Wavefront Refraction Research Articles

Effect of pharmacologic pupil dilation with tropicamide and phenylephrine on wavefront measurements

PurposeTo investigate the influence of tropicamide 1% (as a cycloplegic mydriatic) and phenylephrine 10% (as a noncycloplegic mydriatic) on mydriasis, wavefront refraction, and wavefront aberrations. MethodsIn this prospective study, 31 myopic eyes with a large mesopic pupil size were evaluated with an Allegretto Wave analyzer in a natural dilated state, after instillation of tropicamide 1% or phenylephrine 10%. Aberrations expressed as Zernike polynomials up to the sixth order were analyzed. Wavefront refractions were compared with subjective manifest refraction. ResultsBoth tropicamide and phenylephrine cause significant mydriasis (p < 0.001), but phenylephrine induced a larger pupil size than tropicamide under mesopic conditions (p = 0.029). Compared with the natural state, tropicamide induced a significant hyperopic shift in wavefront refraction (by +0.27 ± 0.09 D; p = 0.002). In contrast, wavefront refraction did not significantly change when using phenylephrine as the mydriatic (+0.03 ± 0.10 D; p = 0.75). Compared with the subjective manifest refraction, wavefront refraction before mydriatics and after phenylephrine showed a significant myopic shift (p < 0.0125), whereas the wavefront refraction after tropicamide was not significantly different from subjective refraction. Zernike coefficient C4 showed a less positive defocus after application of tropicamide (p = 0.0017). Other aberration coefficients of Zernike polynomials up to the sixth order did not change significantly from before to after tropicamide application. There was no significant difference in Zernike coefficients up to C27 before and after phenylephrine. ConclusionPhenylephrine preserves accommodation and provides a larger pupil under mesopic conditions, whereas tropicamide relaxes accommodation and provides an objective wavefront refraction that is closer to the subjective manifest refraction. Neither phenylephrine nor tropicamide causes a significant change in high-order aberrations from the natural state.

Comparisons of wavefront refraction, autorefraction, and subjective manifest refraction

ObjectivesTo compare cycloplegic wavefront refraction, autorefraction, and subjective manifest refraction. Materials and MethodsThirty-one myopic eyes in 17 patients were studied. Subjective manifest refraction was measured and deemed as the true refraction status. After inducing cycloplegia by administering 1% tropicamide, cycloplegic autorefraction was measured using a Topcon autorefractor, and wavefront refraction was measured with an Allegretto wave analyzer. Refraction data were presented as the spherical equivalent and astigmatism. Astigmatism was converted to vector power and analyzed by the Alpins method. ResultsBoth cycloplegic wavefront refraction and autorefraction showed good correlations with subjective refraction. The adjusted R2 value was 0.9726 between cycloplegic autorefraction and subjective manifest refraction, and 0.9693 between cycloplegic wavefront refraction and subjective manifest refraction. Compared with subjective manifest refraction, a myopic shift of −0.14 ± 0.06 D was noted in cycloplegic wavefront refraction (p = 0.0182). However, cycloplegic autorefraction was not different from subjective manifest refraction (p = 0.55). Astigmatism in both wavefront refraction and autorefraction differed from subjective astigmatism (p < 0.0001 for both). The difference of astigmatism vector power from subjective refraction was 0.16 D larger in the cycloplegic wavefront refraction group than in the cycloplegic autorefraction group (p = 0.0039). ConclusionAutorefraction gives a better estimate of subjective manifest refraction than wavefront refraction in both the spherical equivalent and astigmatism.

