Uncertainties due to transport-parameter sensitivity in an efficient 3-D ocean-climate model

Abstract

A simplified climate model is presented which includes a fully 3-D, frictional geostrophic (FG) ocean component but retains an integration efficiency considerably greater than extant climate models with 3-D, primitive-equation ocean representations (20 kyears of integration can be completed in about a day on a PC). The model also includes an Energy and Moisture Balance atmosphere and a dynamic and thermodynamic sea-ice model. Using a semi-random ensemble of 1,000 simulations, we address both the inverse problem of parameter estimation, and the direct problem of quantifying the uncertainty due to mixing and transport parameters. Our results represent a first attempt at tuning a 3-D climate model by a strictly defined procedure, which nevertheless considers the whole of the appropriate parameter space. Model estimates of meridional overturning and Atlantic heat transport are well reproduced, while errors are reduced only moderately by a doubling of resolution. Model parameters are only weakly constrained by data, while strong correlations between mean error and parameter values are mostly found to be an artefact of single-parameter studies, not indicative of global model behaviour. Single-parameter sensitivity studies can therefore be misleading. Given a single, illustrative scenario of CO₂ increase and fixing the polynomial coefficients governing the extremely simple radiation parameterisation, the spread of model predictions for global mean warming due solely to the transport parameters is around one degree after 100 years forcing, although in a typical 4,000-year ensemble-member simulation, the peak rate of warming in the deep Pacific occurs 400 years after the onset of the forcing. The corresponding uncertainty in Atlantic overturning after 100 years is around 5 Sv, with a small, but non-negligible, probability of a collapse in the long term.

Appendix, the sea-ice model

Ocean surface fluxes are everywhere partitioned between ocean- and ice-covered fractions, the total heat flux Q_t into the ocean or ice surface being

$$ Q_{\text{t}} = \left( {1 - C_A } \right)Q_{{\text{SW}}} - Q_{{\text{LW}}} - Q_{{\text{SH}}} - \rho _{\text{o}} LE, $$

(6)

where E is the rate of evaporation or sublimation, calculated as in Weaver et al. (2001), L is the latent heat of evaporation L_v, or sublimation L_s, over ocean or ice respectively and the planetary albedo over sea ice is assumed to decrease linearly with air temperature within a given range following Holland et al. (1993):

$$ \alpha = \max \left( {0.20,\min \left( {0.7,0.40 - 0.04T_{\text{a}} } \right)} \right). $$

(7)

The sea ice has no heat capacity, thus the heat flux Q_t from the atmosphere is assumed to be equal to the vertical heat flux through the ice given by

$$ Q_{\text{t}} = - \frac{{v_i }} {H}\left( {T_{\text{f}} - T_i } \right), $$

(8)

where ν _i is the vertical conductivity of heat within the sea ice and T_f is the local salinity-dependent freezing temperature of sea water, assumed to be the temperature at the base of the sea ice, given by

$$ T_{\text{f}} = {}_{\gamma 1}S + {}_{\gamma 2}S^{3/2} + {}_{\gamma 3}S^2 , $$

(9)

where the values for the constants γ _i , i=1,2,3, are as given by Millero (1978). From Eqs. 6 and 8 we obtain a single equation, Φ(T _i )=0, say, for the ice-surface temperature T _i , which is solved by the Newton–Raphson iteration.

Having obtained the ice-surface temperature, and hence the heat flux from atmosphere to sea ice, we calculate the heat flux from the sea ice into the ocean (normally negative) as

$$ Q_{\text{b}} = \frac{{\rho _i C_{{\text{pi}}} \Delta z}} {{\tau _i }}\left( {T_{\text{f}} - T_{\text{o}} } \right), $$

(10)

where T_o is the temperature at the ocean surface; in the model T_o is the temperature of the uppermost (mixed) layer, Δz is the thickness of this layer, C_pi is the specific heat of sea ice at constant pressure, ρ _i its density and τ _i is a timescale for the relaxation of the ocean surface temperature to the freezing temperature. McPhee (1992) suggests a physical value for the ratio Q_b/(T_f− T_o) although the most appropriate value is liable to depend on model temporal and spatial resolution. However, we retain the value of 17.5 days for τ _i implied by McPhee’s parameterisation with the present value of Δz.

