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Kinetic Theory of the Solar Wind Plasma

Contents

In this problem we are given an input file which contains the solar wind protons velocity space distributions obtained from a particle simulation code for an observer located at 1.0 AU distance from the Sun. Simulated observer collects only the solar wind protons with directional velocity range from 0 to 8 *10^5 m/s and velocity resolution of 5*10^3 m/s in all three directions. The magnetic field is constant +5 nT in a with a magnetic vector along Y axis, and the solar wind flows along X axis.

clc;clear all;close all;

Constants

mp   = 1.6726E-27;      % Mass of a proton [kg]
kb   = 1.3806503e-23;   % Boltzmann constant [J/kg]

Load the file with wind protons velocity space distributions into MATLAB The velocity space distribution values are stored in ‘f’

load data
dv = 5e3;               %Velocity resolution in all directions [m/s]
% Velocity space [m/s]
v_x = -8E5:dv:8E5;
v_y = -8E5:dv:8E5;
v_z = -8E5:dv:8E5;

Solar wind protons number density,Zero moment

ns = sum(f(:))*(dv^3);
fprintf('Solar wind protons number density %4.2d m^-3 \n\n',ns);
Solar wind protons number density 3.00e+006 m^-3

Solar wind protons fluxes in all directions,First moment

Fi_x = 0; Fi_y = 0; Fi_z = 0;
len = length(v_x);
 for i = 1:len
       for j = 1:len
            for k = 1:len
                  Fi_x = Fi_x + v_x(i)*f(i,j,k);
                  Fi_y = Fi_y + v_y(j)*f(i,j,k);
                  Fi_z = Fi_z + v_z(k)*f(i,j,k);
            end
         end
    end
Fi_x = Fi_x*dv^3; Fi_y = Fi_y*dv^3; Fi_z = Fi_z*dv^3;

fprintf('Solar wind protons flux in X direction %4.2d m-2/s\n',Fi_x);
fprintf('Solar wind protons flux in Y direction %4.2d m-2/s\n',Fi_y);
fprintf('Solar wind protons flux in Z direction %4.2d m-2/s\n\n',Fi_z);
Fi = [Fi_x,Fi_y,Fi_z];
Solar wind protons flux in X direction -1.09e+012 m-2/s
Solar wind protons flux in Y direction 7.58e-007 m-2/s
Solar wind protons flux in Z direction -6.91e+004 m-2/s

Solar wind bulk flow velocity

Urt = Fi/ns/1000;       %[km/s]
fprintf('Solar wind bulk flow velocity U [%4.2f %4.2f %4.2f] km/s\n',Urt);
Solar wind bulk flow velocity U [-365.00 0.00 -0.00] km/s

Distribution function plot

Plot the distribution function in the vx – vy plane and compare it with distribution function in the vx – vz plane. The center of the distribution contours should clearly show the bulk velocity in each plane.

v_x = v_x/1000; v_y = v_y/1000; v_z = v_z/1000;
% Maximum of velocity space distribution function
[f_max,ind] = max(f(:));
size_f = [321 321 321];
% Location of maximum
[f_mi,f_mj,f_mk] = ind2sub(size_f,ind);

Distribution function in the vx – vy plane

figure(1);
hold on;grid on;
contour(v_x, v_y, f(:,:,f_mk) );
scatter(Urt(2),Urt(1),'*r');
% Matlab will auto adjust the axis and ellipse will appear as circle
% We adjust the axis manually to see the contour shape
axis([-120,120,-480,-240]);
title('Distribution function in the vx - vy plane');
ylabel('v_x [km/s]');
xlabel('v_y [km/s]');
hold off;

SolarWind

Distribution function in the vx – vz plane

figure(2);
hold on;grid on;
contour(v_x, v_z, squeeze(f(:,f_mj,:)) );
axis([-120,120,-480,-240]);
scatter(Urt(3),Urt(1),'*r');
title('Distribution function in the vx - vz plane');
ylabel('v_x [km/s]');
xlabel('v_z [km/s]');
hold off;

SolarWind

The type of the distribution function is Bi-Maxwellian, because the shape in vx – vy is ellipse.

