UltraLow Frequency ULF waves are critical drivers of particle acceleration and loss in the Earths magnetosphere. While statistical models of ULFinduced radial transport have traditionally assumed that the waves are uniformly distributed across magnetic local time MLT decades of observational evidence show significant MLT localization of ULF waves in the Earths magnetosphere. This study presents for the first time a quasilinear radial diffusion coefficient accounting for localized ULF waves. We demonstrate that even though quasilinear radial diffusion is averaged over drift orbits MLT localization significantly alters the efficiency of particle transport. Our results reveal that when ULF waves cover more than 30 of the MLT the radial diffusion efficiency is comparable to that of uniform wave distributions. However when ULF waves are confined within 10 of the drift orbit the diffusion coefficient is enhanced by 10 to 25 indicating that narrowly localized ULF waves are efficient drivers of radial transport.
['Physics - Space Physics', 'Astrophysics - Earth and Planetary Astrophysics', 'Astrophysics - Solar and Stellar Astrophysics', 'Physics - Plasma Physics']
Radial Diffusion Driven by Spatially Localized ULF Waves in the Earths Magnetosphere
{'Adnane Osmane 1 , Jasmine K. Sandhu 2 , Tom Elsden 3 , Oliver Allanson 4 , 5 , 6 and Lucile Turc 1': '1  \nUniversity of Helsinki, Department of Physics, Helsinki, Finland 2 University of Leicester, Department of Physics and Astronomy, Leicester, UK 3 University of St Andrews, School of Mathematics and Statistics, St. Andrews, UK 4 University of Birmingham, Space Environment and Radio Engineering, School of Engineering,', 'Birmingham, UK': '5 University of Exeter, Environmental Mathematics, Department of Earth and Environmental Sciences,', 'Penryn,UK': '6 University of Exeter, Department of Mathematics, Exeter, UK', 'Key Points:': '- · A radial diffusion coefficient for spatially localized ULF waves is derived for the first time.\n- · ULF waves sampled on less than 10% of the drift orbit provide more efficient diffusion by a factor of 10 to 25 %.\n- · Our results apply to particles transiting through magnetospheric plumes and MLT localised ULF wave packets.', 'Abstract': "Ultra-Low Frequency (ULF) waves are critical drivers of particle acceleration and loss in the Earth's magnetosphere. While statistical models of ULF-induced radial transport have traditionally assumed that the waves are uniformly distributed across magnetic local time (MLT), decades of observational evidence show significant MLT localization of ULF waves in the Earth's magnetosphere. This study presents, for the first time, a quasilinear radial diffusion coefficient accounting for localized ULF waves. We demonstrate that even though quasi-linear radial diffusion is averaged over drift orbits, MLT localization significantly alters the efficiency of particle transport. Our results reveal that when ULF waves cover more than 30% of the MLT, the radial diffusion efficiency is comparable to that of uniform wave distributions. However, when ULF waves are confined within 10% of the drift orbit, the diffusion coefficient is enhanced by 10 to 25%, indicating that narrowly localized ULF waves are efficient drivers of radial transport.", '1 Introduction': "Ultra-Low Frequency waves are key drivers of particle transport, acceleration, and losses in the Earth's radiation belts (Su et al., 2015; Jaynes et al., 2018; George et al., 2022; Olifer et al., 2024). For decades, statistical models of ULF-induced radial transport, typically formulated using Fokker-Planck equations, have assumed that the waves are uniformly distributed across magnetic local time (MLT) (Falthammar, 1965; Elkington et al., 1999, 2003; Lejosne, 2019; Osmane & Lejosne, 2021; Osmane et al., 2023). However, decades of observational evidence have shown that ULF waves are often significantly localized in MLT (Murphy et al., 2020) due to localised source regions and inhomogeneous properties of the magnetospheric plasma. For example, storm-time Pc5 waves  \nwith high azimuthal wave numbers, typically in the poloidal mode, are driven by unstable ring current ion distributions and substorm injections, and are predominantly observed in the pre-midnight sector (Anderson et al., 1990). Statistical studies have further demonstrated that Pc5 fluctuations exhibit strong MLT-dependent variations in wave power and radial distance, as well as pronounced day-night asymmetries-wave power is stronger on the nightside in the inner magnetosphere and on the dayside near the magnetopause (Liu et al., 2009; Sarris et al., 2022; Yan et al., 2023). Additionally, toroidal field line resonances show MLT dependence in amplitude, with larger amplitudes observed on the dawn flank compared to the dusk flank. This pattern has been widely observed both on the ground (Gupta, 1975; Vennerstrøm, 1999; Pahud et al., 2009; Rae et al., 2012) and in situ (Anderson et al., 1990; Kokubun, 2013; Takahashi et al., 2015, 2016; Yan et al., 2023).  \nDespite extensive observational data highlighting MLT-dependent wave localisation, the impact on radial diffusion coefficients D LL remains unknown. Since radial diffusion typically tracks the drift-averaged evolution of the particle distribution function, it is unclear whether the MLT-localized ULF waves are more or less efficient in driving radial transport compared to waves uniformly distributed in MLT. This paper bridges the gap between modeling and observations by deriving, for the first time, a quasi-linear radial diffusion coefficient for MLT-localized ULF waves. In this study, we address two key questions: (1) What is the parametric dependence of the radial diffusion coefficient on MLT localization of ULF waves? and (2) Is radial diffusion more or less efficient when ULF waves are spatially localized compared to when they are uniformly distributed across MLT?  \nIn the following, we consider two scenarios under which magnetically trapped particles in the Earth's magnetosphere can encounter MLT localised ULF waves. These two scenarios are illustrated in Figure (1). In the first scenario, shown in the left panel, the ULF wave is homogeneous across MLT but localized in radial distance. As a result, particles on different drift shells move in and out of the plasma regions where the ULF waves are present. In the second scenario, shown in the right panel, the ULF waves are localized in MLT, meaning that all trapped particles only encounter the waves within specific MLT regions. This second case may occur due to magnetospheric plumes or regions where ULF waves are locally generated in the magnetosphere due to boundary instabilities, e.g., Kelvin-Helmholtz instabilities at the magnetopause (Masson & Nykyri, 2018), the impact of upstream mesoscale transient structures (Hartinger et al., 2013; Kajdiˇc et al., 2024), or particle driven drift or bounce resonant instabilities (Southwood & Kivelson, 1981).  \nAlthough from the particle's perspective, the distinction between MLT localization due to Scenario 1 (left panel of Figure 1) or Scenario 2 (right panel of Figure 1) is irrelevant-since the particle only samples the field along its trajectory-it is important to emphasize that MLT-localized ULF waves might be the norm rather than the exception when treating wave-particle interactions. ULF waves are often radially and MLT-localized, and this spatial localization occurs independently of the drift orbits confined to specific drift shells. The calculations presented here can thus be applied to a wide range of geomagnetic conditions, including both storm and non-storm periods, where ULF waves exhibit spatial localization.  \nThis communication is organized as follows. In Section 2, we first describe the electromagnetic field model that accounts for MLT-localized ULF waves (Section 2.1), followed by a step-by-step quasi-linear derivation of the associated radial diffusion coefficient D LL (Section 2.2). We then compare our results to the general case where ULF waves are uniformly distributed in MLT and provide a physical explanation for the observed discrepancies when ULF waves are MLT-localized (Section 2.3). Finally, in Section 3, we summarize our findings, discuss the key limitations of our work, and outline future steps to further quantify the impact of ULF waves on particle transport.  \n<!-- image -->  \nFigure 1. Illustration of two general scenarios where magnetically trapped particles encounter MLT-localized ULF waves. In the left panel, the ULF wave is homogeneous across MLT but localized in radial distance. As a result, particles on different drift shells move in and out of the plasma regions where the ULF waves are present. In the right panel, the ULF waves are localized in MLT, meaning that all trapped particles only encounter the waves within specific MLT regions.  \n<!-- image -->", '2.1 ULF Wave Model': "We proceed with a derivation of radial diffusion for the special case of an electrostatic poloidal electric field superposed onto a dipolar magnetic field. The background magnetic field is given by:  \nB = -B E R 3 E r 3 ˆ z , (1)  \nin terms of the magnetic moment B E and the Earth's radius R E /similarequal 6300 km. We can ignore the latitude dependence of the background magnetic field because we derive radial diffusion for the special case of equatorially trapped particles, and thus the unit magnetic field vector b = B | B | = -ˆ z always points in the negative z direction in a cyclindrical system of coordinates written in terms of ( r, ϕ, z ), where ϕ is the azimuthal angle. The poloidal electric field due