Ice-shelf vibrations modeled by a full 3-D elastic model (Version 8.2)

1 Ice-shelf vibrations modeled by a full 3-D elastic model: field equations (version 8.2) Y.V. Konovalov ORCID: 0000-0002-8469-9706 Department of Mathematics, MIREA - Russian Technological University, Vernadsky avenue 78, Moscow, Russian Federation, 119454 Correspondence to: Y.V. Konovalov ( [email protected] ) 2 Abstract This version as the previous version 8.1 (https://doi.org/10.5281/zenodo.17069606) considers the modeling of the oscillation of a system consisting of three components: (i) an ice shelf, (ii) seawater in a cavity under the ice shelf and (iii) meltwater lakes filling the valleys on the ice shelf surface. The difference is in the ice shelf geometry, which corresponds to an idealized ice surface morphology with valleys and hills arranged in a checkerboard pattern. As in the previous version 8.1, this model is based on the finite volume method of approximation the momentum equations, as were considered in versions 5 and 6 (https://doi.org/10.5281/zenodo.7697142, https://doi.org/10.5281/zenodo.10252877, https://doi.org/10.1017/aog.2024.47) Both the sub-ice seawater flow and meltwater flow in lakes are treated as a “potential flow”. Accordingly, in this model, as in the previous versions 7 and 8.1, both the subglacial flow of seawater and the flow of meltwater in lakes (long gravity waves on the surface of meltwater lakes) are described by the corresponding wave equations. This version is essentially an extension of the previous version 8.1, and the same experiments were performed with the new ice shelf geometry. As in the previous versions, the oscillations of the three-component system were modeled for harmonic incoming pressure perturbations and for a wide range of incoming wave frequencies. As in the previous versions, the amplitude spectra obtained using the model reveal distinct resonant peaks, as were also observed in previous models. Accounting for the periodic structure of the ice shelf geometry, the obtained dispersion spectra reveal the presence of band gaps (https://doi.org/10.3189/2012AoG60A120) as was observed, in particular, in models applied to an ice shelf corresponding to the geometry of Ward Hunt Ice Shelf, Ellesmere Island, Canadian Arctic (https://doi.org/10.1017/jog.2023.58, https://doi.org/10.1017/aog.2024.47). 3 1. Model description and field equations This item (mathematical model description) fully corresponds (and repeats it) to the item 1 from version 8.1 (https://doi.org/10.5281/zenodo.17069606). 1.1 Basic equations 1.1.1 Momentum equations described the motion of an ice shelf In this model, ice shelf deformations are described using the momentum equations written as the momentum balance equation in the volume 𝑉 (bounded by the surface 𝑆) of an elastically deformable continuous medium have the following form (e.g. [1], [2], [3], [4]) (https://doi.org/10.5281/zenodo.7697142) 𝜕2 𝜕 𝑡2∫𝜌 𝑈𝑖 𝑑𝑉 𝑉=∫(𝜕𝜎𝑖𝑘 𝜕𝑥𝑘+𝜌𝑔𝑖)𝑑𝑉 𝑉 (1.1) or 𝜕2 𝜕 𝑡2∫𝜌 𝑈𝑖 𝑑𝑉 𝑉=∫𝜎𝑖𝑘𝑑𝑆𝑘 𝑆+∫𝜌𝑔𝑖𝑑𝑉 𝑉 (1.2) where 𝝈 is the stress tensor; and 𝜌 is the ice density; 𝑖,𝑘=1,2,3 or in terms of rectangular coordinate (𝑋𝑌𝑍) 𝑖,𝑘 means 𝑥,𝑦,𝑧; 𝑈𝑖 are displacements of an elastic continuous medium (displacements of ice) which are also denoted in rectangular coordinates as 𝑈,𝑉,𝑊: 𝑈,𝑉 and 𝑊 are two horizontal and one vertical ice displacements, respectively. As in previous versions/models (𝑋𝑌𝑍) is a rectangular coordinate system with the 𝑋-axis along the center line, and 𝑍-axis pointing vertically up. The ice shelf has a length L along the center line. The geometry of the ice shelf is assumed to be given by lateral boundary functions 𝑦1,2(𝑥) at sides labeled 1 and 2 and functions for the surface and base elevation, 4 ℎ𝑠,𝑏(𝑥,𝑦), denoted by subscripts s and b, respectively. Thus, the domain, which includes the volume of integration in Eqs. (1), is 𝛺={0<𝑥<𝐿, 𝑦1(𝑥)<𝑦<𝑦2(𝑥), ℎ𝑏(𝑥,𝑦)< 𝑧<ℎ𝑠(𝑥,𝑦)}. 