Errors and Approximations in Thermodynamics

Melting and freezing at the ice shelf base is a thermodynamic process, whereby heat energy is removed from the ocean to melt ice, and vice versa. This occurs at the thin liquid water layer immediately adjacent to the ice interface, which is not resolvable by a standard numerical ocean model. Hence, parameterisations of the thermodynamics must be introduced (see Section 2.4), and along with that, uncertainties and assumptions. The first uncertainty that is implicitly included is in the freezing point of water.

The freezing temperature of water (TB) as employed in the three-equation param-

eterisation (Holland and Jenkins, 1999), is the linearised version of the true, non-

linear relationship. Since TB is integral to numerically solving for the temperature

and salinity characteristics at the ice–ocean interface, it is computationally efficient

to use the linearised equation. Linearisation of TB introduces an error, as shown in

Fig. 1.7, which becomes more important with depth.

The choice of description of the temperature profile through the ice shelf is also important for the melt rate, as heat is conducted from the ice–ocean interface into the ice shelf while melting and freezing are occurring. The ice shelf temperature

profile is constrained at the surface by the ice shelf surface temperature (∼ −20◦C,

CHAPTER 1. INTRODUCTION 14

Figure 1.7: The difference between linearised and non-linearised freezing tempera- ture (TEOS-10; IOC and SCOR and IAPSO, 2010) as a function of depth is shown (with salinity constant at 34.4 psu).

Observations and modelling show the temperature profile between the surface and base is dependent on melt rate (Humbert, 2010). Strong melting causes rapid tem- perature drop to cold internal temperatures over a short length, as ice is being removed before the temperature of the interior can equilibrate. For freezing regions

the curve is reversed, with temperatures close toTB for most of the profile, before

it rapidly asymptotes to the surface temperature. To correctly parameterise heat

flux into the ice shelf, the gradient of ice shelf temperature (TI) evaluated at the

base of the ice shelf (z=b),

∂TI ∂z b (1.1) must be determined. Four approximations for this profile are investigated; Case A, a linear temperature profile, varying from surface to freezing temperature over very

small∂z; Case B, a linear temperature profile, varying from the surface temperature

to the freezing temperature, over large∂z. Both of these are linear profiles; Case C, a

more realistic non-linear profile is achieved by assuming constant vertical advection and vertical diffusion (Holland and Jenkins, 1999). The resulting parameterisation

is valid when thermal forcing is greater than ∼0.2 ◦C and for thick (≥1000 m) ice

shelves; and Case D, insulating ice, i.e. the ice is at basal freezing temperature.

The effect of the 4 parameterisations of ∂TI

∂z

b on the relationship between melt rate

and thermal forcing (T∗; difference between temperature adjacent to the base and

the in situ freezing point at the base), is shown in Fig. 1.8. As different rates of heat conduction into the ice shelf will change the heat available for melting at the ice–ocean interface, the ice shelf thermal profile is expected to be important for ice

shelf melting. High T∗ leads to high melt rates with all parameterisations of ice

temperature profile. At high thermal forcing, A, B and D all over-estimate melt rate, because they do not capture the ability of heat to diffuse quickly upwards through the shelf (Holland and Jenkins, 1999). Only C, which includes the vertical velocity of ice, correctly captures the magnitude of the gradient for high melt rates.

CHAPTER 1. INTRODUCTION 15 T* (°C) -1 0 1 2 3 4 m (myr -1) 0 20 40 60 80 100 120

linear diffusion, small ∂_{z A}

linear diffusion, large ∂_{z B}

advection and diffusion, C no advection, no diffusion, D -0.2 0 0.2 0.4 -5 0 5 10 inset

Figure 1.8: Melt rate (m yr−1) is shown as a function of the thermal forcing, for

different ice temperature profiles. These profiles are linear diffusion (small ∂z, case

A, large ∂z ice case B), advection and diffusion, case C, and the no advection and

no diffusion case D.

forcing. This switch to freezing captures the effect of heat conduction upwards into the ice exceeding the supply of heat into the surface layer from the mixed layer. B and D both have shallow temperature gradients, and thus energy is not removed quickly, and melt rates go to zero with the thermal forcing. The profile with advection and diffusion, C, also does not describe the melt-to-freeze flip accurately as approximations were made which do not hold at low melt rates.

Problems exist in all parameterisations of the ice temperature profile. The most advanced description, with constant vertical advection and diffusion, which describes high melting scenarios well, includes approximations which are not applicable at low melt rates or thin ice shelves. Similarly, the other cases here do not describe high melt rate scenarios well. While there are several ice shelves showing high melting at great depth, there are large areas of thinner ice at low thermal forcing values.

Consequently, the low∂zice, linear temperature profile (which describes low thermal

forcing scenarios well) would be suitable for these low melt ice shelves.