Modelling the sea ice-ocean seasonal cycle in Hudson Bay, Foxe Basin and Hudson Strait, Canada

Abstract

The seasonal cycle of water masses and sea ice in the Hudson Bay marine system is examined using a three-dimensional coastal ice-ocean model, with 10 km horizontal resolution and realistic tidal, atmospheric, hydrologic and oceanic forcing. The model includes a level 2.5 turbulent kinetic energy equation, multi-category elastic-viscous-plastic sea-ice rheology, and two layer sea ice with a single snow layer. Results from a two-year long model simulation between August 1996 and July 1998 are analyzed and compared with various observations. The results demonstrate a consistent seasonal cycle in atmosphere-ocean exchanges and the formation and circulation of water masses and sea ice. The model reproduces the summer and winter surface mixed layers, the general cyclonic circulation including the strong coastal current in eastern Hudson Bay, and the inflow of oceanic waters into Hudson Bay. The maximum sea-ice growth rates are found in western Foxe Basin, and in a relatively large and persistent polynya in northwestern Hudson Bay. Sea-ice advection and ridging are more important than local thermodynamic growth in the regions of maximum sea-ice cover concentration and thickness that are found in eastern Foxe Basin and southern Hudson Bay. The estimate of freshwater transport to the Labrador Sea confirms a broad maximum during wintertime that is associated with the previous summer’s freshwater moving through Hudson Strait from southern Hudson Bay. Tidally driven mixing is shown to have a strong effect on the modeled ice-ocean circulation.

Appendix 1. Model formulation

The equations for the momentum, mass, heat, salt, and turbulent kinetic energy can be written as (the comma subscript denotes partial derivative)

$$\frac{{Du}}{{Dt}} - fv + \rho ^{ - 1} P_{,x} - \left( {K_H u_{,x} } \right)_{,x} - \left( {K_H u_{,y} } \right)_{,y} - \left( {K_{VM} u_{,z} } \right)_{,z} = 0$$

(1)

$$ \frac{{Dv}} {{Dt}} + fu + \rho ^{{ - 1}} P_{{,y}} - {\left( {K_{H} v_{{,x}} } \right)}_{{,x}} - {\left( {K_{H} v_{{,y}} } \right)}_{{,y}} - {\left( {K_{{VM}} v_{{,z}} } \right)}_{{,z}} = 0 $$

(2)

$$\nabla \cdot (u,v,w) = 0$$

(3)

$$\frac{{DT}}{{Dt}} - \left( {K_H T_{,x} } \right)_{,x} - \left( {K_H T_{,y} } \right)_{,y} - \left( {K_{V\sigma } T_{,z} } \right)_{,z} = 0$$

(4)

$$\frac{{DS}}{{Dt}} - \left( {K_H S_{,x} } \right)_{,x} - \left( {K_H S_{,y} } \right)_{,y} - \left( {K_{V\sigma } S_{,z} } \right)_{,z} = 0$$

(5)

$$\frac{{DE}}{{Dt}} - \left( {K_{VM} E_{,z} } \right)_{,z} = K_{VM} \left( {u_{,z}^2 + v_{,z}^2 } \right) + K_{V\sigma } (\frac{g}{\rho }\rho _{,z} ) - \frac{{q^3 }}{{B_1 l}}$$

(6)

in hydrostatic, P _,z = – ρg, and Boussinesq approximations, and where u = (u,v,w) is the velocity along the horizontal axes x and y, and vertical axis z (positive upward), f is the Coriolis parameter (calculated in the β-plane approximation), P is the pressure, T is the temperature, S is the salinity, E is the turbulent kinetic energy, ${\text{q}} = {\sqrt {2{\text{E}}} }$ is the turbulent velocity scale, l(z) is the turbulent length scale, K _H is the horizontal eddy viscosity and diffusivity for momentum, T and S, K _VM is the vertical eddy viscosity (and diffusivity for E), and K _{V σ} is the vertical diffusivity for T and S. The equation of state ρ = ρ (S,T,P) is computed from Unesco (1981). The definitions and values of the different model parameters are given in Table 1.

1.1 Sub-grid mixing

The horizontal viscosity coefficient is described following Smagorinsky (1963) with K _H = γΔx ² [u _,x ² + v _,y ² + 0.5(u _,y + v _,x )²]^0.5. The horizontal diffusivity for scalars is set to 2.5 m² s^–1 . The vertical eddy viscosity and diffusion coefficients are written as K _VM = K _VM0 + lq S _M and K _Vσ = K _Vσ0 + lq S _σ, where S _M and S _σ are stability functions derived by Canuto et al. (2001, first set), as described by Burchard and Bolding (2001). The background viscosity and diffusion coefficients are set to K _VM0 = 10 ^–4 m² s^–1 and k _{V σ 0} = 10^–6 m² s^–1. The turbulent length scale is prescribed following l(z) = min (l _d,l, _u ), between a parabolic law of the wall function, l _d , and the Ozmidov scale, l _u , given by

$$l_d (z) = k(z_{0t} - z)\left[ {1 + h^{ - 1} (z_{0b} + z)} \right]$$

(7)

$$l_u (z) = 0.53qN^{ - 1} $$

(8)

where z _0t and z _0b are the roughness lengths at the surface and bottom computed from Charnock’s formulae (Charnock 1955), z ₀ = g ^–1 αu _* ², where α is the Charnock constant, h is water depth, and N is the Brunt Väisälä frequency.

