Linear Numerical Model Research Articles

Typhoon Induced Extreme Coastal Surge: A Case Study at Northeast Taiwan in 1994

WANG, Y-H.; LEE, I-H., and WANG, D-P., 2005. Typhoon induced extreme coastal surge: a case study at Northeast Taiwan in 1994. Journal of Coastal Research, 21(3), 548‐552. West Palm Beach (Florida), ISSN 0749-0208. During the typhoon season, storm surges often instigate serious damages to the coastal areas of Taiwan. In August 1994, typhoon Doug almost completely destroyed the breakwaters and docking facilities in the Lungtung harbor at the northeast tip of Taiwan. The estimated maximum sea level rise was about 3m. Analyses of tide, wave and meteorological records indicate that the amplitude and spatial pattern of storm surges are related to the local wind speed, direction and duration as well as the local geometry. A linear two-dimensional numerical model is used to quantify the storm surge structure. The model results indicate that sea levels reached a maximum of about 2 m at the peak of the storm. The largest surge, however, does not occur at where the most severe damage was located. This suggests that the wave run-up could be an important contribution to the storm surge.

Linear and nonlinear numerical models of local density perturbation dynamics in a stable stratified medium

Linear and nonlinear numerical models of local density perturbation dynamics in a stable stratified medium

On the spectrum of vertically propagating gravity waves generated by a transient heat source

Abstract. It is commonly believed that cumulus convection preferentially generates gravity waves with tropospheric vertical wavelengths approximately twice the depth of the convective heating. Individual cumulonimbus, however, act as short term transient heat sources (duration 10 to 30min). Gravity waves generated by such sources have broad frequency spectra and a wide range of vertical scales. The high-frequency components tend to have vertical wavelengths much greater than twice the depth of the heating. Such waves have large vertical group velocities, and are only observed for a short duration and at short horizontal distances from the convective source. At longer times and longer distances from the source the dominant wave components have short vertical wavelengths and much slower group velocities, and thus are more likely to be observed even though their contribution to the momentum flux in the upper stratosphere and mesosphere may be less than that of the high frequency waves. These properties of convectively generated waves are illustrated by a linear numerical model for the wave response to a specified transient heat source. The wave characteristics are documented through Fourier and Wavelet analysis, and implications for observing systems are discussed.

Design and analysis of plasma position and shape control in superconducting tokamak JT-60SC

The analyses of the plasma position and shape control in the superconducting tokamak JT-60SC in JAERI are presented. The vacuum vessel and stabilizing plates located closely to the plasma are modeled in 3 dimension, and we can take into account the large ports in the vacuum vessel. The linear numerical model used in the design for the plasma feedback control system is based on Grad–Shafranov equation, which allows the plasma surface deformation. For a slower control of the plasma shape, the superconducting equilibrium field (EF) coils outside toroidal field coils are used, while for a fast control of the plasma position, in-vessel normal conducting coils (IV coil) are used. It is shown that the available loop voltages of the EF and IV coils are very limited, but there are sufficient accuracy and acceptable response time of plasma position and shape control.

On the use of the Galerkin method for 3D numerical modelling of the general circulation: the South Atlantic eperiment

AbstractA linear three‐dimensional hydrodynamical numerical model, with the application of the Galerkin Method for the vertical dependence, is here presented. The spherical coordinate system is used, in order to allow large‐scale simulations. The equations and mathematical development of the model are shown in detail, together with the boundary and initial conditions, and the sequence of equations' solution. The model is applied to the South Atlantic Ocean, for estimating typical seasonal circulations, and the results are summarized in maps of currents at surface and 1000 m depth, and in transport values of the Brazil Current between 30°S and 40°S. Copyright © 2002 John Wiley & Sons, Ltd.

Non linear problems: Introduction

ABSTRACT Solid mechanics problems in general, and specifically in the case of geomaterials, are very often significantly non linear. Although linear numerical models may be used as rough approximations to the problem solution, non linearities are very often too significant to be neglected. To solve these problems special non linear strategies are necessary. A series of non linear analysis techniques are presented in the context of the finite element method. These techniques are based on the Newton-Raphson method, a very well known mot finding algorithm. The different techniques presented are: full Newton-Raphson with load control, modified Newton-Raphson, quasi Newton methods and Newton-Rahson with displacement control. Implementation of the Newton-Raphson method in a finite element program is also covered.