Pressure tensor and scalar pressure

C = V – U and called random velocity

Ux = Urt(1);Uy = Urt(2);Uz = Urt(3);
% To save calculation time we use scalar variables instead of matrix
P11 = 0;P22 = 0; P33 = 0;
P12 =0; P23 = 0; P13 = 0;
for i = 1:len
       for j = 1:len
            for k = 1:len
                  c1 = v_x(i) - Ux;
                  c2 = v_y(j) - Uy;
                  c3 = v_z(k) - Uz;
                  P11 = P11 + c1^2*f(i,j,k);
                  P22 = P22 + c2^2*f(i,j,k);
                  P33 = P33 + c3^2*f(i,j,k);
                  P12 = P12 + c1*c2*f(i,j,k);
                  P23 = P23 + c2*c3*f(i,j,k);
                  P13 = P13 + c1*c3*f(i,j,k);
             end
        end
 end
P = [P11,P12,P13;
     P12,P22,P23;
     P13,P23,P33];
 P = P*mp*dv^3*1e6      % [kg*m^3*m^2/s^2]
P =

  1.0e-011 *

    0.3106   -0.0000    0.0000
   -0.0000    0.9319    0.0000
    0.0000    0.0000    0.3106

Scalar pressure

The scalar pressure ps is defined as one third of the trace of Pij

ps = 1/3*(P(1,1) + P(2,2) + P(3,3));
fprintf('Scalar pressure %4.2d nPa\n',ps*1e9)
Scalar pressure 5.18e-003 nPa

Solar wind thermal velocity and the solar wind kinetic temperature

Kinetic temperature of species can be given in a form of a 3 × 3 matrix

Ts = P/(ns*kb)      % K
Ts =

  1.0e+005 *

    0.7500   -0.0000    0.0000
   -0.0000    2.2500    0.0000
    0.0000    0.0000    0.7500

Thermal Velocity

Vth = ([Ts(1,1),Ts(2,2),Ts(3,3)].*(2*kb/mp)).^0.5;
fprintf('Thermal Velocity Vth [ %4.2f %4.2f %4.2f ] km/s\n',Vth/1000)
V_sp = norm(Vth) ;
Thermal Velocity Vth [ 35.19 60.95 35.19 ] km/s

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Debye length, plasma frequency and cyclotron frequency

The calculation of characteristic Debye length, plasma frequency and cyclotron frequency of electrons in the ionosphere, the plasmasphere, the outer magnetosphere and the solar wind.

clc;clear all;
me   = 9.109E-31;            %[kg]           Electron rest mass
mp   = 1.673E-27;            %[kg]           Proton rest mass
eps0 = 8.8542E-12;           %[A*s/(V*m)]    Permittivity
e    = 1.602E-19;            %[C]            Elementary charge
Re   = 6.37E6;               %[m]            Earth’s radius
Md   = 8E15;                 %[T*m3]         Earth’s magnetic dipole moment

Ionosphere

n   = 1E12;                     %[m-3]  Electron density
kTe = 0.1;                      %[eV]   Thermal energy
r   = Re + 3E5;                 %[km]   Radial distance from the Earth’s center
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 2*Md/r^3;                 %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);
Debye length        2.35e-003 [m]
Plasma frequency    8.98e+006 [Hz]
Cyclotron frequency 1.51e+006 [Hz]

Plasmasphere

n   = 1E7;                      %[m-3]  Electron density
kTe = 1;                        %[eV]   Thermal energy
r   = 5*Re;                     %[km]   Radial distance from the Earth’s center
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 2*Md/r^3;                 %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);
Debye length        2.35e+000 [m]
Plasma frequency    2.84e+004 [Hz]
Cyclotron frequency 1.39e+004 [Hz]

Outer magnetosphere

n   = 1E6;                      %[m-3]  Electron density
kTe = 50;                       %[eV]   Thermal energy
r   = 7*Re;                     %[km]   Radial distance from the Earth’s center
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 2*Md/r^3;                 %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);
Debye length        5.26e+001 [m]
Plasma frequency    8.98e+003 [Hz]
Cyclotron frequency 5.05e+003 [Hz]

Solar wind

n   = 1E6;                      %[m-3]  Electron density
kTe = 5;                        %[eV]   Thermal energy
wp  = (e^2*n/(eps0*me))^0.5/(2*pi);  %[Hz]   Plasma frequency
lD  = (eps0*kTe/(n*e))^0.5;     %[m]    Debye length
B   = 1E-8;                     %[T]    Earth’s magnetic field
Fc  = e*B/(2*pi*me);            %[Hz]   Cyclotron frequency
fprintf('Debye length        %4.2d [m]\n',lD);
fprintf('Plasma frequency    %4.2d [Hz]\n',wp);
fprintf('Cyclotron frequency %4.2d [Hz]\n\n',Fc);
Debye length        1.66e+001 [m]
Plasma frequency    8.98e+003 [Hz]
Cyclotron frequency 2.80e+002 [Hz]

 

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