Method and system for measuring internal wave wake electromagnetic field in density stratified seawater

Abstract

The invention discloses a method and a system for measuring an internal wave wake electromagnetic field in density stratified seawater, and belongs to the technical field of underwater target detection. The invention discloses a method for measuring an internal wave wake electromagnetic field in density stratified seawater, and the method comprises the steps: obtaining the current speed and diving depth of an underwater target, and the density, conductivity and dielectric constant of seawater in a sea area where the underwater target is located; substituting the parameters into an electromagnetic field calculation model to solve and obtain an induced electromagnetic field of the underwater target; the electromagnetic field calculation model is constructed based on a rotation equation of the magnetic field. The problem that an underwater target is difficult to accurately detect by directly calculating an internal wave wake electromagnetic field due to the lack of a direct simulation calculation model at present is solved. The internal wave wake electromagnetic field of the underwater target at different diving depths and speeds is rapidly and accurately calculated, and the underwater target is detected and identified according to the internal wave wake electromagnetic field.

Description

Technical Field

[0001] The present invention belongs to the technical field of underwater target detection, and more specifically, relates to a method and system for measuring the electromagnetic field of internal wave wakes in density-stratified seawater. Background Art

[0002] As a supplementary means to remote sensing and sonar detection technologies, wake electromagnetic field detection technology has attracted increasing attention. This technology is based on the electromagnetic field generated by the movement of underwater targets. The wake electromagnetic field is a part of the marine electromagnetic field. In the field of marine electromagnetic field research, Faraday predicted that conductive seawater moving under the background of the geomagnetic field is similar to a metal conductor moving in the geomagnetic field, and both will generate induced electromagnetic fields. This prediction was later verified by Young's experiment. In the past few decades, research on surges, ocean waves, internal waves, and even tsunamis has been carried out. These studies have shown that the marine electromagnetic field is ubiquitous and can be detected under specific marine conditions.

[0003] For a long time, in the research of wake electromagnetic fields, it has usually been assumed that the seawater density is uniform. Therefore, the research on wake electromagnetic fields has mainly focused on the Kelvin wake electromagnetic field. It should be noted that when an underwater target moves in seawater with uniform density, a Kelvin wake will be generated. The Kelvin wake originates from the perturbation of the free surface (such as the sea surface) caused by the movement of the target and is mainly dominated by surface gravity waves. The moving target compresses the surrounding seawater, resulting in surface undulations, and gravity acts as the restoring force to propagate the waves. Under the influence of the geomagnetic field, the Kelvin wake generates the Kelvin wake electromagnetic field. However, due to the influence of temperature and salinity, the seawater density will continuously change with the increase in depth. When an underwater target moves in stratified seawater, an internal wave wake will be generated. Different from the Kelvin wake, when an underwater target moves in stratified seawater, the target will perturb the density interface, thereby triggering internal gravity waves. The restoring force is a combination of buoyancy and gravity and is determined by the density gradient. Similarly, the internal wave wake generated when an underwater target moves in stratified seawater will generate an internal wave wake electromagnetic field.

[0004] Regarding the research on internal wave wake electromagnetic fields, whether through simulation calculations or experiments, it is extremely rare. The simulation of internal wave wake electromagnetic fields involves the coupling between the velocity field and the electromagnetic field, which is an interdisciplinary multi-physics field simulation problem. Currently, there is a lack of a direct simulation calculation model, resulting in the problem that it is difficult to accurately detect underwater targets by directly calculating the internal wave wake electromagnetic field. Summary of the Invention

[0005] Aiming at the deficiencies of the related technologies, the purpose of the present invention is to provide a method and system for measuring the electromagnetic field of internal wave wakes in density-stratified seawater, aiming to solve the technical problem that there is currently a lack of a direct simulation calculation model, resulting in the difficulty of accurately detecting underwater targets by directly calculating the internal wave wake electromagnetic field.

[0006] To achieve the above object, in a first aspect, the present invention provides a method for measuring the electromagnetic field of internal wave wakes in density-stratified seawater, including:

[0007] Obtain the current speed and diving depth of the underwater target, as well as the density, conductivity, and permittivity of the seawater corresponding to the underwater target;

[0008] Substitute the above parameters into the electromagnetic field calculation model to solve for the induced electromagnetic field of the underwater target;

[0009] The electromagnetic field calculation model is constructed based on the curl equation of the magnetic field, and the expression is:

[0010]

[0011] Among them, a spatial coordinate system is established with the sea level as the xy plane and the direction perpendicular to the sea level upward as the z-axis; the value range of z is -H ≤ z ≤ 0, H represents the density demarcation point of the seawater, and z = 0 represents the sea level; σ0(z) and ε(z) are the conductivity and permittivity of the stratified seawater at z, respectively; is the induced electric field, is the geomagnetic field, is the induced magnetic field, is the internal wave wake velocity field, which is related to the speed and diving depth of the underwater target.

[0012] Optionally, the calculation model of the internal wave wake velocity field is:

[0013]

[0014] Among them, is the volume of the underwater target; L is the length of the underwater target, U is the speed of the underwater target, is the speed of the point source model;

[0015] The calculation formula of the point source model velocity field is:

[0016]

[0017] Among them, θ is the angle between the fluid particle and the x-axis, k is the wave number, and i is the imaginary unit. In the above formula:

[0018]

[0019] W1(-H) = e -kH , W1′(-H) = ke -kH ,

[0020] W2(0) = -1, W2′(0) = -σ,

[0021]

[0022] Among them, ρ0(z) is the density of the stratified seawater at z, W′ represents the differential of W, W1 and W2 are accurately calculated by the fourth-order Runge-Kutta method, and g is the acceleration due to gravity.

[0023] Optionally, the process of solving the electromagnetic field calculation model includes:

[0024] Calculating the curl of the electromagnetic field calculation model to obtain an intermediate equation:

[0025]

[0026] Constructing an expression for the induced magnetic field of the internal wave wake:

[0027]

[0028] Among them, t represents time; ω0 is the oscillation frequency of fluid particles, which is related to the velocity U, wave number k, and included angle θ, and the expression is ω0 = kUcosθ;

[0029] Constructing an expression for the induced electric field of the internal wave wake:

[0030]

[0031] Substituting the expressions of the induced electric field and the induced magnetic field into the intermediate equation, and based on After simplification, the magnetic field integrand in the expression of the induced magnetic field of the internal wave wake is solved And the electric field integrand in the expression of the induced electric field of the internal wave wake

[0032] Substituting the current velocity and diving depth of the underwater target, as well as the density, conductivity, and permittivity of the seawater corresponding to the underwater target into the expressions of the induced magnetic field and induced electric field of the internal wave wake, the calculation results of the internal wave wake electric field and magnetic field are obtained.

