Introduction to Atmospheric Dynamics Chapter 1 Paul A. Ullrich [email protected] Part 3: The Continuity Equation Question: What are the basic physical principles that govern the atmosphere? Conservation of Mass: Mass is not created nor destroyed. The total mass of the atmosphere is (to a very close approximation) constant over time periods of interest. Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Conservation of Mass Conservation of Mass: Mass is not created nor destroyed. The total mass of the atmosphere is (to a very close approximation) constant over time periods of interest. Continuity: The fuid is continuous (it contains no holes). This is a fundamental assumption underlying atmospheric motions. The Continuity Equation is used to describe conservation of atmospheric mass. Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Conservation l There are certain parameters (energy, momentum, mass, air, water, ozone, number of atoms, etc.) that must be conserved. l Conservation means that in an isolated system that parameter remains constant. It’s not created. It’s not destroyed. But it can move around. l Is the Earth’s atmosphere isolated? The Earth’s atmosphere is not exactly isolated (escape to space, escape beneath the surface), but to a close approximation this is the case! Paul Ullrich The Equations of Atmospheric Dynamics March 2014 The Continuity Equation Some region of the atmosphere (not a fuid parcel) Variables we will need ∆z ⇢ Density (mass per unit volume) V = ∆x∆y∆z Volume m = ⇢V ∆y Mass ∆x Paul Ullrich The Equations of Atmospheric Dynamics March 2014 In the Eulerian frame we have a fxed volume and the fuid fows through it. ∆z ∆y ∆x Paul Ullrich The Equations of Atmospheric Dynamics March 2014 In the Eulerian frame we have a fxed volume and the fuid fows through it. Fluid parcels Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What is the rate of change of mass within this region? Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What is the rate of change of mass within this region? Mass fux through face i Change in mass ∆m = ∆t F in ∆A · i i allX faces i Time Area of face i Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What is the rate of change in ∆m = ∆t Fi ∆Ai of mass within this region? · allX faces i The mass fux through this face per unit area is equal to the mass density at the face times the perpendicular velocity in Fw = ⇢w uw Face W Units? Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Defnition: The rate of fow of fuid per unit area is the mass fux. F = ⇢u (x, y, z) ⇢u Mass fux at ( x, y, z ) in the x direction kg m kg 2 ∆z = m− . m3 · s s · Mass flux is mass per unit /me per unit area ∆y @(⇢u) ∆x ⇢u + + h.o.t. ∆x @x 2 x ✓ ◆ Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Defnition: The rate of fow of fuid per unit area is the mass fux. F = ⇢u (x, y, z) ⇢u Mass fux at ( x, y, z ) in the x direction kg m kg 2 ∆z = m− . m3 · s s · Mass flux is mass per unit /me per unit area ∆y @(⇢u) ∆x ∆x ⇢u + h.o.t. − @x 2 x ✓ ◆ Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What is the rate of change of mass within this region? The change of mass in the box is equal to the mass that fows into the box minus the mass that fows out of the box. Flux Area Mass out east (downstream) face h i⇥h i @(⇢u) ∆x F ∆A = ⇢u + ∆y∆z e e @x 2 ✓ ◆ Mass in west (upstream) face @(⇢u) ∆x F ∆A = ⇢u ∆y∆z Volume of box w w − @x 2 ✓ ◆ ∆m @(⇢u) Total change in = F ∆A F ∆A = ∆x∆y∆z mass per unit time ∆t w w − e e − @x Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What is the rate of change of mass within this region? Extend to three dimensions: The change in mass in the box is equal to the mass that fows into the box minus the mass that fows out of the box. ∆m @ @ @ = F in∆A = (⇢u)+ (⇢v)+ (⇢w) V ∆t i i − @x @y @z all faces i X ∆m ∆⇢ = Limit as V ∆t 0 @⇢ ! = (⇢u) @t r · Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Continuity Equation (Eulerian) @⇢ = (⇢u) @t r · Recognize the divergence. Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Continuity Equation: Eulerian @⇢ In the Eulerian frame, a = (⇢u) region of the atmosphere is a @t r · fxed volume and fuid fows through it. ∆z ∆y ∆x Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What is the continuity equation in the Lagrangian frame? In the Lagrangian frame, the point of view follows the fuid parcel which is moving… ∆z ∆y …and changing shape (deforming) ∆x Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Continuity Equation: Lagrangian Starting with the continuity equation in the Eulerian frame @⇢ = (⇢u) @t r · Apply product rule (recall your vector identities) (⇢u)=⇢ u + u ⇢ r · r · · r Collect terms: @⇢ + u ⇢ = ⇢ u What is this? @t · r − r · Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Continuity Equation: Lagrangian @⇢ + u ⇢ = ⇢ u @t · r − r · Material Derivave 1 D⇢ = u ⇢ Dt r · Evolution equation for density in the Lagrangian Frame Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Continuity Equation: Lagrangian 1 D⇢ = u ⇢ Dt r · The change in density following the motion is proportional to the divergence Convergence = increase in density (compression) Divergence = decrease in density (expansion) Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Divergence What is the divergence of this fow? Will density increase or decrease? Calculate it! 1 D⇢ u =(x, y) = u ⇢ Dt r · Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Convergence What is the divergence of this fow? Will density increase or decrease? Calculate it! 1 D⇢ u =( x, y) = u − − ⇢ Dt r · Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Mixed Flow What is the divergence of this fow? Will density increase or decrease? Calculate it! 1 D⇢ u =(x, y) = u − ⇢ Dt r · Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Tracer Transport Passive tracers (chemical species, moisture) are transported with the fow. Their mixing ratio (the ratio of tracer mass to fuid mass) is constant within a fuid parcel: Dqi =0 (Advection equation) Dt To write a conservation law for tracers, we use the continuity equation: @⇢ + u ⇢ = ⇢ u @t · r − r · @qi @⇢ ⇢ + ⇢u qi =0 qi + qiu ⇢ = ⇢qi u @t @t · r − r · · r @ (⇢q )+ (⇢q u)=0 (Flux-Form) @t i r · i Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Thanks! Paul Ullrich The Equations of Atmospheric Dynamics March 2014 .