Advanced Ocean Modelling: Using Open-Source Software

5.13 Advanced Lateral Boundary Conditions 167a time-mean component driving the geostrophic inflow and a superposed timevariablecomponent representing incident waves such as tides. The geostrophic partcan then be described along the boundary as a background distribution, graduallyblended in during the first few days of a simulation, and the time-variable part canbe added via a prognostic equation of the form:∂q b∂t= q o ω sin (ωt) (5.38)where q b is boundary dynamic pressure, and q o and ω are amplitude and frequencyof the time-variable forcing. The reader is encouraged to test this type of forcing fora simplified open channel configuration.Inflow conditions for water properties such as temperature and salinity arestraight forward. These properties are simply prescribed as fixed boundary valuesso that the inflow carries these via the advection scheme into the model domain.Establishment of sharp density gradients near open boundaries can be avoided withthe use of an adjustment method, commonly called Rayleigh damping, of the form:ψ n+1b= ψ n b + ΔtT(ψo − ψ n b)(5.39)where ψ b is the boundary value of either temperature or salinity, T is a prescribedadjustment period, and ψ o is the target boundary value.5.13.4 Outflow ConditionsThe formulation of outflow conditions at open boundaries is not a trivial task giventhat both steady currents and wave signals can simultaneously interfere with sucha boundary. Unwanted partial wave reflection at open boundaries is a commonproblem. Different types of outflow boundary conditions are best demonstratedwith a focus on the propagation of long linear surface gravity waves in a channelof uniform depth h o . The dynamics of such waves can be approximated by theequations:∂u∂t∂η∂t=−g ∂η∂x=−h o∂u∂x(5.40)(5.41)where u is velocity, t is time, g is acceleration due to gravity, η is seasurface elevation,and x is distance along the channel. Assumptions are that the wave periodis short compared with the Coriolis force (so that the latter can be ignored), thatthe wave’s phase speed exceeds the flow speed by far (so that the nonlinear termscan be ignored), that the hydrostatic balance holds (such that the resultant flow is