Solutions to a multi-phase model of sea ice growth

1 Introduction

The formation of sea ice plays an important role in the earth’s climate because it determines the large-scale heat and mass transport delivered to the surface of polar oceans. In the Arctic, temperatures are rising twice as fast as in other parts of the world. As sea ice has begun to melt, see, e.g., [1,2,3], that exposes more of the ocean’s surface to the air, which will be available to power storms. Storms surge into coastal areas, causing erosion and the increased flooding in low-lying areas. Sea ice is usually sandwiched between water at the bottom and a layer of snow at the top. Thus, we will investigate multi-phase dynamics of the sea ice growth. In this paper, we will apply the phase-field method to model the growth of sea ice. Although this method was developed in the 1980s, it has become a theoretical research tool in many fields, see, e.g., [4,5,6, 7,8,9]. As we all know, the application of the multi-phase model proposed in this paper in the study of sea ice growth is the first phase-field model of sea ice evolution.

The classic Stefan problem has been studied, see, e.g., [10]. They need to add appropriate conditions at the interface of the tracking movement. Theoretical analysis is very difficult. The ice formation in turbulent sea water was studied by Boussinesq approximation, see, e.g., [11]. The ice formation in dry snow has been studied, see, e.g., [12]. The temperature gradient is imposed on the snow, the snow microstructure changes due to heat transport, sublimation and resublimation in [12]. The seawater ice model of sea ice growth has been studied, see, e.g., [7]. Ice formation is a complex phenomenon. When ice forms, the geophysical and biological processes occur in the polar ocean and subsequent salt rejection and turbulent convection are ignored [7]. Its construction is inspired by the phase-field model of alloy solidification (see, e.g., [13]). The heat transfer problem plays an important role in the formation of sea ice [7]. Thermodynamics and solidification theory of alloys (see, e.g., [14]) are widely used in metallurgical and geological applications to study the solidification system. In this paper, the boundary evolution of sea water, snow, and ice is studied by using the phase-field theory including heat transfer and microscopic order parameter dynamics. Our construction is inspired by the phase-field model of multi-phase alloy solidification (see, e.g., [14]). When ice forms, the geophysical and biological processes occur in the polar ocean, and subsequent salt rejection and turbulent convection are ignored again. The model is generalization of the one introduced by Steinbach et al. [14], for isothermal solidification/melting process of kinds of alloys. The problem considered in the literature is most closely related to the present work, which is the solidification of alloys, in which the heat transfer needs to be dealt with. Thus, our model is the nonisothermal solidification process in this paper. In this paper, we contribute to only show that in one dimension the initial boundary value problem has solutions and long-time behavior.

According to the standard definition, the order parameter in the phase transition problem represents the non-zero property of the system in a different region of the phase space and 0 otherwise (see, e.g., [14]). The order parameter corresponds to the structural order of the solid-liquid interface on the atomic scale. Our model must satisfy the following system of partial differential equations:

for ( t , x ) ∈ ( 0 , ∞ ) × Ω . The boundary and initial conditions are as follows:

Here, Ω ⊂ R 3 is an open bounded domain. The function θ is the temperature, and the phase-field functions u , v , and w are the respective fractions of ice, snow, and water; thus, physically u + v + w = 1 . Here, the parameters k 1 , k 2 , k 3 , a 1 , a 2 , a 3 , and D are positive. We allow a temperature to be given a priori, but it must also be determined by the physical process that takes place. This means that the model considers the phase changes of sea ice caused by the temperature change. Thus, in order to close the system, it is necessary to include another equation of unknown temperature. The dimensionless temperature θ is scaled so that θ = 0 is the solidification temperature. In the free energy,

where

we choose for ψ ˆ ∈ C 2 ( R , [ 0 , ∞ ) ) , which represents the double-well potential. The function θ satisfies

where h ( u , v , w ) is nondecreasing smooth function satisfying h ( 0 , 0 , 0 ) ≡ 0 and h ( 1 , 1 , 1 ) ≡ 1 , e is the local enthalpy.

Due to the smear out of the phase-field variables p i , at the triple point, all three phase-fields have non-zero values, while at the dual-phase lines, only two phase-fields interact, see, e.g., [14]. In the change of solid-liquid interface, the order parameter changes with the change of material volume fraction. Within a system exhibiting three different phase states, for example, A, B, and liquid phases, the A vanishes at the transition to liquid and it changes to the B at the transition to the B phase. Thereby, we might interpret the B phase and liquid as “non A phase states.” It is easy to see this in the case of dual-phase systems, u = 1 − v , or u = 1 − w , or v = 1 − w and ∂ u ∂ v = − 1 , or ∂ u ∂ w = − 1 , or ∂ v ∂ w = − 1 .

Let us assume that all functions depend on variables x 1 and t . To simplify symbols, denote x 1 by x . The set Ω = ( a , b ) is an open bounded interval with constants a < b . We write Q T ≔ ( 0 , T e ) × Ω , where T e is a positive constant and define

for Z = Ω or Z = Q T e .

Since u + v + w = 1 and u t + v t + w t = 0 , the first three equations are not independent, and in the case of one space dimension, equations can be rewritten in the following form:

The boundary and initial conditions therefore are as follows:

The main results of this article are as follows.

Notation. In the following sections, we employ the letter C to denote any positive constants that can be explicitly computed in terms of known quantities and may change from line to line. The L 2 ( Ω ) -norm is denoted by ‖ ⋅ ‖ .

The paper is organized as follows: we will prove the existence of local solutions for the nonlinear equations (1.7)–(1.11) by using the Banach fixed-point theorem in Section 2. We shall establish a priori estimates for the solution in Section 3. We investigate the long-time behavior of a solution by the a priori estimates in Section 4.

2 Existence of local solutions

In this section, we obtain the existence and uniqueness of solutions by the use of the Banach fixed point theorem. Two theorems easily follow it. Now, we define the operator

which is defined by the following problem:

with the boundary and initial conditions (1.10)–(1.11) and ω = 1 − μ − ν .

We prove the existence and uniqueness of solutions by using the Banach fixed point theorem.

Here, we will prove that the solution space of equations (2.1)–(2.3) belongs to X .

Now, we prove that A is a contraction. We consider arbitrary ( μ 1 , ν 1 , ω 1 , ϑ 1 ) ∈ X , ( μ 2 , ν 2 , ω 1 , ϑ 2 ) ∈ X and denote A ( μ i , ν i , ω i , ϑ i ) = ( u i , v i , w i , θ i ) , for i = 1 , 2 , ( u ˜ , v ˜ , w ˜ , θ ˜ ) = ( u 1 − u 2 , v 1 − v 2 , w 2 , θ 1 − θ 2 ) . Then, ( u ˜ , v ˜ , w ˜ , θ ˜ ) satisfies