), pp.07014001-07014001, ISSN: 0733-9399. DOI: 10.1061/(asce)em.1943-7889.0000698.

Keylock, CJ (2014) Discussion of "Testing Stationarity with Wavelet-Based Surrogates" by Megan McCullough and Ahsan Kareem, JOURNAL OF ENGINEERING MECHANICS, 140(4), ARTN 7014001, ISSN: 0733-9399. DOI: 10.1061/(ASCE)EM.1943-7889.0000698.

Keylock, CJ, Singh, A, Foufoula-Georgiou, E (2014) The complexity of gravel bed river topography examined with gradual wavelet reconstruction, Journal of Geophysical Research: Earth Surface, 119(3), pp.682-700, ISSN: 2169-9003. DOI: 10.1002/2013jf002999.

Keylock, CJ, Lane, SN, Richards, KS (2014) Quadrant/octant sequencing and the role of coherent structures in bed load sediment entrainment, Journal of Geophysical Research: Earth Surface, 119(2), pp.264-286, ISSN: 2169-9003. DOI: 10.1002/2012jf002698.

Keylock, CJ, Ash, M, Vriend, N, Brennan, PV, McEwaine, JN, Sovilla, B (2014) Looking inside an avalanche using a novel radar system, Geology Today, 30(1), pp.21-25, ISSN: 0266-6979. DOI: 10.1111/gto.12033.

Keylock, CJ, Singh, A, Foufoula-Georgiou, E (2013) The influence of migrating bed forms on the velocity-intermittency structure of turbulent flow over a gravel bed, Geophysical Research Letters, 40(7), pp.1351-1355, ISSN: 0094-8276. DOI: 10.1002/grl.50337.

Vriend, NM, McElwaine, JN, Sovilla, B, Keylock, CJ, Ash, M, Brennan, PV (2013) High-resolution radar measurements of snow avalanches, Geophysical Research Letters, 40(4), pp.727-731, ISSN: 0094-8276. DOI: 10.1002/grl.50134.

Eckert, N, Keylock, CJ, Castebrunet, H, Lavigne, A, Naaim, M (2013) Temporal trends in avalanche activity in the French Alps and subregions: from occurrences and runout altitudes to unsteady return periods, Journal of Glaciology, 59(213), pp.93-114, ISSN: 0022-1430. DOI: 10.3189/2013jog12j091.

Keylock, CJ (2012) A resampling method for generating synthetic hydrological time series with preservation of cross-correlative structure and higher-order properties, Water Resources Research, 48(12), ISSN: 0043-1397. DOI: 10.1029/2012wr011923.

Keylock, CJ, Constantinescu, G, Hardy, RJ (2012) The application of computational fluid dynamics to natural river channels: Eddy resolving versus mean flow approaches, Geomorphology, 179, pp.1-20, ISSN: 0169-555X. DOI: 10.1016/j.geomorph.2012.09.006.

Eckert, N, Keylock, CJ, Bertrand, D, Parent, E, Faug, T, Favier, P, Naaim, M (2012) Quantitative risk and optimal design approaches in the snow avalanche field: Review and extensions, Cold Regions Science and Technology, 79-80, pp.1-19, ISSN: 0165-232X. DOI: 10.1016/j.coldregions.2012.03.003.

Keylock, CJ, Nishimura, K, Nemoto, M, Ito, Y (2012) The flow structure in the wake of a fractal fence and the absence of an “inertial regime”, Environmental Fluid Mechanics, 12(3), pp.227-250, ISSN: 1567-7419. DOI: 10.1007/s10652-011-9233-0.

Keylock, CJ, Nishimura, K, Peinke, J (2012) A classification scheme for turbulence based on the velocity-intermittency structure with an application to near-wall flow and with implications for bed load transport, Journal of Geophysical Research: Earth Surface, 117(F1), pp.n/a-n/a, ISSN: 0148-0227. DOI: 10.1029/2011jf002127.

Keylock, CJ, Tokyay, TE, Constantinescu, G (2011) A method for characterising the sensitivity of turbulent flow fields to the structure of inlet turbulence, Journal of Turbulence, 12, pp.N45-N45, DOI: 10.1080/14685248.2011.636047.

Keylock, CJ (2010) Characterizing the structure of nonlinear systems using gradual wavelet reconstruction, Nonlinear Processes in Geophysics, 17(6), pp.615-632, DOI: 10.5194/npg-17-615-2010.