Combining Multimodal Wavefront Examination and Digital Refraction to Create a Rapid and Accurate Approach (XFraction) for a Total Visual System Assessment for Intraocular Lens Selection

Following advances in intraocular lens technology in recent years, greatly increased options are now available to the ophthalmic surgeon. However, there is a need for improved diagnostic technologies to optimize outcomes and select the best treatment strategies. A process combining multimodal wavefront examination and digital refraction, wavefront optimized refraction or ‘XFractionSM,’ has been recently developed to address this requirement. This novel fusion of techniques increases the amount of data that can be captured from each eye by orders of magnitude and is completed in much less time than former separate methods or manual refraction alone. The result is a process that enhances current procedures, provides superior patient diagnosis over previous techniques through the full optical pathway, and provides multiple benefits to both the patient and physician. The process enables the collection of a large amount of relevant information that makes available new levels of understanding about each patient’s unique visual system. Its use in clinical practice has been further associated with increased practice efficiency, enhanced optimized refractions, improved diagnostic surgical outcomes, and higher levels of patient satisfaction.

Multiple regression analysis in myopic wavefront laser in situ keratomileusis nomogram development

To develop a systematic method for quantifying pretreatment adjustments to the treatment sphere in patients having myopic wavefront laser in situ keratomileusis (LASIK). Moorfields Eye Hospital, London, United Kingdom. Comparative case series. Preoperative and 3-month postoperative data from consecutive cases of myopic wavefront LASIK (<10.0 diopters [D] sphere; <4.5 D cylinder) treated in 2008-2009 were tabulated. Multiple regression modeling was used to derive a weighted relationship between the achieved manifest refraction spherical equivalent (MRSE) and several possible influences. A new nomogram was derived from the resultant regression equation in which the percentage treatment boost was based on the partial correlation coefficient for the 4.0 mm pupil wavefront refraction spherical equivalent (SE), and pretreatment adjustment to the treatment sphere was based on the sum of the other weighted independent variables. The MRSE results in eyes treated using the new nomogram in 2009-2010 were compared with those from the original data set. The variance in postoperative MRSE error was significantly reduced using the new nomogram (0.10 D in 2009-2010 versus 0.16 D in 2008-2009; P<.0001, Bartlett test). The R(2) measure of linear fit increased from 0.967 in 2008-2009 to 0.983 in 2009-2010. Multiple regression modeling can be used to take account of several possible influences in enhancing the accuracy of wavefront LASIK. Nomograms developed through multiple regression modeling can be used to derive eye-specific pretreatment adjustments to the treatment sphere. Neither author has a financial or proprietary interest in any material or method mentioned.

Unbiased estimation of refractive state of aberrated eyes

We seek unbiased methods for estimating the target vergence required to maximize visual acuity based on wavefront aberration measurements. Experiments were designed to minimize the impact of confounding factors that have hampered previous research. Objective wavefront refractions and subjective acuity refractions were obtained for the same monochromatic wavelength. Accommodation and pupil fluctuations were eliminated by cycloplegia. Unbiased subjective refractions that maximize visual acuity for high contrast letters were performed with a computer controlled forced choice staircase procedure, using 0.125 diopter steps of defocus. All experiments were performed for two pupil diameters (3mm and 6mm). As reported in the literature, subjective refractive error does not change appreciably when the pupil dilates. For 3mm pupils most metrics yielded objective refractions that were about 0.1D more hyperopic than subjective acuity refractions. When pupil diameter increased to 6mm, this bias changed in the myopic direction and the variability between metrics also increased. These inaccuracies were small compared to the precision of the measurements, which implies that most metrics provided unbiased estimates of refractive state for medium and large pupils. Thus a variety of image quality metrics may be used to determine ocular refractive state for monochromatic (635nm) light, thereby achieving accurate results without the need for empirical correction factors.

