STRAIT.DOC Tracer Flow through Straits 97/10/26 MMST (kg) = mass of sea water in strait MUST (kg/s) = mass flux through strait RMST (kg) = G0MST or S0MST = mean tracer mass in strait RXST (kg) = GXMST or SXMST = horizontal gradient of tracer mass RZST (kg) = GZMST or SZMST = vertical gradient of tracer mass X1,Y1 = vector components [-1,1] where ocean box 1 meets strait DXYP1 (m^2) = area of ocean box 1 MO1 (kg/m^2) = mass per unit area of ocean box 1 MM1 (kg) = MO1*DXYP1 = mass in ocean box 1 RM1 (kg) = G0M1 or S0M1 = mean tracer mass in ocean box 1 RX1 (kg) = GXM1 or SXM1 = west-east gradient of tracer mass RY1 (kg) = GYM1 or SYM1 = south-north gradient of tracer mass RZ1 (kg) = GZM1 or SZM1 = vertical gradient of tracer mass X2,Y2 = vector components [-1,1] where ocean box 2 meets strait DXYP2 (m^2) = area of ocean box 2 MO2 (kg/m^2) = mass per unit area of ocean box 2 MM2 (kg) = MO2*DXYP2 = mass in ocean box 2 RM2 (kg) = G0M2 or S0M2 = mean tracer mass in ocean box 2 RX2 (kg) = GXM2 or SXM2 = west-east gradient of tracer mass RY2 (kg) = GYM2 or SYM2 = south-north gradient of tracer mass RZ2 (kg) = GZM2 or SZM2 = vertical gradient of tracer mass DT (s) = time step AM (kg) = DT*MUST AM > 0 ------ A1 = AM/MM1 FM1 = A1*(RM1 + X1*RX1 + Y1*RY1) = tracer mass flux FZ1 = A1*RZ1 FX1 = -3*AM*FM1 A2 = AM/MMST FM2 = A2*[RMST + (1-A2)*RXST] = tracer mass flux FZ2 = A2*RZST FX2 = AM*(A2*A2*RXST - 3*FM2) RX1 = RX1*[(1-A1)*(1+.5*A1) - 1.5*X1*X1*A1/(1.-A1)] RY1 = RY1*[(1-A1)*(1+.5*A1) - 1.5*Y1*Y1*A1/(1.-A1)] RX2 = RX2 + [RX2*AM*(.5-1.5*X2*X2) + 3*X2*(FM2*MM2old-RM2old*AM)] / MM2new RY2 = RY2 + [RY2*AM*(.5-1.5*Y2*Y2) + 3*Y2*(FM2*MM2old-RM2old*AM)] / MM2new RMST = RMST + FM1 - FM2 RXST = RXST*(1-A2)^3 - 3*FM1 + 3*A2*RMSTnew AM < 0 ------ A1 = AM/MMST FM1 = A1*[RMST - (1+A1)*RXST] = tracer mass flux FZ1 = A1*RZST FX1 = AM*(A1*A1*RXST - 3*FM1) A2 = AM/MM2 FM2 = A2*(RM2 + X2*RX2 + Y2*RY2) = tracer mass flux FZ2 = A2*RZ2 FX2 = -3*AM*FM2 RX1 = RX1 - [RX1*AM*(.5-1.5*X1*X1) + 3*X1*(FM1*MM1old-RM1old*AM)] / MM1new RY1 = RY1 - [RY1*AM*(.5-1.5*Y1*Y1) + 3*Y1*(FM1*MM1old-RM1old*AM)] / MM1new RX2 = RX2*[(1+A2)*(1-.5*A2) + 1.5*X2*X2*A2/(1.+A2)] RY2 = RY2*[(1+A2)*(1-.5*A2) + 1.5*Y2*Y2*A2/(1.+A2)] RMST = RMST + FM1 - FM2 RXST = RXST*(1+A1)^3 - 3*FM2 + 3*A1*RMSTnew RM1 = RM1 - FM1 RZ1 = RZ1 - FZ1 RM2 = RM2 + FM2 RZ2 = RZ2 + FZ2 RZST = RZST + FZ1 - FZ2 Linear Upstream Scheme applied to Strait Flow --------------------------------------------- When AM > 0: RXST = {RXST*MMST - 3*[MMST*(FM1+FM2)-2*AM*RMSTnew] + FX1-FX2} / MMST= = RXST - 3*(FM1+FM2) + 6*A2*RMSTnew - 3*A2*FM1 - A2*(A2*A2*RXST - 3*FM2) = = RXST - 3*FM1 - 3*A2*[RMSTold + (1-A2)*RXST] + 6*A2*RMSTnew - 3*A2*FM1 - A2*A2*A2*RXST + 3*A2*FM2 = = RXST*[1 - 3*A2*(1-A2) - A2^3] - 3*FM1 + 6*A2*RMSTnew - 3*A2*RMSTold - 3*A2*FM1 + 3*A2*FM2 = = RXST*(1-A2)^3 - 3*FM1 + 6*A2*RMSTnew - 3*A2*RMSTnew = = RXST*(1-A2)^3 - 3*FM1 + 3*A2*RMSTnew When AM < 0: RXST = {RXST*MMST - 3*[MMST*(FM1+FM2)-2*AM*RMSTnew] + FX1-FX2} / MMST= = RXST - 3*(FM1+FM2) + 6*A1*RMSTnew + A1*(A1*A1*RXST - 3*FM1) + 3*A1*FM2 = = RXST - 3*A1*[RMSTold - (1+A1)*RXST] - 3*FM2 + 6*A1*RMSTnew + A1*A1*A1*RXST - 3*A1*FM1 + 3*A1*FM2 = = RXST*[1 + 3*A1*(1+A1) + A1^3] - 3*FM2 + 6*A1*RMSTnew - 3*A1*RMSTold - 3*A1*FM1 + 3*A1*FM2 = = RXST*(1+A1)^3 - 3*FM2 + 6*A1*RMSTnew - 3*A1*RMSTnew = = RXST*(1+A1)^3 - 3*FM2 + 3*A1*RMSTnew Formulas for RX1, RY1, RX2 and RY2 are derived in the file PLUMES.777 --------------------------------------------------------------------- AM > 0: X1 = 2*MR/MO AM < 0: X1 = 2*MI/MO X2 = 2*MI/MO X2 = 2*MR/MO AM = MP AM = -MP MM2old = MO MM1old = MO MM2new = MN MM1new = MN RM2old = S0M RM1old = S0M FM2 = SMP FM1 = -SMP RX1 = SZM removal RX1 = SZM insertion RX2 = SZM insertion RX2 = SZM removal A1 = alpha = MP/MO A2 = -alpha = -MP/MO