Shelf/Plume Equations

The basic equations used in ISOFT to describe ice and plume dynamics are similar to those of Sergienko (2013), except that the plume is assumed to be in quasi-steady state. All equations were nondimensionalised. Scales for this were chosen to be able to work with multiple choices of parameterisations. This requirement of flexibility sometimes resulted in otherwise suboptimal choices.

Ice Shelf Equations

After rescaling, the dimensionless ice shelf equations have the form $$\begin{equation} h_{t} + \nabla\cdot(h\vec{u}) = -\lambda m, \label{eq:ice-height-nd} \end{equation}$$ $$\begin{equation} {\left[2\eta h\left(2u_x + v_y\right)\right]}_{x} + {\left[\eta h\left(u_y + v_x\right)\right]}_{y} - \chi{\left(h^2\right)}_x = 0, \label{eq:ice-mom-x} \end{equation}$$ $$\begin{equation} {\left[\eta h\left(u_y + v_x\right)\right]}_{x} + {\left[2\eta h\left(u_x + 2v_y\right)\right]}_{y} - \chi{\left(h^2\right)}_y = 0. \label{eq:ice-mom-y} \end{equation}$$

In these equations $h$ is the ice thickness (scaled by reference thickness $h_0$), $\vec{u} = (u,v)$ is the velocity at which the ice flows (scaled by reference $u_0$), $m$ is the rate at which the ice shelf is melting (rescaled by reference $m_0$, defined below), and $\eta$ is the rescaled ice viscosity. The dimensionless parameters $$\begin{equation}\label{eq:ice-parameters} \lambda \equiv \frac{\rho_0 m_0 x_0}{\rho_i h_0 u_0}, \quad \chi \equiv \frac{\rho_i gh_0x_0}{2\eta_0 u_0}\left(1 - \frac{\rho_i}{\rho_0}\right) \end{equation}$$ represent the ratio of melt versus influx of ice and the stretching rate (ratio of gravitational stresses that drive stretching versus viscous stresses resisting), respectively. The ice density is $\rho_i$, while the reference density for ocean water is $\rho_0$. Otherwise, the subscript nought indicates a typical scale for a variable. The timescale for ice flow is given by $t_0 = x_0/u_0$. Gravitational acceleration is $g$. If viscosity is modelled as being Newtonian, then the dimensionless $\eta$ is set to 1. Alternatively, Glen’s Law can be nondimensionalised to take the form $$\begin{equation}\label{eq:glens-nd} \eta = \frac{1}{2}\xi D^{1/n - 1}_{2}, \end{equation}$$ where $D_2 = \sqrt{D_{i,j}D_{i,j}/2}$ is the second invariant of the strain rate and the dimensionless coefficient $$\begin{equation}\label{eq:glens-coef-nd} \xi \equiv \frac{B}{\eta_0} {\left(\frac{u_0}{x_0}\right)}^{1/n - 1}. \end{equation}$$ Typically, $n=3$.