Wavefront-guided refractive surgery after multifocal lens implantation

To examine whether wavefront-guided refractive surgery has a role in correcting ametropia after implantation of multifocal intraocular lenses (IOLs). This review begins with an oversight of various multifocal IOLs to help understand how a wavefront might return from the IOL and be captured by an aberrometer. The accuracy of the wavefront may be compromised in eyes with multifocal IOLs by the fact that such lenses provide simultaneous vision--that is, a simultaneous projection of in-focus and out-of-focus images of an object on the retina--and hence returning wavefronts theoretically could be captured in a similar manner by an aberrometer. However, in isolated small series, reproducible wavefront refractions--in which the manifest refraction closely approximates the wavefront refraction and good results after refractive surgery--are attained in eyes with diffractive multifocal IOLs. Small studies suggest that it is possible to perform wavefront-guided excimer light amplification by stimulated emission of radiation (LASER) to correct ametropia following multifocal lens implantation. Larger studies with longer follow-up are needed to determine whether eyes with multifocal lenses derive an added benefit from wavefront-guided refractive surgery over conventional surgery. These studies may help determine whether some IOLs are better suited for subsequent refractive surgery. The age of the patient may affect the outcome of refractive surgery after multifocal IOL implantation.

The Alvarez and Lohmann refractive lenses revisited

Alvarez and Lohmann lenses are variable focus optical devices based on lateral shifts of two lenses with cubic-type surfaces. I analyzed the optical performance of these types of lenses computing the first order optical properties (applying wavefront refraction and propagation) without the restriction of the thin lens approximation, and the spot diagram using a ray tracing algorithm. I proposed an analytic and numerical method to select the most optimum coefficients and the specific configuration of these lenses. The results show that Lohmann composite lens is slightly superior to Alvarez one because the overall thickness and optical aberrations are smaller.

Performance of a New Binocular Wavefront Aberrometer Based on a Self-imaging Diffractive Sensor

To compare the performance of the Ophthonix Z-View diffractive aberrometer with two different Hartmann-Shack aberrometers. The Ophthonix Z-View was compared with the Alcon LADARWave and VISX WaveScan using 4 model eyes and 68 human eyes. Comparisons using three fixed, single-surface model eyes each with a different wavefront pattern were used to determine higher order accuracy. Lower order accuracy, linearity, and higher order repeatability were tested with a fourth model eye with a movable retinal surface. Manifest refraction spherical equivalent and the calculated spherical equivalent refraction of the aberrometers were compared in 68 human eyes. The Z-View was more accurate with lower noise compared to the WaveScan and LADARWave systems for higher order measurements of the fixed model eyes. Total root-mean-square difference from surface topography derived average values for all model eye configurations were 0.48, 0.95, and 0.74 microm for the Z-View, WaveScan, and LADARWave aberrometers, respectively. Average inter-measurement standard deviations for the fixed model eyes were 0.024, 0.025, and 0.034 microm for the Z-View, WaveScan, and LADARWave aberrometers, respectively. Results were similar among the systems for measuring the movable retina surface model eye and comparing manifest refraction spherical equivalent of the patients. Data gathered using one variable and several fixed-parameter model eyes showed good correlation to predicted values for all of the aberrrometers with one exception. A significant difference was found in the measurement of one individual fixed model eye with one of the three aberrometers. The wavefront refraction provided by the Z-View correlated well with the results of manifest refraction.

Análise comparativa entre a refração clínica subjetiva e a automatizada obtida por sensor de frentes de onda

To evaluate and compare refractive errors obtained through clinical subjective and automatized wavefront refraction analyses in eyes under cycloplegia. Prospective study of 147 patients, 279 eyes, undergoing preoperative examination for refractive surgery. Clinical subjective refraction was performed first followed by wavefront refraction. Results on astigmatism obtained from refraction were decomposed in power vectors for statistical analyses. Data were first analyzed in one eye and then in both eyes. The mean difference between clinical subjective refraction and automatized wavefront refraction on cycloplegic eyes was of -0.19 SD combined with -0.06 CD in the 15 degrees axis for data in one eye, and -0.17 SD combined with -0.05 CD in the 3 masculine axis for data in both eyes of the same patient. In the present study clinical subjective refraction and automatized wavefront refraction under cycloplegia had similar numerical values.