[0033] In a second aspect, the present invention also provides a measurement system for the electromagnetic field of an internal wave wake in density-stratified seawater, including: a memory and a processor, the memory stores a computer program, and when the processor executes the computer program, it executes the method provided in any one of the first aspects.

[0034] In a third aspect, the present invention also provides an electronic device, including: a memory and a processor, the memory stores a computer program, and when the processor executes the computer program, it executes the method provided in any one of the first aspects.

[0035] Fourthly, the present invention also provides a computer-readable storage medium, which includes a stored computer program. When the computer program is run by a processor, it controls the device where the storage medium is located to execute the method provided in any one of the first aspect.

[0036] Fifthly, the present invention also provides a computer program product, including a computer program / instructions. When the computer program / instructions are executed by a processor, they implement the method provided in any one of the first aspect.

[0037] Through the above technical solutions conceived by the present invention, compared with the prior art, the beneficial effects that can be achieved include:

[0038] The technical solution of the present invention establishes an electromagnetic field calculation model of the internal wave wake generated by the movement of an underwater target in stratified seawater by considering the variations of the density, conductivity, and dielectric constant of seawater with depth. By obtaining the real-time speed and real-time depth of the underwater target, as well as the density, conductivity, and dielectric constant of the seawater layer where the underwater target is located, it can quickly and accurately calculate the electromagnetic field of the internal wave wake of the underwater target at different depths and speeds. Since the frequency of the electromagnetic field of the internal wave wake is extremely low and the attenuation is slow, it still has detectability at a relatively long distance. In engineering applications, a new method for detecting underwater targets is proposed. By analyzing the influence of the speed and depth of the underwater target on the electromagnetic field of the internal wave wake, it provides important data support for optimizing the deployment of sensors and selecting the type of detection signals obtained during underwater target detection and recognition. It solves the technical problem that the current lack of a direct simulation calculation model makes it difficult to achieve precise detection of underwater targets by directly calculating the electromagnetic field of the internal wave wake. It achieves high calculation accuracy, detection accuracy, and practicability. Description of the Drawings

[0039] Figure 1 is a schematic diagram of the mathematical model of the velocity field;

[0040] Figure 2 is a schematic diagram of the mathematical model of the electromagnetic field;

[0041] Figure 3 is a distribution diagram of the density, conductivity, and dielectric constant of the stratified water tank;

[0042] Figure 4 is a schematic diagram of the experimental device;

[0043] Figure 5 is Mode 1 of the internal wave wake frequency, phase velocity, and group velocity varying with the wave number;

[0044] Figure 6 is the phase diagram of Mode 1 of the internal wave wake at different speeds;

[0045] Figure 7 It is the distribution diagram of the internal wave wake velocity field;

[0046] Figure 8 It is the distribution diagrams of the internal wave wake electric and magnetic fields at different speeds;

[0047] Figure 9 It is the distribution diagram of the internal wave wake electric field at different diving depths;

[0048] Figure 10 It is the distribution diagram of the internal wave wake magnetic field at different diving depths;

[0049] Figure 11 It is the diagram of the variation characteristics of the internal wave wake electromagnetic field on the vertical line;

[0050] Figure 12 It is the diagram of the variation characteristics of the internal wave wake electromagnetic field on the horizontal line;

[0051] Figure 13 It is the schematic diagram of the composition of the experimental system;

[0052] Figure 14 It is the picture of the experimental device;

[0053] Figure 15 It is the experimental device diagram of the stratified water tank;

[0054] Figure 16 It is the time domain and frequency domain diagrams of the environmental noise;

[0055] Figure 17 It is the experimental result diagram at different speeds;

[0056] Figure 18 It is the experimental result diagram at different diving depths;

[0057] Figure 19 It is the schematic flow diagram of a method for measuring the internal wave wake electromagnetic field in density-stratified seawater provided by the present invention. Specific implementation manners

[0058] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0059] The following will describe the content involved in the above embodiments in conjunction with a preferred embodiment.

[0060] Embodiment 1

[0061] The present invention provides a method for measuring the electromagnetic field of internal wave wakes in density-stratified seawater, including:

[0062] Obtain the current speed and diving depth of an underwater target, as well as the density, conductivity, and dielectric constant of the seawater corresponding to the underwater target;

[0063] Substitute the above parameters into the electromagnetic field calculation model to solve for the induced electromagnetic field of the underwater target;

[0064] The electromagnetic field calculation model is constructed based on the curl equation of the magnetic field, and its expression is:

[0065]

[0066] Among them, a spatial coordinate system is established with the sea level as the xy plane and the direction perpendicular to the sea level upward as the z-axis; the value range of z is -H ≤ z ≤ 0, where H represents the density demarcation point of the seawater, and z = 0 represents the sea level; σ 0(z) and ε(z) are the conductivity and dielectric constant of the stratified seawater at z respectively; is the induced electric field, is the geomagnetic field, is the induced magnetic field, is the internal wave wake velocity field, which is related to the speed and diving depth of the underwater target.

[0067] When constructing the electromagnetic field calculation model of the internal wave wake, the following two models need to be constructed in sequence: the velocity field model, which is used to calculate the velocity distribution of the internal wave wake; the electromagnetic field model, which is used to calculate the electric and magnetic field distributions of the internal wave wake. The specific steps are as follows.

[0068] Model 1: Velocity field model, which is used to calculate the velocity distribution of the internal wave wake.