1.1.2 Wave equation described sub-ice seawater flow in the cavity Sub-ice water is assumed to be an incompressible inviscid fluid of uniform density. Another assumption is that water depth in the cavity below the ice shelf changes gradually in the horizontal directions. Thus, the ice-front and other such features are not considered here. Moreover, the ice is considered to be a continuous solid elastic plate. Under these three assumptions, sub-ice water flow is independent of z in a vertical column. Manipulating the governing equations of the shallow sub-ice water layer yields the wave equation (https://doi.org/10.1038/274464a0): 𝜕2𝑊𝑠𝑙 𝜕 𝑡2= 1 𝜌𝑤 𝜕 𝜕 𝑥 (𝑑0𝜕 𝑃′ 𝜕 𝑥) + 1 𝜌𝑤𝜕 𝜕 𝑦(𝑑0𝜕 𝑃′ 𝜕 𝑦), (2) where 𝜌𝑤 is sea water density; 𝑑0(𝑥,𝑦) is the depth of the sub-ice water layer; 𝑊𝑠𝑙(𝑥,𝑦,𝑡) is the vertical deflection of the surface of sub-ice water layer and 𝑊𝑠𝑙(𝑥,𝑦,𝑡)= 𝑊𝑏(𝑥,𝑦,𝑡)=𝑊(𝑥,𝑦,ℎ𝑏(𝑥,𝑦),𝑡) (𝑊𝑏(𝑥,𝑦,𝑡) is the vertical deflection of the ice-shelf base) and 𝑃′(𝑥,𝑦,𝑡) is the deviation of the sub-ice water pressure from the hydrostatic value. 1.1.3 Wave equation described gravity waves on surfaces of the meltwater lakes In this model, melt water in lakes is also considered as an incompressible inviscid fluid of uniform density. It is assumed that gravity waves in lakes are described by the wave 5 equation for long gravity waves. The corresponding manipulation of the governing equations for the shallow meltwater layer in the lakes yields the wave equation (e.g. [2]): 𝜕2𝜁 𝜕 𝑡2−𝜕2𝑊𝑠 𝜕 𝑡2=𝑔(𝜕 𝜕𝑥(ℎ0𝜕𝜁 𝜕𝑥)+ 𝜕 𝜕𝑦(ℎ0𝜕𝜁 𝜕𝑦)) , (3) where 𝜁(𝑥,𝑦,𝑡) is the vertical deflection of the water surface in lakes (from the equilibrium surface level); ℎ0(𝑥,𝑦) is the depth of the meltwater lakes; 𝑊𝑠(𝑥,𝑦,𝑡) is the vertical deflection of the ice-shelf surface, and 𝑊𝑠(𝑥,𝑦,𝑡)=𝑊(𝑥,𝑦,ℎ𝑠(𝑥,𝑦),𝑡). 1.2 Boundary conditions The boundary conditions for equations (1) are: (i) stress free ice surface and normal stress exerted by meltwater in regions on the ice surface where there are meltwater lakes; (ii) the normal stress exerted by seawater at the ice-shelf free edges and at the iceshelf base; and (iii) rigidly fixed edges at the grounding line of the ice-shelf. In this model, the boundary conditions are also (as in previous versions) considered as a linear combination 𝛼1 𝐹𝑖(𝑈,𝑉,𝑊)+ 𝛼2 Φ𝑖(𝑈,𝑉,𝑊)=0, 𝑖=1,2,3, (4) where: 6 (i) 𝐹𝑖(𝑈,𝑉,𝑊)=0 is the typical and well-known form of the boundary conditions where, for example, the condition on the ice-shelf surface is expressed as 𝜎𝑖𝑘 𝑛𝑘=0 or 𝜎𝑖𝑘 𝑛𝑘=−𝑃𝑤𝑛𝑖 (𝑛 󰇍 is the unit vector normal to the surface, 𝑃𝑤 is the meltwater pressure); (ii) Φ𝑖(𝑈,𝑉,𝑊)=0 is the approximation based on the application of a finite volume approach to the approximation of the boundary conditions, taking in to account for the typical form of boundary conditions on the boundaries of an elementary volume (https://doi.org/10.5281/zenodo.10252877); and (iii) the coefficients 𝛼1 and 𝛼2 satisfy the condition 𝛼1+ 𝛼2=1. The boundary conditions for the subglacial seawater layer (for equation (2)) correspond to a frontal gravity incident ocean wave. They are (i) at 𝑥=0: 𝜕𝑃′ 𝜕𝑥 =0; (ii) at 𝑦=𝑦1, 𝑦=𝑦2: 𝜕𝑃′ 𝜕𝑦 =0; (iii) at 𝑥=𝐿: 𝑃′=𝐴0 𝜌𝑤𝑔 𝑒𝑖𝜔𝑡, where 𝐴0 is the amplitude of the incident wave. The boundary conditions for equation (3) correspond to the assumption that along the coastlines of the meltwater lakes the vertical deflection of the water surface 𝜁(𝑥,𝑦,𝑡) is approximately equal to the vertical deflection of the ice-shelf surface 𝑊𝑠(𝑥,𝑦,𝑡), i.e. the boundary conditions are 𝜁(𝑥,𝑦,𝑡)=𝑊𝑠(𝑥,𝑦,𝑡) along the coastlines. 1.3 Discretization of the model 7 Numerical solutions were obtained by a finite volume method, which is based on the standard coordinate transformation 𝑥,𝑦,𝑧 →𝑥,𝜂= 𝑦−𝑦1 𝑦2−𝑦1,𝜉=(ℎ𝑠−𝑧)/𝐻, where 𝐻 is the ice thickness (𝐻=ℎ𝑠−ℎ𝑏). The coordinate transformation maps the ice domain Ω into the rectangular parallelepiped Π={0≤𝑥≤𝐿;0≤𝜂≤1;0≤𝜉≤1}, which simplifies the numerical discretization. A detailed description of the approximation of the momentum equations (1) was presented in version 5 (https://doi.org/10.5281/zenodo.7697142). 1.4 Equations for ice-shelf displacements Constitutive relationships between stress tensor components and displacements correspond to Hooke's law (e.g., [3], [4]): 𝜎𝑖𝑗 =𝐸 1+𝜈(𝑢𝑖𝑗 +𝜈 1−2𝜈 𝑢𝑙𝑙𝛿𝑖𝑗) , (5) where 𝑢𝑖𝑗 are the strain components, 𝐸 - Young's modulus, 𝜈 - Poisson's ratio. 