conditionalIceModel.pdf

A Consistent Derivation of the Sea-Ice Model

Using Conditional Averaging

Hrvoje Jasak ¤

10th March 2003

Abstract

In this paper provides consistent derivation of a generic sea ice dy-

namics model using a mathematical technique called conditional averag-

ing [Dopazo, 1977]. The model equations are derived directly from the

mass and momentum balance for a continuum partially ¯lled with ice.

The new form of the model is then compared with the traditional form as

given by Hibler [Hibler, 1979], with the objective of analysing the mathe-

matical behaviour of the two models, speci¯cally regarding the ice fraction

equation. The conditional model implies a di®erent and bounded form of

transport, where the ice fraction equation is in a bounded non-conservative

form. The derivation additionally provides hints on the form of wind forc-

ing and ice-ocean interaction terms and the modelling of sub-grid scale ice

interaction.

1 Introduction

A bulk of sea-ice modelling work today is based on the generic form of the sea

ice transport model as described by Hibler [Hibler, 1979], with a number notable

variants of rheological constitutive laws and ice distribution statistics. In par-

ticular, sea ice models with an elliptical viscous{plastic constitutive relation are

found to reproduce observed ice drift well [Hutchings et al., 2002].

The work presented in this paper is a result of the author's frustration with

the basic set of equations and their mathematical behaviour. In a recent work

[Hutchings et al., 2002,Hutchings, 2000], the Hibler model has been implemented

usingtheboundedandconservativeFiniteVolume(FV)methodofdiscretisation,

where some previously unseen inconsistencies have emerged. Moreover, in spite

of my best e®orts, I have been unable to track down the original derivation

¤ Nabla Ltd, The Mews, Picketts Lodge, Picketts Lane, Salfords, Surrey, RH1 5RG, United

Kingdom

2 Description of the Hibler Sea Ice Model

of the governing equations given in [Hibler, 1979]. Their exact form seems to

have been accepted as standard by most practitioners in the ¯eld, leading to

some mis-conceptions regarding the behaviour of the model and its, necessarily

imperfect, numerical implementation. The basic premise of the work is that it is

¯rst necessary to examine the model in its di®erential form in detail (particularly

in terms of boundedness and interaction between equations) before applying the

necessary numerical modelling techniques.

In this paper, we shall examine the basic form of the Hibler model from a

mathematicalstandpointbyderivingthegoverningequationsfrom¯rstprinciples

usingatechniquecalledconditionalaveraging[Dopazo, 1977]. Thisisdoneinthe

expectation that the conditional model will yield the identical set of equations as

the traditional form [Hibler, 1979].

Conditionalaveragingisarelativelynovelmathematicaltoolusedforthepur-

pose of describing two- or multi-phase systems described as several intertwining

continua. So far, it has been successfully implemented to various areas of con-

tinuum modelling, including multi-phase °ows [Hill, 1998,Hill et al., 1994,Hill

et al., 1995], combustion [Weller, 1993], astrophysics [Grulke et al., 2001] etc.

The rest of this paper is organised as follows. In Section 2 the viscous-plastic

rheologymodelwillbesummarisedandsomegeneralstatementsonitscharacter-

isticswill bemade. Weshallthenstripthemodeltoitsbasicpremises, excluding

the thermodynamic source/sinks of ice and limiting ourselves to generic rheol-

ogy. Section 3 describes the basic premise of the model derivation: a continuum

partially ¯lled with ice with leads of open water between ice °oes and states the

equations governing the behaviour of solid ice. In Section 3.1 we shall introduce

the conditional averaging operators and state the transformations of various dif-

ferential operators. The conditional averaging of ice equations will be performed

in Section 4, ¯nally stating the new conditional model. The model will be com-

pared with the original model by Hibler and the mathematical characteristics of

the two models will be examined in Section 5. The paper is completed with a

short summary in Section 6.

2 Description of the Hibler Sea Ice Model

The viscous{plastic sea ice model was originally developed by Hibler [Hibler,

1979]. For daily time scales the momentum balance is taken to be:

@(mu)

+r ² (muu) = F +r ² ¾;

(1)

where m is the ice mass per unit area, u is ice velocity and F is the sum of

body and surface forces on the ice, including the wind stress and ocean stress on

the ice surface, Coriolis force and force due to gravitational acceleration down

the sea surface slope. The sub-grid scale ice interaction, ¾, is modelled as a

2 Description of the Hibler Sea Ice Model

viscous{plastic material obeying the associated °ow rule and an elliptical yield

criterion [Hibler, 1979].

