Lecture 8: Oceanography PDF

Lecture 8: Oceanography Department of Earth and Environmental Sciences, Indian Institute of Science Education and Research, Bhopal Surface ocean dynamics  Equation of fluid motion If u, v , and w be the velocity...

Lecture 8: Oceanography Department of Earth and Environmental Sciences, Indian Institute of Science Education and Research, Bhopal Surface ocean dynamics  Equation of fluid motion If u, v , and w be the velocity components along x, y, and z directions, and D/Dt is the time derivative  Equation of fluid motion The Coriolis force is non-zero only if the body is in motion, and is important only if the body travels for a significant period of time. In the Northern Hemisphere, the Coriolis force causes a moving body to appear to move to the right of its direction of motion, while in the Southern Hemisphere, it moves to the left. Ω is the rotation rate, ϕ is latitude, u and v are velocity components in the x and y directions Time derivative relationship between the Eulerian and Lagrangian frames is given below 𝜕 𝜕 | = 𝜕𝑡| + 𝜔 ⃗ × 𝜕𝑡 𝐸 𝐿 If a body is moving with instantaneous position and velocity vectors as 𝑟 and 𝑣 in a linear direction in the Eulerian frame then the acceleration in the Lagrangian frame is given as 𝜕𝑟 𝜕( | −𝜔 ⃗ × 𝑟) 𝜕𝑣 𝜕𝑡 𝐸 𝑎𝐿 = | = | 𝜕𝑡 𝐿 𝜕𝑡 𝐿 𝜕𝑟 𝜕( | −𝜔 ⃗ × 𝑟) 𝜕𝑡 𝐸 𝜕𝑟 𝑎𝐿 = | −𝜔 ⃗ ×( | −𝜔⃗ × 𝑟) 𝜕𝑡 𝜕𝑡 𝐸 𝐸 𝜕2𝑟 𝜕𝑟 𝑎𝐿 = 2 | − 2𝜔 ⃗ × | +𝜔⃗ × (𝜔 ⃗ × 𝑟) 𝜕𝑡 𝐸 𝜕𝑡 𝐸 Where the first, second, and third terms are acceleration components due to linear motion, the Coriolis force, and the centrifugal force. The Coriolis force in vector notation from the following picture, where a particle P having mass m is moving with a linear velocity v is given as ⃗⃗⃗⃗⃗⃗⃗⃗ 𝐹𝐶𝑜𝑟 = −2𝑚𝜔 ⃗⃗⃗⃗⃗𝑃 × 𝑣 𝜔 ⃗ cos(90 − 𝜑) = 𝜔 sin 𝜑 𝑟̂ ⃗⃗⃗⃗⃗𝑃 = 𝜔 ⃗⃗⃗⃗⃗⃗⃗⃗ 𝐹 𝐶𝑜𝑟 = −2𝑚𝜔𝑣 sin 𝜑 𝑛 ̂ Where 𝑟̂ 𝑎𝑛𝑑 𝑛̂ point in the directions radially outward and into the plane of the paper, respectively at point P.  Equation of fluid motion where v is the molecular (kinematic) viscosity, while the dynamic viscosity is density times v. where AH is the horizontal eddy viscosity and AV is the vertical eddy viscosity. Both terms have units of kinematic viscosity, m2/sec in mks units. Although eddy viscosity is much larger than molecular viscosity, the ocean is nevertheless nearly inviscid, in the sense that the Reynolds number (Re) is large and the Ekman number (Ek) is small even when eddy viscosities are used.  𝑈𝑈 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 𝐿𝐿 𝑣𝑣 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 2 𝐿𝐿 Reynolds number 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑈𝑈𝑈𝑈 Ek 𝑅𝑅𝑒𝑒 = = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑣𝑣 Ekman number 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑣𝑣 𝐸𝐸𝑘𝑘 = = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓𝐿𝐿2 Rossby number 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑈𝑈 𝑅𝑅𝑂𝑂 = = For Ro