Significance of non-rotational wind stress in the seasonal variation of continental torque

The physical mechanism by which seasonally varying atmospheric wind stress exerted on thesea surface is communicated to the solid earth as oceanic pressure torque (continental torque)and bottom frictional torque is investigated with a linear shallow-water numerical model ofbarotropic oceans. The model has a realistic land’ocean distribution and is driven by a seasonallyvarying climatic wind stress. A novel way to decompose the wind stress into rotational andnon-rotational components is devised. The rotational component drives ocean circulations asclassical theories of wind-driven circulations demonstrate. The non-rotational component doesnot produce ocean circulations within the framework of a barotropic shallow-water model, butbalances with the pressure gradient force due to surface displacement in the steady state. Basedon this decomposition, it is shown that most of the continental torque which plays a majorrole in producing the seasonal variation of length of day (LOD) is caused by the non-rotationalcomponent of the wind stress. Both continental torque due to the wind-driven circulationproduced by the rotational component of the wind stress and the bottom frictional torque areof minor importance.

Stationary Rossby wave propagation in a shear flow along a reflective boundary

This study investigates the impact of lateral boundary conditions on the propagation and dispersion of locally excited Rossby waves in a zonally periodic, barotropic, quasigeostrophic channel model on the β-plane. We use basic flows with either a linear meridional shear or a jet-like profile. On the southern boundary of the channel we impose either a rigid wall or a radiation condition, whereas the northern sidewall is permeable for Rossby waves. We compare the numerical solutions found for a reflecting southern boundary in a weakly dissipative flow to the solutions obtained from a WKB-analysis for the corresponding unforced nondissipative situation. Furthermore, we compare the generalized Eliassen-Palm flux vectors to the ray paths of Rossby wave packets, obtained from WKB ray tracing. In particular, we focus our investigation on the two-dimensional structure of trapped modal waves and wave trains in a simple linear numerical model. Summarizing our results, we find that along the reflective wall, trapped modal wave structures as well as reflected wave trains occur with characteristics (e.g., wave numbers, turning latitudes) similar to the ones computed using asymptotic methods. In a linear sheared flow wave packets are trapped for all zonal wave numbers in contrast to a jet-like mean flow which has a selective effect on the waves; i.e., a turning latitudes phenomenon between the coast and the flow maximum occurs for short waves, while long waves can propagate freely across the zonal mean flow. This comes out clearly when studying the stream lines of the Eliassen-Palm flux vectors of the numerical model simulations. Furthermore, due to the reflected wave activity, the dispersion of Rossby waves is influenced by the southern boundary condition not only in the vicinity of the border but also in regions away from the boundary. These results appear to be important on the one hand for the existence of trapped Rossby waves in large-scale oceanic shear flows along a zonally oriented coast. And, on the other hand for large-scale boundary waves in conceptional atmospheric channel models which can lead to unwanted resonance effects.

El Niño Sea Level and Currents along the South American Coast: Comparison of Observations with Theory

Interannual sea level fluctuations along the South American coast and 7.3-year-long current records at 30°S off central Chile were examined with a low-frequency linear numerical model. The model includes stratification and shelf and slope bottom topography, which varies alongshore. The coastal flow is driven by interannual alongshore wind stress and an interannual alongshore pressure gradient resulting from El Niño/La Niña pressure fluctuations at the equator. Calculations suggest that along the South American coast the coastal interannual variability is primarily due to the remote equatorial forcing rather than interannual coastal alongshore wind stress. Past work using this model has shown that, due to the scattering of lower vertical modes to higher ones by bottom friction, the model predicts a low-frequency jet trapped near the bottom along the continental slope. Until now adequate interannual current time series were unavailable to check this prediction. Analysis of the observations shows that at interannual frequencies the observed alongshore current fluctuated with an amplitude of about 5 cm s−1 at 220 m and 4 cm s−1 at 750 m depth; that is, the observed interannual flow does not have the bottom-trapped continental slope jet predicted by the model. If, however, one takes into account the reduction in bottom friction over the continental slope due to buoyancy forces in the bottom boundary layer, then both model and observed currents at 750 m are small. Although bottom friction is negligible on the continental slope, it should remain strong close to the coast where surface and bottom boundary layers overlap. Inclusion of this bottom friction causes the model interannual sea level to propagate poleward with speeds that vary from 0.5 m s−1 near the equator to 2.9 m s−1 at 37°S. At interannual timescales the phase differences associated with such propagation are small. Model and observed interannual sea levels at six locations along the coast are in basic agreement in the sense that for both the equatorial signal extends all along the South American coast and is nearly in phase. However, the phase differences were too small to check the poleward propagation with the available data. Because model sea level propagates offshore approximately like a Rossby wave at interannual frequencies, the model predicts that coastal sea level should lead the near-surface geostrophic equatorward flow by about 7 months. A lead time of about this size is consistent with coastal sea level and the 7.3-year-long 220-m depth coastal current observations off central Chile. Since the mean flow at 220-m depth is poleward, at about 7 months after the peak of an El Niño (high sea level) the poleward undercurrent is weaker and about 7 months after the peak of a La Niña (low sea level) it is stronger.

