Impact of Polar Ice Layers on the Corrosion-Related Static Electric Field of a Submerged Underwater Vehicle

Abstract

The influence of polar ice-covered environments on the corrosion-related static electric field (CRSE) of underwater vehicles is critical for understanding and applying the characteristics of underwater electric fields in polar regions. This study utilizes a combined methodology involving COMSOL Multiphysics 6.1 simulations and laboratory-simulated experiments to systematically investigate the distribution characteristics of underwater vehicle electric fields under ice-covered conditions. By comparing the electric field distributions in scenarios with and without ice coverage, this study clarifies the effect of ice presence on the behavior of underwater electric fields. The simulation results demonstrate that the existence of ice layers enhances both the electric potential and field strength, with the degree of influence depending on the ice layer conductivity, thickness, and proximity of the measurement points to the ice layer. The accumulation of error analysis and laboratory experiments corroborates the reliability of the simulation results, demonstrating that ice layers enhance electric field signals by modifying the conductive properties of the surrounding medium, whereas the overall spatial distribution characteristics remain largely consistent. These findings offer a theoretical and technical basis for the optimization of stealth strategies in polar underwater vehicles and contribute to the advancement of electric field detection technologies.

1. Introduction

Owing to its unique ice-covered characteristics, the Arctic Ocean has become a strategic area for modern submarine stealth operations [1]. The safe navigation of submarines in polar environments, along with the detection and early warning of enemy submarines, has become a critical aspect of enhancing the operational effectiveness of submarine missions in these regions [2,3]. Nevertheless, the risk of exposure through submarine underwater physical fields—namely, acoustic, magnetic, and electric—continues to pose significant limitations on their combat capabilities. Conducting targeted research on the distribution characteristics and behavioral patterns of these physical fields under polar conditions is essential for advancing technologies related to submarine detection, early warning, tracking, and engagement.

With the rapid advancement of electric field measurement sensor technologies, submarine underwater electric fields have become among the most significant underwater military target characteristics, alongside acoustic, pressure, and magnetic fields [4,5,6]. Regardless of whether a submarine is stationary or in motion, it is consistently enveloped by an electric field within the surrounding seawater. The dominant component of this field is the static electric field (commonly referred to as the corrosion-related static electric field (CRSE), for related abbreviations and symbols, please refer to Appendix A) generated by dissimilar metal corrosion and anticorrosion measures [7,8,9], which accounts for the largest proportion of energy. Furthermore, the extremely low-frequency alternating electric field generated by the modulation of the static electric field constitutes the most prominent and detectable target characteristic of submarine underwater electric fields. Consequently, related research has focused predominantly on the investigation of static electric fields [10,11,12].

The presence of surface ice layers in polar marine environments modifies the spatial domain, thereby inevitably influencing the distribution characteristics of the submarine CRSE. A comprehensive understanding of the mechanisms through which ice layers affect submarine underwater electric fields is essential for the advancement of electric field-based localization and tracking technologies tailored for polar operations. This study specifically examines the impact of ice layers on CRSE by initially employing a COMSOL Multiphysics finite element model to simulate the distribution of CRSE under ice-covered conditions. By comparing scenarios with varying ice layer conductivities, thicknesses, and measurement depths, the influence of ice layers on the CRSE is revealed. The results of laboratory experiments further validate the simulation results. The findings of this research establish a foundation for enhanced military application of polar submarine underwater electric fields, providing theoretical guidance for the optimization of submarine stealth strategies and the advancement of electric field detection technologies in Arctic operations.

2. Physics Problem and Mathematical Description

The CRSE of a submarine primarily originates from the anode-cathode current loop of the impressed current cathodic protection (ICCP) system. To achieve anticorrosion effects, a controlled current of specified intensity is supplied from the anode of the ICCP system and directed toward the submarine’s propeller, which serves as the cathode. This current flow through seawater generates the underwater CRSE. For modeling simplification, the study assumes a pair of square anodes symmetrically positioned on the port and starboard sides of the submarine, with the geometric centers of the anodes and the propeller aligned on the same horizontal plane as the submarine’s central axis. It is further assumed that, aside from the anodes, the submarine hull is entirely covered with sonar-absorbing tiles, with no additional conductive regions exposed.

The submarine operates in ice-covered waters beneath a surface ice layer, as illustrated in Figure 1a. The external environment surrounding the submarine is divided into three distinct domains: the ice layer (Domain I), seawater (Domain II), and seabed (Domain III). In each domain, the potential distribution (with the potential at infinity set to zero), electrical conductivity, and permittivity are denoted by U_i, σ_i, and ε_i, respectively, where i = 1, 2, 3. The ice–water interface, water–seabed interface, water-hull interface, ICCP anode surface, and propeller surface are denoted Σ₁, Σ₂, Σ₃, Σ₄, and Σ₅, respectively. The potential distributions within all three domains satisfy Laplace’s equation. Across interfaces Σ₁, Σ₂, and Σ₃, both the continuity of the electric potential and the continuity of the normal component of the current density are maintained. At interfaces Σ₄ and Σ₅, the total normal current density within the seawater domain corresponds to the output current I_a from the ICCP anodes and the inflow current I_k to the propeller, respectively. For clarity in the subsequent descriptions, a rectangular coordinate system is established, as shown in Figure 1b. The coordinate origin is defined as the midpoint of the line connecting the geometric center of the anode and the midpoint of the line connecting the propeller center. The direction from the bow to the stern of the submarine is designated the positive x-axis, the direction to the right of the submarine is the positive y-axis, and the upward direction is the positive z-axis.