¾ = ´[ru +ru T ]+[(³ ¡ ´)Itr(ru)]¡ I P

2 ;

(2)

where the bulk (³) and shear (´) viscosities are given by

³ = P

2¢ ;

(3)

´ = ³

e 2

(4)

and ¢ is de¯ned as

tr(_") 2 + 2

¢ =

e 2 _" : _"

(5)

is the strain rate, I is the identity

matrix and q : q represents the scalar product of two second-rank tensors.

Ice mass continuity is described by the transport equations for e®ective ice

thickness, h = ½ where ½ is ice density, and A the ice fraction. Here, a two level

thickness model [Hibler, 1979] is used and the sources of ice thickness, S h , and

area, S A , are calculated from climatological growth/melt rates [Thorndike et al.,

1975].

ru +(ru) T

@t +r ² (hu) = S h ;

(6)

@t +r ² (Au) = S A :

(7)

The thermodynamic source terms, S h and S A , are given as [Hibler, 1979]:

S h = AF

+(1¡ A)F (0);

(8)

2h S h ;0

F(0)(1¡ A)

h 0

S A = min

+max

(9)

Here, F is a function describing the thermodynamic rate of change of ice thick-

ness, taken from a look-up table [Thorndike et al., 1975] or modelled in some

other way. Care should be taken to ensure S h and S A do not allow h or A to

fall outside physical bounds [Hutchings, 2000]. Note that neither S h nor S A are

a function u.

To close the system of equations P must be determined. An empirical rela-

tionship for ice strength of the two level model is given by [Hibler, 1979]:

P = P ? he ¡C(1¡A) :

(10)

Here, e is the ratio of semi-major to semi-minor axes of the elliptical yield curve,

P is the ice strength, _" = 2

3 Derivation of the Sea Ice Transport Model

In further discussion, the exact form of the rheology and the thermodynamic

coupling is not of interest, as it does not e®ect the derivation we are about to

present. Without the loss of generality, we shall therefore limit the Hibler model

to the following generic form:

@(mu)

+r ² (muu) = r ² ¾;

(11)

@t +r ² (hu) = 0;

(12)

@t +r ² (Au) = 0:

(13)

The necessary sources/sinks and rheology model may be added later.

3 Derivation of the Sea Ice Transport Model

In this section, we shall attempt to derive the equivalent of Eqs. (11, 12 and 13)

from ¯rst principles. For this purpose, we need to postulate that the behaviour

of sea ice will be examined on the scale larger than the scale of individual ice

°oes and the leads between them. In this way, the lowest-level control volume of

interest is assumed to contain both the ice and the free water and the behaviour

of the system is averaged over it. The limits of \ice-only" and \open water" are

naturally included in the derivation.

We can state the governing equations for the ice phase:

² Conservation of mass (½ = const:):

@½

@t +r ² (½u) = 0;

(14)

where ½ is the density of ice;

² Conservation of momentum:

@(½u)

+r ² (½uu) = r ² ¾:

(15)

Note that u and ¾ are equal to the ice velocity and ice interaction tensor where

the ice is present and unde¯ned in the absence of ice.

3.1 Conditional Averaging

Conditional averaging is applied to the sea ice system, consisting of ice °oes and

open water by considering separately the ice, the open water and the interface

between them. Consider a control volume containing a sea ice fragment and a

lead of open water, Fig. 1.

3.2 Conditional Averaging Operators

Control

Volume

U n

Open water

Ice

Interface

Figure 1: Control volume containing ice and open water.

The velocity of the ice-water interface is denoted u n and we shall denote the

ice velocity with u, keeping them independent at the moment.

Additionally, we will need to describe the distribution of the ice °oes in the

domain, including the sub-grid scale. For this purpose, we shall introduce the

indicator function I(x;t) as

I(x;t) =

(

1 if point (x;t) is covered by ice,

0 if point (x;t) is in open water.

(16)

Equations are conditionally averaged [Weller, 1993] by multiplying by the

indicator function I and then applying conventional averaging techniques. This

approach is a simple extension of that applied to intermittent turbulent °ows by

Dopazo [Dopazo, 1977]. With the use of the indicator function, and by allowing

the phase interface to propagate, the analytical methods developed by Dopazo

can be applied to the present problem [Dopazo, 1977].

The indicator function can be viewed in several ways, the most intuitive of

which is that its average over the control volume represents the area fraction of

ice per unit area, i.e. ° = A from Eqn. (13):

° = I(x;t);

(17)

where the over-bar denotes the ensemble average. ° can also be seen as the

probability of point (x;t) being covered by ice.

3.2 Conditional Averaging Operators

Let Q(x;t)beanyphysicalproperty, scalarortensorofanyrank. Itthenfollows:

rQI = r(QI) = r(°Q ° );

(18)

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: