KARMAN-HOWARTH-MONIN EQUATION FOR COMPRESSIBLE HALL MHD TURBULENCE VICTOR MONTAGUD-CAMPS1, PETR HELLINGER1, ANDREA VERDINI2, SIMONE LANDI2, LUCA FRANCI3, EMANUELE PAPINI2, LORENZO MATTEINI4 1Astronomical Institute , CAS, Prague, Czech Republic, 2Università di Firenze, Italy, 3Queen Mary University of London, UK, 4Imperial College, London, UK Contact e-mail: victor.montagud-camps@asu.cas.cz INTRODUCTION • The rate at which solar wind protons temperature decreases with heliospheric distance (Tp ∝ R−0.9[1]) is slower than the prediction for an ideal gas that expands adiabatically (Tp ∝R−4/3). • The energy from large scale ﬂuctuations driven by plasma turbulence towards smaller scales is one of the proposed mechanisms that could have a heating rate adequate to explain this difference. • The heating rate associated to turbulence is computed as the energy cascade rate, Q, that is, the net energy transfer rate between the scales at which turbulent non-linear interactions take place. • The Kármán-Howarth-Monin (KHM) equation is an accurate estimation of the energy cascade rate that can be directly derived from the compressible MHD equations using structure functions. • Our objective is to extend the estimation of the energy cascade rate given by the KHM equations to scales below the proton inertial length, di and assess the effect of turbulence anisotropy on the energy cascade rate estimation. VERIFICATION OF KHM FOR 2D COMPRESSIBLE HALL-MHD 10-1 100 101 102 -1.0 -0.5 0.0 0.5 1.0 l/di O = −∂S/∂t/4+KMHD+KHall+Ψ−D [Q] • Initial simulation set-up: Simulation box of size 2562d2 i , and 20482 grid-points resolution. B0 along z. Alfvénic ﬂuctuations δBrms/B0 = 0.17 for kdi < 0.2. Zero cross-helicity, turbulent Mach M = 0.26 and η = µ = 10−3. • KHM equations terms are normalized to Hall- MHD heating rate Q ≡⟨η∇B : ∇B + τ : ∇u⟩. • KHM equation holds, but non-linear terms do not show a clear inertial range, i.e., they are not con- stant over an extended range of scales. KHM ANISOTROPY ON 3D COMPRESSIBLE HALL-MHD • Initial simulations set-up: two simulation boxes of size 323d3 i , and 5123 grid-points resolution. B0 along x. One demagnetized |B0| = 0 (3D Iso) and the other one magnetized (3D ani) with B0 along z and Alfvénic ﬂuctuations δBrms/B0 = 0.2. Zero cross-helicity, turbulent Mach M = 0.1 and η = µ = 6 10−4. • Top row: averaging over all increments (top row), demagnetized case is qualitatively similar to the 2D magnetized one. Non constant contribution from non-linear terms. • Bottom row: KMHD and KHall gyrotropized about B0 axis and 1D cuts θ = 0o, 30o, 60o and 90o with respect to B0. Anisotropic behaviour of both terms for |B0| ̸= 0 run. REFERENCES [1] M. et al. Maksimovic. Anticorrelation between the Bulk Speed and the Electron Temperature in the Pristine Solar Wind: First Results from the Parker Solar Probe and Comparison with Helios. ApJS, 246(2):62, February 2020. [2] Petr Hellinger, Andrea Verdini, Simone Landi, Luca Franci, and Lorenzo Matteini. von Kármán-Howarth Equation for Hall Magnetohydrodynamics: Hybrid Simulations. ApJL, 857(2):L19, April 2018. [3] R. Ferrand, S. Galtier, F. Sahraoui, R. Meyrand, N. Andrés, and S. Banerjee. On Exact Laws in Incompressible Hall Magnetohydrodynamic Turbulence. ApJ, 881(1):50, August 2019. DISCUSSION AND CONCLUSIONS • KHM equation for compressible Hall-MHD holds true for 2D and 3D simulations. • Magnetized plasma and 3D non-linear interactions were necessary to develop clear range for constant KMHD. For the same case, B0-aligned KHall contribution is observed above di. Box size effect or inverse cascade? • Higher resolution and smaller dissipation are still needed to show signiﬁcant Hall contributions below di KHM EQUATION FOR COMPRESSIBLE HALL-MHD Derivation of KHM equation[2,3]: evaluate Hall-MHD equations at two different points x′ = x + l and x separated by an increment l. Subtract them and average over all possible increments. Hence, −1 4 ∂S ∂t + KMHD + KH + C + Ψ −D = 0 Second order st