1.2 Lateral boundary conditions

The lateral solid boundary conditions are free-slip for momentum, and isolated for tracers. Bottom friction is parametrized following Cox (1984), τ _b = ρ₀ C _b∣u _b∣u _b, where u _b is the bottom layer velocity. The water level, temperature, salinity and the eddy diffusion and viscosity coefficients are specified at the open boundaries across the mouth of HS and Fury and Hecla Strait. During inflow, a zero gradient condition is applied to all velocity components. Temperature and salinity relax toward prescribed observations, based on a characteristic length scale that a tracer travels normal to the boundary over one time step. During outflow (u > 0), the radiation condition ξ _t + uξ_x = 0 is specified to ξ = u, v, T, S. On open ocean boundaries, horizontal and vertical viscosity and diffusivity coefficients are set to zero. The water level, salinity and temperature are specified to account for river discharge at the appropriate land/sea grid points.

1.3 Surface boundary conditions

Momentum is input at z = 0 from wind stress, K _VM u _,z = ρ _O ^–1 [(1 – A)τ _AO + Aτ _IO ]. The surface traction is described by quadratic drag forms (e.g. Charnock 1955) and τ _AO = ρ _A C _DAO ∣u _I –u _O ∣(u _I –u _O ), u _A , is the wind velocity at 10 m height, u _I is the ice velocity, u _O is the surface layer ocean velocity, C _DAO is the atmosphere-ocean drag coefficient taken from Atakturk and Katsaros (1999), and C _DIO is the atmosphere-ice drag coefficient.

Through the sea surface z = 0, heat is absorbed and released by radiative, sensible and latent heat transfers K _Vσ T _,z = – ρ₀ ^–1 C _pO ^–1 [(1 – A)Q _AO + AQ _IO ], where Q _AO and Q _IO are the net heat fluxes at the atmosphere-ocean and ice-ocean boundaries, respectively. The ocean-atmosphere heat flux is described following Parkinson and Washington (1979) as Q _AO = Q _SAO + Q _LAO – Q _SW – Q _LW ↓+ Q _{LW ↑}, where Q _SAO and Q _LAO are the sensible and latent heat fluxes at the atmosphere-ocean boundary, Q _SW and Q _{LW ↓} are the shortwave and longwave downward incident fluxes, and Q _{LW ↑} is the outgoing longwave radiation.

The sensible heat flux is given by Q _SAO = ρ _A C _pA C _SAO ∣u _A ∣(T _O –T _A ), with air temperature T _A and surface layer temperature T _O (or sea surface temperature SST), where the Stanton number, C _SAO, is taken from Large and Pond (1982). The latent heat flux is given by Q _LAO = ρ _A L _V C _LAO u _A (q _S –q _A ), where C _LAO is the Dalton number, and q _A and q _s are the specific humidity at 10 m height and at the surface, respectively, computed as in Parkinson and Washington (1979).

The shortwave energy flux through the sea surface is written as Q _SW = (1 – α _O )SW where α _O is the albedo of sea water, and

$$SW = S_c^0 \cos ^2 \theta _z \left( {1 - 0.6C_l^3 } \right)\left[ {10^{ - 5} \left( {\cos \theta _z + 2.7} \right)e_A + 1.085\cos \theta _z + 0.1} \right]^{ - 1} $$

(9)

(Laevastu 1960; Zillman 1972), where C _l is the observed cloud fraction, e _A is the near-surface vapor pressure, and θ _z the solar zenith angle.

The longwave incident energy flux is computed from Idso and Jackson (1969), corrected for emission by the cloud base using the model from Bignami et al. (1995)

$$Q_{LW \downarrow } = \sigma T_A ^4 \left( {1 - 0.261\exp ( - 7.77 \times 10^{ - 4} (273.15 - T_A )^2 )} \right)\left( {1 + 0.1762C_l^2 } \right)$$

(10)

The outgoing longwave radiation is computed from the Stefan Boltzmann’s law, $Q_{LW \uparrow } = \varepsilon _o \sigma T_o^4 .$ At the ice-ocean interface, we compute the sensible heat flux following $Q_{IO} = \rho _o C_{po} C_{SIO} |{\mathbf{u}}_l - {\mathbf{u}}_o |(T_o - T_f ).$

Salt is transferred through the sea surface by precipitation, evaporation, and ice growth rate (melt) over open ocean areas f ₀, and ice covered areas f h, using K _V σS _,z = ρ _O ^–1 Q _S = – [(1 – A)(f ₀ + (P – E _v )S _O ) + A f _h S _O], where A is the ice concentration, P is the observed precipitation over open water (over ice and when T _A< 0 °C precipitation accumulates as snow on the ice), E _v = ρ_O ^–1 L _V ^–1 Q _LAO is the evaporation, and S _O = SSS is the sea surface salinity.