The above problem can be formulated as a boundary value problem, as shown in Equations (1)–(8).

The current flows out from the anode, passes through the seawater, ice layer, and seabed regions, and eventually enters the cathode. Apart from this, there are no other sources or sinks of current within the domain of interest we are concerned with. Therefore, the electric potential in the domain satisfies Laplace’s equation, as shown in Formula (1):

$${\nabla }^2{U}_i=0,i=1,2,3$$

(1)

At the sea ice-seawater interface Σ₁, the continuity conditions for both electric potential and current are satisfied:

$${\left.U_1\right|}_{{\Sigma }_1}={\left.U_2\right|}_{{\Sigma }_1}$$

(2)

$${\sigma }_1{\left.{\displaystyle \frac{\mathsf{\partial }{U}_1}{\mathsf{\partial }z}}\right|}_{{\Sigma }_1}={\sigma }_2{\left.{\displaystyle \frac{\mathsf{\partial }{U}_2}{\mathsf{\partial }z}}\right|}_{{\Sigma }_1}$$

(3)

Similarly, at the seawater–seabed interface Σ₂, the continuity conditions for both electric potential and current are satisfied:

$${\left.U_2\right|}_{{\Sigma }_2}={\left.U_3\right|}_{{\Sigma }_2}$$

(4)

$${\sigma }_2{\left.{\displaystyle \frac{\mathsf{\partial }{U}_2}{\mathsf{\partial }z}}\right|}_{{\Sigma }_2}={\sigma }_3{\left.{\displaystyle \frac{\mathsf{\partial }{U}_3}{\mathsf{\partial }z}}\right|}_{{\Sigma }_2}$$

(5)

The submarine hull surface is treated as an insulating interface, where no current flows into or out of the hull:

$${\sigma }_2{\left.{\displaystyle \frac{\mathsf{\partial }{U}_2}{\mathsf{\partial }n}}\right|}_{{\Sigma }_3}=0$$

(6)

In Equation (6), n represents the outward-pointing unit normal vector to the interface.

On the anode surface Σ₄, a constant current I_a flows out from the anode and enters the surrounding seawater:

$${\displaystyle {\int }_{{\Sigma }_4}{\sigma }_2{\left.\frac{\mathsf{\partial }{U}_2}{\mathsf{\partial }n}\right|}_{{\Sigma }_4}dS=I_a}$$

(7)

On the anode surface Σ₅, there is a constant current Ik flowing into it from the cathode:

$${\displaystyle {\int }_{{\Sigma }_5}{\sigma }_2{\left.\frac{\mathsf{\partial }{U}_2}{\mathsf{\partial }n}\right|}_{{\Sigma }_5}dS=-I_k}$$

(8)

3. Finite Element Solution and Simulation Based on COMSOL

As evident from Equations (1)–(8), the physical problem described above involves complex geometries, heterogeneous boundary conditions, and diverse material properties, and it is difficult to obtain its solution by an analytical method. Considering COMSOL software’s ability to handle multiphysics coupling problems, solver stability, and mesh generation adaptability, this paper selects its “Electric Currents” interface for solving the aforementioned problem, with the key parameters considered in the computational process including spatial conductivity, permittivity, current density, and boundary condition configurations.

3.1. Simulation Model Construction and Mesh Generation

Neglecting the effects of air, seabed, and ocean currents, and setting the seawater–seabed interface as an insulating boundary condition, a 3D simulation domain consisting of an ice layer, seawater, and a submarine, along with their corresponding physical models, was established in COMSOL, as shown in Figure 2a.The simulation domain was defined as a cuboid, with the seawater measuring 1500.0 m × 1500.0 m × 400.0 m and the ice layer measuring 1500.0 m × 1500.0 m × 4.0 m. The coordinate system was aligned with that shown in Figure 1b, with the origin set at the geometric center point on the 30.0 m depth plane underwater, following the coordinate origin definition provided in the preceding section. The upper surface of the ice layer was set at z = 30 m, and the seawater–seabed interface was fixed at ${\left.z\right|}_{{\Sigma }_2}=-370.0 \mathrm{m}$.

The submarine’s ICCP system anodes were modeled as platinum sheets with a surface area of 0.5 m². The geometric centers of the two anode plates were located at coordinates of 20.0 m, 0 m, and 0 m. The propeller was represented as a copper circular disk with a radius of 3.9 m, and its geometric center was also positioned at (20.0 m, 0 m, 0 m).

