Meridional Flow Profile Research Articles

Assessing the Observability of Deep Meridional Flow Cells in the Solar Interior

Meridional circulation regulates the Sun’s interior dynamics and magnetism. While it is well accepted that meridional flows are poleward at the Sun’s surface, helioseismic observations have yet to provide a definitive answer for the depth at which those flows return to the equator, or the number of circulation cells in depth. Here, we explore the observability of multiple circulation cells stacked in radius. Specifically, we examine the seismic signature of several meridional flow profiles by convolving time–distance averaging kernels with mean flows obtained from a suite of 3D hydrodynamic simulations. At mid and high latitudes, we find that weak flow structures in the deep convection zone can be obscured by signals from the much stronger surface flows. This contamination of 1–2 m s−1 is caused by extended side lobes in the averaging kernels, which produce a spurious equatorward signal with flow speeds that are 1 order of magnitude stronger than the original flow speeds in the simulations. At low latitudes, the flows in the deep layers of the simulations are stronger (>2 m s−1) and multiple cells across the convection zone can produce a sufficiently strong signal to survive the convolution process. Now that meridional flows can be measured over two decades of data, the uncertainties arising from convective noise have fallen to a level where they are comparable in magnitude to the systematic biases caused by nonlocal features in the averaging kernels. Hence, these systematic errors are beginning to influence current helioseismic deductions and need broader consideration.

A Babcock-Leighton dynamo model of the Sun incorporating toroidal flux loss and the helioseismically inferred meridional flow

Context. Key elements of the Babcock-Leighton model for the solar dynamo are increasingly constrained by observations. Aims. We investigate whether the Babcock-Leighton flux-transport dynamo model remains in agreement with observations if the meridional flow profile is taken from helioseismic inversions. Additionally, we investigate the effect of the loss of toroidal flux through the solar surface. Methods. We employ the two-dimensional flux-transport Babcock-Leighton dynamo framework. We use the helioseismically inferred meridional flow profile, and include toroidal flux loss in a way that is consistent with the amount of poloidal flux generated by Joy’s law. Our model does not impose a preference for emergences at low latitudes; however, we do require that the model produces such a preference. Results. We can find solutions that are in general agreement with observations, including the latitudinal migration of the butterfly wings and the 11 year period of the cycle. The most important free parameters in the model are the depth to which the radial turbulent pumping extends and the turbulent diffusivity in the lower half of the convection zone. We find that the pumping needs to extend to depths of about 0.80 R⊙ and that the bulk turbulent diffusivity needs to be around 10 km2 s−1 or less. We find that the emergences are restricted to low latitudes without the need to impose such a preference. Conclusions. The flux-transport Babcock-Leighton model, incorporating the helioseismically inferred meridional flow and toroidal field loss term, is compatible with the properties of the observed butterfly diagram and with the observed toroidal loss rate. Reasonably tight constraints are placed on the remaining free parameters. The pumping needs to be just below the depth corresponding to the location where the meridional flow changes direction, and where numerical simulations suggest the convection zone becomes marginally subadiabatic. However, our linear model does not reproduce the observed ‘rush to the poles’ of the diffuse surface radial field resulting from the decay of sunspots; reproducing this might require the imposition of a preference for flux to emerge near the equator.

A Potential New Mechanism for the Butterfly Diagram of the Solar Cycle: Latitude-dependent Radial Flux Transport

Abstract The butterfly diagram of the solar cycle is the equatorward migration of the emergence latitudes of sunspots as the solar cycle evolves. Revealing the mechanism for the butterfly diagram is essential for understanding the solar and stellar dynamo. The equatorward meridional flow at the base of the convection zone (CZ) was believed to be responsible for the butterfly diagram. However, helioseismological studies indicate controversial forms of the flow, and even present poleward flow at the base of the CZ, which poses a big challenge to the widely accepted mechanism. This motivates us to propose a new mechanism in this study. Using a data-driven Babcock–Leighton–type dynamo model, we carry out numerical simulations to explore how the latitude-dependent radial flux transport affects the latitudinal migration of the toroidal field, under different meridional flow profiles. The results indicate that when the radial transport of the poloidal field at higher latitudes is sufficiently faster, the toroidal fields of a new cycle at higher latitudes are generated earlier than that at lower latitudes, and vice versa. Thus, the butterfly diagram is suggested to correspond to the time- and latitude-dependent regeneration of the toroidal field due to the latitude-dependent radial transport of the poloidal flux.