We are now in a position to calculate the growth rate G _i of sea-ice height in the ice-covered ocean fraction, which is given by the deficit of heat fluxes into and out of the sea ice, minus the latent heat loss of sublimation. Snow is not considered in the model, and all the precipitation over the ocean or sea ice is added directly to the ocean surface layer. Thus

$$ G_i = \frac{{Q_{\text{b}} - Q_{\text{t}} }} {{\rho _i L_{\text{f}} }} - E\frac{{\rho _{\text{o}} }} {{\rho _i }}, $$

(11)

where L _f is the latent heat of fusion of ice. In the open-ocean fraction we take −Q_b, from Eq. 7, to be the largest possible heat flux out of the ocean. Thus if the ocean-to-atmosphere heat flux is greater than this, the deficit leads to ice growth in the open water fraction. In general the growth rate of sea ice in the open-ocean fraction is

$$ G_{\text{o}} = \max \left( {0,\frac{{Q_{\text{b}} - Q_{\text{t}} }} {{\rho _i L_{\text{f}} }}} \right). $$

(12)

We can thus calculate the net growth rate G and the rate of change of the average sea-ice height H, which is also subject to advection by the surface ocean velocity and a diffusive term, which takes the place of a detailed representation of unresolved sea-ice advection and rheological processes. Note that H represents the height of sea ice averaged over both open ocean and ice-covered fractions.

$$ \frac{{DA}} {{Dt}} - \kappa _{{\text{hi}}} \nabla _{\text{h}}^2 H = AG_i + \left( {1 - A} \right)G_{\text{o}} = G, $$

(13)

where κ_hi is a horizontal diffusivity.

The rate of change of sea-ice area A is given by

$$ \frac{{DA}} {{Dt}} - \kappa _{{\text{hi}}} \nabla _{\text{h}}^2 A = \max \left( {0,\left( {1 - A} \right)\frac{{G_{\text{o}} }} {{H_{\text{o}} }}} \right) + \min \left( {0,AG_i \frac{A} {{2H}}} \right). $$

(14)

The first term on the right-hand side parameterizes the possible growth of ice over open water. The effect of this term is that, if G_o is positive, the open water fraction decays exponentially at the rate G_o/H₀, where H₀ is a minimum resolved sea-ice height. The second term parameterizes the possible melting of sea ice and corresponds (by simple geometry) to the rate at which A would decrease if all the sea ice were uniformly distributed in height between 0 and 2H/A over the sea-ice fraction A. Note that this represents a small modification to the original sea ice-area equation proposed by Hibler (1979) in that the decay term is proportional to the (negative) growth rate AG _i over the sea-ice fraction as opposed to the total (negative) growth rate G. The two formulations differ only if sea ice is forming in the open water fraction and simultaneously melting in the ice-covered fraction, in which case using the total growth rate means that melting affects both terms; a form of “double counting”. This is not as unlikely as it sounds since the ice fraction is normally much colder than the water fraction, thus the heat flux deficit causing sea-ice growth in the model is normally smaller over sea ice and can easily be of opposite sign to the deficit over open water.

We can now define the flux of heat into the ocean as

$$ Q_\theta = \left( {1 - A} \right)\max \left( {Q_{\text{b}} ,Q_{\text{t}} } \right) + AQ_{\text{b}} . $$

(15)

The flux of fresh water into the ocean is given by

$$ F_W = P + R - \left( {1 - A} \right)E_{\text{e}} - G\frac{{\rho _i }} {{\rho _{\text{o}} }} - AE_{\text{s}} . $$

(16)

The first two terms represent precipitation and runoff, the third term evaporation, E_e, over open water and the last two ice melting and freezing. The net fresh water source due to melting being proportional to the sink of average sea-ice height minus the net rate of sublimation, denoted E_s here for clarity. The flux of salinity into the ocean is simply taken to be F _S =-S₀F _W , where S₀ is a constant reference salinity (using the local salinity would complicate the calculation of the global freshwater budget).

Finally, we have to ensure, at each timestep, that the calculated sea-ice height H is positive. In practice we ignore the presence of thin ice and set H=0 whenever the calculated value of H is less than H₀. This means that the heat and freshwater fluxes must be modified accordingly for consistency. For positive or negative H, if H is set to zero, the corresponding amount of heat to be added to the ocean is −Hρ _i L_f, while the corresponding amount of fresh water to be added is Hρ _i /ρ_o. In addition, we have to ensure that numerical truncation error does not result in unphysical values of A, A<0 or A>1, but these latter adjustments carry no further implications for the conservation of heat and fresh water.