The Precision of Wavefront Refraction Compared to Subjective Refraction and Autorefraction

To determine the precision (repeatability) of several methods of calculating refraction from higher-order wavefront aberration data and to compare these wavefront refractions with lower-order (LO) wavefront refraction, subjective refraction, and autorefraction. Four clinicians refracted 16 normal participants aged 23.6 +/- 1.2 years, 69% female with an average spherical equivalent refractive error of -3.03 +/- 2.55 D, median sphere -2.50 D (minimum -7.50, maximum +4.75), and median cylinder -0.50 D (minimum -3.00, maximum 0). Participants were cyclopleged and underwent subjective refraction, autorefraction on two machines (Nidek AR-800, Topcon KR-8000), and wavefront sensing using the Wavefront Sciences Complete Ophthalmic Analysis System. Wavefront error was used to calculate: LO refraction, refractions that incorporated higher-order spherical and astigmatism terms from up to the 4th, 6th, and 10th orders (PCM4, PCM6, and PCM10), and a method based on optimizing image quality metrics [wavefront analysis technology (WAT) refraction]. Within and between examiner agreements for total dioptric difference were determined using Bland-Altman limits of agreement (LOA). The interexaminer LOA for individual measurements for M, J0, J45 were: Topcon (+/-0.18, +/-0.10, +/-0.06), Nidek (+/-0.28, +/-0.16, +/-0.09), LO (+/-0.17, +/-0.10, +/-0.06), PCM4 (+/-0.26, +/-0.09, +/-0.06), PCM6 (+/-0.37, +/-0.17, +/-0.34), PCM10 (+/-0.54, +/-0.32, +/-0.40), WAT (+/-0.28, +/-0.20, +/-0.15), and subjective refraction (+/-0.48, +/-0.20, +/-0.13) and averaging across three measures LOA: Topcon (+/-0.15, +/-0.08, +/-0.05), Nidek (+/-0.21, +/-0.13, +/-0.07), LO (+/-0.12, +/-0.06, +/-0.04), PCM4 (+/-0.16, +/-0.05, +/-0.04), PCM6 (+/-0.23, +/-0.09, +/-0.19), PCM10 (+/-0.29, +/-0.19, +/-0.24), and WAT (+/-0.18, +/-0.12, +/-0.10). The within-examiner LOA for M, J0, J45 were: Topcon (+/-0.08, +/-0.04, +/-0.02), Nidek (+/-0.13, +/-0.07, +/-0.05 D), LO (+/-0.11, +/-0.07, +/-0.04), PCM4 (+/-0.17, +/-0.07, +/-0.04 D), PCM6 (+/-0.28, +/-0.12, +/-0.24 D), PCM10 (+/-0.42, +/-0.24, +/-0.32 D), and WAT (+/-0.19, +/-0.14, +/-0.09 D). All objective refractions except for PCM10 were more repeatable across clinicians than subjective refraction. The precision of all refractions were improved by an expected amount through averaging over multiple measurements. Wavefront refractions were not as precise as standard autorefractions, although not clinically significantly worse.

The Effect of Phenylephrine and Cyclopentolate on Objective Wavefront Measurements

To investigate the impact of phenylephrine and cyclopentolate on wavefront refraction and fourth order spherical aberration C12. This cohort study comprised 151 eyes with sphere up to -10.00 diopters (D) and cylinder -3.75 D. Aberrometry was performed using the ALLEGRO WAVE (WaveLight Laser Technologies AG, Erlangen, Germany) after instillation of phenylephrine 5% yielding objective phenylephrine refraction in accommodated steady-state, as well as after cyclopentolate 0.5% providing objective cyclopentolate refraction in non-accommodated state. Accommodation target fogging was turned off. Wavefront aberrations were expressed by Zernike expansion up to the sixth order, and paraxial curvature matching with Taylor series was used to calculate objective wavefront sphere. Objective wavefront sphere was not influenced by pupil size. Eyes showed substantial accommodation after phenylephrine with a myopic shift of -0.66 D comparing objective to subjective manifest sphere (r=0.942, P<.001). Cycloplegic eyes behaved like a model eye, with a difference of -0.08 D between objective and subjective cycloplegic sphere (r=0.976, P<.001). C12 increased ten-fold from 4.0- to 7.0-mm pupil size, keeping the same sign. Comparing cyclopentolate with phenylephrine, the sign of C12 changed in a positive direction by an average +0.124 +/- 0.109 microm (range: -0.052 to +0.632 microm) at 7.0 mm, whereas the total higher order aberrations changed very little. A good correlation was found between C12 and the change in objective wavefront sphere between cyclopentolate and phenylephrine (r=0.75, P<.001). Fogging of the accommodation target should be used for wavefront measurements. Weaker cycloplegic agents, such as tropicamide, may be used to ensure relaxed but not completely paralyzed accommodation, which would yield "manifest" aberration values close to the natural resting state.