Keylock, CJ (2010) Introduction to special issue: The future of geomorphology, Progress in Physical Geography: Earth and Environment, 34(3), pp.261-264, ISSN: 0309-1333. DOI: 10.1177/0309133310364932.

Clifford, NJ and Keylock, CJ (2010) Dedication, Progress in Physical Geography: Earth and Environment, 34(3), pp.259-259, ISSN: 0309-1333. DOI: 10.1177/0309133310367547.

Keylock, CJ (2009) Evaluating the dimensionality and significance of “active periods” in turbulent environmental flows defined using Lipshitz/Hölder regularity, Environmental Fluid Mechanics, 9(5), pp.509-523, ISSN: 1567-7419. DOI: 10.1007/s10652-009-9127-6.

Keylock, CJ (2008) A criterion for delimiting active periods within turbulent flows, Geophysical Research Letters, 35(11), L11804, ISSN: 0094-8276. DOI: 10.1029/2008gl033858.

Keylock, CJ (2008) Improved preservation of autocorrelative structure in surrogate data using an initial wavelet step, Nonlinear Processes in Geophysics, 15(3), pp.435-444, DOI: 10.5194/npg-15-435-2008.

Keylock, CJ (2007) Identifying linear and non-linear behaviour in reduced complexity modelling output using surrogate data methods, Geomorphology, 90(3-4), pp.356-366, ISSN: 0169-555X. DOI: 10.1016/j.geomorph.2006.10.027.

Keylock, CJ (2007) Withering geomorphology, Earth Surface Processes and Landforms, 32(5), pp.803-804, ISSN: 0197-9337. DOI: 10.1002/esp.1487.

Keylock, CJ (2007) The visualization of turbulence data using a wavelet-based method, Earth Surface Processes and Landforms, 32(4), pp.637-647, ISSN: 0197-9337. DOI: 10.1002/esp.1423.

Keylock, CJ (2007) A wavelet-based method for surrogate data generation, Physica D: Nonlinear Phenomena, 225(2), pp.219-228, ISSN: 0167-2789. DOI: 10.1016/j.physd.2006.10.012.

Keylock, CJ (2006) Reforming AS/A2 physical geography to enhance geographic scholarship, Geography, 91(3), pp.272-279, ISSN: 0016-7487.

Keylock, C (2006) Erratum: 'Hazards resulting from earthquakes: A case study' (Geography Review vol. 19 (5) (14), Geography Review, 20(1), p.31, ISSN: 0950-7035.

Keylock, C, Hirashima, H, Nishimura, K (2006) Hazards resulting from earthquakes: A case study, Geography Review, 19(5), pp.10-15, ISSN: 0950-7035.

Keylock, CJ (2006) Constrained surrogate time series with preservation of the mean and variance structure, Physical Review E, 73(3), 036707, ISSN: 1539-3755. DOI: 10.1103/physreve.73.036707.

Keylock, CJ (2005) An alternative form for the statistical distribution of extreme avalanche runout distances, Cold Regions Science and Technology, 42(3), pp.185-193, ISSN: 0165-232X. DOI: 10.1016/j.coldregions.2005.01.004.

Keylock, CJ, Hardy, RJ, Parsons, DR, Ferguson, RI, Lane, SN, Richards, KS (2005) The theoretical foundations and potential for large-eddy simulation (LES) in fluvial geomorphic and sedimentological research, Earth-Science Reviews, 71(3-4), pp.271-304, ISSN: 0012-8252. DOI: 10.1016/j.earscirev.2005.03.001.

Keylock, CJ (2005) Describing the recurrence interval of extreme floods using nonextensive thermodynamics and Tsallis statistics, Advances in Water Resources, 28(8), pp.773-778, ISSN: 0309-1708. DOI: 10.1016/j.advwatres.2005.02.011.

Keylock, CJ (2005) Simpson diversity and the Shannon-Wiener index as special cases of a generalized entropy, Oikos, 109(1), pp.203-207, ISSN: 0030-1299. DOI: 10.1111/j.0030-1299.2005.13735.x.

Keylock, CJ and Dorling, D (2004) What kind of quantitative methods for what kind of geography?, Area, 36(4), pp.358-366, ISSN: 0004-0894. DOI: 10.1111/j.0004-0894.2004.00237.x.

Keylock, C (2004) Reviewing the Hjulström curve, Geography Review, 17(4), pp.16-20, ISSN: 0950-7035.

Keylock, CJ (2003) Mark Melton's geomorphology and geography's quantitative revolution, Transactions of the Institute of British Geographers, 28(2), pp.142-157, ISSN: 0020-2754. DOI: 10.1111/1475-5661.00084.