[0069] Assume that the free surface of the ocean is a plane located at z = 0, the z-axis is perpendicular upward, the positive direction of the x-axis is opposite to the ship's navigation direction, and the y-axis satisfies the right-hand rule with the xz plane. Assume that the underwater target is a slender body with a length of L, moving at a constant speed U in the opposite direction of the x-axis, located at z = -h (h > 0) below the sea level. When the underwater target moves, it can be regarded as a pair of Havelock point sources. The velocity field of the internal solitary wave is relatively small, so the Euler equation can be linearized. At the same time, it can be considered that the stratified flow on the horizontal plane is irrotational. The motion of the point source satisfies the following three equations:

[0070]

[0071]

[0072] The velocity of the fluid is Among them, u, v, and w are the three components of the perturbation velocity caused by the movement of the underwater target, representing the components in the x, y, and z directions respectively. The density of the fluid is ρ1(z) = ρ0(z) + ρ(z), where ρ1(z) is the density after being perturbed, ρ0(z) is the density before being perturbed, and ρ(z) is the small density change caused by the perturbation.

[0073] At z = 0, the linear free surface boundary condition is adopted instead of the rigid lid assumption. It is assumed that on the sea surface, the wave height without perturbation can be expressed as z = ξ(x, y), and the wave height after perturbation can be expressed as:

[0074] F(x, y, z) = z - ξ(x, y) (4)

[0075] Then the boundary condition for the free surface movement is:

[0076]

[0077] According to Bernoulli's equation, assuming the free surface is linear, its expression is:

[0078]

[0079] After performing small quantity approximation on the above formula, the free surface boundary condition is obtained:

[0080]

[0081] The complete linear free surface boundary condition takes into account the slight perturbation of the sea surface caused by the movement of the underwater target, thus obtaining a boundary condition that is more in line with the actual situation. Among them, g is the acceleration due to gravity.

[0082]

[0083] The bottom boundary condition holds that as the depth gradually increases, the density change will become smaller and smaller. When the depth z > -H (H > 0), it is considered that the density change is relatively significant. When the depth z ≤ -H (H > 0), the density will no longer change.

[0084] ρ0| z≤-H = Const(9)

[0085] In the real ocean environment, the influence of temperature and salinity on seawater at deeper positions is relatively small, and the density is almost unchanged. Therefore, it can be considered that when the depth is greater than the density demarcation point H, the density can be regarded as unchanged. The value of H corresponding to different ocean environments is different; in this scheme, taking the sea area near Dongsha Islands in the South China Sea as an example, the density of seawater is almost unchanged after the depth is greater than 800m, so H = 800m can be selected.

[0086] With the help of the provided method for calculating the internal solitary wave velocity field, formulas (1), (2), and (3) are transformed and solved to obtain the following formula for the velocity field of the point source model:

[0087]

[0088] where θ is the angle between the fluid particle and the x-axis, k is the wave number, and i is the imaginary unit. In the above formula:

[0089]

[0090] W1(-H) = e -kH , W1′(-H) = ke -kH (15)

[0091] W2(0) = -1, W2′(0) = -σ(16)

[0092]

[0093] where ρ0(z) is the density of the stratified seawater at z, W′ represents the differential of W, the above equation (14) is the Sturm-Liouville equation of W with respect to z, W1 and W2 are the solutions of the equations satisfying (15) and (16) respectively, and W1 and W2 can be accurately calculated by the fourth-order Runge-Kutta method.

[0094] The underwater target considered can be regarded as an elongated body. Combining the theory of a pair of Havelock point sources, the velocity field generated by the movement of an underwater target with a length of L can be calculated using (18):

[0095]

[0096] where is the volume of the underwater target. Using Euler's formula, (18) can be transformed into:

[0097]

[0098] When solving the velocity field, it is found that there are many modes of internal waves, and each mode has a dispersion relation, that is, the implicit function k = k(θ) determined by the equation D(k, θ) = 0. Combining the residue theorem, (19) can be simplified to:

[0099]

[0100]

[0101] The summation in the above equation (20) represents the superposition of an infinite number of internal wave modes. The research of fluid mechanics on internal waves shows that the influence of low-order mode internal waves dominates. Therefore, in subsequent simulation and experimental studies, only the first-order mode (mode 1) is considered. The summation in (20) is no longer considered.

[0102] Model 2: Electromagnetic field model, used to calculate the electric and magnetic field distributions of the internal wave wake.

[0103] Equation (20) is based on the carrier coordinate system, with the coordinate origin fixed on the underwater target and moving together with the underwater target. However, for fluid particles in the absolute coordinate system, their perturbation velocity decreases as the underwater target moves. Therefore, in the absolute coordinate system, the velocity of fluid particles oscillates and decays with time. Thus, the velocity field of the internal wave wake of mode 1, which oscillates and decays with time, can be expressed as:

[0104]

[0105] ω o =kUcosθ (23)

[0106] where the oscillation frequency ω0=kUcosθ is related to the velocity U, the wave number k, and the angle θ between the fluid particle and the x-axis. S(θ), D, and have been given in (21), (13), and (11) respectively. k is a function of θ, k=k(θ), which is determined by the equation D(k,θ)=0.

[0107] The mathematical model of the electromagnetic field solution is as Figure 2 shown. Assuming that the geomagnetic field in the region where the underwater target is located can be written in the form of (24), where F is the intensity of the geomagnetic field, I is the magnetic dip angle, and γ is the magnetic declination.

[0108]

[0109] The non-electrostatic field caused by the Lorentz force can be written as is the induced magnetic field, is the geomagnetic field. Since the contribution of the geomagnetic field is much larger than that of the induced magnetic field, the induced magnetic field can be neglected compared with the geomagnetic field and can be approximately written as

[0110] The induced electric field and the induced magnetic field satisfy the following Maxwell equations:

[0111]

[0112] where, is the current density.

[0113]

[0114] where σ0(z) and ε(z) = ε0ε r (z) are the conductivity and permittivity of the stratified seawater at z, respectively, which vary with depth. In stratified seawater, the salinity at different depths is different, resulting in changes in conductivity and permittivity. The permeability μ of stratified seawater is considered a constant because the difference in permeability between saline water and fresh water is very small.

[0115] Observing the formula for the velocity field (22), it can be assumed that the induced magnetic field and the electric field have the following expressions:

[0116]

[0117] where is the integrand of the magnetic field, is the integrand of the electric field.

[0118] In the process of solving partial differential equations, a commonly used method is the Fourier transform combined with polar coordinate transformation. After performing the Fourier transform on the solution of the equation and then substituting it into the original equation, the process of solving the equation can be made simpler to obtain the solution in the Fourier space; after performing the inverse Fourier transform on the solution in the Fourier space, the solution in the original space, that is, the solution in the xyz space we want, can be obtained. In the expressions of H and E, the integration with respect to θ reflects the process of the inverse Fourier transform. Referring to the integral expression of the point source velocity field, the presence of e ik(xcosθ+ysinθ) in its integral indicates that the velocity field oscillates and decays with x and y. At the same time, z can be completely separated from x and y.