Leptocline as a shallow substructure of near-surface shear layer in 3D radiative hydrodynamic simulations

ABSTRACT Understanding the effects driven by rotation in the solar convection zone is essential for many problems related to solar activity, such as the formation of differential rotation, meridional circulation, and others. We analyse realistic 3D radiative hydrodynamics simulations of solar subsurface dynamics in the presence of rotation in a local domain 80 Mm wide and 25 Mm deep, located at 30° latitude. The simulation results reveal the development of a shallow 10 Mm deep substructure of the near-surface shear layer (NSSL), characterized by a strong radial rotational gradient and self-organized meridional flows. This shallow layer (‘leptocline’) is located in the hydrogen ionization zone associated with enhanced anisotropic overshooting-type flows into a less unstable layer between the H and He ii ionization zones. We discuss current observational evidence of the presence of the leptocline and show that the radial variations of the differential rotation and meridional flow profiles obtained from the simulations in this layer qualitatively agree with helioseismic observations.

Optimization of surface flux transport models for the solar polar magnetic field

Context. The choice of free parameters in surface flux transport (SFT) models describing the evolution of the large-scale poloidal magnetic field of the Sun is critical for the correct reproduction of the polar magnetic flux built up during a solar cycle, which is known to be a good predictor of the amplitude of the upcoming cycle. Aims. For an informed choice of parameters it is important to understand the effects of and interplay among the various parameters and to optimize the models for the polar magnetic field. Methods. Here we present the results of a large-scale systematic study of the parameter space in an SFT model where the source term representing the net effect of tilted flux emergence was chosen to represent a typical, average solar cycle as described by observations. Results. Comparing the results with observational constraints on the spatiotemporal variation of the polar magnetic field, as seen in magnetograms for the last four solar cycles, we mark allowed and excluded regions in the 3D parameter space defined by the flow amplitude u0, the magnetic diffusivity η and the decay time scale τ, for three different assumed meridional flow profiles. Conclusions. Without a significant decay term in the SFT equation (i.e., for τ > 10 yr) the global dipole moment reverses too late in the cycle for all flow profiles and parameters, providing independent supporting evidence for the need of a decay term, even in the case of identical cycles. An allowed domain is found to exist for τ values in the 5–10 yr range for all flow profiles considered. Generally higher values of η (500–800 km2 s−1) are preferred though some solutions with lower η are still allowed.

Parameter optimization for surface flux transport models

Accurate prediction of solar activity calls for precise calibration of solar cycle models. Consequently we aim to find optimal parameters for models which describe the physical processes on the solar surface, which in turn act as proxies for what occurs in the interior and provide source terms for coronal models. We use a genetic algorithm to optimize surface flux transport models using National Solar Observatory (NSO) magnetogram data for Solar Cycle 23. This is applied to both a 1D model that inserts new magnetic flux in the form of idealized bipolar magnetic regions, and also to a 2D model that assimilates specific shapes of real active regions. The genetic algorithm searches for parameter sets (meridional flow speed and profile, supergranular diffusivity, initial magnetic field, and radial decay time) that produce the best fit between observed and simulated butterfly diagrams, weighted by a latitude-dependent error structure which reflects uncertainty in observations. Due to the easily adaptable nature of the 2D model, the optimization process is repeated for Cycles 21, 22, and 24 in order to analyse cycle-to-cycle variation of the optimal solution. We find that the ranges and optimal solutions for the various regimes are in reasonable agreement with results from the literature, both theoretical and observational. The optimal meridional flow profiles for each regime are almost entirely within observational bounds determined by magnetic feature tracking, with the 2D model being able to accommodate the mean observed profile more successfully. Differences between models appear to be important in deciding values for the diffusive and decay terms. In like fashion, differences in the behaviours of different solar cycles lead to contrasts in parameters defining the meridional flow and initial field strength.

