The linear sea-breeze circulation and effect of Earth’s rotation ATM 419/563 Spring 2017 Fovell Reference: Rotunno (1983, J. Atmos. Sci.) http://www.atmos.albany.edu/facstaff/rfovell/NWP/

Rough course outline Sea-breeze circulation & 2D simulation of effect of latitude on sea-breeze Downslope windstorms & 2D flow over topography Planetary boundary layer schemes & 1D column model simulations Microphysics schemes & 2D squall line simulations Cumulus parameterizations & real-data simulations of precipitating convection Advanced model initialization, nesting Addressing nonlinear instability and other troubleshooting Verifying model simulations against surface and upper air observations Stochastic perturbations and nudging Final project

Rotunno (1983) Quasi-2D analytic linear model Heating function specified over land Becomes cooling function after sunset Equinox conditions Cross-shore flow u, along-shore v Two crucial frequencies Heating  = 2/day (period 24h) Coriolis f = 2sin (inertial period 17h @ 45˚N) One special latitude… where f =  (30˚N) The two Omegas are the same…

Heating function Rotunno’s analytic model lacks boundary layer mixing and diffusion so horizontal, vertical spreading built into function

Streamfunction 

Circulation C Integrate CCW as shown Take w ~ 0; utop ~ 0 Integrate from ±infinity, from sfc to top of atmosphere

Rotunno’s analytic solution

Rotunno’s analytic solution Generic form Here, B = 0, A Here, B = 0, A > 0, so (f2 – w2) controls behavior Behavior = amplitude and timing of circulation max and magnitude and timing of wind at coastline

Rotunno’s analytic solution If f > w (poleward of 30˚) equation is elliptic • sea-breeze circulation spatially confined • circulation in phase with heating • circulation, onshore flow strongest at noon • circulation amplitude decreases poleward If f < w (equatorward of 30˚) equation is hyperbolic • sea-breeze circulation is extensive • circulation, heating out of phase • f = 0 onshore flow strongest at sunset • f = 0 circulation strongest at midnight Elliptic, just like del^2pi = F. F < omega circ strongest at midnight but onshore flow strongest at SUNSET Elliptic: magnetic field around bar magnet. Hyperbolic: rock-caused wave in infinite ocean; in theory, would propagate forever. What’s changing the behavior is f rel to omega. “What is it about the Coriolis force that can confine the circulation?”

Rotunno’s analytic solution If f = w (30˚N) equation is singular • some friction or diffusion is needed • circulation max at sunset • onshore flow strongest at noon I believe for f=omega, the equation produces a discontinuity that blows up in finite time. Friction/diffusion/damping keeps the solution from blowing up. Rotunno says circ ampl increases as f approaches omega. In reality, Rotunno’s first term is actually (N^2-omega^2), but he neglected omega relative to N. F < omega circ strongest at midnight but onshore flow strongest at SUNSET

Summary Latitude Onshore max (coastline) Circulation max f > 30˚ Noon f = 30˚ Sunset f = 0˚ Midnight

f > w (poleward of 30˚) at noon streamfunction u w Note onshore flow strongest at coastline (x = 0); this is day’s max Stronger onshore than offshore flow, as saw in Part I. coast

f < w (equatorward of 30˚) y at three times sunrise Note circulation max at noon noon (reverse sign for midnight) “windshield wiper” at noon Coastline onshore max SUNSET Circ max NOON sunset Note coastline onshore flow max at sunset

Max |C| noon & midnight Paradox? Why is onshore max wind at sunset but circulation max at midnight/noon? While wind speed at coast strongest at sunset/sunrise, the wind integrated along surface larger at midnight/noon Know coast wind strongest at sunrise/sunset because vert grad of streamfunction largest in mag then. Midnight is onshore, so MAX circ.

Effect of “linear friction” Time of circulation maximum midnight sunset noon As friction increases, the tropical seabreeze circ max becomes much earlier, the 30N circ becomes slightly earlier, the poleward circ becomes later friction coefficient As friction increases, tropical circulation max becomes earlier, poleward circulation max becomes later

Simulation with a linearized numerical model (not WRF)

Solution strategy Model starts at sunrise (6 am) with no flow Integrate for 1 day (too short – why?) Model is linearized Small amount of diffusion applied (acts somewhat like friction) Rotunno’s diurnal heating function used Heating max at noon, zero at sunset Cooling at night, absolute max at midnight

Heating function vs. time Model starts at sunrise - 12 hour day Especially at night:is this realistic heating/cooling function? Is this realistic?

30˚N linear case (5 m2/s diffusion) 400 km x 3 km subdomain depicted 3 km sea land Shaded: vertical velocity; contoured: cross-shore velocity

30˚N linear case Circulation max @ sunset (as expected) Note non-zero C @ 24h… should run several days to spin-up x2000 See something wrong here? Flow at sunrise on second day ≠ initial flow. Need to run LONGER to see if and when it becomes stable w/r/t time.

30˚N linear case Onshore flow max ~ 4pm Note non-calm wind @ 24h… sunset Onshore flow max ~ 4pm (sunset expected) Note non-calm wind @ 24h… should run several days to spin-up (m/s) A little earlier than expected from model, perhaps owing to some finite amt of friction

Two important points Numerical models require “spin-up” time Numerical diffusion is nearly always unavoidable and can influence the results Here, numerical diffusion is intentionally applied, via diffusion terms added to the equations like In models like WRF, the scheme used to solve the equations (Runge-Kutta 3rd order) has implicit diffusion

Variation with latitude

Cross-shore flow and vertical motion at noon (on 1st day) They look quite similar

Cross-shore flow and vertical motion at sunset (on 1st day) Set t 144 Big diffs now. 60N circ pooping out, winds weak across sfc… elliptic… confined. 00N winds strong, extensive… hyperbolic… spreads out unbounded. Keep in mind diurnal forcing is SAME at all three latitudes

Cross-shore flow and vertical motion at midnight (on 1st day) Night… sinking over land at all lat… but at 00N still onshore!

Cross-shore near-surface wind at coastline (linear model, one day) Equator - no offshore flow 30N strongest offshore flow Aperiodicity in 24h indicates need run model longer for it to settle

Circulation vs. time • Circ magnitude decreases w/ latitude (expected) • 30N circ max at sunset (expected) • Poleward circ max later than expected (noon) • Equator circ max earlier than expected (midnight) • Consistent w/ existence of some friction? midnight Eq x2000 30N 60N 90N sunset

Recall: linear friction effect Time of circulation maximum midnight sunset noon As friction increases, the tropical seabreeze circ max becomes much earlier, the 30N circ becomes slightly earlier, the poleward circ becomes later friction coefficient As friction increases, tropical circulation max shifts earlier, poleward circulation max becomes later

t (h after sunrise) x (across-coast) Hovmoller diagrams > 30N behavior is ELLIPTIC, closed, confined < 30N behavior is HYPERBOLIC, spreads w/o end AT 30N, Rotunno’s equation cannot be solved, but we see a transitional structure. Confined but spreading. Influence of the parabolic diffusion term? >30˚ behavior is elliptic (closed, confined) <30˚ behavior is hyperbolic (spreads without end) =30˚ behavior is transitional =0˚ note no land breeze forms

Something is missing in the linearized model. What is it? Next: simulate the 2D sea-breeze using a nonlinear model (WRF) Sea-breeze does not propagate inland!