[0119] Therefore, when constructing the expressions of H and E, H and E should also oscillate and decay with x and y, so there is e ik(xcosθ+ysinθ) . At the same time, h(z) and e(z) are used to represent the influence of z on H and E. Designing the expression in this form makes the partial derivatives with respect to x, y, and t simple to obtain when substituting into the intermediate equation, greatly simplifying the difficulty of solving the partial differential equation.

[0120] For the partial derivatives with respect to x, y, and t: Substituting (28) into (26) gives:

[0121]

[0122] The above formula is the circulation theorem of the magnetic field. The first term on the right side represents the current generated by the electrostatic field, the second term represents the current generated by the Lorentz force, and the third term is the displacement current. Calculating the curl of (31) gives:

[0123]

[0124] Since σ0(z) and ε(z) are functions of z, the curl of (32) is relatively complex. The curls of the three terms on the right side of (32) are respectively:

[0125]

[0126] Since seawater is an incompressible fluid, . Combining (25)-(35), eliminating and only retaining (32) is transformed into:

[0127]

[0128] The above formula is the equation satisfied by the magnetic field in seawater. This is a second-order non-homogeneous differential equation with constant coefficients, where:

[0129]

[0130] Strictly speaking, in equation (36), δ is a function of z because σ0(z) and ε(z) are functions of z. However, since δ is almost completely controlled by k (because the values of ε and μ are very small), to simplify the problem, it is assumed that δ is independent of z.

[0131] Similar to (36), the magnetic field equation in air can be written as:

[0132]

[0133] where: β is also assumed to be independent of z for the same reason as δ. The forms of the solutions of equations (26) and (31) are directly written as:

[0134]

[0135] The magnetic field tends to zero at infinity, that is: Therefore, only the non-divergent part in the exponential term is retained. In (40), is the ocean magnetic field attenuation factor, is the air magnetic field attenuation factor. These two attenuation factors are two constant vectors independent of z. is the particular solution vector function, which is determined only by the non-homogeneous term and equation (36), and is independent of the boundary conditions. Therefore, is a function that can be uniquely determined by the equation. For a second-order non-homogeneous differential equation with constant coefficients in the form of (36), its particular solution can be obtained through (41):

[0136]

[0137] To uniquely determine the constant vectors and , the electromagnetic field boundary conditions and the magnetic field constraint equation need to be used.

[0138] The magnetic field is continuous at the z = 0 interface, and its tangential and normal boundary conditions are respectively:

[0139]

[0140] In (42) and (43), is the normal vector of the interface, is the surface conduction current density. For conductors with finite conductivity (such as seawater and air), there is Therefore, from (42) and (43), it can be seen that at the z = 0 interface:

[0141]

[0142] The magnetic field constraint equation is (27). Substituting the magnetic field in seawater and solving, we get:

[0143]

[0144] Define , and it can be proved that This is determined by the characteristics of the analytical expression of the internal wave velocity field. The proof process is as follows:

[0145] The particular solution vector function can be obtained through formula (41). It can be observed that the non-homogeneous term can be expressed as:

[0146]

[0147] Then The integral expressions of the three components of have a similar form:

[0148]

[0149] The following proof holds due to the particularity of the vector function .

[0150] Substitute the expressions of C x and C y into the first term and perform integration by parts on f′(t), we get:

[0151]

[0152] Differentiate the second term Cz with respect to z, we get:

[0153]

[0154] Add (51) and (52) to obtain:

[0155]

[0156] To prove that (53) is equal to zero, calculate the third term of (53) to obtain (54), that is:

[0157]

[0158] Substitute (54) into (53), and obviously the sum is zero.

[0159] In this case, (45) can be simplified to:

[0160]

[0161] Substitute the magnetic field solution into (26) to obtain the expression of the induced electric field:

[0162]

[0163] The vector X is the general representation after the vector operation (56), and its three components are respectively:

[0164]

[0165] To uniquely determine the values of vectors P and Q in the x, y, and z directions, it is also necessary to introduce the electric field boundary conditions. According to the electric field circulation theorem, the tangential components of the electric field on both sides of different media are always continuous, that is:

[0166]

[0167] Six equations can be obtained from (44), (55), and (58), which are sufficient to uniquely determine the values of the vectors in the x, y, and z directions. Therefore, the solution of the equation exists and is unique. and The expressions of are as follows:

[0168] The formula can be obtained from (58):

[0169]

[0170] For convenience, represent the constants in formula (56) as:

[0171]

[0172] G = σ0(0) - iε0ε r (0)ω0(62)

[0173] F = -iε0ω0 (63)

[0174] Then, (56) can be expressed as:

[0175]

[0176] Combining (59)-(64), we can obtain:

[0177]

[0178] where the subscripts x, y, and z respectively represent the components of the vector in the x, y, and z directions. Combining (44) and eliminating , we can obtain three equations for the three components of

[0179]

[0180] where:

[0181]

[0182] Solving (67)-(70), we can obtain the final expressions of

[0183]

[0184] According to the above method, simulation experiments are carried out, including: designing a stratified water tank experiment to simulate the density-stratified seawater environment; using an electromagnetic field sensor to measure the electromagnetic field of the internal wave wake; processing and analyzing the measured data to verify the accuracy of the calculation model. Then, experiments are carried out to verify the results.

[0185] Construct an experimental device for measuring the electromagnetic field of the internal wave wake in density-stratified seawater, specifically including:

[0186] (1) Design a stratified water tank for simulating the density-stratified seawater environment;

[0187] (2) Design an underwater target simulation device for simulating the movement of an underwater target in the water tank;

[0188] (3) Select a three-axis vector electric field sensor as the electromagnetic field sensor and install it 0.4 meters above the bottom of the water tank for measuring the electromagnetic field of the internal wave wake;

[0189] (4) Connect the electromagnetic field sensor to an external data processor for processing and analyzing the measured data; the data processor uses the variational mode decomposition (VMD) algorithm to process the measured data.

[0190] The specific content of the simulation experiment and the experiment is as follows.