Validation of Spherical Born Approximation Sensitivity Functions for Measuring Deep Solar Meridional Flow

Abstract Accurate measurements of deep solar meridional flow are of vital interest for understanding the solar dynamo. In this paper, we validate a recently developed method for obtaining sensitivity functions (kernels) for travel-time measurements to solar interior flows using the Born approximation in spherical geometry, which is expected to be more accurate than the classical ray approximation. Furthermore, we develop a numerical approach to efficiently compute a large number of kernels based on the separability of the eigenfunctions into their horizontal and radial dependence. The validation is performed using a hydrodynamic simulation of linear wave propagation in the Sun, which includes a standard single-cell meridional flow profile. We show that, using the Born approximation, it is possible to accurately model observational quantities relevant for time–distance helioseismology such as the mean power spectrum, disk-averaged cross-covariance functions, and travel times in the presence of a flow field. In order to closely match the model to observations, we show that it is beneficial to use mode frequencies and damping rates that were extracted from the measured power spectrum. Furthermore, the contribution of the radial flow to the total travel time is found to reach 20% of the contribution of the horizontal flow at travel distances over 40°. Using the Born kernels and a 2D SOLA inversion of travel times, we can recover most features of the input meridional flow profile. The Born approximation is thus a promising method for inferring large-scale solar interior flows.

Predicting the amplitude and hemispheric asymmetry of solar cycle 25 with surface flux transport

AbstractEvidence strongly indicates that the strength of the Sun's polar fields near the time of a sunspot cycle minimum determines the strength of the following solar activity cycle. We use our Advective Flux Transport code, with flows well constrained by observations, to simulate the evolution of the Sun's polar magnetic fields from early 2016 to the end of 2019—near the expected time of cycle 24/25 minimum. We run a series of simulations in which the uncertain conditions (convective motion details, active region tilt, and meridional flow profile) are varied within expected ranges. We find that the average strength of the polar fields near the end of cycle 24 will be similar to that measured near the end of cycle 23, indicating that cycle 25 will be similar in strength to the current cycle. In all cases the polar fields are asymmetric with fields in the south stronger than those in the north. This asymmetry would be more pronounced if not for the predicted weakening of the southern polar fields in late 2016 and through 2017. After just 4 years of simulation the variability across our ensemble indicates an accumulated uncertainty of about 15%. This accumulated uncertainty arises from stochastic variations in the convective motion details, the active region tilt, and changes in the meridional flow profile. These variations limit the ultimate predictability of the solar cycle.

A COUPLED 2 × 2D BABCOCK–LEIGHTON SOLAR DYNAMO MODEL. I. SURFACE MAGNETIC FLUX EVOLUTION

The need for reliable predictions of the solar activity cycle motivates the development of dynamo models incorporating a representation of surface processes sufficiently detailed to allow assimilation of magnetographic data. In this series of papers we present one such dynamo model, and document its behavior and properties. This first paper focuses on one of the model's key components, namely surface magnetic flux evolution. Using a genetic algorithm, we obtain best-fit parameters of the transport model by least-squares minimization of the differences between the associated synthetic synoptic magnetogram and real magnetographic data for activity cycle 21. Our fitting procedure also returns Monte Carlo-like error estimates. We show that the range of acceptable surface meridional flow profiles is in good agreement with Doppler measurements, even though the latter are not used in the fitting process. Using a synthetic database of bipolar magnetic region (BMR) emergences reproducing the statistical properties of observed emergences, we also ascertain the sensitivity of global cycle properties, such as the strength of the dipole moment and timing of polarity reversal, to distinct realizations of BMR emergence, and on this basis argue that this stochasticity represents a primary source of uncertainty for predicting solar cycle characteristics.

