Physical Oceanography - UNAM, Mexico Lecture 2: The Equations of Ocean Circulation and Ocean Modelling Robin Waldman October 15th 2018 A first taste... ¶uh 1 ¶ ¶uh + (u:r)uh + f k × uh = − rhP + rh:(khurh)uh + (kzu ) ¶t r0 ¶z ¶z Z 0 0 P(z) = r0gh + g rdz z Z z 0 w(z) = − rh:uhdz −H ¶h Z h = −rh: uhdz + P + R − E ¶t −H ¶q ¶ ¶q 1 _ + (u:r)q = rh:(khT rh)q + (kzT ) + Θ ¶t ¶z ¶z rcw ¶S ¶ ¶S + (u:r)S = r :(k r )S + (k ) + S_ ¶t h hS h ¶z zS ¶z r = r(q;S;P0(z)) All the physics of an ocean circulation model is here ! Outline The Equations of Ocean Circulation Ocean modelling Outline The Equations of Ocean Circulation Ocean modelling Conservation of mass : continuity In the following : control volume of zonal, meridional and vertical sizes dx, dy and dz and density r on a fixed Cartesian coordinate system (i;j;k) attached to the ground. d(rdxdydz) = 0 dt dr ddx ddy ddz = dxdydz + r(dydz + dxdz + dxdy ) dt dt dt dt dr = dxdydz + r(dydzdu + dxdzdv + dxdydw) dt Hence dividing by dxdydz : dr ¶u ¶v ¶w + r( + + ) = 0 dt ¶x ¶y ¶z dr () + rr:v = 0 dt ¶ ¶ ¶ with r = ( ¶x ; ¶y ; ¶z ) the space derivative operator. Conservation of mass : continuity Mass conservation : Hence dividing by dxdydz : dr ¶u ¶v ¶w + r( + + ) = 0 dt ¶x ¶y ¶z dr () + rr:v = 0 dt ¶ ¶ ¶ with r = ( ¶x ; ¶y ; ¶z ) the space derivative operator. Conservation of mass : continuity Mass conservation : d(rdxdydz) = 0 dt dr ddx ddy ddz = dxdydz + r(dydz + dxdz + dxdy ) dt dt dt dt dr = dxdydz + r(dydzdu + dxdzdv + dxdydw) dt Conservation of mass : continuity Mass conservation : d(rdxdydz) = 0 dt dr ddx ddy ddz = dxdydz + r(dydz + dxdz + dxdy ) dt dt dt dt dr = dxdydz + r(dydzdu + dxdzdv + dxdydw) dt Hence dividing by dxdydz : dr ¶u ¶v ¶w + r( + + ) = 0 dt ¶x ¶y ¶z dr () + rr:v = 0 dt ¶ ¶ ¶ with r = ( ¶x ; ¶y ; ¶z ) the space derivative operator. dr0 0 dt + (r0 + r )r:v ' r0r:v = 0 () r:v = 0 I To a very good approximation, oceanic currents are non-divergent. Conservation of mass : continuity In the ocean, Boussinesq approximation : relative density variations are small : 0 0 r(x;y;z;t) = r0 + r (x;y;z;t) and r << r0 The continuity equation becomes : Conservation of mass : continuity In the ocean, Boussinesq approximation : relative density variations are small : 0 0 r(x;y;z;t) = r0 + r (x;y;z;t) and r << r0 The continuity equation becomes : dr0 0 dt + (r0 + r )r:v ' r0r:v = 0 () r:v = 0 I To a very good approximation, oceanic currents are non-divergent. Conservation of momentum Newton’s 2nd law : the Lagrangian (material) evolution of momentum is determined by the sum of external (gravity) and body (pressure and friction) forces. ¶u ¶u ¶u ¶u du = dt + dx + dy + dz ¶t ¶x ¶y ¶z du ¶u dt ¶u dx ¶u dy ¶u dz () = + + + dt ¶t dt ¶x dt ¶y dt ¶z dt ¶u ¶u ¶u ¶u = + u + v + w ¶t ¶x ¶y ¶z ¶u = + (u:r)u ¶t (u:r)u is the (nonlinear) advection term. Conservation of momentum ¶u The acceleration in the Eulerian (fixed) framework ¶t is far easier to du observe and model than the Lagrangian one dt . Relation between both : Conservation of momentum ¶u The acceleration in the Eulerian (fixed) framework ¶t is far easier to du observe and model than the Lagrangian one dt . Relation between both : ¶u ¶u ¶u ¶u du = dt + dx + dy + dz ¶t ¶x ¶y ¶z du ¶u dt ¶u dx ¶u dy ¶u dz () = + + + dt ¶t dt ¶x dt ¶y dt ¶z dt ¶u ¶u ¶u ¶u = + u + v + w ¶t ¶x ¶y ¶z ¶u = + (u:r)u ¶t (u:r)u is the (nonlinear) advection term. Conservation of zonal momentum over a control volume ∂u ν (z)δ x δ y - w(z) δxδy u(z) ∂ z ∂ u ν ( y)δ x δ z - v(y) δxδz u(y) ∂ y P(x-δx)/ρ δyδz P(x)/ρ δyδz 0 0 u(x-δx) δyδz u(x-δx) -f v δxδyδz ∂u ∂u - u(x) δyδz u(x) −ν (x−δ x)δ y δ z −ν ( y−δ y)δ x δ z ∂ u ∂ x ∂ y ν (x)δ y δ z δz ∂ x v(y-δy) δxδz u(y-δy) δy w(z-δz) δxδy u(z-δz) ∂ u −ν (z−δ z)δ x δ y k j ∂ z i Figure 1 – Conservation of zonal momentum over a control volume : advection (black), viscous forces (red), pressure forces (blue) and Coriolis