[0191] (I) Simulating and calculating the electromagnetic field of the internal wave wake

[0192] The velocity field model and the electromagnetic field model are respectively as Figure 1 and Figure 2 shown, simulating the real experimental conditions. The layered scenario occurs in a water tank with a depth of 0.9 m, with a 0.2 m fresh water layer on top and a 0.7 m salt water layer below, i.e., H = 0.9 m; there is a pycnocline between the salt water and the fresh water. The density stratification is as Figure 3 shown in (a) of

[0193] In the experiment, a vertically movable density probe, conductivity probe, and relative permittivity probe are used to measure the vertical distributions of density, conductivity, and relative permittivity in the water tank respectively. The measurement results are as Figure 3 shown in (b), (c), and (d) of

[0194]

[0195] Table I Parameters of the experimental object

[0196]

[0197]

[0198] The underwater target is a rotating body with a length of L = 0.5 m and a radius of R = 0.05 m, as Figure 4 shown in (a) of Figure 4 shown in (b) of

[0199] The theoretical analysis shows that the internal wave wake velocity field is the high-order mode solution of the partial differential equations (1)-(5), and each mode has a critical velocity. This critical velocity is the maximum phase velocity of each mode. When the velocity U of the underwater target exceeds this critical velocity, the transverse wave system of the internal wave wake velocity field corresponding to this mode will disappear, leaving only the divergent wave system. Figure 5 Shown in (a) of Figure 5 in (b) of Figure 5Figure (c) shows the relationships between the phase velocity Cp, the group velocity Cg and the wave number k respectively. It can be seen that the frequency ω of the internal wave wake increases monotonically with the increase of the wave number k, while the phase velocity and the group velocity decrease monotonically with the increase of the wave number k. After precise calculation, the critical velocity Cp0 of the internal wave wake mode 1 is 0.28 m / s. Since the internal wave wake mode 1 is dominant among all modes, the simulation and experimental results in this paper only focus on the internal wave wake mode 1.

[0200] Figure 6 Figure shows the Keller - Munk phase diagram of the internal wave wake mode 1 when the velocity U ranges from 0.05 m / s to 0.3 m / s. By taking constants φ = 2π, 4π, 6π …… and parameterizing the trajectory of points (x, y) according to the wave number k, the trajectory can be plotted in the (x, y) plane. It can be seen that when U < Cp0 = 0.28 m / s, the phase diagram has both a transverse wave system increasing in the x - direction and a divergent wave system expanding in the y - direction. However, when U = 0.3 m / s > Cp0, the transverse wave system completely disappears. The half - angle θ at different U can be calculated by (73), as shown in Table II. It can be seen that when U does not exceed Cp0 (Cp0 = 0.28 m / s), the fluid is in a sub - critical state, and the half - angle of the internal wave wake increases rapidly with the increase of U. From 0.05 m / s to 0.25 m / s, the half - angle increases from 4.36° to 24.05°. However, when U exceeds Cp0, the fluid is in a super - critical state, and the half - angle of the internal wave wake decreases slowly with the increase of U. When U approaches Cp0 (in Table 2, U = 0.3 m / s), the half - angle reaches the maximum value of 36.99°.

[0201]

[0202] Table 2 Relationship between the half - angle and the velocity

[0203]

[0204] Figure 7 Figure shows the distributions of the three components of the internal wave wake velocity field when the velocity U is 0.2 m / s. The simulation results show that the three components of the internal wave wake velocity field all present a "V" - shaped distribution in the xoy plane. Compared with w, the values of u and v reach from 4×10 - 3 m / s to 0.01 m / s, which are much larger than w. u and v are the main contributing parts of the internal wave wake velocity field. In addition, the biggest difference in the distribution of v relative to u and w is that v is symmetric about the y - axis because the excitation source is located at the center of the y - axis and the excitation direction is opposite to the y - axis.

[0205] According to the calculation method of the electromagnetic field model, the induced electromagnetic field can be obtained. Figure 8Shows the distributions of the total electric field and the total magnetic field at the same submerged depth h = 0.2 m and z = -0.5 m under different velocities U.

[0206] For the total electric field, they exhibit an obvious "V"-shaped distribution. As the velocity U increases, the magnitude of the electric field first increases and then decreases, and the decreasing trend starts from U > Cp0. As can be seen from Figs. 8(a)-(d), when U = 0.15 m / s, 0.2 m / s, and 0.25 m / s < Cp0, the maximum value of the electric field gradually increases from 7.8 μV / m to 10.3 μV / m. Within the same range of measurement, the wavelength of the transverse wave system in the electric field gradually increases, while the frequency gradually decreases. When U = 0.3 m / s > Cp0, the transverse wave system in the electric field distribution disappears, and the maximum value of the electric field decreases compared with that when U = 0.25 m / s. The magnitude of the electric field is roughly on the order of several μV / m, indicating its strong detectability. This means that the electric field generated by the internal wave wake may be detected by sensitive instruments, which is very important for applications such as underwater target detection and ocean research.

[0207] As Figure 8 shown in (e)-(h), for the total magnetic field, its characteristics are similar to those of the electric field. However, the magnitude of the magnetic field is on the order of several pT, which makes its detection more difficult.

[0208] The above simulation results show that the distribution and variation characteristics of the internal wave wake electromagnetic field are completely determined by the internal wave wake velocity field. When U < Cp0, as U increases, the internal wave wake electromagnetic field increases, the wavelength of the transverse wave system increases, and the frequency decreases. On the contrary, when U > Cp0, as U increases, the internal wave wake electromagnetic field decreases, and the transverse wave system disappears. In addition, from the perspective of experimental verification, the electric field is detectable.