MERIDIONAL CIRCULATION IN SOLAR AND STELLAR CONVECTION ZONES

We present a series of 3-D nonlinear simulations of solar-like convection, carried out using the Anelastic Spherical Harmonic (ASH) code, that are designed to isolate those processes that drive and shape meridional circulations within stellar convection zones. These simulations have been constructed so as to span the transition between solar-like differential rotation (fast equator/slow poles) and ``anti-solar' differential rotation (slow equator/fast poles). Solar-like states of differential rotation, arising when convection is rotationally constrained, are characterized by a very different convective Reynolds stress than anti-solar regimes, wherein convection only weakly senses the Coriolis force. We find that the angular momentum transport by convective Reynolds stress plays a central role in establishing the meridional flow profiles in these simulations. We find that the transition from single-celled to multi-celled meridional circulation profiles in strong and weak regimes of rotational constraint is linked to a change in the convective Reynolds stress, a clear demonstration of gyroscopic pumping. Latitudinal thermal variations differ between these different regimes, with those in the solar-like regime conspiring to suppress a single cell of meridional circulation, whereas the cool poles and warm equator established in the anti-solar states tend to promote single-celled circulations. Though the convective angular momentum transport becomes radially inward at mid-latitudes in anti-solar regimes, it is the meridional circulation that is primarily responsible for establishing a rapidly-rotating pole. We conclude with a discussion of how these results relate to the Sun, and suggest that the Sun may lie near the transition between rapidly-rotating and slowly-rotating regimes.

Ensemble Kalman filter data assimilation in a Babcock‐Leighton solar dynamo model: An observation system simulation experiment for reconstructing meridional flow speed

AbstractAccurate knowledge of time variation in meridional flow speed and profile is crucial for estimating the solar cycle's features, which are ultimately responsible for causing space climate variations. However, no consensus has been reached yet about the Sun's meridional circulation pattern observations and theories. By implementing an ensemble Kalman filter (EnKF) data assimilation in a Babcock‐Leighton solar dynamo model using Data Assimilation Research Testbed framework, we find that the best reconstruction of time variation in meridional flow speed can be obtained when 10 or more observations are used with an updating time of 15 days and a ≤10% observational error. Increasing ensemble size from 16 to 160 improves reconstruction. Comparison of reconstructed flow speed with “true state” reveals that EnKF data assimilation is very powerful for reconstructing meridional flow speeds and suggests that it can be implemented for reconstructing spatiotemporal patterns of meridional circulation.

The solar meridional circulation and sunspot cycle variability

AbstractWe have measured the meridional motions of the magnetic elements in the Sun's surface layers since 1996 and find systematic and substantial variations. In general the meridional flow speed is fast at cycle minima and slow at cycle maxima. We find that these systematic variations are characterized by a weakening of the meridional flow on the poleward sides of the active (sunspot) latitudes. This can be interpreted as an inflow toward the sunspot zones superimposed on a more general poleward meridional flow profile. We also find variations in the meridional flow which vary from cycle to cycle. The meridional flow was slower at both the minimum and maximum of cycle 23 compared to similar phases of cycles 21, 22, and 24. Models of the magnetic flux transport by a variable meridional flow suggest that it can significantly modulate the size and timing of the following sunspot cycle through its impact on the Sun's polar magnetic fields. We suggest that the meridional flow variations observed in cycle 23 contributed to the weak polar fields at the end of the cycle which then produced a weak cycle 24 and the extraordinary cycle 23/24 minimum.