to the ULF wave is written as:  \nE = δE ϕ ( r, ϕ, t ) ˆ ϕ = f ( ϕ ; κ ) ∑ m δE ϕ,m R E r e imϕ ˆ ϕ, (2)  \nin terms of the time-dependent Fourier coefficients δE ϕ,m ( t ) and the von Mises distribution:  \nf ( ϕ ; κ ) = e κ cos( ϕ ) 2 πI 0 ( κ ) , = 1 2 π ( 1 + 2 ∞ ∑ n =1 ψ n ( κ ) cos( nϕ ) ) (3)  \nthe latter of which is written in terms of the modified Bessel function of the first kind with degree zero I 0 ( κ ) (Abramowitz & Stegun, 1968), and the parameter κ ∈ [0 , ∞ ). The von Mises distribution is the analogue to the Maxwellian distribution on a circle and normalised. Written in the above form, it is also normalised as can be shown when integrated along ϕ between -π and π . We note that this representation has previously been  \nused to describe coherent interactions between ULF waves and magnetically confined electrons in the Earth's radiation belts (Li et al., 2018; Hao et al., 2020) and that other representations should provide the same transport coefficients, with the stipulation discussed in Section (2.2) and derived in Appendix A to constrain the root mean square fluctuations sampled over a drift orbit.  \nThe choice of the electric field dependence on the radial distance and the azimuthal angle requires an explanation. The term inside the sum of Equation (2) is a Fourier decomposition for coefficients δE ϕ,m R E /r . The 1 /r dependence of the electric field comes from Faraday's equation for an electrostatic field that satisfies ∇× E = 0. In cylindrical coordinates, we are required to keep δE ϕ r constant. The introduction of the von Mises distribution in Equation (2) allows us to confine the ULF electric field to any range of magnetic local times we deem appropriate. ULF modes of azimuthal wave numbers m are also multiplied by the envelope f ( ϕ ; κ ) to confine the mode within a localised MLT region. The range of MLT localisation is parameterised by the coefficient κ . If κ = 0, the ULF field is uniformly spread across all magnetic local times. But as κ becomes much greater than one, the von Mises distribution converges to a Maxwellian:  \nlim κ -→∞ f ( ϕ ; κ ) = √ κ 2 π e -κϕ 2 / 2 . (4)  \nThus, the parameter κ is equivalent to the inverse of the variance of a Maxwellian, and the von Mises distribution appearing in the Fourier decomposition of the poloidal electric field in Equation (2) confines the bulk of the ULF fluctuations around ϕ = ± 2 √ 1 /κ .  \nIn the following, we use a Fourier representation of the poloidal electric field derived from Equation (2). The derivation is provided in Appendix A and given by the following decomposition:  \nE ϕ = R E r ∑ m c m e imϕ , (5)  \nwith the Fourier coefficients:  \nc m = ∑ m ' δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) , (6)  \nwhere we used the Kronecker delta δ m,m ' for indices m and m ' and its complement ε m,m ' = 1 -δ m,m ' . The coefficient ψ n is a ratio of the modified Bessel function of degree n to the modified Bessel function of degree 0:  \nψ n ( κ ) = I n ( κ ) I 0 ( κ ) . (7)  \nNote that the modified Bessel function of the first kind is symmetric with respect to a transformation of the coefficients n ∈ Z with I -n ( κ ) = I n ( κ ), and thus ψ -n = ψ n . With the electric and magnetic field given by Equations (1) and (2), we can compute the guiding center drift velocities (Hazeltine, 2018). For particles trapped in the equatorial plane, we only have two drift contributions, the µ ∇ B drift, for a particle of charge q , relativistic Lorentz factor γ and first adiabatic invariant µ :  \n-µ ∇ B × b qγB = 3 µ qγr ˆ ϕ (8)  \nand the E cross B drift:  \nE × b B = -ˆ r ∑ m c m B E r 2 R E e imϕ (9)  \n= -ˆ r ∑ m ∑ m ' 1 2 π δE ϕ,m ' B E r 2 R E ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e imϕ . (10)  \nThe drift velocity v d can therefore be written as:  \nv d = 3 µ qγr ˆ ϕ -ˆ r ∑ m ∑ m ' 1 2 π δE ϕ,m ' B E r 2 R E ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e imϕ (11)  \nThe radial diffusion equation can be derived from the drift velocity (11) using several approaches. One common method involves calculating the increment ∆ r from Equation (11), applying standard quasi-linear assumptions that the electric field amplitude δE ϕ,m behaves as white noise, and then computing the drift average of ∆ r 2 to estimate the diffusion coefficient (see, e.g., Falthammar (1965); Elkington et al. (2003); Fei et al. (2006); Lejosne (2019)). However, in this work, we derive the quasi-linear radial diffusion coefficient directly from the kinetic equation. This approach, previously applied by Osmane et al. (2023) for ULF waves and by Kennel and Engelmann (1966) for kinetic scale fluctuations, is mathematically more elaborate but offers more clarity in how quasi-linear assumptions are incorporated into the derivation and how higher-order and non-diffusive effects are omitted.", '2.2 Kinetic Equation and Quasi-Linear Derivation': "In the absence of loss processes, the kinetic equation for equatorially trapped guiding centers in planetary radiation belts (see Osmane et al. (2023) and references therein) can be expressed in conservative form as:  \n∂ ∂t ( Bg ) + ∇· ( v d Bg ) = 0 . (12)  \nThe magnetic field amplitude B is included in Equation (12) because it represents the Jacobian of the transformation when the particle's velocity is expressed in field-aligned coordinates. The distribution function of guiding centers is denoted as g = g ( r, ϕ, t ).  \nUtilizing Liouville's theorem  \n∂ ∂t ( B ) + ∇· ( v d B ) = 0 , (13)  \nthe evolution of the distribution function is given by:  \n∂g ∂t + 3 µ qγr 2 ∂g ∂ϕ = ∑ m c m B E r 2 R 2 E e imϕ ∂g ∂r . (14)  \nNote the appearance of the azimuthal drift frequency Ω d = 3 µ/qγr 2 on the left-hand side of Equation (14).  \nThe next step is to decompose the distribution function into two components: one part, denoted as δg , that evolves on fast timescales comparable to or shorter than the drift period, and another part that evolves on much longer timescales, surpassing both the drift period and the characteristic timescales of Pc4 and Pc5 ULF wave frequencies:  \ng ( r, ϕ, t ) = g 0 ( r, t ) + δg ( r, ϕ, t ) = g 0 ( r, t ) + ∑ p δg p ( r, t ) e ipϕ . (15)  \nIn Equation (15), the fast part of the distribution function, which is function of the azimuthal angle ϕ , is decomposed in terms of Fourier modes to simplify our analysis. The slow part of the distribution can be extracted by the following averaging procedure:  \ng 0 ( r, t ) = 〈 g 〉 ϕ = 1 2 πT ∫ π -π d ϕ ∫ T 0 d t g ( r, ϕ, t ) . (16)  \nInserting the decomposition of Equation (15) inside the kinetic equation, multiplying by a factor e -iqϕ , integrating from -π to π , and keeping the terms for the index q = 0 (i.e.  \nkeeping only the MLT averaged part), we find the following evolution equation for g 0 :  \n∂g 0 ∂t = r 2 R 2 E c q =0 B E ∂g 0 ∂r + r 2 R 2 E ∑ m ∂ ∂r ( c ∗ m δg m B E ) (17)  \nThe terms c ∗ m in Equation (17) represent the complex conjugate of the electric field Fourier coefficient and arises after inversing the sum over m with m -→ -m because the electric field is a real quantity, leading to the conditions c -m = c ∗ m .  \nThe quasi-linear equation that tracks transport on timescales longer than the drift period can be retrieved by applying the averaging procedure (16) on Equation (17):  \n∂g 0 ∂t = ∑ m 1 B E r 2 R 2 E ∂ ∂r ( 〈 c ∗ m δg m 〉 ϕ ) (18)  \nWe note that the first term on the right-hand side of Equation (17) cancels after averaging on timescales comparable to the ULF wave period since 〈 δE ϕ,m =0 〉 ϕ = 0.  \nIn order to solve the quasi-linear Equation (18), we need an equation for the perturbed part of the distribution function δg m . The procedure is identical to the one outlined in Appendix B of Osmane et al. (2023) and results in the following equation:  \n∂δg m ∂t + im Ω d δg m ︸ ︷︷ ︸ Azimuthal drifting = r 2 R 2 E c m B E ∂g 0 ∂r ︸ ︷︷ ︸ Linear wave-particle interaction + r 2 R 2 E ∑ m ' c ∗ m ' B E ∂δg m + m ' ∂r ︸ ︷︷ ︸ Higher order wave-particle interaction (19)  \nδg m ( r, t ) = δg m ( r, 0) e -im Ω d t + 1 B E r 2 R 2 E ∫ t 0 d t ' c m ( t ' ) e -im Ω d ( t ' -t ) ∂g 0 ∂r (20)  \nIn the quasi-linear regime, the perturbed part of the distribution function is assumed to evolve linearly. Consequently, higher-order wave-particle interactions in Equation (19) are neglected, reducing the problem to a linear equation with the following general solution:  \nThe first term on the right-hand side of Equation (20) generates drift echoes, and since over long diffusive time, drift echoes experience phase-mixing, we also ignore their contribution to the evolution of g 0 and set δg m ( r, 0) = 0.  \nInserting Equation (20) into Equation (18) we find:  \n∂g 0 ∂t = ∑ m 1 B 2 E r 2 R 2 E ∂ ∂r [ r 2 R 2 E ∫ t 0 d t ' 〈 c m ( t ' ) c ∗ m ( t ) 〉 ϕ e -im Ω d ( t ' -t ) ∂g 0 ∂r ]  \nThe above diffusive equation is a function of the correlator 〈 c m ( t ) c ∗ m ( t ' ) 〉 ϕ for the ULF wave electric field. Within quasi-linear theories, we assume that the fluctuations driving the system are akin to stationary white noise, i.e., 〈 c m ( t ) c ∗ m t ' 〉 /similarequal δ ( t ' -t ), and that the background distribution function g 0 is not significantly affected during an interaction time (Vanden Eijnden, 1997). Moreover, in the absence of a physical model connecting a mode m to a mode m ' , we can simply assume that correlation only exists when m = m ' . This simplification is likely incorrect for ULF waves within plumes, but offers nonetheless a closure to the above equation and a lower bound to the diffusion equation, i.e., accounting for non-zero correlations between m and m ' terms will enhance diffusion.  \nWe proceed to compute the correlator by first taking the product of c m and c ∗ m :  \nc m ( t ' ) c ∗ m ( t ) = ( ∑ q δE ϕ,q 2 π ( δ m,q +2 ε m,q ψ q -m ) )( ∑ p δE ∗ ϕ,p 2 π ( δ m,p +2 ε m,p ψ p -m ) ) = ∑ q ∑ p δE ϕ,q δE ∗ ϕ,p 4 π 2 ( δ m,q +2 ε m,q ψ q -m )( δ m,p +2 ε m,p ψ p -m ) (21)  \nand then take the drift and time average to find:  \n〈 c m ( t ' ) c ∗ m ( t ) 〉 ϕ = ∑ q ∑ p 〈 δE ϕ,q δE ∗ ϕ,p 4 π 2 〉 ϕ ( δ m,q +2 ε m,q ψ q -m )( δ m,p +2 ε m,p ψ p -m ) = δ ( t ' -t ) ∑ q ∑ p 〈 δE 2 ϕ,q 4 π 2 〉 ϕ δ q,p ( δ m,q +2 ε m,q ψ q -m )( δ m,p +2 ε m,p ψ p -m ) = δ ( t ' -t ) ∑ q 〈 δE 2 ϕ,q 4 π 2 〉 ϕ ( δ m,q +2 ε m,q ψ q -m )( δ m,q +2 ε m,q ψ q -m ) = δ ( t ' -t ) ∑ q 〈 δE 2 ϕ,q 4 π 2 〉 ϕ ( δ m,q +4 ε m,q ψ 2 q -m ) (22)  \nwhere in the last line we took advantage of the following identities:  \n/negationslash  \nThus, inserting the correlator 〈 c m ( t ' ) c ∗ m ( t ) 〉 ϕ in the above Equation we find the quasilinear radial diffusion equation for MLT localised waves:  \n- · ∑ q δ 2 m,q = 1 if m = q and zero otherwise. · ∑ q δ m,q ε m,q = 0 for all combinations of m and q . · ∑ q ε 2 m,q = ∑ q ε m,q = 1 if m = q and zero otherwise.  \n∂g 0 ∂t = 1 B 2 E r 2 R 2 E ∂ ∂r [ r 2 R 2 E ∑ m ∑ m ' 〈 δE 2 ϕ,m ' 4 π 2 〉 ϕ ( δ m,m ' +4 ε m,m ' ψ 2 m ' -m ) ∂g 0 ∂r ] = L 2 ∂ ∂L ( D LL L 2 ∂g 0 ∂L ) (23)  \nwith the radial diffusion coefficient given by:  \nD LL = ∑ m ∑ m ' L 4 B 2 E R 2 E 〈 δE 2 ϕ,m ' 4 π 2 〉 ϕ ( δ m,m ' +4 ε m,m ' ) ψ 2 m ' -m = L 4 B 2 E R 2 E ∑ m 〈 δE 2 ϕ,m 4 π 2 〉 ϕ + L 4 B 2 E R 2 E ∑ m ' ∑ m 〈 δE 2 ϕ,m ' 4 π 2 〉 ϕ 4 ε m ' m ψ 2 m -m ' = L 4 B 2 E R 2 E ∑ m 〈 δE 2 ϕ,m 4 π 2 〉 ϕ + L 4 B 2 E R 2 E ∑ m ' 〈 δE 2 ϕ,m ' 4 π 2 〉 ϕ ∑ m 4 ε m ' m ψ 2 m -m ' = L 4 B 2 E R 2 E ∑ m 〈 δE 2 ϕ,m 4 π 2 〉 ϕ ( 1 + 4 ∑ q ε m + q,m ψ 2 q ) (24)  \nwhere we reversed the sums for m and m ' between the second and third line, inverted the indices m ↔ m ' between the third and fourth line, wrote the sum in terms of q = m ' -m and took advantage of the identity ψ q = ψ -q . The quasi-linear radial diffusion in Equation 24 for MLT localised ULF waves is almost identical to the radial diffusion coefficient for the special case of ULF fields uniformly distributed along MLT. The difference is the positive definite factor of  \n1 + 4 ∑ q ε m + q,m ψ 2 q ( κ ) , (25)  \n/negationslash  \nwhich is a function of the parameter κ and originates from the von Mises distribution. If we set κ = 0, we recover the radial diffusion for ULF waves uniformly distributed in magnetic local time: D LL = L 4 B 2 E R 2 E ∑ m 〈 δE 2 ϕ,m 4 π 2 〉 ϕ since ψ q ( κ = 0) = 0 only for q = 0 but ε m + q,q = 0 if q = 0. But if κ = 0, and as the ULF waves become more localised, the additional coefficient increases the value of radial diffusion monotonically.  \n/negationslash  \n/negationslash  \nFigure 2. MLT dependence of ULF waves with m = 1 (left panel) and m = 4 (right panel) with a von Mises envelopes f ( ϕ ; κ ), κ values of [0 , 1 . 77 , 3 . 3 , 8 . 6 , 30], and amplitude rescaling by a fixed root mean square fluctuation f ( ϕ ; κ ) -→ f ( ϕ ; κ ) √ 1+2 ∑ q,q = m ψ q -m given by Equation (A8). The selected κ values correspond to cases where the envelope confines ULF waves to approximately 100, 75, 50, 30 and 15 percent of the drift orbit.  \n<!-- image -->  \n<!-- image -->  \n/negationslash  \nEven though the von Mises distribution function is normalised, the maximum wave amplitude grows by a factor of 10 between κ = 0 and κ = 30. We therefore need to rescale the wave amplitude in a manner that is not affected by the κ parameter. In order to do so, we compute the root mean square fluctuation √ 〈 δE 2 ϕ 〉 , where the brackets denote the drift average, and add the additional requirement that a particle samples the same root mean square fluctuation on a drift orbit. The computation of 〈 δE 2 ϕ 〉 is presented in the Appendix A. We find that for a single mode m , the root mean square fluctuation 〈 δE 2 m,ϕ 〉 needs to be rescaled according to:  \n/negationslash  \nThe impact of this rescaling of the ULF wave is shown in Figure 2 for m = 1 and m = 4 modes. The blue lines in both panels show the ULF fluctuations that are homogeneous in MLT. The radial diffusion coefficient re-scaled to account for fixed root mean square fluctuations along a drift orbit is given by:  \n/negationslash  \nwhereas the diffusion coefficient for MLT homogeneous fluctuations is given by  \n〈 ˜ δE 2 ϕ,m 〉 4 π 2 = 〈 δE 2 ϕ 〉 1 + 2 ∑ p,p = m ψ p -m . (26)  \nD LL ( κ = 0) = L 4 B 2 E R 2 E 〈 δE 2 ϕ 〉 ϕ ∑ m ( 1 + 4 ∑ q ε m + q,m ψ 2 q 1 + 2 ∑ q ε m + q,m ψ q ) (27)  \nD LL ( κ = 0) = L 4 B 2 E R 2 E 〈 δE 2 ϕ 〉 ϕ . (28)  \nIn Figure 3 we show the ratio of the radial diffusion coefficient when κ = 0 to the one when κ = 0 for fixed root mean square fluctuation 〈 δE 2 ϕ 〉 and for a fixed m :  \n/negationslash  \nThe relative increase in D LL due to wave power in the azimuthal wave number m is given by 4 ∑ q ε m + q,m ψ 2 q and is equal to zero when κ = 0, yet increases by about 16% even when the drift orbit encounters ULF waves for less than 20% of its orbit, or equivalently  \n/negationslash  \nD LL ( κ = 0) D LL ( κ = 0) = 1 + 4 ∑ q ε q + m,m ψ 2 q ( κ ) 1 + 2 ∑ q ε q + m,m ψ q ( κ ) . (29)  \nfor κ > 25. This result may seem surprising, as it suggests that even when ULF waves are encountered over a small fraction of the drift orbit, they can still induce radial diffusion comparable to cases where the ULF wave is sampled across all magnetic local times. In the next section we provide mathematical and physical explanation to this result.", '2.3 Explanation for enhanced diffusion despite narrowly localised waves.': "Our analysis indicates that while quasi-linear radial diffusion does not retain information about the magnetic local time, the radial diffusion due to narrowly localised ULF waves is more efficient. This non-trivial result requires a physical explanation. The answer can be found by solving the linear perturbed part of the distribution function g m for a single wave mode of frequency ω m and growth or damping rate γ m :  \nδE ϕ = c m ( t ) e imϕ (30)  \nwith  \nWe also require that γ m ' /lessmuch ω m ' , such that a drifting particle transits the wave before it damps or grows to large-amplitudes where nonlinear effects would become significant. Inserting Equation (30) and (31) into Equation (20) gives:  \nc m = ∑ m ' ˜ δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e ( iω m ' + γ m ' ) t , (31)  \n/negationslash  \nThe mathematical answer to the enhanced radial transport due to narrowly localised MLT waves is in Equation (32). We recover the drift resonance ω m /similarequal m Ω d one would find for MLT homogeneous waves (Osmane et al., 2023). But if the waves are MLT localised, we see the appearance of additional resonant terms when ω m ' /similarequal m Ω d for m ' = m . If m = m ' + n , the additional resonance can be written as ω m ' /similarequal ( m ' + n )Ω d for n = 0. These new drift resonance terms are weighted by the Bessel function ratio ψ m ' -m , which only become significant when κ = 0. Thus, when the drift resonant perturbations are inserted in the quasi-linear diffusion equation, terms with n = 0 result in additional radial diffusion terms. While these terms also have to be re-scaled according to Equation (26) in order to make sure that the particles sample the same root-mean-square fluctuations along a drift orbit, they are positive definite additional sources for radial diffusion.  \n/negationslash  \nMathematically, the above description should be satisfying to the reader since it shows that additional drift resonances are now present when ULF waves are localised in MLT. But physically why is it that confining ULF waves to a narrow range of azimuthal angles results in increased resonant interactions? From the particle's perspective, the sampled wave field fluctuates in time but remains localised in the azimuthal angle ϕ . The more confined the wave is in the azimuthal direction, the larger the corresponding range of m values that the particle samples. As a result, the particle does not encounter a wave  \nδg m ( r, t ) = ∑ m ' ˜ δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e im Ω d t ∫ t 0 d t ' e i ( ω m ' -m Ω d -iγ m ' ) t ' ∂g 0 ∂r = -i ∑ m ' ˜ δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e im Ω d t ( e i ( ω m ' -m Ω d -iγ m ' ) t -1 ω m ' -m Ω d -iγ m ' ) ∂g 0 ∂r = -i ˜ δE ϕ,m 2 π ( e i ( ω m -m Ω d -iγ m ) t -1 ω m -m Ω d -iγ m ) ︸ ︷︷ ︸ Drift Resonance for ω m /similarequal m Ω d ∂g 0 ∂r e im Ω d t -i ∑ m ' = m ψ m ' -m ˜ δE ϕ,m ' π ( e i ( ω m ' -m Ω d -iγ m ' ) t -1 ω m ' -m Ω d -iγ m ' ) ︸ ︷︷ ︸ Additional drift resonances for ω m ' /similarequal m Ω d ∂g 0 ∂r e im Ω d t (32)  \n/negationslash  \n/negationslash  \n/negationslash  \nFigure 3. Top panel (full line) shows the relative change of the radial diffusion coefficient D LL as a function of the κ parameter. The parameter κ determines the spread of ULF MLT range. The higher the value of κ the more localised are the ULF waves. The diffusion coefficient is re-scaled by Equation (A8) to ensure that a particle encounters the same root-mean-square fluctuation during a drift orbit. The relative increase in D LL is zero when κ = 0, and grows by about 15% when κ > 30. The bottom panel (dashed line) shows the relative change of the radial diffusion coefficient D LL as a function of the percentage of MLT coverage for κ = 5 (30%) and κ = 100 (6%).  \n<!