Keylock, CJ (2003) The North Atlantic Oscillation and snow avalanching in Iceland, Geophysical Research Letters, 30(5), pp.n/a-n/a, ISSN: 0094-8276. DOI: 10.1029/2002gl016272.

Barbolini, M and Keylock, CJ (2002) A new method for avalanche hazard mapping using a combination of statistical and deterministic models, Natural Hazards and Earth System Sciences, 2(3/4), pp.239-245, DOI: 10.5194/nhess-2-239-2002.

Keylock, CJ and Barbolini, M (2001) Snow avalanche impact pressure - vulnerability relations for use in risk assessment, Canadian Geotechnical Journal, 38(2), pp.227-238, ISSN: 0008-3674. DOI: 10.1139/t00-100.

Barbolini, M, Gruber, U, Keylock, CJ, Naaim, M, Savi, F (2000) Application of statistical and hydraulic-continuum dense-snow avalanche models to five real European sites, Cold Regions Science and Technology, 31(2), pp.133-149, ISSN: 0165-232X. DOI: 10.1016/s0165-232x(00)00008-2.

Keylock, C and Domaas, U (1999) Evaluation of Topographic Models of Rockfall Travel Distance for Use in Hazard Applications, Arctic, Antarctic, and Alpine Research, 31(3), pp.312-312, ISSN: 1523-0430. DOI: 10.2307/1552262.

Keylock, CJ, McClung, DM, Magnússon, MM (1999) Avalanche risk mapping by simulation, Journal of Glaciology, 45(150), pp.303-314, ISSN: 0022-1430. DOI: 10.3189/s0022143000001805.

Keylock, C (1999) Towards an Interpretation of Historical Droughts in Northern Nigeria: A Comment on a Paper by Aondover Tarhule and Ming-Ko Woo, Climatic Change, 41(2), pp.259-260, ISSN: 0165-0009. DOI: 10.1023/a:1005435611894.

Harbitz, C, Issler, D, Keylock, C (1998) Conclusions from a recent survey of avalanche computational models, Publikasjon - Norges Geotekniske Institutt, (203), pp.128-135, ISSN: 0078-1193.

Keylock, C (1997) Snow avalanches, Progress in Physical Geography: Earth and Environment, 21(4), pp.481-500, ISSN: 0309-1333. DOI: 10.1177/030913339702100401.


Keylock, CJ (2019) Velocity gradient tensor analysis for turbulence reformulated in terms of normal and non-normal contributions. In , 11th International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2019.

Keylock, CJ (2019) Velocity gradient tensor analysis for turbulence reformulated in terms of normal and non-normal contributions. In , 11th International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2019.

Higham, JE, Brevis, W, Keylock, CJ (2016) Turbulent characteristics produced by a blockage of pre-fractal emergent obstacles. In , Proceedings of the 20th Australasian Fluid Mechanics Conference, AFMC 2016,ISBN: 9781740523776.

Higham, JE, Brevis, W, Keylock, CJ (2016) Turbulent characteristics produced by a blockage of pre-fractal emergent obstacles. In , Proceedings of the 20th Australasian Fluid Mechanics Conference, AFMC 2016,ISBN: 9781740523776.

Ash, M, Tanha, MA, Brennan, PV, Kohlert, A, McElwaine, JN, Keylock, CJ (2015) Practical implementation of a 16-channel C-band phased array radar receiver. In 2015 IEEE Radar Conference. DOI: 10.1109/radarconf.2015.7411856.

Ash, M, Ardeshir Tanha, M, Brennan, PV, Kohler, A, McElwaine, JN, Keylock, CJ (2014) Improving the sensitivity and phased array response of FMCW radar for imaging avalanches. In 2014 International Radar Conference (Radar), 2014 International Radar Conference. DOI: 10.1109/radar.2014.7060387.

Ash, M, Brennan, PV, Vriend, NM, McElwaine, JN, Keylock, CJ (2011) FMCW phased array radar for automatically triggered measurements of snow avalanches. In , European Microwave Week 2011: "Wave to the Future", EuMW 2011, Conference Proceedings - 8th European Radar Conference, EuRAD 2011, pp.166-169, ISBN: 9782874870217.

Ash, M, Brennan, PV, Vriend, NM, Mcelwaine, JN, Keylock, CJ (2011) Two-dimensional FMCW radar imaging of entire avalanche events. In , 6th IASME/WSEAS Int. Conf. on Continuum Mechanics, CM'11, 6th IASME/WSEAS Int. Conf. on Water Resources, Hydraulics and Hydrology, WHH'11, 5th IASME/WSEAS Int. Conf. on Geology and Seismology, GES'11, pp.153-157, ISBN: 9789604742752.