DETECTION OF EQUATORWARD MERIDIONAL FLOW AND EVIDENCE OF DOUBLE-CELL MERIDIONAL CIRCULATION INSIDE THE SUN

Meridional flow in the solar interior plays an important role in redistributing angular momentum and transporting magnetic flux inside the Sun. Although it has long been recognized that the meridional flow is predominantly poleward at the Sun's surface and in its shallow interior, the location of the equatorward return flow and the meridional flow profile in the deeper interior remain unclear. Using the first two years of continuous helioseismology observations from the Solar Dynamics Observatory / Helioseismic Magnetic Imager, we analyze travel times of acoustic waves that propagate through different depths of the solar interior carrying information about the solar interior dynamics. After removing a systematic center-to-limb effect in the helioseismic measurements and performing inversions for flow speed, we find that the poleward meridional flow of a speed of 15 m/s extends in depth from the photosphere to about 0.91 R_sun. An equatorward flow of a speed of 10 m/s is found between 0.82 to 0.91 R_sun in the middle of the convection zone. Our analysis also shows evidence of that the meridional flow turns poleward again below 0.82 R_sun, indicating an existence of a second meridional circulation cell below the shallower one. This double-cell meridional circulation profile with an equatorward flow shallower than previously thought suggests a rethinking of how magnetic field is generated and redistributed inside the Sun.

MAGNETOHYDRODYNAMIC SIMULATION-DRIVEN KINEMATIC MEAN FIELD MODEL OF THE SOLAR CYCLE

We construct a series of kinematic axisymmetric mean-field dynamo models operating in the αΩ, α2Ω and α2 regimes, all using the full α-tensor extracted from a global magnetohydrodynamical simulation of solar convection producing large-scale magnetic fields undergoing solar-like cyclic polarity reversals. We also include an internal differential rotation profile produced in a purely hydrodynamical parent simulation of solar convection, and a simple meridional flow profile described by a single cell per meridional quadrant. An α2Ω mean-field model, presumably closest to the mode of dynamo action characterizing the MHD simulation, produces a spatiotemporal evolution of magnetic fields that share some striking similarities with the zonally-averaged toroidal component extracted from the simulation. Comparison with α2 and αΩ mean-field models operating in the same parameter regimes indicates that much of the complexity observed in the spatiotemporal evolution of the large-scale magnetic field in the simulation can be traced to the turbulent electromotive force. Oscillating α2 solutions are readily produced, and show some similarities with the observed solar cycle, including a deep-seated toroidal component concentrated at low latitudes and migrating equatorward in the course of the solar cycle. Various numerical experiments performed using the mean-field models reveal that turbulent pumping plays an important role in setting the global characteristics of the magnetic cycles.

A UNIFIED APPROACH TO THE HELIOSEISMIC INVERSION PROBLEM OF THE SOLAR MERIDIONAL FLOW FROM GLOBAL OSCILLATIONS

Measurements from tracers and local helioseismology indicate the existence of a meridional flow in the Sun with strength in the order of 15 m s −1 near the solar surface. Different attempts were made to obtain information on the flow profile at depths up to 20 Mm below the solar surface. We propose a method using global helioseismic Doppler measurements with the prospect of inferring the meridional flow profile at greater depths. Our approach is based on the perturbation of the p-mode eigenfunctions of a solar model due to the presence of a flow. The distortion of the oscillation eigenfunctions is manifested in the mixing of p-modes, which may be measured from global solar oscillation time series. As a new helioseismic measurement quantity, we propose amplitude ratios between oscillations in the Fourier domain. We relate this quantity to the meridional flow and unify the concepts presented here for an inversion procedure to infer the meridional flow from global solar oscillations.

Observation of oscillation coupling ratios and the meridional flow

Measurements from local helioseismology indicate the existence of a meridional flow in the Sun with a strength up to 15 m/s near the solar surface. The flow profile at depths below ≈ 15 Mm are not accessible. We propose a method using global helioseismic measurements with the prospect to infer the meridional flow profile throughout the solar convection zone and show its performance on simulated data.