[0209] To study the influence of the submerged depth h on the internal wave wake electromagnetic field, the simulation conditions are set as the velocity U = 0.2 m / s, z = -0.5 m, and h takes 0.1 m, 0.2 m, 0.3 m, and 0.4 m respectively. Figure 9 and Figure 10The distributions of the electromagnetic fields of internal wave wakes at different h (h = 0.2 m and h = 0.4 m) on the plane of z = -0.5 m are respectively shown. Due to the velocity of 0.2 m / s, the wavelengths and frequencies of the transverse wave systems are consistent. It can be seen from Table 3 that as the submerged depth h increases, the maximum and average values of the electric and magnetic fields first increase and then decrease. From h = 0.2 m to h = 0.4 m, the values of the electromagnetic fields of the internal wave wakes decrease by approximately 20% - 50%. This trend is consistent with that of the velocity field of the internal wave wakes. When the underwater target is located at the thermocline (h = -0.2 m), according to formulas (10), (11) and (12), it can be known that the maximum value is reached at the thermocline and the velocity of the internal wave wake is the highest, which is the reason why the electromagnetic field value at h = 0.2 m is higher than that at other submerged depths. In addition, as the submerged depth h increases, the hydrostatic pressure on the underwater target increases, and this pressure will hinder the movement of the underwater target, thus resulting in a decrease in the velocity field of the internal wave wake and further leading to a decrease in the electromagnetic field. The order of magnitude of the electric field is several μV / m, and the order of magnitude of the magnetic field is several pT. Since the order of magnitude of the electric field is larger and the current measurement technology is more sensitive to the electric field changes at this level, the electric field is more easily detected than the magnetic field.

[0210] Table 3 The maximum and average values of the electromagnetic fields

[0211]

[0212] The distribution of the electromagnetic fields of the internal wave wakes varies with the depth z. Studying the influence of the depth z on the electromagnetic fields of the internal wave wakes is of great significance for optimizing sensor deployment and detecting signals. In practical applications, sensors need to be arranged below the underwater target to detect the signal. The current problem is how to select the optimal arrangement position to capture the signal more accurately and comprehensively. In this solution, by studying the influence of the depth z on the electromagnetic fields of the internal wave wakes, the variation relationship of the electromagnetic field magnitude with z can be obtained. Guided by this variation relationship, the sensors are arranged at the positions where the signals are the strongest, thereby optimizing the sensor deployment and improving the detection accuracy.

[0213] To study the influence of the depth z, the simulation conditions are set as the velocity U = 0.2 m / s, the submerged depth h = 0.2 m, and the depth z varies from -0.1 m to -0.9 m. Four horizontal lines at y = 0, y = 1 m, y = 2.5 m and y = 5 m, and four vertical lines at x = 1 m, x = 5 m, x = 10 m and x = 15 m are selected in this paper, as Figure 11 shown in (a) and Figure 12As shown in (a). For the internal wave wake electromagnetic field of each calculation result, the maximum value on the corresponding line is selected for comparison. The reason for not choosing certain fixed measurement points is that inappropriate selection may lead to comparing points far from the local peak with points close to the local trough, thus attributing the difference in the internal wave wake electromagnetic field to the periodic distribution rather than the depth z. The comparison method adopted in this scheme effectively avoids the above problems.

[0214] Figure 11 In (b) and Figure 11 In (c) respectively show the variation characteristics of the electric field and magnetic field along the vertical line. It can be seen that as the depth increases (z < 0, that is, the value of z increases), the magnetic field first increases and then decreases, reaching the maximum value at z = -0.4 m. According to formula (40), the attenuation factor in the magnetic field integral has an attenuation effect on the depth z, while the particular solution, density stratification function, and boundary conditions have a local contribution to the increase of the depth z. Under the combined action of the two, the magnetic field shows the Figure 11 variation characteristics shown in (c). Fallah et al. also found that this magnetic field anomaly first increases and then decreases in the vertical direction, reaching the maximum anomaly value at a certain depth below the sea surface. According to formulas (56) and (57), the attenuation factor in the electric field integral is jointly affected by and the internal wave wake velocity field. As the depth z increases, the dominant trend of and is to decrease, so the electric field shows different variation characteristics from the magnetic field. In addition, as the measurement line gradually deviates from the central axis, the values of the electromagnetic field also gradually decrease.

[0215] Figure 12 In (b) and Figure 12 In (c) respectively show the variation characteristics of the electric field and magnetic field along the horizontal line. The influence trend of the depth z on the internal wave wake electromagnetic field is the same as that shown in Figure 11 In (b) and Figure 11 In (c). Slightly different is that on the line of x = 1 m, the maximum value of the magnetic field appears at z = -0.3 m, while the maximum values of the magnetic field on other lines all appear at z = -0.4 m. This is mainly because only nine discrete values of the depth z are selected, and the depth at which the maximum value of the magnetic field appears will naturally be different on different lines, resulting in a certain difference.

[0216] (II) Experimental detection of the internal wave wake electromagnetic field

[0217] To verify the proposed calculation model and the simulation results of Example 1, an experimental verification of the internal wave wake electric field was carried out in this paper. The experimental device is as shown in Figure 13 .

[0218] The experimental facility is a water tank with a length of 20 meters, a width of 0.6 meters, and a height of 1.2 meters. The density stratification in the water tank consists of a 20 - centimeter fresh - water layer and a 70 - centimeter salt - water layer, as shown in Figure 3As shown in (a). The underwater target is made entirely of non-metallic materials and is connected to the upper slide rail by a non-metallic rod. The slide rail is controlled by a motor to ensure that the underwater target moves at a constant speed. The measuring device is a three-axis vector electric field sensor with an accuracy of, and the sensor is installed 0.4 meters above the bottom of the water tank and 0.15 meters away from the nearest water tank wall. A pair of Helmholtz coils, with a diameter of 0.8 meters and a spacing of 0.8 meters, are installed on both sides of the water tank to provide an excitation magnetic field. Figure 15 The experimental equipment is shown, including an electric field sensor, Helmholtz coils, an underwater target, a slide rail, etc.

[0219] Since the previous analysis showed that the internal wave wake magnetic field is too weak to be detected, the experiment verified the effects of different velocities U and submergence depths h on the internal wave wake electric field. The experimental analysis of this scheme mainly selected the measurement results of the electric field sensor in the z direction. The reason is that, as mentioned before, the velocity of the internal wave wake velocity field in the x and y directions is greater than that in the z direction (as Figure 7 shown). The direction of the excitation magnetic field added in the experiment is the y direction. According to the left-hand rule, it can be inferred that the electric field in the z direction is the most significant.

