The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology The Tropical Cyclone Boundary Layer 3. Dynamics: simpler models Jeff Kepert Head, High Impact Weather Research Oct

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Why use simpler models? Hierarchy of models Modern NWP (very realistic, very complex, very hard to understand) Hydrostatic models Filtered models (eliminate gravity waves) Shallow water models (one-layer) Barotropic models (one layer, nondivergent) Quasigeostrophic theory (vertical structure, simplified description e.g. of vertical motion) down to Geostrophic wind equation (somewhat realistic, still useful, very easy to understand)

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Things we would like to understand Why does the boundary-layer depth vary with radius? Why does the surface wind factor vary with radius and azimuth? Why is the frictional updraft near the RMW? Why is the boundary-layer flow sensitive to the storm structure (e.g. peaked, flat)? Why are the strongest winds in the right front quadrant? What causes surface outflow in the left rear quadrant?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Solution Seek a simpler model with an analytical solution. Simplifications: Linearize about the base-state vortex Remove baroclinicity Assume turbulent diffusivity K is constant with height

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Derive linear model for symmetric case

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology

Depth scale depends on I, not f Drag coefficient instead of no-slip condition Radially elongated

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Model intercomparison: Linear vs nonlinear

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Model intercomparison: Linear vs nonlinear Linear model (top), K-W model (bottom). Closely-spaced dots/circles are the BL-mean wind. Widely-spaced are the 10-m wind. Linear model in general agrees with the nonlinear model But does overestimate the surface azimuthal wind Deficiencies in diffusion parameterisation in linear model v gr v -u 100w v 10 -u 10

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology 960 hPa, 40 m/s, RMW = 40 km, b = 1.3 uv v u Blue - linear theory Red - nonlinear model Main reason for difference is vertical advection - can show that updraft => stronger jet, downdraft => weaker (or none). Linear - Nonlinear Comparison in Eyewall

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Weak baroclinicity. Shear-dominated turbulence No diurnal cycle Simple diffusivity profile Shallow boundary-layer. Semi-slip boundary condition. Many of the factors which normally mess up the Ekman spiral in the atmosphere are absent. Is Ekman-like dynamics reasonable?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology It is linear! Neglect of vertical advection weakens jet in core. Neglect of radial advection near surface in core also important in intense storms. Not valid for inertially neutral storms. Only produces wave number 1 asymmetries (although similar solutions for higher wave number asymmetries in the surface forcing - such as at landfall - are easy to get). Surface bc was linearized Model assumes that U t << V. Thus it does not give correct limit at large radii, and does not work for fast moving storms. Storm moves with boundary layer geostrophic flow. Very simple turbulence closure. Linear Model Limitations

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Things we would like to understand Why does the boundary-layer depth vary with radius? Why does the surface wind factor vary with radius and azimuth? Why is the frictional updraft near the RMW? Why is the boundary-layer flow sensitive to the storm structure (e.g. peaked, flat)? Why are the strongest winds in the right front quadrant? What causes surface outflow in the left rear quadrant?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Boundary-layer depth with radius r-z section from linear model White curve is 2.5δ, where δ=sqrt(2K/I).

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Things we would like to understand Why does the boundary-layer depth vary with radius? Why does the surface wind factor vary with radius and azimuth? Why is the frictional updraft near the RMW? Why is the boundary-layer flow sensitive to the storm structure (e.g. peaked, flat)? Why are the strongest winds in the right front quadrant? What causes surface outflow in the left rear quadrant?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Surface wind factor The surface wind factor is given by where

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Things we would like to understand Why does the boundary-layer depth vary with radius? Why does the surface wind factor vary with radius and azimuth? Why is the frictional updraft near the RMW? Why is the boundary-layer flow sensitive to the storm structure (e.g. peaked, flat)? Why are the strongest winds in the right front quadrant? What causes surface outflow in the left rear quadrant?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Ekman pumping in a tropical cyclone Expanding the TC expression: This solution inserted in the continuity equation gives an analytical expression for w above the BL in a TC which can be compared to the classical Ekman expression

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Why the outer RMW has a stronger updraft ς

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Ekman pumping in a tropical cyclone The formula is similar but the “curl of the stress” is not the dominant term! Maximum updraft is dominated by the vorticity gradient term. Maximum updraft is near the RMW. NB Nonlinear model has the updraft at smaller radius than the linear model.