Angular momentum transport in a multicomponent solar wind with differentially flowing, thermally anisotropic ions

Context. The Helios measurements of the angular momentum flux L of the fast solar wind lead to a tendency for the fluxes associated with individual ion angular momenta of protons and alpha particles, Lp and Lα, to be negative (i.e., in the sense of counter-rotation with the Sun). However, the opposite holds for the slow wind, and the overall particle contribution LP = Lp + Lα tends to exceed the magnetic contribution LM. These two aspects are at variance with previous models. Aims. We examine whether introducing realistic ion temperature anisotropies can resolve this discrepancy. Methods. From a general set of multifluid transport equations with gyrotropic species pressure tensors, we derive the equations governing both the meridional and azimuthal dynamics of outflows from magnetized, rotating stars. The equations are not restricted to radial flows in the equatorial plane but valid for general axisymmetric winds that include two major ion species. The azimuthal dynamics are examined in detail, using the empirical meridional flow profiles for the solar wind, constructed mainly according to measurements made in situ. Results. The angular momentum flux L is determined by the requirement that the solution to the total angular momentum conservation law is unique and smooth in the vicinity of the Alfven point, defined as where the combined Alfvenic Mach number MT = 1. MT has to consider the contributions from both protons and alpha particles. Introducing realistic ion temperature anisotropies may introduce a change of up to 10% in L and up to ∼1. 8k m s −1 in azimuthal speeds of individual ions between 0.3 and 1 AU, compared with the isotropic case. The latter has strong consequences on the relative importance of LP and LM in the angular momentum budget. Conclusions. However, introducing ion temperature anisotropies cannot resolve the discrepancy between in situ measurements and model computations. For the fast-wind solutions, while in extreme cases LP may become negative, Lp never does. On the other hand, for the slow solar wind solutions examined, LP never exceeds LM, even though LM may be less than the individual ion contribution, since Lp and Lα always have opposite signs for the slow and fast wind alike.

Meridional flow profile measurements with SOHO/MDI

AbstractWe present meridional flow measurements of the Sun using a novel helioseismic approach for analyzing SOHO/MDI data in order to push the current limits in radial depth. Analyzing three consecutive months of data during solar minimum, we find that the meridional flow is as expected poleward in the upper convection zone, turns equatorward at a depth of around 40 Mm (∼ 0.95 R⊙), and possibly changes direction again in the lower convection zone. This may indicate two meridional circulation cells in each hemisphere, one beneath the other. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of Secondary Flows in Centrifugal Impellers

Secondary flows are undesirable in centrifugal compressors as they are a direct cause for flow (head) losses, create nonuniform meridional flow profiles, potentially induce flow separation/stall, and contribute to impeller flow slip; that is, secondary flows negatively affect the compressor performance. A model based on the vorticity equation for a rotating system was developed to determine the streamwise vorticity from the normal and binormal vorticity components (which are known from the meridional flow profile). Using the streamwise vorticity results and the small shear‐large disturbance flow method, the onset, direction, and magnitude of circulatory secondary flows in a shrouded centrifugal impeller can be predicted. This model is also used to estimate head losses due to secondary flows in a centrifugal flow impeller. The described method can be employed early in the design process to develop impeller flow shapes that intrinsically reduce secondary flows rather than using disruptive elements such as splitter vanes to accomplish this task.

Meridional Flow Field of Axisymmetric Flows in a Rotating Annulus

Abstract Measurements of the flow field were made of the axisymmetric flow in a differentially heated rotating fluid annulus by using a long-term tracking of a tracer particle. Its meridional flow profile is composed of a flow circulating in a large direct (Hadley) cell, which consists of thin boundary and top horizontal layers, and another one in the interior of that cell. Our measurements show that little mixing of the flow between the Hadley cell and the interior occurs and two secondary cells exist in the interior. These cells cause a trajectory of the tracer particle to be chaotic, where the particle follows the main cyclic route but is occasionally trapped in the secondary cells for indefinite times. Very close to the transition to the wave regime, a mixed structure appears consisting of both the axisymmetric and the wave flows in the meridional flow field: an unsteady wave flow occurs in the interior of the Hadley cell.