[0220] To evaluate the effect of velocity U, the experimental conditions were U = 0.15 m / s, 0.2 m / s, 0.25 m / s, and 0.3 m / s, h = 0.2 m, y = -0.15 m, z = -0.5 m. To evaluate the effect of submergence depth h, the experimental conditions were h = 0.1 m, 0.2 m, 0.3 m, and 0.4 m, U = 0.2 m / s, y = -0.15 m, z = -0.5 m. The three-axis vector electric field sensor measures the time variation of the voltage at its position, and the distance between the two electrodes is 0.11 meters. The underwater target is docked at the left end of the stratified seawater tank (as Figure 13 shown). The horizontal distance from the center position of the underwater target to the electric field sensor is 4.75 meters, and the electric field sensor is 15 meters away from the right end of the stratified seawater tank. It should be noted that Figure 13 This is only a schematic diagram of the experiment, and the position of the electric field sensor in the figure does not represent its actual position. The specific experimental details are subject to the introduction in this paragraph.

[0221] The specific experimental steps are as follows:

[0222] Start the motor without outputting power: First, start the motor that drives the underwater target to move, but do not make it output power to prevent the underwater target from moving. At the same time, turn on the electric field sensor and measure the ambient noise for a certain period of time at a sampling rate of 10 Hz. This can evaluate the noise levels of the water tank, motor, acquisition circuit, and other surrounding electrical appliances.

[0223] Start the experiment: Let the motor output power, and the underwater target starts to move. Measure the entire experiment process at a sampling rate of 10 Hz for a duration of 180 seconds. This measurement time is long enough to ensure that the underwater target can move from the left end to the right end of the water tank at the set speed.

[0224] Data interception and analysis: To ensure the accuracy of data analysis and the consistency of result presentation, for each set of experimental results, starting from the moment when the right end of the underwater target passes through the electric field sensor, intercept the subsequent 100 seconds of data for analysis.

[0225] The experimental results will be processed by variational mode decomposition (VMD). Among them, the number of modes K = 6, the bandwidth constraint parameter α = 9000, and the convergence tolerance tol = 1×10 -6 . This algorithm can decompose the signal into a series of intrinsic mode functions (IMFs) with different bandwidths, and the overlap between the bandwidths is minimized. In this paper, the fourth and fifth IMFs after VMD processing are selected as the final results. The experimental conditions are shown in Table 4. For each set of experiments, the experimental results are compared with the simulation results in the time domain and the frequency domain.

[0226] Table 4 TABLE IV Experimental conditions

[0227]

[0228]

[0229] Before the experiment starts, it is necessary to prepare stratified seawater and use the Figure 15 experimental device shown. On the left side of the stratified water tank is a brine tank. During the experiment, first inject tap water into the brine tank, and then add refined salt according to the calculated amount according to the preset salinity density. Use the stirring blade to stir the brine to promote the dissolution of salt, and finally obtain a brine solution with a density of 1030 kg / m 3 . The brine tank is connected to the stratified water tank through a pipeline at the bottom. When the brine tank is full and the stratified water tank is empty, open the pipeline switch and the water pump, and the brine will flow into the stratified water tank through the bottom inlet until the brine height in the stratified water tank reaches 70 cm.

[0230] There is a row of triangular "mushroom"-shaped devices above the stratified water tank, which are used to slowly and evenly add fresh water. These mushrooms are directly connected to the tap water pipeline and slowly release fresh water. During the process of adding fresh water, the lower surface of the mushroom should always be tangent to the current liquid surface. As the fresh water liquid level rises by 1 cm, use the lifting device to lift the mushroom by 1 cm until the fresh water height reaches 20 cm. Since the speed of adding fresh water is slow, the mixing degree between fresh water and brine is small. Correct experimental operations can ensure that the density stratification obtained in each experiment is almost the same.

[0231] After standing in the stratified water tank for a period of time, a vertically movable density probe, conductivity probe, and relative permittivity probe are used to measure the vertical distributions of density, conductivity, and relative permittivity in the water tank respectively. Since the density changes significantly in the pycnocline region, density measurements are taken at 1-cm intervals in the steep density gradient region from 10 cm to 30 cm depth. In the fresh water region (0 - 10 cm) and the salt water region (30 - 90 cm), density measurements are taken at 2-cm intervals. The same intervals are used for conductivity and relative permittivity measurements.

[0232] Figure 16 The time-domain diagram and power spectral density diagram of the ambient noise are shown. The data are from the measurement results in the z-direction of the electric field sensor. Figure 16 The data in (a) are the results after being processed by the VMD algorithm, with only the residuals removed. After calculating the power spectral density, we get Figure 16 in (b). From Figure 16 in (a), it can be seen that within the 300-s test time, the peak-to-peak value of the voltage is about 40 nV. Performing the ADF test on the data, the p-value is 0.001, indicating that the voltage data is stationary and the ambient noise is not strong. From Figure 16 in (b), it can be seen that the power spectral density at 1 Hz is about 4 nV / √Hz, which is a very low noise level and sufficient to meet the test conditions of the internal wave wake electric field described in this paper. The analysis of the ambient noise shows that the ambient noise of the experimental environment designed in this paper is low enough to complete the test experiment.

[0233] Figure 17 The experimental results at different speeds U are shown. Theoretically, when U = 0.15 m / s, 0.2 m / s, 0.25 m / s, and 0.3 m / s, the frequencies of the internal wave wake electric field are about 0.127 Hz, 0.1 Hz, 0.075 Hz, and 0.045 Hz respectively. From Figure 17 it can be seen that the simulation results of the internal wave wake electric field at different U are highly consistent with the experimental results.

[0234] As Figure 17 shown in (a), Figure 17 in (b), and Figure 17 in (c), when U = 0.15 m / s, 0.2 m / s, and 0.25 m / s, the simulation results and the experimental results overlap highly in the time domain, and the theoretical frequency points and the measured frequency points are very close in the frequency domain. Whether it is the trend of oscillation decay, the amplitude of the electric field, or the consistency of the frequency points in the frequency domain, they are all very satisfactory. This indicates that after the VMD processing of the original signal, the internal wave wake electric field signal is strong and significant enough to be classified as the fourth and fifth IMFs, and there is no overlap with the bandwidths of other IMFs. However, in Figure 17In Figure (d), when U = 0.3 m / s, the trends of the simulation results and the experimental results are highly consistent in the time domain, especially in the part after 20 seconds. In the first 20 seconds of the time domain, the experimental results are slightly smaller. In addition, some interfering frequency points appear in the frequency domain of the experimental results. This is mainly because as U increases, the volume wake of the underwater target generates stronger fluid perturbations, resulting in a more chaotic velocity field and increased noise. When U = 0.3 m / s, the noise is significantly enhanced, and some interfering signals are classified into the fourth and fifth IMFs. Fortunately, the overlapping of the frequency points of the simulation results and the experimental results can still be observed at 0.075 Hz.