On the linear and nonlinear resistive tearing‐mode instabilities

The tearing‐mode instabilities of Harris sheet magnetic field configuration are studied based on the numerical simulations of two‐dimensional, resistive, compressible MHD models with isotropic pressure. The linear growth rate and the eigenmode structure are calculated from the linear numerical model for various parameters: magnetic Reynolds number, Rm; the ratio of wavelength to the layer thickness; magnetic By component; and plasma beta, as well as for different equations of state. The linear solutions are then used as initial perturbations of the nonlinear numerical model to allow the full evolution of resistive tearing‐mode instability. The linear calculations show that for the range of Rm = 10–105 the fastest growth rate increases with decreasing Rm and only slightly increases with increasing plasma beta and By but that it is not sensitive to the equation of state. While the nonlinear calculations snow that only for large Rm, where the diffusion time is much larger than the linear growth time, the magnetic island may possibly grow substantially and become saturated. For small Rm the plasmoids either diminish in the late stage or do not have apparent growing; that is, the linear analysis is not meaningful for large resistivity cases. The calculations are compared to the magnetic island structure at Earth's magnetopause reconstructed from the single‐spacecraft data by Hau and Sonnerup [1999].

Improved control algorithm for real‐time substructure testing

AbstractReal‐time substructure testing is a novel method of testing structures under dynamic loading. The complete structure is separated into two substructures, one of which is tested physically at large scale and in real time, so that time‐dependent non‐linear behaviour of the substructure is realistically represented. The second substructure represents the surrounding structure, which is modelled numerically. In the current formulation this numerical substructure is assumed to remain linear. The two substructures interact in real‐time so that the response of the complete structure, incorporating the non‐linear behaviour of the physical substructure, is accurately represented. This paper presents several improvements to the linear numerical modelling of substructures for use in explicit time‐stepping routines for real‐time substructure testing. An extrapolation of a first‐order‐hold discretization is used which increases the accuracy of the numerical model over more direct explicit methods. Additionally, an integral form of the equation of motion is used in order to reduce the effects of noise and to take into account variations of the input over a time‐step. In order to take advantage of this integral form, interpolation of the model output is performed in order to smooth the output. The improvements are demonstrated using a series of substructure tests on a simple portal frame. While the testing approach is suitable for cases in which the physical substructure behaves non‐linearly, the results presented here are for fully linear systems. This enables comparisons to be made with analytical solutions, as well as with the results of tests based on the central difference method. Copyright © 2001 John Wiley & Sons, Ltd.

Modeled Sverdrup flow in the North Atlantic from 11 different wind stress climatologies