It is noteworthy that actual submarine propellers are typically made of nickel-aluminum bronze. However, in this study, we focus on investigating the influence of ice layers on the static electric field distribution related to submarine underwater corrosion, with the primary parameter of interest being the conductivity distribution in the environment. Since copper is more readily available and has a similar conductivity (copper has a conductivity of 5.7 × 10⁷ S/m, while nickel-aluminum bronze has a conductivity of approximately 7.0 × 10⁷ S/m), and both are significantly higher than the conductivities of seawater and sea ice, copper was selected as the material for the submarine propeller in both experiments and simulations.

The conductivity, relative permittivity, and ICCP system parameters, including the anode and cathode currents for seawater, ice layers, submarine hulls, anodes, and propellers, are detailed in

Table 1. Simulation Parameter Settings.

Symbol	Reference Value	Unit	Description
σ1	0.025	S/m	Conductivity of the ice layer
σ2	2.91	S/m	Conductivity of seawater at 0 °C
σsub	1 × 10−5	S/m	Conductivity of submarine hull
σk	5.7 × 107	S/m	Conductivity of propeller material
σa	9.7 × 106	S/m	Conductivity of ICCP anode material
Ia1	22.00	A	Current value of a single anode
Ik	−44.00	A	Current Value of Cathode
εr1	2.8		Relative permittivity of the ice layer
εr2	80		Relative permittivity of seawater
εrsub	2.3		Relative permittivity of submarine hull

.

After completing the modeling and parameter configuration, spatial discretization was performed using free tetrahedral mesh elements for the simulation model shown in Figure 2a, with the resulting mesh distribution illustrated in Figure 2b. The precision of the computational results is governed by the mesh size and quality, with the present study achieving a total of 57,369,058 discretized elements. The submarine meshing strategy employs automatic adaptive refinement based on geometric configuration complexity and physical field gradients, where regions with higher variational intensity receive denser mesh distributions to ensure high-fidelity discretization in geometrically critical zones.

3.2. Simulation of Submarine CRSE Under Ice-Covered Conditions

Based on the parameters defined in Section 3.1, the “AC/DC” module in COMSOL, designed for solving static and dynamic electric field problems, was employed to solve the simulation model constructed in Figure 2. This produced the current and potential distributions on the z = 0 m and y = 0 m planes, as well as on the submarine surface. The results are presented in Figure 3a, where arrowed streamlines indicate the direction and path of the current flow, and color gradients represent the magnitude of the current density. As evident from the figure:

• In the seawater domain, the ICCP currents are primarily distributed between the anodes and the propeller, with the current density demonstrating a pronounced increase in proximity to either the anodes or the propeller.
• A comparison with references [13,14,15] indicates that the potential distributions on the z = 0 m and y = 0 m planes resemble those generated by an electric dipole. On the y = 0 m plane, the potential distribution exhibits a positive maximum (0.082 V) and a negative maximum (−0.481 V). On the z = 0 m plane, the potential distribution shows two positive maxima (corresponding to the two anodes) and one negative maximum.
• The potential distribution on the submarine surface is symmetric along the central axis, with the anodes exhibiting a positive maximum and the propeller reaching a negative maximum.

Additionally, the electric potential and three-component electric field distributions E_x, E_y, and E_z on the z = −20.0 m plane were computed, with the corresponding results shown in Figure 3b, 3c, 3d, and 3e, respectively. A comparison with the literature [13,14,15] indicates that at this depth, the potential distribution retains the essential characteristics of an electric dipole field, with the planar potential distribution displaying spatially symmetric positive and negative potential peaks. However, owing to the two-anode configuration of the ICCP system, the magnitudes of these peaks are not equal: the maximum positive potential on the anode side is 33.90 mV, whereas the maximum negative potential on the cathode side is −39.30 mV. The spatial distributions of the three electric field components also resemble those of a dipole field, but the magnitudes of these positive and negative peaks exhibit unequal quantitative values. For instance:

• The maxima of E_x are 2.23 mV/m (positive) and 0.87 mV/m (negative), respectively.
• The maxima of E_y are 1.12 mV/m (positive) and 1.13 mV/m (negative).
• The maxima of E_z are 3.14 mV/m (positive) and 2.31 mV/m (negative).

These computational results reflect the inherent symmetry of the simulation model structure and are consistent with the established principles governing current field distributions generated by submerged electrodes [16,17,18]. This confirms that the presence of a uniformly conductive, geometrically regular, and sufficiently large ice layer—despite the introduction of an abrupt conductivity boundary—does not modify the overall spatial distribution characteristics of the submarine’s CRSE.

4. Error Sources Analysis and Accumulation Assessment in Simulation Computations

This study employs COMSOL Multiphysics for steady-state field computations, where the primary sources of accumulated error stem from mesh discretization and nonlinear iteration processes (COMSOL’s solver typically employs iterative algorithms to enhance convergence and stability when addressing multiphysics coupling problems. Even if the problem is inherently linear, the software may introduce iterative steps during preprocessing, error control, or convergence evaluation. These steps are classified as “nonlinear iterations” in cumulative error analysis). We first analyze the impact of two critical parameters (mesh partitioning and relative tolerance) on computational accuracy and stability. The reliability of the results is subsequently quantified through error accumulation assessment.