[0235] According to the above experimental and analysis results, the verification experiment of velocity U shows that the experimental results are in good agreement with the simulation results, indicating that the proposed electromagnetic field calculation model of internal wave wake performs well.

[0236] Based on the electromagnetic field calculation model of internal wave wake proposed in this scheme, the real-time velocity and real-time depth of the underwater target, as well as the density, conductivity, and dielectric constant of the seawater layer where the underwater target is located, are obtained. Substituting the above parameters into the electromagnetic field calculation model, the electromagnetic field data of the internal wave wake of the underwater target at different depths and velocities can be calculated quickly and accurately.

[0237] After calculating the electromagnetic field data of the internal wave wake, an electric or magnetic field sensor with a suitable range can be selected according to the magnitude and attenuation law of the electromagnetic field for underwater measurement to monitor unknown underwater vehicles. For example, in a situation where it is necessary to maintain the marine security of our country, an array of electric field sensors can be laid on the seabed in the coastal waters of our country to collect electric field data in real time. When an underwater vehicle of other parties enters the monitoring area, the electric field sensor will detect the internal wave wake electric field with periodic oscillatory attenuation, thus discovering the target.

[0238] The technical solution of the present invention establishes an electromagnetic field calculation model of the internal wave wake generated by the movement of an underwater target in stratified seawater by considering the variations of the density, conductivity, and dielectric constant of seawater with depth. Substituting the depth and velocity of the underwater target obtained by real-time measurement and the parameters of the seawater layer where it is located, the electromagnetic field of the internal wave wake of the underwater target at different depths and velocities can be calculated quickly and accurately. It provides data support for the detection and identification of underwater targets, and improves the accuracy and efficiency of detection; at the same time, the calculated real-time electromagnetic field data of the internal wave wake can also be used to provide guidance for underwater targets to avoid monitoring. It provides an efficient and accurate detection method in the field of underwater target detection and improves the practicality.

[0239] Embodiment 2

[0240] The present invention also provides a measurement system for the electromagnetic field of internal wave wakes in density-stratified seawater, comprising: a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, it executes the method provided in any one of the first embodiments.

[0241] Embodiment III

[0242] The present invention also provides an electronic device, comprising: a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, it executes the method provided in any one of the first embodiments.

[0243] Embodiment IV

[0244] The present invention also provides a computer-readable storage medium, which includes a stored computer program. When the computer program is run by a processor, it controls the device where the storage medium is located to execute the method provided in any one of the first embodiments.

[0245] Embodiment V

[0246] The present invention also provides a computer program product, including a computer program / instructions. When the computer program / instructions are executed by a processor, they implement the method provided in any one of the first embodiments.

[0247] Those skilled in the art can easily understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for measuring the electromagnetic field of internal wave wakes in density-stratified seawater, characterized in that: include: Obtaining the current speed and diving depth of the underwater target, as well as the density, conductivity and dielectric constant of the seawater corresponding to the underwater target; Substituting the above parameters into the electromagnetic field calculation model to solve the induced electromagnetic field of the underwater target; The electromagnetic field calculation model is constructed based on the curl equation of the magnetic field, and the expression is: The spatial coordinate system is established with the sea level as the xy plane and the z axis perpendicular to the sea level. The value range of z is -H≤z≤0, where H represents the density boundary of seawater and z=0 represents the sea level. σ0(z) and ε(z) are the conductivity and dielectric constant of stratified seawater at z, respectively. is the induced electric field, is the Earth's magnetic field, is the induced magnetic field, is the internal wave wake velocity field, Related to the speed and depth of underwater targets.

2. The method according to claim 1, characterized in that The calculation model of the internal wave wake velocity field is: in, is the volume of the underwater target; L is the length of the underwater target, U is the speed of the underwater target, is the velocity of the point source model; The calculation formula of the velocity field of the point source model is: Among them, θ is the angle between the fluid particle and the x-axis, k is the wave number, and i is the imaginary unit. In the above formula: W1(-H)=e -kH ,W1′(-H)=the -kH , W2(0)=-1,W2′(0)=-σ, Where ρ0(z) is the density of stratified seawater at z, W′ represents the differential of W, W1 and W2 are accurately calculated by the fourth-order Runge-Kutta method, and g is the gravitational acceleration.

3. The method according to claim 1, characterized in that The process of solving the electromagnetic field calculation model includes: The curl of the electromagnetic field calculation model is calculated to obtain the intermediate equation: The expression for the induced magnetic field of the internal wave wake is constructed as: Where t represents time; ω0 is the oscillation frequency of the fluid particles, which is related to the speed U, wave number k and angle θ of the underwater target, and the expression is ω0 = kUcosθ; Construct the expression of the induced electric field of the internal wave wake: Substitute the expressions of the induced electric field and the induced magnetic field into the intermediate equation, and based on After simplification, the magnetic field integrand in the expression of the induced magnetic field of the internal wave wake is solved: The electric field integrand in the expression of the induced electric field of the internal wave wake is The current speed and diving depth of the underwater target, as well as the density, conductivity and dielectric constant of the seawater corresponding to the underwater target are substituted into the expressions of the induced magnetic field and induced electric field of the internal wave wake to obtain the calculation results of the electric field and magnetic field of the internal wave wake.

4. A system for measuring the electromagnetic field of internal wave wakes in density-stratified seawater, characterized in that: include: A memory and a processor, wherein the memory stores a computer program, and the processor executes the method provided in any one of claims 1 to 3 when executing the computer program.

5. An electronic device, characterized in that: include: A memory and a processor, wherein the memory stores a computer program, and the processor executes the method provided in any one of claims 1 to 3 when executing the computer program.

6. A computer-readable storage medium, characterized in that: The computer-readable storage medium includes a stored computer program, wherein when the computer program is executed by a processor, the device where the storage medium is located is controlled to execute the method provided in any one of claims 1 to 3.

7. A computer program product, characterized in that The invention comprises a computer program / instruction, which implements the method provided in any one of claims 1 to 3 when executed by a processor.

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