Abstract In studies of large-scale ocean dynamics, often quoted values of Sverdrup transport are computed using the Hellerman–Rosenstein wind stress climatology. The Sverdrup solution varies, however, depending on the wind set used. We examine the differences in the large-scale upper ocean response to different surface momentum forcing fields for the North Atlantic Ocean by comparing the different Sverdrup interior/Munk western boundary layer solutions produced by a 1/16° linear numerical ocean model forced by 11 different wind stress climatologies. Significant differences in the results underscore the importance of careful selection of a wind set for Sverdrup transport calculation and for driving nonlinear models. This high-resolution modeling approach to solving the linear wind-driven ocean circulation problem is a convenient way to discern details of the Sverdrup flow and Munk western boundary layers in areas of complicated geometry such as the Caribbean and Bahamas. In addition, the linear solutions from a large number of wind sets provide a well-understood baseline oceanic response to wind stress forcing and thus, (1) insight into the dynamics of observed circulation features, by themselves and in conjunction with nonlinear models, and (2) insight into nonlinear model sensitivity to the choice of wind-forcing product. The wind stress products are evaluated and insight into the linear dynamics of specific ocean features is obtained by examining wind stress curl patterns in relation to the corresponding high-resolution linear solutions in conjunction with observational knowledge of the ocean circulation. In the Sverdrup/Munk solutions, the Gulf Stream pathway consists of two branches. One separates from the coast at the observed separation point, but penetrates due east in an unrealistic manner. The other, which overshoots the separation point at Cape Hatteras and continues to flow northward along the continental boundary, is required to balance the Sverdrup interior transport. A similar depiction of the Gulf Stream is commonly seen in the mean flow of nonlinear, eddy-resolving basin-scale models of the North Atlantic Ocean. An O (1) change from linear dynamics is required for realistic simulation of the Gulf Stream pathway. Nine of the eleven Sverdrup solutions have a C-shaped subtropical gyre, similar to what is seen in dynamic height contours derived from observations. Three mechanisms are identified that can contribute to this pattern in the Sverdrup transport contours. Along 27°N, several wind sets drive realistic total western boundary current transport (within 10% of observed) when a 14 Sv global thermohaline contribution is added (COADS, ECMWF 10 m re-analysis and operational, Hellerman–Rosenstein and National Centers for Environmental Prediction (NCEP) surface stress re-analysis), a few drive transport that is substantially too high (ECMWF 1000 mb re-analysis and operational and Isemer–Hasse) and Fleet Numerical Meteorology and Oceanography Center (FNMOC) surface stresses give linear transport that is slightly weaker than observed. However, higher order dynamics are required to explain the partitioning of this transport between the Florida Straits and just east of the Bahamas (minimal in the linear solutions vs. 5 Sv observed east of the Bahamas). Part of the Azores Current transport is explained by Sverdrup dynamics. So are the basic path of the North Atlantic Current (NAC) and the circulation features within the Intra-Americas Sea (IAS), when a linear rendition of the northward upper ocean return flow of the global thermohaline circulation is added in the form of a Munk western boundary layer.

Analysis of the stability modes of the non-rigid airship

AbstractThis paper describes and compares various analyses leading to the development of approximate models for the linear stability modes of the non-rigid airship. The progress of the analyses was variously dependent on assumptions made from the detailed scrutiny of linear numerical models for three airships. For each airship studied, the linear models were obtained from non-linear simulation models by linearising about a number of chosen trim speeds representative of a typical speed envelope. The decoupled linear models comprised the longitudinal and lateral-directional state equations of the neutrally buoyant airship, for speeds from the hover to 30m/sec. Since the fidelity of the earliest airship models was not known, the principal purpose of this paper is to re-visit the original analysis using a later airship model of known excellent fidelity.The longitudinal modes of the airship comprise the surge mode, the heave-pitch subsidence mode and the oscillatory pitch-incidence mode. The lateral-directional modes of the airship comprise the sideslip subsidence mode, the yaw subsidence mode and the oscillatory roll pendulum mode. Approximate models for these modes are derived and expressed in terms of concise aerodynamic stability derivatives. The mode characteristics are discussed, and the approximate models are compared with the actual airship modes over the typical airspeed envelope.

The role of sound waves in sea‐breeze initiation

AbstractThe dynamics of the pressure distribution associated with the sea‐breeze circulation is investigated. Analysis of hourly observations of surface pressure reveals that, on a typical sea‐breeze day, large surface‐pressure falls are recorded very far inland. Using a linear numerical model we show that this can only be understood when sound waves are taken into account. Sound waves are excited when air in the boundary layer over land expands due to diabatic heating. the numerical study reveals that vertically travelling sound waves induce a pressure increase through the entire atmosphere above the heated layer within a few minutes, while horizontally travelling sound waves induce a surface‐pressure decrease over land and a surface‐pressure increase over sea, with the resulting horizontal surface‐pressure gradient initiating the sea breeze. Because the signal travels inland at the speed of sound (about 300 m s‐1), points at a distance of more than 1000 km from the coast experience a surface‐pressure decrease within an hour after the initiation of diabatic heating over land. the pressure decrease near the earth's surface over land, as well as the pressure increase aloft, is in accord with the observations at mountain stations and the analyses of the observed daytime surface‐pressure changes in summer over the continent and adjacent seas. the implications of these findings for hydrostatic modelling of sea breezes are investigated. It appears that after one hour the hydrostatic sea breeze is 10% stronger than the non‐hydrostatic sea breeze, while the hydrostatic return current is about 20% weaker.