Accuracy, Repeatability, and Clinical Application of Spherocylindrical Automated Refraction Using Time-Based Wavefront Aberrometry Measurements

To evaluate the clinical utility of spherocylindrical automated refraction (AR) based on time-based wavefront technology compared with subjective manifest refraction (MR) for the purpose of prescribing eyeglasses. Prospective observational case series. A convenience sample of 105 eyes of 53 patients (ages 19 to 87) with corrected acuity no worse than 20/40(-2). Manifest refraction was performed on all subjects followed by three consecutive wavefront measurements at both a 4-mm and 6-mm optical zone using the 3D Wave OPD Scan ARK 10,000 aberrometer. Repeatability of multiple consecutive ARs; difference in the sphere, cylinder, and axis of AR compared with MR; difference between the corrected visual acuity based on AR- and MR-derived acuity. Repeatability analysis showed that the initial wavefront refraction was not significantly different from the mean of the three refractions. At the 4-mm zone, the absolute value (+/- standard deviation) of the difference between the initial AR compared with the MR was 0.34+/-0.28 diopters (D) for sphere, 0.22+/-0.21 D for cylinder, and 12.5+/-15.8 degrees for axis. At the 6-mm zone, these values were 0.32+/-0.29 D for sphere, 0.25+/-0.24 D for cylinder, and 11.8+/-14.6 degrees for axis. When 4-mm AR visual acuity was superior to MR acuity (19.0% of eyes) it was 0.28+/-0.24 Snellen lines better, and when inferior (53.3% of eyes) it was 0.47+/-0.37 lines worse. When 6-mm AR acuity was superior to MR acuity (16.2% of eyes) it was 0.32+/-0.22 Snellen lines better, and when inferior (60.0% of eyes) it was 0.48+/-0.48 lines worse. The absolute value of the difference in spherical equivalent of the first AR measurement at the 4-mm zone compared with the MR value was 0.33+/-0.23 D, and at the 6-mm zone it was 0.30+/-0.27 D. The wavefront-derived AR values reflect the physical optical system, but not necessarily subjectively derived values, and the impact of this disagreement on patient satisfaction is unknown. Future studies evaluating this technology and its role in clinical ophthalmology are warranted.

Accuracy of the WASCA Aberrometer Refraction Compared to Manifest Refraction in Myopia

To evaluate the accuracy of myopic refraction by a single measurement using the Wavefront Supported Custom Ablation (WASCA) aberrometer (Carl Zeiss Meditec AG, Jena, Germany). We retrospectively compared the refractive errors obtained by manifest refraction and wavefront refraction (WASCA) in 50 eyes of 25 consecutive myopic patients undergoing laser refractive surgery. The sphere ranged from -1.00 to -8.25 diopters (D) and cylinder from 0 to -3.75 D. WASCA measurements under cycloplegia were made and WASCA refractions calculated for a 6-mm analysis zone using the Seidel method within the WASCA. We used the manifest refraction as our best estimate of the true refractive error, therefore accuracy was defined as the difference between manifest refraction and that of the WASCA. Correlation coefficients and mean vector errors between manifest and WASCA refraction were calculated. High correlation was shown between manifest and WASCA refractions, with correlation coefficients (R2) of 0.97, 0.85, and 0.79 for M, J180, and J45, respectively. Mean power vector error (standard deviation) was 0.22 D (0.39), +0.03 D (0.21), and +0.03 D (0.13) for M, J180, and J45, respectively. Total dioptric power vector error was 0.43 D with 74% eyes within 0.50 D. When measuring normal myopic eyes, the concordance between manifest and WASCA refractions was found on average to be high; however, outlier measurements occurred.

