# Characteristics of Magnetic Fields Induced by the Wake of an Underwater Vehicle
## 2 Numerical Methods
### Magnetic Fields Induced by the Wake  
The ions in moving seawater are subjected to an electric field force, Lorentz force, friction force, concentration gradient force, and pressure gradient force. The ion motion equation can be obtained using [PERSON]'s second theorem [20]:  
\[m_{p}\frac{dv_{p}}{dt}=e_{p}E+e_{p}(v_{0}\times B_{e})-f_{p}(v_{p}-v_{0})-\frac {kT}{n_{p}}\
abla n_{p}+V_{p}s_{0}\frac{dv_{0}}{dt} \tag{3}\]  
where \(m_{p}\), v\({}_{p}\), \(e_{p}\), \(f_{p}\), \(n_{p}\), \(V_{p}\) are the ion mass, ion motion velocity vector, ion charge, ion friction coefficient, ion concentration, and ion volume, respectively, and the subscript \(p\) represents different ions. \(E\), \(v_{0}\), \(B_{e}\), \(k\), \(T\), \(s_{0}\), \(t\) represent the electric field intensity, vibration velocity of external mechanical excitation, geomagnetic field vector, Boltzmann constant, temperature, electrolyte solution density, and time, respectively.  
According to the principle of mass conservation, the relationship between the ion concentration and velocity is as follows:  
\[\frac{\partial n_{p}}{\partial t}+\
abla\cdot\left(n_{p}v_{p}\right)=0 \tag{4}\]  
The polarization current density generated by ion separation can be expressed as:  
\[J=\frac{\partial P}{\partial t}=\sum_{p}e_{p}n_{p}v_{p} \tag{5}\]  
Owing to the relative movement of the heterotrophic ions, the neutral state of the seawater solution is destroyed, and the space charge density in the solution can be expressed as  
\[\rho_{p}=\sum_{p}n_{p}e_{p} \tag{6}\]The relative movement of the ions produces polarization in the solution, which naturally produces a polarization current density. \(x_{p}\) is the displacement difference generated by ions relative to seawater, and the polarization intensity can be expressed as:  
\[P=\sum_{p}e_{p}n_{p}x_{p} \tag{7}\]  
Therefore, the polarization current density generated by ion separation can be expressed as:  
\[J=\frac{\partial P}{\partial t}=\sum_{p}e_{p}n_{p}v_{p} \tag{8}\]  
The magnetic field was generated by the polarization and displacement currents. The electric field comprises two parts. The first is a Coulomb field, which is generated by a space charge with an uneven distribution of opposite charges. This can be described using the Gauss theorem. The second is the electric field generated by a time-varying magnetic field, which can be described by [PERSON]'s law. Therefore, when the polarization current density and space charge density are clearly defined, the electromagnetic field generated by ion separation can be described by [PERSON]'s equation:  
\[\left\{\begin{array}{c}\
abla\times H=J_{P}+\epsilon\frac{\partial E}{ \partial t}\\ \
abla\times E=-\frac{\partial B}{\partial t}\\ \
abla\cdot E=\frac{\rho_{P}}{\epsilon_{0}}\\ \
abla\cdot B=0\end{array}\right. \tag{9}\]  
After a small amount of neglect, under the assumption of a quasi-stable field, the total magnetic field generated by the seawater ion separation can be described as follows [21]:  
\[\
abla^{2}H-\sum_{p}\frac{\mu{e_{p}}^{2}n_{0}}{f_{p}}\frac{\partial H}{ \partial t}=-\sum_{p}\frac{e_{p}^{2}n_{0}}{f_{p}}\
abla\times v_{0}\times B_{ e}+\sum_{p}\frac{e_{p}n_{0}\big{(}m_{p}-V_{p}s_{0}\big{)}}{f_{p}}\
abla\times \frac{dv_{0}}{dt} \tag{10}\]  
The finite element models were built by ICEM software, flow and magnetic fields are calculated on a common grid. Fluent UDF (user-defined function) was used to compile the wake magnetic field equation [22; 23]. The UDF obtained magnetic field signals according to the wake flow field information attained from the CFD solver.