Ash, M, Chetty, K, Brennan, P, McElwaine, J, Keylock, C (2010) FMCW radar imaging of avalanche-like snow movements. In 2010 IEEE Radar Conference. DOI: 10.1109/radar.2010.5494643.

Brennan, PV, Ash, M, Isa, FM, Keylock, C, Mcelwaine, J (2009) Advanced Radar Imaging of Geophysical Flows. In , GES'09: PROCEEDINGS OF THE 3RD IASME/WSEAS INTERNATIONAL CONFERENCE ON RECENT AD GEOLOGY AND SEISMOLOGY, pp.144-+, ISBN: 978-960-474-058-1.

Higham, JE, Brevis, W, Keylock, CJ (2006) Turbulent characteristics produced by a blockage of pre-fractal emergent obstacles. In , Proceedings of the 20th Australasian Fluid Mechanics Conference, AFMC 2016,ISBN: 9781740523776.


Keylock, CJ (Accepted for publication) Dynamics and Complexity. In The SAGE Handbook of Geographical Knowledge, SAGE Publications Ltd, pp.393-404, DOI: 10.4135/9781446201091.n30.


Keylock, C (2020) Supplementary Information for: "Holder-conditioned Hypsometry: A Refinement To A Classical Approach For The Characterization of Topography" by Chris Keylock et al, published in Water Resources Research, No description supplied.

Keylock, CJ (Accepted for publication) The velocity gradient tensor for homogeneous, isotropic turbulence (HIT), with explicit consideration of local and non-local effects using a Schur decomposition, A Schur decomposition of the velocity gradient tensor (VGT) for homogeneous, isotropic turbulence (HIT) is undertaken and its physical consequences examined. This decomposition permits the normal parts of the tensor (represented by the eigenvalues) to be separated explicitly from the non-normal effects. Given the restricted {E}uler approximation to the VGT dynamics is written in terms of the isotropic part of the pressure Hessian and the invariants of the characteristic equation of the VGT (in turn expressed in terms of the eigenvalues), the non-normal terms are related to the non-local aspects of the dynamics and the anisotropic part of the pressure Hessian. Using a direct numerical simulation of HIT, we show that the norm of the non-normal part of the tensor is of a similar order to the normal part, highlighting the importance of non-local effects. In fact, beneath the discriminant function in a Q-R plot, all enstrophy arises from the non-normal term, meaning that vorticity and intermediate strain eigenvector alignment in this region is an immediate consequence of non-normality. A non-normal term appears in the expressions for both enstrophy and total strain and cancels when calculating the second invariant of the VGT, while the self-amplification of non-normality and the normal straining of non-normality appear in the strain production and enstrophy production equations and cancel when calculating the third invariant. However, these terms are significant for understanding the full VGT dynamics, explaining how flow structures evolve to a disc-like state despite the strain eigenvalues sometimes indicating opposite (rod-like) behaviour, as well as explaining vorticity and strain alignments in HIT..

Keylock, CJ (Accepted for publication) A Schur decomposition reveals the richness of structure in homogeneous, isotropic turbulence as a consequence of localised shear, An improved understanding of turbulence is essential for the effective modelling and control of industrial and geophysical processes. Homogeneous, isotropic turbulence (HIT) is the archetypal field for developing turbulence physics theory. Based on the Schur transform, we introduce an additive decomposition of the velocity gradient tensor into a normal part (containing the eigenvalues) and a non-normal or shear-related tensor. We re-interrogate some key properties of HIT and show that the the tendency of the flow to form disc-like structures is not a property of the normal tensor; it emerges from an interaction with the non-normality. Also, the alignment between the vorticity vector and the second eigenvector of the strain tensor is another consequence of local shear processes..

Keylock, CJ (Accepted for publication) The turbulence velocity gradient tensor formed additively by normal and non-normal tensors, We decompose the velocity gradient tensor for turbulence into normal and non-normal parts, and condition our analysis on the strain eigenvector alignments between these tensors. We identify states that always enhance, and always counteract the axisymmetric expansion state, and give a rationale for decomposing the production balance term into its constituents: complex behavior arises when the dominant strain alignments involve the non-normal tensor. Finally, we develop a topological analysis framework where mathematical bounds on two of the three variables leads to an analysis in two planes. DOI: 10.1103/PhysRevFluids.2.084607.

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