Dependence of Wave Front Refraction on Pupil Size Due to the Presence of Higher Order Aberrations

Propagation of light through the optical pathway within the eye can lead to a deformation of the wave front that might affect objective but also subjective refraction depending on pupil size. The aim of this study was to investigate the change in wave front refraction that is calculated on the basis of second order Zernike polynomials when varying the pupil size from 6 mm to 3 mm. The change was correlated with the amount of fourth and sixth order spherical aberration and fourth and sixth order astigmatism. Wave front aberrations were measured in 130 eyes by means of a Tscherning wave front sensor at a pupil size of 6 mm. Wave front aberrations in terms of Zernike coefficients up to sixth order were approximated for 6 mm and 3 mm pupil size. The wave front refraction was calculated based on the second order Zernike coefficients for both pupil diameters. Resulting differences in wave front refraction (sphere or cylinder) due to the change in pupil size were correlated with the initial higher order aberrations determined for the 6.0 mm pupil by means of a linear regression (Spearman rank correlation coefficient). The correlation between the change in sphere and cylinder on one hand and the spherical aberration and higher order astigmatism on the other hand was found to be highly significant (p<0.001), with a correlation coefficient of R = 0.96 for sphere and R = 0.85 for cylinder. Calculating the wave front refraction on the basis of second order Zernike polynomials is plagued with the influence of the higher order aberration preexisting in the individual eye. This is one reason why this method does not represent precisely enough subjective refraction. Other methods that calculate the refraction based on wave front measurement independent from the pupil size should be established in the ophthalmic community.

Unresolved Issues in the Prediction of Subjective Refraction From Wavefront Aberration Maps

Sources of uncertainty in the conversion of wavefront aberration maps to refractive prescriptions are described. These uncertainties highlight the nature of optimum corrections, the imprecise far point in human eyes, multiple strategies for locating an approximate far point, and unresolved aspects of the gold standard for judging success. The conclusion points to future experiments needed to resolve these uncertainties to achieve more accurate and precise methods of objective wavefront refraction.

Wavefront analysis in post-LASIK eyes and its correlation with visual symptoms, refraction, and topography

Purpose To evaluate the information assessed with the LADARWave wavefront measurement device and correlate it with visual symptoms, refraction, and corneal topography in previously LASIK-treated eyes. Participants One hundred five eyes (58 patients) of individuals who underwent LASIK surgery were evaluated. Design Retrospective, noncomparative case series. Main outcome measures Complete ophthalmologic examination, corneal topography, and wavefront measurements were performed. Correlations were made between the examinations and symptoms. Methods Wavefront measurements were assessed with the LADARWave device. Manifest, cycloplegic refraction, and topographic data were compared with wavefront refraction and higher order aberrations. Visual symptoms were correlated to higher order aberrations in 3 different pupil sizes (5-mm, 7-mm, and scotopic pupil size). Pearson's correlation coefficient and generalized estimating equations were used for statistical analysis. Results In post-LASIK eyes, wavefront refraction components were poorly correlated to manifest and cycloplegic components. The comparison between manifest, cycloplegic, and wavefront refraction with total amount of higher order aberrations showed no strong correlation. The comparison between topography and manifest, cycloplegic, and wavefront refraction did not show strong correlation. Visual symptoms analysis showed correlation of double vision with total coma and with horizontal coma for the 5-mm and 7-mm pupil size; correlation between starburst and total coma for the 7-mm pupil size; and correlation of double vision with horizontal coma, glare with spherical aberrations and with total aberrations, and starburst with spherical aberrations for the scotopic pupil size. Scotopic pupil size had a positive association with starburst and a negative association with double vision. Conclusions The LADARWave wavefront measurement device is a valuable diagnostic tool in measuring refractive error with ocular aberrations in post-LASIK eyes. A strong correlation between visual symptoms and ocular aberrations, such as monocular diplopia with coma and starburst and glare with spherical aberration, suggest this device is valuable in diagnosing symptomatic LASIK-induced aberrations. Horizontal coma was correlated with double vision, whereas vertical coma was not.