1.4 Ice-atmosphere heat flux

The ice-atmosphere heat flux is Q _AI = Q _SAI + Q _LAI – Q _{SW AI} – Q _{LW ↓} + Q _{LW ↑}, where the sensible and latent heat fluxes at the atmosphere-ice interface are written as for the atmosphere-ocean boundary, except that T _O is replaced by T _I , L _v by L _S , and C _SAI = C _SAO and C _LAI = C _LAO . The shortwave incident flux uses the albedo for dry snow, melting snow or bare ice (Table 1).

1.5 Sea ice

The elastic-viscous-plastic (EVP) sea-ice dynamic formulation from Hunke and Dukowicz (1997) (also Hunke 1998) is used in addition with a two-layer ice thermodynamics considering snow thickness over ice based on Semtner (1976). For each grid point sea ice is represented as a thickness distribution g assuming that

$$ {\int\limits_0^{h\max } {g(h,)dh,} } = 1 $$

(12)

where hmax is the maximum allowed sea-ice thickness. The cumulative distribution of ice is defined as

$$G(h) = \int\limits_0^h {g(h)dh} $$

(13)

Ice areas are transported after solving the momentum equation

$$ m\frac{{\partial {\mathbf{u}}_{I} }} {{\partial t}} = - mf{\mathbf{k}} \times {\mathbf{u}}_{I} + {\user2{\tau }}_{{AI}} + {\user2{\tau }}_{{IO}} - mg\nabla \eta + \nabla \cdot {\user2{\sigma }} $$

(14)

where u _I is the two-dimensional ice speed vector, m is the mass per unit area, k is a unit vertical vector, τ _AI is the wind stress vector (same form as τ _AO with coefficient C _DAI ), g is the gravitational acceleration, η is the sea surface elevation, and σ is the two-dimensional Cauchy stress tensor. The mass per unit area is given by

$$m = A\int\limits_0^{h\max } {g(h)(\rho _I h + \rho _S h_S )dh} $$

(15)

ρ _I and ρ _S are the ice and snow density, respectively, and h _s is the snow thickness over ice of thickness h.

The maximum resistance to pressure is defined as

$$p_{\max } = p^* H\exp ( - C(1 - A))$$

(16)

Where H is the mean ice thickness. Sea ice areas are evolving through

$$\frac{{\partial g}}{{\partial t}} + \nabla \cdot (\vec ug) = - \frac{\partial }{{\partial h}}\left[ {\left( {f_I + \frac{{\rho _s }}{{\rho _I }}c_S } \right)g} \right] + \psi $$

(17)

where f _I is the ice growth rate in ms^–1, c _S is the snow compaction rate in ms^–1, and ψ is the ice redistribution term representing ridging. The compaction rate is defined as

$$c_S = \frac{{h_S - h_S 0.5^{dt/\tau _s } }}{{dt}}$$

(18)

where τ _s is the snow compaction time scale (half life period). The term ψ is calculated using the weight function described in Thorndike et al. (1975) for convergence only, no ridging is applied in pure shear. The ridging function can be decomposed as an output and source term representing thinner ridging ice becoming thicker ridged ice such that

$$\psi (h) = - o(h) + s(h)$$

(19)

where

$$\int\limits_0^{h\max } {\psi hdh = 0} $$

(20)

If h _S (h)is the snow thickness over ice of thickness h, and gh _S (h) = g(h)h _S (h),

$$\frac{{\partial (gh_S )}}{{\partial t}} + \nabla \cdot (\vec ugh_S ) = - (f_S - c_S )g + \psi _S $$

(21)

where f _S is the snow growth in ms^–1 from precipitation and thermodynamic melt. The term ψ _S represents the redistribution of snow from ridging and is calculated using ψ in a way that snow volume is conserved

$$\int\limits_0^{h\max } {\psi _S (h)dh = 0} $$

(22)

If ice of thickness h ₁ is ridging into ice of thickness h ₂ in a way that u(h ₁) = s(h ₂)h ₂/h ₁

$$\psi _S (h_1 ) = - \psi _S (h_2 ) = - o(h_1 )h_S (h_1 )$$

(23)

The EVP solution uses two split time steps (Table 1) as discussed in Hunke and Dukowicz (1997).