Plume Equations

The plume equations are scaled according to $$\begin{equation}\label{eq:scaling} \begin{gathered} U_0^2 = \frac{h_0g\Delta\rho}{\rho_0}, \quad e_0=m_0=\gamma_{T0}=\gamma_{S0} = \frac{D_0U_0}{x_0}, \\ \quad \Delta T_0 = \frac{\Gamma^{*}_T x_0}{D_0}(T_a - T_m), \quad \Delta S_0 = \frac{c_0\rho_0\Delta T_0(S_a - S_m)}{\rho_i L}. \end{gathered} \end{equation}$$ As before, the subscript nought indicates the typical scale for a variable. The exception to this is $\rho_0$, which is a representative value for the water density. Because the density difference is used in the plume equation and this difference is quite small, it was found useful to adopt a different scale, $\Delta\rho$, to use when nondimensionalising $\rho_a - \rho$, $\rho_x$, and $\rho_y$. Similarly, temperature and salinity were scaled according to typical differences rather than by their absolute values. An arbitrary point could be set to zero for these two variables and it proved convenient to choose the ambient values as the zero-point. The basal depth, $b$, is scaled by $h_0$. Note that the scale $m_0$ does not correspond to typical physical values of melting but is chosen because it is convenient to have it equal to those of the other variables; as a result, $m \ll 1$. This yields the dimensionless system $$\begin{equation} \nabla\cdot\left(D\vec{U}\right) = e + m, \label{eq:plume-cont} \end{equation}$$ $$\begin{equation} \nabla\cdot\left(D\vec{U}U\right) = D(\rho_a - \rho)\left(b_x - \delta D_x\right) + \frac{\delta D^2}{2}\rho_x + \nu\nabla\cdot\left(D\nabla U\right) - \mu|\vec{U}|U, \label{eq:plume-mom-x} \end{equation}$$ $$\begin{equation} \nabla\cdot\left(D\vec{U}V\right) = D(\rho_a - \rho)\left(b_y - \delta D_y\right) + \frac{\delta D^2}{2}\rho_y + \nu\nabla\cdot\left(D\nabla V\right) - \mu|\vec{U}|V, \label{eq:plume-mom-y} \end{equation}$$ $$\begin{equation} \nabla\cdot\left(D\vec{U}S\right) = eS_a + \nu\nabla\cdot\left(D\nabla S\right) + mS_m - \gamma_S(S-S_m), \label{eq:plume-salt} \end{equation}$$ $$\begin{equation} \nabla\cdot\left(D\vec{U}T\right) = eT_a + \nu\nabla\cdot\left(D\nabla T\right) + mT_m - \gamma_T(T-T_m). \label{eq:plume-temp} \end{equation}$$

These equations were constructed without making any assumptions about the form of $m$, $e$, or $\rho$. For this reason different, more generic, scales are adopted for these values. The velocity scale depends on the density scale, rather than on buoyancy input from subglacial discharge. Instead of scaling the salinity in terms of buoyancy input, its scale is based the level of melt-water input, which is the dominant source of salinity forcing across most of the domain.

The dimensionless parameters $$\begin{equation}\label{eq:plume-parameters} \delta \equiv \frac{D_0}{h_0}, \quad r = \frac{\rho_0}{\rho_i}, \quad \nu \equiv \frac{\kappa}{x_0U_0}, \quad \mu \equiv \frac{C_dx_0}{D_0} \end{equation}$$ represent the dimensionless buoyancy correction, density ratio, turbulent eddy diffusivity, and turbulent drag coefficient, respectively. The latter two depend on the dimensional eddy diffusivity $\kappa$, which is assumed to be equal to the eddy viscosity, and the unscaled turbulent drag coefficient $C_d$.

The simple entrainment parameterisation of Jenkins (1991) can be nondimensionalised to have the form $$\begin{equation} \label{eq:entrainment-jenkins-nd} e = \frac{E_0}{\delta}|\nabla b||\vec{U}|, \end{equation}$$ suggesting it is convenient to take $\delta = E_0$ (or equivalently, $D_0 = E_0 h$). The more complex parameterisation of Kochergin (1987) nondimensionalises to $$\begin{equation} \label{eq:entrainment-koch-nd} e = \frac{K}{S_m}\sqrt{|\vec{U}|^2 + \frac{\delta(\rho_a - \rho)D}{S_m}}, \end{equation}$$ with the dimensionless coefficient $$\begin{equation}\label{eq:ent-koch-coef-nd} K = \frac{c_L^2 x_0}{D_0}. \end{equation}$$ The turbulent Schmidt number, $S_m$, depends on the Richardson number: $$S_m = \frac{Ri}{0.0725(Ri + 0.186 - \sqrt{Ri^2 - 0.316Ri 0.0346})}.$$ With these scales, the Richardson number is given by $Ri = \delta(\rho_a - \rho)D/|\vec{U}|^2$. The simplified melt rate parameterisation taken from Dallaston, Hewitt, and Wells (2015), has the dimensionless form $$\begin{equation}\label{eq:melt-nondim} m = \zeta_1\zeta_2|\vec{U}|(T - T_m), \end{equation}$$ where $$\begin{equation}\label{eq:thermal-zetas} \zeta_1 = \frac{\Gamma^{*}_T x_0}{D_0}, \quad \zeta_2 = \frac{c\Delta T_0}{L}. \end{equation}$$

