Robert S. Pritchard
IceCasting, Inc.
20 Wilson Court
San Rafael, CA 94901-1230
E-mail: pritchardr@asme.org
http://www.icecasting.com
Tel/fax: (415) 454-9899
Introduction
The concept of an oriented thickness distribution that depends on thickness and orientation follows from the assumption that all ice is oriented. Thermal growth is isotropic and it reduces differences between ice having different orientations. Therefore, thermal changes tend to form isotropic ice conditions. Of course, some ice, e.g., open water during the summer, can begin as isotropic ice, rather than result from thickness distributions that become isotropic over time. The oriented thickness distribution can accept all these ideas, including the isotropic ice category introduced by Coon, et al. [1998] to describe the older, thicker ice.
The oriented thickness distribution g( h, 
, x, t) is a probability density function that describes all ice in the
neighborhood R of location x at time t. The fraction of area covered by ice having thicknesses in the range h1 < h <
h2 and orientations in the range 
is
where we have neglected position and time for simplicity. This definition follows from the isotropic definition presented by Thorndike, et al. [1975].
What is the algorithm?
The oriented thickness distribution evolves thermally and mechanically according to the same balance law derived by Thorndike, et al. [1975]. Introducing a Lagrangian description is useful in developing the constitutive law. Thus, changes in thickness distribution satisfy
This description is not completely Lagrangian because the area fractions
are referred to the final total area, not an initial reference area. This
mixture is the reason that the 
term appears in the balance equation. If
the area fraction were normalized by an initial fixed total area, rather than
the instantaneous total area, this term would not appear. The
independent variable q is implicit throughout the governing equation. There is no derivative with respect to q.
The redistributor is not yet well-defined. Pritchard [1998] incorrectly assumed that ice is redistributed only within a single orientation. This is not true because, for example, an isotropic ice cover can be ridged into ice having a single orientation. The redistributor is actively being studied now.
We can introduce the most general form of redistributor as
where K is a kernel that defines the redistribution process. I am working on this definition now, and so there is no algorithm available at this time.
What data are needed?
The anisotropic plasticity constitutive law and the oriented thickness distribution model allow us to define lead formation and evolution explicitly. Ice condition must be defined to initialize a simulation or forecast of ice motion. The oriented thickness distribution needs information on lead direction and width, in addition to the usual thickness.
If the PIPS 3.0 operation assimilates data during its simulations, the ice behavior will improve if data on the oriented thickness distribution are included. The presence and orientation of leads will have a long lasting effect on ice behavior.
Is it practical?
Data on lead orientations are available from SAR imagery. An excellent sample of SHEBA data can be found at the Internet URL: www-radar.jpl.nasa.gov/rgps/radarsat.html.
Pritchard [1998] showed by example how the oriented thickness distribution can be described in eight sectors having 22.5 degree width and centered on the compass points. This angular resolution should be adequate for describing lead orientation.
Introducing an oriented thickness distribution will require computing resources. However, it is necessary if we are to simulate the formation and evolution of leads and ridges. In addition, PIPS 3.0 must include multiple categories in its thickness distribution because strength depends on the amount and thickness of the thinnest ice present. The two-category simplification is not capable of describing strength accurately. We need to have the WMO thickness categories at a minimum: 0, 10, 30, 70, 120, 200, 400 m.
Finally, if computer resources are inadequate to include all desired physics, then the compromises should consider all parameters in the model. It is foolish to dramatically improve ocean model resolution and ice model resolution unless the needed ice physics is included. These tradeoffs can be evaluated, and they must be.
Will it make a difference?
The oriented thickness distribution offers an opportunity to describe the formation and evolution of lead and ridge systems. The presence of leads is important to Navy operations. This practical requirement stands in addition to the more philosophical point that the anisotropic plasticity constitutive law and the orinted thickness distribution better describe the physical processes that effect ice behavior than do other models.
References
Coon, M. D., Knoke, G. S., Echert, D. C., and Pritchard, R. S. (1998) "An Oriented Thickness Distribution for Sea Ice," Submitted to J. Geophys. Res.
Pritchard, R. S. (1998) "Ice Conditions in an Anisotropic Sea Ice Dynamics Model," Int=l. J. Offshore and Polar Engineering, vol. 8, no. 1 pp. 9-15.
Thorndike, A. S., Rothrock, D. A., Maykut, G. A., and Colony, R. (1975) AThe Thickness Distribution of Sea Ice,@ J. Geophys. Res., vol. 80, no. 33, pp.4501-4513.