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Register a new userObtaining Potential Field Solution with Spherical Harmonics and Finite Differences
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Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current and divergence free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: i) Remeshing the magnetogram onto a grid with uniform resolution in latitude, and limiting the highest order of the spherical harmonics to the anti-alias limit; ii) Using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms, and the differences are found to be rather dramatic. We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publically available code, so that other researchers can also use it as an alternative to the spherical harmonics approach.
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Modeling and forecasting the solar wind-driven global magnetic field perturbations is an open challenge. Current approaches depend on simulations of computationally demanding models like the Magnetohydrodynamics (MHD) model or sampling spatially and temporally through sparse ground-based stations (SuperMAG). In this paper, we develop a Deep Learning model that forecasts in Spherical Harmonics space 2, replacing reliance on MHD models and providing global coverage at one minute cadence, improving over the current state-of-the-art which relies on feature engineering. We evaluate the performance in SuperMAG dataset (improved by 14.53%) and MHD simulations (improved by 24.35%). Additionally, we evaluate the extrapolation performance of the spherical harmonics reconstruction based on sparse ground-based stations (SuperMAG), showing that spherical harmonics can reliably reconstruct the global magnetic field as evaluated on MHD simulation.
Bipolar spherical harmonics (BiPoSHs) provide a general formalism for quantifying departures in the cosmic microwave background (CMB) from statistical isotropy (SI) and from Gaussianity. However, prior work has focused only on BiPoSHs with even parity. Here we show that there is another set of BiPoSHs with odd parity, and we explore their cosmological applications. We describe systematic artifacts in a CMB map that could be sought by measurement of these odd-parity BiPoSH modes. These BiPoSH modes may also be produced cosmologically through lensing by gravitational waves (GWs), among other sources. We derive expressions for the BiPoSH modes induced by the weak lensing of both scalar and tensor perturbations. We then investigate the possibility of detecting parity-breaking physics, such as chiral GWs, by cross-correlating opposite parity BiPoSH modes with multipole moments of the CMB polarization. We find that the expected signal-to-noise of such a detection is modest.
We analyze the local field of stellar tangential velocities for a sample of $42 339$ non-binary Hipparcos stars with accurate parallaxes, using a vector spherical harmonic formalism. We derive simple relations between the parameters of the classical linear model (Ogorodnikov-Milne) of the local systemic field and low-degree terms of the general vector harmonic decomposition. Taking advantage of these relationships we determine the solar velocity with respect to the local stars of $(V_X,V_Y,V_Z)=(10.5, 18.5, 7.3)pm 0.1$ kms. The Oorts parameters determined by a straightforward least-squares adjustment in vector spherical harmonics, are $A=14.0pm 1.4$, $B=-13.1pm 1.2$, $K=1.1pm 1.8$, and $C=-2.9pm 1.4$ kmspc. We find a few statistically significant higher degree harmonic terms, which do not correspond to any parameters in the classical linear model. One of them, a third-degree electric harmonic, is tentatively explained as the response to a negative linear gradient of rotation velocity with distance from the Galactic plane, which we estimate at $sim -20$ kmspc. The most unexpected and unexplained term within the Ogorodnikov-Milne model is the first-degree magnetic harmonic representing a rigid rotation of the stellar field about the axis $-Y$ pointing opposite to the direction of rotation. This harmonic comes out with a statistically robust coefficient $6.2 pm 0.9$ kmspc, and is also present in the velocity field of more distant stars. The ensuing upward vertical motion of stars in the general direction of the Galactic center and the downward motion in the anticenter direction are opposite to the vector field expected from the stationary Galactic warp model.
Spherical Harmonics, $Y_ell^m(theta,phi)$, are derived and presented (in a Table) for half-odd-integer values of $ell$ and $m$. These functions are eigenfunctions of $L^2$ and $L_z$ written as differential operators in the spherical-polar angles, $theta$ and $phi$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4pi$ symmetry in the angle $phi$, and hence are not single-valued functions on the Euclidean unit sphere; they are double-valued functions on the sphere, or alternatively are interpreted as having a double-sphere as their domain.
This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned by them using appropriate topologies on our spaces. Finally, we discuss the properties of the functionals that form the continuous basis.
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