Momentum fluxes due to three‐dimensional gravity‐waves: Implications for measurements and numerical modelling

AbstractA three‐dimensional wave‐action equation for linear internal gravity‐waves is derived. When the basic‐state flow does not turn with height, this equation can be used to derive the three‐dimensional version of a theorem of Eliassen and Palm. This states that, for steady flows in the absence of dissipation, the vertical flux of both horizontal components of momentum is independent of height. When the basic‐state flow exhibits turning with height, analysis of the wave‐action equation indicates that, in general, the wave field cannot be regarded as steady and the three‐dimensional Eliassen and Palm theorem does not apply. It is also shown that the wave‐action equation has serious implications for the interpretation of atmospheric measurements of the momentum flux: in order to make accurate measurements of the momentum flux arising from three‐dimensional orographically forced gravitywaves, it is necessary to average ρmu'w' (where ρm is the basic‐state density and u' and w' are the horizontal and vertical wave‐induced velocity‐perturbations respectively) on horizontal planes downstream of the orography, rather than along lines in the downstream direction (as has previously been done with aircraft measurements). Using a three‐dimensional linear numerical model, we demonstrate the importance of this in even the most simple idealized cases.

A stochastic model of IndoPacific sea surface temperature anomalies

It is often desirable to represent a rapidly varying physical process as stochastic forcing of some slower dynamical system. A review of this approximation is presented. The white-noise limit of rapidly varying processes reduces the dynamical description of the affected slower systems to stochastic differential equations, the properties of which are summarized. As an application of stochastic differential equations, we review the evidence that IndoPacific sea surface temperature anomalies (SSTAs) can be represented as a stable linear process driven by spatially coherent stochastic forcing. An “inversemodeling” approach is used. That is, the relevant parameters of the best-fit stable linear process are obtained from observations and, given these parameters, the assumptions of stability and linearity are subsequently tested. An “optimal initial structure” for growth, predicted by the model, is found to occur approximately seven months before most of the major warm and cold (El Niño/La Niña) events in the data record. The optimal structure preferentially occurs in the boreal spring, with the mature phase of the extreme event occurring in the subsequent late fall/winter. The best model to fit the observations, including the dependence of the dynamical system on the annual cycle, is driven by stochastic forcing with periodic statistics. These statistics are inferred using a time-dependent fluctuation-dissipation relation. The variation of the stochastic forcing with the annual cycle is not the same as that of the optimal initial structure. Yet, when a linear numerical model is driven with the derived stochastic forcing, the optimal structure, including its variation with the annual cycle, is recovered. This implies that deterministic dynamics play an important role in setting up the optimal structure and, therefore, a mature phase of El Niño may be predicted before the optimal structure actually appears.

Sea level from temperature profiles in the tropical Pacific Ocean, 1975–1982

We investigate how well the 1975–1992 sea level interannual variability in the tropical Pacific is captured by dynamic height from temperature profiles. For each temperature profile, a surface dynamic height relative to 300 m is estimated, assuming a constant temperature‐salinity relationship. After multiplication by a latitudinally varying factor and the removal of a seasonal cycle, the dynamic height deviations fit the tide gauge sea level variability to within the sampling errors, except at a few sites near the equator west of the date line, where surface salinity variability is large. The dynamic height data are assimilated into a wind‐forced linear numerical model of the sea level in the tropical Pacific, applying a Kalman filter in a space of reduced dimension. A limited number of empirical orthogonal functions of the unfiltered run (1975–1992) define the reduced space, into which the Kalman Filter covariance evolution calculation is done [Cane et al., 1996]. Experiments indicate that results are better with 32 functions than with a smaller number but are not improved by retaining more functions. The resulting analyzed fields of sea level are compared to withheld dynamic height estimates from moorings, sea level data from tide gauges, and sea level analyses made with the same Kalman filter formalism applied to tide gauge measurements. The comparisons to observations suggest that the temperature profiles were usually sufficient to constrain the monthly analyzed fields to be close to the observed sea level with errors typically less than 3 cm near the equator. The comparison to tide gauge sea level reveals that this analysis is often more accurate than the analysis of tide gauge sea level data with which it shares many characteristics. Near the equator west of the date line, salinity variations are large and their neglect in estimating dynamic height has a negative impact on the analysis. The analyzed signal is underestimated in the southwest Pacific and at more than 20° off the equator. The reanalysis of the temperature data done with a primitive equation model at the National Meteorological Center (NMC) [Ji et al., 1995; Enfield and Harris, 1995] does not share this problem. On the other hand, NMC reanalysis (RA4) departs more from the observations elsewhere, although more data were included than in our analysis.