Intraoperative wavefront monitoring during laser thermal keratoplasty.

Low energy laser thermal keratoplasty (LTK) is being proposed as an alternative to conventional holmium LTK in an effort to minimize corneal tissue necrosis, regression, and induced astigmatism. Real-time wavefront monitoring during surgery is also being proposed as a method for regulating induced aberrations and the treatment end point. We evaluated the feasibility of intraoperative wavefront monitoring during low energy LTK. An investigational prototype system combining a Sunrise Hyperion laser with a Shack-Hartmann wavefront sensor (Wavefront Sciences COAS) was used to treat 20 hyperopic eyes with a low energy application of eight spots of 18mJ/shot (144mJ/pulse) at 5 Hz in two rings of 6 mm and 7-mm diameter, while monitoring the treatment with real-time dynamic wavefront sensing up to 10 captures/second of 4th order Zernike data. The integrated wavefront-monitored LTK unit showed good stability in measuring dynamic intraoperative wavefront refractions before, during, and after the low energy LTK treatment. The readings did not interfere with the application of holmium laser pulses, nor did the LTK treatment distort the captured wavefront readings. A steady improvement of the wavefront-derived spherical equivalent refraction followed the laser treatment pattern, with a mean attempted spherical equivalent refractive change of 1.66 D and mean achieved change of 1.71 D. Intraoperative wavefront monitoring was feasible during low energy LTK. With future adaptation, it might be possible to dynamically relay the captured wavefront information to the laser during ongoing treatment, to intraoperatively control the surgical endpoint for multiple aberration terms.

Two‐dimensional simulations of magnetic pulsations upstream of the Earth's bow shock

The development of turbulence upstream of the quasi‐parallel portion of the Earth's bow shock is investigated using two‐dimensional hybrid simulations involving the injection of a very hot ion beam against a cold incident ion flow. Initially, waves are produced in the right‐hand resonant ion beam mode and with wave vectors mainly parallel to the ambient magnetic field B0 as expected in linear theory. The waves attempt to propagate upstream but are convected back toward the shock by the incident flow and strongly grow in amplitude as they encounter larger beam densities. Wave growth is associated with reduction of both incident and beam ion bulk velocities. Differential slowing of the incident flow between the different wave fronts and, when the angle θBn between B0 and the shock normal n is non zero, along the fronts themselves, leads to wave front refraction parallel to the shock and fragmentation into smaller structures. Simultaneously, strong scattering by the waves generates beam ion clumps containing a significant fraction of ions with velocities antiparallel to the beam bulk velocity, which enables the destabilization of the left‐hand resonant ion beam mode. The beam ions ultimately form a diffuse distribution whose density ratio relative to the solar wind is a few percents in average, but increases up to about 20% in the immediate vicinity of the shock. These processes result in the formation of magnetic pulsations whose extension parallel and perpendicular to n is typically 15 and a few tens of ion inertial lengths, respectively. These structures can reach total amplitudes δB of 3 to 4 B0, and exhibit most of the time a left‐handed polarization in the plasma rest frame, in agreement with spacecraft observations upstream of the Earth's bow shock. The simulations show that pulsations induce slowing and moderate heating of the incident flow and play an active role in the quasi‐parallel shock transition.