2.4. Tracers

Numerous tracers are available with the column physics. Several of these are required (surface temperature and thickness, salinity and enthalpy of ice and snow layers), and many others are options. For instance, there are tracers to track the age of the ice; the area of first-year ice, fractions of ice area and volume that are level, from which the amount of deformed ice can be calculated; pond area, pond volume and volume of ice covering ponds; a prognostic floe size distribution; snow density, grain size, and ice and liquid content; aerosols, water isotopes, and numerous other biogeochemical tracers. Most of these tracers are presented in later sections. Here we describe the ice age tracers and how tracers may depend on other tracers, using the pond tracers as an example.

2.4.1. Ice age

The age of the ice, \(\tau_{age}\), is treated as an ice-volume tracer (trcr_depend = 1). It is initialized at 0 when ice forms as frazil, and the ice ages the length of the timestep during each timestep. Freezing directly onto the bottom of the ice does not affect the age, nor does melting. Mechanical redistribution processes and advection alter the age of ice in any given grid cell in a conservative manner following changes in ice area. The sea ice age tracer is validated in [26].

Another age-related tracer, the area covered by first-year ice \(a_{FY}\), is an area tracer (trcr_depend = 0) that corresponds more closely to satellite-derived ice age data for first-year ice than does \(\tau_{age}\). It is re-initialized each year on 15 September (yday = 259) in the northern hemisphere and 15 March (yday = 75) in the southern hemisphere, in non-leap years. This tracer is increased when new ice forms in open water, in subroutine add_new_ice in icepack_therm_itd.F90. The first-year area tracer is discussed in [2].

2.4.2. Tracers that depend on other tracers

Tracers may be defined that depend on other tracers. Melt pond tracers provide an example (these equations pertain to topo tracers; level-ice tracers are similar with an extra factor of \(a_{lvl}\), see Equations (16)–(19)). Conservation equations for pond area fraction \(a_{pnd}a_i\) and pond volume \(h_{pnd}a_{pnd}a_i\), given the ice velocity \(\bf u\), are

(1)\[{\partial\over\partial t} (a_{pnd}a_{i}) + \nabla \cdot (a_{pnd}a_{i} {\bf u}) = 0,\]

(2)\[{\partial\over\partial t} (h_{pnd}a_{pnd}a_{i}) + \nabla \cdot (h_{pnd}a_{pnd}a_{i} {\bf u}) = 0.\]

These equations represent quantities within one thickness category; all melt pond calculations are performed for each category, separately. Equation (2) expresses conservation of melt pond volume, but in this form highlights that the quantity tracked in the code is the pond depth tracer \(h_{pnd}\), which depends on the pond area tracer \(a_{pnd}\). Likewise, \(a_{pnd}\) is a tracer on ice area (Equation (1)), which is a state variable, not a tracer.

For a generic quantity \(q\) that represents a mean value over the ice fraction, \(q a_i\) is the average value over the grid cell. Thus for topo melt ponds, \(h_{pnd}\) can be considered the actual pond depth, \(h_{pnd}a_{pnd}\) is the mean pond depth over the sea ice, and \(h_{pnd}a_{pnd}a_i\) is the mean pond depth over the grid cell. These quantities are illustrated in Figure Melt pond tracer definitions. The graphic on the right illustrates the grid cell fraction of ponds or level ice as defined by the tracers. The chart on the left provides corresponding ice thickness and pond depth averages over the grid cell, sea ice and pond area fractions.

Melt pond tracer definitions

Tracers may need to be modified for physical reasons outside of the “core” module or subroutine describing their evolution. For example, when new ice forms in open water, the new ice does not yet have ponds on it. Likewise when sea ice deforms, we assume that pond water (and ice) on the portion of ice that ridges is lost to the ocean.

When new ice is added to a grid cell, the grid cell total area of melt ponds is preserved within each category gaining ice, \(a_{pnd}^{t+\Delta t}a_{i}^{t+\Delta t} = a_{pnd}^{t}a_{i}^{t}\), or

(3)\[a_{pnd}^{t+\Delta t}= {a_{pnd}^{t}a_{i}^{t} \over a_{i}^{t+\Delta t} }.\]

Similar calculations are performed for all tracer types, for example tracer-on-tracer dependencies such as \(h_{pnd}\), when needed:

(4)\[h_{pnd}^{t+\Delta t}= {h_{pnd}^{t}a_{pnd}^{t}a_{i}^{t} \over a_{pnd}^{t+\Delta t}a_{i}^{t+\Delta t} }.\]

In this case (adding new ice), \(h_{pnd}\) does not change because \(a_{pnd}^{t+\Delta t}a_{i}^{t+\Delta t} = a_{pnd}^{t}a_{i}^{t}\).

When ice is transferred between two thickness categories, we conserve the total pond area summed over categories \(n\),

(5)\[\sum_n a_{pnd}^{t+\Delta t}(n)a_{i}^{t+\Delta t}(n) = \sum_n a_{pnd}^{t}(n)a_{i}^{t}(n).\]

Thus,

(6)\[\begin{split}a_{pnd}^{t+\Delta t}(m) &= {\sum_n a_{pnd}^{t}(n)a_{i}^{t}(n) - \sum_{n\ne m} a_{pnd}^{t+\Delta t}(n)a_{i}^{t+\Delta t}(n) \over a_i^{t+\Delta t}(m) } \\ &= {a_{pnd}^t(m)a_i^t(m) + \sum_{n\ne m} \Delta \left(a_{pnd}a_i\right)^{t+\Delta t} \over a_i^{t+\Delta t}(m) }\end{split}\]

This is more complicated because of the \(\Delta\) term on the right-hand side, which is handled in subroutine icepack_compute_tracers. Such tracer calculations are scattered throughout the code, wherever there are changes to the ice thickness distribution.

Note that if a quantity such as \(a_{pnd}\) becomes zero in a grid cell’s thickness category, then all tracers that depend on it also become zero. If a tracer should be conserved (e.g., aerosols and the liquid water in topo ponds), additional code must be added to track changes in the conserved quantity.

Tracer dependencies and conserved quantities associated with tracers are tracked using the arrays trcr_depend, which defines the type of dependency (area, volume, snow, etc), n_trcr_strata, the number of underlying layers, nt_strata, the indices of the underlying layers, and trcr_base, a mask that is one for the tracer dependency and zero otherwise. These arrays are used to convert between the tracer values themselves and the conserved forms.

More information about the melt pond schemes is in the Melt ponds section.