4.1. Effect of Meshing and Relative Tolerance Configuration

The finite element method (FEM) discretizes a continuous field into nodal values across elements. Consequently, the selection of element types, the simplification of complex geometries, and the settings for mesh size all introduce numerical computation errors. In the simulation presented in this paper, free tetrahedral elements are employed to discretize the spatial domain. An adaptive mechanism is used, wherein the mesh is automatically generated based on the complexity of the geometric structure and physical fields. This approach ensures high-precision discretization in geometrically complex regions. Therefore, the primary source of mesh discretization error in this case stems from the number of elements (abbreviated as Num). While denser meshing clearly improves computational accuracy, excessively fine meshing significantly increases computation time and memory requirements.

In practical modeling and simulation processes, the number of mesh elements is always finite. Insufficient mesh density inevitably introduces some degree of distortion in the field distribution. Moreover, during the solver’s iterative process, errors can accumulate and amplify, potentially leading to invalid computational results. Hence, it becomes essential to control the solver’s iterations by setting convergence thresholds, thereby compelling the solver to perform additional iterations or to terminate appropriately.

Relative tolerance (Rtol) is a dimensionless threshold used to determine whether an iterative solution has converged. Its calculation is based on the relative error norm. Specifically, at the k-th iteration, the solver computes the relative error between the current solution vector u_k and the solution vector from the previous iteration u_k−1:

$$Rtol={\displaystyle \frac{‖u_k-u_{k-1}‖}{‖u_k‖}}$$

(9)

where $‖•‖$ represents the Euclidean norm. When this relative error is less than the set Rtol, the following condition is satisfied:

$${\displaystyle \frac{‖u_k-u_{k-1}‖}{‖u_k‖}}<Rtol$$

(10)

The iterative process is then deemed to have converged, and the solver stops computing.

Setting a smaller Rtol yields higher solution accuracy; however, if Rtol is set too strictly (extremely small), it compels the solver to perform excessive iterations. This can amplify round-off errors and may even lead to divergence of the computational results. Conversely, a Rtol that is too loose (relatively large) fails to sufficiently suppress mesh discretization errors. Therefore, in the numerical simulations presented in this paper, we focus on the dynamic interplay between mesh element quantity and Rtol settings, aiming to enhance computational efficiency while ensuring that solution accuracy meets engineering application requirements.

The following analysis systematically increases mesh density from coarse to fine levels, examining simulation results for four physical parameters—ΔU_max, ΔE_xmax, ΔE_ymax, and ΔE_zmax—under Rtol of 0.1, 0.01, and 0.001. The corresponding results are presented in Figure 4a–d.

As illustrated in the figure, the simulation results of the four physical quantities—ΔU_max, ΔE_xmax, ΔE_ymax, and ΔE_zmax—exhibit the following common pattern: when the same relative tolerance is selected, coarser mesh discretizations lead to more pronounced fluctuations in the results. This indicates that larger discretization errors associated with sparse meshes require more stringent relative tolerances to be effectively suppressed. As the mesh becomes increasingly refined, the accuracy improves, and the simulation results tend to stabilize progressively. A comparative analysis of the results under three different relative tolerances (using ΔU_max as an example) reveals that when the mesh element count exceeds 10 million, the amplitude of fluctuation for a relative tolerance of 0.1 is 0.0140 mV; for 0.01, it is 0.0127 mV; and for 0.001, it is 0.0107 mV. This demonstrates that, under the same mesh density, smaller relative tolerances yield better error suppression and more precise computational outcomes.

From the above analysis, it can be inferred that in the present simulation case, selecting a relative tolerance of 0.001 and a mesh element count exceeding 10 million results in simulation outputs that no longer vary significantly with further mesh refinement. In other words, the numerical solution converges to an accurate value that meets the specified precision requirements. This indicates that the mesh discretization and tolerance settings at this stage are appropriate. Consequently, in the modeling and simulation conducted in Section 3 and Section 5 of this study, a free tetrahedral element mesh was employed to discretize the field domain, incorporating an adaptive mesh refinement mechanism. The total number of mesh elements reached 57,369,058, with a relative tolerance of 0.001. These configurations effectively control the accumulation of computational errors, ensuring the stability and accuracy of the simulation results. Furthermore, these findings validate the high reliability of the simulation-based conclusions.

4.2. Error Accumulation Assessment

Given the well-established nature of the COMSOL software employed, systematic errors arising from the solver’s models, algorithms, and integration strategies can be considered negligible. Consequently, error accumulation is primarily manifested as the progressive buildup of relative errors originating from spatial discretization and propagating through iterative processes [19].