Due to the low efficiency of thermal transfer to the ice shelf, compared to the high rate of entrainment, $\zeta_1 \ll 1$. The large latent heat of ice results in $\zeta_2 \ll 1$, as well. Together, these results mean $m \ll e \sim 1$, so that the mass gain by meltwater input is much smaller than by entrainment. It can be seen that the thermal transfer coefficient nondimensionalises to give $$\begin{equation}\label{eq:therm-trans-nondim} \gamma_T = \zeta_1 |\vec{U}|. \end{equation}$$

The water density is set using a linear equation of state of the form $$\begin{equation} \label{eq:lin-eos} \rho_w = \rho_{\rm ref}[1 + \beta_S(S-S_{\rm ref})-\beta_T(T-T_{\rm ref})]. \end{equation}$$ Here, $\beta_S$ is the haline contraction coefficient, $\beta_T$ is the thermal expansion coefficient, and $S_{\rm ref}$, $T_{\rm ref}$, and $\rho_{\rm ref}$ are reference values for salinity, temperature, and density about which the relation has been linearised.

Typical Scales and Parameter Values

Typical scales and values for ice shelf and plume properties are listed in the table below, along with the values of non-dimensional parameters which result. “Repr. val.” stands for “representative value”. [J11] refers to Jenkins (2011), [B11] to Bindschadler, Vaughan, and Vornberger (2011), [D15] to Dallaston, Hewitt, and Wells (2015), [J91] to Jenkins (1991), [J96] to Jacobs, Hellmer, and Jenkins (1996), [K87] to Kochergin (1987), and [S13] to Sergienko (2013). Scales in the third column are chosen to be comparable to the conditions of the PIG ice shelf and come from the indicated source. Where a scaling is unconstrained it was chosen to provide convenient parameter values (e.g. $x_0$ fixed by $\chi$). The value of $c_L$ was chosen so that its entrainment parameterisation resulted in a similar melt rate as that of Jenkins (1991). Due to an error, $\zeta_2$ is a factor of $r$ too large. $\newcommand{\unit}[1]{{\rm\thinspace #1}} \newcommand{\m}{\unit{m}} \newcommand{\km}{\unit{km}} \newcommand{\kg}{\unit{kg}} \newcommand{\s}{\unit{s}} \newcommand{\yr}{\unit{yr}} \newcommand{\J}{\unit{J}} \newcommand{\K}{\unit{K}} \newcommand{\Pa}{\unit{Pa}} \newcommand{\hPa}{\unit{hPa}} \newcommand{\kPa}{\unit{kPa}} \newcommand{\rad}{\unit{rad}} \newcommand{\psu}{\unit{psu}} \newcommand{\mps}{\m\s^{-1}} \newcommand{\mpss}{\m\s^{-2}} \newcommand{\mpyr}{\m\yr^{-1}} \newcommand{\kmpyr}{\km\yr^{-1}} \newcommand{\C}{\rm ^{\circ}\thinspace C} \newcommand{\kgpmc}{\kg\m^{-3}} \newcommand{\ppsu}{\psu^{-1}} \newcommand{\msps}{\m^{2}\s^{-1}} \newcommand{\Pas}{\Pa\s} \newcommand{\Jpkg}{\J\kg^{-1}} \newcommand{\JpkgpK}{\Jpkg\K^{-1}}$