acceleration (purple). Conservation of momentum Advection of zonal momentum in all faces of the control volume : Conservation of zonal momentum over a control volume ∂u ν (z)δ x δ y - w(z) δxδy u(z) ∂ z ∂ u ν ( y)δ x δ z - v(y) δxδz u(y) ∂ y P(x-δx)/ρ δyδz P(x)/ρ δyδz 0 0 u(x-δx) δyδz u(x-δx) -f v δxδyδz ∂u ∂u - u(x) δyδz u(x) −ν (x−δ x)δ y δ z −ν ( y−δ y)δ x δ z ∂ u ∂ x ∂ y ν (x)δ y δ z δz ∂ x v(y-δy) δxδz u(y-δy) δy w(z-δz) δxδy u(z-δz) ∂ u −ν (z−δ z)δ x δ y k j ∂ z i Conservation of momentum Advection of zonal momentum in all faces of the control volume : Figure 1 – Conservation of zonal momentum over a control volume : advection (black), viscous forces (red), pressure forces (blue) and Coriolis acceleration (purple). Conservation of zonal momentum over a control volume ∂u ν (z)δ x δ y - w(z) δxδy u(z) ∂ z ∂ u ν ( y)δ x δ z - v(y) δxδz u(y) ∂ y P(x-δx)/ρ δyδz P(x)/ρ δyδz 0 0 u(x-δx) δyδz u(x-δx) -f v δxδyδz ∂u ∂u - u(x) δyδz u(x) −ν (x−δ x)δ y δ z −ν ( y−δ y)δ x δ z ∂ u ∂ x ∂ y ν (x)δ y δ z δz ∂ x v(y-δy) δxδz u(y-δy) δy w(z-δz) δxδy u(z-δz) ∂ u −ν (z−δ z)δ x δ y k j ∂ z i Figure 2 – Conservation of zonal momentum over a control volume : advection (black), viscous forces (red), pressure forces (blue) and Coriolis acceleration (purple). Conservation of momentum Zonal pressure force on the western and eastern faces of the control volume : Conservation of zonal momentum over a control volume ∂u ν (z)δ x δ y - w(z) δxδy u(z) ∂ z ∂ u ν ( y)δ x δ z - v(y) δxδz u(y) ∂ y P(x-δx)/ρ δyδz P(x)/ρ δyδz 0 0 u(x-δx) δyδz u(x-δx) -f v δxδyδz ∂u ∂u - u(x) δyδz u(x) −ν (x−δ x)δ y δ z −ν ( y−δ y)δ x δ z ∂ u ∂ x ∂ y ν (x)δ y δ z δz ∂ x v(y-δy) δxδz u(y-δy) δy w(z-δz) δxδy u(z-δz) ∂ u −ν (z−δ z)δ x δ y k j ∂ z i Conservation of momentum Zonal pressure force on the western and eastern faces of the control volume : Figure 2 – Conservation of zonal momentum over a control volume : advection (black), viscous forces (red), pressure forces (blue) and Coriolis acceleration (purple). Conservation of momentum Zonal pressure force on the western and eastern faces of the control volume : FPx = (P(x − dx) − P(x))dydz hence volumic zonal pressure force : F ¶P Px = − dxdydz ¶x Generalizing to other spatial dimensions : F P = −rP dxdydz Conservation of momentum Friction Ft : I Newton’s law of viscosity for a Boussinesq Newtonian fluid : ¶uj ¶ui tij = +nr( + ) ¶xi ¶xj with tij the viscous stress exerted over the coordinate i on velocity component j, xi ;xj = (x;y;z), ui ;uj = (u;v;w) and n = 8:9 × 10−7m2=s the water kinematic viscosity. I The second term vanishes in a Boussinesq fluid : hence friction behaves just like molecular or heat diffusion ! Conservation of zonal momentum over a control volume ∂u ν (z)δ x δ y - w(z) δxδy u(z) ∂ z ∂ u ν ( y)δ x δ z - v(y) δxδz u(y) ∂ y P(x-δx)/ρ δyδz P(x)/ρ δyδz 0 0 u(x-δx) δyδz u(x-δx) -f v δxδyδz ∂u ∂u - u(x) δyδz u(x) −ν (x−δ x)δ y δ z −ν ( y−δ y)δ x δ z ∂ u ∂ x ∂ y ν (x)δ y δ z δz ∂ x v(y-δy) δxδz u(y-δy) δy w(z-δz) δxδy u(z-δz) ∂ u −ν (z−δ z)δ x δ y k j ∂ z i Figure 3 – Conservation of zonal momentum over a control volume : advection (black), viscous forces (red), pressure forces (blue) and Coriolis acceleration (purple). Conservation of momentum Zonal friction force on all faces of the control volume : Conservation of zonal momentum over a control volume ∂u ν (z)δ x δ y - w(z) δxδy u(z) ∂ z ∂ u ν ( y)δ x δ z - v(y) δxδz u(y) ∂ y P(x-δx)/ρ δyδz P(x)/ρ δyδz 0 0 u(x-δx) δyδz u(x-δx) -f v δxδyδz ∂u ∂u - u(x) δyδz u(x) −ν (x−δ x)δ y δ z −ν ( y−δ y)δ x δ z ∂ u ∂ x ∂ y ν (x)δ y δ z δz ∂ x v(y-δy) δxδz u(y-δy) δy w(z-δz) δxδy u(z-δz) ∂ u −ν (z−δ z)δ x δ y k j ∂ z i Conservation of momentum Zonal friction force on all faces of the control volume : Figure 3 – Conservation of zonal momentum over a control volume : advection (black), viscous forces (red), pressure forces (blue) and Coriolis acceleration (purple).