Upon completion of meshing, the accumulation of relative error in one-dimensional scenarios is related to the average ratio of the element size ΔL to the domain size L and the computational accuracy k of the solver, expressed as

$$S\approx {\left({\displaystyle \frac{\Delta L}{L}}\right)}^{\mathrm{k}+1}$$

(11)

The accumulation of error $S_{err}$ in three dimensions can be represented by the superposition of the relative errors accumulated in three directions.

$$S_{err}\approx {\displaystyle \sum _1^3{S}_i}$$

(12)

In Equation (3), the subscripts i = 1, 2, and 3 represent the three directions. The general error accumulation is proportional to the square root of the number of iterations, and if the number of iterations is $n_k$, the total cumulative error S_sum can be quantitatively expressed as $\sqrt{n_k}{S}_{\mathrm{err}}$ [19].

To satisfy numerical accuracy requirements, a tolerance threshold for accumulated error—denoted as the maximum permissible relative error $S^{\max }$ (e.g., $S^{\max }$= 5%)—is typically prescribed. Consequently, the maximum allowable iteration steps can be derived as follows:

$$n_{\max }={\left(S^{\max }/S_{err}\right)}^2$$

(13)

To assess the reliability of the computational results from an error accumulation perspective, the reliability factor is defined as

$$R_s=n_{\max }/n_k$$

(14)

Evidently, $R_s$ ≫ 1 indicates rapid convergence to the target accuracy under given meshing conditions, signifying reliable computational results. Higher $R_s$ values correspond to greater reliability. Conversely, when $R_s$ ≈ 1, the relative error approaches critical thresholds after multiple iterations, implying marginal reliability. This necessitates further verification from alternative perspectives alongside adjustments to computational parameters and processes.

Building on the definitions of symbols in Equations (11)–(14), this study assesses the reliability of the numerical simulation methodology and results from an error accumulation perspective. With the maximum relative error threshold $S^{\max }$ fixed at 5%, the reliability factor $R_s$ is computed across varying mesh configurations. The evaluation results are presented in

Table 2. Error Accumulation Assessment Results.

Num/Million	nk	Ssum	nmax	RS
0.08	344	3.75 × 10−5	1.78 × 106	5.17 × 103
1.00	376	3.00 × 10−6	2.78 × 108	7.39 × 105
6.00	470	5.00 × 10−7	1.00 × 1010	2.13 × 107
30.00	626	1.00 × 10−7	2.50 × 1011	3.99 × 108
53.00	760	5.66 × 10−8	7.80 × 1011	1.03 × 109

. Notably, the COMSOL solver module employs second-order accuracy (k = 2).

• Analysis of tabular data reveals:

• Across all five mesh configurations analyzed, the total accumulated error S_sum remains below the 5% tolerance threshold. The maximum S_sum observed is merely 3.75 × 10⁻⁵, with R_S ≫ 1 consistently, confirming effective error control and result reliability.
• During mesh refinement from 80,000 to 53 million elements, S_sum decreased by three orders of magnitude, and R_s increased by five orders of magnitude, demonstrating that grid refinement significantly enhances solution reliability.
• When the number of mesh elements in the model reached 53 million, S_sum was only 5.66 × 10⁻⁸, R_S increased to 109, and the iteration count n_k doubled. This indicates that at this level, the calculation results are not only reliable but also computationally efficient. This finding confirms both high reliability and the maintenance of computational efficiency. The present simulation utilized 57.369 million elements, further validating the reliability of the results.

5. Impact of Ice Layer Presence on the Submarine CRSE

To investigate the influence of ice layers on the submarine CRSE, the ice layer shown in Figure 2a was replaced with seawater, while all other parameters were held constant. Simulations were then conducted to compute the CRSE distribution of the submarine in the ice-free model, and the results were compared with those obtained from the ice-covered model. The electric potential and electric field components—U, E_x, E_y, and E_z—for the ice-covered and ice-free models are denoted using subscripts 1 and 2, respectively. The difference between the two models is used to represent the influence of the ice layer on the submarine CRSE, which is denoted as ΔU = U₁ − U₂, ΔE_x = E_x1 − E_x2, ΔE_y = E_y1 − E_y2, and ΔE_z = E_z1 − E_z2. The maximum and minimum values of the aforementioned physical quantities in the planar domain are indicated via subscripts max and min, respectively.

5.1. Global Enhancement Effect of Surface Ice Cover on Submarine CRSE

Building on the calculations in Section 3.2, the distributions of ΔU, ΔE_x, ΔE_y, and ΔE_z on the z = −20 m plane were computed, with the results illustrated in Figure 5. The graphical representation is as follows:

• The presence of the ice layer universally amplifies the electric potential and all three electric field components across the plane, with the degree of enhancement dependent on the spatial position of the field points.
• The planar distributions of ΔU, ΔE_x, ΔE_y, and ΔE_z exhibit spatial symmetry akin to that of dipole fields, with the largest magnitude variations occurring at extremum points.
  The parameter settings of this study were as follows: at the positive potential maximum, ΔU_max = 0.651 mV (a 1.95% increase); at the negative potential maximum, ΔU_min = −0.744 mV (a 1.92% increase). For the electric field components, ΔE_xmax = 0.0289 mV/m (a 1.32% increase), ΔE_xmin = −0.00960 mV/m (a 1.10% increase), ΔE_ymax = 0.00645 mV/m (a 0.58% increase), ΔE_ymin = −0.00681 mV/m (a 0.61% increase), ΔE_zmax = 0.0279 mV/m (a 0.90% increase), and Δ E_zmin = −0.0198 mV/m (a 0.86% increase).
  In the above results, the degree of variation observed at the positive and negative extreme value points of the same physical quantity differs slightly, which can be attributed to the asymmetry of the field source structure in the simulation model.