1.5 Ice-shelf harmonic vibrations. The eigenvalue problem. (Essentially, this point repeats the assumptions made in previous versions [6-9]. All assumptions of this point that were valid for the previous models, are also valid for the model considered here) It is assumed that for harmonic vibrations all variables are periodic in time, with the periodicity of the incident wave (of the forcing) given by the frequency 𝜔, i.e., 𝜍(𝑥,𝑦,𝑧,𝑡)=𝜍(𝑥,𝑦,𝑧) 𝑒𝑖𝜔𝑡, (7) where 𝜍={𝑈,𝑉,𝑊, 𝜎𝑖𝑗,𝜁}, 8 where we are interested in the real part of the variables expressed in complex form. This assumption also implies that the full solution of the linear partial differential Eqs. (1), (2), (3) is a sum of the solution for the steady-state flexure of the ice shelf and solution (7) for the time-dependent problem. In other words, solution (7) implies that the deformation due to the gravitational forcing can be separated from the vibration problem, i.e. the term ρg in the third equation from (1) as well as the appropriate terms in the boundary conditions (4) are absent from the final equations formulated for the vibration problem, because a time-independent solution accounting for them applies and is not of interest in this study. The separation of variables in Eq. (7) and its substitution into Eqs. (1), (2), (3) yields the same equations, but with the operator 𝜕2 𝜕 𝑡2 replaced with the constant −𝜔2, i.e. we obtain equation for 𝜍(𝑥,𝑦,𝑧): ℒ 𝜍=−𝜔2𝜍, (8) where ℒ is a linear partial differential operator. The numerical solution of Eq. (8) at different values of 𝜔 yields the dependence of 𝜍 on the frequency of the forcing 𝜔. When the frequency of the forcing converges to the eigenfrequency of the system, we observe the typical rapid increase of deformation/stresses in the spectra in the form of the resonant peaks. Note that here, the term “eigenvalue” refers to the eigenfrequency (𝜔𝑛) of the ice/water system or corresponding periodicity (𝑇𝑛=2𝜋 𝜔𝑛). As mentioned previously, the term “eigenvalue” is employed in the same meaning like in a Sturm-Liouville Eigenvalue Problem (e.g. [10]). Eigenvalues (where resonant peaks would be observed) are denoted by the letters 𝜔𝑛 or 𝑇𝑛 with the subscript 𝑛 (or other), which is integer, because the array of the eigenvalues is a countable set. 9 Letters 𝜔 or 𝑇 without the subscript denote the non-resonant values of frequency or periodicity of the ice/water system. They are defined by the frequency of the incident wave (of the forcing). The eigenvalues can be derived from the equation 𝐷(𝜔)=0, where 𝐷 is the determinant of the matrix, which results from the discretization of Eq. (8) and of the corresponding boundary conditions. However, the probability of the appearance of the forcing at any specific frequency is practically zero. This can be seen when we consider only events within the frequency range (𝜔𝑖−∆𝜔, 𝜔𝑖+∆𝜔). The probability of a forcing that is within the frequency range, is non-zero: 𝑝{𝜔∈(𝜔𝑖−∆𝜔, 𝜔𝑖+∆𝜔)}=2∆𝜔 Ω , (9) where Ω is the width of the range in omega space, which includes all possible frequencies of the forcing. Eq. (9) also assumes that the events have equal probabilities in different parts of Ω. Thus, the probability of the resonant-like motion is higher when the value Δ𝜔, which is defined by the width of the resonant peak, is higher too. Therefore, the width of the resonant peaks is an important parameter, from a practical standpoint, because it defines the probability of the suitable resonant-like motion. Computation of the spectra, such as provided below, thus provides important information about the width of resonant peaks within the likely range of forcing frequencies found in nature. By assessing the widths of such peaks, a better understanding of the probability that any one specific forcing event, at a specific 𝜔 can be assessed. 16 Amplitude of ice thickness oscillations 𝑨𝑯=𝟏𝟐 𝒎 (see Figure 1). 𝜶𝟏=𝟏.𝟎,𝜶𝟐=𝟎.𝟎. Fig. 3a Fig. 3b Fig. 3c Figure 3. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟎𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 17 Fig. 4a Fig. 4b Fig. 4c Figure 4. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟓𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 18 Fig. 5a Fig. 5b Fig. 5c Figure 5. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟎𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 19 Fig. 6a Fig. 6b Fig. 6c Figure 6. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟓𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 20 Fig. 7a Fig. 7b Fig. 7c Figure 7. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 21 Fig. 8a Fig. 8b Fig. 8c Figure 8. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟒𝟎𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 22 Fig. 9a Fig. 9b Fig. 9c Figure 9. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 23 Fig. 10a Fig. 10b Fig. 10c Figure 10. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟔𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 24 Fig. 11a Fig. 11b Fig. 11c Figure 11. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)−𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏=𝟏.𝟎,𝜶𝟐= 𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 25 Fig. 12a Fig. 12b Fig. 12c Figure 12. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟖𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 32 Fig. 19a Fig. 19b Fig. 19c Figure 19. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟓𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 33 Fig. 20a Fig. 20b Fig. 20c Figure 20. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 34 Fig. 21a Fig. 21b Fig. 21c Figure 21. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟓𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 35 Fig. 22a Fig. 22b Fig. 22c Figure 22. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 36 Fig. 23a Fig. 23b Fig. 23c Figure 23. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟒𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 37 Fig. 24a Fig. 24b Fig. 24c Figure 24. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 38 Fig. 25a Fig. 25b Fig. 25c Figure 25. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟔𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 39 Fig. 26a Fig. 26b Fig. 26c Figure 26. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 40 Fig. 27a Fig. 27b Fig. 27c Figure 27. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟖𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 41 Fig. 28a Fig. 28b Fig. 28c Figure 28. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟗𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟐 𝒎. 48 Fig. 35a Fig. 35b Fig. 35c Figure 35. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 49 Fig. 36a Fig. 36b Fig. 36c Figure 36. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟓𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 50 Fig. 37a Fig. 37b Fig. 37c Figure 37. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 51 Fig. 38a Fig. 38b Fig. 38c Figure 38. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟒𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 52 Fig. 39a Fig. 39b Fig. 39c Figure 39. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 53 Fig. 40a Fig. 40b Fig. 40c Figure 40. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟔𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 54 Fig. 41a Fig. 41b Fig. 41c Figure 41. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 55 Fig. 42a Fig. 42b Fig. 42c Figure 42. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟖𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 56 Fig. 43a Fig. 43b Fig. 43c Figure 43. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟗𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 57 Fig. 44a Fig. 44b Fig. 44c Figure 44. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟏𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 64 Fig. 51a Fig. 51b Fig. 51c Figure 51. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟐𝟓𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 65 Fig. 52a Fig. 52b Fig. 52c Figure 52. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 66 Fig. 53a Fig. 53b Fig. 53c Figure 53. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟒𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 67 Fig. 54a Fig. 54b Fig. 54c Figure 54. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 68 Fig. 55a Fig. 55b Fig. 55c Figure 55. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟔𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 69 Fig. 56a Fig. 56b Fig. 56c Figure 56. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 70 Fig. 57a Fig. 57b Fig. 57c Figure 57. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟖𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 71 Fig. 58a Fig. 58b Fig. 58c Figure 58. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟗𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 72 Fig. 59a Fig. 59b Fig. 59c Figure 59. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟏𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 73 Fig. 60a Fig. 60b Fig. 60c Figure 60. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟒 𝒎. 80 Fig. 67a Fig. 67b Fig. 67c Figure 67. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 81 Fig. 68a Fig. 68b Fig. 68c Figure 68. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟒𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 82 Fig. 69a Fig. 69b Fig. 69c Figure 69. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 83 Fig. 70a Fig. 70b Fig. 70c Figure 70. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟔𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 84 Fig. 71a Fig. 71b Fig. 71c Figure 71. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 85 Fig. 72a Fig. 72b Fig. 72c Figure 72. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟖𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 86 Fig. 73a Fig. 73b Fig. 73c Figure 73. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟗𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 87 Fig. 74a Fig. 74b Fig. 74c Figure 74. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟏𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 88 Fig. 75a Fig. 75b Fig. 75c Figure 75. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 89 Fig. 76a Fig. 76b Fig. 76c Figure 76. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟏.𝟎,𝜶𝟐=𝟎.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 96 Fig. 83a Fig. 83b Fig. 83c Figure 83. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟒𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 97 Fig. 84a Fig. 84b Fig. 84c Figure 84. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 98 Fig. 85a Fig. 85b Fig. 85c Figure 85. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟔𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 99 Fig. 86a Fig. 86b Fig. 86c Figure 86. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 100 Fig. 87a Fig. 87b Fig. 87c Figure 87. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟖𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 101 Fig. 88a Fig. 88b Fig. 88c Figure 88. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟗𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 102 Fig. 89a Fig. 89b Fig. 89c Figure 89. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟏𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 103 Fig. 90a Fig. 90b Fig. 90c Figure 90. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟑𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 104 Fig. 91a Fig. 91b Fig. 91c Figure 91. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟓𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 105 Fig. 92a Fig. 92b Fig. 92c Figure 92. a) vertical deflection of the surface of the system: ice shelf and meltwater lakes (𝜁(𝑥,𝑦,𝜔)) b) ice shelf base vertical deflection; c) the difference 𝜁(𝑥,𝑦,𝜔)− 𝑊𝑠(𝑥,𝑦,𝜔). The periodicity of the forcing 𝑻=𝟏𝟕𝟎𝒔. The parameters of the model are 𝜶𝟏= 𝟎.𝟎,𝜶𝟐=𝟏.𝟎; 𝑨𝑯=𝟏𝟔 𝒎. 112 30. Godin, O. A, Zabotin, N. A.: Resonance vibrations of the Ross Ice Shelf and observations of persistent atmospheric waves. J. Geophys. Res. Space Phys., 121, 10157-10171, doi:10.1002/2016JA023226, 2016 31. Goodman, D.J., Wadhams, P., & Squire, V.A.: The flexural response of a tabular ice island to ocean swell. 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