Theoretical Analysis On The Eyewall Sloping In Tropical Cyclones

There are some arguments in the former observation studies on the slantwise structure of the tropical cyclone(TC)eyewall.A slope equation of the TC eyewall is contained in the TC theoretical model which is built up basing on the thermal wind balance.However,some hypotheses of the theoretical model have been doubted.These facts indicate that the theory about the TC eyewall is imperfect,and the mechanism of the eyewall sloping is unclear.This study starts from the volume expanding effect of the pseudo-adiabatically ascending saturated air parcel.The influences of the pressure decrease,the latent heat release and the vapor loss to the volume expansion of the air parcel have been analyzed.The change rules of the vertical depth and the horizontal area of the air parcel have also been learned in varying vertical velocity.It is found that the pressure decrease accounts for more than 80% of the volume expansion.The change of the horizontal area of the air parcel is related to varying vertical velocity,i.e.,the vertical acceleration(deceleration)causes the vertical stretch(compressing),which leads to horizontal shrinking(expending),or suppressing(amplifying)on horizontal shrinkage.By applying this rule to the axisymmetric air parcel,the relationship between the profile of the vertical variation of the air parcel radius and the profile of the vertical velocity can be built up.Similarly,the relationship between the profile of the radial thickness of the tubular air column and the profile of the vertical velocity can also be built,and this relationship can be used to estimate the structure of the equivalent potential temperature in TC eyewall since that the equivalent potential temperature of the air parcel conserved in pseudo-adiabatic process and the trajectory of the air parcel forms the equivalent potential temperature surface.An idealized TC simulated with WRF is used to verify the estimation method.The estimated results are close to the simulation,indicating that the relationship built above is valid and there is relationship between the vertical velocity and the structure of the equivalent potential temperature in TC eyewall.This relationship inspires the question about the existence of the constraint of the continuity equation on the TC eyewall structure.Basing on the reversible adiabatic assumption,the relationship between the secondary circulation and the distribution of the entropy in TC has been derived.It is found that the stream lines coincide with the isentropic surfaces,and the stream function can be expressed with entropy function.With this relationship,a method for diagnosing the secondary circulation from the distribution of the entropy is proposed.The verification is conducted with simulated TC and the diagnosed results reproduce the secondary circulation well.This diagnosis method is simpler than the Sawyer-Eliassen(SE)equation.Again,the relationship between the secondary circulation and the structure of the TC eyewall is proved to be existing.To find out the connection between secondary circulation and the structure of the TC eyewall,derivations are started from the control equations.It is found that the isentropic surfaces are coincident with the M surfaces under the inviscid,non-dissipation,axisymmetric and reversible adiabatic condition;the hydrostatic balance and the gradient wind balance are unable to satisfy at the same time on the curve isentropic surfaces.The vertical variation of the radial component of the horizontal pressure gradient is also found to be caused by the radial gradient of the vertical acceleration and the baroclinicity.Under this mechanism,vertical variation of the radial component of the horizontal pressure gradient in the boundary layer outside the TC eyewall is small since the vertical velocity is close to 0 and the isentropic surfaces are nearly horizontal.This new finding provides theoretical basis to the vertically homogeneous assumption of the horizontal pressure gradient in boundary layer models.The relationship between the thermal wind balance and the secondary circulation is also derived,which indicates that the secondary circulation may break the thermal wind balance to some extent.The slope equation of the isentropic surface is derived from the relation between thermal wind balance and the secondary circulation.Comparing to the slope equation derived in former studies,this new slope equation contains a new term about the secondary circulation,indicating that the secondary circulation is involved in the constraint on the TC eyewall sloping.This is identical with the analytical result above.According to the new slope equation of the isentropic surface,the baroclinicity leads to vertical decrease on the horizontal pressure gradient,making the TC eyewall sloping outward;the secondary circulation affects the TC eyewall slope through the constraint of the continuity equation.