3.

The three components of the electric field intensity exhibit differential susceptibilities to ice layer effects. From the overall view of Figure 5, the longitudinal (E_x) and vertical (E_z) components exhibit significant ice-induced modifications, and the transverse component (E_y) demonstrates minimal susceptibility to ice layer effects. This finding substantiates that the presence of ice layers exerts stronger modulatory effects on longitudinal and vertical current pathways.

Based on the above characteristics, for simplicity, when the influence law of relevant parameters on the CRSE of the submarine is analyzed later, only ΔU_max, ΔE_xmax, and ΔE_ymax ΔE_zmax are taken as the objects of simulation analysis.

5.2. Influence of Ice Layers on Submarine CRSE at Different Depth Planes

With the same parameter settings as those described in Section 3.1, the depth of the measurement plane was varied to simulate and assess the influence of ice layers on the submarine CRSE at different depths. Figure 6a,b illustrate the depth-dependent variations in four CRSE parameters—ΔU_max, ΔE_xmax, ΔE_ymax, and ΔE_zmax—within the sub-ice marine regions both above and below the submarine. The results show that the closer the measurement plane is to the ice layer, the more pronounced the influence of the ice layer on these physical quantities is, indicating a near-field enhancement effect. Notably, in the seawater above the submarine, ΔU_max, ΔE_xmax, and ΔE_ymax are positive, whereas ΔE_zmax is negative. This occurs because the vertical component (E_z) of the electric field strength in the seawater above the submarine is inherently negative, and the presence of the ice surface further increases its magnitude. Additionally, the rate of change with depth for all four physical quantities is greater above the submarine than below, as the measurement points above are closer to the ice layer and thus more significantly affected. As shown in Figure 6b, the transverse electric field component E_y below the submarine is relatively insensitive to changes in measurement depth, indicating that tracking variations in E_y may not be the most effective approach for determining the target position in practical detection scenarios.

5.3. Influence of Ice Layer Conductivity on the Submarine CRSE

By varying the ice layer conductivity (σ₁) while maintaining all other simulation parameters consistent with those outlined in Section 3.1, the influence of σ₁ on the submarine’s CRSE was investigated. When σ₁ was assigned values of 0.025 S/m, 0.5 S/m, 1.0 S/m, 1.5 S/m, 2.0 S/m, and 2.5 S/m, the corresponding values of four physical quantities—ΔU_max, ΔE_xmax, ΔE_ymax, and ΔE_zmax—were computed at the depth plane of z = 20 m above the submarine to assess the effect of the ice layer conductivity. The simulation results are presented in Figure 7a. For comparative analysis, the same physical quantities were also calculated at z = −20 m below the submarine, with the results illustrated in Figure 7b.

As shown in the figures, for both regions above and below the submarine, the ΔU_max on the same measurement plane increases markedly as the ice layer conductivity (σ₁) decreases. The simulation results indicate that when the ice layer conductivity is set to 0.025 S/m, the ΔU_max on the z = 20 m depth plane in the seawater above the submarine reaches 5.521 mV, corresponding to a relative variation rate of 13.19%. At the same depth, ΔE_xmax reaches 0.364 mV/m with a relative change rate of 13.69%, ΔE_ymax reaches 0.163 mV/m with a relative change rate of 12.97%, and ΔE_zmax reaches −0.359 mV/m with a relative change rate of −20.14%. In contrast, on the z = −20 m depth plane below the submarine, ΔU_max reaches 0.651 mV, with a relative variation rate of 1.95%. The corresponding values of ΔE_xmax, ΔE_ymax, and ΔE_zmax are 0.0289 mV/m, 0.00645 mV/m, and 0.0279 mV/m, with relative change rates of 1.32%, 0.58%, and 0.90%, respectively.

Additionally, as shown in Figure 7, for a given ice layer conductivity σ₁, the ΔU_max value increases as the measurement plane approaches the ice surface, indicating a more significant influence of the ice layer. This observation is consistent with the conclusions presented in Section 5.2.

5.4. Influence of Ice Layer Thickness on the Submarine CRSE

With the upper surface of the ice layer fixed at z = 30 m and other parameters consistent with those in Section 3.1, calculations were performed to observe the variations in ΔU_max, ΔE_xmax, ΔE_ymax, and ΔE_zmax at the measurement plane z = −20 m, as the ice layer thickness (H_ice) varied from 2 m to 12 m. The computational results are illustrated in Figure 8.