-- image -->  \n/negationslash  \nwith distinct peaks and troughs, but rather a localized structure confined to part of its drift orbit (as illustrated in Figure 2). Drift resonance can still occur with various harmonics m ' = m if a collection of particles with drift period Ω d experiences a localised but net electric field. Furthermore, since the MLT regions without ULF do not contribute or affect transport radial transport is not precluded even if ULF waves are encountered only for a small portion of the drift trajectory. And thus, even when ULF waves are confined to a narrow range of MLT, radial transport can occur as long as repeated encounters with the ULF region lead to the sampling of a net electric field on average , that is for a collection of particles. We therefore find that radial diffusion remains comparably efficient when particles sample the same root-mean-square ULF fluctuations over a small fraction of the drift orbit, as it does over the entire drift orbit, due to the presence of additional drift resonances.", '3 Conclusions': "For decades, it has been well established that ULF waves in the Earth's magnetosphere exhibit significant localization in magnetic local time (MLT), with clear asymmetries observed across different sectors of the magnetosphere (Gupta, 1975; Anderson et al., 1990; Vennerstrøm, 1999; Liu et al., 2009; Pahud et al., 2009; Rae et al., 2012; Kokubun, 2013; Takahashi et al., 2015, 2016; Murphy et al., 2020; Sarris et al., 2022; Yan et al., 2023). Despite this, quasi-linear radial diffusion coefficients, derived analytically (Falthammar, 1965; Elkington et al., 2003; Fei et al., 2006; Lejosne, 2019; Osmane et al., 2023), typically assume that ULF waves are uniformly encountered across all MLT sectors by magnetically trapped particles. This assumption overlooks the well-documented MLT-dependent variations in ULF wave power and structure.  \nIn this communication, we present, for the first time, a quasi-linear radial diffusion coefficient for ULF wave modes localized in MLT. Our results demonstrate that while the quasi-linear distribution function is averaged over MLT and drift orbits, narrow MLT localization of ULF waves can significantly impact particle transport. We find that when ULF waves cover more than 30% of the MLT, the efficiency of radial diffusion is comparable to scenarios where ULF waves are uniformly distributed across all MLT sectors. However, ULF waves encountered on less than 10% of the drift orbit lead to an enhanced radial diffusion, increasing by 10% to 25%. While these enhancements may seem modest when accounting for other sources of variability in empirically derived radial diffusion coefficients (Thompson et al., 2020; Sandhu et al., 2021), they indicate that even narrowly localized ULF waves can be an efficient driver of radial transport in planetary magnetospheres.  \nWhile our derivation is informative, and indicative of additional drift resonance when ULF waves are MLT localized, it has several limitations that must be addressed to fully account for the role of ULF waves in radial diffusion. Perhaps the most significant limitation is that the ULF field used in our derivation is an electrostatic poloidal field superposed on a dipolar field. In solar wind-driven magnetospheres, a more realistic representation of ULF waves should include both toroidal and poloidal electromagnetic components (see, e.g., Murphy et al. (2020) and references therein) and static non-dipolar contributions (Fei et al., 2006; Cunningham, 2016). Recent numerical modelling works have shown that the 3D inhomogeneous nature of the magnetosphere has a crucial impact on the resulting ULF waves (Degeling et al., 2018; Wright & Elsden, 2020). These works show that azimuthal variations in the plasma mass density, for example from a plasmaspheric plume, fundamentally alter the polarisation of ULF waves, as well as their spatial extents. The classical 1D theory of ULF wave coupling describing field line resonance (FLR) (Southwood, 1974; Chen & Hasegawa, 1974) assuming azimuthal symmetry is no longer suitable in such a 3D varying plasma (Wright & Elsden, 2016). This study represents the first attempt to bring the radial diffusion theory in line with some of the inhomogeneous aspects of ULF waves in observations and simulations.  \n/negationslash  \nA second set of limitations, which also broadly applies to other quasi-linear derivations, is that we focused exclusively on particles with 90-degree pitch angles and assumed that the various azimuthal modes m and m ' (with m ' = m ) are short, uncorrelated impulses. While the former limitation can be easily addressed for arbitrary pitch angles (Schulz & Lanzerotti, 1974), the latter assumption requires a statistical model of ULF waves to account for the correlations between different azimuthal wave modes. A third limitation is that plumes and regions with radially localized ULF waves are time-dependent, and this variability can occur on timescales comparable to drift orbits. For example, the plume structure can change significantly over a few hours (Goldstein et al., 2014), and the radial structure of ULF waves is highly variable in time due to solar wind dynamic pressure variations (Claudepierre et al., 2010) and Alfv'en wave phase-mixing (Mann et al., 1995). We do not address this time dependence in the current communication, but a first step in generalizing the model would be to quantify the impact of a time-varying κ pa-  \nr in the ULF wave field. Addressing these limitations within quasi-linear diffusive and non-diffusive frameworks in future work will provide a more comprehensive understanding of the complex interactions between ULF waves and energetic particles, ultimately leading to more realistic models of radial diffusion in planetary magnetospheres.", 'Appendix A Fourier Representation and Spectra': "In this section we seek a Fourier representation to the MLT localised poloidal electric field in Equation (2)  \nδE ϕ = f ( ϕ ; κ ) ∑ m ˜ δE ϕ,m e imϕ , (A1)  \nδE ϕ ( r, ϕ, t ) = ∑ k c k e ikϕ , (A2)  \nwith the envelope given by Equation (3) and the variable ˜ δE ϕ,m containing the radial dependence. We want instead to have an equation for the electric of the form:  \nwith the Fourier coefficients c k = c k ( r, t ) given by  \nc k = 1 2 π ∫ π -π d ϕ δE ϕ e -ikϕ = 1 2 π ∑ m ˜ δE ϕ,m ∫ π -π d ϕ f ( ϕ ; κ ) e i ( m -k ) ϕ (A3)  \nThe above integral is easily solved with the Fourier convolution theorem and gives the following result for the Fourier coefficients:  \nc m = ∑ m ' ˜ δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) , (A4)  \nwhere we used the Kronecker delta δ m,m ' for indices m and m ' , it's complement ε m,m ' = 1 -δ m,m ' and the function ψ m ' -m as the ratio of the modified Bessel functions I n of order m ' -m and zero. The Fourier representation for the poloidal electric field can therefore be written as:  \nδE ϕ = ∑ m ∑ m ' ˜ δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e imϕ (A5)  \nWhen κ = 0, ε m,m ' ψ m ' -m = 0 for any combinations of m and m ' and the Fourier coefficients reduce to c m = ˜ δE ϕ,m 2 π , as it should.  \nLet's now compute the Fourier spectra 〈 δE ϕ δE ∗ ϕ 〉 ϕ in terms of the time and space average given by Equation (16). The first step is to write the product of δE ϕ to itself:  \nδE ϕ δE ∗ ϕ = ( ∑ m ∑ m ' ˜ δE ϕ,m ' 2 π ( δ m,m ' +2 ε m,m ' ψ m ' -m ) e imϕ )( ∑ p ∑ q ˜ δE ∗ ϕ,q 2 π ( δ p,q +2 ε p,q ψ q -p ) e -iqϕ ) = ∑ m ∑ m ' ∑ p ∑ q ˜ δE ϕ,m ' 2 π ˜ δE ∗ ϕ,q 2 π e i ( m -q ) ϕ ( δ m,m ' +2 ε m,m ' ψ m ' -m ) ( δ p,q +2 ε p,q ψ q -p ) (A6)  \nAfter averaging in phase and time, and assuming the short time correlation assumption, i.e., 〈 ˜ δE ϕ,m ' ˜ δE ∗ ϕ,q 〉 = 〈 ˜ δE 2 ϕ,m ' 〉 δ m ' ,q Equation (A6) yields:  \n〈 δE 2 ϕ 〉 = 1 4 π 2 ∑ m ∑ m ' ∑ p ∑ q 〈 ˜ δE 2 ϕ,m ' 〉 δ q,m ' δ m,q ( δ m,m ' +2 ε m,m ' ψ m ' -m ) ( δ p,q +2 ε p,q ψ q -p )  \nFigure A1. Dependence of m = 1 ULF wave power for a given value of κ when we require that the particle encounters a fixed root mean square fluctuation on the drift orbit. See Equation (A8) and associated text.  \n<!-- image -->  \n= 1 4 π 2 ∑ m ∑ q ∑ p 〈 ˜ δE 2 ϕ,q 〉 δ m,q ( δ m,q +2 ε m,q ψ q -m ) ( δ p,q +2 ε p,q ψ q -p ) = 1 4 π 2 ∑ m ∑ p 〈 ˜ δE 2 ϕ,m 〉 ( δ p,m +2 ε m,p ψ m -p ) = 1 4 π 2 m 〈 δE 2 ϕ,m 〉  1 + 2 p,p = m ψ p -m  (A7)  \n∑ ˜  ∑   \n/negationslash  \nIn Figure (A1) we plot 1 4 π 2 〈 ˜ δE 2 ϕ,m 〉 for a single mode m as a function κ for a fixed root mean square fluctuation 〈 δE 2 ϕ 〉 = 1:  \n/negationslash  \nFigure (A1) shows that the wave power for a given azimuthal wave number m must decrease by a factor of 10 for large values of κ ≥ 25 in order to account for fixed root mean square fluctuations sampled on a drift trajectory.  \n〈 ˜ δE 2 ϕ,m 〉 4 π 2 = 〈 δE 2 ϕ 〉 1 + 2 ∑ p,p = m ψ p -m . (A8)", 'Acknowledgments': 'Support for AO was provided by the University of Helsinki and the Research Council of Finland profiling action Matter and Materials (grant # 318913). OA would like to acknowledge financial support from the University of Birmingham, the University of Exeter, and also from the United Kingdom Research and Innovation (UKRI) Natural Environment Research Council (NERC) Independent Research Fellowship NE/V013963/1 and NE/V013963/2. The work of LT is funded by the European Union (ERC grant WAVESTORMS - 101124500). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. LT also acknowledges support from the Research Council of Finland (grant number 322544).', 'References': "Abramowitz, M., & Stegun, I. A. (1968). Handbook of mathematical functions with formulas, graphs, and mathematical tables (Vol. 55). US Government printing office.  \nAnderson, B. J., Engebretson, M. J., Rounds, S. P., Zanetti, L. J., & Potemra,  \nT. A. 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