Lee waves over the English Lake District

AbstractThis paper presents the results of a field experiment which was conducted for three weeks during November 1991. The aim was to investigate the properties of internal gravity waves in the troposphere and lower stratosphere which were generated by the mountains of the Lake District in north‐west England, UK. The main technique used was that of multiple sounding using radiosondes.Throughout the experiment measurements obtained from groups of five radiosondes launched nearly simultaneously indicate that the waves are quasi‐steady and trapped within the troposphere. This is illustrated by a single detailed case study for 26 November in which the radiosonde ascent‐rate fluctuations are as large as 6.5 m s−1 peak to peak. These measurements are supported by aircraft observations.Simple techniques for analysing the structure of the wave‐field are presented and these include the calculation of a phase‐line tilt and phase speed and comparisons with simple analytical and numerical solutions for trapped gravity waves.A three‐dimensional linear numerical model is used to simulate the waves observed on 26 November and three other cases also. The model predictions are shown to compare well with the observations. This is particularly true in the mid‐troposphere, where the model reproduces both the positions and amplitudes of the peaks and troughs in the vertical‐velocity field.Spectral analysis of the local orography and examination of typical variations of the wind and buoyancy frequency with height show that gravity waves generated by the orography in the Lake District will frequently be trapped in the troposphere.

Investigation of the energy flows for transparent and nontransparent building facade

Based on extensive measurements of energy usage for a variety of building facades, a computer model has been developed for calculating in advance the heating and cooling requirements of test rooms equipped with arbitrary facades. To simplify the predictive calculation, the three-dimensional physical model was transformed to a one-dimensional, linear, computational model. The computer model uses the results of an earlier sensitivity analysis. This permits the precise numerical modeling of the measured results and gives special value to the adaptation of this model for describing the geometric and thermal relations investigated experimentally. The results of the numerical calculations are found to be in good agreement with the measurements for six test rooms. This agreement justifies the linear numerical model and the computer program for predicting the energy flows in different room structures. Further, the computer model not only provides results which follow the course of weather conditions for the test year but yields results as well for a Standard Reference Year for the region.

Alfvén waves in the auroral ionosphere: A numerical model compared with measurements

We solve a linear numerical model of Alfvén waves reflecting from the high‐latitude ionosphere, both to better understand the role of the ionosphere in the magnetosphere/ionosphere coupling process and to compare model results with in situ measurements. We use the model to compute the frequency‐dependent amplitude and phase relations between the meridional electric and the zonal magnetic fields due to Alfvén waves. These relations are compared with measurements taken by an auroral sounding rocket flown in the morningside oval and by the HILAT satellite traversing the oval at local noon. The sounding rocket's trajectory was mostly parallel to the auroral oval, and it measured enhanced fluctuating field energy in regions of electron precipitation. The rocket‐measured phase data are in excellent agreement with the Alfvén wave model, and the relation between the modeled and the measured amplitudes is fair, leading us to conclude that the rocket‐measured fields are dominated by interfering Alfvén waves, forming a standing wave pattern. The field amplitudes measured by HILAT are related by the height‐integrated Pedersen conductivity Σp, indicating that the measured field fluctuations were due mainly to structured field‐aligned current systems. A reason for the relative lack of Alfvén wave energy in the HILAT measurements could be the fact that the satellite traveled mostly perpendicular to the oval and therefore quickly traversed narrow regions of electron precipitation and associated wave activity. Alternatively, the lower velocity and eastward flight path of the rocket lead to Doppler‐shifted frequencies of L shell aligned current structures which are much smaller than HILAT would measure and which in fact may be below our range of interest (0.1–1 Hz), leaving Alfvén waves as the dominant source of fluctuating field energy in the rocket frame of reference.