As shown in the figure, with increasing ice thickness (H_ice), the values of ΔU_max, ΔE_xmax, ΔE_ymax, and ΔE_zmax exhibit nonlinear growth, indicating that thicker ice layers have a more substantial impact on the submarine’s CRSE. The computational results reveal that when the ice thickness reaches 12 m, the relative change rate of the electric potential on the z = −20 m measurement plane beneath the submarine reaches 7.72%. In contrast, the maximum relative change rate among the three electric field intensity components is 7.74%. Moreover, as H_ice increases, the growth rates of the x- and z-components of the electric field intensity significantly exceed that of the y-component. This finding offers important guidance for optimizing underwater detection strategies and is consistent with the conclusions presented in Section 5.2.

6. Laboratory Simulation Tests

6.1. Experimental Setup and Parameter Configuration

To validate the accuracy of the simulation methodology and results presented in this study, a laboratory experimental setup was constructed, as shown in Figure 9a. An artificial sea ice layer was placed over the surface of simulated seawater maintained at a temperature near 0 °C to replicate polar environmental conditions. A submarine model equipped with an ICCP system was deployed in under-ice seawater, and an Ag-AgCl electrode array was used to measure the electric field distribution across the planar region beneath the submarine. During the experiment, the submarine model was driven to move along the x-direction underwater using a system of guide rails, sliders, and drive rods rather than repositioning the detection electrode array, thereby enabling the acquisition of potential variation curves along the x-direction across multiple survey lines. The submarine model is shown in Figure 9b. To simultaneously measure field distributions across two underwater depth planes, two detection electrode arrays were placed at different heights, as illustrated in Figure 9c.

In Figure 9a, the dimensions of the simulated seawater domain are 2.12 m × 1.12 m × 0.485 m, with an artificial ice layer thickness of H_ice = 0.075 m. The coordinate system aligns with that in Figure 1b, and the origin is positioned at the geometric center point on the depth plane 0.065 m below the water surface. Platinum sheet anodes (dimensions: 8 mm × 7 mm × 1 mm) are symmetrically installed on both sides of the submarine model. The propeller cathode is simulated via a circular copper sheet (radius: 31 mm, thickness: 1 mm). The geometric centers of the platinum anodes, the center of the cathode, and the submarine axis lie on the same horizontal plane. The distance between the centers of the anodes is 65 mm, and the distance from the geometric center of an anode to the cathode center is 124.7 mm. The upper surface of the ice layer is positioned at z = 0.14 m. In the experiment, the seawater conductivity is 0.58 S/m, with the current of each anode set to 0.15 A and the cathode current set to 0.30 A. The conductivity and relative permittivity values for the other materials are consistent with the parameters listed in Table 1. Two rows of detection electrodes were arranged along the y direction, with each pair spaced 0.08 m apart, resulting in a total of 10 electrodes. The positions of the electrodes were symmetrically distributed at about y = 0. These electrodes were used to measure the potential distribution along 10 survey lines located at y = ±0.04 m, ±0.12 m, ±0.20 m, ±0.28 m, and ±0.36 m on two depth planes: z = −0.10 m and z = −0.18 m.

It is noteworthy that the seawater conductivity value used in the experiment (0.58 S/m) was significantly lower than the typical polar seawater conductivity (2.91 S/m) adopted in the simulations in Section 3. This parameter configuration was implemented because higher sodium chloride concentrations (corresponding to higher conductivity) inhibit ice crystal formation, making it difficult to achieve the desired ice thickness in laboratory settings. Furthermore, prolonged experimentation with high-concentration brine may induce stratification phenomena, thereby compromising measurement accuracy. Additionally, since both Section 3 and Section 6 employed identical simulation methodologies, the different seawater conductivity settings do not affect the experimental validation of the correctness of the simulation approach.

6.2. Experimental Results and Simulation Analysis

The experiments measured the potential distributions on the planes z = −0.10 m and z = −0.18 m beneath the submarine under both ice-covered and ice-free conditions, with the seawater domain remaining unchanged. The simulation methodology described in Section 2 was employed to model the potential data on these two planes, and the results were compared with the experimental measurements. A comparison of the experimental and simulation results under ice-covered and ice-free conditions is presented in Figure 10, where the subscript e denotes the experimental data and s indicates the simulation results. Owing to the symmetry of the electric field distribution about the x-axis, only the potential distributions along five survey lines in the y > 0 region are shown in Figure 10 for clarity. Subscripts 1–5 correspond to the survey lines located at y = 0.04 m, 0.12 m, 0.20 m, 0.28 m, and 0.36 m, respectively. In accordance with the experimental sampling rate settings, the experimental data points in Figure 10 are spaced 0.16 cm apart, whereas the simulation data points are spaced 5 cm apart. Consequently, each survey line provides a total of 750 experimental data points and 25 simulation data points.

Figure 10a,b compare the experimental and simulated data for the five survey lines on the plane z = −0.10 m under ice-covered and ice-free conditions, respectively. Similarly, Figure 10c,d present the corresponding comparisons for the five survey lines on the plane z = −0.18 m. If the relative deviation δ_i between the experimental and simulation data on the i-th survey line (i = 1, 2, 3, 4, 5) is defined as:

$${\delta }_i={\displaystyle \frac{1}{N}}{\displaystyle \sum _{j=1}^N\left|\frac{U_{sij}/U_{eij}}{U_{eij}}\right|},i=1,2,3,4,5;N=25$$

(15)

δ_i can be used to calculate the relative deviations between the experimental and simulated data for the 10 measurement lines on the two aforementioned planes, as presented in

Table 3. Relative Deviation of the Experimental and Simulation Data.