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology The importance of quadratic friction What happens with a no-slip surface boundary condition? If the eye is in solid-body rotation, then No-slip condition (laminar boundary layer) => w is constant everywhere Quadratic friction (turbulent boundary layer) => w increases linearly with radius So in a vortex with a no-slip condition, friction cannot cause a localised updraft near the RMW See Eliassen (1971, J Met Soc Japan special issue)

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Things we would like to understand Why does the boundary-layer depth vary with radius? Why does the surface wind factor vary with radius and azimuth? Why is the frictional updraft near the RMW? Why is the boundary-layer flow sensitive to the storm structure (e.g. peaked, flat)? Why are the strongest winds in the right front quadrant? What causes surface outflow in the left rear quadrant?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Sensitivity to storm structure: nonlinear Peaked wind profileFlat wind profile

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Sensitivity to storm structure: nonlinear Peaked wind profileFlat wind profile

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Sensitivity to storm structure: linear Peaked wind profileFlat wind profile

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Sensitivity to storm structure: linear Peaked wind profileFlat wind profile

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Momentum budgets in the linear model

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Momentum budgets in the linear model

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Balance between destruction & advection of M a Flat Peaked Flat Peaked At the RMW, the loss of M a is similar in both cases. This loss is approximately balanced by radial advection, –u ∂M a /∂r. Thus inflow must be much stronger in the “peaked” case, because the gradient is weaker.

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Interstorm variability Method: Take flight-level data from these missions Average legs together and calculate azimuthal-mean wind Fit a parametric profile Run nonlinear model Compare Franklin et al. (2003)

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Hurricane Bret Bret had a small RMW and peaked wind profile => strongly supergradient maximum very near surface. No warm core With warm core

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Dennis had a large RMW (~55 km) and strong low-level temperature gradient => weak supergradient effect, deep boundary-layer, strong thermal wind effect. No warm coreWith warm core Hurricane Dennis

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Bonnie had a very large RMW and flat wind profile, diffuse warm core => deep boundary layer, weaker thermal wind effect than Dennis. No warm core With warm core Hurricane Bonnie

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Floyd shows excellent agreement (although there were significant structural changes to other observing days). No warm core With warm core Hurricane Floyd

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Georges shows excellent agreement. No warm core With warm core Hurricane Georges

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Mitch fairly poor performance - possibly due to structure change during observing period? No warm core With warm core Hurricane Mitch

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Things we would like to understand Why does the boundary-layer depth vary with radius? Why does the surface wind factor vary with radius and azimuth? Why is the frictional updraft near the RMW? Why is the boundary-layer flow sensitive to the storm structure (e.g. peaked, flat)? Why are the strongest winds in the right front quadrant? What causes surface outflow in the left rear quadrant?

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Linear asymmetric model With a bit more work, can extend the linear model to include asymmetric friction due to motion (Kepert 2001, JAS). Asymmetric friction introduces two additional components to the solution They are again similar to Ekman spirals One is shallower, and the other deeper, than the symmetric component They scale with the cyclone translation speed They have an azimuthal wavenumber one structure

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Deep component, 1 m/s contours height (2K / (I-V/r)) 1/2 Shallow component, 0.25 m/s contours height (2K / (I+V/r)) 1/2 Total (storm relative), 2 & 5 m/s contours 960 hPa, max wind 40 m/s, RMW = 40 km, b = 1.3, U t = -5 m/s Surface 500 m 1 km 2 km uuvuvv Linear Model Asymmetric Component

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Radial FlowAzimuthal Flow Strongest winds in right forward quadrant Strongest inflow to right 5 m/s Near-Surface Earth-Relative Flow

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Net frictional forcing in the linear model The total upwards mass flux inside a circle of radius s is where we have used Thus the net upwards mass flux inside the circle depends only on conditions at the circumference.

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Net frictional forcing In the linear model, the inflow across s depends only on conditions at radius s. “conditions” = wind speed and vorticity From Stokes’ theorem, the circulation (wind speed) around s equals the total vorticity inside s. So the updraft distribution can be calculated from the vorticity distribution nonand outside of v v

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Slab models “Slab” models simplify the problem by depth-averaging the equations Popular choice, widely used in engineering design studies and for some theoretical studies Two problems with the formulation: Need surface winds to calculate the surface stress, but only have depth- averaged winds The vertical averaging makes calculating the non-linear advection inaccurate Simulations generally do not agree well with those from less approximated models

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Model intercomparison: Slab vs nonlinear Compare BL-mean azimuthal and radial wind between slab model (top) and K-W model (bottom) Slab model has more inflow, bigger departures from gradient flow, updraft eruption Closely-spaced dots/circles are the BL-mean wind. Widely-spaced are the 10-m wind. v gr v -u 100w v 10 -u 10

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Oscillations in slab model Could attempt to resolve excess friction by reducing C D … but then the solution oscillates Noted by Shapiro (1983) and controlled there by horizontal diffusion Analysed in detail by Smith (2003), Smith and Vogl (2008), Kepert (2010a,b) v gr v -u 100w

The Centre for Australian Weather and Climate Research A partnership between CSIRO and the Bureau of Meteorology Summary Linearisation of BL model gives quasi-Ekman pumping Can explain many features seen in the observations, nonlinear model (and full-physics models) Integral constraint on net vertical motion Other simplified models exist – depth-averaged models have been popular, but produce spurious oscillations