Survey Lines	y = 0.04 m	y = 0.12 m	y = 0.20 m	y = 0.28 m	y = 0.36 m
Ice-covered	1.41%	2.13%	3.67%	4.59%	4.71%
Ice-free	5.04%	4.15%	3.71%	2.92%	3.75%

, which demonstrates a relatively small deviation between the two datasets. Given the potential sources of error inherent in experimental testing, the experimental data can be considered highly reliable, thereby validating the accuracy of the numerical simulation model employed in the preceding analysis.

6.3. Influence of the Ice Layer on the Underwater Potential Distribution

Building upon this analysis, the potential difference ΔU on two planes under ice-covered conditions was calculated to investigate the impact of the ice layer on the CRSE field beneath the submarine. The corresponding results are presented in Figure 11.

As shown in the figures:

• The distribution characteristics of ΔU on both measurement planes align with those in Figure 5. The presence of an ice layer increases the potential value, and the degree of increase is related to the position of the field point. This observation is consistent with the conclusions drawn in Section 5.1 of this study.
• In Figure 10a,b, the maximum potential value on the plane z = −0.10 m under ice-covered conditions is 0.1777 V, whereas it is 0.1581 V for no ice layer conditions, such as ΔU_max = 0.0196 V in Figure 11a. Similarly, in Figure 10c,d, the maximum potential value on the plane z = −0.18 m with ice coverage is 0.0946 V, whereas the no ice layer maximum is 0.0756 V, so ΔU_max = 0.0190 V in Figure 11b. These results demonstrate that the closer the measurement plane is to the ice layer, the more significant its influence becomes. This finding aligns with the conclusions presented in Section 5.2 of this study.

6.4. Error Analysis

We believe that the errors in the aforementioned results are primarily attributed to the following two factors:

• Measurement Errors: Ag-AgCl electrodes measure the integrated potential of the surrounding environment. Each electrode exhibits inherent variations, and spatial noise interference during measurement is difficult to accurately replicate in simulations.
• Modeling Errors: Deviations exist in measuring parameters such as the geometric dimensions of the submarine model, precise locations of the anode/cathode, and spatial coordinates of measurement points. These inaccuracies lead to discrepancies between the geometrically constructed simulation model and the actual experimental setup, resulting in errors between the simulated results and actual measurements.

In summary, the experimental results validate the accuracy of the numerical simulation model and confirm the reliability of the simulation methodology, further substantiating the enhancement effect of ice on the electric field. This enhancement effect becomes more pronounced as the measurement depth decreases. However, it must be clarified that the validation of the simulation model’s accuracy and reliability through this experiment is strictly confined to the specific conditions and parameters we established. It does not guarantee perfect agreement of the simulation results under arbitrary changes in temporal increments, spatial increments, or physical parameter values.

7. Conclusions

Submarine concealment and detection technologies in polar ice-covered environments represent a critical frontier in contemporary naval military research. The corrosion-related static electric field (CRSE), as a key signal source within a submarine’s underwater physical fields, directly influences the survivability and operational effectiveness of a vessel through its distribution characteristics. This study establishes a multiphysics coupling model of submarine impressed current cathodic protection (ICCP) systems, integrating COMSOL simulations with laboratory experiments to elucidate the effects of polar ice layers on a submarine CRSE. These findings offer essential theoretical support for advancing submarine stealth and antisubmarine detection technologies in polar regions.

The results demonstrate that polar ice layers enhance the potential and electric field strength of CRSE by altering the conductivity characteristics of the surrounding field. Specifically, lower ice layer conductivity, increased ice thickness, and closer proximity of measurement points to the ice surface result in more pronounced CRSE enhancement. This finding reveals that polar ice layers function not only as physical barriers but also as amplifiers of underwater electric field signals owing to their electrical properties, thereby introducing new challenges for optimizing submarine stealth strategies. This underscores the necessity of reassessing electric field exposure risks in ice-covered regions and implementing targeted improvements in ICCP current regulation schemes to reduce signal leakage. Moreover, this study identifies the asymmetric effects of ice layer conductivity and thickness on electric field components, offering new insights for the development of advanced directional detection technologies based on electric field signatures.

Additionally, this study simplifies the complex three-dimensional electric field distribution into a multidomain coupled Laplace equation. Simulations using COMSOL’s “AC/DC” module validate the numerical model’s applicability to ice–water–submarine multi-interface problems. Laboratory experiments further confirm the reliability of the simulation results. The research method used in this study provides a reproducible technical path for predicting electric fields in complex polar environments in the future.

The research findings presented in this article also offer a quantitative basis for the covert navigation of polar submarines and serve as a reference for the design of electric field-based submarine detection equipment, highlighting their significant military application value.